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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
ç ççççççç ç ç ççç çç ççç çççççççç
ç ç çççç
ç çç çç ç
ç çç çç çç ç çç ç
ç
ó
óó
óó
óóó
óóóóóóóóóó
óóóóóóóó
óóóóóóóóóóóóó
óóóóóóóóóóóóóóóóóóóóóóóóó
óóóóóóóóóóóóóó
óóóóóó óó
óó óó óó
óó óó óó óó ó ó ó ó ó ó
-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
ç ççççççç ç ç ççç çç ççç çççççççç
ç ç çççç
ç çç çç ç
ç çç çç çç ç çç ç
ç
ó
óó
óó
óóó
óóóóóóóóóó
óóóóóóóó
óóóóóóóóóóóóó
óóóóóóóóóóóóóóóóóóóóóóóóó
óóóóóóóóóóóóóó
óóóóóó óó
óó óó óó
óó óó óó óó ó ó ó ó ó ó
(a)
-4 -2 0 2 4Logw
3.5
4
4.5
5
5.5
Log
G'
Log
G''
çç
çç ç ç ç ç
çççç
ç ç ç ç çççç ççç ç
çç ç ç ç ç ç ç ç ç
ó
ó
ó
óóóóóó ó ó ó óóó
óóóóóóó
óóóóóóóóó
óóóóóóóóóóóóóóó
óóóóóóóóóóóóóóó
óóóóóóóóóóóóó
óóóóóóóóó
óó óó óó óó óó óó ó ó ó ó ó
(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