Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation
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Finite size corrections for the Johnson–Mehl– Avrami–Kolmogorov equation L. E. Levine,a) K. Lakshmi Narayan, and K. F. Kelton Department of Physics, Washington University, St. Louis, Missouri 63130 (Received 16 November 1995; accepted 1 October 1996)
The Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation is frequently used to describe phase transformations involving nucleation and growth. The assumptions used in the derivation of this equation, however, are frequently violated when making experimental measurements; use of the JMAK equation for analyzing such data can often produce invalid results. Finite-size effects are among the most serious of these problems. We present modified analytic JMAK equations that correct for the finite-size effects and are roughly independent of both the sample shape and the shape of the growing nuclei. A comparison with computer simulations shows that these modified JMAK equations accurately reproduce the growth behavior over a wide range of conditions.
I. INTRODUCTION
Many first-order phase transitions, including solid state transformations and the crystallization of liquids and glasses, occur by a process of nucleation and growth. A detailed understanding of first-order phase transformations is therefore important in physics, chemistry, metallurgy, ceramics, and materials science. These concepts are also applicable to biological and population studies, however, and are even being applied to cosmological studies of the formation of the universe. Because the physical properties of materials are inextricably linked to their microstructure, an ability to predict and control the microstructural development during transformation is essential. Kinetic data for a phase transformation is often easiest to obtain; with a sufficiently good mathematical model, it is possible to extract values for the nucleation and growth rates. It is also sometimes possible to deduce the thermal history of a system from its resultant microstructure. In the earliest stages of the phase transformation, the small regions of the daughter phase are generally far apart, allowing the total volume transformed to be computed by simply summing the transformed volumes associated with each region. As the transformed regions continue to grow, however, they eventually impinge upon one another, slowing the rate of transformation. Typically this impingement is analyzed following the approach suggested by Kolmogorov,1 Johnson and Mehl,2 and Avrami,3–5 here called JMAK. Though first applied to interface-limited growth and homogeneous nucleation, this approach has been extended to include diffusion-
a)
Present address: Department of Mechanics and Materials, Washington State University, Pullman, Washington 99164-2920.
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http://journals.cambridge.org
J. Mater. Res., Vol. 12, No. 1, Jan 1997
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limited growth6 and spatially nonrandom nucleation, such as on grain boundaries7 and surfaces.8 The validity of the JMAK analysis has been criticized.9,10 Since it is
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