Polymer Gels Boulder ectures in Soft Matter Physics July Yitzhak abin M. ubinstein and.h. Colby, Polymer Physics (Oxford, ), Chapters 6 and 7 P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell, 979), Chapter 5 S. V. Panyukov and Y. abin, Statistical Physics of Polymer Gels, Phys. ep. 69, (996)
Gel polymer network permeated by solvent solvent Cross-linked network polymer volume fraction.- %
Gelation physical bonds chemical (covalent bonds) Weak Strong eacting monomers Crosslinking polymers (Vulcanization) Condensation (Critical Percolation) Addition (Kinetic Growth) End-linking andom Crosslinking Chemical and strong physical gels are soft solids macroscopic elasticity Weak physical gels are viscoelastic liquids
Physical bonds Weak a b + + - - - - + + - - + - Hydrogen bonds Block copolymer nodules Ionic associations Strong + - + c + + + Microcrystal lamellae Glassy nodules Double helices
Here we focus on solid (strong) gels: irreversible crosslinking stable in solvent bath but solvent evaporates when exposed to air Evaporation is diffusion-controlled: -6 cm /s t week, for a cm gel Gelation is a random process no two gels have identical structure (topology) even if identically prepared! Unique speckle patterns in light scattering
Are gels equilibrium solids or glasses? (single vs multiple equilibrium states) Determine uniqueness of eq (r) by temperature cycling and monitoring the speckle patterns T T T 5 C 45 C 5 C Goren et al, Macromolecules, 5757 ().
Gelation: Dilute solution of monomers M (f=) with volume fraction ϕ M and crosslinks C (f>) with volume fraction ϕ C ϕ C < ϕ M < If ϕ C << ϕ M only small connected clusters ( sol ) exist An infinite cluster ( gel ) appears at ϕ C * - Gel point Shear rigidity first appears at ϕ C ** > ϕ C * - much of theory focuses on the sol-gel transition because of the connection to percolation and critical phenomena
Bond percolation: bonds introduced with probability p p < p c p > p c Extent of reaction p fraction of formed bonds out of all possible bonds. Connectivity transition at critical extent of reaction p c (percolation threshold). Only finite clusters (sol) at p<p c. At p>p c infinite cluster (gel), in addition to finite clusters.
Mean-field estimate of the gel point (neglect loops) Condensation polymerization of AB f- monomers: p probability of B to react with A Fraction of reacted A groups: p(f-) Maximum extent of reaction (all A groups reacted): A B B A B B A A B B A B B A B B A B B A A B B B B p c =/(f-) Each branched polymer contains only one unreacted A group # molecules = # unreacted A groups B B = p(f-)= Average polymer size: = diverges at!
Elasticity of networks y y =λ y y z z =λ z z x x =λ x x 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.
Entropy of an ideal chain consisting of N Kuhn segments of length b,,, N S Nb k N S Nb k N S z y x Entropy change upon affine deformation,, Nb k N S N S z z y y x x Entropy change for the whole network consisting of n strands Nb k S n i i z z n i i y y n i i x x net Nb n n i i z n i i y n i i x In undeformed state z y x net net nkt S T F Deformation free energy:
Swelling of Polymer Gels V dry V V dry V volume fraction of polymer in a swollen gel volume fraction in gel preparation state with volume V Polymer amount does not change upon swelling V V V dry Isotropic deformation of uniformly swollen gel: / V /V / / Elastic free energy of a Fel kt kt strand in a swollen gel ref ref (affine deformation) Elastic free energy density G kt b N ref
Swelling in -solvents Size of a free chain in -solvent is independent of concentration Elastic modulus G ref b N kt b N kt b N / kt Nb / / Osmotic pressure in a -solvent: kt b Swelling equilibrium - elasticity is balanced by osmotic pressure: G Swelling ratio: Q V N V dry /8 / 4
Dry Modulus vs. Equilibrium Swelling G()b /kt If network is prepared in the dry state = and -8/ N 8/ Q kt / kt 8/ Network modulus in the dry state G Q Nb b.. / -.75 Q PDMS networks in toluene Patel et al., Macromolecules 5, 54 (99)
Swelling in good solvents Size of a free chain in good solvent depends on concentration ref / bn /8 Elastic modulus G kt b N ref kt b N / / 4 kt Nb 5/ 7 / Osmotic pressure in good solvent kt b 9/ 4 N Equilibrium swelling G with swelling ratio Q / 4 /5 If network is prepared in the dry state: = and N 5/ Q
G(/Q) (Pa) Dry Modulus vs. Equilibrium Swelling kt 5/ kt Network modulus in the dry state G Q Nb b Network modulus at equilibrium swelling kt. G / Q Q b 6 5/ -. 5 4 5 6 7 8 9 Q End-linked PDMS networks in toluene at 5 o C. Urayama et al., J. Chem. Phys. 5, 48 (996).
Deformation of gels What is the difference between gels and rubber?. Gels swell/deswell in response to deformation in a solvent bath. This takes time on short time scales gels respond to deformation like rubber!. Swelling effects on entanglements (melts vs semi-dilute solutions)
Thermodynamics of ubber Helmholtz free energy df =d(u TS) = -SdT pdv + fd Applied force V T V T V T V T S T U TS U F f,,,, Maxwell relation V V T T f S,, is the sum of two contributions f = f E + f S Entropic contribution dominates Force at constant elongation V T V S S T T f T f,, slope = V T f, T f V T E U f, Energetic contribution f E
Uniaxial deformation of incompressible networks V V ; x y z x y z x x y y z z Uniaxial deformation x y z x / y z nkt Free energy change: F net Fnet Fnet Fnet nkt Force fx x x x x Stress xx fx y z nkt nkt xyz V Modulus (affine) G nkt V ckt N
Deviations from Classical Stress-Strain Dependence eng (MPa) 7 6 5 4 eng G Engineering stress eng y fx z true Strain hardening at large elongations due to finite chain extensibility. 4 5 6 7 8 Dangling loop Strain softening at small elongations due to entanglements. polymer entanglement Dangling end
Strain Hardening at arge Elongations -f f Gaussian chain: linear force elongation dependence f kt Nb eal polymers cannot be extended beyond maximum contour length max Force f diverges as max Divergence rate depends on chain flexibility: Flexible chains are described by freely-jointed chain model with f ~ ( max ) - Semi-flexible chains are described by worm-like chain model with f ~ ( max ) -
Mooney ivlin model of incompressible networks: (MPa) Strain invariants: Free energy density: I x y z I x y y z z x F V C ( I ) C( I ) C C Stress true y z F / V F / V C C.4.. C.8.7.8.9. Mooney-ivlin equation true / C C
Deviations from classical elasticity at intermediate deformations: Entanglements? Edwards Tube Model a N x degree of polymerization between crosslinks N e ~ (a/b) degree of polymerization between entanglements Affine modulus G kt c N kt c x N e
Non-Affine Tube Model Confining potential and tube diameter change with network deformation Mooney stress: f * = = G x + G e ubinstein & Panyukov, Macromolecules 997,, 86
Swelling Equilibrium in Charged Gels: charged gel solvent Donnan equilibrium for salt ions (c s ) and counterions (fc) f- degree of ionization; c- monomer concentration Π el - electrostatic contribution to osmotic pressure For fc» c s - ideal gas of counterions: Π el For fc «c s, : Π el Osmotic modulus: Π = Π + Π el osmotic pressure in a in semi-dilute solution (c in ideal and c.5 in good solvent)
Exact solution: Concentration of salt ions outside the gel: c s Concentration of ions inside the gel: c + and c - Equating inside and outside chemical potentials for each type of ions: c + = c s c - = c s Electroneutrality (negatively charged gel):. e(-fc+c + -c - )= Solution: c +/- = Π el +
Elastic modulus for isotropic swelling (ideal chains, no electrostatic effects on conformation): G= λ = Swelling equilibrium: Π = G poly(.75 sodium acrylate.5 acrylic acid) gel in water
Volume Transition and Phase Separation in Charged Gels (change T) surface phase?! homogeneous gel shrunken gel NIPA/AAc (668mM/mM) gel in water at 5 ºC Courtesy M. Shibayama
Salt-free case: Free energy (per monomer) N=5,. 5 phase separation.5.7 f=.5 volume transition continuous shrinking f degree of ionization
Phase transitions in NIPA-AA gels Volume transition (shallow quench) Initial microphase separation followed by volume transition (deep quench).5mm
What about the microscopic structure of gels on length scales < micron? Scattering experiments from mesoscales (S) to nanoscales (SANS, SAXS): Frozen inhomogeneities, topological disorder, thermal fluctuations
log [ I(q) / arb. units ] Monomer concentration inhomogeneities (and therefore scattering from gels) diverge at ϕ C*. Since ordinary solid gels are formed considerably above the gel point, one expects inhomogeneities to decrease with increasing crosslink concentration. Experiment: 6 5 8/5 8/4 S Polyacrylamide gels 4 8/ 8/ 8/ SAXS concentration of cross-links Polyacrylamide solution AAm;.8 w/w to water BIS;. to.5 w/w to AAm -.5 -. -.5 -. -.5 -. log [ q / Å - ] S. Mallam et al., Macromolecules,,, 56
eq ( r, t) ( r) thermal (time) average th (r) eq ( r, t) ( r, t) ( r) spatial (ensemble) average Thermal fluctuations in polymer solutions th ( r, t) 4 - -4 4 6 8 r th ( r, t) instantaneous profile S ( q ) ( q ) ( q ) th th ( q ) ( q ) G(q) Thermal correlator Static inhomogeneities in gels eq (r) C ( q ) eq ( q ) eq ( q ) Static correlator - 4 6 8 r Static inhomogeneities and thermal fluctuations in gels (r) 4 - -4 S ( q ) ( q ) ( q ) C ( q ) G ( q ) 4 r 6 8 The total structure factor is measured by scattering experiments!
Direct evidence for static inhomogeneities: stationary speckle patterns 8 6 without Cross-linker solution time average 8 6 Intensity / cps x - 4 <I> E = <I> T ensemble average Intensity / cps x - with Cross-linker gel time average <I> T 4 <I> E ensemble average 4 6 Sample position 8 4 6 Sample position 8 Pusey et al, 99 Ergodicity broken!
Stretching direction The butterfly effect: scattering from stretched gels Theory of elasticity + statistical mechanics: Thermal fluctuations suppressed along stretching direction and enhanced normal to it SANS experiments observe the reverse! Bastide et al, Macromolecules, 8 (99) These are not thermal fluctuations! Stretching enhances density contrast between heavily and lightly crosslinked regions
Scattering profiles : from replica field theory calculation ( fluctuations around the mean-field solution of the Deam-Edwards model) Structure factor Thermal correlator S(q) G(q) C(q) G G(Q,φ, χ) Na /6 / q reduced scattering vector concentration interaction parameter Static correlator C C Q, φ, χ;φ, χ 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 abin, Phys. ep. 69, (996) Macromolecules 9, 796 (996).
For gels prepared by reacting monomers -crosslinks act as attractions Increasing concentration of crosslinks and/or decreasing solubility during gelation approach spinodal of the monomer-crosslink system P.-G. de Gennes, Scaling Methods in Polymer Physics (979) S.V. Panyukov and Y. abin, Phys. ep. 69, (996) Crosslink saturation threshold C(q ) at the CST ength scale of static inhomogeneities : (effective mesh size) / an away from CST at CST No CST for gels prepared by irradiation crosslinking
Test of CST Correlation length of static inhomogeneities Crosslink concentrration Noritsue et al, Polymer 4, 589 () Inhomogeneities are revealed upon swelling since highly crosslinked regions swell less than those that are poorly crosslinked.
Fitting scattering profiles with P theory: eacting monomer- crosslink mixture Irradiation crosslinking of chains Strong thermal fluctuations: liquid-like Static inhomogeneities dominate: solid-like Noritsue et al, Polymer 4, 589 () Takata et al, Macromolecules 5, 4779 ()
Temperature and cross-link density dependence: SANS vs. theory C BIS C BIS Peak at q* comes from static inhomogeneities, not thermal fluctuations! Structural reorganization subject to crosslink constraints Ikkai et al., Macromolecules, 5 (998)
Crosslink concentration dependence Poor solvent CST Elasticity opposes segregation Good solvent Ikkai and Shibayama, Phys. ev. E apid Commun., 997; Panyukov and abin, Physica A 49, 9 (998)
SANS from deformed NIPA/AAc gel butterfly pattern OBSD. CACD. Static inhomogeneities dominate away from T MST microphase separation T = 4 C =.6 stretching direction =.6 Thermal fluctuations dominate near T MST T = 46 C =.75 Shibayama et al, Macromolecules, 586, (998)