Paul H. E. Meijer

Dynamics of a spin‐1 model with the pair correlation

A spin‐1 or three state system will undergo a first or second order phase transition depending on the ratio of coupling parameters α. Using the pair correlation approximation, the transition temperature is determined in order to establish the unstable, the metastable, as well as the stable regions of this cooperative system. The dynamics of the system is studied by means of the most probable path method and the flow lines and fixed points of the system are given for zero field. The choice of possible initial conditions is discussed. The role of the unstable points in the phase diagram, as separators between the stable and metastable points, is described, and they are computed for a number of cases.

Theory of Relaxation Phenomena near the Second‐Order Phase‐Transition Point

One may expect that the observables connected with nonequilibrium processes will show abrupt changes if a substance undergoes a second‐order phase transition, since the equilibrium thermodynamic quantities appear to do so. In order to study these phenomena in a connected way the assumption is made that the long‐range order parameter and the short‐range order parameter can be treated as fluxes and forces in the sense of Onsagers theory of irreversible thermodynamics. Actual calculations are performed for two cases: an order—disorder system with short‐ and long‐range order and a system with two modes of long‐range order (antiferromagnet). The absorption of sound is calculated and its behavior near the critical temperature is analyzed. The function is continuous with a discontinuity in the slope provided the phenomenological constants are smooth functions of the temperature.

Stable, metastable and unstable solutions of a spin-1 Ising system obtained by the molecular-field approximation and the path probability method ☆

The spin-1 Ising model Hamiltonian with arbitrary bilinear (J) and biquadratic (K) pair interactions has been studied for zero field using the lowest approximation of the cluster variation method. The spin-1 or three-state system will undergo a first- or second-order phase transition depending on the ratio of the coupling parameters α = J/K. The critical temperatures and, in the case of the first-order phase transition, the limit of stability temperatures are obtained for different values of α calculating by the Hessian determinant. The first-order phase transition temperatures are found by using the free-energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves.

Mathematical double points according to the simplified–perturbed‐hard‐chain theory

Critical curves of a binary fluid mixture are usually plotted as a function of pressure and temperature (p,T projection), temperature and composition (T,x projection) or as a function of the reduced densities of the two components (y1,y2 projection). In these three ways of representation, we will show the structure of the critical curves around a mathematical double point. Moreover, it will be shown that mathematical double points can conveniently be divided into two groups: the (meta)stable and unstable mathematical double points. To date, the (meta)stable mathematical double points have not been investigated with a pressure‐explicit equation of state. In this paper, it will be shown how the two types of mathematical double points can be calculated. We will digress on the thermodynamic conditions to be obeyed for the calculation of the two types of mathematical double points.

The three‐state lattice gas as model for binary gas–liquid systems

This paper deals with the three‐state lattice model as applied to binary liquid–gas systems. Schouten, ten Seldam, and Trappeniers [Physica 73, 556 (1974)] used this model to describe the transition between the liquid phases as well as the gas–gas separation when varying the interaction parameters of the model. Our analysis is aimed at the behavior of the system in the vicinity of its tricritical point in the generalized field space. Different shapes of the critical lines are calculated by using the molecular field method as well as by using the lowest approximation of the cluster variation method. The equivalence between the two approaches is demonstrated. In the immediate neighborhood of the tricritical point the shapes of the critical lines are also determined analytically.

The dynamic behaviour of the Pople and Karasz model

Abstract Using the Pople and Karasz model for the solidification of plastic crystals, we construct two different sets of dynamic equations for the translational and rotational order parameters. The first is a straight generalization of the Pople and Karasz model, whereby one coupling parameter is a function of the other, and vice versa. The second generalization is based on the most probable path method of Kikuchi. In order to accomplish this we start with an appropriate transformation of the parameters. It is then shown that it is necessary to incorporate the spacial-angular correlation in order to apply this method. Computations for both systems of equations are given to demonstrate the behaviour of the long range order parameter if it is initially far from equilibrium.

The van der Waals equation of state around the van Laar point

The discovery of van Laar that the van der Waals equation of state for binary mixtures can be solved analytically at the double point on the critical line, provided one introduces the geometric‐mean condition on the interaction parameters, is used to obtain the explicit expression for the critical line for this case. The critical line is expressed as a function of the density variables, and the origin is shifted to this double point: the van Laar point. By doing so it is possible to show that this double point is also a tricritical point. This is different from the lattice gas, where the double point is always a tricritical point. Small deviations from this point in parameter space induce very different phase diagrams. The influence of excursions from the van Laar point is expressed as a function of the state variables. Both the k factor (the deviation from the geometric‐mean rule) and the ‘‘asymmetry’’ coefficient e (the deviation from the crossing point) are introduced. The results are given in the form...

The critical lines of the van der waals equation for binary mixtures around the van Laar point

Abstract In this paper it is pointed out that the differences in critical behavior of the various classes as observed in p-T diagrams are better understood in terms of density-concentration (or equivalent) coordinates. The position of the line of instabilities, which determines the transition from unstable to metastable or stable points on the critical line, is helpful in determining the class of a phase diagram, since its position is shown to be relatively insensitive to changes in the energy parameters. The critical lines and instability lines in p-T coordinates are either tangent or almost to each other, and moreover show several cusps, which makes it very difficult to determine the intersection points which are responsible for the change in character of the critical line. The procedure is illustrated by means of the algebraic equations for the critical lines near the van Laar point.

Influence of the chain length of long molecules on the equation of state in binary gas-liquid mixtures

The simple Flory-Huggins model can be combined with the lattice gas as was done by Tompa and others. Here the same method is used to obtain the van der Waals equation of state for a binary gas-liquid system containing a solute consisting of segmented molecules. The simplified equation of state developed here is useful for the study of the critical properties of such systems, in particular, the dependence of the various parameters on the chain length of the molecules.

Structure of the critical lines for the lattice gas model

Despite its simplicity, the three-component lattice gas model (in which one component is “holes”) for compressible binary mixtures is known to have many important features of the global phase behavior in common with more realistic equations of state. In this paper the underlying structure of the critical lines in a composition-composition projection is investigated by demarking, but maintaining, all unstable segments in the diagrams. Changes in the class of phase behavior with variations of the interaction parameters are then seen to be frequently accompanied by premonitory changes in the critical lines occuring in the unstable regions. A principal transition mechanism is a “crossing point” at which two critical lines meet and exchange branches. An equation for the critical line is given in closed form and used to investigate major regions in the global phase diagram including symmetric sections and the Griffiths shield region.