PHASE TRANSITION IN A ONE-DIMENSIONAL SYSTEM

Abstract

We give an examnple here illustrating a very simple way to obtain a first-order phase transition in a one-dimensional system. We consider discrete cases in this paper (which follow from the use of a step intermolecular potential). Continuous cases (from a smooth intermolecular potential) will be discussed in a second paper.1 One can also introduice length and tension as additional thermodynamic variables (here, tension = 0). Although the basic idea occurred to us in connection with allosteric effects2 in biological macromolecules, it turns out to have been exploited somewhat already in solid-state physics (three-dimensional systems).' The feature of main interest here is not any relation to real one-dimensional systems (the model is quite artificial), but rather the extreme simplicity of the model, which can be treated easily and exactly. Our system consists of a regular one-dimensional lattice (Einstein crystal) of N1 (N1 -- oo) component 1 molecules, which produce a lattice (samne spacing) of N, sites for the adsorption of N2 ? N1 component 2 molecules, at most one per site. The component 2 molecules form a conventional one-dimensional Ising system4 (lattice gas) with nearest-neighbor interactions only. The essential qualitative feature of the model is that the optimal lattice spacing (component 1) does not match the optimal nearest-neighbor spacing (component 2). Hence, adsorption leads to adjustment in the lattice spacing, in amount sufficient to extremalize the appropriate thermodynamic poteintial of the binary system. The artificiality in the model, referred to above and necessary for the phase transition, arises from an imposed restraint of uniform, though variable, lattice spacilig through the whole linear system. As we shall see at the end of the paper, if local (i.e., nonuniform) variations in lattice spacing are allowed-in response to adsorption-no phase transition occurs. The fact that the lattice spacing is restrained to be uniform implies a kind of long-range interaction between component 1 molecules. Statistical Mechanics of the Model.-The nearest-neighbor interaction potential between two adsorbed molecules is denoted by w2(r). For simplicity, we assume that lattice molecules also have, among themselves, nearest-neighbor interactions only with pair potential w1(r). Also, for simplicity, we assume that the partition functions ql(T) and q2(T) (not including e -W1kT and e w2/k17) for individual type 1 and type 2 molecules, respectively, are unaffected by the amount of adsorption or by a change in lattice spacing. The two cases considered in this paper (they can be treated together) are shown