Finite Volume Kolmogorov-Johnson-Mehl-Avrami Theory

Abstract

We study the Kolmogorov-Johnson-Mehl-Avrami theory of phase conversion in finite volumes. For the conversion time we find the relationship tau(con)=tau(nu)[1+f(d)(q)]. Here d is the space dimension, tau(nu) the nucleation time in the volume V, and f(d)(q) a scaling function. Its dimensionless argument is q=tau(ex)/tau(nu), where tau(ex) is an expansion time, defined to be proportional to the diameter of the volume divided by expansion speed. We calculate f(d)(q) in one, two, and three dimensions. The often considered limits of phase conversion via either nucleation or spinodal decomposition are found to be volume-size dependent concepts, governed by simple power laws for f(d)(q).

Water: A Tale of Two Liquids

“…It is also important to point out that both τ x and τ e may depend on the system size. 263 For the mW model the maximum in the compressibility at ambient pressure cannot be reached since water freezes first. This is a clear case where the ratio of τ x to τ e is close to one and one cannot observe some of the water anomalies because water simply freezes first.…”

High pressure phase transformation in iron under fast compression

“…Well known concepts of phase nucleation and growth are being revisited and refined [1][2][3] due to their importance in geophysics, metallurgy, materials science [4][5][6][7] and many other fields, while new experimental techniques, particularly at high pressures, continue to open new vistas of research and exploration [8][9][10][11][12][13][14]. The study of phase transition kinetics under high pressures has a long history, with the earliest reported measurements being performed under static conditions by Bridgman [15].…”

“…depending linearly on the difference between the chemical potentials of the two phases [28], which reproduce some of the dynamic behavior observed in shock experiments but cannot explain more complex experimental features such as the negative acceleration loops recorded in these or the water experiments [18,19]. We adopt here the phenomenological but physical picture proposed by Kolmogorov and others [3], where well defined domains of the growing phase increase their size at the expense of the parent phase through the motion of an infinitely thin interface. The new phase is assumed to start on either a fixed number of initial sites in the case of heterogeneous nucleation, or on sites produced at a certain well defined rate in the case of homogeneous nucleation.…”

“…If the phase interface velocity u is assumed to be constant, as it is perhaps the case if the system, initially in the parent phase, i.e. ϕ(t = 0) = ϕ 0 = 0, is very suddenly quenched (by changing the temperature, pressure or both) into a thermodynamic state of pressure P and temperature T where ϕ 0 > 0, the above model yields a well known evolution equation for ϕ, which has been employed extensively to fit a variety of phenomena [3]. On the other hand more gradual quenching should take into account the dependence of the interface velocity on the changing thermodynamic conditions.…”

Filling of three-dimensional space by two-dimensional sheet growth

“…Extensions towards diffusion-type kinetics, for example, have been proposed by Alekseechkin and Tomellini and Fanfoni [21,22]. Furthermore, KJMA-style growth in finite domains is a field of active research [23,24]. The problem studied in this paper is an extreme case of anisotropic growth.…”