173 Advances in Polymer Science Editorial Board: A. Abe • A.-C. Albertsson • R. Duncan • K. Dusek • W.H. de Jeu J.F. Joanny • H.-H. Kausch • S. Kobyashi • K.-S. Lee • L. Leibler T.E. Long • I. Manners • M. Möller • O. Nuyken • E. M. Terentjev B. Voit • G. Wegner Advances in Polymer Science Recently Published and Forthcoming Volumes Inorganic Polymeric Nanocomposites Polymers and Light and Membranes Volume Editor: Lippert T. K. Vol. 179, 2005 Vol 168, 2004 Polymeric and Inorganic Fibres New Synthetic Methods Vol. 178, 2005 Vol. 167, 2004 Poly(arylene Ethynylenes) Polyelectrolytes with Defined From Synthesis to Application Molecular Architecture II Volume Editor: Weder C. Volume Editor: Schmidt M. Vol. 177, 2005 Vol. 166, 2004 Ring Opening Metathesis Polyelectrolytes with Defined Volume Editor: Buchmeister, M. Molecular Architecture I Vol. 176, 2005 Volume Editor: Schmidt M. Vol. 165, 2004 Polymer Particles Volume Editor: Okubo M. 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Adv.Polym.Sci.(2005)173:1–110 DOI:10.1007/b99426 ©Springer-VerlagBerlinHeidelberg2005 Polymer + Solvent Systems: Phase Diagrams, Interface Free Energies, and Nucleation KurtBinder1,MarcusMüller1,3,PeterVirnau1,4,andLuisGonzálezMacDowell2 1 InstitutfürPhysik,WA331,JohannesGutenberg-Universität, StaudingerWeg7,D-55099Mainz,Germany [email protected] 2 Dpto.deQuimicaFisica,FacultaddeCienciasQuimicas, UniversidadComplutense,Madrid28040,Spain 3 PhysicsDept.,UniversityofWisconsin-Wadison, Madison,WI53706-1390,USA 4 Dept.Phys.,MassachusettsInstituteofTechnology,77Mass.Ave.,Cambridge, MA02139-4307,USA Abstract Sometheoreticalconceptsonpolymer+solventsystemsandMonteCarlosimula- tionsofcorrespondingcoarse-grainedmodelsarebrieflyreviewed.Whilethephasediagram of polymers in bad solvents invoking the incompressibility approximation for the polymer solution has been astandard problem of polymer science for along time, amore complete understanding of compressible polymer solutions, where liquid-liquid phase separation and liquid-vapor transitionscompete, hasemerged onlyrecently. Aftergiving aphenomenolog- ical introduction, we outline and compare three complementary approaches: self-consistent fieldtheory,thermodynamicperturbationtheoryandgrandcanonicalMonteCarlosimulation. Inordertogiveaspecificexample,wefocusonthemixtureofhexadecanewithcarbondiox- ide. Attention is paid to correlate the description of the phase diagram with the properties ofinterfacesandthenucleationbarrierthatneedstobeovercometoformadroplet(orbub- ble,respectively)ofcriticalsize,necessaryforthedecayofthecorrespondingsuper-saturated metastablestate.Particularemphasisisgiventonewtechniquesusedforthecomputer sim- ulation of such phase diagrams where several order parameters compete, and to systematic difficultiesthatstillhamper thepredictionofaccuratenucleationbarriersfromtheobserva- tionofdroplets(orbubbles,respectively)infinitevolumes.Theextenttowhichrealmaterials canbemodeledwillalsobeexamined. Keywords Phaseequilibria,Compressiblemixtures,Nucleation,Interfaceproperties,Flory- Hugginstheory,Self-consistentfieldtheory,Thermodynamicperturbationtheory,Computer simulation,Coarse-grainedmodel,Hexadecane,Carbondioxide 1 Introduction................................................. 3 2 TheFlory-HugginsModelofIncompressiblePolymerSolutions andItsTest.................................................. 6 2.1 TheFlory-HugginsLatticeModelandtheMean-fieldApproximation . 6 2.2 PhaseSeparationBetweenPolymer andSolventBeyondMean-fieldTheory:ScalingPredictions ......... 11 2.3 MonteCarloandMolecularDynamicsMethods.................... 14 2 KurtBinderetal. 2.4 ComparisonBetweenComputerSimulationResults andAnalyticalPredictions...................................... 19 3 Self-consistentFieldTheoryforCompressiblePolymerSolutions.. 21 3.1 HexadecaneandCarbonDioxide: AModelforaCompressiblePolymerMixture..................... 21 3.2 Self-consistentFieldTheory .................................... 24 3.3 ModelingaTypeIII-phaseDiagramofaPolymer+SolventSystem... 32 3.4 ProfilesAcrossInterfaces ...................................... 39 3.5 BubbleNucleation ............................................ 40 3.5.1 ComparisontoClassicalNucleationTheoryandCahn–HilliardTheory 43 3.5.2 NucleationintheVicinityoftheTripleLine....................... 49 3.5.3 TemperatureDependenceand“FoamDiagrams” ................... 52 4 EquationofStateofCompressiblePolymerSolutions ............ 58 4.1 Introduction.................................................. 58 4.2 ThermodynamicPerturbationTheory............................. 60 4.2.1 IntuitiveApproach ............................................ 60 4.2.2 ScalingLawsfortheCriticalProperties........................... 61 4.2.3 GeneralExpressionforPurePolymersandMixtures................ 64 4.2.4 CalculationoftheMonomerandChainContributions............... 67 4.3 ApplicationtoPurePolymers ................................... 70 4.3.1 ApplicationtoaPureModelPolymer ............................ 70 4.3.2 ApplicationtoAlkanes......................................... 74 4.4 ApplicationtoPolymerSolutions................................ 78 4.4.1 ApplicationtoaPolymer–SolventSystem......................... 78 4.4.2 ApplicationtoSolutionsofn-AlkanesinCO ..................... 82 2 5 MonteCarloSimulationsintheGrandcanonicalEnsemble ....... 84 5.1 Technique ................................................... 84 5.2 PhaseDiagrams .............................................. 87 5.3 BubbleNucleation ............................................ 90 6 ConcludingRemarks......................................... 93 References......................................................... 97 List of Abbreviationsand Symbols MC MonteCarlo MD moleculardynamics SCF self-consistentfield TPT1 thermodynamicperturbationtheoryof1storder SAFT selfassociatingfluidtheory MSA meansphericalapproximation RHNC referencehypernettedchain