Order and randomness in Kolmogorov-Johnson-Mehl-Avrami-type phase transitions

The distribution of points on a 2D domain influences the kinetics of its coverage when a growth law is attached at each point. This implies that the kinetics of space filling can be adopted as a descriptor of the degree of order of the initial point distribution. In this paper, the degree of order of an initial array of points has been changed following two paths: (i) from a regular square lattice, through increasing displacement assigned to each point, towards Poissonian disorder; (ii) from a Poissonian distribution, by introducing a hard core potential with increasing correlation lengths, towards a more ordered lattice. A linear growth law has been attached to the points of the initial array and the kinetics X(X(e)), where X(e) is the extended coverage as defined in the Kolmogorov-Johnson-Mehl-Avrami model, has been monitored. The quantitative analysis has been performed by fitting the kinetics to an equation which we propose for the first time and which has proved to be, in fact, highly suitable for the purpose. The results demonstrate that the gross of variation from order to disorder is obtained for point displacements, u, of the order of a, the latter being the constant of a square lattice. Vice versa, the introduction of a correlation distance in a random distribution provokes at most an order limited to the first neighbors and no real order can ever be reached. Others descriptors have been investigated, all confirming our results. We also developed an analytical description based on the use of the f-functions, as have been defined by Van Kampen, up to the second order terms. Such a description has been shown to work well for u/a < 1 within an interval ΔX(e) which depends on the ϵ value.