The influence of temperature gradients on ostwald ripening
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I. INTRODUCTION
AT the end of a number of phase transformation processes, a two-phase system may consist of a dispersed phase and a large amount of interfacial area. Through Ostwald ripening, or coarsening, the system may reduce its interfacial area and, thus, its energy. The origin of this process is the Gibbs–Thomson effect, which alters the concentration at the particle-matrix interface depending on the curvature of the interface. Particles that are smaller than a critical size have an interfacial concentration that is higher than that in the matrix, while particles that are larger have a lower interfacial concentration. The resulting concentration gradients give rise to diffusional transport of material from small particles into the matrix and from the matrix into large particles. Thus, the large particles tend to grow at the expense of the small particles, the average particle size increases, and the total number of particles decreases. A comprehensive theory of particle coarsening in isothermal systems was first developed by Lifshitz and Slyozov[1] and Wagner[2] (LSW). It was found by LSW that, in the regime of self-similar coarsening ¯ (steady state), the cube of the average particle radius (R(t)) increases linearly with time (t): ¯ ¯ R(t)3 2 R(0)3 5 KLSWt [1] ¯ where R(0) is the average radius at t 5 0, and KLSW is the coarsening-rate constant given by Reference 3, KLSW 5
8 T0GD eq 9 mL(Ceq p 2 Cm )
[2]
where T0 is the coarsening temperature, G is the capillary length of the matrix phase, D is the diffusion coefficient of the solute in the liquid, mL is the slope of the liquidus curve eq at T0, and Ceq p and Cm are the compositions of the particle and the matrix at a planar solid-liquid interface at a temperature of T0, respectively. Using the Gibbs–Kanovalov equation[4] in Eq. [2], KLSW can be written as KLSW 5
VmsD 8 eq 2 9 (Cp 2 Ceq m m ) G9
[3]
V.A. SNYDER, Graduate Student, J. ALKEMPER, Research Assistant Professor, and P.W. VOORHEES, Professor, are with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108. N. AKAIWA, Researcher, is with the Computational Materials Division, National Research Institute for Metals, Tsukuba, 305-0047, Japan. Manuscript submitted February 22, 1999.
METALLURGICAL AND MATERIALS TRANSACTIONS A
where Vm is the molar volume of the solid phase, s is the interfacial energy, and G9m is the second derivative of the free energy of the matrix phase with respect to the concentration evaluated at Ceq m . It was also found by LSW that the particle-size distribution becomes time independent when scaled by the time-dependent average radius. However, the theory was developed by solving the diffusion equation for a particle in an infinite matrix. Diffusional interactions which occur between particles in real systems were neglected; thus, the LSW theory is only valid in the limit of zero volume fraction of coarsening phase. A number of theories have since been developed to remove the zero-volume-fraction constraint[5–9]; note also th
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