Polymer Gels - Yale University
Transcript of Polymer Gels - Yale University
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