Transient and Athermal Nucleation of Solids in Rapidly Quenched Liquid Si
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Mat. Res. Soc. Symp. Proc. Vol. 398 01996 Materials Research Society
athermal nucleation and to the possibility that there may be a significant contribution to the total nucleation rates from the mechanism. CLASSICAL NUCLEATION THEORY According to the classical nucleation theory, clusters of a stable new phase arise via heterophase fluctuations. The critical cluster size (i.e., the cluster size above which the clusters tend to grow and below which the clusters tend to shrink) is determined by the interfacial energy and the volumetric free energy difference between the phases. The net rate of formation of the supercritical clusters per unit volume of the metastable phase is defined as the nucleation rate. Hence, one can rigorously define the total nucleation rate, ltotal, in a general fashion as Itotalttl= dt*tnf Nn ,tdn t
(1)
where Nn t is the population of clusters of size n at time t, and n* is the critical size. In our derivatioin, we utilize the conventional continuum approximation and the quasi-steady state treatment that are typically employed [6, 7]; a more thorough treatment of our analysis can be found elsewhere [8]. Now, by utilizing the Zeldovich-Frenkel equation
dt
. dNt 9 +Ne dLNN,_tJ dn n n dn Ne
(2)
where, Ne is the equilibrium population for a cluster of size n and k+ the temperaturedependengrate constant for the addition of a single atom to a cluster of size n, we obtain the equation for the total nucleation rate as
,total
= k+ dNn t n* nt
( +
- N
t
dn*( t
(3)
The above expression identifies that, in general, there are two components that determine the nucleation rate. The first term, which corresponds to the thermal nucleation mechanism, describes the rate of formation of supercritical clusters via growth of critical clusters. This term, Ithermal, represents the familiar expression that corresponds to the current in the size coordinate that is evaluated exactly at the critical value. The second term, in contrast, describes the contribution to the total rate of nucleation arising not from the physical enlargement of critical clusters but from the decrease (increase) in the critical size and the subsequent promotion (demotion) of the critical clusters to the supercritical (subcritical) status. It can be seen that, as long as the population of critical clusters is not zero, the athermal term (Iathermal) takes on a nontrivial value - which can incidentally be either positive or negative whenever the critical size changes as a function of time. An additional point that may be noted from the above analysis is that the athermal mechanism neither requires nor suggests that the population distribution be "frozen" in the initial configuration [9] in order for the mechanism to be relevant. Historically, Fisher et. al. were the first to provide a qualitative description of the process and coined the term "athermal nucleation" [5] (a limited discussion on the subject was briefly alluded by Avrami [10]); the first quantitative analysis of the subject matter and derivation of the expressions that
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