Macromolecular Research, Vol. 16, No. 4, pp 320-328 (2008) Consideration of Long and Middle Range Interaction on the Calculation of Activities for Binary Polymer Solutions Seung Seok Lee, Young Chan Bae*, and Yang Kook Sun Division of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Korea Jae Jun Kim College of Architecture, Hanyang University, Seoul 133-791, Korea Received October 29, 2007; Revised December 26, 2007; Accepted January 2, 2008 Abstract: We established a thermodynamic framework of group contribution method based on modified double lat- tice (MDL) model. The proposed model included the long-range interaction contribution caused by the Coulomb electrostatic forces, the middle-range interaction contribution from the indirect effects of the charge interactions and the short-range interaction from modified double lattice model. The group contribution method explained the com- binatorial energy contribution responsible for the revised Flory-Huggins entropy of mixing, the van der Waals energy contribution from dispersion, the polar force, and the specific energy contribution from hydrogen bonding. We showed the solvent activities of various polymer solution systems in comparison with theoretical predictions based on experimental data. The proposed model gave a very good agreement with the experimental data. Keywords: modified double lattice model, group contribution method, charge interaction, long range interaction, mid- dle range interaction. Introduction et al.,9 Kontogeogis et al.,10 and Bogdanic and Fredenslund.11 Those methods are based on the UNIFAC correlation For engineering purposes, it is often necessary to make which is often successful for estimating phase equlibria in estimate of activity for mixtures where only fragmentary mixtures containing ordinary (nonpolymer) liquids. data, or no data at all, are available. For vapor-liquid equi- The fundamental basis for existing group-contribution libria, such estimates can be made using a group-contribu- methods for polymer solutions is the lattice theory of Flory12 tion method.1 and Huggins.13 To pursue a formal exact solution to the lat- Since activity coefficients calculation using group-contri- tice model using advanced statistical and mathematical butions was suggested by Langmuir,2 the most widely used methods, Freed and coworkers14,15 developed a lattice-clus- and best known of group-contribution method is the UNI- ter theory for polymer-solvent system. This theory provides FAC.3 The acronym UNIFAC denotes the UNIQUAC4 an exact mathematical solution for the Flory-Huggins functional group activity coefficient. The UNIFAC correla- model. tion is based on a semi-empirical model for liquid polymer Hu et al.16 proposed a new theory called the double lat- solutions called UNIQUAC (universal quasi-chemical activity tice model based on Freed’s lattice-cluster theory and Bae coefficient). When compared with the experimental data, et al.17 reported a modified double-lattice model and pro- however, the UNIFAC equation shows deviations too large vided an exact mathematical form for the secondary lattice to satisfactorily explain the polymer solutions. of the double lattice. Hu et al.18 presented the group contri- Oishi and Prausnitz5 modified the UNIFAC model by pro- bution method including a revised Flory-Huggins entropy viding a free volume contribution suggested by Prigogine- and a series expression for excess internal energy as well Flory-Patterson theory for polymer solutions to consider the as a double lattice model to account for specific interac- compressibility and change in density upon isothermal mixing. tions. Later, many modifications of UNIFAC model are reported Debye and Hückel19 presented the theory of inter-ionic by Holten-Andersen and Fredenslund,6,7 Chen et al.,8 Elbro attraction in aqueous electrolyte solutions. This theory first made it possible to calculate the ion activity coefficients at *Corresponding Author. E-mail: [email protected] infinite dilution. In order to apply the Debye-Hückel theory 320 Calculation of Activities of Polymer Solutions Using Group Contribution Method Based on Modified Double Lattice Model to higher electrolyte concentrations, Stokes and Robinson20 polymer solutions. In this study, we calculate the excess proposed the concept of ionic hydration, and derived a two Gibbs energy as the sum of three contributions: parameter equation. Later several variations of the ionic 21 22 E E E E hydration theory are reported: Glueckauf, Robinson et al. G = GLR ++GMR GSR (1) and Stokes and Robinson.23 These theories are used for cal- E culation of the activity of water in concentrated aqueous GLR represents the long-range (LR) interaction contribu- electrolyte solutions. tion caused by the Coulomb electrostatic forces, and mainly 28 E In the early 1970s, the integral techniques based on the describes the direct effects of charge interactions. GMR Ornstein-Zernicke equation have been used to solve the represents the indirect effects of charge interactions. In primitive model of electrolyte solutions. Blum24 used the order to distinguish the indirect effects from the non-charge method proposed by Baxter25 for hard-sphere solutions and short-range interactions of the non-electrolyte solution, and square-well potentials, and obtained a solution for the mean because some of the charge interactions (such as the charge- spherical approximation (MSA). Planche and Renon26 gen- dipole interactions and charge-induced dipole interactions) eralized the model of Baxter by using the formalism of are proportional to r-2 and r-4, we shall refer to this term as Blum, taking into account short-range forces between mole- the middle-range (MR) interaction contribution. (r is a dis- 32,34 E cules in order to get a non-primitive representation of elec- tance with two particles). GSR expresses the contribution trolyte solutions. Pitzer27,28 developed general equations for of the non-charge interactions, which is identical to the the thermodynamic properties of aqueous electrolytes based short-range (SR) interactions in non-electrolyte solutions. on the as Debye-Hückel theory. However, if the Pitzer This model has two ways of usage according to the classi- parameters are applied to the high concentration range (for fication of non-electrolyte and electrolyte: example 10-20 M) very large deviations are obtained in 1. In case of non-electrolyte, only short range interaction most cases. is considered. In last decades, numerous authors have modified the 2. In case of electrolyte, short, mid, and long range are existing gE model and EOS (Equation of State) model for considered. non-electrolyte systems. Mock et al.29 applied the NRTL Long-Range Interaction Contribution. The long-range model to electrolyte systems. Their model provides a con- electrostatic interaction explained interaction between ion- sistent thermodynamic framework for the representation of ion. It gives rise to the Debye-Hückel limiting law, which the phase equilibrium of mixed solvent electrolyte systems. states that the logarithm to the ionic activity coefficient at However, different parameters are required to calculate the high dilution is proportional to the square-root of the ionic VLE at different temperatures. Sander et al.30 proposed an strength. In all the models presented in the literature, this extended UNIQUAC model for mixed-solvent/electrolyte contribution has been described by some or other form of systems. Macedo et al.31 used a modified Debye-Hückel the Debye-Hückel theory. E term derived from the McMillan-Mayer solution theory to In this study, GLR is calculated in terms of the Debye- replace the Debye-Hückel term in the Sander model. Hückel theory as modified by Fowler and Guggenheim34: E The most widely used and best known of the g model for ion 32,33 E –1 2 2 the electrolyte solution is LIQUAC. It consists of a Debye- GLR = –()3D ∑ sizi e τ()ka (2) Hückel term to account for long-range electrostatic interac- i = 1 tions, the UNIQUAC equation for the description of short- where si is the number of i ions in the system, D is the sol- range interactions among all particles, and a middle-range vent dielectric constant, a is the distance of closest approach contribution to include all indirect effects of the charge between two ions and τ (x) is defined as interactions. 2 –3 x In this study, we propose a group-contribution model that τ()x = 3x ln()1 + x – x + ---- (3) 2 can be used to describe solvent activities of polymer solu- tions and phase diagram of perfluorinated SPE/water system. where x = κa and κ is the inverse of the shielding length The proposed model is based on a modified double lattice commonly called Debye length which is a characteristic dis- model, long range interaction and middle range interaction tance of interaction. wherein the Helmholtz function of mixing includes the The expression for the long-range interaction contribution revised Flory-Huggins entropy contribution, the van der Waals to the activity coefficients of solvent, s, follows the appro- energy contribution, and the specific energy contribution. priate derivations of eq. (2). We assume the partial molar volume of the solvent s in the solution is approximated as Model Development the molar volume of the pure solvent. Various types of interactions are taken into account for γ LR 2AMs 12⁄ 1 ()12⁄ ln s = -------------3 1 + bI –21------------------12⁄ – ln + bI (4) describing the activities of electrolyte and non-electrolyte b 1 + bI Macromol.