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Virial Coefficients and Equations of State for Hard Polyhedron Fluids

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Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
§ Institute for Multiscale Simulation, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260, United States
Cite this: Langmuir 2017, 33, 42, 11788–11796
Publication Date (Web):September 15, 2017
https://doi.org/10.1021/acs.langmuir.7b02384
Copyright © 2017 American Chemical Society

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    Abstract

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    Hard polyhedra are a natural extension of the hard sphere model for simple fluids, but there is no general scheme for predicting the effect of shape on thermodynamic properties, even in moderate-density fluids. Only the second virial coefficient is known analytically for general convex shapes, so higher-order equations of state have been elusive. Here we investigate high-precision state functions in the fluid phase of 14 representative polyhedra with different assembly behaviors. We discuss historic efforts in analytically approximating virial coefficients up to B4 and numerically evaluating them to B8. Using virial coefficients as inputs, we show the convergence properties for four equations of state for hard convex bodies. In particular, the exponential approximant of Barlow et al. (J. Chem. Phys. 2012, 137, 204102) is found to be useful up to the first ordering transition for most polyhedra. The convergence behavior we explore can guide choices in expending additional resources for improved estimates. Fluids of arbitrary hard convex bodies are too complicated to be described in a general way at high densities, so the high-precision state data we provide can serve as a reference for future work in calculating state data or as a basis for thermodynamic integration.

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    In fact, Z0 = 1/(1 – 4η) for hard spheres, or Z0 = 1/[1 – (1 + 3α)η] for general HCBs, would give exact behavior to B2 but is obviously unsuitable as the basis for an extrapolation to a higher density due to the pole.

    τ and other simple asphericity metrics are neither clearly orthogonal to α nor as well motivated as quantities of thermodynamic relevance. With the knowledge that R, S, and v0 are all easily mapped to the set of Minkowski tensors, (26) it may be that higher-order information on a shape function could help in understanding HCB thermodynamic behavior, but the authors cannot offer any insight at this time.

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    The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b02384.

    • Additional visualization of data and descriptions of methods (PDF)

    • State data for each simulated shape in columns of pressure, packing fraction, and standard error of the mean packing fraction (ZIP)

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