On the kinetic expression for the growth of precipitate plates
- PDF / 160,265 Bytes
- 2 Pages / 613 x 788.28 pts Page_size
- 56 Downloads / 240 Views
Communications On the Kinetic Expression for the Growth of Precipitate Plates
2
W. P. BOSZE AND R. TRIVEDI The Wldmanstiitten or plateltke morphology is observed in a great number of alloy systems of importance In metallurgy, and the Influence of this morphology on the mechanical properties of alloys is quite significant. Thus, many kinetic studies have been made to understand the physical phenomena controlling the growth of such a precipitate morphology. Of the various kinetic models available so far, *-~ the model proposed by Trtvedt 5 is the most recent one which ass e s s e s the relative contributions of diffusion, surface energy and Interface kinetic p r o c e s s e s during the growth of parabolic shaped precipitates. Unfortunately, the kinetic expression obtained in this model is mathematically quite complex which makes it difficult to visualize the relationship between various growth par a m e t e r s and the supersaturation. The main purpose of this communication is to show that under most experimental conditions the Trivedt expression can be greatly simplified to give a clear insight into the interdependence of the growth rate, radius of curvature and supersaturation. Such an expression simplifies the application of the theory to experimental data. The solution obtained by Trivedi is of the form
I
V~o
Pc ~20S2(p)]
a0 -- i 1 + No(Co- c ~ ) s J p ) + 7
[1]
where I = f ~ e P erfc (~p-) is the Ivantsov solution which assumes an Isoconcentrate interface, ao is the dimensionless supersaturation defined as (C o - C ~)/ (C o- Cp), p is the peclet number equal to Vp/2D in which V is the growth rate, p the radius of curvature at the tip of the plate and D the diffusion coefficient of solute in the matrix. Co and Cp are the equilibrium concentrations in the matrix and precipitate, respectively, at a flat interface, C.o the initial solute concentration In the matrix, Pc the critical radius for nucleation and No the interface kinetic coefficient. The functions Sl(p) and S2(p) are mathematically complicated functions. The above expression can be simplified by taking note of the following two observations: 1) In most solid-solid phase transformations the migration of atoms a c r o s s the interface is believed to be quite rapid, i.e., No is large. Also, when coupled with a small growth rate (V < 10 -3 c m / s ) , the contribution of interface kinetics, the second t e r m on the right hand side of Eq. [1], can be safely neglected. Earlier theories of Zener 1 and Hillert ~ have therefore omitted
W. P. BOSZE, formerly with the Ames Laboratory, USAEC, Ames, Iowa, is now process engineer with the Bourns Inc., Ames, Iowa. R. TRIVEDI is Associate Professor, Department of Metallurgy and Metallurgist, Ames Laboratory, USAEC, Iowa State University, Ames,
Iowa. 50010. Manuscript submitted August 6, 1973. METALLURGICAL TRANSACTIONS
[2]
which matches the function exactly for p < 0.1 and deviates from it by less than 10 pct for values of p as large as 1.0. Utilizing these observations, Eq. [1] can be simplified to give
Data Loading...
