About the relations between finite strain in non-cubic crystals and the related phenomenological P-V Equation of State

Abstract

The notion of scalar strain in minerals is crucial for the formulation of P-V equations of state (EoS). A scalar strain, μ, holding for any crystal symmetry has been derived by a rigorous and general approach, and then used to develop the related phenomenological P-V EoS. μ, which depends on V and the trace of the G^* G⁰ matrix, can be split into two components, M and Φ, where the former takes values close to those of the scalar strain according to Birch. M, providing the main contribution (often larger than 80%) to μ, is appropriate for the formulation of an EoS as ∂M/∂V behaves regularly in the limit of an unstrained configuration. The phenomenological EoS based on M shows the same dependence on the elastic parameters (bulk modulus and derivatives versus pressure) of the usual Birch-Murnaghan EoS, and yields comparable results. Slight deviations occur for low symmetry minerals. This work is meant to contribute (1) to shed light on the relationships between scalar strain and related P-V EoS‘s, and (2) to provide a most general EoS which includes, as a particular case, the Birch-Murnaghan model and explains why this latter is reliable for crystal symmetry other than the cubic one, for which it was originally derived.

Appendices

Appendix 1

Let a _j be the jth - basis lattice vector of a crystal; then, the metric tensor is defined by G _ik = a _i · a _k . It is well known that det(G)= V² and G⁻¹=G*, where G* is the metric tensor of the reciprocal lattice and V the volume of the unit-cell fixed by a₁, a₂ and a₃.

We prove, now, Eq. 6. Matrix R, which transforms ortho-normal co-ordinates into crystallographic co-ordinates, relates a _j vectors to a chosen set of orthogonal versors t _j as follows:

$$ R_{jk}^{\rm T} {\varvec{a}}_{k}= {\varvec{t}}_{j}$$

and therefore

$$ {\varvec{a}}_{j}= R_{jk}^{{\rm T}-1} {\varvec{t}}_{k}.$$

Taking into account that t _j · t _i =δ _ji , it descends that

$$ \begin{aligned} G_{ik} & = {\varvec{a}}_i \cdot {\varvec{a}}_k \cdot = R_{il}^{T - 1} {\varvec{t}}_l \cdot R_{km}^{T - 1} {\varvec{t}}_m = R_{il}^{T - 1} R_{km}^{T - 1} \delta _{ml} = R_{il}^{T - 1} R_{kl}^{T - 1} = R_{il}^{T - 1} R_{lk}^{ - 1} \\ & = ({\varvec{R}}^{T - 1} {\varvec{R}}^{ - 1} )_{ik} = ({\varvec{R}}^{ - 1T} {\varvec{R}}^{ - 1} )_{ik} . \\ \end{aligned} $$

Appendix 2

In the following we report the Lagrangian counterpart of the equations developed through the text for the Eulerian scheme. For Eq. 10:

$${\text{Tr}}({\mathbf{I}} + 2 {\varvec{\eta}} ) = {\text{ Tr}}({\varvec{G}}*^0{\varvec{G}});$$

for Eq. 11

$$\det ({\mathbf{I}} + 2 {\varvec{\eta}} ) = \det ({\varvec{G}}*^0{\varvec{G}}).$$

Replacing 1 + 2μ _L with B, and Tr(G*⁰ G) with λ _L , one has: for Eq. 12

$$3B + 2{\text{ Tr}}({\varvec{\Omega}}_{\text{L}}) = \lambda_{\text{L}};$$

and for Eq. 13

$$ B^3 + 2B^2 Tr({\varvec{\Omega}}_{\text{L}} ) + {\text{o}}({\varvec{\Omega}}_{\text{L}}^2 ) + {\text{o}}({\varvec{\Omega}}_{\text{L}}^2 ) = (V/V_0 )^2 . $$

For Eq. 23

$$ \mu _{\text{L}} = \frac{1} {2}\left( {1 - \frac{{\lambda _{\text{L}} }} {3}} \right) $$

Appendix 3

We aim to calculate the derivatives of μ versus V, assuming

$$ \mu = \frac{1} {2}\left( {\frac{\lambda } {3} - 1} \right) $$

One has ∂μ/∂ V=1/6 (∂λ/∂ V). In the case of the Eulerian strain, which is the one treated in the present work, λ=Σ _i,j=1,3 G^* _ij G⁰ _ji , where G _ij and G^* _ij are the ij-terms of the metric matrices of direct and reciprocal lattice, respectively. We have to analyse ∂ G^* _ji /∂ V. For the sake of clarity, let us focus on the case i=1 and j=2. Then

$$\frac{\partial G^{*}_{12}}{ \partial V}=\frac{\partial G^{*}_{12}}{\partial V^{*}}\frac{\partial V^{*}}{ \partial V}.$$

(26)

Taking into account that G^* ₁₂= a^* ₁ a^* ₂ cosγ*, it follows

$$\left(\frac{\partial G^{*}_{12}}{\partial a^{*}_{1}}\frac{\partial a^{*}_{1}}{\partial V^{*}}+ \frac{\partial G^{*}_{12}}{\partial a^{*}_{2}}\frac{\partial a^{*}_{2}}{\partial V^{*} }+\frac{\partial G^{*}_{12}}{\partial \gamma^{*}}\frac{\partial \gamma^{*}}{\partial V^{*}}\right)\left(\frac{\partial V^{*}}{\partial V}\right).$$

(27)

One has now to keep in mind that, in the present case, the derivatives of the lattice parameters with respect to V^* [i.e. (∂a^* ₁/∂V^*), (∂a^* ₂/∂V^*) and (∂γ^*/∂ V^*)] must be calculated assuming infinitesimal changes of volume produced by pressure. Hence, for a^* ₁, one writes

$$\frac{\partial a^{*}_{1}}{\partial V^{*}}=\frac{\partial a^{*}_{1}}{\partial P}\frac{\partial P}{\partial V^{*}}$$

which can be trivially re-cast into

$$ \frac{{a_{\text{1}}^{\text{*}} }} {{V^{\text{*}} }}\frac{{\beta _{a_1^{\text{*}} } }} {{\beta _{V^{\text{*}} } }} $$

(28)

where β X is the compressibility of X. Expressions similar to Eq. 28 hold for a^* ₂ and γ^* . This all implies that Eq. 27 can be formulated as follows

$$ \left[ {\frac{{G_{12}^* }} {{V^* }}\frac{{\beta _{a_1^* } }} {{\beta _{V^* } }} + \frac{{G_{12}^* }} {{V^* }}\frac{{\beta _{a_2^* } }} {{\beta _{V^* } }} + \frac{{\partial G_{12}^* }} {{\partial \gamma ^* }}\frac{{\partial \gamma ^* }} {{\partial P}}\frac{{\partial P}} {{\partial V^* }}} \right]\left( { - \frac{1} {{V^2 }}} \right). $$

(∂γ^*/∂ P) is in most cases negligible with respect to its counterparts in terms of axial compression, and therefore one can drop off the contribution depending on γ^*. ${{\beta _{a_i^* } } \mathord{\left/ {\vphantom {{\beta _{a_i^* } } {\beta _{V^* } }}} \right. \kern-\nulldelimiterspace} {\beta _{V^* } }}$ involves the axial compressibility of a^* _i , and represents a serious hindrance for the development of a P-V EoS. The simplest way to overcame such a difficulty is to drastically approximate ${{\beta _{a_i^* } } \mathord{\left/ {\vphantom {{\beta _{a_i^* } } {\beta _{V^* } }}} \right. \kern-\nulldelimiterspace} {\beta _{V^* } }}$ with 1/3, which holds rigorously in the case of cubic symmetry only. However, this is not an unrealistic approximation for non-cubic symmetry, too, save the case of structures with a highly anisotropic behaviour under compression. In so doing one obtains

$$\frac{\partial G^{*}_{12}}{\partial V}\approx-\frac{2}{3} \frac{G^{*}_{12}}{V}$$

and therefore

$$\frac{\partial \lambda}{\partial V} \approx -\frac{2}{3}\frac{\lambda}{V}$$

leading finally to

$$\frac{\partial \mu}{\partial V} =-\frac{1}{9}\frac{\lambda}{V} =-\frac{1}{3} \frac{1+2 \mu}{V}.$$

Second and third derivatives are straightforwardly determined by ∂μ /∂ V.