Exact Calculation of Antiferromagnetic Ising Model on an Inhomogeneous Surface Recursive Lattice to Investigate Thermodynamics and Glass Transition on Surface/Thin Film

An inhomogeneous 2-dimensional recursive lattice formed by planar elements has been designed to investigate the thermodynamics of Ising spin system on the surface/thin film. The lattice is constructed as a hybrid of partial Husimi square lattice representing the bulk and 1D single bonds representing the surface. Exact calculations can be achieved with the recursive property of the lattice. The model has an anti-ferromagnetic interaction to give rise to an ordered phase identified as crystal, and a solution with higher energy to represent the amorphous/metastable phase.Free energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function. By analyzing the free energies and entropies of the crystal and supercooled liquid state,we are able to identify the melting and ideal glass transition on the surface. The results show that due to the variation of coordination number, the transition temperatures on the surface decrease significantly compared to the bulk system.Our calculation qualitatively agrees with both experimental and simulation works on the thermodynamics of surfaces and thin films conducted by others. Interactions between particles farther than the nearest neighbor distance are taken into consideration, and their effects are investigated.

Thermodynamic Comparison and the Ideal Glass Transition of Antiferromagnetic Ising Model on Multi-branched Husimi and Cubic Recursive Lattice with the Identical Coordination Number

Two types of recursive lattices with the identical coordination number but different unit cells(2-D square and 3-D cube) are constructed and the antiferromagnetic Ising model is solved exactly on them to study the stable and metastable states. A multi-branched structure of the 2-D plaquette model, which we introduced in this work, makes it possible to be an analog to the cubic lattice. Two solutions of each model can be found to exhibit the crystallization of liquid, and the ideal glass transition of supercooled liquid respectively. Based on the solutions, the thermodynamics on both lattices, e.g. the free energy, energy density, and entropy of the supercooled liquid, crystal, and liquid state of the model are calculated and compared with each other. Interactions between particles farther away than the nearest neighbor distance and multi-spins interactions are taken into consideration, and their effects on the thermal behavior are examined. The two lattices show comparable properties on the thermodynamics, which proves that both of them are practical to describe the regular 3-D case, especially to locate the ideal glass transition, while the 2-D multi-branched plaquette model is less accurate with the advantage of simpler formulation and less computation time consumption.