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Equation of state of MgSiO3-perovskite and MgO (periclase) from computer simulations — reply
PHYSICS O F T H E EARTH ANDPLANETARY INTERIORS
ELSEVIER
Physics of the Earth and PlanetaryInteriors 102 (1997) 293-294
Equation of state of MgSiO3-perovskite and MgO (periclase) from computer simulations - reply Anatoly B. Belonoshko * Theoretical Geochemistry Program, Institute of Earth Sciences, Uppsala University, S-752 36 Uppsala, Sweden
Received 20 March 1997; accepted25 March 1997
I am grateful to Patel et al. (1997) (hereinafter called PPM) for providing me with the opportunity to discuss this important issue once more. Before going into discussion, a few mistakes in the PPM paper should be corrected. First, conlxary to the PPM statement, Belonoshko and Dubrovinsky (1996) (hereinafter called BD) did not apply molecular dynamics (MD) to simulate pressure-volume-temperature (PVT) equation of state (EOS) of MgSiOa-perovskite. It was done by Belonoshko (1994). BD compared the PVT EOS by Belonoshko (1994) with the recent EOS developed by Patel et al., 1996 (hereinafter called PPMBH). This was explicitly stated by BD in the abstract. Second, contrary to the PPM statement, the volume thermal expansivity ( a ) was not simulated in BD. While temperature dependence of volume can indeed be extracted from a simulation at a single temperature (Rickman and Phillpot, 1991), this is not what has been done either by Belonoshko (1994) or by BD. BD calculated ot using the EOS developed by Belonoshko (1994). PPM stated that the difference in a calculated by BD and PPMBH at 0 GPa arises from (a) the particular choice of EOS and (b) the neglect of quantum
* Corresponding author.
correction by BD. BD suggested that this difference appears because PPMBH included in their set of fitted data those MD points which, according to BD, should not be included. In what follows, I show that the PPM statements are not correct. It is necessary to mention that the MD simulations by Belonoshko (1994) and by PPMBH with the Matsui 0 9 8 8 ) model of interatomic interaction are in excellent agreement. The disagreement appears at the 'post-simulation' stage. The choice of particular EOS is important when fitted data are scarce and one also needs to use the fitted EOS for extrapolation. The choice is not important when one can produce as many PVT points as necessary at any P and T, as is the case with the MD method. Therefore, practically any equation of the P = f ( T , V ) kind is suitable as long as it provides accurate description of the data. In this case, EOS is simply a substitution for the initial PVT data set. To check this (which was hardly necessary), I fitted a at 0 GPa using the same EOS but including the MD data up to 2000 K (as PPMBH did) and obtained the result in close agreement with the PPMBH data. This shows that the assertion (a) (see above) made by PPM is not correct. It would be interesting to see if PPM could reproduce the BD data by deleting their points above 1000 K. I believe that this discussion would be unnecessary had PPM done that. The quantum correction to the MD-simulated data
0031-9201/97/$17.00 © 1997 Published by Elsevier Science B.V. All fights reserved. PII S0031-9201(97)00019-8
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A.B. Belonoshko / Physics of the Earth and Planeta O" Interiors 102 (1997) 293 294
set (Belonoshko, 1994) would lead to a decrease of a and the difference between PPMBH and BD a would even be larger. This is because the quantum correction is larger at lower temperatures and smaller at higher temperatures. Therefore, account for the correction leads to a smaller change in volume with temperature and, correspondingly, to a decrease of a . Because PPMBH obtained a , which is higher than the a obtained by BD, the account for the correction, if it had been done by BD, would lead to an increase of the disagreement and not to a better agreement, contrary to the PPM suggestion. Hence, the assertion (b) (see above) made by PPM is incorrect. Even though quantum correction does change the volume, the change is very small and has practically no influence on the magnitude of a . For example, the magnitude of V2000,,-Vs00. ~, according to PPMBH (Appendix A) with and without correction is equal to 2.0947 and 2.0519, respectively. This gives us less than 2.5% change in effective thermal expansion. This is hardly the difference which is worthwhile discussing, because change of 0.05 c m 3 / m o l e when molar volume of a substance is about 25 c m 3 / m o l e is practically within the precision of calculations. Moreover, if interatomic potential was fitted to experimental data without taking quantum correction into account, it would be erroneous to take quantum correction into account afterwards. The potential developed by Matsui (1989) is so called 'effective' potential. It means that whatever the reasons (quantum or classical) for some particular properties (volume, structure, elastic constants, etc.) to be as they are, they will, or, rather, should be 'effectively' taken into account when fitting potential to experimental data. The Eq. (1) (Patel et al., 1997) is empirical, as are the EOSs used in B D and P P M B H . Given that, discussion of values o f fitted parameters, unless they have thermodynamic meaning (for example, volume or ct, which was calculated a s ( 1 / v X o V / O T ) p ...... t has no sense. Therefore, the comment by PPM that 'it is hard to understand' why the coefficient a3 has large negative value, is hard to understand. This
coefficient has no thermodynamic sense and should be treated simply as a coefficient of the empirical EOS only. Even a discussion of the magnitude of n 0 is not acceptable, inasmuch as it was not the purpose to fit PVT data at 0 K, and n 0 could have any value. As I have demonstrated, the comments by PPM are not justified. Hence, the explanation provided by BD should be preferred. When it comes to choosing particular EOS, I can say that this is a matter of purpose for which EOS will be applied. Belonoshko (1994), as it was demonstrated by BD, predicted experimental data by Funamori et al. (1996) nearly perfect. Therefore, if the purpose is to describe results o f M D simulations, including those in the metastable PT range, one should choose the PPMBH EOS. If the intent is to obtain precise PVT data on perovskite under conditions of the deep Earth interior, one should choose the EOS developed by Belonoshko (1994).
References Belonoshko, A.B., 1994. Molecular dynamics of MgSiO3-perovskite at high pressures: Equation of state, structure and melting transition. Geochim. Cosmochim. Acta. 58, 40394047. Belonoshko, A.B., Dubrovinsky, L.S., 1996. Equations of state of MgSiO3-perovskiteand MgO (periclase) from computer simulation. Phys. Earth Planet. Inter. 98, 47-54. Funamori, N., Yagi, T., Utsumi, W., Kondo, T., Uchida, T., Seki, M., 1996. Thermoelastic properties of MgSiO3-perovskitedetermined by in situ X-ray observations up to 30 GPa and 2000 K. K.J. Geophys. Res. 101, 8257-8270. Matsui, M., 1988. Molecular dynamics study of MgSiO3-perovskite. Phys. Chem. Minerals 16, 234-238. Matsui, M., 1989. Molecular dynamics study of the structural and thermodynamic properties of MgO crystal with quantum correction. J. Chem. Phys. 91,489--494. Patel, A., Price, U.D., Matsui, M., Brodholt, J.P., Howarth, R.J., 1996. A computer simulation approach to the high pressure thermoelasticity of MgSiO3-perovskite. Phys. Earth Planet. Inter. 98, 55-63. Patel, A., Price, U.D., Matsui, M., 1997. Equations of state of MgSiO3-perovskiteand MgO (periclase) from computer simulation - discussion. Phys. Earth Planet. Inter., this volume. Rickman, J.M., Phillpot, S.R., 1991. Temperature dependence of thermodynamic quantities from simulations at a single temperature. Phys. Rev. Lett. 66, 349-352.
ELSEVIER
Physics of the Earth and PlanetaryInteriors 102 (1997) 293-294
Equation of state of MgSiO3-perovskite and MgO (periclase) from computer simulations - reply Anatoly B. Belonoshko * Theoretical Geochemistry Program, Institute of Earth Sciences, Uppsala University, S-752 36 Uppsala, Sweden
Received 20 March 1997; accepted25 March 1997
I am grateful to Patel et al. (1997) (hereinafter called PPM) for providing me with the opportunity to discuss this important issue once more. Before going into discussion, a few mistakes in the PPM paper should be corrected. First, conlxary to the PPM statement, Belonoshko and Dubrovinsky (1996) (hereinafter called BD) did not apply molecular dynamics (MD) to simulate pressure-volume-temperature (PVT) equation of state (EOS) of MgSiOa-perovskite. It was done by Belonoshko (1994). BD compared the PVT EOS by Belonoshko (1994) with the recent EOS developed by Patel et al., 1996 (hereinafter called PPMBH). This was explicitly stated by BD in the abstract. Second, contrary to the PPM statement, the volume thermal expansivity ( a ) was not simulated in BD. While temperature dependence of volume can indeed be extracted from a simulation at a single temperature (Rickman and Phillpot, 1991), this is not what has been done either by Belonoshko (1994) or by BD. BD calculated ot using the EOS developed by Belonoshko (1994). PPM stated that the difference in a calculated by BD and PPMBH at 0 GPa arises from (a) the particular choice of EOS and (b) the neglect of quantum
* Corresponding author.
correction by BD. BD suggested that this difference appears because PPMBH included in their set of fitted data those MD points which, according to BD, should not be included. In what follows, I show that the PPM statements are not correct. It is necessary to mention that the MD simulations by Belonoshko (1994) and by PPMBH with the Matsui 0 9 8 8 ) model of interatomic interaction are in excellent agreement. The disagreement appears at the 'post-simulation' stage. The choice of particular EOS is important when fitted data are scarce and one also needs to use the fitted EOS for extrapolation. The choice is not important when one can produce as many PVT points as necessary at any P and T, as is the case with the MD method. Therefore, practically any equation of the P = f ( T , V ) kind is suitable as long as it provides accurate description of the data. In this case, EOS is simply a substitution for the initial PVT data set. To check this (which was hardly necessary), I fitted a at 0 GPa using the same EOS but including the MD data up to 2000 K (as PPMBH did) and obtained the result in close agreement with the PPMBH data. This shows that the assertion (a) (see above) made by PPM is not correct. It would be interesting to see if PPM could reproduce the BD data by deleting their points above 1000 K. I believe that this discussion would be unnecessary had PPM done that. The quantum correction to the MD-simulated data
0031-9201/97/$17.00 © 1997 Published by Elsevier Science B.V. All fights reserved. PII S0031-9201(97)00019-8
294
A.B. Belonoshko / Physics of the Earth and Planeta O" Interiors 102 (1997) 293 294
set (Belonoshko, 1994) would lead to a decrease of a and the difference between PPMBH and BD a would even be larger. This is because the quantum correction is larger at lower temperatures and smaller at higher temperatures. Therefore, account for the correction leads to a smaller change in volume with temperature and, correspondingly, to a decrease of a . Because PPMBH obtained a , which is higher than the a obtained by BD, the account for the correction, if it had been done by BD, would lead to an increase of the disagreement and not to a better agreement, contrary to the PPM suggestion. Hence, the assertion (b) (see above) made by PPM is incorrect. Even though quantum correction does change the volume, the change is very small and has practically no influence on the magnitude of a . For example, the magnitude of V2000,,-Vs00. ~, according to PPMBH (Appendix A) with and without correction is equal to 2.0947 and 2.0519, respectively. This gives us less than 2.5% change in effective thermal expansion. This is hardly the difference which is worthwhile discussing, because change of 0.05 c m 3 / m o l e when molar volume of a substance is about 25 c m 3 / m o l e is practically within the precision of calculations. Moreover, if interatomic potential was fitted to experimental data without taking quantum correction into account, it would be erroneous to take quantum correction into account afterwards. The potential developed by Matsui (1989) is so called 'effective' potential. It means that whatever the reasons (quantum or classical) for some particular properties (volume, structure, elastic constants, etc.) to be as they are, they will, or, rather, should be 'effectively' taken into account when fitting potential to experimental data. The Eq. (1) (Patel et al., 1997) is empirical, as are the EOSs used in B D and P P M B H . Given that, discussion of values o f fitted parameters, unless they have thermodynamic meaning (for example, volume or ct, which was calculated a s ( 1 / v X o V / O T ) p ...... t has no sense. Therefore, the comment by PPM that 'it is hard to understand' why the coefficient a3 has large negative value, is hard to understand. This
coefficient has no thermodynamic sense and should be treated simply as a coefficient of the empirical EOS only. Even a discussion of the magnitude of n 0 is not acceptable, inasmuch as it was not the purpose to fit PVT data at 0 K, and n 0 could have any value. As I have demonstrated, the comments by PPM are not justified. Hence, the explanation provided by BD should be preferred. When it comes to choosing particular EOS, I can say that this is a matter of purpose for which EOS will be applied. Belonoshko (1994), as it was demonstrated by BD, predicted experimental data by Funamori et al. (1996) nearly perfect. Therefore, if the purpose is to describe results o f M D simulations, including those in the metastable PT range, one should choose the PPMBH EOS. If the intent is to obtain precise PVT data on perovskite under conditions of the deep Earth interior, one should choose the EOS developed by Belonoshko (1994).
References Belonoshko, A.B., 1994. Molecular dynamics of MgSiO3-perovskite at high pressures: Equation of state, structure and melting transition. Geochim. Cosmochim. Acta. 58, 40394047. Belonoshko, A.B., Dubrovinsky, L.S., 1996. Equations of state of MgSiO3-perovskiteand MgO (periclase) from computer simulation. Phys. Earth Planet. Inter. 98, 47-54. Funamori, N., Yagi, T., Utsumi, W., Kondo, T., Uchida, T., Seki, M., 1996. Thermoelastic properties of MgSiO3-perovskitedetermined by in situ X-ray observations up to 30 GPa and 2000 K. K.J. Geophys. Res. 101, 8257-8270. Matsui, M., 1988. Molecular dynamics study of MgSiO3-perovskite. Phys. Chem. Minerals 16, 234-238. Matsui, M., 1989. Molecular dynamics study of the structural and thermodynamic properties of MgO crystal with quantum correction. J. Chem. Phys. 91,489--494. Patel, A., Price, U.D., Matsui, M., Brodholt, J.P., Howarth, R.J., 1996. A computer simulation approach to the high pressure thermoelasticity of MgSiO3-perovskite. Phys. Earth Planet. Inter. 98, 55-63. Patel, A., Price, U.D., Matsui, M., 1997. Equations of state of MgSiO3-perovskiteand MgO (periclase) from computer simulation - discussion. Phys. Earth Planet. Inter., this volume. Rickman, J.M., Phillpot, S.R., 1991. Temperature dependence of thermodynamic quantities from simulations at a single temperature. Phys. Rev. Lett. 66, 349-352.