Phase-Field Model Prediction of Nucleation and Coarsening during Austenite/Ferrite Transformation in Steels CHENG-JIANG HUANG and DAVID J. BROWNE A phase-field simulation is performed to study the kinetics of austenite to ferrite ( : ) transformation in a low-carbon steel during continuous cooling. Emphasis is placed on the influence of nucleation, along with ferrite grain coarsening behind the transformation front, on microstructural evolution. Results show that grain coarsening is significant even before all nucleation has been completed and occurs via two different coarsening mechanisms, grain boundary migration and ferrite grain crystallographic rotation, both of which can be clearly observed occurring as the simulated microstructure evolves. For some grains, sudden growth jumps are predicted by the model—a phenomenon that has been observed before by synchrotron X-ray diffraction. This study quantitatively demonstrates the phenomenon that increasing cooling rate leads to nucleation off initial austenite grain boundaries, which is also verified by studying the morphology of ferrite grains as predicted using different nucleation mode assumptions. A relationship between nucleation site distribution and the nucleation rate is demonstrated by computer simulation.
I.
INTRODUCTION
IN recent years, considerable effort has been made to better understand the austenite to ferrite ( : ) transformation in steels,[1–15] which is perhaps the most important transformation in terms of its commercial significance. Important detailed information about the transformation process has been discovered with novel experimental techniques such as synchrotron X-ray diffraction[1] and electron backscattered diffraction.[2] As a consequence of such progress, a few new models have been proposed to describe the growth of ferrite,[3,4] which is important in developing new simulations to describe transformation kinetics more precisely. Compared with experimental observation, progress in computer simulation of the : transformation in steels has been limited. For example, nucleation site saturation is assumed in most simulations,[5–13] which is not very realistic. Moreover, electron-backscatter diffraction (EBSD) observation has confirmed that grain coarsening behind transformation fronts is significant,[2] which is the major factor limiting grain refinement in thermomechanical processing. However, grain coarsening behind these transformation fronts has seldom been considered in simulations, most of which have employed the assumption that every nucleus becomes a grain. In this simulation, the authors abandon the site saturation assumption, and take into consideration grain coarsening behind the transformation fronts, as well as the possibility of nucleation off original austenite grain boundaries. The effects of these factors are studied carefully here. Compared with experimental observations, computer simulation can actually better demonstrate how such grain coarsening influences the process of microstructure evolution (which is difficult or even impossible to observe experimentally). CHENG-JIANG HUANG, Newman Scholar, and DAVID J. BROWNE, Senior Lecturer, are with the School of Electrical, Electronic and Mechanical Engineering, Engineering and Materials Science Centre, University College Dublin, Dublin 4, Ireland. Contact e-mail:
[email protected] Manuscript submitted June 25, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS A
The phase-field method is chosen for this study for its excellent capability of simulating interface evolution of any complexity. Phase-field models have no difficulty in simultaneously simulating nucleation, grain growth, and coarsening. It has been known that there are two possible mechanisms governing grain coarsening: coarsening by grain boundary migration and coalescence by grain rotation. (According to Harris et al.,[15] grain rotation is more significant when the involved grains are small, which is exactly the case at the early stage of : transformation. Thus, it is necessary to take both coarsening mechanisms into consideration.) The phase-field method is capable of simulating both coarsening mechanisms, thanks to the recent development of this method by Kobayashi et al.[16,17] and Warren et al.[18] Another important advantage of the phase-field method is its sophisticated interface model. Major variables controlling interface movement, including solute diffusion coefficient, interface mobility, interface energy, and geometry of the interface, are all taken into consideration. With this sophisticated interface model, the phase-field method can even predict the transition from diffusion-controlled transformation to massive transformation in steels,[11,19] and the very complicated phenomenon of interface acceleration accompanying grain impingement.[20] In previous work,[20] the authors have made the case, in a qualitative fashion, that, for the transformation at high cooling rate, some of the ferrite nucleation occurs away from the grain boundaries of the parent austenite. The current study more quantitatively simulates the nucleation events and provides two new types of microstructural evidence for the nucleation behavior proposed, along with a detailed description of grain coarsening behind transformation fronts. II.
THE PHASE-FIELD MODEL
In the phase-field model, the free energy of a system is postulated to be the sum of bulk energy and interface energy, which in turn arises from gradients of order parameters. For VOLUME 37A, MARCH 2006—589
the purpose of this study, the following energy functional is constructed: F
C f(f,C) 2 ƒ§f ƒ kf
2
kj
ƒ§j ƒ 2
su(j) ƒ§u ƒ tv(j) ƒ§uƒ 2 D dV V
Mf Mp 2
[1]
where f(,C), the bulk free energy density, is independent of , the angle of ferrite orientation, i.e., f(,C) is rotationally invariant; is a variable related to the fraction of austenite, ranging from 0 in to 1 in ; represents the state of ferrite phase, ranging from 0 (ferrite grain boundary) to 1 (ferrite crystal), 0 1 in ferrite within a narrow region adjacent to the middle of a ferrite/ferrite grain boundary; and C is carbon concentration. It is necessary to use the additional independent order parameter to describe the ferrite grain boundary because the boundary energy between ferrite grains is quite different from the / interface energy; k, k, s, and t are constants, and u() v() 2. The phase-field variables evolve in such a way that Gibbs free energy F decreases monotonically in time and the total amount of solute is conserved within the system.[14–19] Hence, the following equations are governing: [2]
s # u(j) u § cMu a
t v(j)b §ud t ƒ§uƒ
[3]
1 j u v kj §2 j s ƒ§u ƒ t ƒ§uƒ 2 Mj t j j
[4]
f D(C,f) C dF § cMC §a bd § 2 b 2 §a t dC C f> C
[5]
where M, M, and M are the mobilities of the phase boundary, ferrite grain boundary, and ferrite grain rotation, respectively; Mc is the mobility of carbon; and D(C,) is the diffusion coefficient of carbon, which depends on both composition C and the parameter . Transportation of carbon atoms, including the rejection of carbon atoms from ferrite to austenite, is governed by Eq. [5]. Bulk energy f (,C) is calculated as follows: f (1 p(f))f g(C ) p(f)f a(C) g(f)w,
[6]
where g(f) f2 11 f 2 2, and
p(f) f3 110 15f 6f2 2 . The term g() is a double-well function, which represents an energy barrier against transformation. The parameter w in Eq. [6] is the height of the barrier; and p() represents the fraction of ferrite phase, which was derived in a thermodynamically consistent way. Here, f (C) and f (C) are the Gibbs free energy of the phase and the phase, respectively, and are calculated according to the model proposed by Gustafson.[21]
2s kf
[7]
where parameter k, and another parameter, the height of energy barrier w, are determined by surface energy and interface half-thickness via the following two equations: s
1kfw 6 12
2l a 12
1kf 1w
[8]
[9]
where is a constant.[20] The physical interface mobility is chosen to be Mp 3.5 exp a
140,000 b mol m J 1 s 1 R T
[10]
which comes from an experimental measurement by Wits et al.[22] III.
f 1 f kf §2f Mf t f
590—VOLUME 37A, MARCH 2006
The phase-field mobility is calculated by Eq. [7]:[20]
NUMERICAL TECHNIQUES AND MODEL DEVELOPMENT
A semi-implicit spectral method is employed to solve the equations governing the evolution of microstructure. Since we use a spectral method to solve the partial difference equations, the simulation domain has a periodical symmetry, i.e., periodic boundary conditions are automatically applied. A fully implicit finite difference method (FDM) is used to solve Eqs. [3] and [5]. In order to save memory, an iterative method rather than a direct method is employed to solve the linear equations generated by the implicit FDM scheme. The temperature within the simulation domain is assumed to be homogeneous because of both the high thermal conductivity of steels and the small size of the simulation domain. Carbon diffusion in both austenite and ferrite is taken into consideration. Following Loginova et al.,[11] the / interface energy is chosen to be 1.0 J/m2, and is assumed to be isotropic. A popular method to model nucleation is via introducing stochastic noise, which is graceful mathematically. Here, the authors use a more direct method proposed by Simmons et al.[23] To introduce a new ferrite nucleus, the desired cell is simply set to be at the ferrite state, i.e., set 1.0, which acts as the center of the new ferrite nucleus, and a gradient of is created surrounding the nucleus center. The authors revealed the relationship between phase-field mobility M and the physical properties of the / interface in a previous analysis[20] of this phase-field model (Eq. [7]), which led to a conclusion that phase-field mobility should be unambiguously determined by interface properties. This implies that phase-field mobility is not a parameter that can be arbitrarily adjusted. On the other hand, the precision of data on austenite/ferrite interface energy is not as reliable as one would hope it to be, nor are the data on austenite/ferrite interface mobility. More importantly, the parameter k in this equation is related to the computational interface thickness 2, which must be larger than a physical interface thickness to make possible larger scale phase-field METALLURGICAL AND MATERIALS TRANSACTIONS A
simulation. Consequently, it is necessary to introduce a parameter kM to compensate for all of these uncertainties. Hence, Eq. [7] becomes Mf kM Mp
2s kf
[11]
where kM is a coefficient to be determined by fitting simulation results with measurement, being adjusted by comparing the velocity predicted by phase-field simulation and experimental measurements. This parameter is different for phase-field models employing different interface parameters. IV.
RESULTS
Before proceeding to phase-field modeling of continuous cooling transformation, it is necessary first of all to validate the model by comparison with the results of simpler, isothermal, transformation experiments. This approach guides the structure of the following subsections.
Fig. 1—Carbon distribution across the austenite/ferrite interface during the isothermal transformation of Fe-1.065 at. pct C at 1050 K, at different times. The initial interface is located at the position 0 m.
A. Simulation of Isothermal Transformation The isothermal transformation experiment done by Bradley et al.[24] is simulated to study the interface behavior and to calibrate parameter kM for this phase-field model. A series of one-dimensional phase-field simulations was carried out to study the behavior of a planar austenite/ferrite interface during isothermal transformation. The initial carbon concentration in austenite is 1.065 at. pct. The total length of the simulation domain is 16.21 m. The temperature of the entire domain is assumed to remain at 1050 K during transformation. The interface moves from left to right with the : transformation. At time t 0 seconds, a very thin layer of ferrite is added to the left end of the specimen to simulate ferrite “nucleation.” Figure 1 shows 25 predicted carbon distribution profiles along the length of the domain during transformation, at different times. The thin layer of material between the and phases is the diffuse / interface. A carbon pileup has been built up next to the interface on the austenite side, caused by rejection of carbon from ferrite to austenite. The peaks of the profiles indicate the position of the / interface. Carbon concentration in the austenite phase keeps rising until the entire system approaches overall equilibrium, with both austenite and ferrite at the thermodynamic equilibrium composition[21] (the last curve from the right in Figure 1, which is at t 296.1 seconds). Note that the space interval between the neighboring curves becomes smaller with time. Since the time intervals between neighboring curves are identical, this indicates that interface movement is decelerating. It is worth emphasizing that local equilibrium is not established on the (/) interface before the entire system reaches thermodynamic equilibrium. The carbon concentration of the (/) interface keeps approaching the equilibrium value, but does not reach it until the end of transformation. In other words, the interface is not at local equilibrium before the end of transformation. Figure 2 shows the carbon concentration profile at an early stage of growth, before t 10.57 seconds. Carbon concentration must increase gradually from initial concentration. At this stage, carbon concentration at the / interface is farMETALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 2—Carbon distribution across the austenite/ferrite interface during the early part of the isothermal transformation. The first curve from the left is the carbon distribution after 1.06 s of transformation. The following curves are carbon distribution profiles at later times, with a time interval between neighboring curves of 1.06 s. The dash line reveals the original / interface and the original carbon concentrations. The curves in this figure clearly show the building up of the concentration profile ahead of the interface.
ther away from local equilibrium. The deviation from local equilibrium is more significant and the interface moves quicker than in the later stages of Figure 1. On the other hand, interface deceleration is already obvious in this figure, owing to the pile up of carbon in front of the interface. The first curve in Figure 2, which represents a carbon profile at t 1.06 seconds, has a nonsmoothness at the position of the initial austenite/ferrite interface. This is due to the jumping of initial composition across the interface, represented by the dash line in this figure. After t 1.06 seconds, the jump is completely smoothed out. From the analysis of Figures 1 and 2, it is clear that both interface mobility and carbon diffusion play a role during the entire process of transformation. Interface movement VOLUME 37A, MARCH 2006—591
Fig. 4—Comparison between the predicted parabolic growth coefficient and the measured ones from Reference 24. Fig. 3—Phase-field simulation of ferrite thickness as a function of square root of time during the isothermal transformation of Fe-1.065 at. pct C at 1050 K.
has never been exclusively interface controlled, or diffusion controlled. Figure 3 shows the growth of ferrite phase as a function of the square root of time. Ferrite thickness increases linearly with the square root of time for most of the transformation, obeying a parabolic growth law. It deviates somewhat from the parabolic growth law during a very brief transient period at the start of transformation and also near the end of the transformation due to the diffusion layer hitting the boundary of the domain. This result is in agreement with the phase-field simulation of Loginova et al.[11] In principle, the parabolic growth law of classical theory is applicable only when local equilibrium is built up at the interface. Simulation results shown in Figures 1 and 2 indicate that local equilibrium cannot be reached before the end of transformation. Then, how can one expect the parabolic growth mode shown in Figure 3? The explanation also comes from study of Figure 1. Although the peak carbon concentrations of all the curves shown in Figure 1 do not reach equilibrium, they are not far from equilibrium. The heights of the peaks are quite close to each other. Hence, ferrite growth still approximately obeys a parabolic law for a good part of the transformation, as is shown by the approximately straight line in Figure 3. We find kM is a function of temperature via comparing calculated parabolic growth coefficients with experiments: kM 0.14T 152.2
[12]
where T is absolute temperature, 960 K T 1130 K. Figure 4 shows the comparison between phase-field prediction and measurement of parabolic growth coefficient for isothermal transformation in the low-carbon steel.[24] Phasefield prediction fits well with measurements within the wide temperature range. The parameter kM calibrated in this section is used to predict growth rate in the simulation of the continuous cooling transformation in another steel, Fe-0.78 at. pct C-0.75 at. pct Mn. This is plausible for two reasons. First, as pointed out 592—VOLUME 37A, MARCH 2006
by Wells et al.,[25] manganese has only insignificant influence on carbon diffusion in steels. Second, the finite reduction of interface mobility caused by manganese has little influence on ferrite growth rate. According to Witts et al.,[22] increasing the manganese concentration by 1.0 at. pct results in interface mobility decreasing by about an order of magnitude. However, a previous study by Enomoto[26] has shown that the growth rate of proeutectoid ferrite will slow by less than 5 pct even if interface mobility of a similar steel decreases tenfold. B. Simulation of Continuous Cooling Transformation 1. The model The continuous cooling transformation experiment on Fe0.78 at. pct C-0.75 at. pct Mn, done by Militzer et al.,[27] was chosen as the object of simulation. The simulation domain is 200 m 200 m, which is discretized by a 512 512, two-dimensional grid. A separate phase-field simulation generated the initial austenite microstructure. The initial mean grain size of austenite is 18 m. As in previous simulations of this experiment,[5,6] the partition of manganese is neglected, since Aaronson and Domain[28] had shown that partitioning of manganese between austenite and ferrite occurs only at higher transformation temperatures. The authors do not assume site saturation for ferrite nucleation, unlike the previous simulations[5,6] of the same experiment. Furthermore, grain coarsening is allowed in this phase-field simulation, i.e., grain coarsening behind transformation is taken into consideration and not necessarily every nucleus becomes a new grain. Ferrite grains are assumed to nucleate preferentially on grain boundary junctions. If all of these junctions are occupied, then new ferrite grains will nucleate on the remaining austenite boundaries. When all of these grain boundaries become populated, subsequent ferrite grains will have to nucleate off the original austenite grain boundaries. New ferrite nuclei are taken to be 1m in diameter in this simulation. Consequently, any new nucleus must be at least 1.0 m away from any existing ferrite grains, which is the result of carbon enrichment in neighboring areas of existing ferrite phase. The current nucleation model is capable of predicting a more realistic nucleation rate since it takes into account the nuclei consumed METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 5—Comparison of phase-field prediction and measurements of the transformation kinetics during the continuous cooling of Fe-0.78 at. pct C-0.75 at. pct Mn. Scatter points are from measurements[27] and solid lines are from phase-field predictions.
Fig. 6—Phase-field prediction of the total number of ferrite grains as a function of temperature during the continuous cooling transformations of Fe-0.78 at. pct C-0.75 at. pct Mn.
during transformation, which is also helpful in reproducing a more realistic microstructure. In this simulation, nucleation rate is determined by fitting simulated kinetic curves with the experimental ones.
of these “disappear” before the end of transformation (Figure 6). At 19 K/s, about 45 pct of the nucleated grains finally disappear, and at 1 K/s, the figure is 43 pct. Cotrina et al.[2] observed that the ferrite grain number declined by 50 to 80 pct during the transformation of deformed steel, Fe-0.082 wt pct C-1.5 wt pct Mn-0.36 wt pct Si. The current simulation is close to that observation. The slightly reduced grain coarsening in this simulation can be reasonably attributed to the fact that no deformation was applied to the steel in the experiments of Militzer et al.[27] Figure 6 shows that ferrite grain coarsening is significant during the transformation. Recently, Cotrina et al.[2] directly observed by EBSD such grain coarsening behind the transformation front during continuous cooling transformation in a carbon steel. Earlier than that, Priestner et al.[12] observed that ferrite grains (not nuclei), which formed earlier on original austenite grain boundaries, were much finer than the final ferrite grains. This observation also led to the conclusion that significant ferrite grain coarsening occurred during transformation. The current simulation result is in accordance with the two preceding experimental observations. About 50 pct of the nuclei have been consumed by the end of transformation. Thus, the assumption that every nucleus becomes a grain, which was employed in previous simulations,[4–7] is not a good approximation.
2. Predicted transformation curves Figure 5 shows the simulated ferrite-fraction/temperature curves for three transformation cooling rates: 1, 19, and 58 K/s. Generally speaking, phase-field predictions fit quite well with measurements. As mentioned previously, the curves in this figure are obtained by fitting the simulated ferrite fraction with measurements, via adjusting the nucleation rate. Parameters influencing ferrite grain coarsening are also adjusted, to ensure that final ferrite grain size is in accordance with experimental observations. Note the greater undercooling for nucleation and growth at high cooling rates. Figure 6 shows the simulation result of the ferrite grain number at different stages of transformation, from which both nucleation and ferrite coarsening can be clearly seen. At all cooling rates, the number of ferrite grains climbs quickly to a maximum due to nucleation, shortly after the start of transformation. The majority of nucleation events occur before 30 pct of austenite is transformed. Note the greater nucleation rate at high cooling rates. Competition between nucleation and grain coarsening is in balance at the peaks of the three curves in Figure 6. During the succeeding transformation process, after the ferrite number reaches a maximum, the decline in total ferrite number is rather quick. But grain coarsening, which is represented by the declining number of ferrite grains, slows significantly thereafter, while the grains grow larger. By the end of transformation, the rate of decline approaches zero. Ferrite coarsening is most significant around the temperatures at which ferrite number reaches peak value. It is less significant near the end of transformation, when nucleation almost stops completely. Based on this observation, it is reasonable to regard the microstructure at the end of the transformation as the final microstructure. The peak number of ferrite grains during continuous cooling at 58 K/s is 3759 in the simulation domain. About 47 pct METALLURGICAL AND MATERIALS TRANSACTIONS A
3. Two mechanisms of coarsening Figure 7 shows four snapshots of the microstructure evolving when grain coarsening behind the transformation front is in progress, during cooling at 1 K/s. The black background represents austenite. The colorful grains are ferrite. Different colors of ferrite grains denote different crystal orientations. The black boundary between ferrite grains represents ferrite grain boundaries, whose width reflects the orientation difference between neighboring grains. In fact, two different mechanisms of ferrite grain coarsening are revealed. The larger white arrow in Figure 7(a) points to a red ferrite grain surrounded by its larger neighbors. It shrinks somewhat when the domain is cooled to 1007.64 K (Figure 7(b)), obviously being consumed by the VOLUME 37A, MARCH 2006—593
Fig. 7—Phase-field simulation of grain coarsening behind the transformation fronts during the continuous cooling transformation of Fe-0.78 at. pct C-0.75 at. pct Mn at 1 K/s. The four snapshots show the simulated microstructure at temperate (a) 1011.7 K (b) 1007.64 K (c) 1003.58 K and (d) 999.52 K.
Fig. 11—(a) Overcrowded ferrite nuclei congregated on original austenite grain boundaries during the continuous cooling transformation, assuming nucleation is exclusively on grain boundary. (b) The resultant final microstructure. The black matrix is austenite. Each colorful spot represents a ferrite nucleus/grain. 594—VOLUME 37A, MARCH 2006
METALLURGICAL AND MATERIALS TRANSACTIONS A
much larger neighboring grains. It keeps shrinking during the transformation (Figures 7(c) and (d)), and will finally disappear. This kind of coarsening is done by the first mechanism of ferrite grain coarsening: grain boundary migration. Reducing the total interface energy is the driving force. There is a second mechanism of grain coarsening, which is also revealed in Figure 7. See the two small grains indicated by the smaller white arrow. These two grains merged into one grain when the simulation domain was at 1007.64 K (Figure 7(b)), without any obvious grain boundary migration. The boundary between them simply disappears. This sort of grain coarsening, or more precisely, coalescence, is the result of the orientation of the two grains becoming identical, via rotation. Hence, the two grains become one. No grain boundary migration is needed for this sort of coalescence. The evolution of crystal orientation, i.e., grain rotation, is commonplace during phase transformation. It is an important mechanism of grain coarsening.[15,16,17] Looking closely at Figure 7, one can see many more examples of grain coarsening via both mechanisms. Figure 8 shows four typical growth curves of different individual ferrite grains. Curve 1 represents the growth of a ferrite grain, which first grows a little bit when cooled from 1030 to about 1012 K. The grain then shrinks continuously in the successive cooling process, until it disappears totally at about 990 K. From the shape of curve 1, it is obvious that this ferrite grain is consumed by its neighboring grains gradually, via grain boundary migration. Curve 2 reveals a step at about 980 K. Obviously, this kind of sudden growth is only possible through the second mechanism of coarsening: grain coalescence. It is the consequence of a smaller neighboring grain rotating to the same orientation of this grain, becoming a new part of the grain. Offerman et al.[1] observed similar sudden growth of ferrite grains during the continuous cooling transformation in a low alloy steel, via real-time X-ray diffraction, which was explained by “growth of ferrite into pearlite.” Our
Fig. 8—Typical growth curves of individual ferrite grains during the transformation of Fe-0.78 at. pct C-0.75 at. pct Mn at 1 K/s; results of phasefield simulation. METALLURGICAL AND MATERIALS TRANSACTIONS A
phase-field simulation unveils another possible mechanism for this phenomenon. Although the rotation of a grain enclosed in a grain aggregate seems hard to understand, such rotation has been observed experimentally more than once.[15] This kind of rotation is made possible by assistance of fast atomic diffusion along grain boundaries. Grain boundary areas are made quite flexible by such localized fast diffusion, and hence can tolerate the rotation of small grains. The driving force of such rotation comes from the reduction of surface energy via the decreasing misorientation.[15] Curve 3 is the continuous growth curve of a common ferrite grain, experiencing neither coarsening nor shrinking during the transformation. It is the impingement with neighboring grains that nearly stops its growth, at about 985 K. Curve 4 has an accelerating growth stage around 1017 K, which is less dramatic than the sudden growth acceleration on curve 2, and occurs, by induction, via interface migration. 4. Carbon redistribution Figure 9(a) shows the predicted carbon distribution field when the simulation is cooled to 1003.58 K during this transformation. The light areas are ferrite grains. The darker areas are austenite. Gray lines between ferrite grains are ferrite grain boundaries. The different gray scale denotes different carbon concentrations. It is obvious that carbon is enriched near austenite/ferrite interfaces. The enrichment of carbon comes from the rejection of carbon atoms from ferrite to austenite, which is more significant in some smaller austenite domains surrounded by ferrite grains. Since ferrite grains were nucleated at different stages of transformation, one cannot expect interface carbon concentration to be identical. Figure 9(b) depicts the carbon distribution along a line shown in Figure 9(a). Carbon distribution in austenite is quite inhomogeneous, while that in ferrite is quite homogeneous, obviously due to the much higher diffusion coefficient in the ferrite phase. It is also worth noting that interface carbon concentration does not reach the equilibrium value at this temperature. Figure 10 shows the final microstructures after different transformations, predicted by phase-field simulation. The larger dark areas are pearlite domains, and the light areas are ferrite grains. Boundaries between ferrite grains are of different width and shade, which represents the different orientation difference between neighboring grains. The predicted final ferrite grain size is 12.7 m for cooling at 1 K/s, 9.1 m at 19 K/s, and 5.0 m at 58 K/s. These predictions are quite close to experimental observation.[27] The value of the parameters s, t, and k, used in Eq. [4], are 0.1, 0.5, and 50, respectively. The measurements of mean ferrite grain size in the simulation results are done with the image analysis software Image-Pro (Media Cybernetics Co., Silver Spring, MD). The fraction of pearlite is approximately 15 pct for all three final microstructures. However, both the size and shape of pearlite domains are quite different from one case to another. The microstructure of slow cooling transformation (Figure 9(a)) has a small number of relatively large pearlite domains. On the contrary, the final microstructure of transformation at 58 K/s (Figure 10(c)) reveals much finer pearlite domains, almost indistinguishable from ferrite grain boundaries. This simulation result is quite close to the optical microstructure photos by Militzer et al.[27] VOLUME 37A, MARCH 2006—595
Fig. 9—(a) and (b) Phase-field simulation of carbon concentration distribution at 1003.58 K during the continuous cooling transformation of Fe-0.78 at. pct C-0.75 at. pct Mn at 1 K/s. (b) depicts the carbon distribution along the line shown in (a).
Fig. 10—Phase-field simulated final microstructure after continuous cooling transformation at (a) 1 K/s, (b) 19 K/s, and (c) 58 K/s.
V.
DISCUSSION
A. Further Evidence of Nucleation Away from Grain Boundary It is well known that faster cooling leads to higher nucleation rate, hence grain refinement. Previous computer modeling studies[6,7,8] have confirmed this phenomenon. The current phase-field simulation predicts an even greater effect of cooling rate on nucleation rate, because grain coarsening behind the transformation front is taken into account. However, this effect is only part of the story of grain refining at increased cooling rates. Increasing cooling rate has another important effect on nucleation: nucleation off grain boundaries. At the low cool596—VOLUME 37A, MARCH 2006
ing rate of 1 K/s, all ferrite grains nucleate on the original austenite grain boundaries. At 19 K/s, the nucleation rate increases about twofold. The original austenite boundaries have all been occupied before the nucleation process is completed. As a result, about 15 pct of the ferrite nuclei have to dock off the austenite grain boundaries. At the higher cooling rate of 58 K/s, because of the even higher nucleation rate (Figure 6), about 70 pct of ferrite nucleation is not on any original grain boundaries. In this case, the original austenite grain boundaries are no longer the dominant sites for nucleation. Nucleation away from the original grain boundaries can also be justified via analysis of the final microstructure. First, such nucleation is necessary so that the predicted shape of METALLURGICAL AND MATERIALS TRANSACTIONS A
ferrite grains is equiaxed, i.e., in agreement with experimental observation. For the transformation at 58 K/s, there must be more than 2050 nuclei in the simulation domain. (The deduction of 2050 ferrite grains comes from the ferrite grain density in the final microstructure, assuming every nucleus becomes a grain). Supposing all of these nuclei originate on grain boundaries exclusively, these boundaries will be overcrowded, as is shown in Figure 11(a). The consequence of this sort of overcrowded nucleation is that the shape of growing ferrite grains cannot remain equiaxed. Testing simulation shows that the resultant microstructure will be like that shown in Figure 11(b): many ferrite grains inevitably become columnar grains rather than equiaxed grains. Second, the diversification of pearlite domains at higher cooling rates provides other evidence of ferrite nucleation off grain boundaries. Supposing no ferrite nucleated off the initial austenite grain boundaries, the size and shape of pearlite domain will be like the dark areas shown in Figure 11(b): larger pearlite domain converging in a few locations, and no diversification of pearlite domains, which is significantly different from the real pearlite microstructure observed by Militzer et al.[27] B. Difference between Measured Parabolic Growth Coefficient and That Predicted Using Local Equilibrium Assumption The validity of local equilibrium was questioned by scholars more than once. According to Hillert,[29] the movement of a practical interface is controlled by two factors: interface mobility and solute diffusion. Sietsma and van der Zwaag referred to this kind of description as “mixed mode,”[30] a mode between diffusion control (local equilibrium) and interface control, and introduced a parameter to quantify the extent of “mixture.” As a matter of fact, Bradley et al.[24] found that the measured parabolic growth coefficients were systematically lower than the predictions based on the local equilibrium assumption. However, no solid evidence was presented to disprove the assumption of local equilibrium. One may also argue that local equilibrium may still hold since the high predicted growth coefficient may also be the result of choosing too high a value of diffusion coefficient. Our simulation reveals a different explanation for this discrepancy, and it is quite possible that the problem is not caused by too high a diffusion coefficient. Even if the diffusivity of carbon is known to high enough accuracy, the predicted growth rate may still be higher than measurements. Observing Figure 1, the / interface is not at local equilibrium because the carbon concentration peaks are lower than the equilibrium carbon concentration, which is illustrated by the last profile. According to Zener,[31] the parabolic growth coefficient in a totally diffusion-controlled transformation is given by a 2
D 1Cp C0 2 2
1Cp Cae 2 1C0 Cae 2
[13]
where is the parabolic growth efficient, D is diffusion coefficient, Cp is the peak solute concentration, C0 is the initial concentration, and Cea is the concentration in the product phase. 1a2 2 D(Cp C0) It can easily be shown that Cp Cp # C0 METALLURGICAL AND MATERIALS TRANSACTIONS A
Cp C0 c d 0, noting Cae is close to 0 and Cp C0. Hence, Cp the parabolic growth coefficient increases with the increasing peak carbon concentration. Thus, the assumption of local equilibrium will always lead to a higher prediction of growth rate than in reality, since Figure 1(b) has indicated peak carbon concentration is always lower than the equilibrium value before the system reaches overall equilibrium.
VI.
CONCLUSIONS
A phase-field simulation is performed to study the kinetics of austenite-to-ferrite transformation in low-carbon steels, with ferrite grain coarsening behind the transformation front taken into consideration, at a comparatively large space scale. Grain coarsening is significant at the early stage of transformation. About 50 pct of ferrite nuclei are consumed before the end of transformation. This sort of ferrite coarsening is realized via the two different coarsening mechanisms, namely, coarsening by grain boundary migration and coalescence by rotation of crystal orientation. Simulated growth curves of individual grains show that a sudden increase of ferrite radius sometimes occurs, which is caused by the second mechanism of grain coarsening (grain rotation). This simulation result provides an alternative explanation for the sudden grain radius increases observed by real-time X-ray diffraction,[1] which was originally explained as the continuous growth of ferrite into pearlite. The ferrite nucleation mode, including nucleation rate and nucleation sites at different stages of transformation, is reproduced by simulation, via fitting the simulated transformation kinetics to measurements. The nucleation rate is found to be about 50 pct higher than the prediction of previous computer simulations, as a result of taking into consideration grain coarsening behind transformation fronts in the current work. It is demonstrated by simulation that there is a close relationship between nucleation density and the shape of grains in final microstructure. The phase-field simulation shows that nucleation off original austenite grain boundaries is necessary in order to keep ferrite grains equiaxed. Such nucleation is also necessary to diversify the pearlite domains, as has been noted in experimental observations. ACKNOWLEDGMENTS C.-J. Huang gratefully acknowledges Enterprise Ireland for supporting this research through a Newman Scholarship at the University College Dublin (UCD), and financial support from the National Science Foundation of China under Contract No. 50371082. The authors are grateful for the insightful suggestions of Shaun McFadden and Dr. Marek Rebow, UCD. REFERENCES 1. S.E. Offerman, N.H. VanDijk, J. Sietsma, S. Grigull, E.M. Lauridsen, and L. Margulies: Science, 2002, vol. 298, pp. 1003-05. 2. E. Cotrina, A. Iza-mendia, B. Lopez, and I. Gutierrez: Metall. Mater. Trans. A, 2004, vol. 35A, pp. 93-101. 3. S.E. Offerman, N.H. VanDijk, J. Sietsma, S. Grigull, E.M. Lauridsen, and L. Margulies: Acta Mater., 2004, vol. 52, pp. 4757-66. VOLUME 37A, MARCH 2006—597
4. J. Sietsma and S. van der Zwaag: Acta Mater., 2004, vol. 52, pp. 4143-52. 5. Y.J. Lan, D.Z. Li, and Y.Y. Li: Acta Mater., 2004, vol. 2, pp. 1721-29. 6. L. Zhang, C.B. Zhang, Y.M. Wang, S.Q. Wang, and H.Q. Ye: Acta Mater., 2003, vol. 51, pp. 5519-27. 7. M. Kuma, R. Sasikumar, and P.K. Nair: Acta Mater., 1998, vol. 46, pp. 6291-93. 8. S. Kumdu, M. Dutta, S. Ganguly, and S. Chandra: Scripta Mater., 2004, vol. 50, pp. 891-95. 9. I. Loginova, J. Odqvist, and G. Amberg: Acta Mater., 2003, vol. 51, pp. 1327-39. 10. D.-H. Yeon, P.-R. Cha, and J.-K. Yeon: Scripta Mater., 2001, vol. 45, pp. 661-68. 11. I. Loginova, J. Agren, and G. Amberg: Acta Mater., 2004, vol. 52, pp. 4055-63. 12. R. Priestner and P.D. Hodgson: Mater. Sci. Technol., 1992, vol. 8, pp. 849-54. 13. H.I. Aaronson, W.F. Lange, and G.R. Purdy: Scripta Mater., 2004, vol. 51, pp. 931-35. 14. G. Pariser, P. Schaffnit, I. Steinbach, and W. Bleck: Steel Res., 2001, vol. 72, pp. 354-60. 15. K.E. Harris, V.V. Singh, and A.H. King: Acta Mater., 1998, vol. 46, pp. 2623-33. 16. R. Kobayashi, J.A. Warren, and W.C. Carter: Physica D, 1998, vol. 119, pp. 415-23.
598—VOLUME 37A, MARCH 2006
17. R. Kobayashi, J.A. Warren, and W.C. Carter: Physica D, 2000, vol. 140, pp. 141-50. 18. J.A. Warren, R. Kobayashi, A.E. Lobkovsky, and W.C. Carter: Acta Mater., 2003, vol. 51, pp. 6035-58. 19. S.G. Kim, W.T. Kim, and T. Suzuki: Phys. Rev. E, 1999, vol. 60, pp. 7186-97. 20. C.J. Huang, D.J. Browne, and S. McFadden: Acta Mater., 2006, vol. 54, pp. 11-21. 21. P. Gustafson: Scand. J. Metall., 1985, vol. 14, pp. 259-97. 22. J.J. Wits, T.A. Kop, Y. van Leeuwen, and S. van der Zwaag: Mater. Sci. Eng., 2000, vol. 283A, pp. 234-41. 23. J.P. Simmons, C. Shen, and Y. Wang: Scripta Mater., 2000, vol. 43, pp. 935-42. 24. J.R. Bradley, J.M. Rigsbee, and H.I. Aaronson: Metall. Trans. A, 1977, vol. 8A, pp. 323-33. 25. C. Wells, W. Batz, and R.F. Mehl: Trans. AIME, 1950, vol. 188, pp. 553-56. 26. M. Enomoto: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 2309-16. 27. M. Militzer, R. Pandi, and E.B. Hawbolt: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 1547-56. 28. H.I. Aaronson and H.A. Domain: Trans. AIME, 1966, vol. 236, pp. 781-88. 29. M. Hillert: Metall. Trans. A, 1975, vol. 6A, pp. 5-18. 30. J. Sietsma and S. van der Zwaag: Acta Mater., 2004, vol. 52, pp. 4143-52. 31. C.J. Zener: Appl. Phys., 1949, vol. 20, pp. 993-1010.
METALLURGICAL AND MATERIALS TRANSACTIONS A
