Polymer Dynamics

Tom McLeish Durham University, UK

(see Adv. Phys., 51, 1379-1527, (2002))

Boulder Summer School 2012: Polymers in Soft and Biological Matter

Schedule

• Coarse-grained polymer physics • Experimental probes of polymer dynamics • Local friction - the Rouse chain • Hydrodynamics - the Zimm chain • Entangled Dynamics

Coarse-grained Polymers

n

R(n,t)

n=0

n=N

R

Polymers as Random Walks

Force, f, from free energy F(R)=-kBTlnP

G0=kBTCmon/Nx

Experiment - techniques that probe polymer dynamics:

• Bulk information orthogonal to rheology • Direct Molecular information

Dielectric Spectroscopy

T1/2-NMR

SANS NSE

PCS

Self-Diffusion

PFGNMR

Simulation

Linear and non-linear rheology

Rheology

tension

log t

log G(t)

τmax

G0

N

N

Shear

Extension

100 101 102 103

time /s104

105

106

107

visc

osity

/ P

a.s

Frequency-dependent Shear rheology

( )

( ) 22

22

0

220

/0

1''

1'

)(

τωτωω

τωωτω

τ

+=

+=

⇒= −

GG

GG

eGtG t

Stress Tensor

( ) ( )n

tnRn

tnR∂

∂∂

∂ ,,~

Linear Polymers

Branched Polymers

104 105

~ Mw3.4

~ exp(ν’Ma/Me)

Span Mw

η (P

oise

)

Fetters and Pearson (1983)

10 -2 10 0 10 2 10 4 10 -3

10 -2

10 -1

10 0

frequency /s

norm

alis

ed re

spon

se fu

nctio

ns

Dielectric Spectroscopy for Z=16 stars (Watanabe 2002) ( ) ( )

nnR

ntnR

∂∂

∂∂ 0,

','

rheology

Neutron Scattering

θ

Ii

I

θλπ sin4

=q

)_()( nscorrelatiodensitynsformFourierTraqI =

Sees inside the chain (“higher q”)

SANS on Deformed Chains

Linears - Muller et al. (1993)

Hs - Heinrich et al. (2002)

[ ]{ }),'(),(.exp'1

002 tntnidndn

N

NN

RRq −∫∫

Neutron Spin Echo

[ ]{ })0,'(),(.exp'1

002 ntnidndn

N

NN

RRq −∫∫

0 100 200 ns

Wischnewski et al. (2002)

Transverse (T2) NMR

Klein et al. (1998)

( ) ( )

∂∂

∂∂∆

∫ ntn

ntndt

b

tb ',..

'','

23cos

02

RLR

Diffusion Measurements (NMR, NR, SIMS..)

Komlosh and Callaghan (1999)

Lodge (1999)

0.1

1

10

100

10 2 10 3 10 4 10 5 10 6

g2

(t), g

3(t)

[σ

2]

t [ τ ]

slope 1/2

slope 0.26

slope 1/2

slope 1

Simulation

Putz et al. (2000)

( )2)0,(),( ntn RR −

(in this case …)

Summary of Probes of Polymer Dynamics

The Rouse Model

n

R(n,t)

n=0

n=N

R(n,t+∆t) ( )

( )

( ) ( ) tRR

RtnD

DtR

eff

≈∆∆⇒

∆≈≈

≈∆

22

2

2

1)(

1

( ) 4/1tR ≈∆⇒

Diffusion in the Rouse Model

kTbN

NkT

NbDR

cmR

22

0

22

≅

≅≅

ζ

τRouse Time

Stress Relaxation in the Rouse Model

21

)()(

≅

≅

tkTc

tnNkTctG

RN

N

τ

Polymer solutions and Hydrodynamics: The Zimm Chain

•Polymers in solution: scaling, correlations •Dynamics: Zimm model, dilute and semi-dilute regimes

M Rubinstein and R Colby, Polymer Physics (2003)

R Larson, The Structure and Rheology of Complex Fluids

References:

Polymers in Solution: excluded volume

• Real polymer chains: excluded volume parameter

Flory: balance contact energy with chain entropy ) Swollen chains

• Phantom coil: Melt, near Θ-point

Semi-dilute regime

• Onset of multi-chain behaviour, interactions, enhanced viscoelasticity.

e.g. Raise T ) swell ) induce overlap increase viscosity!

Chain correlations • Sections of chain only see themselves at short distances:

monomers in a “blob”

blobs fill space

Flory/single chain scaling inside blob

Correlation length (e.g. light scattering)

Small distances: excluded volume negligible

Large distances: Random walk of blobs

1/2

0.588

1/2

Crossover to melt when: (Edwards screening)

Semi-dilute solutions: osmotic pressure

Slope = 1.32

Nocho et al., Macromolecules 1981

Diffusion: Zimm model • Local drag coefficient in Rouse model:

• In solution, include long range hydrodynamic drag:

(Stokes)

• Einstein Relation:

Zimm or Rouse……who cares?!

• Relax by Zimm modes (faster)

• Monomer diffusion up to a Zimm time : Zimm regime

ln(t)

ln<r2>

2/3

1 sub-Fickian diffusion

Fickian diffusion

( )ln Zτ

( )2ln Nb

( )2ln b

( )0ln τMolecular diffusion diffusion

(recall Rouse sub-Fickian t1/4)

Stress relaxation G(t) Count number of unrelaxed chain segments N/n(t)

2/3

[Decalin in Θ-solvent, Hair & Amis, 1989]

After Zimm time…..not much stress left!

Stress relaxation at later times…..

ln(t)

ln(G(t))

-2/3

( )ln Zτ

Intrinsic viscosity (dilute solution)

• Einstein calculation for colloids (spheres),

only due to surrounding hydrodynamics:

• Polymeric contribution in dilute solution:

Dilute solution ) Melt in Good Solvent?

dilute semi-dilute melt

Zimm Rouse

Rouse/Zimm together

ln(t)

ln(G(t)) -2/3

-1/2

Zimm relaxation up to screening length ξ

Rouse relaxation of blobs

Entangled Dynamics

The Problem

The “Solution”

Chain motions

• Linear polymers: reptation

• Branched polymers: arm retraction

lnφ(t)

1/2

1/4

1/4

1

τe τR τd

a2aRg

R2g

Diffusion in the reptation Model

1/2

log t

log G(t)

τmax

G0

N

N

Stress Relaxation in the reptation Model

End-retraction is an “activated process” over a thermal barrier ~M

( )Mντ exp~∴M

Star Polymers

The Star-arm fluctuation potential

1E3

1E4

1E5

1E6

1E7

1.0E-4 1.0E-3 1.0E-2 1.0E-1 1.0E0 1.0E1 1.0E2 1.0E3 1.0E4 1.0E5 1.0E6 1.0E7

Frequency /rad/s

G/Pa

G' linearG'' linear

'G’ starG'' star

Further Topics

• Non-linear Rheology • Quantitative Linear Entangled Dynamics • Rheology, Topology and the Pom-Pom

model • Workshop problems….

A non-linear view of entanglement:

• Bifurcation of Stretch and Orientation relaxation times

lnM

lnτ

lnMe

τstretch

τorient

Startup of extensional flow shows two nonlinearities in rate:

Quantative Theory: New Physical Processes

• Contour length fluctuation • Constraint Release

R(n,t)

•Linear regime - (Likhtman and McLeish, Macromolecules 2002, 35, 6332-6343)

• numerical solution of CLF • CR: Rubinstein and Colby (1988) • longitudinal stress relaxation

++= ∑∑

=

−−

=

− N

Zp

tpZ

p

tp

e

RR epZ

epZ

ctRtMRTctG ττ

νν µρ22 2

2

1

12

11151),()(

54),(

??? µ(t)=L(t)/L(0) - fraction of tube segments survived after time t

Linear Rheology ( ) ( )n

tnRn

tnR∂

∂∂

∂ ,,

Polystyrene, Shausberger et al, 1985, Mw=290K, 750K and 2540K,

Me=13K

1E-5 1E-4 1E-3 0.01 0.1 1 10 100 1000 10000100

1000

10000

100000

1000000

G',

G'' (

Pa)

ω (s-1)

10-2 10-1 100 101 102 103 104 105 106 107103

104

105

106

G',

G'' (

Pa)

ω (s-1)

Polybutadiene, Baumgaertel et al, 1992, Mw=20.7K, 44.1K, 97K and 201K

Rheology and Topology

-3 -2 -1 0 1 2 3 4Logw

4

4.25

4.5

4.75

5

5.25

5.5

5.75

Log

G'

Log

G''

ç

ç

çç

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-3 -2 -1 0 1 2 3 4Logw

4

4.25

4.5

4.75

5

5.25

5.5

5.75

Log

G'

Log

G''

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(a)

-4 -2 0 2 4Logw

3.5

4

4.5

5

5.5

Log

G'

Log

G''

çç

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(b)Figure 8 Experimental and theoretical complex moduli for the twoH-polymers of the table. Fitting parameters of G0 and τe wereconsistent with literature values, the number of entanglementsalong arm and crossbar, sa and sb together with theirpolydispersities εa and εb determined by GPC and SALS.Theoretical curves accounting for polydispersity are dashed; thosewithout are solid.

LCB Polymers in non-linear flow

Pom-pom constitutive equation

• highly entangled branch points

• long flexible backbone sections

• dangling ends

Stretch

Orientation

Stress

Represent a polydisperse (branched) polymer as a spectrum of pom-poms

Linear relaxation spectrum => τbi, gi

‘decorate’ these modes using nonlinear extensional data => qi, τsi

Multi-mode pompom - an example Steady State Viscosity

10-3 10-2 10-1 100 101 102 Strain Rate /s-1

103

104

105

106

107

Vis

cosi

ty /P

a·s

Extension

Shear

Transient Viscosity 0.001 s-1 0.003 s-1 0.010 s-1 0.030 s-1 0.100 s-1 0.300 s-1 1.000 s-1 3.000 s-1 10.000 s-1 30.000 s-1

Rates

10-1 100 101 102 103 104 Time /s 103

104

105

106

107

Vis

cosi

ty /P

a·s

Extension

Shear

First Normal Stress Difference in Shear

10-1 100 101 102 103 Time /s

0.0

0.5

1.0

1.5

2.0

2.5

Stre

ss /1

05 Pa 10 s-1

5 s-1 2 s-1 1 s-1

Rates

Data from Meissner (1972, 1975) and Münstedt and Laun (1979).

100

102

104

106

0.0001 0.01 1 1001

10

100

X riX qiX giA riA qiA gi

τb

g i

r i, qi

IUPAC A and shifted IUPAC X pom-pom parameters

LDPE

Comb 9

Metallocene

Classes of LCB and q-Spectra