- Email: [email protected]
A mixed-mode model for partitioning phase transformations
Scripta Materialia 57 (2007) 1085–1088 www.elsevier.com/locate/scriptamat
A mixed-mode model for partitioning phase transformations C. Bosa,b and J. Sietsmab,* a
b
Netherlands Institute for Metals Research, Mekelweg 2, 2628 CD Delft, The Netherlands Delft University of Technology, Department of Materials Science and Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands Received 5 July 2007; revised 23 August 2007; accepted 26 August 2007 Available online 21 September 2007
A mixed-mode model for the interface velocity during partitioning phase transformations [J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143–4152] is reformulated and rigorously validated. Comparison with a numerical treatment shows that the model gives a near-exact description of the growth kinetics. In addition, the model provides information on the concentration profiles of the partitioning element. The mixed-mode growth model can be combined with a range of different microstructure models. As an example a combination with a Johnson–Mehl–Avrami extended volume model is shown. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase transformation kinetics; Modelling; Precipitation; Mixed-mode kinetics
A number of recent publications [1–5] have shown that traditional interface-controlled [6] and diffusioncontrolled [7] models are often unable to describe correctly the growth kinetics of solid-state partitioning phase transformations in metals over the entire course of the transformation. Consequently, a mixed-mode model that can take both effects, and the varying relative importance of each, into account should lead to a significant improvement in the modelling of the growth kinetics. Most phase-transformation models can be considered to consist of a part that describes the interface velocity and a part that describes the geometrical aspects of the microstructure. In Ref. [1], a concise mixed-mode growth kinetics model was introduced based on widely accepted differential equations for the interface process and the diffusion process. However, a study of the mixed-mode character in phase-field simulations of the isothermal austenite to ferrite transformation in an Fe–C–Mn steel has shown that the mixed-mode model from Ref. [1] overestimates the diffusional character of the transformation [8]. In the present work, an exponential concentration profile is introduced, leading to a significant improvement in the description of the mixedmode character without losing the concise formulation and the efficiency of the original model. In contrast to Ref. [1], a rigorous validation of the mixed-mode growth kinetics model is provided here by means of a compari* Corresponding author. E-mail: [email protected]
son with a fully numerical solution that exactly describes the concentration profile in front of the moving interface according to widely accepted differential equations for the interface velocity and the diffusion [6]. To illustrate the versatility of the mixed-mode description of the interface velocity, it is combined with a Johnson– Mehl–Avrami (JMA) description of the microstructure geometry to give a model for the kinetics of the austenite to ferrite transformation in an Fe–C–Mn steel under para-equilibrium conditions. The model considers a solid-state phase transformation in which partitioning of a single element takes place, say b ! a. The mixed-mode character of the transformation is quantified by the mode parameter S [1], which is defined as S¼
xba xb ; xba x0
ð1Þ
where x0 is the overall concentration of the partitioning element, xb is the concentration in the b phase at the interface, and xba is the concentration in the b phase in equilibrium with the a phase (see Fig. 1, xab is the concentration in the a phase in equilibrium with the b phase). In this model, it is assumed that the concentration in the a phase is homogeneous at the equilibrium value xab (as determined by the temperature). The mode parameter S will have a value between 0 and 1: S = 0 means that the transformation is completely diffusioncontrolled (local equilibrium: xb = xba); S = 1 means
1359-6462/$ - see front matter 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2007.08.030
1086
C. Bos, J. Sietsma / Scripta Materialia 57 (2007) 1085–1088
z0 ¼
Figure 1. The concentration, x, of the partitioning element near the interface.
V a x0 xab : XAa xb x0
ð6Þ
The factor X stems from the z-dependence of A, and thus depends on the particle size ra. X = 1 for systems with a constant surface area (e.g., one-dimensional systems). For growing disks X2D ¼ 1 þ rza0 , and X3D ¼ 1 þ 2 rza0 þ 2ðrza0 Þ2 for spheres. At the interface the rate of partitioning as determined by the interface velocity (Eq. (2)) must be equal to the diffusion flux (as given by Fick’s first law), which combined with Eq. (6) leads to a quadratic equation for the interface concentration xb 2
xb ¼ ðZx0 þ Dx0 xaþb þ ½ðZx0 þ Dx0 xaþb Þ 1
ðZ þ 2Dx0 ÞðZx20 þ 2Dx0 xab xba Þ2 Þ=ðZ þ 2Dx0 Þ
that the transformation is completely interfacecontrolled (xb = x0). In general, the interface velocity can be written as [6]
with Dx0 = x0 x , x
v ¼ MDGI ðT ; xb Þ
Z ¼ 2X
ð2Þ
with M the interface mobility and DGI the driving force. Here DGI is written for the driving force to indicate that only the part of the total driving force that is dissipated in the interface processes are considered in Eq. (2) (see e.g. [9]). The driving force is a function of temperature, T, and the interface concentration, xb. For xab < x0 < xba, at the moving interface partitioning takes place from the a to the b phase with a rate determined by the interface velocity, and diffusion of the partitioning element in the b phase takes place with a flux given by Fick’s law, J = Dox/oz, with D the diffusivity of the partitioning element in the b phase. This moving-boundary problem can be solved by using a variant of the Murray–Landis one-dimensional finite-difference method [10], which has often been applied to purely diffusion-controlled phase transformations [11]. This solution will be referred to hereafter as the fully numerical solution. An expression for the concentration xb can be formulated on the basis of two straightforward assumptions: (1) The driving force, DGI, is proportional to the deviation from the equilibrium concentration DGI ¼ vðxba xb Þ
ð3Þ
with v a proportionality factor which can, for example, be calculated using Thermocalc. This linear proportionality is an approximation that usually holds well within the relevant composition range. (2) The concentration profile in front of the interface as a function of position, z, can be described by z ð4Þ x ¼ x0 þ ðxb x0 Þ exp z0 with z = 0 at the interface (see Fig. 1) and z0 defines the width of the profile. The mass balance for a single a-grain of volume Va and surface area Aa in infinite space, given by Z 1 V a ðx0 xab Þ ¼ AðzÞðx x0 Þ dz ð5Þ 0
with A(z) the surface area of the grain at position z, determines the parameter z0 in Eq. (4)
ab
a+b
ab
=x
+x
ba
ð7Þ
and
a
D A : Mv V a
ð8Þ
The complication that X depends on z0, can easily be solved by determining xb by an iterative procedure over Eqs. (6)–(8) starting with X = 1. Usually, the solution converges after less than 10 iterations within a deviation of less than 0.1% of the true value. The equation for xb (Eq. (7)) is the same equation as in the original model [1,4]; the only difference is the factor 2X in the equation for Z (Eq. (8)). However, as will be shown below, this factor proves to be essential to arrive at the correct mixed-mode interface concentration. The mode parameter S is a direct measure for the kinetics of the transformation: combining Eqs. (1)–(3) gives v ¼ MvSðxba x0 Þ;
ð9Þ
where only S depends on the mixed-mode character. On the basis of Eq. (9) the transformation kinetics can be described at a fraction of the computational cost of the fully numerical method. This equation, however, does not take the effects of soft impingement into account. As the growing grains start to approach each other, the concentration profiles surrounding the grains will start to overlap. This effect can be taken into account in a mean-field approximation. If the partitioning element would redistribute completely in the b phase, the (homogeneous) concentration, xb,h, is given by xb;h ¼
x0 fa xab ; 1 fa
ð10Þ
where fa is the transformed fraction. The mean-field soft-impingement approximation can now be obtained as follows. First S is calculated according to Eqs. (7) and (1). Then, x* is introduced as x ¼ xba Sðxba xb;h Þ; b,h
ð11Þ x*
where x is given by Eq. (10). Subsequently, is taken as the concentration in the b phase for the calculation of the driving force DGI and the interface velocity according to Eq. (2) is used. In order to account for hard impingement the extended volume concept, the basis of JMA models
C. Bos, J. Sietsma / Scripta Materialia 57 (2007) 1085–1088
where V is the total system volume. If N_ ðtÞ is the nucleation rate (i.e., the number of nuclei formed per unit of time), the extended volume can be written as Z t d Z t N_ ðsÞgd vðra ; T Þ ds ds; ð13Þ Ve ¼ 0
s
with gd a geometrical factor, and d the growth dimension. In this work all nuclei (N*) are assumed to form at the start time of the transformation (t = 0, site saturation), which means that Z t d V e ¼ N gd vðra ; T Þ ds ð14Þ 0
for the extended volume. It is possible to use the extended volume approach with the mixed-mode velocity inserted in Eq. (14), but also combined with the fully numerical solution. The integral from Eq. (14) is then replaced with the volume of a growing sphere as obtained from the numerical solution. However, the numerical model is not as flexible as the analytical mixed-mode model combined with the mean-field soft-impingement correction in dealing with growing particles of different sizes as in the JMA extended volume approach. In the numerical solution the maximum grain size determines when soft impingement will start to affect the growth rate. In this work, the maximum grain size is set equal to the average grain size (as determined from the nucleation density). This ensures that soft impingement is described correctly for the early stages of the transformation. However, it also limits the maximum size of the growing grains, which means that the maximum fraction transformed predicted by the combination of the numerical solution with the extended volume concept will be too low. To benchmark the performance of the mixed-mode model, the austenite to ferrite transformation kinetics in an Fe–0.46 C–0.5 Mn (at.%) steel have been studied. Para-equilibrium is assumed, that is, only carbon partitions, and the diffusion of carbon in the austenite phase is described by D = D0 exp(QD/RT), with D0 = 0.15 · 104 m2/s and QD = 142.1 kJ mol1 [12] (R is the gas constant). The interface mobility, M, is given by M = M0 exp(QG/RT), with M0 = 3.5 · 107 m4/Js and QG = 140 kJ mol1 [13]. The driving force is obtained from Thermocalc and described by Eq. (3). Two different geometries have been examined: (i) A single spherical grain in one, two and three dimensions, which has been used with the numerical solution and the mixed-mode growth kinetics model under isothermal conditions (T = 950 K)
with a maximum grain radius of 25 lm. For the fully numerical calculations a grid with 512 nodes was used. With this geometry, no impingement corrections were used with the mixed-mode model. (ii) A microstructure in two dimensions, which has been used with the mixed-mode model with soft and hard impingement corrections according to Eqs. (11) and (12). Here isochronal cooling was used (from T = 1091–920 K with cooling rates of 2, 5 and 15 K s1). The nucleation density was set to 2.8 · 109 m2 (i.e., a grain size of 21 lm). For comparison, the numerical solution combined with the extended volume concept, as described below, has also been applied to this geometry. Figure 2 shows the ferrite particle radius as a function of time obtained from the fully numerical solution and the mixed-mode growth kinetics model. For the onedimensional system Figure 2 also shows the ferrite fraction as predicted by the original mixed-mode model from Ref. [1]. The original mixed-mode model gives a clearly slower transformation. The too slow transformation of the original model indicates that the carbon concentration at the interface is overestimated (a higher carbon concentration at the interface in the austenite phase means a lower driving force). This is confirmed in Figure 3, which shows the carbon concentration at the interface in the austenite phase as a function of time. Note that the mixed-mode character gradually changes from initially interface-controlled (S = 1) to diffusioncontrolled. Also shown in Figure 2 is the width of the (carbon) concentration profile in front of the interface, z0. Since z0 is relatively small, the deviation between the numerical solution (where soft impingement is automatically included) and the mixed-mode model (where no soft impingement correction was applied in this case) remains small until the interface comes close to the edge of the system (at 25 lm). This suggests that, especially for systems with a large average grain size, soft impingement only plays an important role at the very end of the transformation (especially for systems with a relatively low average carbon concentration). Whereas Figures. 2 and 3 concern a single growing grain, in Figure 4 the JMA mixed-mode model results are shown. The numerical solution (combined with the
25
Ferrite radius (µm)
(e.g., [6]), is used. The extended volume Ve is the total volume of all growing particles as if every particle would grow in an infinite medium. Then a correction is applied, which takes into account the impingement of the growing particles, transforming the extended volume to the actual transformed fraction. The well-known impingement equation by Avrami is Ve ; ð12Þ fa ¼ 1 exp V
1087
Mixed-mode 1D Original Mixed-mode 1D Full numerical Mixed-mode 2D Mixed-mode 3D z0
20
15
10
5
0
0
15
30
45
60
75
90
Time (s) Figure 2. The ferrite fraction calculated with the mixed-mode growth kinetics model and the fully numerical solution. The width of the carbon profile in front of the interface, z0, is shown for the onedimensional system.
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C. Bos, J. Sietsma / Scripta Materialia 57 (2007) 1085–1088
Figure 3. The carbon concentration at the interface in the austenite phase (xcc ), for a one-dimensional system. xca c is the equilibrium carbon concentration in austenite and x0 is the average carbon concentration. The mode parameter S decreases from one (interface-controlled) towards zero (diffusion-controlled) as the transformation progresses. 1.0 Cooling rate 2 K/s Cooling rate 5 K/s Cooling rate 15 K/s Numerical
Ferrite fraction
0.8
0.6
0.4
0.2
0.0 920
950
980
1010
1040
1070
1100
Temperature (K) Figure 4. The ferrite fraction according to mixed-mode growth, for a two-dimensional system with cooling rates of 2, 5 and 15 K s1. The dotted lines show the numerical solutions for the same cooling rates.
extended volume concept) predicts a too low final ferrite fraction, as explained before. Combining the numerical solution for growth with the geometry model leads to intrinsic problems, which do not occur while using the mixed-mode model. The JMA model presented here is only a simple example of how the mixed-mode model combined with the soft impingement correction of Eq. (11) can be used in practise. In a more advanced microstructure geometry model the growth of a large set of individual particles can be easily tracked in a computer program because of the minimal computational cost of the mixed-mode model. The development of such models is in progress. Finally, it is worth noting that the transformation curves as shown in Figure 4 are not the only information that can be obtained from the mixed-mode JMA model. Through xb and z0 (Eqs. (7) and (6)) information on the carbon concentration profile is available and an impression of the microstructure can also be obtained from the
Figure 5. The development of the carbon concentration profile taken from the two-dimensional calculation of Figure 2. The equilibrium carbon concentration in austenite is 4.7 at.%.
JMA mixed-mode model. Figure 5 shows that the mixed-mode concentration profiles (labelled MM) as determined by xb and z0 are nearly identical to the profiles from the numerical solution (labelled num). To summarize, a highly efficient, concise mixed-mode growth kinetics model has been formulated that gives a practically exact description of the interface velocity in partitioning phase transformations in case of growth in one, two or three dimensions in an infinite medium. The concise formulation of the mixed-mode model allows a straightforward combination with microstructure geometry models as shown here for the JMA extended volume mixed-mode model for the austenite to ferrite transformation in an Fe–C–Mn steel. [1] J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143– 4152. [2] F. Fazeli, M. Militzer, Metall. Mater. Trans. A 36A (2005) 1395–1405. [3] M.G. Mecozzi, J. Sietsma, et al., Metall. Mater. Trans. A 36A (2005) 2327–2340. [4] J. Sietsma, M.G. Mecozzi, S.M.C. van Bohemen, S. van der Zwaag, Int. J. Mat. Res. 97 (2006) 356–361. [5] C.J. Huang, D.J. Browne, S. McFadden, Acta Mater. 54 (2006) 11–21. [6] J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon Press, Oxford, 2002. [7] C. Zener, J. Appl. Phys. 20 (1949) 950. [8] M.G. Mecozzi, PhD thesis, Delft University of Technology, 2007 (Chapter 7). [9] H.K.D.H. Bhadeshia, Prog. Mater. Sci. 29 (1985) 321– 386. [10] W.D. Murray, F. Landis, Trans. ASME 81 (1959) 106– 112. [11] A. van der Ven, L. Delaey, Prog. Mater. Sci. 40 (1996) 181–264. [12] R.C. Weast, Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1989. [13] M. Militzer, M.G. Mecozzi, J. Sietsma, S. van der Zwaag, Acta Mater. 54 (2006) 3961–3972.
A mixed-mode model for partitioning phase transformations C. Bosa,b and J. Sietsmab,* a
b
Netherlands Institute for Metals Research, Mekelweg 2, 2628 CD Delft, The Netherlands Delft University of Technology, Department of Materials Science and Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands Received 5 July 2007; revised 23 August 2007; accepted 26 August 2007 Available online 21 September 2007
A mixed-mode model for the interface velocity during partitioning phase transformations [J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143–4152] is reformulated and rigorously validated. Comparison with a numerical treatment shows that the model gives a near-exact description of the growth kinetics. In addition, the model provides information on the concentration profiles of the partitioning element. The mixed-mode growth model can be combined with a range of different microstructure models. As an example a combination with a Johnson–Mehl–Avrami extended volume model is shown. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase transformation kinetics; Modelling; Precipitation; Mixed-mode kinetics
A number of recent publications [1–5] have shown that traditional interface-controlled [6] and diffusioncontrolled [7] models are often unable to describe correctly the growth kinetics of solid-state partitioning phase transformations in metals over the entire course of the transformation. Consequently, a mixed-mode model that can take both effects, and the varying relative importance of each, into account should lead to a significant improvement in the modelling of the growth kinetics. Most phase-transformation models can be considered to consist of a part that describes the interface velocity and a part that describes the geometrical aspects of the microstructure. In Ref. [1], a concise mixed-mode growth kinetics model was introduced based on widely accepted differential equations for the interface process and the diffusion process. However, a study of the mixed-mode character in phase-field simulations of the isothermal austenite to ferrite transformation in an Fe–C–Mn steel has shown that the mixed-mode model from Ref. [1] overestimates the diffusional character of the transformation [8]. In the present work, an exponential concentration profile is introduced, leading to a significant improvement in the description of the mixedmode character without losing the concise formulation and the efficiency of the original model. In contrast to Ref. [1], a rigorous validation of the mixed-mode growth kinetics model is provided here by means of a compari* Corresponding author. E-mail: [email protected]
son with a fully numerical solution that exactly describes the concentration profile in front of the moving interface according to widely accepted differential equations for the interface velocity and the diffusion [6]. To illustrate the versatility of the mixed-mode description of the interface velocity, it is combined with a Johnson– Mehl–Avrami (JMA) description of the microstructure geometry to give a model for the kinetics of the austenite to ferrite transformation in an Fe–C–Mn steel under para-equilibrium conditions. The model considers a solid-state phase transformation in which partitioning of a single element takes place, say b ! a. The mixed-mode character of the transformation is quantified by the mode parameter S [1], which is defined as S¼
xba xb ; xba x0
ð1Þ
where x0 is the overall concentration of the partitioning element, xb is the concentration in the b phase at the interface, and xba is the concentration in the b phase in equilibrium with the a phase (see Fig. 1, xab is the concentration in the a phase in equilibrium with the b phase). In this model, it is assumed that the concentration in the a phase is homogeneous at the equilibrium value xab (as determined by the temperature). The mode parameter S will have a value between 0 and 1: S = 0 means that the transformation is completely diffusioncontrolled (local equilibrium: xb = xba); S = 1 means
1359-6462/$ - see front matter 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2007.08.030
1086
C. Bos, J. Sietsma / Scripta Materialia 57 (2007) 1085–1088
z0 ¼
Figure 1. The concentration, x, of the partitioning element near the interface.
V a x0 xab : XAa xb x0
ð6Þ
The factor X stems from the z-dependence of A, and thus depends on the particle size ra. X = 1 for systems with a constant surface area (e.g., one-dimensional systems). For growing disks X2D ¼ 1 þ rza0 , and X3D ¼ 1 þ 2 rza0 þ 2ðrza0 Þ2 for spheres. At the interface the rate of partitioning as determined by the interface velocity (Eq. (2)) must be equal to the diffusion flux (as given by Fick’s first law), which combined with Eq. (6) leads to a quadratic equation for the interface concentration xb 2
xb ¼ ðZx0 þ Dx0 xaþb þ ½ðZx0 þ Dx0 xaþb Þ 1
ðZ þ 2Dx0 ÞðZx20 þ 2Dx0 xab xba Þ2 Þ=ðZ þ 2Dx0 Þ
that the transformation is completely interfacecontrolled (xb = x0). In general, the interface velocity can be written as [6]
with Dx0 = x0 x , x
v ¼ MDGI ðT ; xb Þ
Z ¼ 2X
ð2Þ
with M the interface mobility and DGI the driving force. Here DGI is written for the driving force to indicate that only the part of the total driving force that is dissipated in the interface processes are considered in Eq. (2) (see e.g. [9]). The driving force is a function of temperature, T, and the interface concentration, xb. For xab < x0 < xba, at the moving interface partitioning takes place from the a to the b phase with a rate determined by the interface velocity, and diffusion of the partitioning element in the b phase takes place with a flux given by Fick’s law, J = Dox/oz, with D the diffusivity of the partitioning element in the b phase. This moving-boundary problem can be solved by using a variant of the Murray–Landis one-dimensional finite-difference method [10], which has often been applied to purely diffusion-controlled phase transformations [11]. This solution will be referred to hereafter as the fully numerical solution. An expression for the concentration xb can be formulated on the basis of two straightforward assumptions: (1) The driving force, DGI, is proportional to the deviation from the equilibrium concentration DGI ¼ vðxba xb Þ
ð3Þ
with v a proportionality factor which can, for example, be calculated using Thermocalc. This linear proportionality is an approximation that usually holds well within the relevant composition range. (2) The concentration profile in front of the interface as a function of position, z, can be described by z ð4Þ x ¼ x0 þ ðxb x0 Þ exp z0 with z = 0 at the interface (see Fig. 1) and z0 defines the width of the profile. The mass balance for a single a-grain of volume Va and surface area Aa in infinite space, given by Z 1 V a ðx0 xab Þ ¼ AðzÞðx x0 Þ dz ð5Þ 0
with A(z) the surface area of the grain at position z, determines the parameter z0 in Eq. (4)
ab
a+b
ab
=x
+x
ba
ð7Þ
and
a
D A : Mv V a
ð8Þ
The complication that X depends on z0, can easily be solved by determining xb by an iterative procedure over Eqs. (6)–(8) starting with X = 1. Usually, the solution converges after less than 10 iterations within a deviation of less than 0.1% of the true value. The equation for xb (Eq. (7)) is the same equation as in the original model [1,4]; the only difference is the factor 2X in the equation for Z (Eq. (8)). However, as will be shown below, this factor proves to be essential to arrive at the correct mixed-mode interface concentration. The mode parameter S is a direct measure for the kinetics of the transformation: combining Eqs. (1)–(3) gives v ¼ MvSðxba x0 Þ;
ð9Þ
where only S depends on the mixed-mode character. On the basis of Eq. (9) the transformation kinetics can be described at a fraction of the computational cost of the fully numerical method. This equation, however, does not take the effects of soft impingement into account. As the growing grains start to approach each other, the concentration profiles surrounding the grains will start to overlap. This effect can be taken into account in a mean-field approximation. If the partitioning element would redistribute completely in the b phase, the (homogeneous) concentration, xb,h, is given by xb;h ¼
x0 fa xab ; 1 fa
ð10Þ
where fa is the transformed fraction. The mean-field soft-impingement approximation can now be obtained as follows. First S is calculated according to Eqs. (7) and (1). Then, x* is introduced as x ¼ xba Sðxba xb;h Þ; b,h
ð11Þ x*
where x is given by Eq. (10). Subsequently, is taken as the concentration in the b phase for the calculation of the driving force DGI and the interface velocity according to Eq. (2) is used. In order to account for hard impingement the extended volume concept, the basis of JMA models
C. Bos, J. Sietsma / Scripta Materialia 57 (2007) 1085–1088
where V is the total system volume. If N_ ðtÞ is the nucleation rate (i.e., the number of nuclei formed per unit of time), the extended volume can be written as Z t d Z t N_ ðsÞgd vðra ; T Þ ds ds; ð13Þ Ve ¼ 0
s
with gd a geometrical factor, and d the growth dimension. In this work all nuclei (N*) are assumed to form at the start time of the transformation (t = 0, site saturation), which means that Z t d V e ¼ N gd vðra ; T Þ ds ð14Þ 0
for the extended volume. It is possible to use the extended volume approach with the mixed-mode velocity inserted in Eq. (14), but also combined with the fully numerical solution. The integral from Eq. (14) is then replaced with the volume of a growing sphere as obtained from the numerical solution. However, the numerical model is not as flexible as the analytical mixed-mode model combined with the mean-field soft-impingement correction in dealing with growing particles of different sizes as in the JMA extended volume approach. In the numerical solution the maximum grain size determines when soft impingement will start to affect the growth rate. In this work, the maximum grain size is set equal to the average grain size (as determined from the nucleation density). This ensures that soft impingement is described correctly for the early stages of the transformation. However, it also limits the maximum size of the growing grains, which means that the maximum fraction transformed predicted by the combination of the numerical solution with the extended volume concept will be too low. To benchmark the performance of the mixed-mode model, the austenite to ferrite transformation kinetics in an Fe–0.46 C–0.5 Mn (at.%) steel have been studied. Para-equilibrium is assumed, that is, only carbon partitions, and the diffusion of carbon in the austenite phase is described by D = D0 exp(QD/RT), with D0 = 0.15 · 104 m2/s and QD = 142.1 kJ mol1 [12] (R is the gas constant). The interface mobility, M, is given by M = M0 exp(QG/RT), with M0 = 3.5 · 107 m4/Js and QG = 140 kJ mol1 [13]. The driving force is obtained from Thermocalc and described by Eq. (3). Two different geometries have been examined: (i) A single spherical grain in one, two and three dimensions, which has been used with the numerical solution and the mixed-mode growth kinetics model under isothermal conditions (T = 950 K)
with a maximum grain radius of 25 lm. For the fully numerical calculations a grid with 512 nodes was used. With this geometry, no impingement corrections were used with the mixed-mode model. (ii) A microstructure in two dimensions, which has been used with the mixed-mode model with soft and hard impingement corrections according to Eqs. (11) and (12). Here isochronal cooling was used (from T = 1091–920 K with cooling rates of 2, 5 and 15 K s1). The nucleation density was set to 2.8 · 109 m2 (i.e., a grain size of 21 lm). For comparison, the numerical solution combined with the extended volume concept, as described below, has also been applied to this geometry. Figure 2 shows the ferrite particle radius as a function of time obtained from the fully numerical solution and the mixed-mode growth kinetics model. For the onedimensional system Figure 2 also shows the ferrite fraction as predicted by the original mixed-mode model from Ref. [1]. The original mixed-mode model gives a clearly slower transformation. The too slow transformation of the original model indicates that the carbon concentration at the interface is overestimated (a higher carbon concentration at the interface in the austenite phase means a lower driving force). This is confirmed in Figure 3, which shows the carbon concentration at the interface in the austenite phase as a function of time. Note that the mixed-mode character gradually changes from initially interface-controlled (S = 1) to diffusioncontrolled. Also shown in Figure 2 is the width of the (carbon) concentration profile in front of the interface, z0. Since z0 is relatively small, the deviation between the numerical solution (where soft impingement is automatically included) and the mixed-mode model (where no soft impingement correction was applied in this case) remains small until the interface comes close to the edge of the system (at 25 lm). This suggests that, especially for systems with a large average grain size, soft impingement only plays an important role at the very end of the transformation (especially for systems with a relatively low average carbon concentration). Whereas Figures. 2 and 3 concern a single growing grain, in Figure 4 the JMA mixed-mode model results are shown. The numerical solution (combined with the
25
Ferrite radius (µm)
(e.g., [6]), is used. The extended volume Ve is the total volume of all growing particles as if every particle would grow in an infinite medium. Then a correction is applied, which takes into account the impingement of the growing particles, transforming the extended volume to the actual transformed fraction. The well-known impingement equation by Avrami is Ve ; ð12Þ fa ¼ 1 exp V
1087
Mixed-mode 1D Original Mixed-mode 1D Full numerical Mixed-mode 2D Mixed-mode 3D z0
20
15
10
5
0
0
15
30
45
60
75
90
Time (s) Figure 2. The ferrite fraction calculated with the mixed-mode growth kinetics model and the fully numerical solution. The width of the carbon profile in front of the interface, z0, is shown for the onedimensional system.
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C. Bos, J. Sietsma / Scripta Materialia 57 (2007) 1085–1088
Figure 3. The carbon concentration at the interface in the austenite phase (xcc ), for a one-dimensional system. xca c is the equilibrium carbon concentration in austenite and x0 is the average carbon concentration. The mode parameter S decreases from one (interface-controlled) towards zero (diffusion-controlled) as the transformation progresses. 1.0 Cooling rate 2 K/s Cooling rate 5 K/s Cooling rate 15 K/s Numerical
Ferrite fraction
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Temperature (K) Figure 4. The ferrite fraction according to mixed-mode growth, for a two-dimensional system with cooling rates of 2, 5 and 15 K s1. The dotted lines show the numerical solutions for the same cooling rates.
extended volume concept) predicts a too low final ferrite fraction, as explained before. Combining the numerical solution for growth with the geometry model leads to intrinsic problems, which do not occur while using the mixed-mode model. The JMA model presented here is only a simple example of how the mixed-mode model combined with the soft impingement correction of Eq. (11) can be used in practise. In a more advanced microstructure geometry model the growth of a large set of individual particles can be easily tracked in a computer program because of the minimal computational cost of the mixed-mode model. The development of such models is in progress. Finally, it is worth noting that the transformation curves as shown in Figure 4 are not the only information that can be obtained from the mixed-mode JMA model. Through xb and z0 (Eqs. (7) and (6)) information on the carbon concentration profile is available and an impression of the microstructure can also be obtained from the
Figure 5. The development of the carbon concentration profile taken from the two-dimensional calculation of Figure 2. The equilibrium carbon concentration in austenite is 4.7 at.%.
JMA mixed-mode model. Figure 5 shows that the mixed-mode concentration profiles (labelled MM) as determined by xb and z0 are nearly identical to the profiles from the numerical solution (labelled num). To summarize, a highly efficient, concise mixed-mode growth kinetics model has been formulated that gives a practically exact description of the interface velocity in partitioning phase transformations in case of growth in one, two or three dimensions in an infinite medium. The concise formulation of the mixed-mode model allows a straightforward combination with microstructure geometry models as shown here for the JMA extended volume mixed-mode model for the austenite to ferrite transformation in an Fe–C–Mn steel. [1] J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143– 4152. [2] F. Fazeli, M. Militzer, Metall. Mater. Trans. A 36A (2005) 1395–1405. [3] M.G. Mecozzi, J. Sietsma, et al., Metall. Mater. Trans. A 36A (2005) 2327–2340. [4] J. Sietsma, M.G. Mecozzi, S.M.C. van Bohemen, S. van der Zwaag, Int. J. Mat. Res. 97 (2006) 356–361. [5] C.J. Huang, D.J. Browne, S. McFadden, Acta Mater. 54 (2006) 11–21. [6] J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon Press, Oxford, 2002. [7] C. Zener, J. Appl. Phys. 20 (1949) 950. [8] M.G. Mecozzi, PhD thesis, Delft University of Technology, 2007 (Chapter 7). [9] H.K.D.H. Bhadeshia, Prog. Mater. Sci. 29 (1985) 321– 386. [10] W.D. Murray, F. Landis, Trans. ASME 81 (1959) 106– 112. [11] A. van der Ven, L. Delaey, Prog. Mater. Sci. 40 (1996) 181–264. [12] R.C. Weast, Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1989. [13] M. Militzer, M.G. Mecozzi, J. Sietsma, S. van der Zwaag, Acta Mater. 54 (2006) 3961–3972.