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Subsubsection 4.9.5.1: Constant Osmotic Coefficient Up Chapter 4: Materials Subsection 4.10.1: Guidelines for Multiphasic Analyses

## 4.10 Triphasic and Multiphasic Materials

Triphasic materials may be used to model the transport of a solvent and a pair of monovalent salt counter-ions (two solutes with charge numbers 1 and -1) in a charged porous solid matrix, under isothermal conditions. Multiphasic materials may be used to model the transport of a solvent and any number of neutral or charged solutes; a triphasic mixture is a special case of a multiphasic mixture. Each of the mixture constituents is assumed to be intrinsically incompressible. This means that their true densities are invariant in space and time; this assumption further implies that a multiphasic mixture will undergo zero volume change when subjected to a hydrostatic fluid pressure. Yet the mixture is compressible because the pores of the solid matrix may gain or lose fluid under general loading conditions. Therefore, the constitutive relation of the solid matrix should be chosen from the list of unconstrained materials provided in Section 4.1.3↑. The volume fraction of the solutes is assumed to be negligible relative to the volume fractions of the solid or solvent. This means that the mixture will not change in volume as the solute concentrations change. As the solutes transport through the mixture, they may experience frictional interactions with the solvent and the solid. This friction may act as a hindrance to the solute transport, or may help convect the solutes through the mixture. The distinction between frictional exchanges with the solvent and solid is embodied in the specification of two diffusivities for each solute: The free diffusivity, which represents diffusivity in the absence of a solid matrix (frictional exchange only with the solvent) and the mixture diffusivity, which represents the combined frictional interactions with the solvent and solid matrix. The user is referred to the FEBio Theory Manual for a more detailed description of triphasic and multiphasic theory.
Due to steric volume exclusion and short-range electrostatic interactions, a solute may not have access to all of the pores of the solid matrix. In other words, only a fraction of the pores is able to accommodate solute (. Furthermore, the activity of solute (the extent by which the solute concentration influences its chemical potential) may be similarly altered by the surrounding porous solid matrix. Therefore, the combined effects of volume exclusion and attenuation of activity may be represented by the effective solubility , such that the chemical potential of the solute is given by In this expression, is the solute chemical potential at some reference temperature ; is the solute concentration on a solution-volume basis (number of moles of solute per volume of interstitial fluid in the mixture); is the solute molecular weight (an invariant quantity); and is the universal gas constant. In a triphasic material, a constitutive relation is needed for ; in general, may be a function of the solid matrix strain and the solute concentration. In FEBio, the dependence of the effective solubility on the solid matrix strain is currently constrained to a dependence on .
The solid matrix of a multiphasic material may be charged and its charge density is given by . This charge density may be either negative or positive. The charge density varies with the deformation, increasing when the pore volume decreases. Based on the balance of mass for the solid, where is the solid volume fraction and is the fixed charge density in the reference configuration.
In the multiphasic theory it is assumed that electroneutrality is satisfied at all times. In other words, the net charge of the mixture is always zero (neutral). This electroneutrality condition is represented by a constraint equation on the ion concentrations, where is the charge number of solute . Since the concentrations of the cation and anion inside the triphasic material are not the same, an electrical potential difference is produced between the interstitial and external environments. The electric potential in the triphasic mixture is denoted by and its effect combines with the chemical potential of each solute to produce the electrochemical potential , where In this expression, represents Faraday's constant. It is also possible to rearrange this expression as In a multiphasic material, the interstitial fluid pressure is influenced by both mechanical and chemical environments. In other words, this pressure includes both mechanical and osmotic contributions, the latter arising from the presence of the solutes. The solvent mechano-chemical potential is given by where is the solvent chemical potential at some reference temperature ; is the true density of the solvent (an invariant property for an intrinsically incompressible fluid); and is the osmotic coefficient which represents the extent by which the solute concentrations influence the solvent chemical potential. In a multiphasic material, a constitutive relation is needed for ; in general, may be a function of the solid matrix strain and the solute concentrations. In FEBio, the dependence of the osmotic coefficient on the solid matrix strain is currently constrained to a dependence on .
The solute mechano-electrochemical potential is nearly equal to its electrochemical potential because the solute volume fraction is assumed to be negligible. The solvent mechano-electrochemical potential is the same as its mechano-chemical potential, since the solvent is neutral in a multiphasic mixture. In general, momentum and energy balances evaluated across a boundary surface in a multiphasic mixture require that the mechano-electrochemical potentials of solvent and solutes be continuous across that surface. These continuity requirements are enforced automatically in FEBio by defining the effective fluid pressure and solute concentration as where
is the partition coefficient for solute . The partition coefficient incorporates the combined effects of solubility and long-range electrostatic interactions to determine the ratio of interstitial to external concentration for that solute. Therefore, the effective concentration represents a measure of the activity of the solute, as understood in chemistry. The effective fluid pressure represents that part of the pressure which is over and above osmotic effects.
In FEBio, nodal variables consist of the solid matrix displacement , the effective fluid pressure , and the effective solute concentrations . Essential boundary conditions must be imposed on these variables, and not on the actual pressure or concentrations . (In a biphasic material however, since , the effective and actual fluid pressures are the same, .)
The mixture stress in a triphasic material is given by , where is the stress arising from the solid matrix strain. The mixture traction on a surface with unit outward normal is . This traction is continuous across the boundary surface. Therefore, the corresponding natural boundary condition for a multiphasic mixture is . (In other words, if no boundary condition is imposed on the solid matrix displacement or mixture traction, the natural boundary condition is in effect.)
The natural boundary conditions for the solvent and solutes are similarly and , where is the volumetric flux of solvent relative to the solid and is the molar flux of solute relative to the solid. In general, and are given by
where is the effective hydraulic permeability of the interstitial fluid solution (solvent and solutes) through the porous solid matrix; is the hydraulic permeability of the solvent through the porous solid matrix; is the diffusivity of solute through the mixture (frictional interactions with solvent and solid); and is its free diffusivity (frictional interactions with solvent only). is the solid matrix porosity in the current configuration. The above expressions for the solvent and solute flux do not account for external body forces.
Also see Section 8.6↓ for additional guidelines for running multiphasic materials.

Subsubsection 4.9.5.1: Constant Osmotic Coefficient Up Chapter 4: Materials Subsection 4.10.1: Guidelines for Multiphasic Analyses