Liquid–liquid equilibria of binary polymer solutions with specific interactions Bong Ho Chang and Young Chan Bae* Department of Industrial Chemistry and Molecular Thermodynamics Lab., Hanyang University, Seoul 133-791, Korea (Revised 8 December 1997) In our previous study, we proposed a new expression for the configurational energy of mixing taking into account non-random mixing effect and chain-length dependence of the polymer. But our model can not predict lower critical solution temperature (LCST) behaviours of liquid–liquid equilibria for binary polymer solutions. In this study, we extend our previous model to describe LCST behaviours of binary polymer solutions by employing a secondary lattice concept as a perturbation term to account for oriented interactions (or specific interactions). q 1998 Elsevier Science Ltd. All rights reserved. (Keywords: binary polymer solutions; lower critical solution temperature; specific interactions) INTRODUCTION Many lattice models have been used to correlate the thermodynamic properties of polymer solutions. The most widely used and best known incompressible-lattice models is the Flory–Huggins theory 1–4 which illustrates in a simple way the competition between the entropy of mixing and the attractive forces that produce liquid–liquid phase separation at low temperatures with an upper critical solution temperature (UCST) 5 . However, the Flory–Huggins model cannot describe the lower critical solution temperature (LCST) 5 behaviour of polymer solutions. Phase equilibria in this behaviour are influenced by strong, orientation- dependent interaction forces, such as hydrogen bonds. The Flory–Huggins model does not take into account deviations from random mixing caused by these orientation-dependent interactions. Many theoretical improvements, including Guggenheim’s Quasi-chemical model 6 , have been obtained by various workers to gain the mathematical solution of the lattice model, including chain connectivity and non-random mixing. In recent years, several authors have proposed molecular-thermodynamic models for polymer solutions that attempt to account for non-random mixing 6–10 . These models are based on the local-composition concept where expressions for local composition are obtained either from essentially empirical relations or else are derived from Guggenheim’s traditional quasi-chemical approximation. While an UCST is readily understood in terms of intermolecular forces, interpretation of a LCST is more difficult. Generally a LCST is observed when either of the following conditions prevails: (1) Large differences in thermal expansion of solvent and solute: this situation is often encountered in polymer/ volatile–solvent systems when the system temperature approaches the critical temperature of the solvent. As the temperature rises, the solvent expands more rapidly than the solute; solubility decreases until two separate phases are formed. This behaviour is well described by free-volume theories. Free volume theories for polymer solutions were developed by numerous investigations, notably by Flory 3 and by Patterson and Delmas 11 . These theories were based on a generalized form of the van der Waals partition function, which is the product of two independent partition functions: one accounts for free volume and the other for attractive forces. To account for compressibility and density changes upon isothermal mixing, Sanchez and Lacombe 12–13 and Kleintjens and Koningsveld 14 have derived different forms of a lattice–fluid model based on the Flory–Huggins lattice theory. (2) Order-disorder transitions, as encountered in systems of molecules capable of forming hydrogen bonds. More than 50 years ago, Hirschfelder et al. 15 suggested a qualitative physical picture to explain the occurrence of a LCST in hydrogen bonding systems: mutual solu- bility at temperatures below the LCST is attributed to highly orientation-dependent interactions (hydrogen bonds) between unlike species, which are weaker than those between like species, and so the system splits into two phases. Specific forces between adjacent sites can easily be introduced into the model to account for poly- mers that hydrogen bond. Painter et al. 16,17 developed a Gibbs free energy model for polymers that hydrogen bond using chemical theory to account for the formation of associated species and lattice theory to describe the non-ideal interactions between the associated species. Sanchez and Balazs 18 used a lattice–fluid equation of state to include specific interactions such as hydrogen bonding. In this model, a quantitative description of the spinodal phase diagram, as well as a semiquantitative description of the composition and temperature depen- dence of the x interaction parameter, is possible. Recently, Hu et al. 19,20 reported a new model called ‘double-lattice model’ based on Freed’s lattice–cluster theory 21–28 . In their model, ordinary polymer solutions are described by the primary lattice, while a secondary lattice is introduced as a perturbation to account for POLYMER Volume 39 Number 25 1998 6449 Polymer Vol. 39 No. 25, pp. 6449–6454, 1998 q 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-3861/98/$ - see front matter PII: S0032-3861(97)10386-X * To whom correspondence should be addressed