The number of configurations of a cooperative assembly

Abstract

Formulae are found for the number of configurations of particles on two-and three-dimensional lattices when each particle (a) occupies two closest neighbour sites, and (b) consists of three groups which occupy three sites on the lattice in such a way that adjacent groups in the molecule occupy closest neighbour sites on the lattice. Bethe's method is used to determine the equilibrium conditions of the corresponding order-disorder problem, and the number of configurations is determined from these equilibrium conditions. For the case in which the molecules occupy two closest neighbour sites on the surface the determination of the number of configurations from geometrical considerations is discussed.It is found that for molecules which occupy two closest neighbour sites the number of configurations of particles for a square lattice, a simple cubic lattice and a body-centred cubic lattice respectively are and For molecules which occupy three sites on the lattice the corresponding results are and when the molecules are perfectly flexible and and when the molecules are completely inflexible, Ns being the total number of sites in the lattice.The author wishes to thank Dr J. K. Roberts for suggesting this work; the geometrical treatment given in § 3 was developed from manuscript notes prepared by him. The problem arose during an investigation of the vapour pressure equations of solutions in which the solute molecules are chain molecules consisting of a large number of groups, undertaken as part of the programme of fundamental research of the British Rubber Producers' Research Association, whom the author wishes to thank for the grant of a Research Scholarship.