Monte Carlo Implementation for Simulation of Ostwald Ripening Via Long Range Diffusion in Two-Phase Solids

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 6 Issue 1|| January 2017 || PP. 29-37
www.ijesi.org 29 | Page
Monte Carlo Implementation for Simulation of Ostwald Ripening
Via Long Range Diffusion in Two-Phase Solids
Rifa J. El-Khozondar1
, Hala J. El-Khozondar2
1
Physics Department, Applied Science College/ Al-Aqsa University, Palestine
2
Electrical Engineering Department, Engineering College/ Islamic University of Gaza, Palestine
Abstract: Numerical simulations based on the Monte Carlo Potts model are used to study coupling of grain
growth and Ostwald ripening in two-phase polycrystalline materials. The ratio of the grain boundary energy to
the interphase boundary energy is used as an input parameter. It is shown that the grain growth in two-phase
polycrystalline materials is controlled by long-range diffusion and the change of the mean grain size with time
obeys the growth law, <R>n
= <R>0
n
+kt where n is the grain growth exponent. The value of n is calculated for
a broad series of volume fractions. It is found that the inverse grain growth exponent, 1/n, in agreement with the
theoretical value, 1/n=1/3, noticed during computer simulations for volume fractions between 40% and 90%.
However, the value of 1/n is smaller than 1/3 for volume fractions between 10% and 30%. Furthermore, the
temporal development of the number of grains has been analyzed for the entire range of volume fractions. It is
also seen that the quasi-stationary state is advanced at varied aging times depending on the volume fractions.
Furthermore, it is shown that the simulated size distribution are symmetric and peaked at x=1 for volume
fractions differ between 50% and 90%; however, the simulated size distribution become asymmetric and skew to
smaller grains for lower volume fractions change between 10% and 40%.
Keywords: Monte Carlo, Ostwald Ripening, Polycrystalline Materials, Grain Growth, Volume Diffusion
I. Introduction
The grain size controls the thermal, electrical and mechanical properties of polycrystalline materials. This
emphasizes that the grain size plays an important role of improving materials properties during processing.
Therefore, understanding microstructural evolution and Ostwald ripening in two-phase materials is a key topic
for applications of engineering materials. It is well known that the polycrystalline microstructures do not
comprise grains of one size, but can relatively be defined mean grain size and grain size distribution. The
average grain size varies with time according to the power-law function
 
1/
n
0
R = R + kt
n
(1)
where k is the grain growth constant. The theoretical value of n is 2 for pure metals [1-3]. However, for Ostwald
ripening, n=3 for long-range diffusion [4,5] and n=4 for grain boundary diffusion [6]. Simultaneously, the grain
structure evolves to quasi-stationary state in which the grain size distribution are stationary and
indistinguishable from each other. The total number of grains drops as grain growth continues and can be
described by the following equation
2 2 2
A A A
N
A 4 R 4 π x R
  
       
(2)
where the scaled grain radius, x=R/<R>. Replacing <R> from (1) into (2) gives the fitting relationship
 
α
N (t) = kt + C (3)
where the exponent  has the value of -2/3 when microstructural evolution is controlled by Ostwald ripening via
long-range diffusion. The grain size distribution as a function of the scaled grain radius have been derived by
several authors [6-8].
In two-phase materials such as ceramics and metallic alloys [9, 10], grain growth and Ostwald ripening may
occur simultaneously. These materials have important applications in electronic industries. There have been
experimental attempts [11-17] and numerical simulations [18-21] to study the coupling between grain growth
and Ostwald ripening.

Monte Carlo Implementation for Simulation of Ostwald Ripening Via Long Range Diffusion in Two..
In this work, the Monte Carlo Potts model adjusted by Solomatov et al. [20] to study grain growth in two-phase
polycrystalline materials is developed to study Ostwald ripening via long-range diffusion in two-phase
polycrystalline materials. The model is comparable to one established by El-Khozondar and El-Khozondar [22]
for Ostwald ripening of solid grains in a liquid matrix. However, in the present work the second phase is solid
which has grain boundaries. The Monte Carlo Potts model has shown to be successful to simulate grain growth
in polycrystalline materials for one, two and three phases [23-37]. The modified Monte Carlo model in this work
allows us to display microstructural evolution of polycrystalline materials and gives us information about the
average grain size and the grain size distribution. Therefore, the grain growth exponent can be calculated from
the simulation results. Next section will focus on the simulation model. Section III is dedicated to the Simulation
of Ostwald ripening in two-phase systems. Conclusion is presented in section VI.
II. The Simulation Model
The The Monte Carlo Potts is adapted to simulate Ostwald ripening in two-phase polycrystalline materials. The
simulation is initiated using a quadratic lattice with eight nearest neighbors. Every lattice point is named one
Monte Carlo Unit. In the current simulation, the number of Monte Carlo Units of the simulated microstructure is
taken to be 400×400. The unit of time is given in Monte Carlo steps (MCS) which signifies N orientation
attempts where N is equivalent to the number of Monte Carlo Units. This implies that N=16,000 in our
simulations.
The original microstructure of two-phase system is treated as a two-dimensional square array of Monte Carlo
Units. Each Monte Carlo Unit is given a random number between 1 and Q where Q is the number of
orientations. The value Q=100 is used in our simulations. The Monte Carlo Units of phase A are given positive
numbers. The Monte Carlo Units of phase B are given negative numbers. A grain comprises a group of Monte
Carlo Units with similar orientations. A grain boundary is then an edge, which separates grains with different
orientations.
There are three kinds of boundaries. The grain boundaries between grains in phase A have energy Eaa. The
grain boundaries between grains in phase B have energy Ebb. The interphase boundaries between grains in
phase A and grains in phase B have energy Eab. The values of the interfacial energies are selected such that the
values of the grain boundary energies (Eaa and Ebb) are greater than the value of Eab. Ostwald ripening is
simulated using Eaa=2.5, Ebb=2.5 and Eab=1.
The change of the two-phase microstructure is motivated by the decrease in the entire energy of grain and
interphase boundary. It includes two processes: grain boundary migration and Ostwald ripening by long-range
diffusion. Grain boundary migration happens because of orientation flip in one phase as follows. One Monte
Carlo unit is randomly selected and is flipped to a different randomly selected orientation. Then, the difference
in energy ∆E is calculated. The new orientation is selected using the next rule: If ∆E ≤ 0, the flip is accepted;
however, if ∆E > 0, the flip is rejected. The second process is Ostwald ripening by long-range diffusion that
moves material from one grain to another grain of the same phase through the grains of the other phase.
This is numerically done by using orientation exchange between the phases as follows. One Monte Carlo unit
and its neighbor are selected in a probabilistic way. If they belong to different phases, they are permitted to
exchange their orientations. The energy difference is then calculated. The new orientation of the two Monte
Carlo Units is chosen by the following rule. If ΔE ≤ 0, the exchange is accepted. Conversely, if ΔE > 0, the
exchange is accepted with the following exchange probability function.
Δ E
P = ex p -
k T
 
 
 
(3)
where k is the Boltzman constant and T is the temperature. The temperature has the value of T=1.3. After each
attempt, the time is incremented by 1/N MCS.
III. Simulation of Ostwald Ripening in Two-Phase Systems
Fig. 1 exhibits the temporal microstructural evolution in two-phase system for different volume fractions. The
volume fraction of the B phase (white grains) varies between 10% and 50% as well as the volume fraction of the
A phase (grey grains) varies between 50% to 90%. As can be seen from Fig. 1 that there is an indication of
Ostwald ripening where the number of grains declines and the mean grain size grows with time. Moreover, the
microstructure evolves because of moving atoms from small grains to large grains via volume diffusion.
Consequently, small grains ultimately contract and large grains grow in size.
Fig. 2 displays the time dependence of the number of grains for the whole range of volume fractions. It can be
shown from Fig. 2 that the number of grains drops (solid line) following the quasi-stationary state (4) with the
value of exponent from numeric fit (dashed line) equivalents to -2/3. Additionally, it can be observed from Fig.

2 that the quasi-stationary state is progressed at varied aging times depending on the volume fraction of the case.
The quasi-stationary state is attained near the end of simulation time, t > 50000 MCS when volume fractions
have values of 10%, 20%, and 30%. Whereas the quasi-stationary state is advanced at t > 7000 MCS for volume
fractions with values equivalent to 40%, 50%, and 60%. When volume fractions are equal to 70%, 80%, and
90%, the quasi-stationary state is arrived at t > 1000 MCS. It can be concluded that as the volume fraction
increases, the quasi-stationary state is reached at earlier times and vice versa.
Fig. 3 shows the change of average grain size with time for the entire series of volume fractions. The average
grain size and the standard deviation are calculated after running each case ten times. The standard deviation is
not indicted because it is found to be very small. It is noticed from Fig. 3 that the system goes through a
transitional regime and eventually approach the quasi-stationary state at different times depending on the case of
volume fraction. The value of the growth exponent is calculated by taking the slope of the curves shown in Fig.
3. The slope is calculated in a time gap which moving across the time axis because the growth exponent is very
sensitive to time variation. The logarithmic size of the time gap is equivalent to 5. The slope is plotted against
time in Fig. 4. It can be seen from Fig. 4 that for volume fractions vary between 40% and 90%, the slope in the
quasi-stationary state is close to the theoretical value 1/3 projected for Ostwald ripening via long-range
diffusion. However, the slope is lower than the theoretical value 1/3 for volume fractions differ between 10%
and 30%.
Fig. 5 and Fig. 6 depict the temporal development of the scaled grain size distribution at five different time steps
for a range of volume fractions of phase B varies between 10%-50% for phase B and for a range of volume
fractions of phase A varies between 50%-90%. It is shown from these figures that the size distribution are
identical for volume fractions vary between 50% and 90% and coincide with the normal distribution function.
Additionally, the simulated size distributions are peaked at x=1 and symmetric. While the simulated size
distribution within are asymmetric and skew to smaller grains for lower volume fractions differ between 10%
and 40%.
Fig. 1 Simulated microstructure of a system comprising two-phase polycrystalline material, phase A (grey) and
phase B (white), for different volume fractions of the B phase.

Fig. 2. Variation of the number of grains (N) with time for a varied volume fraction of the B phase (solid line)
with fit to (3) (dashed line).
t
N(t)

Fig. 3 Dependence of grain size on time. The volume fraction of both phases are indicated.

Fig. 4 Dependence of the slope of the curves in Fig. 3 on time for a broad volume fraction range of the B phase.
The horizontal solid lines represent the theoretical value 1/n=0.33 for Ostwald ripening via long-range diffusion.
n
tLog t

Fig. 5 Effect of volume fraction on the temporal development of the relative grain size distribution of the two
phases for five different time steps, comparing with fit of the Gaussian distribution function. The volume
fraction of phase B varies between 10%-30% and the volume fraction of phase A varies between 70%-90%.

Fig. 6 Effect of volume fraction on the temporal development of the relative grain size distribution of the two
phases for five different time steps, comparing with fit of the Gaussian distribution function. The volume
fraction of phase B varies between 40%-50% and the volume fraction of phase A varies between 50%-60%.
IV. Conclusion
The Monte Carlo Potts model has been used to simulate the coupling between grain growth and Ostwald
ripening in two-phase polycrystalline materials. The suitable value of the ratio between the grain boundary
energies to the interphase energies is imbedded as an input parameter in the model. It is found that the
microstructural evolution of the two phases reaches a quasi-stationary state at different aging times depending
on the volume fractions. The value of the grain growth exponent close to the value of n=3 in agreement with the
theoretical value of Ostwald ripening for volume fractions between 40% and 90%. Whereas the quasi-stationary
state is approached very late near the end of simulation time in the cases of volume fractions between 10% and
30%; therefore, the value of n is larger than 3.
The grain size distribution is analyzed for a broad range of second phase volume fractions. It is found that the
grain size distribution vary with the volume fraction. For volume fractions vary between 50% and 90%, the size
distributions in the quasi-stationary state are indistinguishable from each other; furthermore, the simulated size
distribution within the quasi-stationary state can be described very well by the normal distribution function. The
simulated size distributions within the quasi-stationary state are symmetric and peaked at x=1. However, for
lower volume fractions differ between 10% and 40%, the simulated size distribution within the quasi-stationary
state are asymmetric and skew to smaller grains.
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Monte Carlo Implementation for Simulation of Ostwald Ripening Via Long Range Diffusion in Two-Phase Solids

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