9.7 Guidelines for Multiphasic Analyses¶
Initial State of Swelling¶
Under traction-free conditions, a multiphasic material is usually in a state of swelling due to the osmotic pressure difference between the interstitial fluid and the external environment. This osmotic pressure arises from the difference in interstitial versus external concentrations of cations and anions, caused by the charged solid matrix and the requirement to satisfy electroneutrality. An osmotic pressure difference arising from such electrostatic interactions is known as the Donnan osmotic pressure. When the Donnan pressure is non-zero, traction-free conditions do not produce stress-free conditions for the solid matrix, since the matrix must expand until its stressed state resists the swelling pressure.
The Donnan pressure reduces to zero when the fixed charged density is zero, or when the external environment is infinitely hypertonic (having ion concentrations infinitely greater than the interstitial fixed charge density). Since these two conditions represent special cases, it is generally necessary to devise methods for achieving the desired initial state of swelling, in an analysis where loads or displacements need to be prescribed over and above this swollen state. Swelling occurs as a result of the influx of solvent into the porous solid matrix. This influx is a time-dependent process that could require extensive analysis time. Therefore, it is computationally efficacious to achieve the initial state of swelling by using a multi-step analysis (Chapter 4) where the first step is a steady-state analysis (Section Control Parameters). In this steady-state step, the fixed charge density may be ramped up from zero to the desired value using a load curve for that property or, alternatively, the external environmental conditions may be reduced from a very high hypertonic state down to the desired level. In the second step, prescribed displacement boundary conditions (when needed) may be specified to be of type relative (Section Zero Displacement), so that they become superposed over and above the initial swelling state.
Example:
<Module type="multiphasic"/>
<!-- rest of file -->
<Step>
<Control>
<analysis type="steady-state"/>
...
</Control>
</Step>
<Step>
<Control>
...
</Control>
<Boundary>
<prescribe type="relative">
<node id="22" bc="z" lc="4">1</node>
...
</prescribe>
</Boundary>
</Step>
Prescribed Boundary Conditions¶
In most analyses, it may be assumed that the ambient fluid pressure and electric potential in the external environment are zero, thus \(p_{\ast}=0\) and \(\psi_{\ast}=0\), where the subscripted asterisk is used to denote environmental conditions. Since the external environment does not include a solid matrix, the fixed charge density there is zero. For example, for a triphasic analysis, \(c^{+}_{\ast}=c^{-}_{\ast}\equiv c_{\ast}\). It follows that the effective fluid pressure in the external environment is \(\tilde{p}_{\ast}=-R\theta\Phi_{\ast}\sum\nolimits_{\alpha}c^{\alpha}_{\ast}\) and the effective concentrations are \(\tilde{c}^{\alpha}_{\ast}=c^{\alpha}_{\ast}/\hat{\kappa}^{\alpha}_{\ast}\tilde{c}^{\alpha}_{\ast}\). Therefore, in multiphasic analyses, whenever the external environment contains solutes with concentrations \(c^{\alpha}_{\ast}\), the user must remember to prescribe non-zero boundary conditions for the effective solute concentrations and the effective fluid pressure.
Letting \(p_{\ast}=0\) also implies that prescribed mixture normal tractions (Section Mixture Normal Traction) represent only the traction above ambient conditions. Note that users are not obligated to assume that \(p_{\ast}=0\). However, if a non-zero value is assumed for the ambient pressure, then users must remember to incorporate this non-zero value whenever prescribing mixture normal tractions. Similarly, users are not required to assume that \(\psi_{\ast}=0\); when a non-zero value is assumed for the electric potential of the external environment, the prescribed boundary conditions for the effective concentrations should be evaluated using the corresponding partition coefficient, \(\tilde{c}^{\alpha}_{\ast}=c^{\alpha}_{\ast}/\tilde{\kappa}^{\alpha}_{\ast}\).
Prescribed Initial Conditions¶
When a multiphasic material is initially exposed to a given external environment with effective pressure \(\tilde{p}_{\ast}\) and effective concentrations \(\tilde{c}^{\alpha}_{\ast}\), the initial conditions inside the material should be set to \(\tilde{p}=\tilde{p}_{\ast}\) and \(\tilde{c}^{\alpha}=\tilde{c}^{\alpha}_{\ast}\) in order to expedite the evaluation of the initial state of swelling. The values of \(\tilde{p}_{\ast}\) and \(\tilde{c}^{\alpha}_{\ast}\) should be evaluated as described in Section Prescribed Boundary Conditions.
Prescribed Effective Solute Flux¶
The finite element formulation for multiphasic materials in FEBio requires that the natural boundary condition for solute \(\alpha\) be prescribed as \(\tilde{j}^{\alpha}_{n}\equiv j^{\alpha}_{n}+\sum\nolimits_{\beta}z^{\beta}j^{\beta}_{n}\), where \(\tilde{j}^{\alpha}_{n}\) is the effective solute flux. For a mixture containing only neutral solutes (\(z^{\beta}=0,\,\forall\beta\) ), it follows that \(\tilde{j}^{\alpha}_{n}=j^{\alpha}_{n}\).
Prescribed Electric Current Density¶
The electric current density in a mixture is a linear superposition of the ion fluxes,
Since only the normal component \(j^{\alpha}_{n}=\mathbf{j}^{\alpha}\cdot\mathbf{n}\) of ion fluxes may be prescribed at a boundary, it follows that only the normal component \(I_{n}=\mathbf{I}_{e}\cdot\mathbf{n}\) of the current density may be prescribed. To prescribe \(I_{n}\), it is necessary to know the nature of the ion species in the mixture, and how the current is being applied. For example, if the ions in the triphasic mixture consist of Na\(^{\mathrm{+}}\) and Cl\(^{\mathrm{-}}\), and if the current is being applied using silver/silver chloride (Ag/AgCl) electrodes, a chemical reaction occurs at the anode where Ag combines with Cl\(^{\mathrm{-}}\) to produce AgCl, donating an electron to the electrode to transport the current. At the cathode, the reverse process takes place. Therefore, in this system, there is only flux of Cl\(^{\mathrm{-}}\) and no flux of Na\(^{\mathrm{+}}\) (\(j^{+}_{n}=0)\) at the electrode-mixture interface, so that the prescribed boundary condition should be \(j^{-}_{n}=-I_{n}/F_{c}\). Since \(z^{+}=+1\) and \(z^{-}=-1\) in a triphasic mixture, the corresponding effective fluxes are given by \(\tilde{j}^{+}_{n}=2j^{+}_{n}-j^{-}_{n}=I_{n}/F_{c}\) and \(\tilde{j}^{-}_{n}=j^{+}_{n}=0\).
Electrical Grounding¶
If a multiphasic mixture containing ions is completely surrounded by ion-impermeant boundaries it is necessary to ground the mixture to prevent unconstrained fluctuations of the electric potential. Grounding is performed by prescribing the effective concentration of one or more ions, at a location inside the mixture or on one of its boundaries corresponding to the location of the grounding electrode. The choice of ion(s) to constrain may be guided by the type of grounding electrode and the reference electrolyte bath in which it is inserted.