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Structural transformations in silica glass under high pressure
Journal of Non-Crystalline Solids 281 (2001) 205±212
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Structural transformations in silica glass under high pressure Madeleine M. Roberts, Jerey R. Wienho, Kaye Grant, Daniel J. Lacks * Department of Chemical Engineering, Tulane University, New Orleans, LA 70118, USA Received 24 April 2000
Abstract The structural transformations in silica glass that lead to changes in ion coordination under high pressure are examined with molecular simulations. These simulations determine the changes in the potential energy minima upon decreasing volume. The transformations are found to arise from discrete mechanical instabilities associated with disappearances of local minima on the potential energy surface. The ionic displacements arising from the mechanical instabilities are localized to a small number of ions, and involve primarily oxygen motion. The mechanical instabilities appear to be triggered in part by small Si±O±Si angles, which are highly strained. Compression of the glass also leads to a slight ordering of the oxygen ion packing. Ó 2001 Published by Elsevier Science B.V.
1. Introduction Silica glass undergoes interesting transformations under high pressure. The structure of silica glass is characterized by tetrahedrally coordinated silicon ions at low pressure, but octahedrally coordinated silicon ions at high pressure. The tetrahedral coordination is analogous to that in the low-pressure crystal structures quartz and cristobalite, and the octahedral coordination is analogous to that in the high-pressure crystal structure stishovite. The transformation from tetrahedral to octahedral coordination in silica glass occurs at pressures above 8 GPa at room temperature, wherein the ion coordination increases gradually with increasing pressure [1±6]. In contrast, the analogous transformation in crystalline silica is a
* Corresponding author. Tel.: +1-504 862 8258; fax: +1-504 865 6744. E-mail address: [email protected] (D.J. Lacks).
®rst order phase transition that is kinetically hindered at room temperature [7]. Transformations similar to those in silica glass have been observed in many other glasses, [8±12] and also in melts [13,14]. We have recently carried out molecular simulations that show that the pressure-induced transformations in silica glass arise from disappearances of local minima and barriers on the potential energy surface, as shown schematically in Fig. 1 [15]. The disappearances of local energy minima are detected by the decrease of a normal mode frequency to zero (these disappearances of local energy minima were examined in more detail in simpler systems, and we con®rmed that a barrier height decreases to zero coincident with a normal mode frequency decreasing to zero [16,17]). After the local energy minimum that the system is in disappears, the system becomes mechanically unstable and is forced to an alternate energy minimum. The structural rearrangements caused by these mechanical instabilities lead to the
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Fig. 1. Schematic representation of a mechanical instability in a glass.
changes in ion coordination in silica glass. These transformations will occur at any temperature because thermal activation is not necessary for structural rearrangements caused by a mechanical instability. The present investigation focuses on the speci®c ionic displacements that accompany the mechanical instabilities in silica glass. The determination of these displacements will elucidate the mechanism by which the pressure-induced structural transformations occur.
2. Computational method The simulations are carried out for a system composed of 108 silicon ions and 216 oxygen ions, in a cubic simulation cell. The force ®eld of Tsuneyuki et al. [18] is used, with Ewald sum methods to evaluate the coulombic and dispersion energies [19]. This investigation is based on the inherent structure formalism of Stillinger and Weber, [20,21] which separates the dynamics of amorphous systems into vibrational motion within one
`inherent structure' (a local potential energy minimum), and structural relaxations between dierent inherent structures. Thermally activated structural relaxations are relatively rare at temperatures far below the glass transition temperature, and the present simulations focus on the properties of silica glass in the absence of these thermal eects. In particular, the simulations determine the eects of compression on the local energy minima of the glass. Initial local energy minima are generated at a volume of 26 cm3 =mol by cooling a liquid to zero temperature. Cooling is carried out with several stages of molecular dynamics simulations lasting 5±20 ps each; these molecular dynamics simulations are carried out at constant temperature and volume. The ®nal cooling to zero temperature (i.e., potential energy minimization) is carried out with a variable metric minimization algorithm [22]. After an initial local energy minimum is obtained, the volume of the simulation cell is reduced isotropically in steps of 0.016 cm3 =mol. The potential energy is reminimized with respect to the atomic coordinates (at constant volume) after each volume change, and then properties of the system are calculated. We present here results for two such runs (i.e., the compression of structures corresponding to two dierent initial local energy minima). While the present results are obtained from constant-volume simulations, we have shown that the results of constant-pressure simulations (with a fully-variable simulation cell) are very similar [15]. The extent of order in the glass is assessed by the orientational order parameter Q6 [23], !1=2 X 6 4p 2 ; 1 Q6 jhY6m ij 13 m 6 where hY6m i is the average value of the appropriate spherical harmonic for all of the Nbonds `bonds' in the system. We determine Q6 separately for the Si± O, O±O and Si±Si bonds, where we de®ne an Si±O an O±O bond as an Si±O distance less than 2.1 A, and an bond as an O±O distance less than 3.0 A, Si±Si bond as an Si±Si distance less than 3.5 A; these cuto distances correspond to the ®rst minimum in the appropriate radial distribution func-
M.M. Roberts et al. / Journal of Non-Crystalline Solids 281 (2001) 205±212
tion. The Q6 parameter has signi®cant values (0.5) for ordered structures, regardless of the type of ordering (e.g., fcc or hcp), and Q6 ! 1=Nbonds for disordered structures [24].
3. Results The results for the pressure as the glass is compressed are shown in Fig. 2. The pressure usually increases continuously with decreasing volume, but discontinuous pressure drops punctuate these increases. As we have shown previously, the pressure drops are caused by mechanical instabilities associated with the strain-induced disappearance of local energy minima [15]. These mechanical instabilities lead to the increase in the average silicon coordination from 4 towards 6, as shown in Fig. 3 (the coordination is de®ned here as of a silicon the number of oxygen ions within 2.1 A ion). It is seen from Figs. 2 and 3 that the coordination remains essentially unchanged to pressures of 6 GPa, above which the coordination increases continuously with pressure (note that the small changes in coordination at low pressure in Fig. 2(b) are due to slight changes in atomic positions that move ions from just outside to just inside the cuto distance de®ning coordination,
207
and thus do not represent true changes in ion coordination); this result is in good agreement with experiment [5,6]. The distribution of the ionic displacements associated with the mechanical instabilities is shown in Fig. 4; the ionic displacement di is the distance between the positions (with respect to the centerof-mass) of ion i before and after the mechanical instability. Only a small number of ions undergo signi®cant displacement after a mechanical instability. More quantitatively, the number of ions participating in the instability can be estimated by the participation number p
2 Nions X di ; dmax i1
where dmax is the largest displacement undergone by any one ion; the average participation number for these mechanical instabilities is found to be 16. The mechanical instabilities are therefore localized to a small number of ions. The relative magnitudes of the displacements of the oxygen and silicon ions are also shown in Fig. 4. The oxygen ions undergo larger displacements than the silicon ions (on average). We note that previous simulations have shown the relaxational motion in silica glass at ®nite temperatures
Fig. 2. Pressure of the system as a function of volume. Parts (a) and (b) show the results for dierent runs.
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Fig. 3. Average silicon ion coordination number as a function of volume. The silicon ion coordination number is de®ned as the number of oxygen ions within 2.1 A of the silicon ion. Parts (a) and (b) show the results for dierent runs.
Fig. 4. Distribution of ion displacements. (a) Probability of an ion undergoing a given displacement (averaged over all mechanical instabilities). Circles: silicon; squares: oxygen. (b) Ratio of distributions for silicon and oxygen ions.
to be due primarily to the motion of oxygen ions [25]. It therefore appears, based on the present and previous results, that structural changes in silica glass are generally characterized by primarily oxygen motion.
The changes in the Si±O±Si angles during compression are examined to elucidate the speci®c aspects of the local structure that lead to mechanical instabilities. The signi®cance of the Si±O±Si angles is that they provide the primary
M.M. Roberts et al. / Journal of Non-Crystalline Solids 281 (2001) 205±212
means of compression in silica crystals based on tetrahedrally coordinated silicon ions (e.g., quartz and cristobalite) [7]. We focus speci®cally on Si±O±Si angles in which the oxygen ion has only two neighboring silicon ions (this situation corresponds to tetrahedrally-coordinated silicon ions); we do not consider the other Si±O±Si angles because the compression mechanism will likely be dierent for other oxygen ion coordinations. As shown in Fig. 5, the average Si±O±Si angle decreases continuously with decreasing volume (increasing pressure) while the local minimum that the system is in remains stable ± these changes are similar to those that occur in the quartz and cristobalite crystal structures. In contrast, the average Si±O±Si angle increases discontinuously when a mechanical instability occurs. The origin of the discontinuous increases in the average Si±O±Si angle (of 2-fold coordinated oxygen ions) becomes evident from a comparison of the angle distributions just before and just after a mechanical instability, as shown in Fig. 6. These increases in the average Si±O±Si angle are due to decreases in the number of smaller angles (<140°) and increases in the number of larger angles (>140°). The changes in the angle distribution arise from changes in coordination: The decrease in the
209
number of smaller angles occurs because the oxygen ion at the vertex of a smaller angle changes from 2-fold to 3-fold coordination, after which the angle is no longer included in the distribution (note again that this distribution only includes Si±O±Si angles centered at 2-fold coordinated oxygen ions). Similarly, the increase in the number of larger angles occurs because a 3-fold coordinated oxygen ion changes to 2-fold coordinated, and becomes the vertex of a large Si±O±Si angle. Note that the mechanical instabilities cause a net increase in coordination upon compression, but in general some of the ions undergo decreases in coordination. 4. Discussion The above results suggest that the local environment of the oxygen ions, as described by the coordination of the oxygen ions and the Si±O±Si angles, is an important factor in triggering mechanical instabilities. A mechanical instability occurs when the energy at the local minimum becomes greater than the energy at the barrier to another local minimum (at which point the ®rst local minimum no longer exists), as shown in Fig. 1. The energetically favorable environment
Fig. 5. Average Si±O±Si angle at 2-fold coordinated oxygen ions as a function of volume. Parts (a) and (b) show the results for dierent runs.
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Fig. 6. Change in the distribution of Si±O±Si angles (at 2-fold coordinated oxygen ions) due to mechanical instabilities. (a) The angle distribution before and after a speci®c mechanical instability. Circles: before instability; Squares: after instability. (b) Average changes in the angle distribution (averaged over all mechanical instabilities).
for an oxygen ion is 2-fold coordination with a large Si±O±Si angle. However, a local minimum corresponding to a 2-fold-coordinated oxygen ion with a small Si±O±Si angle has a higher energy, which may even be greater than the energy of a barrier to a local minimum corresponding to the 3fold-coordinated ion ± the decrease in an Si±O±Si angle caused by compression could thus lead to a mechanical instability that moves the oxygen ion to a 3-fold coordinated state. The change of an oxygen ion from 2-fold to 3-fold coordination allows more ecient packing around that ion, which can in turn generate local dilatory stresses around other nearby oxygen ions. These local dilatory stresses can allow these other oxygen ions to relax from 3-fold coordination to the energetically more favorable environment of 2-fold coordination with a large Si±O±Si angle. Note, of course, that the potential energy landscape is a function of the coordinates of all ions in the system, and the local environment description of mechanical instabilities described here is only a schematic of the actual behavior. This mechanism for the pressure-induced transformations in silica glass is similar to that put
forth by Stolper and Ahrens [26]. In their model, the compression of silica glass occurs initially by the decrease of Si±O±Si angles; after these angles become small, transitions from tetrahedral to octahedral coordination occur by small ionic displacements. Our results show that the displacements allowing the increase in ion coordination can occur without any thermal activation at all, because they can arise from mechanical instabilities. The present mechanism diers from that of Stolper and Ahrens in that the coordination of speci®c ions increases from 4-fold to 5-fold to 6-fold (i.e., in two steps), rather than from 4-fold directly to 6-fold. The extent of structural order in the glass, and how it changes with compression, is now addressed. The values of the Q6 parameter for the initial glass structure at zero pressure are approximately 0.02 for O±O bonds, 0.05 for Si±Si bonds, and 0.03 for Si±O bonds. These small values of Q6 con®rm that the structure is disordered; the larger value of Q6 for the Si±Si bonds arises only because there are fewer of these bonds, and the value of Q6 1=Nbonds for disordered structures [24]. The changes in structural order as the glass is com-
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Fig. 7. Changes in the structural order parameter Q6 as a function of volume. Parts (a) and (b) show the results for dierent runs.
pressed are shown in Fig. 7. The increase in the value of Q6 for the O±O bonds indicates that the packing of the oxygen ions becomes slightly ordered as the glass is compressed; in contrast, there is no signi®cant increase in order with regard to the silicon ions, as indicated by the nearly constant values of Q6 for the Si±Si and Si±O bonds. This result concurs with the suggestion that the highpressure silica glass structure is based on a distorted close packed array of oxygen ions, with silicon ions distributed randomly among the interstitial positions [27].
5. Conclusions Pressure-induced transformations in silica glass occur by discrete mechanical instabilities associated with the disappearance of local energy minima. The ionic displacements accompanying these instabilities are localized to a small number of ions, and involve primarily displacements of oxygen ions. The mechanical instabilities appear to be triggered by small Si±O±Si angles, which are highly strained; the mechanical instabilities lead to increases in ion coordination that relieve the strain in these angles. Compression of the glass also leads
to a slight ordering of the oxygen ions, but not the silicon ions. Acknowledgements Funding for this project was provided by the National Science Foundation (grant number DMR-9624808) and the DOE/EPSCoR program. References [1] R. Jeanloz, Nature 332 (1998) 207. [2] Q. Williams, R. Jeanloz, Science 239 (1988) 902. [3] C. Meade, R.J. Hemley, H.K. Mao, Phys. Rev. Lett. 69 (1992) 1387. [4] R.J. Hemley, H.K. Mao, P.M. Bell, B.O. Mysen, Phys. Rev. Lett. 57 (1986) 747. [5] C.-S. Zha, R.J. Hemley, H.-K. Mao, T.S. Duy, C. Meade, Phys. Rev. B 50 (1994) 13105. [6] A. Polian, M. Grimsditch, Phys. Rev. B 41 (1990) 6086. [7] R.J. Hemley, C.T. Prewitt, K.J. Kingma, Rev. Mineral. 29 (1988) 41. [8] M. Grimsditch, R. Bhadra, Y. Meng, Phys. Rev. B 38 (1988) 7836. [9] J.P. Itie, A. Polian, G. Calas, J. Petiau, A. Fontaine, H. Tolentino, Phys. Rev. Lett. 63 (1989) 398. [10] G.H. Wolf, D.J. Durben, P.F. McMillan, J. Chem. Phys. 93 (1990) 2280. [11] D.J. Durben, G.H. Wolf, Phys. Rev. B 43 (1991) 2355.
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[12] J.D. Kubicki, R.J. Hemley, A.M. Hofmeister, Am. Mineral 77 (1992) 258. [13] E. Ohtani, F. Taulelle, C.A. Angell, Nature 314 (1985) 79. [14] X. Xue, J.F. Stebbins, M. Kanzaki, R.G. Tronnes, Science 245 (1989) 962. [15] D.J. Lacks, Phys. Rev. Lett. 80 (1998) 5385. [16] D.L. Malandro, D.J. Lacks, J. Chem. Phys. 107 (1997) 5804. [17] D.L. Malandro, D.J. Lacks, Phys. Rev. Lett. 81 (1998) 5576. [18] S. Tsuneyuki, M. Tsukada, H. Aoki, Y. Matsui, Phys. Rev. Lett. 61 (1988) 869. [19] N. Karasawa, W. Goddard, J. Phys. Chem. 93 (1989) 7320.
[20] F.H. Stillinger, T.A. Weber, Science 225 (1984) 983. [21] F.H. Stillinger, Science 267 (1995) 1935. [22] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes, Cambridge University, New York, 1992. [23] P.J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. B 28 (1983) 784. [24] M. Rintoul, S. Torquato, J. Chem. Phys. 105 (1996) 9258. [25] C. Oligschleger, Phys. Rev. B 60 (1999) 3182. [26] E.M. Stolper, T.J. Ahrens, Geophys. Res. Lett. 14 (1987) 1231. [27] D.M. Teter, R.J. Hemley, G. Kresse, J. Hafner, Phys. Rev. Lett. 80 (1998) 2145.
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Structural transformations in silica glass under high pressure Madeleine M. Roberts, Jerey R. Wienho, Kaye Grant, Daniel J. Lacks * Department of Chemical Engineering, Tulane University, New Orleans, LA 70118, USA Received 24 April 2000
Abstract The structural transformations in silica glass that lead to changes in ion coordination under high pressure are examined with molecular simulations. These simulations determine the changes in the potential energy minima upon decreasing volume. The transformations are found to arise from discrete mechanical instabilities associated with disappearances of local minima on the potential energy surface. The ionic displacements arising from the mechanical instabilities are localized to a small number of ions, and involve primarily oxygen motion. The mechanical instabilities appear to be triggered in part by small Si±O±Si angles, which are highly strained. Compression of the glass also leads to a slight ordering of the oxygen ion packing. Ó 2001 Published by Elsevier Science B.V.
1. Introduction Silica glass undergoes interesting transformations under high pressure. The structure of silica glass is characterized by tetrahedrally coordinated silicon ions at low pressure, but octahedrally coordinated silicon ions at high pressure. The tetrahedral coordination is analogous to that in the low-pressure crystal structures quartz and cristobalite, and the octahedral coordination is analogous to that in the high-pressure crystal structure stishovite. The transformation from tetrahedral to octahedral coordination in silica glass occurs at pressures above 8 GPa at room temperature, wherein the ion coordination increases gradually with increasing pressure [1±6]. In contrast, the analogous transformation in crystalline silica is a
* Corresponding author. Tel.: +1-504 862 8258; fax: +1-504 865 6744. E-mail address: [email protected] (D.J. Lacks).
®rst order phase transition that is kinetically hindered at room temperature [7]. Transformations similar to those in silica glass have been observed in many other glasses, [8±12] and also in melts [13,14]. We have recently carried out molecular simulations that show that the pressure-induced transformations in silica glass arise from disappearances of local minima and barriers on the potential energy surface, as shown schematically in Fig. 1 [15]. The disappearances of local energy minima are detected by the decrease of a normal mode frequency to zero (these disappearances of local energy minima were examined in more detail in simpler systems, and we con®rmed that a barrier height decreases to zero coincident with a normal mode frequency decreasing to zero [16,17]). After the local energy minimum that the system is in disappears, the system becomes mechanically unstable and is forced to an alternate energy minimum. The structural rearrangements caused by these mechanical instabilities lead to the
0022-3093/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 3 1 8 - 0
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M.M. Roberts et al. / Journal of Non-Crystalline Solids 281 (2001) 205±212
Fig. 1. Schematic representation of a mechanical instability in a glass.
changes in ion coordination in silica glass. These transformations will occur at any temperature because thermal activation is not necessary for structural rearrangements caused by a mechanical instability. The present investigation focuses on the speci®c ionic displacements that accompany the mechanical instabilities in silica glass. The determination of these displacements will elucidate the mechanism by which the pressure-induced structural transformations occur.
2. Computational method The simulations are carried out for a system composed of 108 silicon ions and 216 oxygen ions, in a cubic simulation cell. The force ®eld of Tsuneyuki et al. [18] is used, with Ewald sum methods to evaluate the coulombic and dispersion energies [19]. This investigation is based on the inherent structure formalism of Stillinger and Weber, [20,21] which separates the dynamics of amorphous systems into vibrational motion within one
`inherent structure' (a local potential energy minimum), and structural relaxations between dierent inherent structures. Thermally activated structural relaxations are relatively rare at temperatures far below the glass transition temperature, and the present simulations focus on the properties of silica glass in the absence of these thermal eects. In particular, the simulations determine the eects of compression on the local energy minima of the glass. Initial local energy minima are generated at a volume of 26 cm3 =mol by cooling a liquid to zero temperature. Cooling is carried out with several stages of molecular dynamics simulations lasting 5±20 ps each; these molecular dynamics simulations are carried out at constant temperature and volume. The ®nal cooling to zero temperature (i.e., potential energy minimization) is carried out with a variable metric minimization algorithm [22]. After an initial local energy minimum is obtained, the volume of the simulation cell is reduced isotropically in steps of 0.016 cm3 =mol. The potential energy is reminimized with respect to the atomic coordinates (at constant volume) after each volume change, and then properties of the system are calculated. We present here results for two such runs (i.e., the compression of structures corresponding to two dierent initial local energy minima). While the present results are obtained from constant-volume simulations, we have shown that the results of constant-pressure simulations (with a fully-variable simulation cell) are very similar [15]. The extent of order in the glass is assessed by the orientational order parameter Q6 [23], !1=2 X 6 4p 2 ; 1 Q6 jhY6m ij 13 m 6 where hY6m i is the average value of the appropriate spherical harmonic for all of the Nbonds `bonds' in the system. We determine Q6 separately for the Si± O, O±O and Si±Si bonds, where we de®ne an Si±O an O±O bond as an Si±O distance less than 2.1 A, and an bond as an O±O distance less than 3.0 A, Si±Si bond as an Si±Si distance less than 3.5 A; these cuto distances correspond to the ®rst minimum in the appropriate radial distribution func-
M.M. Roberts et al. / Journal of Non-Crystalline Solids 281 (2001) 205±212
tion. The Q6 parameter has signi®cant values (0.5) for ordered structures, regardless of the type of ordering (e.g., fcc or hcp), and Q6 ! 1=Nbonds for disordered structures [24].
3. Results The results for the pressure as the glass is compressed are shown in Fig. 2. The pressure usually increases continuously with decreasing volume, but discontinuous pressure drops punctuate these increases. As we have shown previously, the pressure drops are caused by mechanical instabilities associated with the strain-induced disappearance of local energy minima [15]. These mechanical instabilities lead to the increase in the average silicon coordination from 4 towards 6, as shown in Fig. 3 (the coordination is de®ned here as of a silicon the number of oxygen ions within 2.1 A ion). It is seen from Figs. 2 and 3 that the coordination remains essentially unchanged to pressures of 6 GPa, above which the coordination increases continuously with pressure (note that the small changes in coordination at low pressure in Fig. 2(b) are due to slight changes in atomic positions that move ions from just outside to just inside the cuto distance de®ning coordination,
207
and thus do not represent true changes in ion coordination); this result is in good agreement with experiment [5,6]. The distribution of the ionic displacements associated with the mechanical instabilities is shown in Fig. 4; the ionic displacement di is the distance between the positions (with respect to the centerof-mass) of ion i before and after the mechanical instability. Only a small number of ions undergo signi®cant displacement after a mechanical instability. More quantitatively, the number of ions participating in the instability can be estimated by the participation number p
2 Nions X di ; dmax i1
where dmax is the largest displacement undergone by any one ion; the average participation number for these mechanical instabilities is found to be 16. The mechanical instabilities are therefore localized to a small number of ions. The relative magnitudes of the displacements of the oxygen and silicon ions are also shown in Fig. 4. The oxygen ions undergo larger displacements than the silicon ions (on average). We note that previous simulations have shown the relaxational motion in silica glass at ®nite temperatures
Fig. 2. Pressure of the system as a function of volume. Parts (a) and (b) show the results for dierent runs.
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Fig. 3. Average silicon ion coordination number as a function of volume. The silicon ion coordination number is de®ned as the number of oxygen ions within 2.1 A of the silicon ion. Parts (a) and (b) show the results for dierent runs.
Fig. 4. Distribution of ion displacements. (a) Probability of an ion undergoing a given displacement (averaged over all mechanical instabilities). Circles: silicon; squares: oxygen. (b) Ratio of distributions for silicon and oxygen ions.
to be due primarily to the motion of oxygen ions [25]. It therefore appears, based on the present and previous results, that structural changes in silica glass are generally characterized by primarily oxygen motion.
The changes in the Si±O±Si angles during compression are examined to elucidate the speci®c aspects of the local structure that lead to mechanical instabilities. The signi®cance of the Si±O±Si angles is that they provide the primary
M.M. Roberts et al. / Journal of Non-Crystalline Solids 281 (2001) 205±212
means of compression in silica crystals based on tetrahedrally coordinated silicon ions (e.g., quartz and cristobalite) [7]. We focus speci®cally on Si±O±Si angles in which the oxygen ion has only two neighboring silicon ions (this situation corresponds to tetrahedrally-coordinated silicon ions); we do not consider the other Si±O±Si angles because the compression mechanism will likely be dierent for other oxygen ion coordinations. As shown in Fig. 5, the average Si±O±Si angle decreases continuously with decreasing volume (increasing pressure) while the local minimum that the system is in remains stable ± these changes are similar to those that occur in the quartz and cristobalite crystal structures. In contrast, the average Si±O±Si angle increases discontinuously when a mechanical instability occurs. The origin of the discontinuous increases in the average Si±O±Si angle (of 2-fold coordinated oxygen ions) becomes evident from a comparison of the angle distributions just before and just after a mechanical instability, as shown in Fig. 6. These increases in the average Si±O±Si angle are due to decreases in the number of smaller angles (<140°) and increases in the number of larger angles (>140°). The changes in the angle distribution arise from changes in coordination: The decrease in the
209
number of smaller angles occurs because the oxygen ion at the vertex of a smaller angle changes from 2-fold to 3-fold coordination, after which the angle is no longer included in the distribution (note again that this distribution only includes Si±O±Si angles centered at 2-fold coordinated oxygen ions). Similarly, the increase in the number of larger angles occurs because a 3-fold coordinated oxygen ion changes to 2-fold coordinated, and becomes the vertex of a large Si±O±Si angle. Note that the mechanical instabilities cause a net increase in coordination upon compression, but in general some of the ions undergo decreases in coordination. 4. Discussion The above results suggest that the local environment of the oxygen ions, as described by the coordination of the oxygen ions and the Si±O±Si angles, is an important factor in triggering mechanical instabilities. A mechanical instability occurs when the energy at the local minimum becomes greater than the energy at the barrier to another local minimum (at which point the ®rst local minimum no longer exists), as shown in Fig. 1. The energetically favorable environment
Fig. 5. Average Si±O±Si angle at 2-fold coordinated oxygen ions as a function of volume. Parts (a) and (b) show the results for dierent runs.
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Fig. 6. Change in the distribution of Si±O±Si angles (at 2-fold coordinated oxygen ions) due to mechanical instabilities. (a) The angle distribution before and after a speci®c mechanical instability. Circles: before instability; Squares: after instability. (b) Average changes in the angle distribution (averaged over all mechanical instabilities).
for an oxygen ion is 2-fold coordination with a large Si±O±Si angle. However, a local minimum corresponding to a 2-fold-coordinated oxygen ion with a small Si±O±Si angle has a higher energy, which may even be greater than the energy of a barrier to a local minimum corresponding to the 3fold-coordinated ion ± the decrease in an Si±O±Si angle caused by compression could thus lead to a mechanical instability that moves the oxygen ion to a 3-fold coordinated state. The change of an oxygen ion from 2-fold to 3-fold coordination allows more ecient packing around that ion, which can in turn generate local dilatory stresses around other nearby oxygen ions. These local dilatory stresses can allow these other oxygen ions to relax from 3-fold coordination to the energetically more favorable environment of 2-fold coordination with a large Si±O±Si angle. Note, of course, that the potential energy landscape is a function of the coordinates of all ions in the system, and the local environment description of mechanical instabilities described here is only a schematic of the actual behavior. This mechanism for the pressure-induced transformations in silica glass is similar to that put
forth by Stolper and Ahrens [26]. In their model, the compression of silica glass occurs initially by the decrease of Si±O±Si angles; after these angles become small, transitions from tetrahedral to octahedral coordination occur by small ionic displacements. Our results show that the displacements allowing the increase in ion coordination can occur without any thermal activation at all, because they can arise from mechanical instabilities. The present mechanism diers from that of Stolper and Ahrens in that the coordination of speci®c ions increases from 4-fold to 5-fold to 6-fold (i.e., in two steps), rather than from 4-fold directly to 6-fold. The extent of structural order in the glass, and how it changes with compression, is now addressed. The values of the Q6 parameter for the initial glass structure at zero pressure are approximately 0.02 for O±O bonds, 0.05 for Si±Si bonds, and 0.03 for Si±O bonds. These small values of Q6 con®rm that the structure is disordered; the larger value of Q6 for the Si±Si bonds arises only because there are fewer of these bonds, and the value of Q6 1=Nbonds for disordered structures [24]. The changes in structural order as the glass is com-
M.M. Roberts et al. / Journal of Non-Crystalline Solids 281 (2001) 205±212
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Fig. 7. Changes in the structural order parameter Q6 as a function of volume. Parts (a) and (b) show the results for dierent runs.
pressed are shown in Fig. 7. The increase in the value of Q6 for the O±O bonds indicates that the packing of the oxygen ions becomes slightly ordered as the glass is compressed; in contrast, there is no signi®cant increase in order with regard to the silicon ions, as indicated by the nearly constant values of Q6 for the Si±Si and Si±O bonds. This result concurs with the suggestion that the highpressure silica glass structure is based on a distorted close packed array of oxygen ions, with silicon ions distributed randomly among the interstitial positions [27].
5. Conclusions Pressure-induced transformations in silica glass occur by discrete mechanical instabilities associated with the disappearance of local energy minima. The ionic displacements accompanying these instabilities are localized to a small number of ions, and involve primarily displacements of oxygen ions. The mechanical instabilities appear to be triggered by small Si±O±Si angles, which are highly strained; the mechanical instabilities lead to increases in ion coordination that relieve the strain in these angles. Compression of the glass also leads
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