Abstract. This paper studies the influence of temperature 500 K, 1500 K, 2500 K, 3500 K
and 4000 K on the microstructure and phase transition of a CaSiO3 bulk model using the
Molecular Dynamics method with the Born-Mayer pair interaction potential and periodic
boundary conditions. The obtained samples were analyzed through the radial distribution
function (RDF), and the coordination number, angle distribution, size, energy and phase
transition were determined using the relation between temperature and energy. The
calculated results show that temperature influences the microstructure and phase transition
of a CaSiO3 bulk model. In addition, the samples at different temperatures have the
different couplings Si-Si, Si-O, O-O, Si-Ca, O-Ca and Ca-Ca and the different coordination
numbers SiO4, SiO5, SiO6, CaO3, CaO4, CaO5, CaO6, CaO7, CaO8 and CaO9.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0036
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 88-97
This paper is available online at
INFLUENCE OF TEMPERATURE ON THE MICROSTRUCTURE
AND PHASE TRANSITION OF A CaSiO3 BULK MODEL
Nguyen Trong Dung, Nguyen Thi Ha and Nguyen Chinh Cuong
Faculty of Physics, Hanoi National University of Education
Abstract. This paper studies the influence of temperature 500 K, 1500 K, 2500 K, 3500 K
and 4000 K on the microstructure and phase transition of a CaSiO3 bulk model using the
Molecular Dynamics method with the Born-Mayer pair interaction potential and periodic
boundary conditions. The obtained samples were analyzed through the radial distribution
function (RDF), and the coordination number, angle distribution, size, energy and phase
transition were determined using the relation between temperature and energy. The
calculated results show that temperature influences the microstructure and phase transition
of a CaSiO3 bulk model. In addition, the samples at different temperatures have the
different couplings Si-Si, Si-O, O-O, Si-Ca, O-Ca and Ca-Ca and the different coordination
numbers SiO4, SiO5, SiO6, CaO3, CaO4, CaO5, CaO6, CaO7, CaO8 and CaO9.
Keywords: Temperature, microstructure, Molecular Dynamics, CaSiO3 bulk model.
1. Introduction
In recent years, an increasing number of studies on CaSiO3 perovskite have been done.
CaSiO3 is not stable at high pressure [1, 2]. It is an important component which characterizes
the versatility of glass [3]. In studying the microstructure of CaSiO3, many methods can be
used, including X-ray diffraction [4] and EXAFS spectroscopic analysis [5, 6]. In particular, the
couplings 17O and 29Si have been detected in the material using neutron diffraction [7, 8]. The
results show that in this material there are two types of isotopes: A silica tetrahedral lattice with Ca
as the lattice controller and Ca-O with coordination number 6. Neutron diffraction results show
that there are differences in the isotope substitution for the Ca-Ca correlation. X-ray diffraction
and neutron diffraction results [9, 10] for CaSiO3 crystal are similar to the above results.
According to simulation results [11], the Si-O couplings are 1.7 A˚ and the coordination
numbers of Si-O are 4 and 7 [12]. The phase transition in CaSiO3 perovskite was determined
experimentally and theoretically [13]. According to the results for CaO1−x(SiO2)x, the phase
transition temperature depends on the concentration of SiO2 and the research methods. For
example, at x ≈ 0.33, the phase transition temperature is 1978 K; at x ≈ 0.4, the phase transition
temperature is 1709 K and at x ≈ 0.5, the phase transition temperature is 1817 K. [14]. From
experimental data, the phase transition temperature is 1873 K [15]. The distance between couplings
is 1.7 A˚ for Si-O [15], 1.61 A˚ for Si-O, 2.6 A˚ for O-O [16] and 2.48 A˚ for Ca-O [17]. We can see
Received August 29, 2016. Accepted September 26, 2016.
Contact Nguyen Trong Dung, e-mail address: dungntsphn@gmail.com
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Influence of temperature on the microstructure and phase transition of a CaSiO3 bulk model
that it is difficult to determine the property of a material which is not stable in structure. In this
paper, the influence of temperature on the microstructure and phase transition of a CaSiO3 bulk
model was studied using the Molecular Dynamics method.
2. Content
2.1. Calculation method
The CaSiO3 bulk model with 5000 atoms at T = 500 K, 1000 K, 1500 K, 2000 K, 2500
K, 3000 K and 4000 K was studied using the Molecular Dynamics method with Born-Mayer pair
interaction potential (1) and periodic boundary conditions [18].
Uij(r) = Aij exp(−Bijrij)− Cij
r6ij
(2.1)
where Uij(r) is the pair interaction potential in eV units, rij is the distance between atoms in A˚
units, rcut is the disconnect radius in A˚ units and the coefficients Aij , Bij and Cij are determined
experimentally with the use of elastic modules and the lattice constant as shown in Table 1.
Table 1. The coefficients of Born-Mayer pair interaction potential used
in the CaSiO3 bulk model
Si-Si Si-O O-O Si-Ca O-Ca Ca-Ca
Aij (eV) 5006070.785 7363.700 1621.734 39991599 29353.1570 200790000
Bij (A˚−1) 12.5 5.2632 3.333333 10.8696 4.7619 9.6154
Cij (A˚−1) 0 0 30.22 0 0 0
Initially, samples with the density ρ = 7.6 g/cm3 are established by randomly scattering
atoms into a cube with a radius r.
ρ =
N
V
→ r = 3
√
3N
4πρ
= 3
√
3(mSi.nSi +mO.nO +mCa.nCa)
4πρ
(2.2)
with NA = 6.022.1023, mSi = 26.98154, mO = 15.999, mCa = 40.078, nSi, nO, nCa being the
number of atoms of Si, O, Ca, respectively.
After that, the statistical recovery of samples is run through 5.105 steps. They are then
heated with 2.106 NVT steps and the moving step dr = 0.01 to increase temperature to 500 K,
1000 K, 1500 K, 2000 K, 2500 K, 3000 K and 4000 K. The system was continuously run with
4.107 NVE steps until it reached a stable state. Finally, based on the stable state, we determined
the microstructure as well as the phase transition of the model.
2.2. Simulated results and discussions
The CaSiO3 bulk model with 5000 atoms has the shape as shown in Figure 1, the size and
the energy when it reached to a stable state shown in Table 2.
89
Nguyen Trong Dung, Nguyen Thi Ha and Nguyen Chinh Cuong
Figure 1. The shape of the CaSiO3 bulk model at a temperature of 500 K
Table 2. The energy and size of the CaSiO3 bulk model at different temperatures
T (K) 500 1000 1500 2000 2500 3000 3500 4000
Particle
size (nm)
4.25 4.28 4.30 4.35 4.41 4.47 4.54 4.63
Energy
(eV)
-70022.8 -69376.2 -68697.4 -67892.6 -67053.3 -66159.9 -65293.2 -64354.9
We can see from Figure 1 that the CaSiO3 bulk sample with 5000 atoms at a temperature
of 500 K has a cubic shape. It is created from three types of atoms, Ca atoms being blue, the
Si atoms being green and the O atoms being red. Similarly, the results in Table 2 show that the
CaSiO3 bulk samples with 5000 atoms at different temperatures all have a nanometer size. When
the temperature increases from 500 K to 4000 K, the size and the energy of the samples increase.
These results show that temperature influences the size and the energy of the samples.
The calculated results for the radial distribution function are shown in Figure 2, Table 3a
and Table 3b.
Figure 2. The radial distribution functions of the CaSiO3 bulk model
at a temperature of 500 K
90
Influence of temperature on the microstructure and phase transition of a CaSiO3 bulk model
Table 3a. The coupling distance between atoms (molecules) at different temperatures
T (K) rij
Si-Si Si-O O-O Si-Ca O-Ca Ca-Ca
500 3.12 1.60 2.60 3.58 2.36 3.62
1000 3.12 1.60 2.62 3.56 2.34 3.70
1500 3.14 1.60 2.60 3.52 2.34 3.76
2000 3.10 1.60 2.62 3.46 2.34 3.86
2500 3.14 1.60 2.62 3.58 2.34 3.8
3000 3.14 1.60 2.60 3.58 2.32 3.96
3500 3.12 1.58 2.64 3.56 2.30 4.04
4000 3.10 1.58 2.66 3.52 2.30 3.92
experiment
[12]
- 1.70 - -
simulation
[18]
- 1.61 2.6 2.48
Table 3b. The height of the radial distribution function peaks
for the CaSiO3 bulk model at different temperatures
T (K) gij
Si-Si Si-O O-O Si-Ca O-Ca Ca-Ca
500 4.54 25.24 3.78 2.90 5.59 2.09
1000 4.44 19.77 3.38 2.73 4.50 2.09
1500 3.94 16.71 3.08 2.74 3.81 2.00
2000 3.72 14.94 2.86 2.53 3.42 1.82
2500 3.38 13.61 2.64 2.42 3.07 1.69
3000 3.10 12.42 2.49 2.36 2.93 1.60
3500 2.86 11.88 2.37 2.24 2.82 1.50
4000 2.73 11.73 2.27 2.16 2.77 1.44
We can see from Figure 2, Table 3a and Table 3b that in the CaSiO3 bulk sample with
5000 atoms at T = 500 K, the first peak position of couplings Si-O, O-O, Si-Ca, O-Ca, Ca-Ca of
the radial distribution functions isdominant and the length of couplings Si-O and O-Ca are 1.60
and 2.30 respectively. These results are entirely consistent with the experimental data. When the
temperature increases from 500 K to 1000 K, 1500 K, 2000 K, 2500 K, 3000 K, 3500 K and
4000 K, the first peak position of the couplings changes insignificantly; the length of couplings
Si-O and O-Ca decreases (Figure 3a) and the height of the radial distribution function peaks of the
couplings decrease (Figure 3b).
The above results also show that the distance between atoms is not depend temperature
dependent. Thus, only the short-range order exists in the couplings between atoms in the model.
The height of couplings Si-O, O-O, Si-Ca, O-Ca, Ca-Ca for the first peak of the radial distribution
function has the greatest value with the sample at T = 500 K. When the temperature increases, the
height of the couplings for the first peak of the radial distribution function decreases. This result is
entirely consistent with the above conclusion where the size and the energy of the samples increase
91
Nguyen Trong Dung, Nguyen Thi Ha and Nguyen Chinh Cuong
when temperature increases.
The calculated results for the coordination numbers of the couplings of the model are shown
in Figure 3.
Figure 3. The coordination number of the CaSiO3 bulk model at T = 500 K
Table 4a. The coordination number of the CaSiO3 bulk model at different temperatures
T (K) Si-Si Si-O O-O Si-Ca O-Ca Ca-Ca
500 2 4 3 5 2 6
1000 2 4 3 5 2 7
1500 2 4 6 6 2 6
2000 2 4 6 5 2 7
2500 2 4 5 5 2 7
3000 2 4 6 5 2 7
3500 2 4 6 5 2 6
4000 2 4 6 5 2 6
From Figure 3, Table 4a and Table 4b we can see that in the CaSiO3 bulk model, the
coordination numbers of couplings Si-Si, Si-O, O-O, Si-Ca, O-Ca and Ca-Ca are 2, 4, 3, 5, 6 and
6, respectively. When temperature increases, the coordination numbers of couplings Si-Si, Si-O,
Si-Ca and O-Ca remain unchanged and only couplings O-O and Ca-Ca change insignificantly (see
Table 4a). When temperature increases, the coordination number density of couplings Si-O and
O-O decreases, the coordination number density of coupling Si-Ca increases and the coordination
number density of couplings Si-Si, O-Ca and Ca-Ca decreases and then increases steadily (see
Table 4b). Thus, the influence of temperature on the microstructure of the CaSiO3 bulk model
can be explained by the heterogeneity of the model. When temperature increases, the size and the
energy of the samples increase; the length of couplings O-O, Ca-Ca and Si-Ca changes and that
leads to the change in microstructure of the samples.
The simulated results for the coordination number of couplings Si-O and O-Ca are shown
in Figure 4a, Figure 4b and Table 5.
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Influence of temperature on the microstructure and phase transition of a CaSiO3 bulk model
Table 4b. The coordination number density of the CaSiO3 bulk model
at different temperatures
T (K) Si-Si Si-O O-O Si-Ca O-Ca Ca-Ca
500 359.45 979.28 1180.30 225.79 1049.49 251.42
1000 352.63 991.79 908.77 235.59 1018.82 243.66
1500 386.48 990.41 870.84 244.64 1033.29 248.89
2000 360.83 980.27 854.94 247.28 1070.13 248.72
2500 368.82 962.79 790.95 252.63 1149.03 242.95
3000 334.67 929.57 760.55 250.50 1180.81 244.63
3500 341.59 877.95 713.52 276.13 1194.51 258.43
4000 329.58 812.79 689.38 260.50 1163.57 235.58
Figure 4a. The coordination number of the coupling SiOx
with x = 4, 5, 6 and couplings Si-Si through O
Figure 4b. The coordination number of the coupling CaOx
with x = 4, 5, 6, 7, 8, 9 and couplings Ca-Ca through O
We can see from Figure 4a, Figure 4b and Table 5 that the main coordination numbers
of couplings Si-O are SiO4, SiO5, SiO6 and couplings Si-Si through O (Figure 4a). Similarly,
the main coordination numbers of couplings O-Ca are CaO3, CaO4, CaO5, CaO6, CaO7, CaO8,
CaO9 and couplings Ca-Ca through O (Figure 4b). When temperature increases, the coordination
number SiO4 increases, then decreases, and it is discontinued between 1500 K and 2000 K; the
coordination number SiO5 decreases, then increases, and it is discontinued between 1500 K and
2000 K. Coordination number SiO6 appears only at T = 500 K, 2000 K and 3000 K. Coordination
number CaO3 appears and increases gradually from 2000 K. Coordination number CaO4 appears
93
Nguyen Trong Dung, Nguyen Thi Ha and Nguyen Chinh Cuong
and increases gradually from 1500 K. Coordination number CaO5 increases gradually from 500 K.
Coordination number CaO6 decreases gradually from 500 K to 2000 K and is discontinued from
2000 K to 2500 K then decreases gradually from 2500 K. Coordination number CaO7 decreases
gradually from 1000 K. Coordination number CaO8 increases gradually from 500 K to 1500 K,
it is discontinued from 1500 K to 2000 K, and then it decreases gradually from 2000 K to 4000
K. Coordination number CaO9 increases gradually from 500 K to 1000 K, it is discontinued from
1000 K to 1500 K, and it then increases gradually from 1500 K to 4000 K (see Table 5). These
results indicate that the couplings are discontinued in the temperature range of 1500 K to 2000 K.
The question is: Is this the phase transition temperature range of the model?
Table 5. The atoms with coordination numbers of couplings Si-O, O-Ca
at different temperatures
T (K) 500 1000 1500 2000 2500 3000 3500 4000
SiO4 3803 3853 3861 3805 3778 3662 3452 3275
SiOx SiO5 148 72 66 116 182 269 428 417
SiO6 7 - - 7 - 14 - -
CaO3 - - - 8 28 63 127 281
CaO4 105 - 148 290 372 604 722 916
CaO5 882 909 928 1207 1368 1455 1615 1607
CaOx CaO6 2096 2031 1883 1985 1929 1769 1607 1474
CaO7 1864 1895 1867 1709 1489 1275 1138 878
CaO8 883 935 1887 699 671 570 337 255
CaO9 245 168 191 136 147 110 90 70
To answer this question, which is the cause for the change in microstructure, we investigate
the angle distribution of the coordination numbers SiO4, SiO5 and SiO6 and CaO3, CaO4, CaO5,
CaO6, CaO7, CaO8 and CaO9. These results are shown in Table 6.
Table 6. The angles of couplings SiOx, CaOx
in the CaSiO3 model at different temperatures
T (K) 500 1000 1500 2000 2500 3000 3500 4000
SiO4
(degree)
105 105 105 105 105 105 105 100
SiO5
(degree)
90 90 90 90 90 90 90 90
SiO6
(degree)
- - - - - - 85 85
CaO3
(degree)
- - - 105 105 110 105 95
CaO4
(degree)
85 85 90 90 90 90 90 90
CaO5
(degree)
80 80 85 85 85 90 85 90
CaO6
(degree)
80 80 80 80 85 85 85 85
94
Influence of temperature on the microstructure and phase transition of a CaSiO3 bulk model
CaO7
(degree)
80 80 80 80 80 80 55 55
CaO8
(degree)
75 75 80 80 80 80 50 50
CaO9
(degree)
50 50 50 50 50 50 50 50
The results in Table 6 show that the angle distribution of coordination numbers SiO4, SiO5,
SiO6 and CaO3, CaO4, CaO5, CaO6, CaO7, CaO8 and CaO9 changes insignificantly. The angle
distribution is entirely consistent with the coordination number distribution (see Table 5). This
indicates that the change in microstructure is due to the change of couplings O-O and Ca-Ca, and
the coordination number distribution and angle distribution of SiO4, SiO5, SiO6 and CaO3, CaO4,
CaO5, CaO6, CaO7, CaO8, CaO9. To determine the phase transition temperature of the CaSiO3
bulk model, we studied the relation between the temperature and the energy of the samples. These
results are shown in Table 7 and Figure 5.
Table 7. The temperature and the energy of the CaSiO3 bulk model
T (K) 500 1000 1500 2000 2500 3000 3500 4000
Energy
(eV)
-70022.8 -69376.2 -68697.4 -67892.6 -67053.3 -66159.9 -65293.2 -64354.9
Figure 5. The phase transition of the CaSiO3 bulk model
The results in Table 7 and Figure 5 show that when temperature increases, the size of the
bulk model increases (see Table 7). When temperature increases from 500 K to 1500 K and from
2000 K to 4000 K, the energy of the samples increases linearly. The intersection between the two
linear lines is (1837; - 68214) (see Figure 5). This shows that the phase transition temperature of
the model is 1837 K. This result is completely consistent with Refs. [14, 15]. We can see from the
above results that the influence of temperature on the microstructure and phase transition of the
model is significant.
95
Nguyen Trong Dung, Nguyen Thi Ha and Nguyen Chinh Cuong
3. Conclusion
This study examines the influence of temperature on the microstructure and the phase
transition of a CaSiO3 bulk model. The following results have been obtained:
Successfully establish the CaSiO3 bulk model at T = 500 K, 1000 K, 1500 K, 2000 K,
2500 K, 3000 K and 4000 K using the Molecular Dynamics method with Born–Mayer pair
interaction potential and periodic boundary conditions. The results obtained are entirely consistent
with Ref. [12].
Determine that the CaSiO3 bulk model at T = 500 K, 1000 K, 1500 K, 2000 K, 2500 K,
3000 K and 4000 K has a cubic shape and a nanometer size.
Confirm the significant influence of temperature on the microstructure of the CaSiO3 bulk
model. The main cause for this is the size effect. When temperature increases, the size and the
energy of the model increase and the coordination number of the atoms (molecules) decreases.
Determine that the phase transition temperature of the CaSiO3 bulk model is 1837 K. This
result is completely consistent with the results in Refs. [14, 15].
Confirm the significant influence of temperature on the microstructure and phase transition
of the CaSiO3 bulk model.
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