Kinetics of Ordering with Phase Separation
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KINETICS OF ORDERING WITH PHASE SEPARATION T. EGUCHI, K. OKI AND S. MATSUMURA Department of Materials Science and Technology, Sciences, Kyushu University 39, Kasuga-shi 816,
Graduate School of Engineering Japan
ABSTRACT An attempt is made to investigate theoretically the kinetics of ordering accompanying phase separation in an alloy as a continuous medium. A set of coupled equations for the local composition and the degree of order are derived from the first principles of thermodynamics of irreversible processes, under an appropriate assumption as to the free energy of the system. Under certain circumstances a periodic deviation of local order parameter in the disordered state induces that of the composition, leading to a spinodal decomposition, and the resonance conditions for enhancement of such an induction are examined in the perturbation approximation. In a uniform state of imperfect degree of order disturbances of the local composition and degree of order develop in-phase.
INTRODUCTION In the phase diagrams of some alloys there exist phase fielos where the ordering or disordering develops accompanying phase separation. As an example, in the phase diagram of Fe-Al system around stoichiometric composition of Fe 3 AI there is a triangular field of mixed phase, in which the ordered B2 and disordered A2 phases
coexist
in
thermal equilibrium
[1-5].
Previously,
we examined the processes of phase separation of the off-stoichiometric Fe3Al alloys in the mixed phase region of A2+B2 by means of electron microscopy [6,7). The observation by dark field images revealed that the phase separation near the phase boundary between the disorder or order and the mixed phase regions preceeds by nucleation and growth, whereas in the central region of the mixed phase the separation preceeds by spinodal decomposition. We are particularly interested in the process of spinoaal decomposition accompanying the local variation of the degree of order. For analyses of such processes a generalization of Cahn's theory of spinodal decomposition [8] is necessary to include a local variation of degree of order. In the following we shall use the method of thermodynamics of irreversible processes to consider the kinetics of the simultaneous change of local concentration and degree of order, and describe the theoretical implications obtained so far.
GENERAL FORMALISM We consider a system of a substitutional binary alloy consisting of A and B atoms with a composition A(l-x)/2B(l+x)/2. In the continuum theory we introduce two variables depending upon the position • and time t; the one is the local concentration X(T,t), which is conserved as
J X(•,t)
d3r = x
(1)
unit vol.
and the other the local degree of order S(r,t), which is not conserved. The thermodynamical potential of the system is a functional of the variables Mat. Res.
Soc. Symp. Proc.
Vol. 21 (1984) Q Elsevier science Publishing Co.,
Inc.
590
X(P,t) and S(P,t),
given by
F [{X},{S}]
K2(4 =
{f(X,S) + H(h)
+ K(VS) 2
where f(X,S) is the density of the bulk free energy,
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