Quantifying the effect of non-equilibrium vacancies on Bragg–Williams ordering

ABSTRACT

Bragg–Williams (BW) modelling is a mean-field approach to order–disorder phase transformations (ODPT´s) in substitutional alloys. While the BW theory itself is for thermal equilibrium, the relaxation of the alloy to the equilibrium state in terms of the BW approach was studied by Dienes who introduced the chemical balance equation for temporal evolution of the long-range order parameter S. Here, results of solving numerically the Dienes equation are presented, with taking additionally into account that ordering in the alloy occurs through vacancies in atomic lattice. In such a description there are three important parameters that affect the ordering kinetics, namely (1) the interdiffusion coefficient in a disordered alloy, (2) the ratio of initial to equilibrium (thermal) concentration of vacancies, r, and (3) the characteristic timescale τ∝L² for vacancy relaxation, where L is the effective distance between sinks/sources of vacancies in the alloy. With example of Fe-rich Fe aluminides Fe_xAl_1-x (x = 0.6), it is found that, at sufficiently large r, an additional step arises in temporal evolution of S for a time which can be much shorter (scaled as ∝r⁻¹) than the characteristic timescale for ordering at r = 1. The height of this step increases up to unity at sufficient r. The lowest values of r and L are determined, at which non-equilibrium vacancies injected into the alloy can still play the role. This study would be of potential interest for developing the technology of functional alloys (lowering of ordering temperatures) and for obtaining a kind of information about vacancy behaviour in crystals.

KEYWORDS:

• Long-range order parameter
• order–disorder phase transformations
• ordering energy
• interdiffusion in crystalline lattice
• nonequilibrium vacancy
• substitutional alloy

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No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by Russian Science Foundation [grant number 23-22-00044].