Athermal Martensite Transformation Curve

In a recent paper, the present authors [Materials Research. 2015. 18(3): 595-601] demonstrated that the thickness of athermal martensite plates had a thermally activated component. The reason for this was probably the interaction of the athermal glissile martensite-austenite interface with dislocations located within the surrounding austenite. Based on this result, here we derive a new expression for the volume fraction against temperature for athermal martensite transformation. This new expression includes a thermal activated term. The expression obtained here gives good agreement with athermal martensite transformation in Fe-Ni-C and Fe-C alloys.


Introduction
The relevance of martensitic transformation in materials science and technology does not need reiteration.The close relationship of martensite to technology has existed since Osmond's pioneer work 1 .In this context, the rationalization of the martensite microstructure to the transformation curve continues to be of interest.
The current understanding is that martensite transformation is displacive with a defined crystallography.In the present case, we focus on athermal martensite.Athermal martensite transformation comprises two steps.The first step is the springing-up of martensite units.This step is "barrier-less".This means that a martensite unit starts to form without needing to overcome any kind barrier.Clearly, this step does not involve thermal activation.The second step is the growth of those units.This growth normally involves the migration of a glissile interface.This migration takes place without thermal activation.Nonetheless, a glissile interface may interact with dislocations that may exist in its path.Those interactions are dislocation processes that may produce a drag and hinder interface migration.Our previous work suggest that these interactions may result in a thermal activated term for growth.Therefore, a thermal activated component may appear during athermal martensite transformation.This may occur even if the springing-up of its units and interface migration are not themselves thermal activated.This paper proposes a new expression for volume fraction against transformation temperature of athermal martensite that captures this thermal activated component.
The typical microstructure resultant from martensitic transformation comprises a large number of units of decreasing size 2,3 , organized to minimize the shear component of the transformation strain 3 .The mutual impingement of the martensite units, as well as existing austenite obstacles impervious to the displacive mechanism of the transformation such as grain boundaries, delimit the size of the units.It is understood that martensite initially propagates from preferred austenitic defects (PADs), and proceeds by autocatalysis.After springing-up, a martensite unit does not continue to grow or coalesce.The rate of propagation of martensite is very high, which rules-out martensite growth by thermally activated atom transfer at the interface 4 .Instead, current understanding considers that the martensite transformation is nucleation-controlled.Nonetheless, thermally activated ("isothermal") and non-thermal activated ("athermal") aspects can be extracted from the transformation curves [5][6][7] , and the role of dislocation processes in the core of martensite transformation is presently consolidated 8 .
For athermal martensite one may think of a transformation curve to be a volume fraction, V V , as a function of the transformation temperature, T, that is, V V (T).Empirical expressions of this kind are available for martensite transformation.The most well-known is the Koistinen-Marburger 9 ( ) 1 exp( ( )) where M S is the martensite start temperature and α is a positive constant.
Guimarães and Rios 10 have recently derived a more fundamentally based expression, specifically for lath martensite where T M is the highest temperature at which an embryo becomes viable.In practice, it is close to M S martensite start temperature.It is worthy of note the original approach taken by Shankaraiah et al., [11][12][13][14] to simulate the martensite transformation by means of Monte Carlo simulation.

Model derivation
Of special importance to the present work is the martensite transformation of Fe-30wt%Ni particulate reported by Cech and Turnbull 15 that convincingly demonstrated martensite's heterogeneous aspect.Furthermore, their results indicate that the preferred austenite defects (PADs) for martensite nucleation are scarce and randomly distributed.Therefore, the probability of finding a particle with at least one such sites 16 is ( , ) 1 exp( ) where q is the mean particle volume and n V T is the number density of the PADs at temperature T.
We conceive that the propagation of a martensite unit from one PAD yields other plates by autocatalysis which instantaneously propagate and that these units are located within the austenite volume were they formed.Thence, the number of units generated by one PAD, including intergrain spread, is equal to a C .Admitting the extended transformation concept advanced by JMAK [17][18][19] , the extended number density of martensite units generated by n V T PADs is As a consequence, the (extended) volume fraction of martensite is , where v M is the mean volume of a martensite plate supposing that its growth is not hindered by other plates, in other words, v M is an extended martensite mean volume.Therefore, the extended fraction transformed, V VE , is T .Furthermore, one admits that the classical relationship between volume fraction and extended volume fraction [17][18][19][20][21] , V V = 1-exp -V VE , applies here.As a result at a temperature, T ( ) 1 exp( ) Moreover, the present authors 22 reviewing the data from Cech and Turnbull 15 found out that n V T could be fitted to the equation with a high fitting correlation.In this equation n V 0 is the general number density of PADs , ΔS is the transformation entropy change, k is the Boltzmann constant, and T M ≥ M S is the temperature at which those sites become operational.M S is the martensite start temperature.
In a recent work 23 , the present authors examined in detail the temperature dependence of the mean "intrinsic" (or extended) dimensions of the martensite units in Fe-31wt%Ni-0.02wt%C.The conclusions were that the diameter, D, remained roughly temperature independent and scaled with the grain size (the martensite partitions the austenite grains) whereas the thickness, t, decreased with temperature with an Arrhenius like temperature dependence 0 exp( / ) where the activation energy, E A , is about 1.2x10 -20 J/event for the Fe-Ni-C alloy.The present authors attributed this temperature dependence not to a thermally activated martensite growth but rather to a thermally activated arrest of growth owing to the interaction of the moving martensite interface with the surrounding austenite plasticity.In this paper, considering that martensite extended thickness is thermally activated 23 we propose that the martensite extended volume, v M = πtD 2 /4, is also thermally activated 0 exp( / ) Combining the equations above This is the athermal martensite transformation curve where the pre-factor, A, is

Experimental data
This work mainly refers to Fe-Ni-C alloys in the range of 30-33wt%Ni and low carbon (≤0.02wt%).These alloys were used to investigate different aspects of the martensite transformation 3,5,24-29 that have sub-zero transformation.Thence, these alloys are amenable to metallographic analysis at room temperature to characterize the progress of the transformation at different temperatures.1][32] .Complemental, we considered for comparison the data independently obtained typical of Fe-1.86wt%C 33 and of Fe-32wt%Ni-0.04wt%alloys 5 .The values of these microstructure descriptors used in the present work were taken from the referenced publications, by scanning and digitizing the relevant figures, which, unfortunately, did not include error bars.However, our familiarity with the methodology permits ascribing ±10% to volume fraction figures obtained by systematic procedures on a planar random section of the specimen.Notwithstanding that, note that the handling of each material is described in the referenced papers 5,[30][31][32][33] and are not repeated here for brevity sake.Figs. 1 and 2 show typical micrographs of a Fe-Ni-C alloy partially transformed and almost fully transformed to martensite.

Results and discussion
Figures 3-9 show a plot of V V against transformation temperature data obtained for the Fe-Ni-C alloys 5,30 and for the Fe-C alloy 33 .In figures 3-9 the data show error bars.These error bars are not in the original papers and they were drawn estimating the relative error of the volume fraction measurements to be ±10%.
A nonlinear regression of the model, equation 9, to the Fe-Ni-C and Fe-C data gave high coefficients of determination, R 2 , better than 0.93 in all cases except for Fe-Ni-C with 0.049 mm of reference 30 in which R 2 =0.78.Therefore, in all cases the model, equation 9, exhibited good agreement with the experimental data.
It is important to examine the parameters obtained by the fitting.For Fe-Ni-C alloys T M ranged from 221 to 233 K whereas experimental M S ranged from 212 to 226 K.The mean of fitted T M parameters was 227 K whereas the mean M S was 221 K. Considering that T M is the temperature at which the first martensite propagation sites (PADs) become viable, it should be higher but comparable to M S that is the temperature in which the start of martensite transformation was detected.T M is a theoretical concept as it has been experimentally impossible to identify such propagation sites whereas M S is an experimental concept and the detection of martensite start depends on the sensitivity of the technique used.Even using sophisticated the techniques, it is clear that a certain number of propagation sites (PADs) must have already generated martensite units in order for detection to be possible.Hence, T M ≥ M S even though one would expect this difference to be small and 6 K looks like a reasonable value.For the Fe-C alloy 33 T M was equal to 320 K. Mendiratta and Krauss did not measure M S but obtained it by extrapolation of M S from lower C contents.From that extrapolation, they considered M S about 323 K that is consistent with the value of the fitted T M parameter.Therefore, broadly speaking, although T M was obtained by fitting it gave a very good agreement with experimental values.
Another critical parameter is the activation energy, E A .The mean value of the fitted E A values was E A ≅ 10 -20 J/event that is in good agreement with the value obtained in our previous work 23 and compares with the activation energy for dislocation processes in austenite 23 for a detailed discussion of this.
Finally, we acknowledge that it is not possible to make a detailed analysis of the fitted parameter It is apparent that A is a lump factor composed of factors that are not easy to estimate.
Nevertheless, expeditiously taking a C ≈ 10 3 after 34 , admitting that v M 0 is a fraction m (0.01-0.1) of the mean austenite grain volume which is reasonable in the case of the Fe-Ni-C alloys, and 35 S k ∆ =0.55, we estimate the average n V 0 value in the range 1.2 -1.6x10 4 mm -3 which compares with the 23 5.4x10 4 mm -3 typical of the particulate material described by Cech and Turnbull 15 .
Henceforth, we can conclude that the present model gives good agreement with Fe-Ni-C alloys and a Fe-C alloy over a  broad range of grain sizes, from 0.026 mm up to 0.490 mm.Even though at small grain sizes such as 0.026 one observes the presence of martensite spread 36 .
Noteworthy, small values of E A /kT yields This simplification is applicable in two cases.At high temperatures where austenite plasticity is not an issue such as in case of lath martensite transformation in low-medium carbon steels 10 .In addition, if the mechanical autocatalysis/variant selection suffices to accommodate the transformation strains, one may assume that E A ≈ 0. Equation 9 is formally identical to equation 2.

Summary and conclusions
We propose a new model of volume fraction of athermal martensite as a function of transformation temperature.The model includes a thermal activated martensite plate volume.This is supported by the previous finding 23 of a thermal activated martensite thickness.We attributed this thermal activated term to the drag effect of the austenite plasticity on the mobility of the martensite-austenite interface.The dislocations introduced in the adjacent austenite to assist in the relaxation of the transformation strains interact with the mobile athermal martensite-austenite interface.Notice that whereas dislocation processes in the austenite are thermally activated processes, the martensite-austenite interface movement itself is not regarded to be thermally activated 23 .The comparison of the model, equation 9 with experimental data permits the following conclusions: • In all cases examined here, Fe-Ni-C alloys and a Fe-C alloy, the model, equation 9, exhibited good agreement with the experimental data.
• The parameter T M is the temperature at which the first embryo becomes viable.In the Fe-Ni-C alloys, the fitted value of the parameter T M is comparable to M S .
Here it was found T M to be on average 6 K above M S .
• Moreover, under certain circumstances discussed in the text, our model, equation 9, may be simplified yielding a simpler expression than equation 9, namely, equation 10.Equation 10 is formally identical to equation 2 previously obtained by the authors for lath martensite.
• The relaxation of the martensite transformation strains influences the size of the martensite units, tantamount the fraction transformed and the transformation curve of the alloys considered here.

Fig. 2 -Fig. 3 -
Fig. 2 -Light micrograph showing the microstructure of a coarse-grained austenite of a Fe-31wt%Ni-0.02wt%Calloy transformed by cooling into liquid nitrogen.(Courtesy of J. R. C. Guimarães) Fig. 3 -Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-31wt%Ni-0.02wt%Calloy 30 with a mean intercept length of austenite grains equal to 0.142 mm.Solid line corresponds to the model developed here: equation 9. Error bars corresponding to an estimated relative error of ±10% were added to the plot.

Fig. 4 -
Fig. 4 -Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-31wt%Ni-0.02wt%Calloy 30 with a mean intercept length of austenite grains equal to 0.049 mm.Solid line corresponds to the model developed here: equation 9. Error bars corresponding to an estimated relative error of ±10% were added to the plot.

Fig. 9 -
Fig. 9 -Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-1.86wt%C alloy 30 with a mean intercept length of austenite grains equal to 0.110 mm.Solid line corresponds to the model developed here: equation 9. Error bars corresponding to an estimated relative error of ±10% were added to the plot.

Fig. 5 -
Fig. 5 -Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-31wt%Ni-0.02wt%Calloy 30 with a mean intercept length of austenite grains equal to 0.026 mm.Solid line corresponds to the model developed here: equation 9.Error bars corresponding to an estimated relative error of ±10% were added to the plot.Outlier was excluded from the fitting.

Fig. 6 -
Fig. 6 -Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-32wt%Ni-0.004wt%Calloy 5 with a mean intercept length of austenite grains equal to 0.490 mm.Solid line corresponds to the model developed here: equation 9. Error bars corresponding to an estimated relative error of ±10% were added to the plot.

Fig. 7 -
Fig. 7 -Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-32wt%Ni-0.004wt%Calloy 5 with a mean intercept length of austenite grains equal to 0.121 mm.Solid line corresponds to the model developed here: equation 9.Error bars corresponding to an estimated relative error of ±10% were added to the plot.Outlier was excluded from the fitting.

Fig. 8 -
Fig.8-Volume fraction of martensite plates per unit of volume against transformation temperature.Fe-32wt%Ni-0.004wt%Calloy5 with a mean intercept length of austenite grains equal to 0.048 mm.Solid line corresponds to the model developed here: equation 9.Error bars corresponding to an estimated relative error of ±10% were added to the plot.