Multiphase precipitation of carbides in Fe-C system: Part II. Model based on kinetics of complex reactions
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The precipitation of carbides in the Fe-C system aged at constant temperature is simulated considering two transformation kernels of complex reactions, involving different impingement and rate-time factors. The metastable and stable phase are identified by deconvolution of the experimental kinetics. The activation energy of carbides is found to depend, slightly, on the transformation kernel in which it is used.
I.
INTRODUCTION
(5) Yamamoto-Kubota type kinetics tu,~4~
C A R B I D E precipitation processes in supersaturated body-centered cubic (bcc) ferrite can occur in one or more stages, depending on the aging temperature.~ ~-5~In effect, the precipitated phases have been identified as low T carbide at low temperature, the e-carbide and cementite at intermediary temperature, and cementite at high temperature, which defines the Fe-C alloy as one multiphase system. In Part I of this article ~l (referred to herein as Part I), we study multiphase precipitation, proposing a system of differentia/ equations that is based upon the rate of atomic transfer and whose analytical solution generates the transformed fraction of solute as a sum of the simple precipitation kernels. We define precipitation kernels as the mathematical functions that are solutions of the differentia/ equation for the isothermal evolution of the precipitated fraction Y in the monophase system. This differential equation is written as
dY dt
= f ( k , T, Y, t) = K(T)g(Y)
[1]
Where the constant K follows an Arrhenius law with temperature T. The f functions more often referred to in the literature are as follows: (1) Chemical kinetic of first (P = 1) and second (P = 2) order ~61
f ( K , T, Y, 0 = K(l - y)e
I2]
(2) Empirical models of Johnson-Mehl-Avrami 17,sl (P = 0) and Austin-Rickett (P = !) t9~
f ( K , T, Y, O = Kn(1 - Y)P+'(Kt)"-~
[3]
(3) Precipitation kinetics required by diffusion in the matrix (P = 1) and by diffusion in the interphase (P = 2)lml
f ( K , T, Y, t) = 3KY p/3 (1 - Y)
[4]
.'4) Kinetics of the type of Fujita and Damask et a l Y J.~2~ 2K f ( K , T, Y, t) = - ~ (1 - Y) ~v/l - (1 - y)C
[51
~here C is the critical size of clusters with C atoms. NEY J. LUIGGI and ANGEL E. BETANCOURT are with the )epartment of Physics, University of Oriente, Cuman& Venezuela. Manuscript submitted March 24, 1993. ~ETALLURGICAL AND MATERIALS TRANSACTIONS 13
f ( K , T, Y, t) = K~ (l - Y) [1 - (1 - K:t) exp (-K2t)]
{6] where Kj and K2 are rate constants of precipitation and of the formation of precipitation sites, respectively. Theoretical works upon multiphase systems in the literature are limited, I1'~4'15J and the researchers have preferred to split off a temporary region in which each phase can be identified and thereafter to study each phase separately, considering the multiphase system as one monophase. Evidently, these hypotheses will not be valid if there is simultaneous precipitation in each phase. The precipitation process in a-iron, studied in Part I, using simple decomposition kernels, can be deduced using f functions defined by E
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