Ising and regular solution models revisited for 2D open systems

TL;DR

This paper extends the 2D Ising model to open systems with external fluxes, revealing new steady-state morphologies and stability conditions influenced by flux rates, initial conditions, and hysteresis effects.

Contribution

It introduces a generalized Ising model for open systems, analyzing stability and pattern formation under external flux divergence within a mean-field framework.

Findings

Supercritical flux stabilizes the system against decomposition.

Rate-dependent spinodal and binodal domes are identified.

Distinct steady-state morphologies depend on initial conditions and flux rates.

Abstract

The Ising model is well-known for illustrating the fundamental characteristics of phase transitions in closed systems. In this article, we propose a generalization of the two-dimensional Ising model to open systems, considering the divergence of external fluxes within a mean-field approximation, and explore its application for classifying steady states based on external flux (deposition rate), temperature, and composition. We focus on cases with positive mixing energy, which lead to the typical spinodal and binodal domes at the phase diagram under the regular solution approximation for closed systems. For open systems, we demonstrate that a supercritical external flux divergence stabilizes the system, preventing decomposition. We identify rate-dependent spinodal and binodal domes, as well as a subdivision of the instability region at subcritical rates into three distinct steady-state morphologies: spots (gepard-like), layers (zebra-like), and mixed patterns (a combination of gepard and zebra). This morphology map depends on the initial conditions, demonstrating memory effects and hysteresis.