Relaxed Material Classes¶

Overview¶

Some materials have isochemical degrees of freedom, meaning that some macroscopic characteristic of the material can change without changing its overall composition. For example, a solid solution may be able to change how Fe and Mg are distributed amongst its constituent sites, or Fe may be able to change its spin state, or the material itself may be able to change by distortion of the crystal lattice. In a multiphase material, isochemical degrees of freedom may arise from changes in the proportions or compositions of the constituent phases (e.g., Fe-Mg exchange). If these isochemical variables can change on timescales shorter than the timescales of changes in pressure and temperature, then the material is said to be relaxed with respect to these isochemical degrees of freedom.

BurnMan provides classes to model relaxed materials, including burnman.RelaxedSolution, burnman.RelaxedAnisotropicSolution, and burnman.RelaxedComposite. These classes extend their unrelaxed counterparts by adding the capability to minimize the Gibbs energy of the system by varying one or more isochemical variables at fixed pressure and temperature. These modified classes are not simply a combination of the unrelaxed classes with a Gibbs minimiser; they also modify the way thermodynamic properties are calculated to account for the relaxation of the isochemical variables.

The properties of particular interest in relaxed materials are those that depend on the second derivatives of the Gibbs energy, such as the bulk modulus, thermal expansivity, and heat capacity. In unrelaxed materials, these properties are calculated directly from the second derivatives of the Gibbs energy with respect to pressure and temperature, keeping all molar fractions constant. By contrast, in relaxed materials, the second derivatives are affected by simultaneous changes in the isochemical variables. To account for this, BurnMan uses variational calculus to derive expressions for the relaxed second derivatives of the Gibbs energy, which are then used to compute the thermodynamic properties of the relaxed material.

Mathematical Derivation¶

Consider a material whose state is determined by the variables pressure \(P\), temperature \(T\) and amounts of independent components \(C\). Additionally, suppose that \(C\) can be decomposed into a set of variables \(X\), which are externally controlled either by conservation of composition or because they evolve slowly compared to changes in \(P\) and \(T\), and a set of isochemical variables \(\xi\), which can change rapidly at fixed \(P\), \(T\), and \(X\). These variables \(\xi\) are internal variables, and might represent (for example) a change in iron spin state, state of Fe-Mg order, or extent of Fe-Mg exchange between phases.

The driving force for changes in the isochemical variables is the minimization of the Gibbs free energy \(\mathcal{G}(P,T,X,\xi)\). Let us define the partial derivative of the Gibbs energy with respect to the isochemical variables as \(F(P,T,X,\xi) = \partial \mathcal{G}(P,T,X,\xi) / \partial \xi\), and for simplicity, let us also group the pressure, temperature, and composition variables into a single vector \(Z = \{P,T,X\}\), such that

\[F(Z,\xi) = \left( \partial \mathcal{G}(Z,\xi) / \partial \xi \right)_Z.\]

Now, at equilibrium, the Gibbs energy is minimized with respect to the isochemical variables, such that

\[F(Z,\xi) = 0.\]

Let us now define another function \(\phi(Z) = \xi\), which represents the values of the isochemical variables that minimize the Gibbs energy at fixed \(Z\).

This yields the modified function

\[f(Z) = F(Z, \phi(Z)) = \frac{\partial \mathcal{G}}{\partial \xi} (Z, \phi(Z)),\]

where the lowercase \(f\) indicates that this function depends only on \(Z\). Taking the total derivative of this expression with respect to \(Z\), we have:

\[\frac{d f}{d Z} = \frac{\partial F}{\partial Z} + \frac{\partial F}{\partial \xi} \frac{\partial \phi}{\partial Z}\]

Now, since \(f(Z) = 0\) for all \(Z\), we also have \(d f / d Z = 0\), and therefore:

\[\frac{\partial F}{\partial \xi} \frac{\partial \phi}{\partial Z} = - \frac{\partial F}{\partial Z}\]

Note that the matrix \(\partial F / \partial \xi\) is the Hessian matrix of second derivatives of the Gibbs energy with respect to the isochemical variables:

\[\frac{\partial F}{\partial \xi} = \frac{\partial^2 \mathcal{G}}{\partial \xi^2}\]

It is therefore symmetric. Let us define the pseudo-inverse of this matrix as \(R\), so that \(R \partial F / \partial \xi = I\), where \(I\) is the identity matrix. Then:

\[\frac{\partial \phi}{\partial Z} = - R \frac{\partial F}{\partial Z}\]

Finally, define a function \(g(Z) = \mathcal{G}(Z, \phi(Z))\), which gives the Gibbs energy of the system at equilibrium. Note that \(g\) has the same value as \(\mathcal{G}\), but, like \(f\), depends only on \(Z\), since the value of \(\xi\) is determined by minimization. Using the chain rule, we can write the total derivative of this function with respect to \(Z\) as:

\[\frac{d g}{d Z} = \frac{\partial \mathcal{G}}{\partial Z} + \frac{\partial \mathcal{G}}{\partial \xi} \frac{\partial \phi}{\partial Z}\]

Since \(F = \partial \mathcal{G} / \partial \xi = 0\) at equilibrium, the second term on the right-hand side vanishes, and we have:

\[\frac{d g}{d Z} = \frac{\partial \mathcal{G}(Z, \phi(Z))}{\partial Z}\]

Taking the second derivative of \(g\) with respect to \(Z\):

\[\frac{d^2 g}{d Z^2} = \frac{\partial^2 \mathcal{G}}{\partial Z^2} + \frac{\partial^2 \mathcal{G}}{\partial Z \partial \xi} \frac{\partial \phi}{\partial Z}\]

Now, substituting in our expression for \(\partial \phi / \partial Z\):

\[\frac{d^2 g}{d Z^2} = \frac{\partial^2 \mathcal{G}}{\partial Z^2} - \frac{\partial^2 \mathcal{G}}{\partial Z \partial \xi} R \frac{\partial F}{\partial Z}\]

or, equivalently:

\[\frac{d^2 g}{d Z^2} = \frac{\partial^2 \mathcal{G}}{\partial Z^2} - \frac{\partial^2 \mathcal{G}}{\partial Z \partial \xi} R \frac{\partial^2 \mathcal{G}}{\partial \xi \partial Z}\]

This expression gives the relaxed second derivatives of the Gibbs energy with respect to pressure, temperature, and composition. These second derivatives can then be used to calculate the relaxed thermodynamic properties of the system, such as the bulk modulus, thermal expansivity, and heat capacity:

\[K_{P,relaxed} = -V \left(\frac{\partial^2 g}{\partial P^2}\right)^{-1}, \quad \alpha_{relaxed} = \frac{1}{V} \frac{\partial^2 g}{\partial P \partial T}, \quad C_{P,relaxed} = - T \frac{\partial^2 g}{\partial T^2}\]