Johnson Mehl Avrami Kolmogorov Theory Research Articles

Photoinduced growth of the crystalline phase of tellurium on a 1T′-MoTe2 matrix

Due to the growing demand for miniaturization and energy efficiency in modern electronic devices, there is a renewed interest for optoelectronic memories and sensors based on 2D materials. In particular, the molybdenum ditelluride (MoTe2) is one of the most promising materials for applications in nonvolatile phase-change memory devices, as its properties can be controlled by visible-light illumination. Among the several ways to synthesize MoTe2, the molybdenum oxide tellurization through isothermal close space sublimation (CSS) annealing in gas atmosphere is a simple and low-cost effective method for large-scale production of devices based on this layered material. Therefore, the understanding of the physical properties of MoTe2 thin films produced by this technique is crucial for future applications. Surprisingly, our results indicate that there is a photoinduced growth of the crystalline phase of tellurium on the 1T′-MoTe2 matrix even when the power density of the laser is low. From Raman spectroscopy investigations, we were able to show that nanometer-sized tellurium crystallites work as seed sites for the photocrystallization of tellurium. By assuming that the overall crystallization process is described by a kinetic approach that is based on the Kolmogorov–Johnson–Mehl–Avrami theory, our results indicate that the process is governed by an anisotropic organization of the tellurium atoms in helical structures during the crystal growth.

Effect of cold drawing reduction rate on edge-to-center-characterized microstructure and orientation alongside residual stresses in conjunction with magnetic properties of low-carbon high-alloy ferromagnetic steel

In the current research, the effect of cold drawing reduction rate (CDRR) of 15% and 45% and the required subsequent isothermal static recrystallization annealing heat treatment (ISRAHT) on the microstructures, textures, residual stresses, and magnetic properties of ferritic/ferromagnetic stainless steel (FSS), EN 1.4106, are investigated by a series of experimental analyses. The study is carried out by the theoretical well-known model of Johnson-Mehl-Avrami-Kolmogorov (JMAK) in conjunction with aforesaid properties. According to the results, by increasing the CDRR, the recrystallization fractions (RF) become faster in accordance with the JMAK theory. Such an increment also affects more fragmented and elongated grains, which leads to provide smaller grains in size. However, by the effect of slow cooling process (SCP), the grain growth is another noticeable part of study. Likewise, the effects of CDRR and the subsequent ISRAHT find to be beneficial for the evolution of microstructures, textures, and relief of residual stresses, and better performance of magnetic behavior. For instance, higher relative magnetic permeabilities approximately above 1000 causes to reach residual stresses closer to zero. The cold-drawn FSSs are consisted of the α-fibre texture, which is close to {223} 〈110〉 and {111} 〈110〉, with higher intensity while by gradual higher recrystallization, the orientation tendency to {111} 〈011〉 of γ-fibre are formed following to the more distributed texture with lesser intensity. The findings display that while the recrystallization process addresses the formation of new grains, resulting in the more equiaxed grains, more well-aligned textures are also achieved in respect to the lower misorientation uniformity density and even with more distributed clusters.

Insights into the Sodiation Kinetics of Si and Ge Anodes for Sodium-Ion Batteries

Group IVA elements exhibit interesting Na storage capabilities due to the success of their Li alloy analogues. However, beyond hard carbon, they remain poorly understood as anodes for sodium-ion batteries (SIBs). Here, kinetic investigations of the electrochemical sodiation of Si and Ge are conducted using liquid electrolytes and half-cell configurations. Sodiation of Ge is found to be kinetically limited rather than thermodynamically limited. Either increasing temperature or decreasing sodiation rate can facilitate easier transformations from Ge to Na-Ge phases. A critical temperature seems to exist between 50 °C and 60 °C, beyond which a higher sodiation capacity is evident. The phase transformations are analyzed using Kolmogorov–Johnson–Mehl–Avrami theory. Following a one-dimensional growth, the Ge to NaGe4 is determined to be diffusion limited whereas NaGe4 to Na1+x Ge is controlled by reaction speed. Moreover, the Arrhenius equation is employed to investigate the temperature dependence on both phase transformations, giving activation energies of ∼50 kJ·mol−1 and ∼70 kJ·mol−1, respectively. Schematic models are proposed to elucidate the sodiation mechanisms, potentially influencing sought-after advancements in cell formats and classifications. Not only does this work lay the foundation for efforts on the Ge-based anodes, but also provides analogous kinetic information to Si/Sn-based ones for SIBs.

Fokker–Planck equation for the particle size distribution function in KJMA transformations

The Fokker–Planck (FP) equation has been derived for describing the temporal evolution of the particle size probability density function (PDF) for KJMA (Kolmogorov–Johnson–Mehl–Avrami) transformations. The classical case of transformations with constant rates of both nucleation and growth, in 3D space, has been treated. Integration of the equation shows that the PDF is given by the superposition of one-parameter gamma distributions with time-dependent mean size given by the KJMA theory. The asymptotic behavior of the FP solution offers a demonstration of the conjecture, previously proposed by Pineda et al. (2004), according to which the set of crystals formed at the same time are gamma-distributed, with the parameter depending on crystal birth time. Computer simulations of the transformation with constant nucleation and growth rates show that the temporal evolution of the PDF, in volume domain, is in good agreement with the Johnson–Mehl PDF. The approach based on the FP equation provides a particle size PDF that exhibits such behavior.

On the Effect of Nucleation Undercooling on Phase Transformation Kinetics

We carry out an extensive comparison between Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory of first-order phase transformation kinetics and phase-field (PF) results of a benchmark problem on nucleation. To address the stochasticity of the problem, several hundreds of simulations are performed to establish a comprehensive, statistically significant analysis of the coincidences and discrepancies between PF and JMAK transformation kinetics. We find that PF predictions are in excellent agreement with both classical nucleation theory and JMAK theory, as long as the original assumptions of the latter are appropriately reproduced—in particular, the constant nucleation and growth rates in an infinite domain. When deviating from these assumptions, PF results are at odds with JMAK theory. In particular, we observe that the size of the initial particle radius \(r_0\) relative to the critical nucleation radius \(r^*\) has a significant effect on the rate of transformation. While PF and JMAK agree when \(r_0\) is sufficiently higher than \(r^*\), the duration of initial transient growth stage of a particle, before it reaches a steady growth velocity, increases as \(r_0/r^*\rightarrow 1\). This incubation time has a significant effect on the overall kinetics, e.g., on the Avrami exponent of the multi-particle simulations. In contrast, for the considered conditions and parameters, the effect of interface curvature upon transformation kinetics—in particular negative curvature regions appearing during particle impingement, present in PF but absent in JMAK theory—appears to be minor compared to that of \(r_0/r^*\). We argue that rigorous benchmarking of phase-field models of stochastic processes (e.g., nucleation) needs sufficient statistical data in order to make rigorous comparisons against ground truth theories. In these problems, analysis of probability distributions is clearly preferable to a deterministic approach.

Recrystallization model of discrete amorphous regions in C3H5-molecular-ion-implanted silicon substrate surface analyzed by X-ray photoelectron spectroscopy

We investigated the recrystallization of discrete amorphous regions formed in a C3H5-molecular-ion-implanted silicon (Si) substrate surface in the rapid thermal annealing (RTA). The change in the crystalline fraction of the C3H5-molecular-ion-implanted substrate surface after the RTA was obtained from the chemical shifts of Si 2p spectra by X-ray photoelectron spectroscopy. We found that the crystalline fraction increases depending on the RTA temperature after an incubation period. The transformation from the amorphous phase to the crystalline phase was analyzed on the basis of the Johnson–Mehl–Avrami–Kolmogorov theory. It was revealed that recrystallization of discrete amorphous regions proceeded three-dimensionally and activation energy was estimated to be 2.74 ± 0.39 eV, which is approximately equal to 2.70 eV for the solid-phase epitaxy of the continuous amorphous layer in a Si crystal. Therefore, we believe that discrete amorphous regions are recrystallized via solid-phase epitaxy laterally and vertically from the amorphous/crystal interface around them.

On the grain size distribution function in KJMA compliant growth

The temporal evolution of the probability density function (PDF) of grain size in phase transformations by nucleation and growth has been computed by means of Kolmogorov’s second equation. The transformation is modeled in accord with the classical KJMA (Kolmogorov-Johnson-Mehl-Avrami) theory for linear growth of nuclei and either simultaneous or progressive nucleation. For 3D growth, it is shown that for progressive nucleation, second order term can be neglected in the differential equation for the PDF of grain volume. For site-saturated nucleation the variance of the distribution increases with time to attain the value of the Gamma distribution at the end of the transformation. It is demonstrated that in this case the PDF is given by convolution of Gaussian-like solutions of Kolmogorov’s equation with Gamma distribution. The validity of the model is checked by computer simulations available in literature.

Computer simulation of site saturation and constant nucleation rate transformations on a network of Kelvin polyhedra

A crucial step of transformations in solids is that of nucleation. To model nucleation, one must specify nuclei location in space as well as how nuclei appear as a function of time. In the classical theory of Johnson–Mehl–Avrami–Kolmogorov (JMAK), there are two essential nucleation modes. One is site-saturation in which all nucleation sites are exhausted early in the transformation. Another is the so-called constant nucleation rate. JMAK theory considers that nuclei are located uniform randomly within the matrix both in the case of site-saturation and constant nucleation rate. Nonetheless, it is usually common in polycrystals to observe nucleation taking place on the grain surfaces, edges, and vertices. Cahn proposed analytical expression for such situations. In this work, we conduct a computer simulation considering that grains can be represented by a network of Kelvin polyhedra. We restrict nucleation to the grain boundaries of the Kelvin network. Grain boundary nucleation can be either site-saturated or a constant nucleation rate. We compare the effect of these two nucleation modes. JMAK theory and Cahn’s theory are applied to the simulated results. The microstructures are characterized by several quantitative metallographic parameters. We observed that the two nucleation modes exhibited similar behavior.

Relationship between three-dimensional microstructure and Avrami exponent for recrystallization in pure iron

We undertook a three-dimensional analysis of the recrystallization that occurs in pure iron from the viewpoint of consistency between the growth direction of the recrystallized ferrite grains and the Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory. Cold-rolled specimens were heated to 550 °C and, then, held at that temperature for a range of durations. The recrystallization behavior during the annealing was investigated both theoretically and experimentally. The formation of subgrains in non-recrystallized ferrite grains was found to occur before the start of recrystallization. Additionally, the fraction that recrystallized increased with the duration at 550 °C, and the slope became gradual after the annealing duration exceeded 60 s. The values of the Avrami exponent in the early and later stages of annealing were 1.86 and 0.37, respectively. Based on the JMAK theory, the growth direction of the recrystallized ferrite grains was assumed to be one-dimensional. On the other hand, the two- and/or three-dimensional growth of the recrystallized ferrite grains was confirmed by the three-dimensional analysis. Using a modified JMAK equation that takes the effect of recovery in recrystallization during annealing into consideration, the experimental data was found to be in good agreement with the modified JMAK plot. Thus, the inconsistency between the theoretical and experimental results could be attributed to the interaction between the recovery and recrystallization.

Modeling and Simulation of Nucleation and Growth Transformations with Nucleation on Interfaces of Kelvin Polihedra Network

Nucleation is a phenomenon associated to the start of the new phase, from a primary phase, named matrix. Growth is the increase in size of this new phase over time. In metallic materials, the nucleation may take place on the grain boundaries of the primary phase. A network of Kelvin polyhedra was used in this paper to represent the grains. A computer simulation was performed in which nucleation took places at the faces, edges and vertices of this polyhedral network. The Causal cone method was employed in the simulations. The results of the present computational simulations were compared with the classical Johnson Mehl-Avrami-Kolmogorov (JMAK) as well as with Cahn model for nucleation at the grain boundaries. JMAK theory considers nuclei to be uniform randomly located within the matrix. Cahn analytical model specifies that nucleation takes place on random planes. For a small number of nuclei, the simulations approached the JMAK model whereas as the number of nuclei increased the simulation results agree with Cahn's theory. Reasons for this are fully discussed.

On the Use of JMAK Theory to Describe Mechanical Amorphization: A Comparison between Experiments, Numerical Solutions and Simulations

The kinetics of amorphization during ball milling is generally analyzed using two different approaches: the classical Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory and Delogu and Cocco’s model for which a region deterministically transforms after it reaches a certain number of collisions. The application of JMAK analysis to the latter model predicts Avrami exponents to be higher than the experimental ones (typically close to one). We develop simulations based on the probabilistic character of the nucleation phenomenon and concave growth of the amorphous phase in the core of a nanocrystal. The predictions of our simulations are in good agreement with the low Avrami exponents and with the size evolution of the remaining crystallites found experimentally. From these values, the parameters involved in the simulated model (growth rate and probability of nucleation) can be estimated.

A hybrid finite-element and cellular-automaton framework for modeling 3D microstructure of Ti–6Al–4V alloy during solid–solid phase transformation in additive manufacturing

Additive manufacturing such as selective laser sintering and electron beam melting has become a popular technique which enables one to build near-net-shape product from packed powders. The performance and properties of the manufactured product strongly depends on its material microstructure, which is in turn determined by the processing conditions including beam power density, spot size, scanning speed and path etc. In this paper, we develop a computational framework that integrates the finite element method (FEM) and cellular automaton (CA) simulation to model the 3D microstructure of additively manufactured Ti–6Al–4V alloy, focusing on the β → α + β transition pathway in a consolidated alloy region as the power source moves away from this region. Specifically, the transient temperature field resulted from a scanning laser/electron beam following a zig-zag path is first obtained by solving nonlinear heat transfer equations using the FEM. Next, a CA model for the β → α + β phase transformation in the consolidated alloy is developed which explicitly takes into account the temperature dependent heterogeneous nucleation and anisotropic growth of α grains from the parent β phase field. We verify our model by reproducing the overall transition kinetics predicted by the Johnson–Mehl–Avrami–Kolmogorov theory under a typical processing condition and by quantitatively comparing our simulation results with available experimental data. The utility of the model is further demonstrated by generating large-field realistic 3D alloy microstructures for subsequent structure-sensitive micro-mechanical analysis. In addition, we employ our model to generate a wide spectrum of alloy microstructures corresponding to different processing conditions for establishing quantitative process-structure relations for the system.

Analysis of transition from low to high iodide and iodine state in the Briggs–Rauscher oscillatory reaction containing malonic acid using Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory

The oscillatory behavior is not the only interesting nonlinear phenomena that appeared in the Briggs–Rauscher (BR) reaction. The BR reaction containing malonic acid may undergo a sudden transition from low (the state I) to high iodide and iodine (the state II) concentration states. This paper focuses on the mixture with an immutable [CH2(COOH)2]0/[IO3 −]0 = 1.5 value, where state I to state II transition occurs after a time delay and BR reaction ended with a solution abundant of solid iodine. The state I to the state II transition curves obtained at different temperatures were analyzed using the Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory. The KJMA theory was applied for monitoring the crystallization process of isolated solid iodine product at various levels of operating temperatures. At T < 33.5 °C, we have one type of the process and iodine was formed by autocatalysis pathway. On the other hand, at T ≥ 33.5 °C, two processes occur. With the rise in operating temperature, the emergence of inhomogeneous distribution of nuclei was identified and it was established the primary and secondary crystallization processes of iodine. At elevated temperatures, it was also found that the strong influence of impingement mechanism exists. Results obtained are the first step toward elucidation of the complex reaction mechanism of the state I → state II transition.

Modeling the kinetics of consecutive phase transitions in the solid state

A theoretical approach for describing the kinetics of consecutive phase transformations ruled by nucleation and growth is reported. In the considered system, the mother phase (M) transforms to an intermediate phase (α) which, in turn, transforms to the final product (β). The classical Kolmogorov–Johnson–Mehl–Avrami theory is generalized to deal with a finite-size phase with moving boundary. To this end, the statistical method based on the differential critical region has been employed. The exact solution of the kinetics is computed in closed form for the transformation of a spherical α-nucleus growing into the mother phase. By resorting to an approximate expression for the probability function entering the differential critical region method, the consecutive transformation is studied in the case of nucleation and growth of the α-phase. The time dependence of the β/α volume fraction is found to be in very good agreement with the stretched exponential kinetics, and the dependence of Avrami’s exponent on both nucleation and growth rates of the two phases is investigated. Modeling of the non-isothermal kinetics at constant heating rate has also been performed which provides an insight into the shape of the differential scanning calorimetry curves for consecutive phase transitions.

Using dilatometry to study martensitic stabilization and recrystallization kinetics in a severely deformed NiTi alloy

A martensitic NiTi alloy was separately processed by equal-channel angular pressing, repetitive corrugation and straightening by rolling and cold rolling. Measurements by dilatometry revealed a significant increase in the austenitic phase transformation temperature and martensitic stabilization after severe plastic deformation. The release of an excess volume upon recrystallization of the deformed NiTi alloy was measured using a high-precision difference-dilatometer employing non-isothermal and isothermal procedures. The kinetics of the recrystallization process was analyzed according to the Johnson–Mehl–Avrami–Kolmogorov theory and the average effective Avrami exponent was estimated as n ≈ 1. It is concluded that the recrystallization behavior is essentially similar for all three processing modes.

Comparative study of approaches based on the differential critical region and correlation functions in modeling phase-transformation kinetics.

The statistical methods exploiting the "Correlation-Functions" or the "Differential-Critical-Region" are both suitable for describing phase transformation kinetics ruled by nucleation and growth. We present a critical analysis of these two approaches, with particular emphasis to transformations ruled by diffusional growth which cannot be described by the Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory. In order to bridge the gap between these two methods, the conditional probability functions entering the "Differential-Critical-Region" approach are determined in terms of correlation functions. The formulation of these probabilities by means of cluster expansion is also derived, which improves the accuracy of the computation. The model is applied to 2D and 3D parabolic growths occurring at constant value of either actual or phantom-included nucleation rates. Computer simulations have been employed for corroborating the theoretical modeling. The contribution to the kinetics of phantom overgrowth is estimated and it is found to be of a few percent in the case of constant value of the actual nucleation rate. It is shown that for a parabolic growth law both approaches do not provide a closed-form solution of the kinetics. In this respect, the two methods are equivalent and the longstanding overgrowth phenomenon, which limits the KJMA theory, does not admit an exact analytical solution.

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.

Crystallization kinetics of lithium niobate glass: determination of the Johnson–Mehl–Avrami–Kolmogorov parameters

The formation of crystalline LiNbO3 (LN) from LN glass has been studied by means of differential scanning calorimetry and in situ synchrotron X-ray diffraction. The LN glass with no glass former was prepared by the polymerized complex method. The isothermal kinetics of the crystallization process is described using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation and the Avrami exponent n is found to be ~2.0, indicating that the crystallization mechanism is diffusion-controlled growth with a decreasing nucleation rate. The effective activation energy of crystallization calculated from isothermal measurements is 6.51 eV. It is found that the LN glass directly transforms into a rhombohedral LN crystal without any intermediate crystalline phase and most crystal grains are confined within the size of ~40 nm irrespective of different isothermal temperatures. Application of JMAK theory to the non-isothermal thermoanalytical study of crystallization of LN glass is discussed.

Dynamics of crystallization with fractal geometry: Extended KJMA approach in glasses

AbstractAlthough the kinetics of crystallization under isothermal annealing conditions in glassy materials is fairly well understood by the Kolmogorov–Johnson–Mehl–Avrami (KJMA) formalism, there is still a matter of debate on its applicability. By introducing a fractal geometry and the Meyer–Neldel rule on thermally activated processes into the KJMA theory, we can overcome some of the difficulties with the classical KJMA mode, with maintaining its intuitive simplicity. Very reasonable physical parameters are deduced from the extended KJMA theory

Origin and evolution of voids in electroless Ni during soldering reaction

The interfacial voids in Ni2SnP were studied by high-resolution transmission electron microscopy. These voids were found to be responsible for many types of failure in electronic solder joints, but so far, detailed investigation of these voids has been pursued in a only few studies because of the need for advanced analytical techniques. The interaction of Sn3Ag0.5Cu with electroless Ni/immersion Au was investigated in this study. The microstructures during reflow were quenched to understand the evolution of these voids. Different peak reflow temperatures were adopted to analyze the change in the void density. Then, the effect of the number of reflow cycles on the void density was investigated. Two types of interfacial voids were found: one in the Ni3P region and the other in the Ni2SnP region. The voids in Ni2SnP were connected to each other to form a void line. The void density increased with the number of reflow cycles, but not with the peak reflow temperature. These results confirmed that the voids in Ni2SnP were more fragile and responsible for most of the interfacial cracks. Void nucleation and coalescence were modeled on the basis of the Johnson–Mehl–Avrami–Kolmogorov theory. The experimental data agreed well with the model data. The mechanism of the void nucleation and coalescence was then discussed.