The temperature dependence of the isothermal bulk modulus at 1 bar pressure

Transcription

1 he temerature deendence of the isothermal bulk modulus at bar ressure J. Garai a) Det. of Earth Sciences, Florida International University, University Park, PC 344, Miami, FL 3399, USA A. Laugier IdPCES - Centre daffaires PAON, 6, rue Franz Heller, bât. A, F357 Rennes, France It is well established that the roduct of the volume coefficient of thermal exansion and the bulk modulus is nearly constant at temeratures higher than the Debye temerature. Using this aroximation allows redicting the values of the bulk modulus. he derived analytical solution for the temerature deendence of the isothermal bulk modulus has been alied to ten substances. he good correlations to the exeriments indicate that the exression may be useful for substances for which bulk modulus data are lacking. PACS number(s): 62.2.Dc, b, 65.4.De a) Electronic mail: jozsef.garai@fiu.edu I. INRODUCION Any isothermal equation of state (EOS) -7 requires knowing the value of the bulk modulus at the temerature of interest. heoretically the temerature deendence of the elastic constants can be determined as the sum of the anharmonic terms 8; 9. At sufficiently low temeratures the elastic constant should vary as 4. Contrary to this suggestion some metallic substances have been found to show a 2 rather than a 4 deendence at low temeratures ; 2. here is no general rediction for higher temeratures. Exeriments on refractory oxides, conducted at higher than room temerature, show a linear relationshi between the bulk modulus and the temerature 3. he third law of thermodynamics requires that the derivative of any elastic constant with resect to the temerature must aroach zero as the temerature aroaches absolute zero. Combining this criterion with the observed linear relationshi at higher temeratures, Wachtman et al. 4 suggested an equation in the form of be, ()

2 where is the bulk modulus at absolute zero, is the temerature, and b and are arbitrary constants. heoretical justification for Wachtman s Equation was suggested by Anderson 5. ased on shock-wave and static-comression measurements on metals, a linear relationshi between the logarithm of the bulk modulus and the secific volume has been detected for metals 6 : ln ln +, (2) where is a constant deending on the material. he linear correlation is valid u to 4% volume change. Using this linear correlation Jacobs and Oonk 7 roosed a new equation of state. hey rewrite equation (2) as: () () ( ) + b ln, m m (3) ( ) where m denotes molar volume, the reference temerature and the suerscrit refers to standard ressure ( bar). Equation (3) successfully reroduces the available exerimental data 7-9 for MgO, Mg SiO, and Fe SiO In this study, analytical solution for the temerature deendence of the bulk modulus and the ressure deendence of the volume coefficient of thermal exansion has been derived from fundamental thermodynamic equations. he derived theoretical exression for the temerature deendence of the isothermal bulk modulus is comared to exeriments. II. HEORY he isothermal bulk modulus [ ] is defined as:. (4) It is assumed that the solid is homogeneous, isotroic, non-viscous and has linear elasticity. It is also assumed that the stresses are isotroic; therefore, the rincial stresses can be identified as the ressure 2 σ σ 2 σ3. he definition for the volume coefficient of thermal exansion ( ) is given as:. (5) - 2 -

3 oth the volume coefficient of exansion and the isothermal bulk modulus are ressure and temerature deendent; therefore, the universal descrition of solids requires knowing the derivatives of these arameters. ; ; ; (6) he comlete thermo-hysical descrition of an elastic solid requires knowing the two arameters [Eq. (4) and (5)] and their four derivatives [Eq. (6)]. he ressure derivative of the volume coefficient of thermal exansion and the temerature derivative of the isothermal bulk modulus are not indeendent from each other and the relationshi between these derivatives is given 5 : 2. (7) It is worth to note that Eq. (7) reduces the number of indeendent artial derivatives of the fundamental arameters and to three. he definition of the isothermal Anderson-Grüneisen arameter [ δ ] is ln δ. (8) Eq. (8) can be written as: (, ) δ(, ) d e (9) At one bar ressure Eq. (9) reduces to: o ( ) δ o ( ) d o o e, () where subscrit [ o ] refers to bar ressure. Inserting Eq. (7) into Eq. (8) gives: δ 2 ln. () Eq. () can also be written as: where δ( ), o d (, ) e, (2) o o denotes bar ressure. An alternative derivation of Eq. () and (2) is given in the aendix. Relations Eq. () and (2) are generally valid. he difficulty is that the Anderson- Grüneisen arameter in these equations is not constant but rather changes with - 3 -

4 temerature, esecially at low temeratures. Insecting Eq. (8) reveals that it is comosed of the thermal ressure derivative with resect to temerature: th, (3) and the temerature derivative of the isothermal bulk modulus. For substances which the quasi-harmonic aroximation (QHA) model is alicable, the roduct of the volume coefficient of thermal exansion and isothermal bulk modulus is nearly constant above the Debye temerature 2. he constant value for the roduct of the volume coefficient of thermal exansion and the isothermal bulk modulus at temeratures higher than the Debye temerature is also consistent with exeriments Insecting Figure shows that the temerature derivative of the isothermal bulk modulus is nearly constant at temeratures above the Debye temerature. herefore it is reasonable to assume that the Anderson- Grüneisen arameter is aroximately constant at temeratures above the Debye temerature. Assuming that is constant at temeratures higher than the Debye temerature then three arameters, the volume coefficient of thermal exansion, the bulk modulus, and the Anderson-Grüneisen arameter can comletely describe the relationshi between the ressure, volume and temerature. In this study the validity of Eq. () will be investigated by comaring the theoretically derived exression to exeriments. III. RESULS AND DISCUSSION Exeriments with ten or more data oint were chosen from the literature in order to evaluate the theoretically derived temerature deendence of the bulk modulus. Exeriments of Ag, Au, MgO, Al 2O3, MgAl O, Mg SiO, ( Fe.Mg.9 ) 2SiO 4, CaO, NaCl, and KCl were used for the investigation. he integration of d was done numerically by using linear olynomials. he volume coefficient of thermal exansion values at various temeratures were taken from ref. 26 for Ag and Au, and from ref. 2 for the rest of the substances. he olynomial fit of seven exeriments given by Jacobs and Oonk 27 : (,) (4) was also used to determine the values of the volume coefficient of thermal exansion for MgO. If exeriments are not available then the linear correlation between the heat - 4 -

5 caacity and the volume coefficient of thermal exansion can be used calculating the value of the volume coefficient of thermal exansion at the temerature of interest 28. Using least-squares fit the Anderson-Grüneisen arameter and the correlation coefficients were determined. he calculated Anderson-Grüneisen arameters are in very good agreement with exeriments. he correlation coefficients are the lowest for the two noble metals.9972 and.9987, while for the rest of the minerals the values are between.9992 and.. At low temeratures the Anderson-Grüneisen arameter changes significantly as a function of temerature 29, and the constant value aroach for the Anderson-Grüneisen arameter might not be aroriate. he data of the noble metals contains the very low temerature exeriments, which can exlain the slightly weaker correlation. he best fits for the two different data sets of MgO resulted almost identical values for the isothermal bulk modulus at zero temerature and the Anderson-Grüneisen arameters. he results are given in able. he calculated values were lotted against exeriments (Fig. ). It can be seen that the derived theoretical relationshi for the temerature deendence of the bulk modulus can reroduce the exerimental values with high accuracy for the entire temerature range of the solid hase. I. CONCLUSIONS raditionally the bulk modulus has to be measured at the temerature of interest requiring numerous exeriments. Assuming constant value for the roduct of the isothermal bulk modulus and the volume coefficient of thermal exansion allows describing the volume-ressure-temerature relationshi of a solid from three arameters, namely from the zero temerature values of the volume coefficient of thermal exansion, and isothermal bulk modulus, and from Anderson-Grüneisen arameter. he derived theoretical exression, Eq.(), can be emloyed to extraolate data measured at convenient temeratures to the temerature of interest. Our investigation showed that assuming constant value for the Anderson-Grüneisen arameter at temeratures higher than the Debye temerature is reasonable. ACKNOWLEDGEMEN: We would like thank the anonymous reviewer for the constructive comments and suggestions, which heled to imrove the quality of the manuscrit

6 - 6 - Aendix. he temerature deendence of the bulk modulus and the ressure deendence of the volume coefficient of thermal exansion can also be derived from fundamental thermodynamic relationshis, which do not include the definition of the Anderson- Grüneisen arameter. Using the Euler s chain relation. (5) gives the ressure and temerature relationshi at constant volume as:. (6) Combining Eq. (7) and (6) gives. (7) Introducing a arameter (a), which is defined as: 2 a. (8) and substituting this arameter into Eq. (7) gives d a ln d. (9) he integral of Eq. (9) gives the temerature deendence of the bulk modulus as: d a e. (2) he ressure deendence of the volume coefficient of thermal exansion can also be determined from Eq. (7). Rearranging Eq. (7) gives dln. (2) Defining a dimensionless arameter b as: b (22) and integrating Eq. (2) leads to d b e. (23)

7 Eq. (23) describes the ressure deendence of the volume coefficient of thermal exansion. Introducing the symbol th for the artial derivative of the thermal ressure with resect to the temerature th th the arameters a and b can be defined as: a th, and b 2 δ th (24) (25) where δ is the isothermal Anderson-Grüneisen arameter given by: ln δ. (26) Using the definition of the coefficient a [Eq. (25)], and combining with Eq. (7), and the exression of the temerature derivative of the thermal ressure Eq. (24) imlies that the two arameters a and b are equal with each other. a 2 2 b δ (27) th where b was substituted by using Eq. (25). he identity (a b) holds true regards of the temerature. he temerature deendence of the bulk modulus then can be written as: δ d e (28) while the ressure deendence of the volume coefficient of thermal exansion δ d e. (29) Equations (28) and (29) recover equations () and (2) resectively

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9 ALE I. is the bulk modulus at zero temerature, calculated δ averge is the calculated average of Anderson-Grüneisen arameter in the temerature range of interest, δ is the Anderson-Grüneisen arameter, R is the correlation coefficient, and N is the number of exeriments. he data for Ag and Au is from ref while the rest of the data is from ref. 2. he volume coefficient of thermal exansion values for MgO(2) are from ref. 27. he errors reresent the standard deviation. Material calculated δ average δ R N [GPa] Ag 65.2(9) (86) ; Au 23.8(6) 3 5.7(56) ; MgO 65.4(6) 4.9(8) MgO(2) 65.3(5) 4.93(8) Al 2O (8) 5.2(2) MgAl O (7) 6.75(27) Mg SiO (2) 5.46(4) ( Fe 33.8(9) 5.44(25) Mg.9 ) 2SiO 4 CaO 6.(3) 5.6(9) NaCl 26.8(2) 5.96(6) KCl 8.8(2) 5.92(2)

10 FIG. he solid lines are the calculated bulk modulus using Eq.(), while the dots reresent the exerimental values. is the bulk modulus at zero temerature, δ is the calculated average Anderson-Grüneisen arameter, and R is the correlation coefficient. - -