Date Published: July 12, 2018
Publisher: Public Library of Science
Author(s): Jude Simons Bayor, Baohua Teng, Lingli Wang, Nikolaos Fytas.
In this paper we calculate the global phase diagrams with the closed-loop behavior for the phase transition of physical systems by means of the transverse field Ising model with nearest neighbor interaction. The 3D graph plotted by the various physical parameters gives a clear appreciation and qualitative understanding of the reentrant phase behavior of the system. Meanwhile the results show the close correlation between experimental phenomena and our theoretical calculation for the closed-loop behavior for the phase transition of the systems.
Phase transition phenomena occur frequently in the natural world  and they are of special interest in many fields of study . Hence the shapes of the various phase diagrams have been obtained theoretically by various models and they have significantly enhanced the understanding of the underlying physical property changes [3–29]. Campi et al , reports that the lattice gas is viewed as having similar perspective to that of simple fluids, which exhibits a reversed U-shaped phase diagram with the critical point at the top. Also, some complex fluids (examples being certain micro emulsions) with short range interactions exhibit U-shaped phase diagram with the critical point at the bottom. This is observed in the mixture of water and oil in the presence of a surfactant . In fact some more exotic phase diagrams for complex fluids have been reported . For example a solution of nicotine in water, has a closed-loop shaped phase diagram with double critical points . These reentrant phenomena are also reported in various order-disorder systems, such as the composite structure system of potassium , the mixed-spin Ising ferrimagnet system , the transverse Ising nanosystem , and the solvent-mediated protein system . Recently Bayor et al  studied the reentrant phase behavior and obtained closed looped shapes that are coincident with the phenomena exhibited in some proteins, colloids and complex fluid mixtures.
This paper provides a theoretical model for accurately obtaining a topological phase transition sequel with closed looped phase behaviors, commonly associated with some binary fluids in particular, and for other exotic shapes of phase transition phenomena. By using the transverse Ising model and the supposed exponent dependent relations of the effective interaction parameters on temperature, the 3D global phase diagrams were obtained by the mean-field approximation. Cross sections of these global phase diagrams reveal the snap line phase diagrams for systems at the given concentration. Hence four distinct classifications of systems were encountered. These include the phase diagrams of the egg-shaped closed loops, U-shapes, reversed U-shapes and other exotic topologies commonly observed in experiments.