Geometric measures of uniaxial solids of revolution in ${\mathbb{R}}^4$ and their relation to the second virial coefficient

Abstract

We provide analytical expressions for the second virial coefficients of hard, convex, monoaxial solids of revolution in ${\mathbb{R}}^4$. The excluded volume per particle and thus the second virial coefficient is calculated using quermassintegrals and rotationally invariant mixed volumes based on the Brunn-Minkowski theorem. We derive analytical expressions for the mutual excluded volume of four-dimensional hard solids of revolution in dependence on their aspect ratio $\nu$ including the limits of infinitely thin oblate and infinitely long prolate geometries. Using reduced second virial coefficients $B_2^{\ast }=B_2/V_{\mathrm{P}}$ as size-independent quantities with $V_{\mathrm{P}}$ denoting the $D$-dimensional particle volume, the influence of the particle geometry to the mutual excluded volume is analyzed for various shapes. Beyond the aspect ratio $\nu$, the detailed particle shape influences the reduced second virial coefficients $B_2^{\ast }$. We prove that for $D$-dimensional spherocylinders in arbitrary-dimensional Euclidean spaces ${\mathbb{R}}^D$ their excluded volume solely depends on at most three intrinsic volumes, whereas for different convex geometries $D$ intrinsic volumes are required. For $D$-dimensional ellipsoids of revolution, the general parity $B_2^{\ast }(\nu )=B_2^{\ast }({\nu }^{-1})$ is proven.

References in corpus (3)

• Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles
• Exclusion Volumes of Convex Bodies in High Space Dimensions: Applications to Virial Coefficients and Continuum Percolation
• The fourth virial coefficient for hard spheres in even dimension

Cited by in corpus (2)

• Excluded volume geometry and packing fraction in binary convex hyperparticle mixtures
• Relating thermodynamic quantities of convex-hard-body fluids to the body's shape