Properties of solids under the effect of high temperature and pressure- NaCI as an example

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1 ndian Journal of Pure & Applied Physics Vol. 4, March 21, pp Properties of solids under the effect of high temperature and pressure- NaC as an example Munish Kumar Department of Physics, SRMS College of Engineering and Technology, Bareilly ( munish_dixit@yahoo.com) Received 5 September 2, accepted 22 December 2 A simple theoretical method is used to investigate the properties of solids nder varying conditions of temperature and pressure. The case of NaC is discussed as an example for the entire range of temperature and pressure, viz. from room temperature up to the melting temperature ( K) at the pressures varying from atmospheric pressure (referred as P = ) up to the structural transition pressure (P = 3 GPa). The results obtained for the temperature dependence of thermal expansivity are compared with those based on the widely used Suzuki equation. The results of the pressure dependence of thermal expansivity are compared with Anderson-Masuda rel ation. The results of Wang and Reeber [Phys Chem Miner, 23 (1996) 354] are also included for the sake of comparison. The good agreement with the avai lable theoretical as well as experimental data, supports the simple method used in the present paper. 1 ntroduction Kuswaha & Shanker 1 have compiled the methods reported by earlier workers for the determination of thermal expansivity. Such efforts may be used to study the isobaric (P = ) thermal expansion. Suzuki et at. 2 found what became known as the Suzuki equation, which yield the change in volume VV as a function of temperature T along the P = isobar. The detailed analysis of Suzuki equation is available elsewhere 2-4 and the mathematical form reads as follows: [ K - (1-4 KE Th Q ) 112 ] 2Ka... ( ) where K = (8' - 1)12 and Q = B V,[f. Bo, B'o are the isothermal bulk modulus and its first order pressure derivative, respectively, Yo is the Gruneisen ratio. The subscript refers to their value at P = and T = 3 K. ETh is the thermal energy, a equal to one, as discussed in detail by Anderson 3. Using the relation (YoETJV ) = PTh where PTh is the thermal pressure, Eq. () may be rewritten as follows: ~ - = - [ - { 2( B -1 ) B } p Th ] 12 V (B' -1)... (2) Eq. (2) may be used to determine isobaric (P = ) variation of VV versus T or the coefficient of volume thermal expansion a(7). The analytical expression for a(7) found by the differentiation of Eq. () is quite complicated. Therefore, the method of numerical differentiation has been suggested 3. Anderson 3 has discussed the limit of Suzuki equation for its applicability at high temperatures. t has been discussed 3 that it yields imaginary results for NaC at T > 736 K, and therefore not capable of explaining the thermal expansivity of NaC at P = isobar. Anderson & Masuda 5 have developed another method for the determination of a under the effect of pressure P. The formula reads as follows:... (3) where n = VV and ~~ is the value of a at n = for each isotherm. 8-r is the value of &rct) at high T and P =, K = 1.5. The application of Eq. (3) needs the input parameters measured along the temperature axis at P=O. Thus the problem becomes much difficult when one tries to investigate the properties of solids under the varying conditions of P and T. Thus, it is legitimate and may be useful to propose some simple method. n the present paper, an effort is made in this direction by developing a simple method. The method needs the values of input parameters at zero pressure and room temperature to

2 KUMAR: PROPERTES OF SOLDS 23 investigate the behaviour of solids under the varying conditions of pressure and temperature. NaC is a widely studied inorganic compound, because of its simple structure, high melting temperature and structural transition pressure. For this solid a large amount of theoretical as well as experimental data are available. The author has therefore selected NaC for the present study, so that the results may be discussed in the light of earlier studies. 2 Method of Analysis To investigate the properties of solids under varying conditions of pressure and temperature, the theory of equation of state recently developed by Kumar & Bedi 6 is extended. The detailed analysis is available elsewhere and the mathematical form reads as follows: l -=1--n V [ +-{P-a A B (T -T ) }...(4) V A B where P is the pressure, T the temperature and A = (8 T+ ), refers to the value at room temperature and atmospheric pressure. The coefficient of thermal expansion a is defined as a = (V)(dVdT)r. Therefore, from Eq. (4) one can get: a [ 1 AP, - = l--n{ Aa (T -T ) }f a A B { 1 + AP-Aa (T-Tr ) r' B o o -= B [ 1--n{l ArL(T-Tr)}] AP fii. A Bo --u o AP { 1+--An-(T -Tr)} B --u o... (5)... (6) The Anderson-Gruneisen parameter ~ is defined as: 8-1 (dbt J T-- abt dt )P Therefore, from Eq. (6) one can get: 3 Application Results and Discussion... (7) By making use of Eq. (4), the values of VV are investigated as a function of temperature along different isobars and reported in Fig.. The input data 3 are ~= 11.8 X o S K'' Bo = 24 GPa, 8 \ = 5.56 and 8 11 = Using the PTh values from Anderson, the values of VV at different T values are computed from Suzuki relation [Eq. (2)] at P = isobar. These results are reported in Fig. along with the results reported by Anderson 3, and Wang & ReeberR. Most of the results obtained are available up to 7 K. The author, therefore selected 7 K as a comparison point and data for VV are reported in Table. The results obtained in the present work are very close to the results reported by Anderson 3, and Wang & ReeberR. The results obtained by Suzuki formula are slightly low. At higher temperatures, the available data of Wang & ReeberR are used for comparison; a good agreement is obtained (Fig. 1). This demonstrates the effect of pressure on thermal expansion. t is found that thermal expansion is effected by pressure and it decreases as pressure is increased. Moreover, the isobar of P = 3 GPa is almost parallel to the temperature axis. This shows that at high pressure, the effect of temperature ts cancelled by the effect of pressure. Table -Comparison of VV at T = 7 K reported by earlier workers Reported by Anderson 3 Wang and ReeberR Suzuki Formula (Eq. 2) Present work (Eq. 4) VN The values of a calculated using Eq. (5) are reported in Figs 2 and 3 for the entire range of P and T. The results of a as a function of P and T reported by Wang & ReeberR are also given for the sake of comparison. The calculated values of a at P =, are found to lie between the two sets of the results reported by Wang & Reeber and Anderson 3 up to 6 K. Above 6 K, the calculated values of a are found to have a more rapid increase with the values of Wang & ReeberR, but this difference becomes very small as P is increased. This rapid increase of a with Tat P =, near melting temperature is

3 24 NDAN 1 PURE & APPL PHYS, VOL 4, MARCH 22 l.2.-- j~------~r ~ ~ ;e FP~-~D~GPa ~-Q ~ ~ P SGPa. 8 ~ ~ P - OGPa ~ ~P~Q~2GPa ~ FV3UGPa V/Vo Wang and Rceber 8.Andere on ' Suxull<1 for:rnula [ E q. C 21 ) - P resen t iudy Eq. ( 4 l ].2 t-- ol j L ~ ~------~L ~ _J Fig. -- Variation of V/V with temperature at diffe rent pressures ll ~ "-' 1j Wang and Reeber 8 -Present Study [Eq.(5)) P 1G Pa P"-2GPa * 25 ' 2... /.. ~ 15 ll... "-'1 d ',..., Wang and 3 Reebe r 8 + Anderson Suzilkf fomula [Bq,(l)J. - Present Study [Bq. ( 5)) P OGPa ~ * ~ * 2 * * * *' * * * * ~--~--~----L----~--~----~~ 3 ~ lf 9 1 Fig. 2 - Variati on of c1 with temperature consistent with theories of melting of alkali halides as di scussed in detai l by Shanker & Kumar 9 showing that a~ oo at T ~ TM. Anderson and Masuda have reported Eq. (3) for the determination of a as a function of T) along different isotherms. Making the use oft) values reported by Birch 1, ~~ (T) from Suzuki relation, and 8\ (T) from Anderson 3, the values of a are obtained along different isotherms using Eq. (3). The results thus ;j; 5 Fig. 3 - O. L----'-----'--~----"-- ~ ' Variation of a with temperatu re at diffe rent pressures obtained are reported in Fig. 4 along with those obtained using Eq. (5) reported in the present work. A good agreement is found. Here, it is pertinent to mention that the use of Anderson-Masuda relation [Eq. (3)] needs the values of rt along the isotherm considered and &r. a measured as a function of temperature at P =, as input data. On the other hand, Eq. (5) needs the input data at room temperature T = 3 K and P =. Thus, the application of Eq. (5) is simple as compared with Eq. (3).

4 KUMAR: PROPERTES OF SOLDS Anderson-Masuda rela tfon[eq ( 3) Jl. Present study [Eq.(5)] :: Ln ' P (GPa)! Fig. 4-Variation of a with pressure at different temperatures + lang and 3 Reeber 8 Anderson - Present study [Rq.(6)] 16 14r r ~p~ ~3~G~Pa t2 -.. ~ rll 6... P 2GPa Pa!OGPa. BOO 9 1 Fig. 5- Variation of bul k modulus with temperature at di fferent pressures The calculated values of bulk modulus as a function of T at different P using Eq. (6) are reported in Fig. 5. Anderson 3, Wang & ReeberR have 7 * P OGPa _.,_ 6it:=:Z=~=:j*t:::!*::~====~~ ~~~~~~a~ 5 Pt..5G!a ~ ~P~ OGPa.. 4 P 2GPa 2: Fig lang and 3 Reeber Anderson -Present Study [Hq.(7)] ~--~--L---~--~---L---L--~ T(K) Variation of &r with temperature at different pressures reported the variation of bulk modulus under the effect of temperature at P =. These results are al so included in Fig. 5 for the sake of compari son. A good agreement is obtained. The bulk modulus is found to decrease by increasing temperature and it

5 26 NDAN J PURE & APPL PHYS, VOL 4, MARCH 22 increases with increasing pressure. The decreasing behaviour of bulk n1odulus with increasing temperature is found to be less as the pressure increases. Al ong the hi ghest pressure isobar (P = 3 GPa) the percentage decrease the bulk modulus fro m 3 to K is found to be 4.58 %, which is very small as compared with the percentage decrease ( 48.8 % ), along P = isobar. Thus it seems that at hi gh temperatures, the effect of temperature is cancelled by the effect of pressure. The relati on investi gated fo r 8-r as a functi on of P and T [Eq. (7)] is used to predict the values of 8-r for the entire range of P and T. The results obtained are reported in Fi g. 6. The temperature dependence of 8T at P = is compared with the results reported by earl ier workersj.r. At most of the temperatures, the author's predicted values lie in between the two sets of resulth. 8-r is found to decrease by increasing pressure, which is consistent with the recent research in hi gh-pressure physics as di scussed in detail by Anderson 3 To summari se, the author has presented a simple method based on the knowledge of equation of state to in vesti gate the properties of solids under varying conditions of pressures and temperatures. A good agreement with the available theoretical as well as experimental data supports the validity of the simple method. The studies have been presented fo r the entire range of pressure and temperature even in the absence of experimental or theoretical data. Such an analys is may be useful fo r those engaged in the research at high pressures and high temperatures. References Kush wah S S & Shanker J, Phvsica 8, 225 ( 1996) Suzuki, Okaj ima S & Seya K, J Phys Earth, 27 ( 1979) Anderson L, Equation of state for geophysics and ceramic science (Oxford Un iversi ty Pr ss, Oxford), Anderson L, Mas uda K & Guo D, Phys Earth & Plane/a/)' n teriors, 89 ( 1995) Anderson L & Masuda K, Phys t -:arth & Planetary nteriors, 85 ( 1994) Kumar M & Bedi S S, Phys Status Solidi (b), 196 ( 1996) Kumar M, Physica 8, 2 12 ( 1995) 39 1; ( 1996) Wang K & Reeber R R, Phys Chem Miner, 23 ( 1996) Shanker J & Kumar M, Phys Status Solidi (b), 158 ( 199). Birch F, J Geophys Res, 9 1 ( 1986) 49.