Abstract
The evolution and topology of a two-dimensional austenite grain growth of C–Mn steel are simulated by the Monte Carlo (MC) model during the reheating process. Simulated microstructural maps have been generated in a square lattice (400×400) at various MC steps and temperatures. Normalised grain size distribution has also been computed at different reheating temperatures, assuming a uniform temperature distribution in the slab. Two important model parameters, namely, the grain growth exponent and the model constant, have been estimated to predict the average grain size. The activation energy for grain growth has been calculated using the Arrhenius relationship. The grain growth behaviour as a function of both MC and physical time has been computed, and the effect of temperature on the rate of growth has been analysed. The predicted temporal evolutions of grain growth and model parameters have been validated with the published literature and found to be in good agreement.
Keywords
Introduction
Practically all metals, ceramics and rocks are polycrystalline materials. These consist of a mixture of tiny grains that are attached together by interatomic forces. In general, the structure formed by the grains is unstable and evolves with time when the material is heated and deformed. It has been long recognised by researchers that the microstructure of a polycrystalline material directly affects its mechanical properties. Over the years, substantial efforts have been made for a better understanding of the evolutionary mechanism of the microstructure in polycrystalline materials. The underlying mechanism of grain growth is the migration of grain boundaries under the influence of the driving force, which emanates from the reduction in total grain boundary energy of the system. The technological interest in grain growth originates from the important relationships between grain size and properties. Specifically, prediction of mechanical properties, such as strength and toughness, necessitates the development of a material model for better understanding and control of grain growth, which is of considerable importance in the hot rolling of steel. The hot strip mill consists of sequential unit operations, such as reheating furnace, roughing mill, finishing mill, runout table and down coilers (Fig. 1). Selection of an optimal reheat temperature before rolling operation facilitates to achieve an initial uniform grain structure in the slab.
Schematic diagram of hot strip mill
Various numerical models have been developed to simulate the microstructural evolution of materials during processing operations. These simulation methods can be classified into four groups. The first concept has been designated as Voronoi1 and modified Voronoi2,3 methods. The second approach includes curvature driven grain growth,4,5 followed by continuum thermodynamics models such as finite difference solutions of the Cahn–Hillard equation, phase field models and diffusion models.6,7 The last consists of cellular automata8 based simulations. All four simulation methods incorporate the phenomenological features of the system. In comparison, the thermodynamic and kinetic characteristics are inherent to the Monte Carlo (MC) method. There is no need to include material parameters such as grain boundary mobility or the function of free energy into the model.
Modelling of microstructural evolution during thermomechanical processing gained significant importance over the past two decades.9–22 Several investigations have been carried out to characterise the grain growth, recrystallisation and softening kinetics of C–Mn steels during hot strip rolling.9–18 Although such models are reasonably successful in predicting grain size in commercial alloys, further advancement in simulation methods would facilitate better understanding of the basic phenomena of grain growth during thermomechanical treatment. The grain growth models incorporate some of the thermodynamic parameters like grain boundary energy, activation energy of grain boundary diffusion, etc. These parameters are not always easily obtainable from open literature for different alloy systems. In spite of intensive studies in the simulation of microstructure evolution, spatial and temporal evolution of complex microstructure in materials still remains a challenging problem.
Historically, the Potts model based on the MC technique was developed more than four decades ago as an extension to the ferromagnetic Ising model for characterising magnetic domain evolution.23 The Ising model24 envisages a magnetised material as a collection of spins, where only two states are possible, namely, up or down. Although this model has been developed long ago, however, it has found wide applications in the diverse areas of science and engineering in the last few decades.25 It has been generalised subsequently by Potts to assign Q states for each particle in the system and designated as ‘Q state model’. The Potts model has been used extensively to simulate the mesoscopic behaviour of materials, such as recrystallisation and grain growth during material processing operations.26 In particular, it was very successful in describing grain growth kinetics in polycrystalline materials. This is because of a similarity between the ferromagnetic Potts and the grain structures, which are characterised by an array of cells having the same spin or lattice orientation. These cells are separated by boundaries, which evolve in a manner such that the total interfacial energy is minimised.
Initially, Anderson et al.27 proposed the MC method to predict grain growth in two dimensions. Subsequently, several researchers applied this method to model grain growth, static and dynamic recrystallisation and the influences of texture and precipitates on the grain growth process.26–36 Although the method has been applied extensively in materials processing operations, however, some of the issues have not yet been addressed in depth using the MC technique. In particular, the effect of reheating temperature on the austenite grain growth kinetics before hot rolling has not been explored in detail. Nevertheless, the MC study of the influence of deformation temperature on dynamic recrystallisation phenomena has been investigated in generic manner.37 Literature information in this domain is rather scanty. Majority of the reported MC simulation studies primarily focused on topological aspects and kinetics of grain growth at temperature 0 K26–28 and near 0 K.35 However, similar investigations have not been reported yet at elevated temperature for reheating operation.
In the present work, an MC simulation framework has been developed to study the grain growth behaviour of plain C–Mn steel slab during reheating in the furnace before rolling. The microstructure changes continuously during different processing steps of hot rolling operation. Austenite grain growth is the dominant process in the reheating furnace and also subsequent downstream thermomechanical operations. The austenite microstructure after rolling and subsequent cooling on the runout table determines the final ferrite grain size and corresponding mechanical properties of the product. Therefore, the model for the prediction of the grain growth of steel during the slab reheating process is arguably the first step for the development of a comprehensive thermomechanical predictive model of the rolling process. It provides not only knowledge of the initial grain size but also information to control the reheating process parameters.
Semiempirical grain growth models
Few widely referred semiempirical grain growth models in material modelling have been briefly discussed here. On the basis of experimental data, it has been reported38 that grain size under normal isothermal grain growth conditions obeys power law growth kinetics of the form
In the case of d>>d0, the kinetics is reduced to
In his seminal work, Sellars and Whiteman9 arrived at the following general expression for grain growth using previously published data on low carbon–manganese steels
Yoshie et al.15 proposed an empirical approach to predict the grain growth of austenite in as cast C–Mn steels. The model can be briefly described as follows
is the diffusion constant, Qd is the activation energy for diffusion and λ is the thickness of the grain boundary.
The activation energy for grain growth Qg may be determined by estimating the model constant k (equation (2)) at different temperatures and then constructing an Arrhenius plot, which is governed by the following logarithmic relationship
Monte Carlo simulation
The MC method is employed to simulate grain growth of plain C–Mn steel during heating of the slab in the reheating furnace. The temperature of the slab is assumed to be uniform during the grain growth process. In this method, the microstructure is mapped onto a discrete two-dimensional square lattice system. The lattice is initialised by randomly assigning an integer number Si between 1 and Q to each lattice site, where Si is the number of grain orientation in a lattice site, and Q is the total number of possible grain orientations in the system. All sites within a grain have the same orientation number Si, and the grain boundaries are interfaces between two neighbouring sites with different orientation numbers. Periodic grain boundary conditions are used to avoid the singularity of edges in a finite domain. In this boundary condition, a point on the edge of the domain is connected to points on the opposite edges. The simulation steps are as follows:
a lattice site is selected at random
a new trial orientation is chosen at random from one of the other (Q−1) possible orientations and assigned to the selected lattice site
the net energy change ΔE of the system caused by the reorientation is calculated; the total energy E of the system is defined as
where J is a positive constant, which sets the scale of the grain boundary energy, Si and Sj are the orientation numbers of site i and one of its neighbour sites j respectively, n is the total number of neighbour sites and 
is the Kronecker delta. This implies that 
= 1 when Si and Sj are of like orientation and zero otherwise. The sum is taken over all nearest neighbour sites. An isotropic distribution of grain boundary energy is assumed in this model.
if the change in energy associated with the change in orientation is ⩽0, the reorientation is accepted. If the energy change is positive, a random number R between 0 and 1 is generated such that if R⩽P, the change will be accepted. The transition probability P is given by
the above procedure is repeated N number of times, where N is the total number of sites in the lattice system.
Figure 2 shows the block diagram of the MC algorithm for grain growth simulation.
Flowchart for MC simulation of grain growth
In this computational approach, a modified MC technique32 has been incorporated to enhance the computation speed and improve the accuracy of the simulations. According to this methodology, the newly assigned orientation should be limited to the orientations of one of its nearest neighbours. The unit of time is defined as one MC step (MCS) per site, which corresponds to N reorientation attempts. The number of MCS is assumed to be proportional to the physical time, which is implicitly related to temperature through the diffusion coefficient.31 For a square lattice, this can be expressed as
Table 1 shows the numerical values of the model parameters (constants), which have been used in the present simulation. In this model, the average grain size is computed using the following equation26
Numerical values of pertinent model parameters/constants
/m2 s−1The MC algorithm has been implemented in MATLAB code using MATLAB 7·0 software with suitable graphics. The code has been used to predict the grain size evolution and two-dimensional microstructural maps for C–Mn steel. The grain size distribution and the effect of temperature on the growth kinetics have also been computed. The inputs constitute the size of the lattice N, the number of orientations Q, the temperature T and the number of MCS. The area of each grain has been determined by locating and counting all of the sites of adjacent grain orientation. At typical time intervals (MCS), the instantaneous grain size distribution is evaluated by calculating the ratio of each grain size and the average size of the grains in consonance with its frequency of occurrence in the lattice system. The computation time for grain growth simulation in a 400×400 lattice configuration for 50 000 MCS in a 2·0 GHz workstation (2 GB RAM) is ∼125 h.
Results and discussion
In this simulation, the grain structure was initialised with a two-dimensional square lattice of 400×400 size with a lattice parameter (space grid) of 8 μm. The grain orientation has been assigned values in the range from 1 to Q to each lattice site in a random manner. In order to prevent the impingement of grain of similar orientation too frequently, a large value has been assigned to Q (Q = 48).26–31 The value of J/kBT (equations (9) and (10)) has been kept as 2·031,36 in the operating temperature range from 1173 to 1473 K. In Fig. 3, the computed microstructural evolution maps are shown at different MCS, i.e. 1000, 5000 and 10 000 respectively at a uniform reheating temperature of 1173 K. In these maps, multiple colours represent various crystallographic orientations of the lattice. The temporal evolution of grain growth has been clearly elucidated in these figures as a function MC time. Furthermore, coarsening of large grains by absorption of small grains can be observed during the grain growth process with the advancement of MC time. Uniform and isotropic distribution of grains in the simulated microstructures can also be observed. Figure 4 shows the simulated microstructural maps after 10 000 MCS at different discrete temperatures, i.e. 1273, 1373 and 1473 K respectively. Figure 5 depicts the grain size distribution at temperature 1173 K at different MCS, i.e. 1000, 5000 and 10 000. These are generated by plotting the frequency of occurrence as a function of logarithm of the normalised grain size (d/〈d〉). In these figures, it may be observed that the grain size distribution increasingly skewed towards smaller grain sizes at higher MCS. Figure 6 describes the grain size distribution at different temperatures of 1273, 1373 and 1473 K at a given MCS of 10 000. The disappearance of bars in the histogram (Figs. 5c and 6 c ) may be attributed to the probable occurrence of coalescence phenomena of grains of similar orientation. Furthermore, in Figs. 5 and 6, the grain growth appears to be apparently not normal, although the power law growth kinetics is assumed to be valid.
Temporal evolution of grain boundary for 400×400 square lattice at temperature of 1173 K and Q = 48 Potts model
Simulated microstructure for 400×400 square lattice and Q = 48 Potts model after 10 00 MCS at different temperatures
Grain size distribution at temperature 1173 K for different MCS
Grain size distribution at different temperatures at 10 000 MCS
The average grain size as a function of MCS for different values of Q (8, 16, 32, 48 and 64) is shown in Fig. 7. It may be observed that the grain size variation shows a dependence on Q values of 8 and 16. However, for higher Q values (Q⩾32), this dependence sharply diminishes as Q increases. Morphological changes do occur as Q increases from the Ising value (Q = 2) towards the limit where all grain orientations are allowed, i.e. grain growth limit. The low Q configurations consist of irregular and asymmetric grains, and the irregularity is enhanced with reduced Q values. On the other hand, the grains in the high Q configurations are expected to be considerably compact and equiaxed. These predictions and their trend are validated with the published literature27,28,31 and found to be in good agreement. The value of the grain growth exponent n has been extracted from the slope of the plot of logarithm of grain size versus MC time (5×104 MCS). Figure 8 shows the temporal evolution of grain size as a function of MCS at different temperatures, namely, 1173, 1273, 1373 and 1473 K. It may be observed from this figure that the evolution process follows the power growth kinetics.27,31,36 Table 2 shows kinetic parameters such as growth exponent n and model constant k over a temperature range from 1173 to 1473 K. These have been estimated from the kinetic graphs (Fig. 8). The data shown in this table represent values that have been averaged over at least five simulation runs. Hillert39 predicted the value of n to be 0·5 using an analytical model for pure metals. The present MC simulation estimated the value of n in the range of 0·50–0·62 over the temperature range of 1173–1473 K for C–Mn steel. Figure 9 shows the conversion functionality of MC time step (MCS) to physical time using equation (11), and grain growth evolution can be conveniently transformed to physical time using this relationship. Figure 10 depicts the time domain evolution of grain size as a function of physical time at different temperatures, i.e. 1173, 1273, 1373 and 1473 K respectively. It may be observed from this figure that the rate of growth of the grain is faster at elevated temperature, which is reflected by the higher slope of the respective curves at different temperatures. In this figure, the average grain size has been computed over a time range, which is fixed at 1800 s. The growth kinetics in the physical time domain also follows the power law formalism. Figure 11 shows the validation of the current MC prediction with the Yoshie model.9,15 In order to verify the predictive capability of the present model, all the pertinent model parameters used in the simulation have been kept similar to the Yoshie model and shown in Table 1. It may be observed that the current MC calculations are in good agreement with the Yoshie model prediction for C–Mn steel. Figure 12 shows a linear fit between the predicted grain size using MC simulation and the published Yoshie model, which is found to be excellent. Figure 13 shows the variation of model constant k as a function of temperature. It has been found that k increases with increasing temperature, which is consistent with respect to the Arrhenius temperature dependence functionality. Figure 14 shows the variation of grain growth exponent n as a function of temperature in the range of 1173–1473 K. Although several investigators assumed n to be constant over this temperature range, however, it has been found that n is a slowly varying function of temperature and increases by almost 25 within this range. This observation emanates from the prediction of the present model. The scatter in the experimental values of n has also been reported in the literature by Anderson et al.,27 which corroborates the present observation. Figure 15 depicts the logarithmic variation of k as a function of 1/T. As k follows the Arrhenius law with regard to temperature variation, the slope of the line gives the activation energy for grain growth. The value of activation energy (order of magnitude) obtained from this model for C–Mn steel is 78 945 J, which is found to be in close agreement with the value cited in the literature for similar grade of steel.17,40
Grain growth behaviour as function of number of grain orientations Q
Average grain size variation with MC time step at different temperatures
Conversion functionality of physical time and MC time (MCS) at various temperatures
Average grain size variation with physical time at different temperatures
Validation of MC grain growth simulation with Yoshie model15
Comparison of predicted MC grain size with Yoshie empirical model15
Variation of grain growth model constant k as function of reheating temperature
Variation of grain growth exponent n as function of reheating temperature
Logarithmic variation of model constant k as function of reciprocal temperature
Parameter estimation of n and k at various temperatures using MC simulation
Conclusions
The temporal evolution and topology of grain growth pertaining to C–Mn steel have been simulated using the MC method. Time domain two-dimensional microstructural maps have been generated to visualise the growth kinetics at various temperature levels, which also capture the grain coarsening phenomena. In this simulation framework, the effect of temperature on the grain growth behaviour has been studied. The grain size distributions at various temperature levels have also been computed. It has been found that the grain size distribution increasingly gets skewed towards smaller grain sizes at elevated temperature. The average grain size variation with MCS for different values of grain orientation (Q value) is found to be less sensitive to higher Q values (⩾32). However, the average grain size variation shows a dependence on lower Q values. Two important model parameters, namely, the grain growth exponent n and the model parameter k, have been estimated to predict the grain size. The average grain size variation as a function of MCS as well as physical time has also been computed, and the effect of temperature on the growth kinetics has been studied. In the earlier studies by various investigators, the grain growth exponent has been assumed to be a constant over the reheating temperature range. However, the present model demonstrates that the growth exponent is actually an increasing function of temperature, and the value increases almost by 25 in the given temperature range. This may likely have an effect on the accuracy of prediction of grain growth rate at high temperature.
The model predictions pertaining to grain growth exponent, activation energy for grain growth and grain growth evolution have been validated with published literature and found to be in good agreement. The present MC model establishes a predictive methodology to simulate austenite grain growth behaviour for C–Mn steel as a function of temperature of the workpiece in the reheating furnace.
Footnotes
Acknowledgements
The authors would like to express their sincere gratitude to the Director, NML, and Dr I. Cattoraj of NML Jamshedpur for their support and constructive suggestions.