Simulation of grain coarsening in two dimensions by cellular-automaton

Acta mater. 49 (2001) 623–629 www.elsevier.com/locate/actamat

SIMULATION OF GRAIN COARSENING IN TWO DIMENSIONS BY CELLULAR-AUTOMATON ´ SZ2† and P. BARKO ´ CZY2 J. GEIGER1, A. ROO 1

Department of Descriptive Geometry, Faculty of Mechanical Engineering, Miskolc University, H 3515 Miskolc-Egyetemva´ros, Hungary and 2Department of Physical Metallurgy, Faculty of Materials and Metallurgical Engineering, Miskolc University, H 3515 Miskolc-Egyetemva´ros, Hungary ( Received 8 August 2000; received in revised form 11 October 2000; accepted 18 October 2000 )

Abstract—This paper presents a computer simulation model, namely Cellular Automaton (CA), which aims at investigating the behaviour of normal grain coarsening in 2D that corresponds well to the described physical model. The CA model takes into account the following: the energy barrier of cellular transition depends on the energy of the grain boundary; the energy of boundaries depends on the misorientation of the grains and the energy of the cells follows the Maxwell–Boltzmann distribution. The model was tested on both simple geometrical configurations and on real structures. The effect of temperature, orientation difference, activation energy, and boundary energy for the kinetics of grain coarsening were analysed and discussed. The optimal orientation value, qmax, was greater than 64, and at smaller qmax values non-real structures develop. The rate and kinetics of coarsening depend on a qmax value up to 64, the energy barrier and the boundary energy. The rate of coarsening cannot be described by only one exponential function over a wide temperature range.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Grain growth; Kinetics; Computer simulation

1. INTRODUCTION

The physical and chemical properties of alloys are significantly influenced not only by their compositions but also by their microstructures. For example, according to the well known Hall–Petch relation, the tensile strength and hardness depend on the average grain size. A finer grain size gives greater strength and hardness. Also, the grain size of austenite significantly influences the course of the solid-state transformations and microstructures developed in steels. Coarser austenite transforms more slowly and gives a coarser final microstructure. During many heat treatment processes (i.e. austenitisation) the average grain size increases. The driving force for such a microstructure coarsening comes from decreasing of the free energy of the total area of the system. The average grain diameter ¯ ) depends on time (t) as follows [1, 2]: (D ¯ 20 ⫹ k1t ¯2 ⫽ D D

f is a shape factor), the following relationship can be written [2]: A¯ ⫽ A¯0 ⫹ k2t

(2)

In equations (1) and (2) k1 and k2 are constants, ¯ 0 and A¯0 are the initial average diameter and area D of grains at t ⫽ 0. The variation of free energy of an atom jumping from one side of the grain boundary to the other side is shown in Fig. 1. In that case when the atom jumps from grain 1 to grain 2 the energy barrier is ⌬GA, and when the atom jumps back the energy barrier is

(1)

Supposing that the average diameter is proportional ¯ 2 where to the average area (A¯) of the grains (A¯ ⫽ fD

* To whom all correspondence should be addressed. Fax: ⫹36-46-365-924. E-mail address: [email protected]

Fig. 1. Variation of free energy of an atom during jumping through the boundary.

1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 3 5 2 - 9

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GEIGER et al.: SIMULATION OF GRAIN COARSENING IN TWO DIMENSIONS

⌬GA ⫹ ⌬G. (⌬G ⫽ GB1⫺GB2 is the free energy difference of the atom before and after jumping). Inside the grains the energies of the atoms are definitely zero. The probability of an atom jumping from grain 1 to grain 2 is [1, 4]: P1 ⫽ exp(⫺⌬GA/RT)

(3)

and the probability of jumping back is: P2 ⫽ exp(⫺(⌬GA ⫹ ⌬G)/RT)

(4)

where T is the absolute temperature, R is the gas constant. The summarised probability (P) of the transition of an atom from grain 1 to grain 2 through the grain boundary is [12, 13]: P ⫽ P1⫺P2 ⫽ P1(1⫺P3)

(5)

P3 ⫽ exp(⫺⌬G/RT)

(6)

where

It can be seen from equation (3), that the probability of the transition depends on both the activation free energy and the free energy difference of the atoms before and after jumping. The boundary migration velocity (v) depends on the temperature as follows: v ⫽ k3exp(⫺⌬GA/RT)[1⫺exp(⌬G/RT)] ⫽ k3P (7) ¯ and A¯ are determined by the Increasing D migration velocity so the rate of coarsening (k1, k2) depends on the temperature in the same way as the migration velocity. Using equation (5), the numerical methods (CA, MC) found in the literature can be divided to two parts. Most of the authors [3, 5–14] have already used the following transition rule: P ⫽ 1⫺P3

which simultaneously involves the conditions as follows: 앫 the transition of atoms (cells) depends on the energy barrier (activation free energy) and the free energy difference of atoms before and after jumping, 앫 the energy of the grain boundary depends on the misorientation of the grains. The model developed shows the grain coarsening in two dimensions, as this process is generally investigated and modelled on the surface of the sample and also most of the known MC and CA models in 2D [1–3, 5–14]. 2. CA SIMULATION PROCEDURE

In the CA method each cell of the lattice represents a group of atoms, and the evolution of the microstructure is governed by the behaviour of individual cells acting in response to their neighbourhood. The maximum number of lattice sites (N) determines the resolution of the simulation. N lies typically in the range 104 to 5⫻105, depending on the computer power available. Our purpose in the present work is to develop a 2D computer simulation method based on the cellular automaton approach, which will model normal grain coarsening and corresponds well to the physical model described in the Introduction. This new CA model is based on the following: 앫 the transition (jumping) of atoms (cells) depends on the energy barrier and the free energy difference of atoms before and after jumping, 앫 the energy barrier for transitions must depend on the energy of the grain boundary, reflecting the fact that boundaries of different misorientations have different mobilities, 앫 the energy of boundaries depends on the misorientation of the grains, 앫 the thermal energy of the cells follows the Maxwell–Boltzmann distribution, 앫 at the calculation of transition the energy values are directly compared instead of the P probabilities.

(8) 2.1. Model algorithm

In this case the energy barrier (activation free energy) ⌬GA ⫽ 0. In [7] when ⌬G>0 the transition is not accepted which means P3 ⫽ 0. In other papers [15, 16] the authors take into account only the activation free energy, then P ⫽ P1. In some models the energy of the grain boundary does not depend on the misorientation of the grains [3, 6–8, 10–13, 15, 16] but in [5, 9, 14] it does. Reviewing the papers, a model cannot be found

The models in [15, 17] and our CA model have the same basic approaches: 1. definition of the initial grain structure, 2. definition of the geometry of the cells, 3. definition of the cell surroundings (neighbourhood), 4. assigning states to the cells, 5. specifying the cell transition rules,

GEIGER et al.: SIMULATION OF GRAIN COARSENING IN TWO DIMENSIONS

6. calculation of the cell transitions, 7. redefinition of the grain structure. The initial grain structure was generated using a tessellation according to the method in [18]. The data set contains the co-ordinates of the centres and corners of 400–450 grains. An example of an initial structure is shown later in Fig. 10(c). For the calculation 200∗200 ⫽ 40,000 square cells were used. For display, for example on a computer screen, each pixel corresponds to a cell; thus the terms pixel and cell may be used interchangeably. For each cell, its neighbourhood is defined by the von Neumann method; each cell is considered to have four neighbours, being those with which it shares the edges of the square. An identity number, id, and a randomly allocated orientation number, q, were assigned to each grain. Both are integers: 1⬍id⬍idmax, 1⬍q⬍ qmax, where idmax is the number of grains and qmax is the maximum orientation number in the original structure. During the calculation idmax was 400–450, while qmax was 2, 4, 8, 16, 32, 64, 128, 256. The q orientation number indicates primarily the orientation of a cell and the colour code of the pixel. Cells belonging to the same grain have the same id and q values. Each cell in the simulation can be inside a grain or on a grain boundary. The energy of a cell is given by the sum of thermal (GT) and boundary energy (GB). The thermal energy of the cells (as for individual atoms) is presumed to follow the Maxwell–Boltzmann distribution. The energy values are randomly assigned to the cells in each calculation step. The boundary energy along the ith side of the cell (GiB) given by the Hamiltonian, equation (9), taking into account that the boundary energies depend on the misorientation angle between the grains. GBi ⫽ ⫺GB(⌰)⌺(dqi,qj⫺1)

625

In the reviewed models the transition of cells was determined by probabilities shown in the Introduction. In this model the energies will be compared directly. For a cell on a grain boundary, the cell can pass through the boundary if the thermal energy is greater than the difference between the activation free energy and the total boundary energy of the cell: GT>GA⫺⌺GiB

(12)

Note: in this procedure there is only one activation free energy GA, for all cells (see Fig. 1). If equation (12) is satisfied, the boundary passes through the cell, even if the energy of the cell increases. If the cell has more than one possibility to connect to neighbouring grains, then the algorithm chooses the transition which minimises the energy. If a cell is inside a grain, no transition takes place. The CA model checks the cells parameters in each calculation step and makes the transitions. 3. TEST OF THE MODEL ON A SIMPLE SHAPE

Before considering the coarsening of a realistic microstructure it is useful to investigate the behaviour of individual grains. A simple shape (a circle embedded concentricity in a square, Fig. 2) was used to show that the CA model matches equation (2). In general the kinetic of grain coarsening is given by A¯ ⫽ A¯0 ⫹ (kt)n so for the shrinking of the circle: A¯ ⫽ A¯0⫺(kt)n

(13)

Table 1 shows the n-values concerning the test. The values of n in Table 1 show that the shrinking of the circle is linear (n⬇1) with CAS. This linearity can be

(9)

The grain boundary energies are calculated from the Read–Shockley equation as given by Wolf [19]: GB(⌰) ⫽ G0 sin⌰[1⫺ln(sin⌰)]

(10)

where GB(⌰) is the boundary energy as a function of the misorientation angle ⌰, and G0 is the maximum grain-boundary energy seen at high ⌰. The misorientation is obtained from the q numbers of the neighbouring grains ⌰ ⫽ p/2∗兩⌬q兩/qmax

(11)

where ⌬q is the difference between orientation numbers of two adjacent grains and 0ⱕ兩⌬q兩/qmax⬍1. The Read–Shockley equation does not contain the decreasing in energy at the coincidence grain boundary.

Fig. 2. Shrinking of circle after 0, 300, 500, 700 CAS at J ⫽ 1000°C, qmax ⫽ 16.

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GEIGER et al.: SIMULATION OF GRAIN COARSENING IN TWO DIMENSIONS Table 1. The value of the exponent n in equation (13)

Temperature (°C) n

200 0.99

400 1.02

600 1.06

seen in Fig. 3, and this conclusion is in accordance with the simulation results in [3]. In the test the cell number was N ⫽ 40000, the proportion between the area of the square and t the circle was 1:1, and qmax ⫽ 16. 4. RESULTS AND DISCUSSION

For the investigation of coarsening of realistic grain structures, initially the effect of maximum orientation number (qmax) on the initial structure was investigated in order to determine the optimal qmax number for further calculations. For the kinetics of grain coarsening, the effects of maximum orientation number, temperature, boundary energy and activation energy were investigated. 4.1. Effect of maximum orientation number (qmax) on the initial structure The optimal value of qmax is an important problem in the description of real grain structures. The value of qmax was varied between 2 and 256; Fig. 4 shows the structures with qmax ⫽ 2, 4, 128 and 256. The relative grain area distribution curves of these grain structures can be seen in Fig. 5. The relative grain area is the actual grain area (counted in cell numbers) divided by the maximum grain area. For different qmax values significantly different distributions are obtained. At small qmax-values (2, 4, 8) non-real structures develop, as adjacent grains with the same orientation number coalesce. From qmax ⫽ 64 the usual curves develop, and they are consistent. In accordance with the results described in [3], it can be con-

Fig. 3. The area of the circles as a function of CAS at different temperatures.

800 1.06

1000 0.96

1200 0.95

cluded that qmax must be at least 64. In further tests the value of qmax ⫽ 128 was taken. 4.2. Effect of the maximum orientation number At small qmax (2,4,8) the change of average grain area (number of cells) shows great scatter, Fig. 6, and non-real structures develop. With qmax ⫽ 64, 128, 256 the evolution of average grain area is essentially the same in each case. The value of the exponent n (equation 13) depends significantly on qmax (Table 2). It also confirms the conclusion, according to which the qmax value has to be increased only to a definite extent during simulation. The rate of grain coarsening, denoted by k does not change if the value of qmax>64 (Fig. 7, GA ⫽ 10,000 J/mol, G0 ⫽ 3000 J/mol, 600°C). Therefore it is reasonable to use the qmax ⫽ 128 value during the investigation of coarsening of real structures. 4.3. Effect of temperature The effect of temperature on the grain coarsening kinetics is shown in Fig. 8 (qmax ⫽ 128, GA ⫽ 10,000 J/mol, G0 ⫽ 3000 J/mol). The increase of average area is approximately linear as a function of CAS. The rate of grain coarsening (k) increases with temperature. Figure 9 shows the rate of coarsening as a function of temperature as ln(k) vs 1/T. If GA⫺G0 increases, then the curve deviates more and more from a simple exponential curve. In Fig. 10 the change of grain structure can be seen as a function of time, at a temperature of 600°C. The initial structure corresponds to the structure shown in Fig. 4(c). 4.4. Effect of boundary energy Decreasing the maximum boundary energy G0 decreases the growth rate of grain area. The ln(k)苲1/T relationship is not linear (Fig. 9). The Arrhenius curves can be divided into approximately two straight line segments. For example, curve C consists of two linear parts, at 0°CⱕJⱕ200°C with a lower slope and at J>200°C with much more higher slope. Curve B shows similar behaviour but the difference between the two slopes is smaller. For curve A, where GA⫺G0 is smaller, this effect was eliminated. This phenomenon is in good agreement with the practical observation that for a relatively small temperature interval the temperature dependence of rate of coarsening can be described by one exponential function (k ⫽ k0exp(⫺GL/RT), where GL is the apparent activation energy. At high temperatures (above 1000°C) at a constant activation energy, the increase of boundary energy does not have a significant effect on the rate of grain coarsening, while this effect is very significant at

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Fig. 4. Initial grain structures at different qmax values. (a) qmax ⫽ 2, (b) qmax ⫽ 4, (c) qmax ⫽ 128, (d) qmax ⫽ 256.

Fig. 5. Number of grains as a function of relative grain area.

lower temperature. During tests on real grain structures, an n exponent value of n⬇0.95 is obtained in the kinetics of grain coarsening, i.e. the grain coarsening shows a linear relationship with time similar to the simple test structures.

Fig. 6. Average grain area as a function of CAS at different qmax (J ⫽ 600°C, GA ⫽ 10,000 J/mol, G0 ⫽ 3000 J/mol).

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GEIGER et al.: SIMULATION OF GRAIN COARSENING IN TWO DIMENSIONS Table 2. The value of the exponent n as a function of qmax at J ⫽ 600°C

qmax n

2 0.62

4 0.66

8 0.68

16 0.77

32 0.78

64 0.91

128 0.93

256 0.92

Fig. 7. Rate of grain coarsening as a function of qmax.

Fig. 9. Effect of boundary and activation energy on rate of grain coarsening (qmax ⫽ 128). A: G0 ⫽ 3000 J/mol, GA ⫽ 10,000 J/mol; B: G0 ⫽ 1000 J/mol, GA ⫽ 10,000 J/mol.

with the physical model. The present simulation method differs from the previously published approaches in the following aspects:

Fig. 8. Effect of temperature on grain coarsening kinetics (qmax ⫽ 128, activation energy: GA ⫽ 10,000 J/mol, boundary energy: G0 ⫽ 3000 J/mol).

4.5. Effect of activation energy The activation energy has a significant effect on the rate of thermally activated processes. In Fig. 9 (A and C curves) the rates of coarsening are compared. The rate of coarsening decreases by many orders if the activation energy is increased from 10,000 to 20,000 J/mol. The decrease in the rate of coarsening depends on the temperature (at 0°C the difference is about three orders, while at higher temperatures (1200°C) it is only one order). 5. SUMMARY

The aforementioned simulation method is suitable for describing normal grain coarsening. In the course of testing the model it was stated that the kinetics of coarsening change with the parameters in accordance

앫 the cells have a thermal energy, the energy distribution following the Maxwell–Boltzmann distribution, 앫 the sum of the thermal energy of the cells and the boundary energy have to be greater than the activation energy in order that the cell can pass through the boundary; independently the boundary energy of the cell increases or decreases as a result of the migration; 앫 the boundary energy depends on the orientation difference between the grains. In the course of the investigation of real grain structures the following conclusions have been made: 1. In the case of small maximum orientation number (qmax ⫽ 2, 4, 8) non-realistic structures develop, even for initial structures, as a consequence of the frequent coalescence of neighbouring grains, and large, continuous ranges of special morphology develop. It can be stated from the distribution curves, that for qmax ⫽ 64, 128, 256 the usual distribution curves develop in a manner essentially independent of the qmax value. 2. The kinetics of coarsening depend on the

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Fig. 10. Evaluation of microstructure at different CAS. (a) 100 CAS, (b) 250 CAS, (c) 350 CAS, (d) 500 CAS. Parameters: qmax ⫽ 128, J ⫽ 600°C. Initial structure in Fig. 4(c).

maximum orientation number. The tests have established that if the qmax>64, the value of k (rate of coarsening) does not depend on the maximum orientation number. Therefore when using a value of qmax ⫽ 128, the kinetics are surely independent of the value of qmax. 3. The rate of coarsening cannot be described by only one exponential function over a wide temperature range. At the same time, in a smaller temperature interval (in accordance with the practical observations) the ln(k)苲1/T function can be well approximated by a single straight line. 4. The rate of coarsening depends on the energy barrier and the boundary energy.

Acknowledgements—This research work was supported financially by the Hungarian Scientific Academy. The authors would like to thank Dr. A. L. Greer, University of Cambridge (Cambridge, UK) for helpful comments and for reviewing the paper. REFERENCES 1. Porter, D. A. and Easterling, K. E., Phase Transformations in Metals and Alloys. Chapman & Hall, London, 1992. 2. Christian, J. W., The Theory of Transformations in Metals and Alloys, 2nd Ed. Pergamon Press, Oxford, 1975.

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