Treatment of Disordered and Ordered Systems of Polymer Chains by Lattice Methods

Classical lattice theories of systems of long-chain molecules provide estimates of the number Z of random configurations to the exclusion of ordered ones. The decrease of Z thus estimated to values << 1 with decrease in chain flexibility at high densities is genuine, but it does not take account of eligible ordered configurations; the latter are not a subset of the configurations whose numbers are estimated by classical lattice methods. Failure to recognize this fact and the fundamental distinction between disordered and ordered states has engendered misinterpretations and has cast doubt on the validity of lattice-statistical methods. In a system at equilibrium, the decline of Z (disordered) with decrease in chain flexibility must be arrested by a first order transition to an ordered state. The inference that approach of Z (disordered) to value <1 presages a thermodynamic transition of second order is tenable only if the array of ordered configurations, not comprehended by theories in which the mean field of unoccupied lattice sites is random, can be ignored.