2.9 Triphasic and Multiphasic Materials

Multiphasic materials represent an extension of the biphasic-solute material, where the mixture may contain a multitude of solutes. These solutes may be either electrically charged (ionized) or neutral. Similarly, the solid matrix may either carry electrical charge (a fixed charge density) or be neutral. A triphasic material is a special case of a multiphasic material, having two solutes that carry opposite charges. Triphasic and multiphasic materials may be used to model porous deformable biological tissues whose solid matrix may be charged and whose interstitial fluid may contain any number of charged or neutral solutes. When mixture constituents are electrically charged, the response of the tissue to various loading conditions may encompass a range of mechano-electrochemical phenomena, including permeation, diffusion, osmosis, streaming potentials and streaming currents. To better understand multiphasic materials, the reader is encouraged to review the descriptions of biphasic (Section Biphasic Material) and biphasic-solute materials (Section Biphasic-Solute Material).

Governing Equations

In multiphasic materials the solvent is assumed to be neutral, whereas the solid and solutes may carry charge. The mixture is isothermal and all constituents are considered to be intrinsically incompressible. Since the viscosity of the fluid constituents (solvent and solutes) is considered negligible relative to the frictional interactions among constituents, the stress tensor \(\sigma\) for the mixture includes only a contribution from the fluid pressure \(p\) and the stress \(\boldsymbol{\sigma}^{e}\) in the solid,

\[ \begin{equation} \boldsymbol{\sigma}=-p\mathbf{I}+\boldsymbol{\sigma}^{e}\,.\label{eq121} \end{equation} \]

The mechano-chemical potential of the solvent is given by

\[ \begin{equation} \tilde{\mu}^{w}=\mu_{0}^{w}\left(\theta\right)+\frac{1}{\rho_{T}^{w}}\left(p-p_{0}-R\theta\Phi\sum\limits_{\alpha}c^{\alpha}\right)\,,\label{eq122} \end{equation} \]

where \(\mu_{0}^{w}\left(\theta\right)\) is the solvent chemical potential in the solvent standard state, \(\theta\) is the absolute temperature, \(\rho_{T}^{w}\) is the true density of the solvent (which is invariant since the solvent is assumed intrinsically incompressible), \(p\) is the fluid pressure, \(p_{0}\) is the corresponding pressure in the standard state, \(R\) is the universal gas constant, \(\Phi\) is the non-dimensional osmotic coefficient, and \(c^{\alpha}\) is the solution volume-based concentration of solute \(\alpha\). The summation is taken over all solutes in the mixture. The mechano-electrochemical potential of each solute is similarly given by

\[ \begin{equation} \tilde{\mu}^{\alpha}=\mu_{0}^{\alpha}\left(\theta\right)+\frac{R\theta}{M^{\alpha}}\left(\frac{z^{\alpha}F_{c}}{R\theta}\left(\psi-\psi_{0}\right)+\ln\frac{\gamma^{\alpha}c^{\alpha}}{\kappa^{\alpha}c_{0}^{\alpha}}\right),\label{eq123} \end{equation} \]

where \(M^{\alpha}\) is the molar mass of the solute, \(\gamma^{\alpha}\) is its activity coefficient, \(\kappa^{\alpha}\) is its solubility, \(z^{\alpha}\) is its charge number, and \(c_{0}^{\alpha}\) is its concentration in the solute standard state; \(F_{c}\) is Faraday's constant, \(\psi\) is the electrical potential of the mixture, and \(\psi_{0}\) is the corresponding potential in the standard state.

In these relations, \(\Phi\) and \(\gamma^{\alpha}\) are functions of state that describe the deviation of the mixture from ideal physico-chemical behavior; \(\kappa^{\alpha}\) represents the fraction of the pore volume which may be occupied by solute \(\alpha\). The standard state represents an arbitrary set of reference conditions for the physico-chemical state of each constituent. Therefore, the values of \(\mu_{0}^{w}\left(\theta\right)\), \(p_{0}\), \(\psi_{0}\), \(\mu_{0}^{\alpha}\left(\theta\right)\), and \(c_{0}^{\alpha}\), remain invariant over the entire domain of definition of an analysis. Since \(\kappa^{\alpha}\) and \(\gamma^{\alpha}\) appear together as a ratio, they may be combined into a single material function, \(\hat{\kappa}^{\alpha}=\kappa^{\alpha}/\gamma^{\alpha}\), called the effective solubility.

In multiphasic mixtures, it is also assumed that electroneutrality is satisfied at every point in the continuum. Therefore, the net electrical charge summed over all constituents must reduce to zero, and no net charge accumulation may occur at any time. Denoting the fixed charge density of the solid by \(c^{F}\) (moles of equivalent charge per solution volume), and recognizing that the solvent is always considered neutral, the electroneutrality condition may be written as

\[ \begin{equation} c^{F}+\sum\limits_{\alpha}z^{\alpha}c^{\alpha}=0\,.\label{eq124} \end{equation} \]

This condition represents a constraint on a mixture of charged constituents. If none of the constituents are charged (\(c^{F}=0\) and \(z^{\alpha}=0\) for all \(\alpha)\), the constraint disappears.

Each constituent of the mixture must satisfy the axiom of mass balance. In the absence of chemical reactions involving constituent \(\alpha\), its mass balance equation is

\[ \begin{equation} \frac{\partial\rho^{\alpha}}{\partial t}+\divg\left(\rho^{\alpha}\mathbf{v}^{\alpha}\right)=0\,,\label{eq125} \end{equation} \]

where \(\rho^{\alpha}\) is the apparent density and \(\mathbf{v}^{\alpha}\) is the velocity of that constituent. For solutes, the apparent density is related to the concentration according to \(\rho^{\alpha}=\left(1-\varphi^{s}\right)M^{\alpha}c^{\alpha}\), where \(\varphi^{s}\) is the volume fraction of the solid. When the solute volume fractions are negligible, it follows that \(1-\varphi^{s}\approx\varphi^{w}\), where \(\varphi^{w}\) is the solvent volume fraction. The molar flux of the solute relative to the solid is given by \(\mathbf{j}^{\alpha}=\varphi^{w}c^{\alpha}\left(\mathbf{v}^{\alpha}-\mathbf{v}^{s}\right)\), where \(\mathbf{v}^{\alpha}\) is the solute velocity. Using these relations, the mass balance relation for the solute may be rewritten as

\[ \begin{equation} \frac{1}{J}\frac{D^{s}}{Dt}\left(J\varphi^{w}c^{\alpha}\right)+\divg\mathbf{j}^{\alpha}=0\,,\label{eq126} \end{equation} \]

where \(D^{s}\left(\cdot\right)/Dt\) represents the material time derivative in the spatial frame, following the solid; \(J=\det\mathbf{F}\), where \(\mathbf{F}\) is the deformation gradient of the solid. This form of the mass balance for the solute is convenient for a finite element formulation where the mesh is defined on the solid matrix.

The volume flux of solvent relative to the solid is given by \(\mathbf{w}=\varphi^{w}\left(\mathbf{v}^{w}-\mathbf{v}^{s}\right)\), where \(\mathbf{v}^{w}\) is the solvent velocity. When solute volume fractions are negligible, the mass balance equation for the mixture reduces to

\[ \begin{equation} \divg\left(\mathbf{v}^{s}+\mathbf{w}\right)=0\,.\label{eq126b} \end{equation} \]

Finally, the mass balance for the solid may be reduced to the form \(D^{s}\left(J\varphi^{s}\right)/Dt=0\), which may be integrated to produce the algebraic relation \(\varphi^{s}=\varphi_{r}^{s}/J\), where \(\varphi_{r}^{s}\) is the solid volume fraction in the stress-free reference state of the solid.

Differentiating the electroneutrality condition in \eqref{eq124} using the material time derivative following the solid, and substituting the mass balance relations into the resulting expressions, produces a constraint on the solute fluxes:

\[ \begin{equation} \divg\sum\limits_{\alpha\ne s,w}z^{\alpha}\mathbf{j}^{\alpha}=0\,.\label{eq127} \end{equation} \]

Recognizing that \(\mathbf{I}_{e}=F_{c}\sum\nolimits_{\alpha\ne s,w}z^{\alpha}\mathbf{j}^{\alpha}\) is the current density in the mixture, with \(F_{c}\) representing Faraday's constant, the relation of \eqref{eq127} reduces to one of the Maxwell's equation, \(\divg\mathbf{I}_{e}=0\), in the special case when there can be no charge accumulation (electroneutrality).

As described in Section Continuous Variables, the fluid pressure \(p\) and solute concentrations \(c^{\alpha}\) are not continuous across boundaries of a mixture, whereas \(\tilde{\mu}^{w}\) and \(\tilde{\mu}^{\alpha}\)'s for the solutes do satisfy continuity. Therefore, in a finite element implementation, the following continuous variables are used as nodal degrees of freedom:

\[ \begin{equation} \tilde{p}=p-R\theta\Phi\sum\limits_{\alpha\ne s,w}c^{\alpha}\,,\label{eq128} \end{equation} \]

which represents the effective fluid pressure, and

\[ \begin{equation} \tilde{c}^{\alpha}=c^{\alpha}/\tilde{\kappa}^{\alpha}\,,\label{eq129} \end{equation} \]

which represents the effective solute concentration. In the last expression, \(\tilde{\kappa}^{\alpha}\) is the partition coefficient of the solute, which is related to the effective solubility and electric potential according to

\[ \begin{equation} \tilde{\kappa}^{\alpha}=\hat{\kappa}^{\alpha}\exp\left(-\frac{z^{\alpha}F_{c}\psi}{R\theta}\right)\,.\label{eq130} \end{equation} \]

Physically, since \(R\theta\,\Phi\sum\nolimits_{\alpha\ne s,w}c^{\alpha}\) is the osmotic (chemical) contribution to the fluid pressure, \(\tilde{p}\) may be interpreted as that part of the total (mechano-chemical) fluid pressure which does not result from osmotic effects; thus, it is the mechanical contribution to \(p\). Similarly, the effective solute concentration \(\tilde{c}^{\alpha}\) represents the true contribution of the molar solute content to its electrochemical potential.

When using these variables instead of mechano-electrochemical potentials, the momentum equations for the solvent and solutes may be inverted to produce the following flux relations:

\[ \begin{equation} \mathbf{w}=-\tilde{\mathbf{k}}\cdot\left(\mbox{grad}\tilde{p}+R\theta\sum\limits_{\beta\ne s,w}\frac{\tilde{\kappa}^{\beta}}{d_{0}^{\beta}}\mathbf{d}^{\beta}\cdot\mbox{grad}\tilde{c}^{\beta}\right)\,,\label{eq131} \end{equation} \]

and

\[ \begin{equation} \mathbf{j}^{\alpha}=\tilde{\kappa}^{\alpha}\mathbf{d}^{\alpha}\cdot\left(-\varphi^{w}\mbox{grad}\tilde{c}^{\alpha}+\frac{\tilde{c}^{\alpha}}{d_{0}^{\alpha}}\mathbf{w}\right)\,,\label{eq132} \end{equation} \]

where

\[ \begin{equation} \tilde{\mathbf{k}}=\left[\mathbf{k}^{-1}+\frac{R\theta}{\varphi^{w}}\sum\limits_{\alpha\ne s,w}\frac{c^{\alpha}}{d_{0}^{\alpha}}\left(\mathbf{I}-\frac{\mathbf{d}^{\alpha}}{d_{0}^{\alpha}}\right)\right]^{-1}\label{eq133} \end{equation} \]

is the effective hydraulic permeability of the solution (solvent \(+\) solutes) in the mixture. The momentum equation for the mixture is

\[ \begin{equation} \divg\boldsymbol{\sigma}=\mathbf{0}\,.\label{eq134} \end{equation} \]