Interface Versus Diffusion Controlled Growth in Nanocrystallization Kinetic Studies
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transform the data to the forms x=x(T,t) or x=x(T,P3) according to the particular heat treatment applied to the sample. THEORETICAL TREATMENT Nanocrystallization of an amorphous alloy is assumed to proceed up to a fraction x:f of the total volume V when t-ýoo (x---I), with (1-./)V remaining amorphous. When the decrease of the volume available for homogenous nucleation to proceed is considered, the crystallized fraction at time t under isothermal annealing at temperature T is given by: x(T,t) = I - exp[- iI(TT)v(T,t,I){l-x(T, t)}d]
(I)
0
were I(Tj) is the nucleation frequency at temperature T, v(T,t,r) is the volume at time t of nuclei formed at time T and the factor (I-x) is included to account for the compositional change of the remaining matrix.. Similarly, the density of nuclei, N, and their mean grain size, S, are given, respectively, by N(T,t)
f I(T,r){1-x(TT)dr}
(2)
0
Tt N(T,t) I' In the following we assume that the nucleation frequency I is independent of time and takes the usual expression
I = IoexpL- T,6(AG, )32
(4)
with oc and P3 the thermodynamic factors, introduced by Turnbull [9], Tr being the reduced temperature, ACp AG, = (I-Tr)(l-y) - yTrlnT, and y(5) AS,, With regards to the growth mechanism we consider that: i) the initial steps of growth are controlled by the interface between the nanocrystals and the surrounding matrix. The radius of a spherical nuclei is r(T,t,j) = u(T)-(t-T) + r*(T) where r*
2av"3 /AGr
(6)
with u the growth rate
u
-
u0 I - exp(_ AG,,)]~
(7)
and r* the size of the critical nucleus. ii) subsequent growth is limited by diffusion. The rate of growth is dr c*- c I 2R 2 dt - D c*- C.1 -r (R - r)(2R - r)
(8)
Here 2R is the mean distance in between the nanocrystalline grains. When R--oo c*, ct and c are, respectively, the interface, crystal and initial amorphous concentration of the slowest diffusing element and the last factor in the right hand size of equation (8) tends to one. However, when the ratio r/R increases c* and c become a function of r/R. In the present modeling we introduce an apparent diffusion coefficient Dapp, given by
320
Dapp
=Doexp(-ED / RT)qp(x)
(9)
That means that the growth rate is simply
dr .D
app
dt
r
(10)
RESULTS Fig. 1 shows the evolution of the crystallized fraction with time under isothermal annealing at 763 K for an amorphous alloy of overall composition Fe 73.5Si 1 7.5B 5Nb 3CuI. The different curves indicate a comparative study of the experimental results obtained from DSC and Mossbauer spectroscopy [10] and the calculated values obtained from the modelling by use of several assumptions. These are: - No diffusion controlled regime is achieved - Diffusion becomes relevant with increasing nuclei radius with r1=1 or p= I-x. The accuracy of the DSC data is not explicitly indicated in this figure, because of the difficulty to define a base line for isothermal events [II]. Nevertheless, the figure shows that all the calculated curves agree with experimental data at least on the crystallization onset, that is, the assumption of
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