Zero-shear viscosity vs. M (note change of slope)

M > Me

M < Me

Rouse

Berry + Fox, 1968

Entanglements

Edwards deGennes Doi

slope 1

slope 3.4

Question:

Which factors affect the Me:

T, P, M, flexibility, concentration (solution), branching, microstructure?

G(Φ)= G0 Φβ

Dilution: network swells in good solvent

Adam, 1975

Colby, Rubinstein, 1990

Me(φ)=Me F-α

b=2 or 7/3 a=b-1

J. D. Ferry 1912-2002

Viscoelastic properties of polymers

Viscoelastic element:

Viscous element:

Elastic element: G

G G

Characteristic relaxation time

Key relation

in linear viscoelasticity

Obtain time: kT

D

3

R

kT

Einstein 1905

ζ = friction

2

R

D

Fick 1905

R

Stokes 1905

f u

Summary:

Synthesis: N, b

Size: R=N1/2b

Distribution:

Modulus: G=νkT

Diffusion:

Hooke: F=Kx

Newton:

Viscoelasticity:

kT

D

=4.5x1019 Pa.s

granite

Halite (mineral form of salt)

Even some apparent solids “flow” if they are

observed long enough

Importance of the time scale

Question:

How many characteristic relaxation times exist in a polymer?

What are the implications for viscosity?

short time (local motion)

long time (global motion)

A gross classification of viscoelastic materials:

Glass / crystal

Rubber

Melt / solution

Important point: For a solid, macroscopic motion if frozen (long time is infinite)

whereas short-time (local) motion takes place

Glass:

- Local motions: free volume, cooperativity

- Cooling rate dependence

- Glass transition: out of equilibrium (metastable)

- Amorphous (“frozen”) solid

T

-1

Tg

Glass (T1) Crystal T1

- Non-equilibrium – (aging)

- Low temperatures (or short times)

Liquid

T2>T1

The Glass Transition

Tg

heat capacity change cp

specific volume change V

sp

ecific

volu

me V

heat c

apacity

cp

At the glass transition the polymer chain

segments undergo increased local

motions, which consume heat and leads

to an increase in the volume requirement

for the chain.

heat capacity

thermal expansion

Local motion in a dense polymer (melt)

sugar

Vitrification:

Crystal

Cotton candy

Semi-crystalline

polymer: Amorphous polymers:

Journal of Crystal Growth 48, 1980, p. 303/20

T

-1

Tg

Glass: G~109 Pa Crystal: G<107 Pa

Crystallization and Melting

> Tm

< Tc

Question:

Which are the implications of the existence of local short-time motion

in solids?

Think of self-healing (recent group seminar)

Rubber (elastomer) Thermosetting polymers (chemical network)

- solid-like (permanent or temporary)

- network

- T > Tg

G~105 -106 Pa

Vulcanisation:

Latex - liquid

Solid-like (gel)

Natural rubber

Hevea tree

G kT

Indian ‘boot’

Amazon (~500 bC)

Question:

How does vulcanization relate to the Indian boot example?

Think of Goodyear (~1839)

Compare metal vs elastomer

Thermoplastic polymers (physical network):

- Liquid

- temporary network

- Large scale motions

High temperatures (T>Tg),

Low times

Tube model: de Gennes,

Doi and Edwards (1986)

entanglements

0

N

e

R TG

M

Global motion (center of mass) in a flowing polymer melt

Compare permanent and temporary networks

time

0

time

P(t)

or

(t)

rel=

Temporary network

The molecules move in order to recover their stable coil-shape structure

time

(t)

Rubber (permanent network)

( ) expP tt

0 0

Question:

Can the same polymer behave as a glass and as a melt ?

Think of Tg

Important point:

We can tune the state of a polymer (and hence modulus and time) with T

T<Tg solid , glass

T>Tg solid, rubber

T >> Tg liquid

Alternatively, at a given temperature: very short time: glass ; very long time: liquid

This means that there is an interplay of the characteristic time of the material

and the temperature:

principle of time-Temperature Superposition (tTS) - to be discussed below

Importance of the time scale of the material in reference to observation time

time elastic viscous

Characteristic relaxation time

of the material

Use of dimensionless number:

Observation time versus Relaxation time

The same material behaves like a liquid or a solid, depending on the

observation time (experimental interrogation)

Deborah number: (M. Reiner)

exp

Det

“ The mountains flowed before the Lord”

(song by prophetess Deborah, Bible,

Judges 5:5)

•De>>1 Elastic behaviour

•De<<1 viscous behaviour

•De ~ O(1) viscoelastic behaviour

Silly Putty

( t)

( t)

2

20

: angular frequency

: phase angle

t

Amplitude

0

0

0( ) sin( ) t t )cos()(")sin()(' 00 tGtG

Elastic response Viscous response 0(t) sin t

impose

probe

~ Viscous element:

Elastic element: 0~ G

Deborah number in an oscillatory experiment: De De > 1 texp < λ

G and η are material constants (do not depend on external mechanical stimulus)

ωtsinγtγ 0

ωtsinγGγ(t)Gσ(t) 0

0°

90°

180°

270°

0.00 0.25 0.50 0.75 1.00

-2

-1

0

1

2

str

ain

[

1]

str

ess

[P

a]

time t [s]

Ideal elastic solid

ωtsinγtγ 0str

ain

[

1]

str

ess

[P

a]

time t [s]

Ideal viscous liquid

0°

90°

180°

270°

2ωtsinωγηωtcosωγη(t)γησ(t) 00

0.00 0.25 0.50 0.75 1.00

-2

-1

0

1

2

dt

d

G

1

dt

d

dt

d

G

1tcos0

= F = D

tcos1

Gtsin

1

G2222

22

tcos),,G(Gtsin),,G(G

0°

90°

180°

270°

Maxwell-Model

10-4

10-3

10-2

10-1

100

101

10-3

10-2

10-1

100

101

1

2

= 102 s

angular frequency [s-1]

G',

G''

[a.u

.]

G'

G''

Simplest viscoelastic liquid: Maxwell element

= F + D

Importance of time (frequency):

Span time Span responding length scale

Spectroscopy

Kelvin-Voigt-Model

0.00 0.25 0.50 0.75 1.00-2

-1

0

1

2

F

D

270°

0°

90°

180°

time t [s]

str

ain

[

1]

str

ess

[P

a]

= F = D

γηγGσσσ DF

2ωtsinωγηωtsinγGσ(t) 00

Garctan

Simplest viscoelastic solid: Voigt-Kelvin element

Mesoscopic modeling: bead-spring models

Spring (constant k G) : elastic element

Bead (friction ζ η): viscous element

I. Unentangled polymer dynamics (M<Me)

I.1 Diffusion of a small colloidal particle:

2

( ) (0) 6r t r Dt

Diffusion coefficient

Simple diffusive motion: Brownian motion

Friction coefficient:

F vA constant force applied to the particle leads to a constant velocity:

Einstein relationship: kT

D

Friction coefficient

6 R Stokes law: (Sphere of radius R in a Newtonian liquid with

a viscosity )

Stokes-Einstein-Sutherland 6

kTD

R Determination of

6H

kTR

D

random walk

I.2.1. Rouse model:

N beads of length b - Each bead has its own friction

- Total friction: R N R

kTD

N

- Rouse time: 2 2

2

/R

R

R RNR

D kT N kT

t < Rouse: viscoelastic modes t > Rouse: diffusive motion

- Kuhn monomer relaxation time:

32

0sbb

kT kT

2

0R N

sb - Stokes law:

I.2 Diffusion of an unentangled polymer chain:

R

Rouse relaxation modes:

The longest relaxation time: 2

0R N

For relaxing a chain section of N/p monomers:

2

0p

N

p

(p = 1,2,3,…,N) 1 R

1i i

N relaxation

modes

1 R

2 / 4R

4 /16R

0

- At time t = p, number of unrelaxed modes = p.

- Each unrelaxed mode contributes energy of order kT

to the stress relaxation modulus:

3p

kTG p

b N

(density of sections with N/p

monomers: 3)

( / )N p b

- Mode index at time t = p:

1/ 2

0

pp N

1/ 2

3

0

kT tG t

b

, for 0 < t < R

Volume

fraction

Approximation: 1/ 2

3

0

expR

kT t tG t

b

, for t > 0

1/ 2

3

0

kT tG t

b

, for 0 < t < R

1/ 2

30 00

1/ 2

0 0 03 3 30

exp

exp

R

R R

kT t tG t dt dt

b

kT kT kTx x dx N N

b b b b

Viscosity of the Rouse model:

Nb

2

0R N

3( )RG kT

Nb

( )R RG Nb

Or:

31

expN

p p

kT tG t

Nb

Exact solution of the Rouse model:

2 2

2 26p

b N

kTp

with

Each mode relaxes as a Maxwell element

Example: Frequency sweep test:

1/ 2'( ) "( )G G For 1/R < < 1/0

'( )G 2"( )G

For < 1/R

Rouse model: unentangled chains in polymer melts:

For M < Mc : M2

For M > Mc: M3.4

Relaxation times

M

rel

Mc

Main issue with Rouse model: “phantom” chain (no interactions with environment

(hence, it does not work for dilute solutions)

Viscous resistance from the solvent: the particle must drag

some of the surrounding solvent with it.

Hydrodynamic interaction: long-range force acting on solvent

(and other particles) that arises from motion of one particle.

When one bead moves: interaction force on the

other beads (Rouse: only through the springs)

The chain moves as a solid object:

(no friction distribution in chain)

Z sR

Z

Z s

kT kTD

R

Zimm time: 32

3 3/ 2 3/ 2

0s s

Z

Z

bRR N N

D kT kT

(In a theta solvent)

I.2.2. Account for hydrodynamic interactions: Zimm model

- Zimm has a weaker M-dependence than Rouse.

- Zimm < Rouse in dilute solution

- Zimm works better in dilute solution

Rouse time versus Zimm time:

32 2 23/ 2 3/ 2

0/ /

sZ

Z Z s

bR R RN N

D kT kT R kT

32 2 22

/ /

sR

R s

bR R RN

D kT N kT N b kT

Slope: 5/9 to 2/3

(theta condition)

Rheological Regimes: (Graessley, 1980)

Dilute solutions: Zimm model – H.I. – no entanglement

Unentangled chains (Non diluted): Rouse model – no H.I. – no entanglement

Entangled chains: Reptation model – no H.I. – entanglement

III. Polymer: From diluted to concentrated state

c<c* c=c* c>c*

Dilute solutions:

341

* 3

Masseff

cc R

c

Mass ac Nc

M

Semi-dilute solutions:

3

*4

3a

Mc

N R

cMass c*

Concentrated solutions or melts:

cMass > c*: Interpenetration of the chains

c*: overlap concentration

M: [g/mol]

c*: mass per volume

cMass: mass per volume

c: number of molecules per volume