Analysis of volume expansion data for periclase, lime, corundum and spinel at high temperatures

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1 Bull. Mater. Sci., ol. 35, No., August, pp c Indian Academy of Sciences. Analysis of volume expansion data for periclase, lime, corundum and spinel at high temperatures BPSINGH, H CHANDRA, R SHYAM and A SINGH Department of Physics, Institute of Basic Sciences, Khandari, Agra-, India MS received December 9; revised September 11 Abstract. We have presented an analysis of the volume expansion data for periclase (MgO), lime (CaO), corundum (Al O 3 ) and spinel (MgAl O ) determined experimentally by Fiquet et al (1999) from 3K up to 3K. The thermal equation of state due to Suzuki et al (1979) and Shanker et al (1997) are used to study the relationships between thermal pressure and volume expansion for the entire range of temperatures starting from room temperature up to the melting temperatures of the solids under study. Comparison of the results obtained in the present study with the corresponding experimental data reveal that the thermal pressure changes with temperature almost linearly up to quite high temperatures. At extremely high temperatures close to the melting temperatures thermal pressure deviates significantly from linearity. This prediction is consistent with other recent investigations. A quantitative analysis based on the theory of anharmonic effects has been presented to account for the nonlinear variation of the thermal pressure at high temperatures. Keywords. olume expansion; thermal equation of state; thermal pressure; geophysical minerals. 1. Introduction The volume temperature data under isobaric conditions particularly at P = are of basic importance for investigating the thermoelastic properties of minerals at high temperatures and high pressures (Petukhov and Chekovskoi 197; Anderson and Masuda 199; Anderson et al 1995a,b). The high temperature cell parameters of periclase (MgO), lime (CaO), corundum (Al O 3 ) and spinel (MgAl O ) at room pressure have been determined from 3K up to 3K by Fiquet et al (1999) with the help of X-ray diffraction experiments using synchrotron radiation. The volume temperature data measured by Fiquet et al are considered to be more accurate and have been found to agree closely within 1 % with the thermal expansion data reported by Dubrovinsky and Saxena (1997) in case of MgO (Jacobs and Oonk 1). In order to understand the variation of volume with temperature for various minerals, we need a thermal equation of state based on the concept of thermal pressure. Recently, Singh () studied the relationships between thermal pressure and volume expansion with temperature using three different forms of phenomenological equations (Suzuki et al 1979; Shanker et al 1997; Shanker and Kushwah 1). A critical test of these equations can be performed with the help of volume temperature data reported by Fiquet et al (1999). In the present study, we consider the four minerals studied by Fiquet et al for a wide range of temperatures. For performing the calculations conveniently, we Author for correspondence (drbps.ibs@gmail.com) assume that the thermal pressure changes linearly with temperature. However, the results obtained in the present study would indicate that the thermal pressure starts to deviate from linearity at very high temperatures approaching melting temperatures for the minerals under study. The nonlinear variation of thermal pressure at high temperatures can be explained by taking into account the effect of anharmonicity.. Thermal equation of state According to Anderson (1995), the equation of state can be expressed in terms of thermal pressure as follows P(, T ) = P(, T ) + P th, (1) where P(,T ) represents the isothermal pressure volume relationship at T = T, the initial temperature. P th is the difference in the values of thermal pressure at two temperatures, i.e. P th = P th (T ) P th (T ). () At zero pressure, i.e. at P (, T ) =, (1) can be written in the following forms P th = P(, T ), (3) ( ) ( P th = ) P(, T ), () ( ) ( ) P th = P(, T ), (5) 31

2 3 B P Singh et al where is the volume at T = T and P =. The right hand sides of (3) (5) can be expanded in powers of ( ) in a Taylor series. We can expand f, a function of volume, as follows: ( ) d f f = f + ( )+ 1 d ( ) d f ( d ) +... () Terms up to quadratic in ( ) are retained. The subscript represents the values of derivatives at = and T = T. Three functions for f given by the right hand sides of (3) (5) are considered here. Using the expansion given by () we get the following expressions ( ) P th = K 1 1 K ( K + 1 ) ( 1 ), [ ( ) P th = K 1 1 K ( K 1 )( ) ]/ ( 1 P th = [ ( ) K 1 1 K ( K 3 )( ) ]/ ( 1 (7) ), () ), (9) where K and K are isothermal bulk modulus and its pressure derivative both at P =. Equations (7) (9) correspond to (3) (5), respectively and are quadratic equations in [( / ) 1]. On solving these equations we get the following expressions for volume expansion in terms of thermal pressure 1 1 = ( {1 K + 1 ) [1 K ( K + 1 ) ] } 1/ P th, (1) [ 1 = A ( A ) ] 1/ P th A A, (11) 1 where A 1 = K P th and A = K ( K 1 ). [ 1 = B ( B ) ] 1/ P th B B1, () where B 1 = K P th, and B = K (K 3) + P th. Equations (7) and (1) were derived by Shanker et al (1997) by expanding the lattice potential energy in powersof( ) in the form of Taylor series expansion. Equations () and (11) have been obtained by Singh () using the method of expanding the product of pressure and volume in power series of ( ) originally due to Suzuki et al (1979). Equations (9) and () have been obtained by Shanker and Kushwah (1) by expanding the product P( ) in the powers of ( ). In fact, (1) () can be obtained, respectively from the Taylor Series expansion for P, P, and P( ) in powers of change in volume ( ) due to the rise in temperature. At P =, the expansion of volume is controlled by thermal pressure and therefore, (7) () represent different forms of relationship between volume expansion ( / ) and thermal pressure. 3. Thermal pressure and anharmonicity Now we can calculate the values of volume expansion, /, with the help of (1) () provided the values of thermal pressure are known. It is evident from (1) that P (, T ) is a function of volume only along an isotherm T = T, we can write ( ) ( ) dp dpth = (13) dt dt where P = P(, T ). Now using the thermodynamic identity ( ) dp = αk T, (1) dt where α is thermal expansivity and K T the isothermal bulk modulus. From (13) and (1), we have ( ) dpth = αk T. (15) dt It is known that the product αk T remains nearly constant for most of the solids at higher temperatures T θ D, the Debye temperature (Anderson 1995). We can, therefore, integrate (15) to obtain P th = T θ D αk T dt = αk T (T θ D ), (1) As an empirical finding, P th is linear in T down to much lower temperatures than θ D, and we usually find empirically that the data satisfy (Anderson and Isaak 1995) P th = α K T (T T ), (17) where T = 3K. Equation (17) can be used to calculate the values of P th at different temperatures starting from room temperature provided the values of α and K T corresponding to Debye temperature are taken for each solid as input. The values of α and K T for the four solids under study at temperatures close to θ D are given in table 1. The calculated values of P th are given in tables 5. alues of volume expansion / are calculated with the help of (1) ()usingthe

3 Analysis of volume expansion data for MgO, CaO, Al O 3 and MgAl O 33 values of P th. The input data on K and K at T = 3K are also given in table 1. The results for MgO, CaO, Al O 3 and MgAl O are given in tables 5. We note from these tables that the values of volume expansion / calculated in the present study are in good agreement with the experimental data reported by Fiquet et al (1999) for the four minerals under study. In case of MgO and CaO, (1) of Shanker et al (1997) yields imaginary values at very high tempe- ratures close to the melting temperature T m. This point has already been noted earlier by Shanker et al (1999) and found it useful for developing a criterion of melting. The Singh Suzuki (11) and Shanker Kushwah () yield the results for / in fair agreement with the experimental data. Equations (7) (9) provide a method for calculating the values of thermal pressures, P th, at different temperatures using the experimental data on / reported by Fiquet et al (1999). Table 1. alues of input data (Anderson 1995) for MgO, CaO, Al O 3 and MgAl O. Thermal expansivity, α and isothermal bulk modulus, K T botharetakenatt = θ D, the Debye temperature. MgO CaO Al O 3 MgAl O K (GPa) (T = 3 K) K (T = 3 K) α (1 5 /GPa) 3 (T = 9 K) 3 9 (T = 7 K) 73 (T = 1 K) 5 (T = 9 K) K T (GPa) 1 3 (T = 9 K) 3 (T = 7 K) 31 (T = 1 K) 17 3 (T = 9 K) Table. alues of thermal pressure based on (1) and values of / calculated from (a) (1), (b) (11), (c) () and(d)(1) of MgO P th (GPa) / Table 3. alues of thermal pressure based on (1) and values of / calculated from (a) (1), (b) (11), (c) () and(d)(1) of CaO P th (GPa) /

4 3 B P Singh et al Table. alues of thermal pressure based on (1) and values of / calculated from (a) (1), (b) (11), (c) () and(d)(1) of Al O 3 P th (GPa) / Table 5. alues of thermal pressure based on (1) and values of / calculated from (a) (1), (b) (11), (c) () and(d)(1) of MgAl O P th (GPa) / Conclusions alues of P th thus calculated for MgO, CaO, Al O 3 and MgAl O are compared in figures 1 along with the values of P th estimated from (17) according to which P th varies linearly with temperature. In case of Al O 3 (figure 3) and MgAl O (figure ), the variations of P th with temperature are quite linear. The range of temperatures considered for these two solids is only up to 3K. However, in case of MgO and CaO for which the maximum temperature goes up to about 3K, the plots showing the variations of P th with T become nonlinear at temperatures T > K (figures 1 and ). It is evident from these figures that the thermal pressure deviates substantially from linearity at very high temperatures approaching the melting temperatures (T m = 373K for MgO and T m = 53K for CaO). This deviation from linearity is such that the actual values of thermal pressures become considerably less than the corresponding values predicted from the linear relationship. This prediction is consistent with the recent studies (Wang and Reeber 199; Jacobs and Oonk 1; Taravillo et al ; ( Popov/ and) Borodai 5). Wang and Reeber, have found that Pth T does not remain constant, i.e. the thermal pressure does not change linearly with T at high temperatures. They have shown that [ (P th ) / T ] remains almost constant with temperature at high T. This is related to the obser- vation that the product of thermal expansivity, α and isothermal bulk modulus, K T, decreases slowly but significantly at high T (Shanker et al 1999; Jacobs and Oonk 1). According to Wang and Reeber (199), the product αk T remains nearly constant at high T. Since the volume,, increases by about 1 15% in case of MgO and CaO (tables and 3) in the temperature range 5 3K, the values of αk T should decrease by the same amount for this temperature range. This would imply from (17) that the actual values of thermal pressure should be smaller by about 1 15%. It is worth mentioning here that when we use the smaller values of P th in the temperature range 5 3K, the Shanker (1) yields real values for / up to the temperatures approaching T m.att T m,(1) yields imaginary values for /. Equation (1) has been used by Shanker et al (1999) and Wang et al (, 1) for studying the high pressure melting of ionic solids and geophysical minerals. An analysis of the non-linearity of thermal pressure at high T can be presented by taking into account the effect of anharmonicity. The first three anharmonic terms arising from the strong temperature-induced motions contribute to the Helmholtz energy as follows (Landau and Lifshitz 195; Wallace 197) F anh = A 1 ( ) T + A ( ) + A ( ) T, (1) where the coefficients A 1, A and A are the functions of volume only. The anharmonic contribution to the thermal pressure is found by using the relationship P = ( F ), (19) T

5 Analysis of volume expansion data for MgO, CaO, Al O 3 and MgAl O 35 which gives the following additional terms to thermal pressure [ A1 (P th ) anh = T + A + A ] T. () These additional terms should be added in the linear relationship for thermal pressure given by (17) to obtain P th at high T. The last term in () is negligibly small at high T and the second term is independent of T. Only the first term in () contributes to the non-linearity of thermal pressure at high T. This quadratic dependence of P th on T has been found by Wallace (197). The following quadratic expression is obtained in the high temperature limit of the quasi-harmonic approximation (Anderson and Isaak 1995) P th = α K T (T T ) [1 α (T T )], (1) where αk T is the value of product taken at temperature close to the Debye temperature (table 1). Equation (1) can be used to estimate the values of P th provided we know the values of thermal expansivity, α, at high temperatures. alues of α in case of MgO and CaO have been reported by Singh and Chauhan () up to very high temperatures close to their melting temperatures. alues of P th calculated from (1) are shown in figures 1 and for MgO and CaO, respectively. 1 MgO Eq. () Eq. (1) Eq. (1) Figure 1. Plots between thermal pressure, P th and temperature, T. alues of thermal pressure are obtained from (7) (9) using experimental data on /, and compared with the values based on (1) CaO 1 Eq. () Eq. (1) Eq. (1) Figure. Plots between thermal pressure, P th and temperature, T. alues of thermal pressure are obtained from (7) (9) using experimental data on /, and compared with the values based on (1)

6 3 B P Singh et al 1 Al O 3 1 Eq. () Eq. (1) Figure 3. Plots between thermal pressure, P th and temperature, T. alues of thermal pressure are obtained from (7) (9) using experimental data on /, and compared with the values based on (1) MgAl O 1 Eq. () Eq. (1) Figure. Plots between thermal pressure, P th and temperature, T. alues of thermal pressure are obtained from (7) (9) using experimental data on /, and compared with the values based on (1) It is revealed from these figures that the nonlinear variation of thermal pressure, P th, predicted in (1) is very similar to that obtained from the thermal equation of state (7). This provides strong evidence that the non-linearity of thermal pressure at high T originated from the anharmonic effects. Acknowledgements The author is grateful to Prof. Jai Shanker for his valuable guidance and constructive discussions. He is also thankful to Mrs Sudha Singh for her help in the computational work. References Anderson O L 1995 Equations of state of solids for geophysics and ceramic sciences (New York: Oxford University Press) Anderson O L and Masuda K 199 Phys. Earth Planet. Inter. 5 7 Anderson O L and Isaak D G 1995 Elastic constants of mantle minerals at high temperature. Mineral physics and crystallography, A handbook of physical constants, AGU Reference Shelf Anderson O L, Masuda K and Guo D 1995a Phys. Earth Planet. Inter Anderson O L, Masuda K and Isaak D G 1995b Phys. Earth Planet. Inter. 91 3

7 Analysis of volume expansion data for MgO, CaO, Al O 3 and MgAl O 37 Dubrovinsky L S and Saxena S K 1997 Phys. Chem. Miner. 57 Fiquet G, Richet P and Montagnae G 1999 Phys. Chem. Miner Jacobs M H G and Oonk H A J 1 Phys. Chem. Chem. Phys Landau L D and Lifshitz E M 195 Statistical physics (London: Pergamon Press Ltd) Petukhov A and Chekovskoi Ya 197 High Temp. High Press. 59 Popov A G and Borodai E 5 Russ. Phys. J. 5 Singh K S High Temp. High Press Shanker J and Kushwah S S 1 High Temp. High Press Shanker J, Kushwah S S and Kumar P 1997 Physica B 3 7 Shanker J, Sharma M P and Kushwah S S 1999 J. Phys. Chem. Solids 3 Suzuki I, Okajima S and Seya K 1979 J. Phys. Earth 7 3 Taravillo M, Baonza G and Rubio J E F, Nunez J and Caceres M J. Phys. Chem. Solids Wallace D C 197 Thermodynamics of crystals (New York: John Wiley and Sons) Wang K and Reeber R R 199 J. Mater. Res Wang Z, Mao H and Saxena S K J. Alloys Compd 99 7 Wang Z, Schott B, Lazor P and Saxena S K 1 J. Alloys Compd