4-nm liquid film: the structuralized liquid film
Figure
3 shows snapshots of molecular dynamics simulations perpendicular to the a-axis of forsterite using the visualization software VESTA (Momma and Izumi
2011). The liquidus temperature appears to be 1,873 K in Figure
3 because the crystal structure of the forsterite is altered. However, equilibrium is not achieved in a 4.5-ns run at 1,873 K (Figure
1c). The liquidus temperature of the simulated system is approximately 150 K lower than expected from the phase diagram (Jung et al.
2004). Figure
4a shows a distribution of elements perpendicular to the (010) surface of forsterite at 1,773 K. Figure
4b gives a distribution of Q-species of Si atoms perpendicular to the forsterite surface at 1,773 K. The threshold length of Si-O bonding for calculating the Q-speciation is 0.22 nm. The altered crystal surfaces exist at z = 2 to 2.5 nm and 7 to 7.5 nm. The abundance of Mg atoms is higher than that of Si atoms in this region. The nesosilicate structure is disturbed, and 20% to 40% of the SiO4 tetrahedra form Si-O-Si bridging in this region. In the near crystal surface, the Si-rich layer exists in the liquid domain. The Mg-rich layer exists on the inside of Si-rich layer. Figure
5 shows a distribution of the direction of the Si-O-Si bridging perpendicular to the forsterite surface as a square of the cosine between the z-axis and the vector Si-Si calculated as an inner product. The orientation of the Si-O-Si bridging in the sandwiched liquid film is clearly parallel to the crystal surface. Figure
6b gives a distribution of the two-dimensional self-diffusion coefficients of oxygen parallel to the forsterite surface, and Figure
6c shows the calculated viscosities determined using the Einstein-Stokes relation with the effective radius obtained by Lacks et al. (
2007). The two-dimensional self-diffusion coefficients of oxygen are consistently lower than that of the bulk MgSiO3 liquid (dotted line) by an order of magnitude. The two-dimensional self-diffusion coefficients of oxygen are strongly related to the Mg/Si ratio of the layer to which the oxygen atoms belong (Figure
6a). In the high Mg/Si ratio layer, the self-diffusion coefficients of oxygen show higher values than those in the low Mg/Si ratio layer.
16-nm liquid film
Figure
7 shows snapshots of molecular dynamics simulations perpendicular to the a-axis of forsterite. The volume of the melting part increased at 1,773 K, and the system is completely melted at 1,873 K. Figure
8a shows the distribution of elements perpendicular to the (010) surface of forsterite at 1,673 K. Figure
8b gives the distribution of Q-species perpendicular to the forsterite surface at 1,673 K. The altered crystal surfaces exist at z = approximately 3 and 20 nm. The nesosilicate structure is altered, and approximately 20% of the SiO4 tetrahedra formed Si-O-Si bridging in this region. In the near crystal surface, the Si-rich layer exists in the liquid domain. In the domain between the disturbed crystal surface and the Si-rich layer, Si-O-Si bridging is formed, and Q1 becomes the dominant species. The Mg-rich layer exists on the inside of the Si-rich layer. The composition of the sliced layer of the MgSiO3 liquid is heterogeneous and oscillates, whereas the sliced layer is distant from the crystal surface (Figure
8a). Figure
9 shows the distribution of the direction of the Si-O-Si bridging perpendicular to the forsterite surface. The orientation of the Si-O-Si bridging in the sandwiched liquid film is clearly parallel to the crystal surface. However, the orientation in the disturbed crystal surface is strongly perpendicular to the crystal surface. Figure
10b gives the distribution of two-dimensional self-diffusion coefficients of oxygen parallel to the forsterite surface, and Figure
10c shows the calculated viscosities. The two-dimensional self-diffusion coefficients of oxygen are all slightly lower than those of bulk MgSiO3 liquids (dotted line). The two-dimensional self-diffusion coefficients of oxygen are strongly related to the Mg/Si ratio of each layer in which the oxygen atoms belong in the liquid region. In the high Mg/Si ratio layers, the self-diffusion coefficients of oxygen show larger values than in the low Mg/Si ratio layers (Figure
10a). However, coefficients in altered crystal surface regions are lower than those in liquid regions, in contrast to the case of the 4-nm liquid film.
Effect of film thickness
We show the result of simulation of the system containing films of 7- and 28-nm thickness. The layered structure is also found in this system. However, the global decrease of self-diffusion coefficients is much smaller than the interface system with the 4-nm liquid film (Figure
11). Figure
12 shows the distribution of the Mg/Si ratio, elements, and the two-dimensional self-diffusion coefficients of oxygen of the system with a 28-nm liquid film. The layered structure is similar to the case of the 16-nm liquid film. The two-dimensional self-diffusion coefficients of oxygen in the middle part (z = 12 to 23 nm) of the liquid films are the same as those of bulk liquids. This means that the MgSiO3 liquid film sandwiched by the forsterite (010) surface acts like a bulk liquid in the region 7 nm apart from the crystal surface. Figure
13 shows the calculated viscosities for various film thicknesses. The average viscosity approaches the viscosity of a bulk liquid with increasing film thickness. The average viscosity in the liquid films of 28-nm thickness is nearly the same as that of a bulk liquid. The average viscosity in the thin (<5 nm) liquid film is much higher than that of a bulk liquid.
General discussion
The layered structure that exists in the liquid films in these simulations is caused by the difference in bond strength between Si-O semi-covalent bonds and Mg-O ionic bonds. Si-O-Si bridgings are excessively formed between the altered crystal surface and the Si-rich layer because the strength of Si-O bonding is much higher than that of Mg-O bonding. The excess formation of Si-O-Si bridging in these compositions produce free oxygen atoms. Formation of Si-O-Si bridging depletes the non-bridging oxygen as an acceptor of magnesium atoms and concentrates silicon atoms. The excess free oxygen atoms concentrate magnesium atoms. Consequently, an Si-rich layer is formed in the nearest surface region of the liquid domain, and an Mg-rich layer is formed on the inside of the Si-rich layer on the liquid film (Figures
6a and
10a). The Q-speciation reflects the Mg/Si ratio on the Si-rich layer because Q4 species increase in the case of the 4-nm liquid film. However, the Q-speciation does not reflect the Mg-rich layer. The abundance of Q0 is too small in the case of the 4-nm film (Figure
4), meaning free oxygen atoms are produced. Clustering of the -Si-O- network also occurs. The Q-speciation does not reflect the Mg/Si ratio because the -Si-O- corner shared network makes the cluster. The clustering and layering weaken with increasing film thickness. The weakening can be explained by the degree of freedom in the film structure. The excess free oxygen and magnesium atoms from altered surfaces could not be accepted in the liquid film in the case of a thin film because the degree of freedom in the film structure is small. However, they can be accepted in a liquid film with increasing film thickness. Consequently, the degree of clustering decreases with increasing film thickness (Figures
6,
10,
11 and
12).
The layered structure is also reported by Gurmani et al. (
2011). They investigated forsterite-MgSiO3 liquid with a film thickness of up to 8 nm by the classical molecular dynamics method. However, the contrast of composition and decrease of self-diffusion coefficients by the interface was much smaller than in our study. The difference between Gurmani et al. (
2011) and our studies is caused by the area of the surface cross section. Horbach et al. (
1996) reported the finite-size effects in simulations of silicate glass. According to their study, the system should contain more than 8,000 atoms (4 to 5 nm for each side of the periodic cell) to avoid the size effects. Gurmani et al. (
2011) simulated the same system with approximately 2-nm length sides for the cross section of the interface. The length of the cross section of our simulation is approximately 5 nm to avoid the size effects reported by Horbach et al. (
1996).The structure of the altered crystal surface seems to be different because of the thickness of the liquid films. The concentration of magnesium in the altered crystal surface with the 4-nm liquid film is much higher than that with the 16-nm liquid film (Figures
4a and
8a). In addition, the orientation of Si-O-Si bridging in the altered crystal surface with the 16-nm liquid film is strongly perpendicular to the crystal surface compared with the surface with the 4-nm liquid film (Figures
5 and
9). These differences might be explained by the degree of freedom of the configuration of the liquid films. There might be no structural flexibility in the liquid domain to accept the excess magnesium atoms in altered crystal surfaces because there are only five layers in the 4-nm liquid film. In contrast, the concentration of magnesium in the altered crystal surface with the 16-nm liquid film is lower than that with the 4-nm liquid film because of the structural flexibility of the liquid film. The surface becomes well-ordered by acceptation of magnesium in liquid and disordered by existence of excess magnesium atoms.
The two-dimensional self-diffusion coefficients parallel to the crystal surface are dependent on their distance from the altered crystal surface. Those coefficients show larger values in Mg-rich layers and smaller values in Si-rich layers. The regional dependences are simply explained by the composition of the sliced layer. Self-diffusion coefficients of network-forming elements in binary silicate liquids depend on their SiO2 contents (Keller et al.
1982; Keller and Schwerdtfeger
1979). Self-diffusion coefficients of Si and O atoms decrease with increasing SiO2 contents in bulk silicate liquids because of the difference in bond strength between Si-O covalent bonds and ionic bonds. Consequently, the self-diffusion coefficients of oxygen atoms decrease in Si-rich layers and increase in Mg-rich layers. The two-dimensional self-diffusion coefficients of oxygen in the 4-nm liquid film are lower than those in the 16-nm liquid film by an order of magnitude (Figures
4b and
8b). The difference in the self-diffusion coefficients might be explained by the structural flexibility of liquid films. The variety of Mg/Si ratios in the 4-nm liquid film is larger than that in the 16-nm liquid film. The degree of freedom of the configuration in the 4-nm liquid film is lower than that in the 16-nm liquid film. The liquid film is strongly structured by the crystal surface in the case of thin liquid films. Consequently, the flexibility of motion of each atom in the 4-nm liquid film is lower than that in the 16-nm liquid film, and mobility is decreased by the lower flexibility.
The oxygen diffusivity is related to shear viscosity by the Einstein-Stokes relation. The Einstein-Stokes relation is stated as
where D is the self-diffusion coefficient, k is the Boltzmann constant, T is the temperature, r is the radius of diffusing particles, and η is the shear viscosity. This equation has been used with remarkable success in a variety of silicate liquids to relate diffusivity of network forming atoms and shear viscosity (e.g., Oishi et al.
1975; Dunn
1982; Shimizu and Kushiro
1984; Rubie et al.
1993; Lacks et al.
2007). Applying the Einstein-Stokes relation to the results of this study, the viscosity of thin film silicate liquid might be considered to have higher viscosity than bulk liquid by 1 order of magnitude (Figure
6c). In the 16-nm film liquid, the viscosity of the liquid is slightly lower than that of bulk liquid in the near surface region, the same as with bulk liquid in the Mg-rich layer, and oscillates with distance from the crystal surface (Figure
10c).
Implications
The melt segregation velocity in partially molten rock is inversely proportional to the melt viscosity (McKenzie
1989). In texturally equilibrated partially molten rocks, melts reside at the grain corners and edges (see Figure one of Zhu and Hirth
2003). A melt channel exists on the grain edge, and its cross-sectional area is determined by the dihedral angle, the melt fraction, and grain size. Von Bargen and Waff (
1986) showed that the minimum channel cross-sectional area of partially molten rock consists of tetrakaidecahedron grains of varying melt fraction and dihedral angle θ. We calculated the viscosity at melt channels using the results of our simulations (Figure
13) and geometrical calculation by von Bargen and Waff (
1986). Figure
14 shows the logarithm of normalized viscosity of melts in melt channels versus melt fraction with two grain sizes and two dihedral angles. Waff and Bulau (
1979) showed that the distribution of dihedral angles in partially molten ultramafic rocks is between 30° and 47°. The viscosity becomes higher than bulk liquids by 1 order of magnitude at lower melt fractions and 1-μm grain. However, viscosity seems to be constant with varying melt fractions in the case of 10-μm grain size. The effect of the silicate crystal-liquid interface is negligible in melt flow in the mantle conditions, postulating the textural equilibrium, because the range in grain size of the mantle is 1 to 50 mm (Faul and Jackson
2005). However, a melt film of 0.6- to 3.0-nm thickness was found in deformed partially molten olivine-orthopyroxene rocks (de Kloe et al.
2000). Consequently, the crystal-liquid interface can affect the melt flow in deformed samples.