First-principles calculation of microstructural processes in alloys

Abstract

By combining Cluster Variation Method with FLAPW electronic structure total energy calculations and Phase Field Method, time evolution of Anti Phase Boundary associated with L1₀ ordering process in Fe–Pd was calculated from the first-principles. The theoretical framework of these calculations is reviewed, and it is pointed out that the introduction of the local lattice relaxation effects is indispensable to achieve higher accuracy. Preliminary calculations based on Continuous Displacement Cluster Variation Method are attempted on two-dimensional square lattice to examine the significance of the local lattice relaxation effects.

Introduction

Cluster Variation Method (hereafter CVM) [1] has been recognized as one of the most reliable theoretical tools to calculate configurational entropy and free energy of an alloy system. The level of the CVM approximation is specified by a largest cluster explicitly considered in the free energy formula, termed basic cluster, and it has been amply demonstrated [2], [3] that the calculated transition temperatures approaches a correct value obtained by other methods such as Monte Carlo simulation or high temperature expansion by increasing the size of the basic cluster. Although the employment of a bigger basic cluster demands heavy computational burden, recent development of high performance computer resolves such a difficulty.

The power of the CVM is not limited to the accuracy of the calculated results, but also the expandability and connectivity with other theoretical means to perform first-principles calculation of phase equilibria and phase transition dynamics are unique advantageous feature. In fact, the author have been attempting the first-principles calculations of phase diagrams [4], [5], [6], [7], [8], [9] by combining CVM with the electronic structure total energy calculations such as FLAPW method and reproduced the experimental phase diagram with high accuracy. The key to such an expandability of the CVM is ascribed to the correlation functions [2], [3], [10] which describe the atomic configurations of an alloy, and correlation functions are common variables shared by various theoretical methods in other realm of alloy theories including energetics and dynamics as will be discussed in this article.

Recently, Phase Field Method (hereafter PFM) [11] has been attracting broad attention as a powerful theoretical tool to predict and analyze microstructure evolution process of alloys. The key to the PFM is to define appropriate order parameters of which spatial distribution represents microstructure of interests, and the evolution process is described by Time Dependent Ginzburg Landau equation [12] and/or Cahn Hilliard equation [13]. The applicability of the PFM is surprisingly versatile, which can be ascribed to the phenomenological nature of the PFM in which the free energy is efficiently parametrized. It is, however, noted that the microstructure in the PFM is defined in a continuum medium, indicating that the order parameter is a continuous quantity of which atomistic origin is obscured. Hence, the length scale is not uniquely assigned based on the discrete nature of a lattice. This is regarded as a drawback of PFM to extend it to more quantitative and atomistic calculations.

The author’s group attempted [14], [15], [16], [17], [18], [19], [20] to combine PFM with the CVM by assigning correlation functions appearing in the CVM as order parameters in the PFM. Since the correlation functions are defined on a discrete lattice, it is necessary to perform the coarse graining operation in order to incorporate correlation functions in a coherent manner in the PFM which is defined in the continuum medium. Ohno [18] developed a unique procedure of the coarse graining operation by extending the traditional work by Kikuchi and Cahn [21], and performed multi-scale calculation for the growth process of Anti Phase Boundary (hereafter APB) associated with ordering reactions. Later, Mohri et al. further included the electronic structure total energy calculations and performed the first-principles microstructure evolution calculations for Fe–Pd [20] and Fe–Pt systems. The first half of the present paper is attributed to the introduction of the theoretical framework of these calculations.

However, there still remain room for improvement for these first-principles calculations in view of the recent development of the CVM. In particular, the lattice dealt with by the conventional CVM is allowed to deform only in a uniform manner and local lattice distortion is by no means introduced. This is deemed a serious drawback to achieve high accuracy in calculated results, since the system may still stay in the excited state without considering the local lattice relaxation. However, a fully satisfactory calculation of even static phase equilibria incorporating the local lattice relaxation effects is still pre-matured, and the implementation of such a scheme into PFM is far away. In the later half of the present report, preliminary calculations [22] for phase equilibria with local lattice relaxation effects are introduced and the significance of the effects are pointed out.

Furthermore, a brief discussion on the interface structure by the CVM is offered at the end, since the evolution kinetics of an APB is affected by the detailed atomistic structure and atomistic calculations of the interface should be more seriously explored. In fact, in the multi-scale calculation, interface falls in the medium scale range and it is a difficult task to reflect the atomistic structure of the interface efficiently into the microstructural formation and evolution processes.

The organization of the present report is as follows. In the next section, theoretical frameworks to perform the first-principles microstructure evolution calculation in the previous studies are reviewed and main results are reproduced. Two problems to be settled in the future calculations are pointed out in the third section and preliminary results are demonstrated. Finally the brief summary follows in the last section. Throughout this report, a particular focus is placed on the discussion of how the CVM free energy has been modified towards the first-principles microstructure calculations and how the CVM free energy should be further revised for accurate calculations.