An Order-Wave Description of the Kinetics of Spinodal Ordering
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AN ORDER-WAVE DESCRIPTION OF THE KINETICS OF SPINODAL ORDERING
R. C. CAMMARATA, A. L. GREER, AND C. J. LOBB Div. of Applied Sciences, Harvard University,
Cambridge,
MA 02138,
U.S.A.
INTRODUCTION Several authors have modeled the order-disorder transformation as a Their continuous process, not occurring by nucleation and growth. approaches are based either on treating the transformation as a second-order chemical reaction [1,2], or on discrete formulations of the phenomenological diffusion equations [3-8]. Extending the work of Gorsky [9] and of Bragg and Williams [10], Dienes [2] used the chemical reaction model with absolute reaction rate theory to derive numerically the time-dependence of the longrange order parameter. As his thermodynamic model he used the BraggWilliams (B-W) approximation, assuming that the solid was uniformly ordered and ignoring inhomogeneity effects such as the excess interface and coherency strain energies that arise if antiphase domains exist. Hillert [3,4] modeled the kinetics of both the phase separation and ordering transformations, employing a modified diffusion equation for a He used discrete lattice with a one-dimensional concentration variation. the B-W approximation to obtain the free energy as a function of the local composition, but also included a gradient energy term to account for the Hillert numeriexcess free energy due to inhomogeneity in concentration. cally solved the difference equations for diffusion, and calculated the time-dependence of an arbitrary concentration variation (which could be described by a Fourier series of concentration waves). Cahn [11] developed Hillert's work into a three-dimensional continuum model of spinodal decomposition. In spinodal decomposition the concentration waves of interest have By conwavelengths sufficiently long for a continuum model to be valid. trast, in the order-disorder transformation, the wavelengths of the relevant concentration waves are on the order of the interatomic spacing [12] and are too small for a continuum model. Cook, DeFontaine, and Hilliard [7,8] generalized Hillert's discrete lattice model for three-dimensional concentration variations, and obtained an analytical solution for the early stages of the transformation. The presence of, and effects due to, antiphase domains are, however, not easily discernible in a concentration-wave formulation. In this paper we present a continuum model for the order-disorder Instead, we transformation, which is not based on concentration waves. extend the chemical reaction approach by including the effects of inhomogeneities in the B-W long-range order parameter. The temporal and spatial development of the order parameter in a solid is obtained analytically for the early stages of ordering. The solution is expressed in terms of order waves that describe the origin and evolution of antiphase domains. The formulation and results of our model for spinodal ordering facilitate detailed comparison with the sister transformation, spinodal decomposition. Due to space limitations, we consi
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