Cluster integrals contributing to the fourth virial coefficient of hard convex bodies

The fourth-order virial expansion represents an important tool in the description of the equilibrium behaviour of pure fluids and mixtures in the vicinity of their critical point/critical region. Dependences of cluster integrals D(4)(HCB), D(5)(HCB) and D(6)(HCB) of hard convex bodies on the geometric characteristics (i.e. the volume, surface area and the mean curvature integral of the given body) form the basic information necessary for the evaluation of the fourth virial coefficient, D, of Kihara non-spherical molecules. We determined D(4)(HCB), D(5)(HCB) and D(6)(HCB) for pure prolate and oblate hard spherocylinders with the non-sphericity parameter alpha is an element of (1, 3). A Monte Carlo integration technique was employed and the individual contributions D(4)(HCB)/V(3), D(5)(HCB)/V(3) and D(6)(HCB)/V(3) were expressed as quadratic functions of alpha, with coefficients (integral quantities) obtained by a three-step fitting procedure. Values of the HCB fourth virial coefficient (obtained as an algebraic sum of m(i)D(i)(HCB)) for the individual types of molecules agree well with the pseudo-experimental data from the literature. The expressions for D(i)(PS) and D(i)(OS) as well as that for the total fourth virial coefficient for prolate and oblate spherocylinders differ considerably; none of the one-parameter equations of state (proposed for HCB systems) yields an expression predicting correctly the fourth virial coefficient of HCBs in the considered range of alpha. An attempt is made to express the fourth virial coefficient in terms of two non-sphericity parameters; different results for prolate and oblate hard spherocylinders were obtained.