(207d) Accurate Characterization of Intramolecular Contributions to the Thermodynamic Perturbation Terms of Chain Fluids

Previous reports have established that thermodynamic perturbation theory (TPT) can accurately characterize the thermodynamics of vibrating bond models of n-alkane chains, including vapor pressure and density. These molecular models are composed of hard sphere united atom sites with diameters of roughly 0.36 nm, bond lengths of 0.154 nm, and bond angles of 110°.¹ The bonded interactions are defined by vibrating potential wells centered at 0.154 nm, 0.265 nm, and by an intramolecular diameter that controls how closely atoms can approach if they are more than two bonds apart, thus controlling the dihedral angle distribution.² The bonded interactions apply to intramolecular sites that are seven or fewer bonds apart. The non-bonded interactions in this work are the same for intermolecular interactions or intramolecular interactions more than seven bonds apart. They are described by a sequence of four wells with varying depth, resembling steps in a staircase. The non-bonded interactions of CH3 are distinct from those of CH2, but the same potential models are transferred to chains of all lengths from C2 to C30. The simulations were performed at packing fractions ranging from 0.01 to 0.56. Each state point was simulated for 5 ns with 100 molecules. TPT decomposes the description of the molecular simulation results into a repulsive contribution (A0), and first and second order perturbations (A1, A2). Higher order perturbations are found to be negligible for packing fractions greater than 0.3. A0 is obtained by integrating the reference compressibility factor (Z0), which in turn is computed directly from the pressures simulated by molecular dynamics. A1 and A2 are computed by sums of distributions of sites within the simulated reference fluids.

Note that the TPT contributions are not necessarily computed with direct regard to the molecular connectivity. For example, the Z0 contribution is computed from the sum of all collisions, regardless of whether they are intramolecular or intermolecular collisions. Similarly, the A1 and A2 contributions are summed over site-site distribution functions that may involve sites on the same molecule as well as those on different molecules. For the A1 term, this includes only the non-bonded intramolecular interactions in addition to intermolecular interactions. The density dependence of the intramolecular A1 term has been investigated in this study. For the A2 term, we must recognize that there are both inter- and intra- molecular contributions to the pair correlations. For example, suppose N_ijm designates the number of i, j pairs of sites a distance corresponding to the m^th well. N_ijm may refer to a bonded ij pair, and N_kln may refer to an intermolecular pair. This pairing gives rise to an intra/inter coupling in the fluctuations that must be taken into account. For molecules with 12 bonds or more between sites, the density dependence of intramolecular correlations becomes significant. We show that the intramolecular contribution to A1 is a weak function of density but it must be taken into account because A1/T ~ 10x A2/T² in the liquid region. The A2 contribution must be decomposed into intra/intra, inter/inter, and intra/inter contributions. The intra/inter contribution is has not been explicitly treated previously, whereas the behavior of the other contributions is qualitatively similar to what has been discussed.³ While both the intra/intra and inter/inter contributions are negative, the intra/inter contribution is positive and monotonically increasing with respect to increasing density. This behavior can be understood as a reflection of screening, the same effect that leads to the correlation hole in the radial distribution function. By accounting for these effects, a smooth, systematic trend is observed in the residuals when compared to experimental values of the vapor pressures of n-alkanes. The reduction of random variability in this trend indicates that the proper accounting is necessary, but also indicates that the four-well description has limitations that should be addressed if further reductions in model error are to be achieved.

Keywords: Physical properties, molecular dynamics simulation, vapor pressure, density, phase equilibria, thermodynamic perturbation theory, intramolecular correlations, screening.

(1) Cui, J.; Elliott Jr., J. R. Phase Diagrams for Multi-Step Potential Models of n-Alkanes by Discontinuous Molecular Dynamics/Thermodynamic Perturbation Theory. J. Chem. Phys. 2002, 116, 8625.

(2) Unlu, O.; Gray, N. H.; Gerek, Z. N.; Elliott, J. R. Transferable Step Potentials for the Straight Chain Alkanes, Alkenes, Alkynes, Ethers, and Alcohols. Ind. Eng. Chem. Res. 2004, 43, 1788-1793.

(3) Elliott, J. R.; Gray, N. H. Asymptotic Trends in Thermodynamic Perturbation Terms. J. Chem. Phys. 2005, 123, 184902.