Phase diagram of silica from computer simulation

We evaluate the phase diagram of the “BKS” potential [van Beest, Kramer, and van Santen, Phys. Rev. Lett. 64, 1955 (1990)], a model of silica widely used in molecular dynamics (MD) simulations. We conduct MD simulations of the liquid, and three crystals ($\beta$-quartz, coesite, and stishovite) over wide ranges of temperature and density, and evaluate the total Gibbs free energy of each phase. The phase boundaries are determined by the intersection of these free energy surfaces. Not unexpectedly for a classical pair potential, our results reveal quantitative discrepancies between the locations of the BKS and real silica phase boundaries. At the same time, we find that the topology of the real phase diagram is reproduced, confirming that the BKS model provides a satisfactory qualitative description of a silicalike material. We also compare the phase boundaries with the locations of liquid-state thermodynamic anomalies identified in previous studies of the BKS model.

Figure 1

Location of points in the $V\text{−}T$ plane at which we conduct simulations of the liquid (circles), stishovite (squares), coesite (triangles), and $\beta$-quartz (crosses). The large symbols locate reference states $(V_R,T_R)$ at which the entropy of each phase is evaluated directly.

Figure 2

Examples of fitted and interpolated data for the liquid phase: (a) values of $E$ along the $V=6.655{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$ isochore, fitted with a cubic polynomial (line); (b) values of $P$ for $V=6.655{\mathrm{cm}}^3{\mathrm{mol}}^{-1}$, fitted with a quartic polynomial (line); (c) interpolated values of $E$ along the $T=4000\mathrm{K}$ isotherm, fitted with a cubic spline (line); (d) interpolated values of $P$ for $T=4000\mathrm{K}$, fitted with a cubic spline (line).

Figure 3

$E_{anh}\left(T\right)$ for (a) $\beta$-quartz, (b) coesite, and (c) stishovite along their respective reference isochores. In (a) the solid line is the fit to the data given by Eq. (3) with $N_{max}=2$. The fit for $N_{max}=3$ is not visible as it overlaps with the $N_{max}=2$ curve on the scale of this plot. In (b) the fits for both $N_{max}=2$ (solid) and $N_{max}=3$ (dashed) are shown. In (c) the fits for $N_{max}=2$ (solid) and $N_{max}=4$ (dashed) are shown, while the curve for $N_{max}=3$ is not visible as it overlaps with the $N_{max}=4$ curve.

Figure 4

(a) $\Delta G$, the Gibbs free energy relative to that of coesite $\left(C\right)$ for the liquid $\left(L\right),\beta$-quartz $\left(Q\right)$, and stishovite $\left(S\right)$ phases, at constant $P=2\mathrm{GPa}$. The intersections locate points on the stable and metastable coexistence lines that cross this isobar. (b) Potential energy $U$ as function of $V$ at $T=0$ for various crystal phases. Thin straight lines represent “common tangent constructions,” the slopes of which yield the $T=0$ coexistence pressures plotted in (b).

Figure 5

(a) Experimentally determined coexistence lines of silica in the $P\text{−}T$ plane. Stability fields for the stishovite $\left(S\right)$, coesite $\left(C\right),\beta$-quartz $\left(Q\right)$, and liquid $\left(L\right)$ phases are shown. Both stable (solid) and metastable (dashed) coexistence lines are shown. The inset shows the stability fields of cristobalite and tridymite, not considered in this work. Adapted from Ref. 3. (b) Phase diagram of BKS silica in the $P\text{−}T$ plane. Solid lines are stable coexistence lines. Dotted lines show error estimates for the crystal-liquid coexistence lines, as described in the text. Metastable coexistence lines (dashed) are also shown that meet at the metastable $S\text{−}L\text{−}Q$ triple point. The locations of the $S\text{−}C$ (filled square) and $C\text{−}Q$ (filled circle) coexistence boundaries at $T=0$, determined from Fig. 4b, are also shown.

Figure 6

Phase diagram of BKS silica in the $V\text{−}T$ plane. The notation and symbols used have the same meaning as in Fig. 5. Note that in this projection, both one-phase stability fields as well as two-phase coexistence regions are located. The projections of the metastable coexistence lines (dashed) shown in Fig. 5 are also presented.

Figure 7

BKS phase boundaries in (a) the $P\text{−}T$ plane and (b) the $V\text{−}T$ plane, in relation to density maxima (filled circles) and $C_V$ maxima (squares) in the liquid phase. Also located is the state point (star) at which evidence of liquid-liquid phase separation was reported in Ref. 5.