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A molecular dynamics simulation of the effect of high pressure on fast-ion conduction in a MgSiO3-perovskite analogue; KCaF3
PHYSICS OFTHE EARTH AND PLANETARY INTERIORS ELSEVIER
Physics of the Earth and Planetary Interiors 89 (1995) 137-144
A molecular dynamics simulation of the effect of high pressure on fast-ion conduction in a MgSiO3-perovskite analogue; KCaF 3 G.W. Watson
*, A . W a l l , S . C . P a r k e r
School of Chemistry, Universityof Bath, Claverton Down, Bath BA2 7AY, UK
Received 14 June 1994; revision accepted 16 August 1994
Abstract
Analysis of the geomagnetic field estimates the electrical conductivity of the Earth's lower mantle to range from 1 to 100 S m -1. However, measurements of the electrical conductivity of (Mg, Fe)SiO3-perovskite and magnesiowustite range from less than 10 -3 S m -1 to as high as 70 S m -1. The presence of water or iron in the lower mantle may account for the observed high conductivity, but alternatively, the perovskite phase may become a fast-ion conductor at lower mantle temperatures and pressures. We have used a constant pressure-constant temperature molecular dynamics simulation to investigate the effect of pressure on fast-ion conductivity in the perovskite KCaF3, a structural analogue of MgSiO3-perovskite. Although increased pressure decreases the ionic conductivity, increasing the pressure also increases the melting point and the high-conductivity regime is extended to a lower fraction of the melting temperature. However, if (Mg, Fe)SiO3-perovskite follows the behaviour of the structural analogue and does become a fast-ion conductor at high temperature, most of the lower mantle may not be hot enough for (Mg, Fe)SiO3-perovskite to be within its fast-ion regime.
1. Introduction
The conductivity of the important lower mantle phase MgSiO3-perovskite at mantle conditions has been investigated by both experiment and computer simulation. However the experiments are very difficult and often are subject to high degrees of uncertainty (Heinz, 1991; Kerr, 1991). An alternative approach for gaining insight into the behaviour of MgSiO3-perovskite is to use analogue systems at conditions more accessible to experiments. The fluoride perovskites are com-
* Corresponding author.
monly employed (O'Keeffe and Bovin, 1979; Poirier et al., 1983; Ridou et al., 1986; Z h a o et al., 1993) because with their lower charge they have melting points which are much lower than the oxide perovskites and are more compressible, but by choosing appropriate cations they are isostructural with the oxides. This allows experiments to investigate key properties close to their melting points and compression to similar V / V o values as experienced by MgSiO3-perovskite in the mantle, without the need for a laser-heated high-pressure diamond anvil cell (as used by Li and Jeanloz 1987, 1990, 1991a, b). This p a p e r describes work on the effect of pressure on the ionic conductivity of KCaF3-perovskite, which
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G.W. Watson et al. / Physics of the Earth and Planetary Interiors 89 (1995) 137-144
shows the same structural distortion a s M g S i O 3. We believe this will give insight in to the relative effects of temperature and pressure on the ionic conductivity of an important perovskite analogue. In addition, we aim to provide information relating to the effect of temperature and pressure on ionic conductivity for a system which can be compared directly with future unambiguous experimental data using established experimental techniques. This direct comparison will give confidence in the molecular dynamics technique when applied to systems where unambiguous data are not available (e.g. MgSiO3-perovskite itself (Matsui and Price, 1991)). The electrical conductivity of the Earth's lower mantle is estimated from analysis of the transient and secular variations of the geomagnetic field to range from about 1 S m -1 at 1000 km depth to about 100 S m -1 at the core-mantle boundary (Achache et al., 1981; Parkinson, 1988). However, measurements of the electrical conductivity of the major lower mantle minerals. (Mg, Fe)SiO3-perovskite and (Mg, Fe)O-magnesiowustite, at high temperature and pressure have produced widely different results (Heinz, 1991; Kerr, 1991). Li and Jeanloz (1987, 1990) measured the electrical conductivity of silicate perovskite and a perovskitemagnesiowustite assemblage at pressures of up to 80 GPa and temperatures up to 3500 K. They found the conductivity to be less than 10 -3 S m -1. Peyronneau and Poirier (1989) and Shankland et al. (1993) have also studied the conductivity of both a pure silicate perovskite phase and a mixed perovskite-magnesiowustite assemblage at the lower temperature of 673 K and pressures of up to 40 GPa. On extrapolating their results to temperatures and pressures appropriate for the lower mantle, they predict the conductivity to be to 1 S m -1 at 1100 km depth rising to about 70 S m -1 at the core-mantle boundary. In addition, following a high-temperature study of the electrical conductivity of magnesiowustite, Wood and Nell (1991) extrapolated their results to lower mantle conditions and found good agreement with the work of Peyronneau and Poirier (1989). The studies of Peyronneau and Poirier, Shankland et al. and Wood and Nell predict that an assemblage of magnesiowustite and perovskite which has the
same composition as the upper mantle could produce a high electrical conductivity in agreement with the geophysical measurements. In contrast, the low conductivity observed by Li and Jeanloz is not consistent with the geophysical observations. In an attempt to resolve the differences Li et al. (1993) performed larger volume measurements at low pressure but found no evidence for high electrical conductivity. They suggest that the lower mantle must either be enriched in iron relative to the upper mantle--increasing iron content is observed to increase the electrical conductivity (Li and Jeanloz, 1991b; Peyronneau and Poirier, 1989) or that the high electrical conductivity is caused by the presence of water (Li and Jeanloz, 1991a). Hence an understanding of electrical conduction in important lower mantle minerals will contribute towards the debate on the compositional structure of the Earth. One point of agreement between these studies is that electronic conduction is the predominant mechanism for the observed conductivity. However, estimations using ion porosity (Gautason and Muehlenbachs, 1993) and molecular dynamics simulation studies have suggested that at high temperature and pressure, MgSiO3-perovskite may become a fast-ion conductor (Kapusta and Guillope, 1988; Wall and Price, 1989; Matsui and Price, 1991) with an electrical conductivity consistent with the geophysical measurements. Ionic conduction is dismissed as an important mechanism within the lower mantle by Li and Jeanloz (1990) and Heinz (1991) because the activation energy for ionic conduction for many systems increases with increasing pressure (Samara, 1984). However, Samara also notes that some fast-ion conductors show enhanced electrical conductivity with increasing pressure. This is the result of the dependence of the fast-ion conductivity on the diffusion mechanism. In perovskite, the diffusion mechanism is probably dependent on vacancy formation (Watson et al., 1992) which becomes more energetically unfavourable with increasing pressure. Thus it is likely that, as assumed by, for example, Li and Jeanloz (1990) and Heinz (1991), pressure will reduce any ionic conduction in the perovskite phase. Therefore, if fast-ion conduction occurs in silicate perovskite, whether or not
G.W. Watson et al. /Physics of the Earth and Planetary Interiors 89 (1995) 137-144
it makes an important contribution to the electrical conductivity of the lower mantle will depend on the opposing effects of increasing temperature and increasing pressure with depth. In a previous simulation study (Watson et al., 1992), we investigated the occurrence of fast-ion conductivity in fluoride perovskites at high temperature and found that fast-ion conduction was only predicted to occur in the perovskite KCaF3, although both the melting temperature and temperature of the fast-ion regime were overestimated because of the lack of a term in the potential model to describe the electronic polarisability (Watson et al., 1992). In this work, we have extended the calculations to investigate the effect of increasing pressure on the fast-ion conductivity of KCaF3.
2. Molecular dynamics technique The molecular dynamics technique involves the solution of Newton's Laws of Motion numerically for a fixed number of ions. This enables us to investigate the effect of temperature and pressure on solids and liquids at the atomic scale. The forces between the ions are calculated at each time interval of 10 -~5 s using the Born model of solids. The potential model contains simple analytical functions which model the interaction between charges, electron cloud repulsions and van der Waals attraction. The method currently is confined to the rigid ion model owing to the constraints of c.p.u, time and does not yet model electronic polarisibility. The lack of electronic polarisibility results in a reduction in the dielectric constants and hence an increase in the defect formation energies. This causes the simulation to overestimate the melting point as insufficient defects are created at the appropriate temperature to form the melt. We believe this argument also extends to the fast ionic conductivity regime which is also the result of defect formation. Thus both the melting point (Tm) and the temperature at which fast ionic conductivity occurs (Tc) will be raised to a similar degree by the use of the rigid ion model (Watson et al., 1992). Therefore we describe the effect of pressure on the fractional
139
temperature ( T / T m) to allow direct comparison with experiment. We have used a well-established technique which enables us to simulate constant temperature (Nose, 1984, 1990) and constant pressure (Parrinello and Rahman, 1981) conditions. This is available through the molecular dynamics code QCTPMD (Matsui, 1989; Watson et al., 1992). These conditions allow the simulation of thermal transfer with the extended system and dynamic change in both the lattice vectors and the angles of the periodically repeating box, giving rise to simulation in the isothermal-isobaric (constant NPT) ensemble (Allen and Tildesley, 1987). Newton's Laws of Motion, with a time step of 10 -15 s, are solved numerically using a fifth-order predictor-corrector method (Gear, 1971). The forces calculated from interatomic potentials of the form E l = ~ " --qiqj i~j ri2
A i J e x p ( - r-ij]+ Pij ~ *°ij ]
6Cij r7
where A, p and C are potential parameters (see Watson et aL, 1992), calculated to a short-range cut-off of 9 A. The long-range Coulombic forces were calculated using the Ewald sum (see Kittel, 1963) with the cut-offs defined by the methods of Catlow and Norgett (1976). Simulations were performed using a box size of 996 ions (4 x 4 × 3 orthorhombic cells) using periodic boundary conditions. Equilibration of the system was performed for between 5000 and 10000 time steps (5-10 ps) depending on the magnitude of the fluctuations. Data, including the mean square deviation (MSD), were collected during 15000 time-step runs (15 ps). A linear increase in MSD with time, for a given ion, indicates that there is ionic diffusion. The gradient of the plot of the MSD with time represents the diffusion coefficient, which in turn yields the conductivity from the Nernst-Einstein equation, assuming a correlation factor of 1 (Watson et al., 1992). Melting points for the various pressures were evaluated by increasing the temperature in steps of either 50 or 100 K until the MSD of all the ion types increased with time, indicating that melting had occurred.
G.W. Watson et aL / Physics of the Earth and Planetary Interiors 89 (1995) 137-144
140
12.5t
3. Results The molecular dynamic simulations predicted that at all four pressures considered (0, 2.5, 5 and 10 GPa), fast-ion conduction occurs in KCaF 3perovskite at high temperatures. As in our previous zero pressure study, the fast-ion conduction is predicted to occur because of rapid diffusion of fluoride ions whereas the calcium and potassium ions only vibrate about their lattice sites. This is illustrated in Fig. 1 which shows the MSD of the potassium, calcium (which are constant) and fluoride ions in KCaF3-perovskite at 3000 K and 10 GPa. The mechanism was identified by animating the trajectories (Watson et al., 1992) and involves the formation of anion Frenkel defects. The vacancies thus formed move through the systems via correlated motion of the fluoride ions (between 1 and 5 ions). This mechanism is further confirmed by the introduction of pseudo-Schottky defects (a K and F ion removed) which had no significant effect on the conductivity. A least-squares regression of the MSD with time was used to calculate the diffusion coefficient at each pressure and temperature which in turn gave the conductivities using the NernstEinstein equation. The natural logarithms of the product of conductivity and temperature with re-
c3
i=o I !"I 151 [" 11.o!
'~'~10.5
m
~
i 10.0 0 9.5
p]
-
.
2.5 GPa
P
I 10 GPa 9.0-0.30
o.~5
0~
o.~
0.50
1 0 0 0 / T ( 1 / K) Fig. 2. Comparison of linear fit and data for natural logarithm of the product of conductivity and temperature as a function of the inverse of temperature.
Table 1 Parameters for the linear fit of the natural logarithm of the product of conductivity and temperature as a function of the inverse of temperature Pressure (GPa)
Gradient (m) (1000 K)
Intercept (C)
Activation energy (kJ mol- 1)
0.0 2.5 5.0 10.0
-23.2145 -24.2139 -25.7827 -28.9665
21.1447 20.6933 20.8792 21.6541
193 201 214 241
30.0
70
25.0"
~-~
~, 20.0< 15.0-
•
I0.05.0-
0"(~0
[]
50'00
10000
15000
Time (fs) Fig. I. Mean square deviation (MSD) for KCaF 3 at 3000 K and I0 GPa.
20100 1800
2000
220D
24'00
26~
2~0
3000
3200
Temperature (K) Fig. 3. Variation in conductivity as a function of pressure and temperature.
G. IV.. Watson et at/Physics of the Earth and Planetary Interiors 89 (1995) 137-144 70¸
I
/
¢~ 30
~ 2o 10
o 0.84 0.86 0.~8 019 o.b2 0.84 0.88 0.88 o.s2 r/Tin Fig. 4. Variation in conductivity as a function of fraction of the melting point and pressure.
spect to reciprocal temperature are shown in Fig. 2. The data were fitted to equations of the form
ln(trT)=m(~-~)+C with the parameters for each pressure shown in Table 1. Table 1 also gives the intrinsic activation energies resulting from the fit which represent an upper limit owing to the use of the rigid ion model. The conductivities at each pressure up to the simulated melting points calculated from the pa-
3200----ll--
M~eltlng Point 3OOO-
T 120 S/m)_~
2800-
~, 2600-
240C
2201
0.00
2,50
5.0o 7.50 Pressure (GPa)
1o.oo
Fig. 5. Variation in the temperature that 20 S m -~ is achieved in comparison with the melting point as a function of pressure.
141
rameters in Table 1 are shown in Fig. 3. This illustrates that, as expected, for a particular temperature, the conductivity decreases with increasing pressure and that the melting point increases with increasing pressure. However, the conductivity at a particular fraction of the melting temperature increases with increasing pressure as shown by a plot of conductivity against fraction of the melting point (Fig. 4). If we follow the, albeit arbitrary, conductivity of 20 S m -1, at 0 GPa 20 S m -~ is achieved at 0.98 of Tm whereas at 10 GPa this is reduced to 0.87 of Tm. This is illustrated in Fig. 5 which shows the increase in the fast-ion region at high pressure. In summary, as the pressure is increased the stability range of the fastionic phase also increases.
4. Discussion and conclusions
KCaF3-perovskite has the same structural distortion as MgSiOa-perovskite - both have the G d F e O 3 structure characterised by rotation of the CaF 6 or SiO 6 octahedra. Our previous work on fluorides, Watson et al. (1992), demonstrated that the MD technique is able to distinguish those perovskites that become fast-ion conductors at high temperature from those which do not. Hence we agree with the predictions of Matsui and Price that MgSiO3-perovskite will also be a fast-ion conductor at high temperature. However, it is interesting to consider whether any other oxide perovskites have also been observed to show such behaviour. In a search for some systematic trends amongst perovskite structured materials, Beauchesne and Poirier (1990) note that, on the basis of high-temperature creep results, the oxide perovskites BaTiO 3, KTaO 3 and KNbO 3 are unlikely to be fast-ion conductors at temperatures approaching their melting points. However, all these perovskites are related in that they distort from the ideal cubic structure by a ferroelastic phase transition to the BaTiO3-structure caused by displacement of cations from the centre of the octahedra rather than by octahedral rotation. Measurements of the electrical conductivity of the perovskites CaTiO 3 and SrTiO3, which both
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G.W. Watson et al. / Physics of the Earth and Planetary Interiors 89 (1995) 137-144
show structural distortions caused by octahedral rotation, have only been made up until about 1300 K (Poirier, 1991) and hence probably have not sampled temperatures sufficiently near to their melting points (2248 K and 2353 K for CaTiO 3 and SrTiO 3, respectively) to detect any fast-ion conductivity should it occur. There has also been extensive work on the electrical conductivity of substitutionally mixed oxide perovskites, usually containing transition metal cations, which may have important technological applications as electrodes in fuel cells, oxygen sensors and oxygen permeable membranes (see, for example, Khandkar et al., 1992). Such work has, therefore, focused on achieving high oxygen ion mobility at relatively low temperatures by doping to create mobile vacancies or interstitials. Beside having high oxygen ion mobility, some of these perovskites have high mixed electronic and ionic conductivity. The major factors that appear to control oxygen ion conductivity at a particular temperature are the extent of cation substitution and the partial pressure of oxygen (Worell, 1992). A similar phenomenon may be responsible for the high electrical conductivity of (Mg, Fe)SiO3-perovskite in the lower mantle, but this is difficult to evaluate, particularly as the oxygen partial pressure is unknown. Although subject to some uncertainty because both Tm and the temperature of the fast-ion regime are overestimated by our MD simulations (see Watson et al., 1992), we predict fast-ion behaviour to occur in KCaF3-perovskite at about 0.85 Tm at zero pressure decreasing to about 0.77 Tm at 10 GPa. If (Mg, Fe)SiO3-perovskite does become a fast-ion conductor at high temperature, whether or not fast-ion conduction makes a significant contribution to the conductivity in the lower mantle will depend on the temperature profile. We have studied the behaviour of KCaF3-perovskite at a similar range of relative compressions ( V / V o of 0.9-0.7) as are probably applied to (Mg, Fe)SiO3-perovskite in the lower mantle (Knittle and Jeanloz, 1987). The lower mantle temperature profile and the melting curve of (Mg, Fe)SiO3-perovskite are poorly constrained to plus or minus 800 K and plus or minus 300 K, respectively (Heinz and Jeanloz, 1987;
Knittle and Jeanloz, 1989; Poirier, 1991). In addition, more recent experiments by Zerr and Boehler (1993) determined much higher melting points, although there is considerable dispute (Zerr and Boehler, 1994; Heinz et al., 1994). However, despite this uncertainty, we suggest that it is unlikely that (Mg, Fe)SiO3-perovskite in the lower mantle will be close enough to its melting temperature to be within its fast-ion conduction region. A possible exception to this is in the D" layer at the base of the lower mantle where (if the silicate perovskite phase is still stable in this region, Knittle and Jeanloz, 1991) there is probably a high thermal gradient (Poirier, 1991). Therefore, although the electrical conductivity of much of the lower mantle is probably due to electronic conduction (Peyronneau and Poirier, 1989), perhaps requiring an increased iron or water content (Li and Jeanloz, 1991a, b), there will be a contribution from fast-ion conduction in the perovskite phase if the temperature becomes high enough, for example, in the D" layer. Further experiments and simulations to study the elevated temperature and high-pressure electrical conductivity in (Mg, Fe)SiO3-perovskite and perovskite structural analogues, with the GdFeO 3 distortion (for example CaTiO 3 requiring temperatures above 1700 K) which pay particular attention to the role of impurities would help to further investigate the source of lower mantle electrical conductivity.
Acknowledgements We would like to thank Matsanori Matsui for providing us with the original molecular dynamics code and G.D. Price for reviewing the manuscript. We would also like to thank the NERC for research grant GR3/6970 and G.W.W. would like to thank the NERC for a studentship.
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Physics of the Earth and Planetary Interiors 89 (1995) 137-144
A molecular dynamics simulation of the effect of high pressure on fast-ion conduction in a MgSiO3-perovskite analogue; KCaF 3 G.W. Watson
*, A . W a l l , S . C . P a r k e r
School of Chemistry, Universityof Bath, Claverton Down, Bath BA2 7AY, UK
Received 14 June 1994; revision accepted 16 August 1994
Abstract
Analysis of the geomagnetic field estimates the electrical conductivity of the Earth's lower mantle to range from 1 to 100 S m -1. However, measurements of the electrical conductivity of (Mg, Fe)SiO3-perovskite and magnesiowustite range from less than 10 -3 S m -1 to as high as 70 S m -1. The presence of water or iron in the lower mantle may account for the observed high conductivity, but alternatively, the perovskite phase may become a fast-ion conductor at lower mantle temperatures and pressures. We have used a constant pressure-constant temperature molecular dynamics simulation to investigate the effect of pressure on fast-ion conductivity in the perovskite KCaF3, a structural analogue of MgSiO3-perovskite. Although increased pressure decreases the ionic conductivity, increasing the pressure also increases the melting point and the high-conductivity regime is extended to a lower fraction of the melting temperature. However, if (Mg, Fe)SiO3-perovskite follows the behaviour of the structural analogue and does become a fast-ion conductor at high temperature, most of the lower mantle may not be hot enough for (Mg, Fe)SiO3-perovskite to be within its fast-ion regime.
1. Introduction
The conductivity of the important lower mantle phase MgSiO3-perovskite at mantle conditions has been investigated by both experiment and computer simulation. However the experiments are very difficult and often are subject to high degrees of uncertainty (Heinz, 1991; Kerr, 1991). An alternative approach for gaining insight into the behaviour of MgSiO3-perovskite is to use analogue systems at conditions more accessible to experiments. The fluoride perovskites are com-
* Corresponding author.
monly employed (O'Keeffe and Bovin, 1979; Poirier et al., 1983; Ridou et al., 1986; Z h a o et al., 1993) because with their lower charge they have melting points which are much lower than the oxide perovskites and are more compressible, but by choosing appropriate cations they are isostructural with the oxides. This allows experiments to investigate key properties close to their melting points and compression to similar V / V o values as experienced by MgSiO3-perovskite in the mantle, without the need for a laser-heated high-pressure diamond anvil cell (as used by Li and Jeanloz 1987, 1990, 1991a, b). This p a p e r describes work on the effect of pressure on the ionic conductivity of KCaF3-perovskite, which
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(94)02989-X
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G.W. Watson et al. / Physics of the Earth and Planetary Interiors 89 (1995) 137-144
shows the same structural distortion a s M g S i O 3. We believe this will give insight in to the relative effects of temperature and pressure on the ionic conductivity of an important perovskite analogue. In addition, we aim to provide information relating to the effect of temperature and pressure on ionic conductivity for a system which can be compared directly with future unambiguous experimental data using established experimental techniques. This direct comparison will give confidence in the molecular dynamics technique when applied to systems where unambiguous data are not available (e.g. MgSiO3-perovskite itself (Matsui and Price, 1991)). The electrical conductivity of the Earth's lower mantle is estimated from analysis of the transient and secular variations of the geomagnetic field to range from about 1 S m -1 at 1000 km depth to about 100 S m -1 at the core-mantle boundary (Achache et al., 1981; Parkinson, 1988). However, measurements of the electrical conductivity of the major lower mantle minerals. (Mg, Fe)SiO3-perovskite and (Mg, Fe)O-magnesiowustite, at high temperature and pressure have produced widely different results (Heinz, 1991; Kerr, 1991). Li and Jeanloz (1987, 1990) measured the electrical conductivity of silicate perovskite and a perovskitemagnesiowustite assemblage at pressures of up to 80 GPa and temperatures up to 3500 K. They found the conductivity to be less than 10 -3 S m -1. Peyronneau and Poirier (1989) and Shankland et al. (1993) have also studied the conductivity of both a pure silicate perovskite phase and a mixed perovskite-magnesiowustite assemblage at the lower temperature of 673 K and pressures of up to 40 GPa. On extrapolating their results to temperatures and pressures appropriate for the lower mantle, they predict the conductivity to be to 1 S m -1 at 1100 km depth rising to about 70 S m -1 at the core-mantle boundary. In addition, following a high-temperature study of the electrical conductivity of magnesiowustite, Wood and Nell (1991) extrapolated their results to lower mantle conditions and found good agreement with the work of Peyronneau and Poirier (1989). The studies of Peyronneau and Poirier, Shankland et al. and Wood and Nell predict that an assemblage of magnesiowustite and perovskite which has the
same composition as the upper mantle could produce a high electrical conductivity in agreement with the geophysical measurements. In contrast, the low conductivity observed by Li and Jeanloz is not consistent with the geophysical observations. In an attempt to resolve the differences Li et al. (1993) performed larger volume measurements at low pressure but found no evidence for high electrical conductivity. They suggest that the lower mantle must either be enriched in iron relative to the upper mantle--increasing iron content is observed to increase the electrical conductivity (Li and Jeanloz, 1991b; Peyronneau and Poirier, 1989) or that the high electrical conductivity is caused by the presence of water (Li and Jeanloz, 1991a). Hence an understanding of electrical conduction in important lower mantle minerals will contribute towards the debate on the compositional structure of the Earth. One point of agreement between these studies is that electronic conduction is the predominant mechanism for the observed conductivity. However, estimations using ion porosity (Gautason and Muehlenbachs, 1993) and molecular dynamics simulation studies have suggested that at high temperature and pressure, MgSiO3-perovskite may become a fast-ion conductor (Kapusta and Guillope, 1988; Wall and Price, 1989; Matsui and Price, 1991) with an electrical conductivity consistent with the geophysical measurements. Ionic conduction is dismissed as an important mechanism within the lower mantle by Li and Jeanloz (1990) and Heinz (1991) because the activation energy for ionic conduction for many systems increases with increasing pressure (Samara, 1984). However, Samara also notes that some fast-ion conductors show enhanced electrical conductivity with increasing pressure. This is the result of the dependence of the fast-ion conductivity on the diffusion mechanism. In perovskite, the diffusion mechanism is probably dependent on vacancy formation (Watson et al., 1992) which becomes more energetically unfavourable with increasing pressure. Thus it is likely that, as assumed by, for example, Li and Jeanloz (1990) and Heinz (1991), pressure will reduce any ionic conduction in the perovskite phase. Therefore, if fast-ion conduction occurs in silicate perovskite, whether or not
G.W. Watson et al. /Physics of the Earth and Planetary Interiors 89 (1995) 137-144
it makes an important contribution to the electrical conductivity of the lower mantle will depend on the opposing effects of increasing temperature and increasing pressure with depth. In a previous simulation study (Watson et al., 1992), we investigated the occurrence of fast-ion conductivity in fluoride perovskites at high temperature and found that fast-ion conduction was only predicted to occur in the perovskite KCaF3, although both the melting temperature and temperature of the fast-ion regime were overestimated because of the lack of a term in the potential model to describe the electronic polarisability (Watson et al., 1992). In this work, we have extended the calculations to investigate the effect of increasing pressure on the fast-ion conductivity of KCaF3.
2. Molecular dynamics technique The molecular dynamics technique involves the solution of Newton's Laws of Motion numerically for a fixed number of ions. This enables us to investigate the effect of temperature and pressure on solids and liquids at the atomic scale. The forces between the ions are calculated at each time interval of 10 -~5 s using the Born model of solids. The potential model contains simple analytical functions which model the interaction between charges, electron cloud repulsions and van der Waals attraction. The method currently is confined to the rigid ion model owing to the constraints of c.p.u, time and does not yet model electronic polarisibility. The lack of electronic polarisibility results in a reduction in the dielectric constants and hence an increase in the defect formation energies. This causes the simulation to overestimate the melting point as insufficient defects are created at the appropriate temperature to form the melt. We believe this argument also extends to the fast ionic conductivity regime which is also the result of defect formation. Thus both the melting point (Tm) and the temperature at which fast ionic conductivity occurs (Tc) will be raised to a similar degree by the use of the rigid ion model (Watson et al., 1992). Therefore we describe the effect of pressure on the fractional
139
temperature ( T / T m) to allow direct comparison with experiment. We have used a well-established technique which enables us to simulate constant temperature (Nose, 1984, 1990) and constant pressure (Parrinello and Rahman, 1981) conditions. This is available through the molecular dynamics code QCTPMD (Matsui, 1989; Watson et al., 1992). These conditions allow the simulation of thermal transfer with the extended system and dynamic change in both the lattice vectors and the angles of the periodically repeating box, giving rise to simulation in the isothermal-isobaric (constant NPT) ensemble (Allen and Tildesley, 1987). Newton's Laws of Motion, with a time step of 10 -15 s, are solved numerically using a fifth-order predictor-corrector method (Gear, 1971). The forces calculated from interatomic potentials of the form E l = ~ " --qiqj i~j ri2
A i J e x p ( - r-ij]+ Pij ~ *°ij ]
6Cij r7
where A, p and C are potential parameters (see Watson et aL, 1992), calculated to a short-range cut-off of 9 A. The long-range Coulombic forces were calculated using the Ewald sum (see Kittel, 1963) with the cut-offs defined by the methods of Catlow and Norgett (1976). Simulations were performed using a box size of 996 ions (4 x 4 × 3 orthorhombic cells) using periodic boundary conditions. Equilibration of the system was performed for between 5000 and 10000 time steps (5-10 ps) depending on the magnitude of the fluctuations. Data, including the mean square deviation (MSD), were collected during 15000 time-step runs (15 ps). A linear increase in MSD with time, for a given ion, indicates that there is ionic diffusion. The gradient of the plot of the MSD with time represents the diffusion coefficient, which in turn yields the conductivity from the Nernst-Einstein equation, assuming a correlation factor of 1 (Watson et al., 1992). Melting points for the various pressures were evaluated by increasing the temperature in steps of either 50 or 100 K until the MSD of all the ion types increased with time, indicating that melting had occurred.
G.W. Watson et aL / Physics of the Earth and Planetary Interiors 89 (1995) 137-144
140
12.5t
3. Results The molecular dynamic simulations predicted that at all four pressures considered (0, 2.5, 5 and 10 GPa), fast-ion conduction occurs in KCaF 3perovskite at high temperatures. As in our previous zero pressure study, the fast-ion conduction is predicted to occur because of rapid diffusion of fluoride ions whereas the calcium and potassium ions only vibrate about their lattice sites. This is illustrated in Fig. 1 which shows the MSD of the potassium, calcium (which are constant) and fluoride ions in KCaF3-perovskite at 3000 K and 10 GPa. The mechanism was identified by animating the trajectories (Watson et al., 1992) and involves the formation of anion Frenkel defects. The vacancies thus formed move through the systems via correlated motion of the fluoride ions (between 1 and 5 ions). This mechanism is further confirmed by the introduction of pseudo-Schottky defects (a K and F ion removed) which had no significant effect on the conductivity. A least-squares regression of the MSD with time was used to calculate the diffusion coefficient at each pressure and temperature which in turn gave the conductivities using the NernstEinstein equation. The natural logarithms of the product of conductivity and temperature with re-
c3
i=o I !"I 151 [" 11.o!
'~'~10.5
m
~
i 10.0 0 9.5
p]
-
.
2.5 GPa
P
I 10 GPa 9.0-0.30
o.~5
0~
o.~
0.50
1 0 0 0 / T ( 1 / K) Fig. 2. Comparison of linear fit and data for natural logarithm of the product of conductivity and temperature as a function of the inverse of temperature.
Table 1 Parameters for the linear fit of the natural logarithm of the product of conductivity and temperature as a function of the inverse of temperature Pressure (GPa)
Gradient (m) (1000 K)
Intercept (C)
Activation energy (kJ mol- 1)
0.0 2.5 5.0 10.0
-23.2145 -24.2139 -25.7827 -28.9665
21.1447 20.6933 20.8792 21.6541
193 201 214 241
30.0
70
25.0"
~-~
~, 20.0< 15.0-
•
I0.05.0-
0"(~0
[]
50'00
10000
15000
Time (fs) Fig. I. Mean square deviation (MSD) for KCaF 3 at 3000 K and I0 GPa.
20100 1800
2000
220D
24'00
26~
2~0
3000
3200
Temperature (K) Fig. 3. Variation in conductivity as a function of pressure and temperature.
G. IV.. Watson et at/Physics of the Earth and Planetary Interiors 89 (1995) 137-144 70¸
I
/
¢~ 30
~ 2o 10
o 0.84 0.86 0.~8 019 o.b2 0.84 0.88 0.88 o.s2 r/Tin Fig. 4. Variation in conductivity as a function of fraction of the melting point and pressure.
spect to reciprocal temperature are shown in Fig. 2. The data were fitted to equations of the form
ln(trT)=m(~-~)+C with the parameters for each pressure shown in Table 1. Table 1 also gives the intrinsic activation energies resulting from the fit which represent an upper limit owing to the use of the rigid ion model. The conductivities at each pressure up to the simulated melting points calculated from the pa-
3200----ll--
M~eltlng Point 3OOO-
T 120 S/m)_~
2800-
~, 2600-
240C
2201
0.00
2,50
5.0o 7.50 Pressure (GPa)
1o.oo
Fig. 5. Variation in the temperature that 20 S m -~ is achieved in comparison with the melting point as a function of pressure.
141
rameters in Table 1 are shown in Fig. 3. This illustrates that, as expected, for a particular temperature, the conductivity decreases with increasing pressure and that the melting point increases with increasing pressure. However, the conductivity at a particular fraction of the melting temperature increases with increasing pressure as shown by a plot of conductivity against fraction of the melting point (Fig. 4). If we follow the, albeit arbitrary, conductivity of 20 S m -1, at 0 GPa 20 S m -~ is achieved at 0.98 of Tm whereas at 10 GPa this is reduced to 0.87 of Tm. This is illustrated in Fig. 5 which shows the increase in the fast-ion region at high pressure. In summary, as the pressure is increased the stability range of the fastionic phase also increases.
4. Discussion and conclusions
KCaF3-perovskite has the same structural distortion as MgSiOa-perovskite - both have the G d F e O 3 structure characterised by rotation of the CaF 6 or SiO 6 octahedra. Our previous work on fluorides, Watson et al. (1992), demonstrated that the MD technique is able to distinguish those perovskites that become fast-ion conductors at high temperature from those which do not. Hence we agree with the predictions of Matsui and Price that MgSiO3-perovskite will also be a fast-ion conductor at high temperature. However, it is interesting to consider whether any other oxide perovskites have also been observed to show such behaviour. In a search for some systematic trends amongst perovskite structured materials, Beauchesne and Poirier (1990) note that, on the basis of high-temperature creep results, the oxide perovskites BaTiO 3, KTaO 3 and KNbO 3 are unlikely to be fast-ion conductors at temperatures approaching their melting points. However, all these perovskites are related in that they distort from the ideal cubic structure by a ferroelastic phase transition to the BaTiO3-structure caused by displacement of cations from the centre of the octahedra rather than by octahedral rotation. Measurements of the electrical conductivity of the perovskites CaTiO 3 and SrTiO3, which both
142
G.W. Watson et al. / Physics of the Earth and Planetary Interiors 89 (1995) 137-144
show structural distortions caused by octahedral rotation, have only been made up until about 1300 K (Poirier, 1991) and hence probably have not sampled temperatures sufficiently near to their melting points (2248 K and 2353 K for CaTiO 3 and SrTiO 3, respectively) to detect any fast-ion conductivity should it occur. There has also been extensive work on the electrical conductivity of substitutionally mixed oxide perovskites, usually containing transition metal cations, which may have important technological applications as electrodes in fuel cells, oxygen sensors and oxygen permeable membranes (see, for example, Khandkar et al., 1992). Such work has, therefore, focused on achieving high oxygen ion mobility at relatively low temperatures by doping to create mobile vacancies or interstitials. Beside having high oxygen ion mobility, some of these perovskites have high mixed electronic and ionic conductivity. The major factors that appear to control oxygen ion conductivity at a particular temperature are the extent of cation substitution and the partial pressure of oxygen (Worell, 1992). A similar phenomenon may be responsible for the high electrical conductivity of (Mg, Fe)SiO3-perovskite in the lower mantle, but this is difficult to evaluate, particularly as the oxygen partial pressure is unknown. Although subject to some uncertainty because both Tm and the temperature of the fast-ion regime are overestimated by our MD simulations (see Watson et al., 1992), we predict fast-ion behaviour to occur in KCaF3-perovskite at about 0.85 Tm at zero pressure decreasing to about 0.77 Tm at 10 GPa. If (Mg, Fe)SiO3-perovskite does become a fast-ion conductor at high temperature, whether or not fast-ion conduction makes a significant contribution to the conductivity in the lower mantle will depend on the temperature profile. We have studied the behaviour of KCaF3-perovskite at a similar range of relative compressions ( V / V o of 0.9-0.7) as are probably applied to (Mg, Fe)SiO3-perovskite in the lower mantle (Knittle and Jeanloz, 1987). The lower mantle temperature profile and the melting curve of (Mg, Fe)SiO3-perovskite are poorly constrained to plus or minus 800 K and plus or minus 300 K, respectively (Heinz and Jeanloz, 1987;
Knittle and Jeanloz, 1989; Poirier, 1991). In addition, more recent experiments by Zerr and Boehler (1993) determined much higher melting points, although there is considerable dispute (Zerr and Boehler, 1994; Heinz et al., 1994). However, despite this uncertainty, we suggest that it is unlikely that (Mg, Fe)SiO3-perovskite in the lower mantle will be close enough to its melting temperature to be within its fast-ion conduction region. A possible exception to this is in the D" layer at the base of the lower mantle where (if the silicate perovskite phase is still stable in this region, Knittle and Jeanloz, 1991) there is probably a high thermal gradient (Poirier, 1991). Therefore, although the electrical conductivity of much of the lower mantle is probably due to electronic conduction (Peyronneau and Poirier, 1989), perhaps requiring an increased iron or water content (Li and Jeanloz, 1991a, b), there will be a contribution from fast-ion conduction in the perovskite phase if the temperature becomes high enough, for example, in the D" layer. Further experiments and simulations to study the elevated temperature and high-pressure electrical conductivity in (Mg, Fe)SiO3-perovskite and perovskite structural analogues, with the GdFeO 3 distortion (for example CaTiO 3 requiring temperatures above 1700 K) which pay particular attention to the role of impurities would help to further investigate the source of lower mantle electrical conductivity.
Acknowledgements We would like to thank Matsanori Matsui for providing us with the original molecular dynamics code and G.D. Price for reviewing the manuscript. We would also like to thank the NERC for research grant GR3/6970 and G.W.W. would like to thank the NERC for a studentship.
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