VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000

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Interfacial Properties of Polymeric Liquids

Miguel Aubouy,1 Manoel Manghi,1 and Elie Raphaël21Service des Interfaces et des Matériaux Moléculaires et Macromoléculaires,* D.R.F.M.C., CEA-Grenoble,

38054 Grenoble cedex 9, France2Laboratoire de Physique de la Matière Condensée,† Collège of France, 75231 Paris cedex 5, France

(Received 22 February 2000)

We show that the variations of the surface tension of polymeric solutions g with the molecular weightof the sample Mn, the temperature T , and the volume fraction of monomers fb are explained by theself-organization of the long flexible chains that are present in the immediate vicinity of the interface.Within a scaling functional theory we find a simple law gfb, Mn, T which accounts quantitatively forthe experimental data available in the literature with a very high accuracy.

PACS numbers: 61.25.Hq

Concentrated polymer solutions made of long flexiblechains belong to complex liquids because their proper-ties are dominated by the self-organization of the chains.This is especially relevant at an interface, where the pres-ence of a sharp boundary condition modifies the three-dimensional structure of the fluid and induces the presenceof a self-structured inhomogeneous layer of chains. Thislayer is responsible for the interfacial properties of the liq-uid which can differ from the corresponding properties inthe bulk [1]. Most significantly, the interfacial tension ofthe polymeric phase g is a function of the molecular weight(MW) Mn of the macromolecules. In many applicationsof polymeric materials (coating, adhesion, colloidal stabi-lization, etc.), g is a crucial parameter, and this feature hasimportant consequences. For polydisperse samples, e.g., afraction of the chains will adsorb preferentially and mod-ify the composition of the interface in comparison to thatof the bulk. Polymeric liquids play a special role amongcomplex liquids, not only because they are widely spreadin the industry, but also because they have been studied ata fundamental level for many years now. It is clear thatunderstanding the interfacial properties of these will bringnew, broader insights into the interfacial properties of softcondensed matter in general. In this Letter, we present thefirst theory for the surface tension of polymeric solutionswhich is able to reproduce the variations of g with MW,temperature, T , and volume fraction of monomers, fb.

The variations of g with MW for molten polymer liq-uids, gfb 1, Mn, T , have received much experimen-tal attention. Data are usually fitted by the empirical lawg g` 2 kMy

n , with y 23 and y 1 for low MW

and high MW samples, respectively [2]. In the theoreti-cal literature, one may find three different approaches toexplain the empirical law: (i) Dee and Sauer were ableto fit the experimental data with few parameters throughan empirical equation of state for the bulk phase [2].However, the overall treatment needed to reach an agree-ment remains quite complicated. (ii) Using scaling argu-ments based on the idea of a specific tensioactivity of thechain ends, de Gennes recovers the empirical equation, but

0031-90070084(21)4858(4)$15.00

with y 12 [3]. (iii) Sophisticated self-consistent field

(SCF) approaches yield y 1 [4–7], and thus lead to noimprovement. Furthermore, g is predicted to be an in-creasing function of T , unlike what is reported in the ex-perimental literature. Much less is known about semidilutesolutions f

b # fb # 1. To the best of our knowledge,we know only about a modified Cahn-Hilliard theory [8]which yields Dg 2f

54b (good solvent), with no MW

dependence. However, this conclusion differs from the ex-perimental observations [9,10].

Here, we estimate gfb, Mn, T using a scaling func-tional theory (SFT) initially proposed to describe thestructure of adsorbed polymer layers [11]. This approachis a modern (simplified) version of an early theory byHoeve, Silberberg, and Di Marzio where the interface isdescribed as a thermodynamical ensemble of loops andtails very polydispersive in size [12,13]. The structureof the layer in the immediate vicinity of the interfaceresults from the competition between (a) the interactionsbetween the chain segments: two-body interactions (goodsolvent) or stretching (melt); (b) the entropy associatedto the large polydispersity in size of the loops that areformed at the interface; and (c) a specific attraction ofchain ends towards the interface. This entropy (b) was notconsidered in the previous theories for the surface tensionof a polymeric solution. On general grounds, we expectthat it contributes to g: The longer the chains are, thewider is the range of loop sizes accessible for adsorbedloops, and thus the larger is the entropy of the interface.Our theory which combines the three contributions yieldsa simple law gfb, Mn, T which accounts quantitativelyfor the experimental data. This law is first estimated inthe melt limit, and then generalized to an arbitrary volumefraction.

In what follows, the melt is a set of linear, neutral, ho-mogeneous, flexible, monodisperse chains of N monomersof size a. (Mn NMn0, where Mn0 is the MW of themonomer.) We assume that the air/melt interface is sharp,and fb 1. We distinguish between those chains whichhave at least a monomer in direct contact with the surface,

© 2000 The American Physical Society

VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000

“adsorbed” chains, and the others, “free” chains. Adsorbedchains form loops and tails of various sizes. Trains (chainsegments lying on the surface) are considered as a succes-sion of “loops of two monomers.” In the spirit of Ref. [11],we make the following simplifications: (i) Each loop of2n monomers is considered as two “pseudotails” of lengthn. In other words, we describe the interface as a thermo-dynamical set of tails, polydisperse in size, end attached

to the surface. (ii) The tails behave identically: Theirconformation is described by a single function nz, wheren is the curvilinear index along the tail, and z is the dis-tance to the surface na 1. (iii) The free chains do notpenetrate the layer formed by the adsorbed chains.

Our main tool is the “loop size profile,” S, such thatSn a22

RNn Pu du, where P is the statistical distri-

bution of tail sizes in monomer units. The free energy (percm2) of the adsorbed chains may be written as [11]

FadsS g0 1kBTa2

Z N

1

Ωk1a2Sn3 1 2a2S0n ln2a2S0n 2

2D

Na2Sn

ædn , (1)

where k1 1 is a constant, kBT is the thermal energy, andS0 ≠S≠n. The quantity g0 is the van der Waals energy(per cm2), which accounts for monomer/air interactions.The second term in the right-hand side (rhs) of Eq. (1) isthe elastic energy per unit surface:

RSn dzdn2 dn

rewritten using the local conservation of monomers:dzdn a3Sn. The third term is the usual entropyassociated with a set of polydisperse objects. The last termaccounts for chain-ends attraction. This attraction arisesbecause the last monomers have a polarizability slightlydifferent from that of a monomer in the middle of thebackbone [14]. The number (per cm2) of adsorbed chainsis GN

RSn dnN . Assuming an attractive energy

DkBT per chain ends in direct contact with the surface,and two adsorbed ends per polymer, this gives a contri-bution 22DNSn to the functional. We expect that D

should be of order unity and the third term in Eq. (1) doesnot contribute to the equilibrium profile. This argumentcan be made quantitative. Suppose D 0, minimizingEq. (1) yields the equilibrium loop-size profile: Seqn 2k1212a2n12 1 # n # N. Our approxima-tion is valid whenever a4Seqn3 ¿ 2DNSeqn, orn ø N4k1D in the domain of integration. If D is large,the solution Seqn n212 is not valid. In this strongadsorption regime, the adsorbed chains do not keep theirGaussian shape: They stretch away from the interface toincrease the number of adsorbed ends [11]. However, forsmall values of D (weak adsorption regime), N4k1D ¿N , the layer is not affected by this contribution to thefree energy. In physical terms, this means that there isa range of values 0 # D 6 14k1 where all the chainends are captured by the surface without disturbing theGaussian shape of the chains. In practice, there is afurther restriction to our model [3]: At very low D, onlya fraction of ends are captured. This means that Eq. (1) isvalid only for values of D close to 14k1. In what follows,we will assume that both conditions are fulfilled, andwe will use Eq. (1) together with the solution Seqn n212. The extension of the layer can then be estimated:H

RSeq dn aN12.

Let us now calculate the surface tension. The quan-tity g is the penalty for the presence of the free surface:g FadsSeq 2 Fbulk, where Fbulk is the free energyof our reference state, the bulk of the melt. The quan-

tity Fbulk has only one contribution: the elastic “energy”of Gaussian chains. Counting kBT per chain, we findFbulkkBTa2 HaN N212. Then, g can becomputed, yielding

gfb 1, Mn, T g` 1 DM12n0

kBTa2

lnMnMn

Mn12 ,

(2)

where g` g0 2 CkBTa2. The quantityM

n Be83DMn0 which appears in Eq. (2) playsthe role of a characteristic MW of the polymer underconsideration. With reasonable values D 1, we findN M

nMn0 50. (B, C, and D are numerical coef-ficients of order unity.) Qualitatively, Eq. (2) is in verygood agreement with the empirical law which has beenproposed. Indeed, when Mn , M

n , Eq. (2) is equivalentto g g` 2 kMy

n with an apparent exponent y, whichis both larger than 1

2 and an increasing function of theMW. Concerning the dependence with temperature, itis found that ≠g≠T is (a) negative, (b) of the order ofkBa2, (c) almost independent of the particular polymerstudied, and (d) a function of the MW of the sample [1].This is what we find with Eq. (2).

It is simple to generalize our findings to the semidiluteregime (good solvent) in the limit where the monomersare adsorbed at the interface [15]. This involves twosteps. First, Seq has to be renormalized: 1a2n12 !1j2nja5312, where j af

234b is the “blob”

size [16]. Thus Seqn f78b a2n12. Second, we

have to account for the increase of the volume fractionin the immediate vicinity of the surface. The bulk chem-ical potential per monomer is kBTf

54b . Assuming that

fz za243, as is expected in the dilute limit [8], wefind an extra contribution kBT a2f54

b

Rja fz dz

2kBT a2f54b to g. After some calculations, we find

Dgfb, Mn, T 2EkBTa2 f

54b

1 DM12n0

kBTa2

f78b

Mn12 ln

√Mn

f712b M

n

!,

(3)where E 1 is a numerical coefficient.

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VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000

FIG. 1. Surface tension of molten polymer solutions fb 1vs MW for three different polymers: PE at 150 ±C [17], PDMSat 20 ±C [18], and PTFE at 100 ±C [19]. The solid lines corre-spond to the theoretical law [Eq. (2)] fitted with two adjustableparameters D, N. See text and Table I for details.

Although we are working at a scaling level, we haveperformed a quantitative comparison with the data.Figure 1 displays the surface tension gMn for threedifferent polymers: polyethylene (PE), poly(dimethyl-siloxane) (PDMS), and poly(tetrafluoroethylene) (PTFE).We have taken g` g0 1 T≠g≠T jMn!` as given bythe experimental data (Table I). The data have thus beenfitted with Eq. (2) with two adjustable parameters (D, N,see Table I). The quantitative agreement is remarkablygood. For a given polymer, the validity of Eq. (2) canalso be tested by varying the temperature. Figure 2displays experimental data for PDMS at two differenttemperatures. Inserted in Fig. 2, we display two differentseries of experiments on PDMS, where the variationsgT at fixed Mn have been measured. In both cases, thesolid lines represent Eq. (2) with the values of N andD that we have found previously, i.e., with no adjustableparameters this time. Figure 3 displays the variations ofgfb for semidilute solutions of PDMS of different MW.Most significantly, we observe a splay of the differentcurves when fb increases, indicating that g is not simplythe sum of a pure MW and a pure fb contribution. Thissplay is well accounted for by the second term of the rhsof Eq. (3) with no adjustable parameter. The value ofE 0.46 6 0.01 is determined once for a given MW.The quantitative agreement with data is again very goodwithin, plausibly, a systematic experimental uncertainty.

We have spent some time in collecting and fitting theexperimental data for two reasons. The first reason is tosee whether our predictions would represent an improve-ment upon the different theories proposal before. Thesecond reason is to check whether our predictions arean artifact of our theory. There is an important concep-tual difficulty which is not explicit. Suppose g0 0 andD 0, and consider a piece of melt divided into twosemi-infinite spaces: in case (A) the separating surfaceis impenetrable; in case (B) the dividing plane is imagi-nary. Obviously, there is no interfacial tension in case

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TABLE I. Experimental values of the quantities a, Mn0 [20],g0, and 2≠g≠T jMn!` [1] for PDMS, PE, and PTFE. These,together with the fits, allow a precise determination of Dand N.

PE PDMS PTFE

a nm 1.08 0.73 3.7Mn0 gmol 98 185 2400g0 mNm 53.71 37.78 43.962≠g≠T jMn!`

1022 mNmK 6.187 5.786 6.377D 0.85 6 0.04 0.14 6 0.03 5.6 6 0.02N 27 6 3 95 6 5 2 6 1

(B). We find a tension in case (A) which has two ori-gins. Let us call (c), the monomers in contact with theplane, and (n), the others. Because the segments have to goback to the melt, each monomer (c) loses an orientationalchoice. This is the contribution kBT

RS3

eq dn kBTa2.We find a second contribution which we think is relatedto a broken symmetry. Because they experience a localforce due to the impenetrable wall, (c) monomers are dif-ferent. They become distinguishable. As we see it, thesituation is quite similar to the Gibbs paradox. In case(B), only (n) monomers are present: There is no physicalreason why we should distinguish those monomers whichtouch the imaginary plane from the others. In case (A), wemay distinguish two types of monomers, as if we were inthe presence of two different chemical species (c) and (n).Then, an entropy of mixing should be accounted for thisdifference. This is in our theory the origin of the contribu-tion kBT

R2S0

eq ln2S0eq dn kBTa2lnNN12 to g.

FIG. 2. Effect of temperature on the surface tension of moltenpolymer solutions fb 1. g versus Mn for PDMS at T 20 ±C (data taken from [18]) and T 30 ±C [21]. The insetdisplays the variations gT at fixed Mn for three different val-ues of the MW: 770 gmol (data taken from [22]), 3900 and75 000 gmol [23]. The solid line is the theoretical law [Eq. (2)]with no adjustable parameter.

VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000

FIG. 3. Surface tension of semidilute solutions of PDMS intoluene vs the monomer volume fraction, fb , for several differ-ent MW. The different symbols correspond to experimental dataof [9] (T 24 ±C, Mn: 500 kgmol) and [10] (T 19.5 ±C,Mn: 1 kgmol, 1.6 kgmol, 3.7 kgmol). Full lines are Eq. (3)(see text for details).

A careful comparison of our SFT approach with SCF the-ories is out of the scope of this Letter and will be doneelsewhere [15]. We speculate that this second contributionto g is lost in a description where the only relevant field,fz, does not account for chain connectivity. This wouldexplain why we get only the next to leading order contri-bution Dg N21 with a SCF theory.

Our model involves several simplifications. Asimple calculation shows that cutting a Gaussian loopof n monomers into two symmetrical, independent,grafted tails amounts to DFcut kBT 3

2 lnn 2 an 1

b per loop a, b 1. We expect an error DFads R212S0

eqDFcut dn, which can be rewritten as

212

R2S0

eq ln2S0eq dn 1 ON212. Because this

error does not bring any new leading contribution (anddoes not cancel any of the terms) in Eq. (2), we areconfident that hypothesis (i) does not modify our essentialresults (see also [11]). Once this is assumed, hypothesis(ii) is reasonable and has been used in SCF theories ofthe same problem [24]. Relaxing hypothesis (iii) wouldcertainly modify Seq, but we do not expect that the scalinglaw will be affected. A direct confirmation of that pointis provided by experiments where S is measured afterthe free chains have been removed [25]. We have alsoassumed an incompressible melt a const, as is usualin first approximation [3–7]. This may not always bethe case [26]. Of course, the air/liquid interface is arather restricted situation. Equations (2) and (3) hold for

mixtures of polymers and colloidal solids, and for blendsof polymers in the strong segregation limit.

We are grateful to Pierre-Gilles de Gennes, MohamedDaoud, Sanat Kumar, and Jeff Koberstein for useful discus-sions. We thank Jean-François Legrand, Thomas Waigh,and Avi Halperin for a critical reading of the manuscript.

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