Fcc/Hcp Martensitic Transformation in the Fe-Mn System: Part II. Driving Force and Thermodynamics of the Nucleation Process

This article, which continues the series started by Cotes et al. ( Metall. Mater. Trans. A , 1995, vol. 26A, p. 1957-69), presents a study of the energetics of the heterogeneous nucleation of hcp martensite in the fcc matrix of the Fe-Mn system. A major goal of the work is the exploration of the various qualitative and quantitative consequences of applying the Olson and Cohen approach (OCA) to the most reliable information on the Fe-Mn system. To this end, an extensive assessment of the quantities involved in the OCA is performed. The selected Gibbs energy and the lattice parameter descriptions for fcc and hcp are based on recent experimental studies. Explicit calculations are presented of the key quantities in the OCA, including those quantities that have not yet been measured. In particular, a probable range of values is established for the surface energy parameter (cid:1) , which plays a crucial role in the treatment of the nucleation size. On these bases, values are derived for the probable number of atomic planes ( n C ) of the embryo, which, according to the OCA, is at the condition of spontaneous growth. Predictions of the Mn content upon n C are also reported. The present attempt to calculate in detail the energetics of the martensitic transformation also revealed the lack of thermo-physical data for Fe-Mn alloys. The need for experimental studies of various relevant quantities is emphasized. fcc (cid:65) hcp


I. INTRODUCTION
3][4][5][6][7][8][9][10] The transformation in this system has essentially athermal characteristics, and the nucleation of the martensite phase is generally considered as heterogeneous. [11]Recently, the present authors [4] performed a thermodynamic analysis of the MT, which yielded values for the Gibbs energy difference, referred to as the resistance-to-start-thetransformation-energy (RSTE).However, in those previous works, a detailed analysis of the energetics of the nucleation process in terms of classical nucleation theory (CNT) was not reported.These aspects are dealt with in the present article, in the framework of the Olson and Cohen approach (OCA). [12]In this approach, the nucleation of martensite is described in terms of a mechanism in which the formation of an embryo occurs through a faulting process that originates in a pre-existing crystalline defect.Furthermore, the Gibbs energy of the embryo is described by invoking two main contributions, namely, a dislocation and a fault energy contribution.In this way, the spontaneous growth of the embryo is related to the vanishing of the fault energy contribution at the M S temperature.In the present article, detailed fcc hcp fcc hcp predictions of the composition dependence of the size of an embryo of hcp martensite able to grow spontaneously will be performed, by applying the OCA to an extensive database comprising assessed Gibbs energy functions [4] and information [13,14] on the stacking-fault energy in Fe-Mn alloys.
The remainder of the present article is organized as follows.In Section II, we summarize the key thermodynamic aspects of the OCA, introducing the physical properties which will be dealt with in the article.In Section III, we review the information on the fcc/hcp relative phase stability in the Fe-Mn system.Section IV is devoted to the calculation of the strain energy associated with embryo formation.In Section V, we discuss the structural and elastic properties.In Section VI, we present and discuss the results and predictions of the work.The article ends with Section VII with a summary and remarks.

A. General Considerations
When discussing the possible defect originating a martensitic embryo for the MT, Olson and Cohen [12] hypothesized a finite symmetric tilt boundary segment in which a group of dislocations spaced each two planes apart dissociate simultaneously.After dissociation, the defect may be visualized as a pileup of m intrinsic stacking faults lying on n (111) compact planes, with m ϭ n/2, and bounded by partial dislocations with a length l separated at a distance r, as shown schematically in Figure 1.The Gibbs energy per unit length of dislocation of the embryo, G(r), is expressed as the sum of a term describing the repulsion energy between the partial dislocations, E rep (r), which only depends upon the separation distance, r, and a second term describing fcc hcp (⌬G m hcp / fcc ) the total Gibbs energy of the fault with n planes thickness, ⌫(n); i.e., [1]   Since E rep (r) decreases monotonically with increasing r, the restraining force necessary to provide a stable configuration of the partial dislocations arises from a positive fault energy (⌫ Ͼ 0).Consequently, there is an equilibrium separation of the partial dislocations where their repulsion is balanced by the attractive effect associated with the fault energy.As explained in Section B, ⌫(n) depends on temperature, and when the temperature is lowered making ⌫(n) ϭ 0, the repul- sion causes the equilibrium separation to increase indefinitely, and the defect becomes unstable independently of the details of the short-range repulsive dislocation interaction, E rep (r).Further lowering of the temperature makes the fault energy become negative, which contributes to push the dislocations apart.In such a situation, the embryo formation can occur, and therefore, the process is athermal.

B. Total Gibbs Energy of the Fault and Spontaneous Growth Condition
In the OCA, the ⌫(n) quantity in Eq. [1] is modeled by considering the fault as a second-phase particle, and accounting for surface and volume contributions to the Gibbs energy, using the following equation [2]   where is the difference between the chemical Gibbs energy per mole between the product and the parent phases, is the molar strain energy, (n) is a Gibbs energy per unit area of the particle/matrix interphase for a fault n planes in thickness, and is the density of atoms in a close-packed plane in moles per unit area.In the case of an fcc matrix, the latter quantity is related to the lattice parameter of the fcc phase, a fcc , as follows: [3]   where N A is Avogadro's number.The usual stacking-fault energy (SFE) is obtained from Eq. [2] for the case in which n ϭ 2.
What is usually referred to as the "critical size" in CNT is obtained by differentiating a Gibbs energy of formation of the nucleus with respect to the variables describing its dimensions.In this way, a size is identified that corresponds to the maximum height of the energy barrier associated with the formation of a nucleus in the parent phase.In the OCA, however, the idea of a "critical condition" does not involve the energy cost of forming an embryo-which is assumed to be pre-existing-but to the onset of its spontaneous growth.In particular, Eq. [1] shows that such growth is possible if the condition ⌫ ϭ 0, describing the vanishing of a Gibbs energy barrier, holds.More specifically, this critical condition of the OCA is expected to be reached at T ϭ M S , in which case Eq. [2] yields [4]   The number of planes, n C , in the defect reaching this critical condition in the OCA will be referred to in the following as critical size.Since and are positive, n C will have physical meaning only when the M S temperature is low enough to make -and the denominator in Eq. [4]-sufficiently negative.

III. ASSESSMENT OF THE DRIVING FORCE
Various thermodynamic analyses of the martensitic transformation temperatures (MTTs) in the Fe-Mn system have been reported. [15,16]The thermodynamic bases of the present

A. General Considerations
The driving force (DF) is defined as the difference for the forward (i.e., fcc hcp) transformation, and for the reverse (i.e., hcp fcc) transformation.These quantities, which become positive at M S and A S , respectively, are represented in Figure 2, where we indicate the so-called T 0 temperature [19,20] obtained at each composition by solving the equation [5]   We assume that the fcc hcp MT can start on cooling only when the DF reaches a certain negative value, , where is the posi- tive quantity we have previously referred to as the "resistanceto-start-the-transformation" energy, RSTE. [1,4,17,18]In the same way, we define the RSTE for the reverse transformation (Figure 2), where A S is the corresponding hcp fcc MTT.
In principle, there is no exact thermodynamic relation between the T 0 given by Eq. [5] and the experimentally determined M S and A S .If, however, the vs T function remains linear between M S and A S , and , it is found [17,18] that T 0 ϭ (A S ϩ M S )/2.Since this relation involves a linear approximation (Figure 2(c)), it cannot be exact if there is a significant heat-capacity difference between fcc and hcp.Such a difference is expected for the Fe-Mn and related Fe-Mn-X systems in the temperature range where the fcc phase orders antiferromagnetically. [21]This fact was taken into account in the assessment procedure (as subsequently discussed in Section B).

B. Gibbs Energy Modeling
The G m function of the fcc or hcp phase in the Fe-Mn system was described as the sum of various contributions, using the following expression: [6]   In Eq. [6]  represents the part of the nonmagnetic Gibbs energy of the phase, which is accounted for by the ideal solution model, viz.[7]   where x i is the atomic fraction of the element i (i ϭ Fe, Mn), and is the Gibbs energy of the element i with the structure of the phase , in a nonmagnetic state. [22,23,24]The term in Eq. [6] is the nonmagnetic excess Gibbs energy.It was modeled using the Redlich-Kister formalism, [25] viz.[8]   Here, and are composition-independent parameters, which in general are allowed to vary linearly with temperature.The last term in Eq. [6], , is the magnetic contribution to , which was described by means of the Inden, [26] Hillert, and Jarl [27] phenomenological model, using the expression [9]   In Eq. [9] ␤ is a composition-dependent parameter related to the total magnetic entropy, i.e., the quantity , represents the polynomials of Hillert and Jarl, [27] and is defined as , where is the Néel temperature of the phase (fcc or hcp).When applying the model to alloys, ␤ and are expressed as functions of composition, which was done by adopting low-order polynomials in the composition variables. [4]inally, we refer to the composition dependence of the RSTEs.The low-order polynomials and ⌬G* m (A S ) ϭ a 2 ϩ b 2 • x Mn have been shown empirically [4] to account for the experimental trends, where a i and b i (i ϭ 1,2) are constants determined in the optimization procedure described subsequently in Section D.

C. Experimental Database
In a previous review of the available MTTs, [1] a significant scatter was detected, which was partly attributed to the sensitivity of the detection methods used, to factors such as purity and grain size (not always reported), and to the various thermomechanical treatments applied to the samples.In view of these problems, a new experimental database was developed, in which each one of the alloys was prepared, treated, and measured using the same procedure, thus diminishing the systematic differences.The homogeneity and composition of the samples were determined by microprobe analysis and corroborated by other techniques.The MTTs were determined by means of dilatometric and electrical resistivity experiments on samples subjected to controlled thermal cycles, after quenching from 1000 °C. [1,3]sing symbols, in Figure 3, we plot the experimental M S and A S determined in this way by Cotes et al. [1] The dotted and solid lines in this graphic describe the assessed M S and A S , respectively, to be discussed subsequently, and the dashed line describes the composition dependence of the Néel temperature of the fcc phase, T fcc N .Three different types of behavior may be distinguished in alloys with 0 Յ wt pct Mn Յ 30.a. Alloys with less than 13 wt pct Mn: the MT does not occur, which is due to the formation upon quenching of the more stable bcc phase.b.Alloys with Mn content between 13 and 23 wt pct Mn: the fcc hcp MT is easily detected and both M S and A S decrease smoothly with increasing Mn content until the MTTs approach the T fcc N vs composition line.c.Alloys with more than 23 wt pct Mn: since T fcc N for these alloys is higher than M S and even than A S , the transformation is affected by the stabilization of fcc caused by the magnetic ordering reaction.As a consequence, the fcc hcp transformation was not detected in alloys with more than 30 wt pct Mn.
The experimental database adopted for the present assessment comprised measurements on 12 Fe-Mn alloys [1] and 4 4 fcc hcp those on 37 Fe-Mn-Si alloys with 10 Ͻ wt pct Mn Ͻ 35 and up to 6.5 wt pct Si.

D. Computerized Optimization and Recalculation of the MTTs
The evaluation of the various Gibbs energy parameters from experimental information was performed by using a computer optimization program.The program [28] is able to optimize the parameters that are allowed to vary by minimizing the sum of the squares of the differences between experimental and calculated values.This computerized assessment was carried out in a stepwise fashion.In a first step, the expression T 0 ϭ (A S ϩ M S )/2 was used as an approximation for those Fe-Mn alloys whose MTTs are higher than of the fcc phase.In the final optimization step, the experimental M S and A S values were directly used as input data, and the following parameters were determined: the and excess Gibbs energy parameters of hcp, the ␤ parameter of fcc phase, , as well as the RSTEs, and .The optimized composition dependence of the RSTE for the hcp/fcc MT is given by [10]   Figure 3 demonstrates that the M S and A S temperatures recalculated using the optimum parameters reproduce very well the experimental MTTs included in the fitting procedure.In the remainder of the present article, the values calculated in this way will be referred to as "assessed" MTTs.

IV. ACCOUNT OF THE STRAIN-ENERGY CONTRIBUTION
The strain energy contribution in Eq. [2] was considered as the sum of a dilatational and shear contribution: [11]   The evaluation of the contributions in Eq. [11] was made in terms of isotropic linear elasticity theory. [29]In this case, the strain energy of a coherent particle in a matrix where the stress-free transformation strain of the particle is of pure dilatation character is found to be independent of the particle shape. [12]In particular, the dilatational part depends on the volume change, the shear modulus, and the Poisson's ratio, as follows: [12]   In Eq. [12], v is the Poisson ratio, is the shear modulus, and and and are the molar volumes of fcc and hcp, respectively.In this expression, the parameters v and are assumed to be the same for the particle and the matrix.
The strain energy per mole of a coherent particle whose stress-free transformation strain is of pure shear character is shape dependent and may be expressed as follows: T N fcc Fig. 3-Experimental [1] and calculated [4] fcc hcp MT temperatures M S and A S in the Fe-Mn system.Eqs.[17] and [18] and the lattice parameters determined by Marinelli et al. [30,31]  In Eq. [13], is the energy per unit volume necessary to pull the particle back to its original shape in the absence of the matrix or, alternatively, the strain energy per unit volume that it would have if it had transformed in a perfectly rigid matrix.The value of depends upon particle shape, the matrix elements of the strain tensor ( i,j ), and the shear modulus.[12] The parameter in Eq. [13] is a particle-shapedependent factor, representing the fraction to which the total energy per unit of the volume particle is reduced by accommodation of the transformation strain in the matrix.
In the early stages of embryo formation, due to the dissociation of a set of dislocations in an inhomogeneous edge tilt boundary, it is often assumed that the particle is rodlike or spherical. [2]For a rod with axis in the shear direction, we have ϭ 1/2, and for a spherical particle, is given by the relation [12] [14] For the Poisson ratio v ϭ 1/3, the values of of a sphere and a rod are practically identical, viz.Ϸ 1/2.
In the present work, an estimate of the strain energy at the early stage of nucleation was made by assuming that the particle adopts a spherical shape, an approximation also used by Olson and Cohen. [12]In such case, the strain energy is independent of the shear direction, and we have [15]   Since the relevant deformation for calculating the shear energy of the MT in Fe-Mn alloys is the contraction along the c-axis, Eq. [15] was simplified, by expressing the shear component of the strain energy as [16]   Equation [16] was used as an approximation in the calculations discussed in the remainder of the present analysis.

A. Lattice Parameters and Volume per Atom
Recently, Marinelli et al. [30,31] reported X-ray measurements of the lattice parameters (LPs) of the fcc and hcp phases in the Fe-Mn system.Their expressions describing the composition dependence of the LPs were adopted in the present work to evaluate the molar volumes and , the density of a close-packed plane in fcc, (Eq.[3]), and the contraction along the c-axis direction, 33 , as functions of the Mn content.The procedure was as follows.First, the quantities [17]   and where N A is Avogadro's number, were evaluated.The resulting molar volumes of the individual phases, as well as the ⌬V/V fcc quantity, are plotted as a function of the Mn content in Figures 4(a) and (b), respectively.Next, the composition dependence of the quantity 33 , representing the contraction along the c-axis, was calculated as [19]   where is the distance between (111) planes.The result is plotted in Figure 4(c).

B. Elastic Properties
The strain energy associated with the formation of the nucleus was calculated using Eqs.[12], [13], and [17], with c fcc ϭ 2a fcc / 13 33 ϭ (c fcc Ϫ c hcp ) / c fcc Fig. 5-Composition dependence of the bulk modulus of the fcc phase in the Fe-Mn system, according to the present estimations.Fig. 6-SFE for fcc Fe-Mn alloys as a function of temperature, according to experimental data by Volosevich et al. [13] Fig. 7-SFE for fcc Fe-Mn alloys, as a function of Mn content, according to results by Schumann. [14]e value v ϭ 1/3, and estimating the shear modulus from the bulk modulus B as [32] [20] No experimental information on the bulk modulus of the Fe-Mn fcc phase was found.Therefore, a tentative linear composition dependence of B was assumed, viz.[21]   whose coefficients were determined as follows.The value of B 0 was directly taken from the thermodynamic assessment of pure Fe by Fernández Guillermet and Gustafson, [22] which yielded B 0 ϭ (6.2951 • 10 Ϫ12 ϩ 6.5152 • 10 Ϫ17 • T) Ϫ1 where T is the temperature in Kelvin, and the units are 10 11 N/m 2 .The value of B 1 was determined by combining B 0 and Eq.[21] with the single value B ϭ 1.339 ϫ 10 11 N/m 2 for a Fe-22.6 pct Mn alloy, which was measured at 295 K by Lenkkeri and Levoska. [33]The estimated B vs at.pct Mn function for T ϭ 298 K arrived at in this way is plotted in Figure 5.

A. Analysis of SFE Data
No measurements of the parameter entering into Eq.[2]  seem to be available, but some indirect information was obtained by analyzing SFE data. [13,14]The procedure was as follows.First, using Eq.[3] with n ϭ 2, and neglecting the possible variation of with n, we obtained [22]  s ϭ g/2 Ϫ r (⌬G m hcp/fcc ϩ E str ) Second, two sources of SFE information were considered.Volosevich et al. [13] determined the SFE of various Fe-Mn alloys with Mn contents between 16 and 30 wt pct Mn and temperatures between 100 and 525 K, using TEM observation of partial dislocation nodes.Their values are presented in Figure 6.In Figure 7, we plot the SFE values for Fe-Mn alloys determined by Schumann [14] at room temperature on the basis of transmission electron microscopy (TEM) measurements and observation of partial dislocation nodes.
Third, the Gibbs energy difference and the strain energy contribution in Eq. [22] were calculated using the results of Sections III through V.In Figures 8 and 9, we plot the and the vs composition relations predicted by the present description for the alloys in references 13 and 14, respectively.In Figure 10, we plot the various contributions to as a function of Mn content. m hcp/fcc   [13] Finally, in Figures 11 and 12, we plot the values of the parameter obtained by applying Eq. [22] to the experimental data in references 13 and 14.

B. Assessment of the Parameter
The present evaluation indicates that decreases with increasing temperature for various Mn contents.According to Figure 12, both sources of information [13,14] suggest a similar behavior for alloys with up to about 22 at.pct Mn, namely, a decrease of with increasing Mn content.How-ever, a systematic difference is observed in the roomtemperature values for , which amounts to 4 to 8 mJ/m 2 .In addition, the values extracted from reference 13 increase for alloys with more than about 22 at.pct Mn, but this trend has not been corroborated.
In summary, the values of the parameter, which are consistent with the information available on the Fe-Mn fcc hcp martensitic transformation, belong to the interval 0.013 J/m 2 Ͻ Ͻ 0.020 J/m 2 .The single value ϭ 0.014 J/m 2 obtained by Ghosh [34] from experimental data on a Fe-Mn-Cr-C alloy [35] also falls into the range established in the present work.In the following, we explore the consequences of adopting these values in predicting the size of the embryo at the critical condition of spontaneous growth.

C. Composition Dependence of the Critical Size of the Embryo
In this section, we turn to the problem of predicting the "critical size" of the martensite embryo.This was dealt with using Eq. .The assessed RSTE values for Fe-Mn alloys are plotted in Figure 13 as a function of the Mn content.
Finally, in Figure 14, we plot the n C values predicted by Eq. [4], when the limits of the probable range of values are adopted, viz ϭ 0.013 J/m 2 (dotted line) and ϭ 0.020 J/m 2 (dashed line).The present work indicates that n C decreases with increasing Mn content.In particular, we find that n C decreases from a value of the order of 35 to 50 planes for alloys with 13 at pct Mn, down to about 10 to 15 planes for alloys with 23 at.pct Mn.

VII. SUMMARY AND CONCLUDING REMARKS
In this work, an attempt has been made to explore in depth the consequences of adopting the OCA to nucleation of hcp martensite in the Fe-Mn system.More specifically, the present work aimed at obtaining reliable predictions of those quantities that enter into the key equations of the OCA but have not been established experimentally, viz. the surfaceenergy parameter and the number of planes n C characterizing an embryo of critical size.To this end, an assessment has been performed of the various physical properties involved in the OCA.In some cases, the information needed is taken from very recent studies performed in our group, viz. the chemical Gibbs energy differences between fcc and hcp, [4] and the lattice parameters of these phases. [30,31]Other properties are poorly known from experiments and were estimated.In particular, a tentative interpolation scheme was adopted, which assumes a linear dependence of the bulk modulus of the fcc Fe-Mn phase upon the Mn content.In  [13] (dashed lines ϩ symbols) and Schumann [14] (solid line).
spite of this lack of information, a range of probable values of was established for the fcc hcp MT in the Fe-Mn system, which is consistent with the SFE values and the rest of the data included in the assessment.With this new information, explicit predictions of the number of planes n C characterizing the embryo of critical size have been performed.A relatively strong decrease of n C with increasing Mn content has been obtained.For instance, for the alloy with 23 at.pct Mn, the predicted n C values are roughly 30 pct of those for the 10 at.pct Mn alloy.Hopefully, the present results will stimulate further experimental studies of the thermophysical properties of Fe-Mn alloys, as well as TEM investigations of the nucleation of martensite in Fe-Mn alloys.

Fig. 1 -Fig. 2 -
Fig. 1-Figure extracted from Ref. 4 showing possible particle shapes at the critical stage of nucleation, where a tilt boundary segment originates an hcp martensitic embryo.(a) Rod-shape particle, corresponding to the dissociation of a group of infinite dislocations and (b) Spherical shape particle, corresponding to the bowing out of pinned dislocations with spacing of pinning points (l ) comparable to the boundary segment height (n).

Fig. 4 -
Fig.4-(a) Molar volumes of the fcc and hcp phases, calculated using Eqs.[17] and[18] and the lattice parameters determined by Marinelli et al.[30,31](b) Volume change corresponding to the full fcc hcp transformation, referred to the volume of fcc.(c) Composition dependence of the strain tensor element, 33 , according to Eq.[19].It represents the contraction along the c-axis associated with the fcc hcp transformation.
Fig.4-(a) Molar volumes of the fcc and hcp phases, calculated using Eqs.[17] and[18] and the lattice parameters determined by Marinelli et al.[30,31](b) Volume change corresponding to the full fcc hcp transformation, referred to the volume of fcc.(c) Composition dependence of the strain tensor element, 33 , according to Eq.[19].It represents the contraction along the c-axis associated with the fcc hcp transformation.

Fig. 11 -
Fig.11-Values of the surface energy parameter , obtained in the present work from SFE data by Volosevich et al.[13]
[4].Most of the calculated quantities entering into this equation have already been discussed in Sections A and B. It remains to consider the difference evaluated at the known M S temperatures, which is directly related to what we have called the RSTE for the fcc hcp MT, viz.

Fig. 12 -
Fig.12-Values of the surface energy parameter obtained in the present work from SFE data by Volosevich et al.[13] (dashed lines ϩ symbols) and Schumann[14] (solid line).