Monte-Carlo simulations of spinodal ordering and decomposition in compositionally modulated alloys
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I. INTRODUCTION
Ordering kinetics in binary alloys have been the subject of many studies. The first theory was developed by Dienes1 and was based on a quasi-chemical approach. More recent studies2"4 employ the cluster variation method and assume a vacancy mechanism for atomic diffusion. The history of these theories has been reviewed by Cahn.5 Most investigations of diffusion in multilayered materials employ the theory of Cook et al.6 It is a discrete version of the Hillert and Cahn and Hilliard theory,7-8 originally developed to describe spinodal decomposition. The results of Ref. 6 have been used extensively to describe the kinetics of ordering in bulk alloys and of interdiffusion in compositionally modulated thin films with short-wavelength modulations. Theories of the migration of anti-phase boundaries9 assume local equilibrium and apply therefore only to later stages. In the present study, atomic diffusion at small composition-modulation amplitudes is simulated in a kinetic Ising model. Both ordering and phase-separating alloys are studied and compared to theory and experiment. In contrast to other studies,1011 an atomic interaction potential is assumed rather than a Ginzburg-Landau free energy functional with composition dependence only. This results in a time dependence of the free-energy functional, through its dependence on short-range order. The present study is the first simulation examining the application of the Cahn-Hilliard theory to ordering (negative heat of mixing, AH). While the theory and simulations are in qualitative agreement for spinodal decomposition, they disagree for ordering alloys below the order-disorder transition temperature. In some crucial cases theory and simulation differ in the sign of the amplification factor. The linear approximation leads to an averaging of the composition over the coordinates perpendicular to the modulation direction. This makes the theory 92
J. Mater. Res., Vol. 5, No. 1, Jan 1990
one-dimensional and therefore not applicable to higherdimensional systems. The temporal evolution of single-phase binary alloys with composition fluctuations have been studied experimentally and theoretically by many investigators. The theoretical treatment is based on a postulate by Hillert7 and Cahn and Hilliard,8 according to which the free energy functional is expressed as a sum of a composition-dependent bulk term/0(c) and a gradient energy term, proportional to the square of the composition gradient. This approach is based on a continuum assumption; i.e., the wavelength of composition variations is greater than the interatomic spacing. The total free energy is then given by: (1)
where K is the gradient energy coefficient. The diffusional flux is set proportional to the gradient of the chemical potential, and the resulting diffusion equation can be linearized and solved analytically for smallamplitude composition modulations. If the initial composition is expressed as a Fourier series, then each term evolves independently according to: A(t) • cos j8 • r = cfi exp -
j
(2
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