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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−
(σ
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−
(σ
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−
(σ
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
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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).