Polymer Gels - Boulderschool.yale.edu

Polymer GelsBoulder Lectures in Soft Matter PhysicsJuly 2012Yitzhak Rabin 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)

Gel – polymer network permeated by solventsolventCross-linked networkpolymer volume fraction 0.1-10 %

Gelationphysical bondsWeakStrongchemical (covalent bonds)Reacting monomersCondensation(Critical Percolation)Crosslinking polymers(Vulcanization)Addition(Kinetic Growth) End-linkingChemical and strong physical gels are soft solids– macroscopic elasticityWeak physical gels are viscoelastic liquidsRandomCrosslinking

Physical bondsWeak --- - - - aa bondsHydrogenbbBlock copolymernodules- - - -- - --ccIonic associationsStrongMicrocrystal lamellaeGlassy nodulesDouble helices

Here we focus on “solid” (strong) gels: irreversible crosslinking stable in solvent bath but solvent evaporates when exposed to airEvaporation is diffusion-controlled:10-6 cm2/st 1 week, for a 1cm3 gelGelation is a random process –no two gels have identical structure (topology) even ifidentically prepared!Unique speckle patterns in light scattering

Are gels equilibrium solids or glasses?(single vs multiple equilibrium states)Determine uniqueness ofeq(r ) by temperature cyclingT1T2T1and monitoring the speckle patterns35 C45 C35 CGoren et al, Macromolecules 33, 5757 (2000).

Gelation:Dilute solution of monomers M (f 2) with volume fraction ϕMand crosslinks C (f 2) with volume fraction ϕCϕC ϕM 1If ϕC ϕM only small connected clusters (“sol”) existAn infinite cluster (“gel”) appears at ϕC* - Gel pointShear rigidity first appears at ϕC** ϕC*- much of theory focuses on the sol-gel transition because ofthe connection to percolation and critical phenomena

Bond percolation: bonds introduced with probability pp pcp pcExtent 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(neglect loops)ABBBACondensation polymerizationof ABf-1 monomers:BABABAABBABAp – probability of B to react with ABBBABBBBFraction of reacted A groups: p(f-1)Maximum extent of reaction (all A groups reacted): pc 1/(f-1)Each branched polymer contains only one unreacted A group –# molecules # unreacted A groups 1 – p(f-1) Average polymer size: diverges at!BB

Elasticity of networksRR0 RLy λy Ly0Ly0Lz λz Lz0Lz0Lx0Lx λx Lx0Affine network modelDeformation of the network strand is proportionalto the macroscopic deformation of the network asif the ends of the strand are nailed to the elasticnon-fluctuating background. R0

Entropy of an ideal chain consisting of N Kuhn segments of length b 2222 RRR3 R3 xyzS N, RkSN,0kS N ,0222 Nb2NbEntropy change upon affine deformation2222221R1R1R3xx0yy0zz0k2Nb 2Entropy change for the whole network consisting of n strands S N, RS net S N , R03k22xn1i 12yn1i 12R y20in2zRz201i 1iNbnIn undeformed statei 1Deformation freeenergy:Rx20 iRx20nii 1FnetR y20niRz20i 1T S netnkT2n 2Nb3i2x2y2z3

Swelling of Polymer GelsVdryVdryVvolume fraction of polymer ina swollen gelV0 – volume fraction in gel preparation statewith volume V0Polymer amount doesnot change upon swellingV0Isotropic deformation of uniformly swollen gel:Elastic free energy of a Felstrand in a swollen gelElastic free energy density GR2kT 2RrefkTR02RrefR0 2kT2b3 N Rref0VV /V0Vdry1/ 30 /1/ 32(affine deformation)

Swelling in -solventsSize of a free chain in -solvent is independent of concentration2RrefElastic modulus GR02kTb3 Nb2 N2kTb3 N2/30kTb3Osmotic pressure in a -solvent:kTNb 32 / 3 1/ 303Swelling equilibrium - elasticity is balanced by osmotic pressure:GSwelling ratio:Q1VVdryN 3/81/ 40

Dry Modulus vs. Equilibrium SwellingIf network is prepared in the dry state0 1 and NkTNb3Network modulus in the dry state G 12/30Q8 / 3kTQ3b8/31G(1)b3/kT-8/30.1-1.751/0.01110QPDMS networks in toluenePatel et al., Macromolecules 25, 5241 (1992)

Swelling in good solventsSize of a free chain in good solvent depends on concentrationRrefElastic modulusR0 2kTG2b3 N RrefR0kTb3 NbN 1/ 21/ 42/30Osmotic pressure in good solventEquilibrium swelling G1/ 80kTb3kTNb 39/ 4with swelling ratio QIf network is prepared in the dry state:5 / 12 7 / 1200 1 and N1N 3/ 51/ 40Q5 / 3

Dry Modulus vs. Equilibrium SwellingNetwork modulus in the dry state G 1kTNb3Network modulus at equilibrium swelling5 / 120G 1/ QkTQ3bkTQ3b5/32.3G(1/Q) (Pa)106-2.31053456789 10QEnd-linked PDMS networks in toluene at 25 oC.Urayama et al., J. Chem. Phys. 105, 4833 (1996).

Deformation of gelsWhat 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 likerubber!2. Swelling effects on entanglements (melts vs semi-dilute solutions)

Thermodynamics of RubberHelmholtz free energy dF d(U – TS) -SdT – pdV fdLFLApplied force fU TSLT ,VT ,VULTT ,VSLT ,Vis the sum of two contributions f fE fSSLMaxwell relationForce at constant elongationfEntropic contribution dominatesslope 00T ,VfTfSV ,LTfETfTTV ,LSLT ,VEnergetic contributionUfEL T ,VfTV ,L

Uniaxial deformation of incompressible networksV0V ; Lx 0 L y 0 Lz 0Lx L y Lz1/ 2Uniaxial deformationyxFnetFree energy change:ForceStressxxFnetLxfxL y LzLx 0 y L y 0 z Lz 01x y zfxxFnetLx 01Lx 0nkTLx 0 L y 0 Lz 0Modulus (affine)GznkT2Fnet12nkTVckTN223nkTLx 0nkTV1221

Deviations from Classical Stress-Strain Dependence7Engineering stress(MPa)5eng63eng4eng22101234trueStrain hardening atlarge elongationsdue to finite chainextensibility.1GfxL y 0 Lz 056Dangling loop78Strain softeningat small elongationsdue to entanglements.polymer entanglementDangling end

Strain Hardening at Large Elongationsf-fRGaussian chain: linear force –elongation dependence f3kT R2NbReal polymers cannot be extendedbeyond maximum contour length RmaxForce f diverges as R RmaxDivergence rate depends on chainflexibility:Flexible chains are described byfreely-jointed chain model withf (Rmax – R)-1Semi-flexible chains are describedby worm-like chain model withf (Rmax – R)-2

Mooney – Rivlin model of incompressible networks:Strain invariants: I12x2y2z2 2x yI22 2y z2 2z xFree energy density:FVC1 ( I1 3) C2 ( I 2 3)C1223C2 2223Stress(MPa)true1L y LzF /VF /V1.42C1212C2Mooney-Rivlin equation1.22C21.0true0.820.70.80.91.01/2C12C212

Deviations from classical elasticity at intermediate deformations:Entanglements?Edwards Tube ModelaNx – degree of polymerization between crosslinksNe (a/b)2 – degree of polymerization between entanglementsAffine modulusGkTcNxkTcNe

Non-Affine Tube ModelConfining potential and tube diameter change with network deformationMooney stress:f* Gx Ge1Rubinstein & Panyukov, Macromolecules 1997, 30, 8036

Swelling Equilibrium in Charged Gels:charged gelsolventDonnan equilibriumfor salt ions (cs) and counterions (fc)f- degree of ionization;c- monomer concentrationΠel - electrostatic contribution to osmotic pressureFor fc » 2cs - ideal gas of counterions:ΠelFor fc « 2cs, : ΠelOsmotic modulus: Π Π0 Πelosmotic pressure in a in semi-dilute solution (c2 in idealand c2.25 in good solvent)

Exact solution:Concentration of salt ions outside the gel: csConcentration of ions inside the gel: c and cEquating inside and outside chemical potentials for each type of ions:c c- csElectroneutrality (negatively charged gel):3. e(-fc c -c-) 0Solution:Πelc /- cs

Elastic modulus for isotropic swelling (ideal chains, no electrostaticeffects on conformation):G λ2 Swelling equilibrium: Π Gpoly(0.75 sodium acrylate 0.25 acrylic acid) gelin water

Volume Transition and Phase Separation in Charged Gels(change T)surface phase?!homogeneous gelshrunken gelNIPA/AAc (668mM/12mM) gel in waterat 35 ºCCourtesy M. Shibayama

Salt-free case:Free energy(per monomer)N 50,00.05phase separation0.025volume transition0.017f 0.005f– degree of ionizationcontinuous shrinking

Phase transitions in NIPA-AA gelsVolume transition(shallow quench)0.15mmInitial microphase separationfollowed by volume transition(deep quench)

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 scatteringfrom gels) diverge at ϕC*. Since ordinary “solid” gels are formedconsiderably above the gel point, one expects inhomogeneities todecrease with increasing crosslink concentration.6log [ I(q) / arb. units ]Experiment:Polyacrylamide solutionLS8/558/44 8/33Polyacrylamide gelsSAXS8/28/1concentration of cross-links21AAm; 0.08 w/w to waterBIS; 0.001 to 0.005 w/w to AAm0-3.5-3.0-2.5 -2.0 -1.5log [ q / Å-1 ]S. Mallam et al., Macromolecules,-1.0, 22, 3356

eq(r , t )(r )thermal (time) averageth(r , t )(r , t )(r ) spatial (ensemble) averageeq(r )Thermal fluctuations in polymer solutions400th200th(r , t )(r , t )00S(q )-200-400020406080( q ) ( q)th(q)th( q)G (q)instantaneous profilerThermal correlatorStatic inhomogeneities in gelseq(r ) 200C(q )0eq(q )eq( q)Static correlator-200020406080rStatic inhomogeneities and thermal fluctuations in gels(r )4002000-200-4000S(q )204060r(q ) ( q)C(q ) G( q)80The total structure factor is measured by scattering experiments!

Direct evidence for static inhomogeneities:stationary speckle patterns6without Cross-linkersolution timeaverage I E I T42008Intensity / cps x10-3Intensity / cps x10-38ensembleaverage20406080Sample positionPusey et al, 19911006with Cross-linkergeltime average I T I E4ensembleaverage20020406080Sample positionErgodicity broken!100

The butterfly effect:scattering from stretched gelsTheory of elasticity statistical mechanics:Thermal fluctuations suppressed along stretching direction and enhanced normal to itStretching directionSANS experiments observe the reverse!Bastide et al, Macromolecules 23, 1821 (1990)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 factorThermal correlatorNa 2 /61/2S(q) G(q) C(q)G G(Q, φ, χ)qconcentration interaction parameterreduced scattering vectorStatic correlatorC C Q, φ, χ; φ 0 , χ 0parameters at state of preparationA gel “remembers” not only its topology but also the concentrationand temperature at which it was prepared!Panyukov and Rabin,Phys. Rep. 269, 1 (1996)Macromolecules 29, 7960 (1996).

For gels prepared by reacting monomers -crosslinks act as attractionsIncreasing concentration of crosslinks and/or decreasing solubilityduring gelation – approach spinodal of the monomer-crosslink systemP.-G. de Gennes, Scaling Methods in Polymer Physics (1979)S.V. Panyukov and Y. Rabin, Phys. Rep. 269, 1 (1996)Crosslink saturation thresholdC(q 0)at the CSTaN 1/ 2Length scale of static inhomogeneities :(effective mesh size)away from CSTat CSTNo CST for gels prepared by irradiation crosslinking

Test of CSTCorrelation length ofstatic inhomogeneitiesCrosslink concentrrationNoritsue et al, Polymer 43, 5289 (2002)Inhomogeneities are revealed upon swelling since highly crosslinkedregions swell less than those that are poorly crosslinked.

Fitting scattering profiles with PR theory:Reacting monomer- crosslink mixtureIrradiation crosslinking of chainsStrong thermal fluctuations:liquid-likeStatic inhomogeneitiesdominate: solid-likeNoritsue et al, Polymer 43, 5289 (2002)Takata et al, Macromolecules 35, 4779 (2002)