Configurational kinetics studied by Path Probability Method

Abstract

Cluster Variation Method (CVM) has been widely employed to calculate alloy phase diagrams. The atomistic feature of the CVM is consistent with first-principles electronic structure calculations, and the combination of CVM with electronic structure calculation enables one to formulate a free energy from the first principles. In the conventional CVM, however, local lattice distortion cannot be introduced into a free energy since the local lattice distortion alters the lattice symmetry from one lattice point to another and entropy formula is not justified. Hence, the phase equilibrium determined by the CVM is still under an excited state. In order to circumvent such a deficiency of the conventional CVM, Kikuchi devised the Continuous Displacement Cluster Variation Method (CDCVM) which is capable of introducing local lattice distortion into the theoretical framework of the CVM by viewing an atom displaced to a quasi-lattice point as hypothetical atomic species located at the Bravais lattice point. Then, additional freedom due to the local atomic displacements is converted to configurational freedom of a multicomponent alloy on a rigid lattice.This idea of conversion of a freedom of local atomic displacements into configurational freedom is extended to other internal freedoms to study collective atomic displacements leading to displacive phase transition and magnetic transition. As an example of the former case, we investigated cubic-tetragonal transition of ZrO2, where two kinds of oxygen atoms, one undergoes upward shifting and the other downward shifting associated with the tetragonal transition, are distinguished as different atomic species. This way, the cubic-tetragonal transition is treated as the binary phase equilibria of the two kinds of oxygens on a cubic lattice surrounded by fcc lattice of Zr. In the latter case of magnetic transition, we calculated Curie temperature of collinear magnetic system by viewing various magnetic spin moments of up and down spins as different atomic species. Hence, Curie temperature is determined as the temperature at which long range order of a multicomponent system vanishes. For both the cases, first-principles calculations of the transition temperature are attempted by introducing electronic structure calculations into the free energy, and reasonable values are obtained.Path Probability Method (PPM) has been known as a natural extension of the CVM to a time domain. PPM is utilized to investigate transition, relaxation and steady state kinetics upon the temperature change. Hence, the CVM and PPM are coherent to perform synthetic study from initial non-equilibrium to final equilibrium states. One of the serious problems of PPM is the fact that the number of variables involved in a Path Probability Function, which is corresponding to a free energy of the CVM, becomes formidably large with higher order approximations. This is because PPM deals with transition path from one configuration to another, hence one needs to consider all atomistic paths which connect two configurations at time t and t+Δt. For kinetics studies, three mechanisms should be distinguished; spin-flipping kinetics, direct exchange kinetics and vacancy mediated kinetics. Among these, most studies of PPM have focused on spin flipping kinetics but vacancy mediated kinetics is most commonly observed for metals and alloys. The number of variables to formulate vacancy mediated kinetics is much larger than that of spin flipping and direct exchange kinetics.The authors developed a new formalism of time transition variables, known as path variables, and this facilitates to apply PPM for various vacancy mediated kinetics phenomena. The basic idea behind the new formalism originates from the spirit of CDCVM to convert internal freedom to configurational freedom. And the numerous transition paths from one configuration to another are dealt in the framework of configurational freedom. Among various calculations of relaxation processes, two examples are introduced. One is associated with a cubic-tetragonal displacive phase transition in ZrO2 for which equilibrium states are determined by CDCVM as described above. For this, we employed a high order approximation with spin flipping kinetics. The other is the relaxation processes associated with up- and down-quenching in a Ni3Al ordered phase within the tetrahedron approximation. The vacancy mechanism is fully considered in this case, and the effects of vacancy concentration on the relaxation behavior are investigated. In the present article, recent progress of CDCVM and PPM are reviewed.