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Print version ISSN 1516-1439
Mat. Res. vol.13 no.1 São Carlos Jan./Mar. 2010
Paulo Rangel RiosI,II,*; José Roberto Costa GuimarãesI,III
IEscola de Engenharia Industrial Metalúrgica de Volta Redonda, Universidade Federal Fluminense UFF, Av. dos Trabalhadores, 420, CEP 27255-125 Volta Redonda - RJ, Brazil
IIRWTH Aachen University, Institut für Metallkunde und Metallphysik, D-52056 Aachen, Germany
IIIMal. Moura 338H/22C, CEP 05641-000 São Paulo - SP, Brazil
Modeling the martensite reaction requires reckoning with spatial aspects of the reaction. For that, we used formal kinetics, more specifically, the microstructural path method (MPM) to analyze the microstructure observed in a burst. The microstructural path analysis revealed that the size of the spread cluster in extended space, characterized by the Vandermeer and Juul-Jensen's impingement compensated mean intercept length, λG, remained constant, independently of the parent austenite grain size. Moreover, current analysis introduced a purely formal description of the reaction progress by taking the parent austenite grain size as the progress variable. This description worked very well and resulted in a relationship between the volume fraction of partially transformed austenite, VVG, and austenite grain size, λG. The significance of these findings in the light of the advantages and disadvantages of formal kinetics is discussed.
Keywords: martensitic phase transformation, kinetics, nucleation, microstructure, analytical methods
Martensite is a displacive phase transformation in which transformation shear strains are significantly higher than the transformation homogenous strains. This characteristic prompts microstructural heterogeneity, influences plate's shape, and can lead to a diversity in kinetics with a corresponding diversity of spatial arrangement of the transformation product. Since a martensite unit quickly reaches its final size, further transformation requires additional nucleation events that are, frequently, correlated with previous ones. In other words, martensite is a nucleation controlled reaction with a significant autocatalytic component.
One of the unique and intriguing features that athermal martensite presents in some alloys is the so called "martensite burst" or simply "burst". As soon as the alloy reaches the martensite start temperature, Ms, the burst may even be triggered, even by a single nucleation event1, resulting in a substantial volume fraction of martensite, up to 80 vol.(%) in some cases. The size of the burst depends on the driving force (burst temperature), but the size of the plates formed is that is controlled by the austenite grain size and by the arrangement of the plates. The propagation of martensite occurs rather quickly, so that in a burst spread and fill-in are completed almost instantaneously. An audible sound may accompany the burst.
Concerning spatial distribution of transformed regions, it is important to recall that the martensite reaction does not start simultaneously in all austenite grains as the experiment of Cech and Turnbull2 with particulate Fe-30 wt.(%) Ni demonstrates. The probability that a particle transforms depends upon the propagation of nucleation at the reaction temperature. The fraction of particles transformed at a temperature T has been described by Cohen and Olson3 (Equation 1).
where q is the particle volume and nV the density of nucleation sites. On the other hand, the fill-in of these particles with the martensite units is an autocatalytic process. The volume fraction of martensite in the particles is determined by the quenching temperature or driving force2.
In the case of bulk material, the connectivity of the grains allows the transformation in one grain to induce the propagation of a nucleation site in an untransformed neighbor, resulting in a cluster of partially transformed grains. This cluster of partially transformed grains is normally designated as a "spread event" or 'spread-cluster'. The collection of these single spread events can be defined as the 'spread' or 'martensite spread'. This process is restarted by another pioneer nucleation elsewhere as well as by further transformation within the partially transformed grains, in what is normally called the 'fill-in'.
Thus, for convenience, we describe a spread event comprising three steps: 1) The prime nucleation event by propagation of a pre-existent nucleation site; 2) The autocatalytic fill-in of the grain with martensite units, and 3) The impingement of some of these plates onto the GB. The latter step causes the spreading of the reaction from the grain where the prime event occurs into a neighbor and so on until exhaustion. Let the average number of martensite per grain be qNV. Therefore, the analog of Equation 1 applicable to the burst transformation in bulk polycrystal is (Equation 2):
where PI is the probability that a martensite unit impinges on the GB. This expression of VVG has been validated before5-7, and Pi ≈ 0.1 in the case of lenticular martensite forming in Fe-31 mass% Ni-0.02 mass% C.
Figure 1 shows clusters of partially transformed grains formed in bursting Fe-31% Ni-0.02% C cooled to the martensite start temperature that coincides with the burst temperature, MB. It is conspicuous that grain boundaries are impervious to martensite propagation and blocks the propagation of the midrib, the reaction's fast leading step. The sudden stoppage of a midrib at a grain boundary generates a peak perturbation that can interfere with the transfer of transformation strains to the matrix as well as with subsequent transformation events8-11. The martensite units forming zigzags are observed in every grain where the transformation occurred. The zigzags result from the coupling of the transformation shape strain, a form of autocatalysis that helps to optimize the discharge of transformation strain12-14. In bulk polycrystalline material we expect grain boundaries to provide a suitable environment for the initial nucleation of martensite10,14-18. Hence, the impingement of a martensite unit onto a grain boundary should provide stimulus for the propagation of a nucleation site in the next grain. It is notable that the balance of these two effects results in the spread of the reaction. It is a contention that in absence of martensite units impinging onto grain boundaries, a burst would spread sluggishly as in isothermal martensite.
These experimental facts indicate that modeling martensite burst requires reckoning with the spatial aspects of the reaction. For this purpose, formal kinetic modeling and the microstructural path method, briefly described in the following, have shown to be useful tools19-23.
The basis of formal kinetic modeling is the early work of Johnson-Mehl24, Avrami25 and Kolmogorov26 (JMAK), which used only a single microstructural descriptor, VV. JMAK's work was subsequently extended by DeHoff and Gokhale27,28, who proposed the use of an additional microstructural descriptor, SV, and the associated concept of microstructural path. Vandermeer and coworkers29 extended the microstructural path concept and crystallized it in an all round theoretical treatment covering variable nucleation and growth rates as well as non-spherical regions: the microstructural path method (MPM) or microstructural path analysis (MPA). Recently, Rios and Villa30,31 have extended previous work using stochastic geometry methods.
The microstructural path method was originally developed to deal with recrystallization and later applied also to diffusional transformations and grain growth but it is in fact quite general and has been used to model a diversity of heterogeneous transformations in both metallic and non-metallic materials.
Recently, Rios and Guimarães have successfully used MPM to analyze martensite spread of both athermal19 and isothermal martensite20. In this work the same MPM methodology is applied to the martensite burst.
2. Experimental Description of the Burst in a Fe-31 mass% Ni-0.02 mass% C Alloy
The overall spread of martensite transformation has been conveniently described by the volume fraction of clusters of partially transformed grains, VVG, and by the area per unit volume of clusters' boundaries, SVG5. These two stereological parameters, obtainable by point and lineal counting on a planar surface, have been used in formal analyses and in computer-modeling the martensite spread6. VVG increases with the volume fraction transformed and readily responds to a variation in the austenite grain size, whereas SVG goes through a maximum at an intermediate value of VVG5. In this work we analyze published quantitative metallographic data from the burst in a Fe-31 mass% Ni-0.02 mass% C alloy with different grain sizes4-7. Table 1 provides a quantitative description of the microstructure of the initial burst in bulk Fe-31% Ni-0.02% C . It shows the aforementioned quantities VVG, and SVG as well as: austenite grain size, λγ, and area per unit of volume, SVγ, and martensite volume fraction, VV. The burst temperatures, MB, are also included. The data in Table 1 were compiled from the original publications5-6. Non-conspicuous data points were ignored. Small variations from the compilation were averaged out by reiteration. Originally, the compiled data did not bear error bars. However the quantitative stereological methods used by the original sources suggest that it is not unreasonable ascribing a relative error of ±10%, although errors could be larger in the case of parameters derived from the ratio of two measured quantities.
It can be seen in Table 1 that the range of burst temperatures, MB, is narrow (212-222K). However, the mean intercept length, λγ, ( or the area per unit volume of grain boundaries, SVγ) exhibit a significant variation. It is noticeable that VVG varies significantly with SVγ, compared with the volume fraction of martensite in the clusters given by VV/VVG, likewise observed in particulate Fe-30%Ni material5.
3. Microstructure Path Method Applied to the Martensite Burst
The classical expressions for the clusters of partially transformed grains located in space according to a homogeneous Poisson point process30,31 are well-known: Equations 3, 4.
where and are extended quantities, i. e. expressed without considering impingement.
Furthermore, weighing the concepts in32-34, we use the impingement compensated cluster's mean intercept, , to describe the cluster of partially transformed grains (Equation 5),
Combining the equations we obtain a relationship between SVG and VVG, the microstructural path: Equation 6.
where has the same units as (SVG)-1.
Figure 2 shows that the plot of the microstructural path is an almost perfect straight line, the correlation coefficient is R2 = 0.99 with = 287 µm. It is interesting to point out that the mean intercept length, λG, for a "real" cluster is λG = 4VVG / SVG. For small volume fractions, VVG<< 1, approaches λG because the impingement effect is small. This can be easily checked noticing that VVG = 0.07 and SVG =1.04 mm1 (Table 1) result in λG = 269 µm, which is comparable to = 287 µm.
In previous work19,20, also using MPM, the present authors have described the spread event as a cluster of tetrakaidecahedron grains. Clearly, one model can be easily converted into another, if necessary. It can be shown that the average number, g, of grains comprising such a cluster in extended volume is related to by = γ1/3λγ.
It is worthy of note that the progress of athermal martensitic spread could be formally described by assuming that mean number of grains of the spread-cluster in extended space remained constant throughout the reaction, γ ≈ 68. Moreover, this number of partially transformed austenite grains constituting a spread-cluster did not change with grain size. Since γ was independent of the austenite grain size, the mean size of the spread cluster in extended space was larger for the coarser grained austenite.
The influence of the austenite grain size on the behavior of the burst is very different from the behavior of the athermal spread just described. In the burst the size of the spread-cluster is independent of the austenite grain size. The material with coarse grain size has fewer grains per cluster, that is, γ decreases with increasing the austenite grain size at nearly constant (burst) temperature. For a spherical cluster containing tetrakaidecahedral grains the number of grains located within the cluster, γS, is19,10.
For a tetrakaidecahedral cluster containing tetrakaidecahedron grains the number of grains located within the cluster, γ, is20:
Equations 7 and 8 imply that a significant number of grains, roughly 900, are involved in the spread-cluster when the grain size is small, 26-28 µm. This number significantly decreases to about 200 grains when the grain size is about 49 µm. The figures indicated that the number of parent austenite grains contained in a spread-event in a burst may be significantly larger than the number of parent austenite grains in a spread event induced by further cooling the alloy below the burst temperature.
4. Formal Analysis of Spread Kinetics Taking the Austenite Grain Size as the Progress of Reaction Variable
Formal analysis using the classical JMAK approach implies that evolution of the transformed region is related to nucleation and growth. In the case of the martensite spread, the transformed regions, that is, the spread-clusters, quickly reach their final size because the reaction is largely nucleation controlled and growth is very fast. Further martensite nucleation in athermal martensitic transformation requires further cooling, so that 'rate' is a 'temperature rate' that was found constant19.
In the realm of the MPM, the burst is a nearly instantaneous event, so that the observed microstructure provides only one reaction point along its microstructural path. Nevertheless, by utilizing alloys with different grain sizes other reaction points along their respective microstructural paths can be found, provided that the martensite units maintain their morphology. These points can be described by a common vs. curve, as shown in Figure 2, if remains approximately constant with grain size. This is what has been found here: in extended space the cluster-spread size during martensite burst is roughly independent of the austenite grain size. Bearing this in mind one may write (Equations 9, 10).
where nV is the number of spread-clusters per unit of volume and is the area of a spread-cluster, which also remains constant with grain size since the spread-cluster size remains approximately constant with grain size.
For formal purposes, we can define a 'spread rate', IB, with reference to the austenite grain size, assuming that temperature effects can be neglected owing to the narrow temperature range of the burst, 212-222 K. (Equations 11, 12, 13, 14).
A list of symbols used in this paper may be found in Table 3.
And from Equations1, 6-7
Figure 3 shows the plot of vs. λγ. It gives a straight line with a correlation coefficient is R2 =0.91. Thus, the equation above satisfactorily describes the fact that the volume fraction of the spread does depend on austenite grain size even though its size in extended space is approximately independent of grain size.
A summary of fitted parameters of can be found in Table 2. For a grain size λγ0 = 18 µm it is predicted that VVG= 0. This value of λγ0 = 18 µm is remarkably close to the value of 5-15 µm calculated from a previous work from one of the authors11.
In previous papers by the same authors19,20, formal kinetics methodology was able to successfully describe athermal and isothermal martensite. Likewise, the same methodology could also successfully describe the geometric and kinetic characteristics of the martensite burst. It is important to stress that in all three cases, isothermal, athermal and burst, the methodology was rigorously applied.
The microstructural path analysis displayed in Figure 2 revealed that the size of the spread cluster in extended space, characterized by the impingement compensated cluster's mean intercept, , remained constant, independently of the parent austenite grain size. This in contrast with the results previously obtained for athermal and isothermal martensite in which the number of austenite grains in the spread-cluster remained constant but the size of the spread-cluster increased with parent austenite grain size.
Moreover, current analysis introduced a purely formal description of the reaction progress by taking the parent austenite grain size as the progress variable. This description worked very well and resulted in a relationship between the volume fraction of partially transformed austenite, VVG, and austenite grain size, λγ.
However successful formal kinetics has been in modeling these situations it is important to highlight the limitations of the results obtained here. Formal kinetics is a general geometric/kinetic approach. It does not make any assumptions concerning the mechanisms or the causes whereby the relationships found here are established. In other words, the methodology reveals very well the general kinetic/geometric relationships among relevant measurable stereological quantities. Yet, it cannot explain on its own the physical mechanisms causing these relationships to occur in the first place.
In order to investigate the physical causes behind the increase in VVG with λG or why remains constant during the burst it is necessary to carefully consider several factors pertaining to the martensite plate formation. These factors are, for example, austenite grain partition, induction of martensite transformation by a plate impinging on an austenite grain boundary, i. e. autocatalysis, among others. Thus, it is necessary considering the spatial aspects of the martensite plate formation, which would be a step further, beyond the scope of this paper.
In spite of the limitations discussed above, formal kinetics reveals important geometric/kinetic facts about the martensite burst that constitute a solid foundation upon which further understanding can be built upon.
6. Summary and Conclusions
In summary, the utilization of formal kinetics methodology to analyze the microstructure observed in a burst clarified the spatial aspects of the martensite burst.
The microstructural path analysis displayed in Figure 2 revealed that the size of the spread cluster in extended space, characterized by the impingement compensated cluster's mean intercept, , remained constant, independently of the parent austenite grain size.
Moreover, the analysis carried out in this work introduced a purely formal description of the reaction progress by taking the parent austenite grain size as the progress variable. This description worked very well and resulted in a relationship between the volume fraction of partially transformed austenite, VVG, and austenite grain size, λγ.
P. R. Rios is grateful to Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, and to Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, for financial support. P. R. Rios would also like to thank the Humboldt Foundation for the Humboldt Research Award and Professor Günter Gottstein for his hospitality during the author's stay at Institut für Metallkunde and Metallphysik of RWTH-Aachen where part of this work has been done.
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Received: November 9, 2009;
Revised: February 10, 2010