Martensite’s Logistic Paradigm

This work introduces a deterministic approach to the martensite transformation curve. Martensite is a nucleation-controlled transformation that has two characteristics: autocatalysis and auto-accommodation. Only a small number of martensite units initially form owing to primary nucleation. These new units may cause the transformation of other units by autocatalysis. We call this kind of transformation chained autocatalysis. Moreover, as the transformation progresses, the auto-accommodation influences the arrangement of new units. This work assumes that the transformation-saturation relates to the exhaustion of the chained autocatalysis, which underlines the microstructure. To compare, we considered the KJMA’s extended-transformation concept that implies assuming exhaustion by impingement. Data from isothermal martensite transformations and anisothermal martensite transformations are used to validate the model. Those data comprised different grain sizes and carbon contents. The model is based upon Verhulst’s logistic concept. We propose that the model’s high fitting-capability stems from its deterministic aspect combined with martensite’s self-similarity. Additionally, we suggest that chained autocatalysis controls the rate of martensite transformation. Therefore, the relaxation of transformation strains by plasticity assisted by mutual accommodation determines the transformation’s martensite volume in the absence of post-propagation coarsening/coalescence.


Introduction
The transformation curve, that is, the volume fraction transformed, V V , against time, t , is a tool in research and process development and industrial operations. Modeling the transformation curve is an issue that has been studied for decades. In the late thirties and early forties of the last century, Kolmogorov, Johnson-Mehl, and Avrami 1-5 , KJMA, published seminal papers on this subject. KJMA used a geometrical model to obtain transformation curves. KJMA supposed that the growing regions were spherical, that their growth rate was a constant, that the nuclei were uniform randomly located in space, and that the nucleation took place in two ways: site-saturation and constant nucleationrate. Their most important contribution was how to consider impingement 6 . As Liu et al. 7 put it, "KJMA's model consists of nucleation, growth, and impingement." KJMA model was generalized in different directions. One direction was to obtain more KJMA-like expressions using mathematically exact methods when nucleation and growth took place in a way distinct from KJMA's. Recently, Rios and Villa 8 used mathematical methods for this purpose. The disadvantage of such an approach is that a limited number of situations admit an exact expression. Another possibility, suggested by Avrami herself, is to employ the well-known "Avrami's equation," which is an expression containing two adjustable parameters: k and n ( ) Focusing the transformations in steels, the present authors have proposed an alternative to Avrami's equation [9][10][11] In the equation above, x is an "advance" variable, which in previous works [9][10][11] was equal to temperature, magnetic field, mechanical deformation, and time. The x i is the first datum in a dataset. Vi V is the integration constant resulting from the process of obtaining Equation 2. We denote this as Vi V . This constant is a small volume fraction transformed when the martensite transformation starts. In this work, one uses Vi V as a fitting parameter. x * and K ϕ are also fitting parameters. Throughout the text, one discusses the meaning of these parameters. This equation showed excellent agreement when fitted to transformations ranging from martensite to pearlite 10,11 . *e-mail: andrealves@metalmat.ufrj.br However, another way to approach formal kinetics is possible. Abramov´s idea 12 of using Verhulst's logistic equation 13 as the basis to describe transformation kinetics represents a significant shift of paradigm. For the derivation of the transformation curve, the kinetic ideas are expressed directly by mathematics instead of mediated by the transformation's geometry, as did KJMA. The purpose of using such an approach for modeling transformations is not new. In 1938, Austin and Rickett 14 took the logistic equation as their starting point to obtain the so-called "Austin-Rickett equation": In 1938 not all KJMA's papers had already been published. The so-called Austin-Rickett equation is seldom applied today, superseded by KJMA's developments.
The description/rationalization of the fundamental aspects of martensite transformations [15][16][17][18][19][20] constructed the present understanding of martensite. That is, the martensite is a diffusionless and nucleation-controlled transformation. This understanding has been a particular venue to develop steels with optimized characteristics to suit the engineering demand. Martensite bears a lattice-correspondence with the austenite matrix. It also possesses a notable shape-change whose relaxation influences the geometric aspects of its constricted microstructure. Moreover, martensite-units do not coarsen or coalesce after propagation.
Consequently, the austenite grains confine the transformation because impingement on high-angle boundaries disassembles the reaction mechanism. However, martensite impingement on the grain-boundaries raises a stress field that can stimulate further intragrain and intergrain transformations to optimize transformation strains' accommodation. Thus, the first units can induce the formation of other units through autocatalysis. We call this kind of transformation chained autocatalysis. Chained autocatalysis occurs after the initial heterogeneous nucleation events in a scarce number of randomly scattered austenite grains 20 .

Verhulst's Logistic Equation
The nucleation-controlled aspect of the martensite transformation is compatible with the original Verhulst equation 13 . Verhulst analyzed the sustainability of populationgrowth based on his "logistic equation" where ( ) N t means the population, means the time, r stands for the population-intrinsic growth-rate and MAX N stands for the maximum population, which can be maintained by available resources. Thus, Equation 4 is consistent with martensite's autocatalytic kinetic. Equation 4 also agrees with the view that the transformation process may be studied in terms of propagation-events since the transformation is nucleation-controlled, and the martensite units do not grow/coalesce after propagation. Furthermore, Equation 4 implies that the transformation-saturation is determined by nucleation-exhaustion instead of the matrix's volumetric exhaustion. Indeed, experimental results show that saturation may occur for a volume fraction transformed 1 V V << 21,22 . Therefore assuming that post-incubation autocatalysis controls the transformation, we substituted ( ) ϕ ∆ is a time-independent transformation-intrinsic factor referred to the external process variable, D, e.g., driving force, temperature, or an applied field. This substitution is equivalent to admitting the pertinence of self-similarity 23 . Besides, both the morphology and the auto-accommodated of the martensite units suggest self-similarity. See Figure 1 in ref 24 . Thus, we recast Equation 4 to describe the martensite transformation curves, where x is the experimental "advance" variable and the subscript " V " indicates per unit volume of material, is the incubation delay. Since we cannot calculate or ( ) ϕ ∆ from first principles, they are treated here as fitting parameters.
Then, acknowledging that transformation curves are usually described in terms of the fraction transformed, N ξ is the mean volume of the martensite units. We calculate Introducing these relationships into Equation 5 includes the influence of the relaxation of the transformation strains, which affects the growth of the martensite units, into the logistic model. The ( ) ϕ ∆ refers to this crucial process, is not available. Thus, we considered two approximations. The invariance of the mean martensite units proposed by Magee 25 and the KJMA's approach assumes transformation in extended space [1][2][3][4][5] . In the first case , so that the integration of Equation 6 yields a formal analog of the "Austin-Rickett equation," where ξ i refers to the value of x at the beginning of the transformation detected in the experimental dataset. We , what is reasonable in the absence of an initial transformation-burst. To use KJMA's impingementcorrection, we set that is analog to Equation 2. Summing up, we have obtained two logistic equations where autocatalytic nucleation advances the transformation, but the volume fraction transformed depends on the relaxation of the transformation strains. Noteworthy the parameter ( ) ϕ ∆ refers to the relaxation of the transformation strains which influences the growth of the martensite units, whereas the transformation exhaustion described by depends on the arrangement of the martensite in the austenite grains and the spread of the transformation over the austenite grains 26 .

Experimental Data
As in the previous work, we imported databases from papers found in peer-reviewed scientific journals to validate the proposed equations. To fit the analytical expression to the experimental values, one calculated the sum-of-squares, ΣSQ, between experimental and calculated values of The ΣSQ gives a "global" idea of the fitting quality. One may also define the relative distance, δ , between the experimental data and the analytical solution predictions x means volume fraction imported from experimental data and the ( ) VA V x means the volume fraction predicted by the analytical equations. As already established, experimental procedures may be subject to errors. One can consider a reasonable error of 5% for metallurgical experiments. Regarding the error of 5%, Tables 1-5 show the percentage of the number of points below the error of 5%. This number can help to give a quantitative basis for the fitting besides ΣSQ and visual inspection.

Isothermal Transformation
and Equation 9 becomes where τ is the incubation time, i t is the first transformationtime datum of the dataset, and T means the temperature. The isothermal-martensite database, Fe-23.2wt%Ni, 2.8wt%Mn, 0.009wt%C, grain intercept 0.048 mm, was initially described in ref 22 . The isothermal-martensite database, Fe-12wt%Cr, 9wt%Ni maraging steel, was presented in ref 27 . Since the imported data did not allow a precise determination of the incubation time, we assumed τ = λ ⋅ i t , and fitted λ until the sum-of-squares, ΣSQ, between experimental and  Figure 1 shows the FeNiMn database asfitted. Figure 2 shows the maraging curves as-fitted.
The values of ΣSQ point out that Equation 11 performed slightly better than Equation 12. Visual inspection is consistent with ΣSQ values. That is, both expressions provided a good fit despite their different formal-approaches to transformationsaturation. The behavior of the parameter δ confirms this.
Concentrating on the FeNiMn alloy, Table 1 and Figures   3 and 4, at the high transformation temperatures, ( ) ϕ T refers to a thermally activated process. By contrast, at the lower temperatures, 163K -77K, the anti-thermal variation in ( ) ϕ T points to the mechanical autocatalysis, which feeds back strain energy 28 . Phenomenologically, we propose, where 0 ϕ is a proportionality factor, Ea , and ∆Ga are apparent energies, T is the reaction temperature and B k is the Boltzmann constant. The charts in Figure 3 yield Ea ≈ 5 kJ/mol -13 kJ/mol, which is compatible with dislocation processes, and ∆Ga ≈ 0.9 -1.3 kJ/mol, which has the same magnitude as the elastic free-energy (0.9 kJ/mol) of an oblate spheroid with a typical 0.05 aspect-ratio in a constrained matrix 29 . The FeNiMn isothermal martensite undergoes a substructure change at low transformation temperatures 22 . Thus, we propose that the variation in ( ) ϕ T refers to changes in the relaxation of transformation strains 30 . The variation in the Vi V corroborate the variation in ( ) ϕ T . However, the variations in 1/ τ show the opposite trends, see Figure 4. Such specific behavior point to differences in the martensite propagation. Martensite propagation at incubation depends on the probability that austenite defects sustain coordinated atomic groups to cross the nucleation path 31,32 . By contrast, the nucleation's postincubation is determined by a previously formed martensite unit (autocatalysis feedback) 28,33 . At high-temperature thermal agitation hamper atomic groups' stability, creating an entropic barrier for converting such groups into nuclei. Thence the chemical driving force controls the incubation. Instead, at low temperatures (higher driving forces), a thermal barrier controls the martensite incubation/nucleation. In this regard, it is noteworthy that the apparent activation energy obtained from the incubation time, ~ 6 kJ/mol, compare with the ~ 5 kJ/mol obtained from the parameter ( ) ϕ T , which refers to the accommodation of the shape strain at high transformation temperatures. This comparison says that dislocation processes are present in both processes (relaxations of lattice-misfit and the shape strain). At this time, the analysis of the temperature variation in the parameter Vi V was not conclusive. Lastly, mind that impingement of martensite on the austenite grain boundary generates a stress-field capable of fostering martensite propagation into the next grain 26,34 . However, such an "intergrain-spread" is hindered if the austenite plasticity halts the radial propagation of a martensite unit 35 . Such a possibility is comparable to "soft-impingement."

Martensite "Athermal" Transformation
To describe the transformation curve of time-independent, driving-force induced ("athermal") martensite, one replaces temperature for the advancing variable in Equations 11 and 12 that gives: and ( ) where * T is the upper temperature for martensite nucleation, and i T is the highest experimental temperature in a data set.
The variables, Vi V , * T and ϕ G are fitting parameters. We quantities seem to be inverted when compared with The reason for this is that time increases and temperature decreases. Thus, the terms are inverted so that the subtractions remain positive.
In FeMnSiMo, the transformation took place in a dilatometer. In addition to allowing the models' validation, the database permits to characterize the austenite grain size's influence on the transformation curve. Bearing scatter in * T , we expressed * = λ ⋅ i T T, and fitted λ until the sum-of-squares, ΣSQ, between experimental and calculated values of V V became invariant, see Figure 5. Table 3 lists the values of the obtained parameters of Equations 14 and 15. Inspection of the values of ΣSQ indicates that Equation 14 provides the best fittings with a minor variation in ΣSQ. By contrast, the values of ΣSQ, which characterize the fittings with Equation 15, increase with increasing the austenite grain-size.
A similar procedure was used to fit the Fe-18wt%Cr, 8wt%Ni data. The results are shown in Figure 6 and Table 4. The fit is excellent.
Concentrating on the FeMnSiMo, Figures 7 and 8  , are due to the severe effect of crystallographic-variance in fine-grain austenite. This crystallographic-variance affects the microstructure's localrandomicity, which is a requirement for utilizing the KJMA's methodology [1][2][3][4][5] . Again, the behavior of the parameter δ indicated a better agreement between Equations 14 and 15, which assumes exhaustion by nucleation and by impingement, respectively.
Lastly, we consider the influence of the carbon in the martensite, transformed by continuous cooling. Typical plain carbon-steels with similar austenite grain-sizes were considered: Fe46C(0.46wt%C, 0.71wt%Mn, 0.26wt%Si, 0.1wt%Ni, 0.2wt%Cr), Fe66C(0.66wt%C, 0.69wt%Mn, 0.30wt%Si, 0.1wt%Ni, 0.2wt%Cr), and Fe80C(0.80wt%C, 0.61wt%Mn, 0.41wt%Si, 0.2wt%Ni, 0.3wt%Cr). These databases were imported from ref 40 . The fittings with Equations 14 and 15 are       shown in Figure 9, and the values of the respective modelparameters are listed in Table 5. Again, visual inspections of the charts and the variations in ΣSQ indicate that the Equation 14 provided better fittings, especially concerning the transformation-charts' progressive induction. These fittings were consistent with the behavior of the parameter .

δ
We ascribe the variation in * T to the influence of the carbon on the austenite stability. The variation in G − ϕ is related to the influence of carbon content on the transformation microstructure since increasing carbon enhances the partitioning of the austenite grains into finer packets and blocks 41 .
Like the isothermal transformation curves above analyzed, the different modes of considering the transformationsaturation provided proper fittings of the data. Nonetheless, the values of the fitting parameters are model-dependent, as might be expected.

Discussion
The classical Verhulst's logistic equation, Equation (4), proposed to describe constrained population growth, provides a venue to express transformation curves 12  Notably, the variations in the parameters obtained from the FeMnSiMo database typical of martensite transformation by cooling ("athermal") exhibit similarity and are in qualitative agreement with the results reported in the referenced paper 36 . Thence, the experimentalists may choose the more appropriate expression to analyze their data and describe the transformation under consideration 12 . Nonetheless, the meaning of the physical parameters obtained from formal models depends on the models' premises. We assert that autocatalysis and transformation-saturation by nucleationexhaustion are realistic premises to model martensite transformation curves as provided by Equations 11 and 14. It is worth discussing the fitting parameters displayed in Tables 1-5.
First, we would like to offer some background on the use of phenomenological equations and fitting parameters to describe a specific kinetic curve. In the present case, to fit a ( ) curve. The first possible approach to describe experimental measurements by an analytical expression is to employ an arbitrary function to fit the experimental curve. This fit may be useful if one has an analytical theory that takes a continuous function as its input, for example, Ref 42 . On another extreme, one may fit an expression derived from fundamental theories. Generally, these are not easy to come by. An intermediary approach is to use the so-called formal kinetics. These provide exact expressions when one specifies the nucleation and growth rates. The pioneering work is, of course, KJMA theory [1][2][3][4][5]  From the equation employed here, one expects: I) that they give a good fit; II) that we can extract some information from the fitting parameters. Notice that the functional form is different for Equations 14 and 15. Therefore it comes as no surprise that the absolute value of the fitting parameters differs. Nonetheless, Tables 6, 7, and 8 show that they do not differ by the same magnitude. In the case of Table 6, the differences between the fitting parameters were calculated as follows: (Equation 12 parameter -Equation 11 parameter)/ (mean value of Equation 11 and Equation 12 parameters). The same reasoning was adopted for Tables 7 and 8, but with Equations 11 and 12 replaced by Equations 14 and 15.
The parameters that mark the beginning of the transformation, such as initial transformation temperature and incubation time, are physical parameters. Tables 7 and 8 demonstrates that the values of * T lie quite close when Equations 14 and 15 determine them. Table 6 shows the values of τ , obtained from Equations 11 and 12, behave similarly but with an apparent discrepancy at the highest and lowest temperatures.
The absolute values of the other parameters have a significantly higher difference. This behavior is unavoidable as the proper functions are different. This result suggests that the function Table 6. Difference between the fitting parameters shown in Table 1  parameters, such as, Vi V and ϕ . One cannot expect Equations 11, 12, 14 and 15 to be more than they are. They are equations with a physical or mathematical background, but they are still approximations. And it is well-known that fitting parameters carry the error made by assuming a certain approximation. But, as shown above, the parameters are not influenced in the same way.
Here, parameters that have a direct physical interpretation tend to be almost independent of the fitting expression. By contrast, parameters that are more directly related to the functional form of the fitting expressions tend to have more considerable differences.

Conclusions
1. The utilization of the logistic formalism to describe isothermal and continuous cooling martensite transformations yielded quality-fittings of experimental data. These quality-fittings are consistent with current views regarding martensite's nucleation-controlled, autocatalytic kinetics, and self-similarity. 2. The apparent activation energies obtained from Equation 13, 5 kJ/mol -13 kJ/mol, compares with the activation energies for martensite nucleation reported in refs 46,47 . Therefore, one may suggest that there are two kinds of active dislocation processes. One dislocation process acts in the conversion of coordinated atomic groups into nuclei. The other, intrinsically different dislocation process relates to the relaxation of the martensite shape-strain 30 . 3. The incorporation of self-similarity into Verhulst's logistic formalism allowed good descriptions of the martensite transformation curves as well as characterizations of kinetic aspects of isothermal or "athermal" transformations.