WIXLIB

Polymers PhysicsMichael RubinsteinUniversity of North Carolina at Chapel Hill

Outline1. “Real” Chains2. Thermodynamics of Mixtures3. Polymer Solutions

Summary of Ideal ChainsIdeal chains: no interactions between monomers separated by many bondsMean square end-to-end distance of ideal linear polymerR 2 Nb 22NbMean square radius of gyration of ideal linear polymer Rg2 6 3 3 / 2 3R 2 exp Probability distribution function P3d N , R 2 2 2 Nb 2 Nb 23RFree energy of an ideal chain F kT2 Nb 2 3kT REntropic Hooke’s Law f 2Nb3 1g(r) Pair correlation function rb2

Real ChainsInclude interactions between all monomersShort-range (in space) interactionsProbability of a monomer to be in contact with another monomerfor d-dimensional chaind N * b dRIf chains are ideal R bN 1/ 2 * N 1 d / 2small for d 2Number of contacts between pairs of monomers that are far alongthe chain, but close in spaceN * N 2 d / 2 small for d 4For d 4 there are many contacts between monomers in an ideal chain.Interactions between monomers change conformations of real chains.

Life of a Polymer is a Balanceof entropic and energetic parts of free energyEntropic part “wants”chains to have ideal-likeFentconformationsFengEnergetic part typically“wants” something else(e.g. fewer monomermonomer contactsin a good solvent).Chain has to find a compromise between these two desiresand optimize its shape and size.R

Flory TheoryNumber density of monomers in a chain is N/R3Probability of another monomer being within excludedvolume v of a given monomer is vN/R3Excluded volume interaction energy per monomer kTvN/R3Excluded volume interaction energy per chain kTvN2/R3Entropic part of the free energy is of order kTR2/(Nb2)Flory approximation of the total free energy of a real chain N 2 R2 F kT v 3 2 Nb RRFree energy is minimum at R v1/ 5b 2 / 5 N 3 / 5Universal relation R N - Flory scaling exponentMore accurate estimate 0.588

1000Rg (nm)1000. 5 91/21011 041 051 061 071 08Mw ( g/mole)Radius of gyration of polystyrene chains in a q-solvent (cyclohexaneat 34.5oC) and in a good solvent (benzene at 25oC).Fetters et al J. Phys. Chem. Ref. Data 23, 619 (1994)

Polymer Under TensionIdeal chainReal chainUnperturbed sizeRF bN 3 / 5R0 bN 1/ 2xTension blobs (Pincus blobs) of size xcontain g monomersRfChains are almost unperturbed on length scales up to x3/ 51/ 2x bgx bgOn larger length scales they are stretched arrays of Pincus blobsN Nb 2 R02Rf x gxxR02x RfN Nb5 / 3 RF5 / 3R f x 2/3 2/3gxxSize of Pincus blobsRF5 / 2x 3/ 2Rf

Polymer Under TensionIdeal chainx R02Real chainSize of Pincus blobsRF5 / 2x 3/ 2RfRfFree energy cost for stretching a chain is on order kT per blob2Rf RfN F kT kT kT gx RF Tension force is on order kT divided by blob size xRf RfNF kT kT kT gx R0kT 3 / 2 kT R f f 5/ 2 R f x RFRF RFkT kTkT R ff 2 Rf x R0R0 R0linear elasticityFor b 1nm at room TkT/b 4pN kTkT/bf013/2RfR0 RF Rmax5/ 2 3/ 2non-linear elasticityeven at low forcesLog – log plot

Polymer Under TensionReal chainIdeal chainf kTR R f20linear elasticitykT/bf1R0 RFf kT RF5 / 2 R 3f / 23/2Rfnon-linear elasticityRmaxChallenge Problem 1: Pulling a RingConsider a pair of forces applied to monomers 1 and 1 N/2i. Show that the modulus of an ideal ringis twice the modulus of a linear N/2-mer f f 2 f 2ii. How does modulus of a ring in a good solvent Grcompare to twice the modulus of linear N/2-mer GN/2?a. Gr 2GN/2 similar to ideal caseb. Gr 2GN/2 because the entropy of sections of a ring is lowerc. Gr 2GN/2 because ring sections reinforce each other f f 2 f 2

Biaxial CompressionIdeal chainR Real chainDR DOn length scales smaller than compression blob of size D chain is almost5/32unperturbedD D g g b b Occupied part of the tube1/ 22/3Nb N R D NbR D bN 1/ 2 g D g Free energy of confinement5/32Nb Nb Fconf kT kTN Fconf kT kTN g D g D R Fconf kT 0 D 2FconfRF kT D 5/3

Challenge Problem 2:Biaxial Confinement of a Semiflexible ChainConsider a semiflexible polymer – e.g. doublestranded DNA with Kuhn length b 100nm andcontour length L 16mm.Assume that excluded volume diameter of doublehelix is d 3nm (larger than its actual diameter2nm due to electrostatic repulsions).R DRFCalculate the size RF of this lambda phage DNA in dilute solutionand the length R occupied by this DNA in a cylindrical channelof diameter D (for d D RF).

Uniaxial CompressionFree energy of confinement in a slit is thesame as in the cylindrical poreDLongitudinal size of an ideal chain in a slit is the same as for anunperturbed ideal chain. R bN 1/ 2 2-dimensional Flory theory for real chain confined in a slitD2 is the excluded areaof a confinement blobwith g monomers2 2 ( N / g )2 R F kT D 22 R(N/g)D N R D Longitudinal size of a real chain in a slit g 3/ 4Fractal dimension of real chains in 2-d is D 4/31/ 4b bN 3 / 4 D

Scaling Model of Real ChainsThermal blob - length scale at which excluded 2gTvolume interactions are of order kTkT v 3 kTxT1/ 2Chain is ideal on length scales smaller than thermal blob xT bgTNumber of monomers in a thermal blob gT b6 v 2b4Size of a thermal blob xT vgood solvent v 0poor solvent v 0RxTRglxT N R xT gT 3/ 51/ 5 v b 3 b N3/ 5 NR xT gT1/ 3 b2v1/ 3N 1/ 3

Flory Theory of a Polymer in a Poor Solvent N 2 R2 In poor solvent v 0 and R 0F kT v 3 2 Nb RCost of ConfinementR g b Compression blob of size Rwith g monomersNNb 2Confinement free energy Fconf kT kT 2gR N 2 R 2 Nb 2 F kT v 3 2 2 For v 0NbR RR Nb 2Three Body Repulsion23 R 2 Nb 2Size of a globuleNNF kT 2 2 v 3 w 6 1/ 3 wN RRR Nb Rgl v 2R 0

Real Chains in Different SolventsxTathermalRRathermal bN 3 / 53/5xTRgood1gT3/ 5Rq bN 1/ 2good3/5 q-solvent1/2poor1/3xTnon-solvent1/3Rnon solvent bN 1/ 3b N xT gT N1/ 3R poor N xT gT

Temperature Dependence of Chain SizeMayer f-function{U r f r exp 1 kT Excluded volume -1for r b where U(r) kTU r for r b where U(r) kTkT4 q 3 2v 4 f r r dr 4 r dr U r r dr 1 b kT b T 002b2Interaction parameter z is related to number of thermal blobs per chainz Nv 1/ 2 T q 1/ 2 3N NgT bTChain contraction in a poor solventR 1 / 3 z1/ 2bNb6gT 2vChain swelling in a good solventR2 1 zbN 1/ 2

2.52.52.02.01.51.522Rg /Rq22Rg /RqUniversal Temperature Dependenceof Chain Size1.00.51.00.50.00.0-40 -30 -20 -100102030N (1 - q/T)1/2Monte-Carlo simulationsGraessley et.al., Macromolecules32, 3510, 1999 & I. Withers-505101520N1/2(1 - q/T)Polystyrene in decalinBerry, J. Chem. Phys.44, 4550, 1966

Summary for Real ChainsSize of Linear ChainsRq bN 1/ 2Rgoodnearly ideal in q-solvents v b 3 b 2 1 v N b 3 b Rathermal bN bN 0.588R poor v 1 / 3 2b N 1/ 30.18Excluded volumeT q v b3 T N 0.588 swollen in good solventsin athermal solventscollapsed into a globule in poor solventsIn good solventsStretching a real chain1 /(1 ) R R F kT kT RF RF non-Hookean elasticityConfinement of a real chain2.431 / RF kT F D 1.7R kT F D

Outline1. Real Chains2. Thermodynamics of Mixtures3. Polymer Solutions

Binary MixturesHomogeneous – if components are intermixed on a molecular scaleHeterogeneous – if there are distinct phasesAssume, for simplicity, no volume change on mixingConsider a mixture with total number n of monomers (sites)and total volume v0n where v0 – volume of a lattice siteAssume that a monomer of each species occupies the same volume v0 VAVolume fractionsVA A B 1 VA VB VBVA VB A n n monomers of type A and B nB (1- n monomers of type B

Entropy of Mixing - volume fraction of A nv0 1 nv0 1 - volume fraction of Bnv0v0 – volume of a lattice siteNumber of translational states of a moleculein a mixture is the number of sites n AB V A VB nv0VA A n v0Number of states of molecule A in a pure A phaseEntropy change upon mixing of a molecule A AB 1 k ln k ln S A k ln AB k ln A k ln A SnA/NA and nB/NB – number of A and B moleculesTotal entropy change upon mixing nA nAnBnB S S A S B k ln ln 1 NANBNB NA Infiniteslopes0 1

Energy of a Homogeneous MixtureMean Field TheoryAverage energy per A monomeruij – interaction of i with jA monomer with probability ?B monomer with probability 1 – ? A? ?zz – coordination numberu A u AA 1 u AB 2(number of neighbors)zAverage energy per B monomer u B u AB 1 u BB ?2Average energy per monomer in homogeneous mixtureUn u n uuf AABB u A 1 uB nz 22 u AA 2 1 u AB 1 uBB2 zu AA2umzuBB2 um1Average energy per monomer of pure components 0before mixing ui z u AA 1 uBB 2Energy of mixing um u f ui

Energy of Mixingu AA uBB U m num n 1 z u AB n 1 kT 2 Flory interaction parameter Umz u AA u BB u AB kT 2Probability of AB contact 1 Energy change on mixing per site01 um kT (1 )Free Energy of Mixing Fm U m T S mper site Fm1 kT ln ln 1 (1 ) nNB NA Regular solution: NA NB 1Polymer solution: NA N 1, NB 1Polymer blend: NA 1, NB 1

Flory-Huggins Free Energy of Mixing Fmix1 kT ln ln 1 1 nNB NA At lower T for 0 repulsiveinteractions are important andthere is a composition range ' ''with thermodynamically stablephase separated state.This composition range, calledmiscibility gap, is determined bythe common tangent line. Fmix/(kn)At high T entropy of mixing dominates, Fmix is convex and homogeneous0mixture is stable at all compositions. -0.1250o K-0.2-0.3300o K-0.4-0.5-0.6-0.7350o K0 ’ 0.20.40.60.8 ” 5o KNA 200 NB 100 T1

Phase Diagrams Fmix1 kT ln ln 1 1 nNB NA Critical composition cr NBN A NBFor a symmetric blend NA NB N cr 2/N cr 1/2For polymer solutions NA N, NB 111 11 cr cr N2N 2N0 N 2.7N FmixkTnthere is a miscibility gap for ' ' 'NA NB ’ sp1-0.1 sp243N cr2Two Phases10Single PhaseN For cr mixture is stable at allcompositions.2 1 11 Forcr 2 NANB 00.2 0.4 0.6 0.8 cr ”1

Phase Diagram of Polymer SolutionsPolymer solutions phase separate upon decreasingsolvent quality below q-temperatureB A TUpper critical solution temperature B 0binodalSolution phase separates belowthe binodal in poor solventregime into a dilute supernatantof isolated globules at andconcentrated sediment at .dilutesupernatantconcentratedsediment Single Phase spinodal ”unstable(spinodaldecomposition)metastableTwo Phases(nucleationand growth)M 53.3kg/molePolyisoprene in dioxaneTakano et al., Polym J. 17, 1123, 1985

Intermolecular InteractionsOsmotic pressure c 2P RT A2c . Mn A2 – second virial tyrene)in toluene at 25 oCNoda et al, Macromol.16, 668, 198130106 P/cRT (moles/g)solution2520Mn 70800 g/mole1510Mn 200000 g/mole500.00Mn 506000 g/mole0.01c (g/ml)0.02

Mixtures at Low Compositions Fmix1 kT ln ln 1 1 nNB NA Expand ln(1- ) in powers of composition 2 1 3 Fmix1 kT ln 2 . nNB 2 NB 6N B NA Osmotic pressure Fmix P Fmix V 2 3 n kT 1 3 3 2 . b b N A NB 2 3N B 2nA cn v 2 Virial expansion in powers3P kT c wc . nnof number density cn /b3 NA 2 1 3b6 2 bExcluded volume v 3-body interaction w 3N B NB vT q 2 A2 M 02 1 2 3In polymer solutions NB 13Tbb N Av

Polymer MeltsConsider a blend with a small concentration of NA chains in a meltof chemically identical NB chains.No energetic contribution to mixing 0. 1 3 b3is very small for NB 1Excluded volume v 2 b NB NB Flory Theorem6b2xT b gT bN Bg NThermal blobTB2vChains smaller than thermal blob NA NB2 are nearly ideal.In monodisperse NA NB and weakly polydispersemelts chains are almost ideal.RA3/52In strongly asymmetric blends NA NBbNAlong chains are swollen1/23/ 51 / 103/ 5 NA NA NA 1/2bR A xT bN B 2 bN A 2 2 N1NNN gT BA B B

Challenge Problem 3:RA2/(b2NA)Long NA-mer in a 3-d Melt of NB-mersNBNBNBNBNBNBNB1/ 5 NA R 2 2b N A NB 2ANA/NB2M. LangWhy doesn’t Flory Theorem work?

Challenge Problem 4:Mixing of Polymers with Asymmetric MonomersNA – monomersper A chainNB – monomersper B chainv0 – volume of a lattice site volume of a small B monomermv0 – volume of an A monomer m times larger than B monomerDerive the free energy of mixing Fmix of A and B polymers.Calculate the size RA of dilute Achains in a 2-d of B-chains form 1 in the case of 0.