A Review of Different Models Derived from Classical Kolmogorov, Johnson and Mehl, and Avrami (KJMA) Theory to Recover Physical Meaning in Solid-State Transformations

A Review of Different Models Derived from Classical Kolmogorov, Johnson and Mehl, and Avrami (KJMA) Theory to Recover Physical Meaning in Solid-State Transformations Javier S. Blázquez,* Francisco J. Romero, Clara F. Conde, and Alejandro Conde 1. Introduction The classical KJMA theory of transformation in solid state assumes nucleation and growth processes which consider the geometrical impingement between different growing regions. This idea is summarized in the well-known Kolmogorov, [1] Johnson and Mehl, [2] and Avrami [3] (KJMA or JMAK) equation X¼1expðK·ðtt0ÞnÞ(1) where Xis the transformed fraction, Kis a prefactor (in order to consider a dimensionally correct frequency factor, K¼kncan be used instead), t0is the induction time, and nis the Avrami exponent. Kinetic data are generally analyzed in the frame of Equation (1) using the so-called KJMA-plot which represents lnð lnð1XÞÞ as a function of lnðtt0Þ. The slope of KJMA-plot is the Avrami exponent. It is worth mentioning that Equation (1) can be obtained after the useful definition of the extended transformed volume, X. This magnitude corresponds to the fraction transformed but neglecting the overlapping between growing regions. On the one hand, Xhas no direct physical meaning as it reaches values above 1 and, in fact, it never saturates. On the other hand, X can be easily calculated when the nucleation and growth laws are known. In 1937, Kolmogorov mathematically developed the following equation correlating the actual and the extended transformed fractions for transformations in which randomly distributed nuclei geometrically compete for the available space in the formation of a new product phase. [4] X¼1expðXÞ(2) Kolmogorov found Equation (1) as a particular solution to Equation (2) in two cases [4] : 1) constant nucleation rate and constant linear growth rate (with n¼4); 2) instantaneous nucleation and constant linear growth rate (with n¼3). Independently from Kolmogorov, in 1939 Johnson and Mehl were able to describe the pearlite formation from austenite by using Equation (1) with n¼4. [2] From then, and soon later further developed by Avrami, KJMA theory has been widely used in materials science [5–9] and in many other fields of research including chemical reactions, [10–14] medicine, [15–17] biology, [18–22] genetics, [23–25] food research, [26–30] sociology, [31] etc. Figure 1 shows the distribution of research works containing “Avrami” in the different subject areas and years since 1960. J. S. Blázquez, F. J. Romero, C. F. Conde, A. Conde Dpto. Física de la Materia Condensada ICMSE-CSIC Universidad de Sevilla P.O. Box 1065, 41080 Sevilla, Spain E-mail: [email protected] The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/pssb.202100524. © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution-NonCommercialNoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. DOI: 10.1002/pssb.202100524 The classical theory of solid-state transformation based on nucleation and growth processes, developed by Kolmogorov, Johnson and Mehl, and Avrami (KJMA theory), is widely used in many different fields of research. In KJMA theory, two parameters (the frequency factor and, particularly, the Avrami exponent) can supply information about the mechanisms involved in the transformation. Despite its apparent simplicity, on the one hand, the results derived from this theory can be strongly affected by the indetermination of experimental data (e.g., onset of the transformation). On the other hand, KJMA theory is developed for isothermal polymorphic transformations in which randomly distributed nuclei grow in convex shapes. However, several procedures have extended KJMA theory to nonisothermal regimes and to many different processes deviating from those premises. Herein, the requirements of KJMA theory and the expected deviations for these approximations are briefly discussed. In addition, some strategies are proposed for recovering physical meaning from the effective parameters deduced in several transformations including nanocrystallization and martensitic transformations for which results can be interpreted under the approximation of instantaneous growth. REVIEW 60 years of pss www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (1 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH Therefore, the use of KJMA analysis has been extended beyond its rigorous limits. However, despite this lack of rigorousness, deviations from KJMA requisites can be negligible allowing for an effective interpretation of the mechanisms involved in the transformation process. The aim of the present article is to show some examples in which an effective KJMA analysis still preserves some physical information of the transformation process, although the strict conditions are somehow not fulfilled. The article is structured as follows: Section 1 presents KJMA equation and its wide use in a broad range of scientificareas. In the next section, the requisites of KJMA equation, summarized in the five postulates of Kolmogorov as collected by Burbelko et al., [4] are briefly described and negligibility of certain deviations is discussed. Section 3 shortly describes the development of the extended transformed fraction from known laws of nucleation and growth. Section 4 describes the important role of accurate determination of experimental data such as the induction time and the final transformed fraction even for ideal KJMA processes. Section 5 reviews some extensions of KJMA theory to nonisothermal regimes. Melting of pure In is described as an example. Section 6 describes some of our proposed strategies to afford the results from effective KJMA analysis applied to systems for which deviations from some of the postulates of Kolmogorov are no longer negligible. Some simple simulations are performed to evaluate the importance of deviations of ideal systems from the strict requirements of KJMA theory, including diffusion controlled growth and instantaneous growth approximation. Crystallization of amorphous alloys, mechanical amorphization of ball-milled powders, and martensitic transformation are described as examples. Finally, some conclusions are briefly summarized. 2. Requirements to Obtain the KJMA Equation Burbelko et al. [4] summarized the requisites for KJMA theory to be valid in the five postulates of Kolmogorov: 1) The initial parent phase is progressively and completely replaced by a product phase. 2) The volume of any transformed region is tiny with respect to the whole volume of the system. 3) Nucleation is random. 4) The shape of the growing phase is convex. 5) The linear growth rate can be expressed as a product of a timedependent function and a direction-dependent function. To require a progressive and complete replacement of the initial phase by the product phase would restrict the use of KJMA theory to none but polymorphic transformations. However, KJMA theory is broadly applied beyond this limit as we will show in Section 6. Postulates number 2 and 3 are not very restrictive. In the first case, we are considering the formation of polycrystalline systems and assuming an infinite limit for the space to be transformed in comparison with the size of an individual crystal. In the third postulate, we are not restricted to homogeneous nucleation but the heterogeneous nucleation centers (defects, grain boundaries, etc.) must be randomly distributed. Concerning postulates number 4 and 5, they can be reformulated following the requisites appearing in the review article from Korobov [32] on KJMA theory in which it is pointed that all nuclei must have a common shape and orientation, and that the growth rate of a nucleus must not depend on its growing time. Both conditions prevent overestimation of transformed fraction avoiding the overgrowth contribution from phantom nuclei, which are those contributing to Xbut formed in an already transformed region. When these requisites are fulfilled, all the contribution to the extended volume corresponding to the growth of a phantom nucleus will overlap with the already transformed region of the parent crystal where the phantom nucleus appears. Therefore, KJMA equation is valid to describe the actual transformed fraction. However, when linear growth rate decreases with the growing time, a phantom nucleus (close enough to the interface of the transformed region into which it appeared) can overcome the limit of this transformed region as the boundary of the former moves faster than the latter, leading to an overgrowth region (see Figure 2a). This is the case of diffusion controlled growth, for which the linear growth rate of a crystal decreases with the growing time as ∝ðtτÞ0.5, where tis the observation time and τis the nucleation time. In Section 6, we will show some simple simulations to account for the importance of the effect of overgrowth regions. Moreover, when anisotropic growth occurs and orientation is not the same for all the crystallites, nonoverlapped regions of the extended transformed fractions can be physically banned (schematic representations can be found in Figure 2b,c). These problems were afforded in two dimensions by Tomellini and Fanfoni [33] calculating the fraction transformed without the contribution of the phantom nuclei. On the other hand, Kooi, [34] using Monte Carlo simulations, modifies the extended transformed fraction and finds the time at which the transformation deviates from normal KJMA to a blocking regime in systems for which dimensionality of growth is smaller than that of the corresponding space. 3. Development of KJMA Equation Extended transformed fraction, X, can be easily calculated after adding the expected volume of each transformed region Materials Science Physics Chemistry Engineering Crystallography Polymer Science Metallurgy Met. Engineering Science Technology Other Topics Thermodynamics Spectroscopy Mechanics Instruments/Instrumentation Biochemistry Molecular Biology Energy Fuels Mathematics Food Science Technology Environmental Sciences Ecology Agriculture Electrochemistry Business Economics Nutrition Dietetics Mathematical Comput. Biology Pharmacology Pharmacy Optics Toxicology OTHERS 0 1000 2000 3000 4000 5000 number of papers 1960 1970 1980 1990 2000 2010 2020 0 100 200 300 400 500 Figure 1. Distribution of research papers containing “Avrami”as a function of the area of research (bars, left axis) and year of publication (symbols, right axis). Data obtained from Web of Science (September 20, 2021). www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (2 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH whether it was nucleated at a certain time or it existed before the transformation started XðtÞ¼Zt 0 IðτÞVðτ,tÞdτþZ ∞ 0 dN0 dr0 Vðr0,tÞdr0(3) The first integral depends on the nucleation rate per unit volume IðtÞ¼dNðtÞ=dt(being NðtÞthe number of nuclei per unit volume) and corresponds to the contribution to XðtÞat the observation time tof those regions nucleated at time τ(0 <τ<t) which should have grown (neglecting any impingement) to a volume Vðτ,tÞ¼4π 3"rðτÞþZt τ uðt0,TÞdt0#3 (4) For simplicity, spherical regions are assumed and r¼2γSLTE ΔT·ΔH is the critical radius for a nucleus to be stable, [35,36] with γSL is the surface energy, TEis the equilibrium temperature between the initial and final phases, ΔT¼TTEis the thermal span from the equilibrium temperature, and ΔHis the latent heat ascribed to the transformation. Moreover, uðt0,TÞis the linear growth rate at time t0and depends on temperature T. The second integral corresponds to the contribution to XðtÞ of the growth of dN0preexisting regions per unit volume of the product phases, with an initial radius between r0and r0þdr0,to a volume: Vðr0,tÞ. Vðr0,tÞ¼4π 3"r0þZt τ uðt0,TÞdt0#3 (5) Equation (4) and (5) are valid for spherical crystals and, in general, the size of the nuclei is small enough and can be neglected. KJMA theory is a global theory of the transformation and many characteristics of growth processes (e.g., cooperative movement of atoms in coherent interfaces) are overseen in the KJMA model. However, anisotropic growth can be considered but, as described above, is only valid when shape and orientation are the same for all the crystals, preventing overgrowth regions. [7,32,34] For example, particularizing to an ellipsoidal crystal, in the case of Equation (4) Vðτ,tÞ¼4π 3"Zt τ uxðt0,TÞdt0#"Zt τ uyðt0,TÞdt0#"Zt τ uzðt0,TÞdt0# (6A) where ux,uy, and uzare the linear growth on the corresponding axes of the ellipsoid that should be common for all crystallites. A general approach is to consider absence of growth in one or two dimensions to represent the formation of needle-like crystals (1D growth) or platelets (2D growth). Several authors have studied this limitation in the growth dimensionality. Kooi [34] using Monte Carlo simulations studied the formation of 1D crystals in 2D and 3D spaces, founding that KJMA describes the process at the initial state of the transformation. In that article, a well-defined transition time was obtained after which the transformation progresses in a blocking regime. Following the interpretation of Korobov, [32] and assuming that the shape and orientation of the crystals are common, Equation (6A) can be approached to Vðτ,tÞ¼π 4D2"Zt τ uðt0,TÞdt0#(6B) for needle-like crystals of diameter D, and Vðτ,tÞ¼πλ"Zt τ uðt0,TÞdt0#2 (6C) for thin discs of thickness λ. Similar equations could be obtained for Vðr0,tÞ, neglecting the initial radius of those preexisting regions. Therefore, Equation (3) can be explicitly stated as XðtÞ¼Zt 0 IðτÞCI"Zt τ uðt0,TÞdt0#dI dτþN0C0"Zt τ uðt0,TÞdt0#d0 (7) where dIand d0are the dimension of growth for the new nucleated regions and for the preexisting ones, respectively, Figure 2. Schematic pictures showing that: a) decreasing growth rate with growing time, as it occurs in diffusion controlled processes, may lead to the appearing of overgrowth regions (red area). b) Anisotropic growth can also lead to overgrowth regions when orientation and shape is not share by all the crystallites. [7,32,34] c) When orientation and shape are the same for all the crystallites, overgrowth does not occur. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (3 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH and CIand C0are geometrical factors with dimension of length at the power ð3dIÞand ð3d0Þ, respectively. Concerning nucleation phenomenon, two simple cases already analyzed by Kolmogorov are considered when analyzing in the frame of KJMA theory. On the one hand, assuming constant nucleation rate, neglecting the presence of already transformed regions, Equation (7) becomes XðtÞ¼IðTÞZt 0 CI"Zt τ uðt0,TÞdt0#dI dτ(8) but only in the case of isothermal conditions, as nucleation rate, and generally the transformation rate, [37] strongly depends on temperature following an Arrhenius law IðTÞ¼I0exp ðQþWÞ kBT  (9) where Qis the activation energy for an atom to be incorporated to the new phase and Wis the work needed to form a stable nucleus. [35] On the other hand, in the case of neglecting the formation of new nuclei and assuming that the transformation is only due to the growth of preexisting nuclei, the second addition term in Equation (7) is the only one to be considered. Concerning growth phenomena, analysis in the frame of KJMA theory considers two extreme cases: interface controlled growth, when diffusion of heat and atoms is fast enough compared to the time required for an atom to jump into the product phase; and diffusion controlled growth, when the progress of the transformation is governed by this mechanism. The latter mechanism, as described above, is not strictly valid in KJMA frame as it leads to overgrowth of phantom nuclei (see Figure 2a). In fact, KJMA theory accounts for the phantom nuclei but when linear growth decreases as the crystal grows (as it occurs in diffusion controlled growth) this implies that a phantom nucleus, which is formed inside an already transformed region, grows faster than its host region, and can go beyond its limit. We will show in Section 6 that deviations due to this overgrowth are expected to be negligible and, in agreement with other authors, [38,39] KJMA theory can be used as a fair approach to describe diffusion controlled growth processes. On the one hand, assuming an interface controlled growth leads to time independent linear growth rate uðTÞ, which implies a volume of the growing particle Vðτ,tÞ¼CI"Zt τ uðTÞdt0#d ¼K1ðTÞðtτÞd(10) As the linear growth rate depends on temperature, only for isothermal transformations it should be properly written the final identity with K1ðTÞ¼CIuðTÞ½ d. An Arrhenius law can be approximated to the linear growth in the case of interface controlled growth [37] uðTÞ¼u0exp Q kBT  (11) which is constant for a constant temperature. On the other hand, a diffusion controlled growth is characterized by a linear growth rate that depends on the observed time, t, the time the nucleus was formed, τ, and the temperature, T uðT,t,τÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðTÞ ðtτÞ s(12) where DðTÞis the diffusion coefficient, and the volume of the growing particle (in isothermal conditions) would be Vðτ,tÞ¼CI2 6 4Zt τffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðTÞ t0τ rdt03 7 5 d ¼K2ðTÞðtτÞd=2(13) In this case, the temperature-dependent prefactor is K2ðTÞ¼2CIffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðTÞ p, with an Arrhenius law describing the temperature dependence of the diffusion coefficient. Therefore, in isothermal conditions and taking into account the different possibilities described above, the extended volume fraction can be expressed as XðtÞ¼KðTÞðtt0Þn(14) where Avrami exponent n¼nIþd·nG, with nI¼1 for constant nucleation process or nI¼0 for absence of nucleation, and nG¼1 for interface controlled growth or nG¼0.5 for diffusion controlled growth. Restricted possibilities with physical meaning (0.5 ≤n≤4) are collected in Table 1. As it is observed in this table, ambiguity can be found for some values of nand, Table 1. Possible physical meaningful values of Avrami exponent nin the frame of KJMA theory. Avrami exponent n Nucleation exponent nI Growth exponent nG Dimension of growth d 0.5 No nucleation Diffusion controlled 1 1 Constant nucleation rate No growth 0 No nucleation Interface controlled 1 No nucleation Diffusion controlled 2 1.5 Constant nucleation rate Diffusion controlled a) 1 No nucleation Diffusion controlled 3 2 Constant nucleation rate Interface controlled 1 Constant nucleation rate Diffusion controlled a) 2 No nucleation Interface controlled 2 2.5 Constant nucleation rate Diffusion controlled a) 3 3 Constant nucleation rate Interface controlled 2 No nucleation Interface controlled 3 4 Constant nucleation rate Interface controlled 3 a) Diffusion controlled processes for which nucleation is extended along the transformation are affected by overgrowth of phantom nuclei. The Avrami exponent would correspond to a maximum limit when this overgrowth is negligible. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (4 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH therefore, it is important to combine this kinetic analysis with microstructural observations to elucidate which are the mechanisms involved in the transformation. In a semiquantitative way, values of nI<1 are ascribed to a decreasing nucleation rate and nI>1 to an increasing nucleation rate. This can be implemented in Equation (7) assuming a power expression for the number of nuclei as a function of time, [34] leading to a nucleation rate of the form: IðtÞ¼tawith 1<a<0 for a decreasing nucleation rate process and a>0 for an increasing nucleation rate process. The relationship between actual transformed fraction, X, and the extended one, X, is obtained statistically as [40] dX dX¼ð1XÞη(15) In KJMA theory, the impingement exponent η¼1 but other kinetic theories are developed assuming a stronger impingement exponent. For example, Austin–Rickett model [41] can be developed using η¼2. Tagami and Tanaka [42] proposed an intermediate value of ηdefining the overlap factor γ¼2η to describe nucleation and halt in growth processes. These are processes in which the crystals grow to a fixed size. The overlap factor was developed to account for the phantom crystals from phantom nuclei which should be accounted in X. The value of γ¼0.75 corresponds to needle-like crystals, γ¼0.5865 to planar disc-like crystals, and γ¼0.48675 to spherical crystals. Starink [40] also proposed a variable impingement parameter to approximate several deviations from KJMA requirements. The relation dX=dX¼ð1XÞbetween the extended and actual transformed fractions described by Equation (15) with η¼1 was obtained mathematically by Kolmogorov under rigorous requirements (as we qualitatively collected in Section 2). However, as an approximation, it can be understood as the simplest relation between two magnitudes being equal when they are close to zero but describing the saturation behavior of Xat 1 whereas Xnever saturates. This may explain why KJMA kinetics is a fairly good approximation even when the systems depart from its strict requirements and why different kinetic approaches lead to similar equations to KJMA one. [32] Integration of Equation (15) leads to ZX 0 dX ð1XÞη¼ZX 0 dX!8 < : η¼1!lnð1XÞ¼X η6¼ 1!1ð1XÞ1η 1η¼X(16) Substitution of Equation (14) in (16) for η¼1 leads to Equation (1). Therefore, experimentally, we will consider the value of lnð1XÞ¼Xand, in the following, we will discuss on the deviations of the different models from theoretical X. Therefore, the KJMA-plot represents lnðlnð1XÞÞ ¼ lnðXÞ versus lnðtt0Þ. 4. Effects of Indetermination of Experimental Data in KJMA Analysis 4.1. Effects of Indetermination in the Induction Time In order to apply Equation (1), we need to obtain experimental data on transformed fraction Xas a function of the time of transformation t. In this sense it is very important to take into account that we also need an accurate estimation of the induction time, t0. As we will see below, a wrong estimation of the induction time leads to important deviation effects. Figure 3 shows how the effective Avrami exponent becomes reduced when experimental t0delays with respect to the theoretical value at which the transformation starts (i.e., Xðt0Þ¼0 but we normally detect the onset once the magnitude we use to follow transformation appreciably changes from its baseline value). It is particularly important to notice that the steepest slope method to obtain the onset, which is widely used, supplies a correct value only for n¼1. The induction time measured from steepest slope method, tslope, can be obtained as a function of Avrami exponent for theoretical curves, taking into account that the time at which the transformation rate is maximum corresponds to tinf ¼1 k n1 n  1=n: tslope ¼tinf Xðtinf Þ dXðtinf Þ dt (17) Results are shown in Figure 4 for the different values of n along with the transformed fractions at this induction time XðtslopeÞ, which can be 10% for high values of n. Taking into account theoretical results from Figure 3 and 4, it is important to know that this effect can lead to erroneous values. Definitely, the steepest slope method is only valid when n¼1. However, we can estimate the correct induction time, t0, from the proportionality shown in Figure 4c between time spans: α¼ðtslopet0Þ ðtinf tslopeÞ, which linearly depends on Avrami exponent: α¼ð0.578 0.009Þþ ð0.564 0.03Þn. Therefore, the correct induction time can be -6 -4 -2 0 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -6 -4 -2 0-6 -4 -2 0 n=1 X=0.1 ln(-ln(1-X)) ln(t-t 0 ) X=0.9 n=4 ln(t-t 0 ) n=2.5 ln(t-t 0 ) t correct t from steepest slope t =t(X=0.05) t =t(X=0.01) Figure 3. Effect of indetermination of induction time on the KJMA-plot for three different values of the Avrami exponent. Limit values corresponding to X¼0.1 and X¼0.9 are shown as horizontal lines. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (5 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH obtained as t0¼tslope αðtinf tslopeÞ(18) Something similar would be obtained when applying the steepest slope method to determine the induction time (or more generally, the onset of the transformation) using dX=dtcurves. In such a case, the value obtained from steepest slope is generally used as the calibration onset temperature in differential scanning calorimetry. Although KJMA assumes simple processes with constant kinetic parameters through the transformation, complexities of the actual process can be better evidenced by obtaining the so-called local Avrami exponent, first proposed by Calka and Radlinski: [43] nðXÞ¼dlnðlnð1XÞÞ dlnðtt0Þ(19) Figure 5 shows how the effective local Avrami exponent deviates from the theoretical values due to a wrong determination of t0. Qualitatively, it is observed that overestimation of t0leads to an underestimation of nand vice versa. This behavior could be used to detect errors in the determination of the onset of simple transformations. In the case of n¼1, deviations are almost negligible for X>0.2. 4.2. Effects of Indetermination in the Final Transformed Fraction in Nonpolymorphic Transformations Although first postulate of Kolmogorov assumes a complete transformation of the system, this restriction is generally overseen after normalizing the transformed fraction to 1 at the end of the considered process by dividing the actual transformed fraction xðtÞby the final one, X¼xðtÞ=xend. In fact, KJMA analysis was widely applied to primary crystallization, [44–51] precipitation, [52–58] eutectoid, [59,60] and quasicrystals formation [61] using this normalization. This yields an extra source of error in the data, as the final transformed fraction must be measured. Figure 6 shows the effect of indetermination in the final transformed fraction xend. Underestimating the final transformed fraction is not seriously affecting nbut at very high values of X. However, overestimation can lead to two slopes artifact but ncan be obtained from the linear part (for such a high value as 30% overestimation, Avrami exponent only changes 10%). In any (a) (b) (c) Figure 4. Variation with the Avrami exponent of a) the onset of the transformation estimated from the steepest slope method, tslope, b) the transformed fraction at this time, and c) the ratio between ðtslope t0Þ and ðtinf tslopeÞspans. Figure 5. Effect of the indetermination of the induction time on the effective local Avrami exponents for different values of n. -1 0 1-4 -2 0 2 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1 0 1 ln(t-t 0 ) n=2.5 ))X-1(nl-(nl X=0.1 ln(t-t 0 ) X=0.9 X=x/xend n=1 ln(t-t 0 ) x correct overestimated x 10 % x 20 % x 30 % underestimated x 10 % x 20 % x 30 % n=4 Figure 6. Effect of indetermination in the final transformed fraction on KJMA-plots for different values of n. Limit values corresponding to X¼ 0.1 and X¼0.9 are shown as horizontal lines. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (6 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH case, it is important to be aware of using Xdata in a correct range, avoiding the extreme values. When compositions of initial and final phases are not the same, a side effect can be derived from the compositional change of the environment of the growing crystal. This may affect the growth rate and even lead to soft impingement, [39,40,51,62,63] which is generally used to distinguish between the pure geometrical impingement considered in KJMA and that produced by the overlapping of concentration gradients. 5. Extensions of KJMA Theory to Nonisothermal Conditions As described above, KJMA theory is strictly valid for isothermal conditions. However, the advantages of constant heating rate experiments [37] made the extension of KJMA to nonisothermal regimes a desired goal since earlier times. [5,6,64–68] In this section, we will briefly discuss on the consequences of a constant heating rate regime for a hypothetical process following KJMA kinetics. We will consider the case with no preexistent regions (no quenched in nuclei) in nonisothermal conditions. In this case, neither IðTÞnor uðt0,TÞcan be extracted from the integral in Equation (7) and thus, for nonisothermal cases, using Equation (8), (11), and (12), it is possible to write XðTÞ¼CII0Zt 0 exp QI kBT  2 4Zt τ u0exp QG kBT  dt03 5 d dτ (20A) for interface controlled growth, and XðTÞ¼CII0Zt 0 exp QI kBT  2 4Zt τffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D0 ðt0τÞ sexp QG kBT  dt03 5 d dτ (20B) for diffusion controlled growth. In both cases we will assume that I0,D0, and u0as well as the activacion energies do not change during the transformation due to, for example, neglected temperature dependence in our approximations, compositional changes of the matrix, or activation of new mechanisms of nucleation. Assuming a constant heating/cooling rate β¼dT=dt, some further simplification can be obtained: XðTÞ¼CII0u0dZT T0 exp QI kBθ  ZT θ exp QG kBT0  dT0 β 2 43 5 d dθ β (21A) for interface controlled growth, and XðTÞ¼CII0ð2D0Þd 2ZT T0 exp QI kBθ  ZT θffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðT0θÞ sexp QG kBT0  dT0 β0.5 2 43 5 d dθ β (21B) for diffusion controlled growth (neglecting overgrowth of phantom nuclei). We can have a more general equation to describe both cases and taking out of the integral the heating rate dependency XðTÞ¼ C βd·nGþ1ZT T0 exp QI kBθ  ZT θ ðT0θÞnG1exp QG kBT0  dT0 2 43 5 d dθ (22) which can be simplified to XðTÞ¼ZðT,T0Þ βd·nGþ1(23) where ZðT,T0Þis the crystallization function. Ozawa [64] proposed a method in which, for a fixed temperature, a plot of lnðlnð1XÞÞ versus lnðβÞmight supply a straight line with the Avrami exponent as the slope. However, Ozawa’smethod implicitly assumes that all the kinetic parameters are constant along the transformation process, as the transformation is compared at different stages of the transformation (i.e., for a given temperature, ZðT,T0Þis independent of Xor β). Although this is apparently consistent with KJMA theory, it prevents any analysis on the local Avrami exponent and neglects any effect of T0. Ozawa’s method was applied in the present study to the melting of an indium sample used as standard for calibration of the differential scanning calorimeter (Perkin-Elmer DSC7). This equipment has two independent furnaces which supply heat to the sample and the reference, respectively, to keep them at the same temperature. This minimizes the errors due to differences in temperatures [37] between reference and sample in other thermal analyzers. The mass of the sample was 21.65 mg and it was heated at heating rates between 5 and 80 K min 1 (Figure 7a) after correcting the thermal inertia of the equipment. Transformed fraction Xwas obtained from the integral of the heat flux (DSC signal) normalized to the area of the peak ΔH¼3302 24J mol1. It is worth noticing that as the peak shifts to higher temperatures, Ozawa’s method is comparing the transformation in very different stages. Therefore, we limit our analysis to the range 0.1 <X<0.9, which corresponds to 2.25 <lnðlnð1XÞÞ <0.834. This prevents the use of all the curves in a single analysis. Figure 7b shows the plot of lnðlnð1XÞÞ for temperatures between 432 and 438 K (1 K span) and Figure 7c shows the plot of lnðlnð1XÞÞ versus lnðβÞin the range for which at least three βcurves can be used. The resulting Avrami exponent from Ozawa analysis is n¼1.3 0.2. Although the order of this value is correct, absence of nucleation or low-dimensional growth is hardly expected in the formation of the liquid. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (7 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH Returning to the Equation (23), it is worth mentioning that, for any transformation, the temperature range of interest is limited and when the transformation occurs in a sufficiently reduced temperature range, substituting the exponential for an effective average value leads to XðTÞ¼ C βdnGþ1nGdðdnGþ1Þexp QIþdQG kBT  ðTT0ÞdnGþ1 (24) If we started from the second integral of Equation (3), we will recover something similar but dnGwould appear instead of dnGþ1. Therefore, the direct application of KJMA-plot in the general nonisothermal case assuming an average temperature in the exponential leads to lnðXÞ¼ QIþdQG kBTnlnðβÞþlnðBÞ  þnlnðTT0Þ(25) where Bwould be independent of temperature (although as it was pointed above for Ozawa’s method, this implies that we have to neglect also any dependence of B¼ln C nGdðdnGþ1Þ  on Xor β). On the one hand, the simple relationship n¼nGdþnIcan still be valid. Moreover, the intercept of KJMA-plot, lnðXintÞ¼  QIþdQG kBT DE nlnðβÞþlnðBÞ hi , depends on the heating rate and on the average temperature used to simplify the integrals. This direct application of KJMA-plot to the melting of the indium standard sample was also performed in the present study. However, due to the high sensitivity of the results to the indetermination of the induction time (temperature in the case of nonisothermal regime) the procedure was done as follows: 1) KJMA-plots were generated using the estimated T0 values from the measured Tslope using Equation (18). 2) The average slope of KJMA-plots in the range 0.1 <X<0.9 results n¼4.3 0.6, which is consistent with an interface controlled growth and constant nucleation rate. However, the plot of 430 440 450 /1( ).u.a(td/Hd) T (K) (K/min) 5 10 20 40 60 80 (a) 1.5 2.0 2.5 3.0 3.5 4.0 4.5430 432 434 436 438 -2.0 -1.5 -1.0 -0.5 0.0 0.5 )X-1(nl-(nl T (K) (K/min) 5 10 20 40 60 80 (b) (c) T (K) 432 433 434 435 436 437 438 ln( ) Figure 7. a) DSC signal of In standard sample in the melting region. b) lnð lnð1XÞÞ as a function of temperature, and c) lnð lnð1XÞÞ as a function of lnðβÞ. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (8 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH lnðXintÞþ Q kBT DE (Q¼790 50 kJ mol from Kissinger’s method [69] ) versus lnðβÞwith T hi TP(the peak temperature at which the transformation rate is maximum) yields an inconsistent n2, which is due to the strong effect of indetermination of T0. 3) Fine-tuning of the value of Toallows us to find the corresponding optimum T0values for n¼4 in KJMA-plots for each β. 4) Finally, using these values we obtain n¼3.4 (n¼4.1 neglecting low heating rates) as the slope of lnðXintÞþ Q kBT DE versus lnðβÞ.Figure 8a,b show the dependence of T0and lnðXintÞwith n; Figure 8c shows the Kissinger plot to determine the activation energy Q; Figure 8d shows the corresponding values of estimated T0, optimum T0,TP, and Tslope for each β; and Figure 8e shows the plot of lnðXintÞþ Q kBT DE versus lnðβÞ. Figure 9 shows the local Avrami exponents (a) and KJMA-plots (b) for the different heating rates in the range 0.1 <X<0.9. In 1972, Nakamura et al. [67] proposed an approximation for the Avrami equation to be extended to nonisothermal regimes, in which, XðTÞ, for constant heating rate, is expressed as XðTÞ¼ 1 βnZT T0 k0exp Q kBT0  dT0 2 6 43 7 5 n (26) This approximation, developed for isokinetic transformations, is coherent with Ozawa’s but comparing Equation (26) with (22) shows that the former is an approximation compiling the temperature dependence in just one exponential Arrhenius-type dependence. k0nexp nQ kBT  ZT T0 d dθ Cexp QI kBθ  ZT θ ðT0TRÞαexp QG kBT0  dT0 2 43 5 d 8 < :9 = ; dθ (27) However, Equation (26) is strictly equal to Equation (22) for transformations for which the growth is so fast that new crystals are only observed with its final size. These conditions define the instantaneous growth approximation (d¼0) [70,71] and explains why a direct approach to nonisothermal conditions works so well in the case of nanocrystallization processes. [72] This latter approximation, derived from Nakamura et al. work, assumes an effective onset temperature T0 0TP=2, where TPis the peak temperature (i.e., the temperature at which the transformation rate is maximum), to simplify Equation (26) to [73] XðTÞ¼ 1 βnk0 0exp Q0 kBT  ðTT0 0Þ  n (28) The application of expression (19) to determine the local Avrami exponent is thus modified after taking into account the temperature dependence of the frequency factor 0 1020304050607080 424 426 428 430 432 434 436 438 440 442 444 3,5 4,0 4,5 5,0 424 425 426 427 428 429 3,5 4,0 4,5 5,0 -10 -8 -6 -4 229 230 231 -10 -9 -8 T P T slope T 0 estimated T 0 optimum )K(erutarepmeT (K/min) 5 K/min 10 20 40 60 80 T0 (K) n ln(Xint*) n (e)(d) (c)(b) ln( /TP2) 1/TP (10-5 K-1) (a) 12345 204 206 208 210 212 214 216 218 ln(X *int )+Q/R<T> ln( ) Figure 8. KJMA analysis applied to the melting of In standard. Relationship between Avrami exponent and a) the induction temperature, and b)lnðXintÞ. c) Kissinger plot to determine the activation energy Q. d) Peak temperature, TP, and onset temperatures estimated from the steepest lope, Tslope, and from Equation (18), and optimum T0value chosen as the value leading to n¼4 in (a). e) lnðXintÞþ Q kBT DE versus lnðβÞusing the intercepts in (b) for the optimum T0and TPas the average temperature. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (9 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH exponents obtained from linear regression of KJMA-plot in the range 0.05 <X<0.5 lead to n0.93, which is close to the theoretical prediction of KJMA for a constant nucleation process without growth (n¼1). However, deviations appear at high transformed fractions and n0.6 is found in the range 0.5 <X<0.95. 6.3.3. Instantaneous Growth Process Applied to Nanocrystallization The instantaneous growth model was applied to nanocrystallization processes. [70,71] Equation (42) and (43) were applied to the nanocrystallization of melt-spun amorphous ribbons of Fe 60 Co 18 Nb 6 B 16 composition using η¼1. This alloy shows, after nanocrystallization, a microstructure formed by spherical nanocrystals of 5 nm diameter but agglomerated in groups of 20 nm embedded in a residual amorphous matrix (Figure 16). Figure 17a,b shows, for as-melted ribbons, the isothermal and nonisothermal DSC scans recorded in a Perkin-Elmer DSC7 calorimeter calibrated using the melting temperatures of lead and K 2 CrO 4 standards, and Figure 17c shows the local Avrami exponents obtained using Equation (19) and (29), respectively. Average value of n0.25 from Ozawa’s method is also shown. Combining isothermal and nonisothermal experiments is highly recommended to determine kinetic properties as concluded Figure 16. Bright field image (left), selected area diffraction pattern (center), and high resolution transmission electron image (right) of a nanocrystallized Fe 60 Co 18 Nb 6 B 16 alloy (further details in ref. [71]). 1 10 100 1000 10000 700 750 800 850 900 950 (c) (b) 853 813 K 784 K 763 K 753 K 743 K ).u.a(td/Hd t (s) (a) 2.5 K/min 5 K/min 10 K/min 20 K/min 80 K/min 40 K/min (1/ )·dH/dt (a.u.) T (K) 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 Ozawa 80 K/min 40 K/min 20 K/min 10 K/min 5 K/min 2.5 K/min Non-Isotherms 853 K 813 K 784 K 763 K 743 K 753 K n X Figure 17. a) Isothermal DSC scans, b) nonisothermal DSC scans, and c) local Avrami exponent from Equation (19) and (29) for the nanocrystallization process of Fe 60 Co 18 Nb 6 B 16 alloy. The horizontal line in (c) corresponds to the value of nobtained from Ozawa’s method. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (16 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH by the International Confederation for Thermal Analysis and Calorimetry (ICTAC). [37] Figure 18a shows the nucleation rate Ias a function of transformed fraction Xand, finally, Figure 18b shows the value of I0ðXÞassuming an Arrhenius dependence for IðX,TÞ. Kissinger method was used to obtain the activation energy Q¼380 20kJ mol1(Q¼3.9 0.2eV at1). [106] From KJMA analysis, isokinetic behavior can be observed and low nvalues should be interpreted qualitatively as characteristic of a strongly impinged process. In the frame of the instantaneous growth approximation, it can be observed that isothermal nucleation rate is not constant along the process but it initially increases and then decreases. This behavior is coherent with the values of nðXÞ, which initially increases above 1 and then decreases below 1. The early increase in IðXÞ(nðXÞ>1) is ascribed to a second nucleation mechanism in the surface of already formed nanocrystals (leading to the formation of agglomerates). After the maximum, the continuous decrease in IðXÞ (nðXÞ<1) was understood due to the depletion of the amorphous matrix in Fe. Although this semiquantitative analysis agrees with the isothermal values of IðXÞ, it does not correspond to the IðXÞvalues obtained in nonisothermal conditions (see an almost constant value in Figure 18a). This is due to the strong effect of the exponential dependence on temperature, which enhances nucleation rate. Moreover, the decrease in the Avrami exponent as the transformation progresses agrees with the previous simulations shown here. It is worth reminding that the assignment of decreasing nucleation rate as nI<1 (or increasing Ias nI>1) is based on a functional form of the nucleation rate IðtÞ¼ta. If this is not the case, and nucleation rate changes during the transformation, the relationship between Avrami exponent and evolution of IðtÞis no longer straightforward. However, in Figure 18b a fairly good agreement is observed for I0ðXÞfor isothermal and nonisothermal curves. This is found despite the simple Arrhenius approximation used and the strong effect of Q(see dashed lines in Figure 18b). 6.3.4. Instantaneous Growth Process Applied to Martensitic Transformation As an example of martensitic transition, we present the martensite to austenite transformation of a Ni 50.53 Mn 33.65 In 15.82 metamagnetic shape memory alloy. This system has a change at 285 K from paramagnetic martensite to ferromagnetic austenite on heating. Therefore, the transformation can be registered from the change in specific magnetization σðH,TÞ (see upper inset in Figure 19) as well as from calorimetric measurements. [107] Isothermal curves as a function of the applied magnetic field, μ0H, were obtained in a Lakeshore 7407 vibrating sample magnetometer (VSM) with LN2 cryostat. [107] Specific heat, cP, measurements were obtained at a very slow rate (β0.1 K h1) [107] in a home-made calorimeter [108] following a protocol that prevents the measurement of latent heat ascribed to the transformation. [109] Figure 19 shows Xðμ0HÞcurves from the isothermal curve obtained at 285 K. In order to extract Xðμ0HÞ, the specific 1E-6 1E-5 1E-4 1E-3 0.0 0.2 0.4 0.6 0.8 1.0 1E17 1E18 1E19 1E20 1E21 1E22 1E23 (b) non-Isotherm (10 K/min) 743 K 753 K 763 K 784 K 813 K 853 K I (s-1 nm-3) (a) I0s( -1 mn -3) X 10 K/min Q=3.9 eV/at Q=4.1 eV/at Q=3.7 eV/at 743 K 753 K 763 K 784 K 813 K 853 K non-Isotherm Figure 18. a) Nucleation rate from isothermal and non-isothermal (at 10 K min 1 ) DSC scans as a function of the transformed fraction and b) prefactor of the nucleation rate to compare the curves independently of the temperature. Dashed lines corresponds to the nonisothermal scans calculated using the extreme values of the activation energy taking into account its error bar (0.2 eV at1). [106] 0.0 0.5 1.0 1.5 0.0 0.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 Martensite (emu/g) 0H (T) Austenite X 0 H (T) 250 300 350 0.0 0.1 )g/ume( T (K) Figure 19. Main panel: austenite fraction from specific magnetization as a function of magnetic field at 285 K for a Ni–Mn–In alloy. Upper inset: temperature dependence of specific magnetization at 25 mT (black) and 1.5 T (red). Lower inset: isothermal magnetization curve at 285 K (red symbols) along with the estimated magnetization curves of the pure martensite (blue dashed line) and the austenite (orange dashed line) phases. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (17 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH magnetization of both pure martensite and pure austenite phases was estimated at 285 K. In the case of pure martensite phase, a linear field dependence is assumed which slope was extrapolated from the values of the slopes of the magnetization curves in the range 265 ≤T≤275K. In the case of pure austenite phase, the magnetization curve measured at 290 K was rescaled to the saturation value of the isothermal curve at 285 K. The corresponding estimated values of specific magnetization along with the experimental isothermal curve at 285 K are shown in the lower inset of Figure 19. On the other hand, the ferromagnetic behavior of austenite phase leads to an extra contribution to the specific heat with respect to the paramagnetic martensite (see inset of Figure 20). Therefore, an abrupt change in cPis observed in the transition. Above 310 K, cPfalls due to the Curie transition of the austenite phase. This allows us to understand the change of cPduring the transition as a sum rule of the weighted contribution of each phase: cP¼ð1XÞcM PþXcA P(superindex M and A identify the martensite and austenite phases, respectively). Therefore, XðTÞcan be obtained and it is shown in Figure 20. Despite the controversy [110,111] about the athermal (nonthermally activated) or isothermal (thermally activated) character of martensitic transformations, an effective KJMA analysis can be performed. However, the meaning of the resulting parameters may be out of the frame of the KJMA theory. In fact, several kinetic models developed for martensitic transformations yield equivalent expressions for the transformed fraction to KJMA with effective Avrami exponents n¼1 [103] and n¼2. [104] The corresponding KJMA-plots for isothermal and nonisothermal curves are shown in Figure 21 for the 0.1 <X<0.9 range. The effective Avrami exponents are n3.5 for isothermal magnetization curve and n5 for nonisothermal specific heat curve. In both cases, the onset values were estimated to be H0¼0Tand T0¼279.5 K, respectively, using analogous equations to Equation (18) and assuming n¼4. It has been discussed above how indetermination in the onset affects, particularly at high values of n. Moreover, we have neglected any field and/ or temperature dependence of the kinetic parameters, using an average exponential as it was done in the case of the melting of indium. For athermal processes, this would not be a limitation but it should be important in the case of thermally activated processes. In the frame of KJMA theory, a 3D interface controlled growth with constant nucleation will be roughly in agreement with the resulting values of n. However, taking into account the high speed of growth in martensitic transformations, in the frame of an instantaneous growth, these results should just mean that nucleation rate is increasing during the process as it is shown in Figure 22. This figure shows the transformation rate (although with respect to field change and temperature change, respectively) divided by the untransformed fraction, i.e., a magnitude proportional to IðXÞonce field change and heating rate are constant, 275 280 285 290 295 300 0.0 0.2 0.4 0.6 0.8 1.0 1.2 260 280 300 320 30 35 40 X T (K) ~10 J/molK cp (J/molK) T (K) ~5 J/molK Figure 20. Austenite fraction estimated from specific heat measurements. Inset: specific heat as a function of temperature. 8.08.28.48.68.8 -2 -1 0 1.4 1.6 1.8 2.0 2.2 ))X-1(nl-(nl ln(H-H0) From magnetization T=285 K H0=0 T From specific heat 0H=0 T T0=279.5 K ln(T-T0) Figure 21. KJMA-plots for austenite formation from isothermal magnetization curves at 285 K (left) and from specific heat measurements (right). 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 )X-1( -1 (d/Xd 0 T()H(I~)H -1 ) (a) (b) )X-1( -1 K()X(I~Td/Xd -1 ) X Figure 22. Nucleation rate (arbitrary units) as a function of the transformed fraction a) from isothermal magnetization curves at 285 K and b) from specific heat measurements. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (18 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH respectively. This increase, approximately linear, should indicate that new nucleation sites appear in contact with the already transformed regions. This leads to an expression of the transformation rate: dX dt ¼kMXð1XÞ, which corresponds to the kinetic equation of an autocatalytic transformation of the first order [35] which has been used to describe the kinetics of martensitic transformation of Ni–Fe–Ga alloys. [105] In the case of the martensite-to-austenite transformation studied here, a KJMA process with n¼4 and a first-order autocatalytic transformation yields similar and apparently satisfactory descriptions of the transformation curve. However, the interpretation of the data yields strongly different mechanisms depending on the model assumed. In the former case, although an interface controlled growth is feasible in martensitic transformation, n¼4 also implies a constant nucleation rate without no activation of new nucleation sites and a 3D growth despite the lath structure of martensitic samples. Moreover, the problem arises when we consider the growth rate of martensitic transformations, which is in the order of hundreds of meters per second. This high speed implies that our acquisition time (20 and 720 s in the magnetic and specific heat measurements) is much larger than the time an austenite region grows to its final size. Therefore, the interpretation of instantaneous growth, which has been classically used [112–114] would be more convenient to describe the process assuming that new nucleation sites are activated as the transformation progresses, probably ascribed to the boundary of the formed regions where new dislocations are formed to store the elastic energy and those dislocations are expected to act as heterogeneous nucleation sites. [110,113] In fact, reported slow growth rates extracted from kinetic analysis could be an artifact from imposing an inappropriate kinetic model such as KJMA. 7. Conclusions KJMA theory is broadly used to interpret the kinetics of transformation in solid state in many different cases. However, a considerable number of them are beyond the applicability conditions. Understanding the restrictions allows us to interpret whether a physical meaning of the Avrami exponent is straightforward or not. Even for those ideal processes following KJMA theory, determination of experimental parameters, particularly the induction time, is critical to obtain the correct Avrami exponent. Taking into account an Arrhenius-like temperature dependence for the frequency factors, a direct extension to nonisothermal regimes works well for processes with Avrami exponent close to 1. For processes with a small temperature span, assuming an average frequency factor may lead to fair local values of n at low transformed fractions. Despite decelerated growth processes, such as diffusion controlled one, are out of the KJMA theory and lead to overgrowth artifact, our simulations show that deviations are small and KJMA theory is a fairly good approximation to describe diffusion controlled growth processes. In addition, some strategies for recovering parameters with physical meaning from KJMA analyses are proposed. Finally, when growth process is not registrable, interpretation of the data in the frame of an instantaneous growth approximation allows us to analyze the nucleation rate. Acknowledgements This work was supported by AEI/FEDER-UE (Projects US-1260179 and P18-RT-746) and the PAI of the Regional Government of Andalucía. VI-PPUS from University of Seville is also acknowledged. Conflict of Interest The authors declare no conflict of interest. Keywords Avrami exponent, kinetics, KJMA theory, JMAK theory, JMA theory, solidstate transformations Received: October 13, 2021 Revised: February 3, 2022 Published online: [1] A. N. Kolmogorov, Bull. Acad. Sci. USSR, Phys. Ser. 1937,1, 355; Selected Works of A. N. Kolmogorov (Ed: A. N. Shiryayev), Vol. 2, Kluwer, Dordrecht 1992, p. 188, English translation. [2] W. A. Johnson, R. F. Mehl, Trans. Am. Inst. Mining Met. Engrs. 1939, 135, 416. [3] M. Avrami, J. Chem. Phys. 1941,9, 177. [4] A. A. Burbelko, E. Fras, W. Kapturkiewicz, Mater. Sci. 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Rios, Metall. Mater. Trans. A 2012,44,2. Javier Sebastián Blázquez Gámez is a full professor at the Department Physics of Condensed Matter, Faculty of Physics, University of Seville. He is ascribed to the Non-Crystalline Solids Research Group (FQM-121 Research Group at the University of Seville). His fields of interest include metastable soft magnetic materials, crystallization kinetics, and magnetocaloric effect. Francisco Javier Romero Landa is an associate professor at the Department Physics of Condensed Matter, Faculty of Physics, University of Seville. He is ascribed to the Thermal and Dielectric Properties of Solids Research Group (FQM-130 Research Group at the University of Seville). His fields of interest include phase transitions, metastability, avalanches, calorimetry, and shape-memory alloys. Clara Francisca Conde Amiano is a full professor at the Department Physics of Condensed Matter, Faculty of Physics, University of Seville. She is the head of the Non-Crystalline Solids Research Group (FQM-121 Research Group at the University of Seville). Her fields of interest include metastable soft magnetic materials, crystallization kinetics, and magnetocaloric effect. Alejandro Conde Amiano is an emeritus professor at the Department Physics of Condensed Matter, Faculty of Physics, University of Seville. He is ascribed to the Non-Crystalline Solids Research Group (FQM-121 Research Group at the University of Seville). His fields of interest include metastable soft magnetic materials, crystallization kinetics, magnetocaloric effect. www.advancedsciencenews.com www.pss-b.com Phys. Status Solidi B 2022, 2100524 2100524 (21 of 21) © 2022 The Authors. physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH