Effective atom-atom forces [panel (a)] and corresponding potentials [panel (b)] in liquid SiO2 generated through the FM method as functions of interatomic separation: O–O (black), Si–O (red), and Si–Si (green). The dashed line corresponds to the model without the repulsive shoulder (FM-ns model). In panel (b) the dotted (FM-ws model) and dotted-dotted-dashed lines indicate the variation of the O–O repulsion in FM with block averaging along reference trajectories as discussed in the text. In panel (c) a comparison of the BKS (solid), Pedone (solid/circles), and FM (dashed) potentials is given. Due to the aphysical behavior of the BKS exp-6 at small interatomic separations, we modified the function to generate the BKS curve shown in this figure as follows: For r < r infl , where r infl is the position of the inflection point [u ″(r infl ) = 0] on the repulsive wall of the original BKS exp-6 potential, the curve is described by an exponential function whose energy and energy first derivative are the same as the original BKS exp-6 function at r infl . The vertical dashed line marks the location of the repulsive shoulder in the O–O FM potential.
Panel (a): Errors ΔF vs. time. Instant values of ΔF are shown for oxygen ions (black) and silicon ions (cyan). White lines show corresponding running time averages 〈ΔF〉 t . Panel (b): Distributions of errors P(ΔF) for oxygen (solid/thin), silicon (dashed), and total (solid/thick).
King's ring size distribution [panel (a)] and all-particle RDF, g(r), (solid) with corresponding running integration number, , (dashed) [panel (b)] for the (FM-l) (red), (FM-h) (blue) structures using the FM model and for the (P-l) structure using the Pedone model (green).
Density vs. pressure at T = 298 K (solid lines) and along the Hugoniot locus (dashed lines) from simulations of (FM-l) structure using the FM (cyan), FM-ns (green) [panel (a)], Pedone (blue) [panel (b)], and FM-ws (magenta) [panel (c)] models. Simulations were carried out at the same pressure points (given in the text) as marked by filled circles on the cyan line. Experimental EOS obtained by cold compression (black squares) and by shock compression (red circles) is from Refs. 6 and 65, respectively. Inset to panel (c) compares the 298 K (solid) and Hugoniot (dashed) EOS from simulations of the (FM-l) (cyan) and (FM-h) (black) structures using the FM model.
Temperature along the Hugoniot locus, T Hg(P), simulated using the FM (red), FM-ns (green), and Pedone (blue) models at the same pressure points as in Fig. 4.
Pressure dependence of g X(r) at T = 298 K [panel (a)] and along the Hugoniot locus [panel (b)] for the (FM-l) structure using the FM model. The inset is a magnification of the region of the first peak. Arrows show the direction of changes in peak positions with pressure.
Panel (a): T = 298 K (solid) and shocked (dashed) O–O coordination number within the shell r < 0.153 nm, , for the (FM-l) structure using the FM model. The inset is an enlargement of the low pressure region. Panel (b): T = 298 K (solid) and shocked (dashed) Si–O coordination number, , from FM (red, filled squares), FM-ns (blue, empty squares), and Pedone (green, empty diamonds) models. Experimental cold compression data obtained by X-ray adsorption measurements (black, filled squares) are from Refs. 7 and 9.
Comparison of the probability density distributions of O–Si–O angles at T = 298 K and different pressures by the FM model for the (FM-l) structure [panel (a)] and the Pedone model for structure (P-l) [panel (b)]. The panels (c) and (d) show similar comparison for Si–O–Si angle distributions.
g X(r) (solid) and g OO(r) (dashed) from T = 298 K simulations at P = 74 GPa for the (FM-l) structure using FM (blue) and FM-ns (red) models.
Compression ρ0/ρ, where ρ0 is ambient density vs. pressure curves for the (FM-l) structure decompressed using FM (solid) and FM-ns (dashed) models, which was previously compressed with the FM model under cold (diamonds, blue) and shock (red, triangles) conditions. Densities in samples compressed under cold (solid black, empty circles) and shock (solid red, empty squares) conditions are shown for a comparison.
g X(r) for the decompressed (FM-l) sample from selected simulations using the FM model from Fig. 10. Panel (a): For samples compressed at pressures 1, 2, 3, 5, and 8.118 GPa under cold conditions. Panel (b): For samples compressed at pressure 74 GPa under cold (solid green) and shock (dashed red) conditions. The cyan lines show the original g X(r) in the sample compressed at 74 GPa under cold (solid cyan) and shock (dashed cyan) conditions. In both panels the ambient structure is shown by the black solid line.
Coefficients of the least-squares fit for the BLYP FM SiO2 force field f αβ(r) using the expansion in Eq. (2) with n max = 16. Atomic units are used. At small separations r < r core , the f αβ(r) is extrapolated as . The following core radii are used: a.u., a.u., and a.u. The cutoff of 0.7038 nm must be applied to this expansion. The original numerical forces and potentials are provided in the supplementary material.66
Structural parameters, a, b, c, β (length in nm and angle in deg), density ρ (kg/m3), cohesive energy per SiO2 E c (eV), elastic constants C ij (GPa), and bulk modulus B (GPa), for silica polymorphs.
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