Strong temperature dependence of the first pressure derivative of isothermal bulk modulus at zero pressure

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jb004865, 2007 Strong temperature dependence of the first pressure derivative of isothermal bulk modulus at zero pressure Yigang Zhang, 1 Dapeng Zhao, 2 Masanori Matsui, 3 and Guangjun Guo 1 Received 20 November 2006; revised 6 June 2007; accepted 29 August 2007; published 27 November [1] The first pressure derivative of isothermal bulk modulus at zero pressure (K 0 0T) isa key equation of state (EOS) parameter. Its variation with temperature is poorly constrained and may influence the inference of composition and temperature of the deep Earth. In the present study, molecular dynamics simulations are performed to derive the K 0 0T of MgO from 300 to 3000 K using directly the definition of K 0 0T and without relying on any EOS. The most important finding of the present study is that K 0 0T depends strongly on temperature. The cross 2 K 0T /@P@T is found to be 5.1 ± K 1 at 300 K, passing to 9.9 ± K 1 at 1600 K, and reaches 15.1 ± K 1 at 3000 K. The value at 300 K agrees with 3.9 ± K 1 of Isaak (1993) within the uncertainty. The experimental adiabatic cross derivative of Chen et al. (1998), which is 2.7 ± K 1 and often considered too high, is not far from our value of 1.1 ± K 1 at 1600 K when the experimental and calculation uncertainties are considered. The obtained K 0 0T are further compared with those from EOS fittings to infer the valid temperature domains of several commonly used EOSs. With this knowledge, the consistency of zero-pressure isothermal bulk modulus (K 0T ) from different experimental techniques (direct resonance measurements and fitting isothermal P-V data by an EOS) is demonstrated. Citation: Zhang, Y., D. Zhao, M. Matsui, and G. Guo (2007), Strong temperature dependence of the first pressure derivative of isothermal bulk modulus at zero pressure, J. Geophys. Res., 112,, doi: /2006jb Introduction [2] Equations of state (EOSs) are used widely in studies of the deep Earth for making extrapolations and inferences. Validity of the extrapolations and inferences requires that EOS parameters (zero-pressure volume V 0T, bulk modulus K 0T, and its first pressure derivative K 0 0T, etc.) represent real properties of the material under consideration instead of merely fitting parameters. If fitting of the lower mantle pressure-volume (P-V) or bulk modulus-volume (K-V) data by an EOS does not give real material properties in the fitting parameters, subsequent comparison of the fitting parameters with experimentally derived real material properties will lead to incorrect inference of lower mantle composition. Similarly, if parameters of an EOS are not representative of real material properties, putting real material properties into the EOS will lead to wrong predictions. It may be argued that fitting parameters can be obtained from a data set and then used in the same EOS for making extrapolations. As will be shown in the present study, the fitting parameters in such a case have a strong dependence 1 State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China. 2 Department of Geophysics, Tohoku University, Sendai, Japan. 3 School of Science, University of Hyogo, Kamigori, Hyogo, Japan. Copyright 2007 by the American Geophysical Union /07/2006JB004865$09.00 on the pressure range of the data set, which makes the extrapolations again uncertain. In all cases, the real meaning of EOS parameters needs to be considered carefully. [3] The way to check whether EOS parameters obtained by fitting P-V or K-V data are consistent with real material properties is certainly to compare the parameters with the corresponding properties measured directly, such as by sound velocity or resonance measurements. In practice, however, verification of the consistency is difficult, due to (1) scarcity of experimental data points, especially at high temperatures, (2) inaccuracy of experimental data caused by pressure scales, (3) trade-offs among fitted parameters, especially when pressure range of the experimental data is limited, and (4) the assumptions used in the fittings such as that K 0 0T is independent of temperature. Consequently, parameters derived using different techniques are often found to be inconsistent (e.g., Dewaele et al. [2000] for the temperature dependence of K 0T ). [4] The purpose of the present study is to obtain K 0 0T by using directly its definition and without relying on any EOS. The K 0 0T values are obtained from 300 to 3000 K, from which the temperature dependence of K 0 0T is derived. For applications, these K 0 0T values are compared with those from EOS fittings to infer the temperature domains where EOS parameters are representative of real material properties. With the information available, the K 0 0T are further used to establish the consistency of different experimental techniques. 1of8

2 ZHANG ET AL.: TEMPERATURE DEPENDENCE OF K0 0 [5] P-V data used for deriving K 0 0T and for making EOS fittings are from molecular dynamics (MD) simulations. The mineral periclase (MgO) is used as an example. As shown by Matsui et al. [2000] and Matsui and Nishiyama [2002], MD simulations can predict very accurately the pressure-volume-temperature (P-V-T) relation and other thermo-elastic properties of MgO. [6] The paper is organized as follows. In the next section, technical details of the molecular dynamics simulations are given, including how K 0 0T is derived by using directly its definition. In the Results and Discussion section, first of all, the accuracy of MD simulations is further demonstrated using data recently available from experiments and thermodynamic calculations. Next, K 0 0T is calculated at different temperatures by using directly its definition and compared with experimental data wherever possible. From this, the temperature dependence of K 0 0T is derived. For applications of the K 0 0T values obtained in the present study, they are first compared with those from EOS fittings to constrain the temperature ranges where EOSs can give correct K 0 0T. Two more applications are given in the final sections to show the strong dependence of EOS parameters on the pressure range of the P-V data used and the consistency of different experimental techniques. 2. Calculation Methods [7] The pairwise interatomic potential model, the breathing shell model of oxygen, and the quantum correction employed in the present study have been described in detail in the previous investigations [Matsui, 1989, 1998; Matsui et al., 2000]. [8] The pairwise interatomic potential model has the following form: V r ij ¼ q i q j r ij C ic j r 6 ij þ f B i þ B j A i þ A j r ij exp B i þ B j where f is 1 kcalå 1 mol 1, q the charge, A the repulsive radius, B the softness parameter, and C the van der Waal s coefficient. The model parameters for Mg and O are given by Matsui et al. [2000]. [9] The breathing shell potential model of Oxygen has the exponential form [Matsui et al., 2000]: U bre ¼ X i ð1þ fb i exp½ða i0 A i Þ=b i Š ð2þ where b i is the breathing parameters of ion i, and A i0 is the repulsive radius of ion i at which the derivative of the breathing bre /@A i,isf. [10] The quantum correction to pressure is estimated based on the Wigner-Kirkwood expansion of free energy in terms of Planck constant and the differentiation of the free energy with respect to volume at constant temperature [Matsui, 1989]. Calculations employing the quantum correction reproduce very well the true low-temperature behaviors of the thermal expansion and heat capacity of MgO [Matsui, 1989]. [11] MD simulations are performed in the constanttemperature and constant-pressure (NPT) ensemble employing the Nosé thermostat and Parrinello and Rahman constantstress method. Control parameters of the simulations, such as the size of the system (512 ions), length of the time step (0.4 fs), the fictitious mass constant for breathing oxygen (60 g/mol), are the same as those by Matsui et al. [2000]. The scaling and equilibrating periods are both 100,000 steps, and 600,000 steps are used to obtain the average pressures and volumes. [12] K 0T and K 0 0T are calculated directly by their definitions, K 0T K0T 0 T T at P ¼ 0 [13] Simulations at pressures from 4 and 4 GPa with an interval of 0.5 GPa are made to obtain the volumes. These V-P data are fitted with 2nd-order polynomials to give K T as a function of P. The K T -P data are then fitted with 2nd-order polynomials to calculate K 0 0T. Checks are made with more intense data points (e.g., with a pressure interval of 0.25 GPa), different pressure ranges (e.g., 2 to 2 GPa), and different polynomial degrees to ensure that the obtained results are not influenced by these factors. 3. Results and Discussions 3.1. Accuracy of Molecular Dynamics Simulations [14] Matsui et al. [2000] demonstrate that MD simulations predict accurately the P-V-T properties of MgO. For example, molar volume is different by less than 0.3% from the experimental data of Dubrovinsky and Saxena [1997] between 300 and 3000 K at 0 GPa. The MD simulated volume compression at 300 K agrees very well up to 100 GPa with the experimental data of Fei [1999] and Duffy et al. [1995]. This is further confirmed in the lowpressure range by the recent experimental work of Li et al. [2006], which uses an experimental technique independent of any pressure standard. The pressure difference between the data of Li et al. [2006] and those of Matsui et al. [2000] is only 0.15 GPa at 21 GPa and 300 K. [15] Matsui and Nishiyama [2002] further show that the relative volumes (V/V 0 ) obtained by MD calculations employing ab-initio variational induced breathing potential [Cohen, 2000] are in close agreement with those using Matsui et al. [2000] empirical potential model along 300, 1000, and 2000 K isotherms. [16] Recently, Dorogokupets and Oganov [2007] derive the P-V-T relations of MgO using thermodynamic and thermoelastic properties at zero pressure and shock wave experimental data. Their data are also in close agreement with those of Matsui et al. [2000] along the 300 K isotherm up to 120 GPa, the highest pressure used in the present study. Along high-temperature isotherms, the agreement is generally very good below 40 GPa, but some discrepancy is seen at high pressures. For example, at 3000 K and 120 GPa, the difference in volume is 0.6%. This is equivalent to a pressure difference of 2.7% if the comparison is made at a volume of cm 3 /mol and the same temperature. The discrepancy will not influence our calculation of K 0 0T since only low-pressure data are used in deriving K 0 0T, while its ð3þ 2of8

3 ZHANG ET AL.: TEMPERATURE DEPENDENCE OF K0 0 Table 1. Thermoelatic Properties of MgO at 0 GPa T, K V 0T,cm 3 /mol Polynomial Fit 2 K 0T/@P@T, 10 4 K 1 Low-P EOS Fit b K 0T, GPa K 0 0T K 0T, GPa K 0 0T (3) 4.11(5) 5.1(1.6) 160.5(1) 4.17(3) (2) 4.24(1) 6.2(1.9) 151.3(1) 4.27(4) (5) 4.51(2) 7.0(2.0) 145.2(1) 4.50(4) (4) 4.48(2) 7.7(2.2) 138.8(1) 4.50(5) (7) 4.71(1) 8.4(2.3) 132.5(1) 4.68(5) (6) 5.00(1) 9.2(2.5) 126.3(1) 4.98(5) (6) 5.21(1) 9.9(2.7) 120.0(1) 5.21(5) (5) 5.24(1) 10.6(2.8) 113.7(1) 5.25(3) (4) 5.43(4) 11.4(3.0) 107.4(2) 5.44(6) (5) 5.75(3) 12.1(3.2) 100.9(2) 5.78(6) (4) 5.92(9) 12.9(3.3) 94.5(2) 5.97(7) (6) 6.24(5) 13.6(3.5) 88.1(1) 6.23(10) (2) 6.72(14) 14.3(3.6) 81.5(2) 6.72(15) (1) 6.78(16) 15.1(3.8) 74.4(2) 6.80(14) a Uncertainties of K 0T and K 0 0T are estimated by the differences of 2nd- and 3rd-order polynomial fittings. b Listed are the average values from different EOSs. Fittings with each EOS give uncertainties of the obtained parameters K 0T and K 0 0T. Then the standard deviations of the parameters from different EOS are calculated. The sum of the uncertainty and the standard deviation is given in the parentheses as the error of the parameters. influence on the inference of the validity range of EOSs will be discussed in the corresponding section K 0T [17] Matsui et al. [2000] calculated K 0T up to 2000 K by fitting P-V data using the 4th-order Birch-Murnaghan EOS. The K 0T derived in such a way might depend on the particular EOS used. Trade-offs between fitting parameters may also influence the results. In this study, K 0T is calculated up to 3000 K by using directly its definition and without relying on any EOS. [18] The calculated K 0T are given in the third column of Table 1 and plotted in Figure 1 together with those from previous investigations. It can be seen that our data are in close agreement with those of Isaak et al. [1989] at low temperatures, but diverge slightly at high temperatures. The difference reaches 2% (2.4 GPa) at 1800 K. The reason for the discrepancy may be two-fold. On the one hand, it may be due to the uncertainty of MD simulations. However, note that the isothermal bulk moduli of Isaak et al. [1989] are not direct experimental measurements but calculated from adiabatic ones using K T =K S /(1 + agt). Errors in the higherorder thermoelastic parameters, i.e., the thermal expansion (a) and the Grüneisen parameter (g) will lead to errors in the K T at high temperatures. No matter what the reason is, the 2% difference is regarded as the uncertainty of our K 0T and considered in subsequent calculations. [19] At further higher temperatures close to 3000 K, our K 0T is larger than those of Karki et al. [2000] with a difference of 7% (5.1 GPa) at 3000 K. The discrepancy is expected since the first-principle data are derived using the quasi-harmonic approximation, which is known to be invalid at high temperature and low pressure. At these high temperatures, our K 0T are in better agreement with those of Dorogokupets and Oganov [2007]. [20] Variation of K 0T with temperature can be described by a straight line from 300 to 3000K: K 0T ¼ 0:0317ð1ÞT þ 170:91ðÞ 2 where numbers in the parentheses give errors in the last digit. The slope ( GPa/K) is slightly smaller than ð4þ the value ( GPa/K) given by Isaak et al. [1989]. This is expected, as our K 0T are slightly smaller than those of Isaak et al. [1989] at high temperatures. Note the formula cannot be extended to lower temperatures where K 0T become nonlinear with temperature as can be seen clearly by the first-principles data of Karki et al. [2000] and the experimental data of Sumino et al. [1983] (not drawn in Figure 1 for clarity) K 0 0T Derived by Using Polynomials [21] Before presenting the results of K 0 0T, we discuss several factors that may influence the calculation of K 0 0T. The effect of the order of polynomial is first considered. As shown in Figure 2, the effect is minimum except at the highest temperatures, where the difference in K 0 0T obtained using the 2nd- and 3rd-order polynomials can reaches 0.2. The total average of the differences for all temperatures is, however, only Figure 1. Comparison of K 0T obtained in the present study with those from previous investigations. The thick solid line is a linear least squares fit of the present results (equation (4)). 3of8

4 ZHANG ET AL.: TEMPERATURE DEPENDENCE OF K0 0 the sign of its change with temperature is uncertain. The discrepancies reflect the fact that the cross derivative is poorly constrained; However, our result at 300 K (5.1 ± K 1 ) agree, within the estimated uncertainty, with that of Isaak et al. [1993] (3.9 ± K 1 ) derived using experimental data and thermodynamic analyses. The highest value from experimental data, 2.7 ± K 1, given by Chen et al. [1998] is 2 K S /@P@T. Using the equations (3.26), (3.30), and (1.47) of Anderson [1995] which are: KS 0 ¼ K0 T ð1 þ agt Þ agt ð d T þ qþ q ¼ d T KT 0 þ 1 d T ¼ ak P ð6þ Figure 2. Comparison of K 0 0T obtained by using 2nd-order and 3rd-order polynomials. The thin straight line has a slope of 1. Filled squares are obtained by using P-V data in the pressure range from 4 to 4 GPa. Open circles are calculated using the 3rd-order polynomial but with data in the ranges from 2 to 2 GPa and from 4 to 4 GPa, respectively. [22] Next, the effects of data intensity and range are considered. At 2600, 2800, and 3000 K, additional calculations are made with a pressure interval of 0.25 GPa and in the pressure range from 2 to 2 GPa. As shown in Figure 2, the difference with those calculated by using data between 4 and 4 GPa is small (0.08 at both 2600 and 2800 K); however, at 3000 K, the difference reaches The scatters of K 0 0T in Figure 2 shows clearly that K 0 0T at 2800 K and 3000 K are less accurate than those calculated at other temperatures. [23] The MD calculated K 0 0T are given in the fourth column of Table 1 and plotted in Figure 3a. Our K 0 0T at 300 K, which is 4.11, is within the range of previous results as can be seen when compared with data in the Table 2 of Li et al. [2006]. Further comparison with experimental data at higher temperatures is difficult due to the facts that direct sound measurement of the parameter at high temperatures are still lacking and that previous high-temperature K 0 0T are often derived using an EOS, which may depend on the data range, the specific EOS used, and the potential trade-offs among fitting parameters. [24] Figure 3a shows clearly that K 0 0T varies strongly and nonlinearly with temperature. Fitting K 0 0T with temperature gives: K0 0 ðtþ ¼ 4:0 ð 1 Þþ4:0 ð 14 Þ10 4 T þ 1:8ð4Þ10 7 T 2 ð5þ where numbers in the parentheses give errors in the last digit. Differentiation of the function with respect to temperature gives the cross 2 K 0T /@P@T. It is 5.1 ± at 300 K, passing to 9.9 ± at 1600 K, and finally reaches 15.1 ± at 3000 K. Thus the cross derivative also depends strongly on temperature. [25] There are large discrepancies in the previous inferences of the cross derivative. Li et al. [2005] shows that even Figure 3. Comparison of K 0 0T obtained using its definition with those from EOS fittings using (a) MD and (b) Dorogokupets and Oganov [2007] P-V data from 0 to 120 GPa. The thick solid line is from equation (5). 4of8

5 ZHANG ET AL.: TEMPERATURE DEPENDENCE OF K0 0 our K 0 0T can be transformed into K 0 0S. The thermoelastic properties needed for the transformation can be estimated using MD simulations, including the thermal expansion coefficient a, bulk modulus K 0T, its variation with temperature (@K 0T /@T) P, and the Grüneisen parameter estimated by g = g 0 (V/V 0 ) q with g 0 taken to be 1.54 [Isaak et al., 1989]. The cross 2 K 0S /@P@T estimated from 2 K 0T /@P@T value at 1600 K is 1.1 ± K 1. This value is not too far from the experimental datum of Chen et al. [1998] when the experimental and calculation uncertainties are both considered. All these data indicate a strong temperature dependence of K 0 0T. [26] Many studies [e.g., Bina and Helffrich, 1992] have used the so-called high-temperature Birch-Murnaghan EOS in which K 0 0T is assumed to be independent of temperature. Our results show that this approximation is clearly incorrect. This may help us to understand discrepancies observed in previous experimental studies. For example, Dewaele et al. [2000], assuming K 0 0T independent of temperature, find it difficult to reconcile their dk 0T /dt value ( GPa/K) with that of Isaak et al. [1989] ( GPa/K). With our results, it can be seen clearly that the larger K 0T at high temperatures shown by Dewaele et al. [2000] is caused by the smaller K 0 0T they use, which should increase with temperature instead of remaining constant. [27] Neglecting the variation of K 0 0T with temperature will have a strong effect on the inference of lower mantle composition and temperature [e.g., Chen et al., 1998; Aizawa and Yoneda, 2006]. As K 0 0T describes the variation rate of bulk modulus with pressure, using a smaller K 0 0T in an EOS will lead to lower estimate of bulk modulus of a mineral at the Earth s lower mantle condition. To match with seismological models, the underestimated bulk modulus would have to be increased, either by lowering temperature or increasing perovskite content of the lower mantle, leading to errors in these two parameters K 0 0T Derived by Using EOSs [28] In the previous section, K 0 0T is derived using polynomials. It may be argued that K 0 0T derived in such a way may depend on the functional form. In this section, we demonstrate that this is not the case by deriving K 0 0T using EOSs. The P-V data used in the fittings are kept the same as in the previous section. The EOSs used include the 3rdorder Birch-Murnaghan and Logarithmic, the Vinet EOS. The functional forms of these EOSs can be found, for example, in the study by Vinet et al. [1987], Poirier and Tarantola [1998], and Stacey and Davis [2004]. K 0T and K 0 0T are both relaxed during the fittings. Tests are also made to see the effect of using experimental K 0T on the derived K 0 0T. At 1800 K, using the experimental K 0T (116.6 GPa) instead of MD K 0T (114.2 GPa) creates a difference of only 0.06 in K 0 0T, which is similar, in magnitude, to the uncertainties we give to the K 0 0T based on the errors of the fitting processes and the standard deviations among different EOSs. [29] The obtained K 0T and K 0 0T are given in the 6th and 7th columns of Table 1. Using different EOSs makes no difference in their values, as can be seen by the uncertainties of data from different EOSs. More importantly, K 0T and K 0 0T derived from polynomial and EOS fittings are almost identical. This agreement demonstrates clearly that K 0 0T obtained in the present study is independent of the functional forms used for its derivation Valid Temperature Domains of EOSs [30] In this section, MD P-V data from 0 to 120 GPa are fitted with different EOSs to obtain K 0 0T; then, the K 0 0T values are compared with those derived in the previous section to constrain the temperature range in which an EOS can give correct K 0 0T. The chosen EOSs include, in addition to the ones used in the previous section, the Keane EOSs [Stacey and Davis, 2004]. The 4th-order Birch-Murnaghan and logarithmic EOSs are left out because the potential trade-off between K 0 and the second pressure derivative of bulk modulus (K 00 0T) prevents us from examining the validity of these EOSs. For these EOSs, calculation of K 00 0T is even more difficult than that of K 0 0T and requires a future investigation. For the Keane EOS, all P-V data along different isotherms are fitted together to take the advantage 0 that all isotherms have a common K 1 [Stacey and Davis, 2004]. [31] During the fittings, K 0T values are fixed at the values given in Table 1 to minimize the effect of the trade-offs between EOS parameters and to compare K 0 0T from different EOSs on the same basis. The potential error of K 0T (2%) is also considered. When the experimental K 0T at 1800 K is used in the fittings, the resulted K 0 0T is different, on average, by 0.17 from that when MD K 0T is used. The value 0.17 is used as a criterion for constraining the validity ranges of EOSs. In other words, if the difference between real K 0 0T and that from an EOS is smaller than 0.17, the EOS is considered valid in the temperature domain. [32] The K 0 0T derived using different EOSs and those from polynomial fittings are plotted in Figure 3a. Obviously, K 0 0T of different EOSs are already different at 300 K, and they diversify largely at higher temperatures. [33] In the low-temperature domain, the 3rd-order Birch- Murnaghan EOS gives the lowest K 0 0T and also the slowest change of K 0 0T with temperature. Thus fitting experimental data with the EOS at the assumption of constant K 0 0T may work approximately if the temperature is not too high. The Vinet EOS works in the temperature range from 300 to 1200 K. In comparison, the 3rd-order logarithmic and the Keane EOSs tend to overestimate K 0 0T in the low-temperature domain. [34] In the intermediate temperature range from 800 to 1400 K, the Keane EOS works and can give the real K 0 0T. Between 1400 and 2200 K, it is the 3rd-order logarithmic EOS that can give the most correct EOS parameter. [35] To check whether above observations are influenced by the MD P-V data used, P-V data from Dorogokupets and Oganov [2007] are also fitted with the EOSs to obtain K 0 0T. For the fittings, K 0T is fixed at the values given by Dorogokupets and Oganov [2007, Figure 7]. K 0T at 3000 K is not presented in the figure so our MD value is used. The resulted K 0 0T is plotted in Figure 3b. Obviously, the general trends and the valid ranges for the three EOSs are quite similar to those observed in Figure 3a, except for one feature that the 3rd-order logarithmic EOS is valid from 1400 K all the way up to 3000 K. [36] Observations made above are important because they allow us to choose the right EOS to extrapolate correctly measured properties (bulk modulus, density, etc.) from low 5of8

6 ZHANG ET AL.: TEMPERATURE DEPENDENCE OF K0 0 Figure 4. Dependence of K 0T on the pressure range of P-V data. The horizontal axis represents that the pressure of the data set goes from 0 GPa to the one marked on the axis. 3BM denotes the 3rd-order Birch-Murnaghan EOS, V the Vinet EOS, and 3LOG the 3rd-order logarithmic EOS. Error bars give the uncertainties of K 0T from the fitting processes. to high pressures. In other words, we can use K 0T and K 0 0T, measured at zero pressure and different temperatures, in an EOS, chosen appropriately based on the above information, and make simple and straightforward extrapolations to very high pressures, at which experimental measurements are difficult to perform. This is the purpose an EOS is invented for. Also, our evaluation of EOSs in different temperature domains makes the purpose realizable Dependence of K 0T on Pressure Range [37] The previous section uses K 0 0T to examine the valid temperature domains for several commonly used EOSs. There are other applications of the K 0 0T obtained in the present study. In this section, we consider how EOS parameters depend on the pressure range of the P-V data used for their derivation. This is an important issue since experimental P-V data are often limited to a certain pressure range, especially at high temperatures. A clearer understanding of the dependence will help us to obtain and apply EOS parameters more appropriately. [38] MD P-V data along the 1800 K isotherm are used as an example. The K 0 0T is fixed at the real value (5.29, calculated by equation (5)). P-V data in the ranges [0,P i ] with P i = 4, 10, 20,..., 120 GPa are fitted sequentially with the 3rd-order Birch-Murnaghan, 3rd-order logarithmic, and Vinet EOSs. The values of K 0T from the fittings are plotted in Figure 4. It shows that K 0T values derived using the 3rd-order logarithmic EOS have negligible dependence on the pressure range of the P-V data, with a maximum difference of 1 GPa. In contrast, K 0T derived from the 3rdorder Birch-Murnaghan and the Vinet EOS show strong dependence on the pressure range, with the maximum differences of K 0T reaching 14 and 9 GPa, respectively for the two EOSs. As can be seen from Figure 3, the 3rdorder logarithmic EOS can give the real K 0 0T at 1800 K while the two other EOSs give lower K 0 0T than the real value. Care thus must be exerted to make sure that an EOS is applied in its valid temperature domain; otherwise, the strong dependence of EOS parameters on the pressure range of the data will make both the EOS parameters and the extrapolations using the parameters highly uncertain. Consideration of EOS parameters simply as fitting parameters is clearly insufficient for reliable predictions Consistency of Experimental Data [39] With the help of data and observations made in previous sections, we demonstrate in this section that directly measured K 0T, such as those by the resonance method [Isaak et al., 1989], and indirectly obtained K 0T, such as those from fitting P-V data using an EOS are consistent if EOSs are applied appropriately. [40] The experimental P-V data sets chosen for obtaining K 0T, their pressure range, and the temperatures of the isotherms are listed in Table 2. To avoid the trade-offs of fitting parameters, which is particularly severe when the number and the pressure range of the P-V data points are limited, V 0T and K 0 0T are both fixed, leaving K 0T as the only parameter to be defined. V 0T are obtained by fitting the experimental data of Dubrovinsky and Saxena [1997] and Fiquet et al. [1999] together, V 0T ¼ exp 2:414ðÞþ2:9 3 ðþ T þ 0:5 1:3ðÞ T 2 1:2ð8Þ=T ; and K 0 0T by the values from equation (5). For ferropericlase, only K 0 0T is fixed because the thermal expansion data at zero pressure are not available. [41] The obtained K 0T are listed in Table 2, and plotted in Figure 5, together with those of Isaak et al. [1989] and the present study. Within uncertainties, K 0T derived from fitting P-V data by EOSs are in general very consistent with those from direct experimental measurements [Isaak et al., 1989] and the present MD calculations, and the consistency extends up to high temperatures. However, there is one Table 2. Pressure-Volume Datasets Used for Obtaining K 0T T, K P Range, GPa ndp a Material Ref b EOS c K 0, GPa MgO [1] V 164.3(6) MgO [1] K 135.9(7) (Mg 0.6 Fe 0.4 )O [2] K 132(1) (Mg 0.64 Fe 0.36 )O [3] K 139(3) (Mg 0.64 Fe 0.36 )O [3] 3log 124(3) (Mg 0.64 Fe 0.36 )O [3] 3log 109(3) (Mg 0.64 Fe 0.36 )O [3] 3log 113(8) 2000 d MgO [4] 3log 107.6(7) 2200 d MgO [4] 3log 100.6(8) MgO [5] K 136(1) MgO [5] K 124.8(7) MgO [5] 3log 102.0(3) MgO [6] V 160.9(3) a Number of data points available. b References: [1] Fei [1999]; [2] Zhang and Kostak [2002], [3] van Westrenen et al. [2005]; [4] Dewaele et al. [2000]; [5] Utsumi et al. [1998]; [6] Speziale et al. [2001]. c K denotes the Keane EOS, V the Vinet EOS, and 3log the 3rd-order logarithmic EOS. The choice of EOS for the fittings is based on the information from Figure 3, i.e., the best EOS that is valid at the temperature of the isotherm. For the Keane EOS, the bulk modulus at infinite compression is 2.54, calculated in the present study. d P-V data in the temperature range 2000 ± 50 or 2200 ± 50 K are collected for fittings along the two isotherms. ð7þ 6of8

7 ZHANG ET AL.: TEMPERATURE DEPENDENCE OF K0 0 parameters and for making extrapolations based on the parameters. [47] (4) Establishment of the consistency of experimental data measured by different techniques is possible once an appropriate EOS is chosen based on the validity of its parameters in the specific temperature domain under consideration. [48] Acknowledgments. This work is supported by a grant ( and ) to Y. G. Zhang from the National Natural Science Foundation of China, grants (Kiban-B , Kiban-A ) to D. Zhao, and a grant ( ) to M. Matsui from Grant-in-Aid for Scientific Research from the Japanese Ministry of Education and Science (Monkasho), Japan. The first author (YGZ) appreciates the visiting professorship at Geodynamics Research Center, Ehime University. D. Zhao received a special grant for Center of Excellence (COE) from the President of Ehime University. We also thank A. Yamada and Liu Hua for their helps with computer facilities. Insightful comments by Yingwei Fei, Prof. Lars Stixrude, and two anonymous reviewers have greatly improved the manuscript. Figure 5. Comparison of K 0T directly measured [Isaak et al., 1989] and calculated (the present study) with those from fitting experimental P-V data using EOSs. Symbols with no error bar signify that errors are smaller than the size of the symbols. exception. The data sets of Utsumi et al. [1998] give lower estimates of K 0T, in particular at high temperatures, which may be caused by the narrow pressure range of the P-V data available. [42] The consistency, as demonstrated above, of K 0T values obtained by fitting experimental P-V data with those from experimental resonance method and our MD simulations demonstrates the coherence, up to very high temperatures, of the experimental zero-pressure volume measurements, the experimental P-V measurements along isotherms, the experimental direct resonance measurements of K 0T,andcertainly our MD calculations of both K 0T and K 0 0T. This can only be done when the real meaning of EOS parameters is well understood. 4. Conclusions [43] Molecular dynamics simulations are made in the present study to calculate the first pressure derivative of bulk modulus at zero pressure from 300 to 3000 K. The calculations are made using directly the definition of K 0 0T and without using any EOS. Several observations are made in the present study: [44] (1) K 0 0T has a strong and nonlinear dependence on temperature. 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