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Subsubsection 4.8.3.1: Starling Equation Up Chapter 4: Materials Subsection 4.9.1: Guidelines for Biphasic-Solute Analyses

4.9 Biphasic-Solute Materials

Biphasic-solute materials may be used to model the transport of a solvent and a solute in a neutral porous solid matrix, under isothermal conditions. Each of these 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 biphasic-solute 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 solute 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 concentration changes. As the solute transports through the mixture, it 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 solute through the mixture. The distinction between frictional exchanges with the solvent and solid is embodied in the specification of two diffusivities for the 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 the biphasic-solute theory.
Due to steric volume exclusion and short-range electrostatic interactions, the 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 a solute of a particular size (. Furthermore, the activity of the 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 biphasic-solute 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 .
In a biphasic-solute 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 solute. 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 concentration influences the solvent chemical potential. In a biphasic-solute 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 osmotic coefficient on the solid matrix strain is currently constrained to a dependence on .
The solute mechano-chemical potential is nearly equal to its chemical potential because the solute volume fraction is assumed to be negligible. In general, momentum and energy balances evaluated across a boundary surface in a biphasic-solute mixture require that the mechano-chemical potentials of solvent and solute be continuous across that surface. These continuity requirements are enforced automatically in FEBio by defining the effective fluid pressure and solute concentration as Therefore, nodal variables in FEBio consist of the solid matrix displacement , the effective fluid pressure , and the effective solute concentration . Essential boundary conditions must be imposed on these variables, and not on the actual pressure or concentration . (In a biphasic material however, since , the effective and actual fluid pressures are the same, .)
The mixture stress in a biphasic-solute 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 biphasic-solute 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 solute 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 solute) through the porous solid matrix; is the hydraulic permeability of the solvent through the porous solid matrix; is the solute diffusivity through the mixture (frictional interactions with solvent and solid); and is the solute 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.
The governing equations for a biphasic-solute material are the momentum balance for the mixture, Eq.(4.8-3), the mass balance for the mixture, which reduces to Eq.(4.8-4) under the assumption of dilute solutions, and the mass balance for the solute,
Subsubsection 4.8.3.1: Starling Equation Up Chapter 4: Materials Subsection 4.9.1: Guidelines for Biphasic-Solute Analyses