Articular cartilage with intra- and extrafibrillar waters - mass transfer and generalized diffusion

The present analysis complements the chemo-mechanical model of articular cartilage developed in Loret and Simoes [Loret, B., Simoes, F.M.F., 2004. Articular cartilage with intra- and extrafibrillar waters. A chemo-mechanical model. Mech. Mater. 36 (5-6), 511-541; Loret, B., Simoes, F.M.F., 2005a. Mechanical effects of ionic replacements in articular cartilage. Part I - The constitutive model. Biomech. Model. Mechanobiol. 4 (2-3), 63-80. Part II - Simulations of successive substitutions of NaCl and CaCl2 81-99], where only equilibria were considered, and therefore time was absent. The focus here is, first, to present how transport phenomena are aggregated to the porous media framework, and, second, to detail the constitutive equations of these transports. Indeed, these equations are developed in the context of a three-phase multi-species electro-chemo-mechanical model that accounts for the effects of two water compartments, namely intrafibrillar water stored between collagen fibrils and extrafibrillar water covering the negatively charged proteoglycans. The electrolyte circulating the two fluid phases contains ions sodium Na+, calcium Ca2+ and chloride Cl-. Species diffuse within their phase. They transfer from one fluid phase to the other. The various sources of dissipation are built in a thermodynamic framework, segregated and decoupled via the Clausius-Duhem inequality. Linear and non-linear equations of mass transfer are proposed along an onsagerist approach. The generalized diffusion in the extrafibrillar compartment accounts for Darcy's law of seepage through the porous solid skeleton. Fick's law of ionic diffusion, and Ohm's law of electric flow. An original derivation of the constitutive equations of generalized diffusion is proposed. Indeed, the dissipation inequality is written in two forms, which are required to be equivalent. This approach has the advantage of delivering the general structure of the diffusion matrix. It also displays in explicit form the degrees of freedom for possible refinements. Simple assumptions, phrased in terms of entities that are standard in transport of porous media, allow to recover arrowhead diffusion matrices. Comparison with an earlier proposal is detailed. An osmotic coefficient is found to be hidden in the equations, and anomalous negative osmosis is observed to take place for both sodium chloride and calcium chloride electrolytes. Finally, an experimental setup to measure transport properties is analyzed. The model describes correctly the increase and leveling of the experimental diffusion coefficient, and no additional ad hoc constitutive assumptions are needed in contrast to some suggestions in the literature. The results are presented for sodium chloride NaCl and calcium chloride CaCl2. (c) 2007 Elsevier Masson SAS. All rights reserved.