BraWl: Simulating the thermodynamics and phase stability of multicomponent alloys using conventional and enhanced sampling techniques

TL;DR

BraWl combines a Bragg-Williams energy model with conventional and enhanced sampling techniques to enable thermodynamic analysis and phase diagram construction for binary and multicomponent alloys. By implementing Metropolis--Hastings MC, Wang-Landau sampling, and Nested Sampling, it efficiently explores configurational space defined by fixed lattices and EPIs, enabling extraction of $E$, $C_V$, ASRO, ALRO, and related observables across temperature and composition. The open-source package facilitates phase-equilibrium studies, visualization-ready configurations, and integration with downstream modelling, while demonstrating ordering and decomposition phenomena in systems such as Fe–Ni and high-entropy alloys. Limitations include neglect of vibrational, magnetic, and electronic entropies and the fixed-lattice assumption, which users should consider when interpreting transition temperatures and lattice distortions.

Abstract

We present BraWl, a Fortran package implementing a range of conventional and enhanced sampling algorithms for exploration of the phase space of the Bragg-Williams model, facilitating study of diffusional solid-solid transformations in binary and multicomponent alloys. These sampling algorithms include Metropolis-Hastings Monte Carlo, Wang-Landau sampling, and Nested Sampling. We demonstrate the capabilities of the package by applying it to some prototypical binary and multicomponent alloys, including high-entropy alloys.

Figures (4)

• Figure 1: Illustrations of potential states of a substitutional alloy in thermal equilibrium. A solid solution (a) is a state where lattice sites are occupied at random by elements of different chemical species. An ordered intermetallic compound (b) has an identifiable regular, repeating motif of atoms. A system may also undergo phase decomposition (c), where pairs of elements phase segregate from one another. In the multicomponent setting (d) there can be many possible competing phases.
• Figure 2: Evolution of the simulation internal energy (top panel) and conditional pair probabilities (bottom panel) for an Fe$_{0.5}$Ni$_{0.5}$ alloy as a function of the number of Metropolis-Hastings sweeps at a simulation temperature of $T=300$ K. One 'sweep' is one trial move per atom in the system. Beyond approximately 100 sweeps, the system can be seen to have reached equilibrium, with L1$_0$ order established.
• Figure 3: Plots of energy probability distributions, Warren-Cowley ASRO parameters ($\alpha^{pq}_n$) and simulation specific heat ($C_V$) as a function of temperature for AlTiCrMo obtained using lattice-based Monte Carlo simulations employing Wang-Landau sampling. Here, show $\alpha^{pq}_n$ only for $n$ = 1. The zero of the energy scale for the energy histograms is set to be equal to the average internal energy of the alloy obtained at a simulation temperature of 3000 K.
• Figure 4: Internal energy, $E$, and isochoric specific heat, $C_V$, obtained using the Nested Sampling algorithm applied to the equiatomic, fcc, AlCrFeCoNi high-entropy alloy. The simulation cell contained 108 atoms. Upon cooling, the initial peak in the specific heat is associated with an L1$_2$ ordering driven by Al, with subsequent peaks indicating eventual decomposition into multiple competing phases.