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pearlite microstructure
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ScienceDirect Acta Materialia 88 (2015) 302–313 www.elsevier.com/locate/actamat
A mixed-mode model for the ferrite-to-austenite transformation in a ferrite/ pearlite microstructure ⇑
M.G. Mecozzi,a, C. Bosb and J. Sietsmaa a
Delft University of Technology, Department of Materials Science and Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands b Tata Steel Europe, Wenckebachstraat 1, 1951 JZ Velsen-Noord, The Netherlands Received 19 December 2014; revised 22 January 2015; accepted 25 January 2015
Abstract—A concise semi-analytical mixed-mode model is proposed to describe the ferrite-to-austenite transformation kinetics. The initial microstructure for the model consists of a ferrite matrix with supersaturated austenite grains. The carbon supersaturation of austenite grains resulted from the rapid dissolution of pearlite colonies present in the initial microstructure. A similar approach to the one used for describing the ferrite growth kinetics (Bos and Sietsma, 2007) was used, employing a sharp interface between the phases. In comparison to ferrite growth, the carbon concentration profile of the growing austenite contains an additional parameter, which is the carbon concentration at the centre of the grain. This extra parameter is accounted for by assuming a parabolic carbon profile in the austenite. A comparison with a numerical solution of the differential equations shows that the developed semi-analytical mixed-mode model gives a good description of the ferrite-to-austenite transformation, significantly more accurate over the entire course of the transformation than the classical interface-controlled and diffusion-controlled models. Significant deviations from the numerical solution, as a result of the parabolic approximation, are limited to the initial stage of the transformation. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Austenite growth kinetics; Modelling; Mixed-mode kinetics
1. Introduction In the past years, there has been a significant increase in the use of high strength Dual-Phase (DP) steels in the automotive industry. The favourable combination of strength and formability of DP steels makes these materials suitable for the construction of several parts of the car body. The microstructure of cold-rolled and annealed DP steels, consisting of a dispersion of hard martensite in a soft ferrite matrix, is typically obtained by annealing in the intercritical temperature range to produce an austenite-ferrite microstructure, followed by quenching to room temperature to transform austenite to martensite. Since the properties of DP steels markedly depend on the fraction, distribution and carbon content of the martensite phase, the understanding of the mechanism of austenite formation during heating and holding at the intercritical temperature is of primary importance. Despite this fact, the kinetics of austenite formation during heating has not been studied in such detail as the austenite-to-ferrite transformation on cooling. The explanations for this are several: first, the austenite formed during heating is transformed back to different product phases during cooling and therefore direct observation of the austenite formed on heating is very difficult; second, the austenite formation occurs very rapidly
⇑ Corresponding author; e-mail: [email protected]
as the temperature increases and consequently conventional techniques, like dilatometry and metallography, usually employed to study the transformation on cooling, are less adequate for studying the austenite formation. Another aspect makes the study of the austenite formation kinetics on heating difficult: the distribution and morphology of phases present in the starting microstructure affect the process of austenite formation, which then must be studied starting from different initial microstructures, whereas the ferrite formation usually occurs from homogenous austenite. This increases the number of required experiments. For similar reasons physically based models for the austenitisation formation are relatively underdeveloped. From experiments it is known that during intercritical annealing of steel having an initial ferrite plus pearlite microstructure the austenite formation occurs in two steps: the pearlite transforms to carbon-supersaturated austenite and subsequently ferrite transforms to austenite. The first step is controlled by the carbon diffusion within the pearlite regions. Since the diffusion distances involved are short, it is much more rapid than the second step. Therefore most models in the literature restrict the analysis of the austenitisation process to the ferrite-to-austenite transformation, starting from a microstructure of carbon-supersaturated austenite grains in a ferrite matrix [1–4]. In the literature, the ferrite-to-austenite transformation is often modelled assuming that the carbon diffusion in austenite controls the rate of transformation and the lattice transformation is
http://dx.doi.org/10.1016/j.actamat.2015.01.058 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
2. Theory In this work the ferrite-to-austenite phase transformation is assumed to occur with the partitioning of a single element as it happens in binary alloys or in ternary alloys in which one of the elements has a negligible diffusivity with respect to the effective interface motion. This is the case of the Fe–C–Mn system, in which the transformation can be assumed to take place without Mn partitioning since the Mn diffusion is a slow process compared to the motion of the interface as a consequence of the large difference in diffusivity of Mn and C in both austenite and ferrite: the transformation is said to occur under para-equilibrium conditions. This section will present a physical model of the interface motion of a sharp austenite/ferrite interface during the austenite-to-ferrite phase transformation under mixedmode conditions, governed by carbon partitioning and diffusion and the interface mobility. This approach leads to analytic expressions for the interface motion and the carbon fractions at the interface and within the austenite. 2.1. Semi-analytical model A ferritic matrix of dimension L is assumed as the initial microstructure, in which an austenite grain of width 2s0 is present, produced from the rapid transformation of a
pearlite colony at a temperature T above A1, when austenite becomes thermodynamically stable. If z is the distance from the centre of the austenite grain, the carbon mole fraction xC ðt; zÞ at t = 0 is given by xC ð0; zÞ ¼ xpC forjzj < s0 and xC ð0; zÞ ¼ xaeq C forjzj > s0
ð1Þ
xpC
where is the average pearlite carbon mole fraction and xaeq C is the equilibrium carbon mole fraction in ferrite at T (Fig. 1). The origin of the z-axis is in the centre of the grain. Because of the symmetry in this system, in the remainder we will consider only z > 0. The model focusses on hypo-eutectoid systems, for which the equilibrium carbon mole fraction in austenite p (c) in equilibrium with ferrite, xceq C , is lower than xC at intercritical temperatures. Therefore the austenite/ferrite interface experiences a driving force, DG, to move into the ferrite phase. This driving force DG depends on the carbon mole fraction at the interface in the austenite, xiC , which develops during the phase transformation. In the present approach DG is assumed to be proportional to the deviation of the carbon fraction at the interface from the equilibrium fraction in austenite, that is DG ¼ vðxiC xceq C Þ;
ð2Þ
xiC
with the carbon mole fraction in austenite at the interface position s, xC ðt; sÞ ¼ xiC . Although it has been shown [20–21] that the chemical potential of interstitial elements should always be equal at the interface of a sharp interface model here it is assumed that ferrite forms at the equilibrium carbon fraction, also when the interface carbon fraction in austenite is different from the equilibrium value. The differences in the ferrite carbon fractions caused by this simplification are negligibly small. The interface velocity v is proportional to this driving force, i.e. v ¼ MDG
ð3Þ
with M the interface mobility. For finite carbon diffusivity, a carbon gradient is established within the austenite. In line with the analytical description of the austenite-to-ferrite transformation [9] [19], the carbon-fraction profile in austenite (z < s) is assumed to be exponential in z (see Fig. 1) and given by expðz=z0 Þ þ expðz=z0 Þ 2 expðs=z0 Þ þ expðs=z0 Þ 2 coshðz=z0 Þ 1 ¼ xhC þ ðxiC xhC Þ coshðs=z0 Þ 1
xC ðt; zÞ ¼ xhC þ ðxiC xhC Þ
ð4Þ
p
Carbon concentration, xC
such a fast process that the carbon fractions in ferrite and austenite at the interface are equal to the equilibrium fractions [5–6]. The transformation is therefore diffusion controlled. In the more general case of finite interface mobility, carbon fractions at the interface differ from the equilibrium values and the transformation has a mixedmode character, which means that it is controlled by both carbon diffusivity and interface mobility. While the mixed-mode character of the austenite dissolution on cooling was extensively analysed in the literature, e.g. [7–13], few studies can be found on the mixed-mode character of austenite formation on heating [14–15]. In terms of modelling the phase transformation, phase-field modelling (e.g. [16–18]) does take the mixed-mode character into account in a diffuse-interface approach, but the method is too computationally intensive to allow 3D microstructural simulations. A more efficient modelling approach is therefore needed, similar to the concise analytical model that has been formulated recently to describe the austenite-to-ferrite transformation [19]. The model in Ref. [19] simply considers the conservation of carbon at the interface and the dependence of the driving force on the deviation from equilibrium to derive the carbon fraction in austenite at the interface, the width of carbon profile within the austenite and the interface velocity. The comparison with a fully numerical solution of the differential equations describing the carbon flux through the moving interface has shown that this analytical model is able to quite accurately describe the mixed-mode character of the austenite-to-ferrite transformation [19]. In the present paper the ferrite-to-austenite transformation during holding at an intercritical temperature is modelled by using a mixed-mode semi-analytical model similar to that formulated for the austenite-to-ferrite transformation in Ref. [19]. The results of the model and its limitations are discussed. A comparison with numerical solutions of the equations involving the movement of the interface and carbon diffusion is also presented. The obtained results are compared with those obtained for diffusion-controlled kinetics and interface-controlled kinetics.
303
xC
t=0 t
h
xC
i
xC
s0
s
α eq
xC
Distance from the center of γ grain, z Fig. 1. Carbon mole fraction in austenite (z < s) and in ferrite (z > s); the initial carbon mole fraction is given by the dashed line, the fraction at a certain stage during the transformation by the solid line.
304
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
where xhC is the carbon mole fraction at the centre of the austenite grain, xhC ¼ xC ðt; 0Þ, and z0 represents the width of the fraction profile. The expression given as Eq. (4) fulfils the boundary conditions xC ðt; sÞ ¼ xiC and a zero fraction gradient at the centre of the grain due to its symmetry, i.e. @xC ðxiC xhC Þ sinhðz=z0 Þ ¼ z0 coshðs=z0 Þ 1 @z
ð5Þ
is zero at z = 0. Since no carbon fraction gradients develop in the ferrite, the total amount of carbon in the austenite grain is, when it has grown to an interface position s, equal to the amount of carbon in the region 0 < z < s for the initial configuration (see Fig. 1), i.e. Z s xC ðt; zÞ dz ¼ s0 xpC þ ðs s0 Þxaeq ð6Þ C 0
Using Eqs. (4) and (6) becomes sðxiC xhC Þ
ð7Þ
which expresses the interdependence of the interface position s and the carbon-fraction profile (z0, xiC ; xhC ). To simplify the equations, all the carbon fractions are related to the equilibrium carbon fraction in ferrite by defining ¼
xiC
xaeq C ;
~xhC
¼
xhC
xaeq C ;
~xpC
¼
xpC
xaeq C :
ð8Þ
Then Eq. (7) becomes s0~xpC s~xhC ¼ sð~xiC ~xhC Þ
½ðz0 =sÞ sinhðs=z0 Þ 1 : coshðs=z0 Þ 1
ð9Þ
Interface motion causes a flux of carbon given by the product of the interface velocity and the difference in carbon fraction between austenite and ferrite at the interface. This flux, which causes a depletion of carbon at the interface, is compensated by the diffusional flux of carbon in the austenite towards the interface (i.e. at z = s): c vðxiC xaeq C Þ ¼ DC
dxC j : dz z¼s
ð10Þ
with DcC the carbon diffusivity in austenite. Eq. (10) is the central equation for this mixed-mode model, since it expresses the relation between the interface motion and the carbon diffusion. Using Eqs. (2), (3), (5) and (10) becomes aeq i MvðxiC xceq C ÞðxC xC Þ ¼
ðxi DcC C
xhC Þ
z0 sinhðs=z0 Þ : coshðs=z0 Þ 1
ðs0 =sÞ~xpC ð1 BÞ ~xiC : ð14Þ B These two quantities are crucial for the mixed-mode character of the phase transformation and govern the further development of the transformation through carbon diffusion and the driving force at the interface. The additional parameters E, B and F in the Eqs. (13) and (14) are combinations of previously introduced parameters and given by
~xhC ¼
ð~xiC ~xhC Þ sinhðs=z0 Þ z0 coshðs=z0 Þ 1
Mv : DcC
B¼1
F ¼
z0 ½sinhðs=z0 Þ ðs=z0 Þ : coshðs=z0 Þ 1 s
1 sinhðs=z0 Þ : z0 coshðs=z0 Þ 1
ð15Þ
ð16Þ
ð17Þ
The parameter E has a direct and interesting physical meaning. Its numerator gives the product of interface mobility and the proportionality factor v (see Eq. (2)), which represents the interface kinetics; its denominator is the diffusivity and thus represents the diffusion kinetics. A high value of E indicates the transformation character to be close to diffusion control, a low E indicates a transformation character that is closer to interface control. Note that the evolution of the carbon fraction during the transformation is related to the variable parameters s and z0 in the Eqs. (13) and (14). The mixed-mode approach should represent the conditions for diffusion-controlled transformation for infinite mobility and the conditions for interface-controlled transformation for infinite diffusivity. For the diffusion-controlled mode (DCM), i.e. M ! 1, the factor E ! 1, and it follows from Eq. (13) that xiC jDCM ¼ xceq C
ð18Þ
and from Eq. (14) that ðs0 =sÞðxpC xaeq C Þ ð1 BÞDxC : ð19Þ B In the diffusion-controlled mode a carbon-fraction gradient develops and the carbon mole fraction at the centre of the austenite grain, xhC jDCM , depends on the position of the interface, s, and on the width of carbon profile z0 through the factor B (Eq. (16)). Eq. (18) expresses the well-known local equilibrium for diffusion-controlled transformation. Under the assumption of interface mobility controlled mode (ICM), which means D ! 1, the condition xhC jDCM ¼ xaeq C þ
ð11Þ
Relating to the equilibrium carbon fraction in ferrite, Eqs. (8) and (11) become Mvð~xiC DxC Þ~xiC ¼ DcC
and
E¼
½ðz0 =sÞ sinhðs=z0 Þ 1 coshðs=z0 Þ 1
aeq h ¼ s0 ðxpC xaeq C Þ sðxC xC Þ
~xiC
used to implement the boundary condition of a constant total amount of carbon: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 ½BDxC F =E2 þ 4s s0~xpC B F =E s½BDx C F =E þ i ~xC ¼ 2sB ð13Þ
ð12Þ
aeq with DxC ¼ xceq C xC . From Eq. (12), analytic expressions for the interface carbon fraction and the carbon fraction in the centre of the austenite grain, ~xiC and ~xhC , can be derived when Eq. (9) is
~xiC ¼ ~xhC should be fulfilled.
ð20Þ
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
Using Eq. (20) in Eq. (14) yields xhC jICM
¼
xiC jICM
¼
xaeq C
þ
ðxpC
xaeq C Þs0 =s
ð21Þ
If L is the domain dimension, s0 is given by s0 ¼ L
x0C xaeq C xpC xaeq C
ð22Þ
and Eq. (21) becomes aeq 0 xiC jICM ¼ xaeq C þ ðxC xC ÞL=s:
ð23Þ
The ratio s/L is equal to the austenite fraction fc and therefore Eq. (23) becomes x0 ð1 f c Þxaeq C xhC jICM ¼ xiC jICM ¼ C ð24Þ fc x0C
with the alloy carbon mole fraction and f c the fraction of austenite. This equation is simply the lever rule for two phases with homogeneous carbon fraction. The Eqs. (18) and (24) show that the carbon fractions resulting from the proposed mixed-mode model are, in the extremes, consistent with both the diffusion-controlled and interface-controlled model. Eq. (13) gives the interface velocity v when it is combined with the Eqs. (2) and (3). However, in order to evaluate Eq. (13), an additional assumption on the value of z0 is needed. At the start of transformation (t = 0) and also when the equilibrium is reached the value of the carbon fraction inside the austenite phase is constant and therefore z0 ¼ 1; during transformation z0 has a finite value greater than 0. If it is assumed that during transformation z0 >> s > s0 , the carbon profile in austenite can be approximated by a parabola, since Eq. (4) then becomes z2 xC ðt; zÞ ¼ xhC þ ðxiC xhC Þ : ð25Þ s where xhC and xiC can be determined by Eqs. (13) and (14) in sinhðyÞy the limit s=z0 ! 0. Knowing that lim yðcoshðyÞ1Þ ¼ 13 and y!0
y sinhðyÞ y!0 coshðyÞ1
lim
¼ 2, from Eq. (16)it follows that B ! 2=3 and
from Eq. (17) that F ! 2=s. Under this parabolic approximation, ~xiC and ~xhC then become respectively qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ~xiC ¼ ð26Þ ½ðEsDxC 3Þ þ ðEsDxC 3Þ2 þ 12s0 E~xpC 2Es and ~xhC ¼ ð3s0 =2sÞ~xpC ~xiC =2: ð27Þ aeq with E given by Eq. (15) and DxC ¼ xceq C xC . The analytical model for the carbon distribution that has been presented here can be used to predict the development of the phase transformation as a function of time, albeit not analytically. At the starting time, t = 0, s ¼ s0 and the velocity v0 is calculated by using Eqs. (3) and (2) with xiC ¼ xpC . For the following steps in an iterative process with time step dt, the carbon fraction at the centre of austenite grain and at the interface are calculated by using Eqs. (26) and (27) with s calculated from the interface position and velocity from the previous time step by
sðtÞ ¼ sðt dtÞ þ vðt dtÞdt:
ð28Þ
The proposed model thus results in a scheme of a few analytical expressions that can be used in a simulation in which the motion of the austenite/ferrite interface is traced in time steps dt, taking the carbon
305
diffusivity D and the interface mobility M as input parameters. Given the interface position s from the previous time step, the interface carbon fraction in the austenite can be calculated by Eq. (26). This fraction can be employed in Eq. (2) to calculate the driving force DG. The proportionality parameter v in Eq. (2) is derived by Thermo-CalcÒ [22], by calculating the driving force for austenite formation as a function of carbon fraction in austenite at the interface for a given set of temperatures; in this work the driving force is calculated under para-equilibrium condition. The interface velocity, calculated from Eq. (3), is then used to derive the new interface position from Eq. (28). 2.2. Numerical model The validity of the analytic description of the interface velocity in mixed-mode transformation conditions is tested by comparing the obtained results with a fully numerical solution of the differential equations for the interface velocity and the carbon diffusion, which will be described in this section. The numerical model is based on the Murray–Landis finite-difference approach [23] and is essentially the same as used in Ref. [19] for the austenite-to-ferrite transformation. For the description of the ferrite-to-austenite transformation only the diffusion of carbon in austenite is taken into account. The carbon fraction in ferrite is assumed to be equal to the para-equilibrium fraction for ferrite in equilibrium with austenite. The Murray–Landis finite-difference approach has been implemented by the following algorithm: 1. perform a diffusion step according to Fick’s second law; 2. calculate the maximum driving force DG based on the austenite interface carbon fraction and the temperature; 3. calculate the interface velocity, v, with Eq. (3); 4. move the interface a distance dz = vdt, where dt is the time step; 5. scale the finite difference grid (i.e. the Murray–Landis approach of variable grid-spacing); 6. calculate the new austenite carbon interface fraction based on the mass balance; 7. repeat from step 1. Like in the semi-analytical model, the driving force is derived as a function of the carbon fraction in austenite at the interface from Eq. (2), with the proportionality parameter v derived by Thermo-calc. 2.3. Diffusion-controlled model In order to investigate the effect of the mixed-mode approach as compared to the more generally used approximations of diffusion control or interface control, in the present study the ferrite-to-austenite transformation is also modelled by assuming that the carbon diffusion in austenite is the rate controlling process. Under the assumption of a fast interfacial reaction, thermodynamic equilibrium can be assumed to be maintained locally at the interface (local equilibrium). The simulation of ferrite-to-austenite transformation is performed using the DIffusion Controlled TRAnsformations (DICTRA) software [24] in combination with Thermo-calc software [22] and Calphad database [25], which provides all necessary thermodynamic data and the diffusivities to perform the simulations. The treatment of
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
1.0
0.05
0.8
0.04 xC (mole fraction)
0.6 0.4 0.2
0.03
γ line
0.02 0.01
γ
ferrite fraction
306
α line
0.00 0.0 650
700
750 T (ºC)
800
850
a
650
700
750 T (ºC)
800
850
b
γ line
c Fig. 2. (a) Ferrite fraction, (b) carbon mole fraction in austenite and ferrite, (c) driving force for austenite formation as a function of temperature and carbon mole fraction in austenite, calculated under the assumption of immobile substitutional elements.
the moving boundary problem in Dictra software is outlined in Ref. [24]. The principal equation in the diffusion-controlled model is the mass balance at the interface, expressed by c x xa vca kc ka ¼ J ck J ak ; k ¼ 1; 2 . . . n 1 ð29Þ Vm Vm where vca denotes the interface migration rate, V cm and V am are the molar volume of the c and a phase, xck and xak are the mole fractions of component k in c and in a, and J ck and J ak are the corresponding diffusional fluxes, expressed as n1 @xj 1 X Dckj ; c V m j¼1 @z c
J ck ¼
J ak ¼
n1 @xaj 1 X a D : V am j¼1 kj @z
ð30Þ
where Dckj and Dakj denote the kj-component of the diffusivity matrix in the phases a and c respectively. The summation is performed over (n–1) independent fractions, as the dependent nth component may be taken as the solvent. The temporal profile of the diffusion species in each phase is given by solving the diffusion equations c 1 @xj @ ¼ J ck ; V cm @t @z
1 @xaj @ ¼ J ak : V am @t @z
ð31Þ
Under the assumption of constant carbon mole fraction in ferrite, equal to the equilibrium value at the selected temperature, and equal molar volume of austenite and ferrite, Eq. (29) with carbon as the k-component, reduces to
Eq. (10). In the diffusion-controlled model Eq. (29) is used to calculate the interface velocity since Eq. (3), utilised in the numerical and semi-analytical model, cannot be applied when the interface mobility is infinite. The diffusivities in Eq. (30) can be obtained as a product of a thermodynamic factor, which is the second derivative of the molar Gibbs energy with respect to the fractions, and a kinetic factor, which contains the atomic mobilities. This approach considerably reduces the number of independent variables necessary to calculate the diffusional flux [24]. There are three main modules in the DICTRA software, one for solving the diffusion equations inside each phase, i.e. Eq. (31), and one for calculating the local equilibrium at the interface. The third module sets up and solves the flux-balance equations, i.e. Eq. (30), in order to obtain the transformation rates. The position of the phase interface is subsequently obtained by integration. 2.4. Interface-controlled model Also a comparison with the interface-controlled transformation mode is made. In the interface-controlled approach it is assumed that the ferrite-to-austenite transformation kinetics is controlled by the rate of lattice transformation (reflected by the interface mobility M) and that the carbon diffusion in ferrite and austenite is fast enough to maintain a homogenous carbon fraction in each phase. Just like in ferrite, the carbon fraction in austenite is
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
307
600
-1
ΔG (J mol )
800
400 200 0 0.00
0.02 0.04 0.06 γ γ,eq xC - xC (mole fraction)
0.08
800
600
600
ΔG (J mol )
800
-1
-1
ΔG (J mol )
a
400 200 0 0.00
0.02 0.04 0.06 γ γ,eq xC - xC (mole fraction)
b
0.08
400 200 0 0.00
0.02 0.04 0.06 γ γ,eq xC - xC (mole fraction)
0.08
c
Fig. 3. Driving force as a function of the deviation of the carbon mole fraction in austenite at the interface from the equilibrium mole fraction at (a) 760 °C, (b) 780 °C and (c) 800 °C.
assumed to be homogenous through the whole phase and its value, xcC , is given by Eq. (24). It should be noted that while the carbon fraction in ferrite remains constant during isothermal holding, equal to the equilibrium at the considered temperature, xa;eq C , the carbon fraction in austenite changes during transformation since it depends on the fraction of austenite. As for the semi-analytical model, at the starting time, t = 0, s ¼ s0 and the velocity v0 is calculated by using Eqs. (2) and (3) with xiC ¼ xpC . For the following steps the carbon composition at the interface is calculated using Eq. (24), and the resulting value is used to calculate the driving force and velocity with the Eqs. (2) and (3), using xiC ¼ xcC . 3. Simulation conditions All results in this paper are obtained for an Fe–0.45 C– 1.5 Mn (at.%) alloy. The thermodynamic data are calculated by ThermoCalc software [22]. Fig. 2a and b shows the equilibrium ferrite phase fraction and carbon mole fraction in ferrite and austenite as a function of temperature under para-equilibrium conditions. The driving force for the ferrite-to-austenite transformation is shown in Fig. 2c as a function of temperature and carbon fraction in the austenite at the interface. The intersection of this surface with the plane DG ¼ 0 gives the equilibrium carbon fraction in austenite as a function of temperature (c line in Fig. 2b). In order to calculate the factor v in Eq. (2) for three selected transformation temperatures, 760 °C, 780 °C and
800 °C, the driving force is plotted as a function of the deviation from the equilibrium fraction DxcC ¼ xiC xcCeq and fitted with a straight line for 0 < DxcC < 3 at:%, as shown in Fig. 3. The equilibrium carbon mole fraction in austenite and the factor v, reported in Table 1, are used to calculate the carbon profile in austenite and the kinetics of austenite formation in the semi-analytical model. The equilibrium ferrite phase fraction under para-equilibrium conditions at the three selected temperatures are also reported in Table 1. The interface mobility is assumed to obey an Arrhenius relationship ac M ¼ M ac 0 expðQ =RT Þ
ð32Þ
with the gas constant R = 8.31 JK1 mol1, the activation energy Qac ¼ 140 kJ=mol [7]; the pre-exponential factor 1 1 M ac 0 is varied between 0.05 and 0.5 mol m J s , which is the range of interface mobility values reported in different studies of the kinetics of austenite formation or dissolution [14,26–28]. The mobility is varied to study its effect on the accuracy of the semi-analytical model, as compared to the numerical model, and on the mixed-mode character of the transformation with respect to the interface-controlled model and diffusion-controlled mode. For the carbon diffusivity in austenite the data from [29] are taken, i.e. 8339:9 K DcC ¼ 4:53 107 1 þ y C ð1 y C Þ T 1 4 1 2:221 10 K Þð17767 K y C 26436K m2 s1 exp T ð33Þ
308
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
Table 1. Thermodynamic and kinetic factors governing the phase transformation at the three considered temperatures. Simulations were performed for two different values of the interface mobility. xceq C ðmol%Þ 1
vðJ mol mol%1 Þ f eq c DcC ðm2 s1 Þ 1 1 Eðm1 mol%1 Þ for M ac s 0 ¼ 0:05 m mol J 1 1 Eðm1 mol%1 Þ for M ac ¼ 0:5 m mol J s 0
760 °C
780 °C
800 °C
1.27 105 0.336 1.1 1012 3.8 105 3.8 106
0.85 102 0.516 1.4 1012 4.1 105 4.1 106
0.48 101 0.937 1.7 1012 4.5 105 4.5 106
where y C ¼ xC =ð1 xC Þ, with xC the atomic fraction of carbon and T the temperature in Kelvin. Since the results of the mixed-mode model are evaluated in comparison with those of the diffusion-controlled model, for which local equilibrium condition at the interface ðxiC ¼ xceq C Þ is assumed, the carbon diffusivity is calculated from Eq. (33) with xC ¼ xceq C . The carbon diffusivity at the different temperatures is reported in Table 1. The domain size, L, is set depending on the initial austenite grain size, s0, according to the relation L¼
xpC xaeq C s0 : x0C xaeq C
equilibrium for different transformation temperatures. The results of the semi-analytical model are compared with results from the fully numerical model, the diffusion-controlled model and the interface-controlled model. In order to assess the effect of austenite/ferrite interface mobility on the nature of the transformation, two values of the pre-exponential factor of the 1 1 interface mobility are selected: M ac s 0 ¼ 0:05 mol m J 1 1 and M ac ¼ 0:5 mol m J s . 0 Figs. 4–6 report the obtained results for M ac 0 ¼ 0:05 mol m J1 s1 at the three temperatures 760 °C, 780 °C, 800 °C. It is evident that the character of the transformation is in between the diffusion-controlled mode and the interfacecontrolled mode independently on the transformation temperature. For a normalised austenite fraction of 0.5 (the transformation is halfway to full equilibrium) the deviation from local equilibrium predicted by the interface-controlled model is significantly higher than the value calculated in the semi-analytical and numerical model at any transformation temperature (see Figs. 4a, 5a and 6a). This means that the interface-controlled model gives an overestimation of the carbon fraction at the interface. Consequently, the interface-controlled model underestimates the carbon fraction at the centre of the austenite grain (see Figs. 4b, 5b and 6b). On the other hand, the diffusion-controlled model gives an underestimation of the carbon fraction at the interface, which results in an overestimation of the carbon fraction at the centre of the austenite grain. The deviations between the fractions given by the semi-analytical model and the numerical model are less than one tenth of the deviations found for the two extreme models. Given that the numerical solution can be regarded as the most accurate
ð34Þ
where x0C is the alloy average carbon composition. 4. Results and discussion The mixed mode character of the ferrite-to-austenite transformation with respect to diffusion-controlled mode and interface-controlled mode is evaluated by plotting the deviation from the local equilibrium of the carbon fraction at the interface, ðxiC xcCeq Þ. This deviation is equal to zero in the diffusion-controlled mode. Also, the difference between the carbon fraction at the interface and at the centre of the austenite grain, ðxhC xiC Þ, is considered, which is equal to zero in the interface-controlled mode. These quantities are depicted as a function of the austenite fraction normalised with respect to the equilibrium austenite fraction. The use of the normalised austenite fraction in the plots allows to evaluate the nature of the transformation at the same stage of the transformation with respect to
0.020 0.015
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.020 0.015 0.010
h
0.010
0.025 xC - xC (mole fraction)
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.005
i
0.005
i
x C - xC
γ eq
(mole fraction)
0.025
0.000 0.3
0.4
0.5
0.6 0.7 eq fγ / fγ
a
0.8
0.9
1.0
0.000 0.3
0.4
0.5
0.6 0.7 eq fγ / fγ
0.8
0.9
1.0
b
Fig. 4. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium, and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 760 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:05 mol m J
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
0.015 0.010
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.020 0.015 0.010
i
0.005
0.005
h
i
x C - xC
0.025 xC -xC (mole fraction)
0.020
Semi-analytical Numerical Diffusion-controlled Interface-controlled
γ eq
(mole fraction)
0.025
309
0.000
0.000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
b
Fig. 5. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 780 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:05 mol m J
0.020 0.015
γ eq
0.010
0.030 (mole fraction)
0.025
Semi-analytical Numerical Diffusion-controlled Interface-controlled
xC - xC
x C - xC
γ eq
(mole fraction)
0.030
0.020
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.015 0.010 0.005
i
i
0.005
0.025
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
b
Fig. 6. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 800 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:05 mol m J
representation of the actual phase transformation, this should be considered as a very satisfactory performance of the semi-analytical model for the low interface mobility. Figs. 7–9 report the obtained results for 1 1 M ac s . For this increased interface mobil0 ¼ 0:5 mol m J ity, the character of the transformation moves towards the diffusion-controlled mode. Thus, the diffusion-controlled model consequently performs better than for the low mobility; in this condition the disagreement between the numerical and the interface-controlled model, which was clearly seen for the low mobility, is even more evident at high mobility. For all chosen conditions, at the initial stage of austenite growth, where the increment of the fraction f c =f eq c with respect to the initial value is lower than 5%, the semi-analytical model overestimates the carbon fraction at the centre of the austenite grain. This overestimation is due to the fact that the parabolic approximation of the carbon fraction profile is less accurate, since the condition z0 s is not fulfilled. This can be seen from the carbon profiles in the austenite grain calculated by the numerical model, shown for the three different temperatures in Fig. 10. This figure clearly shows that the carbon-fraction profile significantly deviates from a parabola in the initial stage of transformation. Figs. 11 and 12 show the kinetics of the austenite formation at different temperatures, using M ac 0 ¼ 0:05 1 1 mol m J1 s1 and M ac s , respectively. 0 ¼ 0:5 mol m J
These figures show that not only in terms of fractions, but also in terms of transformation kinetics, the semi-analytical mixed-mode model produces accurate results. For all the chosen conditions the diffusion-controlled and interfacecontrolled models give the fastest kinetics, which is a direct consequence of the assumptions M = 1 and D = 1, respectively. For the high value of the interface mobility the semianalytical model predicts a slightly faster transformation kinetics than in the diffusion-controlled model (see Fig. 12). That is due to the overestimation of the deviation of the carbon fraction at the interface from the equilibrium (see Figs. 7a, 8a and 9a) and then of the driving force for the transformation in the semi-analytical model. The interface-controlled model gives faster growth kinetics than predicted by the other models as a consequence of the higher deviation from the local equilibrium in austenite at the interface (see Figs. 4–9). The overestimation of the transformation kinetics is more pronounced at higher interface mobility. The diffusion-controlled model also gives an overestimation of the growth kinetics, especially at low interface mobility when the carbon fraction in austenite at the interface significantly deviates from the equilibrium. The comparative analysis between the different models shows that both the diffusion-controlled model and the interface-controlled model are less accurate than the
310
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0.025
0.020 0.015
xC - xC (mole fraction)
Semi-analytical Numerical Diffusion-controlled Interface-controlled
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.020 0.015 0.010
i
0.010
0.005
h
0.005
i
xC - xC
γ eq
(mole fraction)
0.025
0.000 0.3
0.4
0.5
0.6
0.7
fγ / fγ
0.8
0.9
0.000 0.3
1.0
eq
a
0.4
0.5
0.6 0.7 eq fγ / fγ
0.8
0.9
1.0
b
Fig. 7. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 760 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:5 mol m J
0.020 0.015
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.025 0.020 0.015 0.010
i
0.010
0.030 xC - xC (mole fraction)
0.025
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.005
h
0.005
i
x C - xC
γ eq
(mole fraction)
0.030
0.000
0.000
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
b
Fig. 8. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 780 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:5 mol m J
0.020 0.015
0.025 0.020
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.015 0.010
i
0.010
0.030 xC - xC (mole fraction)
0.025
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.005
h
0.005
i
xC - xC
γ eq
(mole fraction)
0.030
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
b
Fig. 9. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 800 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:5 mol m J
mixed-mode model in describing the ferrite-to-austenite transformation kinetics over the entire course of the transformation. For the austenite-to-ferrite transformation kinetics, the mixed-mode character of the transformation was found
to be interface controlled in the initial stage of the transformation, related to the nucleation event. In the ferrite-toaustenite transformation nucleation does not play a role, since it takes place in the pearlite and an austenite grain is already present at the start of the ferrite-to-austenite
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
γ
xC (at %)
γ
xC (at %)
3.2 0.04 s 3.0 1.87 s 1.02 s 2.8 2.6 2.4 5.33 s 2.2 2.0 10.26 s 1.8 20.40 s 1.6 49.66 s 1.4 -6 -6 0.0 2.0x10 4.0x10 s (m)
-6
6.0x10
-6
8.0x10
3.2 3.0 1.02 s 2.8 2.6 1.98 s 2.4 2.2 2.0 5.15 s 1.8 10.10 s 1.6 1.4 19.97 s 1.2 50.60 s 1.0 0.0
311
0.04 s
-6
-5
5.0x10
1.0x10
s (m)
a
b
0.04 s
3.0 1.01 s
γ
xC (at %)
2.5 2.31 s
2.0 5.20 s
1.5 1.0
10.92 s 20.22 s 49.78 s
0.5 0.0
-6
-5
-5
1.0x10 s (m)
5.0x10
1.5x10
c
1.0
0.8
0.8
0.6
0.6
eq
1.0
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
fγ / f γ
fγ / f γ
eq
Fig. 10. Carbon profile in austenite at different holding times at (a) 760 °C, (b) 780 °C, (c) 800 °C calculated by the numerical model; 1 1 M ac s . 0 ¼ 0:05 mol m J
0
50
100
150
200
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2
250
300
time (s)
0.0
0
50
100
150
200
250
300
time (s)
a
b
1.0
fγ / f γ
eq
0.8 0.6 Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
0
50
100
150
200
250
300
time (s)
c 1 1 Fig. 11. Ferrite to austenite transformation kinetics during holding at (a) 760 °C, (b) 780 °C, (c) 800 °C for M ac s . 0 ¼ 0:05 mol m J
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
1.0
0.8
0.8
0.6
0.6
eq
1.0
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
fγ / f γ
fγ / f γ
eq
312
0
50
100
150
200
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2
250
300
0.0
0
50
100
150
time (s)
time (s)
a
b
200
250
300
1.0
fγ / fγ
eq
0.8 0.6 Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
0
50
100
150
200
250
300
time (s)
c 1 1 Fig. 12. Ferrite to austenite transformation kinetics during holding at (a) 760 °C, (b) 780 °C, (c) 800 °C for M ac s . 0 ¼ 0:5 mol m J
transformation. The fraction gradients are more pronounced at the early stages of transformation than at high holding times. This can be seen from the carbon profiles in the austenite grain calculated by the numerical model, shown for the three different temperatures in Fig. 10.
5. Conclusions A semi-analytical mixed-mode model is developed to describe the austenite growth kinetics at constant temperature in the intercritical range (after pearlite transformation into supersaturated austenite). The model uses the carbon diffusivity in austenite and the interface mobility as physical parameters. The comparison with a fully numerical model, requiring the numerical solution of the differential equations for the interface velocity and the carbon diffusion in austenite, shows that the semi-analytical mixed-mode model gives a good description of the ferrite-to-austenite transformation kinetics, although deviations appear in the initial stage of the transformation. Transformation conditions and kinetics predicted by the interface-controlled model and the diffusion-controlled model are considerably less accurate over the entire course of the transformation. The interface-controlled model gives faster austenite growth kinetics than that obtained from the numerical model, especially at the highest interface mobility. The overestimation of the carbon fraction at the interface, which is constant within the austenite grain, causes an increase of the deviation with respect to the equilibrium
carbon fraction and therefore of the driving force of the transformation. The diffusion-controlled model provides a satisfactory description of the ferrite-to-austenite transformation kinetics only at high interface mobility. At low interface mobility the diffusion-controlled model gives an overestimation of the austenite growth kinetics, especially at the early stage of transformation; the development of a strong carbonfraction gradient within the austenite grain causes faster transformation kinetics than the numerical model. Acknowledgement This research was carried out under project number MC5.06257 in the framework of the Strategic Research programme of the Materials Innovation Institute for Metals Research in the Netherlands (http://www.m2i.nl).
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[7] G.P. Krielaart, S. van der Zwaag, Mater. Sci. Eng. A 237 (1997) 216. [8] Y. van Leeuwen, J. Sietsma, S. van der Zwaag, ISIJ Int. 43 (2003) 767. [9] J. Sietsma, S. van der Zwaag, Acta Mater. 52 (2004) 4143. [10] J. Svoboda, F.D. Fischer, P. Fratzl, E. Gamsja¨ger, N.K. Simha, Acta Mater. 49 (2001) 1249. [11] E. Gamsja¨ger, F.D. Fischer, J. Svoboda, ASME J. Eng. Mater. Technol. 125 (2003) 22. [12] E. Gamsja¨ger, M. Militzer, F. Fazeli, J. Svoboda, F.D. Fischer, Comput. Mater. Sci. 37 (2006) 94. [13] H. Chen, S. van der Zwaag, Acta Mater. 72 (2014) 1. [14] C. Zheng, D. Raabe, Acta Mater. 61 (2013) 5504. [15] E. Gamsja¨ger, H. Chen, S. van der Zwaag, Comput. Mater. Sci. 83 (2014) 92. [16] M.G. Mecozzi, J. Sietsma, S. van der Zwaag, M. Apel, P. Schaffnit, I. Steinbach, Metall. Mater. Trans. A 36A (2005) 2327. [17] M. Militzer, M.G. Mecozzi, J. Sietsma, S. van der Zwaag, Acta Mater. 54 (2006) 3961.
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ScienceDirect Acta Materialia 88 (2015) 302–313 www.elsevier.com/locate/actamat
A mixed-mode model for the ferrite-to-austenite transformation in a ferrite/ pearlite microstructure ⇑
M.G. Mecozzi,a, C. Bosb and J. Sietsmaa a
Delft University of Technology, Department of Materials Science and Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands b Tata Steel Europe, Wenckebachstraat 1, 1951 JZ Velsen-Noord, The Netherlands Received 19 December 2014; revised 22 January 2015; accepted 25 January 2015
Abstract—A concise semi-analytical mixed-mode model is proposed to describe the ferrite-to-austenite transformation kinetics. The initial microstructure for the model consists of a ferrite matrix with supersaturated austenite grains. The carbon supersaturation of austenite grains resulted from the rapid dissolution of pearlite colonies present in the initial microstructure. A similar approach to the one used for describing the ferrite growth kinetics (Bos and Sietsma, 2007) was used, employing a sharp interface between the phases. In comparison to ferrite growth, the carbon concentration profile of the growing austenite contains an additional parameter, which is the carbon concentration at the centre of the grain. This extra parameter is accounted for by assuming a parabolic carbon profile in the austenite. A comparison with a numerical solution of the differential equations shows that the developed semi-analytical mixed-mode model gives a good description of the ferrite-to-austenite transformation, significantly more accurate over the entire course of the transformation than the classical interface-controlled and diffusion-controlled models. Significant deviations from the numerical solution, as a result of the parabolic approximation, are limited to the initial stage of the transformation. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Austenite growth kinetics; Modelling; Mixed-mode kinetics
1. Introduction In the past years, there has been a significant increase in the use of high strength Dual-Phase (DP) steels in the automotive industry. The favourable combination of strength and formability of DP steels makes these materials suitable for the construction of several parts of the car body. The microstructure of cold-rolled and annealed DP steels, consisting of a dispersion of hard martensite in a soft ferrite matrix, is typically obtained by annealing in the intercritical temperature range to produce an austenite-ferrite microstructure, followed by quenching to room temperature to transform austenite to martensite. Since the properties of DP steels markedly depend on the fraction, distribution and carbon content of the martensite phase, the understanding of the mechanism of austenite formation during heating and holding at the intercritical temperature is of primary importance. Despite this fact, the kinetics of austenite formation during heating has not been studied in such detail as the austenite-to-ferrite transformation on cooling. The explanations for this are several: first, the austenite formed during heating is transformed back to different product phases during cooling and therefore direct observation of the austenite formed on heating is very difficult; second, the austenite formation occurs very rapidly
⇑ Corresponding author; e-mail: [email protected]
as the temperature increases and consequently conventional techniques, like dilatometry and metallography, usually employed to study the transformation on cooling, are less adequate for studying the austenite formation. Another aspect makes the study of the austenite formation kinetics on heating difficult: the distribution and morphology of phases present in the starting microstructure affect the process of austenite formation, which then must be studied starting from different initial microstructures, whereas the ferrite formation usually occurs from homogenous austenite. This increases the number of required experiments. For similar reasons physically based models for the austenitisation formation are relatively underdeveloped. From experiments it is known that during intercritical annealing of steel having an initial ferrite plus pearlite microstructure the austenite formation occurs in two steps: the pearlite transforms to carbon-supersaturated austenite and subsequently ferrite transforms to austenite. The first step is controlled by the carbon diffusion within the pearlite regions. Since the diffusion distances involved are short, it is much more rapid than the second step. Therefore most models in the literature restrict the analysis of the austenitisation process to the ferrite-to-austenite transformation, starting from a microstructure of carbon-supersaturated austenite grains in a ferrite matrix [1–4]. In the literature, the ferrite-to-austenite transformation is often modelled assuming that the carbon diffusion in austenite controls the rate of transformation and the lattice transformation is
http://dx.doi.org/10.1016/j.actamat.2015.01.058 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
2. Theory In this work the ferrite-to-austenite phase transformation is assumed to occur with the partitioning of a single element as it happens in binary alloys or in ternary alloys in which one of the elements has a negligible diffusivity with respect to the effective interface motion. This is the case of the Fe–C–Mn system, in which the transformation can be assumed to take place without Mn partitioning since the Mn diffusion is a slow process compared to the motion of the interface as a consequence of the large difference in diffusivity of Mn and C in both austenite and ferrite: the transformation is said to occur under para-equilibrium conditions. This section will present a physical model of the interface motion of a sharp austenite/ferrite interface during the austenite-to-ferrite phase transformation under mixedmode conditions, governed by carbon partitioning and diffusion and the interface mobility. This approach leads to analytic expressions for the interface motion and the carbon fractions at the interface and within the austenite. 2.1. Semi-analytical model A ferritic matrix of dimension L is assumed as the initial microstructure, in which an austenite grain of width 2s0 is present, produced from the rapid transformation of a
pearlite colony at a temperature T above A1, when austenite becomes thermodynamically stable. If z is the distance from the centre of the austenite grain, the carbon mole fraction xC ðt; zÞ at t = 0 is given by xC ð0; zÞ ¼ xpC forjzj < s0 and xC ð0; zÞ ¼ xaeq C forjzj > s0
ð1Þ
xpC
where is the average pearlite carbon mole fraction and xaeq C is the equilibrium carbon mole fraction in ferrite at T (Fig. 1). The origin of the z-axis is in the centre of the grain. Because of the symmetry in this system, in the remainder we will consider only z > 0. The model focusses on hypo-eutectoid systems, for which the equilibrium carbon mole fraction in austenite p (c) in equilibrium with ferrite, xceq C , is lower than xC at intercritical temperatures. Therefore the austenite/ferrite interface experiences a driving force, DG, to move into the ferrite phase. This driving force DG depends on the carbon mole fraction at the interface in the austenite, xiC , which develops during the phase transformation. In the present approach DG is assumed to be proportional to the deviation of the carbon fraction at the interface from the equilibrium fraction in austenite, that is DG ¼ vðxiC xceq C Þ;
ð2Þ
xiC
with the carbon mole fraction in austenite at the interface position s, xC ðt; sÞ ¼ xiC . Although it has been shown [20–21] that the chemical potential of interstitial elements should always be equal at the interface of a sharp interface model here it is assumed that ferrite forms at the equilibrium carbon fraction, also when the interface carbon fraction in austenite is different from the equilibrium value. The differences in the ferrite carbon fractions caused by this simplification are negligibly small. The interface velocity v is proportional to this driving force, i.e. v ¼ MDG
ð3Þ
with M the interface mobility. For finite carbon diffusivity, a carbon gradient is established within the austenite. In line with the analytical description of the austenite-to-ferrite transformation [9] [19], the carbon-fraction profile in austenite (z < s) is assumed to be exponential in z (see Fig. 1) and given by expðz=z0 Þ þ expðz=z0 Þ 2 expðs=z0 Þ þ expðs=z0 Þ 2 coshðz=z0 Þ 1 ¼ xhC þ ðxiC xhC Þ coshðs=z0 Þ 1
xC ðt; zÞ ¼ xhC þ ðxiC xhC Þ
ð4Þ
p
Carbon concentration, xC
such a fast process that the carbon fractions in ferrite and austenite at the interface are equal to the equilibrium fractions [5–6]. The transformation is therefore diffusion controlled. In the more general case of finite interface mobility, carbon fractions at the interface differ from the equilibrium values and the transformation has a mixedmode character, which means that it is controlled by both carbon diffusivity and interface mobility. While the mixed-mode character of the austenite dissolution on cooling was extensively analysed in the literature, e.g. [7–13], few studies can be found on the mixed-mode character of austenite formation on heating [14–15]. In terms of modelling the phase transformation, phase-field modelling (e.g. [16–18]) does take the mixed-mode character into account in a diffuse-interface approach, but the method is too computationally intensive to allow 3D microstructural simulations. A more efficient modelling approach is therefore needed, similar to the concise analytical model that has been formulated recently to describe the austenite-to-ferrite transformation [19]. The model in Ref. [19] simply considers the conservation of carbon at the interface and the dependence of the driving force on the deviation from equilibrium to derive the carbon fraction in austenite at the interface, the width of carbon profile within the austenite and the interface velocity. The comparison with a fully numerical solution of the differential equations describing the carbon flux through the moving interface has shown that this analytical model is able to quite accurately describe the mixed-mode character of the austenite-to-ferrite transformation [19]. In the present paper the ferrite-to-austenite transformation during holding at an intercritical temperature is modelled by using a mixed-mode semi-analytical model similar to that formulated for the austenite-to-ferrite transformation in Ref. [19]. The results of the model and its limitations are discussed. A comparison with numerical solutions of the equations involving the movement of the interface and carbon diffusion is also presented. The obtained results are compared with those obtained for diffusion-controlled kinetics and interface-controlled kinetics.
303
xC
t=0 t
h
xC
i
xC
s0
s
α eq
xC
Distance from the center of γ grain, z Fig. 1. Carbon mole fraction in austenite (z < s) and in ferrite (z > s); the initial carbon mole fraction is given by the dashed line, the fraction at a certain stage during the transformation by the solid line.
304
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
where xhC is the carbon mole fraction at the centre of the austenite grain, xhC ¼ xC ðt; 0Þ, and z0 represents the width of the fraction profile. The expression given as Eq. (4) fulfils the boundary conditions xC ðt; sÞ ¼ xiC and a zero fraction gradient at the centre of the grain due to its symmetry, i.e. @xC ðxiC xhC Þ sinhðz=z0 Þ ¼ z0 coshðs=z0 Þ 1 @z
ð5Þ
is zero at z = 0. Since no carbon fraction gradients develop in the ferrite, the total amount of carbon in the austenite grain is, when it has grown to an interface position s, equal to the amount of carbon in the region 0 < z < s for the initial configuration (see Fig. 1), i.e. Z s xC ðt; zÞ dz ¼ s0 xpC þ ðs s0 Þxaeq ð6Þ C 0
Using Eqs. (4) and (6) becomes sðxiC xhC Þ
ð7Þ
which expresses the interdependence of the interface position s and the carbon-fraction profile (z0, xiC ; xhC ). To simplify the equations, all the carbon fractions are related to the equilibrium carbon fraction in ferrite by defining ¼
xiC
xaeq C ;
~xhC
¼
xhC
xaeq C ;
~xpC
¼
xpC
xaeq C :
ð8Þ
Then Eq. (7) becomes s0~xpC s~xhC ¼ sð~xiC ~xhC Þ
½ðz0 =sÞ sinhðs=z0 Þ 1 : coshðs=z0 Þ 1
ð9Þ
Interface motion causes a flux of carbon given by the product of the interface velocity and the difference in carbon fraction between austenite and ferrite at the interface. This flux, which causes a depletion of carbon at the interface, is compensated by the diffusional flux of carbon in the austenite towards the interface (i.e. at z = s): c vðxiC xaeq C Þ ¼ DC
dxC j : dz z¼s
ð10Þ
with DcC the carbon diffusivity in austenite. Eq. (10) is the central equation for this mixed-mode model, since it expresses the relation between the interface motion and the carbon diffusion. Using Eqs. (2), (3), (5) and (10) becomes aeq i MvðxiC xceq C ÞðxC xC Þ ¼
ðxi DcC C
xhC Þ
z0 sinhðs=z0 Þ : coshðs=z0 Þ 1
ðs0 =sÞ~xpC ð1 BÞ ~xiC : ð14Þ B These two quantities are crucial for the mixed-mode character of the phase transformation and govern the further development of the transformation through carbon diffusion and the driving force at the interface. The additional parameters E, B and F in the Eqs. (13) and (14) are combinations of previously introduced parameters and given by
~xhC ¼
ð~xiC ~xhC Þ sinhðs=z0 Þ z0 coshðs=z0 Þ 1
Mv : DcC
B¼1
F ¼
z0 ½sinhðs=z0 Þ ðs=z0 Þ : coshðs=z0 Þ 1 s
1 sinhðs=z0 Þ : z0 coshðs=z0 Þ 1
ð15Þ
ð16Þ
ð17Þ
The parameter E has a direct and interesting physical meaning. Its numerator gives the product of interface mobility and the proportionality factor v (see Eq. (2)), which represents the interface kinetics; its denominator is the diffusivity and thus represents the diffusion kinetics. A high value of E indicates the transformation character to be close to diffusion control, a low E indicates a transformation character that is closer to interface control. Note that the evolution of the carbon fraction during the transformation is related to the variable parameters s and z0 in the Eqs. (13) and (14). The mixed-mode approach should represent the conditions for diffusion-controlled transformation for infinite mobility and the conditions for interface-controlled transformation for infinite diffusivity. For the diffusion-controlled mode (DCM), i.e. M ! 1, the factor E ! 1, and it follows from Eq. (13) that xiC jDCM ¼ xceq C
ð18Þ
and from Eq. (14) that ðs0 =sÞðxpC xaeq C Þ ð1 BÞDxC : ð19Þ B In the diffusion-controlled mode a carbon-fraction gradient develops and the carbon mole fraction at the centre of the austenite grain, xhC jDCM , depends on the position of the interface, s, and on the width of carbon profile z0 through the factor B (Eq. (16)). Eq. (18) expresses the well-known local equilibrium for diffusion-controlled transformation. Under the assumption of interface mobility controlled mode (ICM), which means D ! 1, the condition xhC jDCM ¼ xaeq C þ
ð11Þ
Relating to the equilibrium carbon fraction in ferrite, Eqs. (8) and (11) become Mvð~xiC DxC Þ~xiC ¼ DcC
and
E¼
½ðz0 =sÞ sinhðs=z0 Þ 1 coshðs=z0 Þ 1
aeq h ¼ s0 ðxpC xaeq C Þ sðxC xC Þ
~xiC
used to implement the boundary condition of a constant total amount of carbon: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 ½BDxC F =E2 þ 4s s0~xpC B F =E s½BDx C F =E þ i ~xC ¼ 2sB ð13Þ
ð12Þ
aeq with DxC ¼ xceq C xC . From Eq. (12), analytic expressions for the interface carbon fraction and the carbon fraction in the centre of the austenite grain, ~xiC and ~xhC , can be derived when Eq. (9) is
~xiC ¼ ~xhC should be fulfilled.
ð20Þ
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
Using Eq. (20) in Eq. (14) yields xhC jICM
¼
xiC jICM
¼
xaeq C
þ
ðxpC
xaeq C Þs0 =s
ð21Þ
If L is the domain dimension, s0 is given by s0 ¼ L
x0C xaeq C xpC xaeq C
ð22Þ
and Eq. (21) becomes aeq 0 xiC jICM ¼ xaeq C þ ðxC xC ÞL=s:
ð23Þ
The ratio s/L is equal to the austenite fraction fc and therefore Eq. (23) becomes x0 ð1 f c Þxaeq C xhC jICM ¼ xiC jICM ¼ C ð24Þ fc x0C
with the alloy carbon mole fraction and f c the fraction of austenite. This equation is simply the lever rule for two phases with homogeneous carbon fraction. The Eqs. (18) and (24) show that the carbon fractions resulting from the proposed mixed-mode model are, in the extremes, consistent with both the diffusion-controlled and interface-controlled model. Eq. (13) gives the interface velocity v when it is combined with the Eqs. (2) and (3). However, in order to evaluate Eq. (13), an additional assumption on the value of z0 is needed. At the start of transformation (t = 0) and also when the equilibrium is reached the value of the carbon fraction inside the austenite phase is constant and therefore z0 ¼ 1; during transformation z0 has a finite value greater than 0. If it is assumed that during transformation z0 >> s > s0 , the carbon profile in austenite can be approximated by a parabola, since Eq. (4) then becomes z2 xC ðt; zÞ ¼ xhC þ ðxiC xhC Þ : ð25Þ s where xhC and xiC can be determined by Eqs. (13) and (14) in sinhðyÞy the limit s=z0 ! 0. Knowing that lim yðcoshðyÞ1Þ ¼ 13 and y!0
y sinhðyÞ y!0 coshðyÞ1
lim
¼ 2, from Eq. (16)it follows that B ! 2=3 and
from Eq. (17) that F ! 2=s. Under this parabolic approximation, ~xiC and ~xhC then become respectively qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ~xiC ¼ ð26Þ ½ðEsDxC 3Þ þ ðEsDxC 3Þ2 þ 12s0 E~xpC 2Es and ~xhC ¼ ð3s0 =2sÞ~xpC ~xiC =2: ð27Þ aeq with E given by Eq. (15) and DxC ¼ xceq C xC . The analytical model for the carbon distribution that has been presented here can be used to predict the development of the phase transformation as a function of time, albeit not analytically. At the starting time, t = 0, s ¼ s0 and the velocity v0 is calculated by using Eqs. (3) and (2) with xiC ¼ xpC . For the following steps in an iterative process with time step dt, the carbon fraction at the centre of austenite grain and at the interface are calculated by using Eqs. (26) and (27) with s calculated from the interface position and velocity from the previous time step by
sðtÞ ¼ sðt dtÞ þ vðt dtÞdt:
ð28Þ
The proposed model thus results in a scheme of a few analytical expressions that can be used in a simulation in which the motion of the austenite/ferrite interface is traced in time steps dt, taking the carbon
305
diffusivity D and the interface mobility M as input parameters. Given the interface position s from the previous time step, the interface carbon fraction in the austenite can be calculated by Eq. (26). This fraction can be employed in Eq. (2) to calculate the driving force DG. The proportionality parameter v in Eq. (2) is derived by Thermo-CalcÒ [22], by calculating the driving force for austenite formation as a function of carbon fraction in austenite at the interface for a given set of temperatures; in this work the driving force is calculated under para-equilibrium condition. The interface velocity, calculated from Eq. (3), is then used to derive the new interface position from Eq. (28). 2.2. Numerical model The validity of the analytic description of the interface velocity in mixed-mode transformation conditions is tested by comparing the obtained results with a fully numerical solution of the differential equations for the interface velocity and the carbon diffusion, which will be described in this section. The numerical model is based on the Murray–Landis finite-difference approach [23] and is essentially the same as used in Ref. [19] for the austenite-to-ferrite transformation. For the description of the ferrite-to-austenite transformation only the diffusion of carbon in austenite is taken into account. The carbon fraction in ferrite is assumed to be equal to the para-equilibrium fraction for ferrite in equilibrium with austenite. The Murray–Landis finite-difference approach has been implemented by the following algorithm: 1. perform a diffusion step according to Fick’s second law; 2. calculate the maximum driving force DG based on the austenite interface carbon fraction and the temperature; 3. calculate the interface velocity, v, with Eq. (3); 4. move the interface a distance dz = vdt, where dt is the time step; 5. scale the finite difference grid (i.e. the Murray–Landis approach of variable grid-spacing); 6. calculate the new austenite carbon interface fraction based on the mass balance; 7. repeat from step 1. Like in the semi-analytical model, the driving force is derived as a function of the carbon fraction in austenite at the interface from Eq. (2), with the proportionality parameter v derived by Thermo-calc. 2.3. Diffusion-controlled model In order to investigate the effect of the mixed-mode approach as compared to the more generally used approximations of diffusion control or interface control, in the present study the ferrite-to-austenite transformation is also modelled by assuming that the carbon diffusion in austenite is the rate controlling process. Under the assumption of a fast interfacial reaction, thermodynamic equilibrium can be assumed to be maintained locally at the interface (local equilibrium). The simulation of ferrite-to-austenite transformation is performed using the DIffusion Controlled TRAnsformations (DICTRA) software [24] in combination with Thermo-calc software [22] and Calphad database [25], which provides all necessary thermodynamic data and the diffusivities to perform the simulations. The treatment of
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
1.0
0.05
0.8
0.04 xC (mole fraction)
0.6 0.4 0.2
0.03
γ line
0.02 0.01
γ
ferrite fraction
306
α line
0.00 0.0 650
700
750 T (ºC)
800
850
a
650
700
750 T (ºC)
800
850
b
γ line
c Fig. 2. (a) Ferrite fraction, (b) carbon mole fraction in austenite and ferrite, (c) driving force for austenite formation as a function of temperature and carbon mole fraction in austenite, calculated under the assumption of immobile substitutional elements.
the moving boundary problem in Dictra software is outlined in Ref. [24]. The principal equation in the diffusion-controlled model is the mass balance at the interface, expressed by c x xa vca kc ka ¼ J ck J ak ; k ¼ 1; 2 . . . n 1 ð29Þ Vm Vm where vca denotes the interface migration rate, V cm and V am are the molar volume of the c and a phase, xck and xak are the mole fractions of component k in c and in a, and J ck and J ak are the corresponding diffusional fluxes, expressed as n1 @xj 1 X Dckj ; c V m j¼1 @z c
J ck ¼
J ak ¼
n1 @xaj 1 X a D : V am j¼1 kj @z
ð30Þ
where Dckj and Dakj denote the kj-component of the diffusivity matrix in the phases a and c respectively. The summation is performed over (n–1) independent fractions, as the dependent nth component may be taken as the solvent. The temporal profile of the diffusion species in each phase is given by solving the diffusion equations c 1 @xj @ ¼ J ck ; V cm @t @z
1 @xaj @ ¼ J ak : V am @t @z
ð31Þ
Under the assumption of constant carbon mole fraction in ferrite, equal to the equilibrium value at the selected temperature, and equal molar volume of austenite and ferrite, Eq. (29) with carbon as the k-component, reduces to
Eq. (10). In the diffusion-controlled model Eq. (29) is used to calculate the interface velocity since Eq. (3), utilised in the numerical and semi-analytical model, cannot be applied when the interface mobility is infinite. The diffusivities in Eq. (30) can be obtained as a product of a thermodynamic factor, which is the second derivative of the molar Gibbs energy with respect to the fractions, and a kinetic factor, which contains the atomic mobilities. This approach considerably reduces the number of independent variables necessary to calculate the diffusional flux [24]. There are three main modules in the DICTRA software, one for solving the diffusion equations inside each phase, i.e. Eq. (31), and one for calculating the local equilibrium at the interface. The third module sets up and solves the flux-balance equations, i.e. Eq. (30), in order to obtain the transformation rates. The position of the phase interface is subsequently obtained by integration. 2.4. Interface-controlled model Also a comparison with the interface-controlled transformation mode is made. In the interface-controlled approach it is assumed that the ferrite-to-austenite transformation kinetics is controlled by the rate of lattice transformation (reflected by the interface mobility M) and that the carbon diffusion in ferrite and austenite is fast enough to maintain a homogenous carbon fraction in each phase. Just like in ferrite, the carbon fraction in austenite is
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
307
600
-1
ΔG (J mol )
800
400 200 0 0.00
0.02 0.04 0.06 γ γ,eq xC - xC (mole fraction)
0.08
800
600
600
ΔG (J mol )
800
-1
-1
ΔG (J mol )
a
400 200 0 0.00
0.02 0.04 0.06 γ γ,eq xC - xC (mole fraction)
b
0.08
400 200 0 0.00
0.02 0.04 0.06 γ γ,eq xC - xC (mole fraction)
0.08
c
Fig. 3. Driving force as a function of the deviation of the carbon mole fraction in austenite at the interface from the equilibrium mole fraction at (a) 760 °C, (b) 780 °C and (c) 800 °C.
assumed to be homogenous through the whole phase and its value, xcC , is given by Eq. (24). It should be noted that while the carbon fraction in ferrite remains constant during isothermal holding, equal to the equilibrium at the considered temperature, xa;eq C , the carbon fraction in austenite changes during transformation since it depends on the fraction of austenite. As for the semi-analytical model, at the starting time, t = 0, s ¼ s0 and the velocity v0 is calculated by using Eqs. (2) and (3) with xiC ¼ xpC . For the following steps the carbon composition at the interface is calculated using Eq. (24), and the resulting value is used to calculate the driving force and velocity with the Eqs. (2) and (3), using xiC ¼ xcC . 3. Simulation conditions All results in this paper are obtained for an Fe–0.45 C– 1.5 Mn (at.%) alloy. The thermodynamic data are calculated by ThermoCalc software [22]. Fig. 2a and b shows the equilibrium ferrite phase fraction and carbon mole fraction in ferrite and austenite as a function of temperature under para-equilibrium conditions. The driving force for the ferrite-to-austenite transformation is shown in Fig. 2c as a function of temperature and carbon fraction in the austenite at the interface. The intersection of this surface with the plane DG ¼ 0 gives the equilibrium carbon fraction in austenite as a function of temperature (c line in Fig. 2b). In order to calculate the factor v in Eq. (2) for three selected transformation temperatures, 760 °C, 780 °C and
800 °C, the driving force is plotted as a function of the deviation from the equilibrium fraction DxcC ¼ xiC xcCeq and fitted with a straight line for 0 < DxcC < 3 at:%, as shown in Fig. 3. The equilibrium carbon mole fraction in austenite and the factor v, reported in Table 1, are used to calculate the carbon profile in austenite and the kinetics of austenite formation in the semi-analytical model. The equilibrium ferrite phase fraction under para-equilibrium conditions at the three selected temperatures are also reported in Table 1. The interface mobility is assumed to obey an Arrhenius relationship ac M ¼ M ac 0 expðQ =RT Þ
ð32Þ
with the gas constant R = 8.31 JK1 mol1, the activation energy Qac ¼ 140 kJ=mol [7]; the pre-exponential factor 1 1 M ac 0 is varied between 0.05 and 0.5 mol m J s , which is the range of interface mobility values reported in different studies of the kinetics of austenite formation or dissolution [14,26–28]. The mobility is varied to study its effect on the accuracy of the semi-analytical model, as compared to the numerical model, and on the mixed-mode character of the transformation with respect to the interface-controlled model and diffusion-controlled mode. For the carbon diffusivity in austenite the data from [29] are taken, i.e. 8339:9 K DcC ¼ 4:53 107 1 þ y C ð1 y C Þ T 1 4 1 2:221 10 K Þð17767 K y C 26436K m2 s1 exp T ð33Þ
308
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
Table 1. Thermodynamic and kinetic factors governing the phase transformation at the three considered temperatures. Simulations were performed for two different values of the interface mobility. xceq C ðmol%Þ 1
vðJ mol mol%1 Þ f eq c DcC ðm2 s1 Þ 1 1 Eðm1 mol%1 Þ for M ac s 0 ¼ 0:05 m mol J 1 1 Eðm1 mol%1 Þ for M ac ¼ 0:5 m mol J s 0
760 °C
780 °C
800 °C
1.27 105 0.336 1.1 1012 3.8 105 3.8 106
0.85 102 0.516 1.4 1012 4.1 105 4.1 106
0.48 101 0.937 1.7 1012 4.5 105 4.5 106
where y C ¼ xC =ð1 xC Þ, with xC the atomic fraction of carbon and T the temperature in Kelvin. Since the results of the mixed-mode model are evaluated in comparison with those of the diffusion-controlled model, for which local equilibrium condition at the interface ðxiC ¼ xceq C Þ is assumed, the carbon diffusivity is calculated from Eq. (33) with xC ¼ xceq C . The carbon diffusivity at the different temperatures is reported in Table 1. The domain size, L, is set depending on the initial austenite grain size, s0, according to the relation L¼
xpC xaeq C s0 : x0C xaeq C
equilibrium for different transformation temperatures. The results of the semi-analytical model are compared with results from the fully numerical model, the diffusion-controlled model and the interface-controlled model. In order to assess the effect of austenite/ferrite interface mobility on the nature of the transformation, two values of the pre-exponential factor of the 1 1 interface mobility are selected: M ac s 0 ¼ 0:05 mol m J 1 1 and M ac ¼ 0:5 mol m J s . 0 Figs. 4–6 report the obtained results for M ac 0 ¼ 0:05 mol m J1 s1 at the three temperatures 760 °C, 780 °C, 800 °C. It is evident that the character of the transformation is in between the diffusion-controlled mode and the interfacecontrolled mode independently on the transformation temperature. For a normalised austenite fraction of 0.5 (the transformation is halfway to full equilibrium) the deviation from local equilibrium predicted by the interface-controlled model is significantly higher than the value calculated in the semi-analytical and numerical model at any transformation temperature (see Figs. 4a, 5a and 6a). This means that the interface-controlled model gives an overestimation of the carbon fraction at the interface. Consequently, the interface-controlled model underestimates the carbon fraction at the centre of the austenite grain (see Figs. 4b, 5b and 6b). On the other hand, the diffusion-controlled model gives an underestimation of the carbon fraction at the interface, which results in an overestimation of the carbon fraction at the centre of the austenite grain. The deviations between the fractions given by the semi-analytical model and the numerical model are less than one tenth of the deviations found for the two extreme models. Given that the numerical solution can be regarded as the most accurate
ð34Þ
where x0C is the alloy average carbon composition. 4. Results and discussion The mixed mode character of the ferrite-to-austenite transformation with respect to diffusion-controlled mode and interface-controlled mode is evaluated by plotting the deviation from the local equilibrium of the carbon fraction at the interface, ðxiC xcCeq Þ. This deviation is equal to zero in the diffusion-controlled mode. Also, the difference between the carbon fraction at the interface and at the centre of the austenite grain, ðxhC xiC Þ, is considered, which is equal to zero in the interface-controlled mode. These quantities are depicted as a function of the austenite fraction normalised with respect to the equilibrium austenite fraction. The use of the normalised austenite fraction in the plots allows to evaluate the nature of the transformation at the same stage of the transformation with respect to
0.020 0.015
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.020 0.015 0.010
h
0.010
0.025 xC - xC (mole fraction)
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.005
i
0.005
i
x C - xC
γ eq
(mole fraction)
0.025
0.000 0.3
0.4
0.5
0.6 0.7 eq fγ / fγ
a
0.8
0.9
1.0
0.000 0.3
0.4
0.5
0.6 0.7 eq fγ / fγ
0.8
0.9
1.0
b
Fig. 4. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium, and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 760 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:05 mol m J
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
0.015 0.010
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.020 0.015 0.010
i
0.005
0.005
h
i
x C - xC
0.025 xC -xC (mole fraction)
0.020
Semi-analytical Numerical Diffusion-controlled Interface-controlled
γ eq
(mole fraction)
0.025
309
0.000
0.000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
b
Fig. 5. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 780 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:05 mol m J
0.020 0.015
γ eq
0.010
0.030 (mole fraction)
0.025
Semi-analytical Numerical Diffusion-controlled Interface-controlled
xC - xC
x C - xC
γ eq
(mole fraction)
0.030
0.020
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.015 0.010 0.005
i
i
0.005
0.025
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
b
Fig. 6. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 800 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:05 mol m J
representation of the actual phase transformation, this should be considered as a very satisfactory performance of the semi-analytical model for the low interface mobility. Figs. 7–9 report the obtained results for 1 1 M ac s . For this increased interface mobil0 ¼ 0:5 mol m J ity, the character of the transformation moves towards the diffusion-controlled mode. Thus, the diffusion-controlled model consequently performs better than for the low mobility; in this condition the disagreement between the numerical and the interface-controlled model, which was clearly seen for the low mobility, is even more evident at high mobility. For all chosen conditions, at the initial stage of austenite growth, where the increment of the fraction f c =f eq c with respect to the initial value is lower than 5%, the semi-analytical model overestimates the carbon fraction at the centre of the austenite grain. This overestimation is due to the fact that the parabolic approximation of the carbon fraction profile is less accurate, since the condition z0 s is not fulfilled. This can be seen from the carbon profiles in the austenite grain calculated by the numerical model, shown for the three different temperatures in Fig. 10. This figure clearly shows that the carbon-fraction profile significantly deviates from a parabola in the initial stage of transformation. Figs. 11 and 12 show the kinetics of the austenite formation at different temperatures, using M ac 0 ¼ 0:05 1 1 mol m J1 s1 and M ac s , respectively. 0 ¼ 0:5 mol m J
These figures show that not only in terms of fractions, but also in terms of transformation kinetics, the semi-analytical mixed-mode model produces accurate results. For all the chosen conditions the diffusion-controlled and interfacecontrolled models give the fastest kinetics, which is a direct consequence of the assumptions M = 1 and D = 1, respectively. For the high value of the interface mobility the semianalytical model predicts a slightly faster transformation kinetics than in the diffusion-controlled model (see Fig. 12). That is due to the overestimation of the deviation of the carbon fraction at the interface from the equilibrium (see Figs. 7a, 8a and 9a) and then of the driving force for the transformation in the semi-analytical model. The interface-controlled model gives faster growth kinetics than predicted by the other models as a consequence of the higher deviation from the local equilibrium in austenite at the interface (see Figs. 4–9). The overestimation of the transformation kinetics is more pronounced at higher interface mobility. The diffusion-controlled model also gives an overestimation of the growth kinetics, especially at low interface mobility when the carbon fraction in austenite at the interface significantly deviates from the equilibrium. The comparative analysis between the different models shows that both the diffusion-controlled model and the interface-controlled model are less accurate than the
310
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0.025
0.020 0.015
xC - xC (mole fraction)
Semi-analytical Numerical Diffusion-controlled Interface-controlled
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.020 0.015 0.010
i
0.010
0.005
h
0.005
i
xC - xC
γ eq
(mole fraction)
0.025
0.000 0.3
0.4
0.5
0.6
0.7
fγ / fγ
0.8
0.9
0.000 0.3
1.0
eq
a
0.4
0.5
0.6 0.7 eq fγ / fγ
0.8
0.9
1.0
b
Fig. 7. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 760 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:5 mol m J
0.020 0.015
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.025 0.020 0.015 0.010
i
0.010
0.030 xC - xC (mole fraction)
0.025
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.005
h
0.005
i
x C - xC
γ eq
(mole fraction)
0.030
0.000
0.000
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
b
Fig. 8. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 780 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:5 mol m J
0.020 0.015
0.025 0.020
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.015 0.010
i
0.010
0.030 xC - xC (mole fraction)
0.025
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.005
h
0.005
i
xC - xC
γ eq
(mole fraction)
0.030
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
a
0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 eq fγ / fγ
b
Fig. 9. (a) Deviation of the carbon mole fraction in austenite at the interface from the equilibrium and (b) difference between the carbon mole fraction at the interface and at the centre of the austenite grain during holding at 800 °C predicted by different models for interface mobility 1 1 M ac s . 0 ¼ 0:5 mol m J
mixed-mode model in describing the ferrite-to-austenite transformation kinetics over the entire course of the transformation. For the austenite-to-ferrite transformation kinetics, the mixed-mode character of the transformation was found
to be interface controlled in the initial stage of the transformation, related to the nucleation event. In the ferrite-toaustenite transformation nucleation does not play a role, since it takes place in the pearlite and an austenite grain is already present at the start of the ferrite-to-austenite
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
γ
xC (at %)
γ
xC (at %)
3.2 0.04 s 3.0 1.87 s 1.02 s 2.8 2.6 2.4 5.33 s 2.2 2.0 10.26 s 1.8 20.40 s 1.6 49.66 s 1.4 -6 -6 0.0 2.0x10 4.0x10 s (m)
-6
6.0x10
-6
8.0x10
3.2 3.0 1.02 s 2.8 2.6 1.98 s 2.4 2.2 2.0 5.15 s 1.8 10.10 s 1.6 1.4 19.97 s 1.2 50.60 s 1.0 0.0
311
0.04 s
-6
-5
5.0x10
1.0x10
s (m)
a
b
0.04 s
3.0 1.01 s
γ
xC (at %)
2.5 2.31 s
2.0 5.20 s
1.5 1.0
10.92 s 20.22 s 49.78 s
0.5 0.0
-6
-5
-5
1.0x10 s (m)
5.0x10
1.5x10
c
1.0
0.8
0.8
0.6
0.6
eq
1.0
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
fγ / f γ
fγ / f γ
eq
Fig. 10. Carbon profile in austenite at different holding times at (a) 760 °C, (b) 780 °C, (c) 800 °C calculated by the numerical model; 1 1 M ac s . 0 ¼ 0:05 mol m J
0
50
100
150
200
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2
250
300
time (s)
0.0
0
50
100
150
200
250
300
time (s)
a
b
1.0
fγ / f γ
eq
0.8 0.6 Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
0
50
100
150
200
250
300
time (s)
c 1 1 Fig. 11. Ferrite to austenite transformation kinetics during holding at (a) 760 °C, (b) 780 °C, (c) 800 °C for M ac s . 0 ¼ 0:05 mol m J
M.G. Mecozzi et al. / Acta Materialia 88 (2015) 302–313
1.0
0.8
0.8
0.6
0.6
eq
1.0
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
fγ / f γ
fγ / f γ
eq
312
0
50
100
150
200
Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2
250
300
0.0
0
50
100
150
time (s)
time (s)
a
b
200
250
300
1.0
fγ / fγ
eq
0.8 0.6 Semi-analytical Numerical Diffusion-controlled Interface-controlled
0.4 0.2 0.0
0
50
100
150
200
250
300
time (s)
c 1 1 Fig. 12. Ferrite to austenite transformation kinetics during holding at (a) 760 °C, (b) 780 °C, (c) 800 °C for M ac s . 0 ¼ 0:5 mol m J
transformation. The fraction gradients are more pronounced at the early stages of transformation than at high holding times. This can be seen from the carbon profiles in the austenite grain calculated by the numerical model, shown for the three different temperatures in Fig. 10.
5. Conclusions A semi-analytical mixed-mode model is developed to describe the austenite growth kinetics at constant temperature in the intercritical range (after pearlite transformation into supersaturated austenite). The model uses the carbon diffusivity in austenite and the interface mobility as physical parameters. The comparison with a fully numerical model, requiring the numerical solution of the differential equations for the interface velocity and the carbon diffusion in austenite, shows that the semi-analytical mixed-mode model gives a good description of the ferrite-to-austenite transformation kinetics, although deviations appear in the initial stage of the transformation. Transformation conditions and kinetics predicted by the interface-controlled model and the diffusion-controlled model are considerably less accurate over the entire course of the transformation. The interface-controlled model gives faster austenite growth kinetics than that obtained from the numerical model, especially at the highest interface mobility. The overestimation of the carbon fraction at the interface, which is constant within the austenite grain, causes an increase of the deviation with respect to the equilibrium
carbon fraction and therefore of the driving force of the transformation. The diffusion-controlled model provides a satisfactory description of the ferrite-to-austenite transformation kinetics only at high interface mobility. At low interface mobility the diffusion-controlled model gives an overestimation of the austenite growth kinetics, especially at the early stage of transformation; the development of a strong carbonfraction gradient within the austenite grain causes faster transformation kinetics than the numerical model. Acknowledgement This research was carried out under project number MC5.06257 in the framework of the Strategic Research programme of the Materials Innovation Institute for Metals Research in the Netherlands (http://www.m2i.nl).
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