Adv. Polym. Sci. (2005) 173:1–110 DOI:10.1007/b99426 © Springer-Verlag Berlin Heidelberg 2005 Polymer + Solvent Systems: Phase Diagrams, Interface Free Energies, and Nucleation Kurt Binder1, Marcus Müller1,3, Peter Virnau1,4, and Luis González MacDowell2 1 Institut für Physik, WA331, Johannes Gutenberg-Universität, Staudinger Weg 7, D-55099 Mainz, Germany [email protected] 2 Dpto. de Quimica Fisica, Facultad de Ciencias Quimicas, Universidad Complutense, Madrid 28040, Spain 3 Physics Dept., University of Wisconsin-Wadison, Madison, WI53706-1390, USA 4 Dept. Phys., Massachusetts Institute of Technology, 77 Mass. Ave., Cambridge, MA 02139-4307, USA Abstract Some theoretical concepts on polymer + solvent systems and Monte Carlo simula- tions of corresponding coarse-grained models are briefly reviewed. While the phase diagram of polymers in bad solvents invoking the incompressibility approximation for the polymer solution has been a standard problem of polymer science for a long time, a more complete understanding of compressible polymer solutions, where liquid-liquid phase separation and liquid-vapor transitions compete, has emerged only recently. After giving a phenomenolog- ical introduction, we outline and compare three complementary approaches: self-consistent field theory, thermodynamic perturbation theory and grandcanonical Monte Carlo simulation. In order to give a specific example, we focus on the mixture of hexadecane with carbon diox- ide. Attention is paid to correlate the description of the phase diagram with the properties of interfaces and the nucleation barrier that needs to be overcome to form a droplet (or bub- ble, respectively) of critical size, necessary for the decay of the corresponding super-saturated metastable state. Particular emphasis is given to new techniques used for the computer sim- ulation of such phase diagrams where several order parameters compete, and to systematic difficulties that still hamper the prediction of accurate nucleation barriers from the observa- tion of droplets (or bubbles, respectively) in finite volumes. The extent to which real materials can be modeled will also be examined. Keywords Phase equilibria, Compressible mixtures, Nucleation, Interface properties, Flory- Huggins theory, Self-consistent field theory, Thermodynamic perturbation theory, Computer simulation, Coarse-grained model, Hexadecane, Carbon dioxide 1 Introduction ................................................. 3 2 The Flory-Huggins Model of Incompressible Polymer Solutions and Its Test .................................................. 6 2.1 The Flory-Huggins Lattice Model and the Mean-field Approximation . 6 2.2 Phase Separation Between Polymer and Solvent Beyond Mean-field Theory: Scaling Predictions . ...... 11 2.3 Monte Carlo and Molecular Dynamics Methods. ................... 14 2 Kurt Binder et al. 2.4 Comparison Between Computer Simulation Results andAnalyticalPredictions...................................... 19 3 Self-consistent Field Theory for Compressible Polymer Solutions .. 21 3.1 Hexadecane and Carbon Dioxide: AModelforaCompressiblePolymerMixture..................... 21 3.2 Self-consistentFieldTheory.................................... 24 3.3 Modeling a Type III-phase Diagram of a Polymer + Solvent System . 32 3.4 Profiles Across Interfaces . ................................... 39 3.5 Bubble Nucleation ............................................ 40 3.5.1 Comparison to Classical Nucleation Theory and Cahn–Hilliard Theory 43 3.5.2 NucleationintheVicinityoftheTripleLine....................... 49 3.5.3 TemperatureDependenceand“FoamDiagrams”................... 52 4 Equation of State of Compressible Polymer Solutions ............ 58 4.1 Introduction . ................................................. 58 4.2 Thermodynamic Perturbation Theory. ......................... 60 4.2.1 IntuitiveApproach............................................ 60 4.2.2 ScalingLawsfortheCriticalProperties........................... 61 4.2.3 GeneralExpressionforPurePolymersandMixtures................ 64 4.2.4 Calculation of the Monomer and Chain Contributions . .............. 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 Application to Solutions of n-Alkanes in CO2 ..................... 82 5 Monte Carlo Simulations in the Grandcanonical Ensemble ....... 84 5.1 Technique................................................... 84 5.2 PhaseDiagrams .............................................. 87 5.3 Bubble Nucleation ............................................ 90 6 Concluding Remarks ......................................... 93 References ......................................................... 97 List of Abbreviations and Symbols MC Monte Carlo MD molecular dynamics SCF self-consistent field TPT1 thermodynamic perturbation theory of 1st order SAFT self associating fluid theory MSA mean spherical approximation RHNC reference hypernetted chain Polymer + Solvent Systems 3 P-RISM polymer reference interaction site model RMPY/HTA reference molecular Percus Yevick / high temperature approximation RIS rotational isomeric state CO2 carbon dioxide C16H34 hexadecane 1 Introduction Understanding the properties of solutions of polymers in solvents of variable quality has been an outstanding problem in polymer science for decades [1, 2, 3, 4, 5]. The- ses systems are of fundamental interest as model systems for the statistical mechanics of fluid binary mixtures [6] and also are of enormous practical interest for predict- ing processing properties of various plastic materials in the chemical industry [7]. A good example is the production of solid polystyrene foam from polystyrene+ carbon dioxide mixtures [8, 9, 10, 11]. While in this system phase separation between the polymer and the small, molecular solvent is induced by a pressure jump, other sys- tems show phase separation when the solvent quality is changed by variation of tem- perature (see e.g. [12]). Fig. 1. (a) Schematic phase diagram of an incompressible polymer solution. Temperature T and molar fraction x of the solvent are variables, pressure p is constant. The position of the critical point (xcrit, Tcrit) is indicated by a filled circle. The coexistence curve is drawn as a full curve, while the (mean-field) spinodal is shown as a dotted curve. It separates metastable states which can only decay via nucleation and growth from unstable states which decay via spontaneous growth of concentration fluctuations (“spinodal decomposition”). A temperature quench from the one-phase region into the two-phase region is also indicated. (b) Schematic isothermal slice of a phase diagram of a compressible polymer solution, using pressure p and molar fraction x of the solvent as variables. A pressure jump from the one-phase region into the two-phase region is indicated. From [16] Actually, the latter case is conceptually most simple, since it can already be addressed using a special case of the Flory-Huggins theory [1, 2, 13, 14, 15]of 4 Kurt Binder et al. binary polymer mixtures (see Sect. 2). A schematic phase diagram is drawn in Fig. 1 (a)[16]. Note that the critical molar fraction of the solvent xcrit → 1 when the chain length N of the polymer tends to infinity. In this limit the critical temperature Tcrit of the polymer solution converges to the Θ temperature. We shall summarize the main facts for this case in Sect. 2. When one works at temperatures far below the liquid-vapor critical point of the solvent, treating the polymer solution as incompressible is a reasonable approxima- tion. Often, however, many solvents are used in their super-critical region [7], and then it would be completely inappropriate to assume incompressibility: Rather then, pressure is a useful control variable to trigger the phase separation (Fig. 1 (b)). While in Fig. 1 (a) an equilibrium state in the two phase coexistence region consists of poly- mer with some dissolved solvent molecules (on the left branch of the coexistence curve) coexisting with essentially pure (x = 1) dense solvent of high density in the liquid phase, Fig. 1 (b) actually may represent an equilibrium of polymer (with some dissolved solvent molecules) with the solvent (x = 1) in the vapor phase, having a much lower density than the polymer-rich phase. Fig. 2. (a) Phase diagrams of pure CO2 (lower two curves) and pure hexadecane (C16 H34, upper two curves) in the temperature-density plane. Experimental data were taken from [17, 18, 19, 20, 21], simulation data from [22] (see Sect. 5) (b)Sameas(a) but in the pressure- temperature plane. Curves are experimental results [17, 18, 19, 21], simulation data were taken from Virnau et al. [22] Of course, Fig. 1 only focuses on the simplest cases of the phase behavior. Even when we disregard the solid (crystalline or glassy) phases of the polymer and the solvent from the outset, it is necessary to discuss the global phase diagram of the binary fluid mixture consisting of polymer and solvent in more detail. As is well- known [6, 23, 24], in the space of control parameters – pressure p, temperature T , and molar fraction x of the solvent – several distinct types of phase diagrams may oc- cur, due to different ways in which liquid-vapor transitions and liquid-liquid unmix- ing interfere with each other.