Use of stereological measurements for the study of grain boundary diffusion controlled precipitate growth kinetics
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dV~ = B 9 J~ 9 (2zrR~ 9 x) 9 dt
[1]
where Ji is the grain boundary diffusion flux at c~a/3 triple line, x is the grain boundary thickness, and B is the stoichiometric factor. The parameter B converts the solute mass flux accumulated by the precipitate (i.e., J~ (2zrR,x) - tit) in the time interval t to (t + dt) into the change in volume of the precipitate dV~. Thus, B depends on the composition of the precipitate, density of the precipitate, etc. However, B is independent of time t. The change in the volume fraction dVv of the precipitate phase in the time interval t to (t + dt) can be obtained by adding the contribution dV~ from all the grain boundary precipitates in a unit volume of microstructure. Thus, Nv d W v = 2"17"Bx " Z J i
~ Ri " dt
[2]
i=1
where Nv is the number of grain boundary precipitates per unit volume. The grain boundary diffusion flux J~ is given by:
A.M. GOKHALE is Assistant Manager, R&D Centre, Hindustan Brown Boveri Ltd., P.O. Box 284, Baroda, India. Manuscript submitted June 12, 1984.
[3]
where the parameter A is a function of grain boundary diffusion coefficient, equilibrium concentration, etc. Combining Eqs. [2] and [3] yields:
dVv
Solid state transformations involving nucleation and growth of precipitates often lead to the formation of the precipitates along the matrix grain boundaries. Diffusion along the matrix grain boundaries can contribute significantly to the solute flux to the grain boundary precipitates during their growth; ~ under certain conditions the grain boundary diffusion can be the rate controlling mechanism for the grain boundary precipitate growth. Thus, the experimental verification of the grain boundary diffusion controlled mechanism of precipitate growth from experimentally measured global microstructural properties is of importance. It is the purpose of this paper to develop the procedure for the study of the kinetics of the grain boundary diffusion controlled precipitate growth from the experimentally measured global microstructural properties. The analysis is valid for the grain boundary diffusion controlled precipitate growth as well as dissolution. Let a be the matrix, and/3 be the precipitate phase. One aa/3 triple line (line of contact between two adjacent matrix grains and grain boundary precipitate) is associated with each grain boundary precipitate. The a a f l triple line encloses the grain boundary area occupied by a grain boundary precipitate. Assume that the aa/3 triple lines have a circular shape. Let V, be the volume of the i th grain boundary precipitate at the process time t, and let Ri be the radius of the circular ac~/3 triple line associated with it. Assume that the precipitate growth kinetics are controlled by the grain boundary diffusion. The change in the volume of ith precipitate dV~ in the time interval t to (t + dt) is given by the following mass balance condition:
456--VOLUME 16A, MARCH 1985
A Ri
J , -- -
dt
-
C
9 Nv
[4]
where C = 2 z r - A "B "x
[5]
Note that the parameter Cis not sensitive to t, and it depends on temp
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