Kolmogorov Johnson Mehl Avrami Kinetics Research Articles

Effects of the Glass-Forming Ability and Annealing Conditions on Cocrystallization Behaviors via Rapid Solvent Removal: A Case Study of Voriconazole.

The kinetic entrapment of molecules in an amorphous phase is a common obstacle to cocrystal screening using rapid solvent removal, especially for drugs with a moderate or high glass-forming ability (GFA). The aim of this study was to elucidate the effects of the coformer’s GFA and annealing conditions on the nature of amorphous phase transformation to the cocrystal counterpart. Attempts were made to cocrystallize voriconazole (VRC) with four structural analogues, namely fumaric acid (FUM), tartaric acid (TAR), malic acid (MAL), and maleic acid (MAE). The overall GFA of VRC binary systems increased with decreasing glass transition temperatures (Tgs) of these diacids, which appeared as a critical parameter for predicting the cocrystallization propensity such that a high-Tg coformer is more desirable. A new 1:1 VRC-TAR cocrystal was successfully produced via a supercooled-mediated re-cocrystallization process, and characterized by PXRD, DSC, and FTIR. The cocrystal purity against the annealing temperature displayed a bell-shaped curve, with a threshold at 40 °C. The isothermal phase purity improved with annealing and adhered to the Kolmogorov–Johnson–Mehl–Avrami kinetics. The superior dissolution behavior of the VRC-TAR cocrystal could minimize VRC precipitation upon gastric emptying. This study offers a simple but useful guide for efficient cocrystal screening based on the Tg of structurally similar coformers, annealing temperature, and time.

Interplay between Kolmogorov–Johnson–Mehl–Avrami kinetics and Poisson–Voronoi tessellation

In this paper we investigate the connection between Voronoi tessellation and the KJMA approach of space filling. In particular, we study how nuclei, in their growth, cover a given Voronoi cell. This approach leads to an integral equation for the cell-size distribution function. Starting from the 1D case, that is solved exactly, we extend the results to the dD case. The analysis allows to find a rationale to the phenomenological parameter entering the Gamma distribution function and to improve the description of the transformation through the knowledge of the kinetics of grain formation. Moreover, the nucleus size distribution function has been calculated as a function of the transformed fraction.

Generalized additivity rule for the Kolmogorov–Johnson–Mehl–Avrami kinetics

The validity of the classical additivity rule for the non-isothermal KJMA kinetics is studied. In the case of linear temperature variation, it is demonstrated that, with an exception for the isokinetic range, the kinetics is not additive. Yet, a generalized form of the additivity rule holds true with a weighting function, W, which depends upon actual temperature, heating rate, and time. The exact expression of the weighting function is computed, in closed form, for integer and half-integer Avrami exponents. The behavior of W strongly depends on the ratio between activation energies for nucleation and growth. Its effect on the rate of the transformation, as well as on transformed volume fraction, is investigated in connection with the classical additivity rule. The present results are also discussed in relation to approximate computations previously proposed in the literature.

Kolmogorov–Johnson–Mehl–Avrami kinetics for non-isothermal phase transformations ruled by diffusional growth

We report on the functional form of the rate of the transformed volume fraction in non-isothermal phase transitions occurring by nucleation and diffusional growth. The microscopic growth rate is computed by solving the diffusion problem for time-dependent diffusion coefficient. The growth law is further employed in the Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory for describing the time dependence of the transformed volume at constant heating rate. It is demonstrated that the transformation rate separates in the product of volume fraction and actual temperature functions. In the framework of the KJMA approach this factorization is exact. It is also shown that for real systems (due to the high values of the reduced activation energies for nucleation and growth), the kinetics is in excellent agreement with the stretched exponential function appropriate for isothermal transformations.

Coupling of Crystal Plasticity Finite Element and Phase Field Methods for the Prediction of SRX Kinetics after Hot Working

The conventional approach to model static recrystallization (SRX) kinetics is to use the well‐known Kolmogorov–Johnson–Mehl–Avrami (KJMA) equation. However, the validity of KJMA kinetics depends on whether the real recrystallization process follows the assumptions made in the derivation of the KJMA kinetics, which are a random distribution of nucleation sites, isotropic growth and a constant growth velocity. When these assumptions are violated the KJMA kinetics loose their physical meaning and turn into an empirical formula with limited predictive capability. In such cases, phase field (PF) method can be an alternative to the KJMA model, since it is based on irreversible thermodynamics. An effective strategy to apply the PF method to static recrystallization problems is coupling it to a deformation model though which the prior deformation process is simulated. For that purpose crystal plasticity finite element method (CPFEM) can be employed. In this work, an example of a weak coupling strategy was applied to a steel grade of commercial interest (X60Mn23 TWIP steel) for which KJMA equation is invalid. Deformation of a Voronoi type representative volume element was modeled using a Crystal plasticity finite element formulation after determining the material parameters of a strain hardening model from macroscopic experiments. Afterwards, the deformed structure was transferred on to a finite difference grid with a dedicated mapping scheme, which was then used as input for phase field simulation. Finally, the simulation results were compared with the results of stress relaxation tests. It was found that although the grain morphology of the simulation differed from the experimental results, the coupled simulation strategy was able to mimic the main features of the static recrystallization (SRX) kinetics.

Functional form of the Kolmogorov–Johnson–Mehl–Avrami kinetics for non-isothermal phase transformations at constant heating rate

An analytical computation is presented for determining the functional form of the Kolmogorov–Johnson–Mehl–Avrami (KJMA) kinetics in the case of non-isothermal transformations under linear temperature variation. Differently to previous modeling, in the present computation no approximate expression of the exponential integral is employed. The mathematical derivation holds true independently of the activation energy values for nucleation and growth and, in the limiting case of zero heating rate, reduces to the isothermal kinetics. The present computation demonstrates that the transformation rate factorizes in the product of temperature-dependent and volume fraction-dependent functions. In the framework of the KJMA theory this result is exact and implies that the KJMA kinetics is fully consistent with the requirement for the applicability of the multiple-scan analysis techniques, widely employed in thermochemistry. In the limit of large values of the reduced activation energies, the non-isothermal kinetics yields the KJMA transformation rate equation.

Cell dynamics simulation of Kolmogorov–Johnson–Mehl–Avrami kinetics of phase transformation: One and two-dimensional cases

In this study, we use the cell dynamics method to test the validity of the Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory of phase transformation. This cell dynamics method is similar to the well-known phase-field model, but it is a more simple and efficient numerical method for studying various scenarios of phase transformation in a unified manner. We find that the cell dynamics method reproduces the time evolution of the volume fraction of the transformed phase predicted by the KJMA theory. Specifically, the cell dynamics simulation reproduces a double-logarithmic linear KJMA plot and confirms the integral Avrami exponents n predicted from the KJMA theory. Our study clearly demonstrates that the cell dynamics approach is not only useful for studying the pattern formation but also for simulating the most basic properties of phase transformation.

On the equations describing the grain size distribution change for KJMA kinetics

A partial differential equation is proposed governing the particle size probability density function during grain growth. The equation is derived by using a mean field hypothesis under the same assumptions as the classical Kolmogorov–Johnson, Mehl–Avrami (KJMA) model for nucleation and growth kinetics. The resulting set of equations should give the final grain size distribution after an arbitrary nucleation and growth procedure, provided that the time dependence of the nucleation and growth rates are known, as for example in nanocrystalline materials. Results for the simplest case of initial nucleation and constant growth rates (pre-existing nuclei) are shown.

Model for crystallization kinetics: Deviations from Kolmogorov–Johnson–Mehl–Avrami kinetics

We propose a simple and versatile model to understand the deviations from the well-known Kolmogorov–Johnson–Mehl–Avrami kinetics theory found in metal recrystallization and amorphous semiconductor crystallization. We analyze the kinetics of the transformation and the grain-size distribution of the product material, finding a good overall agreement between our model and available experimental data. The information so obtained could help to relate the mentioned experimental deviations due to preexisting anisotropy along some regions, to a certain degree of crystallinity of the amorphous phases during deposition, or more generally, to impurities or roughness of the substrate.

Kinetics of crystallization preceded by metastable-phase formation

A theoretical description is proposed of the kinetics of overall crystallization when the appearance of the stable crystalline (SC) phase is preceded by the formation of a metastable phase. A general formula is derived for the fraction αs of volume of the SC phase crystallized at time t. This formula is used for obtaining the αs(t) function in the case of Kolmogorov–Johnson–Mehl–Avrami kinetics of overall crystallization. The time t1/2 at which half of the transformation of the parent phase into the SC one is accomplished is determined and it is found that this time is the sum of the times for half completion of the two successive phase transformations of the parent phase into the metastable one and of the latter into the SC phase. A general condition for formation of a long living metastable phase is formulated in terms of the nucleation and growth rates of the metastable and the SC phases. The general results are applied to melt crystallization and an expression for the temperature dependence of t1/2 is obtained. It is found that the presence of a metastable phase in the melt is easy to observe only below a certain critical temperature, because then the transformation of the metastable phase into the SC one is sufficiently delayed. In a supplementary analysis, an answer is given to the classical question about the possibility for formation of a metastable phase prior to the stable one in accordance with Ostwald’s Rule of Stages.