Applicability of isothermal three-parameter equations of state of solids—a reappraisal

Abstract

The objective of the present study, an extension of a recent one, is to inter-comparethe utility of the various isothermal three-parameter equations of state (EOSs)of solids, considered viable at different stages in the development of the EOSfield, spanning over a period of about a century now. In a recent paper we havecompared our isothermal three-parameter equation of state of solids with seventhree-parameter isothermal EOSs—five corresponding to the regression curves ofV/V0 onP, and twoto those of P on V/V0. In this study, we investigate the relative utility of 21, i.e., virtually all, of theviable three-parameter EOSs—for the purposes of smoothing and interpolationof pressure–volume data, and extraction of accurate values of isothermal bulkmodulus and its pressure derivative—corresponding to the regression curves ofP on V/V0. We have applied the EOSs, with no constraint on the parameters, to accurateand model-independent isotherms of nine solids, and assessed the goodnessof the fitting accuracy; goodness of the stability of the fit parametersB0,B0′,and B0′′ with variation in the pressure/compression ranges; and goodness of the agreement of the fit parametersB0 and B0′ with experiment. Further, an additional test of goodness of randomization of the datapoints about the fit curves, quantified in terms of the number of wiggles of the datadeviation curves about the fits, is also applied in the present study. The EOSs subjected tothese seven tests are the three-parameter extensions of the EOS models formulated byBridgman (1929), Murnaghan (1937), Birch (1938), Slater (1939), Davis and Gordon(1967), Macdonald (1969), Holzapfel (1991), Poirier and Tarantola (1998), and theso-called ‘universal EOS’ promoted by Vinet et al (1986). Also included for theinter-comparison purposes are the three-parameter EOSs proposed by Keane (1954), Mao(1970), Thomsen (1970), Huang and Chow (1974), Luban (1983), Freund andIngalls (1989), Kumari and Dass (1990), Hama and Suito (1996), and Bose Royand Bose Roy (1999), and also the EOS based on a modified Eulerian strain assuggested by Sushil et al (2004). Interestingly, the three-parameter Mie–GruneisenEOS, built on the century-old Mie potential (Partington 1957, Stacey and Davis2004), is also tested. The present study leads to some remarkable findings. Mostnotably, some of the old EOS models like those of Birch and of Keane, and also theMie–Gruneisen EOS, are observed to be in better agreement with experiment than most ofthe EOSs that appeared much later in the literature. An inter-comparison ofthe overall results inferred from all the 21 EOSs—based on the most stringentdiscrimination technique comprising seven important tests ever applied in theliterature—demonstrates that, while most of the EOSs marginally differ from each otherconstituting a narrow band of applicability, our model is decisively superior to all of them.