Starting with version 1.5, triphasic materials may now be analyzed in FEBio. A triphasic material consists of a mixture of a porous solid, an interstitial solvent, and two monovalent counter-ions (solutes whose electrical charges are +1 and -1). The solid matrix may be electrically charged.

Essential Boundary Conditions

As explained in the User's Manual, the nodal degrees of freedom for triphasic materials are the solid displacement components (ux, uy, uz), the effective fluid pressure (pe), and the effective solute concentrations (ce, or more specifically, cep for the positive cation and cem for the negative anion). In general, pe and ce are continuous across the boundaries of triphasic materials, or at the boundary between a triphasic material and a biphasic material or a fluid.

Understanding the meaning of pe and ce is necessary for the proper application of essential boundary conditions (boundary conditions applied to nodal degrees of freedom). The actual fluid pressure is denoted by p and the actual solute concentration is c (or cp and cm for the two ions). p and c are the familiar variables for fluid pressure and solute concentration. In particular, in a charged triphasic mixture, p includes the contribution of osmotic effects (osmotic pressure).

For an ideal mixture where the osmotic coefficient is equal to 1, the effective fluid pressure is given by

pe = p - R*T*osm

where osm is the osmolarity of the fluid, R is the universal gas constant (8.314e-6 mJ/nmol.K) and T is the absolute temperature (293 K). The osmolarity is equal to the sum of all solute concentrations in that fluid. In particular, in a triphasic material, osm = cp + cm (the sum of the ion concentrations). From this expression, it becomes apparent that pe represents only the mechanical contribution to the fluid pressure (total pressure minus osmotic pressure).

For an ideal mixture where the solute solubility and activity coefficient are equal to 1, the effective solute concentration is given by

ce = c*exp(z*Fc*psi/R*T)

where z is the solute charge number (e.g., +1 and -1 for cation and anion, respectively), psi is the electric potential, and Fc is Faraday's constant (96500e-9 C/nmol). This expression shows that the effective solute concentration accounts for the effects of electrical charge and potential. For a neutral solute (z=0), or in the absence of electric potential (psi=0 mV), we find that ce=c.

Electrophoresis

For illustration, let's consider the problem of one-dimensional electrophoresis. Electrophoresis represents the phenomenon of ion transport in response to an externally applied electric potential gradient. This condition may be achieved by prescribing a potential gradient across a slab of triphasic material (1 mm thick) sandwiched between an upstream (u) and a downstream (d) bath. Both baths are under ambient pressure (to preclude permeation), which means that the actual pressure p is p_u=0 MPa (upstream) and p_d=0 MPa (downstream). Both baths contain ions at the same concentrations (to preclude diffusion and also enforce bath electroneutrality), thus cp_u=cm_u=150 mM (upstream) and cp_d=cm_d=150 mM. To produce an electric potential gradient, let the electric potential psi in the upstream bath be set to psi_u=10 mV, and that in the downstream bath to psi_d=0 mV.

Given all these parameters, and using the above formulas, we can now evaluate the essential boundary conditions at the interfaces between the upstream bath and triphasic material, and between the triphasic material and downstream bath. On the upstream side:

pe_u = p_u - R*T*osm = 0 MPa - (8.314e-6 mJ/nmol.K)*(293 K)*(150 mM + 150 mM) = -0.731 MPa

cep_u = cp_u*exp(z_p*Fc*psi_u/R*T) = (150 mM)*exp{ (+1)*(96500e-9 C/nmol)*(10 mV)/ [(8.314e-6 mJ/nmol.K)*(293 K)] } = 223 mM

cem_u = cm_u*exp(z_m*Fc*psi_u/R*T) = (150 mM)*exp{ (-1)*(96500e-9 C/nmol)*(10 mV)/ [(8.314e-6 mJ/nmol.K)*(293 K)] } = 101 mM

On the downstream side, since psi_d=0, these expressions simplify to

pe_d = p_d - R*T*osm = 0 MPa - (8.314e-6 mJ/nmol.K)*(293 K)*(150 mM + 150 mM) = -0.731 MPa

cep_d = cp_d*exp(z_p*Fc*psi_d/R*T) = 150 mM

cem_d = cm_d*exp(z_m*Fc*psi_d/R*T) = 150 mM

These values can now be prescribed on the upstream and downstream faces of the triphasic material, to produce the phenomenon of electrophoresis.

(Download the attached file, execute it with FEBio and display the results in PostView)

To implement this problem in FEBio, let the triphasic material be represented by a cube (1 mm on each side), with electrophoresis occurring along the z-axis. The solid matrix is modeled as a neo-Hookean material with Young's modulus equal to 1 MPa and Poisson's ratio equal to 0. The triphasic material is negatively charged, with a fixed charge density in the reference configuration given by cFr=-200 mEq/L. When exposed to the upstream and downstream baths whose osmolarity is 300 mM, a Donnan osmotic pressure will be produced in the interstitial fluid, causing the triphasic material to swell. Since this swelling may be significant, the analysis is split into two steps: In the first step, the upstream and downstream baths are both kept at 0 mV (thus, pe_u=pe_d=-0.731 MPa, and cep_u=cem_u=cep_d=cem_d=150 mM) while cFr is ramped from 0 to -200 mEq/L. This swelling step is analyzed under steady-state conditions. In the second step, the upstream bath potential is ramped from 0 to 10 mV, which is effectively achieved by ramping up cep_u from 150 mM to 223 mM, and ramping down cem_u from 150 mM to 101 mM. This step is also analyzed under steady-state conditions in this example.

At the end of the analysis, we find that the cation flows from upstream (10 mV) to downstream (0 mV), whereas the anion flows in the opposite direction. These ionic flows in response to an electric potential gradient represent electrophoresis. The net current density resulting from these flowing charges is also directed from upstream to downstream, with a magnitude of 8.1 uA/mm^2. The flow of current in response to an electric potential gradient represents electric conduction. The solvent also flows from upstream to downstream in response to the electric potential gradient, and this phenomenon is known as electroosmosis. Finally, note that the electric potential in the triphasic material is not continuous across the boundaries with the external baths, due to the negative fixed charge density inside. Whereas the external potential is 10 mV upstream and 0 mV downstream, the internal potential is -3.9 mV upstream and -13.9 mV downstream.

This example illustrates a subset of electrokinetic phenomena in charged hydrated materials. By prescribing only an electric potential gradient, we observe electrophoresis, electroosmosis and electrical conduction (or electroconduction). Diffusion (solute transport in response to its concentration gradient) and osmosis (solvent transport in response to solute concentration gradient) were precluded by maintaining the same solute concentrations upstream and downstream. Similarly, permeation (solvent transport in response to a pressure gradient) and barophoresis (solute transport in response to a pressure gradient) were precluded by maintaining the same fluid pressure upstream and downstream.