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ＡｂｉｎｉｔｉｏＭｏｌｅｃｕｌａｒ-ＤｙｎａｍｉｃｓＳｔｕｄｙｏｆＳｔｒｕｃｔｕｒａｌａｎｄＥｌｅｃｔｒｏｎｉｃ

Ｐｒｏｐｅｒｔｉｅｓ　ｏｆ　Ｌｉｑｕｉｄ　ＭｇＳｉＯ３　ｕｎｄｅｒ　Ｐｒｅｓｓｕｒｅ

ＲｙｏＹＯＳＨＩＭＵＲＡ、ＳａｔｏｓｈｉＯＨＭＵＲＡａｎｄＦｕｙｕｋｉＳＨＩＭＯＪＯ Department of Physics, Kumamoto University, Kumamoto 860-8555

(Received September 30, 2010)

The structural and electronic properties of liquid MgSiC>3 under pressure are investigated by ab initio molecular-dynamics simulations. At ambient pressure, most Si atoms have the same coordination even in the liquid state as in the crystalline phase, i.e., each Si atom is bonded to two bridging oxygens twofold-coordinated to Si, and two nonbridging oxygens onefold-coordinated to Si. It is found that the structural defects, such as fivefold- or threefold- coordinated Si atoms, are always formed with the rearrangement of Si-0 covalent bonds in the atomic diffusion processes. The population analysis clarifies that maximum of the diffusivity in the pressure dependence is originated from the increasing of the number of defects under compression.

§1. Introduction

The structural and dynamic properties of liquid silicates have been attracted the interest of many scientists because of the importance in the earth's mantle. In addition, the magnesium-rich silicate minerals are the primary constituents of the

earth mantle,1) and are also present as major phases in chondrite meteorites and interplanetary dust.2)'3)

In ambient conditions, crystalline magnesium silicate (MgSiOa) has a SiC>4 tetra- hedral unit in which each silicon atom is coordinated to four oxygen atoms with

single bonds.4) A high-pressure phase appears at a pressure of about 25 GPa under

compression. There is a SiC>6 octahedral unit in this structure. Each Mg atom is

coordinated to 12 oxygens with bond lengths ranging from 2.014 to 3.12 A and O- Mg-0 angles ranging from 52.19° to 70.89°.5) The ideal value of an O-Mg-0 angle

for a regular MgOi2 polyhedron is 60°. The small variations in the Si-0 distances and the small deviation of the O-Si-0 angles from 90° (88.5° - 91.5°) show that the SiO6 octahedron is much more regular than the MgOi2 polyhedron.

Since it would be difficult to clarify the diffusion mechanism in liquid MgSiC>3 experimentally, theoretical studies would be needed. However, only a few theoretical

studies of atomic diffusion have been reported so far,6) while their crystalline phases have been extensively investigated.7) Molecular dynamics (MD) simulations6) with

empirical interatomic potentials have been used for the diffusion mechanism of oxy gen ions in Me2OSi02 (Me = Li, Na, K, Cs) melts at 2000 K. Although effects of alkali ions have been examined on the structure of transient complexes temporarily formed in alkali silicate systems, the effects of electronic properties are still unclear.

In this paper, we carry out ab initio MD simulations of liquid MgSiC>3 under

pressure. We obtain the atomic forces quantum mechanically from the electronic-

structure calculations. The purpose of our study is to clarify the microscopic mech-

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294 R. Yoshimura, S. Ohmura and F. Shimojo

Table I. Pressure P, temperature T and number density p used in this study.

P (GPa) 0.1 9.5 21.1 48.9 97.8 149.1 198.2

T(K) 2500 2900 3300 4500 5500 6000 7000

9 (A"3) 0.068 0.086 0.099 0.113 0.132 0.146 0.156

anism of atomic diffusion in the liquid state from first principles. We are unaware of investigations of the diffusion mechanism in liquid silicates based on first-principles theories though some first principles studies of liquid MgSiC>3 under pressure have already been reported.8)'9) We discuss how Si-0 bonds are exchanged, accompa nying the diffusion of atoms in detail. Such findings are important not only for imderstanding the equilibrium properties of liquid MgSiC>3 but also for properties of basic liquids in the earth's mantle.

§2. Method of calculation

In our MD simulations, a system of 120 (24 Mg + 24 Si + 720) atoms in a cubic supercell is used under periodic boundary conditions. The equations of mo

tion for atoms are solved via an explicit reversible integrator10) with a time step

of At = 1.2 fe. The atomic forces are obtained from the electronic states cal

culated by the projector-augmented-wave method11)'12) within the framework of density-functional theory. The generalized gradient approximation13) is used for

the exchange-correlation energy. The plane wave cutoff energies are 30 and 250 Ry for the electronic psedo-wave-functions and the pseudo-charge-density, respectively.

The energy functional is minimized using an iterative scheme.14)'15) Projector func

tions are generated for the 3s, 3p and 3d states of Mg and Si, and the 2s and 2p states of O.

For each pressure (0 - 200 GPa), the temperature is set to be the melting point plus about 500 K. To determine the density of the liquid state under compression,

constant-pressure MD simulation16) is performed for 2.4 ps at given pressure. The

static and diffusion properties are investigated by MD simulations in the canonical

ensemble.17)'18)

§3. Results and discussion

3.1. Structure factor

Figure 1 shows the pressure dependence of the structure factor of liquid MgSiC>3.

The solid and dashed lines display the X-ray and neutron structure factors, Sx(k)

and Sn(fc), respectively. Sx(fc) is obtained from the partial structure factors 5Q^(fc),

shown in Fig. 2, with the X-ray scattering factors, and Sn(k) is calculated from

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Sap{k) with the neutron-scattering lengths. At pressures below 10 GPa, clear peaks

exist at about k = 2.0 and 4.4 A""1 in the profiles of Sx{k) and Sn(k). Note that the

overall profile of Sx(k) at 0 GPa is consistent with the experimental Sx(k) of the

vitreous state.19) While only <Sn(A;) has another peak at 3.0 A""1 at lower pressures,

the corresponding peak grows in Sx(k) at higher pressures. With increasing pressure,

the peak at about A; = 2.0 A"1 in both Sx(k) and Sn(k). A remarkable feature exhibited in Sx(k) and Sn(k) is the small peak at about A; = 1.2 A"1 at 0 GPa, which means that the spatial correlation exists on an intermediate distance ~ 6 A.

This peak disappears under pressure.

3-

— X-ray(simulation, liquid) --- Neutron(simulation, liquid)

• X-ray(experiment, glass)

200 GPa (+6.0) ^{— Mg-Mg} _{- Si-Si}

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0 2 4 6 8 10 0 2 4 6 8 10

Fig. 1. Pressure dependence of the total struc- Fig. 2. Pressure dependence of the partial struc

() (d) S

ture factors Sap(k). (a) 5m8m8 (solid), (dashed) and Soo (dotted), (b) SMgSi (solid), SMgo (dashed) and Ssio (dotted).

ture factors. The solid and dashed lines indi- cate the X-ray and neutron structure factors, Sx(fc) and Sn(fc), respectively. The open cir- cles at 0 GPa are the experimental values for glass MgSiO3. The curves are shifted verti cally as indicated by the figures in parenthe

ses.

Figure 2 shows the Ashcroft-Langreth partial structure factors Sap(k). The pres sure dependence of the profiles of Sx(k) and Sn(k) is well understood from Sap{k).

At 0 GPa, Ssisi(fc) has Peaks at about k = 1.2 A"1, which gives the corresponding

peak in Sx(k) and Sn(k). At lower pressures, SMgMg(fc), SSiSi(*0 ^d Soo(k) have

the peak at about k = 2.3 A"1. Because of the cancellation due to the existence

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296 R. Yoshimura, S. Ohmura and F. Shimojo

of a negative dip in Sugo(k) and Ssio(0> no clear Peak appears in Sx(k) aroimd k = 2.3 A"1. With increasing pressure, all peaks shift to larger k in SMgMg(0>

SsiSi(fc) and Soo(k), whereas the negative dip changes only a little in SMgo(fc) and Ssio{k). Therefore, the peaks of Sx(k) and Sn(k) grow at the corresponding k when the pressure increases. At higher pressure, the main peak of SuSMg(k) is as high as that of SsiSi(k).

3.2. Diffusivity

The diffusion coefficients Da for a = Mg, Si, and 0 atoms are defined as

(3.1) where r»(*) is the position of the i-th atom at time t. Figure 3 shows Da as a function of pressure. Whereas Ds» Do and DMg have maxima at 50 GPa, DMg has a minimum at 20 GPa. While DMg has larger values than Do under pressures up to about 10 GPa, Do becomes comparable with £>Mg at about 20 GPa.

P (GPa)

Fig. 3. Pressure dependence of the diffusion coefficients Da for a = Mg (circles), Si (squares) and O (diamonds) atoms.

3.3. Mechanism of atomic diffusion

It is found, from the distribution of coordination numbers, that Si atoms are

mainly coordinated to four O atoms; two O atoms bridge two adjacent Si atoms,

and the other two O atoms are coordinated to only one Si atom, as in the crystalline

phase even though atoms diffuse in the liquid state. However, it is unclear how Si-0

bonds are exchanged with the diffusion of atoms in the liquid state while retaining

the covalent bonds. To clarify the mechanism of atomic diffusion, we investigate the

time evolution of the bonding nature by utilizing population analysis. The bond-

overlap population Ojj(t), which gives a semiquantitative estimate of the strength of

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evolution of the bond-overlap populations between atoms. The number of structural defects increases gradually with compression. Therefore, atomic diffusion events re lated to overcoordinated Si occur easily. As a result, Dsi and Do increase up to 50 GPa.

Acknowledgments

The authors thank Research Institute for Information Technology, Kyushu Uni versity for the use of facilities. The computation was also carried out using the computer facilities at the Supercomputer Center, Institute for Solid State Physics, University of Tokyo.

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