TESIS DOCTORAL Quantitative approach to unstable non ...

UNIVERSIDAD CARLOS III DE MADRID

TESIS DOCTORAL

Quantitative approach to unstable non-conserved growth with fluctuations

Autor: Matteo Nicoli

Director/es:

Rodolfo Cuerno Rejado y Mario Castro Ponce

DEPARTAMENTO DE MATEMÁTICAS

Leganés, julio de 2009

TESIS DOCTORAL

Quantitative approach to unstable non-conserved gro wth with fluctuations

Autor: Matteo Nicoli

Director/es: Rodolfo Cuerno Rejado y Mario Castro Ponce

Firma del Tribunal Calificador:

Firma Presidente:

Vocal:

Vocal:

Vocal:

Secretario:

Calificación:

Leganés, de de

Quantitative Approaches to UnstableNon-Conserved Growth with Fluctuations

Matteo Nicoli

July 2009

Ringraziamenti

La realizzazione di questa tesi dottorale e stata possibile grazie al lavoro e, so-pratutto, grazie all’appoggio di molte persone.

Innanzitutto, voglio ringraziare Mario Castro e Rodolfo Cuerno per avermidato la possibilita di lavorare in un ambiente dinamico e dipoter vivere a Madridfacendo ricerca (cosa non trascurabile al giorno d’oggi). Spero che il dottorato siasolamente l’inizio di un lungo rapporto di lavoro e di proficua collaborazione.

Ringrazio tutti i membri del GISC per avermi fatto sentire parte integrante delgruppo e per aver sopportato tutte le miemisas(che spesso si sono rivelate unavera sola).

Ringrazio Luis Vazquez per il tempo perso a cercare di spiegarmi la fisica dellesuperfici da un punto di vista sperimentale e, anche, per il rispetto dimostrato neiconfronti dei kernel non-locali.

Ringrazio Mathis Plapp per avermi accolto all’Ecole Polytechnique di Paris eper credere che vale la pena lavorare con me (e per darmi l’opportunita di conti-nuare a fare ricerca).

Ringrazio l’Universita di Bologna, la Universidad CarlosIII de Madrid, il Mi-nisterio de Ciencia e Inovacion, la Fundacion Carlos III ela Fondazione Angelodella Riccia per gli aiuti economici ricevuti in questi anni.

Infine, dedico questa tesi alla mia famiglia, a Cristina, ai miei amici di Bologna,Madrid, Paris e a tutti quelli dispersi per il mondo.

Contents

1 Introduction 11.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Published articles and preprints . . . . . . . . . . . . . . . . . . .5

2 Interfacial height equations 72.1 Continuum equations . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Anomalous scaling . . . . . . . . . . . . . . . . . . . . . 112.1.2 Scaling of anisotropic surfaces . . . . . . . . . . . . . . . 132.1.3 Universality classes in surface growth phenomena . . .. 152.1.4 Morphological instabilities . . . . . . . . . . . . . . . . . 21

2.2 Dynamical renormalization group technique . . . . . . . . . .. . 262.2.1 The Forster-Nelson-Stephen theory . . . . . . . . . . . . 27

2.3 Numerical solution of the growth equations . . . . . . . . . . .. 322.3.1 Spectral representation with discrete Fourier transform . . 332.3.2 Pseudo-spectral numerical integration . . . . . . . . . . .37

3 Moving boundary description of diffusive growth 393.1 Unified model of diffusive growth . . . . . . . . . . . . . . . . . 41

3.1.1 Chemical vapor deposition . . . . . . . . . . . . . . . . . 413.1.2 Continuum model of galvanostatic electrodeposition. . . 433.1.3 Role of fluctuations . . . . . . . . . . . . . . . . . . . . . 46

3.2 Small slopes approximation . . . . . . . . . . . . . . . . . . . . . 473.2.1 Non-instantaneous surface kinetics . . . . . . . . . . . . . 503.2.2 Instantaneous surface kinetics . . . . . . . . . . . . . . . 513.2.3 Full dispersion relation . . . . . . . . . . . . . . . . . . . 533.2.4 Non-linear evolution equation . . . . . . . . . . . . . . . 59

3.3 Comparison with experiments . . . . . . . . . . . . . . . . . . . 613.3.1 Mullins-Sekerka and Kuramoto-Sivashinsky behaviors . . 613.3.2 Beyond the analytic approximation . . . . . . . . . . . . 64

3.4 Comparison with discrete models . . . . . . . . . . . . . . . . . . 683.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

iv CONTENTS

4 Non-local growth equations 714.1 The dynamics of cauliflower-like growth . . . . . . . . . . . . . .72

4.1.1 Continuum description . . . . . . . . . . . . . . . . . . . 734.1.2 Chemical vapor deposition growth . . . . . . . . . . . . . 754.1.3 Numerical and analytical estimation of growth exponents . 76

4.2 A class of non-local equations . . . . . . . . . . . . . . . . . . . 814.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Ripple rotation in the anisotropic Kuramoto-Sivashinsky equation 955.1 Pattern formation in ion-beam sputtering . . . . . . . . . . . .. . 955.2 An interface equation for ripple rotation . . . . . . . . . . . .. . 975.3 DRG analysis of the aKS equation . . . . . . . . . . . . . . . . . 1025.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Phase-field model of diffusive growth 1096.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 Numerical convergence . . . . . . . . . . . . . . . . . . . . . . . 1156.4 Multivalued interfaces in electrodeposition . . . . . . . .. . . . . 1196.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7 Conclusions 1257.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

A Non-local transforms 129

B DRG calculations for non-local interface equations 133B.1 Propagator renormalization . . . . . . . . . . . . . . . . . . . . . 133B.2 Noise variance renormalization . . . . . . . . . . . . . . . . . . . 136B.3 Vertex Renormalization . . . . . . . . . . . . . . . . . . . . . . . 136B.4 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C Irrelevant relaxation terms 145C.1 DRG calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 145C.2 Irrelevance of thek3 term . . . . . . . . . . . . . . . . . . . . . . 151C.3 Irrelevance of thek4 term . . . . . . . . . . . . . . . . . . . . . . 154

D DRG for the anisotropic Kuramoto-Sivashinsky equation 159D.1 Propagator renormalization . . . . . . . . . . . . . . . . . . . . . 159D.2 Noise variance renormalization . . . . . . . . . . . . . . . . . . . 168

E Thin interface limit 169E.1 Equations non-dimensionalization . . . . . . . . . . . . . . . . .169E.2 Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . 172

E.2.1 Fieldφ at orderO(1) . . . . . . . . . . . . . . . . . . . . 172

CONTENTS v

E.2.2 Fieldu orderO(1/ǫ2) . . . . . . . . . . . . . . . . . . . 173E.2.3 Fieldφ orderO(ǫ) . . . . . . . . . . . . . . . . . . . . . 174E.2.4 Fieldu orderO(1/ǫ) . . . . . . . . . . . . . . . . . . . . 175E.2.5 Fieldφ orderO(ǫ2) . . . . . . . . . . . . . . . . . . . . . 176E.2.6 Fieldu orderO(1) . . . . . . . . . . . . . . . . . . . . . 178

E.3 Corrections, function involved and relations between the parameters 180

F Resumen en Castellano 183F.1 Metodologıa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183F.2 Aportaciones originales . . . . . . . . . . . . . . . . . . . . . . . 184F.3 Artıculos publicados y preprints . . . . . . . . . . . . . . . . . .186F.4 Conclusiones alcanzadas . . . . . . . . . . . . . . . . . . . . . . 186

Introduction

In the last century the study of interfacial phenomena has attracted an increasingnumber of scientists from the fields of physics, chemistry, biology and engineering.In the nanotechnology era scientists have started to investigate systems character-ized by a considerable surface to volume ratio. This is merely a consequence of ascience that is interested in ever smaller length scales. Infact, at these micro, oreven, nano-metric scales the surface plays an essential role for the dynamics of thesystem and it cannot be neglected anymore.

In recent years the growth of surfaces at these scales has become possible withnew production techniques. Thin films production by growth,erosion or etchingonto solid substrates has unveiled unexpected interestingphysical properties of thegrown interfaces.

From a technological point of view, these properties can affect device perfor-mance, a prominent example being represented by the surfaceroughness. Indeed,during the production process of these devices, the controlof the interface rough-ness is necessary to obtain e.g. good electric conductivityor a good mechanicalcontact. Thus, the Materials Science community has a paramount interest in deter-mining the phenomena that control the pattern formation of nanoscale structures,such as the order of quantum dot arrays or ripple formation under ion beam erosion.

In contrast with other well-established techniques, the majority of growth tech-niques at the nanoscale are instances of out-of-equilibrium systems: the evolutionof the surface is driven by a physical or chemical deposition(or etching) processpreventing relaxation towards equilibrium. This is an additional difficulty both for

2 Introduction

the experimentalist and the theorist.However, these out-of-equilibrium systems are often described by a powerful

set of tools borrowed from Statistical Mechanics as scale invariance, multiscaling,morphological instabilities, pattern formation, or pattern coarsening.

A remarkable consequence of the universality of these phenomena is their ubiq-uity in experimental situations, which in principle enables the use of those com-mon tools to describe physical processes taking place at length scales differing byseveral orders of magnitude. Hence, in Statistical Physicsthe subject of surfacedynamics has become increasingly important as a paradigmatic field to study out-of-equilibrium phenomena in an unified perspective.

Furthermore, from the seminal work of Benoıt Mandelbrot itis well knownthat many interfaces in nature display a statistically self-affine structure over anextended, but finite, scale range. Typical examples are coastlines, snowflakes, rivernetworks, cauliflowers, and blood vessels. Other instancesof statistically self-affine interfaces originate from the growth techniques usedby materials scientists,such as colloidal aggregation, electrochemical deposition, chemical vapor deposi-tion, molecular beam epitaxy, and other.

In Mathematics, a self-affine object is exactly (or at least statistically) similarto a part of itself, meaning that the whole object has the sameshape as one or moreof its parts. Hence, an anisotropic magnification of a smaller piece of a self-affineobject reveals the entire original structure.

Generically, the property of self-affinity implies irregularities at all scales and,for this reason, it cannot be described within the frameworkof standard analyti-cal methods. Among mathematical structures, fractals are the best candidates todescribe self-affinity, and indeed in many cases interfacescan be effectively rep-resented by stochastic self-affine fractals. For instance,the first growth modelbased on a well-defined physical process was diffusion-limited aggregation. Thisarchetypical model for Laplacian growth has a clear-cut connection with severalphenomena like dendritic growth, dielectric breakdown, orviscous fingering, andis able to describe (at lengths scales that depend on the system) all these phenom-ena with a small number of ingredients. This is only one example of the greatflexibility of models built using a coarse-grained approachin which the atomisticdetails of the system are only qualitatively incorporated.

In this thesis we use the tools of Statistical Mechanics to study the kineticroughening of the surfaces described by a continuum model ofdiffusive growth.Our approach is expected to apply to a large class of systems in which surfacegrowth takes place in the absence of conservation laws. An important characteris-tic of these systems is the presence of pattern formation (due to a morphologicalinstability) at small lengths scales and disorder (kineticroughening) at large lengthsscales.

By using an unified moving boundary formulation of diffusivegrowth, we areable to describe in the same framework the interfaces produced by diverse thinfilm production techniques, such as chemical vapor deposition and electrochemicaldeposition. The study of this model can be performed with twocomplementary

1.1 Outline of the thesis 3

strategies:

(i) we can perform an approximate analysis in the limit of small slopes of thesurface height, or

(ii) we can integrate numerically the complete moving boundary problem.

Following the first strategy we obtain a closed effective equation for the evolu-tion of the surface height in which the interplay between morphological instability,non-locality, non-linearity, and fluctuations leads to unexpected features. Althoughthe ensuing equation has been derived from a model for diffusive growth, its rangeof applicability is very large. In fact, the morphologies arising from this equationresemble the hierarchical structures on the surface of cauliflower plants, and thus,the study of this equation can give insight on the physical mechanisms that pro-duce these cauliflower-like structures in nature. Moreover, this equation is onlyone instance of a large class of non-local equations that canbe derived as effec-tive descriptions of very different physical systems. By employing the dynamicalrenormalization group formalism, we classify the criticalbehavior of these equa-tions finding a continuum of new universality classes.

As we are interested in both qualitative and quantitative descriptions of growthsystems, the latter approach cannot be used as a predictive tool for the experi-mentalist. Consequently, only by employing a multiscale integration method, aquantitative comparison between the morphologies obtained in e.g. electrodeposi-tion experiments and the moving boundary description of thesystem. So, to matchthe length scales between the diffusive transport and the structures at the interfacewe use a multigrid phase-field model that includes fluctuations. This methodol-ogy allow us to finally study quantitatively kinetic roughening phenomena in elec-trodeposition and, as a final result of the thesis, provide a sound and satisfactorydescription of non-conserved unstable growth.

1.1 Outline of the thesis

After this short introduction, the manuscript is divided into six chapters, each onewith its own conclusions section (due to its character, we have omitted the conclu-sions section in chapter 2). For the sake of readability we have collected the lengthycalculations and minor results into appendixes. In the finalchapter we summarizeour conclusions from an unified perspective and present, briefly, an outline of fu-ture research lines and developments that follow naturallyfrom the results of thisthesis. Let us now explain more concisely the content of the five main chapters:

• In chapter 2, we introduce most of the concepts and tools usedin the rest ofthe thesis. First of all we define the observables that quantify the rougheningprocess and we express how they behave for scale invariant surfaces. Wedo not only consider the Family-Vicsek scaling ansatz but wealso consider

4 Introduction

anomalous scaling and the scaling of anisotropic surfaces.Moreover, we in-troduce the notion of universality class and we analyze how scale invariancecan coexist with morphological instabilities and, thus, with pattern forma-tion. Finally, we give some details about the analytical technique broadlyused in this thesis, i.e. the dynamic renormalization group, and on the nu-merical scheme employed in the integration of the interfaceequations.

• In the first half of chapter 3, we summarize briefly the features of the movingboundary model of diffusive growth partially studied in thePhD thesis of M.Castro [1]. This model unifies in the same framework the diffusive growthof interfaces by chemical vapor deposition and by galvanostatic electrodepo-sition. In the second part of the chapter, the reader can find the first originalcontribution of the present thesis. We focus our study on thelinear and non-linear effective interface equations arising from the perturbative expansionof the moving boundary model under a small slope condition. Alarge partof this analysis has been published in(1) 1 and (2), while the remainingresults are contained into the preprint(3).

• In chapter 4, we use extensively the dynamical renormalization group tech-nique in order to obtain the asymptotic (critical) behaviorof the non-local in-terface equation found in the previous chapter. We generalize this equationto an entire family that is characterized by non-locality, non-linearity andmorphological instability. Due to the universality of thiscontinuum descrip-tion we can predict the critical exponents of the whole class. By numericalintegration of the equations, we support our analytical predictions with nu-merical estimates of the critical exponents. The most important results of thischapter are contained in the letter(4) and the preprint(5), while the com-plete details of the dynamical renormalization group analysis of non-localequations can be found in(6).

• In chapter 5, we depart from the main physical system of our investigations(although falls in the same theoretical framework) and focus our attentionon an unexpected new result observed in surface patterning by ion-beamerosion. The anisotropic Kuramoto-Sivashinsky equation is an importantinterfacial equation derived from first principles, that reproduces many ex-perimental features in this context. Once more, by employing the dynamicalrenormalization group we propose a theoretical explanation of the time evo-lution of ripple patterns obtained in numerical simulations of this equation,giving a formal argument for the interpretation of some experimental obser-vations. This theoretical result is an integral part of preprint (7).

• In chapter 6, we introduce a multigrid phase-field model withfluctuationsin order to integrate numerically the moving boundary model, and, conse-quently, to overcome the limitations imposed by our previous small slopes

1References in parenthesis are provided in the next section.

1.2 Published articles and preprints 5

approximation. In this chapter, after a first numerical calibration of themodel, we are able to study the late times morphologies produced by electro-chemical deposition in a multiscale perspective. In fact, the complex inter-actions between the growing structures at the surface and the long range dif-fusion field lead to an effective interface composed by all the active sites ofthe growing aggregate. Encompassing all the relevant macroscopic scales ofthe electrodeposition process we can investigate the kinetic roughening pro-cess of this effective interface. The analysis of kinetic roughening in surfacegrowth by electrodeposition has been published in(2), while the phase-fieldformulation is the central theme of preprint(3).

1.2 Published articles and preprints

(1) M. Nicoli, M. Castro, and R. Cuerno,Unified moving-boundary model withfluctuations for unstable diffusive growth, Physical Review E78, 021601 (2008).

(2) M. Nicoli, M. Castro, and R. Cuerno,Kinetic roughening in a realistic modelof non-conserved interface growth, Journal of Statistical Mechanics: Theory andExperiment P02036 (2009).

(3) M. Nicoli, M. Plapp, M. Castro, and R. Cuerno,Surface kinetics in a phase-field model of diffusive growth, preprint (2009).

(4) M. Nicoli, R. Cuerno, and M. Castro,Unstable nonlocal interface dynamics,Physical Review Letters102, 256102 (2009).

(5) M. Castro, R. Cuerno, M. Nicoli, J. G. Buijnsters, and L. Vazquez,The physicsof cauliflower-like growth, preprint (2009).

(6) M. Nicoli, R. Cuerno, and M. Castro,Dynamical renormalization group analy-sis of nonlocal unstable interface equations, preprint (2009).

(7) A. Keller, M. Nicoli, R. Cuerno, S. Facsko, and W. Moller,Pattern rotation inthe anisotropic stochastic Kuramoto-Sivashinsky equation, preprint (2009).

Interfacial height equations

2.1 Continuum equations

Many growing surfaces can be studied on a coarse grained scale in the hydrody-namic limit (long wavelength and long times). Effective descriptions include themost relevant physical mechanisms involved in the dynamics, even if they act atdifferent length scales separated by several orders of magnitude. For example, tworelaxational mechanisms such as surface tension and surface diffusion: the formeris more relevant at large scales and the latter is at smaller ones.

The dynamics of non-equilibrium interfaces can be cast intoa continuum stochas-tic differential equation for the surface heighth(r, t) at timet above pointr on ad-dimensional substrate [2; 3], see figure 2.1. This description allows to increasethe length and time scales that are accessible by analyticalor numerical methods(with the limitation of being only capable to describe single-valued interfaces).

Thus, the whole dynamical process is described by a stochastic differentialequation (Langevin equation) as

∂th(r, t) = Ξ(r, h, t) + η(r, t), (2.1)

whereΞ is the deterministic part of the equation and usually depends on the localheight function and its spatial derivatives. The second term on the right hand sideof equation (2.1),η, is a noise term that represents all the fluctuations involvedin the growth process (due to erosion, growth, diffusion, . .. ) and, frequently, isassumed to be Gaussian and uncorrelated.

8 Interfacial height equations

x1

x2

h(!r, t)

L

L

Figure 2.1 – Continuum representation of a surface in real space for a two-dimensionalsubstrate.

The exact functional form of equation (2.1) can be obtained systematicallyconsidering the symmetries of the system [3], or, alternatively, can be derived fromatomistic models [4; 5; 6; 7; 8] or from a moving boundary description of theprocess [9] (see chapter 3).

Starting from an ideally flat interface, equation (2.1) produces a rough surface.Different points of the interface are correlated as a resultof the interplay betweenthe fluctuations (modeled byη) and the deterministic part. The correlation lengthgrows in time until the whole system is correlated and reaches a stationary state(also referred to as saturation). This time evolution of thestatistical features ofrough interfaces is also known as kinetic roughening.

Kinetic roughening can be characterized quantitatively byseveral observables:the mean height and the global interface width. Themean heightof the surface ish(t) =

∑

h(r, t)/L whereL is the lateral size of thed-dimensional substrate, seefigure 2.1. Theglobal interface widthcharacterizes the fluctuations on the heightof the interface around its mean value (the standard deviation)

W (L, t) =

⟨[

1

L

∑

r

[

h(r, t) − h(t)]2

]1/2⟩

η

, (2.2)

where〈. . . 〉η means average over noise realizations.

Rough surfaces display regimes where the width increases asa power law oftime, namely,W (t) ∼ tβ. Thus, in a plot of the global interface width versus time(in log-log scale) we can find a characteristic exponentβ, calledgrowth exponent.When the correlations between the points onto the surface,ξ‖, reach the systemsizeL, the width no longer grows with time. This happens whent1/z ∼ L, wherez is thedynamical exponent. Finally, at saturation, the width scales with the systemsize asWsat ∼ Lα, where another exponent appears,α, known as theroughnessexponent.

2.1 Continuum equations 9

In summary,

W (L, t) ∼ tβ, ξ‖ ∼ t1/z for t ≪ Lz,

W (L, t) ∼ Lα, ξ‖ ∼ L for t ≫ Lz.(2.3)

This simplified picture, known as theFamily-Vicsek(FV) scaling Ansatz, is usu-ally only valid in certain regimes neglecting transients and subdominant mecha-nisms. Moreover, it allows to relate the three exponents through z = α/β andprovides a simple way to check the consistency of the ansatz by collapsing differ-ent curves obtained for different system sizes. Specifically,

W (L, t) ∼ Lαf1 (t/Lz) ∼ tβf2(L/t1/z), (2.4)

such that curvesW (L, t) obtained for different system sizes collapse by rescalingthe global interface width by its saturation value, and timeby tsat (see figure 2.2for a schematic representation). In this framework the scaling functions behave inthe following way

f1(u) ∼ uβ, f2(u) ∼ uα when u≪ 1,

f1(u) = const., f2(u) = const. when u≫ 1.(2.5)

The existence of critical exponents and the functionsf1 and f2 agrees with theconcept of Universality borrowed from the theory of equilibrium phase-transitions.

tx

β L

log(t)

log

(W(L

,t)

)

log(t)

log

(W(L

,t)/

Lα

)

log

(W(L

,t)/

Lα

)

log (t/Lz)

Figure 2.2 – Collapse of the global roughnessW .


Experimentally, the global roughness is usually not accessible because of thelarge system size that can easily exceed the maximum range ofinspection of thedevices employed in the measurement. Additionally, the saturarion regime may befar from the times that can be explored in a laboratory. In this case, the interfacecan be characterized by thelocal interface width (or local roughness)

w(l, t) =

⟨

1

n

n∑

i=1

1

|l|∑

r∈Ii

[

h(r, t) − h(Ii, t)]2

1/2⟩

η

, (2.6)

calculated over a partition of the system withn observation windows,Ii ≡ I(r0, l),of sizel ≪ L, and the mean height is calculated inside the window. In the pres-ence of FV dynamic scaling the local interface width of a self-affine surface, i.e.a surface without any characteristic length scale besides the system size, shouldexhibit the same scaling behavior as the global interface width. Then, at lengthscales smaller than the correlation length,w(l, t) ∼ lα. At long distances, whenl ≫ ξ‖, the local width scales with time asw(l, t) ∼ tβ. But, for surfaces thatare not self-affine, the local and the global interface widthmay exhibit differentscaling behaviors (namely, different roughness exponents).

Two additional important observables are the height-height correlation function

C(l, t) =

⟨

1

L

∑

r

[h(l + r, t) − h(r, t)]2⟩

η

, (2.7)

and thesurface structure factor(or power spectral density), defined in Fourier spaceby

S(k, t) = 〈hk(t) h−k(t)〉η . (2.8)

In this last expression, we apply the Fourier transform (FT ) to the fluctuations ofthe height around its mean value, defining in this case

hk(t) =∑

r

eik·r[

h(r, t) − h(t)]

. (2.9)

From the definitions above, we can find some useful relations among the obsev-ables:

W 2(L, t) =∑

k

S(k, t), (2.10)

C(l, t) ∼ w2(l, t), (2.11)

C(l, t) ∼∑

k

[1 − cos(k · l)]S(k, t). (2.12)

Equation (2.10) shows that we can estimate the roughness exponent from the plotof S(k, t) as a function of the wavenumber (in log-log scale) when the growth


processes has reached the saturation regime. In fact, the FVdynamic scaling hy-pothesis can be written for the structure factor as

S(k, t) = k−(2α+d)s1(kzt), (2.13)

where the scaling functiong behaves as

s1(u) ∼ u(2α+d)/z when u≪ 1,

s1(u) = const. when u≫ 1.(2.14)

Frequently, the growth exponent is estimated from the powerlaw behavior of theglobal roughnessW , whereas the roughness exponent is obtained from the struc-ture factorS. Besides, the consistency of the estimated exponents is checked bymeans of data collapses of theW (L, t) andS(k, t) curves. A comprehensive re-view about the various procedures that have been employed toestimate the scalingexponents can be found in [3].

2.1.1 Anomalous scaling

Scale invariance implies that there is no characteristic length scale in the surface(except for the system size) so that all scales obey the same physics. In particularthe local interface width, defined by equation (2.6), shouldscale according to theFV scaling Ansatz, hence

w(l, t) ∼ tβ when t1/z ≪ l,

w(l, t) ∼ lα when l ≪ t1/z.(2.15)

This scaling behavior for the local width is not guaranteed in general when theFV scaling holds for the global width, since the self-affinity of the interface is anadditional independent condition [10].

Many growth models display a roughness exponent for the global width that islarger than one,α > 1. When this condition is met, the model produces surfacesthat aresuperroughand the usual assumption of the equivalence between the globaland local description of the surface is no longer valid [11; 12; 13]. In the presenceof anomalous scaling,αl andα differ, hence not all length scales are equivalent inthe system. For this reason, the usual FV scaling for the local width (2.15) has tobe replaced by

w(l, t) ∼ tβ when t1/z ≪ l,

w(l, t) ∼ lαltβl when l ≪ t1/z ≪ L,

w(l, t) ∼ lαlLzβl when l ≪ L≪ t1/z ,

(2.16)

whereαl = 1 is called local roughness exponent andβl = (α − αl)/z. Thisbehavior has been termedanomalous scalingin the literature [13; 14; 15] and


different models have been studied in whichαl andβl take values different form1andβ − 1/z, respectively [16].

If we consider only the short and the long time behavior of theheight-heightcorrelation function (2.16), we observe a scaling behaviorsimilar to the FV scaling

w(l, t) = tβfa

(

l/ξ‖(t))

, (2.17)

but with an anomalous scaling function

fa(u) ∼ uαl when u≪ l,

fa(u) ∼ const. when u≫ 1,(2.18)

with a new independent exponentαl [17]. Besides, it has been shown that there aresurfaces whose scaling is called intrinsically anomalous,for which αl can differfrom 1 and the global roughness exponent can be actually anyα > αl.

A detailed and comprehensive study about the different types of anomalousscaling can be found in [10; 17]. As in the work of Ramascoet al.we restrict theanalysis to the one-dimensional case (generalization tod-dimensions is straightfor-ward). In order to investigate all the possible forms that the scaling functions canexhibit when solely the existence of generic scaling is assumed, we only assumethat

S(k, t) = k−(2α+1)s2(kt1/z), (2.19)

where the scaling function has the general form

s2(u) ∼ u2(α−αs) when u≫ 1,

s2(u) ∼ u2α+1 when u≪ 1,(2.20)

and the exponentαs is the so-calledspectral roughness exponent. This scalingansatz is a natural generalization of the scaling proposed by Lopezet al. in [10;18] for anomalous scaling.

Despite the apparent simplicity of the generalization ofS for the intrinsic case,the scaling of the local width in much more involved. Thus, ifwe write

w(l, t) ∼√

C(l, t) ∼ tβg(l/ξ‖), (2.21)

the scaling functiong(u) is not unique [17] because of the no-commutativity of thelimits involved in the calculation of (2.12)1.

We can now summarize all the scaling behavior reported in theliterature thathave been obtained from the generic dynamic scaling Ansatz (2.20). Two major

1Ford = 1, thek variable in the sum of equation (2.12) varies between the lowest wavenumber inthe system,2π/L (given by the system size), to the highest,π/∆x (given by the space discretization).Substituting equations (2.19) and (2.20) into (2.12), one can see that when we try to obtain thebehavior of this function for large length scales in the continuum limit, i.e. taking the limits∆x → 0,ξ‖/L → ∞ andL → ∞, these limits do not commute [10; 18] resulting into a different scalingbehavior ofg(u) depending on the value of the spectral exponentαs [17].


cases can be distinguished, namely,αs < 1 andαs > 1. Forαs < 1 the integral inequation (2.12) (note that the sum becomes an integral in thecontinuum limit) hasbeen computed in [10; 18]

gαs<1(u) ∼ uαs when u≪ 1,

gαs<1(u) ∼ const. when u≫ 1.(2.22)

In this case, the corresponding scaling functiongαs<1 is proportional tofa, seeequation (2.18), and the spectral and the local roughness exponents are equal, i.e.αs = αl, the surface displaying an intrinsic anomalous scaling. Furthermore, theinterface would satisfy the FV scaling (for the local as wellas for the global width)only if α = αs were satisfied for the particular growth model under study.

A different type of anomalous dynamic shows up for growth models in whichαs > 1. In this case, one finds that, in the large scale limitL→ ∞, the integral inequation (2.12) has a divergence arising from the lower integration limit. In orderto avoid the divergence, the integral is computed keepingL fixed. We then obtainthe scaling function

gαs>1(u) ∼ u when u≪ 1,

gαs>1(u) ∼ const. when u≫ 1.(2.23)

So, in this case one always getsαl = 1 for anyαs > 1. Thus, for growth modelsin whichα = αs, one recovers the superrough scaling behavior [10; 18].

However, there is no restriction about the values ofα andαs, hence whenαs > 1 andα 6= αs a different anomalous scaling can be possible. Ramascoet al. [17]. have shown that a model displaying this faceted scaling behavioris the Sneppen model of self-organized depinning [19]. Morerecently an ex-perimental realization has been reported [20]. The main feature of this type ofanomalous roughening is that it can be detected only by determining the scalingof the structure factor [17]. In fact, the stationary regimeof a surface exhibitingthis kind of anomalous scaling will be characterized in the saturation regime byW (L) ∼ Lα andw(l, L) ∼

√

C(l, L) ∼ lLα−1, however, the structure factorscales asS(k, L) ∼ k−(2αs+1)L2(α−αs), where the spectral roughness exponentαs appear as a new and independent exponent.

From the picture depicted so far, it is clear that the situation of a model display-ing anomalous roughness has to be considered carefully. Moreover, taking into ac-count the reasons exposed above, we consider the structure factor as a fundamentalobservable in order to estimate the global or the spectral roughness exponents andto classify the scaling behavior of the model considered. Finally, we conclude thissection by summarizing the four different classes of scaling in table 2.1.

2.1.2 Scaling of anisotropic surfaces

Anisotropic surfaces may exhibit an even more complex dynamic scaling behaviorthan isotropic ones since the anisotropy of the surface is reflected in the correlation


αs < 1 ⇒ αl = αs

αs = α Family-Vicsek

αs 6= α Intrinsic

αs > 1 ⇒ αl = 1αs = α Superrough

αs 6= α Faceted interfaces

Table 2.1 – Summary of the four possible scaling behaviors in terms of the three differentroughness exponents: globalα, localαl and spectralαs [17].

function (2.7). Therefore, different roughness exponentsαx,y can be expected cor-responding to correlations computed in thex andy directions. Rescaling along thex direction in the saturation regime, i.e. when the time dependence can be ignored,leads to

C(x, y) ∼ bαxC(b−1x, b−χxy), (2.24)

with the anisotropy exponentχx = αx/αy accounting for the different rescalingfactors along the two directions that are required in order to have scale invariance.Equivalently, the scaling behavior of the anisotropic surface in real space is deter-mined by the two independent exponents,αx andαy.

However, in momentum space, we can apply similar definitionsto the scalingof the structure factor of one-dimensional slices along thex and they directions.This procedure leads to [21; 22]

S(kx) ∼ k−(2αx+2−χx)x , (2.25)

S(ky) ∼ k−(2αy+2−χy)y , (2.26)

with two independent roughness exponents in momentum spaceαx,y. Then, therelation between the roughness exponents in momentum and inreal space is givenby [22]

αx = αx − 1 − χx

2= αx − 1 − αx/αy

2, (2.27)

αy = αy −1 − χy

2= αx − 1 − αy/αx

2. (2.28)

Therefore, the dynamic scaling behavior of the surface is described by four differ-ent roughness exponents that characterize the surface along thex or they directionin momentum or real space. Only in the case ofχx = χy = 1, i.e. isotropic scaling,we haveαx = αy, αx = αx andαy = αy.

Equations (2.24)-(2.28) imply that the roughness exponents in momentum spacecan be calculated from the real space exponents determined from real space mea-surements. However, substituting (2.27) and (2.28) into equations (2.25) and (2.26)


gives [23]

S(kx) ∼ k−(2αx+1)x , (2.29)

S(ky) ∼ k−(2αy+1)y . (2.30)

This last result shows that, ifkx,yt1/z ≫ 1, the real space exponents can be de-

termined from momentum space measurements by evaluating the one-dimensionalstructure factors in thex and they directions, respectively.

2.1.3 Universality classes in surface growth phenomena

Universality is an attractive feature in statistical physics because a wide range ofmodels and experimental systems can be classified in terms oftheir scaling expo-nents, a universality class being defined by the complete setof critical exponentsand their dependence on system dimension. As stated in the first chapter, oneexpects that different modeling approaches provide the same universal behaviorwhere the microscopic details are not relevant. In this section we report on thedifferent discrete and continuum models that have been usedin the description ofsurface growth phenomena trying to elucidate the idea of universality class in outof equilibrium systems.

Let us start with the simplest stochastic surface growth process: random depo-sition (RD). Mainly, a RD process models a growth condition in which particlesare adsorbed at surface sites independently. In fact, in itsdiscrete formulation, RDis modeled by a “rain” of particles falling onto the surface which are added to thegrowing aggregate when they reach it. The rules of RD do not include any relax-ation mechanism, so columns grow independently, linearly,and without bound. InRD the roughness exponentα and correspondinglyz are not defined because themodel does not reach the saturation regime, and only the growth exponent is welldefined. In fact, this regime is reached when the correlationlengthξ‖ is equal tothe system sizeL, but, in this model,ξ‖ is exactly zero for all times. The modelcan be solved exactly to find that [3]W ∼ t1/2, henceβ = 1/2. The continuumdescription of RD is the following Langevin equation

∂th(r, t) = F + η(r, t), (2.31)

in which the particle flux at each position onto the surfacer is not constant butfluctuates around a mean valueF . Due to the uncorrelated nature of the process,the fluctuation term has zero mean, i.e.〈η(r, t)〉 = 0, and variance

〈η(r, t) η(r′, t′)〉 = 2Π0δd(r − r′) δ(t − t′). (2.32)

The solution of this equation gives a global widthW =√

2Π0t that grows withoutbound withβ = 1/2, in accordance with the discrete version of RD [3].

Going back to the discrete version of RD, for instance a dynamical modifica-tion in order to reach a saturation regime requires the introduction of a relaxation


mechanism. In fact, we can include surface relaxation if we allow the depositedparticles to e.g. diffuse along the surface until they reacha position with lowerheight.

L

Figure 2.3 – Schematic representation of the rule in random deposition with surfacerelaxation.

For example, the diffusion length considered in the figure 2.3 is equal to onelattice spacing. This relaxation process generates correlations among neighbor-ing sites and, ultimately, all the points of the interface are correlated. Simulationsin one dimensional substrates result into scaling exponents β = 0.24 ± 0.01 andα = 0.48± 0.02 [24]. In the continuum equation language, this discrete relaxationmechanism can be reproduced introducing a term in equation (2.1) that smoothesout height fluctuations. Taking into account that RD with relaxation respects sev-eral symmetries as translational invariance along the surface, in the growth direc-tion and in time, i.e. under

r → r + ∆r, t→ t+ ∆t, h→ h+ ∆h, (2.33)

and additionally, up/down symmetry forh and rotation/inversion symmetry aboutthe growth direction, i.e. under

h→ −h, r → −r, (2.34)

we can write down the simplest stochastic equation with all these symmetries

∂th(r, t) = F + ν∇2h+ η(r, t). (2.35)

Hereν is called a surface tension coefficient; in fact, the termν∇2h is indeedresponsible for the relaxation mechanism that leads to saturation. This term actson the irregularities of the surface height redistributingthe “mass” from the peaksto the valleys, while maintaing the average height unchanged. Thus the surface


tension acts as a conservative relaxation mechanism. equation (2.35) is calledEdwards-Wilkinson (EW) equation.

Equation (2.35) is linear, so we can calculate exactly the exponents

α = (2 − d)/2, β = (2 − d)/4, z = 2. (2.36)

In d = 1 these exponents are compatible with those of the RD model with surfacerelaxation, so they belong to the same universality class. Further support for thisconclusion is given by numerical simulations ford = 2 [25], in fact they reveal avery weak divergence of global width, consistent with logarithmic scaling (α = 0),as predicted by (2.36). Also, from the works by Vvedensky et al. [4; 5; 6], anexplicit link between the two models can be shown.

Let us consider now discrete models with lateral growth. In the ballistic de-position (BD) model particles fall following a straight vertical trajectory until theyreach the surface, whereupon they stick. There are many different variants of thissticking rule, for example, the nearest-neighbor (NN) model in which the particlesstick to the first nearest neighbor on the aggregate (see leftpanel of figure 2.4) ornext-nearest neighbor (NNN) model (right panel of the same figure). In this lastmodel particles can furthermore stick to a diagonal neighbor as well.

Figure 2.4 – Schematic representation of ballistic deposition with nearest-neighbor (leftpanel) and next-nearest neighbor (right panel) rules.

In figure 2.5 we show a typical interface after the fall of several (∼ 105) parti-cles, note the difference between BD and RD with surface relaxation. The lateralsticking rule of BD leads to a morphology characterized by lateral growth in whichlarge voids (due to overhangs) can be observed. Ind = 1 numerical simulationsgive the exponentsα = 0.47 ± 0.02 andβ = 0.33 ± 0.01 [26] that are differentfrom those of the EW equation.

In order to model lateral growth, we need to include non-linear terms to theevolution equation. By considering the Pythagorean theorem to compute the localincrease in the surface height induced by a particle that sticks on the surface andusing a small slopes expansion [3], i.e.|∇h| ≪ 1, the resulting equation with the


Figure 2.5 – Surface generated by ballistic deposition with nearest neighbor rule (leftpanel) and by random deposition with surface relaxation (right panel) [3]. Note the lateralgrowth of the aggregate in BD.

most significant terms is

∂th(r, t) = F + ν∇2h+λ

2(∇h)2 + η(r, t). (2.37)

In this equation the non-linear, quadratic term is the one responsible for lateralgrowth. In fact, it generates an increase in height by addingmaterial to the pointsof the interface with large slope, typically the lateral sides of the bumps on thesurface, resulting in a growth that is conformal to the interface. Clearly this non-linear mechanism does not conserve the value of the mean height and generatesan additional growth velocity proportional to the parameter λ. equation (2.37) wasfirst proposed by Kardar, Parisi and Zhang (KPZ) [27] as a non-linear extension ofthe EW equation.

Due to the non-linear character of the KPZ equation a closed solution cannot beobtained. Kardaret al. [27] applied a powerful approximate method, the dynami-cal renormalization group (DRG), to obtain insight into thescaling properties andexponents of this equation (a detailed description of this technique can be found insection 2.2). DRG gives the exact values of the critical exponents ford = 1 butfails whend > 1 [3].

In d = 1 the exponents areα = 1/2, β = 1/3 and z = 3/2, compatiblewith the exponents found in the numerical simulations of ballistic deposition withthe NN and NNN rules. Nevertheless, non-universal parameters, such as the meangrowth velocity, are different [28]. In fact, this parameter is proportional to thestrength of the non-linearity introduced by the lateral sticking rule of the model


[3]. Changing this rule we thus change the “microscopic” details but we observethe same critical behavior.

So far we have shown that, when a discrete model has the same critical behaviorof a certain continuum equation, we can conclude that they belong to the sameuniversality class. The modeling approach is then a matter of convenience. Forinstance, to estimate the critical exponents of the KPZ equation for d > 1 it is moreefficient to study a solid-on-solid model in the same universality class [29] than tosimulate the KPZ proper equation. On the other hand, a continuum equation isproposed considering the model symmetries or, sometimes, derived from a movingboundary or a discrete model, and typically, it represents adescription of the systemat larger scales.

We provide a classification of the universality classes thatcan be found in theliterature considering only continuum equations. In general, a natural and standardclassification of the universality classes consider the existence of conservation lawsof the surface height in the representative equation [30]. Thus, we have:

1. Conserved systems: The representative Langevin equation has the shape of acontinuity equation, reflecting conservation law on the mass of aggregatingunits at the surface. Such kind of symmetry may arise dynamically in aneffective way, or be explicitly manifest in the physical mechanisms leadingto growth. The three main classes are:

(a) EW universality class: There are surfaces whose dynamics can beasymptotically described by an evaporation-condensationeffect (com-bined with fluctuations) of the Gibbs-Thomson type. Note that in thiscase, apart from the external flux, the surface evolves as around anequilibrium state. equation (2.35) is simply model A for theGinzburg-Landau functional in the Gaussian approximation. The critical expo-nents characterizing this universality class can be determined exactlyfor any dimension, see (2.36).

(b) Linear Molecular Beam Epitaxy (MBE) universality class: In the sameway as equation (2.35) can be thought of, in the simplest representa-tion, as the equilibrium dynamics of a surface minimizing its area, animportant class of rough surfaces can be described effectively as mini-mizing their mean curvature. The Langevin equation representing thisuniversality class reads

∂th(r, t) = F −K∇4h+ η(r, t). (2.38)

Again (as reflected in the previous equation being linear) the scalingexponents are exact for any value ofd

α = (4 − d)/2, β = (4 − d)/8, z = 4. (2.39)

Although the thermodynamic limit of a system like (2.38) is not welldefined, this does not prevent real (and finite) systems from displayingthe referred scaling behavior [31; 32].


(c) Non-linear MBE universality class: To some extent in analogy withthe relationship between the KPZ and EW equations, much workhasbeen devoted to elucidating the appropriate non-linear version of (2.38)leading to a well defined universality class. Its Langevin equation, theso-called Lai-Das Sarma-Villain (LDV) equation [33; 34; 35], is

∂th(r, t) = F −K∇4h+ λ∇2(∇h)2 + η(r, t). (2.40)

In this case the critical exponents are close to [34; 36]α = 1, β = 1/3andz = 3 for d = 1 while they areα = 2/3, β = 1/5 andz = 10/3for d = 2.

2. Non conserved systems: If the height evolves in time in absence of a con-servation law on the mass of aggregating units at the interface, for instanceif bulk vacancies are significant, or conformal growth occurs [37; 38], thenthe expected asymptotic behavior is that of the KPZ equation(2.37). Thisuniversality class is exceedingly important in the theory of non-equilibriumprocesses, however, there are few experimental realizations of this scalingbehavior.

These universality classes have attracted much work in the kinetic rougheningfield. But the above list is far from complete. Generalizing the hypothesis of local-ity and the correlations of the noise un equation (2.1), someadditional universalityclasses have also been proposed in the literature.

• Noise properties: It is well known theoretically that the introduction of par-ticular features in the noise distribution can modify the scaling exponents[3; 2; 39]. A somewhat trivial one is considering conserved noise (as inmodel B [40] for conserved dynamics of an order parameter), instead ofnon-conserved noise (2.32). Usually conserved noise,ηc, used in this con-text has zero mean but a variance that depends on the wavenumber, readingin real space

〈ηc(r, t) ηc(r′, t′)〉 = −2Π2∇2δd(r − r′) δ(t − t′), (2.41)

and after Fourier transform

〈ηck(t) ηc

k′(t′)〉 = 2Π2k2δd(k + r′) δ(t − t′). (2.42)

Other exotic modifications include power-law correlationsfor the noise [41],or a noise amplitude which is itself power-law distributed [42; 43], which hasbeen proven to be useful for the description of spatio-temporal chaos [44].These modifications provide non-trivial scaling exponentsthat depend con-tinuously on the parameters that characterize the properties of the noise.

• Non-locality: Another way to obtain scaling exponents depending continu-ously on model parameters is by considering long range effects on the sur-face, as represented in the dynamic equation by integral kernels that couple


the height field values at all points on the surface [45; 46; 47; 48]. The pa-rameters mentioned characterize in this case the spatial decay (typically asa power law) of such integral kernels. If non-locality appears as a conse-quence of a diffusing field being coupled to the height field, then frequentlyscaling exponents are also different from those of the universality classessummarized above [49; 50; 51; 52; 53; 54; 9].

• Quenched vs time dependent noise: Another issue obscuring the identifica-tion of universality classes is the fact that two surfaces may share the samevalues of the exponents, but still be physically rather different, the noisedistribution being quenched in one case and time dependent in the other.Such a situation occurs for example when comparing (ford = 1) an EWtype equation with a diffusion coefficient which is a random variable with aquenched columnar distribution [55; 10; 18], with the linear MBE equation.Incidentally, for surface growing in a quenched disorderedmedium, like aHele-Shaw cell with randomly varying gap spacing, scaling exponents canalso depend continuously on parameters characterizing thedisorder [56; 57].

table 2.2 summarize the universality classes for some Langevin equations thathave been used in kinetic roughening [3]. The validity of these exponents is re-stricted to the asymptotic state of these equations. Note that the exponents for theKPZ universality class have been estimated using a restricted solid-on-solid model,for more details see [29].

2.1.4 Morphological instabilities

An issue that has already arisen in the first chapter is the existence of instabilitiesin the context of models and experiments of kinetic roughening. Already sincethe first systematic studies of kinetic roughening, physical instabilities have beenwell known to take place in the systems studied. The paradigmatic example inthis context is the Mullins-Sekerka instability arising incrystal growth from anundercooled melt, and in other growth systems (see [58; 59]). More recently theyhave been seen to play a key role in the dynamics e.g. of crystal growth by atomicor molecular beams [60]. In these systems, the Ehrlich-Schwoebel effect hinderingthe crossing of steps by adatoms leads−in the case of growth onto a high symmetrysurface− to development of mound formation.

But, in spite of their ubiquity, the effect of physical instabilities on the prop-erties of the surface roughness has been overlooked to a large extent. Following asimilar scheme to the preceeding section, we explore some discrete and continuummodels where instabilities play a key role.

Multiparticle biased diffusion limited aggregation (MBDLA) is a discrete modelintroduced to describe quantitatively experiments of amorphous electrochemicaldeposition [61]. MBDLA maps the physics of a typical ECD experiment into aset of discrete rules that model the diffusion and the aggregation processes of thecations onto the cathode (for more detalis see section 3.1.2). This minimal model


Type Langevin equation α β z

RD ∂th = η − 1/2 −

LCN2 ∂th = ν∇2h+ η2 − d

2

2 − d

42

(EW)

NNN2 ∂th = ν∇2h+ λ2 (∇h)2 + η

(KPZ) d = 1 1/2 1/3 3/2

d = 2 0.39 0.25 1.61

d = 3 0.31 0.19 1.69

d = 4 0.26 0.15 1.75

LCN4 ∂th = −K∇4h+ η4 − d

2

4 − d

84

(MBE)

LCC2 ∂th = ν∇2h+ ηc −d/2 −d/4 2

LCC4 ∂th = −K∇4h+ ηc 2 − d

2

2 − d

84

NCN4 ∂th = −K∇4h+ λ∇2(∇h)2 + η4 − d

3

4 − d

8 + d

8 + d

3(LDV)

NCC4 ∂th = −K∇4h+ λ∇2(∇h)2 + ηc

d ≤ 12 − d

3

2 − d

10 + d

10 + d

3

d > 12 − d

2

2 − d

84

Table 2.2 – Summary of the universality classes in continuum growth equations [3].The three letters stand forL inear/Non-linear equation driven by aConservative/Non-conservative dynamic with aConservative/Non-conservative noise term. The number2or 4 refers to the degree of the linear part of the equation.

displays interesting features such as morphological instabilities. In fact, the Lapla-cian character of the model leads to a transient regime associated with the onsetof the instability in which the effective scaling is intrinsically anomalous and de-scribed by non-universal exponents [62; 63]. An additionalexample of a discretemodel in which a morphological instability has been observed has been introducedby Cuernoet al. in order to describe surface erosion via ion beam sputtering[64;


65]. Moreover, from this model, Lauritsenet al. derived an effective continuumequation for the height of the surface, called Kuramoto-Sivashinsky (KS) equa-tion, that is characterized by a linear morphological instability, analogous to theMullins-Sekerka instability [65].

Recently, instabilities have been put forward as a possibleexplanation for thedifficulty in observing experimentally the ideal universality classes discussed insection 2.1.3. In fact, in many equations with morphological instabilities crossoverphenomena can hinder the observation of the asymptotic scaling of the equation.This effect is due to the difference between the typical scales associated with theoperators that contribute to generating the instability and those controlling theasymptotic behavior. Hence, at short times or small system sizes we can estimatethe critical exponents associated with a given operator butthey can be differentfrom the actual asymptotic growth exponents. A paradigmatic example is providedby the (noisy) Kuramoto-Sivashinsky equation

∂th(r, t) = F − ν∇2h(r, t) −K∇4h(r, t) +λ

2(∇h)2 + η(r, t), (2.43)

whereν,K > 0. This equation has been actually derived form constitutiveequa-tions−not merely based on scaling and/or symmetry arguments− as a descriptionof kinetically roughened surfaces arising in a number of physical contexts. To namea few of them, we can mention directional solidification of dilute binary alloys [66],solidification of a pure substance at large undercooling including interface kinetics(see [67] and references therein), erosion (sputtering) byion bombardment (IBS)[64; 68], dynamics of steps on vicinal surfaces under MBE conditions [69; 70; 71],or growth by CVD or ECD [53; 54; 9]. In our context, a distinguishing featureof this continuum model is the negative, and therefore linearly unstable, surface-tension-like term. As shown below, competition between this term and the stablebiharmonic one leads to the existence of a finite band of linearly unstable Fouriermodes in the surface.

In general, we can study the behavior of the surface generated by equation(2.1). We consider a flat interface as initial condition and ageneric uncorrelatednoise with zero mean and variace

〈ηk(t) ηk′(t′)〉 = 2Π(k)δd(k + r′) δ(t − t′), (2.44)

where in case ofΠ(k) = Π0 the noise is non-conserved while it is conserved whenΠ(k) = Π2k

2. The deterministic functionΞ(r, t) can be decomposed in threemain parts

Ξ(r, t) = F + σ(r)h(r, t) + N [h(r, t)], (2.45)

whereF is average addition of mass andσ(r) is a factor named (linear) dispersionrelation, such as

ν∇2h(r, t) ⇒ σk = −νk2, (2.46)

−K∇4h(r, t) ⇒ σk = −Kk4, (2.47)


for the surface tension and surface diffusion respectively. Finally, N contains allthe non-linearities of the evolution equation. After Fourier transform, equation(2.45) reads

Ξk(t) = Fδd(k) + σkhk(t) + N [h]k(t). (2.48)

From this expression it is evident that theF term modifies only the zero mode i.e.the value of mean heighth. In a frame of reference comoving with the interface ata velocity equal toF , this term can be removed fromΞ.

Due to the initial condition (hk(0) = 0, ∀|k| 6= 0) the evolution of equa-tion (2.1) is characterized at short times by the dominance of the noise term overthe deterministic part. The non-conserved noise introduces fluctuations that areindependent of the wavenumber while conserved noise startsto correlate the shortscales (large wavenumbers). In fact, considering only the the linear part of equation(2.1)

∂thk(t) = σkhk + ηk(t), (2.49)

all the Fourier modes are uncoupled and the solution is readily obtained after timeintegration

hk(t) =

∫ t

0ds eσk(t−s)ηk(s). (2.50)

The structure factor can be obtained immediately from the previous result

S(k, t) = Π(k)e2σk t − 1

σk

. (2.51)

Evidently, in the limit of very short times (t → 0) the structure factor is onlyproportional to the noise varianceS(k, t) ∼ 2Π(k)t. In the general case thatΠ(k)is a polynomial ofk, the large wavelength fluctuations are due to the non-conservedpart of the noise term, i.e.Π(0) = Π0 (in case of a pure conserved noiseΠ0 = 0).For this reason, through (2.10) the global widthW grows in time initially witha power law with exponent1/2, as in the random deposition case [see equation(2.10)]. At some point the space variations (derivatives) of the height build up andthe part of equation (2.49) that is proportional toσk cannot be neglected anymore.

Whenσk is negative for all the values ofk (stable dispersion relation as in theblue dashed curve of figure 2.6), fluctuations are smoothed sothat the interfacedoes not develop any pattern with a preferred wavelength. Inthis case the Fouriermodes continue to grow according to (2.51) and the interfacedisplays genuinescale invariance. Usually the non-linear terms in equation(2.1) are the last onesto contribute to the height evolution. The interplay between these and the otherdeterministic terms leads to saturation of the global widthand determines the valueof the critical exponents.

On the other hand, if the dispersion relation has a band of unstable modes, i.e.an interval ofk in whichσk takes positive values (see the red curve in figure 2.6),all the modes included in the unstable band grow exponentially in time whereasthose with a negative value ofσk are smoothed out as in the stable case. The


k

σk

Figure 2.6 – Examples of dispersion relations in Fourier space: the bluedashed curve isstable for everyk whereas the red solid curve has a band of unstable modes.

Fourier modekm that maximizes the structure factor grows exponentially fasterthan the other, therefore it is the responsible for the characteristic wavelength of thearising pattern. If the noise variance is independent on thewavenumber, as for thenon-conserved case,km is also the Fourier modes that maximizes the dispersionrelation.

A simple calculation will illustrate the connection between the dispersion re-lation and the maximum observed in the structure factor. Considering, for sim-plicity, the one-dimensional case, thekm mode is found solving the condition∂kS(k, t) = 0 that reduces to the equation

Π0σ′kσ2

k

[

e2σkt (2σkt− 1) + 1]

= 0, (2.52)

in case of non-conserved noise. The modes that satisfy the condition σ′k = 0satisfy equation (2.52) as well and, clearly, are time independent. Moreover, foreveryσk > 0 it is easy to check that the part of equation (2.52) between the squarebrackets is positive for everyt > 0: In the limit t → 0 the exponential in equation(2.52) can be expanded

limt→0

∂kS(k, t) = 0, ∼ 4Π0t2σ′k = 0. (2.53)

Consequently, the mode that maximizes the structure factordoes not changewith time. Additionally, when the noise variance depends onk the situation ismore complicated andkm can change with time. If we consider conserved noise,in this case the condition∂kS(k, t) = 0 reads

Π2k

σ2k

2σk

(

e2σkt − 1)

+ kσ′k[

e2σkt (2σk − 1) + 1]

= 0, (2.54)

and, for smallt, it reduces to the equation

limt→0

∂kS(k, t) = 0, ∼ 4Π2k t[

1 + t(

2σk + kσ′k)]

= 0. (2.55)


The solution of equation (2.55) usually differs from the maximum ofσk and changeswith time. When this occurs, the wavelength of the generatedpattern typically in-creases with time. Then, we refer to this regime ascoarsening. This behaviorusually arises from non-linear terms, for example the conserved KPZ non-linearity∇2(∇h)2, but in this case it is also found in the linear regime.

After the creation of the pattern, scale invariance is broken, since the systemselects a typical wavelength. A linearly unstable Langevinequation needs, at least,a non-linear term to asymptotically stabilize its behaviorand prevent blow up ofthe structure factor. For example, in the case of the Kuramoto-Sivashinsky equa-tion (2.43), the KPZ non-linearity acts, once the interfacedevelops slopes withsignificative magnitude, by disordering the pattern. The non-linearity correlatesdifferent wavelengths breaking the validity of relation (2.51) and its characteris-tic conformal growth stabilizes the surface [72]. Therefore, in this equation thelinear pattern wavelength, i.e.lKS = 2π

√

−2K/ν, separates two regions: atwavelengths belowlKS the interface shows a well defined cellular structure, i.e.an ordered pattern, whereas above this value the interface displays scale invariance[72]. Actually in the hydrodynamical limit even the deterministic KS equation isscale invariant and belongs to the KPZ universality class (at least ford = 1 [73;74]).

Note that, at large scales, equations (2.37) and (2.43) are indistinguishable fromthe point of view of symmetries, hence derivations based on the latter typically endup by proposing the KPZ equation as the continuum description even in systemswhere the KS equation is a better representation, due to the existence of physicalinstabilities, see for example [75] for the case of erosion by IBS. Note that thecrossover from the pattern-formation transient to the asymptotic kinetic rougheningcan be exceedingly large in the KS system [72]. Hence, it is conceivable thatwhen studying experimentally systems described by the KS equation and whoseasymptotic state is effectively unaccessible, fits to some effective (and improper)exponents are attempted that lead to incorrect conclusionson the scaling propertiesof the system.

At any rate, the long crossover can even hinder the actual experimental ob-servation of the asymptotic KPZ scaling in system describedby equation (2.43).Analogous conclusions can be drawn for systems in which physical instabilities areexpected, even if the relevant dispersion relation differsfrom the KS one. An openavenue for research in this connection is undoubtedly the interplay of noise andinstabilities and its implication for the interface morphology, and the formulationof generic continuum descriptions incorporating these phenomena.

2.2 Dynamical renormalization group technique

The exact solution of non-linear equations like KPZ or KS is far beyond the currentmathematical possibilities. Physicists and mathematicians have adoptd a variety ofapproximation methods in order to shed some light on their asymptotic behav-

2.2 Dynamical renormalization group technique 27

ior: weakly non-linear analysis, multiple scale analysis and renormalization group,have been used successfully in the context of out of equilibrium systems.

Renormalization group is a procedure introduced in quantumfield theory toeliminate divergences, and only subsequently became famous when it was suc-cessfully applied to equilibrium critical phenomena in theearly 1970s [76; 77].This invaluable contribution of Kenneth G. Wilson to theoretical and statisticalphysics was awarded by the Noble Prize in 1982. In the next section we will presentthe Forster-Nelson-Stephen renormalization group scheme[78] in the language ofgrowth equations. The FNS renormalization group was first used in the study ofthe hydrodynamic limit of Burgers equation and, later, has been used widely in thecontext of out of equilibrium phenomena (see, for example, Refs. [27; 41; 79; 73;74]).

2.2.1 The Forster-Nelson-Stephen theory

The FNS theory of renormalization adapts the approach due toMa and Mazenko[80] for dynamical critical phenomena to the determinationof the critical expo-nents of the noisy Burgers equation [78]

∂tv + λ(v · ∇)v = ν∇2v −∇η(r, t). (2.56)

Burgers equation describes a randomly stirred vorticity-free fluid and can be mappedinto the KPZ equation by using the change of variablev = −∇h. Thus, the Burg-ers equation and the KPZ equation should have related scaling exponents. In thisthesis we investigate the properties of stochastic equations in which the KPZ non-linearity appears, so that, from now on we consider only thisnon-linear term andwe use the FNS formalism.

First of all, we identify the cut-off wavenumber of the problem Λ that is in-versely proportional to the system discretization unitΛ = 2π/∆x [for example thelattice spacing for the Ising model, or the operator discretization in the numericalintegration of equation (2.1)]. For a surface growth system, the Fourier transformhk(ω) is truncated fork > Λ (often referred to as an ultraviolet cut off) and theheight and the noise fields are splitted into two disjoint intervals

hk,ω =

h<k,ω for 0 < k < Λ/b,

h>k,ω for Λ/b ≤ k ≤ Λ,

(2.57)

ηk,ω =

η<k,ω for 0 < k < Λ/b,

η>k,ω for Λ/b ≤ k ≤ Λ,

(2.58)

whereω is the frequency (the Fourier conjugated variable of time) and the scalingparameterb is taken larger that one. Now, in the FNS scheme we eliminate the high(or fast) modes by solving the growth equation perturbatively for h>

k,ω, and substi-tute the solution into the equation forh<

k,ω. In Fourier space, the non-linear term


of equation (2.1) induces a convolution between modes, so that the solution forh>

will contain terms proportional toh< (sometimes, non-linear terms are referred toas mixing in this context). We average the resulting equation over realizations ofthe fast mode part of the noise termη>

k,ω. The next step consists in rescaling thevariables and the fields in equation (2.1), (i.e.k, ω, h< andη<) and calculatingthe renormalized parameters. Finally, we can use the group property of the scalingtransformation in order to write the parameters renormalization in differential form.Using the identityb = eδl and considering an infinitesimal shellS ≡ [Λ/b,Λ] wecan perform a perturbative expansion in the limitδl → 0. This procedure allowsus to obtain the final differential form for the RG flow in parameter space, and tostudy the associated fixed points and their stability.

We begin our DRG analysis by considering a Langevin equationwith a genericdispersion relationσk and the KPZ non-linearity

[−σk − iω] hk,ω = ηk,ω − λ

2

∫

|p|≤Λ

dp

(2π)d

∫ +∞

−∞

dΩ

2πp · (k − p)×

× hp,Ω hk−p,ω−Ω,

(2.59)

where the noise has zero mean〈ηk(ω)〉 = 0 and is delta correlated

〈ηk,ω ηk′,ω′〉 = 2Π(k)(2π)d+1 δk+k′ δω+ω′ . (2.60)

Using relations equations (2.57)-(2.58) we are able to write the evolution equationsfor slow

G−10 (k, ω)h<

k,ω = η<k,ω − λ

2

∫

|p|≤Λ

dp

(2π)d

∫ +∞

−∞

dΩ

2πp · (k − p)×

×[

h<p,Ω h

<k−p,ω−Ω + 2h<

p,Ω h>k−p,ω−Ω + h>

p,Ω h>k−p,ω−Ω

]

,

(2.61)

and for fast modes

G−10 (k, ω)h>

k,ω = η>k,ω − λ

2

∫

|p|≤Λ

dp

(2π)d

∫ +∞

−∞

dΩ

2πp · (k − p)×

×[

h<p,Ω h

<k−p,ω−Ω + 2h<

p,Ω h>k−p,ω−Ω + h>

p,Ω h>k−p,ω−Ω

]

.

(2.62)

In equations (2.61) and (2.62) thebare propagatorG0(k, ω) = [−σk− iω]−1 is thestarting point of the perturbative expansion; in fact, whenthe non-linearity is equalto zero, equation (2.59) reduces to

h0 k,ω = G0(k, ω)ηk,ω. (2.63)

The expansion parameter in the FNS scheme is the coefficient of the non-linearity,λ0; substitution of equation (2.63) for fast modes in the perturbative series leads to

h>(k) = h>0 (k) +

λ0

2h>

1 (k) +

(

λ0

2

)2

h>2 (k) +

(

λ0

2

)3

h>3 (k) + . . . , (2.64)


where the compact notationh(k) meanshk,ω. Plugging this expansion into theevolution equation for the fast modes (2.62) we are able to obtain the functionsh>

n (k) at the desired order

n = 0 : h>0 k,ω = G0(k, ω)η>

k,ω,

n = 1 : h>1 k,ω = G0(k, ω)

∫

|p|≤Λ

dp

(2π)d

∫ +∞

−∞

dΩ

2πp · (k − p) ×

×[

h<(p)h<(k − p) + 2h<(p)h>0 (k − p) + h>

0 (p)h>0 (k − p)

]

,

n = 2 : h>2 k,ω = G0(k, ω)

∫

|p|≤Λ

dp

(2π)d

∫ +∞

−∞

dΩ

2πp · (k − p) ×

×2[

h<(p)h>1 (k − p) + h>

0 (p)h>1 (k − p)

]

, (2.65)

and so on. Now we write the equation for slow modes (2.61) where the heightfunction for the fast modes is replaced by (2.64). The resulting equation is quitecomplicated, and a diagrammatic expansion is very useful. Once the whole set ofdiagrams is determined we can average out the effect of high frequencies accordingto the following assumptions:

1. The low-frequency components are statistically independent of the high-frequency components and are invariant under the averagingprocess:〈η<〉 =η< and〈h<〉 = h<.

2. Averages involvingh>0 can be evaluated using (2.63) and the statistics of the

noiseη>, asG0 is a deterministic function and is statistically sharp.

3. The noise has zero mean; hence〈h>0 〉 = 0 and the same result holds for all

averages involving an odd number ofh>0 .

4. We drop terms containing higher order products ofh< (for instance,h<h<h<)because they are irrelevant with respect toh<h< in the hydrodynamic limit(i.e. k → 0 andω → 0).

In the diagrammatic expansion of equation (2.61) only one term is not null andcontributes to the propagator renormalization, see figure 2.7.The DRG procedure leads to an effective equation in which thefast modes are thusintegrated out

[−σk − Σ(k, 0) − iω]h<k,ω = η<

k,ω − λ

2

∫

|p|≤Λ/b

dp

(2π)d

∫ +∞

−∞

dΩ

2π×

× p · (k − p)h<p,Ω h

<k−p,ω−Ω.

(2.66)

In this equation the renormalized propagator is given by

G<0 (k, ω) = [−σk − Σ(k, 0) − iω]−1 , (2.67)


k kk − q

q −q

Σ = 4×

Figure 2.7 – Diagram of the propagator renormalizationΣ(k, ω). The arrows with thevertical bar represent the bare propagators of fast modes. For the corresponding integralsee equation (B.1).

and, from this expression, we can write the propagator renormalization as a sumof the renormalized parameters. For instance, in the KS equation (2.43) the propa-gator renormalization can be written as a sum of the surface tensionν and surfacediffusionK renormalizations

Σ(k, 0) = Σνk2 + ΣKk

4. (2.68)

Taking into account that the dispersion relation for the KS equation isσk = νk2 −Kk4, we use equation (2.67) in order to obtain the renormalized parameters of theequation for the slow modes:ν< = ν + Σν andK< = K − ΣK .

The renormalization of the noise varianceΠ(k) is different from the propagatorrenormalization. As a matter of fact,Π(k) is renormalized according to the rulesstated above, but the starting point of the whole procedure is the equation

⟨

h<k,ωh

<−k,−ω

⟩

= 2Π<(k)G(k, ω)G(−k,−ω), (2.69)

that leads, with a little algebra, to

〈η<k,ω η

<k′,ω′〉 = 2 [Π(k) + Φ(k, 0)] (2π)d+1 δk+k′ δω+ω′ . (2.70)

The diagram that survives the elimination of the fast modes is shown in figure 2.8.Hence, the noise variance in the equation for the slow modes isΠ< = Π + Φ.

Finally, the KPZ vertex renormalization is obtained from the two diagramsshown in figure 2.9. However, in the one-loop renormalization scheme the KPZvertex does not renormalize in case of a dispersion relationthat can be written as apolynomial ofk (see appendix B.3 for a proof), so thatλ< = λ.

At this point we can apply the scale transformation

real space:

r → brt → bz t

h → bαh

, Fourier space:

k → k/bω → ω/bz

h → bαh

, (2.71)

that allows us to write the parameter renormalization flow ina differential form,where the independent variable isδl, which is related to the scale transformation


q − k

−kk

k − q

q −q

Φ = 2×

Figure 2.8 – Diagram of the noise variance renormalizationΦ(k, ω). The arrows withthe vertical bar represent the bare propagators of fast modes. For the corresponding integralsee equation (B.14).

k1

q

k1 − q

k1

2+ k2

q −

k1

2− k2

k1

2+ k2 − q

k1

2− k2

k1

q

k1 − q

k1

2+ k2

q −

k1

2− k2

k1

2− k2

q − k1

Γb = 8×Γa = 4×

Figure 2.9 – Diagrams of the KPZ vertex renormalizationΓa(k, ω) andΓb(k, ω). Thearrows with the vertical bar represent the bare propagatorsof fast modes. For the corre-sponding integrals see equations (B.17) and (B.23).

through the identityb = eδl (specific examples about this procedure can be foundin appendices B and C).

Finally, the set of equations of the DRG flow can be written as an autonomousdynamical system employing the appropriate coupling variables. The study of thefixed points and their stability of this autonomous system gives insights about thecritical behavior of equation (2.1) in the hydrodynamic limit. At the fixed pointswe can calculate the critical exponents by substituting thevalues of the couplingvariables into the equations of the parameters flow. The linear stability of eachfixed point gives a valuable information about the true asymptotic behavior of thegrowth equation. In fact, in case of a stable and an unstable fixed points the DRGflow trajectory can pass near the unstable fixed point, so that, the growth equationwill probably experience the influence of this fixed point [6]. When this condi-tion occurs the route to the asymptotic state usually goes through one (or more)crossovers with a number of consequences for the estimationof the critical expo-nents. Moreover, the DRG technique is based on a perturbation expansion and itcan fail to predict the exponents of equation (2.1) (for the example of the KPZ


equation ford = 2 see [79]). Taking into account that DRG gives valuable insightabout the critical behavior of out-of-equilibrium models but requires approxima-tions, it is very useful for the understanding of the systemsdescribed by equationssuch as (2.1), to combine DRG analysis with direct numericalintegration of theseequations. In next section we discuss in detail the method wehave used for thenumerical estimation of the critical exponents.

2.3 Numerical solution of the growth equations

In many cases an exact solution of the growth equations can not be obtained whileanalytical approximation techniques fail in predicting the scaling exponents. Atypical example is the DRG analysis of the KPZ equation ford = 2, in which theKPZ fixed point belongs to a strong coupling regime and perturbative techniquesfail [79]. This often makes direct numerical integration ofgrowth equations anextremely useful and necessary tool as a reliable source of precise values for thecritical exponents and preasymptotic behavior.

In the literature there is a zoo of possible numerical methods, ranging form thestraightforward finite-differences (FD) method on a lattice to sophisticated adap-tive mesh finite elements. In most cases FD does an excellent job and provides aprecise estimation of the critical exponents, at least in dimensions of experimentalrelevance. The balance between precision of the numerical estimates and the cod-ing effort is a very important issue in the development of an integration code. Incase of FD, the accuracy of the numerical results mainly depends on the discretiza-tion rules that are chosen, Lamet al.having shown that the exact critical exponentsof the continuum version of KPZ can not be obtained by using a conventional dis-cretization [81].

Moreover, the use of FD schemes sometimes poses important problems [82; 83;84]. In particular Dasguptaet al. have shown by means of numerical simulationsthat discretized version of commonly studied non-linear growth equations exhibitan instability in the sense that single pillars (grooves) become unstable when theirheight (depth) exceeds a critical value. In some cases theseinstabilities are notpresent in the corresponding continuum equations, indicating that the behavior ofthe discretized version is indeed different from their continuum counterparts [85;86]. It is important to remark that this pillar-groove instability is actually genericto the FD discretization of a large class of non-linear growth equations, includingthe KPZ or the Lai-Das Sarma-Villain equations [33; 34; 35].This is a puzzlingresult because the corresponding continuum equations are not unstable.

In many situations, like for instance in KPZ, the existence of this instabilityis of little significance for the estimation of the critical exponents, and one canactually carry out a correct numerical integration by usingFD schemes. The reasonis that one is mostly interested in growth from an initially flat surface (or at leastwith small slopes) and common relaxation mechanisms do not favor the formationof large pillars or grooves. In these cases the FD instability only occurs if the initial

2.3 Numerical solution of the growth equations 33

condition is prepared in such a state that there is a pillar orgroove of a size abovea certain threshold, which is highly artificial and usually uninteresting for practicalpurposes. However, as already pointed out in [86], there is alarge class of systemsfor which the instability of any FD scheme is unavoidable. Specifically, discreteversions of models exhibiting anomalous kinetic roughening [87; 13; 88; 16; 10;18; 89; 17; 90] will certainly show this kind of instability at sufficiently long times.Besides, for continuum growth equations with a linear morphological instability,as for the KS equation, this spurious instability could be a major numerical issuein numerical integration.

Given the unreliability of the standard FD, several authorsproposed variantsbased on regularization of FD, as in [85; 86; 91], or have tried different numericalschemes. Spectral methods are a class of techniques widely used in fluid dynamicsand in applied mathematics to solve partial differential equations. These methodswork in Fourier space and their strength resides in their exponential convergencefor smooth solutions. In Fourier space, the KPZ non-linearity is transformed intoa convolution product. Convolution is a costly operation, hence the main ideaof the Pseudo-Spectral (PS) method resides in calculating the linear deterministicand the stochastic parts of equation (2.1) in Fourier space whereas the non-linearKPZ part is computed in real space as a simple multiplication[92; 93; 94; 95]. Aclear and detailed performance comparison between FD and the PS methods canbe found in [96]. In this work, Gallegoet al.show how FD underestimate the non-linear part of KPZ and LDV equations giving an stationary amplitude width of thediscrete numerical interfaces smaller than that of the continuum equation (in theKPZ case there is an exact relation in terms of the equation parameters). Besides,the PS method is numerically more stable than FD and, for thisreason, it needs lesscomputational effort in order to reach the hydrodynamic limit of these equationsand estimate the critical exponents.

2.3.1 Spectral representation with discrete Fourier trans form

If we assume that the height functionh(r, t) satisfies periodic boundary conditionsin thed-dimensional cube[0, L]d, it could be represented using a Fourier series

h(r, t) =

+∞∑

|k|=−∞hk(t)eiq·r, (2.72)

whereq = 2πk/L and the functionshk(t) are the Fourier coefficients ofk

hk(t) =1

Ld

∫

dr e−iq·rh(r, t). (2.73)

The representation given by equations (2.72)-(2.73) is unfeasible in a numericalscheme, due to the requirement of a complete knowledge ofh(r, t) in the wholereal domain. In case of a band limited function, i.e. in case that hk(t) = 0 for


every|k| > N/2, we can truncate the series (2.72) according to this upper limit

hN (r, t) =∑

k∈ΓN

hk(t)eiq·r, (2.74)

the setΓN over which the sum is taken being given by

ΓN = (k1, k2, . . . , kd) | −N/2 ≤ ki ≤ N/2 − 1, i = 1, . . . , d . (2.75)

The second approximation of this representation consists in discretizing the realspace withN collocation pointsrj in each direction. These points are chosenuniformly distributed in thed-dimensional cube

rj = ∆x(j1, j2, . . . , jd), (2.76)

where∆x = L/N andji = 0, . . . , N − 1 for each dimension. The coefficientshk(t) are now approximated through the discrete Fourier transform (DFT) of thediscrete height functionhj evaluated at these collocation points

hk(t) = FD [hj]k =1

Nd

∑

j

hj(t)e−iq·rj , (2.77)

where the coefficients arehj(t) = h(rj , t) = F−1D [hk]j =

∑

k∈ΓNhke

iq·rj . Weadopt these normalization factors for the discrete transform because they coincidewith those that are used in the Fast Fourier transform algorithms that can be foundin standard numerical libraries.

The Fourier basiseiq·rj is orthogonal and satisfies the following periodicityrelation

eiq·rj = ei2πN

k·rj = ei2πN

(k+Ni)·rj = eiq·rj , (2.78)

where all components ofNi are zero except for theith component that is equalto N . From equation (2.72) and using this periodicity relation,it is clear thatthe relation between the complete Fourier transform ofh(r, t) and the one that iscomputed considering only the collocation points can be obtained equating (2.72)to (2.74). Explicitly, in each one of thed independent directions we have

h(rj , t) =

+∞∑

kl=−∞hkl

(t)eiqlrj =

N/2−1∑

kl=−N/2

eiqlrj

(

+∞∑

m=−∞hkl+mN (t)

)

, (2.79)

from which we have that

hkl(t) =

+∞∑

m=−∞hkl+mN (t). (2.80)

This equation shows that thek-th mode of thel direction of the DFT ofh(rj , t)depends on this mode and all the modes whichalias it on the grid of collocation


pointsrj. The(kl +mN)th frequency aliases thekthl frequency on the grid, and

they are indistinguishable at the collocation points due torelation (2.78). Duringthe numerical implementation of the PS method we have to takeinto account thispoint, in particular when we have to compute the non-linear term.

If we want to estimate the error arising from a interpolated representation ofthe Fourier transform of the height function we can write equation (2.74) as a sumof equation (2.77) and a residue

INh =

N/2−1∑

kl=−N/2

hkleiqr =

N/2−1∑

kl=−N/2

hkleiqr +

N/2−1∑

kl=−N/2

+∞∑

m=−∞m6=0

hkl+mNeiqr =

= PNh+RNh. (2.81)

In other words, equation (2.81) shows how theN/2-degree trigonometric interpo-lating polynomial ofh can be expressed as the truncation of equation (2.73) and analiasing errorRN . This error is orthogonal to the truncation errorh−PNh, so that

‖h− INh‖2 = ‖h− PNh‖2 + ‖RNh‖2 , (2.82)

hence, the error due to the interpolation is actually alwayslarger than the error dueto the truncation of the Fourier series. It can be shown that the aliasing error isasymptotically of the same order as the truncation error andit does not spoil thestability and accuracy of the spectral method [95].

Let us consider now the derivative of the height function in Fourier space. Dueto the periodic boundary conditions of the problem considered, the Fourier seriesof the derivative of a functionh is easily computed from equation (2.72)

∂xlh(r, t) =

+∞∑

|k|=−∞iqlhk(t)eiq·r, (2.83)

where the subscriptl refers to one of thed possible directions in real space. How-ever, since our knowledge is limited only to the coefficientshk, we have to approx-imate equation (2.83) with the Fourier collocation derivative

∂xlh(rj , t) ∼ ∂xl

(INh) =

N/2−1∑

|k|=−N/2

iqlhk(t)eiq·rj , (2.84)

which differs because of the aliasing error in the coefficients, and does not evencorrespond to theN/2-degree trigonometric interpolating polynomial of∂xl

h, i.e.∂xl

(INh) 6= IN (∂xlh). However, this error is of the same order of the truncation

error [95], and we can say that collocation differentiationis spectrally accurate.Finally we have to take into account the computational cost and the aliasing

problem of the convolution operator in Fourier space. A general non-linear termwritten as a multiplication of two functionsu andv

w(x) = u(x)v(x), (2.85)


can be Fourier transformed, and takes the form of a standard convolution sum

wk =∑

m+n=k

umvn, (2.86)

where the indicesm andn belong toZ. The FTum and vn are calculated fromequation (2.73). Whenu, v, andw are approximated by their respective truncatedFourier series of degreeN/2, equation (2.86) translates into

wk =∑

m+n=k|m|+|n|≤N/2

umvn, (2.87)

with k ∈ [−N/2, N/2 − 1]. Clearly this sum requires a number of operationsof orderN2 and, compared with FFT, is computationally too costly. In order tospeed up the algorithm one could apply the inverse DFT tov andu, compute theirproduct, and use again the DFT to getw. In this way we obtain

wk =1

N

N−1∑

j=0

v(rj)u(rj)e−iqrj =

1

N

N/2−1∑

m,n=−N/2

N−1∑

j=0

vmuneirj(n+m−q) =

=

N/2−1∑

m=−N/2

1∑

l=−1

vmuk−m+Nl = wk +

N/2−1∑

m=−N/2

vmuk−m±N ,

(2.88)

after using the orthogonality relation

1

N

N−1∑

j=0

eiqrj =1

N

N−1∑

j=0

e2πiN

krj =

1 if k = lN, l ∈ Z,

0 otherwise.(2.89)

Note that the range ofl in equation (2.88) arises from the constraintsm,n ∈[−N/2, N/2 − 1]. The last sum in equation (2.88) is the aliasing error for thenon-linear term.

There are mainly two ways to remove completely this error, using the zeropadding or by phase shift. A typical choice is zero padding because of its straight-forward extension to higher dimensions, while the phase shift method requiressome modifications. Zero padding consists in calculating the DFT with a num-berM > N of modes. The extra modes with|k| > N/2 are initially set to zero.In this way equation (2.88) reads

wk =

M/2−1∑

m=−M/2

vmuk−m +

M/2−1∑

m=−M/2

vmuk−m±M , (2.90)

with k ∈ [−M/2,M/2 − 1]. Since we need only the firstN modes, it is sufficientto chooseM such that the second term on the right-hand side vanishes forthesek.Sinceum andvn are zero for|m|, |n| > N/2, the worst case condition is

−N2

− N

2≤ N

2− 1 −M, (2.91)


so we expand the DFTs to at leastM ≥ 3N/2 − 1 components. This methodrequires the computation of two FFT of size3N/2, which is obviously not a powerof 2 if N is. There are, however, FFT algorithms that scale asn log(n) also whenn is not a power of2. A rough estimate of the number of operations required tocalculate a non-linear term with this spectral method is(45/4)N log2(3N/2) [95].

2.3.2 Pseudo-spectral numerical integration

Now we are ready to explain how we have implemented the pseudo-spectral (PS)method for the numerical integration of growth equations. In our scheme we startwith an initial conditionh(rj , 0) at the collocation points (usually a flat interface)and we transform it into the Fourier domain using the FFT. Theevolution equationin momentum space reads

∂thk(t) = σkhk(t) + N [h]k + ηk(t), (2.92)

where the variance of the noise term after the DFT transformsinto

⟨

ηk(t) ηk′(t′)⟩

=2(Π0 + Π2k

2)

(N∆x)dδk+k′ δ(t− t′). (2.93)

In equation (2.92) we do not write the explicit form of the non-linear term becausewe do not want to calculate it in the Fourier domain. Moreover, the solution of thelinearized equation suggests the following change of variable

hk(t) = eσk tzk(t) + eσk t

∫ t

0ds e−σksηk(s) = eσk tzk(t) +Gk(t), (2.94)

where the integrated noise term is

Gk(t) = eσk t

∫ t

0ds e−σksηk(s). (2.95)

The new variablezk is simply related tohk through the equation

zk(t) = e−σk t(

hk(t) −Gk(t))

, (2.96)

and its temporal evolution depends only on the non-linear term

∂tzk(t) = e−σk tN [h]k. (2.97)

The set of ODEs in equation (2.97) can be solved using one of the several al-gorithms available in literature (Euler, Runge-Kutta, predictor-corrector methods,etc.). Considering the stochastic nature of the problem, wehave chosen a one stepEuler’s method to integrate equation (2.97)

zk(t+ ∆t) = zk(t) + ∆t e−σk tN [h]k, (2.98)


and going back from this result to original variables the updated values for theheight function are

hk(t+ ∆t) =eσk (t+∆t)zk(t+ ∆t) +Gk(t+ ∆t) =

= eσk∆t[

hk(t) + ∆tN [h]k

]

+ gk(t).(2.99)

The stochastic part of equation (2.99) is contained in the new noise term

gk(t) = Gk(t+ ∆t) − eσk∆tGk(t) = eσk(t+∆t)

∫ t+∆t

tds eσksηk(s), (2.100)

with variance

⟨

gk(t) gk′(t′)⟩

=

(

Π0 + Π2k2

∆xd

)

e2σk∆t − 1

σk

Nd δk+k′ δ(t − t′). (2.101)

We conclude this section with the details of the numerical algorithm we usedfor the PS integration. Mainly the steps of the algorithm are:

1. we start the integration with an initial condition that can be a flat interface,hk(0) = 0, or an initial condition considered as an input for the program.

2. By using the integration parameters, the algorithm computes the constantvectors of the numerical scheme, i.e.σk, eσk∆t, and the noise variance

Vk =

[(

Π0 + Π2k2

∆xd

)

e2σk∆t − 1

σk

]1/2

. (2.102)

3. Successively, it evaluates the non-linear termN [h]k by Fourier transformingthe zero padded height derivativeiq · hk(t) to real space and by calculatingits square value. After this operation, it Fourier transforms to wave-vectorspace and removes the redundant modes. The padding procedure is verydependent on the numerical algorithm which performs the Fourier transform.In our case, the transform has been implemented by the fftw3 library (furtherdetails can be found atwww.fftw.org).

4. The next step is the generation of noise. In fact, the algorithm calls a routinein order to generate in real space a vector of Gaussian uncorrelated randomnumbersur with unit variance and, successively, Fourier transforms this vec-tor in order to obtainuk. Note that the variance of the transformed vectoris not equal to one, in fact,〈ukuk′〉 = Nd δk+k′. The noise vectorgk ofequation (2.99) is obtained by multiplyingVk by uk.

5. The state of the interface is updated according to equation (2.99). Afterthis last step, the algorithm jumps back to point 3 until it reaches the finalintegration time.

Moving boundary description ofdiffusive growth

Many natural and technological growth systems exist whose dynamics results fromthe interplay between diffusion and aggregation. Aggregating units are transporteddiffusively and, in contact with a surface, they attach producing a condensed grow-ing cluster. Examples abound particularly within Materials Science, e.g. in theproduction of both amorphous (through Chemical Vapor Deposition [97] or Elec-trochemical Deposition [98]), and of epitaxial (by Molecular Beam Epitaxy [59])thin films. Still, a similar variety of surface morphologiescan be found in diversesystems, such as e.g. bacterial colonies [99], to the extentthat universal mecha-nisms are expected to govern the evolution of these non-equilibrium systems [100].Moreover, many times it is not matter, but e.g. heat that is transported, but obeyingprecisely the same constitutive laws (one-sided solidification) [101].

In particular, electrochemical deposition (ECD) has been widely studied mainlydue to the apparent simplicity of the experimental setup andthe capability to tuneup different relevant physical parameters. Most of the studies in ECD address thegrowth of quasi-two dimensional fractal aggregates. This can be achieved by con-fining a metallic salt between two parallel plates with a small separation betweenthem.

The formulation of the problem can be accurately posed to include the physicalmechanisms involved. But the solution (even for the deterministic case) cannot beobtained. Besides, the numerical integration of the equations often involves the use

40 Moving boundary description of diffusive growth

of complex algorithms that take into account the motion of the boundary and which,in practice, are difficult to implement (for instance, the role of fluctuations and theirinclusion in numerical schemes is still an open question). In the present context,the interplay among different transport or electrochemical mechanisms is not clearand the predictive ability of different models cannot go beyond the description ofthe concentration of cations in the solution or the qualitative morphologies of thefractal aggregates [62; 63].

In this scenario, two possibilities arise from a theoretical point of view. On theone hand, one can try to obtain as much information as available from customaryperturbation methods. Thus, linearization, Green function projection techniques orweakly nonlinear expansions provide important information in both short (linear)and asymptotic times [9]. Briefly, this approximations allow to explore the physicsof the problem in the so-called small slopes approximation.On the other hand, onecan try to integrate numerically an alternative and computationally efficient versionof the moving boundary problem in order to accurately determine the validity ofthe approximations and also gain some insight in the intermediate regimes (wheresmall slopes and interfacial pattern formation should emerge).

This second approach can be achieved by the introduction of an equivalentphase-field model [102] version of the moving boundary problem. Phase-fieldmodels have been successfully introduced in the last few years to understand equiv-alent moving boundary problems which are harder to integrate numerically (or incombination with discrete lattice algorithms [103]). Theyhave become a paradig-matic representation of multi-phase dynamics whose application ranges from so-lidification [104; 105] or material science [106; 107; 108; 109] to Fluid Mechan-ics [110; 111] or even Biology [112].

In this chapter we provide a quantitative picture of non-conserved growth froma diffusive field. We formulate the physical problem employing a moving boundarymodel and we will focus on two important and complementary tasks: the validityof the effective small-slopes (interfacial) theories and the emergence of unstableaggregates. More importantly, we will provide a close connection between theoryand ECD experiments and will show that our model provides both a qualitative anda quantitative picture of electrochemical deposition in the small slopes expansionregime.

In anticipation, in the next chapter we will introduce a phase-field model thatis equivalent to the moving boundary formulation in the thininterface limit. Thisphase-field model will allow us to study the asymptotic experimental morpholo-gies. Additional benefits of the model are the capability to reproduce quantita-tively many conditions studied in literature and infer the kinetic roughening of theeffective single-valued interfaces studied in experiments.

3.1 Unified model of diffusive growth 41

3.1 Unified model of diffusive growth

Our diffusive growth theory is, mainly, a stochastic version of long studied modelsin the context of CVD, with some additional mechanisms. Thus, we first reviewthe classic constitutive equations of CVD [113; 114; 115], and then, we writethe equations of ECD growth in a form that unifies the description of this growthtechnique with that of CVD [1; 53; 54; 9]. Subsequently, we consider the effect ofnoise due to the fluctuations of the relaxation mechanisms involved in these growthprocesses.

3.1.1 Chemical vapor deposition

A stagnant diffusion layer of infinite vertical extent is assumed to exist above thesubstrate upon which an aggregate will grow, see a sketch in figure 3.1.1. This

Figure 3.1 – Schematic representation of a model CVD growth system [9]. Black pointsrepresent aggregation units that diffuse in the dilute phase. The different transport mecha-nisms (bulk diffusion, attachment and surface diffusion) are indicated in the figure by theircorresponding equations, see (3.1)-(3.3). For the definitions of the noise terms(q,p, χ)see section 3.1.3.

approach implies that the length of the stagnant layer (typically of the order ofcm) is much larger than the typical thickness of the deposit (in the range of mi-crons). The particles within the vapor diffuse randomly until they arrive at thesurface, react and aggregate irreversibly to it. The concentration of these particles,


c(x, z, t) ≡ c(r, t), obeys the diffusion equation

∂tc = D∇2c. (3.1)

In experiments, the mean concentration at the top of the stagnant layer is chosen tobe a constant, equal to the initial average concentrationca.

Mass is conserved at the aggregate surface, so that the localnormal velocity atan arbitrary point on the surface is given by

Vn = ΩD∇c · n− Ω∇s · Js, (3.2)

whereΩ is the molar volume of the aggregate andn is the local unit normal, exte-rior to the aggregate. Henceforth, we will use indifferently the operators∇ · n and∂n for the directional derivative of a scalar field along the normal direction of theinterface.

Equation (3.2) expresses that growth takes place along the local normal direc-tion (usually referred to asconformal growthin the CVD literature) and is due tothe arrival of particles from the vapor [the first term on the rhs in equation (3.2)] andvia surface diffusion (Js stands for the diffusing particle current over the aggregatesurface and∇s is the surface gradient).

Finally, the particle concentration,c, and its gradient at the surface are relatedthrough the mixed boundary condition

kD(c− c0eq + Γκ)∣

∣

∣

ζ(x,t)= D∇c · n

∣

∣

∣

ζ(x,t), (3.3)

wherec0eq is the local equilibrium concentration of a flat interface incontact withits vapor, andζ(x, t) is the local surface height. This equation is closely related tothe probability of a particle to stick to the surface when it reaches it (see below).

In summary, equations (3.1) and (3.2) describe diffusive transport in the vaporphase and the way that particles attach to the growing aggregate. Let us concentrateon the physical meaning of equation (3.3). The parameterΓ is related to tempera-ture [116; 117] asΓ = γc0eqΩ/(kBT ), whereγ is the surface tension (that will beassumed a constant), andκ is the surface curvature

κ =∂xxζ

[1 + (∂xζ)2]3/2

. (3.4)

The boundary condition (3.3) can be obtained analyticallye.g. from kinetic theoryby computing the probability distribution for a random walker close to a partiallyabsorbing boundary. If the particles have a sticking probability, s, of aggregatingirreversibly (i.e., attachment is not deterministic) the mass transport coefficient,kD, is related to this probability through [118]

kD = DL−1f

s

2 − s, (3.5)


Lf being the mean free path of the vapor molecules. Assuming that Lf is suffi-ciently small we find two limits in equation (3.3): If the sticking probability van-ishes (s = 0) then∇c = 0 at the boundary, so that the aggregate does not grow.On the contrary, if the sticking probability is close to unity (providedLf is smallenough), thenkD takes very large values. In this case equation (3.3) reducesto thewell-known Gibbs-Thomson relation [116; 117], which incorporates the fact thatconcentration is different in regions with different curvature.

In summary, equation (3.3) gives a simple macroscopic interpretation of a mi-croscopic parameter, the sticking probability, and allowsto quantify the efficiencyof the chemical reactions leading to species attachment at the interface.

3.1.2 Continuum model of galvanostatic electrodeposition

In an electrochemical experiment, dynamics is more complexthan in the CVDsystem as represented above, due to the existence of two different species subjectto transport (anions and cations) [119] and an imposed electric field E. For avisual reference, see figure 3.1.2. Although more elaboratetreatments of these canbe performed [120; 121; 122] the morphological results are qualitatively similar tothe more simplified description we will be making in what follows. The virtues ofthe latter include an explicit mapping to the CVD system, andexplicit experimentalverification.

Thus, in ECD, mass transport is not only due to diffusion in the dilute phase,but also due to electromigration and convection. Nonetheless, an explicit mappingof the corresponding dynamical equations can be made to the CVD system, bymeans of the following assumptions[1; 53; 54; 9]:

(i) zero fluid velocity in a thin electrochemical cell;

(ii) anion annihilation at the anode, whose position is located at infinity, namely,the aggregate height is much smaller than the distance separating the elec-trodes;

(iii) electroneutrality away from the electrodes;

(iv) zero anion flux at the cathode

In order to better understand the way in which the particles evolve in the cell,we need to follow their dynamics when the external electric field is switched on.Thus, the cation and anion concentrations are initially constant and uniform acrossthe cell and then, once the electric field is applied, anions move towards the anodeand cations move towards the cathode. Cations reduce at the cathode thus formingan aggregate of neutral particles. On the contrary, anions do not aggregate, rather,they merely pile up at the anode, which is dissolving at the same rate as the cationsaggregate in the cathode. Hence, the number of cations remains a constant.


Anode

Cathode

Figure 3.2 – Schematic representation of a model ECD growth system [9]. Cations mi-grate towards the cathode (lower side) while anions migratetowards the anode (upper side)in an infinite cell. The different transport mechanisms (bulk diffusion and drift, cation re-duction) are indicated in the figure by their corresponding equations.A andC correspondsto the anions and cations concentration, respectively, and, the external electric field is rep-resented byE.

Mathematically, this mechanism of aggregation can be expressed as a boundarycondition for the cation concentration. Before introducing such a boundary condi-tion we will simplify the set of diffusion equations for the anions and cations con-centration following Refs. [123; 124]. Let us consider thatthe deposit moves with agiven constant velocityV , in such a way that, in the frame of reference co-movingwith the surface,z = 0 is the position of the mean height andz → ∞ representsthe position of the anode (thus, we are dealing with the case in which the heightof the aggregate is negligible with respect to the electrodeseparation). Moreover,we are assuming that the system is under galvanostatic conditions, namely, that thecurrent density at the cathode,J , is maintained constant. Thus, the problem can beseparated into two spatial regions: far enough from and close to the cathode.

At distances larger than the typical diffusion length,lD = D/V , the net chargeis zero, and we can write a diffusion equation like equation (3.1) where the dif-fusing species,c, becomes the cation concentrationC, rescaled by a quantityRc

that depends on the diffusivity and the mobility of the two species. Specifically,Rc = Dc/D(1 − tc) where the ambipolar diffusion coefficientD and the constant


tc are defined by

D =µcDa + µaDc

µc + µa, tc =

µc

µc + µa, (3.6)

andµa(µc),Da(Dc) are the anion (cation) mobility and diffusivity, respectively. Inthis region the physics of the electrodeposition process simplifies into an effectivediffusion problem in which the net characteristics of the diffusing species dependson the mobility and diffusivity of the anions and cations.

So far we have shown how (under some restrictions) diffusionand electro-migration can be merged into a diffusive term with an effective diffusivity. Tocomplete our program to unify ECD and CVD, we still need to write down theboundary conditions for the ECD case.

Right at the interface, by imposing mass conservation we canwrite down anequation similar to (3.2) in which the surface diffusion is not taken into account(this term can be added later). In fact, the local velocity ofthe aggregate surface isproportional to the flux of particle arriving to it, therefore

Vn = − Ω

zcFJ, (3.7)

whereF is the Faraday constant,zc is the number of electrons involved in theelectrochemical reaction, andΩ is the molar volume, here defined as the ratio of themetal molar mass,M , and the aggregate mean density,ρ. This explicit dependenceof the molar volume onρ can be avoided considering that for a flat frontVn = Vand imposing: (i) that the current density at the aggregate surface is only due to thecations; (ii) that the charge is proportional to the cation current [125; 126]; thus wecan relateΩ to simpler model parameters

Ω =M

ρ=

1 − tcCa

. (3.8)

This relationship has been previously proposed theoretically [126] and experimen-tally verified [127], thus supporting the hypotheses made inthis section.

In this context, the second boundary condition at the interface, i.e. equation(3.3), arises from the Butler-Volmer equation [125; 98; 120; 128]. This equation(well-known in the field of electro-chemistry) relates the transport of charge at theinterface and its local properties, such as curvature. The parameters in equation(3.3) are mapped to electrochemical quantities through thefollowing relations [1;53; 54; 9]

c0eq = RcCa exp [zcF∆ψ/(RT )] , Γ = γ/(RT ), (3.9)

andkD = J0 exp[−bzcF∆ψ/(RT )]/(zcFCaRc), (3.10)

whereCa is the initial cation concentration, held fixed at infinity,T is temperature,γ is the aggregate surface tension,R is the ideal gas constant,J0 is the exchange


current density in equilibrium, and∆ψ is the overpotential for which a surfacecurvature contribution in the corresponding Butler-Volmer boundary condition [97;98; 128] has been singled out,b estimating the asymmetry of the energy barrierrelated to cation reduction.

Note that the rescaled mass transport coefficientkD is (again) related with thesticking probability for cations: if aggregation is very effective (large sticking prob-ability) the overpotential is a large negative quantity andthenkD grows exponen-tially. In addition, the rescaled concentrationc0eq decreases. In the limit whenevery particle that arrives at the surface sticks irreversibly, the solution cannot sup-ply enough particles,c = 0, and the current density takes its maximum value. Thisvalue of the current is called limiting current density [129]. On the other hand,if the sticking probability is small, the system is always close to equilibrium and∇c ≃ 0, so that the net current is zero.

With these redefinitions, equations (3.1)-(3.3) describe (under the physical as-sumptions made above) the evolution of both types of diffusive growth systems,CVD and ECD.

3.1.3 Role of fluctuations

The set of equations presented in the previous section describes the evolution ofthe mean value of the concentration so that, formally, we cantrack the positionof the interface at any instant. Their main merit is that unifies ECD and CVDin a simple theoretical framework. However, it explicitly ignores the (thermal)fluctuations related to the different transport and relaxation mechanisms involved.In order to account for these, we define the stochastic functionsq, p andχ as thefluctuations in the flux of particles in the dilute phase (−D∇c), in the surface-diffusing particle current (Js), and in the equilibrium concentration value at theinterface, respectively.

Thus, the stochastic moving boundary problem we propose to describe diffu-sive growth has the form

∂tc = D∇2c−∇ · q, (3.11)

D∂nc∣

∣

∣

ζ= kD(c− c0eq + Γκ+ χ)

∣

∣

∣

ζ+ q · n, (3.12)

Vn = Ω[

D∂nc−∇s · Js − q · n −∇s · p]

, (3.13)

limz→∞

c(x, z; t) = ca. (3.14)

In (3.13) the surface diffusion term,∇s ·Js, is proportional to the surface diffusioncoefficientDs and the surface concentration of particlesνs; moreover, this term isrelated to the local surface curvature [116; 117] as

∇s · Js = B∇2sκ = Ω2νsDsγ(kBT )−1∇4

sζ. (3.15)

3.2 Small slopes approximation 47

For ECD, all parameters in the above equations can, in principle, be estimatedfrom experiments. This is also the case for the noise terms: indeed, their amplitudescan be shown to be functions of the physical parameters above, once we take thenoises as Gaussian distributed numbers that are uncorrelated in time and space, andmake a local equilibrium approximation [1; 53; 54; 9].

We choose these noise terms to have zero mean values and correlations givenby

〈qi(r, t) qj(r′, t′)〉 = 2Dca δijδ(r − r′)δ(t− t′), (3.16)

〈pi(r, t) pj(r′, t′)〉 = 2Dsνs δij

δ(r − r′)δ(t − t′)√

1 + (∂xζ)2, (3.17)

〈χ(r, t)χ(r′, t′)〉 = 2c0eqkD

δ(r − r′)δ(t − t′)√

1 + (∂xζ)2, (3.18)

wherei, j denote vector components. The different noise variances have been de-termined from the equilibrium fluctuations following [130;71], for further detailssee [1; 53; 54; 9]. Finally, the factor

√

1 + (∂xζ)2 in (3.17), (3.18) ensures that thenoise strength is independent of the (local) surface orientation. Notep andχ areonly evaluated at the aggregate surface.

3.2 Small slopes approximation

Equations (3.11)-(3.14) are very hard to handle for practical purposes. In this sec-tion we will reformulate it into an integro-differential form that will allow us toderive (perturbatively) an approximate evolution equation for the interface heightfluctuation,ζ(x, t). In this respect, we will use a technique based on the Greenfunction theorem. For brevity, we show here the main resultsleaving the technicaldetails for the interested reader in [9].

Green function projection techniques have been successfully used in othersimilar problems, such as solidification [130] or epitaxialgrowth on vicinal sur-faces [71], and is based on the use of the Green theorem [131] in order to convertan integral extended over a certain two-dimensional domainonto another one eval-uated precisely at the moving boundary (the aggregate surface). In our case, thedomain is the region between the electrodes (see figure 3.3).

Let us consider that the distance separating the electrodesand the lateral sizeof the system are both infinite, so that the only part of the dashed line in figure 3.3whose contribution is non vanishing is the aggregate surface. The Green functionrelated to equation (3.11) is the solution of

(

∂

∂t′+D∇′2 − V

∂

∂z′

)

G(r − r′, t− t′) = −δ(r − r′)δ(t − t′), (3.19)

where we have made a change of coordinates to a frame of reference moving withthe average growth velocity,V . The functionG is easily evaluated in Fourier


ζ(x,t) ds

A

Figure 3.3 – Integration domain,A, and its boundary (solid line) used for Green’s theo-rem. The infinitesimal arc length along the moving boundaryζ(x, t) is given byds.

domain

Gkω =1

Dk2 + iω − iV kz, (3.20)

with k2 = k2x + k2

z . Therefore, using the inverse Fourier transform we obtain

G(r − r′, τ) =Θ(τ)

4πDτexp

[

−(x− x′)2

4Dτ−[

z − z′ + V τ]2

4Dτ

]

, (3.21)

Θ(τ) being the Heaviside step function andτ = t− t′. After some calculations weare able to obtain the integro-differential equation

c(r, t)

2= ca −

∫ t

−∞dt′[

∫ ∞

−∞dx′(

V +∂ζ ′

∂t′

)

c′G−

−D

∫

ζ′ds′(

c′∂G

∂n′−G

∂c′

∂n′

)

]

z′=ζ′

− S(r, t),

(3.22)

where the noise termS reads

S(r, t) =

∫ t

−∞dt′∫ ∞

−∞dx′∫ ∞

ζ′dz′G∇′ · q′ (3.23)

The single equation (3.22) relates the concentration at theboundary with the sur-face height, and is shown in [1; 9] to be actually equivalent to the full set of equa-tions (3.11)-(3.14), providing essentially the so-calledGreen representation for-mula for our system [131]. Unfortunately, equation (3.22) is still highly nonlinearand has also multiplicative noise (through the noise termS). Notwithstanding, itwill allow us to perform a perturbative study in a simpler way.


First, let us consider solutions of equation (3.22) that areof the formc = c0 +c1, wherec0 stands for the part associated with the flat (i.e.,r-independent) frontsolution, andc1 is a small perturbation of the same order as the height fluctuationζ(x, t). Hence, to lowest order in the latter and its derivatives [9],

c0 =V ca + kDc

0eq

V + kD. (3.24)

Thus, the concentration at a flat surface is the mean between its local equilibriumvaluec0eq and that at the top of the stagnant layerca, weighted by the two charac-teristic velocities of the moving boundary model,V andkD.

One remarkable feature of the Green function representation is that, from theknowledge of the concentration at the boundary, we can extrapolate the value ofthe particle concentration everywhere. Thus, from equation (3.22) and using theproperties of the Green function [9], we find

c0(z) = ca + (c0 − ca) e−zV/D. (3.25)

It is remarkable that equation (3.25) has been theoretically obtained and experi-mentally verified by Legeret al. [126]. Physically, it shows how, close the surface,the concentration grows up to a length on the order ofD/V (the diffusion length).Far beyond this length the exponential term is negligible and the concentration isalmost identical to the concentration at the stagnant layer.

We now proceed with the next order of the expansion. At this order we alreadyobtain a proper (albeit linear) evolution equation for the interface, which moreovercontains all the noise terms contributions, that is

∂tζk(t) = σkζk(t) + ηk(t), (3.26)

whereσk is the dispersion relation as function of wave-vectork whose form maychange with the values of the phenomenological parameters (see below). Note that,being linear, equation (3.26) can be exactly solved.

From the first order expansion of equation (3.22) we find an implicit equationfor the interface perturbations (after time Fourier transform) [9]

Tkω = ζkωΠkω, (3.27)

with

Tkω =

(

1

2G0kω

+V

2+ kD

)(

iω

ΩkD+ Γk2

)

−

− V

ΩD

(

1

2G0kω

− V

2

)

− kDΓk2,

(3.28)

where the functionG0kω is introduced in the calculation of the zero order expansion

of equation (3.22) see [9] and is

G0kω =

(

4iωD + 4D2k2 + V 2)−1/2

. (3.29)


The stochastic termΠkω leads to the effective noise that appears in the equation(3.26) and can include both non-conserved and conserved contributions (at leastat lowest order). The connection between the noise varianceand the physical pa-rameters of CVD and ECD has been established in [9] and we report this resultbelow.

From the Fourier transform of the deterministic part of equation (3.26) (i.e. forηk(t) ≡ 0) we can replaceiω in equation (3.28) byσk and we obtain the explicitform of the dispersion relation from the zeros ofTkω. It turns out that the behaviorof σk can be most significantly studied as a function of the values of the kineticcoefficientkD. Specifically, in the next two sections we analyze separately the casein which surface kinetics is instantaneous (that is, the sticking probability is high)and all other cases in which the attachment rate is finite. Finally, in the last sectionwe study the numerical solution of the equationTkω = 0 in order to quantify theaccuracy of the analytical approximations that have been used in the two limitingvalues ofkD, and to study theexactlinear behavior of the moving boundary model.

3.2.1 Non-instantaneous surface kinetics

For a finite value of the kinetic coefficientkD, we cannot obtain (even in the zero-noise limit) the linear dispersion relationσk in a closed analytic form, unless weperform a large scale (k → 0) approximation. Thus, we can analyze implicitlythe zeros of the functionTkω, which yield the required form ofσk as a function ofwave-vector [132]. In the large scale limit we find [1; 9]

σk = a2k2 − a4k

4, (3.30)

where

a2 =DkD

V∆, a4 =

DkDlDd0∆

V

[

1 −√

V d0

D

] , (3.31)

and∆ = 1 − d0/lD. The two constants appearing in∆ are the capillarity length,d0 ≡ ΓΩ, and the diffusion length,lD. If ∆ < 0, thenk = 0 is the only zero ofσk,and sincea2 < 0 all Fourier modesζk(t) of the height fluctuation are stable, sincethey decay exponentially in time within linear approximation.

On the contrary, if∆ > 0, thena2 anda4 are both positive and there is a bandof unstable modes for allk ∈ (0, k∗), with

k∗ =

[

V

Dd0

(

1 −√

V d0

D

)]1/2

. (3.32)

For these values of the wave-vector,ζk(t) grows exponentially in time within linearapproximation. A maximally unstable mode exists corresponding to the maximumpositive value ofσk, whose amplitude dominates exponentially all other and leadsto the formation of a periodic pattern. Under these parameter conditions, the linear


dispersion relation (3.30) is that of the Kuramoto-Sivashinsky equation (see figure3.4) [133].

Althoughσk in equation (3.30) containsO(k4) terms, typical of relaxation bysurface diffusion [116; 117], these originate as higher order contributions in whichdiffusion (D), aggregation (kD), and surface tension (Γ) become coupled. We canalso include proper surface diffusion into the analysis, for which we simply haveto replacea4 by a4 +BΩ(V + kD)/V , withB as in (3.15), which merely shiftsk∗

closer to zero. In this case, the band of unstable modes shrinks, which is consistentwith the physical smoothing effect of surface-diffusion atshort length scales.

0 0.2 0.4 0.6 0.8 1.0 1.2

k / k*

-1.0

-0.5

0

0.5

1.0

σ k / σ

m

Figure 3.4 – Linear dispersion relations given by equation (3.30) (dashed line) and (3.34)(solid line), normalized by the growth-rate of the most unstable mode, vs spatial frequencyk normalized byk∗. Both axes are in arbitrary units.

3.2.2 Instantaneous surface kinetics

If the sticking probability is essentially one, as seen above kD → ∞. This fastattachment condition occurs in many irreversible growth processes [39].

Following a similar procedure (and long wave-length approximation) as theone that led us to the KS dispersion relation in the previous section, we now get

σk = D

(

Γ2Ω2

2− BΩ

D

)

k4 − 3ΓΩV k2

2+ |k|(V − ΓΩDk2)×

×[

1 − ΓΩV

D+

(

Γ2Ω2

4− BΩ

D

)

k2

]1/2

.

(3.33)

This expression has several interesting limits. For instance, if we neglect surfacetension and surface diffusion terms (that is, forΓ = B = 0), thenσk = V |k|,the well-known dispersion relation of the Diffusion Limited Aggregation (DLA)model [39]. In this case, every spatial length scale is unstable, the shorter ones(largek values) growing faster than the larger ones. Thus, in such a case the aggre-gate consists of wide branches plenty of small tips. Moreover, there is actually no


characteristic length scale in the system, hence the aggregate has scale invariance(that is, it is self-similar).

If we only neglect the surface diffusion term and, sinceΓΩ ≡ d0 (the capillaritylength) is typically in the range10−6 − 10−5 cm, andD/V ≡ lD (the diffusionlength) is close to1 − 10−2 cm, we can write

σk ≃ V |k|(1 − d0lDk2), (3.34)

which is the celebrated Mullins-Sekerka (MS) dispersion relation [134; 135] (seefigure 3.4), ubiquitous in growth systems in which diffusiveinstabilities (inducedby shadowing of large branches over smaller surface features) compete with relax-ation by surface tension. This dispersion relation has beenexperimentally verifiedin several ECD systems [136; 137; 138], and has actually beentheoretically pro-posed before for ECD by Barkeyet al. [139] (although under non-galvanostaticconditions). The appearance of this dispersion relation isnot a coincidence. Ac-tually, it is a central idea of the present thesis to show why (as we show in thefollowing chapters).

However, in many diffusive growth systems both surface tension and surfacediffusion are non-negligible; considering again the physical hypothesisd0 ≪ lDand a long wave-length approximation, we get

σk = V |k|[1 − (d0lD +BΩ/2D)k2] −BΩk4. (3.35)

Nevertheless, there aree.g. some CVD conditions [37; 38], for which the vaporpressure in the dilute phase is so low that relaxation by evaporation/condensationis negligible in practice. In such a case, the dispersion relation is provided by (3.35)with an effective zero value ford0.

In summary, we see that, in reducing the efficiency of attachment from com-plete (kD → ∞) to finite (kD < ∞), the symmetry of the dispersion relationchanges so that non-local terms like odd powers of|k| are replaced by local (lin-ear) interactions. For instance,−k2ζk(t) is the Fourier transform of the local term∂2

xζ(x, t), while |k|ζk(t) cannot be written as (the transform of) any local differen-tial operator acting onζ(x, t). In fact, the terms proportional to odd powers ofkhave the following form in real space

F−1

k2p−1hk

∝ P.V.

∫

R2

∇′2ph(r′)|r − r′| , (3.36)

where the principal value of the integral is to be taken. Thisis nothing but theconvolution of the surface derivatives field,∇2ph, with the long-range kernel1/r(for d = 1 the right hand side of equation (4.2) can be written as the Hilbert trans-form of ∇2ph, for further details see appendix A). Hence, as for thek2p−1 termsin (3.34), the value of the local growth velocity at a given surface point dependson the values of the height derivatives at all other surface points. Moreover, thisresult can be understood heuristically: if the sticking probability is small, then par-ticles arriving at the interface do not stick the first time they reach it, but they can


explore other regions of the aggregate. This attenuates thenon-local shadowingeffect mentioned above, so that growth becomes only due to the local geometry ofthe surface.

Although thekD → ∞ limit is a mathematical idealization, for practical pur-poses we can determine under which conditions it is physically attained. Thus, ifwe introduce equation (3.25) describing the concentrationfield for a flat interfaceinto the boundary condition (3.12), the latter takes the form

c− c0eqD/kD

=c− c0D/V

. (3.37)

The termD/V is the diffusion length; hence, analogously, we can defineD/kD asasticking length. Physically, this length can be seen as the typical distancetraveledby a particle between its first arrival at the interface and its final sticking site. Wecan neglect this length scale if the sticking probability isclose to unity. On thecontrary, if kD → 0, the distance that the particle can explore before attachingis infinite. Therefore, taking equation (3.37) into account, we can say that thekD → ∞ limit describes accurately the problem whenkD ≫ V , and in sucha case the diffusion length and the capillarity length determine the characteristiclength-scale of the system.

3.2.3 Full dispersion relation

One step further in the analysis of the dispersion relation can be taken by solvingnumerically the equationTkω = 0. As seen before, the dispersion relation can onlybe derived from this equation after neglecting some terms (in the MS case) or byperforming a perturbative expansion (in the KS case).

In order to manage the function (3.28) it is convenient to employ non-dimensionalthe space and time variables, re-rescalingr andt by 2lD and4lD/V , respectively.In the new coordinates the non-dimensional wave number and angular frequencyare k = 2Dk/V , ω = 4Dω/V 2, and, after some calculations (recall that wehave defined the dispersion relation through the equationiω = σk), the equationTkω = 0 simplifies into

(

1 +[

1 + σk + k2]1/2

)(

2 − V

2kDσk − d0

2lDk2

)

= σk + 4. (3.38)

This implicit equation depends only on two length scales andtwo velocities which,hopefully, can be experimentally measured. As said before,in electrodepositionthe capillarity lengthd0 is about the same order of magnitude of the lattice spac-ing [101], i.e. between10−6 − 10−5 cm, while the diffusion length ranges fromthe whole lateral size of the electrochemical cell, about1 cm, down to10−2 cm,depending on the mean growth velocity. This velocity,V , typically ranges from10−6 cm s−1, for very slow growth [137], to10−4 cm s−1 [126]. Differently, themass transport coefficientkD can be adjusted changing the overpotential∆ψ, sothat, the ratioV/kD can range from0 to approximately the unit.


Clearly, equation (3.38) provides the dispersion relation. In particular, it canbe easily checked thatk equal to zero always givesσk = 0, as expected due to theinvariance of the physics under uniform translations of theform ζ → ζ + const.Another interesting feature of equation (3.38) arises whenwe look for the modeswith σk equal to zero. Namely, the functionσk has a band of unstable modes onlyif the parameterrl = d0/lD is smaller than one. In fact, the reduced equation isindependent onkD, that is the surface kinetics does not modify the width of theunstable band of modes. In this case equation (3.38) reads

√

1 + k2 =4 + rlk

2

4 − rlk2, (3.39)

where the equality holds for non-zerok only if rl < 1: it easy to see that the lefthand side of equation (3.39) can equal the right hand side only if the hyperbola hasa smaller concavity than the square root (see figure 3.5).

0 2 4 6 8 10k

-9

-6

-3

0

3

6

9

Figure 3.5 – Second zero condition forrl = 0.2. Red curve is the left hand side ofequation (3.39) while the blue curve is its right hand side.

The two dispersion relations (3.30) and (3.34) can satisfactorily approximatethe full dispersion relation depending on certain ratios ofthe parameters of themodel (3.11)-(3.14). The numerical solution of (3.38) could be used to identifywhich parameters influence the accuracy of the approximations made leading to(3.30), (3.34) and to quantify the error of such approximations.

First, we consider the fast kinetics regime,kD → ∞, in which the MS disper-sion relation is derived under the assumptiond0 ≪ lD, and is only valid fork suchthatd0 ≪ lDk. For these reasons, the ratio between the capillarity and diffusionlengths,rl, allows us to assess the error for this approximation. On theleft panel offigure 3.6 we show the convergence of the MS approximation to the full dispersionrelation whenrl goes to zero. In this comparison we keep fixed both the capillaritylength and the diffusion coefficient, whereas we tune the velocity V in order tochangerl. The figure shows that equation (3.34) is a poor approximation of thefull dispersion relation whenrl → 1.


0 0.5 1

k/k0

-0.5

0

0.5

1

σ k/σk m

V = 10-4

V = 1V = 10MS with V = 10

-4

MS with V = 1MS with V = 10

10-4

10-2

100

V

10-1

100

101

102

∆km [%

]

5.3 V 0.507

10-2

100

102

V

10-1

100

101

102

∆σk m

[%]

9.4 V 0.522

Figure 3.6 – Comparison between the analytical approximation equation(3.34) and thefull dispersion relation in the fast kinetic regime, i.e.k−1

D = 0. Left panel: The curves arethe MS approximation equation (3.34) while the discrete points are calculated numericallyfrom the implicit equation for the full dispersion relationequation (3.38). The velocityVchanges across several order of magnitude while other parameters are kept fixed:d0 =5 × 10−4, D = 0.1. The axes are rescaled in order to collapse the points onto the sameuniversal function. Right panel: Errors of the analytical approximation relative to the fulldispersion relation. We plot the relative error for the maximum wavenumber (left) and forthe maximum of the dispersion relation (right) (in percentage) as functions ofV . All axesare in arbitrary units.

Besides, we can estimate numerically the errors in the position of the mostrelevant unstable modekm, and its heightσkm

. On the right panel of figure 3.6 weshow that relative errors scale approximately as the squareroot ofV .

On the other hand, when the attachment kinetics is not instantaneous, i.e. incase of a mass transport coefficientkD that is not very large compared to the aver-age growth velocity, we have employed a perturbative expansion in powers of inkin order to obtain the KS dispersion relation, see equation (3.30). In order to vali-date this approximation we recall that it has been derived within two constraints:

(i) that the band of unstable modes has the proper size and

(ii) that the long wave-length modes, i.e. smallk, have to recover thek2 behaviorof the full dispersion relation.

As stressed above, the width of the unstable band is independent of kD and canbe exactly obtained from equation (3.38). Moreover, the large wave-length scalingof equation (3.26) is mainly determined by the smallk behavior ofσk. For thesereasons, equation (3.30) approximates poorly the full dispersion relation in theregion around its maximum. This discrepancy gives rise to a different characteristicgrowth time for the aggregate in the linear regime. In fact, the most unstable mode,km, is related to the pattern wave-lengthλm when the mean value of the surfaceslope is small (‖∇ζ‖ ≪ 1) and its characteristic growth time is proportional toσ−1

km.In figure 3.7 we show the convergence of the approximation (3.30) to the full

dispersion relation by fixing the mass transport coefficientand changing the cap-


00

10-2

2x10-2-5x10

-6

0

5x10-6

1x10-5

1.5x10-5

2x10-5

σ k

00

3x10-3

6x10-3

0

2x10-7

4x10-7

6x10-7

8x10-7

00

3x10-4

6x10-4 -5x10

-11

0

5x10-11

1x10-10

1.5x10-10

00

10-3

2x10-3

0

1.5x10-7

3x10-7

4.5x10-7

6x10-7

σ k

d0 = 10

00

10-3

2x10-3

0

1x10-7

2x10-7

3x10-7

4x10-7

d0 = 50

3x10-4

4.5x10-4

6x10-4

9x10-11

1.05x10-10

1.2x10-10

d0 = 99

Figure 3.7 – Comparison between the analytical approximation equation(3.30) (solidline) and the full dispersion relation (blue points) when the kinetics is not instantaneous.The figures in the upper row show the convergence of the approximation changing thecapillarity lengthd0 by one order of magnitude. Parameters in this case areV = 10−3,D = 0.1 andkD = 10−3. The figures in the second row are zooms of the graphs above.The first two highlight the convergence in the smallk region while the third one shows theaccuracy around the most unstable modeσkm

. All axes are in arbitrary units.

illarity length. The long scale limit behavior is well captured for all values of thecapillarity but, ford0 ≪ lD, the convergence occurs only for systems that are muchlarger than the wave-length of the most unstable modekm (see the first two insetsof figure 3.7). This condition holds for ECD and other experiments, hence, the ob-servation of the KS behavior is usually hindered by the finitesize effects of thesesystems [53; 54]. As shown, whenrl increases, also does the region where theKS approximation is accurate. In the unusual parameter condition of comparablevalues for the capillarity and the diffusion lengths,d0 ≃ lD, equation (3.7) resem-bles very closely the full dispersion relation and KS-type behavior is expected inthis case (see the third column of figure 3.7). On the other hand, for d0 ≪ lDconvergence occurs only for very small values ofk.

Beyond the two analytical expressions derived for the dispersion relation, wecan study the dispersion relation that satisfies equation (3.38) by decreasing theattachment kinetics. The reduction of the mass transport coefficient changes dras-tically the system behavior: (i) the position and the value of the most unstablemode moves; (ii) the dispersion relation goes to zero in a different way. This lastdifference with (3.34) is remarkable, because a change in the functional depen-


dence of the lowest order term inσk, from |k| to k2, gives rise to a different scaling[9]. Moreover, we have to differentiate between two parameter conditions: whenthe capillarity is much smaller than the diffusion length, i.e. the physically morerelevant regime, and when these two length scales are comparable.

0 10 20 30 40 50k

-1.0

-0.5

0

0.5

1.0

σ k/σk m

1/kD = 0, σ

km = 1.6770

1/kD = 0.5, σ

km = 0.7645

1/kD = 1, σ

km = 0.5106

1/kD = 5, σ

km = 0.1467

00

10-2

2x10-2 0

5x10-4

10-3

1.5x10-3

Figure 3.8 – Full dispersion relation for differentkD in the physically relevant regime,i.e. whend0 ≪ lD. Parameters areV = 0.1,D = 0.1 andd0 = 5 × 10−4. In the inset weshow the convergence to equation (3.30) changing the mass transport coefficient. All axesare in arbitrary units.

In figure 3.8 we show how the full dispersion relation changeswith the attach-ment kinetics when the capillarity length is negligible compared to the diffusionlength. The black points in the figure represent the limit of instantaneous surfacekinetics, i.e. the MS limit. DecreasingkD we observe the same behavior forkm,hence the linear pattern wave-length increases and, as expected,σkm

decreases (seethe legend of figure3.8). Nevertheless, the inset of figure 3.8 shows how decreas-ing the mass transport coefficient increases the interval inwhich the full dispersionrelation has a KS-type (∼ k2) smallk behavior.

When the system is in the other parameter condition, that is whend0 ≃ lD, asmaller value ofkD corresponds to a larger value ofkm and, consequently, an in-stability pattern with a smaller wave-length, see figure 3.9. As in the previous case,the full dispersion relation changes its functional dependence and takes a KS-typebehavior for wave-numbers that are of the same order of magnitude askm. In thehypothetical case of a system satisfying this condition (d0 ≃ lD), the transitionfrom the MS to the KS behavior can be observed at length scalescomparable withthe wave-length of the most unstable mode of the dispersion relation,km. Never-theless, this is not the case of electrodeposition, in whichthe separation of thesetwo characteristic lengths is usually very large.

Finally, in figure 3.10 we summarize the result of the change of the most un-


0 0.2 0.4 0.6 0.8k

-0.5

0

0.5

1.0

σ k/σk m

1/kD = 0, σ

km = 1.461x10

-2

1/kD = 1, σ

km = 1.103x10

-2

1/kD = 5, σ

km = 5.631x10

-3

1/kD = 10, σ

km = 3.506x10

-3

1/kD = 25, σ

km = 1.647x10

-3

Figure 3.9 – Full dispersion relation for differentk−1

D in the regime of comparable cap-illarity and diffusion lengths. Parameters areV = 0.1, D = 0.1 andd0 = 0.1. All axesare in arbitrary units.

stable wave-lengthλm and the maximum of the dispersion relationσkmwith kD

considering the two opposite behaviors. The parameters used in this figure are thesame as in figures 3.8 and 3.9. Notice that in the two regimesσk decreases exactlyin the same way whereasλm displays an opposite behavior. Decreasing the stick-ing coefficient, i.e. decreasingkD, the attachment process slows down leading toa drop-off of theσkm

. Hence, forV ∼ kD the growing time of the most unsta-ble linear mode,σ−1

km, begins to increase when the sticking coefficient decreases,

and, within the regimeV ≫ kD, we observe a power law behavior which doesnot dependent on the capillarity or the diffusion length, see second row of figure3.10. On the contrary, decreasing the sticking coefficient,the pattern wavelengthformed during the linear growth regime,λm, increases or decreases depending onthe ratio between the capillarity and the diffusion length,see first row of figure3.10. Ford0 ≪ lD we observe an increase ofλm up to three times its value in theMullins-Sekerka condition (the red line of figure 3.10).

Microscopically, the conditiond0 ≪ lD means that the particles have a meanfree path much larger than the typical structures formed on the surface. For thisreason, a decrease in the sticking coefficient allows particles to explore the mostunexposed regions of the surface, i.e. the spots with minimum height, leading toa linear instability with a larger pattern wavelength. On the contrary, in the otherparameters regime, i.e. ford0 ∼ lD, we observe a decrease ofλm when we de-crease the sticking coefficient. There is not an intuitive explanation for this result,but, at least microscopically, the particles can not explore a region of the surfacelarger than the extent of one structure formed by the linear instability. This restric-tion on the mean free path of the particles and the constraintof an asymptotick2

behavior fork → 0 result in a decrease ofλm. Thus, the non-dimensional numberd0/lD gives both a mathematical and a physical criterion to distinguish which are


10-4

10-2

100

102

1040.2

0.3

0.4

0.5

0.6

0.7

λ m

λm(1/k

D=0)

10-4

10-2

100

102

104

13

14

15

16

λm(1/k

D=0)

10-4

10-2

100

102

104

1/kD

10-4

10-2

100

σ k m

σk

m(1/k

D=0)

10-4

10-2

100

102

104

1/kD

10-6

10-4

10-2

σk

m(1/k

D=0)

Figure 3.10 – Pattern wave-length and characteristic growth time as function ofkD. Leftpanel: physically relevant cased0 ≪ lD. Right panel: case of comparable capillarity anddiffusion lengthsd0 ≃ lD. The red lines correspond to the limit of instantaneous surfacekinetics, i.e. the MS condition. All axes are in arbitrary units.

the most relevant parameters.

3.2.4 Non-linear evolution equation

In this section we proceed one step further with our perturbative approach by in-cluding the lowest order nonlinear contributions to equation (3.26). If we evaluatethe Green function of the problem at the boundary we find

G(r − r′, τ) =Θ(τ)

4πDτexp

[

−(x− x′)2

4Dτ−(

ζ − ζ ′ + V τ)2

4Dτ

]

, (3.40)

whereτ = t− t′. Expanding the last term in the argument of the exponential as aseries inζ we get(ζ − ζ ′)2 + 2V (ζ − ζ ′)τ + V 2τ2. The second and third termswere already taken into account before in the linear analysis, so the only nonlinearcontribution in equation (3.40) is related to the first term,(ζ − ζ ′)2, that introducesa correcting factorexp[−(ζ − ζ ′)2/(4Dτ)] which is only significant whenζ ′ ≃ ζ,hence we can replaceζ ′ − ζ by the lowest term in its Taylor expansion, to get

exp

(

−(ζ − ζ ′)2

4Dτ

)

≃ 1 − (∂xζ)2(x′ − x)2

4Dτ. (3.41)


By incorporating this contribution into the Green functionexpansion [9] the con-sequence can be readily seen to be the addition of a mere term equal to the (Fouriertransform of)V (∂xζ)

2/2 to the right hand side of equation (3.26), resulting intoan evolution equation with the form

∂tζk(t) = σkζk(t) +V

2N [ζ]k + ηk(t), (3.42)

whereN [ζ]k is the Fourier transform ofN [ζ] = (∂xζ)2. Note that the non-linear

term obtained is precisely the characteristic KPZ non-linearity, that appears herewith a coefficient equal to half the average growth velocity,agreeing with standardmesoscopic arguments [27]. Physically, we can attribute its presence to the fact thatgrowth is conformal under the assumptions made in the derivation of the model (forinstance, whenever Butler-Volmer equation is applicable).

The KPZ non-linearity(∂xζ)2 is also recognized as the (asymptotically) most

relevant non-linear term that can be obtained for a non-conservative growth equa-tion such as equation (3.22). Hence, any other non-linear term will not change thelong time, large length-scale behavior of the system[3]. Aswe have shown above,for small sticking probabilityσk is given by equation (3.30) while for large stick-ing it is given by the MSKPZ equation studied in section 4.1 [see equation (4.4)].In all cases the noise correlations involve constant terms,as well as terms that areproportional to increasingly higher powers ofk. By retaining only the lowest-ordercontributions in a long wave-length and quasi-static approximation [9], we obtain

〈ηkωηk′ω′〉 =(

Π0 + Π2k2)

δk+k′ δω+ω′ , (3.43)

where

Π0 =

V c0

(

1 +2V

kD

)

, kD <∞V c0eq, kD → ∞

, (3.44)

and

Π2 =

2Π0 l2D + 2Dsνs

(

1 +V

kD

)2

, kD <∞2Π0 l

2D + 2Dsνs, kD → ∞

, (3.45)

where equation (3.24) forc0 is to be used in the case of finite kinetics. In general,parameterΠ0 provides the strength of non-conserved noise, whileΠ2 measures thecontribution of conserved noise [140] to the interface fluctuations.

We can briefly interpret the physical content of equation (3.44) and (3.45).Thus, in the case of very high sticking conditions,kD ≫ V , aggregate growth isdiffusion limited because of the very efficient reaction kinetics, and the varianceof nonconserved fluctuations is simply proportional to the particle concentration atthe surface. The opposite scenario occurs when the kinetic coefficient is negligi-ble as compared to the mean interface growth velocity,kD ≪ V , and the processbecomes reaction limited. In this case, the concentration at the interface,c0, is

3.3 Comparison with experiments 61

proportional to the particle concentration at the infinity,ca, and nonconserved fluc-tuations are enhanced. Finally, in the case of comparable average velocity andkinetic coefficient,V ∼ kD, the reaction kinetics at the surface and the diffusionin the bulk have a similar relevance,Π0 being proportional to a weighted meanbetweenc0eq andca.

On the other hand, the variance of the conserved noise,Π2, is composed by twoterms. The first one arises as a higher-order contribution ofthe nonconserved noise,in which the typical diffusion lengthlD appears. The second one (proportional toDsνs) originates in the conserved mechanism of surface diffusion. The presenceof conserved noise can naturally modify some short length and time scales of thesystem but, in the presence of nonconserved noise, it is known to be irrelevant to thelarge-scale behavior. Thus,Π2 will be neglected in the numerical study performedin chapter 4.

For the case of finite kinetic coefficient, the evolution equation (3.42) is thestochastic generalization of the KS equation [64; 73; 74]. In the case of fast at-tachmentkD → ∞, the resulting interface equation combines the linear dispersionrelation of MS with the KPZ nonlinearity, two ”ingredients”that seem ubiquitousin growth systems (see section 4.1).

3.3 Comparison with experiments

In this section, we focus on the applications of the model equations to understandand explain some experimental results, qualitatively and quantitatively and, more-over, compare our results from continuum theory to relevantdiscrete models ofdiffusion limited growth.

First we consider the two analytical approximations of the full dispersion rela-tion and we use them in order to explain some relevant features that characterizethe ECD experiments. Besides, one step further in the determination of the disper-sion relation of these experiments can be done solving numerically equation (3.38)and plugging this result into equation (3.42). In fact, our model provides a fulldispersion relation that interpolates quite well the data from the estimation ofσk

in Cu ECD. Employing a inverse method to estimate the mass transport coefficientwe are able to collect all the parameters needed by the model.Finally, we con-struct an effective dispersion relation to study the scaling of compact aggregates ingalvanostatic electrodeposition.

3.3.1 Mullins-Sekerka and Kuramoto-Sivashinsky behavior s

From the experimental point of view, it is very difficult to determine if the disper-sion relation characterizing a specific physical system is the MS or, rather, the KSone, because they are both very similar except very neark = 0 (see figure 3.4).In principle, such a distinction would be very informative,since it could providea method to assess whether the dynamics is diffusion limited(kD → ∞), or elsereaction limited (kD ≪ V ). Several previous theoretical works in this field [141;


142] predict, qualitatively, a KS dispersion relation, while other [139], predict adispersion relation similar to the one of MS. All these studies are not necessarilyincompatible with one another because they are consideringdifferent experimentalconditions. Moreover, to our knowledge, there are only a fewexperimental reportsin which the dispersion relation is measured, and in most cases the authors presumethat it can be accounted for by MS [136; 137; 138]. Notwithstanding, within ex-perimental uncertainties that are specially severe at small wave-vectors, they couldall equally have been described by the KS dispersion relation.

Despite these difficulties, we can sill try to interpret someexperimental resultsreported in the literature. Legeret al. [143; 124; 126] have presented several ex-haustive works dealing with ECD of Cu under galvanostatic conditions. In additionto properties related to the aggregate, they also provide detailed information aboutthe cation concentration. Their main result in this respectis that such concentra-tion obeys experimentally equation (3.25) as anticipated.It can also be seen thatthe product aggregates are branched and that the topmost site at every positionxdefines a rough front growing with a constant velocity.

b) c) d)a)

Figure 3.11 – Influence ofCa at fixed diffusion length on the morphology of copperdeposit obtained from Cu(CH3COO)2 solutions [126]. Experimental parameters: a)Ca =0.2 mol l−1, j = 22 mA cm−2, lD ∼ 235 µm, λm ∼ 210 µm. b)Ca = 0.165 mol l−1,j = 20 mA cm−2, lD ∼ 275 µm, λm ∼ 245 µm. c)Ca = 0.135 mol l−1, j = 16 mAcm−2, lD ∼ 290 µm, λm ∼ 325 µm. d)Ca = 0.1 mol l−1, j = 12 mA cm−2, lD ∼ 220µm,λm ∼ 432 µm [126].

In figure 3.11 we show the results obtained by Legeret al. about the dependenceof the branch width from the diffusion length: It is clear that λm and lD are notlinearly related. Moreover, they seem to better agree with the present predictionfrom either of our effective interface equations,λm ∝ l

1/2D [1; 9], rather than with

the linear behavior argued for in [126].Another important experimental feature is the relation betweenλm andCa.

From equation (3.8) andd0 = ΩΓ = Ωγ/RT , the characteristic length scale,λm,

will be proportional toC−1/2a . Thus, one would expect the branches to be narrower

as we increase the initial concentration, consistent with the patterns obtained byLegeret al. [1; 9].

We will try also to interpret the ECD experiments of Schilardi et al.[144]. They


have measured the interface global width considering that the topmost heights ofthe branches provide a well defined front. Thus, the time evolution of the globalwidth, or roughness, presents three different well defined regimes: A short initialtransient, which cannot be accurately characterized by anypower-law due to thelack of measured points, is followed by an unstable transient. We consider thesystem unstable in the sense that the average interface velocity is not a constantbut, rather, grows with time. Finally, the system reaches a regime characterizedby exponent values that are compatible with those of KPZ the equation, while theaggregate grows at a constant velocity. These regimes againresemble qualitativelythe behavior expected for the noisy KS equation.

Moreover, we can check whether the order of magnitude of the experimentalparameters is compatible with our predictions. First, we estimate km from themean width of the branches within the unstable regime. This width is about0.05mm [144], hencekm ≃ 1.3 × 103 cm−1. Besides, the mean aggregate velocity atlong times isV ≃ 2 × 10−4 cm s−1, and the diffusion coefficient isD ≃ 10−5

cm2 s−1, so that the diffusion length is aboutlD = D/V ≃ 0.05 cm. Furthermore,the instability appears at times of order1/σkm

. In the experiment this time is about6 min. Thus,σkm

≃ 3 × 10−3 s−1 a value incompatible with a MS dispersionrelation. In fact, considering that a typical capillary length ranges within10−6 −10−5 cm, from equation (3.34) we obtain thatσkm

∼ 4 × 10−2 − 2 × 10−1 s−1.Hence, from these estimation we can conclude that the growthprocess is kineticallylimited and the KS-type behavior can be observed at large wave-length.

As a final example, let us perform some comparison with the experiments ofPastor and Rubio [145; 138]. At short times, they obtain compact aggregates withexponent values [138]α = 1.3 ± 0.2, αl = 0.9 ± 0.1, z = 3.2 ± 0.3, andβ = 0.4 ± 0.08. This means that the interface is superrough (α > 1). Afterthis superrough regime, the aggregate becomes unstable andthe dispersion rela-tion has the MS form. The fact that the aggregates are compact(and not rami-fied) seems to show that surface diffusion is an important growth mechanism inthese experiments (which is also consistent with the small value of the velocity,V ≃ 4 µm/min). Consequently, we can use equation (3.42) with the MSdis-persion relation and an additional surface diffusion contribution [−Bk4ζk(t)] tounderstand the behavior of the experiments. We choose the parametersν = 0.25,K = Π0 = 1 andλ = Π2 = 0 (because the velocity is small and we are onlyinterested in the short time regime), for several values ofB between0 and1 in or-der to determine the influence of surface diffusion in the growth exponents. Otherparameters only change the characteristic length and time scales of the experiment.The exponents thus obtained numerically are (withB = 0.75) β ≃ 0.39 ± 0.02,andα ≃ 1.3 ± 0.1, which are (within error bars) equal to the experimental ones.Clearly, in this linear equation (note thatλ is zero) the regime before the onsetof the morphological instability is dominated by the stabilizing surface diffusionterm and the exponents are compatible with those found in linear Molecular BeamEpitaxy model (β = 3/8 andα = 3/2) [1; 9]. As we have pointed out above,this superrough regime is followed by an unstable transientcharacterized by the


MS dispersion relation, as has been also observed in other ECD experiments by deBruyn [136], and Kahandaet al. [137].

More recently, additional ECD experiments have been reported under galvano-static conditions.E.g. in Ref. [146] three growth regimes can be distinguished: afirst one at short times, in which a Mullins-Sekerka like instability is reported, isfollowed by a regime in which anomalous scaling (see section2.1.1) takes place,and finally at long times ordinary Family-Vicsek scaling [3]is recovered. Similartransitions to and from anomalous scaling behavior have been also reported in Ref.[147]. Our present theory does not predict the anomalous scaling regimes reportedby these authors. This may be due to the small slope conditionemployed in thederivation of equation (3.42). However, we want to emphasize two points in thisrespect:(i) In Ref. [147] the authors report a transition from rough interface behav-ior to mound formation. These mounds can be obtained by numerical integrationof equation (4.4) in 2+1 dimensions, see section 4.1.(ii) As we will show in thenext section, the theory is in good agreement with a discretemodel of growth inwhich anomalous scaling is clearly reproduced. Hopefully,a numerical integrationof the full moving boundary problem (3.22) would capture theanomalous scalingregime, and this will be the subject of chapter 6.

3.3.2 Beyond the analytic approximation

As we have said before, several experimental studies of ECD tried to estimate thefunctional form of the dispersion relation in compact aggregates. An example takenfrom the literature is the growth of quasi-two-dimensionalCu and Ag branchesby de Bruyn [148; 136]. In [136], the branches grow at the cathode due to theelectrodeposition of ions created from the CuSO4 aqueous solution. The analysisof the early stages of the branch growth allows the author to fit the experimentaldispersion relation to

σk =q|k|(1 − rk2)

1 + sk, (3.46)

whereq, r, ands depend on the properties of the electrolyte and the depositedmetal, the ion concentration, and the applied current. Fromthe reported data andfigures in Ref. [136] we can extract the information summarized in table 3.1, wherewe also include data from the experiments in [137; 124].

We need to supplement these values with reasonable estimates for the capillarylength,d0 and the mass transfer coefficient,kD. We estimated0 from the experi-ments by Kahandaet al.[137] and we rescale this value by the cation concentrationto d0 = 2 × 10−5 cm. Finally,kD can be obtained from the location of the max-imum in the dispersion relation. Thus we can estimatekD to be in the range (dueto the experimental uncertainty)4.44 − 5.56 × 10−4 cm s−1.

These data are compatible with the assumptions made in the derivation of themoving boundary problem introduced in this chapter. For instance, the diffusionlength is much larger than the branch width (typically0.1 mm in these experiments)but the mean growth velocity is comparable to the mass transport coefficient, and


Parameter Reference [136] Reference [137] Reference [124]

Ca (mol cm−3) 1 × 10−4 2 × 10−3 5 × 10−4

I (mA) 1.4225 NR NR

J0 (mA cm−2) 3 × 10−2 3 × 10−2 3 × 10−2

J (mA cm−2) 11.16 NR 65.00

Dc (cm2 s−1) 0.720 × 10−5 0.720 × 10−5 0.720 × 10−5

Da (cm2 s−1) 1.065 × 10−5 1.065 × 10−5 1.902 × 10−5

zc 2 2 2

V (cm s−1) 3.45 × 10−4 ≈ 1.00 × 10−6 4.87 × 10−4

lD (cm) 2.49 × 10−2 ≈ 8.59 2.15 × 10−2

Table 3.1 – Experimental parameter values estimated from [136] (second column), [137](third column), and [124] (fourth column). NR stands for notreported.

0 50 100 150k [1/mm]

-5.0×10-3

0.0

5.0×10-3

1.0×10-2

1.5×10-2

2.0×10-2

2.5×10-2

σ k[1

/s]

KD= 5.56x10

-4 cm/s, ∆ψ= -0.142 V

KD= 5.26x10

-4 cm/s, ∆ψ= -0.140 V

KD= 5.00x10

-4 cm/s, ∆ψ= -0.139 V

KD= 4.44x10

-4 cm/s, ∆ψ= -0.136 V

Figure 3.12 – Linear dispersion relation,σk, of Cu electrodeposition. Squares providethe experimental values taken from [136], and the red line isthe fit to (3.46) made in thisreference. The other lines are numerical solutions of (3.38) using the parameters in thesecond column in table 3.1 for different values ofKD = RckD as indicated in the legend.

surface kinetics becomes relevant. For this reason, the growth process is not diffu-sion limited and the dispersion relation is very different from the Mullins-Sekerkaone. Similarly, the estimate forkD implies that the applied overpotential∆ψ variesbetween−0.160 and−0.154 V, which are plausible enough. With this informationwe can now proceed to recast the experimental results in [136] as shown in figure3.12. The agreement between our theory (3.38) and experiment is remarkable, tak-ing into account that we are not performing any fit of the data.Moreover, the long


wave-length analytical approximations, equation (3.30) and (3.34), are not able tofit the experimental data.

We have performed a numerical integration of an improved version of equation(3.42) in which the full linear dispersion relation (3.38) is employed, using thestochastic pseudo-spectral scheme (see section 2.3.2). Results are shown in figure3.13 (left panel) for the time evolution of the surface roughness and the surfacestructure factor of the surface height. As we see, the roughness increases very fast(close toW (t) ∼ t1.8) within an intermediate time regime, after which saturationto a stationary value is achieved.

10-1

100

101

102

k [1/mm]

100

103

106

S(k,

t)2α+d = 3.5

10-1

100

101

102

103

t [s]

10-2

10-1

100

101

W(t

)

β = 1.80

10-2

10-1

100

101

102

103

t

10-2

100

102

104

W(t

)

β = 1.8510-2

10-1

100

101

k

100

105

1010

1015

S(k,

t)

2α+d = 3.5

Figure 3.13 – Time evolutions of the surface roughness and PSD (insets) using (3.42)with σk given by (3.38) (left panel) and (3.47) forµ = 0.75 (right panel).

As for the time evolution of the PSD curves, they show a standard Family-Vicsek behavior, leading to a global roughness exponentα = 1.25, from theasymptotic behaviorS(k) ∼ k−(2α+d) for a 1D (d = 1) interface. The timeevolution of the interface height in real space can be appreciated in movie 1 in theaccompanying CDROM. As is clear from the images, an initially disordered inter-face develops mound-like structures that coarsen with time, leading to a long timesuper-rough morphology. This is inconsistent with the expectation of KPZ scalingfor the present case since, taking into account the values ofV andkD, a KS behav-ior is expected for (3.38). In the inset of figure 3.14 we show the full dispersionrelation (3.38) for sufficiently small values of the wavevector at which behavior isindeed of the KS type. This apparent contradiction can be explained taking intoaccount that such a KS shape (that holds for length scales in the cm range) can-not be observed in practice for the physical length scales that are accessible in theexperiment.

As seen in the figure, the effective behavior of the dispersion relation is notof the KS type but, rather,σk behaves (sub)linearly for the smallest accessiblekvalues. This is made explicit in the main panel of figure (3.14), where a straight lineis shown for reference at the smallest wavevector. Thus, forall practical purposesthe linear dispersion relation that is experimentally operative doesnot behave as


0 0.1 0.2k [1/mm]

0

1.5×10-5

3.0×10-5

4.5×10-5

σ k [1/s

]

20 40 60 80 100 120k [1/mm]

-0.005

0

0.005

0.01

0.015

σ k [1/s

]

Figure 3.14 – Full linear dispersion relation as shown in figure 3.12 forkD = 5× 10−4

cm s−1. For the meaning of the solid line see the main text. The insetis a zoom showingtheσk ∼ k2 behavior for sufficiently smallk values.

∼ k2 but is, rather, closer to a form such as

σk(µ) = νµ|k|µ −Kk2, (3.47)

whereνµ andK are positive, effective parameters, and0 < µ ≤ 1.Taking into account the previous paragraphs, we see that (3.47) summarizes

the main features of the linear dispersion relation for manyexperimental systems,namely,

(i) fast growth at the smallest availablek values,σk ∼ |k|µ,

(ii) single finite maximum at intermediatek values, and

(iii) sufficiently fast decay at large wavevectors, at leastof the formσk ∼ −k2 orfaster (i.e. with a larger exponent value).

Thus, a simple way to explore the dynamics as predicted by thesmall slope ap-proximation with an improved dispersion relation consistsin considering equation(3.42), but using (3.47) rather than the analytical approximations, i.e. (3.34) or(3.30). We have carried out such type of study for several values ofµ. For the sakeof reference, movie 2 in the accompanying CDROM shows the evolution of an in-terface obeying the KS equation, while movie 3 displays the analogous evolutionbut as described by equation (3.42) using the effective dispersion relation (3.47)for µ = 0.75.

It is apparent that the latter resembles much more closely the dynamics associ-ated with the full dispersion relation (3.38). Quantitative support is provided by theright panel of figure 3.13, where the evolution of the roughness and PSD is shownfor equation (3.42) with (3.47) forµ = 0.75. In the inset, we can appreciate theformation of a periodic surface (signaled as a peak in the PSD) at initial times that


grows very rapidly in amplitude, as confirmed by the exponential evolution of theglobal surface roughnessW (t), shown in the main panel of figure 3.13, for timest . 15. For long enough times, this peak smears out, once the lateral correlationlength is larger than the pattern wave-length. That marks the onset for non-lineareffects that induce power-law behavior (kinetic roughening) both in the roughnessand in the PSD, approximately asW (t) ∼ t1.85 andS(k) ∼ 1/k3.5, from which weconclude that the associated critical exponents have valuesβ = 1.85 andα = 1.25.

From the exponents so far reported, we can obtainz = α/β = 0.68, notfar from z = µ = 0.75, an exponent relation which holds for (3.47). In fact,equation (3.47) belongs to a class of unstable non-local growth equation that westudy in section 4.2. The critical behavior of these equations is characterized by tworelations between exponents:z = µ andα + z = 2, the last relation is associatedto Galilean invariance (note that these relations hold onlyin case ofµ smaller thatthe dynamic exponent of the KPZ equation in the dimension considered, furtherdetails can been found in section 4.2).

As stressed before, we find it remarkable that the KPZ non-linearity is able tostabilize the initial morphological instability of equation (3.42) even in the case ofa dispersion relation such asωµ=0.75 that induces such a fast rate of growth. Noticez = 0.68 < 1 corresponds effectively to correlations building up at a rate that iseven faster than ballistic. Moreover, the height profile develops values of the slopeat long times that are no longer compatible with the|∇ζ(t)| ≪ 1 that led to thederivation of equation (3.42) in the first place. This requires us to reconsider theoriginal moving boundary problem from a different perspective, as will be done inchapter 6.

3.4 Comparison with discrete models

Multiparticle Biased Diffusion Limited Aggregation (MBDLA) model has been in-troduced in order to study the dynamics of the galvanostaticECD growth [61; 62;63]. Mainly, MBDLA is a cellular automaton defined on a two-dimensional gridin which many random walkers (cations) are randomly distributed with fixed con-centration. In the model the anion dynamic is not taken into account, but particlesare created in order to maintain the electroneutrality condition [62; 63]. At everytime step a walker is chosen and moved to one of its four neighboring sites withprobability1/(4+p) to move left, right, or upward, and probability(1+p)/(4+p)to move down toward the cathode. The parameterp is referred to as the bias and itranges between0 and∞. In galvanostatic ECD, the bias is quantitatively related tothe electric current density in the physical system [61]. After a destination site hasbeen chosen, the particle moves if that node of the lattice isempty; if not, anotherparticle is selected and the destination selection procedure is repeated. Once theparticle has been moved, if the new position has any nearest neighbor site belong-ing to the aggregate, the present position of the walker is added to the aggregatewith probability s; otherwise it stays there (and is able to move again) with prob-

3.4 Comparison with discrete models 69

ability 1 − s. We have already introduced the sticking parameters, see equation(3.5). As particles are added to the aggregate, others are created at the top of thelattice, keeping the mean cation concentration constant [62; 63].

10-2

10-1

100

101

k

10-4

10-3

10-2

10-1

100

101

S(k,

t)

Figure 3.15 – Power spectrum obtained from MBDLA simulations (solid line) withr = 0.5, p = 0.5, s = 1, c = 1 (taken from figure 15 in [63], by permission), and equation(??) with ν = 1, K = 1/4, λ = 40, Π0 = 10−2, L = 512 (dashed line), averaged over105 realizations. For the sake of clarity, the latter has been vertically offset. Straight bluelines are guides to the eye having slope−4. Axes are in arbitrary units.

As mentioned above,kD is related with the sticking probability through equa-tion (3.5). This probability acts as a noise reduction parameter [149] in discretegrowth models,e.g. in [61; 62; 63]. In particular, MBDLA has been seen todescribe quantitatively the morphologies obtained in [144]. In MBDLA, by re-ducing the sticking probability the asymptotic KPZ scalingis indeed more readilyachieved, reducing the importance of pre-asymptotic unstable transients, as illus-trated by figure 8 of [63]. Hence, noise reduction is not a merecomputationaltool for discrete models but, rather, it can be intimately connected with the surfacekinetics via equation (3.5) and (3.12).

MBDLA with surface diffusion also predicts the existence ofa characteristicbranch width. This is shown in figure 3.15 in which the power spectral density isplotted and compared with the one obtained from the noisy KS equation, provingthe equivalence between both descriptions of ECD.

These results are reinforced by the fact that, as shown in Ref. [63], the cationconcentration obeys equation (3.25), and the branch width dependence on thecation concentration is consistent with the relationλm ∝ C

−1/2a . Moreover, in

simulations of MBDLA, an unstable transient was found before the KPZ scalingregime, being characterized by intrinsic anomalous scaling, as recently observedin the experimental works by Huo and Schwarzacher [150; 151]. As mentioned,probably the absence of such an anomalous scaling transientin our continuummodel is related to the small slope approximation, and we expect to retrieve it from


a numerical integration of the full moving boundary problem, see chapter 6.

3.5 Conclusions

In this chapter we have introduced a moving boundary model with fluctuationsin order to describe diffusive growth. The model unifies in the same framework,through a mapping between its parameters, the surface growth by chemical vapordeposition and by galvanostatic electrodeposition and emphasizes the role of thefluctuations in all of the transport mechanisms.

By employing the Green function projection technique we have derived an ef-fective interface equation for the surface height in the small slopes approximation.Depending on the model parameters we can approximate the full equation with theKuramoto-Sivashinsky equation or, alternatively, with anequation composed bythe Mullins-Sekerka dispersion relation and the Kardar-Parisi-Zhang non-linearity,the MSKPZ equation.

One step further in the understanding of the full interface equation has beendone by solving numerically the full dispersion relation (3.38) and by character-izing the behavior of the wavelength of its most unstable mode as function of themodel parameters. The analytical approximations can explain some qualitative be-havior of the electrodeposition experiments found in the literature, while the fulllinear dispersion compares quantitatively with the data ofde Bruyn [136]. Thiscomparison strengthens the validity of the hypothesis madein the derivation of themodel.

As we mentioned at the beginning of the chapter, two complementary ap-proaches are possible: we can study the critical exponents of the MSKPZ equa-tion or, alternatively, we can introduce a numerical model in order to integrate themoving boundary model.

In chapter 4 we study the asymptotic behavior of a class of non-local equationwhich generalizes the MSKPZ equation. We estimate the critical exponents ofthese equations in one and two dimensions by numerical integration (employing thepseudo-spectral scheme) and we compare them with the analytical results obtainedfrom a dynamic renormalization group analysis.

Finally, in chapter 6 we introduce a phase-field model which converges to themoving boundary model in the thin interface limit. This model allows us to obtaincomplex morphologies overcoming the limitations due to thesmall slopes approx-imation. Through the phase-field formulation, we are able tostudy the transientstates and the asymptotic behavior of the moving boundary model and, conse-quently, complete the theoretical description of non-conserved diffusive growth.

Non-local growth equations

Fractal geometry has been recognized to encode the behaviorof self-similar sys-tems, namely those whose structure looks the same with independence of thescale of observation [152]. Many of the best well-known fractals —like, for in-stance, computational models of biological morphogenesis— are geometrical pat-terns constructed deterministically by iteration of a simple initial motif. However,many natural patterns are not exactly regular but, rather, present statistically someself-similar or hierarchical structure in a characteristic sea of randomness [153;154]. The trouble with purely geometric (static) characterizations of these struc-tures is that they do not reveal any information on the growthprocess that has ledto the final structure. This prevents us from elucidating a guiding principle as towhy this type of hierarchical morphologies can be observed across length scalesthat range from hundreds of nanometers (surfaces of amorphous thin films) up tocentimeters (the familiar cauliflower plants; turbulent combustion fronts) or evenup to kilometers (clouds).

In the formulation of a continuum dynamical description (inorder to accessto the large scale behavior of the system) of these growth processes, we have toconsider that interface dynamics of many non-equilibrium systems arises from theinterplay between non-local interactions and morphological instabilities. Examplesrange from flame front propagation to thin film growth [58].

Often, although the basic physical interactions are short-ranged and local, theevolution of the system is driven by non-local effects implicitly (through projec-tion of the overall dynamics on the interface) or explicitly(as in elastic media or

72 Non-local growth equations

viscous flow) [155]. These non-localities appear in many fields, diffusion-limitedgrowth being a prominent example that provides a simple language to illustratethe main nonlinear effects. Thus, although diffusion events of aggregating unitsare local, the morphology of a growing cluster is dominated by shadowing of themost prominent surface features over less exposed ones. Hence, thelocal growthvelocity depends on theglobal surface shape. This moreover leads to the clas-sic Mullins-Sekerka (MS) morphological instability [58; 155], whereby prominentfeatures grow faster. Naturally, non-locality and morphological stability are inde-pendent properties. Thus, we may consider diffusion-limited erosion (DLE) [156;2], which is qualitatively relevant to experiments on e.g. ion irradiation smoothing[157]. In this case the most exposed surface features are eroded faster, leadingto non-local stable interface evolution in which differences in surface height aresmoothed out. These are only two examples of experimental systems that can bemodeled by effective interfacial equations.

In the first section of this chapter we report on typical morphologies that arisefrom a non-local growth equation that has been derived for diffusive growth inthe context of CVD experiments (see section 4.1). The large scale description ofthe this system is provided by a combination of the linear MS instability, the non-linear KPZ term and a generic Gaussian uncorrelated noise, and, for this reason,we have named it MSKPZ equation. This equation produces a kind of hierarchi-cal interfaces that we have called cauliflower-like structures for their resemblancewith the familiar self-similar surface of the cauliflowers.Structures with similarfeatures have been observed in different growth processes and, probably, there is acommon ingredient in an asymptotic description of these systems that leads to thismorphology.

In the second section of this chapter we generalize the MSKPZequation to afamily of non-local growth equations. Introducing an appropriate control parame-ter we are able to tune the dynamics of the equation from a super to a sub ballisticregime. This family also includes the local case, the Kuramoto-Sivashinsky equa-tion, as upper limit of the exponent. By means of the DRG analysis and the PSnumerical integration we have studied the asymptotic (and preasymptotic) behav-ior of this family of equations identifying a continuum of new out of equilibriumuniversality classes.

4.1 The dynamics of cauliflower-like growth

In the search for an universal description, it is instructive to identify themain com-mon phenomenaoccurring in the production of the hierarchical structuresconsid-ered in the introduction. For instance, in the case of cauliflower plants, intuitivelythere is an interplay between thetransportof nutrients to cells and light absorption,inducing competitive cell replication; moreovermass is non conservedprovidedthe cauliflower is growing from a seed;and fluctuationsare intrinsic to the biolog-ical underlying processes taking place both at the level of the cell metabolism and

4.1 The dynamics of cauliflower-like growth 73

the interaction with the environment. An extra stabilizingingredient is necessaryin order to guaranty the stability of the surface.

4.1.1 Continuum description

The main ingredients of systems that develop cauliflower-like structures are non-locality (from the competition among the various system parts to growth resources),non-conservation (i.e., the system is growing by incorporating mass to the surface),non-linearity, and fluctuations. The non-trivial balance of these processes resultsgenerically in a morphology that, albeit disordered, presents a self-similar, hierar-chical structure.

Figure 4.1 – Morphology: Comparison between experiments and theory. CVD exper-iments at times: A)t = 40 min, B) t = 2 h and C)t = 6 h. In all cases, the lateraldimension is1 µm. Panels D-F, integration of equation (4.1) for the same times.

As stated above, we expect the physical principles to be largely independentof the scale of observation. Thus, here we adopt a continuum description for theinterface in the spirit of fluctuating hydrodynamics [158].In order to make a quan-titative comparison between theory and experiments we firstfocus on the dynamicsof a surface that grows by the incorporation of atoms or molecules from a vapor.This system is a paradigmatic case of non-conserved growth far from equilibrium:the evolution of the growing surface is due to the nontrivialcoupling among va-


por diffusion, particle attachment, surface diffusion, and fluctuations. Besides, theexperimental technique is well-known and some experimental parameters, such asgrowth velocity or vapor concentration, can be controlled accurately.

Despite the overall equations having been well-known for decades [159; 115],only recently [53; 54] has a closed evolution equation for the surface been obtained(see chapter 3). The equation describes (under some approximations) the currentposition of the surface,h(r, t). The equation takes a simpler form in Fourier spaceas

∂thk =

4∑

j=1

Cjkj

hk +λ

2F(∇h)2k + ηk, (4.1)

wherehk = Fh(r, t) is the space Fourier transform of the height field,k isthe two-dimensional wave-vector of magnitudek = |k|, ηk is a noise term thataccounts for external fluctuations in the system, andCj are constants that dependon the physical conditions of the experiment. Despite the apparent simplicity ofthe above equation some emphasis must be done in its real space representation.Thus, the terms proportional to odd powers ofk have the following form in realspace

F−1

k2p−1hk

∝ −∫

R2

∇′2ph(r′)|r − r′| , (4.2)

where the principal value of the integral is to be taken. Thisis nothing but theconvolution of the surface derivatives field,∇2ph, with the long-range kernel1/r(for d = 1 the right hand side of equation (4.2) can be written as the Hilbert trans-form of ∇2ph, for further details see appendix A). Hence, as for thek2p−1 termsin (4.1), the value of the local growth velocity at a given surface point dependson the values of the height derivatives at all other surface points. Physically thisnon-local coupling arises from the competition of the different parts of the systemover the resources for growth. For instance, for a solid surface growing out ofparticles that aggregate from a vapor phase, it is induced bythe geometrical shad-owing of prominent surface features that are more exposed todiffusive fluxes, overmore shallow ones. In general, this non-locality is one of the main features of thesystems described in this chapter.

Pursuing the physical example provided by aggregate growthfrom a vaporphase, two important limits can be distinguished in equation (4.1) related to therelative values of the two velocity scales in the system: thesurface mean velocity,V , and the mass transfer ratekD (see Ref. [54] for details), that is related to theprobability of vapor molecules to become permanently attached to the surface (orsticking probability,s) through relation (3.5), see chapter 3. As previously see insection 3.2, when the attachment probability is small (namely, kD ≪ V ) dynam-ics is slow for which the long wavelength behavior of the system is described bythe (local) noisy Kuramoto-Sivashinsky equation [9] [hence C1 = C3 = 0, andequation (4.1) takes a local form in real space]

∂th = −C2∇2h− C4∇4h+λ

2|∇h|2 + η. (4.3)


Note that the celebrated Kardar-Parisi-Zhang equation is aparticular case of equa-tion (4.3) forC2 < 0 andC4 = 0 [27]. The Kuramoto-Sivashinsky equation hasbeen recognized as a paradigmatic example of spatio-temporal chaos. On the con-trary, when the attachment probability is high, competition between high peaksand deep valleys is key to the dynamics, see section 3.2. Then, the kinetics is fast(kD ≫ V ) and the surface height evolves according to

∂thk = C1khk − C3k3hk +

λ

2F|∇h|2k + ηk, (4.4)

with ηk being a Gaussian white noise term with zero mean and correlations givenby

〈ηk(t)ηk′(t′)〉 = 2Π0δk+k′ δ(t − t′), (4.5)

where the symbol〈. . .〉 stands for averages over different realizations of the noise.The linear part of equation (4.4) —containing the so-calledMullins-Sekerka

dispersion relation— is ubiquitous in nonlinear science and governs the evolu-tion of fronts that are morphologically unstable to perturbations due to preferentialgrowth at prominent surface features.

Moreover, the KPZ non-linearity appearing in (4.4) has beenargued to appeargenerically in the continuum description of surfaces that grow irreversibly in theabsence of conservation laws [27] and, indeed, has been derived from constitutivelaws for some physical systems. However, in spite of the generality of the variousterms in (4.4), the coupling among them and their interplay with noise has remainedpoorly understood. Moreover, no explicit quantitative connection has been madebetween such a theoretical description and three-dimensional experiments. Thishas motivated us to check the validity of the approximationsmade in the derivationof equation 4.1 by performing experiments of amorphous hydrogenated carbon (a-C:H) thin films by CVD.

4.1.2 Chemical vapor deposition growth

In the CVD experimental set up carried out by the group of Prof. L. Vazquez atICMM (Instituto de Ciencia de Materiales de Madrid, CSIC) films were grownby electron cyclotron resonance chemical vapor deposition(ECR-CVD) on siliconsubstrates

In this growth system, the main growth species are ions and radicals. The lattercan be distinguished into two main groups,i.e., C1Hx and C2Hx radicals. Withinthe first subgroup, C1H2,3 radicals have sticking coefficients of about10−4−10−2,whereas C1H and C1 have a sticking parameter close to unity [160]. The C2Hx

radicals generally have a high sticking coefficient (s ∼ 0.4 − 0.8) [161]. In fact,for a pure methane plasma, an overall sticking of0.65 ± 0.15 was found [162].Moreover, when methane is diluted with argon, the impingingof argon ions onthe film growing surface generates dangling bonds at the surface leading to aneffective increase ofs for the different growth species. Thus, we can assume thatthe effective sticking coefficient is close to unity in this system,s ≃ 1.


In figure 4.1A-C we show three snapshots of the surface for timest = 40 m,t = 2 h andt = 6 h, respectively. Morphological analysis of the surface, aswell asthe configurational parameters of the experimental set, will allow us to extract someinteresting features of the system, the main ones being summarized in table (4.1).However, many microscopic details of the experimental setup cannot be measured

Observable Value

Growth velocity (V ) 864 ± 43 nm/hMean substrate temperature (T ) 343 KMean free path (Lf) 0.45 ± 0.05 cmPartial pressure (methane) 3.75 ± 0.25 × 10−3 mbar

Table 4.1 – Experimental parameters in CVD growth.

or even estimated from data. Unfortunately, some of these parameters are crucialin order to determine the quantitative values of the coefficients in Eq (4.4). To citea few, mean lattice spacing at the surface, surface tension or the capillarity length,mainly due to the limited resolution of the experimental measurements but also dueto the coexistence between species both in the vapor and at the very surface. Thus,we need to combine the experimental values in table (4.1) with an inverse methodas we illustrate below.

4.1.3 Numerical and analytical estimation of growth expone nts

As we mentioned above, the coefficientsC1−4, λ andΠ0 in equations (4.3) and(4.4) depend on the physical magnitudes of the overall problem, namely, the diffu-sivity at the vapor,D; the velocity of the surface,V ; the capillarity length,d0 orthe mass transfer coefficient,kD to cite a few, see chapter 3.

The details of the theoretical model depend on the precise regime at which theexperiments were carried out. As discussed above, the sticking probability can beassumed to be large in the experiments. From data in table (4.1) and equation (3.5)we can estimate thatkD ∼ 0.75 cm/s, which is considerably larger thanV = 864nm/h hence, actually, the system can be assumed to be in the fast kinetics regimeand can, consequently, be well described by equation (4.4) with

C1 = λ = V, C3 = Dd0, Π0 = V c0eq/2. (4.6)

As we stressed above, the values of the capillarity length,d0 or the equilibriumconcentration at the interface,c0eq, cannot be easily determined experimentally.Thus, in order to proceed on we must extract that informationfrom inspectionof the morphologies in figure 4.1A-C. First of all, we render equations (4.4) and(4.6) non-dimensional by fixing appropriate time, length and height scales so thatour main equation is simply equation (4.4) withC1 = C3 = λ/2 = 1, and


Π0 = c0eq/(16D3d3

0V )1/2. Note that, in simulations our system size must be fixedaccording to the length scale previously chosen.

10-1

100

101

102

103

k (1/µm)

100

104

108

S(k,

t)

Figure 4.2 – Power spectral density comparison between theory and experiments. Cir-cles: experimentt = 40 min. Solid red line: theoryt = 40 min. Squares: experimentt = 6 h. Dashed blue: theoryt = 6 h.

At short times we expect that non-linear terms are negligible and the systemwill evolve according to the linear terms, which has important consequences on themorphology. For instance, the skewness of the height distribution in figure 4.1Ais negligible, so that we expect that non-linear effects notto be important, giventhat the non-linear term|∇h|2 breaks down the up-down symmetry of the surface.Thus we can assume that the system is in the linear regime for the correspondingtimes in which, according to our theory, the characteristiclength scale is given by(for the non-dimension equation)

lc = 2π√

3. (4.7)

Thus, the ratio betweenlc and the system sizeLx must agree with the ratio betweenthe experimental value oflc, few nanometers, and the experimental AFM window,Lc = 1 µm. Thus, we obtain approximatelyLx ≃ 512 (we have rounded thisvalue up to an exact power of2 in order to optimize the numerical integration ofthe equation by means of the pseudo-spectral algorithm). Finally, we determine thevalue of the noise amplitudeΠ0 (the only unknown of the non-dimensional equa-tion) by means of the time scale of the experiments (using a numerical trial-and-error inverse method). The value that seems to reproduce better the experiments isΠ0 = 0.9.

Numerical simulations of the non-dimensional equation forthis choice of pa-rameters are shown in figure 4.1D-F, at simulation timest = 10, t = 30, andt = 70. Note that the remarkable visual agreement with the experimental mor-phologies. More quantitative measurements have been done in order to test the


validity of equation (4.4), for instance through the power spectral density (see fig-ure 4.2)

As described in chapter 2, from the power spectral density wecan also deter-mine with high accuracy the global interface width throughW (t)2 =

∫

S(k, t) dk.The critical exponents have been estimated numerically from these two observ-ables [9] (see figure 4.3 and figure 4.4), their valuesα = 1.00 ± 0.05, z =0.95 ± 0.05 being equal, within numerical error bars, to the valuesα = z = 1predicted by the DRG calculations (see section 4.2). In order to check the consis-tency of our numerical estimates, in the inset of figure 4.4 weshow the collapse ofthe structure factor using these exponent values. Collapses are satisfactory, includ-ing the behavior of the scaling function for the global width, indicated by a solidline in the inset of figure 4.3. The discrepancies in the collapsed curves for largekt1/z values are due to the existence of a short-distance scale scaling different fromthe asymptotic one.

100

101

102

103

t

10-1

100

101

102

W(t

)

10-2

100

102

t1/z

/L

10-4

10-3

10-2

10-1

W(t

)/Lα

Figure 4.3 – Global width vs time for a system withC1 = C3 = λ = 1, Π0 = 10−2

obtained numerically for equation (4.4), for increasing system sizes,L = 32, 64, 128,256, 512 and1024, bottom to top, averaged over103 realizations. The red dashed line isa guide to the eye with slope1.05 suggesting the late time value ofβ. Inset: Collapse ofW (t) usingα = 1.00, β = 1.05, andz = 0.95. The blue line has slope1.00, showingthe consistency of our estimate forα. Axes in the main panel and in the inset are all inarbitrary units.

With the aim of stressing the peculiarity of this values of the roughness expo-nents, we can check the scaling hypothesis through the relation

h(r, t) ≈ b−αh(br, bzt), (4.8)

where the symbol≈ means that the equivalence is only true statistically.Given (4.8) and the implied scaling for the surface roughness (W ∼ Lα, see

chapter 2), the valueα = 1 has strong implications on the final morphology. Notice


10-2

10-1

100

101

102

k

100

103

106

109

S(k

,t)

100

101

102

103

104

k t1/z

100

101

102

103

S(k

,t)

k2α+1

Figure 4.4 – Structure factor vs spatial frequencyk with the same parameters as in figure4.3 andL = 1024. Different curves stand for different times (the ones at thebottom is forthe earliest time). The dashed line is a guide to the eye with slope−3, compatible withα = 1. Inset: Collapse ofS(k, t) usingα = 1.00, β = 1.05, andz = 0.95. Axes in themain panel and in the inset are all in arbitrary units.

that the relative value of the fluctuations obeys,

W

L∼ Lα−1 as L→ ∞, (4.9)

whereL is the scale of observation of the system. Ifα < 1 the surface is apparentlyflat on the macroscopic scale, although it is rough at short enough scales. Similarly,for α > 1 the fluctuations diverge indicating that this kind of behavior would thenbe observed only as a transient. However, ifα = 1 the system is self-similar andits geometrical features remain invariant in the macroscopic limit L→ ∞.

Also the dynamic exponent,z is equal to unity. This reflects a more profoundfact about the fractality of the system: the system is self-similar also in time. Inother words, equation (4.8), reflects the fact that in the present system time behavessimilarly to space so that, if we observe the system at two different times,z = 1implies that we cannot distinguish (statistically) the second one from an uniformspatial zoom performed in the earlier one. In figure 4.5 we illustrate this fact bynumerical integrating equation (4.4). In more mundane terms z = 1 is indicativeof ballistic transport (or, rather, relaxation) at the surface.

Note that, as long asC1 6= 0, the value of the critical exponents are robust forany stabilizing mechanism of the formCik

i with i ≥ 2, as the DRG shows (seeappendix C). Thus, the exact form of the stabilizing term(s)in equation (4.1) isirrelevant asymptotically, and from the point of view of theroughness exponents.Consequently, we are confident that this type of behavior canbe found in otherexperimental systems for which equation (4.1) holds.

Our DRG analysis also puts forward another important aspectof equation (4.1)


Figure 4.5 – The fractality in space and time is reflected in the inabilityto distinguishimage zooms and time shifts. Top: Simulation fort1 = 2 h; Bottom right: Simulation fort2 = 4 h; Bottom left: Spatial amplification of top panel a quantityt2/t1 = 2. The zoomedarea is indicated in the top panel with a white dashed line square.

in comparison with its particularlocal caseC1 = C3 = 0 (Kuramoto-Sivashinskyequation). Namely, in the KS case, asymptotically the dominant term of the equa-tion is precisely the non-linear term. Thus, the spatio-temporal chaos is a purelynon-linear effect (external noise being somehow redundant[72]). In contrast, whenC1 > 0 the dominant term is preciselyC1khk (which induces a dynamical expo-nentz = 1). However, the expected unstable behavior of the equation is controlledby the subdominant term(∇h)2 through a non-trivial restoration of the so-calledGalilean invariance. Galilean invariance induces the scaling relationα + z = 2and thus leads to the exponentsα = z = 1 whose importance we have discussedabove. Numerical integration of equation (4.1) with different subdominant linearterms shows, within numerical precision, that the value of the exponents are theones predicted by the theory. It is important to note that thelinear part of theequation only provides information at short times but the fully non-linear regimeis characterized, precisely, by the absence of a well definedlength scale (due to thefractal character of the surface).

However, we can obtain some information from the other system experimen-tally available for us is that of the biological (food) cauliflowers. As the readermight have noted by inspection of figure 4.5, the resemblanceamong simulatedand real cauliflowers is remarkable although it cannot serveas a proof of the uni-versality of the phenomena. In order to investigate this connection numerically, wehave computed the power spectral density of the interface profiles of several slicesfrom a cauliflower (an improved analysis might be done with a profilemeter). Theroughness exponentα can be determined by analyzing the power spectral density

4.2 A class of non-local equations 81

of these slices. In figure 4.6 we show the averaged power spectral density over 10different slices of a cauliflower. The slope for small valuesof k is approximately3 = 2α + 1 confirming that, within the experimental uncertainty, the value ofα = 1.02±0.03 ≃ 1 is the same as the one provided by our theory. Moreover, thisresult is compatible with the work of Kim, in which he estimated experimentallythe fractal dimension of sliced cauliflowers finding a value of df = 2.85 ± 0.03[163]. Through the relation between the roughness exponentof a self-affine inter-face and its fractal dimensiondf = 2 − α (for d = 2 [3]), we estimate from theKim’s work an exponentα = 0.85 ∼ 1.

10-3

10-2

10-1

100

k (1/mm)

10-4

100

104

S(k,

t)

Figure 4.6 – Power spectral density obtained by averaging 10 cauliflowerslices (obtainedfrom two different individuals). The slope is a power-law fitwith slope−3.04 ± 0.09 =−(2α+1), which provides a roughness exponentα = 1.02±0.03, consistent with theory.

4.2 A class of non-local equations

Equation (4.1) can be easily generalized to a whole class of non-local equations bysimply replacing thekhk term in the dispersion relation bykµhk. More generally,assuming translational and rotational symmetry, we propose the following equa-tion (after space Fourier transform) as a general description of interface systemsin which dynamics arises as the interplay between morphological instabilities andnon-conserved growth

∂thk(t) = (−νkµ −Kkm −Nkn)hk(t) +λ

2F [(∇h)2]k + ηk(t). (4.10)

Here,µ,m, n, K, andN are positive parameters (0 < µ ≤ 2,m ≥ 2, andn > m,see below), the linear dispersion relationσk ≡ −νkµ − Kkm − Nkn providingthe amplification rate for periodic disturbances (with wave-vectork) of a planarinterface. The−Nkn term in the dispersion relation is a higher order correctionto −Kkm; it will be only considered when needed for technical reasons. As seen


in the previous section, the term proportional toλ is the KPZ non-linearity, whileηk(t) is Gaussian uncorrelated noise.

Equations like (4.10) have been derived from constitutive laws, and sometimesfrom symmetry arguments, as weakly nonlinear, long wavelength (k = |k| ≪ 1)descriptions of a variety of systems [58; 155; 2; 3; 133]. Locality holds wheneverσk is a polynomial ink2, as for e.g. the celebrated Kuramoto-Sivashinsky equa-tion [133] (µ = 2, m = 4). In contrast, non-locality is associated withodd or,generically, non-integer powers ofk, equation (4.10) depending on slowly decay-ing kernels in real space [164]. Important examples are the MS or Saffman-Taylorinstabilities (µ = 1, m = 3) [9]. But also the Darrieus-Landau (DL) instabilityoccurring in the propagation of a premixed laminar flame [165], for which the gasexpansion produced by heat induces wrinkles on the flame front, and for whichµ = 1,m = 2 [166; 167].

Growing interfaces controlled byballistic transport are described by an iden-tical dispersion relation [168], which is a remarkable fact. In all these cases theinstability is induced by theνkµhk term. In general, the unstable (ν < 0) nonlocalequation (4.10) has been proposed as an universal description of systems in whichlong-range interactions persist at the level of amplitude descriptions [164], while itsstable (ν > 0) counterpart, frequently studied in the context of kineticrougheningfor theλ = 0 case [2], has been proposed [169] as a description of surfacegrowthmediated by long-range surface hopping mechanisms like Levy flights. Indeed,kµhk is actually a fractional power of the Laplacian operator that appears in thedescription of anomalous diffusion [170]. Moreover, unstable dispersion relationsof the precise same form as in equation (4.10) have been seen to describe effec-tively (for 0 < µ < 2, m = 2) experiments of surface growth by electrodeposition[171].

While the stable version of equation (4.10) (ν > 0) leads to scale invariantinterfaces whose critical exponents follow from dimensional analysis forµ = 1[156], and can be obtained analytically for genericµ [172], the behavior for theunstable cases has remained poorly understood. In this section, we show that theyare novel and unexpected, having a number of remarkable features. While, as in theKS system, the KPZ non-linearity stabilizes the short time instability, now itdoesnot control the asymptotic scaling for a whole parameter range.This is, rather,controlled by the−νkµhk term, but differ from those of the stable (DLE-type)case because: (i) they do not follow from dimensional analysis; (ii) they ared-independent, and (iii) they are associated with a Galilean invariance that is hidden(non-explicit) in equation (4.10).

Specifically, we will consider the cases for whichm = 2, and0 < µ ≤ 2. Asin the case of equation (4.1), we have checked that the long-distance properties areunmodified for stabilizing terms withm > 2 and for non-zeroN . Note that, whiletheµ = 2 limit of equation (4.10) (in the unstable case) is the (noisy) KS equation,cases withµ < 2 correspond to super-diffusive surface relaxation. A conspicu-ous instance isµ = 1, for which relaxation is ballistic, equation (4.10) becominga stochasticgeneralization of the Michelson-Sivashinsky (SMS) equation. As an


stochastic interface equation, it has been derived for reactive infiltration of chemi-cals in porous media [173]. In the deterministic case (η = 0), it was obtained [166;167] as a weakly nonlinear description of premixed laminar flames near onset ofinstability. Now the linear dispersion relation incorporates the DL instability, withK being related with the Markstein length in the combustion problem [165].

We can thus gain insight into the dynamics described by equation (4.10) byfirst considering theµ = 1, SMS case. For convenience, notice that, forN =0, equation (4.10) depends on a single independent parameterafter appropriaterescaling ofh, k, andt; we choose it to be the noise amplitudeΠ0. We show infigure 4.7 the evolution of the global surface roughnessW (t) as computed froma numerical integration of equation (4.10) withµ = 1, using the pseudo-spectralmethod described in section 2.3. As we see [inset of panel figure 4.7(a)], in the

10-4

10-3

10-2

10-1

100

k / kmax

100

102

104

106

108

1010

1012

1014

1016

S(k

,t)

2α+1 = 1.002α+1 = 3.10

10-4

10-3

10-2

10-1

100

k / kmax

10-3

100

103

106

109

1012

1015

1018

2α+1 = 0.502α+1 = 4.04

10-1

100

101

102

103

t

10-2

100

102

W(t

)

β = 1.14

10-2

100

102

104

0,3

0,6

0,9

1,2

1,5

10-2

100

102

t

10-4

10-2

100

102

104

W(t

)

β = 3.45

10-2

100

102

0,15

0,30

0,45

0,60

a) b)

Figure 4.7 – S(k, t) curves for equation (4.10) (d = 1) with µ = 1 (times betweent = 200 and t = 2000, bottom to top) (a), andµ = 1/2 (times betweent = 20 andt = 100) (b). In each panel the stable case is the (single color) lower set of curves and theunstable case is the (two-color) upper set. Insets provideW (t) vs t for the unstable (lowercurve, lower/right axes) and stable (upper curve, upper/left axes) conditions. Parametersare|ν| = K = λ = 1 andΠ0 = 10−2 (a), 10−4 (b) (Π0 = 1 for the stable conditions).System size isL = 214, except for the unstable casesL = 210 (a),213/5 (b). All axes arein arbitrary units. In each casekmax = π/∆x with ∆x the space discretization unit.

unstable case an initial transient is followed by a fast increase ofW due to thelinear instability, that is followed by power law growth of the formW (t) ∼ tβ withβ = 1.14. Qualitatively, and similarly to the KS system, nonlinear effects stabilizethe morphological instability, although they do not operate in exactly the same


fashion, as suggested by the noiseless limits: while nonlinear cell interaction leadsto spatio-temporal chaos in the KS case [2; 133], the (long time) non-linear shapereached for the MS equation is a giant cusp that responds to noisy perturbations bycreation/annihilation of smaller cusps [174]. Quantitatively, the value ofβ signalsvery fast growth in the context of surface kinetic roughening [3; 2], but figure 4.7proves it to be associated with genuine surface scale invariant behavior. The powerspectral density or surface height structure factorS(k, t) is shown in panel (a)for different times (two-color set of curves). While short times are dominated bythe peak associated with the linear instability, for longertimes theS(k, t) curvesactually fulfill the FV scaling Ansatz [2; 3], leading to asymptotic scaling of theform S(k, t → ∞) ∼ 1/k2α+d with a value of the roughness exponent that weestimate from data collapse [2; 3] asα = 1.05 ± 0.05. Likewise we estimate adynamic exponent valuez = 0.92±0.05. These exponent values contrast stronglywith those obtained for the stable case [156; 2], which areα = 0 and z = 1,implying β = α/z = 0, namely, logarithmic increase of the roughness, see theupper curve in the inset of figure 4.7(a).

10-3

10-2

10-1

100

k / kmax

103

106

109

1012

1015

1018

S(k

,t)

2α+2 = 1.002α+2 = 4.20

10-3

10-2

10-1

100

k / kmax

100

103

106

109

1012

1015

1018

2α+2 = 0.502α+2 = 5.10

10-1

100

101

102

103

t

10-2

100

102

W(t

)

β = 1.22

10-2

10-1

100

101

102

0,1

0,2

0,3

100

101

102

t

10-2

100

102

W(t

)

β = 3.44

10-2

10-1

100

101

102

0,1

0,2

0,3

a) b)

Figure 4.8 – S(k, t) curves for equation (4.10) (d = 2) with µ = 1 (a), andµ = 1/2(b). In each panel the stable case is the black lower curve andthe unstable case is the(two-color) upper set of curves. Insets provideW (t) vs t for the unstable (lower curve,lower/right axes) and stable (upper curve, upper/left axes) conditions. Parameters are|ν| =K = λ = 1 andΠ0 = 0.01. System size isL = 512× 512, exceptL = 256× 256 for theunstable case in (b). All axes are in arbitrary units. In eachcasekmax = π/∆x with ∆xthe space discretization unit.

Numerically, for the unstable condition we obtain exponentvalues compatible


with α+ z = 2, andz = µ. Moreover, these equalities hold in higher dimensions,as can be checked in figure 4.8(a) for the (µ = 1) 2+1 dimensional case, wherewe obtainα = 1.10 ± 0.05, z = 0.90 ± 0.05. In general, the large value of theroughness exponent implies a strong persistence in the height fluctuations, leadingto suppression of small surface features. Actually, as in section 4.1, forα = 1 thesurface height field is a self-similar (rather than self-affine) fractal, related with thedisordered but somehow hierarchical morphologies that areproduced (small cuspmotion [174]). The morphological evolution of the SMS and MSKPZ equationsare shown in figure 4.9 and in the movies 4 and 5 in the accompanying CDROM.

t

Figure 4.9 – Left column: Morphological evolution of the MSKPZ equation(µ = 1andm = 3). Pseudo-spectral integration of equation (4.10) on a square lattice withL =512 × 512 and parametersν = K = λ = 1, Π0 = 10−2. Right column: Morphologicalevolution of the SMS equation (µ = 1 andm = 2). The pseudo-spectral integrationhas been performed for the same parameter condition of the MSKPZ equation. Time isincreasing from top to bottom. Note the creation/annihilation of the smaller cups and thecoarsening of the structure that leads to the final (disordered) giant cusp. All axes are inarbitrary units.

On the other hand, also as in section 4.1z = 1 relates with the ballistic natureof the surface relaxation mechanism implied by theνkhk term in the evolutionequation. Experimental realizations of these scaling relations are available, e.g. forplasma-etched Si(100) interfacesα = 0.96 andz = 1.05 [175].

We have also studied equation (4.10) for other values ofµ, both for 1+1 and


for 2+1 dimensional systems. Results are shown in figure 4.7(b) and figure 4.8(b)for µ = 1/2 < 1, that corresponds to an unexplored (to our knowledge) instanceof unstable super-ballistic (Levy flight type) interface relaxation [169]. As we seein the figures, the qualitative behavior is similar to the SMS(µ = 1) case, speciallyat large length scales. Quantitatively, we obtainα = 1.52± 0.03, z = 0.44± 0.03in d = 1 andα = 1.55 ± 0.05, z = 0.45 ± 0.05 in d = 2.

As a reflection of the strong relaxation mechanism note, however, howS(k, t)increasingly approaches for long times and highk the power law that sets up atlarge distances, leading eventually to a single power law describing the whole rangeof scales in the system. This contrasts markedly with the time evolution of thepower spectrum for local interfaces (even in the unstable case) displaying crossoverphenomena [2; 3], and even with the SMS behavior.

Conversely, the roughness and dynamic exponents change quantitatively ascompared with the latter, albeit still fulfilling thed-independent scaling relationsz = µ andα + z = 2. In our simulations this seems to be the case for all values0 < µ < zKPZ(d), wherezKPZ(d) is the dynamic exponent of the KPZ equationfor d-dimensional substrates (zKPZ3/2 for d = 1, zKPZ1.61 for d = 2 [29]). Ateachd(= 1, 2), we obtainα andz exponents that take (d-dependent) KPZ valuesfor zKPZ(d) ≤ µ ≤ 2, as shown in figure 4.10. Note,µ→ 2− is not a well definedlimit for fixed m = 2, N = 0. In this case, we allow e.g. forN 6= 0 with n = 4,theµ = 2 limit becoming the stochastic KS equation. In the noisy case, the scalingproperties are indeed expected to be those of the KPZ equation [64; 74].

10-2

10-1

100

101

k

100

102

104

106

108

1010

S(k,

t)

2α+d = 2.02α+d = 1.5

100

102

104

t10

-1

100

101

102

W(t

)

β = 1/3β = 0.14

10-2

10-1

100

101

k

104

106

108

1010

1012

1014

S(k,

t)

2α+d = 2.782α+d = 2.50

100

101

102

103

t

100

101

102

W(t

)

β = 0.24β = 0.14

Figure 4.10 – Left: S(k, t) for equation (4.10) with dispersion relationσk = k7/4 −k2 − k3 − k4, d = 1, Π0 = 1, andL = 2048. The extra relaxation terms in the dispersionrelation have been added in order to reach the asymptotic state faster. Inset:W (t) for thesame system. The slopes of the green lines correspond to the KPZ exponents (α = 1/2,β = 1/3), while the Galilean exponents (α = 1/4, β = 1/7) correspond to the violetlines. Right: Same as left panel ford = 2, Π0 = 100, andL = 512× 512. KPZ exponentsareα = 0.39, β = 0.39/1.61 = 0.24 (green lines) and Galilean exponents areα = 1/4,β = 1/7 (violet lines). Note the lastS(k, t) curve ford = 2 is not completely saturated.All curves are averaged over100 noise realizations.

The exponent relationα + z = 2 is well known to be associated with the


Galilean invariance of the KPZ equation [3; 176]. In principle, in our case it is abit surprising since power counting arguments [156; 2] suggest the irrelevance ofthe KPZ non-linearity. Indeed, if we rescale according to equation (4.8), thenhsatisfies equation (4.10) with modified parametersν = bz−µν, K = bz−2K, λ =bα+z−2λ, andΠ0 = bz−2α−dΠ0. Sinceν is the most relevant parameter, scale in-variance is expected with exponents fulfillingz = µ andz = 2α+d. The latter re-lation is hyper-scaling, associated with non-renormalization of the noise amplitude,given that theK = λ = 0 limit of equation (4.10) is variational, and has (in thestable case) the asymptotic height distributionPh ∝ exp[(ν/2)

∫

kµ|hk|2dk].Indeed, all these properties are fulfilled forν > 0, the ensuingnegativevalues

of α = (µ−d)/2 signalling rather flat interfaces ford ≥ 1 (see insets of figure 4.7,and figure 4.8 [156; 169; 172]. Moreover, these results for the morphologicallystable condition suggest that equation (4.10) isnotGalilean invariant [9]. This canalso be explicitly checked by performing the usual tilt transformation [3].

In order to account for the scaling properties of the unstable condition we needto improve on the previous dimensional analysis. To this end, we perform a Dy-namic Renormalization Group study of equation (4.10) form = 2 andN = 0.We focus our analysis on this case due to the asymptotic irrelevance of relaxingterms withn > m, see appendix C. Following the standard procedure [176; 79](see section 2.2.1 for a general introduction to the FNS renormalization schemeand appendix B for the details of the calculation), we arriveat the following oneloop RG flow of the parameters for arbitraryd

dν

dl= ν [z − µ] ,

dλ

dl= λ [α+ z − 2] , (4.11)

dKdl

= K[

z − 2 − λ2Π0Kd

4d

(d− 2)K + (d− µ)ν

K (K + ν)3

]

, (4.12)

dΠ0

dl= Π0

[

z − 2α− d+λ2Π0Kd

4 (K + ν)3

]

, (4.13)

where the coarse graining is performed in an infinitesimal shell k ∈ [Λ(1− dl),Λ]within the band of linearly stable (large) wave-vector values, we fix the latticecut-off, Λ = 1, andKd = 2/[(4π)d/2Γ(d/2)]. Naturally, equations (4.11)-(4.13)generalize the KPZ flow for non-zeroµ and inherit the known analytical limitationsof the latter [2; 3; 79].

Still, they carry valuable information. In fact, introducing the coupling param-eters [6]

g =λ2Π0Kd

4(K + ν)3, f =

KK + ν

, (4.14)

we can calculate their flow using equations (4.11)-(4.13)

df

dl= (1 − f)

(µ− 2)f − g

d[(d− 2)f + (d− µ)(1 − f)]

, (4.15)


dg

dl= g

6f − 4 − d+ g + 3µ(1 − f) +

+3g

d

[

(d− 2)f + (d− µ)(1 − f)]

. (4.16)

Thus, a detailed analysis yields the same fixed point structure as for the KPZ equa-tion, with the addition of two new nontrivial fixed points. One is associated withhyper-scaling (we call it Smooth fixed point and its couplings aref = 0, g = 0,hencez = µ, z = 2α + d) and corresponds to the stable interfaces, while thesecond one implements Galilean invariance (we call it Galilean fixed point, in factz = µ andα + z = 2), and is the one found in our simulations for the morpho-logically unstable condition. The values off andg at this fixed point depend onthe exponentµ and the dimensionalityd [see equation (B.34)], in such a way thatit moves in the(f, g) plane as shown in figure 4.11 ford equal to one and two.The detailed analysis of these flow equations ford = 1 andd = 2 is reported inappendix B, here we stress only the main results.

0 0.5 1 1.5 2 2.5 3f

0

0.5

1

1.5

2

2.5

3

g

unstablestablesaddleEWSmoothKPZ

0 <

µ <

1

4/3 < µ < 3/2

3/2 < µ < 5/3

0 1 2 3 4 5 6f

0

1

2

3

4

5

6

g

saddleEWSmooth

0 < µ < 4/3

Figure 4.11 – Position of the Galilean fixed point in the plane(f, g) for d = 1 (leftpanel) andd = 2 (right panel). Arrows and colors represent the direction ofthis fixedpoint increasingµ and its stability. The intervals ofµ near the fixed point trajectories showexplicitly the values ofµ at which the stability changes.

Ford = 1 we have three fixed points whose coordinates do not depend on theexponentµ, i.e. the Smooth, EW and KPZ fixed points (see left panel of figure4.11). On the contrary, the Galilean fixed point moves by changing the value ofµ.For0 < µ ≤ 1 this fixed point is unstable and moves according the red line of theleft panel of figure 4.11. Qualitatively, within this interval of µ the flow behavesas in figure 4.13(a), in which the DRG flow is shown for the caseµ = 1. Forµwithin (1, 4/3] the Galilean fixed point is not an admissible solution of the DRGflow equations (4.15) and (4.16) and it disappears untilµ is larger than4/3, forexample see figure B.2(d). For0 < µ < 4/3 the only stable fixed point is theSmooth fixed point and it attracts only some trajectories with f < 1. The basin ofattraction of this fixed point changes as function ofµ, in figure B.2(a)-(c) we showthe casesµ = 1/4, 1/2, 1.

In addition, within the region4/3 < µ ≤ 3/2 the Galilean fixed point becomes


stable and attracts all the trajectories withf > 1, this condition is shown in figure4.13(b). The remaining trajectories are attracted by the Smooth fixed point, so thatthe vertical linef = 1 (i.e. ν = 0) separates the plane of the couplings into twodisjoint regions. Note that on this line equation (4.10) reduces to KPZ equation.Increasingµ from 4/3 to 3/2, the Galilean fixed point merges with the KPZ fixedpoint at exactlyµ = zKPZ(d = 1) [see left panel of figure 4.11 and figure 4.13(c)],loosing stability in favor of the latter for larger values,zKPZ(1) ≤ µ ≤ 2, justifyingour numerical observation of KPZ scaling in this range ofµ. For3/2 < µ ≤ 5/3the Galilean is a saddle fixed point and it is located on the line of disjunction of thebasins of attraction of the two stables fixed points for the DRG flow, the Smoothand the KPZ fixed points, see figure 4.13(d). Forµ > 5/3 this fixed point is notanymore an admissible solution of the coupling equations and disappears.

Finally, the only stable fixed point for this exponent condition is the KPZ fixedpoint [see figure B.2(d)], and, as said before, in numerical integrations we observeits critical exponents. In the first row of figure 4.2 we summarize the stabilityof the four fixed point as function ofµ. Our simulations disagree with the DRGflow when0 < µ < 4/3 for the unstable case, probably due to the assumptionsmade in the perturbative expansion of the DRG scheme. Nevertheless, the DRGanalysis gives several hints about the critical behavior ofthis non-local class ofgrowth equations.

1 20 3/2 5/34/3

µ

EW

KPZ

Smooth

Galilean

d = 1

20 4/3

µ

EW

Smooth

Galilean

d = 2

Figure 4.12 – Stability of the fixed points ford = 1, 2. As before the red color standsfor unstable, blue for saddle and green for stable fixed point.

For d = 2, as it is well known, the KPZ fixed point is at infinity while theGalilean fixed point remains finite only up toµ = 4/3 < zKPZ(2) (for µ = 4/3this fixed point is atf = ∞ andg = 2), see the right panel of figure 4.11. As


for d = 1 the position of the Galilean fixed point depends on the value of µ, asshown in the right panel of figure 4.11. Moreover, the stability of the three fixedpoints is not modified whenµ changes, see the second row of figure 4.2. Mainlywe recognize two main behaviors of the DRG flow, the first one isrepresented infigure 4.14(a). In this case the Galilean fixed point is present and influences thetrajectories characterized byf > 1, in fact, it is located on the manifold which di-vides two class of trajectories: Asymptotically, one type of trajectories flow towardthe pointf = 1 andg → ∞, and, the other type flow toward the pointf → ∞andg = 2, see figures B.4(a)-(c). Within the region of the plane withf < 1, theSmooth fixed point attracts some trajectories while those trajectories outside thebasin of attraction of this fixed point flow asymptotically tof = 0 andg → ∞.

Forµ ≥ 4/3 the Galilean fixed point is not an admissible solution of the cou-pling equations. In this case all the trajectories withf > 1 flow to f → 1 andg → ∞, attracted by the KPZ fixed point, see figure 4.14(b). The region withf < 1does not display any qualitative change compared to the caseµ < 4/3. In spite ofthe DRG analysis, our simulations of the unstable case of equation (4.10) indicatethe Galilean scaling forµ < zKPZ(2) and the KPZ scaling forzKPZ(2) < µ ≤ 2.

In general ford = 1, 2, the competition between the noise and the non-linearterms [177] induces the KPZ value for the roughness exponentwhich, combinedwith Galilean invariance, yieldsν irrelevant forµ ≥ zKPZ(d) (albeit still withoutν renormalization, at variance with the KS case [2]). Given that zKPZ(d) ≤ 2 allthe way up to the upper critical dimension [2], we hypothesize that, for equation(4.10), KPZ scaling occurs forzKPZ(d) ≤ µ ≤ 2. We summarize the asymptoticproperties of the family of unstable non-local equations (4.10) for 0 < µ ≤ 2 infigure 4.15, marking cases for which simulations are provided.

Although the fixed point structure of the RG flow agrees with our numericalsimulations, several questions arise. On the one hand, there is the interference [2]between the typical length-scale set in by the morphological instability and theanalytical structure of the flow [the pole of equations (4.12), (4.13) forK+ ν = 0][64; 74]. On the second hand, the role of the KPZ nonlinearityin equation (4.10)is very special. Thus, while here it does not control the scaling properties for allparameter values, in marked contrast with the KS case (at least for d = 1) [2], itis definitely required as in the latter in order to stabilize the system dynamicallyfor intermediate to long times. In the RG language, as long asµ < zKPZ(d),the KPZ non-linearity needs to renormalise somehow to zero in infinite RG flow“time”. This may be related with the peculiar fact thatλ is a singular perturbationto the linear equation in (4.10); and while the exponents arenot those of the KPZuniversality class for theseµ values, they do satisfy its Galilean invariance, eventhough the dynamical equation does not. Notice that such a symmetry allows,through equation (4.11), to have a non-zeroλ at the fixed point. Thus, in someway the asymptotic dynamics preserves the nonlinearity required for dynamicalstability.

By using a scaling argument based on Flory theory we can estimate the mag-nitude of individual terms present in (4.10) [177]. We assume that at long times


0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

f f

gg

(a) (b)

(c) (d)

G

K

E E

GK

S

S E G E

K

S

S

G ≡ K

Figure 4.13 – Numerical integration of equations (4.15)-(4.16) ford = 1 and differentvalues of the exponentµ: (a) µ = 1, (b) µ = 1.4, (c) µ = 3/2, and, (d)µ = 1.6.The value of the coupling variables of the different fixed points (in the graphs E standsfor Edwards-Wilkinson, K for KPZ, S for Smooth and G for Galilean) and the associatedcritical exponents are reported in table B.1 of appendix B. The color of each fixed pointrepresent its stability (red stands for unstable, blue for saddle and green for stable). Notethat the Galilean fixed point moves whenµ is changed and merging with the KPZ fixedpoint forµ = 3/2.

t ≫ tl, and averaged over length scalesl, the height-height correlation functionscales asC(l, t) ∼ h2

l , and that at long times the relaxation time of these fluctua-tions is of the ordertl. Note thattl andhl depend on the length scalel. The variousterms in equation (4.10) can be estimated as

〈|∂th|〉l ∼hl

tl, ν

⟨∣

∣F−1[kµhk]∣

∣

⟩

l∼ ν

hl

lµ, λ

⟨∣

∣(∇h)2∣

∣

⟩

l∼ λ

h2l

l2. (4.17)

For white noise we can estimate its means-square fluctuations on length scalesland times scalestl asηl ∼ (Π0/Sltl)

1/2, whereSl is the average surface area ofthe interface on length scalesl [177], and we can estimateSl ∼ (h2

l + l2)d/2. For


0 2 4 60

2

4

6

0 1 2 30

1

2

3

ff

g

(a) (b)

SS

G

EE

Figure 4.14 – Numerical integration of equations (4.15)-(4.16) ford = 2 and differentvalues of the exponentµ: (a)µ = 1, (b) µ = 3/2. The value of the coupling variables ofthe different fixed points (in the graphs E stands for Edwards-Wilkinson, K for KPZ, S forSmooth and G for Galilean) and the associated critical exponents are reported in table B.2.The color of each fixed point represent its stability (red stand for unstable, blue for saddleand green for stable). Note that the Galilean fixed point is not present forµ = 3/2.

smooth surfaces thel2 term inSl dominates the height fluctuations, instead, forrough surfaces the dominant term ish2

l . The ratio between the scaling of the non-local term and the KPZ term (4.17) allows us to find the condition of the relativedominance of each term of equation (4.10). In fact, ifhl ≫ λlµ−2/ν its scalingis determined by the non-local term. In this case, by equating this term with theinertial term we obtain a simple condition for the characteristic time of the heightfluctuationstl ∼ lµ/ν, and consequently, the relationz = µ. When equation (4.10)has stable dispersion relation, i.e. forν > 0, it produces smooth interfaces, hencewe estimateηl ∼ (Π0/l

dtl)1/2. Moreover, the stabilization mechanism of the non-

local term is dominant on the KPZ term, thus, by comparing theinertial term withnoise fluctuations we obtain

hl ∼(

Π0

ν

)1/2

l(µ−d)/2. (4.18)

This last expression give us the value of the roughness exponentα = (µ − d)/2,which combined withz = µ gives the exponents of the Smooth fixed point, asobserved in our numerical simulations. However, in the unstable case the equationproduces rough interfaces so that the noise terms scales asηl ∼ (Π0/h

dl tl)

1/2.The exponents found for the Galilean fixed point can not result by equating theinertial term with the noise fluctuations as for the stable case. In fact, for thisfixed point, thed-independency ofα andz can not arise from the scaling of noisefluctuations. On the contrary, if we assume that the KPZ term is dominant, i.e. forhl ≫ νl2−µ/λ, Galilean invariance is given by the relationtl ∼ l2/λhl. These tworelations between exponents allows our RG flow to succeed in describing the results

4.3 Conclusions 93

of the numerical simulations, in a way that is reminiscent from the “accidental”success of the DRG for thed = 1 KPZ equation [79]. Besides, the irrelevance ofνfor µ ≥ zKPZ(d) is not due to the renormalization of this parameter from negativeto positive values, as occurs for the Kuramoto-Sivashinskyequation [65; 74], butfor the irrelevance of the non-local term compared to the non-linearity. In any case,one may think of the Galilean symmetry as hidden: even thoughthe underlyingequation does not have it, the hydrodynamic behavior displays it. Explicit Galileaninvariance of the dynamical equation seems sufficient but not necessary for it to bea feature of the large-scale properties.

Figure 4.15 – Summary of scaling properties of equation (4.10) forµ ∈ (0, 2], extrapo-lated to generald. MSKPZ is the equation studied in section 4.1, and havingµ = 1,m = 3[9], is asymptotically equivalent to the SMS equation. KS denotes thenoisyKS equation.3/2 ≤ zKPZ(d) increases for increasingd ≥ 1 [29].

4.3 Conclusions

In this chapter we have focused on non-local growth and its implications. In thefirst section, we have introduced a continuum dynamical description of non-localgrowth (4.1) that is able to give rise to cauliflower-like morphologies characterizedby two apparently contradictory features: a hierarchical (fractal) structure and dis-order (randomness). One of the reasons why fractals are so popular is the promisethat, knowing their generating rules, we can infer the underlying physical or bio-logical mechanisms. But, in contrast to those geometrical descriptions of fractals,the virtue of our continuum dynamical formulation is that itallows us to extractwhich are the most relevant mechanisms [178] whose interplay gives rise to theseappealing structures. These are non-locality, non-conservation (at lowest nonlinearorder), and noise.

This conclusion is expected to guide the inference of the relevant mechanismsat play in specific physical or biological systems where cauliflower-like structuresare identified. Furthermore, in order to test our theory, we have compared it withexperiments on thin film production by Chemical Vapor Deposition establishing


quantitatively the domain of validity of the equation. By employing analyticalarguments based on the Dynamical Renormalization Group we have justified therobustness of our theory as a continuum description of a large class of systems,providing an explanation for the ubiquity of cauliflower-like structures in manydomains of nature and technology.

From a more general point of view, our theory also brings up the long-standingquestion about why natural evolution favors self-similar structures. The so-calledallometric scaling relations [179] explain (and predict) the branching structure ofliving bodies. The central idea behind these theories is that biological time scalesare limited by the rates at which energy can be spread to the places where it is ex-changed with the tissues. Thus, the space-filling structure[180] required to supplymatter and energy to a living system can be accounted for. Focusing in more spe-cific systems (cauliflowers, pine trees, forests, etc.), albeit with a large degree ofuniversality, our work suggests the self-similar featuresthat the canopy atop suchbranched structures may have. Besides, it is remarkable that a such simple equa-tion like (4.1) can be able to capture this non-trivial dynamics, to the extent that,by means of pseudo-spectral integration, the system is capable to produce realisticpatterns resembling turbulent flame fronts or the texture ofcauliflower plants witha minimal computational effort.

In the second section of this chapter, we have generalized equation (4.1) byreplacing the termkhk in its dispersion relation bykµhk. This replacement can bejustified not only by the challenge of a more general interface equation but, also,because it can be an effective version of a non-polynomial dispersion relation whenfinite-size effects are accounted for.

This, apparently, small change in the dispersion relation allowed us to con-sider a whole class of non-local interface equations. Beyond their role as realisticdescriptions of experimental interfaces (additional examples exist for electrodepo-sition [171; 136] as shown in chapter 3, and for unstable flamepropagation [181]),these equations are interesting because they are among the simplest non-trivial in-stances of non-local interfaces subject to fluctuations, that, to date, have remainedpoorly understood [164].

Regarding the connection with experiments, the obtained irrelevance of theKPZ non-linear term for the asymptotic scaling may account for the difficulty toobserve KPZ scaling in surface growth experiments, not onlyin the large class offilm production techniques that are based on diffusive transport, and that featuremorphological instabilities [9], but also in the case that material transport to thesurface is ballistic [168]. Moreover, the equations we havestudied provide an inter-esting playground in which to analyze the interplay betweenanomalous diffusion(Levy flights), as described by the fractional Laplacian, and non-linearities thatdrive the system strongly away from equilibrium. While analytical tools from Sta-tistical Mechanics, like DRG, allow us to indeed retrieve important information onthe scaling properties, still several theoretical puzzles(as e.g. thed-independenceof the exponents) remain as a challenge for future research.

Ripple rotation in the anisotropicKuramoto-Sivashinsky equation

5.1 Pattern formation in ion-beam sputtering

In this chapter we will apply some of the techniques and methods exposed so farto the ion-sputtering induced nanopattern formation. Thissort of systems haverecently gained considerable interest, both experimentally [182] and theoretically[183]. The main reason is their promising applications in microelectronic devicefabrication [184], magnetic data storage [185], and the growth of functional films[186; 187; 188; 189].

These nanopatterns have been found on various kinds of surfaces includinginsulators [189], metals [186], and semiconductors [184] and, depending on theexperimental conditions, consist of periodic ripple [188]and hexagonally ordereddot structures [185], respectively.

The history of ion-sputtering pattern formation goes back to 1962 since thework of Navezet al. [190]. Navez showed that, that under oblique incidence of theions, often washboard-like ripple patterns oriented normally to the beam directioncan be observed on the surface [191; 192; 68].

In this long term history, it is striking that until 1999 ordered patterns werenot found under normal incidence (namely, when the flux of ions arrives in thedirection perpendicular to the sample substrate). Thus, until the works of Facskoetal. [193; 194], just a short rough, unstructured surface evolution had been expected.

96 Ripple rotation in the anisotropic Kuramoto-Sivashinsky equation

In those works, the authors produced quite regular ordered,hexagonally arrangeddot structures for semi-conducting GaSb under normal incidence of low energeticAr+-ions. Moreover, Frostet al. [195; 196; 197] found for rotated InP, InAs, InSband GaSb targets under oblique ion incidence a variety of distinct pattern such asdot or square structures and even rather flat or smooth surfaces.

Recently, experiments under oblique incidence without target rotation have re-vealed an even more puzzling picture. Using Si targets and ion energies≤ 2 keV,Ziberi et al. [198] experimentally detected transitions from ripples oriented normalto the beam direction to rather smooth surfaces by increasing the incidence angle.Similar results have also been obtained for Ge and Si targets[199; 200], where inthe latter case almost perfectly straight ripples have beenidentified. Using GaSbtargets and low ion fluxes, Allmerset al. [201] identified a transition from hexag-onal dot patterns to ripples oriented tangentially to the beam direction for smalloff-normal incidence.

In recent high-temperature sputter experiments on Si(111)surfaces, Brown andErlebacher observed a rotation of the initial ripple pattern with ion fluence [202;203]. At low fluences, a periodic ripple pattern oriented normal to the direction ofthe incident ion beam formed on the surface. At intermediatefluences, however,another ripple pattern rotated by 90 overlaid the initial one, resulting in a patternof dot-like features. At even higher fluences, the initial pattern vanished and onlythe rotated pattern with a significantly larger wavelength remained.

The standard interface equation used in literature for the description of this ex-periment on S(111) is the anisotropic version of the Kuramoto-Sivashinsky equa-tion (aKS) [68] [see equation (5.1)]. In fact, under these experimental conditions,step-edge barriers, anisotropic diffusion, and other non-isotropic surface effectscan be neglected [202; 203]. Unfortunately, the experimentally observed rotationof the ripple pattern by 90 does not agree with the predicted angle for cancel-lation modes (see below) which under these experimental conditions is expectedto be∼ 25 [202]. Therefore, the observed ripple rotation is not related to theappearance of cancellation modes and must be of a different origin.

In order to overcome this discrepancy between theory and this experimentalresult, Brownet al. proposed a kinetic Monte Carlo model as a minimal modelfor sputter patterning [202] obtaining good agreement between simulations and theexperimental behavior.

In spite of the theoretical proposition of Brown and Erlebacher, in this chap-ter we report the observation of a rotation of the initial ripple pattern by 90 atlong times in numerical integrations of the anisotropic KS equation under param-eter conditions that prevent cancellation modes. Comparison with analytical pre-dictions indicates that the observed rotated ripple pattern arises from anisotropicrenormalization properties of the aKS equation. This result might offer an expla-nation for the observed 90 ripple rotation in the high-temperature experiments ofBrown et al. [202; 203]. This success of the comparison between theory and ex-periments represents a considerable validation of the kindof models presented inthis theory, both at a qualitative and a quantitative level.

5.2 An interface equation for ripple rotation 97

5.2 An interface equation for ripple rotation

Although the theory of ion-beam erosion is beyond the scope of the present thesis,we will here summarize the state-of-the-art in the field. Thequest for a minimalmodel able to produce all the different patterns observed inion-beam experimentsis a challenging theoretical task. Thus, it is commonly assumed that the appearanceof ripple patterns during ion sputtering can be described bythe linear continuumequation derived by Bradley and Harper (BH) [204]. In this framework, the for-mation of periodic structures during ion sputtering is explained as resulting fromthe interplay between curvature dependent roughening and smoothing by surfacediffusion.

The BH equation can successfully reproduce the experimentally observed align-ment of the ripple structures with respect to the ion beam: the ripples are orientednormal and parallel to the beam direction for near-normal and grazing ion inci-dence, respectively [204].

However, for longer sputter times, however, non-linear effects start to dominatethe surface evolution and non-linear continuum models haveto be considered. Thesimplest non-linear generalization of the BH equation is ananisotropic version ofthe noisy Kuramoto-Sivashinsky equation given by

∂th = −v0+vl∂sh+νx∂2xh+νy∂

2yh−D∇4h+

λx

2(∂xh)

2+λy

2(∂yh)

2+η. (5.1)

This equation has been derived by Cuerno and Barabasi in order to describe the dy-namics of the interface produced by ion beam erosion [205]. Moreover, molecular-beam epitaxy (MBE) is another context in which the anisotropic KS equation hasbeen obtained from first principles [206]. The equation has been used to describethe non-linear evolution of steps on vicinal surfaces subject to MBE in the stepflow regime.

In the model of ion beam sputtering provided by equation (5.1) the projectionof the incident ion beam points in the positivex direction.v0 is the erosion veloc-ity of a planar surface andvl induces a lateral velocity for the ripples. The surfaceinstability due to curvature dependent roughening is incorporated by the linear co-efficientsνx,y with at least one of them being negative. The terms proportional toλx andλy are non-conserved Kardar-Parisi-Zhang [27] non-linearities that repre-sent the local slope dependence of the erosion velocity. Theterm proportional toDis a surface diffusion-like smoothing coefficient that might incorporate thermallyactivated and ion-induced contributions [182] andη is a delta-correlated noise termthat accounts for the stochastic nature of the sputtering process. The celebrated BHequation corresponds to the deterministicλx = λy = 0 limit of equation (5.1).

A variant of the Kuramoto-Sivashinsky equation in two dimensions has beenproposed by Facskoet al. [207] in order to improve the temporal evolution of dotpatterns produced by the isotropic version of equation (5.1). In fact, the two dimen-sional KS equation fails in two decisive points: (i) It can not reproduce the hexag-onal ordering of dots under normal incident ion sputtering,(ii) It do not predict the


stabilization of the periodic patterns for long sputteringtimes [207]. Facskoet al.proposed the so-calleddampedKuramoto-Sivashinsky (dKS) equation, which, inthe co-moving frame of reference, reads

∂th = −v0 − αh+ ν∇2h−D∇4h+λ

2(∇h)2 + η. (5.2)

Here, the term−αh introduces an additional dissipation suppressing spatiotem-poral chaos under certain conditions [208]. This additional term in the equationallow one to produce a richer variety of patterns and to overcome the limitation ofthe KS equation about the dots ordering. Unfortunately, this term also introducea preferred height into (5.2) and breaks down the translation invariance of the KSequation. Subsequently, Vogel and Linz tried to fix the physical inconsistency dueto the−αh term of (5.2) relating the non-local equation

∂tH = −v0 + b(

H −H)

+ ν∇2H −D∇4H +λ

2(∇H)2 + η, (5.3)

with (5.2) through the non-local transformation [209]

h = H −H − v0b

(

1 − ebt)

+ ebt∫ t

0dt′ ∂t′He

−bt′ . (5.4)

The functionH in equation (5.3) and (5.4) is the erosion depth averaged over thesample area and the termH −H might be interpreted as an approximation of theredeposition effect of the sputtered target particles [209]. The anisotropic versionto the dKS (adKS) equation has been studied by Vogel and Linz as model for ero-sion under normal and oblique incidence [210]. The numerical study of the adKSshowed that in the damping term acts as an order-disorder parameter, i.e. it cangradually change the pattern form the maximally disorderedlimit (α = 0 the aKSlimit) to a quite regular ripple or dot pattern, depending onthe equation parame-ters. Hence, the adKS equation allow one to observe a varietyof distinct transi-tions between patterns such as normal-ripples↔ smooth surfaces↔ transverse- ornormal-ripples, normal-ripples↔ dot structures↔ transverse- or normal-ripplesas well as more disordered states [210]. Indeed, the adKS equation seems to be apromising candidate as model for the description of the distinct patterns that arisesfrom ion-beam erosion but, nowadays, it still missing a complete derivation fromfirst principles such as for the aKS equation. Moreover, due to the fact that adKSequation is an extension of the aKS equation, the minimal model which describesthe rotation of ripple pattern found in [202; 203] seems to bethe aKS equation. Forthis reason we restrict our analysis to this equation.

A more promising approach has been proposed by M. Castro, R. Cuerno andcollaborators in the so-called hydrodynamic theory of erosion [211; 212]. This the-ory describes accurately some phenomena as pattern coarsening or the dependenceof the pattern lengthscale with physical parameters as temperature or energy flux.This model contains aKS equation as a particular (simplified) case which, itself,explains the ripple rotation phenomena under consideration in this chapter.


In order to study the influence of anisotropy on the surface morphology, it isconvenient to transform equation (5.1) into a minimal equation. As a first step,by introducing a co-moving frame of reference, the erosion velocity of the flatsurfacev0 is omitted. Then, the termvl∂h/∂x which just causes a lateral drift ofthe pattern can be eliminated by the transformationh(x, y, t) → h(x − vlt, x, t).Finally, rescaling the time by(ν2

x/D)t → t, the lateral scales by√

|νx|/Dr → r,the height by−(λx/2νx)h→ h, and the noise by−(λxD/2ν

3x)η → η leads to the

equation

∂th = −∂2xh− aν∂

2yh−∇4h+ (∂xh)

2 + aλ (∂yh)2 + η (5.5)

with the minimal number of independent coefficientsaν = νy/νx andaλ = λy/λx.These two coefficientsaν andaλ control the strength of the linear and the non-linear anisotropies, respectively. Rost and Krug studied the deterministic versionof equation (5.1) finding several interesting behaviors [213].

In equation (5.5), foraν > 0 andaλ > 0 all terms in the equation have thesame sign as in the isotropic case, and the two equations behave almost in the samemanner (apart from the loss of rotational symmetry in the plane for the anisotropicversion), displaying a cellular chaotic state. The caseaν < 0 andaλ > 0 is moreinteresting, since then the linear instability occurs onlyin thex direction. Hence,the surface evolution is dominated by a pattern of ripples aligned parallel to theyaxis with spacing determined bykm, i.e. the maximum mode of the linear disper-sion relationσk. However, numerical integrations show that the homogeneity inthey direction is subsequently lost due to a pinching off of the ripples, and the sys-tem evolves into an anisotropic cellular chaotic state as inthe previous parametercondition. Finally, in case ofaλ < 0 cancellation modescan appear.

Although most experimental studies on ion-induced patternformation havebeen performed under oblique ion incidence [182], only few atheoretical stud-ies [213; 214; 215] have focused on the corresponding anisotropic KS equation.These have shown that at a certain transition timetc, the surface enters a kineticroughening regime in which the initial ripple pattern vanishes.

For even longer times, however, the appearance of rotated ripple structureswas observed in the special case ofλxλy < 0 [213; 214; 215]. These ripplesrepresent one-dimensional solutions of equation (5.1) forwhich the non-linearitiesprecisely cancel one another and are, therefore, called cancellation modes. In orderto see this, we substitute solutions of the formh(x, y, t) = f(x− uy, t), which areconstant along the linesx = s+ uy, into equation (5.5) withη = 0

∂tf = −(

1 + aνu2)

∂2sf −

(

1 + u2)2∂4

sf +1

2

(

1 + aλu2)

(∂sf)2 . (5.6)

Thus, the non-linear part of this equation vanishes foruc = ±√

−1/aλ, and thesolution grows exponentially unlessaν < aλ < 0. Solutions withu ≡ uc arethe cancellation modes and are dynamically relevant only inthe unstable regimeaλ < aν . In thek plane the cancellation modes are composed of wave-vectorsforming angles± tan−1

√

−1/aλ with thekx axis.


Thus, the cancellation mode with the largest growth rate forms a ripple pat-tern of wavelength2π

√

2(1 − aλ)/(aν − aλ), with the ripples aligned at an an-gle ± tan−1 √−aλ relative to thex axis. Rost and Krug showed that in the un-stable regimeaλ < min(aν , 0) the asymptotic morphology is always dominatedby fastest growing cancellation modes resulting into a rotated ripple pattern thatcoarsens in time. Finally, the asymptotic pattern is formedby a single ripple thatsaturates the system [213]. Mainly, the deterministic aKS equation is characterizedby these three behaviors [213; 214] but the noisy aKS equation has not been thesubject of exhaustive investigations.

For this reason, Kelleret al.have integrated numerically equation (5.5) for thecaseaλ > 0, i.e. in the parameter condition that avoids cancellation modes [216].The numerical integration has been carried out on a grid composed by200 × 200lateral nodes (∆x = ∆y = 1) with periodic boundary conditions. They used thespatial discretization introduced by Lam and Shin for the non-linear terms [81].The integration step was∆t = 10−2 and the noise amplitude was fixed atΠ0 =10−2.

Figure 5.1 shows the simulated morphologies for constantaν = 0.1 and twovalues ofaλ at three different times. At short times [figure 5.1(a,d)], aperiodicripple pattern forms for bothaλ values. The amplitude of the ripples grows ex-ponentially until the surface enters the non-linear regimeand undergoes kineticroughening. Here, the initial ripple patterns vanish. Foraλ = 0.5, the resultingstationary morphology in this regime is rather isotropic (see figure 5.1(b,c)). For amuch loweraλ value of10−4, however, one observes an anisotropic morphologyat t = 200 [figure 5.1(e)]. Att = 103, a new ripple pattern with larger wavelengthhas developed and now dominates the surface morphology [figure 5.1(f)]. Thispattern is rotated by90 with respect to the initial pattern.

Figure 5.2(a) depicts the evolution of the global two-dimensional interfacewidth W for different values ofaλ at constantaν . During the initial ripple pat-tern formation, the ripple amplitude and, therefore, alsoW grows exponentiallywith time. When entering the kinetic roughening regime at a time tc1,W saturates.

In fact, when the non-linear effects begin to be relevant in the dynamics of (5.5),the surface width stabilizes rather abruptly [214]. For a rather largeaλ of 0.5,Wremains rather constant after this saturation. Foraλ = 0.1, however, a short fastincrease can be again observed fort & 100, indicating the formation and growthof the rotated pattern. Attc2 ≈ 200 > tc1, however, the growth is once moreinterrupted andW saturates at a slightly higher value. With decreasingaλ, thissaturation is further delayed astc2 increases. A similar behavior is observed for thedependence ofW on aν given in figure 5.2(b), withtc2 increasing for decreasingaν .

In these simulations,aλ was always positive. Therefore, the observed rotatedripple patterns cannot be explained by cancellation modes.In addition, they appearonly for a rather strong non-linear anisotropyaλ ≪ 1.

The crossover time from the linear to the non-linear behavior along a given di-rection, sayy, can be estimated by comparing the strength of the two corresponding


eb

fc

da

Figure 5.1 – Surface morphologies foraν = 0.1 andaλ = 0.5 (a-c) andaλ = 10−4

(d-f). Snapshots taken att = 20 (a,d),200 (b,e), and103 (c,f) [217].

terms in the dynamical equation. LetWc be the typical roughness at the crossovertime tc. Then, from the linear equation we obtainWc ∼ exp(aνtc/l

2y), while from

∂th ∼ Wc/tc ∼ aλ(∂h)2 we estimateWc/tc ∼ aλW2c /l

2y, where the wavelength

of the rotated pattern should given by [204]

ly ∼√

1

aν. (5.7)

Using the value of the time derivative ofh at the crossover time, i.e. when the linearterm equals the non-linear contributions, we obtain

tc ∝1

aνln

(

aν

aλ

)

. (5.8)

A comparison is given in the inset in figure 5.2(b) between thevariation of thewavelengthly of the rotated ripple pattern withaν as obtained from simulations andthe correspondingly values as calculated from equation (5.7). The dependenceof tc2 on aλ and aν is shown in figure 5.2(a) and (b), for theory and numericsrespectively. Here, the square symbols give the value oftc2 as determined from thesimulations and the solid lines represent fits of the data according to equation (5.8).


In both cases, the simulatedtc2 values are in good agreement with the analyticalprediction of equation (5.8).

t

W(t

)W

(t)

aν =

aλ =

aν

aν

l y

tc2

tc2

aλ

Figure 5.2 – Left panel: Evolution of the global two-dimensional interface widthW for(a) different values ofaλ with aν = 0.1 and (b) different values ofaν with aλ = 10−3.Right panel: Transition timetc2 vs. (a)aλ for aν = 0.1 and (b)aν for aλ = 10−3. Thesolid lines represent fits according to equation (5.8) with aprefactor as the fitting parameter.The inset in (b) gives the wavelengthly of the rotated pattern as a function ofaν determinedfrom the simulations foraλ = 10−3 (squares, error bars are smaller than the symbol size)and calculated using equation (5.7) (circles). All axes arein arbitrary units.

5.3 DRG analysis of the aKS equation

Numerical integration of aKS has shown a good agreement withexperiments.However, one can ask the question about the generality of this result for a widerrange of parameters. In this section we will interpret the latter results under thelight of the dynamical renormalization group.

In the asymptotic case of long times, the one-dimensional KSequation has beenshown to renormalize to the one-dimensional KPZ equation with a positive linearcoefficientν [73; 74]. Then, the surface instability is lost and the morphology un-dergoes kinetic roughening. The same is assumed to occur forthe two-dimensionalisotropic KS equation [73].

In the case of the aKS equation withaλ close to zero, one can expect that thisrenormalization is still valid in thex-direction, resulting in a positiveνx. In they-direction, however, the non-linearity is so weak that the transition to the non-linear regime is strongly delayed. Therefore, at the time when thex-direction isalready renormalized to a valueνx > 0, they-direction does not yet renormalize,

5.3 DRG analysis of the aKS equation 103

and νy = νy remains negative. Then,aν = νy/νx < 0 leads to a new linearinstability in they-direction that causes the formation of another ripple patternrotated by 90. These ripples then grow exponentially in time until also the y-direction renormalizes and the ripple amplitude saturates.

The data presented in the previous section support this argument. However, inorder to confirm this hypothesis, analytical evidence is needed. Therefore, we haveperformed a DRG analysis of the totally anisotropic aKS equation (with anisotropicsurface diffusion)

∂th = νx∂2xh+ νy∂

2yh−Dx∂

4xh − 2Dxy∂

2x∂

2yh−Dy∂

4yh+

+λx

2(∂xh)

2 +λy

2(∂yh)

2 + η.(5.9)

As seen in what follows, even if the “bare” surface diffusionterm has the isotropicform used in (5.1), renormalization spoils it and breaks thefourth order deriva-tive term into the different three contributions appearingin (5.9). After Fouriertransform in time and space variables, equation (5.9) reads

[iω − σk]hk,ω = ηk,ω − λx

2

∫∫

dq

8π3dΩ qx(kx − qx) hq,Ω hk−q,ω−Ω −

−λy

2

∫∫

dq

8π3dΩ qy(ky − qy) hq,Ω hk−q,ω−Ω,(5.10)

where the dispersion relation is

σk = −νxk2x − νyk

2y −Dxk

4x − 2Dxyk

2xk

2y −Dyk

4y . (5.11)

Equation (5.10) shows that each non-linearity is associated with a different vertexcontribution, so that in the renormalization procedure we have to take into accountall the possible combinations of different vertices. In figure 5.3 and 5.4 we show thefour diagrams for the renormalization of the propagator andof the noise variance,respectively.

For anisotropic equations the DRG scheme changes with respect to the isotropiccase. In fact, ford = 2, the anisotropy introduces two sets of critical exponents,each one associated with one particular direction in real space, thex direction orthe y direction (further detail can be found in section 2.1.2). Therefore, we haveto introduce at least one additional exponentχx in the scaling function in orderto take into account the anisotropy in the DRG scheme. This so-called anisotropyexponent accounts for the different rescaling factors along the two directions, seeequation (2.24) in chapter 2 and has not been deeply analyzedin the literature.

Thus, the re-scaling procedure changes with respect to the isotropic case, read-ing now

real space:

x → bxy → bχxyt → bzxth → bαxh

, Fourier space:

kx → kx/bky → y/bχx

ω → ω/bzx

h → bαxh

. (5.12)


k kk − q

q −q

k kk − q

q −q

k kk − q

q −q

k kk − q

q −q

Σxx = 4× Σyy = 4×

Σxy = 4× Σyx = 4×

Figure 5.3 – Diagrammatic contributions to the propagator renormalization Σ(k, ω).The blue circles stand for aλx vertex while the red circles represent aλy vertex. The arrowswith the vertical bar represent the bare propagators of fastmodes. For the correspondingintegrals, see equation (D.1).

In our calculations we have chosen thex direction as preferred direction for there-scaling, so that in this case the anisotropy exponent isχx. The exponents cor-responding to they direction are obtained through the relations reported in section2.1.2.

q − k

−kk

k − q

q −q

q − k

−kk

k − q

q −q

q − k

−kk

k − q

q −q

q − k

−kk

k − q

q −q

Φxx = 2× Φyy = 2×

Φxy = 2× Φyx = 2×

Figure 5.4 – Diagrammatic contributions to the noise variance renormalizationΦ(k, ω).The blue circles stand for aλx vertex while the red circles represent aλy vertex. The arrowswith the vertical bar represent the bare propagators of fastmodes. For the correspondingintegrals, see equation (D.52).

Here, we present only the main results of the renormalization procedure andwe defer the detailed calculations to appendix D. The renormalization flow of theparameters reads

dλx

dl= λx (zx + αx − 2) ,

dλy

dl= λy (zx + αx − 2χx) , (5.13)

dνx

dl= νx (zx − 2) − Σνx,

dνy

dl= νy (zx − 2χx) − Σνy , (5.14)

5.3 DRG analysis of the aKS equation 105

dDx

dl= Dx (zx − 4) − ΣDx − Σs

Dx, (5.15)

dDy

dl= Dy (zx − 4χx) − ΣDy − Σs

Dy, (5.16)

dDxy

dl= Dxy

[

zx − 2(1 + χx)]

− ΣDxy − ΣsDxy

, (5.17)

dΠ0

dl= Π0 (zx − χx − 1 − 2αx) + Φ, (5.18)

where the functionsΣνx, . . . ,ΣsDxy

andΦ are given in equations (D.42) and (D.54)of appendix D. Unfortunately these functions depend on integrals of the parameters(νi,Dij , λi,Π0,Λ) and, due to their complexity, we are not able to solve themanalytically.

Using the renormalization flow (5.13)-(5.18) we now try to explain the90

rotation of the ripple pattern. Therefore, we introduce a set of couplings

rν =νx

νy, rλ =

λy

λx, g =

λ2xΠ0

π2ν3y

,

dx =Dx

νy, dy =

Dy

νy, dxy =

Dxy

νy,

(5.19)

for which the RG trajectories remain bounded during the pattern rotation. In fact,the numerical integration of (5.13)-(5.18) that is presented below has been per-formed with parametersνx = −1, νy = −0.1, λx = 1, λy = 10−4, Dx = Dy =Dxy = 1, andΠ0 = 10−2, hence, the conditionλx ≫ λy imposes a differentrenormalization velocity for the two surface tensions. If our conjecture about theparameter renormalization were true, the surface tension along thex direction,νx,would renormalize from negative values to positive values earlier than the surfacetension in the other direction,νy, so that the couplings (5.19) remain bounded. Therenormalization flow for the couplings is

rλ = 2rλ (1 − χx) , (5.20)

rν = ν−1y

(

Σνy − Σνx

)

+ 2rν (χx − 1) , (5.21)

dx = ν−1y

(

Σνy − ΣDx

)

+ 2dx (χx − 2) , (5.22)

dx = ν−1y

(

Σνy − ΣDy

)

− 2dyχx, (5.23)

dxy = ν−1y

(

Σνy − ΣDxy

)

− 2dxy, (5.24)

g = ν−1y

(

3Σνy + Φ)

+ 5g (χx − 1) , (5.25)

where the dot over the lhs of the equations stands for derivative with respect tol(the infinitesimal re-scaling factor). From equation (5.20), i.e. from the Galilean


invariance of the two non-linearities, it is clear that onlythe choiceχx = 1 (rλ = 0)preserves such an invariance for the two KPZ terms at the fixedpoint.

In the case ofχx < 1, i.e. when the rescaling is larger in thex than in they direction, the rhs of equation (5.20) is positive and the coupling rλ increasesexponentially. Therefore, in the case of an anisotropy exponent smaller than unity,the renormalization flow transforms equation (5.9) into a degenerate equation inwhich only one non-linear term survives, namely,λx. On the contrary, in case thatχx > 1, the renormalization procedure leads to an equation which is non-linear inthey direction and is linear in thex direction. In this chapter we study only thecase of an isotropic rescaling, i.e.χx = 1, leaving the two other cases for futurework.

As stated before, equations (5.20)-(5.25) cannot be solvedanalytically so thatwe have integrated them numerically employing a variable step, fourth order Runge-Kutta scheme. For each “time” step of the numerical scheme wehave evaluatednumerically the integrals inΣνx, . . . ,Σ

sDxy

, andΦ. Unfortunately, these equationsare stiff, in fact the typical evolution scale of the coupling associated to the diffu-sion terms,dx, dy, and,dxy, is several orders of magnitude shorter than the evolu-tion scale of the other couplings. Note that, if we neglect the non-linear parametersfrom equations (5.15)-(5.17), the surface diffusion termshave a characteristic re-laxation “frequency” constant equal tozx − 4, i.e.Dx ∼ exp[(zx − 4)l]. This rateis much faster than the relaxation rate associated with the surface tension terms,which is equal tozx − 2. Hence, in the dynamics of the flow (5.13)-(5.18) thereare two separated time scales. For this reason, the fastest parameters, the surfacediffusion terms, become enslaved [218; 219] to the dynamicsof the slowest param-eters. As a first approach to this multiple scale problem we have chosen to simplifyequations (5.20)-(5.25) by considering the reduced set of equations

rν = ν−1y

(

Σνy − Σνx

)

, (5.26)

g = ν−1y

(

3Σνy + Φ)

, (5.27)

in which the fast terms are kept constant. In figure 5.5 we showthe DRG flowfor the remaining couplingsrν andg. The flow starts from a condition of unstablesurface tensions withrν > 0 andg < 0, i.e. νx, νy < 0, and, due to the relationνx < νy < 0 the surface develops a ripple pattern with crests parallel to they axis[see figure 5.1(d)].

The flow trajectories cross over the vertical axis while keeping the couplinggnegative, i.e.νy < 0. Whenrν is negative the surface tension in thex directionhas renormalized to positive values. At this point the first ripple pattern disappears[see figure 5.1(e)] and the rotated one begins to develop [seefigure 5.1(f)]. Atlater renormalization time,g flows to−∞, a signature of the renormalization ofνy

towards positive values. Finally, in the quadrant withνx, νy > 0 the renormaliza-tion flow goes toward a region in which the strong coupling regime, i.e.g → ∞,controls the dynamic of the equation.

5.4 Conclusions 107

λ2

xΠ0

π2ν3y

νx

νy

Figure 5.5 – Dynamic renormalization flow of the couplingsrν andg. The numericalintegration of equation (5.26) and (5.27) has been carried out in the conditionνx = −1,νy = −0.1, λx = 1, λy = 10−4, Π0 = 10−2, and,Dx = Dy = Dxy = 1.

Notably, these results support the qualitative argument put forward at the be-ginning of this section. A more rigorous treatment would require a perturbativemultiple scale expansion of flow (5.13)-(5.18) but we do not expect substantialqualitative differences from the main conclusions just reached.

5.4 Conclusions

Comparison of the above numerical integration of (5.5) withthe experimental re-sults of Brownet al. reveals striking similarities [202; 203] like the observed90

rotation and the larger wavelength of the rotated patterns.Thus, the presence ofa strong non-linear anisotropy withaλ ≪ 1 can be assumed for the experimentalsystem investigated in Ref. [202] and [203].

Transient morphologies of two-dimensional features similar to those observedin the experiments can also be achieved in the simulations ofthe aKS equation bytuning theaν andaλ coefficients in a way that the growth of the rotated ripples setsin before the initial pattern has fully vanished (not shown)[217].

One experimental finding that cannot be explained by these simulations is thecoarsening of both the initial and the rotated ripple pattern. Wavelength coarseningis generally considered as a truly non-linear feature. However, in the aKS simula-tions, the initial ripple pattern forms in the linear regimeand gets destroyed whentheλx non-linearity becomes active. Therefore, no coarsening ofthe initial pattern


can be expected. The rotated pattern also forms in a linear regime without the pos-sibility of wavelength coarsening. When also theλy non-linearity becomes active,the ripple amplitude saturates but the rotated pattern doesnot immediately get de-stroyed by kinetic roughening. However, no coarsening of the remaining patternhas been observed.

In summary, we have shown in numerical simulations that the aKS equationmay exhibit a 90 rotation of the initial ripple pattern at long times. This rotation isonly observed in the presence of a strong non-linear anisotropy. Comparison withanalytical predictions suggests that this rotation results from anisotropic renormal-ization properties of the aKS equation. This result offers an explanation for therecently observed ripple rotation in high-temperature experiments on silicon sur-faces [202].

Phase-field model of diffusivegrowth

6.1 Introduction

The experimental morphologies obtained in diffusive growth system are often verycomplex: branches form abruptly, grow exponentially, coarsen and eventually mergeforming a smooth interface. Specifically, in electrodeposition the small slopes ap-proximation is broken out after a very short initial transient. In fact, after the onsetof the non-linear growth regime, the interface develops fast growing bumps intothe diluted phase. These bumps evolve into columns until coarsening and surfacetension (and fluctuations) make them to lose their stabilityand the branching pro-cess begins to disorder the pattern, leading to a final ramified morphology. Clearly,after the bumps evolution into columns the condition|∇h| ≪ 1 is not valid anymore and all the formalism developed in previous sections can not be used in or-der to investigate the kinetic roughening phenomena of the late time morphologiesobtained in diffusive growth.

For this reason, in order to study the set of equations (3.11)-(3.14) beyond thesmall slopes approximation we have to integrate it numerically. Due to the sharpboundary between the growing region and the vapor this task is not trivial. Froma computational point of view, the integration domain has tobe divided into twodisjoint regions and, during the growth of the surface, the algorithm has to trackdown the evolution of the interface and impose the boundary condition (3.12)-

110 Phase-field model of diffusive growth

(3.13).Because of the complexity of this procedure, there have not been many at-

tempts to integrate such a moving boundary problems in the literature (either intwo or in three dimensions), by employing the boundary integral method. A dif-ferent way to tackle this type of interfacial problems is through a diffuse interfaceformulation, e.g. a phase-field (PF) model. The main characteristic of PF models isthe introduction of the supplementary order parameterφ(r, t) that varies smoothlybetween the two phases. This scalar field is coupled with the physically relevantfield and tracksimplicitly the boundary position.

Recently, PF models have become popular as numerical tools to integrate sys-tems in which two or more phases are separated by moving boundaries. Histori-cally, Langer and Fix were the first to introduce phase field models for first-orderphase transitions [220; 221]. In fact, in the field of Materials Science there aremany investigations in which a PF model has been used to integrate numericallythe Stefan problem (a moving boundary formulation of the solidification process)in two and three dimensions. Subsequently, many researchers from this scientificcommunity have employed these models to study different physical phase transfor-mation, for example binary alloys solidification, directional solidification and grainboundary growth, for a review about PF models in material science see [222].

Nowadays, the range of applicability of PF models is broad, and they have beenused effectively in contexts ranging from fluid dynamics to biology. In physic offluids, PF model have been extensively employed to study Saffman-Taylor fingersin a Hele-Shaw cell [223]. Besides, Folchet al. formulated a phase-field modelto simulate the dynamics of the interface between two inmiscible fluids with ar-bitrary contrast in this experimental set up [224]. Vendantam and Panchagnuladescribed the wetting of a liquid drop on a planar solid surface by using a non-conserved phase field variable to distinguish between the wetted and non-wettedregions [225], while, Borciaet al.derived a PF model from a Navier-Stokes equa-tion to investigate the stability of a thin liquid film on a flatsolid support withvariable wettability [226]. Another field in which PF modelshave been success-fully employed is in propagation of fractures [227; 228]. In[227] the continuumdescription of the crack phenomena has been based on simple physical considera-tion but, nevertheless, the authors were able to reproduce the different behavior ofcracks and to describe oscillatory cracks found in experiments [227]. In the workof Spatscheket al., the authors formulated a continuum theory including the elas-todynamic effects in order to study the phase transitions induced by elasticity infast crack propagation [228]. In biology, it is remarkable the number of researchgroups who have investigated the behavior of membranes and vesicles using PFmodels [229; 230; 231; 112; 232]. By employing a formulationbased on a Hel-frich free energy, Campeloet al.studied the curvature-driven pearling instability inmembranes. Finally, we remark that, in last years, many efforts have been devotedto couple phase-field models with physical equations, such as the Navier-Stokesequation [233; 234], or equations for the stress field [235; 236].

After this introduction, we can discuss briefly how to formulate a phase-field

6.2 Model formulation 111

model. Traditionally, there are different approaches to construct a PF model and thechoice mainly depends on the considered phenomena. Many models have been for-mulated in order to recover a given moving boundary problem describing a systemin which a phase transformation takes place. In spite of these models, for physicalsystems in which a free energy is not defined, the PF formulation has been used asan effective front tracking method.

With respect to PF models of phase transformation, we can classify these mod-els into two main groups:

(i) models that are thermodynamically consistent [221], inwhich, by employingconcepts of irreversible thermodynamics, all the model equations are derivedfrom an unique Lyapunov functional representing a phenomenological freeenergy , and

(ii) isothermal PF models [237], where only the phase-field equation is derivedfrom the functional, while the equation of the physical field(temperaturefield, concentration field, . . . ) is postulated a hoc.

Although these two distinct model formulations are equivalent in the limit of van-ishing interface thickness, in fact, they converge to the same MB model. For apractical perspective, the isothermal models have less constraints on the choiceof functions in the Lyapunov free energy than the thermodynamically consistentPF models. This difference results in a great computationaladvantage, leadingto a better convergence of the isothermal models with respect to the others [237].Taking into account the nature of our problem, we have decided to employ anisothermal PF model in order to take advantage of its computational efficiency.

In the rest of the chapter we formulate the phase-field model we have used tointegrate the moving boundary problem. We tested the convergence properties ofthe numerical scheme by changing the interface thickness (the only free parameterleft in the model) and the spatial discretization. After thetest procedure, we haveobtained several complex morphologies and we have studied how these morpholo-gies behave by changing the particle concentration. In lastsection, by using thetypical parameters of ECD growth, we have reproduced the asymptotic morpholo-gies obtained in experiments and, besides, we have studied the kinetic rougheningof these surfaces.

6.2 Model formulation

Let us start the formulation of our phase field model with someconventions. Theorder parameterφ(r, t) defines the state of the system in space and time. In fact,the growing aggregate and the vapor phase are defined byφ = 1 andφ = −1,respectively and, thus, the interface position is determined by the level-setφ =0. This method allow us to track down implicitly the interface, avoiding manynumerical difficulties. The domain in which the order parameter varies is calledinterior regionΩi, whereas the complementary region is called exterior region Ωe,


being composed by two parts,Ω+ andΩ−, in which the phase-field variable isconstant, i.e.φ(r, t) = ∓1, ∀r ∈ Ω±. The extent of the region where the modelchanges phase is the fundamental difference between movingboundary and phase-field models: in the former it is zero (physically it is formedby atomic layers),whereas in the latter it is

√2W (whereW is a model parameter). For this reason,

PF models are also named diffuse interface models.In order to formulate our PF model, it is convenient to re-write the system of

equations (3.11)-(3.14) for the non-dimensional concentration fieldu; this field canbe chosen according to two normalizations,u1 = Ω(c− c0eq) or u2 = (c− c0eq)/c0eq(for the definition ofΩ and c0eq see chapter 3). Depending on the choice of thenormalization, the re-scaled MB model reads

∂tui = D∇2ui, (6.1)

D∂nui

∣

∣

+= kD (ui − liK) , (6.2)

Vn = LiD∂nui

∣

∣

+, (6.3)

limz→∞

ui = uai, (6.4)

wherei = 1, 2 corresponds to the different normalized concentration fields,ui

∣

∣

+

is the concentration inΩ+, and we have omitted the fluctuations. The constantlidefines a characteristic length scale, andLi is a non-dimensional number. When thenon-dimensional fieldu1 is chosen,l1 = ΓΩ ≡ d0,L1 = 1 andua1 = Ω(ca−c0eq),while in case ofu2 the constants arel2 = Γ/c0eq, L2 = (Ωc0eq)

−1, andua2 =ca/c

0eq − 1. Using equations (6.2) and (6.3), we obtain the generalizedGibbs-

Thomson condition for this moving boundary model

ui = liK +k−1

D

LiVn. (6.5)

After the rescaling procedure, we have to chose a PF model converging toequations (6.1)-(6.4) in the sharp (W → 0) [221] or in the thin (W/lD → 0) in-terface limit [237]. In these two limits the phase-field parameters become relatedto the macroscopic constants of the MB model, so that the PF model provides anaccurate approximation of equations (6.1)-(6.4). Taking into account the numericalconstraints of the phase-field model, the sharp interface limit is less efficient thanthe thin interface limit. Besides, only through the thin interface limit we can attainthe condition of vanishing interface kineticsk−1

D = 0 [237], which corresponds tothe case of fast attachment kinetics (kD ≫ Vn) (for more informations see chapter??).

Considering the nature of our problem, we employ an isothermal PF model totake advantage of its computational efficiency. The equation of the order parameterφ is derived from

τ∂tφ = −δLδφ, (6.6)

6.2 Model formulation 113

whereL is the Lyapunov functional

L =

∫

d3r

(

W 2

2|~∇φ|2 + f(φ) − λug(φ)

)

, (6.7)

andf is a double well function in which the minima, located atφ = ±1, stand forthe equilibrium states of the functional, each one associated to a different phase.The functiong is odd in its argument, and is used to drive the interface out-of-equilibrium: foru different from zero (its equilibrium value) the two double wellminima have different free energies (the black solid curve in figure 6.1); finally,λis the coupling between the order parameter and the concentration field.

For u > 0, i.e. c > coeq, the solid phase has less energy than the vapor phase,so that the aggregate grows into the diluted phase (see the colored curves in figure6.1). The moving boundary model (3.11)-(3.14) is not symmetric with respect tothe two phases, in fact its diffusivity is discontinuous across the interface, i.e. itis zero inside the solid aggregate andD within the vapor phase. Diffuse interfacemodels interpolate between these two limit values with a continuous function. Theinterpolation introduces a spurious effect absent in the sharp interface model. Infact, the particles inside the interior region (the solid aggregate)Ωi diffuse whenφ > 0, whereas their diffusivity is less thanD whenφ < 0, i.e. into the dilute part.

R. Almgren performed a perturbative analysis (up to second order) of a phase-field model describing a solidification problem with asymmetric diffusion coeffi-cient [238]. In his work he showed the limitations of isothermal models for thecase of a diffusivity jump across the interface. Later on, Echebarriaet al.proposedan isothermal PF model converging to three different movingboundary modelswith unequal diffusivities in the various phases: the binary alloy problem, the so-lidification problem with zero diffusion in the solid phase (termed one-side model)and the directional solidification problem [239]. In order to compensate for thespurious effects that prevent the model convergence, a flux∇ · jat is subtracted tothe concentration equation. Echebarriaet al.showed that this anti-trapping currentmust be proportional to the width of the interior region,W , to the local growthvelocity of the surface,∂tφ, and, it has to be normal to the interface, i.e. directedalongn = −∇φ/|∇φ|. By combining these three requirements, the anti-tranpingcurrent reads [239]

jat = −a(φ)W∂tφ∇φ|∇φ| , (6.8)

wherea(φ) depends on the other functions of the PF model. Due to the term∂tφ∇φ/|∇φ|, the anti-trapping current is localized into the interior region withpositive flux (∇ · jat) inside the vapor component ofΩi, being negative into thesolid component.

Besides, the functionq(φ) has been used to interpolate the diffusivity between0 andD, settingq(1) = 0 into the solid aggregate andq(−1) = 1 in the va-por phase. By including fluctuations into the asymmetric phase-field model of


Echebarriaet al. [239], we obtain

τ∂tφ = W 2∇2φ− ∂φ

[

f(φ) − λug(φ)]

+ σφ, (6.9)

∂tu = ∇ · [Dq(φ)∇u− jat] −1

2∂th(φ) −∇ · σu, (6.10)

whereσφ andσu are noises whose variances are related to those ofq andχ in(3.11)-(3.14),τ is the characteristic time for the particle attachment to the aggre-gate,L is a numerical constant related with the growth kinetics, and,h(φ) concernsthe energy production at the interface coupling the concentration field with the or-der parameter.

-2 -1 0 1 2φ

-1

0

1

2

3

4

f(φ)

-λu

g(φ)

λu = 0λu = 0.2λu = 0.4λu = 0.6

Figure 6.1 – Driving term of the equation (6.9) for different values ofλu. Note that thesolid black curve represents the equilibrium conditionλu = 0. In this figure we have usedthe functions (6.11) and (6.12).

The phase-field convergence imposes only the parity and the values of the PFfunctions (f , g, q, h, and,a) at φ = ±1, so that the choice of these functions canspeed up the convergence with a significant increase in numerics. The functionsthat we have chosen in our PF model are [239]

f(φ) =φ4

4− φ2

2, (6.11)

g(φ) = φ− 2

3φ3 +

1

5φ5, (6.12)

q(φ) =1 − φ

2, (6.13)

h(φ) = φ and a(φ) = − Li

2√

2. (6.14)

Kim et. al. have verified that many thermodynamic and isothermal phase-fieldmodels converge to the same MB model when they operate in the correct limit

6.3 Numerical convergence 115

[240]. In the thin interface limit (see appendix E) this PF model converges asymp-totically to equations (3.11)-(3.14), the connection between the models parametersbeing

li = a1W

λ, (6.15)

k−1D = a1

(

τ

λWLi− a2

W

D

)

, (6.16)

wherea1 = 0.8839 anda2 = 0.6267 depend on the functions (6.11)-(6.14) (furtherdetails can be found in appendix E).

6.3 Numerical convergence

In this section we emphasize the importance of the convergence of our model andwe describe the conditions under which it occurs. The diffuse interface formula-tion of the moving boundary model can introduce undesired effects when it is notoperating in the appropriate regime. These “errors” are notdue to the numericaldiscretization of equations (6.9)-(6.10) and, for this reason, they cannot be ruledout by increasing the numerical accuracy of the simulation scheme.

First, we have studied how the model (6.9), (6.10) behaves byvarying the in-terface thicknessW , and we have checked our results in several parameter condi-tions. As noted by Kimet al. [240], the fundamental convergence parameter is theinterfacial Peclet numberp = W/lD = VnW/D, estimating the systematic errorintroduced by the diffuse interface formulation.

Usually, a standard PF model converges accurately whenp is quite small, typ-ically on the order of10−2. In fact, a negligible interfacial Peclet number meansthat the interface thickness, and consequently, the grid spacing are very small ascompared to the diffusion length. Actually, the concentration profile in the interiorregion can be expanded in powers ofp, and, in the thin interface limit, it corre-sponds to the expansion parameter.

The second important convergence parameter is the non-dimensional interfacevelocityν = Vnτ/W , which depends on the mass transport coefficientkD throughτ , see (6.16). The velocity of a planar front is uniquely related to the driving forceand, in the regime of convergence, it is proportional to the termλu calculated at theinterface. In equation (6.9), only the productλu contributes to the stationary phase-field solution, and the linear relation between the front velocity Vn and the productkDu in (6.5) breaks down for small velocity ifλ exceeds a limiting threshold.

When this breakdown occurs, Bragardet al. proposed to replace the couplingbetween the phase-field and the concentration equation in (6.9) by hv(λu)g(φ),wherehv is a monotonously increasing function of its argument [241]. Moreover,they showed that the dependence ofV from the driving force turns out to be non-linear, and, typically, the linear relation is not verified for λu larger than one, see


0 0.5 1 1.5 2λu

00.5

11.5

22.5

3

h v

0 2 4 6 8λu

0

10

20

30

40

50

60

h v

Figure 6.2 – Non-linear relation betweenλu andhv calculated from (6.17).

figure 6.2. The functionhv has been calculated numerically solving the one di-mensional problem

ν∂xφ+ ∂2xφ− f ′(φ) + hvg

′(φ) = 0, (6.17)

for different values ofhv. In equation (6.17) the spatial variablex has been rescaledby the interface thicknessW . In this way we can estimate numerically the de-pendence ofν on hv, and the linearity can be restored by employing the inversefunctionν−1. Actually, we can re-write equation (6.16) in the followingway

k−1D =

1

k0D

(1 − ǫkD) , k0

D = LiλW

a1τ, ǫkD

= a2k0D

λd0

D, (6.18)

and we obtain a Gibbs-Thomson boundary condition for a planar interface as thesum of a most relevant part and a perturbation

Vn =[

k0D + O(ǫkD

)]

u ≃ k0Du. (6.19)

If ǫkDcan be considered as a small perturbation,ǫkD

≪ 1, and the PF modelconverges correctly, thus the linear relation betweenV andλu simplifies, and thenon-dimensional velocity isν = λuL2

i /a1. Under this approximation the numeri-cal estimation ofν−1 can be used to recover the right interface velocity [241]

Vn =W

τν (hv) =

W

τν ν−1

(

λuL2i

a1

)

= Lik0Du. (6.20)

The estimation of the convergence errors due top andν has been performedby comparing the solution of equations (6.1)-(6.4) with theasymptotic state of thePF model for the propagation of a plane front in the stationary state with velocityVn (for clarity we consider only the caseu ≡ u1). In the comoving frame ofreference of the interface, the stationary state of equation (6.1) is found by solving

6.3 Numerical convergence 117

0 0.03 0.06 0.09p

0

2

4

6

8

10∆V

n [%]

0 0.2 0.4 0.6ν

0 0.03 0.06 0.09p

100

101

102

103

104

Cpu

Tim

e [s]

0 0.01 0.02 0.03p

100

101

102

Cpu

Tim

e [s]

0 0.02 0.04ν

0 0.01 0.02 0.03p

10-1

100

101

102

∆Vn [%

]

Figure 6.3 – Convergence of the PF model to the planar solution (6.21) in the degeneratecase ofk−1

D = 0 and initial velocityVn = 0.1. Blue circles denote the values ofp,whereas red squares denote the values ofν of the numerical integrations. Left column:Parametersd0 = 0.1, D = 0.1, and,tf = 126. Cased0 . lD. Right column: Parametersd0 = 5 × 10−4, k−1

D = 0, and,tf = 5.96. Cased0 ≪ lD. Simulations have beenperformed with∆x = W/2 and∆t < ∆x2 min(τ/W 2, D)/2.

Vn∂zu +D∂2zu = 0. By imposing boundary conditions (6.2)-(6.4), we obtain the

solution of the rescaled moving boundary model

u(z) = ua − e−zVn/D, (6.21)

where the velocity is given byVn = kD(ua−1). This relation between the velocityand the kinetics coefficient is valid only ifk−1

D 6= 0. Moreover, the solution (6.21)represents a stationary state only forua > 1, while, for ua ≤ 1, the plane frontslows down asV ∼ t−1/2.

On the contrary, in the degenerate case ofk−1D = 0, the solution (6.21) is stable

only for ua = 1. Here, the velocity is undetermined and the front propagatesaccording to its initial velocity [only if the initial stateis already the stationarystate (6.21)]. Forua > 1 (ua < 1) the front increases (decreases) its velocity withtime.

As a first test, we have estimated the relative error between the PF front propa-gation velocity and the plane solution of the moving boundary model (6.21) in caseof k−1

D = 0, by integrating equations (6.9)-(6.10) with an explicit finite differencenumerical scheme and the Euler’s method. By changing the interface thicknessWwe are able to adjustp andν. Besides, due to the time step restrictions imposed


0 0.01 0.02 0.03p

0

3

6

9

∆Vn [%

]

Echebarria et al. modelBragard et al. correction

0 2 4 6ν

0

3

6

9

0 0.01 0.02 0.03p

100

101

102

103

Cpu

Tim

e [s]

Figure 6.4 – Convergence of the PF model to the planar solution (6.21) in case of kinet-ics, i.e.k−1

D 6= 0, with velocityVn = 0.1. Blue circles denote the results from the model(6.9)-(6.10), while green triangles denote the results from the same model with correctionproposed by Bragardet al.(see main text). The simulation parameters are:d0 = 5×10−4,D = 0.1, k−1

D = 1, and,tf = 19.6. Simulations have been performed with∆x = W/2and∆t < ∆x2 min(τ/W 2, D)/2.

by the stability condition of the numerical scheme, the CPU time required to reachthe final simulation timetf depends on the interface thickness through the spatialdiscretization. We have chosentf = 10/σkm

, whereσkmis the maximum of the

linear dispersion relation of the moving boundary model. Inleft column of figure6.3 we show the results for the conditiond0 ∼ lD, while in the right column ford0 ≪ lD. In the first case,d0 ∼ lD, we observe an almost linear dependence of∆Vn on the Peclet number forp < 0.1. In the case ofd0 ≪ lD, the convergence ofthe PF model is very sensitive top. As shown in figure, for values ofp above thethreshold1.5×10−2 the error grows very fast, and the PF model does not convergefor p > 2 × 10−2. If of k−1

D 6= 0 the convergence is more difficult due to an addi-tional error produced by the non-linear response of PF modelto the driving force,see equation (6.17). In the left panel of figure 6.4 we show theblow up of the errorfor the model (6.9)-(6.10) whenp andν are increased. By using the method pro-posed by Bragardet al.we have been able to extend the region of PF convergenceto a larger parameter range (see the green triangles in figure6.4). Finally, in fig-ures 6.3 and 6.4 we show the drastic decrease of the CPU time when the interfacethickness is increased. In fact, in simulations of complex morphologies the choiceof the interface thickness is determinant to achieve a good model convergence andto observe an asymptotic pattern in a reasonable simulationtime.

In the second test we have studied the linear dispersion relation of the PF modelunder the small slopes approximation. We have prepared many“almost flat” (withamplitude∼ 10−2∆x) sinusoidal initial condition with wavenumberk and we havetraced the amplitude evolution these sinusoids with time.

In figure 6.3 we show the convergence of the linear dispersionrelation of the

6.4 Multivalued interfaces in electrodeposition 119

0 10 20 30 40 50k

-2

-1

0

1

2

σ kFull dispersion relationMS approximation

W = 1.00 x10-2

W = 1.25 x10-2

Figure 6.5 – Convergence of the PF model to the full dispersion relation of the movingboundary model (3.38) for two conditions of the interface thickness. Black solid line standsfor the Mullins-Sekerka approximation (3.34). The simulation parameters are:d0 = 5 ×10−4,D = 0.1, k−1

D = 0, Vn = 0.1, and,∆x = W/2.

PF model to the full dispersion relation (3.38) for the case of instantaneous attach-ment kinetic coefficient, i.e.k−1

D = 0. By changing the interface thickness, and,consequently, the Peclet number, we are able to bound the errors due to the diffuseinterface approach.

Moreover, we have performed a study on the dependence of the PF convergenceon the spatial discretization. Figure 6.3 shows that the PF convergence, at lestfor the linear growth regime, is not sensitive to the spatialdiscretization of theoperators in (6.9)-(6.10). In fact, within the rangeW/2 ≤ ∆x < W we observethat the numerical results fall on the same curve. As can be seen into the insetof figure 6.3, for∆x = W the large wavelengths (smallk) behavior ofσk isnot well captured, and, for this reason, we set this value as upper limit for oursimulations. Finally, in figure 6.7 we show that the Bragardet al.correction at themodel (6.9)-(6.10) is mandatory when the parameterkD decreases. Note that themass transport parameter for the simulation in left panel offigure 6.7 is higher thatfor the simulation in right panel. However, as pointed out byVetsigianet al.in caseof a kinetic coefficientkD lower that10−1, the correction due to Bragardet al. isnot effective anymore, and a different PF formulation has tobe employed [242].This parameter regime is not considered in this thesis and wewill not discuss thisissue in detail.

6.4 Multivalued interfaces in electrodeposition

There have been previous proposals of phase-field formulations of diffusive [243;244; 245] (and ballistic [246; 245]) growth systems of the type of our movingboundary problem. However, in these works the phase-field equations are pro-


0 2 4k

-0.2

0

0.2

0.4

σ k0 10 20 30 40 50 60

k

-1

0

1

2

σ k

Full dispersion relationMS approximation∆x = 0.50 W∆x = 0.75 W∆x = 0.90 W∆x = 1.00 W

Figure 6.6 – Convergence of the PF model to the full dispersion relation of the movingboundary model (3.38) by changing the spatial discretization ∆x. Black solid line standsfor the Mullins-Sekerka approximation (3.34). The simulation parameters are:d0 = 5 ×10−4,D = 0.1, k−1

D = 0, Vn = 0.1, and,W = 1.25 × 10−2.

0 10 20 30 40 50k

-1

-0.5

0

0.5

1

σ k

0 10 20 30 40 50k

-0.1

0

0.1

0.2

0.3

σ kFigure 6.7 – Convergence of the PF model to the full dispersion relation (black circles)of the moving boundary model (3.38) in case ofk−1

D 6= 0. Simulation parameters:d0 =5 × 10−4, D = 0.1, Vn = 0.1, and,W = 10−2. Left panel: PF model (6.9)-(6.10)for k−1

D = 1/2. Right panel: PF model (6.9)-(6.10) with Bragardet al. correction fork−1

D = 2.5.

posed on a phenomenological basis, rather than being connected with a physicalmoving boundary problem as in our case. Thus, quantitative comparison is harderto achieve, although many qualitative morphological features can indeed be re-trieved.

We have performed numerical simulations of (6.9)-(6.10) using physical pa-rameter values that correspond to the experiments on Cu ECD by Kahandaet al.[137] (see parameters in table 3.1), for different surface kinetics conditions. Dueto the different length scales present in the problem, we have employed a multigridscheme for the resolution of the concentration profileu. By using several levels

6.4 Multivalued interfaces in electrodeposition 121

of square grids with increasing lateral size of their elements, we can decrease thecomputational cost of our simulations. As shown in the left panel of figure 6.8 thelateral size of the elements of each grid double from a level to the next. In the rightpanel we show how this scheme interpolates a typical exponential concentrationprofile such as (6.21). As shown in this figure, moving away from the interface,the density of the grid points decreases according to the decrease of the gradientof u. This method allows us to achieve a high numerical accuracy with a reducednumber of grid points.

0 2 4 6 8

z

0

0.2

0.4

0.6

0.8

1

u

z

Figure 6.8 – Left panel: Representation of the multigrid scheme used in the simulations.The thick black line is the growing interface. Right panel: Multigrid interpolation of thenon-dimensional concentration profileu = 1−0.9 exp(−z/2). In this figure we have usedthree grids as in the left panel.

Representative morphologies are shown in figure 6.9, where the overpotentialvalue is employed to tune the value of the kinetic coefficientkD. Notice that in ref-erence [137] the average growth velocity is overestimated,having been measuredin the nonlinear regime, which also overshoots the estimated value of the capillarylength. For this reason, the value of the capillary length employed in our simula-tions (and provided in table 3.1) is estimated as smaller than that in [137]. As wecan see, for the faster kinetics case (upper row in figure 6.9), indeed the interfacebecomes multivalued already at early times, leading to a relatively open, branchedmorphology. As the overpotential increases (thereby decreasing the value ofkD,lower row), the morphology is still multivalued, but more compact. We wouldlike to stress the large degree of quantitative agreement between the simulated andexperimental morphologies.

Our phase-field model (and the corresponding moving boundary problem that itrepresents) has moreover a large degree of universality, inthe sense that it can alsoreproduce quantitatively other ECD systems. An example is provided by figure6.10, which shows simulations for ECD growth from a Cu(NO3)2 solution, as inreference [124], see experimental parameters in table 3.1.This dense branching


Figure 6.9 – Comparison of morphologies from the experiments on CuSO4 in [137](left column, taken from the original paper; reference barsrepresent 0.5 mm) and phase-field simulations using parameters in table 3.1 (right column). Top row corresponds to fastkinetics (∆ψ = −0.50 V), and bottom row corresponds to intermediate kinetics (∆ψ =−0.35 V).

morphology corresponds to a fast kinetics condition for which the single valuedapproximation breaks down already for short times. Figure 6.11 collects the timeevolution of the surface roughness and of the PSD function for the same parametersas in figure 6.10.

As shown in figure 6.11, the roughness grows rapidly within a transient regime,that is later followed by stabilization and rapid saturation to a stationary value. Thisresult is similar to those obtained within the small slope approximation. However,the behavior of the PSD differs dramatically. Thus, for verysmall times the linearinstability shows up in the finite maximum of the curves. For intermediate timesthe PSD’s shift upwards with time, a behaviour that is associated with anomalouskinetic roughening properties [62; 150] that contrasts with the standard FV scal-ing found above in which the PSD values are time independent for length scalesthat are below the correlation length. Note that at these intermediate times thesingle valued approximation does not describe the featuresof the interface at all.Actually, the single valued approximation to the height field (not shown) presentsabundant large jumps that are possibly responsible for the effective anomalous scal-ing [247]. However, for sufficiently long times branches grow laterally closing theinter-branch spacing, and the validity of the single valuedapproximation is re-stored.

Thus, the PSD curves for the longest times again display Family-Vicsek scalingbehavior for the largestk values. A crossover betweenαs = 0.5 for small scalesandαs = 0 (log) for intermediate scales can be read off from the figure.

Finally, note that, for the longest distances in our system,scale invarianceseems to break down, since no clear power law can be distinguished for, say,k < 30 mm−1 ≃ l−1

D . In order to properly characterize the behavior of the in-

6.5 Conclusions 123

1 mm

1 mm

Figure 6.10 – Simulated morphology using parameters as in the experimentof reference[124] on Cu(NO3)2, corresponding to a fast kinetics condition. Lower panel: zoom of aportion of the upper panel.

terface at these scales, simulations are required for system sizes that are muchlarger the diffusion length.

6.5 Conclusions

In this chapter we have studied the physics of non-conservedgrowth from a diffu-sive source with the help of a computationally efficient phase-field model.

We have focused on the emergence of a long time structure arising from awell characterized linearly unstable regime. We have constructed a phase-fieldmodel to perform a systematic analysis of the experiments reported in [137]. Ourcomparison not only reproduces qualitatively well the morphologies obtained inthose experiments, but also predicts the existence of a scaling regime characterizedby anomalous scaling. This anomalous scaling deserves further analysis but, pre-liminarily, can be understood in terms of the emergence of large slopes for timesfollowing the appearance of overhangs and multivalued interfaces.

As we have shown, our model sheds some light on the characterization of theexperiments reported in [137; 124] and the role of the different mechanisms in-volved. The quantitative estimation of the parameters appearing in our theoreticalframework leads also to remarkable agreement with the morphological structure of


10-2

10-1

100

101

102

t [s]

10-4

10-3

10-2

10-1

100

101

W(t

)

101

102

103

k [1/mm]

10-4

10-2

100

102

104

106

S(k,

t)

2αs+d=2

2αs+d=1

Figure 6.11 – Numerical time evolution of the surface roughness (left panel) and PSD(right panel) for the same parameters as in figure 6.10. The PSD curves are for increasingtimes from bottom to top. Notice the time shift of the PSD curves for high values ofk(small length scales) signalling anomalous scaling, see the text for details. For reference,the dashed line has slope−2 while the dash-dotted line has slope−1. Note that in case ofintrinsic anomalous scalingαs ≡ αl, see table 2.1.

the observed deposits.Finally, we want to emphasize that, although the formulation of the electro-

chemical deposition process with a phase-field theory has been addressed beforein the literature [243; 244; 246; 245], our model has been notconstructed phe-nomenologically but, rather, as an equivalent formulationof the physical equations(diffusion, electrochemical reduction at the cathode, . . .) that reduces to them inthe thin interface limit.

In addition, our formulation allows us to make a quantitative comparison withexperiments that is far from the capabilities of the aforementioned theories. Notwith-standing, the promising results presented above still needmore detailed compar-ison and analysis (through more extensive numerical experiments), that will bethe aim of future work. This numerical work will focus, specifically, on the ap-pearance of the anomalous scaling regime. Also,from the morphological point ofview, future work will deal with on the influence, cooperation and/or competitionof different physical mechanisms that are naturally included in our theory.

Conclusions

In this thesis we have shown that continuum modelling is an useful tool for theunderstanding of scale invariance, pattern formation and kinetic roughening in thesurface growth processes of many out-of-equilibrium systems. The versatility ofthis theoretical approach allow one to formulate a model by starting from constitu-tive equations or by only including into an effective interfacial equations the termscompatible with the symmetries of the system under study.

From the moving boundary model introduced by M. Castro and R.Cuerno inorder to describe diffusive growth we have obtained severaltheoretical results. Infact, the numerical study of the full linear dispersion relation of the model hasrevealed a strong dependence of the functional form of this linear operator on theattachment kinetic coefficient. We have shown that this dependence has importantconsequences for the pattern formation process and for the asymptotic behavior ofthe model. Moreover, our linear and non-linear theory compares well quantitativelywith data from galvanostatic electrodeposition experiments of de Bruyn. We haveremarked that the actual hydrodynamical behavior of the electrodeposition processis typically beyond the length scales accessible in those experimental set ups, sothat the observed scaling exponents are due to finite size effects.

By means of the dynamical renormalization group analysis and the numericalintegration of the effective interface growth equations obtained from the movingboundary model, we have addressed two interesting theoretical issues. First, wehave analyzed the dynamical behavior of the MSKPZ equation and we have shownhow the interplay between non-locality, non-linearity andfluctuations gives rise

126 Conclusions

to a class of hierarchical morphologies that can be found in nature, the so-calledcauliflower-like structures. These structures display self-similarity in space andtime, being characterized by the relationα = z = 1 between the roughness and thedynamical exponent. Besides, from a more general point of view, our continuumtheory also brings up the question about why natural dynamics favors self-similarstructures.

Subsequently, we have performed a study on a class of non-local equationsthat generalize the MSKPZ equation. This study is not only motivated by a merelymathematical curiosity, in fact, many instances of these equations can be derivedfrom first principles for different physical phenomena, such as in fluid imbibition,combustion, solidification and many others. By means of dynamic renormalizationgroup predictions and numerical estimates of critical exponents we have shownthat the competition between the anomalous “surface diffusion” term and the KPZnon-linearity controls the scaling behavior of these equations. Moreover, regardingthe connection with experiments, the irrelevance of the KPZnon-linear term for awhole parameter range may account for the difficulty to observe KPZ scaling inmany surface growth experiments.

The dynamical renormalization group of the anisotropic Kuramoto-Sivashinskyequation is a further example of the usefulness of this technique for the compre-hension of pattern formation and critical phenomena. In this framework we wereable to explain the90 rotation of ripple pattern observed in numerical simulationsof this equation. Moreover, through the connections between the aKS equation andthe process of ion-beam erosion we have proposed a theoretical explanation for theexperimental findings of Brownet al.

Finally, the formulation of a multigrid phase-field model allowed us to in-tegrate numerically the moving boundary model of diffusivegrowth beyond thesmall slopes approximation. After the formal matching expansion and the calibra-tion of the numerical scheme, we have used our model to study the formation ofthe complex morphologies observed in surface growth by elctrodeposition. Afterthis numerical study we can assert that the anomalous scaling of the surface re-ported in experiments is due to a columnar pattern that develops after the onsetof the non-linear growth regime and of overhang formation. At saturation timesthese interfaces are formed by branches which lead to an almost compact aggre-gate. At lengths scales comparable with the diffusion length the roughness of theseinterfaces doesnot display power law behavior.

7.1 Future work

Following the investigations described in this thesis, we plan to pursue four mainprojects immediately related with this work. An open issue,that can be easilyaddressed with our phase-field model, is the characterization of kinetic roughen-ing phenomena in growth by electrodeposition in case of a system size that is evenlarger (at lest one order of magnitude) than the diffusion length. The second project

7.1 Future work 127

requires a computational improvement of our phase-field model and concerns thepattern formation of three dimensional structures by electrodeposition. A very de-bated topic related with the investigations contained in this thesis is scale invarianceof surface grown on a circular (or spherical) substrate. In fact, to date the relationsbetween universality classes and system geometry are not settled. Finally, in afourth project we plan to study the scaling behavior of growth models for stronglyanisotropic surfaces.

Non-local transforms

The Hilbert transform of a functionf(x) can be defined in several ways

H[f ](x) =1

π−∫ +∞

−∞

f(s)

s− xds =

1

πx⋆ f(x), (A.1)

where the integral is calculated by the Cauchy pricipal value. Clearly, this is anon-local transform, being a convolution betweenf and the long range kernel1/x.Using the definition (A.1) we can show the relation betweenH in real space andthe sign ofk (sgn(k)) in Fourier space. We write the Fourier transform of(πx)−1

using the Cauchy principal value

F(

1

πx

)

=1

π

∫ +∞

−∞

e−ikx

xdx =

1

π

∫ +∞

−∞

[

cos(kx) − i sin(kx)

x

]

dx =

=1

πlim

R→∞

[

limǫ→0

∫ R

ǫ

cos(kx)

xdx+

∫ −ǫ

−R

cos(kx)

xdx− i

∫ +∞

−∞

sin(kx)

xdx

]

=

= −2i

π

∫ ∞

0

sin(kx)

xdx = −2i sgn(k)

π

∫ ∞

0

sin(s)

sds = −i sgn(k). (A.2)

130 Non-local transforms

Using this result, we calculate the inverse transform of theMullins-Sekerka disper-sion relation; writing explicitly the terms associated to|k| and|k|3

F−1 (|k| ζ(k, t)) = F−1 (sgn(k)) ⋆F−1 (k ζ(k, t)) =

=i

πx⋆

(

1

i∂xζ(x, t)

)

= H[∂sζ(s, t)](x), (A.3)

F−1(

|k|3 ζ(k, t))

=i

πx⋆

(

1

i3∂3

xζ(x, t)

)

= −H[∂3sζ(s, t)](x), (A.4)

finally we obtain the expression of the MS instability in realspace. These relationsare valid only ford = 1 and in case ofd > 1 we need to introduce a differenttransform. Thed-dimensional Hilbert transform is defined by

H(d)(f)(r) =1

πd−∫ +∞

−∞

dx′1x1 − x′1

−∫ +∞

−∞

dx′2x2 − x′2

. . .−∫ +∞

−∞

dx′dxd − x′d

f(r′), (A.5)

and it consists ofd applications of the one-dimensional Hilbert transform, eachone associated with a different component of ther coordinate. For example, intwo dimensions the Hilbert transform reads

H(f)(x, y) =Hy

(

Hx(f))

(x, y) =1

π−∫ +∞

−∞

dy′

y − y′Hx(f)(x, y′) =

=1

π2−∫ +∞

−∞

dy′

y − y′−∫ +∞

−∞

dx′

x− x′f(x′, y′).

(A.6)

Unfortunately, for instance the Fourier transform of (A.5)applied to function∂xζis (for d = 2)

F(

H(∂xζ))

(kx, ky) = F[

1

πy⋆

(

1

πx⋆ ∂xζ(x, y)

)]

(kx, ky) =

= isgn(ky)|kx|ζ(kx, ky).

(A.7)

This equation shows that ford > 1 there is not a direct correspondence between theFourier transform ofH and terms of the form|k|2p+1ζ(k) (herep is an integer).

Ford = 2 we can introduce an integral transformG in order to write the terms(|k| − |k|3)ζ(k) as Fourier transforms of the function

−∫ +∞

−∞dr

f(r′)|r − r′| , (A.8)

wheref is a function related toζ. In fact, we use

G(2n)(ζ)(r) =1

2π−∫

R2

dr∇′2nζ(r′)|r − r′| =

1

2π

[

1

r⋆∇2nζ

]

(r), (A.9)

131

wherer = |r|. The Fourier transform ofG(2n) is

F[

G(2n)(ζ)]

(k) =1

2πF(

1

r

)

(

−k2)nζ(k), (A.10)

so that we need the explicit form of the Fourier transform ofr−1 in order to obtainthe final expression. Hence

F(

1

r

)

(k) =

∫

R2

dr

re−ik·r =

∫ +∞

0dr

∫ 2π

0dθ e−i|k|r cos θ =

= 2π

∫ +∞

0dr J0(−|k|r) =

2π

|k| ,(A.11)

in which we have used the integral representation of the Bessel function of firstkind Jn(x)

Jn(x) =1

2πin

∫ 2π

0dθ eix cos θeinθ, (A.12)

and its normalization property

∫ +∞

0dx Jn(x) = 1 ∀n > −1/2. (A.13)

Note thatJ0 is a even function; in fact, it can be represented as the series

J0(x) =∞∑

l=0

(−1)l

l!(m+ l)!

(x

2

)2l. (A.14)

Finally, we obtain

F[

G(2n)(ζ)]

(k) = |k|2n−1ζ(k), (A.15)

that leads to

|k|ζ(k) = F[

1

2π−∫

R2

dr∇′2ζ(r′)|r − r′|

]

(k), (A.16)

|k|3ζ(k) = F[

1

2π−∫

R2

dr∇′4ζ(r′)|r − r′|

]

(k). (A.17)

DRG calculations for non-localinterface equations

For simplicity we consider equation (4.10) withσk = −νkµ−Kk2 as the simplestdispersion relation representative for the class of non-local equations studied insection 4.2. In case of a dispersion relation with extra stabilizing relaxation terms,for example including any contribution that is less relevant (i.e. higher powers)thank2, we will obtain the same results in the hydrodynamic limit [see appendixC for the renormalization of equation (4.1) forN 6= 0 andd = 1].

As sketched in section 2.2.1, the DRG calculation is dividedinto three mainparts: the renormalization of the propagator, of the noise and of the vertex. Finally,we will study the flow of the renormalized parameters in orderto find its fixedpoints and the associated critical exponents.

B.1 Propagator renormalization

In the equation for the slow modes (2.66) the contribution from the elimination ofthe fast modes is contained in the functionΣ that arises from the four diagramsof figure 2.7. We obtain this contribution by solving perturbatively the following

134 DRG calculations for non-local interface equations

integral

Σ(k, ω) = 2λ2Π0

∫ > dq

(2π)d

∫

dΩ

2πk(k − q)q(−q)×

×Go(q)Go(−q)Go(k − q),

(B.1)

where the integral inp is calculated only for the modes within the shell (fastmodes), and the noise is uncorrelated:Π(k) = Π0. Note that, in this expres-sion, we have used a compact notation for the variables in thebare propagatorGo(q) = Go(q,Ω). In order to calculate the terms in (B.1) we symmetrize theintegral with respect to the shell by introducing the standard substitution(q,Ω) →(j + k/2,Ω + ω/2) [78]. We begin the calculation by considering separately thevarious contributions to the wave-vector integral.

The wave-vector contribution due to the vertex is readily obtained in the newsymmetrized variables as

k(q − k)q2 = j3 cos(θ)k +j2

2k2 +O(k3), (B.2)

whereθ is the angle betweenk andj. Next, we expand the bare propagator con-tribution by taking the long time limit (i.e.ω → 0)

limω→0

|Go(q)|2Go(k − q) =

=

(

νjµ(

1 + x cos(θ) +x2

4

)µ/2

+ Kj2(

1 + x cos(θ) +x2

4

)µ/2)2

+ Ω2

−1

×

×[

νjµ(

1 − x cos(θ) +x2

4

)µ/2

+ Kj2(

1 − x cos(θ) +x2

4

)µ/2

+ iΩ

]−1

≃

≃[

1 +

(

1

∆ + iz− 2∆

∆2 + z2

)

C cos(θ)k

j

]

×

×[

(

Kj2)3 (

∆2 + z2)

(∆ + iz)]−1

, (B.3)

where we have introducedz = Ω/Kj2, the small expansion variablex = k/j, andthe functions∆ = 1 + ν/K|j|2−µ andC = 1 + νµ/2K|j|2−µ in order to shortenthe notation. Now we can multiply this last result by (B.2) and split the integral inp into ad− 1 angular integral and a one dimensional radial integral. Thus,

∫ > dq

(2π)d=

Sd−1

(2π)d

∫ >

qd−1dq

∫ π

0sind−2(θ)dθ, (B.4)

where

Sd =2πd/2

Γ(d/2). (B.5)

B.1 Propagator renormalization 135

is the surface area of thed-dimensional unit sphere.We can rewrite (B.1) up to second order

Σ(k, 0) = 2λ2Π0

KSd−1

(2π)d+1

∫ Λ

Λ/bjd−3dj

∫ π

0sind−2(θ)dθ×

×[

j cos(θ)I11k +

(

1

2I11 + C cos2(θ) (I12 − 2∆I21)

)

k2

]

,

(B.6)

where the integrals

Ilm =

∫ +∞

−∞dz(

∆2 + z2)−l

(∆ + iz)−m , (B.7)

are easily calculated through the residue theorem asI11 = π/2∆2, I12 = π/4∆2,I21 = 3π/8∆4. So far, we have considered that the function∆ is positive whenjis equal toΛ. In fact, the dispersion relation fork = Λ can be written as

σΛ = −KΛ2(

1 +ν

KΛµ−2)

= −KΛ2∆Λ, (B.8)

where∆Λ is the function∆ for j = Λ. Considering thatσΛ has to be negative inorder to have a well behaved dispersion relation and that, for the same reason,Khas to be positive, the equation above shows that∆Λ is a positive quantity.

Integrating (B.6) inθ and using the following relations

∫ π

0sind−2(θ) cos(θ)dθ = 0, (B.9)

∫ π

0sind−2(θ) cos2(θ)dθ =

1

d

∫ π

0sind−2(θ)dθ, (B.10)

Kd =Sd−1

(2π)d

∫ π

0sin(θ)d−2dθ, (B.11)

we obtain

Σ(k, 0) =λ2Π0Kd

4dK2

∫ Λ

Λ/bdjjd−3

∆3(d∆ − 2C) k2. (B.12)

Finally, we perform a perturbative expansion up to first order in 1/b = e−δl =1 − δl + O(δl2), that leads to

Σ(k, 0) =λ2Π0Kd

4d

Λd+2−2µ

(KΛ2−µ + ν)3[

(d− 2)KΛ2−µ + (d− µ)ν]

δl k2, (B.13)

up to first order inδl.


B.2 Noise variance renormalization

The contraction of two noise diagrams of figure 2.8 gives

Φ(k, ω) = λ2Π20

∫ > dq

(2π)d

∫

dΩ

2πq2|k − q|2|Go(q)|2|Go(k − q)|2. (B.14)

We proceed as for the propagator using the symmetrized variables

limω→0

|Go(q)|2|Go(k − q)|2 =1

(Kj2)4[

(

∆ + C cos(θ)x+ z2)

×

×(

∆ − C cos(θ)x+ z2)

]−1∼

(

Kj2)−4

(∆2 + z2)2.

(B.15)

Putting this last result into the expansion of (B.14) up to zero order in powers ofk,we obtain

Φ(k, 0) =λ2Π2

0

(2π)d+1

∫ Λ

Λ/bjd−1dj

∫ +∞

−∞dz

j4

(Kj2)3Sd−1

(∆2 + z2)2

∫ π

0sin(θ)d−2dθ =

=λ2Π2

0Kd

4K3

∫ Λ

Λ/b

jd−3

∆3dj ∼ λ2Π2

0

KdΛd+4−3µ

4 (KΛ2−µ + ν)3δl. (B.16)

B.3 Vertex Renormalization

In this section we show that the vertex originated by the KPZ non-linearity does notrenormalize in the one-loop DRG scheme. In order to obtain this result we considera generic polynomial dispersion relationσk and we retain the perturbative expan-sion up to zero order ink. We show that the one-loop perturbative expansion leadsto two different classes of diagrams and their sumalwaysgives a null contribution.

Let us start withΓa (see figure 2.9)

Γa(k1, ω) = λ3Π0

∫ >dq

(2π)d

∫

dΩ

2πq(k1 − q)

(

q − k2 −k1

2

)(

k2 +k1

2− q

)

×

×Go(q)Go(k1 − q)Go(q − k2 − k1/2)Go(k2 + k1/2 − q). (B.17)

Using the change of variablej = q − k1/2, we get

q(k1 − q)

(

q − k2 −k1

2

)(

k2 +k1

2− q

)

∼ j4 +O(k1) +O(k2). (B.18)

The polynomial dispersion relation can be written as

σk = −n∑

i=1

cikri , (B.19)

B.3 Vertex Renormalization 137

where the setri is formed by a collection of positive real exponents that increasewith the index,r1 < r2 < · · · < rn, and coefficientsci are generic. As statedbefore, we stress thatcn andσΛ must be negative in order to have a meaningfulcontinuum behavior.

The contribution due to the bare propagator is a little involved, but we haveto retain only the zero order contribution of the perturbative expansion, whichstrongly simplifies the calculation. Thus,

limω→0

Go

(

j +k1

2

)

Go

(

k1

2− j

)

Go

(

j − k2

)

Go

(

k2 − j)

=

=

(

n∑

i=1

cijri

)2

+O(k1) + Ω2

−1

×

(

n∑

i=1

cijri

)2

+O(k2) + Ω2

−1

=

= (cnjrn)−4

(

∆2 + z2)−2

, (B.20)

where

z =Ω

cnjrn, ∆ = 1 +

n−1∑

i=1

cicnjri−rn = − σj

cnjrn. (B.21)

As a generalization of (B.8) function∆ evaluated atj = Λ takes a positive value∆Λ due to the signs ofσΛ andcn. Hence, after the integration inz, this contributionto vertex renormalization reduces to

Γa(k1, 0) =2λ3Π

(2π)d+1

∫ > dj

(cnjrn)3π

2∆3. (B.22)

We stop the calculation at this point because the remaining diagrams cancel thecontribution arising fromΓa. Indeed, as seen in figure 2.9, the other diagram in thevertex renormalization is

Γb(k1, ω) = 4λ3Π0

∫ > dq

(2π)d

∫

dΩ

2πq(k1 − q)(q − k1)

(

q − k2 −k1

2

)

×

×Go (q)Go

(

k1 − q)

Go

(

q − k1

)

Go

(

q − k2 − k1/2)

, (B.23)

where

q(k1 − q)(q − k1)

(

q − k2 −k1

2

)

∼ −j4 +O(k1) +O(k2), (B.24)

and

limω→0

Go

(

j +k1

2

)

Go

(

k1

2− j

)

Go

(

j − k1

2

)

Go

(

j − k2

)

∼

∼ 1

(cnjrn)4

[

(∆2 + z2)(∆ − iz)2]−1

.

(B.25)


The integral inz is exactlyI12 and is equal toπ/4∆3Λ, so that

Γb(k1, 0) = − 4λ3Π0

(2π)d+1

∫ > dj

(cnjrn)3π

4∆3. (B.26)

TheΓb contribution cancels exactlyΓa and the non-linear termλ does not renor-malize at one-loop order for any polynomial dispersion relation of the form (B.19).

B.4 Flow equations

The renormalized parameters are obtained from equation (B.13) and equation (B.16)

ν< = ν, λ< = λ,

K< = K

1 − λ2Π0Kd

4d

Λd+2−2µ

(KΛ2−µ + ν)3[

(d− 2)KΛ2−µ + (d− µ)ν]

δl

,

Π<0 = Π0

1 + λ2Π0KdΛ

d+4−3µ

4 (KΛ2−µ + ν)3δl

. (B.27)

Now we are ready to perform the rescaling according to the usual DRG trans-formation. In Fourier space the momentum and the frequency are rescaled inthe following mannerk = bk and ω = bzω. The rescaled height functionis h(k, ω) = h<(k, ω)/bα, and the renormalized parameters areν = bz−µν<,K = bz−2K<, Π0 = bz−d−2αΠ<

0 ; as an example, we show the procedure tocalculate the flow equation forν in the limit of vanishingδl

dν

dl= lim

δl→0

ν − ν

δl= lim

δl→0

[(z − µ)δl + O(δl2)]ν

δl= [z − µ] ν. (B.28)

Finally the parameter flow reads

dν

dl= ν [z − µ] ,

dλ

dl= λ [α+ z − 2] ,

dKdl

= K

z − 2 − λ2Π0Kd

4d

Λd+2−2µ

(KΛ2−µ + ν)3[

(d− 2)KΛ2−µ + (d− µ)ν]

,

dΠ0

dl= Π0

z − 2α− d+ λ2Π0KdΛ

d+4−3µ

4 (KΛ2−µ + ν)3

. (B.29)

The set of equations (4.11)-(4.13) in the main text follows from the previous oneby imposing an unit wave-vector cut-off, i.e.Λ = 1.

Now we are ready to study the flow equations (B.29) using the couplings

g =λ2Π0Kd

4(K + ν)3, f =

KK + ν

. (B.30)

B.4 Flow equations 139

The ratio between the (possibly) unstable parameter,ν, and the stable one,K,follows from the exact relation1 − f = ν/(ν + K). The condition∆Λ ≥ 0 issatisfied as long these two couplings are positive (or at least equal to zero); for thisreason we can restrict the DRG flow analysis to first quadrant of the (f, g) plane.The flow equations for the couplings are easily obtained fromequations (B.29)after the substitutions shown in equations (B.30)

f =df

dl= (1 − f)

(µ− 2)f − g

d[(d− 2)f + (d− µ)(1 − f)]

, (B.31)

g =dg

dl= g

6f − 4 − d+ g + 3µ(1 − f) +

+3g

d

[

(d− 2)f + (d− µ)(1 − f)]

. (B.32)

These are equations (4.15), (4.16) of the main text. The values (f, g) fulfillingequationsf = 0 and g = 0 simultaneously are the fixed points of the flow. It iseasy to check that the pointsf = 1 andg = 0 satisfies the first and the secondequations, respectively. The two other conditions are morecomplicated and can beobtained with a little algebra

f =g(d− µ)

(µ− 2)(d − g), g =

3µ(f − 1) + 4 + d− 6f

3 [(d− 2)f + (d− µ)(1 − f)] /d+ 1. (B.33)

If we combine these two equations we obtain the Galilean fixedpoint with coordi-nates(f∗, g∗) as

f∗ =g∗(d− µ)

(µ− 2)(d − g∗), g∗ = d+ 4 − 3µ. (B.34)

Besides, this fixed point has to satisfy the conditionsf∗ ≥ 0 andg∗ ≥ 0, hence it isnot defined for some values ofµ in the(0, 2] interval. The conditiong∗ > 0 leads toan admissible value for this coupling in case ofµ < (d+4)/3. The other condition,i.e. f∗ > 0, has to be analyzed carefully. In fact, ford < 4/3 the Galilean fixedpoint is present forµ within the intervals

(

0, d]⋃(

4/3, (d + 4)/3]

(for d = 1 seethe left panel of figure 4.11). In the case of4/3 ≤ d < 2 the admissibility intervalsare(

0, 4/3)⋃[

d, (d + 4)/3]

. Finally, for d ≥ 2 we have only one condition forthe existence of the Galilean fixed pointµ < 4/3, as in the right panel of figure4.11.

In table (B.1) we summarize all the fixed points that exist ford = 1. Letus now examine how the critical exponents change due to the different values off andg. Whenf is equal to one, evidently theν parameter is zero, and (4.10)reduces to the KPZ equation. Moreover, a value ofg equal to zero means that thenon-linear termλ is vanishing and (4.10) becomes a linear equation. In this case,the equation is variational and the hyperscaling relation is fulfilled (hyperscaling,i.e. z − 2α− d = 0, is related to non-renormalization of the noise amplitude [79]).On the other hand, wheng is different to zero, the non-renormalization of the non-linear term leads to Galilean invariance, i.e.z + α− 2 = 0.


f g z α β Relation Name

1 0 2 1/2 1/4 2α + d = z EW

1 1/2 3/2 1/2 1/3 α+ z = 2 KPZ

0 0 µµ− 1

2

µ− 1

2µ2α + d = z S

(5 − 3µ)(1 − µ)

(µ− 2)(3µ − 4)5 − 3µ µ 2 − µ

2 − µ

µα+ z = 2 G

Table B.1 – Fixed points in1 + 1 dimensions (d = 1). In the last column G stands forGalilean and S for Smooth, according to the labels defined in the main text. The Galileanfixed point is not defined in the intervals1 < µ ≤ 4/3 and5/3 < µ ≤ 2.

In order to study the linear stability of these fixed points wehave to computede derivative of (B.31)-(B.32) with respect to the couplings

∂

∂f

df

dl= (1 − 2f)(µ− 2) +

g

d[(1 − f)(d− 2µ+ 2) + (d− 2)f ] , (B.35)

∂

∂g

df

dl=f − 1

d[(d− 2)f + (d− µ)(1 − f)] , (B.36)

∂

∂f

dg

dl= 3

g

d(µ− 2)(g − d), (B.37)

∂

∂g

dg

dl= 6f + 8g − d− 4 − 1

d[12fg + 3µ(d− 2g)(f − 1)] . (B.38)

Ford = 1 the linear stability matrices

S =

(

∂f/∂f ∂f/∂g∂g/∂f ∂g/∂g

)

, (B.39)

of the fixed points are

S(1, 0) =

(

2 − µ 0

0 1

)

, S(1, 1/2) =

(

3/2 − µ 0

34(2 − µ) −1

)

,

S(0, 0) =

(

µ− 2 µ− 1

0 3µ− 5

)

, S(f∗, g∗) =

(

3 − 2µ P1(µ)

P2(µ) 5−3µ4−3µ

)

,

(B.40)

wheref∗ andg∗ are the values of couplings at the Galilean fixed point and

P1(µ) =(1 − µ)(2µ− 3)

(µ− 2)(3µ − 4)2, P2(µ) = 3(3µ− 4)(3µ − 5)(µ − 2). (B.41)


0 0.5 1 1.5 2µ

-4

-2

0

2

Re

(vi)

1.4 1.5 1.6µ

-2

-1

0

Re

(vi)

v1

v2

unstable

stable saddle

Figure B.1 – Real part of the eigenvalues of the linear stability matrix of the Galileanfixed point ford = 1. In the inset we show a zoom of the region4/3 < µ < 5/3 withinthe fixed point changes from stable to saddle behavior forµ equal to3/2.

The two eigenvalues of the Galilean fixed point have been obtained numerically;we show in figure B.4 how they change as functions ofµ. In figure B.2 we showeight representative snapshot of the flow evolution ford = 1. This figure adds fourµ cases with respect to figure 4.13, and, for further information about the fixedpoint evolution see section 4.2.

The situation changes drastically ford = 2; as expected [27] the KPZ fixedpoint disappears (it is at infinity). This being a well known limitation of the presentperturbative DRG approach (common even to field-theoretical approaches [248]).In table (B.2) we summarize the results for the fixed points inthis case. The stabil-

f g z α β Relation Name

1 0 2 0 0 2α+ d = z EW

0 0 µµ− 2

2

µ− 2

2µ2α+ d = z S

3(µ− 2)

3µ− 43(2 − µ) µ 2 − µ

2 − µ

µα+ z = 2 G

Table B.2 – Fixed points in2 + 1 growth dimensions (d = 2). In the last column Gstands for Galilean and S for Smooth, according to the labelsdefined in the main text. TheGalilean fixed point is not defined forµ ≥ 4/3.


ity matricies of these fixed points are

S(1, 0) =

(

2 − µ 0

0 0

)

, S(0, 0) =

(

µ− 2 µ−22

0 3(µ− 2)

)

,

S(f∗, g∗) =

µ− 2 2(µ−2)(3µ−4)2

92(µ− 2)2(3µ− 4) 6(2−µ)

3µ−4

.

(B.42)

In figure B.3 we show the eigenvalues of the Galilean fixed point as function ofµ.In figure B.4 we represent the flow evolution increasing the value of µ. This

figure complete figure 4.14 in main text with two additional snapshots. For thefurther informations about the fixed point evolution see section 4.2.


0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4

0

1

2

3

4

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

1

(a) (b)

(c) (d)

(e) (f)

(g) (h)

gg

gg

ff

S

K

E

G

SK

E

G

SK

E

G

S

K

E S

K

E

G

S

K

E

G

S

K

E

G

S

K

E

Figure B.2 – Parameter flow ford = 1 and different values ofµ. In (a)µ = 1/4, (b)µ = 1/2, (c)µ = 1, (d)µ = 1.15, (e)µ = 1.4, (f) µ = 3/2, (g)µ = 1.6, (h)µ = 1.8.


0 0.5 1 1.5 2µ

0

0.5

1

1.5

2

v 1-50

-40

-30

-20

-10

0

v 2

v1

v2

Figure B.3 – Real part of the eigenvalues of the linear stability matrixS(f∗, g∗) for theGalilean fixed point ford = 2 as function ofµ. Note the different scales in the left (forv1)and the right (forv2) vertical axes of the graph.

0 1 2 3 4 5

0

1

2

3

4

5

0 2 4 60

2

4

6

0 5 10 15 20 25

0

5

10

15

20

25

0 1 2 30

1

2

3

f

g

f

g

G

S E

GS E S E

G

S E

(a) (b)

(c) (d)

Figure B.4 – Parameter flow ford = 2 and different values ofµ. In (a)µ = 1/2, (b)µ = 1, (c)µ = 1.3, (d)µ = 3/2.

Irrelevant relaxation terms

In this appendix we present the DRG calculations for equation (4.1) with the com-plete dispersion relation for 1+1 growth dimensions. We will see that the linearterms parametrized byN 6= 0, see equation (4.10), and exponentn larger than2are irrelevant and do not change the hydrodynamic scaling behavior of this equa-tion. This result can be generalized straightforwardly to the class of non-localequations considered in the previous section, although forthe sake of concretenesswe are going to show the detailed DRG calculations for the case µ = 1. Thisresult is not surprising as those terms are linear and simpledimensional analysisconfirms this result. However, we have included for completeness and to show howthe relevant fixed points are already contained in the surface tension term (the oneproportional tok2).

C.1 DRG calculations

To begin with, we consider a linear dispersion relation of the form

σk = −C1|k| − C2k2 − C3|k|3 − C4k

4, (C.1)

so that in the propagator renormalization we have to retain up to fourth order. Laterwe will consider as particular cases the choicesC3 = 0 andC4 = 0. We proceed asin appendix B.1, using the standard symmetrization considering the various con-tributions to the integrand one to one. The first contribution is readily obtained

146 Irrelevant relaxation terms

as

k(q − k)q2 = k

(

j +k

2

)(

j2 − k2

4

)

= j3k +j2

2k2 − j

4k3 − 1

8k4. (C.2)

The product of the first two terms in the bare propagator part of equation (B.1) isevaluated considering that|k| ≪ |j|

|j + k/2| = |j| + s(j)k/2, (C.3)

|j + k/2|3 = |j|3 + 3j2s(j)k/2 + 3|j|k2/4 + s(j)k3/8, (C.4)

wheres(j) is the sign ofj. By employing these identities we are ready to expandin (B.1)

limω→0

|Go(q)|2 =

∣

∣

∣

∣

Go

(

j +k

2, 0

)∣

∣

∣

∣

2

=

=1

(C4j4)2

[

(

∆ + c1x+ c2x2 + c3x

3 +O(x4))2

+ z2]−1

∼

∼ 1

(C4j4)2

1

∆2 + z2

1 − 2c1∆x

∆2 + z2+

1

∆2 + z2

(

(2c1∆)2

∆2 + z2− 2c2∆ − c21

)

x2 +

+1

∆2 + z2

(

4c1∆

∆2 + z2

(

2c2∆ + c21)

− 2(c3∆ + c1c2) −(2c1∆)3

(∆2 + z2)2

)

x3

,(C.5)

where

∆ =1 +C3

C4|j|+

C2

C4j2+

C1

C4|j|3,

c1 =2 +3

2

C3

C4|j|+

C2

C4j2+

C1

2C4|j|3,

c2 =3

2+

3

4

C3

C4|j|+

C2

4C4j2,

c3 =1

2+

C3

8C4|j|,

(C.6)

and we have introduced two new variables: the non-dimensional expansion pa-rameterx = k/j and the rescaled frequency variablez = Ω/(C4j

4). The same

C.1 DRG calculations 147

procedure applied to the remaining factor in the integrand of (B.1) leads to

limω→0

Go(k − q) =1

C4j4[

∆ − c1x+ c2x2 − c3x

3 +O(x4) + iz]−1 ∼

∼ 1

C4j41

∆ + iz

1 +c1

∆ + izx+

1

∆ + iz

(

c21∆ + iz

− c2

)

x2+

+1

∆ + iz

(

c3 −2c1c2

∆ + iz+

c31(∆ + iz)2

)

x3

.

(C.7)

After multiplication of these two expansions we can integrate out the rescaled fre-quency, i.e. after the change of variableΩ → z = Ω/(C4j

4). We have to take intoaccount that the frequency integral in the new variable is

∫ +∞

−∞dΩ → C4j

4

∫ +∞

−∞dz. (C.8)

We can use the compact notation introduced before for the integrals involved in thebare propagator expansion [see equation (B.7)] and with a little algebra we obtainthe contribution to equation (B.1)∫ +∞

−∞dΩ

∣

∣

∣

∣

∣

Go

(

j +k

2

)∣

∣

∣

∣

∣

2

Go

(

k

2− j

)

∼ 1

(C4j4)2

I11 + (I12 − 2∆I21) c1x+

+

[

c21I13 − c2I12 − 2c21∆I22 + (2c1∆)2 I31 − 2c2∆I21 − c21I21

]

x2 +

+

[

c3I12 − 2c1c2I13 + c31I14 − 2c1∆(

c21I23 − c2I22)

+ (C.9)

+(

(2c1∆)2 I32 − 2c2∆I22 − c21I22

)

c1 +

+4c1∆(

2c2∆ + c21)

I31 − 2 (c3∆ + c1c2) I21 − (2c1∆)3 I41

]

x3

.

TheIlm integrals are easily calculated using the residue theorem,giving the con-tributions listed below

I11 =π

2∆2, I12 =

π

4∆3, I13 =

π

8∆4, I14 =

π

16∆5,

I21 =3π

8∆4, I22 =

π

4∆5, I23 =

5π

32∆6,

I31 =5π

16∆6, I32 =

15π

64∆7,

I41 =35π

128∆8.

(C.10)


Using these relation and multiplying all contributions (C.2), (C.9) together we canwrite Σ as sum of four functions

Σ(k, 0) =λ2Π0

4π

∫ >

dj[

f1(j)k + f2(j)k2 + f3(j)k

3 + f4(j)k4]

, (C.11)

where

f1(j) =1

(C4∆)2 j5,

f2(j) =1

(C4∆j3)2

(

1

2− c1

∆

)

,

(C.12)f3(j) =

1

(C4∆)2 j7

[

1

∆

(

c21∆

− 2c2

)

− c12∆

− 1

4

]

,

f4(j) =1

(C4∆j4)2

[

1

2∆

(

c21∆

− 2c2

)

+c14∆

− 1

8− 1

∆

(

c3 −3

∆c1c2 +

c31∆2

)]

.

Considering the parity of the functions (C.6) we can simplify expression (C.11).In fact, ci and∆ in (C.12) are even inj: for this reason the functionsf1, f3 areodd whereasf2 andf4 are also even. The momentum shell is a symmetric intervalaroundj equal to zero, hence the integrals off1(j) andf3(j) vanish.

Another important point to address is the contribution of the shell to the prop-agator renormalization. This calculation is only requiredwhenk4 terms (or higherorder ones) are present because the lowest shell contribution is, precisely, at thatorder. Writing explicitly the limits of integration in the positive part of the momen-tum integral and expanding for smallk [249]

∫ >

+f(j) dj =

1

2

[

∫ Λ−k/2

Λ/b+k/2f(j)dj +

∫ Λ+k/2

Λ/b−k/2f(j)dj

]

∼∫ Λ

Λ/bf(j)dj +

+k

4

[

−f(

Λ − k

2

)

− f

(

Λ

b+k

2

)

+ f

(

Λ +k

2

)

+ f

(

Λ

b− k

2

)]

∣

∣

∣

∣

∣

k=0

+ (C.13)

+k2

16

[

f ′(

Λ − k

2

)

− f ′(

Λ

b+k

2

)

+ f ′(

Λ +k

2

)

− f ′(

Λ

b− k

2

)]

∣

∣

∣

∣

∣

k=0

+ . . . ,

wheref is any function ofj (the symbol+ in the integral means that integrationinterval is restricted to positivej). With this simple calculation we see that everyfunction in equation (C.11) contributes at ordern and at every ordern+2iwith i apositive integer. In our calculation we retain only the contribution of f2 to surface

C.1 DRG calculations 149

diffusion, i.e. the term proportional tok4

∫ >

+f2(j) dj ∼ F2(j)

∣

∣

∣

Λ

Λ/b+k2

8

[

f ′2(Λ) − f ′2(Λ/b)]

∼

∼[

f2(Λ) +k2

8f ′′2 (Λ)

]

Λδl.

(C.14)

Here, the functionF2 is the primitive off2. We now introduce three new couplingvariables that help us to write (C.11) in a more compact way

A(j) =C1

C4|j|3, F (j) =

C2

C4j2, B(j) =

C3

C4|j|, (C.15)

and we can expressf2, its second derivative andf4 using these variables

f2(j) = − 1

2 (C4j3)2

3 + 2B + F

(1 +A+B + F )3

f4(j) =1

8 (C4j4)4

[

11 + 40A + 27B + 29F + 2A2 + 19B2 + 11F 2 + 47AB +

+21AF + 35BF + (2B + F )(

A2 + 2B2 + F 2 + 7AB + 3AF + 4BF) ]

×

× (1 +A+B + F )−5 , (C.16)

f ′′2 (j) = − 1

(C4j4)4

[

63 − 90A + 119B + 27F + 9A2 + 82B2 + 15F 2 −

−95AB − 27AF + 55BF + 3F 3 +A(

2AB − 3F 2)

+

+B(

20B2 + 14F 2 − 26AB − 17AF + 25BF) ]

× (1 +A+B + F )−5 .

Considering that the shell is composed of positive and negative wave-vectors wehave to count twice the previous contribution

Σ(k, 0) =λ2Π0

4π

[

2f2(Λ)k2 +

(

2f4(Λ) +1

4f ′′2 (Λ)

)

k4

]

Λδl =

=(

Σ2k2 + Σ4k

4)

δl,

(C.17)

whereΣ2 andΣ4 are

Σ2 = −λ2Π0

4π

(3 + 2B + F )

C24Λ5 (1 +A+B + F )3

, (C.18)

Σ4 = − λ2Π0

16πC24Λ7

[

52 − 130A + 92B − 2F + 7A2 + 63B2 + 4F 2 +

+16B3 + 2F 3 − 142AB − 48AF + 20BF − 30ABF −

−40AB2 − 6AF 2 + 8BF 2 −A2F + 15B2F]

×

× (1 +A+B + F )−5 . (C.19)


In our DRG program, the next step is calculating the contributions arising fromthe renormalization of the noise variance. We start by considering equation (B.14)for d = 1. On the other hand,

[q(k − q)]2 =

(

k2

4− j2

)2

= j4 − j2

2k2 +

1

16k4, (C.20)

while the first factor due to the bare propagator has been already obtained, seeequation (C.5). The other contribution is exactly the square of the absolute valueof equation (C.13)

limω→0

∣

∣

∣

∣

∣

Go

(

k

2− j

)∣

∣

∣

∣

∣

2

∼ 1

(C4j4)2

1

∆2 + z2

1 +2c1∆

∆2 + z2x+

+1

∆2 + z2

(

(2c1∆)2

∆2 + z2− 2c2∆ − c21

)

x2 + (C.21)

+1

∆2 + z2

(

2(c3∆ + c1c2) +(2c1∆)3

(∆2 + z2)2− 4c1∆

∆2 + z2

(

2c2∆ + c21)

)

x3

.

The renormalization of the non-conserved noise involves only the zero order termof this expansion, hence

Φ(k, 0) = λ2Π20

∫ > dj

2π

∫

dz

2πj4

1

(C4j4)3

1

(∆2 + z2)2. (C.22)

Using the residue theorem we calculate the frequency integral∫ +∞

−∞dz

1

(∆2 + z2)2=

π

2∆3, (C.23)

and, due to the parity of functions in (C.22), we can finally write the noise variancerenormalization in the following form

Φ(k, 0) =λ2Π2

0

4πC34

∫ Λ

Λ/b

dj

j8∆3=λ2Π2

0

4πC34

δl

Λ7∆3. (C.24)

As seen in section B.3 the vertex renormalization for the KPZnonlinearity isequal to zero for any dispersion relation that can be writtenas a polynomial ink,at least for the one-loop expansion. Now we are ready to applythe usual rescalingand to calculate the renormalized parameters

C<1 = C1, C<

3 = C3, (C.25)

C<2 = C2

[

1 + λ2Π0P2(A,B,F )

4πC2C24Λ5

δl

]

, (C.26)

C<4 = C4

[

1 + λ2Π0P4(A,B,F )

16πC34Λ7

δl

]

, (C.27)

Π<0 = Π0

[

1 +λ2Π0

4πC34

(1 +A+B + F )−3 δl

Λ7

]

, (C.28)

C.2 Irrelevance of the k3 term 151

where the two functionsP2 andP4 are

P2 =3 + 2B + F

(1 +A+B + F )3, (C.29)

P4 =[

52 − 130A + 92B − 2F + 7A2 + 63B2 + 4F 2 +

+16B3 + 2F 3 − 142AB − 48AF + 20BF −

−30ABF − 40AB2 − 6AF 2 + 8BF 2 −A2F + 15B2F]

×

× (1 +A+B + F )−5 . (C.30)

In the next two sections we want to show that for a dispersion relation like inequation (4.1) forn = 2, the extra relaxation terms, withn > 2, do not change thehydrodynamical properties of the system as obtained in appendix B.

C.2 Irrelevance of the k3 term

As a first case we are going to consider the dispersion relation is a third orderpolynomial in|k|, σk = −C1|k| − C2k

2 − C3|k|3. The ensuing equation (4.1) isan interpolation of the SMS and MSKPZ equations. Using the results from sectionC.1 we can write the RG flow equations

dC1

dl= C1 [z − 1] ,

dC3

dl= C3 [z − 3] , (C.31)

dC2

dl= C2

[

z − 2 +λ2Π0Λ

2

4πC2

(

2C3Λ + C2

(C1 + C2Λ + C3Λ2)3

)]

, (C.32)

dλ

dl= λ [α+ z − 2] , (C.33)

dΠ0

dl= Π0

[

z − 2α− 1 +λ2Π0Λ

2

4π

(

C1 + C2Λ + C3Λ2)−3]

. (C.34)

Introducing the coupling variables

a3 =C1

C3Λ2, f3 =

C2

C3Λ, g3 =

λ2Π0

4πC33Λ4

, (C.35)

the resulting flow reads

da3

dl= 2a3, (C.36)

df3

dl= f3

[

1 +g3(2 + f3)

f3 (1 + a3 + f3)3

]

, (C.37)

dg3dl

= g3

[

4 +g3

(1 + a3 + f3)3

]

. (C.38)


In this case the condition∆Λ ≥ 0 that we have used in order to calculate the inte-gralsIlm requires that the sum1+a3 +f3 be positive. In fact, when the dispersionrelation is a polynomial of third order, the coefficientC3 has to be positive and

σΛ = −C3Λ3(1 + a3 + f3), (C.39)

has to be negative for a reasonable short distance behavior,hence the positive signfor the parenthesis in (C.39). For this reason we do not consider fixed points ofequations (C.36)-(C.38) that do not satisfy this restriction.

a3 f3 g3 z α β Name

0 0 0 3 1 1/3 Fp3,0

Table C.1 – Fixed point of (C.36)-(C.38) in the(a3, f3, g3) plane.

The only admissible fixed point in the(a3, f3, g3) parameter space is reportedin table C.1, having as stability matrix

S(Fp3,0) =

2 0 0

0 1 2

0 0 4

. (C.40)

This fixed point is associated with the linear limit of (4.1) in whichC1 = C2 =λ = 0. Evidently all the eigenvalues of this matrix are positive hence this fixedpoint is linearly unstable in every direction.

A second set of couplings that can be considered is

a2 =C1

C2Λ, f2 =

C3Λ

C2, g2 =

λ2Π0

4πC32Λ

, (C.41)

their flow reading

da2

dl= a2

[

1 − g2(2f2 + 1)

(1 + a2 + f2)3

]

, (C.42)

df2

dl= −f2

[

1 +g2(2f2 + 1)

(1 + a2 + f2)3

]

, (C.43)

dg2dl

= g2

[

1 − 2g2(3f2 + 1)

(1 + a2 + f2)3

]

. (C.44)

The corresponding fixed points and stability matrices are listed in table C.2 andequation (C.45), whereby they are seen to correspond to the EW (C1 = C3 =λ = 0) and KPZ (C1 = C3 = 0) fixed points. Note they have stable and unstabledirections.



0 0 0 2 1/2 1/4 EW

0 0 1/2 3/2 1/2 1/3 KPZ


S(EW) =

1 0 0

0 −1 0

0 0 1

, S(KPZ) =

1/2 0 0

0 −3/2 0

3/2 0 −1

, (C.45)

A third convenient set of couplings is

a1 =C3Λ

2

C1, f1 =

C2Λ

C1, g1 =

λ2Π0Λ2

4πC31

, (C.46)

their flow equations reading

da1

dl= −2a1, (C.47)

df1

dl=

g1(2a1 + f1)

(1 + a1 + f1)3 − f1, (C.48)

dg1dl

= g1

[

g1

(1 + a1 + f1)3 − 2

]

. (C.49)

In this parameter space: the Smooth and the Galilean fixed points that we foundin the previous section, arise naturally, with features as in table C.3 and equation(C.50).


0 0 0 1 0 0 Smooth

0 0 2 1 1 1 Galilean


S(0, 0, 0) =

−2 0 0

0 −1 0

0 0 −2

, S(0, 0, 2) =

−2 0 0

4 1 0

12 12 2

. (C.50)


Note, the Smooth fixed point is completely stable in every direction.By substituting the exponentsα andz calculated for every fixed point we can

show the irrelevance ofC3 compared toC2. After the rescaling procedure, equation(4.10) reads

∂thk =(

−C1bz−1|k| − C2b

z−2k2 − C3bz−3|k|3

)

hk+

+λ

2bα+z−2F

[

(∇h)2]

+ b(z−1−2α)/2ηk,(C.51)

and we consider now the case ofC2, C3 > 0, i.e. when surface tension and|k|3are two relaxation terms. Clearly, from linear analysis theratio between these twoterms isC2b/C3, and, after every rescaling, the surface tension term becomes morerelevant than the|k|3 term. Moreover, in the EW fixed point the linear stability ma-trix (C.45) shows that this fixed point is stable along the directionf2 = C3Λ/C2.The same situation occurs for the KPZ fixed point (C1 = 0), in fact the stabilitymatrix (C.45) has two stable directionsf2 andg2. For the Galilean fixed point theonly stable direction isa1 = C3Λ

2/C1 while for the Smooth fixed point everydirection is stable. The linear stability of these fixed points shows that the Fp3,0

can be reached only ifC1 = C2 = λ = 0 and is unstable in every direction, see(C.40), hence we can conclude that the|k|3 term is irrelevant compared tok2 (forC2, C3 > 0). We need this term only in case of a negative surface tension, i.e.C2 < 0, in order to pose a well defined dispersion relation in the limit k → ∞.

C.3 Irrelevance of the k4 term

In the last part of this section we want to show that the parameterC4 is irrelevantwhen the dispersion relation contains a relaxation term with an exponent smallerthat four. Let us consider the dispersion relation of this form σk = −C1|k| −C2k

2 − C4k4: in this case expressions (C.18) and (C.19) reduce to

Σ2 =λ2Π0

4πC24Λ5

[

3 + F

(1 +A+ F )3

]

, (C.52)

Σ4 =λ2Π0

16πC24Λ7

[

52 − 130A + 4F 2 − 2F + 2F 3 − 48AF +

+7A2 − 6AF 2 − FA2]

× (1 +A+ F )−5 . (C.53)

As before, we consider a first set of couplings

a4 =C1

C4Λ3, f4 =

C2

C4Λ2, g4 =

λ2Π0

4πC34Λ7

, (C.54)


for which the renormalization flow is

da4

dl= a4

[

3 − g44

[

52 − 130a4 + 4f24 − 2f4 + 2f3

4 − 48a4f2 + 7a24 −

−6a4f24 − f4a

24

]

× (1 + a4 + f4)−5

]

, (C.55)

df4

dl= 2f4 +

g44

[

12 − 24f4 + 22f24 − 2f4

4 + 24a4 + 12a24 + 56a4f

24 +

+162a4f4 − 3f4a24 + a2

4f24 + 6a4f

34

]

× (1 + a4 + f4)−5 , (C.56)

dg4dl

= g4

[

7 +g4

(1 + a4 + f4)3

(

1 − 3

4

[

52 − 130a4 + 4f24 − 2f4 + 2f3

4 −

−48a4f4 + 7a24 − 6a4f

24 − f4a

24

]

× (1 + a4 + f4)−2

)]

. (C.57)

The different fixed points in the(a4, f4, g4) parameter space are given in table


0 0 0 4 3/2 3/8 Fp4,0

0 13.1868 1064.4 1.54 0.46 0.299 KPZ

7.17 −6 20.4 1 1 1 Fp4,1

Table C.4 – Fixed points of (C.55)-(C.57) in the(a4, f4, g4) plane.

C.4, where the coordinates of the KPZ fixed point has been determined solving theequationx4 − 8x3 − 63x2 − 68x − 42 = 0 and using the only admissible realsolution of this equation for component of vector(0, x0, P (x0)), whereP (x) =(686+ 969x+ 846x2 − 25x3)/97. Likewise, the other fixed point, Fp4,1, has beenobtained solving7y2 +2y− 374 = 0 and using the only admissible solutiony0 forthe components of vector(y0,−6, Q(y0)), whereQ(y) = (13014y − 92286)/49.The Fp4,0 fixed point is associated to the lineal MBE equation (C1 = C2 = λ =0) while Fp4,1 has the same exponents of the Galilean fixed point. The stabilitymatrices of these points are dense, so we report only their eigenvalues in table C.5.All these points are unstable or at least hyperbolic in the three dimensional spacebut if we consider the planeC1 = 0, the KPZ fixed points is stable in this subspace.

An informative set of coupling parameters is

a1 =C4Λ

3

C1, f1 =

C2Λ

C1, g1 =

λ2Π0Λ2

4πC31

, (C.58)


Name λ1 λ2 λ3 Stability

Fp4,0 3 2 7 Unstable

KPZ 0.54 −2.90 −1.06 Saddle

Fp4,1 88.4 7.33 −0.24 Saddle

Table C.5 – Eigenvalues of the fixed points of table C.4.

whose flow is

da1

dl= −3a1 +

g14

[

52a31 − 2f1a

21 + 4f2

1a1 + 2f31 − 130a2

1 − 48a1f1 +

+7a1 − 6f21 − f1

]

/ (1 + a1 + f1)5 , (C.59)

df1

dl= −f1 +

g1 (3a1 + f1)

(1 + a1 + f1)3 , (C.60)

dg1dl

= g1

[

g1

(1 + a1 + f1)3 − 2

]

. (C.61)

In this parameter space the fixed points and their exponents are in table C.6, and


0 0 0 1 0 0 Smooth

0 0 2 1 1 1 Galilean

0.1395 −0.8371 0.05534 1 1 1 Fp4,2

Table C.6 – Fixed points of (C.59)-(C.61) in the(a1, f1, g1) plane.

the associated eigenvalues in table C.7. The Fp1,1 fixed point is obtained bysolving equation374x2 − 2x − 7 = 0 and using the only admissible solutionx = (1 +

√291)/374 for the components of vector(x0,−6x0, L(x0)), where

L(x) = (167238 − 1184895x)/34969.As for the|k|3 term, we can show the irrelevance of the surface diffusion, i.e.

thek4 term, compared to the surface tension forC2, C4 < 0. After the rescalingof equation (4.10), the ratio between these two terms is equal to C2b

2/C4. In caseof the surface diffusion term we obtain several fixed points which are not presentwithout this term. However, the only fixed point with exponents different from thefixed points found in section B.4 is Fp4,0, i.e. the lineal MBE fixed point, whilethe others have the same exponents of the Galilean fixed pointz = 1 andα = 1.The stability matrices of these fixed points show that they are unstable, as for the


Name λ1 λ2 λ3 Stability

Smooth −1 −3 −2 Stable

Galilean 2 34 + i

√474

34 + i

√474 Unstable

Fp4,2 1.09 2.28 −1.98 Saddle

Table C.7 – Eigenvalues of the fixed points of table C.6.

Galilean and Fp4,0, or they have one stable direction, as for Fp4,1 and Fp4,2. TheKPZ fixed point has two stable directions and if we restrict our analysis at the caseC1 = 0, i.e.a4 = 0, this is the stable fixed point for the DRG flow. Finally, as forthe |k|3 term, the surface diffusion does not introduce any unexpected behavior incase ofC1 6= 0 and a positive surface tension, hence, we conclude that thisterm isrequired only whenC2 < 0, in order to pose a well defined dispersion relation inthe limit k → ∞.

In summary, we have shown that within the limitations of the DRG technique,the main conclusions addressed for a linear dispersion relation of the formk −k2 can be extracted for higher order polynomial contributions(as long as theyhave the proper sign and, consequently, they are only stabilizing). Only whenthe surface tension term,k2 is destabilizing, those higher order terms are requiredfor physical consistency. This irrelevance (and also the self-consistency of theDRG calculations) stresses the main results of this thesis:a few ingredients provideuniversal equations that contain all the relevant physicalmechanisms.

DRG for the anisotropicKuramoto-Sivashinsky equation

D.1 Propagator renormalization

As stated in section 5, we have four types of diagrams for the propagator renor-malization (see figure 5.3), each one with multiplicity equal to four. Here, forsimplicity we report the whole functionΣ(k, ω) after taking the limitω → 0 andafter the change to symmetrical momentaq → j + k/2.

Σ(k, 0) = 2Π0

∫ +∞

−∞dΩ

∫ > dj

8π3

[

λ2xkx

(

j3x +j2x2kx − jx

4k2

x − 1

8k3

x

)

+

+λxλy

(

jxj2ykx + j2xjyky +

j2y2k2

x +j2x2k2

y − jy4k2

xky −jx4kxk

2y − 1

4k2

xk2y

)

+

+λ2yky

(

j3y +j2y2ky −

jy4k2

y − 1

8k3

y

)]

limω→0

∣

∣

∣

∣

∣

G0

(

k

2+ j

)∣

∣

∣

∣

∣

2

G0

(

k

2− j

)

. (D.1)

In (D.1) the vertex contribution of each diagram in figure 5.3is easy to obtain.However, the expansion of the bare propagators is more involved and it has to beexpanded up to third order in the two variableskx andky. The expansion of thefirst two propagators with respect to the small variablesx = kx/jx andy = ky/jy

160 DRG for the anisotropic Kuramoto-Sivashinsky equation

leads to

limω→0

∣

∣

∣

∣

∣

G0

(

k

2+ j

)∣

∣

∣

∣

∣

2

=

[

νx

(

j2x + jxkx1

4k2

x

)

+ νy

(

j2y + jyky1

4k2

y

)

+

+Dx

(

j4x + 2j3xkx +3

2j2xk

2x

)

+ 2Dxy

(

j2xj2y + jxj

2ykx + j2xjyky +

+j2y4k2

x + jxjykxky +j2x4k2

y

)

+Dy

(

j4y + 2j3yky +3

2j2yk

2y

)

]2

+ Ω2

−1

∼

∼ 1

∆− 2

∆2a0bxx− 2

∆2a0byy + g2xx

2 + gxyxy + g2yy2 + g3xx

3 +

+g2xyx2y + gx2yxy

2 + g3yy3, (D.2)

while the expansion of the third propagator gives

limω→0

G0

(

k

2− j

)

=

[

νx

(

j2x − jxkx1

4k2

x

)

+ νy

(

j2y − jyky1

4k2

y

)

+

+Dx

(

j4x − 2j3xkx +3

2j2xk

2x

)

+ 2Dxy

(

j2xj2y − jxj

2ykx − j2xjyky +

+j2y4k2

x + jxjykxky +j2x4k2

y

)

+Dy

(

j4y − 2j3yky +3

2j2yk

2y

)

+ iΩ

]−1

∼

∼ 1

Θ+bxΘ2

x+byΘ2

y +

(

b2xΘ3

− b2x

Θ2

)

x2 +

(

2

Θ3bxby −

bxy

Θ2

)

xy +

+

(

b2yΘ3

− b2y

Θ2

)

y2 +

(

b3xΘ4

+b3x

Θ2− 2

Θ3bxb2x

)

x3 +

(

3

Θ4byb

2x +

b3xy

Θ2−

− 2

Θ3(bxbxy + byb2x)

)

x2y +

(

3

Θ4bxb

2y +

b3xy

Θ2− 2

Θ3(bybxy + bxb2y)

)

xy2 +

+

(

b3yΘ4

+b3y

Θ2− 2

Θ3byb2y

)

y3. (D.3)

D.1 Propagator renormalization 161

For the sake of simplicity, in (D.2) and (D.3) we have introduced the functions

a0 = j2xνx + j2yνy + j4xDx + 2j2xj2yDxy + j4yDy, (D.4)

bx =jx2∂jxa0 = j2xνx + 2j4xDx + 2j2xj

2yDxy, (D.5)

by =jy2∂jya0 = j2yνy + 2j4yDy + 2j2xj

2yDxy, (D.6)

b2x =j2x8∂2

jxa0 =

j2x4νx +

3

2j4xDx +

1

2j2xj

2yDxy, (D.7)

b2y =j2y8∂2

jya0 =

j2y4νy +

3

2j4yDy +

1

2j2xj

2yDxy, (D.8)

bxy =jxjy4∂jx∂jya0 = 2j2xj

2yDxy, (D.9)

b3x =j3x48∂3

jxa0 =

j4x2Dx, b3y =

j3y48∂3

jya0 =

j4y2Dy, (D.10)

b3xy =jxj

2y

16∂jx∂

2jya0 =

j2xjy16

∂2jx∂jya0 =

j2xj2y

2Dxy, (D.11)

∆ = a20 + Ω2, (D.12)

Θ = a0 + iΩ, (D.13)

and

g2x =4

∆3a2

0b2x − 1

∆2(2a0b2x + b2x), (D.14)

gxy =8

∆3a2

0bxby −2

∆2(a0bxy + bxby), (D.15)

g2y =4

∆3a2

0b2y −

1

∆2(2a0b2y + b2y), (D.16)

g3x =4

∆3a0bx(2a0b2x + b2x) − 8

∆4a3

0b3x − 2

∆2(a0b3x + bxb2x), (D.17)

g2xy =8

∆3a0bx(a0bxy + bxby) +

4

∆3a0by(2a0b2x + b2x) −

− 24

∆4a3

0b2xby −

2

∆2(a0b3xy + bxbxy + byb2x), (D.18)

gx2y =8

∆3a0by(a0bxy + bxby) +

4

∆3a0bx(2a0b2y + b2y) −

− 24

∆4a3

0bxb2y −

2

∆2(a0b3xy + bybxy + bxb2y), (D.19)

g3y =4

∆3a0by(2a0b2y + b2y) −

8

∆4a3

0b3y −

2

∆2(a0b3y + byb2y). (D.20)


Multiplying equations (D.2) and (D.3) we can integrate in the frequency variableΩ. All the integrals involved in this calculation are exactlythe same functions,Ilm, found in appendix C so we omit further details of this step. Another importantissue that helps to simplify the calculation is related to the parity of the functionsin the perturbative expansion. In fact, considering that intwo dimensions the shellis a square interval of wave-vectors that is symmetric around the origin, everyfunction that is odd in one wave-vector component gives a null contribution to thepropagator renormalization. Hence, after some algebra, weobtain

Σ(k, 0) =Π0

16π2

∫ >

dj

λxfx

a30

(

λxj2x + λyj

2y

)

k2x + λy

fy

a30

(

λyj2y + λxj

2x

)

k2y +

+λxf3x

a50

[

λx

(

1 +a2

0bx2

− a30

4

)

+ λy

j2yj2x

]

k4x +

+1

a50

[

λ2xf2y

j2xj2y

+ λxλy

(

f2x +a2

0

2(bx + by − a0) + f2y

)

+ λ2yf2x

j2yj2x

]

k2xk

2y +

+λyf3y

a50

[

λy

(

1 +a2

0by2

− a30

4

)

+ λxj2xj2y

]

k4y

, (D.21)

where we have introduced the following functions

fx = a0 − 2bx, (D.22)

fy = a0 − 2by, (D.23)

f2x = a0b2x − 2a2

0b2x + 6a0bxbxy − 6b2xby + 6a0b2xby − 2a20b3xy, (D.24)

f2y = a0b2y − 2a2

0b2y + 6a0bybxy − 6bxb2y + 6a0bxb2y − 2a2

0b3xy, (D.25)

f3x = a0b2x − 2a2

0b2x + 6a0bxb2x − 2b3x − 2a20b3x, (D.26)

f3y = a0b2y − 2a2

0b2y + 6a0byb2y − 2b3y − 2a20b3y. (D.27)

At this point, we are ready to write down explicitly the non-null contributions tothe integration of the wave-vectorsjx andjy in the two dimensional shell.

In the present case, due to anisotropy, we cannot approximate the shell usingpolar coordinates [see figure D.1(a)]. Rather, we have to useits real square form,as shown in figure D.1(b). Due to the parity of the functions, we consider onlythe sector of the whole region with positive wave-vectors. The shell integral iscomposed of two strips as shown in D.1(c), and, can be explicitly decomposed into


twelve different integrals as follows

∫ >

dj =

Λy− ky2

∫

Λybχx +

ky2

djy

Λx− kx2

∫

kx2

djx +

Λx+ kx2

∫

− kx2

djx

+

Λy+ky2

∫

Λybχx − ky

2

djy

Λx− kx2

∫

kx2

djx +

Λx+ kx2

∫

− kx2

djx

+

+

Λx− kx2

∫

Λxb

+ kx2

djx

Λy− ky2

∫

ky2

djy +

Λy+ky2

∫

− ky2

djy

+

Λx+ kx2

∫

Λxb− kx

2

djx

Λy− ky2

∫

ky2

djy +

Λy+ky2

∫

− ky2

djy

−

−Λy− ky

2∫

Λybχx +

ky2

djy

Λx− kx2

∫

Λxb

+ kx2

djx +

Λx+ kx2

∫

Λxb− kx

2

djx

−Λy+

ky2

∫

Λybχx − ky

2

djy

Λx− kx2

∫

Λxb

+ kx2

djx +

Λx+ kx2

∫

Λxb− kx

2

djx

. (D.28)

Note that the anisotropick dependence on the integration limits, that generalizes

Λx

Λx/b

Λy/bΛy

kx

kyky

kx

ΛΛ/b

kx

a) b) c)

ky

Figure D.1 – Two dimensional shell in the wave-vector plane. In panel (a)we representthe spherical approximation used in the case of an isotropicequation, as in appendix B.The fast modes are shown in grey and the slow modes in white. Panel (b) depicts theactual square shell with the explicit dependence on the two components of the vectork. Inpanel (c) we show the part of the shell with positive wave-vectors differentiating betweenthe contributions due to the two strips. Note that the small square[Λx/b,Λx] × [Λy/b,Λ]enters twice into the integration; in order to obtain the correct result, we have to subtractits contribution from (D.28).

an analogous feature in the isotropic case (see appendix B).The first two lines ofintegrals in (D.28) contribute at first order inδl while the third one only contributesto second order and can be neglected. The explicit contribution of each integral isreadily obtained first through an expansion inkx andky, and finally consideringthat the shell has an infinitesimal extension, i.e.δl → 0. From now on we considera square shell withΛx = Λy = 1. For every functionf of the wave-vectors wecan calculate its expansion forkx, ky ≪ 1 (in the following formulaesx andsy are


two constants) as

1−sxkx∫

1

eδl +sxkx

djx

1−syky∫

syky

djyf(jx, jy) ∼1∫

1

eδl

djx

1∫

0

djyf(jx, jy) −

−sx

1−syky∫

syky

djy

[

f(1 − sxkx, jy) + f(1/eδl + sxkx, y)]

∣

∣

∣

∣

kx=0

ky=0

kx −

−sy

1−sxkx∫

1

eδl+sxkx

djx

[

f(x, 1 − syky) + f(x, syky)]

∣

∣

∣

∣

kx=0

ky=0

ky +

+s2x

1−syky∫

syky

djy

[

∂jxf(1 − sxkx, jy) − ∂jxf(1/eδl + sxkx, y)]

∣

∣

∣

∣

kx=0

ky=0

k2x

2+

+s2y

1−sxkx∫

1

eδl+sxkx

djx

[

∂jyf(x, 1 − syky) − ∂jyf(x, syky)]

∣

∣

∣

∣

kx=0

ky=0

k2y

2+

+sxsy

[

f(1 − sxkx, 1 − syky) + f(1 − sxkx, syky) +

+f(1/eδl + sxkx, 1 − syky) + f(1/eδl + sxkx, syky)]

∣

∣

∣

∣

kx=0

ky=0

kxky. (D.29)

In the δl → 0 limit we obtain the complete contribution of integral in (D.28) toorderδl

∑

sx,sy=±1/2

1−sxkx∫

e−δl+sxkx

djx

1−syky∫

syky

djy f(jx, jy) +

1−syky∫

e−χxδl+syky

djy

1−sxkx∫

sxkx

djx f(jx, jy)

∼

∼ 4

[

χx

∫ 1

0djxf(jx, 1) +

∫ 1

0djyf(1, jy)

]

δl +

+1

2

[∫ 1

0djy∂

2jxf(1, jy) + χx [∂jxf(1, 1) − ∂jxf(0, 1)]

]

δl k2x +

+1

2

[

χx

∫ 1

0djx∂

2jyf(jx, 1) + ∂jyf(1, 1) − ∂jyf(1, 0)

]

δl k2y. (D.30)

Using the result from equation (D.30) into the expression ofthe propagator renor-malization, equation (D.21), we can calculate the shell contribution to the surface


diffusion terms due to the functions involved in the renormalization of the two dif-ferent surface tensions. In order to do that, it is convenient to introduce two func-tions,Gx andGy and their derivatives, as differential operators acting on1/a2

0:

Gx =fx

a30

(

j2x +λy

λxj2y

)

, wherefx

a30

=

[

1 +jx2∂jx

]

a−20 , (D.31)

∂jxGx =

[

2jx +

(

5

2j2x +

3

2

λy

λxj2y

)

∂jx +jx2

(

j2x +λy

λxj2y

)

∂2jx

]

a−20 , (D.32)

∂jyGx =

[

jyλy

λx(2 + jx∂jx) +

(

j2x +λy

λxj2y

)(

∂jy +jx2∂jx∂jy

)]

a−20 , (D.33)

∂2jxGx =

[

2 + 7jx∂jx +

(

4j2x + 2λy

λxj2y

)

∂2jx

+

+jx2

(

j2x +λy

λxj2y

)

∂3jx

]

a−20 , (D.34)

∂2jyGx =

λy

λx

[

2 + jx∂jx + 4jy∂jy +

(

λx

λyj2x + j2y

)

∂2jy

+ 2jxjy∂jx∂jy +

+jx2

(

j2xλx

λy+ j2y

)

∂jx∂2jy

]

a−20 , (D.35)

and

Gy =fy

a30

(

j2y +λx

λyj2x

)

, wherefy

a30

=

[

1 +jy2∂jy

]

a−20 , (D.36)

∂jyGy =

[

2jy +

(

5

2j2y +

3

2

λx

λyj2x

)

∂jy +jy2

(

j2y +λx

λyj2x

)

∂2jy

]

a−20 , (D.37)

∂jxGy =

[

jxλx

λy

(

2 + jy∂jy

)

+

(

j2y +λx

λyj2x

)(

∂jx +jy2∂jx∂jy

)]

a−20 , (D.38)

∂2jyGy =

[

2 + 7jy∂jy +

(

4j2y + 2λx

λyj2x

)

∂2jy

+

+jy2

(

j2y +λx

λyj2x

)

∂3jy

]

a−20 , (D.39)

∂2jxGy =

λx

λy

[

2 + jy∂jy + 4jx∂jx +

(

λy

λxj2y + j2x

)

∂2jx

+ 2jxjy∂jx∂jy +

+jy2

(

j2yλy

λx+ j2x

)

∂2jx∂jy

]

a−20 . (D.40)

The propagator renormalization can be decomposed into different contributions


according with the renormalization of each parameter

Σ(k, 0) = δl[

Σνxk2x + Σνyk

2y +

(

Σ0Dx

+ ΣsDx

)

k4x+

+ 2(

Σ0Dxy

+ ΣsDxy

)

k2xk

2y +

(

Σ0Dy

+ ΣsDy

)

k4y

]

,

(D.41)

contributions being listed below:

Σνx =λ2

xΠ0

4π2

∫ 1

0ds

(

1 +λy

λxs2)

fx

a30

∣

∣

∣

∣

∣

jx=1

jy=s

+ χx

∫ 1

0ds

(

s2 +λy

λx

)

fx

a30

∣

∣

∣

∣

∣

jx=sjy=1

,

Σνy =λ2

xΠ0

4π2

λy

λx

∫ 1

0ds

(

λy

λxs2 + 1

)

fy

a30

∣

∣

∣

∣

∣

jx=1

jy=s

+ χx

∫ 1

0ds

(

s2 +λy

λx

)

fy

a30

∣

∣

∣

∣

∣

jx=sjy=1

,

Σ0Dx

=λ2

xΠ0

4π2

∫ 1

0ds

1

a50

[

f3x

(

1 +λy

λxs2)

+a2

0

2bx − a3

0

4

]

∣

∣

∣

∣

∣

jx=1

jy=s

+

+χx

∫ 1

0ds

1

a50

[

f3x

(

1 +λy

λxs2

)

+a2

0

2bx − a3

0

4

]

∣

∣

∣

∣

∣

jx=sjy=1

,

Σ0Dy

=λ2

xΠ0

4π2

λy

λx

∫ 1

0ds

1

a50

[

f3y

(

λy

λx+

1

s2

)

+λy

λx

(

a20

2by −

a30

4

)]

∣

∣

∣

∣

∣

jx=1

jy=s

+

+χx

∫ 1

0ds

1

a50

[

f3y

(

λy

λx+ s2

)

+λy

λx

(

a20

2by −

a30

4

)]

∣

∣

∣

∣

∣

jx=sjy=1

,

Σ0Dxy

=λ2

xΠ0

8π2

λy

λx

∫ 1

0ds

1

a50

[

f2xλy

λx

(

1 +λy

λxs2)

+ f2y

(

1

s2+λy

λx

)

+

+λy

2λxa2

0 (bx + by − a0)

]

∣

∣

∣

∣

∣

jx=1

jy=s

+ χx

∫ 1

0ds

1

a50

[

f2xλy

λx

(

1 +λy

λxs2

)

+

+f2y

(

s2 +λy

λx

)

+λy

2λxa2

0 (bx + by − a0)

]

∣

∣

∣

∣

∣

jx=sjy=1

,

ΣsDx

=λ2

xΠ0

32π2

[∫ 1

0ds ∂2

jxGx(1, s) + χx [∂jxGx(1, 1) − ∂jxGx(0, 1)]

]

,


ΣsDy

=λ2

xΠ0

32π2

(

λy

λx

)2 [

χx

∫ 1

0ds ∂2

jyGy(s, 1) + ∂jyGy(1, 1) − ∂jyGy(1, 0)

]

,

ΣsDxy

=λ2

xΠ0

64π2

[

χx

∫ 1

0ds ∂2

jyGx(s, 1) + ∂jyGx(1, 1) − ∂jyGx(1, 0) +

+

∫ 1

0ds ∂2

jxGy(1, s) + χx [∂jxGy(1, 1) − ∂jxGy(0, 1)]

]

. (D.42)

The functionsΣs... are the contributions of the shell toΣ; in order to obtain them

explicitly, we finally need all the derivatives of the function 1/a20:

∂jx

1

a20

= − 2

a30

∂jxa0, (D.43)

∂jy

1

a20

= − 2

a30

∂jya0, (D.44)

∂2jx

1

a20

=1

a50

[

6a0 (∂jxa0)2 − 2a2

0∂2jxa0

]

, (D.45)

∂2jy

1

a20

=1

a50

[

6a0

(

∂jya0

)2 − 2a20∂

2jya0

]

, (D.46)

∂jx∂jy

1

a20

=1

a50

[

6a0 (∂jxa0)(

∂jya0

)

− 2a20∂jx∂jya0

]

, (D.47)

∂3jx

1

a20

=1

a50

[

18a0 (∂jxa0)(

∂2jxa0

)

− 2a20∂

3jxa0 − 24 (∂jxa0)

3]

, (D.48)

∂3jy

1

a20

=1

a50

[

18a0

(

∂jya0

)

(

∂2jya0

)

− 2a20∂

3jya0 − 24

(

∂jya0

)3]

, (D.49)

∂2jx∂jy

1

a20

=1

a50

[

12a0 (∂jxa0)(

∂jx∂jya0

)

+ 6a0

(

∂jya0

) (

∂2jxa0

)

−

−2a20∂

2jx∂jya0 − 24

(

∂jya0

)

(∂jxa0)2]

, (D.50)

∂jx∂2jy

1

a20

=1

a50

[

12a0

(

∂jya0

) (

∂jx∂jya0

)

+ 6a0 (∂jxa0)(

∂2jya0

)

−

−2a20∂jx∂

2jya0 − 24 (∂jxa0)

(

∂jya0

)2]

. (D.51)

The equations in (D.42) are directly incorporated into the RG flow equations of themain text (5.13)-(5.18) and are in principle amenable to numerical integration.


D.2 Noise variance renormalization

As for the propagator, the renormalization of the noise variance is due to all thecombinations of the two different vertices, hence the four diagrams in figure??lead to

Φ(k, ω) =

+∞∫

−∞

dΩ

2π

∫ > dj

4π2

[

λ2x

(

k2x

4− j2x

)2

+ 2λxλy

(

k2x

4− j2x

)

(

k2y

4− j2y

)

+

+λ2y

(

k2y

4− j2y

)2]

⟨

ηj+k/2 η−(j+k/2)

⟩ ⟨

ηk/2−j ηj−k/2

⟩

×

×∣

∣

∣

∣

G0

(

j +k

2,Ω +

ω

2

)∣

∣

∣

∣

2 ∣∣

∣

∣

G0

(

k

2− j,

ω

2− Ω

)∣

∣

∣

∣

2

. (D.52)

In our case we are interested only in the zero order expansionof equation (D.52), infact we are neglecting the conserved noise contribution in the diagram contractions.Thus, the integral in the frequency domain of the two bare propagators is readilyevaluated using equations (D.2) and (D.3)

∫ +∞

−∞dΩ lim

ω→0

∣

∣

∣

∣

G0

(

j +k

2,Ω +

ω

2

)∣

∣

∣

∣

2 ∣∣

∣

∣

G0

(

k

2− j,

ω

2− Ω

)∣

∣

∣

∣

2

∼ π

2a30

. (D.53)

Using this last result in equation (D.52) we immediately obtain the noise variancerenormalization

Φ(k, 0) =λ2

xΠ20

π2

∫ > dj

a30

(

j2x +λy

λxj2y

)2

=λ2

xΠ20

4π2

[

∫ 1

0

ds

a30(1, s)

(

1 +λy

λxs2)2

+

+χx

∫ 1

0

ds

a30(s, 1)

(

s2 +λy

λx

)2]

δl ≡ Φδl. (D.54)

FunctionΦ is directly incorporated into the RG flow equation (5.18) of the maintext.

Thin interface limit

E.1 Equations non-dimensionalization

In this appendix we report in detail the calculations which allow us to derive the re-lations between the phase-field (6.9),(6.10) and the movingboundary model (6.1)-(6.4) parameters in the thin interface limit (in this appendix we consider the deter-ministic limit of these models).

First we introduce some useful constants which allow to non-dimensionalizeequations (6.9) and (6.10). Moreover, we will not write explicitly the functionswhich forms the PF model unless we explicitly need them.

Considering the length scaleli, we can identify the first parameter relationbetween the two models

li = a1W

λ, with a1 =

I

J, (E.1)

whereI andJ are two constants that depends on the functional forms off andg.In order to choose a small parameterǫ for the perturbative expansion we considerthe width of the inner region to be small as compared to the characteristic lengthof the pattern which we want to explore, and writeǫ = W/li.

Using (E.1) we can re-write the coupling constant asλ = a1ǫ. Thus, the per-turbative expansion in this variableǫ is different to the original thin interface limitoriginally proposed by Karma [237]. However, Echebarriaet al. [239] employed

170 Thin interface limit

physical arguments to justify the model convergence in the most favorable situa-tion in which the widthW is negligible compared to the diffusion lengthlD, thatrepresent the original condition for the thin interface limit.

We begin the calculations by re-scaling the spatial and temporal variables usingthe capillarity length and the diffusion constant

r′ =r

li, t′ =

D

l2it. (E.2)

Employing these relations we can non-dimensionalize the equations (6.1)-(6.4).The diffusion constant, the Stefan condition and the generalized Gibbs-Thomsonequation read, respectively,

∂tu = ∇2u, (E.3)

∂nu∣

∣

+= Livn, (E.4)

u = κ+ βLivn, (E.5)

whereκ, vn andβ are the non-dimensional curvature, velocity, and kinetic coeffi-cient. Their explicit expressions are

κ = liK, (E.6)

vn =liDVn, (E.7)

β =D

lik−1

D . (E.8)

Re-scaling also (6.9),(6.10), we obtain

αǫ2 ∂tφ = ǫ2 ∇2 φ− f ′(φ) + ǫa1 ug′(φ), (E.9)

∂tu = ∇ ·(

q(φ)∇u+ ǫ a(φ)∂tφ∇φ|∇φ|

)

− Li

2∂th(φ), (E.10)

whereα = τD/W 2.

In this new system of equations the interface becomes sharp in the limit. Noticehow theǫ2 term multiplies the higher order derivative of equation (E.9).

Mathematically, this kind of problem is often referred to asa singular perturba-tion problem. In order to solve it we have to partition the domain into two regions,one in which∇2φ is small and another one in which this term is large. Evidently,this partition corresponds to the exterior regionΩe ≡ Ω+ ∪ Ω− and to the interioroneΩi, respectively. Using the notationφ, u for the variables inΩi andΦ, U for

E.1 Equations non-dimensionalization 171

those inΩe, we can expand them as

Ωi

φ = φo + ǫφ1 + ǫ2φ2 + . . .

u = uo + ǫu1 + ǫ2u2 + . . .

(E.11)

Ωe

Φ = Φo + ǫΦ1 + ǫ2Φ2 + . . .

U = Uo + ǫU1 + ǫ2U2 + . . .

In order to obtain a unique solution, we have to impose matching conditions bywhich these functions and their derivatives are equal at thefrontier between thetwo regions.

The two expansions (E.11) can be written in curvilinear coordinates (see ap-pendix??), and their expressions are different depending on the region: we denotethe normal coordinate asr in the exterior and isη = r/ǫ in the interior region.In this manner, the matching condition corresponds to ther → 0 limit in Ωe andη → ±∞ in Ωi. Re-writing them explicitly up to first order inǫ we obtain

limη→±∞

uo(η, s) = limr→0

Uo

∣

∣

±(r, s), (E.12)

limη→±∞

u1(η, s) = limr→0

[

U1

∣

∣

±(r, s) + η ∂rUo

∣

∣

±(r, s)

]

, (E.13)

limη→±∞

u2(η, s) = limr→0

[

U2

∣

∣

±(r, s) + η ∂rU1

∣

∣

±(r, s) +

η2

2∂2

rUo

∣

∣

±(r, s)

]

. (E.14)

At every stage of this calculation we assume that the functionsUn(r, s) and theirderivatives are finite.

The equations (E.9) and (E.10) of the functionsΦ andU assume a very simpleform in the exterior region: the order parameter is constantin Ω+ andΩ−. Thephase-field equation reads

f ′(Φ) = −ǫa1 Ug′(Φ), (E.15)

which at zero order reduces tof ′(Φo) = 0, and its solutions areΦo = ±1. More-over, sinceg′(Φo) = 0, the first order expansion in this equation is

f ′′(Φo)Φ1 = −a1Uo g′(Φo) = 0, (E.16)

and we obtainΦ1 = 0. The second order

f (3)(Φo)Φ1

2+ f ′′(Φo)Φ2 = −a1

[

Uo g′′(Φo)Φ1 + U1 g

′(Φo)]

= 0, (E.17)

givesΦ2 = 0, etc. Evidently,Φ = ±1 are the stable solutions of (E.15) at everyorder inǫ and for every concentration valueU . Finally we conclude that the solu-tion of the phase-fieldΩe is a step function. The equation (E.10) inΩe reduces tothe diffusion equation

∂tUn = q(±1)∇2Un, (E.18)

at every order inǫ.


E.2 Asymptotic expansion

Employing the local curvilinear coordinates and expandingthe phase-field equa-tion up to second order we obtain

∂2ηφ− f ′(φ)+ǫ

[

(αvn + κ) ∂ηφ+ a1ug′(φ)

]

+

+ ǫ2[

∂2sφ+ αvs∂sφ− α

dφ

dt− κ2η∂ηφ

]

+O(

ǫ3)

= 0.(E.19)

Similarly, we expand the anti-trapping termjat ≡ (jr, js) up to zero order

−ǫ∇ · jat = −∂ηjr − ǫ (∂sj

s + κjr) + O(ǫ2) =

= −∂η

[

a(φ)

(

−vn

ǫ∂ηφ+

dφ

dt− vs∂sφ

)]

+ κvna(φ)∂ηφ+O(ǫ), (E.20)

and using it in equation (E.10) we obtain the most relevant contribution

1

ǫ2∂η(q∂ηu) +

1

ǫ

(vn + κq)∂ηu+ vn∂η(a∂ηφ) +vn

2Li∂ηh

+

+

∂s(q∂su) − κ2qη∂ηu+ +avnκ∂ηφ−−∂η

[

a

(

dφ

dt− vs∂sφ

)]

−

−(

d

dt− vs∂s

) (

Li

2h− u

)

+O(ǫ) = 0. (E.21)

In the next section we will show how the equations (E.19) and (E.21) can be solvedby imposing the matching condition. The zero order expansion of the fieldφ givesthe stationary solution,φo, and we will see that this function scales with the interiorregion widthΩi.

In order to obtain the zeroth order solution ofu , uo, we will solve equation(E.21) at order−2 and, using the first order solution of (E.19), we will find theintegration constant that arises at this order. In this way we can solve the fieldu atthe lowest order and calculateUo in the exterior region.

Similarly, the next order of (E.21) gives the first order correction tou, u1, andemploying the second order of (E.19) we will set the integration constant ofu1,obtaining the generalized Gibbs-Thomson condition at firstorder. Besides, we willcalculate∂ηu1, and, using the matching condition, we will find the expression of∂rUo in Ωe, obtaining the Stefan condition at zero order. The last order in (E.21),namely, the zero order, allows to calculate∂rU1 and the corrections to the Stefancondition at first order.

E.2.1 Field φ at order O(1)

At this order, in (E.19) we have the equation∂2ηφ − f ′(φ) = 0 and, considering

the expansion ofφ, we obtain

∂2ηφo − f ′(φo) = 0, (E.22)

E.2 Asymptotic expansion 173

that can be re-written as

∂η (∂ηφo)2 = 2 f ′(φo)∂ηφo. (E.23)

Integrating this equation from−∞, we obtain

(∂ηφo)2 (η) − (∂ηφo)

2 (−∞) = 2[

f(φo) − f(1)]

. (E.24)

Due to the double well functionf(φ) we havef(1) = −14 and, considering that

the phase-field does not change in the exterior region∂ηφo(±∞) = 0, we obtain

(∂ηφo)2 (η) =

φ4o

2− φ2

o +1

2, ⇒

√2 ∂ηφo

√

(1 − φ2o)

2= ±1; (E.25)

integrating the last equation from0 to η, using the conditionφo(0) = 0 and impos-ing the boundary conditions

limη→±∞

φo(η) = ∓1, (E.26)

we find the zero order expression of the phase-field

∫ φo(η)

φo(0)

dx

1 − x2= − η√

2, ⇒ φo(η) = − tanh

(

η√2

)

. (E.27)

The functionφo is a kink and varies in a characteristic region of widthη/2. Sincethe variableη is inversely proportional to the constantW , which defines the interiorregion width, the variation ofφ is localized inΩi.

E.2.2 Field u order O(1/ǫ2)

Considering (E.21) and the two field expansion (E.11), we obtain

∂η (q(φo) ∂ηuo) = 0 ⇒ ∂ηuo =Ao(s)

q(φo), (E.28)

whereAo is constant inη. Deriving the matching condition (E.12), and considering

that∂rUo

∣

∣

∣

±(r, s) is finite, we obtain an expression for theη derivative ofuo

limη→±∞

∂ηuo(η, s) = limǫ→0

ǫ∂r

(

Uo

∣

∣

∣

±(r, s)

)

= 0, (E.29)

thusAo(s) = 0 anduo = uo(s). In order to fix this constant we have to use thenext order in the phase-field equation (E.19).


E.2.3 Field φ order O(ǫ)

Considering the first order expansion off(φo + ǫφ1 + . . . ), equation (E.19) reads

(

∂2η − f ′′(φo)

)

φ1 = −[

a1 g′(φo) uo(s) + (αvn + κ) ∂ηφo

]

. (E.30)

Deriving the equation (E.22) inη

∂η

(

∂2ηφo − f ′(φo)

)

= ∂2η (∂ηφo) − f ′′(φo) (∂ηφo) = L (∂ηφo) = 0, (E.31)

where we have introduced the linear differential operatorL = ∂2η − f ′′(φo), for

which∂ηφo is an eigenfunction with zero eigenvalue. We can re-write the equation(E.30) using this operator

Lφ1 = −[


]

. (E.32)

In order to solve this equation we can chooseg′(φo) = − 1a1∂ηφo, so thatLφ1 = 0.

The main problem with this choice is that it introduces an undesired constraintbetween the functionsf andg. A different way to solve it is to consider that thisproblem is inhomogeneous and its solution will be not trivial only if the right handside of equation (E.32) satisfies the solvability condition(Fredholm alternative),namely, if it is orthogonal to the kernel of the self-adjointoperatorL [131],

∫ +∞

−∞

[


]

∂ηφo dη = 0. (E.33)

Using the following definitions

I =

∫ +∞

−∞(∂ηφo)

2 dη, (E.34)

J =

∫ −∞

+∞g′(φo) ∂ηφo dη = g(1) − g(−1), (E.35)

we can check that, by defininga1 according to (E.1), the following equation

−J a1uo(s) + I (αvn + κ) = 0, (E.36)

leads touo(s) = αvn + κ. (E.37)

Finally, employing the matching condition (E.12) we obtainthe lowest order of thenon-dimensional Gibbs-Thomson condition (E.5)

Uo = Uo

∣

∣

±= uo(s) = αvn + κ. (E.38)

In order to calculate the corrections to (E.38) we need to solve the orderǫ−1 in(E.21).


E.2.4 Field u order O(1/ǫ)

At this order, equation (E.21) reads

∂η [q(φo) ∂ηu1] = −vn

2∂η [Li h(φo) − 2a(φo) ∂ηφo] , (E.39)

and, integrating inη, we obtain

q(φo) ∂ηu1 = −vn

2[Li h(φo) − 2a(φo) ∂ηφo] +A1(s). (E.40)

We find the constantA1(s) using the values of the functions in the limit of the solidregionΩ−

limη→−∞

lhs. (E.39)= q(1) ∂ηu1(−∞) = 0, (E.41)

limη→−∞

rhs. (E.39)= −vn

2[Li h(1) − 2a(1) ∂ηφo(−∞)] +A1(s) =

−vn

2Li +A1(s) =, (E.42)

and we obtain

A1(s) =vn

2Li. (E.43)

With this result equation (E.39) reads

∂ηu1 = − vn

2q(φo)[Li(h(φo) − 1) + 2a(φo) ∂ηφo] = −vn

2p(φo), (E.44)

where we have introduced the function

p(φ) =1

q(φ)

[

Li(h(φ) − 1) + 2a(φ) ∂ηφ]

. (E.45)

Integrating (E.44) inη we obtain the expression ofu1

u1(η) = u1 −vn

2

∫ η

op(

φo(x))

dx, (E.46)

depending on an integration constantu1. In order to obtain the convergence of thislast integral we have to impose the following condition

limη→−∞

Lih(φo) − 1

q(φo)= 0, (E.47)

sinceq(

φo(−∞))

= 0. The constantu1 is determined from the solution of theǫ2

order in the equation (E.19).


E.2.5 Field φ order O(ǫ2)

Considering thatφo depends only onη, we have

∂2sφo = ∂sφo =

dφo

dt= 0, (E.48)

and the next contribution in (E.19) reads

Lφ2 = f (3)(φo)φ2

1

2−(αvn + κ) ∂ηφ1 − a1 u1g

′(φo)−

− a1uoφ1 g′′(φo) + κ2η ∂ηφo.

(E.49)

In order to solve it we can use the parity of the functions in (E.49):

• For anyf that is an even double well,f(−φ) = f(φ), from (E.22) weobtain an equilibrium profile that is odd inη: φo(−η) = −φo(η). Thus, itsderivative∂ηφo is even.

• For anyg that is odd we have eveng′(φ) and oddg′′(φ).

• L being an even operator,φ1 has to be even to not obtain a trivial solution.Its derivative∂ηφ1 will be odd.

By imposing again the solvability condition to equation (E.49), we obtain the fol-lowing integral

∫ +∞

−∞

[

f (3)(φo)φ2

1

2−(αvn + κ) ∂ηφ1 − a1 u1g

′(φo)−

− a1uoφ1 g′′(φo) + κ2η ∂ηφo

]

∂ηφo dη = 0.

(E.50)

This expression can be considerably simplified by considering that there is anunique even function in the integral

∫ +∞

−∞a1 u1 g

′(φo)∂ηφo dη = 0. (E.51)

Inserting this result into the implicit expression ofu1 found in (E.46), we obtain

u1

∫ +∞

−∞g′(φo) ∂ηφo dη −

vn

2K = 0, (E.52)

where

K =

∫ +∞

−∞g′(φo) ∂ηφo

(∫ η

0p(

φo(x))

dx

)

dη. (E.53)

We have already identified the integral in (E.52) with the constantJ , so that thisequation gives us the constantu1

u1 = −vn K

2J, (E.54)


the form ofu1(η) and its derivative

u1(η) = −vn

2

(

K

J+

∫ η

0p(

φo(x))

dx

)

, (E.55)

∂ηu1(η) = −vn

2p(

φo(η))

. (E.56)

If we differentiate the matching condition (E.13) inη, the zeroth order of the Stefancondition (E.4) is

limη→±∞

∂ηu1 = ∂rUo

∣

∣

±; (E.57)

calculating (E.56) in the limit of the solidΩ− and the dilutedΩ+ phases

∂rUo

∣

∣

+= lim

η→∞∂ηu1 = −vn

2p(−1) = vn Li, (E.58)

∂rUo

∣

∣

−= lim

η→−∞∂ηu1 = −vn

2p(1) = 0. (E.59)

Using the condition (E.13), we obtain

U1

∣

∣

±= lim

η→±∞

[

−vn

2

(

K

J+

∫ η

0p(

φo(x))

dx

)

+vn

2p(

φo(η))

]

=

= −vn

2

(

K

J+

∫ ±∞

0

(

p(φo) − p(φ±o ))

dη

)

= −vn

(

K + JF±

2J

)

, (E.60)

where we have introduced a new constant that depends on the functions in thephase-field model

F± =

∫ ±∞

0

(

p(φo) − p(φ±o ))

dη, (E.61)

withp(φ±o ) = lim

η→±∞p(

φo(η))

. (E.62)

Now, we can write the concentration field in the exterior region

U∣

∣

±= Uo

∣

∣

±+ ǫU1

∣

∣

±+O(ǫ2) = κ+

[

α− ǫ

(

K + JF±

2J

)]

vn =

= κ+ β± Li vn,

(E.63)

where

β± =α

Li− ǫ

(

K + JF±

2JLi

)

. (E.64)

If we choose the functionp in order to equateF+ toF−, we guarantee the continu-ity of U across the normal component of the interface. Finally, to obtain the Stefancondition we have to solve the last order of the equation for the concentration field.


E.2.6 Field u order O(1)

Finally we can calculate∂rU1

∣

∣

±that gives us the first order corrections inǫ to the

Stefan condition. We re-write the equation (E.21) at this order

∂η

(

q(φo)∂ηu2 + q′(φo)φ1∂ηu1

)

+(

vn + κ q(φo))

∂ηu1 +

+vn ∂η

(

a(φo) ∂ηφ1 + a′(φo)φ1 ∂ηφo

)

−

−vn

2Li ∂η

(

h′(φo)φ1

)

+ q(φo) ∂2s uo + vnκa(φo) ∂ηφo = 0. (E.65)

Now we identify the terms in (E.21) that contribute to this equation:

1.1

ǫ2∂η

(

q(φ)∂ηu)

= ∂η

(

q(φo)∂ηu2 + q′(φo)φ1∂ηu1

)

, (E.66)

using∂ηuo = ∂ηuo(s) = 0,

2.1

ǫ

(

vn + κ q(φ))

∂ηu =(

vn + κ q(φo))

∂ηu1, (E.67)

3.1

ǫvn∂η

(

a(φ) ∂ηφ)

= vn ∂η

(

a(φo) ∂ηφ1 + a′(φo)φ1 ∂ηφo

)

, (E.68)

4.−vn

2ǫLi ∂ηh(φ) = −vn

2Li ∂η

(

h′(φo)φ1

)

, (E.69)

5. finally, considering the equation (E.27), that∂ηuo = 0, and retaining thezero order of the expansion that is missing, we obtain the last term

∂s

[

q(φo) ∂suo

]

+vnκa(φo) ∂ηφo +

(

duo

dt− vs∂suo

)

=

= q(φo) ∂2s uo + vnκa(φo) ∂ηφo,

(E.70)

due to ∂sh(φo) = h′(φo) ∂sφo = 0 and that the tangential componentof the velocityvs can be neglected because it arises from an interface re-parameterization. The temporal derivative ofuo can be omitted using a phys-ical argument. The generalized Gibbs-Thomson condition (E.5) shows thatthe temporal variations in the concentration fieldu depend on the curvatureand velocity variation. The characteristic time scale of these variations is(K Vn)−1, namely, the time that the interface needs to move a distanceequalto the local spatial scale of the growing pattern. Considering thatκ andvn are two small quantities, the temporal variations ofuo are of the ordervn κ (κ + β vn), which are neglible compared with the term of ordervn κappearing in (E.70).


Finally, collecting all the contributions we obtain an equation in which all the termsinvolving the functions∂ηφo orφ1 can be neglected: theirη → ±∞ limits are zero(whereas we cannot omit their integral contributions). Integrating this result inηwe can write

q(φo) ∂ηu2 + vn

∫ η

0∂xu1 dx+ κ

∫ η

0q(φo) ∂xu1 dx+

+vn κ

∫ η

0a(φo) ∂xφo dx+

(

∂2s uo

)

∫ η

0q(φo) dx = A2(s). (E.71)

Summing the third and the fourth terms on the left hand side ofequation (E.71) weobtain the following integral

−vn

2κLi

∫ η

0

(

h(φo(x)) − 1)

dx, (E.72)

whose limits in the exterior region can be written as

limη→−∞

(E.72)= −vn

2κLi

∫ −∞

0

(

h(φo) − h(φ−o ))

dx+ κη ∂rUo

∣

∣

−, (E.73)

limη→+∞

(E.72)= limη→+∞

−vn

2κLi

∫ η

0

(

h(φo(x)) + 1)

dx+ κη vnLi =

= −vn

2κLi

∫ +∞

0

(

h(φo) − h(φ+o ))

dx+ κη ∂rUo

∣

∣

+, (E.74)

usingh(φ±o ) = ∓1 and the results (E.58), (E.59). We write the two limits in oneexpression

limη→±∞

(E.72)= κη ∂rUo

∣

∣

± − vn

2κLiH

±, (E.75)

where

H± =

∫ ±∞

0

(

h(φo) − h(φ±o ))

dη, (E.76)

is a numerical constant of the PFM.Integrating the second term on the left hand side of (E.71), collecting the con-

stantvnu1(0) in A2(s), considering the equation (E.59), and using the matchingcondition (E.13)

limη→±∞

vnu1(η) = vn

(

U1

∣

∣

±+ η ∂rUo

∣

∣

±)

, (E.77)

we can calculate the constantA2(s) employing the limit in the solid region

A2(s) = limη→−∞

q(φo) ∂ηu2 + (E.72)+ vn u1(η) +(

∂2s uo

)

∫ η

0q(φo) dx =

= vn U1

∣

∣

− − vn

2κLiH

− +(

∂2s uo

)

∫ −∞

0q(φo) dx. (E.78)


The derivative inη of the third matching condition (E.14)

∂rU1

∣

∣

+= lim

η→+∞

[

∂ηu2(η, s, t) − η ∂2rUo

∣

∣

+(s)]

, (E.79)

gives the correction at the first order of the Stefan condition (E.4). Considering thatthe functionUo satisfies the diffusion equation in curvilinear coordinates, that is

[

∂2r + (vn + κ) ∂r + ∂2

s

]

Uo

∣

∣

+= 0, (E.80)

from∂2

rUo

∣

∣

+= −(vn + κ) ∂rUo

∣

∣

+ − ∂2sUo

∣

∣

+, (E.81)

we obtain

∂rU1

∣

∣

+= A2(s) − κη ∂rUo

∣

∣

++vn

2κLiH

+ − vn U1

∣

∣

+ − vnη ∂rUo

∣

∣

+ −

−(

∂2s uo

)

∫ +∞

0q(φo) dη + (vn + κ)η ∂rUo

∣

∣

++ η ∂2

sUo =

=vn

2κLi

(

H+ −H−)− vn

(

U1

∣

∣

+ − U1

∣

∣

−)−

−(

∂2s uo

)

(∫ +∞

0

(

q(φo) − 1)

dη −∫ −∞

0q(φo)dη

)

=

=vn

2κLi

(

H+ −H−)+v2n

2

(

F+ − F−)

− ∂2sUo

(

Q+ −Q−) , (E.82)

where the last constants in the phase-field model are

Q± =

∫ ±∞

0

(

q(φo) − q(φ±o ))

dη. (E.83)

Summing this result to (E.58), we can finally find the Stefan condition

∂r (Uo + ǫ U1)∣

∣

+=Livn + ǫ

[

vn

2κLi

(

H+ −H−)+

+v2n

2

(

F+ − F−)

− ∂2sUo

(

Q+ −Q−)]

,

(E.84)

and choosing the phase-field functions in order to equate thetwo constants withsuperscript+ to those with−, we obtain the right result up to first order inǫ.

E.3 Corrections, function involved and relations be-tween the parameters

The different diffusivities of the two phases and its smoothvariation within theinterior region, generates three corrections at the first expansion order.

E.3 Corrections, function involved and relations between the parameters 181

The term proportional toH+ − H− is due to the geometrical contributionto the generalized Gibbs-Thomson condition (E.5). Considering that a positivelycurved interface moves into the diluted phase, the area of the diluted part of theinterior region is slightly larger than the solid one and, for this reason, more activein particle aggregation.

The differenceH+−H− is the integral of this activity multiplied by the activeregion area. If this quantity is not zero the Stefan condition (E.4) acquires a spu-rious part when the local interface length changes, that is when the productvnκ isnot zero.

The second correction concerns the discontinuity ofU across the interface.This correction is proportional to the differenceF+ − F− and to the velocityvn.Note how the velocity is included in the Stefan and in the Gibbs-Thomson condi-tions. This correction is the trapping term, since in each side of the interface theconcentration changes with the velocity. By introducing the anti-trapping currentwe obtain more freedom in the choice of the phase-field functions in order to cancelthis correction.

Finally, the third termQ+ −Q− arises from the mass flux difference betweenthe solid and diluted phases and the domain of the interior region whereφ 6= ±1.Clearly in the moving boundary model this difference does not exist, since thewidth of Ωi is zero. Due to the pre-factor of this correction in (E.84), whenΩ+ 6=Ω− we need to put an additional surface diffusion term to obtainthe correct Stefancondition.

When the functionh is odd andq(φ) = 1− q(−φ), two corrections cancel out,namely,H+ = H− andQ+ = Q−. In order to have the same integral constant(calculated imposing the solvability conditions) found inthe symmetric diffusionphase-field model [237], it is enough to choose the functionsq anda so thatF+ =F−. Considering

∂ηφo =φ2

o − 1√2

, (E.85)

choosingq, h anda, the equation (E.45) reads

p(φo) =2

1 − φo

[

Li(φo − 1) +Li

2(1 − φ2

o)

]

= Li (φo − 1), (E.86)

complying the limits ofh andq. Inserting this function into the first integral con-stant (E.62) and sinceφo is odd with limitsφ±o = ∓1,

F± = Li

∫ ±∞

0(φo ± 1) dη = Li F

± = Li F, (E.87)

whereF± = F are the constants calculated in the symmetric diffusivity model


[237]. Inserting these functions into equation (E.53), we have

K =

∫ +∞

−∞g′(φo) ∂ηφo

(∫ η

0Li(φo − 1) dη

)

=

= Li

(

K −∫ +∞

−∞ηg′(φo) ∂ηφo dη

)

= LiK,

(E.88)

sinceηg′(φo) ∂ηφo is an odd function.In this last part we derive the third relation between the parameters of the two

models. With the constantα = τD/W 2 and due to the equation (E.64), we canwrite β as function of the parameters involved in the phase-field model

β =τD

W 2Li− λ

(

K + JF

2I

)

=τD

W 2Li− λa2, (E.89)

wherea2 = (K + JF )/2I is the same constant found in the symmetric diffusionmodel [237]. Finally, using the third equation in (E.8), we find the mass transportkinetic coefficient

k−1D =

a1W

Dλβ = a1

(

τ

λWLi− a2

W

D

)

. (E.90)

Resumen en Castellano

F.1 Metodologıa

Esta tesis se centra en fenomenos de invariancia de escala yde formacion de pa-trones en modelos de crecimiento de superficies. Utilizandometodos propios de laMecanica Estadıstica, hemos estudiado un modelo de crecimiento no-conservadoque ha sido utilizado para describir la produccion de superficies con tecnicas ex-perimentales de gran relevancia tecnologica.

Una caracterıstica importante de estos sistemas es la formacion de patrones (de-bido a inestabilidades morfologicas) a pequenas escalascoexistiendo con desorden(rugosidad cinetica) a grandes escalas. Utilizando un modelo de frontera movilde crecimiento difusivo, hemos podido describir, a travesde una misma formu-lacion, superficies producidas con diferentes tecnicas de crecimiento de pelıculasdelgadas, como por ejemplo el crecimiento por deposito qu´ımico de vapor y pordeposito electroquımico. El estudio de este modelo se ha llevado a cabo siguiendodos estrategias opuestas:

(i) a traves de un analisis aproximado suponiendo que el perfil de la superficiemuestra pendientes pequenas, o

(ii) integrando numericamente el problema de frontera movil sin ninguna re-striccion sobre su dinamica.

Siguiendo la primera estrategia hemos obtenido una ecuaci´on efectiva para la evolucion

184 Resumen en Castellano

de la altura de la superficie en la que la interrelacion entrela inestabilidad mor-fologica, la no-localidad, la no-linealidad, y las fluctuaciones origina propiedadesinesperadas.

A pesar de que esta ecuacion se haya derivado desde un modelode crecimientodifusivo, su aplicabilidad es muy amplia. Debido a que las morfologıas producidaspor esta ecuacion tienen mucha semejanza con las estructuras jerarquicas tıpicasde las superficies de las plantas de coliflores, el estudio de esta ecuacion es impor-tante para poder comprender los mecanismos que producen este tipo de estructurascomplejas en la naturaleza.

Por otra parte, esta ecuacion representa solamente un ejemplo de una vastaclase de ecuaciones no locales utilizadas como descripcion efectiva de fenomenosfısicos muy diferentes. Utilizando el grupo de renormalizacion dinamico, hemospodido clasificar el comportamiento crıtico de estas ecuaciones, encontrando uncontinuo de nuevas clases de universalidad.

No obstante, una comparacion cuantitativa entre las morfologıas obtenidas enlos experimentos de crecimiento por deposito electroquımico y el modelo de fron-tera movil es posible solamente utilizando un metodo multiescala para su inte-gracion numerica.

A fin de de resolver numericamente todas las escalas de longitud entre el trans-porte difusivo y las estructuras de la intercara, hemos utilizado un esquema multi-malla para un modelo de frontera difusa con fluctuaciones, que converge al mod-elo de frontera movil en el lımite de intercara delgada. Esta metodologıa nos hapermitido estudiar cuantitativamente el fenomeno de la rugosidad cinetica en elcrecimiento por deposito electroquımico.

F.2 Aportaciones originales

A continuancion presentamos la estructura del la tesis y sus aportaciones origi-nales. La tesis se ha dividido en varios capıtulos, y cada uno tiene sus propriasconclusiones (debido a su caracter, hemos omitido las conclusiones del capıtulo 2).Para mejorar la legibilidad de la tesis hemos recogido en ap´endices los detalles delos calculos y los resultados menores. En el capıtulo finalhemos resumido nuestrasconclusiones desde una prespectiva unificada y presentamosbrevemente las lıneasde investigacion relacionadas con los resultados de esta tesis que abordaremos enel futuro.

En el segundo capıtulo, introducimos la mayorıa de los conceptos y las herra-mientas que utilizaremos a lo largo de la tesis. En primer lugar definimos losobservables que cuantifican el proceso de rugosidad cinetica y su comportamientopara las superficies con invariancia de escala. En esta parteno examinamos so-lamente superficies que satisfacen el ansatz de Family-Vicsek sino que considera-mos tambien las que se caracterizan por scaling anomalo y por scaling anisotropo.Ademas, introducimos la nocion de clase de universalidady analizamos como lainvariancia de escala puede coexistir con inestabilidadesmorfologıcas y, por tanto,

F.2 Aportaciones originales 185

con la formacion de un patron. Finalmente, damos algunos detalles sobre lastecnicas analıticas que utilizamos esta tesis, a saber elgrupo de renormalizaciondinamico, y sobre el esquema numerico que utilizamos paraintegrar las ecuacionesinterfaciales.

En la primera parte del tercer capıtulo, resumimos las propiedades del mode-lo de frontera movil que M. Castro estudio parcialmente ensu tesis doctoral paradescribir el crecimiento difusivo [1]. Este modelo unifica en la misma descripcionteorica el crecimiento difusivo de intercaras obtenidas por el proceso de depositoquımico de vapor y el proceso de deposito electroquımico. En la segunda parte deeste capıtulo, el lector encuentra la primera aportacionoriginal de la presente tesis.De hecho, focalizamos nuestro estudio sobre las ecuacionesinterfaciales efectivas(lineales y no-lineales) que obtenemos desde un desarrolloperturbativo del modelode frontera movil en el lımite de pendientes pequenas. Gran parte de este analisisha sido publicado en(1)1 y (2), mientras que los restantes resultados son parte delpreprint(3).

En el cuarto capıtulo, utilizamos extensivamente la tecnica del grupo de renor-malizacion dinamico para obtener el comportamiento asintotico (crıtico) de la ecua-cion interfacial no local que hemos derivado en el capıtulo anterior. Generalizamosesta ecuacion a una familia entera cuyos miembros se caracterizan por ser ecua-ciones no-locales, no-lineales, y, ademas, tener una inestabilidad morfologica. De-bido a la universalidad de esta descripcion continua, podemos predecir los expo-nentes crıticos de la familia entera. Ademas, gracias a laintegracion directa de lasecuaciones interfaciales, corroboramos nuestas predicciones analıticas con la es-timacion numerica de los exponentes crıticos. Los resultados mas importantes deeste capıtulo estan contenidos en el artıculo(4) y el preprint(5), mientras que losdetalles del estudio con el grupo de renormalizacion dinamico de las ecuacionesno locales se pueden encontrar en(6).

En el quinto capıtulo, nos alejamos del sistema fisico principal de nuestras in-vestigaciones (aunque las herramientas empleadas son las mismas) y examinamosun resultado novedoso observado en experimentos recientesde formacion de pa-trones por erosion ionica. La version anisotropa de la ecuacion de Kuramoto-Sivashinsky es una importante ecuacion interfacial derivada desde primeros prin-cipios, que reproduce muchos comportamientos experimentales tıpicos de estecontexto. Una vez mas, utilizando el grupo de renormalizacion dinamico pro-ponemos una explicacion teorica de la evolucion temporal del patron de ondula-ciones obtenido en las simulaciones numericas de esta ecuacion, lo que propor-ciona un argumento formal para la interpretacion de algunas observaciones exper-imentales. Este resultado teorico es parte del preprint(7).

En el sexto capıtulo, gracias a un esquema numerico multimalla, utilizamos unmodelo de intercara difusa con fluctuaciones para integrar el modelo de fronteramovil, y, consecuentemente, superar las limitaciones della aproximacion de pen-dientes pequenas. En este capıtulo, despues de una primera calibracion numerica

1Las referencias entre parentesis se proporcionan en las siguiente seccion.

186 Resumen en Castellano

del modelo, estudiamos las complejas morfologıas obtenidas por crecimiento electro-quımico a largos tiempos. La interaccion entre las estructuras que crecen en la su-perficie y el campo difusivo de largo alcance produce una intercara efectiva com-puesta por las regiones activas del agregado. Abarcando todas la escalas macro-scopicas relevantes para el proceso de deposito electroquımico podemos investigarel fenomeno de la rugosidad cinetica de esta intercara. Este analisis ha sido publi-cado en(2), mientras que la formulacion del modelo de intercara difusa es el temacentral del preprint(3).

F.3 Artıculos publicados y preprints

(1) M. Nicoli, M. Castro y R. Cuerno,Unified moving-boundary model with fluc-tuations for unstable diffusive growth, Physical Review E78, 021601 (2008).

(2) M. Nicoli, M. Castro y R. Cuerno,Kinetic roughening in a realistic modelof non-conserved interface growth, Journal of Statistical Mechanics: Theory andExperiment P02036 (2009).

(3) M. Nicoli, M. Plapp, M. Castro y R. Cuerno,Surface kinetics in a phase-fieldmodel of diffusive growth, preprint (2009).

(4) M. Nicoli, R. Cuerno y M. Castro,Unstable nonlocal interface dynamics, Phys-ical Review Letters102, 256102 (2009).

(5) M. Castro, R. Cuerno, M. Nicoli, J. G. Buijnsters y L. Vazquez, The physicsof cauliflower-like growth, preprint (2009).

(6) M. Nicoli, R. Cuerno y M. Castro,Dynamical renormalization group analysisof nonlocal unstable interface equations, preprint (2009).

(7) A. Keller, M. Nicoli, R. Cuerno, S. Facsko y W. Moller,Pattern rotation in theanisotropic stochastic Kuramoto-Sivashinsky equation, preprint (2009).

F.4 Conclusiones alcanzadas

El estudio del modelo de frontera movil introducido por M. Castro y R. Cuerno haevidenciado que la forma funcional de la relacion de dispersion lineal depende delcoeficiente cinetico de transporte de masa, con importantes consecuencias para elproceso de formacion de patrones y su comportamiento asintotico. Ademas, nues-tra teorıa lineal y no-lineal explica cuantitativamente los datos obtenidos por deBruyn en experimentos de deposito electroquımico; a la luz de nuestros resultados,podemos afirmar que los exponentes crıticos encontrados enla literatura se debena efectos de tamano finito. El estudio dinamico de la ecuacion MSKPZ ha eviden-ciado que la interaccion entre la no-localidad, la no-linealidad y las fluctuacionesproduce morfologıas auto-similares (en el espacio y el tiempo) caracterizadas porla relacionα = z = 1 entre el exponente de rugosidad y el exponente dinamico.

F.4 Conclusiones alcanzadas 187

Ademas, hemos podido estimar los exponentes crıticos de la familia de ecuacionesno locales que generaliza la MSKPZ. Hemos observado que la competicion entreel termino de “difusion superficial” anomala y la no-linealidad de KPZ controlael comportamiento de escala de estas ecuaciones, y hemos aportado un argumentoteorico mas para explicar la dificultad en observar el scaling de KPZ en experimen-tos de crecimiento. Por otra parte, hemos demonstrado que larotacion de90 delpatron de ondulaciones observada en experimentos recientes de erosion ionica sepuede explicar a traves de la aplicacion del grupo de renormalizacion dinamico ala version anisotropa de la ecuacion de Kuramoto-Sivashinsky.

Finalmente, utilizando una formulacion de intercara difusa del modelo de fron-tera movil, hemos podido estudiar la formacion de morfologıas complejas similaresa las que se obtienen en experimentos de deposito electroquımico. Conforme anuestro modelo, el scaling anomalo observado en los experimentos se debe al de-sarrollo de un patron de columnas y a que la intercara deja deser univaluada. Atamanos comparables con la longitud de difusion, la rugosidad de estas intercarasno sigue una ley de potencias.

References

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[5] C. A. Haselwandter and D. D. Vvedensky, “Scaling of ballistic depositionfrom a Langevin equation,”Phys. Rev. E, vol. 73, p. 040101, 2006.

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[8] C. A. Haselwandter and D. D. Vvedensky, “Renormalization of stochasticlattice models: Epitaxial surfaces,”Phys. Rev. E, vol. 77, p. 061129,2008.

[9] M. Nicoli, M. Castro, and R. Cuerno, “Unified moving-boundary modelwith fluctuations for unstable diffusive growth,”Phys. Rev. E, vol. 78,p. 021601, 2008.

[10] J. M. Lopez, M. A. Rodrıguez, and R. Cuerno, “Superroughening versusintrinsic anomalous scaling of surfaces,”Phys. Rev. E, vol. 56, p. 3993,1997.

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