Modelling crystallization: When the normal growth velocity depends on the supersaturation

The crystallization proceeds by the advance of the crystal faces into the disordered phase at the expense of the supersaturation which is not sustained in our model. Using a conservation constraint for the transformation ratio and a kinetic law, we derive a general equation for the rate of transformation. It is integrated for the six combinations of the three spatial dimensions D = 1, 2, 3 and the two canonical values of the growth order (1 and 2). The same equation with growth order 1 is obtained when taking only the linear term from the Taylor's expansion around 0 transformation of the model of Johnson-Mehl-Avrami-Kolmogorov(JMAK). We verify our model by fitting it with JMAK. We start the validation of our model in 2D with published results.Comment: 30 pages, 13 figures, 53 reference