Analytical treatment of the volume and surface area of molecules formed by an arbitrary collection of unequal spheres intersected by planes

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Abstract
A general algorithm has been developed for the analytical determination of the volume and exposed surface area of a solid body formed by a collection of arbitrarily sized intersecting spheres and delimited by a set of arbitrarily directed planes. The algorithm is useful for analysing molecules represented as fused hard spheres, sections of such molecules, as well as void or available spaces formed among such molecules. The multisphere-multiplane problem is decomposed into a set of problems involving the intersection of a single sphere by an arbitrary collection of planes. The volume and exposed area of the convex body formed by such an intersection are found using simple principles of analytical geometry. Applications of the new method are presented for the determination of the volume, excluded volume, and surface area of long-chain molecules and of the void volume and internal surface area of a zeolite crystal. It is found that the method is faster, more efficient, more versatile, and more accurate than other analytical and numerical methods. As a result of the decoupling strategy used, the new algorithm scales better with system complexity and can readily provide the exposed areas and volumes contributed by individual spheres in the system.

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