On homogenization of a binary alloy after dissolution of planar and spherical precipitates
- PDF / 157,458 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 88 Downloads / 172 Views
I. INTRODUCTION
IN the past, problems of particle dissolution in the matrix were analyzed theoretically by various analytical and numerical methods.[1–12] The central assumption of these studies was that the rate of dissolution is diffusion limited, and, therefore, equilibrium is always established at the interface between the precipitate and matrix. Generally, in these analyses, the change in matrix concentration due to the homogenization treatment following dissolution of the precipitate was not considered. In this work, the effect of homogenization treatment on concentration uniformity following precipitate dissolution is analyzed. We limit our analysis to spherical and plate-like precipitates only.
interface. From a flux balance at the precipitate/matrix interface, (CP 2 CI) (dR/dt) 5 D( C/r)r5R
[3]
where CP , the composition of the precipitate, is considered to be a constant independent of r and t. The term R0 denotes the value of R at t 5 0. The analytical solution of Eq. [1] with conditions [2] and [3] for the dissolution of a plate-like precipitate is known.[5] Using this solution, the time of precipitate dissolution (tdis) and the concentration field (C(x,tdis)) can be obtained as tdis 5 S20 /4l2D C(x,tdis) 2 CM 5 (CI 2 CM)
erfc((x/S0 2 1)l) erfc(2l)
[4] [5]
II. THE MODEL
p1/2lel erfc (2l) 5 g
The dissolution of an isolated precipitate in an infinite matrix in one or three dimensions is considered. The influence of the interface reaction and the curvature of the precipitate particle on the dissolution are neglected. Assuming the diffusion-limited character of the dissolution of the precipitate, the following equation can be written:[7]
Here, 2S0 is the thickness of the plate-like precipitate at t 5 0 and g [ (CI 2 CM)/(CP 2 CI). The term g is a dimensionless parameter that describes the character of the dissolution of the precipitate. It is the only dimensionless parameter being used, and it is valid for the plate and the spherical precipitates.[8,10,12] No exact analytical solution of Eqs. [1] through [3] for a spherical geometry is available for arbitrary values of g. Therefore, numerical calculation is performed for the kinetics of spherical precipitate dissolution. The algorithm of the numerical solution is based on an analytical solution describing the dissolution of a spherical precipitate at a constant diminishing rate of the radius of the precipitate (dR/dt 5 constant , 0).[13] For the calculations, the time of complete dissolution of the particle is divided into equal intervals. The value of dR/dt in each of these intervals is assumed to be constant. The term dR/dt is determined by the concentration gradient in an earlier time interval through Eq. [3]. In the initial interval, the dissolution of the precipitate is calculated on the basis of the exact analytical solution for a plate.[5,10] Data related to the algorithm are described in Appendices A and B. If the concentration field after precipitate dissolution is known, then the redistribution of the solute may be calculated on the bas
Data Loading...
