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Competition between nucleation and early growth of ferrite from austenite—studies using cellular automaton simulations
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Acta mater. Vol. 46, No. 17, pp. 6291±6303, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00243-2 1359-6454/98 $19.00 + 0.00
COMPETITION BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE FROM AUSTENITEÐSTUDIES USING CELLULAR AUTOMATON SIMULATIONS M. KUMAR1, R. SASIKUMAR1{ and P. KESAVAN NAIR2 Regional Research Laboratory (CSIR), Trivandrum 695019, India and 2Department of Metallurgical Engineering, Indian Institute of Technology, Madras 600 036, India
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(Received 29 January 1998; accepted 18 June 1998) AbstractÐA model for the nucleation of ferrite on austenite grain boundaries and the growth of these nuclei along the grain boundary and into the grain, is developed. A cellular automaton algorithm, with transformation rules based on this model, is used to simulate the decomposition of austenite into ferrite. When performed under continuous cooling conditions, the simulations give an insight into the competition between nucleation and early growth, which determines the variation of ferrite grain size with the cooling rate and with austenite grain size. The number of ferrite grains per austenite grain, ferrite grain size and the kinetics of ferrite formation are obtained as a function of the cooling rate and austenite grain size. Contour plots of the volume fraction of ferrite in the cells at dierent times, enables visualization of the ferrite growth process. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
In recent years, computer-based predictive tools for linking processing conditions to microstructure and properties of the processed materials, have been getting a lot of attention from researchers and software developers. The basis for such a tool is a model for the process which describes the major phenomena that control the process and which can be implemented on a computer. Predicting the transformation behaviour of steel components during heat treatment is an area of considerable industrial importance. One successful route is the semi-empirical approach which combines numerical heat transfer analysis of heat-treated components with empirical time±temperature± transformation (TTT) diagrams to calculate the amount of various transformation products [1±6]. The limitations of this approach are that extensive empirical data are required for this and that only average phase fractions can be calculated and not the size or shape of the phases. Also these simulations do not lead to any deeper understanding of the controlling phenomena. Therefore, a need for more fundamental models of the transformation has been felt lately [7] both for obtaining deeper understanding as well as for more con®dent predictions. In this paper we present a model for nucleation and growth of ferrite from austenite which could form the basis for a predictive tool for the number, size and shape of ferrite grains in austenite as a {To whom all correspondence should be addressed.
function of the austenite grain size and cooling rate. We adopt a cellular automaton scheme for implementing this model in two dimensions. The twodimensional simulations are naturally not quantitatively precise, but it is possible to make a numerical study of the process to understand the various factors that control the transformation. With higher computing power, it is possible to implement the model in three dimensions and make quantitative predictions. Recently, Militzer et al. [7] studied the kinetics of austenite decomposition under continuous cooling conditions for dierent cooling rates and austenite sizes. The ferrite grain size decreased with the cooling rate while the undercooling for the start of transformation (the undercooling at 5% transformation) increased as the cooling rate increased. The number of ferrite grains nucleated per austenite grain was determined and it was found that as the cooling rate increased, the number was more than what could have nucleated at the grain corners. Therefore, it was proposed that as the undercooling increases, nucleation sites at the grain edges and surfaces must be becoming active. At low cooling rates, these sites are deactivated by growth of nuclei which originated earlier at the corners at lower undercoolings, into these sites. At higher cooling rates, higher undercoolings can be reached before the early nuclei have had time to inactivate the nucleation sites on the edges and surfaces by growing into them. Thus the number of ferrite grains in an austenite grain is governed by the competition between nucleation and early growth of the ferrite. The transformation start temperature also depends
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on this phenomenon. A detailed model describing the complex interaction of nucleation and growth, which can predict these eects, has not yet been reported. In our attempt to develop a model for describing the nucleation and early growth of ferrite we noticed that the competition between nucleation and growth which occurs here is very similar to the phenomenon of grain structure formation during solidi®cation of alloy melts. During solidi®cation the cooling curve exhibits a recalescence whose depth is indicative of the grain size or the number density of activated nuclei. The higher the cooling rate, the higher the undercooling reached and the ®ner the grain size. This phenomenon can be explained on the basis of a continuous nucleation model and competition between nucleation and early growth. It is assumed that nucleation sites with a continuous range of associated undercoolings exist in the melt and that each nucleating site becomes instantaneously active when the undercooling becomes equal to the undercooling associated with it. The grains nucleated earlier keep growing and releasing latent heat and at a certain point of time recalescence occurs. Recalescence marks the highest undercooling reached and, usually, the end of nucleation. (Nucleation does not take place during recalescence because all the nucleation sites which could have become activated at these temperatures have already been activated during the cooling up to the recalescence temperature. Nucleation can, in principle, start again when the material cools down, after recalescence, below the recalescence temperature; however, this does not happen often because, by this time, the growth of the early nucleated grains would have consumed these sites.) The point of recalescence is controlled by the competition between nucleation and growth. During slow cooling the grains nucleated earlier have time to grow and liberate sucient amount of latent heat to produce recalescence before high undercoolings are reached, whereas during fast cooling large undercoolings are reached before the early nuclei grow and liberate sucient latent heat to produce recalescence. Solidi®cation grain structure can thus be seen to be governed by phenomena very similar to the one described by Militzer et al. for ferrite grain structure in heat-treated steels. The continuous nucleation model proposed by Rappaz and co-workers [8] has proved to be very useful for developing a predictive tool for grain structure in castings. We have adapted this nucleation model in our model for ferrite formation from austenite. Potential nucleation sites with a de®nite undercooling range associated with them, are distributed randomly on the grain boundary. When the grain is cooled, the nucleation sites become active when the associated undercooling is reached. All the potential sites cannot get activated because, (1) they are consumed by growth of ferrite
nucleated earlier and (2) the carbon concentration at these sites increases and this decreases the undercooling. The parameters that describe the distribution of nucleation sites are determined by ®tting the results to the experiment. This semi-empirical approach to nucleation naturally leaves out the details of the nucleation process, but enables the study and understanding of the competition between nucleation and growth at a level that is inaccessible in the simulations using the TTT diagrams. Growth of ferrite in austenite is assumed to be diusion controlled. The nucleated ferrite grows along the grain boundary as well as into the bulk. The cellular automaton scheme helps to visualize the growth process and the ®nal ferrite morphology. Numerical simulations using the model are performed to obtain a deeper understanding of the factors controlling the transformation and ®nal microstructure.
2. THE MODEL
2.1. Nucleation Following the approach of Rappaz and coworkers for nucleation of solid from melts, we assume that the grain boundary has a number of groups of potential nucleating sites, each of which can become instantaneously active when an associated undercooling is reached. A continuous distribution of nucleating sites with respect to the undercooling is assumed as shown in Fig. 1. The choice of the gaussian form for dN=dDT is purely arbitrary. The three parameters describing the gaussian (Nmax, DTmax and DTs) have to be determined by ®tting the results to experiment. In the case of nucleating sites on the austenite grain boundary, we can assume three separate gaussian distribution functions for the grain corners, the grain edges and the grain surfaces. The present simulations are two-dimensional and the results could be interpreted as what happens in a cross-section of the austenite grain. The corners of the cross-section correspond to the grain edges and the edges of the cross-section correspond to the grain surfaces. Therefore, we assume two distributions of nucleation sites in our simulations. The ``nucleation'' on the corners of the two-dimensional cross-section could correspond to an actual nucleation event at that point of the three-dimensional grain edge, or to the capture of that point on the edge by growth of ferrite nucleated earlier at a three-dimensional grain corner. The DTmax of the distribution of the corner sites is smaller than that of the edge (surface in three-dimensional) sites. 2.2. Growth The growth of ferrite is assumed to be controlled by the diusion of carbon in austenite. Equilibrium
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Fig. 1. Continuous distribution of nucleation sites.
conditions are assumed at the austenite±ferrite interface and the interface movement is controlled by solute balance at the interface. Growth continues until the diusion becomes much too slow due to a fall in temperature or when carbide nucleates in front of the growing ferrite±austenite interface and prevents further growth. 2.3. Nucleation of carbide As the temperature goes down, more and more ferrite forms and the concentration of carbon builds up in front of the growing interface. Many locations become undercooled with respect to carbide formation. Nucleation of carbide would stop the growth of proeutectoid ferrite at these locations. The undercooling required for nucleation of carbides is arbitrarily assigned in this model. This also has to be ®tted by comparison to experiment. 3. CELLULAR AUTOMATON SIMULATION OF THE TRANSFORMATION
In cellular automaton simulations the system is divided into cells characterized by state variables which get transformed according to speci®c rules. The rules for transformation of the state variables, in these simulations are obtained from the above model applied at the local level of the cell. 3.1. The implementation of the nucleation model The boundary of the computational domain is the grain boundary region and nucleation is assumed to take place in this region only. The boundary is adiabatic to diusion. The simulations reported in this paper have been performed in two dimensions on a square grid. The grain boundary is the one-dimensional outline of the square; but it is not implied by this that the shape of the grain boundary section is square. The square has been chosen so that it is easy to divide the grain into square cells for performing ®nite dierence solution of the diusion equation. Equidistant nodes distributed on the grain boundary line depict potential nucleation sites. A certain number of nodes can be chosen to be ``corner sites'' (most preferred nucleation sites) and these need not be the corners of the square. The nodes divide the
grain boundary into one-dimensional cells for simulating the growth of the nuclei along the grain boundary. The ®rst step in the simulation is the random choice of nucleating sites and their associated activation undercooling from among the potential nucleating sites (nodes) on the grain boundary. For each value of the undercooling, a loop in the potential sites is executed and sites are chosen with a probability that depends on the N(DT) curve. This procedure results in a certain number of nodes being chosen on the grain boundary for acting as nucleating sites at speci®ed undercoolings. During simulation of the transformation during continuous cooling, the temperature is reduced at a speci®ed rate and whenever the undercooling associated with a node is reached, ferrite is allowed to grow from this node to the one-dimensional cells on either side of it unless the node has already been ``captured'' by growth of ferrite from some other node. 3.2. Implementation of the growth model The growth algorithm that we use here is very similar to the one followed for cellular automaton simulation of dendritic alloy growth [9] which is also a diusion controlled process assuming equilibrium at the phase interface. Every cell in the system is characterized by a phase-state variable, carbon concentration in austenite, carbon concentration in ferrite, average concentration, and fraction of ferrite. The phase-state variable can be 1, 2, 3, or 4 representing austenite, austenite + ferrite, ferrite, and austenite + ferrite + carbide, respectively. The initial state of the system has phase-state variable = 1, carbon concentration = C0, ferrite fraction = 0, and average carbon concentration =C0. When the temperature is lowered, some of the nodes on the grain boundary get ``nucleated'' and these start growing immediately with a velocity given by vD
dCaust 1 * 1 ÿ k dn C aust
where dCaust =dn is the concentration gradient perpendicular to the interface, and, the interface car-
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KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE Table 1. Diusion coecient Initial carbon concentration of the steel Maximum carbon solubility in ferrite at eutectoid temperature Partition coecient of carbon between austenite and ferrite Slope of hypoeutectoid line Slope of hypereutectoid line
bon concentration C* is given by the phase diagram value for austenite concentration corresponding to the current temperature, k is the partition coecient of carbon between austenite and ferrite, and D is the diusion coecient for carbon in austenite. The carbon gradient into the austenite at the interface is calculated using the dierence formula, knowing the interface carbon concentration and the carbon concentration in the neighbouring austenite cell ahead of the interface. The phase-state variable of a cell in which there is a ferrite±austenite interface is given a phase-state variable of 2. The position of the interface is tracked knowing the velocities in each transforming cell. Knowing the position of the interface, the fraction of ferrite in each transforming cell can be calculated. The average carbon concentration in a cell varies due to diusion and can be calculated by solving the ®nite dierence form of Fick's equation. The average concentration, concentration in ferrite, and concentration in austenite are related by Caver Cferr Va 1 ÿ Va Caust and Cferr k Caust : Knowing Caver t dt from the diusion equation and Va t dt in every cell, Caust t dt and Cferr t dt are calculated. When the ferrite ®lls a cell, its phase-state variable is changed to 3. As the carbon concentration in austenite increases and the temperature decreases, it becomes undercooled with respect to the formation of carbide. The undercooling with respect to carbide formation is calculated as DTcarbide Tac3 mhyper C *aust ÿ T where mhyper is the slope of the hypereutectoid line in the Fe±Fe3C equilibrium diagram and T is the current temperature. The ferrite growth is stopped once the DTcarbide exceeds the speci®ed undercooling for the nucleation of carbide and at this point the phase-state variable is changed to 4. All transformation stops on crossing the martensitic transformation start temperature {Ms}. The interface movement on a square grid implies an anisotropy in the growth velocity which exists in reality, but cannot be controlled or quanti®ed in these simulations.
1.75 10ÿ5 exp(ÿ143 320/RT) 0.17% 0.0925 0.025 233.75 340
4. SIMULATION PARAMETERS
The grid size used in these simulations is 1 mm. The constants used are given in Table 1. The experimental variables are the cooling rate and the austenite grain size. 4.1. The adjustable parameters In addition to these we have the six parameters (Nmax1, DTmax1, DTs1, Nmax2, DTmax2, DTs2) describing the two gaussian functions which have to be determined by ®tting the experimental results to the simulated results. Of these Nmax1, the density of ``corner nucleation sites'' should have a value that is realistic considering the number of corners an austenite grain can have. An austenite grain is assumed to be a polyhedron having approximately 20 corner points, each to be shared with four neighbouring grains; therefore, the number of corners per austenite grain is approximately 5. Thus Nmax1 should be around 5/Vg where Vg is the volume of an austenite grain. In these simulations we keep Nmax1 as 5/Vg. We assume that the grain boundary diusivity is greater than the bulk diusivity by a factor which is also adjustable. The undercooling for carbide nucleation and the number of preferred nucleation sites on a grain boundary section are the other adjustable parameters. There are thus nine uncertain parameters in these simulations; however, they are not free parameters in the sense that they can take any value. They are all quantities with de®nite physical meaning. 4.2. Interpretation of the two-dimensional simulation results For comparison with experimental results threedimensional interpretations have to be made from the results of the two-dimensional simulations. If we take the two-dimensional simulations as depicting the microstructure evolution on a two-dimensional cross-section we can apply stereological relations to convert the two-dimensional quantities to three-dimensional quantities. The two-dimensional quantities we obtain are, the number of ferrite grains in the two-dimensional austenite grain cross-section, N, the average area occupied by the ferrite grains in the cross-section Aa, and the area of the austenite grain cross-section, Ag. The equivalent sphere diameters of the austenite and ferrite grains are calculated as d3D 4 d2D =p: The number of ferrite grains per austenite grain is
KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE
calculated as
Na Va d3g =d3a
where Va is the volume fraction of ferrite, and is taken as equal to the area fraction in the cross-section, Aa/Ag. 5. RESULTS AND DISCUSSIONS
In the following, we try to reproduce the results of Militzer et al. [7] in our simulation by adjusting
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the nucleation parameters and grain boundary diffusion coecient. The undercooling for carbide nucleation does not have much eect on the competition between nucleation and initial growth and we keep this quantity as 100 K in all the simulations. The steel used in the experiments of Militzer et al. was A-36 with a composition C 0.17%, Mn 0.74%, P 0.009%, S 0.009%, Si 0.012%, Cu 0.016%, Ni 0.01%, Cr 0.019%, Al 0.04% with an Ac3 temperature of 8248C. Since the model at this stage handles
Fig. 2(a) and (b).
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only binary alloys our simulation was performed with plain carbon steel of C = 0.17%. Since alloying elements do play a signi®cant role in transformation kinetics the results we present here only have a demonstrative value. In order to handle multicomponent alloys, the growth model has to be combined with thermodynamic software and diusion of all the components has to be included [10]. This exercise, however, helps to understand better the nature of the competition between nucleation and
initial growth and the physical factors that can aect it. The results we try to reproduce are: 1. variation of the transformation start temperature as a function of cooling rate; 2. variation of number of ferrite grains per austenite grain as a function of cooling rate; 3. variation of ferrite grain size as a function of cooling rate; 4. variation of ferrite grain size as a function of austenite grain size;
Fig. 2(c) and (d).
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Fig. 2. Sensitivity of the undercooling for the start of transformation to the nucleation parameters: (a) sensitivity to DTmax1; (b) sensitivity to DTs1; (c) sensitivity to Nmax2; (d) sensitivity to DTmax2; (e) sensitivity to DTs2.
5. variation of the transformation start temperature as a function of austenite grain size.
5.1. Sensitivity of the results to the adjustable parameters
All these variations are studied with many sets of adjustable parameters and the attempt is to determine the set which best reproduces all the experimental results.
The sensitivity of the undercooling for the start of transformation to the nucleation parameters is shown in Figs 2(a)±(e). At low cooling rates, the transformation start temperature is aected signi®-
Fig. 3. Sensitivity of number of ferrite grains per austenite grain to DTmax2.
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Fig. 4. Sensitivity of the undercooling for the start of transformation to the grain boundary diusion coecient (Dgb).
cantly only by DTmax1 [Fig. 2(a)], the undercooling at which the maximum preferred (corner) nucleation sites become active. This con®rms that at low cooling rates nucleation from the preferred sites accounts for all the ferrite grains. Decrease of DTmax1 means that the nucleation of the preferred (corner) sites occurs at higher temperatures, there-
fore the initial growth rate is faster and thus the start of perceptible transformation occurs at a lower undercooling. At higher cooling rates, the results are very sensitive to the parameters of the second gaussian showing that the less preferred sites on the grain boundary are activated at the higher cooling rates [Figs 2(c)±(e)]. The larger the number
Fig. 5. Variation of the undercooling for the start of transformation with cooling rate for austenite grain size of dg=17 mm: r, experimental points; *, simulated results. (Adjustable parameters Nmax1 5=Vg , DTmax1 5, DTs1 80, Nmax2 1:57 106 , DTmax2 5, DTs2 95, Dgb 6Dbulk .)
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Fig. 6. Variation of the number of ferrite grains per austenite grain with cooling rate for austenite grain size of dg=17 mm: r, experimental points; *, simulated results. Adjustable parameters same as that of Fig. 5.
of the less preferred sites (higher Nmax2), or, the lower the undercooling at which they nucleate (lower DTmax2), the start of perceptible transformation occurs at higher temperatures. The number of ferrite grains per austenite grain shows an interesting dependence on DTmax2. At
the higher cooling rates, as DTmax2 is increased, the number of ferrite grains ®rst increases and then decreases (Fig. 3). Initially, when DTmax2 is increased nucleation takes place later and therefore the growth is slower and therefore the other potential surface sites get activated before they
Fig. 7. Variation of ferrite grain size with cooling rate for austenite grain size of dg=17 mm: r, experimental points; *, simulated results. Adjustable parameters same as in Fig. 5.
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Fig. 8. Variation of the undercooling for the start of transformation with austenite grain size: r, experimental points; q simulated results with Dgb Dbulk ; r, simulated results with Dgb 6Dbulk .
are captured. However, when the undercooling for surface site nucleation is increased further, the growth from the preferred ``corner'' sites deactivates all the potential surface sites before any nucleation of the surface sites takes place and
therefore the number of ferrite grains decreases again. The increase of the grain boundary diusion coecient makes little dierence to the start of transformation at low cooling rates, but makes a
Fig. 9. Variation of ferrite grain size with austenite grain size: q, simulated results with Dgb Dbulk ; r, simulated results with Dgb 6Dbulk .
KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE
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Fig. 10. Variation of number of ferrite grains per austenite grain with austenite grain size: q, experimental points; r, simulated results with Dgb Dbulk .
large dierence at higher cooling rates (Fig. 4). This is because the extent to which the potential surface sites are deactivated depends greatly on the grain boundary diusivity, which plays a key role in the competition between nucleation and initial growth. The higher the diusivity, the more the number of the less preferred (surface) sites deactivated by growth, the lower the number of ferrite grains per
austenite grain, and the earlier the start of transformation. 5.2. Comparison with experimental results The adjustable parameters were varied (within limits) to ®t one set of experimental results and it was examined whether all the other experimental results could be reproduced with the same set of parameters. The set which could reproduce most of
Fig. 11. Kinetics of ferrite transformation for austenite grain size dg=17 mm at cooling rate of: (a) 1 K/ s; (b) 5 K/s; (c) 10 K/s; (d) 25 K/s; (e) 50 K/s; (f) 80 K/s. Adjustable parameters same as that of Fig. 5.
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Fig. 12. Contour plots of volume fraction of ferrite at various stages of transformation for a cooling rate of 10 K/s for an austenite grain size of dg=17 mm. (Adjustable parameters Nmax1 5=Vg , DTmax1 5, DTs1 95, Nmax2 1:57 106 , DTmax2 5, DTs2 95, Dgb 6Dbulk .) (a) At the beginning of transformation. (b) At 50% transformation. (c) At the end of transformation.
the experimental results has been chosen to obtain the following. Figure 5 shows the variation of undercooling for the start of transformation with cooling rate. The solid line is ®tted to experimental points (the sym-
bol, r) and the * symbols denote the simulated results. Figure 6 shows the variation of number of ferrite grains per austenite grain with cooling rate for the same set of adjustable parameters as in Fig. 5.
KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE
There is reasonable agreement with the experimental results. Figure 7 shows the dependence of ferrite grain size on the cooling rate. We have ®tted the experimental points of Ref. [7] to a straight line (instead of making a log±log plot as in Fig. 3 of Ref. [7]). The simulated results show a close resemblance to the experimental variation. The ®tted linear relation between the ferrite size and cooling rate can be written as da ÿ0:1033F 10:6982 simulated da ÿ0:1085F 11:2473 experimental: When it comes to the variations with austenite grain size, the simulated results show results that are very dierent from the experimental results. When the adjustable parameters, which ®tted well with the experimental results for the variations with cooling rate, were used the agreement was very poor. However, on assuming that the grain boundary diusion coecient is equal to the bulk diusion coecient, better agreement was obtained (Figs 8±10). The dependence of ferrite grain size on austenite grain size can be written as da 1:37613d0:831 Dgb 6Dbulk g da 7:578d0:26922 Dgb Dbulk : g Dierent authors report dierent experimental results for the variation of ferrite grain size with austenite size, the exponents varying from 0.1 to 0.6 [7, 11±13]. Figure 11 shows the kinetics of ferrite transformation at dierent cooling rates. As the cooling rate increases, the ®nal amount of transformed ferrite decreases considerably whereas Militzer et al. [7] report that up to a cooling rate of 80 K/s the ferrite fraction is around 80%. Figures 12(a)±(c) show the simulated ferrite growth for a cooling rate of 10 K/s. Four grains nucleate at dierent stages of cooling and grow into the grain. The ®gures are contour plots of the volume fraction of ferrite which give a visualization of the nucleation and growth process.
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6. CONCLUSION
A model for nucleation and growth of ferrite from austenite is presented. Results from a twodimensional cellular automaton simulation based on the model are reported to demonstrate the possibilities of the model. The simulations provide insight into the competition between nucleation and growth which controls the number and size of the ferrite grains. The results of the simulation could reproduce the experimental results of Militzer et al. for the variation of transformation start temperature, number of ferrite grains per austenite grain, and ferrite grain size, as a function of cooling rate, but showed poor agreement with the results for the variation of these quantities with austenite grain size.
REFERENCES 1. Agarwal, P. K. and Brimacombe, J. K., Metall. Trans. B, 1981, 12B, 121. 2. Hougardy, H. P. and Yamazaki, K., Steel Res., 1986, 57(9), 466. 3. Buza, G., Hougardy, H. P. and Gergely, M., Steel Res., 1986, 57(12), 650. 4. Campbell, P. C., Howbolt, E. B. and Brimacombe, J. K., Metall. Trans. A, 1991, 22A, 2769. 5. Kamat, R. G., Howbolt, E. B., Brown, L. C. and Brimacombe, J. K., Metall. Trans. A, 1991, 23A, 2469. 6. Denis, D., Farias, D. and Simon, A., ISIJ Int., 1992, 32(3), 316. 7. Militzer, M., Pandi, R. and Hawbolt, E. B., Metall. Mater. Trans. A, 1996, 27A, 1547. 8. Thevoz, Ph., Desbiolles, J. L. and Rappaz, M., Metall. Trans., 1989, 20A, 311. 9. Sasikumar, R., Jacob, E. and George, B., Scripta mater., 1998, 38, 693. 10. Inden, G. and Neumann, P., Steel Res., 1996, 67(10), 401. 11. Tamura, I., Trans. Iron Steel Inst. Japan., 1987, 27, 763. 12. Coquet, P., Fabregue, P., Guisti, J., Chamont, B., Pezani, J. N. and Blanchet, F., in Mathematical Modelling of Hot Rolling of Steels, ed. S. Yue. Hamilton, Ontario, 1990, p. 34. 13. Abe, T., Honda, T., Ishizaki, S., Wada, H., Shikanai, N. and Okita, T., in Mathematical Modelling of Hot Rolling of Steels, ed. S. Yue. Hamilton, Ontario, 1990, p. 66.
Acta mater. Vol. 46, No. 17, pp. 6291±6303, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00243-2 1359-6454/98 $19.00 + 0.00
COMPETITION BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE FROM AUSTENITEÐSTUDIES USING CELLULAR AUTOMATON SIMULATIONS M. KUMAR1, R. SASIKUMAR1{ and P. KESAVAN NAIR2 Regional Research Laboratory (CSIR), Trivandrum 695019, India and 2Department of Metallurgical Engineering, Indian Institute of Technology, Madras 600 036, India
1
(Received 29 January 1998; accepted 18 June 1998) AbstractÐA model for the nucleation of ferrite on austenite grain boundaries and the growth of these nuclei along the grain boundary and into the grain, is developed. A cellular automaton algorithm, with transformation rules based on this model, is used to simulate the decomposition of austenite into ferrite. When performed under continuous cooling conditions, the simulations give an insight into the competition between nucleation and early growth, which determines the variation of ferrite grain size with the cooling rate and with austenite grain size. The number of ferrite grains per austenite grain, ferrite grain size and the kinetics of ferrite formation are obtained as a function of the cooling rate and austenite grain size. Contour plots of the volume fraction of ferrite in the cells at dierent times, enables visualization of the ferrite growth process. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
In recent years, computer-based predictive tools for linking processing conditions to microstructure and properties of the processed materials, have been getting a lot of attention from researchers and software developers. The basis for such a tool is a model for the process which describes the major phenomena that control the process and which can be implemented on a computer. Predicting the transformation behaviour of steel components during heat treatment is an area of considerable industrial importance. One successful route is the semi-empirical approach which combines numerical heat transfer analysis of heat-treated components with empirical time±temperature± transformation (TTT) diagrams to calculate the amount of various transformation products [1±6]. The limitations of this approach are that extensive empirical data are required for this and that only average phase fractions can be calculated and not the size or shape of the phases. Also these simulations do not lead to any deeper understanding of the controlling phenomena. Therefore, a need for more fundamental models of the transformation has been felt lately [7] both for obtaining deeper understanding as well as for more con®dent predictions. In this paper we present a model for nucleation and growth of ferrite from austenite which could form the basis for a predictive tool for the number, size and shape of ferrite grains in austenite as a {To whom all correspondence should be addressed.
function of the austenite grain size and cooling rate. We adopt a cellular automaton scheme for implementing this model in two dimensions. The twodimensional simulations are naturally not quantitatively precise, but it is possible to make a numerical study of the process to understand the various factors that control the transformation. With higher computing power, it is possible to implement the model in three dimensions and make quantitative predictions. Recently, Militzer et al. [7] studied the kinetics of austenite decomposition under continuous cooling conditions for dierent cooling rates and austenite sizes. The ferrite grain size decreased with the cooling rate while the undercooling for the start of transformation (the undercooling at 5% transformation) increased as the cooling rate increased. The number of ferrite grains nucleated per austenite grain was determined and it was found that as the cooling rate increased, the number was more than what could have nucleated at the grain corners. Therefore, it was proposed that as the undercooling increases, nucleation sites at the grain edges and surfaces must be becoming active. At low cooling rates, these sites are deactivated by growth of nuclei which originated earlier at the corners at lower undercoolings, into these sites. At higher cooling rates, higher undercoolings can be reached before the early nuclei have had time to inactivate the nucleation sites on the edges and surfaces by growing into them. Thus the number of ferrite grains in an austenite grain is governed by the competition between nucleation and early growth of the ferrite. The transformation start temperature also depends
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on this phenomenon. A detailed model describing the complex interaction of nucleation and growth, which can predict these eects, has not yet been reported. In our attempt to develop a model for describing the nucleation and early growth of ferrite we noticed that the competition between nucleation and growth which occurs here is very similar to the phenomenon of grain structure formation during solidi®cation of alloy melts. During solidi®cation the cooling curve exhibits a recalescence whose depth is indicative of the grain size or the number density of activated nuclei. The higher the cooling rate, the higher the undercooling reached and the ®ner the grain size. This phenomenon can be explained on the basis of a continuous nucleation model and competition between nucleation and early growth. It is assumed that nucleation sites with a continuous range of associated undercoolings exist in the melt and that each nucleating site becomes instantaneously active when the undercooling becomes equal to the undercooling associated with it. The grains nucleated earlier keep growing and releasing latent heat and at a certain point of time recalescence occurs. Recalescence marks the highest undercooling reached and, usually, the end of nucleation. (Nucleation does not take place during recalescence because all the nucleation sites which could have become activated at these temperatures have already been activated during the cooling up to the recalescence temperature. Nucleation can, in principle, start again when the material cools down, after recalescence, below the recalescence temperature; however, this does not happen often because, by this time, the growth of the early nucleated grains would have consumed these sites.) The point of recalescence is controlled by the competition between nucleation and growth. During slow cooling the grains nucleated earlier have time to grow and liberate sucient amount of latent heat to produce recalescence before high undercoolings are reached, whereas during fast cooling large undercoolings are reached before the early nuclei grow and liberate sucient latent heat to produce recalescence. Solidi®cation grain structure can thus be seen to be governed by phenomena very similar to the one described by Militzer et al. for ferrite grain structure in heat-treated steels. The continuous nucleation model proposed by Rappaz and co-workers [8] has proved to be very useful for developing a predictive tool for grain structure in castings. We have adapted this nucleation model in our model for ferrite formation from austenite. Potential nucleation sites with a de®nite undercooling range associated with them, are distributed randomly on the grain boundary. When the grain is cooled, the nucleation sites become active when the associated undercooling is reached. All the potential sites cannot get activated because, (1) they are consumed by growth of ferrite
nucleated earlier and (2) the carbon concentration at these sites increases and this decreases the undercooling. The parameters that describe the distribution of nucleation sites are determined by ®tting the results to the experiment. This semi-empirical approach to nucleation naturally leaves out the details of the nucleation process, but enables the study and understanding of the competition between nucleation and growth at a level that is inaccessible in the simulations using the TTT diagrams. Growth of ferrite in austenite is assumed to be diusion controlled. The nucleated ferrite grows along the grain boundary as well as into the bulk. The cellular automaton scheme helps to visualize the growth process and the ®nal ferrite morphology. Numerical simulations using the model are performed to obtain a deeper understanding of the factors controlling the transformation and ®nal microstructure.
2. THE MODEL
2.1. Nucleation Following the approach of Rappaz and coworkers for nucleation of solid from melts, we assume that the grain boundary has a number of groups of potential nucleating sites, each of which can become instantaneously active when an associated undercooling is reached. A continuous distribution of nucleating sites with respect to the undercooling is assumed as shown in Fig. 1. The choice of the gaussian form for dN=dDT is purely arbitrary. The three parameters describing the gaussian (Nmax, DTmax and DTs) have to be determined by ®tting the results to experiment. In the case of nucleating sites on the austenite grain boundary, we can assume three separate gaussian distribution functions for the grain corners, the grain edges and the grain surfaces. The present simulations are two-dimensional and the results could be interpreted as what happens in a cross-section of the austenite grain. The corners of the cross-section correspond to the grain edges and the edges of the cross-section correspond to the grain surfaces. Therefore, we assume two distributions of nucleation sites in our simulations. The ``nucleation'' on the corners of the two-dimensional cross-section could correspond to an actual nucleation event at that point of the three-dimensional grain edge, or to the capture of that point on the edge by growth of ferrite nucleated earlier at a three-dimensional grain corner. The DTmax of the distribution of the corner sites is smaller than that of the edge (surface in three-dimensional) sites. 2.2. Growth The growth of ferrite is assumed to be controlled by the diusion of carbon in austenite. Equilibrium
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Fig. 1. Continuous distribution of nucleation sites.
conditions are assumed at the austenite±ferrite interface and the interface movement is controlled by solute balance at the interface. Growth continues until the diusion becomes much too slow due to a fall in temperature or when carbide nucleates in front of the growing ferrite±austenite interface and prevents further growth. 2.3. Nucleation of carbide As the temperature goes down, more and more ferrite forms and the concentration of carbon builds up in front of the growing interface. Many locations become undercooled with respect to carbide formation. Nucleation of carbide would stop the growth of proeutectoid ferrite at these locations. The undercooling required for nucleation of carbides is arbitrarily assigned in this model. This also has to be ®tted by comparison to experiment. 3. CELLULAR AUTOMATON SIMULATION OF THE TRANSFORMATION
In cellular automaton simulations the system is divided into cells characterized by state variables which get transformed according to speci®c rules. The rules for transformation of the state variables, in these simulations are obtained from the above model applied at the local level of the cell. 3.1. The implementation of the nucleation model The boundary of the computational domain is the grain boundary region and nucleation is assumed to take place in this region only. The boundary is adiabatic to diusion. The simulations reported in this paper have been performed in two dimensions on a square grid. The grain boundary is the one-dimensional outline of the square; but it is not implied by this that the shape of the grain boundary section is square. The square has been chosen so that it is easy to divide the grain into square cells for performing ®nite dierence solution of the diusion equation. Equidistant nodes distributed on the grain boundary line depict potential nucleation sites. A certain number of nodes can be chosen to be ``corner sites'' (most preferred nucleation sites) and these need not be the corners of the square. The nodes divide the
grain boundary into one-dimensional cells for simulating the growth of the nuclei along the grain boundary. The ®rst step in the simulation is the random choice of nucleating sites and their associated activation undercooling from among the potential nucleating sites (nodes) on the grain boundary. For each value of the undercooling, a loop in the potential sites is executed and sites are chosen with a probability that depends on the N(DT) curve. This procedure results in a certain number of nodes being chosen on the grain boundary for acting as nucleating sites at speci®ed undercoolings. During simulation of the transformation during continuous cooling, the temperature is reduced at a speci®ed rate and whenever the undercooling associated with a node is reached, ferrite is allowed to grow from this node to the one-dimensional cells on either side of it unless the node has already been ``captured'' by growth of ferrite from some other node. 3.2. Implementation of the growth model The growth algorithm that we use here is very similar to the one followed for cellular automaton simulation of dendritic alloy growth [9] which is also a diusion controlled process assuming equilibrium at the phase interface. Every cell in the system is characterized by a phase-state variable, carbon concentration in austenite, carbon concentration in ferrite, average concentration, and fraction of ferrite. The phase-state variable can be 1, 2, 3, or 4 representing austenite, austenite + ferrite, ferrite, and austenite + ferrite + carbide, respectively. The initial state of the system has phase-state variable = 1, carbon concentration = C0, ferrite fraction = 0, and average carbon concentration =C0. When the temperature is lowered, some of the nodes on the grain boundary get ``nucleated'' and these start growing immediately with a velocity given by vD
dCaust 1 * 1 ÿ k dn C aust
where dCaust =dn is the concentration gradient perpendicular to the interface, and, the interface car-
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KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE Table 1. Diusion coecient Initial carbon concentration of the steel Maximum carbon solubility in ferrite at eutectoid temperature Partition coecient of carbon between austenite and ferrite Slope of hypoeutectoid line Slope of hypereutectoid line
bon concentration C* is given by the phase diagram value for austenite concentration corresponding to the current temperature, k is the partition coecient of carbon between austenite and ferrite, and D is the diusion coecient for carbon in austenite. The carbon gradient into the austenite at the interface is calculated using the dierence formula, knowing the interface carbon concentration and the carbon concentration in the neighbouring austenite cell ahead of the interface. The phase-state variable of a cell in which there is a ferrite±austenite interface is given a phase-state variable of 2. The position of the interface is tracked knowing the velocities in each transforming cell. Knowing the position of the interface, the fraction of ferrite in each transforming cell can be calculated. The average carbon concentration in a cell varies due to diusion and can be calculated by solving the ®nite dierence form of Fick's equation. The average concentration, concentration in ferrite, and concentration in austenite are related by Caver Cferr Va 1 ÿ Va Caust and Cferr k Caust : Knowing Caver t dt from the diusion equation and Va t dt in every cell, Caust t dt and Cferr t dt are calculated. When the ferrite ®lls a cell, its phase-state variable is changed to 3. As the carbon concentration in austenite increases and the temperature decreases, it becomes undercooled with respect to the formation of carbide. The undercooling with respect to carbide formation is calculated as DTcarbide Tac3 mhyper C *aust ÿ T where mhyper is the slope of the hypereutectoid line in the Fe±Fe3C equilibrium diagram and T is the current temperature. The ferrite growth is stopped once the DTcarbide exceeds the speci®ed undercooling for the nucleation of carbide and at this point the phase-state variable is changed to 4. All transformation stops on crossing the martensitic transformation start temperature {Ms}. The interface movement on a square grid implies an anisotropy in the growth velocity which exists in reality, but cannot be controlled or quanti®ed in these simulations.
1.75 10ÿ5 exp(ÿ143 320/RT) 0.17% 0.0925 0.025 233.75 340
4. SIMULATION PARAMETERS
The grid size used in these simulations is 1 mm. The constants used are given in Table 1. The experimental variables are the cooling rate and the austenite grain size. 4.1. The adjustable parameters In addition to these we have the six parameters (Nmax1, DTmax1, DTs1, Nmax2, DTmax2, DTs2) describing the two gaussian functions which have to be determined by ®tting the experimental results to the simulated results. Of these Nmax1, the density of ``corner nucleation sites'' should have a value that is realistic considering the number of corners an austenite grain can have. An austenite grain is assumed to be a polyhedron having approximately 20 corner points, each to be shared with four neighbouring grains; therefore, the number of corners per austenite grain is approximately 5. Thus Nmax1 should be around 5/Vg where Vg is the volume of an austenite grain. In these simulations we keep Nmax1 as 5/Vg. We assume that the grain boundary diusivity is greater than the bulk diusivity by a factor which is also adjustable. The undercooling for carbide nucleation and the number of preferred nucleation sites on a grain boundary section are the other adjustable parameters. There are thus nine uncertain parameters in these simulations; however, they are not free parameters in the sense that they can take any value. They are all quantities with de®nite physical meaning. 4.2. Interpretation of the two-dimensional simulation results For comparison with experimental results threedimensional interpretations have to be made from the results of the two-dimensional simulations. If we take the two-dimensional simulations as depicting the microstructure evolution on a two-dimensional cross-section we can apply stereological relations to convert the two-dimensional quantities to three-dimensional quantities. The two-dimensional quantities we obtain are, the number of ferrite grains in the two-dimensional austenite grain cross-section, N, the average area occupied by the ferrite grains in the cross-section Aa, and the area of the austenite grain cross-section, Ag. The equivalent sphere diameters of the austenite and ferrite grains are calculated as d3D 4 d2D =p: The number of ferrite grains per austenite grain is
KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE
calculated as
Na Va d3g =d3a
where Va is the volume fraction of ferrite, and is taken as equal to the area fraction in the cross-section, Aa/Ag. 5. RESULTS AND DISCUSSIONS
In the following, we try to reproduce the results of Militzer et al. [7] in our simulation by adjusting
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the nucleation parameters and grain boundary diffusion coecient. The undercooling for carbide nucleation does not have much eect on the competition between nucleation and initial growth and we keep this quantity as 100 K in all the simulations. The steel used in the experiments of Militzer et al. was A-36 with a composition C 0.17%, Mn 0.74%, P 0.009%, S 0.009%, Si 0.012%, Cu 0.016%, Ni 0.01%, Cr 0.019%, Al 0.04% with an Ac3 temperature of 8248C. Since the model at this stage handles
Fig. 2(a) and (b).
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only binary alloys our simulation was performed with plain carbon steel of C = 0.17%. Since alloying elements do play a signi®cant role in transformation kinetics the results we present here only have a demonstrative value. In order to handle multicomponent alloys, the growth model has to be combined with thermodynamic software and diusion of all the components has to be included [10]. This exercise, however, helps to understand better the nature of the competition between nucleation and
initial growth and the physical factors that can aect it. The results we try to reproduce are: 1. variation of the transformation start temperature as a function of cooling rate; 2. variation of number of ferrite grains per austenite grain as a function of cooling rate; 3. variation of ferrite grain size as a function of cooling rate; 4. variation of ferrite grain size as a function of austenite grain size;
Fig. 2(c) and (d).
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Fig. 2. Sensitivity of the undercooling for the start of transformation to the nucleation parameters: (a) sensitivity to DTmax1; (b) sensitivity to DTs1; (c) sensitivity to Nmax2; (d) sensitivity to DTmax2; (e) sensitivity to DTs2.
5. variation of the transformation start temperature as a function of austenite grain size.
5.1. Sensitivity of the results to the adjustable parameters
All these variations are studied with many sets of adjustable parameters and the attempt is to determine the set which best reproduces all the experimental results.
The sensitivity of the undercooling for the start of transformation to the nucleation parameters is shown in Figs 2(a)±(e). At low cooling rates, the transformation start temperature is aected signi®-
Fig. 3. Sensitivity of number of ferrite grains per austenite grain to DTmax2.
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Fig. 4. Sensitivity of the undercooling for the start of transformation to the grain boundary diusion coecient (Dgb).
cantly only by DTmax1 [Fig. 2(a)], the undercooling at which the maximum preferred (corner) nucleation sites become active. This con®rms that at low cooling rates nucleation from the preferred sites accounts for all the ferrite grains. Decrease of DTmax1 means that the nucleation of the preferred (corner) sites occurs at higher temperatures, there-
fore the initial growth rate is faster and thus the start of perceptible transformation occurs at a lower undercooling. At higher cooling rates, the results are very sensitive to the parameters of the second gaussian showing that the less preferred sites on the grain boundary are activated at the higher cooling rates [Figs 2(c)±(e)]. The larger the number
Fig. 5. Variation of the undercooling for the start of transformation with cooling rate for austenite grain size of dg=17 mm: r, experimental points; *, simulated results. (Adjustable parameters Nmax1 5=Vg , DTmax1 5, DTs1 80, Nmax2 1:57 106 , DTmax2 5, DTs2 95, Dgb 6Dbulk .)
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Fig. 6. Variation of the number of ferrite grains per austenite grain with cooling rate for austenite grain size of dg=17 mm: r, experimental points; *, simulated results. Adjustable parameters same as that of Fig. 5.
of the less preferred sites (higher Nmax2), or, the lower the undercooling at which they nucleate (lower DTmax2), the start of perceptible transformation occurs at higher temperatures. The number of ferrite grains per austenite grain shows an interesting dependence on DTmax2. At
the higher cooling rates, as DTmax2 is increased, the number of ferrite grains ®rst increases and then decreases (Fig. 3). Initially, when DTmax2 is increased nucleation takes place later and therefore the growth is slower and therefore the other potential surface sites get activated before they
Fig. 7. Variation of ferrite grain size with cooling rate for austenite grain size of dg=17 mm: r, experimental points; *, simulated results. Adjustable parameters same as in Fig. 5.
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Fig. 8. Variation of the undercooling for the start of transformation with austenite grain size: r, experimental points; q simulated results with Dgb Dbulk ; r, simulated results with Dgb 6Dbulk .
are captured. However, when the undercooling for surface site nucleation is increased further, the growth from the preferred ``corner'' sites deactivates all the potential surface sites before any nucleation of the surface sites takes place and
therefore the number of ferrite grains decreases again. The increase of the grain boundary diusion coecient makes little dierence to the start of transformation at low cooling rates, but makes a
Fig. 9. Variation of ferrite grain size with austenite grain size: q, simulated results with Dgb Dbulk ; r, simulated results with Dgb 6Dbulk .
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Fig. 10. Variation of number of ferrite grains per austenite grain with austenite grain size: q, experimental points; r, simulated results with Dgb Dbulk .
large dierence at higher cooling rates (Fig. 4). This is because the extent to which the potential surface sites are deactivated depends greatly on the grain boundary diusivity, which plays a key role in the competition between nucleation and initial growth. The higher the diusivity, the more the number of the less preferred (surface) sites deactivated by growth, the lower the number of ferrite grains per
austenite grain, and the earlier the start of transformation. 5.2. Comparison with experimental results The adjustable parameters were varied (within limits) to ®t one set of experimental results and it was examined whether all the other experimental results could be reproduced with the same set of parameters. The set which could reproduce most of
Fig. 11. Kinetics of ferrite transformation for austenite grain size dg=17 mm at cooling rate of: (a) 1 K/ s; (b) 5 K/s; (c) 10 K/s; (d) 25 K/s; (e) 50 K/s; (f) 80 K/s. Adjustable parameters same as that of Fig. 5.
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Fig. 12. Contour plots of volume fraction of ferrite at various stages of transformation for a cooling rate of 10 K/s for an austenite grain size of dg=17 mm. (Adjustable parameters Nmax1 5=Vg , DTmax1 5, DTs1 95, Nmax2 1:57 106 , DTmax2 5, DTs2 95, Dgb 6Dbulk .) (a) At the beginning of transformation. (b) At 50% transformation. (c) At the end of transformation.
the experimental results has been chosen to obtain the following. Figure 5 shows the variation of undercooling for the start of transformation with cooling rate. The solid line is ®tted to experimental points (the sym-
bol, r) and the * symbols denote the simulated results. Figure 6 shows the variation of number of ferrite grains per austenite grain with cooling rate for the same set of adjustable parameters as in Fig. 5.
KUMAR et al.: COMPETITON BETWEEN NUCLEATION AND EARLY GROWTH OF FERRITE
There is reasonable agreement with the experimental results. Figure 7 shows the dependence of ferrite grain size on the cooling rate. We have ®tted the experimental points of Ref. [7] to a straight line (instead of making a log±log plot as in Fig. 3 of Ref. [7]). The simulated results show a close resemblance to the experimental variation. The ®tted linear relation between the ferrite size and cooling rate can be written as da ÿ0:1033F 10:6982 simulated da ÿ0:1085F 11:2473 experimental: When it comes to the variations with austenite grain size, the simulated results show results that are very dierent from the experimental results. When the adjustable parameters, which ®tted well with the experimental results for the variations with cooling rate, were used the agreement was very poor. However, on assuming that the grain boundary diusion coecient is equal to the bulk diusion coecient, better agreement was obtained (Figs 8±10). The dependence of ferrite grain size on austenite grain size can be written as da 1:37613d0:831 Dgb 6Dbulk g da 7:578d0:26922 Dgb Dbulk : g Dierent authors report dierent experimental results for the variation of ferrite grain size with austenite size, the exponents varying from 0.1 to 0.6 [7, 11±13]. Figure 11 shows the kinetics of ferrite transformation at dierent cooling rates. As the cooling rate increases, the ®nal amount of transformed ferrite decreases considerably whereas Militzer et al. [7] report that up to a cooling rate of 80 K/s the ferrite fraction is around 80%. Figures 12(a)±(c) show the simulated ferrite growth for a cooling rate of 10 K/s. Four grains nucleate at dierent stages of cooling and grow into the grain. The ®gures are contour plots of the volume fraction of ferrite which give a visualization of the nucleation and growth process.
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6. CONCLUSION
A model for nucleation and growth of ferrite from austenite is presented. Results from a twodimensional cellular automaton simulation based on the model are reported to demonstrate the possibilities of the model. The simulations provide insight into the competition between nucleation and growth which controls the number and size of the ferrite grains. The results of the simulation could reproduce the experimental results of Militzer et al. for the variation of transformation start temperature, number of ferrite grains per austenite grain, and ferrite grain size, as a function of cooling rate, but showed poor agreement with the results for the variation of these quantities with austenite grain size.
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