Polymer Dynamics and Rheology

Polymer Dynamics and Rheology

2

Polymer Dynamics and Rheology Brownian motion

Harmonic Oscillator

Damped harmonic oscillator Elastic dumbbell model

Boltzmann superposition principle Rubber elasticity and viscous drag

Temporary network model (Green & Tobolsky 1946) Rouse model (1953)

Cox-Merz rule and dynamic viscoelasticity Reptation

The gel point

The Gaussian Chain Boltzman Probability

For a Thermally Equilibrated System

Gaussian Probability

For a Chain of End to End Distance R

By Comparison The Energy to stretch a Thermally Equilibrated Chain Can be Written

Forc e

Forc e

Assumptions:

-Gaussian Chain

-Thermally Equilibrated

4

Stoke’s Law

F = v V

V = 6 ph

R

F

Creep Experiment

6

Boltzmann Superposition

8

Stress Relaxation (liquids)

Creep (solids) ^{J t}

( )

( )

s Dynamic Measurement

Harmonic Oscillator: = 90° for all except = 1/ where = 0°

d w w t d

Hookean Elastic Newtonian Fluid

10

Brownian Motion

For short times For long times 0

12

http://mathworld.wolfram.com/First-OrderOrdinaryDifferentialEquation.

html

14

16

The response to any force field

18

20

22

Both loss and storage are based on the primary response function, so it should be possible to express a relationship between the two.

The response function is not defined at t =∞ or at ω = 0

This leads to a singularity where you can’t do the integrals

Cauchy Integral

24

W energy = Force * distance

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Parallel Analytic Technique to Dynamic Mechanical

(Most of the math was originally worked out for dielectric relaxation)

Simple types of relaxation can be considered, water molecules for instance.

Creep:

Instantaneous Response

Time-lag Response Dynamic:

e₀ Free Space e Material

e_u Dynamic material

D = e

E + P = e

e E

Rotational Motion at Equilibrium

A single relaxation mode, ^t relaxation

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Creep Measurement

Response

K = 1t

http://mathworld.wolfram.com/First-OrderOrdinaryDifferentialEquation.

html

Apply to a dynamic mechanical measurement

dg₁₂

( )

dt = iws₁₂⁰ J ^*

( )

( )

Single mode

Debye Relaxation Multiply by

(

)

iwt -1

( )

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Single mode

Debye Relaxation

Symmetric on a log-log plot

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Single mode

Debye Relaxation

More complex processes have a broader peak

Shows a broader peak but much narrower than a Debye relaxation

The width of the loss peak indicates the difference

between a vibration and a relaxation process Oscillating system displays a moment of

inertia

Relaxing system only dissipates energy

Equation for a circle in J’-J” space

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dg

dt = -g

( )

t ^{+ DJ}s₀

Equilibrium Value

Time

Dependent Value

Lodge Liquid

Boltzmann Superposition

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Boltzmann Superposition

Rouse Dynamics Flow

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42

w

w

w

w

Newtonian Flow

Entanglement Reptation

Rouse Behavior

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w

w

w

w

Newtonian Flow

Entanglement Reptation

Rouse Behavior

Lodge Liquid and Transient Network Model

Simple Shear Finger Tensor

Simple Shear Stress

First Normal Stress

Second Normal Stress

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For a Hookean Elastic

For Newtonian Fluid

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Dumbbell Model

x t

( )

( )

x æ

èç

ö

ø÷ g t

( )

-¥

ò

Dilute Solution Chain Dynamics of the chain

Rouse Motion

Beads 0 and N are special For Beads 1 to N-1

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Dilute Solution Chain Dynamics of the chain

Rouse Motion

The Rouse unit size is arbitrary so we can make it very small and:

With dR/dt = 0 at i = 0 and N

Reflects the curvature of R in i,

it describes modes of vibration like on a guitar string

x, y, z decouple (are equivalent) so you can just deal with z

For a chain of infinite molecular weight there are wave solutions to this series of differential equations

V

^dz

dt = b

(z

- z

) + b

(z

- z

)

z

~ exp - t t æ èç ö

ø÷ exp ( ) il d

t

= b

2 - 2cos d

( ) ^{= 4b}

^sin

^d

Phase shift between adjacent beads

Use the proposed solution in the differential

equation results in:

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t

= b

z

( ^{2 - 2cos} d ) ^{= 4b}

z

^sin

2

d 2

Cyclic Boundary Conditions:

z

= z

R

N

d = m2 p

N^R values of phase shift

d

= 2 p

N

m; m= - N

2 -1 æ èç ö

ø÷ ,..., N

2

For N_R = 10

t

= b

z

( ^{2 - 2cos} d ) ^{= 4b}

z

^sin

2

d 2

Free End Boundary Conditions:

z

- z

= z

R-1

- z

R-2

= 0

N

-1

( ) ^{d = m p}

N^R values of phase shift

dz

dl ( l = 0 ) ^{= dz}

dl ( l = N

-1 ) ^{= 0}

For N_R = 10

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t

= 1 3 p

z

a

æ èç

ö ø÷

kT R

Lowest order relaxation time dominates the response

This assumes that ^zR a_R² æ èç

ö ø÷

is constant, friction coefficient is proportional to number of monomer units in a Rouse segment This is the basic assumption of the Rouse model,

z_R ~ a_R² ~ N

N_R = n_R

t

= 1 3 p

z

a

æ èç

ö ø÷

kT R

Lowest order relaxation time dominates the response

Since

R

= a

N

t

~ N

kT

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The amplitude of the Rouse modes is given by:

Z

= 2 3 p

R

m

The amplitude is independent of temperature because the free energy of a mode is proportional to kT and the modes are distributed by Boltzmann statistics

p Z ( )

^{= exp - F}

kT æ

èç

ö ø÷

90% of the total mean-square end to end distance of

the chain originates from the lowest order Rouse-modes so the chain can be often represented as an elastic

dumbbell

Rouse dynamics (like a dumbell response)

dx

dt = -

dU dx æ èç ö

ø÷

z ^{+ g(t) = -}

k

x

z ^{+ g(t)} x t ( ) ^{= dt'exp - t- t'}

t

æ èç ö

-¥

ø÷

ò

^{g t ( )}

Dumbbell Rouse

t

= z

4 b

sin

d 2 d = p

N

-1 m , m=0,1,2,...,N

-1

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Rouse dynamics (like a dumbell response)

g t ( )

^{g t} ( )

^{= 2D d ( ) t} where t = t

- t

and d ( ) is the delta function whose integral is 1

Also,

D = kT z x t ( ) ^{x 0} ( ) ^{= kT exp -}

t t æ èç ö

ø÷

k

t = z

k

x

= kT

k

Predictions of Rouse Model

G t ( ) ^{~ t}

G' ( ) w ^~ ( ^wh

)

h

= kT r

t

p

12 ~ N

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w

w

w

w

Newtonian Flow

Entanglement Reptation

Rouse Behavior

Dilute Solution Chain Dynamics of the chain

Rouse Motion

Rouse model predicts

Relaxation time follows N² (actually follows N³/df)

Diffusion constant follows 1/N (zeroth order mode is translation Predicts that the viscosity will follow N which

is true for low molecular weights in the melt and for fully draining polymers in solution

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Dilute Solution Chain Dynamics of the chain

Rouse Motion

Rouse model predicts

Relaxation time follows N² (actually follows N³/df)

Predicts that the viscosity will follow N which is true for low molecular weights in

the melt and for fully draining polymers in

solution

Chain dynamics in the melt can be described by a small set of

“physically motivated, material-specific paramters”

Tube Diameter dT

Kuhn Length l^K Packing Length p

Hierarchy of Entangled Melts

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Quasi-elastic neutron scattering data demonstrating the existence

of the tube

Unconstrained motion => S(q) goes to 0 at very long times Each curve is for a different q = 1/size

At small size there are less constraints (within the tube) At large sizes there is substantial constraint (the tube)

By extrapolation to high times a size for the tube can be obtained

d^T

There are two regimes of hierarchy in time dependence Small-scale unconstrained Rouse behavior

Large-scale tube behavior

We say that the tube follows a “primitive path”

This path can “relax” in time = Tube relaxation or Tube Renewal

Without tube renewal the Reptation model predicts that viscosity follows N³ (observed is N^3.4)

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Without tube renewal the Reptation model predicts that viscosity follows N³ (observed is N^3.4)

Reptation predicts that the diffusion coefficient will follow N² (Experimentally it follows N²)

Reptation has some experimental verification

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Reptation of DNA in a concentrated solution

Simulation of the tube

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Simulation of the tube

Plateau Modulus

Not Dependent on N, Depends on T and concentration

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Kuhn Length- conformations of chains <R²> = l^KL

Packing Length- length were polymers interpenetrate p = 1/(ρ^chain <R²>) where ρ^chain is the number density of monomers

this implies that d ~ p

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McLeish/Milner/Read/Larsen Hierarchical Relaxation Model

http://www.engin.umich.edu/dept/che/research/larson/downloads/Hierarchical-3.0-manual.pdf

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