Thermodynamic modeling and computational predictions of NbC precipitation in Fe–Mn–Si-based shape memory alloys by the classical nucleation and growth theories

NbC precipitation in Fe–Mn–Si-based alloys is an effective method to improve the shape memory effect. In this study, the precipitation behavior was investigated using a thermodynamical model to understand the mechanism and optimize the precipitates for a better performance of Fe–Mn–Si-based shape memory alloys. The influence of alloying elements can be considered in the model by introducing interaction parameters. The precipitate size distribution, mean size, precipitate volume faction, and number density of three typical Fe–Mn–Si-based alloys with different NbC addition amounts were calculated. The results indicated that the mean size could be decreased significantly as the NbC addition increased from 0.5% to 1.0%, while the precipitate volume fraction and number density showed obvious increments. The Fe–28Mn–6Si–5Cr alloys exhibited smaller mean sizes and higher number densities than the Fe–14Mn–6Si–9Cr–5Ni and Fe–21Mn–6Si–9Cr–5Ni alloys. It was also found that the precipitate size distribution showed no evident change as the aging time increased from 0.5 h to 2 h except for the Fe–28Mn–6Si–5Cr–0.5NbC alloy in which the precipitates began to coarsen after about 1.25 h.


Introduction
Fe-Mn-Si-based shape memory alloys have recently attracted great interest for their good machinability, weldability, and lower cost [1][2][3][4][5][6]. However, the recovery strain of the polycrystalline Fe-Mn-Si-based alloy obtained by conventional manufacturing process consisting of forging, rolling, and subsequent solution treatment is only 2%-3%, which cannot meet the engineering application requirements. Modification of alloying elements [7,8], heat treatment [9][10][11], plastic deformation [12][13][14], thermomechanical processing [15], training treatment [16], and second-phase precipitation [17][18][19][20] have been employed to improve the shape recovery strain. The precipitation of nanosized second-phase was first proposed by Kajiwara as an effective method to improve the shape memory effect in Fe-Mn-Si-based alloys [17,21]. It has been reported that the shape recovery rate of the Fe-28Mn-6Si-5Cr alloys with NbC precipitates produced by aging treatment at 1070 K for 2 h could be increased to 90%. However, the shape recovery rate of the solutionized alloys was only about 50%. Other research works also showed that NbC precipitates could improve the shape memory effect of Fe-Mn-Si-based alloys [22,23]. The improvement of shape memory effect by NbC precipitation can be ascribed to several factors [17,[23][24][25]. On the one hand, fine NbC precipitates can strengthen the austenite and prevent unrecoverable plastic deformation by dislocation slip. On the other hand, the NbC precipitates are considered to provide preferential nucleation sites for the stress-induced martensite. Meanwhile, a back-stress generated at the martensite plate tip stopped by precipitates can help the reverse movement of the Shockley partial dislocations when the reverse transformation takes place. In addition, stacking faults appear to associate with NbC precipitates, which will benefit the martensite transformation. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Precipitate characteristics, such as particle size, amount, and precipitation location, can influence the shape memory effect. The precipitation behavior of second-phase particulate is determined by the chemical composition, austenite state, and aging treatment. Many efforts, including heat treatment, chemical composition design, and pre-deformation, have been made to control the precipitation behavior in Fe-Mn-Sibased shape memory alloys [21,26,27]. In addition to the experimental investigations, thermodynamic modeling is also feasible to explore the underlying precipitation mechanism. The precipitation predictions in some alloys by classical nucleation and growth theories have shown good agreement with experimental observations [28][29][30][31]. However, investigation of the precipitation behavior in Fe-Mn-Si-based shape memory alloys by thermodynamic modeling has not been reported. In this study, interaction parameters were introduced into a thermodynamic model to account for the influence of alloying elements. Precipitate size distribution, precipitate volume fraction, and number density in three typical Fe-Mn-Si-based shape memory alloys with different NbC concentrations were predicted using the model. Based on the simulation, the precipitation behavior of the Fe-Mn-Si-based shape memory alloys was discussed.

Thermodynamical model of NbC precipitation 2.1. Nucleation
The classical nucleation and growth theory was described for precipitation in detail in references [28][29][30][31][32]. Here we only present the main feature.
Assuming a spherical nucleus of radius R, the change of chemical free energy is given by: where G v ∆ is the driving force for nucleation per unit volume of NbC precipitate, which is given by [32]: Here, R g is the gas constant, T is the absolute temperature, V NbC is the NbC molar volume, W Nb and W C are the weight fractions of Nb and C in the solid solution respectively, W e Nb and W e C are their equilibrium fractions. Considering the density of NbC is comparable with that of austenite, the effect of mismatch stain is ignored. Therefore, the free energy change associated with the NbC precipitate formation in a supersaturated solid solution is given by: where g is the specific interface energy between the NbC precipitate and the solid solution. The critical radius R cri can be expressed by: The corresponding critical free energy change is: The nucleation rate is given by [28]: at which stable precipitates nucleate. N 0 the number of nucleation sites per unit volume, which can be estimated using the austenite lattice parameter [33], N is the number of nuclei formed in the solid solution within time t, k B is the Boltzmann constant, Z is the Zeldovich factor, b* is the condensation rate, and t is the incubation time which is determined by Z and . b*

Growth and coarsening
The nucleated precipitates can grow by consuming solute atoms in the supersaturated solid solution. In this process, the volume fraction of precipitates increases. As the precipitation continues, the solution supersaturation and the nucleation rate decrease. The volume fraction of precipitates gradually stabilizes and keeps constant. When the large precipitates grow with the dissolution of small ones, precipitate coarsening happens. In this simulation, the radius and number of the nucleated precipitates at every time step were recorded. The evolution of these radiuses with time at both growing and coarsening stages can be simulated using the same equation given by [28]: is the ratio of mean volume per atom of the matrix to NbC precipitates, X NbC is mole composition of precipitates, X is the mole fraction of solute atoms, X R i ( )is the equilibrium mole fraction at the interface between matrix and precipitate with radius R, which is given by [34]: where X eq is the equilibrium mole fraction of solute atoms at temperature T. From equation (7), it can be found that the radius R increases with time t during the growth stage because the concentration of solute atoms X is high. However, in the coarsening stage, the radius R of the small precipitates decreases because X X R , i < ( ) considering the Gibbs-Thomson effects. As the radius R decreases to a value less than R , nucl * the number of these precipitates is set to zero.

Solubility product
The NbC solubility product is an important thermodynamic parameter in modeling the precipitation behavior, which is given by a general expression: In this simulation, A 0.13 = and B 7900 = were used for NbC in austenite [28]. The alloying elements in shape memory alloys, including Mn, Si, Cr, and Ni, can affect the solubility through their effect on the activities of C and Nb [35,36]. The solubility product KSP of NbC in Fe-based shape memory alloys is estimated by: The relation between equilibrium weight fractions in equation (2) and solubility product is expressed as: Considering the stoichiometric ratio of NbC precipitates, we can get: where A Nb and A C are the atomic mass of Nb and C, respectively. Using equations (11) and (12), W e Nb and W e C can be calculated.

Simulation procedure
Given an initial composition of the Fe-Mn-Si-based shape memory alloys, the solubility product KSP of NbC at temperature T is determined through equation (10). The equilibrium weight fractions W e Nb and W e C are solved by equations (11) and (12). Then, the driving force G v D for nucleation can be calculated by equation (2). Thus, the critical radius R cri and critical free energy change G cri D can be obtained by equations (4) and (5). The nucleation rate of NbC at time t can be calculated according to equation (6). The nucleation number within time step t D is obtained. The change in radius of all NbC precipitates is calculated using equation (7).
After that, the new solution concentration can be obtained. If the ratio of change in the solution concentration to the concentration before nucleation is larger than 0.01, a smaller time step is used, and the above calculations are repeated. Otherwise, the precipitation is modeled successfully for a time step t. D A bigger time step can be employed for the next computation. By repeating the calculation procedure, the precipitation behavior of NbC in Fe-Mn-Si-based alloys can be modeled. The parameters used in this simulation are given in table 2.
Predictions were compared with the experimental results reported in the literature [17,25] to validate the model. The simulated precipitate size distributions are shown in figure 1. The predicted mean diameter was 41.3 nm for the Fe-15Mn-5Si-9Cr-5Ni-0.5NbC alloy aged at 1070 K for 2 h (figure 1(a)), which agreed well with the TEM observations showing an average precipitate size of about 40 nm [25]. The predicted size was 18.8 nm for the Fe-28Mn-6Si-5Cr-0.5NbC alloy aged at 1070 K for 2 h ( figure 1(b)), while the experimental measurements gave a mean size of about 20 nm [17]. The predicted size range also shows agreement the TEM observations.
The present model can be used to examine the influence of temperature, duration, and concentration of alloying element on the NbC precipitation behavior in Fe-Mn-Si-based shape memory alloys. The influence of heating rate and cooling rate can be investigated as well. However, it should be noted that the crystal defects of the supersaturated solid solution, such as dislocations, grain boundaries, and lattice distortion, are not considered in the present model. Therefore, this model is only applicable to the simulation of NbC precipitation in solutionized Fe-Mn-Si-based shape memory alloys. The free energy change described by equation (3) and the number of nucleation sites per unit volume N 0 should be modified if the influence of these crystal defects is taken into account.  3. Results and discussion

Precipitate size distribution
Three typical Fe-Mn-Si-based shape memory alloys with Mn mass fraction of 0.14, 0.21, and 0.28 were examined in this study. The NbC concentrations varied from 0.5% to 1.0% to investigate their influence on precipitation behavior. The aging temperature in the simulation was 1073 K, at which most of the experiments were carried out [25,26]. Precipitate size distribution of NbC in Fe-14Mn-6Si-9Cr-5Ni alloys is shown in figure 2. It can be seen that the mean size increased from 32.1 to 33.1 nm as the aging time increased from 0.5 to 2 h for the alloy with 0.5% NbC. The standard deviation of precipitate size slightly increased from 1.38 to 1.53 nm. As the NbC addition increased to 1.0%, the mean size was almost invariable with aging time increased from 0.5 to 2 h. The corresponding standard deviation showed a very small increment. The simulation implies that the influence of aging time on the precipitate size distribution of Fe-Mn-Si-based alloys, especially for the alloys containing 1.0% NbC, was weak as it varied in the range from 0.5 to 2 h.
With increasing the NbC addition, the mean size and standard deviation decreased significantly, as shown in figure 2. It can be seen that the mean size decreased from 33.1 nm to 19.8 nm as the NbC addition increased from 0.5% to 1.0% for the Fe-14Mn-6Si-9Cr-5Ni alloys aged at 1073 k for 2 h. The standard deviation was decreased from 1.53 to 0.75 nm as well. In addition, it is noted that for the alloys with 1.0% NbC, the peaks of precipitate size distribution were symmetric, which is different from those of the alloys with 0.5% NbC.
Precipitate size distributions of the Fe-21Mn-6Si-9Cr-5Ni alloys are illustrated in figure 3. It can be found that the change of size distribution was very small as the aging time increased from 0.5 to 2 h, which is similar to the variation shown in figure 2. As the NbC addition increased from 0.5% to 1.0%, both the mean size and standard deviation decreased obviously, and the peaks became symmetric. Precipitate size distributions of the Fe-28Mn-6Si-5Cr alloys are shown in figure 4. It is seen that the mean size was quite small compared with that of the alloys with 14% and 21% Mn content. As the aging time increased from 0.5 to 2 h, the mean size increased from 15.4 to 17.2 nm, and the standard deviation increased from 1.03 to 4.48 nm for the alloy with 0.5% NbC. As the NbC addition increased to 1.0%, the precipitate size distribution showed little difference with the aging time.

Precipitate volume fraction and number density
In the simulation, the number and radius of the precipitates nucleated at different moments are recorded sequentially. The total number N t of precipitates at aging time t is given by: where n i is the number of precipitates nucleated at the i-th time in the simulation. Number density is the total number per unit volume, which is equal to N t in the simulation. The precipitate volume fraction f V can be obtained by: where R i is the radius of the precipitates nucleated at the i-th time. The evolution of precipitate volume fraction and number density of the Fe-14Mn-6Si-9Cr-5Ni alloys is illustrated in figure 5. It can be seen that the precipitate volume fraction of the alloy with 1.0% NbC increased quickly as soon as the aging treatment began. The fraction approached 1.0% within approximately 500 s. On the contrary, the precipitate volume fraction of the alloy with 0.5% NbC was increased very slowly, which approached the stable value after about 2500 s. Compared with the alloy with 0.5% NbC, the alloy with 1.0%   NbC exhibited a very high nucleation rate for its high supersaturation. After the number density achieved the maximum value, it was almost stable for both of the two alloys within the simulation time.
When the Mn content increased to 21%, the evolution of precipitate volume fraction and number density showed no obvious variation, as shown in figure 6. By comparing with the Fe-14Mn-6Si-9Cr-5Ni and Fe-21Mn-6Si-9Cr-5Ni alloys, it can be found that the Fe-28Mn-6Si-5Cr alloys exhibited the highest increase rate of precipitate volume fraction because they achieved the stable value in a shorter time as shown in figure 7(a). This can be ascribed to the higher nucleation rate in the growth stage resulted from the chemical composition variation. It is seen that the number density of Fe-28Mn-6Si-5Cr alloys could be increased to 2-11 × 10 21 m −3 in a very short time, which is far more than those of the Fe-14Mn-6Si-9Cr-5Ni and Fe-21Mn-6Si-9Cr-5Ni alloys with the same NbC addition amount.
The NbC nucleation and growth behavior is influenced by the degree of supersaturation. By combing equations (9)- (12) ) can be obtained, which are illustrated in figure 8. It is shown that the degree of supersaturation is quite high for the three alloys. For the alloys with 1.0% NbC addition, the Nb element exhibits higher degree of supersaturation compared with that of C element. While for the alloys with 0.5% NbC addition, the C element shows a higher degree of supersaturation. It can be seen that the degree of supersaturation for the Fe-14Mn-6Si-9Cr-5Ni and Fe-21Mn-6Si-9Cr-5Ni alloys is almost the same. However, it is increased obviously for the Fe-28Mn-6Si-5Cr  alloys. Correspondingly, they exhibit a higher nucleation rate and a higher increase rate of precipitate volume fraction shown in figure 7.
Another difference in the Fe-28Mn-6Si-5Cr alloy with 0.5% NbC is that the number density decreased obviously after about 4500 s, as shown in figure 7(b). In the beginning stage, the number density and precipitate volume fraction increased quickly because of the high degree of supersaturation. The mean diameter also exhibited significant increase, as shown in figure 9(a). With increasing the ageing time, the degree of supersaturation decreased which can be seen in figure 9(b). It is found that the variation of degree of supersaturation was very slight after about 1000 s. The mean diameter also showed slight variation after that. However, the mean diameter increased quickly again after about 4500 s. Meanwhile, the number density decreased resulted from the dissolution of small NbC particulates. This indicated that the precipitates begin to coarsen. During the coarsening stage, the large particulates grew and the small ones dissolved. Therefore, the standard deviation of diameter increased obviously, as shown in figure 9(a).
In the present study, we developed a model based on the classical nucleation and growth theories to predict the precipitation behavior in Fe-Mn-Si-based shape memory alloys. Influence of alloy chemical composition, aging time, and other parameters on the features of NbC particulates, such as mean diameters, standard deviation of diameters, number density, and precipitate volume fraction, can be captured by this model. A problem naturally arises as to which precipitate feature is better to improve the shape memory effect of Fe-Mn-Si-based alloys. Some research [17,25] implied that fine NbC precipitates can improve the shape memory effect remarkably because of the hardening effect on austinite, promotion on nucleation of martensite, and help of reverse transformation. However, other research [38] indicated that larger particles increase the shape memory  effect more significantly than smaller particles because the fine precipitates can pin the martensite and inhibit its reversion. As for the precipitate volume fraction, it is thought that the improvement is slight with a low precipitate volume fraction, but the alloys are embrittled with a high precipitate volume fraction. Therefore, a suitable amount of the second phase is beneficial [24]. Though research has been carried out on this issue, the relationship between precipitate features and properties of the Fe-Mn-Si-based alloys is still unclear. Further investigation is required to reveal the influence of precipitate features on the shape memory effect of these alloys. The model of present study can help to understand the precipitate behavior and to control the precipitate features, thus help to reveal the relationship.

Conclusion
A thermodynamical model was used to simulate the precipitation behavior of three typical Fe-Mn-Si-based shape memory alloys containing NbC at 1073 K for different durations. This model introduced interaction parameters to account for the influence of alloying elements. Precipitate size distribution, mean size, precipitate volume fraction, and number density of the Fe-Mn-Si-based shape memory alloys were calculated by the model. Based on the simulation, the following conclusions were drawn: (1)With increasing the aging time from 0.5 to 2 h, the precipitate size distribution showed no obvious variation except for the Fe-28Mn-6Si-5Cr-0.5NbC alloy, whose size standard deviation showed a significant increase. The Fe-14Mn-6Si-9Cr-5Ni and Fe-21Mn-6Si-9Cr-5Ni alloys exhibited the similar precipitate size distribution. The mean size of the Fe-28Mn-6Si-5Cr alloys was smaller than that of the Fe-14Mn-6Si-9Cr-5Ni and Fe-21Mn-6Si-9Cr-5Ni alloys. The mean size was obviously decreased by increasing the NbC addition from 0.5% to 1.0%.
(2)In the initial aging treatment stage, the precipitate volume fraction increased quickly. It was stable as it achieved the maximum fraction. With increasing the NbC addition from 0.5% to 1.0%, the precipitate volume fraction could achieve the maximum value in a shorter time because of the marked increment of nucleus number. The Fe-28Mn-6Si-5Cr alloys showed the highest increase rate of precipitate volume fraction in the initial state among the three typical alloys.
(3)The precipitate number density increased at a fast rate in the initial aging treatment stage. It approached the maximum and almost was stable until the precipitate coarsening occurred. In the simulation, the precipitates in Fe-28Mn-6Si-5Cr-0.5NbC began to coarsen after about 4500 s. With increasing NbC from 0.5% to 1.0%, the number density was increased obviously because of the higher supersaturation. The Fe-28Mn-6Si-5Cr alloys achieved the highest number density among the three alloys.