Crystal Growth Simulations To Establish Physically Relevant Kinetic Parameters from the Empirical Kolmogorov–Johnson–Mehl–Avrami Model

Abstract

A series of simulations was performed to enable interpretation of the material and physical significance of the parameters defined in the Kolmogorov, Johnson and Mehl, and Avrami (KJMA) rate expression commonly used to describe phase boundary controlled reactions of condensed matter. The parameters k, n, and t 0 are shown to be highly correlated, which if unaccounted for seriously challenge mechanistic interpretation. It is demonstrated that rate measurements exhibit an intrinsic uncertainty without precise knowledge of the location and orientation of nucleation with respect to the free volume into which it grows. More significantly, it is demonstrated that the KJMA rate constant k is highly dependent on sample size. However, under the simulated conditions of slow nucleation relative to crystal growth, sample volume and sample anisotropy correction affords a means to eliminate the experimental condition dependence of the KJMA rate constant, k, producing the material-specific parameter, the velocity of the phase boundary, v pb .

From Rate Measurements to Mechanistic Data for Condensed Matter Reactions: A Case Study Using the Crystallization of [Zn(OH2)6][ZnCl4]

“…This defines the dimensionality of crystal growth to be n = 3, and allows the geometric factor to be modeled as g = 1. However, as demonstrated by our previous simulations, the anisotropy of the sample container can impose a lower apparent dimensionality on a phase transition [9]. For example, a 3-D growth process constrained by a capillary sample geometry will exhibit approximately 1-D growth, whereas growth in a disk geometry will approximate 2-D growth.…”

“…And while, as noted above, there may be some kinetic isotopic effect difference between the rates of crystallization of the H2O hydrates by x-ray and DSC measurements and the D2O hydrates measured by neutron diffraction, KIE cannot explain the dramatically different rate constants observed. Analytical simulations of crystal growth experiments demonstrated a similar distribution of rate constants when analyzed using the KJMA model [9]. In that work, it was observed that the largest samples exhibit significantly smaller KJMA rate constants than was observed for smaller samples, and single crystallite rate constants appear to be faster than bulk rate constants.…”

“…It is not practical to conduct a set of aspect ratio and nucleation rate simulations for each type of material to obtain anisotropy corrections. Instead, to correlate these DSC experiments to the previous analytical simulation [9], we fixed t0 to be the time at which crystallization heat flow was observed to first increase above the baseline, then fit 0% < α(t) < 60% to Equation (1) The loss of the liquid and emergence of the crystalline phase is again observed to cross at α(t) = 50% indicative of the A→B reaction mechanism. The time-resolved data of the fraction of the sample crystallized 0% < α(t) < 60% was fit to the KJMA model, fixing n =3, with t 0 and k A as fitting parameters.…”

“…It is not practical to conduct a set of aspect ratio and nucleation rate simulations for each type of material to obtain anisotropy corrections. Instead, to correlate these DSC experiments to the previous analytical simulation [9], we fixed t 0 to be the time at which crystallization heat flow was observed to first increase above the baseline, then fit 0% < α(t) < 60% to Equation (1) to obtain n values for each crystallization reaction conducted with the same sample. Those n values for each respective sample were superimposed on the fixed volume simulation from reference [9] (Figure 5) to evaluate the actual anisotropy of the sample.…”

“…These comparative data are evaluated using the modified KJMA expression (M-KJMA), Equation (2) [2,9], demonstrating the corrections necessary to ensure rate constants are intrinsic to the sample and reaction conditions, not the specific technique of measurement.…”