5.9 Solute Diffusivity

Diffusivity materials provide a constitutive relation for the solute diffusivity in a biphasic-solute material. In general, the diffusivity tensor \(\mathbf{d}\) may be a function of strain and solute concentration.

Constant Isotropic Diffusivity

When the permeability is isotropic,

\[ \begin{equation} \mathbf{d}=d\,\mathbf{I}\,.\label{eq510} \end{equation} \]

For this material model, \(d\) is constant. This assumption is only true when strains are small. Note that the user must specify \(d\leqslant d_{0}\), where \(d_{0}\) is the solute diffusivity in free solution, since a solute cannot diffuse through the biphasic-solute mixture faster than in free solution.

Constant Orthotropic Diffusivity

When the permeability is orthotropic,

\[ \begin{equation} \mathbf{d}=\sum\limits_{a=1}^{3}d^{a}\mathbf{V}_{a}\otimes\mathbf{V}_{a}\,,\label{eq511} \end{equation} \]

where \(\mathbf{V}_{a}\) are orthonormal vectors normal to the planes of symmetry. For this material model, the \(d^{a}\) are constant. Therefore this model should be used only when strains are small. Note that the user must specify \(d^{a}\leqslant d_{0}\), where \(d_{0}\) is the solute diffusivity in free solution, since a solute cannot diffuse through the biphasic-solute mixture faster than in free solution.

Referentially Isotropic Diffusivity

This material uses a strain-dependent diffusivity tensor that is isotropic in the reference configuration and accommodates strain-induced anisotropy:

\[ \begin{equation} \mathbf{d}=\left(d_{0r}\mathbf{I}+\frac{d_{1r}}{J^{2}}\mathbf{b}+\frac{d_{2r}}{J^{4}}\mathbf{b}^{2}\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)e^{M\left(J^{2}-1\right)/2},\label{eq512} \end{equation} \]

where \(J\) is the jacobian of the deformation, i.e. \(J=\det\mathbf{F}\) where \(\mathbf{F}\) is the deformation gradient, and \(\mathbf{b}=\mathbf{F}\cdot\mathbf{F}^{T}\) is the left Cauchy-Green tensor. Note that the diffusivity in the reference state (\(\mathbf{F}=\mathbf{I}\)) is isotropic and given by \(\mathbf{d}=\left(d_{0r}+d_{1r}+d_{2r}\right)\mathbf{I}\).

Referentially Orthotropic Diffusivity

This material uses a strain-dependent diffusivity tensor that is orthotropic in the reference configuration and accommodates strain-induced anisotropy:

\[ \begin{equation} \mathbf{d}=d_{0}\mathbf{I}+\sum\limits_{a=1}^{3}d_{1}^{a}\mathbf{m}_{a}+d_{2}^{a}\left(\mathbf{m}_{a}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}_{a}\right)\,,\label{eq513} \end{equation} \]

where

\[ \begin{equation} \begin{aligned}d_{0} & =d_{0r}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{^{0}}}e^{M_{^{0}}\left(J^{2}-1\right)/2}\,,\\ d_{1}^{a} & =\frac{d_{1r}^{a}}{J^{2}}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{^{a}}}e^{M_{^{a}}\left(J^{2}-1\right)/2}\,,\\ d_{2}^{a} & =\frac{d_{2r}^{a}}{2J^{4}}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{^{a}}}e^{M_{a}\left(J^{2}-1\right)/2}\,. \end{aligned} \quad a=1,2,3\label{eq514} \end{equation} \]

Here, \(J\) is the Jacobian of the deformation, i.e. \(J=\det\mathbf{F}\) where \(\mathbf{F}\) is the deformation gradient. \(\mathbf{m}_{a}\) are second order tensor representing the spatial structural tensors describing the orthogonal planes of symmetry, given by

\[ \begin{equation} \mathbf{m}_{a}=\mathbf{F}\cdot\left(\mathbf{V}_{a}\otimes\mathbf{V}_{a}\right)\cdot\mathbf{F}^{T},\quad a=1-3\,,\label{eq515} \end{equation} \]

where \(\mathbf{V}_{a}\) are orthonormal vectors normal to the planes of symmetry. Note that the permeability in the reference state (\(\mathbf{F}=\mathbf{I}\)) is given by \(\mathbf{k}=k_{0r}\mathbf{I}+\sum\limits_{a=1}^{3}\left(k_{1r}^{a}+k_{2r}^{a}\right)\mathbf{V}_{a}\otimes\mathbf{V}_{a}\).