Formal Analysis of Isothermal Martensite Spread

A formal kinetic approach was applied to the spread of isothermal martensite over the neighboring austenite grains in a Fe-23.2 wt. (%) Ni-2.8 wt. (%) Mn alloy. The number of grains in a spread event changed with parent austenite grain size. However, isothermal martensite spread formed from fine-grained parent austenite and athermal martensite from a Fe-31 wt. (%) Ni-0.02 wt. (%) C alloy studied in a previous work followed the same microstructural path. The number of grains per spreadevent found in the present study was shown to be consistent with the number of neighbors of the grain originating the spread event by means of a simple geometrical model of the parent austenite grain network. The study of the kinetics of isothermal martensite spread showed that the nucleation rate of the spread-event in isothermal martensite remained constant during the transformation. This result parallels the constant nucleation rate of the spread-event also found using the same methodology in athermal martensite formed in a Fe-31 wt. (%) Ni-0.02 wt. (%) C alloy studied in a previous work.


Introduction
Martensite is a displacive phase transformation with significant transformation strains which prompts microstructural heterogeneity, influences plate's shape, and can lead to diversity in kinetics.Usually, the martensite reaction does not start simultaneously in all austenite grains.In bulk polycrystals, the first nucleation event in a single grain may induce transformation in neighboring grains, resulting in a cluster of partially transformed grains.This cluster of partially transformed grains is normally designated as a 'spread event'.The collection of these single spread events can be defined as the 'spread' or 'martensite spread'.The martensite reaction then proceeds by nucleation of additional martensite units within the partially transformed grains, in what is normally called the 'fillin-in'.Martensite nucleation events may take place in a number of grains leading to a certain number of clusters or spreads.In athermal martensite this nucleation takes place very fast at a certain temperature, as a consequence spread and fill-in are completed almost instantaneously.By contrast, in isothermal martensite, nucleation of new martensite plates in untransformed grains and the consequent spread event takes place as a function of reaction time.As a consequence, in isothermal martensite, martensite spread can be followed as function of time.
The microstructure of the martensitic spread is described by the volume fraction of material in grains partially transformed, V v g (not the martensite volume fraction) and by the area of interface between a cluster of partially transformed grains and the untransformed parent matrix, per unit volume of material, S v g .All symbols used in this paper are defined in Table 1 for convenience.The martensite burst is the classical example of autocatalytic spread of martensite over a large number of neighboring austenite grains in a single event.An optical photomicrograph of a Fe-31 wt.(%) Ni-0.01 wt.(%) C alloy partially transformed by cooling to its burst temperature, M B ≈ 220 K, is shown in Figure 1.Clusters of partially transformed grains in a matrix of untransformed grains are conspicuous.
The martensitic spread in materials that isothermally transform to plate martensite at sub-zero temperatures is the topic of this communication.The current analysis is based on plate martensite formation in alloys of high Ni and high Ni plus Mn contents with sub-zero transformation temperatures.The present analysis does not apply to lath martensite formation in Fe-C alloys or steels with high M s temperatures.
In a recent paper 1 , the present authors have shown that the methodology of formal kinetics was useful to deal with spread in athermal martensite.The basis of formal kinetic modeling is the early work of Johnson-Mehl 2 , Avrami 3 and Kolmogorov 4 (JMAK ), which used only a single microstructural descriptor, V V .JMAK's work was subsequently extended by DeHoff and Gokhale's microstructural path method (MPM) [5][6][7][8] , who proposed the use of an additional microstructural descriptor, S V , and the associated concept of microstructural path.Vandermeer and coworkers 9 extended DeHoff's 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).
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 can in principle be used to model any heterogeneous transformation.
In this work, the evolution of spread during isothermal martensite transformation is modeled with the help of the formal kinetics methodology.The theoretical results will be checked with experimental data obtained from Fe-23.2 wt.(%) Ni-2.8 wt.(%) Mn 10,11 isothermally transformed.
Martensite reaction is nucleation-controlled.Preferred nucleation sites are postulated to exist dormant in the material and propagate the pioneer nucleation event in untransformed grains under the available chemical driving force.Their potency, however, is not uniformly distributed.The reaction inherits this heterogeneity and it is firstly observed only in a few austenite grains 12 .Nonetheless, the pioneer nucleation in a grain has influence over a certain spread volume, v sp , comprising a number γ of austenite grains.where q is the average austenitic grain volume , q = 1/ N v m where N v m is the number of austenite grains per unit volume.The transformation continues by subsequent "spread events" that spread the reaction to the untransformed grains.

Experimental Data
The data used here were obtained by quantitative metallographic techniques to determine volume fraction and interface area per unit of volume from planar sections.The highly pure FeNiC and FeNiMn alloys were reacted under strict control as described in the referenced papers.The authors 10,11 have described and discussed their experimental conditions at length and that will not be repeated here for brevity.However, for the present work, the data were compiled by scanning and digitizing their graphs with relevant metallographic quantities.Non-conspicuous data points were dismissed.These data were consolidated by reiteration to average out small variations.In the following, FeNiC refers to Fe-31 wt.(%) Ni-0.02 wt.(%) C and FeNiMn refers to Fe-28 wt.(%) Ni-3 wt.(%) Mn. S v m was obtained from the mean intercept length with the well-known 13 stereological relation-ship S v m = 2/λ.The values of q were calculated from λ assuming that parent austenite grains can be approximated by tetrakaihedra 13 (See Figure 4 and discussion in Section 5.): q = 2.34 λ 3 ; d, the diameter of a sphere with a volume equal to the grain volume, can be found by d = (6 q/π) 1/3 .The values of λ, S v m ,q and d are listed in Table 2 for the FeNiMn alloy.In his original work, Ghosh 10,11 gives the mean intercept length, λ, for each alloy.

Microstructural Path Analysis
The MPM describes the transformation in extended space.The extended quantities are transformed into real measurable quantities by means of the fundamental relationships where S v g ,ext and V v g ,ext are extended quantities.Rios and Guimarães 1 have derived an expression for the microstructural path of the athermal martensite spread.They assumed a relationship between the volume and surface area of the individual spread, v sp and s sp , respectively.This relationship is 9 S sp = kv s where k is a shape factor.For a tetrakaihedron shape 13 one obtains k ≅ 5.31 whereas for a sphere: k ≅ 4.8.In extended space, n v is defined as the total number per unit volume of nuclei which have become active during cooling to the temperature of interest.Using n v , the extended quantities are given by V v g ,ext = v sp n v and S v g ,ext = s sp n v .Inserting these into Equation 4 results in (5)   Equation 5 is the microstructural path in extended space.It can be converted to the microstructural path in real space with the help of Equations 2 and 3 (6)   where S v m , the total area of austenite grain boundaries per unit of volume of material.The constant can be simplified assuming that the parent austenite grains can be approximated by tetrakaihedra 13 ) -1 .The parameter a in Equation 6 depends only on γ, which on its turn depend on the parameter k.The parameter k is a function of the shape of the spread volume.One possible assumption, adopted in our earlier paper was that the spread volume had a tetrakaidecahedral shape, thus leading to a = 2 γ -1/3 (7)   Another possibility would be to assume a spherical shape and in this case (8)   where the subscript 's' in γ s is meant to emphasize that it was obtained from the assumption of a spherical shape.A relationship between γ s and γ can be easily found: γ s ≅ 0.73 γ.We will use γ throughout this paper for consistency with previous work but for comparison purposes it is also interesting to have γ s .
It is worthy of note that the derivation of Equation 6did not make any specific kinetic assumptions concerning athermal martensite but only assumptions regarding the spread volume, Equation 1.Therefore, Equation 6should also be applicable for isothermal martensite and possibly to mechanically induced martensite as well.Figure 2 confirms that the experimental data can be well described by the microstructural path, Equation 6.
Both isothermal sets of data plotted after Equation 6 can be described by a best-fit straight line passing through the origin with a slope a equal to 0.49 for d = 0.031 mm and a equal to 0.75 for d = 0.079 mm, with a high correlation coefficient, R = 0.98 and 0.99, respectively.The values of γ, calculated using Equation 7, were equal to 68 (γ S ≈ 50) for the fine-grained austenite and 19 (γ S ≈ 14) for the coarse-grained austenite and are listed in Table 2. Previous analysis of athermal martensite in Fe-31 wt.(%) Ni-0.02%C showed that the parameter a was also equal to 0.49 and consequently γ was also equal to 68 and independent of grain size.By contrast, in the isothermal transformation analyzed here the values of γ depended on the grain size.The reason for this is discussed in section 5 below.For the fine-grained matrix, d = 0.031 mm, the values of a = 0.49 and γ = 68 obtained by MPM were identical to those obtained for the athermal martensite in Fe-31 wt.(%) Ni-0.02%C 1 .The data corresponding to the athermal martensite of Fe-31 wt.(%) Ni-0.02%C 1,14,15 are also included in Figure 2. Notice that the theoretical MPM line is common to both fine-grained isothermal and athermal martensite data since their values of a were identical as already mentioned.Therefore, it is evident from Figure 2 that the fine-grained isothermal and the athermal martensite spreads follow the same microstructural path    6, is compared with experimental data from a Fe-23.4wt.(%) Ni-2.8 wt.(%) Mn alloy obtained by Ghosh 10,11 .The correlation coefficient, R, is also shown.The fine-grained isothermal martensite and the athermal martensite follow the same microstructural path.
regardless of their significant differences in kinetics particularly with regard to the number of martensite units per grain in a spread event 10,16 .This result is remarkable and indicates a basic formal similarity between the special aspects of martensite spreads in such apparently different reactions.

Formal Analysis of Spread Kinetics
Whereas pioneer isothermal martensite nucleation takes place at the most potent pre-existent nucleation sites, autocatalytic nucleation promoted by the martensite plate in its vicinity has a clear effect on filling-in with martensite units the partially transformed grains and also on poking the reaction into a next grain.In extended space, the number per unit volume of pre-existent nucleation sites which have become active as a function of time at temperature of interest,n v , is given by dn v (t) = I v (t) dt (9)   or (10)   where I v is rate of that process.It is important to stress that no assumptions are made here concerning the specific time dependence of either n v or I v .However, one must consider that there is an incubation time, t 0 , at which a pioneer nucleation event takes place at one of the pre-existent nucleation sites.The initial density of these pre-existent, dormant sites is .Thereafter, follows Eqution 10.Therefore Equation 10 can be rewritten as (11) Here, γ is considered to remain constant independent of transformation temperature, time or fraction transformed but may depend on grain size.The spread event originated by a pioneer nucleation event propagates until it reaches a volume, v sp , Equation 1.As a result, v sp , is assumed to remain constant throughout the transformation.Therefore, the extended volume of spread can be calculated by (12)   defining (13)   and inserting Equation 13 into Equation 12gives (14)   introducing equation 14 into equation 2 obtains (15)   Eqution 15 can be rearranged and be put in a form more convenient for comparison with experimental data, see below, (16)   Figure 3 shows that plotting the left hand side of Equation 16as a function of time results in straight lines of the form (17)   In these plots, the values of γ used, were obtained from MPM analysis, see Table 2.
From these straight lines, β can be immediately recognized to be equal to the nucleation rate, I v .Unfortunately, one cannot extract the values of n v i,T and t 0 from this regression.This problem can be better understood rewriting the right hand side of Equation 16, using, I v = β and comparing it with Equation 11 It can be seen from Equation 18that there are two unknowns: n v i,T and t 0 and only one experimental parameter, α.Under these circumstances, one may obtain only a lower bound for the incubation time, t 0,lower , by setting the volume fraction of partially transformed grains equal to zero The actual t 0 will be larger than t 0,lower because n v i,T > 0. The calculated values of t 0,lower are given in Table 2.In diffusional reactions, a nucleation event generates a very small volume of the product phase which then grows relatively slowly.However, in martensitic reactions, once a nucleus is activated a plate grows nearly instantaneously to its final size.As a consequence, a finite volume fraction of martensite and also of partially transformed grains become instantaneously greater than zero at t ≥ t 0 whereas the volume fraction of martensite is equal to zero for t < t 0 .

Discussion
As early as 1887, Thomson (Lord Kelvin) 17 proposed that the truncated octahedron or tetrakaidecahedron network would provide a space-filling arrangement of similar cells of equal volume with a minimal interface area.This problem is still of interest to this day 18 .In Metallurgy one often adopts Kelvin's network as an approximation for the polycrystalline structure.We will use such an approximation here to offer a geometric interpretation of the values of γ found from the above analysis.))/( q) (mm ) Figure 3. Isothermal martensite experimental data from a Fe-23.wt.(%) Ni-2.8 wt.(%) Mn alloy obtained by Ghosh 10,11 plotted after Equation 17.The correlation coefficients, R, are also shown.
The basic unit of Kelvin's network is the truncated octahedron that is depicted in Figure 4.It is a polyhedron possessing fourteen faces, eight hexagonal faces and six square faces.In order to fill space they must be packed in a body centered cubic structure.The BCC unit cell with nine truncated octahedra is shown in Figure 5.The polyhedron located at the center of the BCC cell shares a hexagonal face with each polyhedra located at the vertices.A full cluster consisting of 15 truncated octahedral is shown in Figure 6.The other six polyhedral are attached in such a way that they share a square face with the polyhedron located at the center of the BCC cell.These six polyhedra occupy the center of adjacent BCC cells.If all polyhedra on these adjacent cells were counted one would have a cluster with 39 polyhedra.If one went further and completed a cube consisting of twenty seven BCC cells, the number of polyhedra in the cluster would raise to 91.
These numbers are quite close to the values of γ obtained experimentally by means of the MPM methodology.For the isothermal martensite with large grain size, γ was about 14-19, suggesting that the spread was essentially limited to the grains in direct contact with the grain in which nucleation started.On the other hand, for fine grained isothermal martensite and also to athermal martensite γ was about 50-68, indicating that the spread went beyond the nearest neighbors grains.In this case, plates forming in grains adjacent to the grain containing the pioneer plate might also have been able to induce nucleation on their first neighbors thus increasing the number of grains of the spread.Therefore, the experimental result and its geometrical interpretation are quite consistent.Based on this one might suggest that the number of grains in the spread should be somewhere between the cluster formed by the first neighbors with 15 grains and the cluster formed by the cube of 27 BCC cells with 91 grains.
In order to explain the different microstructural path of coarsegrained isothermal martensite when compared to the fine-grained isothermal and athermal martensite, consideration must be given to the fact that plates that outright run across and transform a larger volume fraction of a grain have a more significant effect on its surroundings, as it is observed with the "bursting" FeNiC alloy 15 .This is seen in Figure 7 where the number of grains comprising a spread event, γ, is plotted against the grain volume fraction transformed by the first plate to form, m = v/q, where v is the volume of a grain's pioneer plate.This is in apparent contempt with the fact in isothermally transformed FeNiMn, a coarser grain size has an adverse effect on m 16 , hence on γ, Table 2.However, when plate and grain size are tuned, such as promoted in FeNiMn by a finer grained condition, the values of γ in the "isothermal spread" become similar to FeNiC's "athermal spread" as shown in Figure 2.This also supports the view that plate impingement on grain boundaries has a key role in the spread of the martensite reaction 16 .
In spite of the obvious differences between athermal and isothermal martensite, the former transforms as a function of temperature and   .γ, i. e., the number of grains of the spread event or cluster of partially transformed grains of the parent matrix plotted against m, the ratio of the volume of a pioneer plate, v, to the grain volume, q 16 .Values of γ for Fe-23.4 wt.(%) Ni-2.8 wt.(%) Mn, isothermal martensite, were obtained from Table 2.For Fe-31 wt.(%) Ni-0.02 wt.(%) C, athermal martensite, γ was taken from previous work 1 .The solid line depicted in the figure is just a guideline, not a theoretical result.latter as a function of time, the present formalism revealed interesting similarities with regard to the martensitic spread in both cases.
The first similarity concerns the microstructural path of both reactions.Both follow Equation 6, moreover, fine-grained isothermal and the athermal martensite spreads followed the same microstructural path, a remarkable result, considering their very different kinetics particularly with regard to the number of martensite units per grain in a spread event.This suggests a basic geometric likeness between the martensitic spread in both reactions.
The second similarity pertains to the kinetics of spread.In a previous work, we analyzed the spread kinetics with a formalism identical to that used in section 4 and obtained a relationship that closely resembles Equation 16(20)

Where now n v
Ms pertains to the martensite start temperature, and n v ∆T represents the number of pre-existent nucleation sites that become available as the transformation temperature decreased from M s to T. Notice that athermal martensite starts with a burst at M s .Therefore M s , that is normally known, is our 'incubation temperature', which eliminates the issue of incubation time.When Equation 20 was plotted against temperature, straight lines like Equation 17with temperature substituted for time where obtained (See Figure 2 of Rios and Guimarães) 1 .
(21) From Equation 21 Γ v could be identified as a constant nucleation temperature rate of the athermal spread event (increase in number of nuclei or pioneer plates per unit of volume per unit of temperature ).The meaning of both athermal and isothermal spread events having a constant nucleation rate requires further investigation.

Summary and Conclusions
The formalism proposed here was able to account for the microstructural path and the evolution of the volume fraction of partially transformed grains, V v g , as a function of time for the isothermal martensite transformation in the Fe-23.4wt.(%) Ni-2.8 wt.(%) Mn alloy 10,11 analyzed here.
From the microstructural path itself, Equation 6and Figure 2, it could be shown that the number of grains of the spread event or cluster of partially transformed grains of the parent matrix, γ, depends on the grain size of the parent austenite.However, as shown in Figure 2, the spread of isothermal martensite of the Fe-23.4wt.(%) Ni-2.8 wt.(%) Mn alloy from fine-grained austenite and the spread of athermal martensite transformation in a Fe-31 wt.(%) Ni-0.02 wt.(%) C alloy followed the same microstructural path.This indicated a basic geometric similarity of the spread in both reactions.
The analysis of the evolution of the volume fraction of partially transformed grains, V v g , as a function of time, has shown that the rate of nucleation of pioneer plates per unit of volume, I v , remained constant during isothermal transformation.Such constant nucleation rate was also observed in athermal martensite analyzed by an identical formalism in a previous work 1 .
A geometrical interpretation was provided for the values of the number of grains of the spread event or cluster of partially transformed grains of the parent matrix, γ, showing that the numbers obtained in the analysis are consistent with the number of neighbor grains present in the parent matrix.
In summary, the formalism used here, likewise previous analysis conducted for athermal martensite transformation 1 , was able to describe quite well the isothermal martensite spread.Striking similarities between athermal and isothermal martensite with regard to the microstructural path and nucleation rate of the spread event in both reactions were revealed.

Figure 2 .
Figure 2. Microstructural path of isothermal martensite spread.The theoretical result, Equation6, is compared with experimental data from a Fe-23.4wt.(%) Ni-2.8 wt.(%) Mn alloy obtained by Ghosh10,11  .The correlation coefficient, R, is also shown.The fine-grained isothermal martensite and the athermal martensite follow the same microstructural path.

Figure 4 .
Figure 4. Truncated octahedron or tetrakaidecahedron, the basic unit for a space-filling network, proposed by Thomson17 .This polyhedron has 14 faces: 8 hexagons and 6 squares.

Figure 5 .
Figure 5. Body centered cubic arrangement of truncated octahedra or tetrakaidecahedra.Space-filling is achieved through BCC packing.The polyhedron located at the center of the BCC cell shares a hexagonal face with each one of the eight polyhedra located at the vertices.

Figure 6 .Figure 7
Figure 6.Cluster of 15 truncated octahedra or tetrakaidecahedra, the central polyhedron is surrounded by 14 nearest neighbors.The 9 polyhedra depicted in Figure5are joined by 6 polyhedra, which are attached in such a way that they share a square face with the polyhedron located at the center of the BCC cell.Other grains joining this cluster will not have contact with the central grain, which contains the pioneer plate.

Table 1 .
List of symbols used in this paper and their definitions.