(222bi) A Statistical Mechanically-Based Cubic Equation of State for Athermal Hard-Sphere Chains

Traditional engineering equations of state such as the Soave-Redlich-Kwong or Peng-Robinson EOS are desirable owing to their cubic in molar volume form. On the other hand these equations of state generally give poor results when applied to systems at extreme conditions or which contain polar and/or associating components. Conversely, molecularly-based equations of state (i.e., statistical associating fluid theory EOS and its many variants) are much more accurate when successfully applied to such systems (Muller and Gubbins, 2001); but they are sufficiently computationally complex on account of  several molar volume roots, which make physically meaningful solutions often difficult to obtain. One approach to overcome this trade-off between accuracy and complexity is to incorporate molecular structure and behavior into a form of equation of state which has the attractive cubic in molar volume form.

We present here the athermal backbone for constructing a new class of statistical mechanically-based cubic equations of state. Specifically, we incorporate molecular simulation data for hard-sphere interaction, covalent bond formation and hard-dimer interaction into a simple expression which retains the cubic in molar volume form  of the EOS while accurately describing the compressibility factor versus packing fraction behavior for homonuclear hard-sphere chains up to 201 segments.  Firstly, molecular simulation data for hard-sphere fluids (Wu and Sadus, 2005) were nearly exactly fit to a simple two-parameter second-order polynomial in packing fraction which ensures the ideal gas limits of the compressibility factor Z=1 and for the radial distribution function at contact g^HS(s)=1 as packing fraction h=1. Next, Werthiem’s first-order perturbation theory (TPT1) (Wertheim, 1987) is applied to obtain a simple, highly accurate and consistent expression for the compressibility factor of chain formation which retains the overall cubic  in molar volume form of the EOS while introducing no new parameters. Finally, as shown  in earlier published work (Ghonasgi and Chapman, 1994; Chang and Sandler, 1995),  that the performance of equations of state for athermal hard-sphere chains is improved  by including a contribution for the interaction between athermal hard-dimers.  We refit g^HD(s) versus h data produced from a simple theoretical expression based on the Percus-Yevick approximation (Chiew, 1990) a form which was derived from same simple second-order polynomial in packing fraction used for Z^HS  which yielded two additional parameters. The combined equation of state including all three contributions yielded excellent results in representing molecular simulation data for Z versus h for athermal homonuclear  hard-sphere chains up to 201 segments (Yu et. al., 1994).. Furthermore, decent results comparable with those obtained from considerably more complex equations of state (Chang and Sandler, 1995) were obtained for the second-virial coefficient B versus segment number m up to 128 segments. Finally, since the four temperature-independent parameters are universal(the same for all real fluids), the EOS is easily extended to mixtures. Results for Z versus h were also generally very good when compared with available molecular simulation data from the literature (Yu et. al., 1994).

In sum, a simple and accurate molecularly-based EOS was developed for athermal homonuclear hard-sphere chains using molecular simulation data for three key effects. The equation contains four universalparameters, is easily applied to mixtures and serves as a basis for developing fundamentally sound cubic equations of state including dispersion, multipolar interactions and association.

REFERENCES

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Chiew, Y. C., Intermolecular Site-Site Correlation Function of Athermal Hard-Sphere Chains: Analytic Integral Equation Theory, J. Chem. Phys., 93, 5067-5074 (1990).

Ghonasgi D. and Chapman, W. G., A New Equation of State for Hard Chain Molecules, J. Chem. Phys., 100, 6633-6639 (1994).

Muller, E. A. and Gubbins, K. E., Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches, Ind. Eng. Chem. Res., 40, 2193-2211 (2001).

Wertheim, M. S., Thermodynamic Perturbation Theory of Polymerization, J. Chem. Phys., 87, 7223-7231 (1987).

Wu, G.-W. and Sadus, R. J., Hard Sphere Compressibility Factors for Equation of State Development, AIChE J., 51, 309-313 (2005).

Yu, Y.- X, Lu, J.-F., Tong, J.- S. and Li, Y.- G., Equation of State for Hard-Sphere Chain Molecules, Fluid Phase Equilibria, 102, 159-172 (1994).