https://doi.org/10.1140/epjb/e2018-90452-6

## The modified van der Waals equation of state

### Part IV: Crystalline materials

Laboratoire de Physique des Solides, CNRS, Université de Paris-Sud,
91405 Orsay, France

^{a} e-mail: jacques.rault@yahoo.fr

Received:
11
July
2018

Received in final form:
19
September
2018

Published online: 28 January 2019

*PVT* data of crystallizable materials (CM), minerals, alkali, alkali halides, metals, mineral oxides and hydroxides, rare gas, water, organic compounds and polymers, published in the literature are reanalyzed. It is shown that all these materials under pressure verify the modified van der Waals equation of state (mVW-EOS), discussed recently [J. Rault, Eur. Phys. J. E **40**, 82 (2017)]. The characteristic parameters *P*^{*} *V* ^{*} of this EOS depend only on the nature of the material and not on its state (liquid, glassy, solid of different structure) and whatever are its conductivity and magnetic properties (insulator, conductor, superconductor, paramagnetic, ferromagnetic). This EOS explains the following properties: (a) the fan structure of the isobars *V*(*T*), and of the tangents to the isotherms *V*(*P*); (b) the superposition principle of the isotherms *V*(*P*); (c) the *αB* rule: the constancy of the thermal pressure coefficient (*dP*/*dT*)_{V} = *αB*, product of the thermal expansion coefficient *α* and the bulk modulus *B*; (d) its relation with the Slater conjecture: (*dP*/*dT*)_{V} ~ *dP*/*dT*_{m} in crystallized materials, *T*_{m} being the melting temperature. The characteristic pressure *P*^{*} (*T* and *V* independent) is compared with the various pressures: (i) *P*_{coh} = *E*_{coh}/*V*, the cohesive energy density; (ii) *P _{L}*

_{m}=

*L*

_{m}/

*ΔV*

_{m},

*L*

_{m}and

*ΔV*

_{m}being the enthalpy and volume jumps at the melting, respectively; (iii)

*P*

_{D}=

*ΔH*

_{a}/D

*V*

_{a}, ratio of the activation parameters of the autodiffusion coefficient; (iv)

*P*

_{X}=

*X*∕

*X*′,

*X*being the bulk modulus

*B*, shear modulus

*G*, elastic constants

*C*

_{ij}, and the yield stress

*σ*

_{y}of the CM,

*X*′ their pressure derivative (at ambient conditions). All the elastic constants

*B*,

*G*,

*C*

_{ij}and the yield stress

*σ*

_{y}are linear functions of

*P*at low pressure (

*P*<

*P*

^{*}) and extrapolate to zero at the same negative pressure −

*P*

_{X}= −

*P*

^{*}. (e)

*P*

_{Bγ}=

*B*∕

*γ** ratio of the bulk modulus

*B*and Grüneisen parameter

*γ** at zero pressure. (f)

*PΔV*

_{m}is the pressure deduced from the linear relation between the volume jump

*ΔV*

_{m}(

*P*) at the transition (melting or crystalline transition) and the pressure. The universal relation

*P*

^{*}=

*P*

_{coh}=

*P*

_{L}_{m}=

*P*

_{D}=

*P*

_{B}=

*P*

_{G}=

*P*

_{Cij}=

*P*

_{σy}=

*P*Δ

*V*

_{m}is observed and discussed. In molecular compounds such as H

_{2}O, H

_{2}, and polymers with different intra- and intermolecular interactions, the compression involves two different processes, at low and high pressures, verifying the mVW-EOS with characteristic pressures

*P*

_{1}

^{*}and

*P*

_{2}

^{*}. The ratio of these pressures is about the ratio of the weak intermolecular and strong intramolecular bond energies. The generalized modified equation of state (gmVW-EOS) describes the two-step process of compression in materials having two (or several) types of bonds.

Key words: Solid State and Materials

*© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2019*