Abstract
Monte Carlo (MC) technique is becoming a very effective simulation method for prediction and analysis of the grain growth kinetics at mesoscopic level. It should be noted that MC models have no real time of physical systems due to the probabilistic nature of this simulation technique. This leads to difficulties when converting simulated time, the Monte Carlo steps , to real time. The correspondence between Monte Carlo steps and real time should be proposed for comparing the kinetics of MC models with the experiments. In this work, the conversion of Monte Carlo steps to real time is attempted. The lattice sites spacing Δ and the temperature cannot be ignored in the Monte Carlo simulation of grain growth. Real time will be associated with , , and Δ.
1. Introduction
Grain growth in polycrystals is achieved by decreasing the total number of grains as result of the small grains vanishing. As a grain grows, atoms just outside the boundary change the lattice arrangement from that of the neighboring shrinking grain to the growing grain. To model the microstructure evolution, there are some analytical theories that adequately describe grain growth kinetics [1–4]. Many efforts have been devoted to study real materials in the presence of particles and texture effects by using these theories [5–7]. Grain growth in a polycrystal is driven by the curvature of the grain boundaries. As a result, the grain boundary moves with a velocity that is proportional to its curvature. According to classical grain growth theory [3] based on the assumption that the driving pressure on a grain boundary arises only from its curvature, grain growth kinetics are represented bywhere is the mean grain size at time , is the initial mean grain size at , and is a constant.
In the limit where , we getThe simulation from experimental data demonstrates that the grain size increases with time in accordance with a power law , where [8, 9].
With the progress of computational technology, significant progress has been made in quantitative understanding of grain growth by using computer simulation techniques, including Monte Carlo (MC) Potts model [10, 11], vertex model [12, 13], cellular automata [14], phase field approaches [15, 16], and molecular dynamics for nanocrystalline [17, 18]. Among these numerical methods, the MC Potts technique has become a very effective simulation method for prediction and analysis of microstructure evolution in polycrystalline materials at mesoscopic level. The MC Potts model is a discrete statistical simulation technique, which does not sample the properties in a deterministic way but stochastically. It treats the evolution of nonequilibrium discrete ensemble, which represents the microstructure. This model is based on the classical works of the Exxon group [10, 11]. The advantage of MC method relies on its simplicity and flexibility to implement it in 2 and 3 dimensions. One of the most serious problems of Monte Carlo simulation for grain growth is that the correspondence between Monte Carlo steps and real time is not well understood. The time progression of the sites position proceeds randomly. So MC procedure suffers from the inability to deal with the physical mechanisms characteristics of grain growth. Several works modeled grain growth by considering a linear relationship between simulated time and real time. Ling et al. [19] have proposed the following relation between the real time and :Other research groups [20, 21] have suggested the relationwhere is constant, is the activation energy, is the temperature, and is the gas constant.
The lattice sites spacing Δ plays an important role in the Monte Carlo simulation of grain growth. In the present study, the relation between real time and simulated time is derived from the assumption that evolution of the mean matrix area , that is, the mean number of sites contents in this area, is an invariant between the simulation and the experimental. Therefore, was used to establish the relation between the real time and . A correlation between the real time, , Δ, and is established.
2. Conversion of Monte Carlo Steps to Real Time
The variation of versus the time will be derived from the theory (real time) firstly and from Monte Carlo simulation secondly. If the microstructure contains cells (lattice sites) (Figure 1), the total matrix area is given by where is the cell area:Then (5) becomesThe matrix average areas and , which contain, respectively, and sites, verify also relation (7):where and are the matrix average areas at times and , respectively.
With the circular grains hypothesis, the matrix average areas are given byThen (1) will bewhere is the real time in seconds.
By substituting (8) into (10), the real time will be Equation (11) gives the real time as function of and Δ. On the other hand, can be obtained from Monte Carlo simulation. When is plotted as a function of the simulated time for grain growth simulation without particles consideration, one obtains by linear fitting a line with the equation [22]where “” and “” are obtained by regression analysis of the data generated from MC simulation.
Equation (11) will beEquation (13) gives the real time as function of the simulated time and the lattice sites spacing Δ.
3. Grain Growth Monte Carlo Algorithm
The grain growth Monte Carlo simulation is performed on a two-dimensional triangular lattice, where for each lattice site “” a number is assigned which corresponds to the grain orientation [10, 11]. The neighboring lattice sites are spaced of a distance Δ. A grain is defined by neighboring lattice sites with the same orientation, while neighboring sites with another orientation belong to a neighboring grain. A grain boundary lies between two adjacent lattice sites with different orientations. The energy of a lattice site “” is given bywhere is the Kronecker delta function with if and 0 otherwise and is a positive constant that represents the grain boundary (km) energy.
The grain growth MC algorithm is as follows:(1)Randomly select a lattice site “” .(2)Calculate the site energy (see (14)).(3)Assign to this site a new orientation among its near neighboring area “.”(4)Calculate the new site energy (see (14)).(5)Calculate the net energy change due to the reorientation(6)Reorientation is accepted with the transition probability (TP):where is the simulation temperature and is a constant.
The number of reorientation attempts , that is, the number of lattice sites, is defined as one Monte Carlo step (MCS). Starting material used in this study is a real Fe-3%Si microstructure ( µm2) obtained by orientation imaging microscopy () (Figure 2). This microstructure corresponds to a hexagonal grid with µm and sites. The MC simulation is done for the case of isotropic grain boundaries (all energies and mobilities were set to unity) at K. The parameter in (1) has been supposed to be equal to 1 for simplification.
4. Results and Discussion
From MC simulation results, Figure 3 depicts the MC time dependence of the matrix mean number of sites . One obtains by linear fitting a line with the equationSubstituting (17) into (13) gives the relation between real time and :Figure 4 illustrates the simulated time dependence of the real time for the matrix simulation.
Equation (18) permits us to represent the MC simulation results as function of real time; for example, Figure 5 shows the variation of the matrix mean sites number according to the real time. By linear fitting, one obtains
4.1. Effect of Lattice Sites Spacing Δ on Real Time
The lattice sites spacing Δ plays an important role in the Monte Carlo simulation of grain growth. To see the influence of Δ on the real time, three cases will be considered for the second simulation: µm ( sites and ), µm ( sites and ), and µm ( sites and ). Figure 6 shows the MC time dependence of the matrix mean number of the sites for different values of the space step . One obtains by linear fitting a line with the equationSubstituting (20) into (13) gives a relation between real time and simulated time :The real time sensitivity to the space step Δ can be seen from plotting the variation of the real time versus for different Δ values. At constant , the real time increases with increasing Δ as shown in Figure 7. For example, Figure 8 shows the microstructure evolution after MCS for three cases: (a) min for Δ = 2 µm, (b) min for Δ = 1.5 µm, and (c) min for Δ = 1 µm. It is clear that the grain growth is faster for the case where Δ = 2 µm. The movement of grain boundaries controls grain growth process. Based on the equivalence of grain boundary migration and single-site switches in the Potts model, during one Monte Carlo step, the grain boundary between two adjacent grains is displaced over a distance Δ. This can explain why for constant MCS the growth of grains is faster for the case where Δ = 2 µm than the case where Δ = 1 µm. From (13), it is obvious that the real time at constant MCS becomes smaller with decreasing Δ.
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4.2. Introduction of Temperature in the Monte Carlo Simulation Method
In addition to the problem of the real time conversion, the mechanism of the temperature influence on the calculated results using MC simulation for grain growth is not well understood. The influence of temperature can be introduced usually in the Monte Carlo simulation through the transition probability in reorientation attempts (see (16)). Based on isothermal experimental data and the grain growth regression analysis, Burke and Turnbull [1] deduced the following parabolic law for isothermal grain growth:where is a constant, is the time, is the temperature, is the gas constant, and is the activation energy for grain growth. Both and are obtained from experimental data.
While using relation (22) instead of relation (1) in the previous calculations, (13) will beEquation (23) gives the real time as function of the simulated time , the temperature , and the lattice sites spacing Δ. The last simulation is done for the case where Δ = 1 µm and with the transition probability in reorientation attempts:According to (23), the parameters that will be used in the calculation are as follows: cal·K/mol, , and cal/mol. Influence of the temperature on real time can be seen from plotting the real time versus for different values. At constant , the real time decreases when increases as shown in Figure 9. Equation (23) permits us to introduce the influence of the temperature in the MC simulation instead of using the transition probability (see (16)). Figures 10 and 11 show the real time dependence of the square matrix mean-radius and the evolution of grain growth after 6 min by using MC simulation for different values of the temperature .
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5. Conclusion
In addition to the problem of the correspondence between the simulated time and the real time, the mechanism of the temperature’s influence on the calculated results in the MC simulation for grain growth is not well understood. The lattice sites spacing Δ cannot be ignored in the Monte Carlo simulation. A new equation that gives the real time as function of simulated time , the temperature , and the lattice sites spacing Δ has been derived. The simulation results show that, for modeling grain growth, the relation between real time and is achieved linearly. The influence of the temperature can be introduced in the Monte Carlo simulation through the proposed equation instead of using the transition probability.
Conflicts of Interest
The author declares that there are no conflicts of interest.