Kumar, P

Kushwah, S S

Shanker, J

IACS Institutional Repository

Irldian J. Phys. 71A (4), 429-442 (1997)I J P  A— an international journalAnalysis of bulk modulus and thermal expansivity for six minerals at high temperaturesS S Kushwah*, P Kumar** and J ShankerDepartment o f  Physics, Institute o f  Basic Sciences, Khandan, Agra-282 002, Uttar Pradesh, IndiaR e c e iv e d  16 J u ly  1 9 9 6 , a c c e p te d  6  F e b r u a r y  1 9 9 7A bstract : The data on isothermal bulk modulus and thermal expansivity tor six minerals (M gO, M g2 SiC>4 , A b O }, Grossular Garnet, Pyrope and CaO) at high temperatures are analysed in order to obtain sim ple empirical relationships Various thermodynamic formulations reported in the recent literature, have been discussed and evaluated cnticully The thermodynamic model due to Xia and Xiao has been found to yield good agreement with experimental data for thermal expansivity as a function o f temperature for all the minerals under studyK eyw ords : Bulk modulus, thermal expansivity, geophysical mmeials PA C S N os. : 62.20.D c, 65 70 +y1. IntroductionOne of the most widely used thermodynamic approximations |1~4| for studying the high lempcrature-high pressure behaviour of solids is that the product of volume thermal expansion coefficient, i.e. the thermal expansivity a  and the isothermal hulk modulus B remains constant. Thus, we can writea B  = K, (1)where the product K has been assumed to remain constant under the variation of pressure or lemperature. If this assumption docs not hold, then cq. ( 1) yields the following relationship lor the isobaric variation of a , B and K with the change in volume under the effect of lemperature,Permanent Address ■ Rishi Galav C ollege. Morena-476 001, Madhya Pradesh. IndiaNarain College, Shikohabod-205 195, Uttar Pradesh. India ^  1 9 9 7  IACS430 S S Kushwah, P Kumar and J ShankerThe data reported by Anderson et al [5] for six minerals at high temperatures, are used in the present study to assess the validity of basic assumptions regarding the variations of a, B and K with temperature. In particular, we study the variation of a  and volume with temperature using the thermodynamic model due to Xia and Xiao [6]. The method of analysis is presented in Section 2. Results are discussed and compared with experimental data in Section 3.2. Method of analysisTaking the high temperature data from Anderson et al [5], we made an attempt to plot a, B and K , each as a function of WV0» where V0 is the volume V at 300 K. Values of WV0 corresponding to different temperatures were estimated from the data on dinsity 151. Systematic linear relationships are obtained only for In B versus In VfV{) for all the six minerals [Figures l(a-f)]. Plots for lna or In K versus In V7V0 arc found to be unsystematic and non-linear. Thus, a  or K depends on the change in volume in a complicated manner. On the other hand, the slopes of the linear plots in Figures I (a—0 yield values of the Anderson-Grunciscn parameter <5/ which is definedThus, represents the slope of the plots of In B versus In (VtV0). Values of Sj determined from the plots given in Figures l(a-f) are reported in Table 1.If the product K = aB  is assumed to remain constant with the variation in temperature then eq. (2 ) yieldsas [5,7]Using a  = y in eq. (3), wc get(4)which can also be written in the following form [7]Anderson et al |7] have used eq. (6) to obtain the relationship given belowAnalysis o f bulk modulus and thermal expansivity for six minerals etc 431Eq. (7) has been derived by the integration of (6) taking $j- as constant. Under the same ihcrmodynamic assumptions, the following relationships can also be obtained from eqs. (5 ) and (7)and ~n~ — [ l  a o ^ r ^ ~  ^0^]Ml (9)[ q (7) is equivalent to (8) in view of (9). As pointed out by Anderson et al [7], the assumptions implied in the derivation of the relationships (5 to 9) arc valid only in the high tcinpcialurc region above the Debye temperature.Figure 1(a). Plots o f In B versus In (V/Vq) at high temperaturesFigure 1(b). Same as in Figure 1(a).432 S S Kushwah, P Kumar and J ShankerRecently Xia and Xiao [6] have developed a thermodynamic model for investigating a  of pcriclase (MgO) as a function of temperature. The pressure derivative of heat capacity Cp can be expressed by the following thermodynamic identity• E AdP (10)Using the definition of thermal expansivity a, we can rewrite eq. (10) as followsT — (aV)„.d T \ )p (IDFigure 1(c). Same as in Figure 1(a)Figure 1(d). Same as in Figure 1(a).Analysis of bulk modulus and thermal expansivity for six minerals etc 433It has been found lhal in the range of clastic Debye temperature, the best fit to the e x p e r im e n ta l data is represented by the following relationship |6]c r = c ° - np  ( i 2 )Figure 1(e). Same as in Figuie 1(a)Figure 1(f). Sam e as in Figuie 1(a)elicit* C() is the heat capacity at zero pressure and 77 is a positive constant independent ot pressure and temperature. Eqs. (II)  and (12) yield( 13)436 5 S Kushwah, P Kumar and J ShankerAttempts have also been made [1,8] to develop models for predicting a  as a function ol temperature by incorporating the variation of 5T with volume and temperature.Figure 2(c)- Same as in Figure 2(a)Figure 2(d). Same as in Figuie 2(a).Thus, Tallon [ 11 has considered that 8j  varies as VIV« and obtained the following expression lor thermal expansivitya  = a 0 cxp( 18)where 5'/ is the value of ST at initial temperature. Chopelas and Boehler [91 have shown that ($, + I ) varies as V!V0 under the effect of pressure. Considering the similar variation under the effect of temperature also, Kumar and Bcdi [8] obtained the followingexpression( 11)Hqs ( 18) and (19) have been derived incorporating the variation of 5r in the following lunnulaAnalysis o f bulk modulus and thermal expansivity for six minerals etc 437Wo have used eqs. (18) and (19) to calculate a  as a function of V7Kn and temperature. It is I,wild that both the equations yield almost identical results which are smaller than the expetimcntal data by about 15 to 40 per cent.Figure 2(e). Same as in Figure 2(a)Figure 2(f). Same as in Figure 2(a).Kumar and Bedi [8] have used the following relationship also for the temperature dependence of djST -  , ( 0 .where x is an empirically fitted parameter. Using eqs. (6) and (21) they obtained(21)—  = [l _ a 0^r /yr+l jx+ l)]« 0 [  r i ( x  + i ) F  > ■ (22)Table 2. Values of VW0 and thermal expansivity a (l(T6 K'1) for six minerals (a) Values calculated from eqs (15) and (16) due to Xia and Xiao [6], and (b) experimental values [5]438 S S Kushwah. P Kumar and J Shanker2C/3ft2r4 m <N p oo On O — <N m p r- ON o(N 8 rvi<*i »*im 3 irim rn 38 39 § 3 5 £ornON OOR 3O — O oo «nO) Q -  -  rim ^  T* T*$3 3ON ^  ONin o> m 3 1 oom r* s mi£> & —  oom  on r-8  ©  O o p © no mp mo 3 3 3 «noON —fN NO8 8 m  —  o \  c h  m  no ooo 3 8 mm NOr- CN 8 C l  O n —  mP O O s no S mo mp 3 3 • n  i no  o§  3-SOc433ONin3— O r~ oo rt rfO n3•rf o  m£  3  8  Si g g s S S S S S S S . S S S S Sm  o  S P: 1 8 8 8p  g- on «n•A) O' rn oo© © © S S n  m  n«*i oo mm rno  o 38 8r- oo 8 8 8 8 8— <n m mm r- ?  !8 8r-~ ooGrossular GomelAnalysis o f  bulk modulus and thermal expansivity fo r six minerals etc 439oosi aON —  — si n  vo «nOn —^—  CM INO  IN■ri voO oo Ov On IN  IN©S©^  r*~,oo vo g> in ir, 2 m r i'T - tr-8  © © ©INO s s %§  £00 Om r-s sIN Ov rn © O n VO ro © OO vO r*-i O n © 1r ,vd O n IN ■*r ■ri v i VO 06 OO ON © © _ i n i nIN IN IN IN (N IN M i n IN P-, r*-, r^ , r r ,I N r - ON CN •r> v o • O _, OO -*T © © N -v d 0 0 I N m VO r ~ OO O v O ' © — — « n r iI N I N I N I N I N I N I N I N IN r r - pr-,«3' 8m ■<* © op moo —  ^  vO on8  o  © o  ©©mo-tNO ON ■*-, pp, rr,O  Ov «rt moo ©  m  vo8 0 0 0 0 S s 3i s 8  § 8 8 8 8— w ms8 8Table 2. (Corn'd)440 S S Kushwah, P  Kumar and J ShankerW h en  we apply this formula to the minerals for reproducing a  at high temperatures ( j 200-1800 K), the values of x  are required to remain between 0.4 and 1.4. This would imply from ccl- (^l) that Sp increases Very fast with temperature and becomes 2 to 6 limes iis value at room temperature. Such a large variation in the value of 5p is not consistant with ihe siudies reported in literature T5-71.It should be mentioned here that a number ot relationships showing linear and non­linear dependences of a  on temperature, have been reviewed and studied by Plymate and Smut 1101. These relationships when applied to minerals [11], yield results which deviate much hom the experimental data. The thermodynamic model which has been found to yield Jose agreement with experimental data is that due to Xia and Xiao [6]. The basic .issumption used in this model is given by eq. (12), in which rj remains nearly constant at diMerent temperatures. The validity of this assumption has been demonstrated by Xia and Xiao by calculating the thermal expansivity of MgO upto 1800 K in good agreement with e x p e r im e n ta l  data. The minerals under study remain in the solid phase for the temperature range considered here. Their thermoelastic properties in the solid phase for the entire temperature range have been determined experimentally by Anderson et al [5], No phase tiansition to liquid or gaseous state has been observed in this temperature range [7]. We can therefore, assume the approximate validity of eq. (12) m the same solid phase for the mineralsIt is clear from the Figures 2(a-f) that the relationship given by eq. (16) is satisfied appioximatcly well, within a few per cent, with the experimental data for the minerals under study. T o a good approximation we can therefore, take the proportionality constant A ~ to be independent of temperature. Using this assumption, we have derived the relationship (eq. 15) between V/V0 and temperature. We have calculated the values of ihennal expansivity a  and V/Vq as a function of temperature using eqs. (15) and (16). The limits are compared with the experimental data in Table 2. The good agreement between ihe calculated and experimental values of a  and V/V0 support the validity of the assumption used Thus, the thermodynamic model due to Xia and Xiao developed recently for MgO if*, 111 is also applicable for other minerals under study and provides a simple method for calculating a  and V7V0 in agreement with the experimental data (Table 2).AcknowledgmentsOne ol the authors (SSK) is grateful to the University Grants Commission, Bhopal, Madhya Pradesh, lor providing financial assistance.ReferencesMl J N Tallon J P h y s . Chem. S o l id s  41 837 (1980)I2| F Birch J. O e o p h y s .  R e s  91 4949 (1986)IM N Hass and M Kuman P h y s . S la t  S o l  (b) 127 103 (1 9 8 5 ); 133 101 (1986)Analysis of bulk modulus and thermal expansivity for six minerals etc 441\A\ J Shanker and M Kumar Phys. Slat. Sol. (b) 179 351 (1993)[5] O L Anderson, D G Isaak andH O d a / Geophys Res. 96 18037 (1991)[6] X Xia and J Xiao J Phys. Chem Solids 54 629 (1993)[7] O L Anderson, D G Isaak and H Oda Rev Geophys 30 57 (1992)[8] M Kumar and S S Bedi Phys. Stat. Sohdi (b) 188 645 (1995)[9] A Chopelas and R Boehler Geophys. Res. Lett 19 1983 (1992)110] T G Plymale and J H Stour J. Geophys. Res 94 9477 (1989)111] S S Kushwah and J Shanker Phystca B225 283 (1996)442 S S Kushwah, P Kumar and J Shanker

Analysis of Bulk Modulus and Thermal Expansivity for Six Minerals at High Temperatures

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Analysis of Bulk Modulus and Thermal Expansivity for Six Minerals at High Temperatures

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