The excluded volume for a given object is defined as that volume surrounding and including a given object, which is excluded to another object [53,54]. A similar definition of the excluded area holds in 2-D. This functional is always defined for a pair of objects. The "excluded volume" terminology comes from the statistical mechanics of gases, where this functional arises in the leading order concentration expansion ("virial expansion") for the pressure in the case of gas particles that repel each other with a hard-core volume exclusion [43].
Isihara [43] gives a general expression for the excluded volume of two convex objects, involving the surface area and mean radius of curvature of each, and then derives the explicit formula for ellipsoids of revolution. The elegant and general form for < Vex > for two convex objects, denoted 1 and 2, is expressed in terms of their surface areas Fi, the average radii of curvature on their surfaces, Ri, and their individual volumes, V>i. For convex objects the general formula is:
Isihara [43] has proven that a sphere minimizes the excluded volume of all convex bodies of finite volume. One should note that when the two particles are identical, as below, the usual convention is to use one half the quantity shown in eq. (10). Balberg does not normalize by this factor of one half in his definition of the excluded volume < Vex > [28,38]. The excluded volume for ellipsoids of revolution is then obtained by inserting the expressions for A and R, from eqs. (2-5). The value of < Vex > normalized by the excluded volume for a sphere with equal volume is invariant to this factor of 1/2 and results in the final expression for ellipsoids of revolution:
where z = a/b for prolate ellipsoids, z = b/a for oblate ellipsoids, and ε2 = 1 − 1/(z2) [55].