Polymer Gels Boulder Lectures in Soft Matter Physics

July 2012

• M. Rubinstein and R.H. Colby, “Polymer Physics” (Oxford, 2003), Chapters 6 and 7

• P.-G. de Gennes, “Scaling Concepts in Polymer Physics” (Cornell, 1979), Chapter 5

• S. V. Panyukov and Y. Rabin, “Statistical Physics of Polymer Gels”, Phys. Rep. 269, 1 (1996)

Yitzhak Rabin

Gel – polymer network permeated by solvent

solvent

Cross-linked network

polymer volume fraction ≈ 0.1-10 %

Gelation

physical bonds chemical (covalent bonds)

Weak Strong Reacting monomers Crosslinking polymers

(Vulcanization)

End-linking Random

Crosslinking

Addition

(Kinetic Growth)

Condensation

(Critical Percolation)

Chemical and strong physical gels are soft solids

– macroscopic elasticity

Weak physical gels are viscoelastic liquids

cba

+-

++-- - -

--

-

+

+ +

+

+

Weak

cba

+-

++-- - -

--

-

+

+ +

+

+

cba

+-

++-- - -

--

-

+

+ +

+

+

Strong

Hydrogen bonds Block copolymer nodules Ionic associations

Microcrystal lamellae Glassy nodules Double helices

Physical bonds

Here we focus on “solid” (strong) gels:

• irreversible crosslinking

• stable in solvent bath but solvent evaporates when exposed to air

Gelation is a random process –

no two gels have identical structure (topology) even if

identically prepared!

Evaporation is diffusion-controlled: 10-6 cm2/s t

1 week, for a 1cm3 gel

Unique speckle patterns in light scattering

C35 C45 C35

Determine uniqueness of by temperature cycling )(req

121 TTT

and monitoring the speckle patterns

Goren et al, Macromolecules 33, 5757 (2000).

Are gels equilibrium solids or glasses?

(single vs multiple equilibrium states)

Gelation:

Dilute solution of monomers M (f=2) with volume fraction ϕM

and crosslinks C (f>2) with volume fraction ϕC

If ϕC << ϕM only small connected clusters (“sol”) exist

ϕC < ϕM < 1

An infinite cluster (“gel”) appears at ϕC* - Gel point

Shear rigidity first appears at ϕC** > ϕC

*

- much of theory focuses on the sol-gel transition because of

the connection to percolation and critical phenomena

p < pc p > pc

Bond percolation: bonds introduced with probability p

Extent of reaction p – fraction of formed bonds out of all possible bonds.

Connectivity transition at critical extent of reaction pc

(percolation threshold). Only finite clusters (sol) at p<pc.

At p>pc – “infinite” cluster (gel), in addition to finite clusters.

Mean-field estimate of the gel point

Condensation polymerization

of ABf-1 monomers: A

B

B

A B

B

A B

B

A B

B

A B

B

A B

B

A B

B

A B

B

A B

B

pc=1/(f-1)

p – probability of B to react with A

Fraction of reacted A groups: p(f-1)

= 1 – p(f-1)=

Maximum extent of reaction (all A groups reacted):

Each branched polymer contains only one unreacted A group –

# molecules = # unreacted A groups

Average polymer size:

=

diverges at

!

(neglect loops)

Elasticity of networks

R 0

Ly=λy Ly0

R

Lx=λx Lx0

Lz=λz Lz0

Affine network model

Deformation of the network strand is proportional

to the macroscopic deformation of the network as

if the ends of the strand are nailed to the elastic

non-fluctuating background.

0RR

Ly0

Lz0

Lx0

Entropy of an ideal chain consisting of N Kuhn segments of length b

0,2

30,

2

3,

2

222

2

2

NSNb

RRRkNS

Nb

RkRNS

zyx

Entropy change upon affine deformation

2

20

220

220

2

0

111

2

3,,

Nb

RRRkRNSRNS

zzyyxx

Entropy change for the whole network consisting of n strands

21

20

2

1

20

2

1

20

2 111

2

3

Nb

RRR

kS

n

iizz

n

iiyy

n

iixx

net

2

1

20

1

20

1

20

3Nb

nRRR

n

iiz

n

iiy

n

iixIn undeformed state

32

222zyxnetnet

nkTSTFDeformation free

energy:

Swelling of Polymer Gels

volume fraction of polymer in

a swollen gel V

VdryVdry

V 0 – volume fraction in gel preparation state

with volume V0

Polymer amount does

not change upon swelling dryVVV 00

Isotropic deformation of uniformly swollen gel: 3/1

0

3/1

0 //VV

Elastic free energy of a

strand in a swollen gel 2

20

2

2

refref

elR

RkT

R

RkTF

Elastic free energy density

(affine deformation)

2

20

3refR

R

Nb

kTG

Swelling in -solvents

Size of a free chain in -solvent is independent of concentration

NbRRref22

02

Elastic modulus 3/13/203

3/2

03

2

3 Nb

kT

Nb

kT

Nb

kTG

Osmotic pressure in a -solvent: 3

3b

kT

4/1

0

8/31 N

V

VQ

dry

Swelling equilibrium - elasticity is balanced by osmotic pressure:

G

Swelling ratio:

Q

1 10

0.01

0.1

1

1/

-8/3

-1.75

G(1

)b3/k

T

Dry Modulus vs. Equilibrium Swelling

PDMS networks in toluene

Patel et al., Macromolecules 25, 5241 (1992)

If network is prepared in the dry state 0 = 1 and 3/8QN

Network modulus in the dry state 3/8

3

3/203

1 Qb

kT

Nb

kTG

Swelling in good solvents

Size of a free chain in good solvent depends on concentration

Elastic modulus

4/9

3b

kT

Equilibrium swelling 4/10

5/31 NQG with swelling ratio

If network is prepared in the dry state: 0 = 1 and 3/5QN

8/12/1

0 bNRRref

12/712/503

4/1

0

3/2

032

20

3 Nb

kT

Nb

kT

R

R

Nb

kTG

ref

Osmotic pressure in good solvent

Q

10

10 5

10 6

3 4 5 6 7 8 9

-2.3

G(1

/Q)

(Pa)

End-linked PDMS networks in toluene at 25 oC.

Network modulus in the dry state 3/5

3

12/503

1 Qb

kT

Nb

kTG

Network modulus at equilibrium swelling 3.2

3/1 Q

b

kTQG

Urayama et al., J. Chem. Phys. 105, 4833 (1996).

Dry Modulus vs. Equilibrium Swelling

Deformation of gels

What is the difference between gels and rubber?

1. Gels swell/deswell in response to deformation in a solvent bath.

This takes time – on short time scales gels respond to deformation like

rubber!

2. Swelling effects on entanglements (melts vs semi-dilute solutions)

Thermodynamics of Rubber

Helmholtz free energy dF =d(U – TS) = -SdT – pdV + fdL

Applied force VTVTVTVT L

ST

L

U

L

TSU

L

Ff

,,,,

Maxwell relation LVVT T

f

L

S

,,

is the sum of two contributions f = fE + fS

Entropic contribution dominates

Force at constant elongation

VTLVS

L

ST

T

fTf

,,slope =

LVT

f

,

0 0

T

f

VTE

L

Uf

,

Energetic contribution fE

Uniaxial deformation of incompressible networks

0000000 ; zzyyxxzyxzyx LLLLLLLLLVV

1zyx

xUniaxial deformation 2/1

zy

Free energy change: 32

2

2nkTFnet

Force 2

000

11

x

net

xx

net

x

netx

L

nkTF

LL

F

L

Ff

Stress 11 2

2000 V

nkT

LLL

nkT

LL

f

zyxzy

xxx

Modulus (affine) N

ckT

V

nkTG

1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

eng (

MP

a)

Engineering stress

true

zy

xeng

LL

f

00

2

1Geng

Strain hardening at

large elongations

due to finite chain

extensibility.

Strain softening

at small elongations

due to entanglements.

polymer entanglement

Dangling loop

Dangling end

Deviations from Classical Stress-Strain Dependence

Strain Hardening at Large Elongations

-f f

R

Gaussian chain: linear force –

elongation dependence RNb

kTf

2

3

Real polymers cannot be extended

beyond maximum contour length Rmax

Force f diverges as R → Rmax

Divergence rate depends on chain

flexibility:

Flexible chains are described by

freely-jointed chain model with

f ~ (Rmax – R)-1

Semi-flexible chains are described

by worm-like chain model with f ~ (Rmax – R)-2

Mooney – Rivlin model of incompressible networks:

Strain invariants: 2221 zyxI 222222

2 xzzyyxI

32

232

)3()3(22

2

12211 CCICICV

F

Free energy density:

Stress

22

2

1

12

12

//1CC

VFVF

LL zy

true

Mooney-Rivlin equation

212

22

/1

CCtrue

0.7 0.8 0.9 1.0 0.8

1.0

1.2

1.4

(MP

a)

2C2

Deviations from classical elasticity at intermediate deformations:

Entanglements?

a

Edwards Tube Model

ex N

ckT

N

ckTGAffine modulus

Ne ~ (a/b)2 – degree of polymerization between entanglements

Nx – degree of polymerization between crosslinks

Non-Affine Tube Model

Confining potential and tube diameter change with network deformation

Mooney stress: f* = = Gx + Ge

1

Rubinstein & Panyukov, Macromolecules 1997, 30, 8036

Swelling Equilibrium in Charged Gels:

charged gel solvent

Donnan equilibrium

for salt ions (cs) and counterions (fc)

For fc » 2cs - ideal gas of counterions: Πel

Πel - electrostatic contribution to osmotic pressure

f- degree of ionization; c- monomer concentration

For fc « 2cs, : Πel

Osmotic modulus: Π = Π0+ Πel

osmotic pressure in a in semi-dilute solution (c2 in ideal

and c2.25 in good solvent)

Concentration of salt ions outside the gel: cs

Concentration of ions inside the gel: c+ and c-

Equating inside and outside chemical potentials for each type of ions:

c+=

cs

c-=

cs

Electroneutrality (negatively charged gel):

3. e(-fc+c+-c-)=0

c+/-=

Πel

+

Exact solution:

Solution:

Swelling equilibrium: Π = G

Elastic modulus for isotropic swelling (ideal chains, no electrostatic

effects on conformation):

G=

λ2=

poly(0.75 sodium acrylate−0.25 acrylic acid) gel

in water

NIPA/AAc (668mM/12mM) gel in water

at 35 ºC

homogeneous gel

shrunken gel

surface phase?!

Courtesy M. Shibayama

Volume Transition and Phase Separation in Charged Gels

(change T)

0.017

f=0.005

volume transition 0.025

continuous shrinking

phase separation

f– degree of ionization

Free energy

(per monomer)

N=50, 05.00

Salt-free case:

Volume transition

(shallow quench)

Initial microphase separation

followed by volume transition

(deep quench)

Phase transitions in NIPA-AA gels

0.15mm

What about the microscopic structure of gels on length scales < 1 micron?

Scattering experiments from mesoscales (LS)

to nanoscales (SANS, SAXS):

Frozen inhomogeneities, topological disorder, thermal fluctuations

Monomer concentration inhomogeneities (and therefore scattering

from gels) diverge at ϕC*. Since ordinary “solid” gels are formed

considerably above the gel point, one expects inhomogeneities to

decrease with increasing crosslink concentration.

6

5

4

3

2

1

0-3.5 -2.5-3.0 -2.0 -1.5 -1.0

log [ q / Å-1 ]

log [

I(q

) /

arb.

unit

s ] 8/5

8/4

8/3

8/28/1

LS

SAXS

S. Mallam et al., Macromolecules, , 22, 3356

Polyacrylamide gels

AAm; 0.08 w/w to water

BIS; 0.001 to 0.005 w/w to AAm

concentration of cross-links

Polyacrylamide solution

Experiment:

)(),(),( rtrtr eqth

thermal (time) average spatial (ensemble) average )(),( rtr eq)(r

Static inhomogeneities in gels

200

0

-200

80 60 40 20 0

r

C ( q ) eq ( q ) eq

( q ) Static correlator )(req

80 60 40 20 0

r

Thermal fluctuations in polymer solutions

-400

-200

0

200

400

),( trth

0),( trth

instantaneous profile

S ( q ) ( q ) ( q ) th ( q ) th

( q ) )(qG

Thermal correlator

-400

-200

0

200

400

80 60 40 20 0

r

Static inhomogeneities and thermal fluctuations in gels

S ( q ) ( q ) ( q ) C ( q ) G ( q ) )(r

The total structure factor is measured by scattering experiments!

8

6

4

2

0

Inte

nsi

ty /

cps

x10-3

100806040200Sample position

<I>E = <I>T

without Cross-linker

time

average

ensemble

average

solution

8

6

4

2

0

Inte

nsi

ty / c

ps

x1

0-3

100806040200

Sample position

<I>E

<I>T

with Cross-linker

ensemble

average

time averagegel

Direct evidence for static inhomogeneities:

stationary speckle patterns

Ergodicity broken! Pusey et al, 1991

The butterfly effect: scattering from stretched gels

Theory of elasticity + statistical mechanics:

Thermal fluctuations suppressed along stretching direction and enhanced normal to it

These are not thermal fluctuations!

Stretching enhances density contrast between heavily and lightly crosslinked regions

Str

etch

ing d

irec

tio

n

SANS experiments observe the reverse!

Bastide et al, Macromolecules 23, 1821 (1990)

Scattering profiles : from replica field theory calculation

Structure factor C(q)G(q)S(q)

χ)φ,G(Q,GThermal correlator

reduced scattering vector

q/6Na1/22

concentration interaction parameter

Static correlator 00 χ,φχ;φ,Q,CC

parameters at state of preparation

A gel “remembers” not only its topology but also the concentration

and temperature at which it was prepared!

Panyukov and Rabin,

Phys. Rep. 269, 1 (1996)

Macromolecules 29, 7960 (1996).

(“fluctuations” around the mean-field solution of the Deam-Edwards model)

at the CST 0)C(q

Length scale of static inhomogeneities :

(effective mesh size)

2/1aN away from CST

at CST

For gels prepared by reacting monomers -crosslinks act as attractions

Crosslink saturation threshold

Increasing concentration of crosslinks and/or decreasing solubility

during gelation – approach spinodal of the monomer-crosslink system

No CST for gels prepared by irradiation crosslinking

P.-G. de Gennes, Scaling Methods in Polymer Physics (1979)

S.V. Panyukov and Y. Rabin, Phys. Rep. 269, 1 (1996)

Test of CST

Correlation length of

static inhomogeneities

Inhomogeneities are revealed upon swelling since highly crosslinked

regions swell less than those that are poorly crosslinked.

Noritsue et al, Polymer 43, 5289 (2002)

Crosslink concentrration

Noritsue et al, Polymer 43, 5289 (2002)

Takata et al, Macromolecules 35, 4779 (2002)

Fitting scattering profiles with PR theory: Irradiation crosslinking of chains Reacting monomer- crosslink mixture

Strong thermal fluctuations:

liquid-like

Static inhomogeneities

dominate: solid-like

CBIS

CBIS

Temperature and cross-link density dependence: SANS vs. theory

Ikkai et al., Macromolecules 31, 151 (1998)

Peak at q* comes from static inhomogeneities, not thermal fluctuations !

Structural reorganization subject to crosslink constraints

Ikkai and Shibayama, Phys. Rev. E Rapid Commun., 1997; Panyukov and Rabin, Physica A 249, 239 (1998)

Elasticity

opposes segregation

Poor solvent

Good solvent

CST

Crosslink concentration dependence

SANS from deformed NIPA/AAc gel

T = 42 °C

T = 46 °C

CALCD.

= 0.60

= 0.75

OBSD.

stre

tch

ing

dir

ecti

on

=

1.2

6

butterfly

pattern

microphase

separation

Shibayama et al, Macromolecules 31, 2586, (1998)

Static inhomogeneities dominate

away from TMST

Thermal fluctuations dominate

near TMST