Uncertainty Analysis for Ferromagnetic-Paramagnetic Phase Transition Behavior of Magnetic Materials

Abstract

This study aims to model the phase transition of ferromagnetic materials using two- (2D) and three-dimensional (3D) Ising models, incorporating long-range magnetic spin-to-spin interactions and the influence of an external magnetic field. The 2D Ising model is investigated mainly for a 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} domain, while its extension to larger domains is also explored. For the 3D Ising model, the selection of representative volume element is discussed in terms of free energy. An uncertainty quantification formulation is integrated into the Ising model to capture the uncertainties related to the external magnetic field and evaluate their effects on the phase transition from a ferromagnetic to a paramagnetic state. Despite the common use and known effectiveness of the Ising model, there is still no universally accepted exact solution for the 3D domains, making it an ongoing research focus. The research also acknowledges the complexities and limitations arising from experimental methods, despite the insights they provide on the microscopic dynamics of ferromagnetic materials during phase transitions.