KINETICS OF ORDER-DISORDER TRANSFORMATIONS

Relaxation of order (both long range and short range) in bcc binary alloys is worked out theoretically based on the vacancy mechanism of atomic migration using the pair approximation of the Path Probability method. Concepts of relaxation times and relaxation modes in states far from the equilibrium are examined. A peculiar isothermal relaxation process in three distinct steps is explained in terms of coexistence of many relaxation modes and their intercouplings. Transitions into and out of frozen-in state due to cooling and heating of the system with finite rates of temperature change are discussed in analogy to phenomena which accompany the glass transition. Introduction. The order-disorder transformation in alloys is one of the simplest and best understood phase transitions. In ordinary alloys, atomic migration takes place through vacancies and the phase transition proceeds in a gradual fashion, in many cases in the form of the second-order transition. Therefore, the order-disorder transformation in alloys can be regarded as a representative one among many diffusioncontrolled phase transitions, and a detailed understanding of its mechanism is quite useful. However, detailed relaxation processes, especially in the region far from equilibrium, are theoretically not well understood yet. The main aims of the present research are therefore : 1. To investigate the kinetics of order-disorder transformation based on a realistic model of binary alloys. 2. To utilize a method of non-equilibrium statistical mechanics by the use of approximation which is well understood. 3. To apply the results obtained to the elucidation of the nature of glass transitions. (*) Work supported in part by the National Science Foundation under Contract No. DMR 76-21497. The treatment is based on the vacancy mechanism of atomic migration. The paper helps understand the role of a small number of vacancies (which we introduce in the crystal) in the diffusion-controlled phase transitions. We utilize the Path Probability method [l] (PPM) to formulate the problem. The PPM is known to be valid for dealing with the problem far from the equilibrium state. Furthermore, the PPM formulation reduces to results of the known Cluster Variation (CV) theory [I, 21 in the limit as the time variation ceases. Therefore, the nature of the approximation in the PPM can.be understood by comparing it with the CV, which has been explored extensively. The glass transition is essentially a relaxation process in the non-equilibrium state and it is understood to be the transition to and from a frozen-in state. Although real glass systems are very complex, it is expected that a detailed statistical mechanical understanding of the frozen-in state would be quite useful in elucidating the nature of glass transitions. In the paper we summarize the key processes of the treatment [3] first and then show numerical examples of several types of relaxation phenomena [3, 41. Article published online by EDP Sciences and available at http://dx.doi. org/10.1051/jphyscol:1977769 C7-3% H. SATO. K. GSCHWEND A N D R. KlKUCHl The model. The system is composed of A atoms, B atoms and vacancies distributed over the bcc lattice. The bcc lattice is chosen because we use the pair approximation of the PPM, and the pair approximation is known to be quite suitable for describing, at least qualitatively, all physical features of phase transitions in the bcc lattice [5]. The three constituent species will be denoted by subscripts either A, B, and v, or with i o r j with i, j = 1,2 and 3. In the ordered state we distinguish two sublattices, I and 11. There are to be occupied preferentially by A and B atoms, respectively, in the ordered state. We assume interactions only between nearest neighbors and designate these as &AA, E, and cAB. Although no interaction is assumed between v-A and v-B, these are taken care of indirectly through differences among cil)s. Instead of using values of individual E'S defined above, we use E, U and E*B/E In usual discussions of equilibrium properties U = 0 is assumed. However, in kinetics the value of U plays an important role since this is a measure of which bond, A-A or B-B, is more easily broken [6]. The value of U can be estimated from the energy of formation of vacancies. The sign of E determines whether the system forms an ordered state or decomposes at low temperatures, and the magnitude of E determines the order-disorder transition temperature as (in the pair approximation) in the absence of vacancies. The inclusion of a small number of vacancies only slightly lowers T,. The temperature scale is thus normalized with respect to E. Under the above condition, the state of the alloy at time t is specified by nine variables in the pair approximation. These are called the state variables and are defined as follows. The probability that a species i is found on a I lattice point at t is written as xi(t) for i = 1, 2 and 3 and ui(t) for the corresponding quantities on a 11 lattice point. The probability of finding an i-j pair at t, i on a 1 point, and j on an adjacent I1 point is written as yi,{t). Among these variables, the following geometrical relations exist : variables to six. Besides x, u, and y, it is convenient to define the long-range order parameters ti(t) E xi(t) ui(t) for i = 1, 2, 3 , (4) which vanish in the disordered state. The equilibrium properties of the alloy are now determined by the pair approximation of the CV method in terms of these six independent variables [3]. The path probability method. The PPM is based on the contention that the macroscopic change of state in a system occurs in the most probable direction, that is in mathematical terms in a direction that maximizes the probability of the path in At (a short time interval) to be explained in more detail below. We define the path variables which describe how the system changes in At, and then write the path probability function in terms of the path variables. For the pair approximation in the bcc lattice, we use the path variables Yij,,,(t, t + At) defined in table I. These are Yij,,, used in eq. (7.1) of the accompanying paper in this issue [I]. The meanings of these variables are the following : For example, the third one Yij,3j is the probability of finding such a bond which is i-j at time t, and changes into 3-j (i.e., u j ) within At. Using the path variables in table 1, we can write the change of the state variables in At as Ati = 16(Y3i.i3 Yi3,3i) i = 1 and 2 (5) and Ayij = 7(Yi3,ij + Y3j,ij Yij,i3 Yij,Jj) i, j = 1 and 2 . (6)