Structure and Dynamics of Polymer Filmsperso.univ-rennes1.fr/denis.morineau/main-frame... · Introduction to Polymer Physics Generic Polymer Models Molecular Simulation Methods Polymer

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






Summary

Outline

Introduction to Polymer Physics

Generic Polymer Models

Molecular Simulation Methods

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






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






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)






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






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






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−

(σ

r

)6]

LJε b0

bond

Ubond(b) =k2

(b − b0

)2






Summary




b = 2,√

5,√

6, 3,√


b

Monte Carlo simulation

Bead-Spring Model


ULJ(r) = 4ε

[(σ

r

)12−

(σ

r

)6]

LJε b0

bond

Ubond(b) =k2

(b − b0

)2






Summary




b = 2,√

5,√

6, 3,√


b

Monte Carlo simulation

Bead-Spring Model


ULJ(r) = 4ε

[(σ

r

)12−

(σ

r

)6]

LJε b0

bond

Ubond(b) =k2

(b − b0

)2

Monte Carlo orMolecular Dynamics






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))






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






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 )])






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 )])






Summary





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






Summary





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






Summary

Flory and Silberberg Hypotheses . . .

BFM (J. Wittmer)

Re = beNν

I single chain:ν = 0.588

I melt: ν = 12






Summary


BFM (J. Wittmer)

Re = beNν


I melt: ν = 12

ideal behavior?






Summary


BFM (J. Wittmer)

Re = beNν


I melt: ν = 12

ideal behavior?

Swelling in dilute solutionb

`0θ

φ

excludedvolume






Summary


BFM (J. Wittmer)

Re = beNν


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






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






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






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






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)]






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)]






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






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






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)






Summary






Uw(z) = εw

[(1z

)9

− fw

(1z

)3 ](εw = 3, fw = 1)








Summary






Uw(z) = εw

[(1z

)9

− fw

(1z

)3 ](εw = 3, fw = 1)








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)






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)






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)






Summary


Tg(h) =Tg


!= γ/E)






I low Mw(N = 20)







Summary


Tg(h) =Tg


!= γ/E)






I low Mw(N = 20)







Summary


Tg(h) =Tg


!= γ/E)






I low Mw(N = 20)







Summary

Layer-Resolved DynamicsMean-Square Displacements

Supported Films Free-Standing Films






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






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).

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