Johnson Mehl Avrami Research Articles

A critical review on applications of the Avrami equation beyond materials science.

The Johnson-Mehl-Avrami-Kolmogorov (JMAK) formalization, often referred to as the Avrami equation, was originally developed to describe the progress of phase transformations in material systems. Many other transformations in the life, physical and social sciences follow a similar pattern of nucleation and growth. The Avrami equation has been applied widely to modelling such phenomena, including COVID-19, regardless of whether they have a formal thermodynamic basis. We present here an analytical overview of such applications of the Avrami equation outside its conventional use, emphasizing examples from the life sciences. We discuss the similarities that at least partially justify the extended application of the model to such cases. We point out the limitations of such adoption; some are inherent to the model itself, and some are associated with the extended contexts. We also propose a reasoned justification for why the model performs well in many of these non-thermodynamic applications, even when some of its fundamental assumptions are not satisfied. In particular, we explore connections between the relatively accessible verbal and mathematical language of everyday nucleation- and growth-based phase transformations, represented by the Avrami equation, and the more challenging language of the classic SIR (susceptible-infected-removed) model in epidemiology.

Kinetics of Transformation, Border of Metastable Miscibility Gap in Fe–Cr Alloy and Limit of Cr Solubility in Iron at 858 K

The study was aimed at determination of the position of the Fe-rich border of the metastable miscibility gap (MMG) and of the solubility limit of Cr in iron at 858 K. Towards this end, a Fe73.7Cr26.3 alloy was isothermally annealed at 858 K in vacuum up to 8144 hours and Mössbauer spectra were recorded at room temperature after every step of the annealing. Three spectral parameters viz. the average hyperfine field, 〈B〉, the average isomer shift, 〈IS〉, and the probability of the atomic configuration with no Cr atoms in the two-shell vicinity of the probe Fe atoms, P(0,0), gave evidence that the transformation process took place in two stages. All three parameters could have been well described in terms of the Johnson–Mehl–Avrami–Kolmogorov equation, yielding kinetics parameters. The first stage, associated with the phase decomposition, proceeded much faster than the second stage, associated with the alpha-to-sigma phase transformation. The most reliable estimation of the position of the MMG and that of the value of the Cr solubility limit was obtained from the annealing time dependence of 〈B〉, namely 24.5 at. pct Cr for the former and 20.3 at. pct Cr for the latter. A comparison of these figures with the recent phase diagrams pertinent to Fe–Cr system was done.

Direct comparison of surface crystal growth kinetics in chalcogenide glass measured by microscopy and DSC

AbstractSurface crystallization in fine powder Se70Te30 chalcogenide glass was studied by differential scanning calorimetry (DSC) and optical microscopy. A complex kinetic analysis of these experimental data reveals that the contracting sphere mechanism (R3 model) is the rate determining step of crystal growth, and the conventional Johnson–Mehl–Avrami–Kolmogorov model cannot be used in this case. Moreover, it is clearly shown that the particle size distribution should be considered in crystallization studies. Actually, when the particle size effect is taken into account, the simulated DSC curves for the R3 model agree very well with the experimental data over the entire temperature range. The crystallization kinetics determined from the nonisothermal DSC data are consistent with previously reported isothermal crystallization data for the same powder fraction. The crystal growth rate calculated from isothermal and nonisothermal DSC data agrees very well with the microscopically measured surface and bulk crystal growth rate.

Short-range order patterns in Mg66Zn29Ca5 metallic glass: Phase transformations and crystallization kinetics

Thanks to outstanding mechanical and chemical properties, metallic glasses have been recently alluring for biomedical applications. However, inadequate knowledge of atomic rearrangements as well as kinetics and mechanisms of crystallization prevents further microstructure improvement controlling the properties. In this work, we reveal that hierarchical transformations happen during the crystallization of Mg66Zn29Ca5 metallic glass ribbons which were investigated using differential scanning calorimetry (DSC), electron microscopy (SEM and TEM), and X-ray diffraction (XRD). The activation energies corresponding to the crystallization of the glassy ribbons were evaluated by different thermodynamic models, i.e., Kissinger, and Augis and Bennett analysis. Our findings prove the formation of short-range order patterns in Mg66Zn29Ca5 metallic glass with a unit size of ∼ 0.247 nm. Furthermore, the applied heating rate affects the glass transformation only slightly but accelerates the growth process quickly. The phase transformations occurring in annealed samples were studied by XRD, proving that crystallization starts with the formation of α-Mg + metastable Mg51Zn20. The metastable Mg51Zn20 gradually transfers to Mg7Zn3 with increasing temperature, and also Ca2Mg5Zn13 precipitates. Close to the end of crystallization, Ca2Mg6Zn3 precipitates by consuming Ca2Mg5Zn13. The Johnson–Mehl–Avrami–Kolmogorov (JMAK) model was employed to obtain the Avrami exponent of the crystallization reaction. The average slope of the Avrami plots is close to 2, indicating that crystallization progresses with diffusional growth and decreasing nucleation rate.

Precipitation kinetics and mechanical properties of Mg–Y–Zn and Mg–Y–Ni alloys containing long-period stacking ordered (LPSO) structures

Microstructures, mechanical properties by tensile deformation, and precipitation kinetics during elevated-temperature homogenization/aging heat treatment of Mg95.5Y3Zn1.5 and Mg95.5Y3Ni1.5 (at.%) magnesium alloys were systematically compared in the present work. The microstructure of both alloys in the as-cast state consisted of α-Mg and 18R long-period stacking ordered (LPSO) phase. For peak aging at 473 K (maximum hardness), the 18R phase in Mg95.5Y3Zn1.5 alloy partly transformed to lamellar 14H-LPSO phase, and hence, the ultimate tensile strength (UTS) and total elongation increased from 160.6 MPa to 2.6% to 210.5 MPa and 8.5%, respectively. After similar aging treatment of the Mg95.5Y3Ni1.5 alloy, the βʹ-Mg7Y particles precipitated in the structure, and as a result of precipitation hardening effect, the UTS and total elongation increased from 143.4 MPa to 1.9% to 201.6 MPa and 9.2%, respectively. It was observed that the formation of LPSO and precipitating phases during thermomechanical processing might lead to an improvement of strength-ductility synergy, for which overaging at hot temperatures might also be considered for improving plasticity and work-hardening behavior. The investigation of the formation kinetics of 14H-LPSO in Mg95.5Y3Zn1.5 alloy and βʹ-Mg7Y in Mg95.5Y3Ni1.5 alloy based on the Johnson–Mehl–Avrami–Kolmogorov (JMAK) analysis led to the activation energy (Q) of 104 kJ/mol and 89 kJ/mol, respectively. The former is similar to that of the diffusion of zinc in the matrix and the latter is close to that for the diffusion of yttrium in the matrix. Accordingly, the present work can shed light on the phase transformations and strengthening mechanisms of high-strength LPSO-containing Mg alloys.

Crystallization kinetics analysis of Ge18BixSe82-x chalcogenide glasses alloys by using non-isothermal methods

The melt quench method is used to prepare the chalcogenide glass Ge18BixSe82-x (x: 0.5, 2, 4, 6, and 8 at percent). The samples are characterized using X-ray diffraction (XRD) and a scan electron microscope (SEM). The kinetics of crystallization of the chalcogenide glasses under consideration are investigated by using Differential Thermal Analysis (DTA) under non-isothermal conditions. Different models are used to calculate the activation energy of crystallization (Ec). The Kissenger-Akahira-Sunose (KAS) method is used to calculate the change of activation energy with crystallization volume fraction (Ecα). As well, the activation energy of glass transitions is measured by using different models. The results of fitting the experimental DTA data to the computed DTA curves show that the Johnson-Mehl-Avrami (JMA) model cannot represent the crystallization process of Ge18BixSe82-x glass. Our findings show that the Sestak-Berggren (SB) model is better suited to describe the crystallization process for the glass under investigation.

Influence of powder layer thickness on microstructure and T5 heat treatability of F357 alloy fabricated by laser powder bed fusion process

Laser powder bed fusion (LPBF) is one of the most widely used additive manufacturing (AM) methods to produce metal parts with new functionalities, but it still suffers from a low built rate. Increasing the powder layer thickness is an efficient way to augment LPBF productivity. The objective of this study is to investigate the influence of the powder layer thickness during the LPBF of F357 alloy on the microstructure and age-hardening response during T5 heat treatment. Scanning electron microscope (SEM) and transmission electron microscope (TEM) observations show that microstructure differences are negligible between samples printed at 30 µm and 50 µm layer thicknesses in the as-built condition. Samples built with both layer thicknesses and subjected to T5 heat treatments at 150 °C, 160 °C, and 170 °C also show similar microstructures and their α-Al cells contain large Si precipitates and small Si clusters and no β’’-Mg2Si precipitates. Therefore, β’’-Mg2Si precipitation hardening is not the dominant mechanism, and strengthening appears to be mainly due to the small Si clusters. With the help of the Johnson-Mehl-Avrami (JMA) model, the activation energy of the precipitation process of the samples built at 30 and 50 µm layer thicknesses is calculated to be 142 kJ/mol and 139 kJ/mol, respectively. These comparable values suggest a similar evolution of the precipitation hardening for both layer thicknesses, which is consistent with the observed microstructures. Consequently, the production efficiency of LPBF-printed F357 alloys can be enhanced by using higher layer thickness and the T5 heat treatment.

Shear strain rate sensitivity and crystallisation kinetics investigation in melt spun Cu64Zr36 binary metallic glass

Abstract Shear strain rate effect on crystallisation behaviour and characteristic temperatures of the three well-known glass formers in Cu–Zr binary amorphous system, namely Cu50Zr50, Cu56Zr44 and Cu64Zr36, were investigated. The crystallisation behaviour of Cu64Zr36 was uniquely found to be heavily dependent on shear strain rate. Crystallisation kinetics were studied through Isochronal transformation and isothermal transformation. The activation energy of crystallisation of each case was compared and contrasted. Johnson-Mehl-Avrami (JMA) analyses were also employed to study its kinetics behaviour. Finally, high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) combined with laser-assisted local electrode atom probe (LEAP) investigation revealed no detectable clustering or phase separation.

Atomistic study on reaction kinetics and reactivity of Ni/Al clad particles composites under shock loading.

In prior research on shock-induced reaction, the interfacial crystallization of intermetallics, which plays an important role in solid-state reaction kinetics, has not been explored in detail. This work comprehensively investigates the reaction kinetics and reactivity of Ni/Al clad particle composites under shock loading with molecular dynamics simulations. It is found that the reaction acceleration in a small particle system or the reaction propagation in a large particle system breaks down the heterogeneous nucleation and continuous growth of B2 phase at the Ni/Al interface. This makes the generation and dissolution of B2-NiAl show a staged pattern consistent with chemical evolution. Importantly, the crystallization processes are appropriately described by the well-established Johnson-Mehl-Avrami kinetics model. With the increase in Al particle size, the maximum crystallinity and growth rate of B2 phase decrease and the value of the fitted Avrami exponent decreases from 0.55 to 0.39, showing a good agreement with the solid-state reaction experiment. In addition, the calculations of reactivity reveal that the reaction initiation and propagation will be retarded, but the adiabatic reaction temperature can be elevated when Al particle size increases. An exponential decay relationship is found between the propagation velocity of the chemical front and the particle size. As expected, the shock simulations at non-ambient conditions indicate that elevating the initial temperature significantly enhances the reactivity of large particle systems and results in a power-law decrease in the ignition delay time and a linear-law increase in the propagation velocity.

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.

Promotion of thermomechanical processing of 2-GPa low-alloyed ultrahigh-strength steel and physically based modelling of the deformation behaviour

A low-alloyed ultrahigh-strength steel comprising CrNiMoWMnV was designed based on thermodynamic calculations and by controlling the microalloying elements to promote various strengthening mechanisms upon processing. The hot deformation behaviour and mechanism were correlated with the processing parameters, that is, strain rate and temperature. The fine features of the deformed microstructures were analysed using electron backscatter diffraction (EBSD) and MATLAB software, combined with the MTEX texture and crystallographic analysis toolbox. The flow stress behaviour at high temperatures was modelled using the dislocation density-based Bergström's model, which could be applied up to the peak strain. However, the diffusional transformation (i.e. recrystallisation)-based Kolmogorov–Johnson–Mehl–Avrami model has been applied to fit the flow stress over a wide deformation strain. The effective grain size (EGS) of martensite and prior austenite grain size (PAGS) were correlated with the deformation temperature and strain rate. Because the PAGS was significantly refined from 16 μm in the initial microstructure to 6 μm after processing at 850 °C/0.01 s−1, the corresponding martensite EGSs were 1.38 and 1.01 μm, respectively. Therefore, these fine-controlled characteristics of the processed microstructures at high temperatures help to enhance the mechanical properties, such as the strength and toughness, of the designed ultrahigh-strength steel.

Kinetics of gold-induced layer exchange crystallization of amorphous silicon suboxide films: Experimental and theoretical study

The kinetics of gold-induced crystallization (AuIC) of amorphous silicon suboxide (a-SiOx, x = 0.2) films in the layer exchange mode was detailed studied by variation of the annealing temperature in the range of 220–250 °C for the first time. In situ optical microscopy performed during annealing made it possible to study the processes of the nucleation and the growth of Si crystal structures. A decrease in the nucleation rate during annealing was shown, which is probably associated with coarsening of grains in the Au film which is confirmed by scanning electron microscopy. Obtained experimental data along with the simulations carried out within the framework of the Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory made it possible for the first time to obtain the values of the activation energies for the nucleation, the growth, as well as the overall process of AuIC of a-SiO0.2 which amounted to 2.51 ± 0.80, 0.31 ± 0.10, and 1.62 ± 0.15 eV, respectively.

Crystallization of LiNbO3 and NaNbO3 in niobiosilicate glass–ceramics

AbstractAlthough glass–ceramics have been widely explored for their thermal stability and mechanical properties, they also offer unique symmetry‐dependent properties such as piezoelectricity and pyroelectricity through controlled crystallization of a polar phase. This work examines crystallization of LiNbO3 in a 35SiO2–30Nb2O5–35Li2O mol% composition and crystallization of LiNbO3 and NaNbO3 in a 35SiO2–30Nb2O5–25Li2O–10Na2O mol% composition. Crystallization kinetics are examined using the Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory where the Avrami exponent, n, is calculated to be 1.0–1.5. Microscopical analysis shows dendritic morphology, which when combined with the JMAK analysis, suggests diffusion‐controlled one‐dimensional growth. Adding Na2O to the glass composition increases the inter‐diffusivity of ions which causes LiNbO3 to crystallize faster and lowers the activation energy of transformation from 1054 ± 217 kJ/mol in the ternary composition to 882 ± 212 kJ/mol. Time‐temperature‐transformation diagrams are presented which show that the temperature for maximum rate of transformation for LiNbO3 is ∼650°C and for NaNbO3 is ∼715°C.

Some novel perspectives of iso-conversional analysis in the study of Meyer–Neldel energy for thermally governed crystallization by using Johnson–Mehl–Avrami (JMA) theory

Some novel perspectives of iso-conversional analysis in the study of Meyer–Neldel energy for thermally governed crystallization by using Johnson–Mehl–Avrami (JMA) theory

Growth Mechanism of High-Quality hBN Monolayers on Cu through Chemical Vapor Deposition with Inductively Coupled Plasma

The growth mechanism associated with the nucleation and growth kinetics of hexagonal boron nitride (hBN) on Cu through chemical vapor deposition (CVD) has been studied. A fully covered hBN film with a large grain size has been obtained by lowering the density of the grain boundary during the thermal CVD. The Cu substrate is first annealed under the H2 ambiance in order to efficiently reduce the copper oxide on the surface and promote the Cu surface with the (111) plane. Next, the nucleation and growth kinetics of CVD hBN are investigated by the analysis of scanning electron microscopy (SEM) images and the Johnson–Mehl–Avrami–Kolmogorov model. The hBN growth is overall dominated by nucleation under a low flow rate of H2, while it is initially dominated by grain growth under a high flow rate of H2 and the grain size of hBN can be larger than 25 μm. The competition between nucleation and grain growth is reversed at higher temperatures, resulting from the wavier Cu surface. Furthermore, the hBN film is grown by the inductively coupled plasma-enhanced CVD (ICP-CVD) and its growth mechanism is compared with that of CVD hBN. The growth rate of ICP-CVD hBN is 50 times faster than that of CVD hBN due to more energetic B and N species caused by plasma. The growth mechanism of ICP-CVD hBN without a long incubation time is dominated by nucleation.

Effects of Pressure on Homogeneous Nucleation and Growth during Isothermal Solidification in Pure Al: A Molecular Dynamics Simulation Study

Effects of different pressures on the isothermal-solidification process of pure Al were studied by molecular dynamics (MD) simulation using the embedded-atom method (EAM). Al was first subjected to a rapid-cooling process, and then it was annealed under different pressures conditions. Mean first-passage times (MFPT) method, Johnson-Mehl-Avrami (JMA) law, and X-ray diffraction (XRD) simulation analysis method were used to qualify the solidification- kinetic processing. Nucleation rate, critical-nucleus size, Avrami exponent, growth exponent, and crystallite size were calculated. Results show that the nucleation rate increases as the pressure increases. The change of critical-nucleation size is not obvious as the pressure increases. With the pressure increasing, growth exponent decreases, indicative of decreased grain-growth rate. It was also found that with the pressure increasing, the Avrami exponent decreases, indicating that the increased pressure has an effect on growth modes during solidification, which changes from three-dimensional growth to one-dimensional growth. Results of XRD simulation shows that with pressure increasing, crystallite size decreases.

Effect of Cr on Isothermal Structure Transformation of Iron Oxide Scale

Herein, the effect of different contents of Cr on the isothermal structure transformation of FeO is systematically studied by a thermogravimetric analyzer, and it is found that the temperature range of eutectoid transformation of FeO in Fe–Cr alloy is 300–500 °C. Combined with the experimental data, the kinetic model of isothermal phase transformation of FeO is established based on Johnson–Mehl–Avrami–Kolmogorov equation, and the time–temperature–transformation curve of isothermal transformation of FeO is predicted, which follows the law of C curve, and the C curve shifts backward with the increase of Cr content. Therm‐calc thermodynamic software calculates the Fe–O phase diagrams of different Cr contents. Combined with the experimental results, the effect mechanism of Cr content on FeO eutectoid transformation is clarified. With the increase of Cr content, the stability of FeO decreases, and under the same isothermal temperature and time, high Cr content has greater undercooling, a higher driving force of phase transition, and the starting time of eutectoid transformation is earlier than that of lower Cr content.

Crystal nucleation in Au49Ag5.5Pd2.3Cu26.9Si16.3 glass and undercooled melt

Crystal nucleation is an indispensable step in the crystallization of metallic glasses. With the increase of undercooling, different nucleation kinetics from that predicted by the classical nucleation theory (CNT) could be revealed. A prerequisite to addressing the nucleation kinetics is building a well-identified amorphous structure, which is yet arduous. Moreover, due to the limited temporal and spatial resolution of the conventional experimental methods, nucleation during the crystallization of metallic glass cannot be well recognized. In this study, nanocalorimetry with ultrahigh sensitivity and ultrafast scanning rate is used to address the nucleation kinetics in glass and undercooled melt. With the increase in cooling rate, crystallization, homogeneous nucleation, and heterogeneous nucleation are successively distinguished. Accordingly, the critical cooling rates suppressing crystallization and nucleation are estimated, based on which a well-identified amorphous phase for nucleation studies is produced. By applying Tammann’s two-stage crystal nuclei development method, nucleation in glass and undercooled melt is investigated. According to the evolution of crystallization heat and overall heat on the reheating after the annealing, the underlying kinetic mechanism is revealed with the Johnson-Mehl-Avrami (JMA) method. The nucleation half-time indicates that homogeneous nucleation is dominant for the crystallization below Tg. For the annealing at and above Tg, crystallization completes by the inevitable heterogeneous nucleation followed by crystal growth.

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.