Abstract. The diffusion mechanism in liquid MgO under pressure up to 25 GPa at
a temperature of 3800K have been studied using molecular dynamics simulation
(MD). The results show that each atom undergoes a series of stages while
associated with the unchanged structural unit MgOx or OMgy. The diffusivity
strongly depends on the rate of transition in MgOx → MgOx±1 and OMgy →
OMgy±1. Under low pressure, diffusion proceeds due to the transition with x = 3,
4, 5 and y = 3, 4, 5 but mainly x = 4 and y = 4. Under high-pressure, diffusion in
the sample proceeds due to a transition with x = 4, 5, 6 and y = 4, 5, 6 but mainly
x = 5 and y = 5. Investigating the movement of atoms in liquid MgO shows the
spatially heterogeneous dynamics. The diffusion coefficient of Mg and O atoms is
also examined through mean square displacement. Structural stability (the life time
of basic structural units) is investigated in detail in this work.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 124-133
This paper is available online at
DIFFUSIONMECHANISM IN LIQUID MgO UNDER HIGH PRESSURE
Mai Thi Lan, Nguyen Van Hong and Pham Khac Hung
Department of Computational Physics, Institute of Engineering Physics,
Ha Noi University of Science & Technology
Abstract. The diffusion mechanism in liquid MgO under pressure up to 25 GPa at
a temperature of 3800K have been studied using molecular dynamics simulation
(MD). The results show that each atom undergoes a series of stages while
associated with the unchanged structural unit MgOx or OMgy . The diffusivity
strongly depends on the rate of transition in MgOx → MgOx±1 and OMgy →
OMgy±1. Under low pressure, diffusion proceeds due to the transition with x = 3,
4, 5 and y = 3, 4, 5 but mainly x = 4 and y = 4. Under high-pressure, diffusion in
the sample proceeds due to a transition with x = 4, 5, 6 and y = 4, 5, 6 but mainly
x = 5 and y = 5. Investigating the movement of atoms in liquid MgO shows the
spatially heterogeneous dynamics. The diffusion coefficient of Mg and O atoms is
also examined through mean square displacement. Structural stability (the life time
of basic structural units) is investigated in detail in this work.
Keywords: MD simulation, transition, diffusion mechanism, heterogeneous
dynamics.
1. Introduction
In recent years, some research on microstructure and diffusion mechanisms in
disordered materials has shown that the spatially heterogeneous dynamics (SHD), i.e.
the atom displacement and local structural relaxation, are non-uniformly distributed
through space. Numerous approaches have been employed to study the dynamics in
liquid including the Mode Coupling theory [1] and the Adam-Gibbs theory [2]. However,
the atomic mechanism behind those phenomena has not been successfully identified
in these studies. SHD and the relation between dynamics and diffusion mechanism in
liquids and glassy polymers have been shown in [3-5] and it is also predicted that the
SHD phenomenon also occurs for non-strong (or fragile) liquids (e.g. liquid MgO).
The study of SHD is essential because it provides detailed information on the diffusion
Received August 2, 2012. Accepted September 4, 2012.
Physics Subject Classification: 62 44 01 01.
Contact Mai Thi Lan, e-mail address: lanmt.iep@gmail.com
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Diffusion mechanism in liquid MgO under high pressure
mechanism and phase transitions in disordered material systems. Because it is difficult to
perform experimental studies at a high temperature and pressure, the simulation method
is still an effective tool for investigating the structure and dynamics of liquids and glassy
polymers [6, 7]. Molecular dynamics simulation is one of the most widely used methods
[8]. It can provide more information about physical processes in conditions that other
methods would be impossible or very difficult to apply. While structure and dynamics
of compounds containing Si and Mg in the liquid state (e.g. MgSiO3 and MgSiO4) were
studied extensively using both experimental and simulation methods, studies of liquid
MgO are very limited. Furthermore, the microstructure, dynamics, diffusion mechanism
and relation between microstructure and dynamics as well as diffusion mechanism in
liquidMgO is still not clear. In our study, we used molecular dynamics simulation (MD) to
study diffusion mechanism in liquid MgO at a temperature of 3800K at a pressure ranging
from 0 to 25 GPa. The calculated results and analysis of dynamical properties, spatially
heterogeneous dynamics and diffusion mechanisms in MgO liquids will be reported in
detail in this work.
2. Content
2.1. Calculation method
The molecular dynamics simulation of liquid MgO was done in a cubic box
containing 2000 atoms (1000 Mg and 1000 O) with periodic boundary conditions using
Lewis and Catlow potentials. The form of this potential is:
Uij =
qiqj
rij
+Aijexp(− Bijrij)− Cij
r6ij
here qi = +2 and qj = -2 is the ionic charges of Mg and O. rij is the distance between the
ith and jth atoms.
The Mg-Mg, Mg-O and O-O interaction is described by parameters Aij , Bij , Cij
which can be found in [9]. The initial model is generated by randomly placing 2000 Mg
and O atoms in a simulated cubic box with an experimental density ρ = 3, 58g/cm3 and
then heated to 5000K at ambient pressure and relaxed over 100,000 MD steps to remove
possible memory effects. This model is then cooled to 3800K at a rate of 0.25 K/ps and
relaxed for 20,000 MD steps to obtain a good equilibrium P0 model (3800K at ambient
pressure). Next, the model P0 is compressed to obtain five liquid MgO models, each at
a different pressure. By this way, six models (P0, P5, P10, P15, P20, P25) have been
constructed at 3800K at pressures from 0 to 25 GPa. Afterward, these models are relaxed
in ensemble NPT (constant temperature and pressure) by 100,000 MD steps to reach
the equilibrium. The structural characteristics of the considered models are determined
by averaging over 2000 configurations during the last 10,000 MD steps. To calculate
the coordination number we use cutoff distance RMg−O = 2.92 A˚, the chosen minimum
after the first peak of the pair radial distribution function (PRDF). To study the dynamic
properties as well as the diffusion mechanism in liquid MgO, the obtained models are
125
Mai Thi Lan, Nguyen Van Hong and Pham Khac Hung
relaxed in ensemble NVE (constant volume and energy) over 40,000 MD steps. The mean
square displacement of atoms allows calculation of the diffusion coefficient of Mg, O
atoms via Einstein equation:D = lim
t→∞
6t
, where is mean square displacement
over time t.
2.2. Result and discussion
2.2.1. Diffusion mechanism
Table 1. Structural characteristics of MgO liquid
Model P0 P5 P10 P15 P20 P25
P,GPa 0.08 4.65 9.51 14.79 19.54 25.2
ZMg−O 4.08 4.63 4.83 5.12 5.21 5.57
ZO−Mg 4.08 4.63 4.83 5.12 5.21 5.57
RMg−Mg, A˚ 3.10 3.02 2.96 2.88 2.90 2.80
RMg−O, A˚ 1.84 1.86 1.86 1.86 1.86 1.86
rO−O, A˚ 3.12 3.00 2.96 2.88 2.88 2.88
Mg3 0.20 0.06 0.03 0.01 0.01 0.00
Mg4 0.54 0.39 0.31 0.20 0.17 0.08
Mg5 0.23 0.42 0.47 0.48 0.48 0.39
Mg6 0.03 0.12 0.18 0.27 0.31 0.43
O3 0.20 0.06 0.03 0.01 0.01 0.00
O4 0.53 0.39 0.31 0.20 0.17 0.08
O5 0.23 0.42 0.47 0.48 0.47 0.38
O6 0.03 0.12 0.18 0.28 0.31 0.44
, degree 90 85 85 80 80 80
, degree 90 85 85 85 85 80
Note. rlk: positions of first peak of PRDF; Zlk: the mean coordination number;
Mgx and Oy: the fraction of unitMgOx and OMgy, respectively; P: pressure
Table 2. The variation of coordination number of ith Mg atoms over stages
P, Gpa Stage 1 Stage 2 Stage 3 Stage 4 Stage 5
Zi n Zi n Zi n Zi n Zi n
0.08 4 372 5 81 4 9 3 46 4 134
25.20 6 64 5 30 6 37 5 49 6 81
Note. Zi: coordinated oxygen number; n: MD steps; P: pressure
The microstructure of liquids MgO can be characterized by the radial distribution
function, the distribution of coordination number and bond Mg-O-Mg and O-Mg-O angle
distribution. To calculate this we use the cutoff distance of RMg−O = 2.92 A˚ which is
chosen as a minimum after first peak of the PRDF. From Table 1 and Figure 1 one can see
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Diffusion mechanism in liquid MgO under high pressure
that the PRDF and structural characteristics of liquid MgO are very close to calculated
results in ref [10, 11]. This shows that the constructed models are reliable. Then we
continued in order to clarify the diffusion mechanism in liquid MgO.
To clarify the diffusion mechanism at the atomic level we track the movements and
changes in coordination number of each atom during the simulation. The results show
that diffusion occurs only when there is an exchange of O (Mg) coordination between the
structural units MgOx (OMgy) (x = y = 2, 3, 4, 5, 6) i.e. the transition from structural
unit MgOx (OMgy) stage to structural unit MgOx (OMgy) other stage. To understand
this, we illustrate the exchange of oxygen coordination among units ofMgOx structure in
a short time as given in Table 2, Figure 2 at low pressure (0.08 GPa) and high pressure
(25.20 GPa) (only show 5 stages, about several hundred MD steps).
Figure 1. The pair radial distribution functions of Mg-Mg, Mg-O and O-O at different pressure
Figure 2. The replacement of neighboring atoms by others in network units of MgO4
The while and black circle represent the oxygen and magie;
the dotted-line circle shows the region where neighboring atoms locate
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Mai Thi Lan, Nguyen Van Hong and Pham Khac Hung
Table 3. The lifetime of structural units MgOx and OMgy
Structural units P, Gpa MgO2 MgO3 MgO4 MgO5 MgO6 MgO7
0.08 6.58 202.13 525.22 235.33 29.88 0.85
04.65 0.54 55.21 394.49 420.76 120.37 8.47
MgOx 09.51 0.21 29.57 314.98 467.39 173.97 13.61
014.79 0.05 11.06 205.10 479.43 271.32 32.09
019.54 0.01 7.16 167.81 472.83 311.21 39.70
025.20 0.00 1.36 73.61 379.27 439.53 100.29
P, Gpa OMg2 OMg3 OMg4 OMg5 OMg6 OMg7
00.08 6.64 201.14 526.71 235.01 29.69 0.80
04.65 0.54 54.41 396.00 419.80 120.96 8.13
OMgy 09.51 0.20 29.17 316.35 465.84 174.66 13.51
014.79 0.05 10.78 205.90 477.92 273.07 31.32
019.54 0.02 7.04 168.03 471.72 313.18 38.81
025.20 0.00 1.29 73.02 378.56 442.90 98.44
Note. P: pressure
Table 2 shows that at 0.08 GPa, one Mg of jth atom undergoes 5 stages and
in turn has coordination number 4, 5, 4, 3, 4 and lifetime 372, 81, 9, 46, 134 MD
steps respectively. Specifically as follows, the initial MgO4 consists of one Mg and four
coordinated oxygen atoms (in Figure 2a - labeled as 1, 2, 3, 4). We have a MgO4 unit at
stage 1. This stage exists in 372 MD steps (lifetime), there is oxygen movement inside
the unit (Figure 2b - labeled as 5) so a transition MgO4 → MgO5 occurs, i.e. MgO4
at stage 1 transforms into MgO5 at stage 2 which consists of the 5 coordinated oxygen
atoms, 1, 2, 3, 4, 5. This stage 2 exists in 81 MD steps, and there is an oxygen atom which
leaves the unit of MgO5 at stage 2 (e.g. oxygen 4) so MgO5 at stage 2 transforms into
MgO4 at stage 3 with 4 coordinated oxygen atoms 1, 2, 3, 5. Next, stage 3 exists in 9 MD
steps, one oxygen atom leaves it, and MgO4 at stage 3 transforms MgO3 at stage 4 with 3
coordinated oxygen atoms. This stage 4 exists in 46 MD steps, there is oxygen movement
insite, MgO3 at stage 4 transforms MgO4 at stage 5 with 4 coordination oxygen atoms
and it comes to exists in 134 MD steps. Thus, in a short time, we have all 5 stages of
the Mgjth atom which occurs 4 times in the transition from MgO4 → MgO5, MgO5 →
MgO4, MgO4 → MgO3 and MgO3 → MgO4. Similarly, at high pressure (25.20 GPa), it
undergoes 5 stages and in turn has coordination number 6, 5, 6, 5, 6 and lifetime 64, 30,
37, 49, 81 MD steps, respectively. The exchange of coordinated Mg between OMgy units
(y = 2, 3, 4, 5, 6, 7) are also similar happening. In this work, we track the movements
and changes in coordination number of each atom during 20,000 MD steps. The results
show that each atom undergoes a sequence of stages and that during the simulation time
it exists in that state, its coordination number remains unchanged. Each stage can exist for
a certain period of time. In other words, each stage has a certain lifetime. Table 3 shows
the pressure dependence on the lifetime of structural units.
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Diffusion mechanism in liquid MgO under high pressure
Figure 3. The pressure dependence of the rate of transition
For structural units MgOx (x = 2 - 7), at low pressure, most of time, Mg atoms
have coordination number 3, 4 and 5 (build up structural units MgO3, MgO4, MgO5). The
lifetime of MgO4 is the longest (525.55 MD steps). Conversely, at high pressure, most of
time, Mg atoms have coordination number 5, 6 and 7 (build up structural units MgO5,
MgO6, MgO7). The lifetime of MgO6 is the longest (439.53 MD steps). With increasing
pressure up to 25 GPa, the lifetime of MgO2, MgO3 and MgO4 deceases strongly, and
conversely the lifetime of MgO5, MgO6 and MgO7 increases strongly. For structural units
OMgy (y = 2 - 7), with increasing pressure, the lifetime of OMg2, OMg3 and OMg4
decreases but the lifetime of OMg5, OMg6 and OMg7 increases. From Table 3 we see that
the lifetime of MgOx is longer than one of OMgy (OMgy relates to the structural order
in immediate range). This demonstrates that the short range order is more stable than the
immediate range order. Figure 3 shows the pressure dependence of the transition rate. The
rate of transition vtrans is defined as follows: vtrans = mtrans/n; mtrans is the number of
transitions (1) to (4) occurs in the system for n MD steps.
MgOx →MgOx−1 x = 3, 4, 5, 6 (1)
OMgy → OMgy−1 y = 3, 4, 5, 6 (2)
MgOx →MgOx+1 x = 2, 3, 4, 5 (3)
OMgy → OMgy+1 y = 2, 3, 4, 5 (4)
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Mai Thi Lan, Nguyen Van Hong and Pham Khac Hung
where MgOx means Mg atom has x O atoms as the nearest neighbors, OMgy means O
atom has y Mg atoms as nearest neighbors.
Because the fraction ofMgOx (at temperature T and pressure P) is nearly constant
(fluctuating round about a constant), transitions (1) (2) (and transitions (3) (4)) happen
at the same time. In addition, a transition (1) always combines with a transition (2)
and a transition (3) always combines with a transition (4). From Figure 3 we see that
the rate of transition for (MgOx → MgOx−1) is closed to one of OMgy → OMgy−1
(MgOx → MgOx+1), and this shows that despite the change, the fraction of MgOx and
OMgy units with increasing pressure but the shape of the structural units are unchanged
and independent on pressure. The transition (1) - (4) results in the movement of atoms
and this is the diffusion mechanism in liquid MgO. Figure 3 shows that at low-pressure
the diffusion proceeds due to the transition with x = 3, 4, 5 and y = 3, 4, 5 but mainly x =
4 and y = 4 and at high-pressure the diffusion proceeds due to transition of x = 4, 5, 6 and
y = 4, 5, 6 but mainly x = 5 and y = 5.
Figure 4. The dependence of mean square displacement on number of MD steps (left)
and on number of transition (right)
Figure 4 (left) shows the dependence of mean-squared displacement of all Mg, O
atoms on the number of MD steps at pressure 0.08 and 25.20 GPa and Figure 4 (right)
shows the dependence of mean-squared displacement on number of transitionsMgOx →
MgOx±1 and OMgy → OMgy±1 at pressure 0.08 and 25.20 GPa. The results show
that the dependence of mean-squared displacement is not only linear with the number
of MD steps but also with the number of transitions. This result demonstrates that the
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Diffusion mechanism in liquid MgO under high pressure
diffusion mechanism in liquid MgO is caused only by the transitionMgOx → MgOx±1
and OMgy → OMgy±1 which results in exchanging the O (Mg) atom between MgOx
(OMgy) structural units (see Figure 2). The diffusion coefficient of Mg and O atoms
is also examined through the mean square displacement. Figure 5 shows the pressure
dependence of the diffusion coefficient in liquid MgO. It can be seen that the diffusion
coefficient of an Mg atom is similar to that of an O atom. At pressure 0.08 GPa,
the diffusion coefficient of Mg and O is about 1.26.10−4 cm2/s and 1.23.10−4 cm2/s,
respectively. With increasing pressure up to 25 GPa, the diffusion coefficient of Mg and
O falls to 0.6.10−4 cm2/s and 0.55.10−4cm2/s, respectively.
Figure 5. The dependence of the diffusion coefficient on pressure
2.2.2. Heterogeneous dynamics
To investigate the spatially heterogeneous dynamics, we have calculated the
distribution of the number of transitions MgOx → MgOx±1 over atoms. Figure 6
shows the distribution of transitions over atoms for 20,000 MD steps. This distribution
of transitions has the Gaussian form. The result shows that when pressure increases from
0.08 Gpa to 25.20 Gpa, 1.4% to 1% of Mg atoms undergo 140 transitions (mtrans), 0.4%
to 0.3% ofMg atoms undergo 260 mtrans and 36.2% - 40.1% ofMg atoms undergo mainly
200 mtrans. From this distribution it can be seen that there are units that undergo a very
large number of transitions. Conversely there are units that undergo a very small number
of transitions. In the space in which the number of transition is large, the atoms at that
location have higher mobility and vice versa, i.e. mobile atoms heterogeneous distribution
in the space. Mobile atoms assist their neighbors to become mobile and vice versa. This
distribution is the evidence of the spatially heterogeneous dynamics in liquid MgO.
131
Mai Thi Lan, Nguyen Van Hong and Pham Khac Hung
Figure 6. The distribution of transitions MgOx ←→MgOx±1
3. Conclusion
In this article, we clarify the diffusion mechanism in liquid MgO. The results show
that each atom undergoes a sequence of stages where it presents in unchanged structural
unit MgOx or OMgy. The lifetime of MgOx or OMgy units and the rate of transition
of MgOx → MgOx±1 and OMgy → OMgy±1 depend strongly on pressure. The
dependence of mean-squared displacement is linear with the number of transitions. The
diffusion coefficient of an Mg atom is similar to that of an O atom and it decreases with
increasing pressure. The diffusion proceeds due to the transitionMgOx → MgOx±1 and
OMgy → OMgy±1 lead to the nearest neighbor atom exchange among structural units
MgOx or OMgy. At low pressure, the diffusion mechanism is dominated by transitions
MgO3 ←→ MgO4 and MgO4 ←→ MgO5 (OMg3 ←→ OMg4 and OMg4 ←→ OMg5) (x
= 4 and y = 4). Meanwhile, at high pressure, the diffusion mechanism is dominated by
transitions MgO4 ←→MgO5 and MgO5 ←→MgO6 (OMg4 ←→ OMg5 and OMg5 ←→
OMg6) (x = 5 and y = 5). Distribution of the number of transitions over atoms has the
Gaussian form. It also shows that dynamics in liquid MgO are spatially heterogeneous.
Acknowledgments. This research was funded by the National Foundation for Science and
Technology Development (NAFOSTED) of Vietnam. Grant number: 103.99-2011.22.
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