An Equation for Melting and Freezing Transition Rates
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AN EQUATION FOR MELTING AND FREEZING TRANSITION RATES PHILIP H. BUCKSBAUM* AND MICHAEL 0. THOMPSON** * AT&T Bell Laboratories, Murray Hill, NJ 07074 "**Department of Materials Science, Cornell University, Ithaca, NY 14853 ABSTRACT Statistical thermodynamics is used to derive the reaction rate for melting and freezing by considering an atomically sharp interface with or without an activated intermediate state. The resulting predictions differ substantially from those of the classical kinetic rate theory at large deviations from equilibrium. The model may be appropriate for the analysis of pulsed laser melting experiments where large deviations are expected.
This paper describes a simple model of melting and freezing kinetics based on the statistical mechanics of an atomically sharp interface. The model has been
motivated by recent results of laser-induced melting experiments, particularly in silicon, which measure the melting and freezing rates as a function of the temperature
at the solid-liquid interface [1-7]. Such velocity-temperature data are usually analyzed using equations derived from classical kinetic rate theory.[8] Although these classical theories have been successful in predicting the behavior of viscous systems, they encounter difficulties in low-viscosity melts or when the temperature excursions from equilibrium become extreme. For example, the simplest model predicts melting velocities which exceed the sound of speed at very high tempera-
tures unless an additional entropic term is included.
The statistical approach
adopted here, while it produces equations that are formally quite similar to the classi-
cal theories, avoids such catastrophes in a natural manner. This approach also provides a natural way to include the effects of a transition state on transformation rates.
Classical rate theories describe melting as a balance between the entropy S which drives solid-liquid systems toward the less ordered liquid phase L, and the enthalpy H
which drives them toward the lower energy solid phase S. Melting and freezing rates are thus taken to be proportional to
rS-L oc
(eAS/R
-
eAH(T)/RT) OC(1 -
eAG(T)/RT),
where AG(T) is the free energy release of the transformation at temperature T. These ideas, first discussed by Wilson at the turn of the century [9], have been refined over the years to accommodate the specialized requirements of the solidliquid interface [8,10]. The classical equation embodying these ideas relates the velocity of the interface to its temperature by: v(T) = XfVe-AC 5 /RT[ 1I
e-AG(T)/RT].
(1)
Here
X = lattice spacing in the solid; v = transition attempt frequency, or impingement rate per interface site; f = geometrical limiting factor equal to the fraction of interface sites where transitions are possible; AGa = activation barrier due to a transition state on the interface. Two additional important contributions to the theory bear mentioning. Turnbull suggested that for low viscosity elemental systems, the product Xi, is approximately the sound velocity v8 in the liqu
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