Understanding liquid-liquid immiscibility and LCST behaviour in polymer solutions with a Wertheim TPT1 description

The aim of the work presented in this paper is to help in the understanding of the lower critical solution temperature (LCST) fluid phase behaviour exhibited by polymer solutions. It is well recognized that the LCST in polymer solutions is a consequence of density (compressibility) effects; the solvent is much more compressible than the polymer and the increasing difference in compressibility when the temperature is increased leads to a negative volume of mixing. The separate roles that the repulsive and attractive intermolecular interactions play in this regard are less well understood. In this study we use the Wertheim first-order thermodynamic perturbation theory (TPT1) [Wertheim, M. S., 1987, J. chem. Phys., 87, 7323; Chapman, W. G., Jackson, G., and Gubbins, K. E., 1988, Molec. Phys., 65, 1057] to describe the phase equilibria of model polymer solutions of hard spheres and hard-sphere chains where the diameter of the solvent and the polymeric segments are the same (symmetrical system). The thermodynamic functions (volume, enthalpy, entropy and Gibbs function) of mixing are determined to assess the possibility of a demixing instability in such a system. No fluid-fluid phase separation is found for the purely repulsive (athermal) system, regardless of the chain length of the polymer. The role of the attractive interactions is then investigated by incorporating attractive interactions at the mean-field level; the simplest system with equivalent (symmetric) solvent-solvent, solvent-polymer segment, and polymer segment-polymer segment interaction energies is examined. The attractive interactions are found to be essential in describing the liquid-liquid phase separation; LCST behaviour is found for mixtures with ‘polymer’ chains of seven segments or more. In this case we show that the phase behaviour is driven by an unfavourable (negative) entropy of mixing due to an increase in the density of the solvent on addition of small amounts of polymer. We also determine the thermodynamic properties of mixing for a system of spherical molecules of the same size with directional interactions that give rise to LCST and closed-loop behaviour. As expected the mechanism for phase separation in such systems is very different to that in polymer solutions.

[1]  J. Prausnitz,et al.  Equation of State for Mixtures of Hard-Sphere Chains Including Copolymers , 1994 .

[2]  C. Vega,et al.  Packing effects on the conformational equilibrium of alkanes , 1995 .

[3]  M. Huggins.  Solutions of Long Chain Compounds , 1941 .

[4]  W. Stockmayer.  Problems of the statistical thermodynamics of dilute polymer solutions , 1960 .

[5]  J. Prausnitz,et al.  Representation of vapor–liquid and liquid–liquid equilibria for binary systems containing polymers: Applicability of an extended flory–huggins equation , 1993 .

[6]  R. Scott,et al.  Static properties of solutions. Van der Waals and related models for hydrocarbon mixtures , 1970 .

[7]  K. Khanna,et al.  SAFT-D theory for hard sphere and hard disc chain fluids , 1999 .

[8]  Yuhua Song,et al.  A Perturbed Hard-Sphere-Chain Equation of State for Normal Fluids and Polymers , 1994 .

[9]  J. Perram,et al.  Percus-Yevick theory of correlation functions and nucleation effects in the sticky hard-sphere model , 1976 .

[10]  A. Galindo,et al.  Predicting the High-Pressure Phase Equilibria of Binary Mixtures of Perfluoro-n-alkanes + n-Alkanes Using the SAFT-VR Approach , 1998 .

[11]  J. S. Rowlinson,et al.  The thermodynamic properties of aqueous solutions of polyethylene glycol, polypropylene glycol and dioxane , 1957 .

[12]  Fumio Oosawa,et al.  On Interaction between Two Bodies Immersed in a Solution of Macromolecules , 1954 .

[13]  M. S. Yeom,et al.  Equation of state based on the thermodynamic perturbation theory of sequential polymerization for associating molecules and polymers , 2002 .

[14]  R. Koningsveld,et al.  Mean-field lattice equations of state: 4. Influence of pressure on the phase behaviour of the system polystyrene/cyclohexane , 1992 .

[15]  M. S. Yeom,et al.  A new equation of state for the hard-sphere chain fluids based on the thermodynamic perturbation theory and the multidensity integral equation , 2000 .

[16]  R. J. Baxter.  Percus-Yevick Equation for Hard Spheres with Surface Adhesion , 1968 .

[17]  George Jackson,et al.  New reference equation of state for associating liquids , 1990 .

[18]  D. Patterson.  Polymer compatibility with and without a solvent , 1982 .

[19]  C. Hall,et al.  Generalized Flory equations of state for hard heteronuclear chain molecules , 1996 .

[20]  C. Vega,et al.  The second virial coefficient of hard alkane models , 1998 .

[21]  C. Hall,et al.  Generalized Flory equations of state for copolymers modeled as square-well chain fluids , 1998 .

[22]  A. Galindo,et al.  Predicting the High-Pressure Phase Equilibria of Water + n-Alkanes Using a Simplified SAFT Theory with Transferable Intermolecular Interaction Parameters , 1996 .

[23]  A. Galindo,et al.  Global fluid phase behavior in binary mixtures of rodlike and platelike molecules , 2002 .

[24]  Tomáš Boublı́k,et al.  Hard‐Sphere Equation of State , 1970 .

[25]  A. Vrij,et al.  Polymers at Interfaces and the Interactions in Colloidal Dispersions , 1976 .

[26]  C. Hall,et al.  A new equation of state for athermal chains , 1989 .

[27]  R. Koningsveld,et al.  Liquid-liquid phase separation in multicomponent polymer systems , 1980 .

[28]  P. H. van Konynenburg,et al.  Critical lines and phase equilibria in binary van der Waals mixtures , 1980, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[29]  Yaoqi Zhou,et al.  Chemical association in simple models of molecular and ionic fluids. III: The cavity function , 1992 .

[30]  David Chandler,et al.  Derivation of an integral equation for pair correlation functions in molecular fluids , 1973 .

[31]  G. Jackson,et al.  Theory and computer simulation of hard-sphere site models of ring molecules , 1994 .

[32]  R. Koningsveld,et al.  Coalescence of upper and lower miscibility gaps in systems with concentration-dependent interactions , 1992 .

[33]  Frenkel,et al.  Evidence for entropy-driven demixing in hard-core fluids. , 1994, Physical review letters.

[34]  George Jackson,et al.  Statistical associating fluid theory for chain molecules with attractive potentials of variable range , 1997 .

[35]  J. Hansen,et al.  Influence of polymer-excluded volume on the phase-behavior of colloid-polymer mixtures. , 2002, Physical review letters.

[36]  D. Tildesley,et al.  An equation of state for hard dumbell fluids , 1980 .

[37]  Schweizer,et al.  Integral-equation theory of the structure of polymer melts. , 1987, Physical review letters.

[38]  Jane E. G. Lipson,et al.  Collapse of a polymer chain: A Born–Green–Yvon integral equation study , 1996 .

[39]  Clare McCabe,et al.  SAFT-VR modelling of the phase equilibrium of long-chain n-alkanes , 1999 .

[40]  M. Wertheim,et al.  Fluids with highly directional attractive forces. I. Statistical thermodynamics , 1984 .

[41]  S. Andrews,et al.  A Born–Green–Yvon integral equation treatment of a compressible fluid , 1992 .

[42]  A. Ben-Naim.  Statistical Mechanical Study of Hydrophobic Interaction. I. Interaction between Two Identical Nonpolar Solute Particles , 1971 .

[43]  Gabriele Sadowski,et al.  Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules , 2001 .

[44]  D. Patterson.  Free Volume and Polymer Solubility. A Qualitative View , 1969 .

[45]  W. Stockmayer.  Liquids and Liquid Mixtures. , 1961 .

[46]  K. G. Honnell,et al.  Local Structure of Semiflexible Polymer Melts , 1990 .

[47]  George Jackson,et al.  Study of the demixing transition in model athermal mixtures of colloids and flexible self-excluding polymers using the thermodynamic perturbation theory of Wertheim , 2003 .

[48]  Walter G Chapman,et al.  Henry's law constant for diatomic and polyatomic Lennard-Jones molecules , 1993 .

[49]  M. McHugh,et al.  Phase Behavior of Polymers in Supercritical Fluid Solvents. , 1999, Chemical reviews.

[50]  Isaac C. Sanchez,et al.  An elementary molecular theory of classical fluids. Pure fluids , 1976 .

[51]  I. R. Mcdonald,et al.  Theory of simple liquids , 1998 .

[52]  G. Habermehl,et al.  ReviewPure appl. Chem: Rinehart, K. L., et al. Marine natural products as sources of antiviral, antimicrobial, and antineoplastic Agents. 53, 795 (1981). (K. L. Rinehart, University of Illinois, Urbana, IL 61801, U.S.A.) , 1983 .

[53]  Meijer,et al.  Computer simulation of polymer-induced clustering of colloids. , 1991, Physical review letters.

[54]  P. Flory,et al.  Statistical Thermodynamics of Chain Molecule Liquids. I. An Equation of State for Normal Paraffin Hydrocarbons , 1964 .

[55]  M. Taylor,et al.  LATTICE VERSUS CONTINUUM MODELS OF A POLYMER CHAIN , 1999 .

[56]  J. Prausnitz,et al.  Perturbed hard chain theory for fluid mixtures: Thermodynamic properties for mixtures in natural gas and petroleum technology , 1978 .

[57]  P. B. Warren,et al.  Phase Behaviour of Colloid + Polymer Mixtures , 1992 .

[58]  J. Prausnitz,et al.  Perturbed hard-chain theory: An equation of state for fluids containing small or large molecules , 1975 .

[59]  L. Mcmaster.  Aspects of Polymer-Polymer Thermodynamics , 1973 .

[60]  B. Eichinger,et al.  EXISTENCE OF TWO CRITICAL CONCENTRATIONS IN BINARY PHASE DIAGRAMS , 1991 .

[61]  J. Lipson.  A Born–Green–Yvon integral equation treatment of incompressible lattice mixtures , 1992 .

[62]  G. Jackson,et al.  Closed-loop phase equilibria of a symmetrical associating mixture of square-well molecules examined by Gibbs ensemble Monte Carlo simulation , 2000 .

[63]  C. Hall,et al.  Theory and simulation of hard‐chain mixtures: Equations of state, mixing properties, and density profiles near hard walls , 1991 .

[64]  T. Boublı́k.  Equation of state of linear fused hard-sphere models , 1989 .

[65]  S. Sandler,et al.  Equation of state for the square-well chain fluid based on the dimer version of Wertheim's perturbation theory , 1995 .

[66]  B. Eichinger,et al.  Phase diagrams of binary polymer solutions and blends , 1991 .

[67]  G. Stell,et al.  Equations of state for hard chain molecules , 1993 .

[68]  P. Gennes.  Scaling Concepts in Polymer Physics , 1979 .

[69]  P. Cummings,et al.  Statistical mechanical models of chemical reactions , 1987 .

[70]  Phase behaviour of mixtures of colloidal spheres and excluded-volume polymer chains , 2002 .

[71]  V. Srivastava,et al.  Equation of state of long chain molecules and rings , 2002 .

[72]  W. Bruns,et al.  Molecular dynamics study of a single polymer chain in solution , 1981 .

[73]  Y. Sheng,et al.  Mixing properties of model polymer/solvent systems , 1995 .

[74]  George Jackson,et al.  Mixtures of associating spherical and chain molecules , 1989 .

[75]  G. Jackson.  Theory of closed-loop liquid-liquid immiscibility in mixtures of molecules with directional attractive forces , 1991 .

[76]  K. Schweizer,et al.  Theory of Polymer Melts: An Integral Equation Approach' , 1987 .

[77]  D. Chandler,et al.  Theory of the hydrophobic effect , 1977 .

[78]  A. Ravve,et al.  Principles of Polymer Chemistry , 1995 .

[79]  Shao-Tang Sun,et al.  Coil-Globule Phase Transition in a Single Polystyrene Chain in Cyclohexane , 1980 .

[80]  C. Vega,et al.  Virial coefficients and equation of state of hard alkane models , 1994 .

[81]  S. Sandler,et al.  The correlation functions of hard‐sphere chain fluids: Comparison of the Wertheim integral equation theory with the Monte Carlo simulation , 1995 .

[82]  W. Bruns,et al.  Molecular dynamics study of a single polymer chain in solution. II. Bead–spring model , 1981 .

[83]  M. Rosinberg,et al.  A perturbation density‐functional theory for polyatomic fluids. I. Rigid molecules , 1992 .

[84]  I. Prigogine,et al.  Statistical thermodynamics of r-MERS and r-MER solutions , 1953 .

[85]  R. Dickman,et al.  High density Monte Carlo simulations of chain molecules: Bulk equation of state and density profile near walls , 1988 .

[86]  K. Gubbins,et al.  An Equation of State for Water from a Simplified Intermolecular Potential , 1995 .

[87]  J. Johnson,et al.  Perturbation theory and computer simulations for linear and ring model polymers , 1996 .

[88]  Walter G Chapman,et al.  A new equation of state for hard chain molecules , 1994 .

[89]  K. E. Starling,et al.  Equation of State for Nonattracting Rigid Spheres , 1969 .

[90]  J. S. Rowlinson,et al.  Lower critical points in polymer solutions , 1960 .

[91]  Joseph M. Brader,et al.  Phase behaviour and structure of model colloid-polymer mixtures , 1999 .

[92]  M. Wertheim.  Fluids of dimerizing hard spheres, and fluid mixtures of hard spheres and dispheres , 1986 .

[93]  Joachim Gross,et al.  Application of perturbation theory to a hard-chain reference fluid: an equation of state for square-well chains , 2000 .

[94]  Walter G Chapman,et al.  TPT2 and SAFTD equations of state for mixtures of hard chain copolymers , 2000 .

[95]  G. Jackson,et al.  Predicting the High-Pressure Phase Equilibria of Methane + n-Hexane Using the SAFT-VR Approach , 1998 .

[96]  S. Sandler,et al.  An equation of state for the hard-sphere chain fluid: theory and Monte Carlo simulation , 1994 .

[97]  M. Wertheim,et al.  Fluids with highly directional attractive forces. IV. Equilibrium polymerization , 1986 .

[98]  M. Zuckermann,et al.  Simulation of short-chain polymer collapse with an explicit solvent , 2002 .

[99]  J. Pablo,et al.  Monte Carlo simulation of polymer chain collapse in an athermal solvent , 1997 .

[100]  J. Lipson.  Polymer blends and polymer solutions: a Born-Green-Yvon integral equation treatment , 1991 .

[101]  Clare McCabe,et al.  Examining the Adsorption (Vapor-Liquid Equilibria) of Short-Chain Hydrocarbons in Low-Density Polyethylene with the SAFT-VR Approach , 2001 .

[102]  I. Sanchez,et al.  Statistical Thermodynamics of Polymer Solutions , 1978 .

[103]  I. Sanchez,et al.  Monte Carlo simulation of polymer chain collapse in athermal solvents , 1996 .

[104]  J. Prausnitz,et al.  Liquid-liquid Phase Diagrams for Binary Polymer Solutions from a Perturbed Hard-Sphere-Chain Equation of State , 1994 .

[105]  C. Hall,et al.  Square-well diatomics: Bulk equation of state, density profiles near walls, virial coefficients and coexistence properties , 1991 .

[106]  George Jackson,et al.  THE THERMODYNAMICS OF MIXTURES AND THE CORRESPONDING MIXING RULES IN THE SAFT-VR APPROACH FOR POTENTIALS OF VARIABLE RANGE , 1998 .

[107]  L. F. Rull,et al.  Simulation study of the link between molecular association and reentrant miscibility for a mixture of molecules with directional interactions , 1999 .

[108]  J. Pablo,et al.  Polymer chain collapse near the lower critical solution temperature , 1997 .

[109]  The Fluid Phase , 2022 .

[110]  S. Sandler,et al.  Local composition model for chainlike molecules: A new simplified version of the perturbed hard chain theory , 1986 .

[111]  A Simplified Thermodynamic Perturbation Theory-Dimer Equation of State for Mixtures of Hard-Sphere Chains , 1996 .

[112]  K. Gubbins,et al.  Simulation of hard triatomic and tetratomic molecules. A test of associating fluid theories , 1993 .

[113]  David Chandler,et al.  Optimized Cluster Expansions for Classical Fluids. II. Theory of Molecular Liquids , 1972 .

[114]  Phase separation in mixtures of colloids and long ideal polymer coils. , 2000, Physical review letters.

[115]  G. Sadowski.  A square-well based equation of state taking into account the connectivity in chain molecules , 1998 .

[116]  M. Radosz,et al.  Phase equilibria in high-pressure polyethylene technology , 1995 .

[117]  Toyoichi Tanaka,et al.  First observation of the coil–globule transition in a single polymer chain , 1979, Nature.

[118]  M. Wertheim,et al.  Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations , 1984 .

[119]  R. Dickman,et al.  Equation of state for chain molecules: Continuous‐space analog of Flory theory , 1986 .

[120]  Y. Chiew,et al.  Percus-Yevick integral-equation theory for athermal hard-sphere chains: Part I: Equations of state , 1990 .

[121]  K. Gubbins,et al.  A modified thermodynamic perturbation theory equation for molecules with fused hard sphere cores , 1990 .

[122]  P. Cummings,et al.  Structural and thermodynamic properties of a multicomponent freely jointed hard sphere multi-Yukawa chain fluid , 2002 .

[123]  L. Kleintjens.  Developments in the thermodynamics of polymer systems , 1989 .

[124]  T. Boublík.  Background correlation functions in the hard sphere systems , 1986 .

[125]  A. Galindo,et al.  Recent advances in the use of the SAFT approach in describing electrolytes, interfaces, liquid crystals and polymers , 2001 .

[126]  C. Vega,et al.  Equation of state for hard n‐alkane models: Long chains , 1996 .

[127]  M. Wertheim,et al.  Thermodynamic perturbation theory of polymerization , 1987 .

[128]  L. Vega,et al.  Thermodynamic properties and phase equilibria of branched chain fluids using first- and second-order Wertheim’s thermodynamic perturbation theory , 2001 .

[129]  K. Gubbins,et al.  Phase equilibria of associating fluids : spherical molecules with multiple bonding sites , 1988 .

[130]  M. Wertheim,et al.  Fluids with highly directional attractive forces. III. Multiple attraction sites , 1986 .

[131]  K. E. Starling,et al.  Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres , 1971 .

[132]  N. Chatterjee.  Equations of State for Fluids and Fluid Mixtures , 1991 .

[133]  S. Toxvaerd.  Solution of the generalized Langevin equation for a polymer in a solvent , 1987 .

[134]  C. J. Grayce.  The conformation of hard-sphere polymers in hard-sphere solution calculated by single-chain simulation in a many-body solvent influence functional , 1997 .

[135]  M. Taylor,et al.  Effects of solvent on polymer chain dimensions: a Born–Green–Yvon integral equation study , 1997, cond-mat/9708122.

[136]  C. Vega,et al.  The virial coefficients of the pearl-necklace model , 2000 .

[137]  D. Frenkel,et al.  Phase separation in binary hard-core mixtures , 1994 .

[138]  C. Vega,et al.  Equation of state of chain molecules , 1990 .