A complete integral equation theory for accurate thermodynamics of chain molecules

Abstract

An integral equation theory is developed for n-alkane chains ranging in carbon number from eight to infinity through an adaptation of the polymer referenced interaction site model (PRISM). Thermodynamic perturbation theory (TPT) is adapted as the basis for expressing the equation of state through the coupling parameter expansion (CPE) as an extension of the theory developed by Ramana et al. for spherical molecules. The key feature of PRISM theory is the assumption that all pair correlations for a given site type are equivalent, regardless of whether they appear at the end of the chain or in the middle. The present analysis assumes that the PRISM result represents the average over all sites in the chain. The site-site interaction energies are then implemented through a weighted average based on their frequency of occurrence. The CPE is demonstrated through the fourth order in temperature, although higher order terms are accessible in principle. An advantage of the TPT representation is that convergence need be achieved only once for each chain length and density, then the TPT coefficients can be tabulated and interpolated for future reference. This approach moderates difficulties with molecular simulation of fully equilibrated long chains and uncertainty in the estimation of higher order TPT coefficients. Step potentials are used as the basis for validating and refining the theory using published simulation results for n-alkanes through 80 carbons. We find that the first order TPT contribution is fairly accurate regardless of conformational details but the second order TPT contribution is remarkably sensitive to conformation. We can obtain accurate characterization of the second order contribution by varying the stiffness parameter of the semi-flexible chain model. Higher order TPT contributions follow from the second order result. The theory would not be “complete,” however, without accurate characterization of the TPT reference contribution, which simultaneously impacts the first order contribution. We combine the compressibility route with a Yukawa closure to accurately characterize the reference fluid thermodynamics as a function of density, chain length, and bond length. Altogether, the result is an accurate model of all contributions to chain molecular thermodynamics, including the underlying fluid structure. A sample application is demonstrated by generating the phase diagrams through 80 carbons. It is observed that the flatness near the critical region is reproduced by including the higher order TPT contributions, but the impact of higher order contributions in the critical region is diminished for 80 carbon chains.