George M. Bell

Algebraic Methods in Statistical Mechanics

There are rather few models in statistical mechanics which have been solved exactly and many of these can be formulated in vertex form (see Chap. 5 and Baxter 1982a). In the absence of such solutions, two alternative methods of attack have been to determine ‘rigorous’ conditions for the existence of phase transitions, or bounds on the regions of phase space within which phase transitions can exist (Ruelle 1969, Griffiths 1972, Sinai 1982). A common feature of many of these approaches is that information about the properties of an infinite system is obtained by investigating an equivalent finite system (or system of reduced dimensionality). This can be taken to be the unifying theme of the methods described in this chapter. In Sects. 4.3–4.8 it is convenient, although not essential, to described the ideas in terms of a lattice fluid model. The equivalence between the spin-½ Ising model with nearest-neighbour interactions and a simple lattice fluid with nearest-neighbour pair interactions between particles was first shown by Lee and Yang (1952). This relationship holds between any one-component lattice fluid of particles of chemical potential μ and a spin-½ system on the same lattice in a magnetic field 𝓗. It follows that the methods and results described here can be ‘translated’ into a spin-½ formulation.

Theory of cooperative phenomena in monolayers of hydroxyhexadecanoic acid isomers

Kellner and Cadenhead have obtained surface pressure/area isotherms for monolayers of several hydroxyhexadecanoic acid isomers. They find considerable differences in behaviour, depending on the position of the hydroxyl group on the carbon chain, and divide the isomers into “monopolar” and “bipolar” types. A lattice fluid model is introduced here with monomer states to represent erect conformations of the molecules, a dimer state to represent conformations parallel to the interface and vacant sites or “holes”. Using a Flory–Huggins type approximation the main behaviour types are reproduced. There are second-order (slope-discontinuity) transitions on some isotherms for all cases and first-order (liquid–gas) transitions in addition for the “monopolar” case with the monomer states energetically favoured. For “bipolar” cases with the dimer state energetically favoured the isotherms are more extended with nearly horizontal portions on the high-density side of the transition at some temperatures. With appropriate interaction parameters there is a region of negative thermal expansion on the low-density side of the transition similar to that found experimentally in 9-HHA monolayers.

Second-order phase changes in phospholipid monolayers at the oil/water interface and related phenomena

Surface pressure against area isotherms for phospholipid monolayers at the oil/water interface display second-order phase transitions which resemble the ones between “condensed” and “expanded” monolayers at air/water interfaces. Behaviour at lower surface densities indicates a strong repulsive interaction between the phospholipid molecules at the oil/water interface. A statistical mechanical model is developed in which the monolayer molecules occupy sites on a two-dimensional lattice and adopt one of two possible orientational states. By appropriate choice of the interaction energies of the molecules in these states and the use of order–disorder statistics the features characteristic of a second-order phase transition are obtained for isotherms at both interfaces. From the point of view of our present theory what is termed the “expanded” state is in fact a fluid phase whose degree of order is lower than that of the “condensed” state. The choice of a large overall attractive interaction energy between nearest-neighbours yields the liquid–vapour phase transition which has been observed in experiments at the air/water interface whereas with an overall repulsive interaction the liquid–vapour transition is absent, as in the oil/water case.

Theory of anomalous temperature dependence in spin-labelled lipid monolayers

Certain spin-labelled monolayers, for instance 12-nitroxide stearic acid, have the remarkable property that (pressure against area) isotherms lie above those obtained at higher temperatures, due to density increasing with temperature over quite large intervals. The qualitative explanation is that each molecule has both “extended” conformations with the spin-label as well as the head-group bonded to the water surface and “upright” conformations with only the head-group bonded. In the present paper a statistical-mechanical theory embodying these ideas is developed. The constant pressure partition function for a one-dimensional lattice model is evaluated exactly by a matrix method and (pressure against density) isotherms calculated. “Isotherm reversal” proves to be a characteristic of the model and is found for a range of interaction energy parameters. The isotherms are similar in shape to the experimental ones, with a high-density “plateau” region of reduced slope at lower temperatures. However the pressure range over which anomalous behaviour is found is smaller relative to the “plateau” pressure than that found experimentally.

The Mean-Field Approximation, Scaling and Critical Exponents

We consider the Ising model on a regular lattice (see Appendix A.1) where each interior site has the same number of nearest-neighbour sites. This is called the coordination number of the lattice and will be denoted by z. We shall denote the dimension of the lattice by the symbol d. It is assumed that, in the thermodynamic limit, boundary sites can be disregarded and that, with N sites, the number of nearest-neighbour site pairs is 1/2zN. Since the Ising model is restricted in the sense of Sect. 2.3.4, the Helmholtz free energy A depends on the two extensive variables M and N, the latter being equal to the number of spins and also to the reduced volume V if the volume μ0 per site is taken as the standard volume (see discussion after (2.55)). The configurational Helmholtz free energy per site a depends on T and the (relative) magnetization per site m, defined by (2.70), so that n = 2.

Solid Mixtures and the Dilute Ising Model

We consider an isotropic solid mixture of M A atoms of component A and M B of component B in volume \(\tilde V\). This is modelled by a regular lattice of N sites, the volume υ0 per lattice site being as usual taken as constant and used as the standard volume in (1.54). Each site is occupied by an atom so that

Exact Results for Two-Dimensional Ising Models

M = {M_A} + {M_B} = \tilde V/{v_0} = V = N.

The Eight-Vertex Model

(6.1) .