Structure and Dynamics of Polymer Filmsperso.univ-rennes1.fr/denis.morineau/main-frame... ·...

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Introduction to Polymer Physics Generic Polymer Models Molecular Simulation Methods Polymer Films I: Chain Extension Polymer Films II: Glass Transition Summary Structure and Dynamics of Polymer Films J. Baschnagel, S. Peter, H. Meyer, J. P. Wittmer Institut Charles Sadron Université Louis Pasteur Strasbourg, France NanoSoft Nanomatériaux, Surfaces et Objets FoncTionnalisés 21–25 May 2007 / Roscoff

Transcript of Structure and Dynamics of Polymer Filmsperso.univ-rennes1.fr/denis.morineau/main-frame... ·...

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Structure and Dynamics ofPolymer Films

J. Baschnagel, S. Peter, H. Meyer, J. P. Wittmer

Institut Charles SadronUniversité Louis Pasteur

Strasbourg, France

NanoSoftNanomatériaux, Surfaces et Objets FoncTionnalisés

21–25 May 2007 / Roscoff

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Outline

Introduction to Polymer Physics

Generic Polymer Models

Molecular Simulation Methods

Polymer Films I: Chain Extension

Polymer Films II: Glass Transition

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Polymer: Definition and ConformationPolymer := macromolecule of N monomers

conformation:x = (~r1, . . . ,~rN) (monomer positions)x = (~r1, ~b1 . . . , ~bN−1) (bond vectors)

Example: polyethylene (monomer = CH2)

persistence length `p ∼ 5 Å

bond length `0 ∼ 1 Å

θ

φ

end-to-enddistance

radius of gyration Rg

N =104: Re∼103Å

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Polymer PhysicsPolymer physics = calculate macroscopic properties

A(x ) from microscopic interactions U(x )

〈A〉 =1Z

∫dx A(x ) exp

[− βU(x )

], β =

1kBT

Assumption about U:

U(x ) =N−1∑i=1

U0(~bi , ~bi+1, . . . , ~bi+imax)︸ ︷︷ ︸“short range”: `,θ,φ

+U1(x , solvent)︸ ︷︷ ︸“long range”

`p

`0

θ

φRg

Re

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Ideal ChainsIdeal polymer := U1 = 0

`p := “memory” of orientation along the chain

`p = `0

∞∑k=0

〈b̂i · b̂i+k 〉 (b̂i = unit vector)

R2e =

N−1∑i=1

N−1∑j=1

〈~bi · ~bj〉 = 2`20

N−1∑i=1

N−1∑j=i

〈b̂i · b̂j〉 − (N − 1)`20

' 2`20

N∑i=1

∞∑k=0

〈b̂i · b̂i+k 〉︸ ︷︷ ︸=`p/`0

−N`20 = N

[2`0`p − `2

0

]︸ ︷︷ ︸

= b2e (effective bond length)

Result:N ↔ t and Re =̂ distance covered in time t

Re ∝ N1/2 (Brownian motion or random walk)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Real ChainsReal polymer := U0 + U1, good solvent: U1 = repulsive

Re ∝ Nν 12

< ν < 1 (ν = 0.588)

`p

`0

θ

φ

local propertiesdepend on chemistry

RgRe

global properties = universal:

polymer ↔ critical system1/N ↔ (T − Tc)/Tc = τ

Re ∝ Rg ∼ Nν ↔ ξ ∼ τ−ν

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Coarse-Graining the Model

b

Θ

I spatial continuum

I `, θ, φ, . . . =continuous variables

I realistic potentials

self-avoiding walk

b

Θ

I lattice (e.g. simple cubic)

I b = lattice constantΘ = 90◦, 180◦

I connectivity, excluded volume

When can this be a viable model?

I no long-range (e.g., electrostatic) orspecific (e.g., H-bonds) interactions

I generic properties

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

A Lattice and a Continuum Model

Bond-Fluctuation Model

BFM = lattice modelI ~b ∈ B with:

b = 2,√

5,√

6, 3,√

10I hard-core interaction

b

Bead-Spring Model

BSM = continuum model

ULJ(r) = 4ε

[(σ

r

)12−

r

)6]

LJε b0

bond

Ubond(b) =k2

(b − b0

)2

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

A Lattice and a Continuum Model

Bond-Fluctuation Model

BFM = lattice modelI ~b ∈ B with:

b = 2,√

5,√

6, 3,√

10I hard-core interaction

b

Monte Carlo simulation

Bead-Spring Model

BSM = continuum model

ULJ(r) = 4ε

[(σ

r

)12−

r

)6]

LJε b0

bond

Ubond(b) =k2

(b − b0

)2

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

A Lattice and a Continuum Model

Bond-Fluctuation Model

BFM = lattice modelI ~b ∈ B with:

b = 2,√

5,√

6, 3,√

10I hard-core interaction

b

Monte Carlo simulation

Bead-Spring Model

BSM = continuum model

ULJ(r) = 4ε

[(σ

r

)12−

r

)6]

LJε b0

bond

Ubond(b) =k2

(b − b0

)2

Monte Carlo orMolecular Dynamics

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Molecular Dynamics (MD) SimulationsMD: numerical solution of Newton’s equations of motion

d2~ri

dt2 =1m

~Fi ⇐⇒

d~ri

dt=

1m

~pi =∂H∂~pi

d~pi

dt= ~Fi = −∂H

∂~ri

discretization−→~ri(tµ + h) = ~ri(tµ) +

~pi(tµ)

mh +

~Fi(tµ)

2mh2

~pi(tµ + h) = ~pi(tµ) + ~Fi(tµ) h

Observables:

A != lim

t→∞

1t

∫ t

0dt A(x (t))

M�1≈ 1

M

M∑µ=1

A(x (tµ))

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

From MD to Monte Carlo Simulations . . .MD = deterministic dynamics in phase space

x = (~r1, . . . ,~rN ; ~p1, . . . , ~pN)

d~ri

dt=

1m

~pi

d~pi

dt= ~Fi

⇒microcanonical ensemble

%(x ) ∝ δ(H(x )− E

)MD with noise := introduce a weak stochastic damping

and a random force =̂ “Langevin thermostat”

d~ri

dt=

1m

~pi

d~pi

dt=

[~Fi −

ζ

m~pi

]+~fi(t)

⇒canonical ensemble

%(x ) ∝ e−βH(x )

〈~f (t)〉 = 0

〈fα(t)fβ(t ′)〉 = 2kBT ζ δαβδ(t − t ′

)random force

←→ − ζ

m~pi

friction

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

. . . From MD to Monte Carlo Simulations

Brownian dynamics := |�

~pi | � |ζ~pi/m| ⇒ stochasticdynamics in configuration space x = (~r1, . . . ,~rN)

ζd~ri

dt=

ζ

m~pi = ~Fi +~fi = −∂U(x )

∂~ri+~fi

stationary distribution =canonical distribution

%(x ) ∝ e−βU(x )

MC := generation of a sequence of correlatedconfigurations x via a Markov process

· · · xW (x→x ′)−→ x ′ · · ·

present state future state

accept transition according to the Metropolis criterion

W (x → x ′) = min(

1, e−β[U(x ′)−U(x )])

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Monte Carlo (MC) Simulations

Example (local moves BFM)

I choose a monomer andjump direction at random

I accept jump ifI excluded volume is satisfiedI ~b ∈ B

b

Example (local moves BSM)

~ri

~r ′i

x = (~r1, . . . ,~rN) : U(x )

displacement: ~ri → ~r ′i = ~ri + ∆~r

⇔ x → x ′ ⇒ U(x ′)

accept according to

min(

1, e−β[U(x ′)−U(x )])

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Monte Carlo (MC) Simulations

Example (local moves BFM)

I choose a monomer andjump direction at random

I accept jump ifI excluded volume is satisfiedI ~b ∈ B

b

Example (nonlocal moves)~b

~b ′

1. slithering-snake movescut ~b at one chain end andpaste ~b ′ at the other end

2. double-pivot movesconnectivity-altering move

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Monte Carlo (MC) Simulations

Example (local moves BFM)

I choose a monomer andjump direction at random

I accept jump ifI excluded volume is satisfiedI ~b ∈ B

b

Example (nonlocal moves)

1. slithering-snake movescut ~b at one chain end andpaste ~b ′ at the other end

2. double-pivot movesconnectivity-altering move

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Flory and Silberberg Hypotheses . . .

BFM (J. Wittmer)

Re = beNν

I single chain:ν = 0.588

I melt: ν = 12

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Flory and Silberberg Hypotheses . . .

BFM (J. Wittmer)

Re = beNν

I single chain:ν = 0.588

I melt: ν = 12

ideal behavior?

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Flory and Silberberg Hypotheses . . .

BFM (J. Wittmer)

Re = beNν

I single chain:ν = 0.588

I melt: ν = 12

ideal behavior?

Swelling in dilute solutionb

`0θ

φ

excludedvolume

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Flory and Silberberg Hypotheses . . .

BFM (J. Wittmer)

Re = beNν

I single chain:ν = 0.588

I melt: ν = 12

ideal behavior?

Swelling in dilute solutionb

`0θ

φ

excludedvolume

What changes in a melt?

sphere of size R(s)

s monomers

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

. . . Flory and Silberberg HypothesesFlory: “Chains in a melt are (nearly) ideal.”

size R(s)

s monomersI strong overlap:

ρchain ∝ s/R3(s)� ρ (1� s ≤ N)I potential of mean force:

1− ρchain/ρ = e−U ' 1− U

⇒ U(s) ∝ sρR3(s)

R∼√

s∼ 1√s� 1

Silberberg: “Parallel chain extension is bulk-like.”

R2 = R2x + R2

y + R2z

independentcomponents

for ideal chains

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Test of Silberberg’s Hypothesis

Radius of gyration:

h ' 4Rbulkg

Special case: 2D melt

I radius of gyration

N ∼ R2g = Rd

g (compact)

I segregated chainsI irregular shape

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Why is Silberberg’s Hypothesis Violated?“3D”: h� Rbulk

g

“2D + ε”: Rbulkg � h

Chain overlap:

Bulk: ρchain ∝N

Rbulk 3g

� ρ

Film: ρchain ∝N

Rbulk 2g h

� ρ

I h� Rbulkg : 3D behavior

I h� Rbulkg , but non-vanishing

overlap: “2D + ε”

Corrections to ideality in “2D + ε” films:(Semenov, Johner, EPJE 2003)

R2e(N) ' Nb2

e

[1 + f (h)

4b2

eln N

], f (h) =

14πρ h

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Corrections to Ideality

Theory for “2D + ε”:

R2e(h)

Rbulk 2e

− 1

=4b2

e

ln N4πρ h

Further implications:I chain relaxation = exponentially slow with N

[A. N. Semenov, PRL 80, 1908 (1998)]

I there are also deviations from ideality in the 3D bulk[J.P. Wittmer et al, EPL 77, 56003 (2007)]

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Experimental ResultsThickness Dependence of the Glass Transition Temperature

Example

SiOx

air

polystyrene thicknessmonomer-surface

interaction≈ weak

103 <N <3·104 ⇔ 7.5 nm<Rg <42 nm

PALS

grafted chains

ellipsometry, BLS, X-ray

T 0g = 373 K

[J. A. Forrest, K. Dalnoki-Veress (2001)]

N = 20 ⇔ Rg = 1.3 nm

T 0g = 327 K

[S. Herminghaus et al (2001)]

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Introduction to the Glass TransitionGlass := frozen liquid⇔ amorphous solid

crystal(ordered solid)

liquid

glass(amorphous solid)

T < Tm

T � Tg

T ≤ Tgt = t0

t � t0

Properties on cooling toward T g :I structure = liquid-likeI relaxation time τ � than in a liquid

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

BSM in the Bulk: Structure and RelaxationBSM

w(q) =1N

N∑a,b=1

⟨e−i~q·

[~r a

i −~r bi

] ⟩S(q) = w(q) + ρh(q)

intra inter

rsc ' 0.1(Lindemann)

Mode-coupling theory:

τ ∼ (T − Tc)−γ

Result: Tc ' 0.405

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Modeling Polymer Films

I polymer = BSM goto BSM

I constant pressure MD:p = 0 (,1)

I N = 10, . . . (Ne ≈ 35)I different types of films:

I supported films: smooth,attractive walls

Uw(z) = εw

[(1z

)9

− fw

(1z

)3 ](εw = 3, fw = 1)

I free-standing filmsI supported + capped films:

smooth, repulsive walls(εw = 1, fw = 0)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Modeling Polymer Films

I polymer = BSM goto BSM

I constant pressure MD:p = 0 (,1)

I N = 10, . . . (Ne ≈ 35)I different types of films:

I supported films: smooth,attractive walls

Uw(z) = εw

[(1z

)9

− fw

(1z

)3 ](εw = 3, fw = 1)

I free-standing filmsI supported + capped films:

smooth, repulsive walls(εw = 1, fw = 0)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Modeling Polymer Films

I polymer = BSM goto BSM

I constant pressure MD:p = 0 (,1)

I N = 10, . . . (Ne ≈ 35)I different types of films:

I supported films: smooth,attractive walls

Uw(z) = εw

[(1z

)9

− fw

(1z

)3 ](εw = 3, fw = 1)

I free-standing filmsI supported + capped films:

smooth, repulsive walls(εw = 1, fw = 0)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

BSM: Bulk versus Film DynamicsMean-square displacements

ga(t) =⟨[~r a(t)−~r a(0)]2

⟩→

{gN/2(t)

g0(t) = 1N

∑Na=1 ga(t)

Example (supported + capped film)

rsc ' 0.1(Lindemann)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

BSM: Bulk versus Film DynamicsMean-square displacements

ga(t) =⟨[~r a(t)−~r a(0)]2

⟩→

{gN/2(t)

g0(t) = 1N

∑Na=1 ga(t)

Example (supported and free-standing films)

Relaxation time

gx(τ)!= 1

Mode-coupling theory

τ ∼ (T − Tc)−γ

⇒ Tc(h)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Thickness Dependence of T c(h)

Tg(h) =Tg

1 + h0/h(Tg = bulk-Tg, h0

!= γ/E)

[S. Herminghaus et al. EPJE (2001)]

[S. Peter et al., J. Polym. Sci. 44, 2951 (2006)]

I Tc from MD(N = 10)

I Tg(PP): atomisticMC simulations offree-standingpolypropylene films(N = 50)

I Tg(PS): supportedPS films

I low Mw(N = 20)

I high Mw(N = 30000)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Thickness Dependence of T c(h)

Tg(h) =Tg

1 + h0/h(Tg = bulk-Tg, h0

!= γ/E)

[S. Herminghaus et al. EPJE (2001)]

[S. Peter et al., J. Polym. Sci. 44, 2951 (2006)]

I Tc from MD(N = 10)

I Tg(PP): atomisticMC simulations offree-standingpolypropylene films(N = 50)

I Tg(PS): supportedPS films

I low Mw(N = 20)

I high Mw(N = 30000)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Thickness Dependence of T c(h)

Tg(h) =Tg

1 + h0/h(Tg = bulk-Tg, h0

!= γ/E)

[S. Herminghaus et al. EPJE (2001)]

[S. Peter et al., J. Polym. Sci. 44, 2951 (2006)]

I Tc from MD(N = 10)

I Tg(PP): atomisticMC simulations offree-standingpolypropylene films(N = 50)

I Tg(PS): supportedPS films

I low Mw(N = 20)

I high Mw(N = 30000)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Thickness Dependence of T c(h)

Tg(h) =Tg

1 + h0/h(Tg = bulk-Tg, h0

!= γ/E)

[S. Herminghaus et al. EPJE (2001)]

[S. Peter et al., J. Polym. Sci. 44, 2951 (2006)]

I Tc from MD(N = 10)

I Tg(PP): atomisticMC simulations offree-standingpolypropylene films(N = 50)

I Tg(PS): supportedPS films

I low Mw(N = 20)

I high Mw(N = 30000)

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Layer-Resolved DynamicsMean-Square Displacements

Supported Films Free-Standing Films

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Layer-Resolved DynamicsLocal Relaxation Times

Example (supported films)

I center:

τ ' τbulk

I free surface andinterface withsubstrate have:

τ < τbulk

I surface effectspenetrate moredeeply on cooling

Introduction toPolymer Physics

Generic PolymerModels

MolecularSimulationMethods

Polymer Films I:Chain Extension

Polymer Films II:Glass Transition

Summary

Summary

I Coarse-grained models := generic modelsI monomers: LJ-spheres or unit cells on a latticeI connectivity, LJ- or hard-core interactions

I Case study: BFMI MC simulations: fast equilibration of long chains via

nonlocal movesI chain conformations: large-scale features are

universalI deviations from chain ideality in thin films and

in the bulkI Case study: BSM

I MD simulations: natural choice for studying dynamicsI glass transition: local phenomenonI faster dynamics in supercooled polymer films due to

surface effects

AppendixFor Further Reading

For Further Reading

R. A. L. Jones and R. W. Richards.Polymers at interfaces and surfaces.Cambridge University Press, 1999.

E. Donth.The glass transition.Springer, 2001.

A. Cavallo et al.Single chain structure in thin polymer films:corrections to Flory’s and Silberberg’s hypotheses.J. Phys.: Condens. Matter 17, S1697 (2005).

S. Peter et al.Thickness-dependent reduction of the glass transitiontemperature in thin polymer films with a free surface.J. Polym. Sci.: Part B 44, 2951 (2006).