2.11 Equilibrium Swelling¶
When the interstitial fluid of a porous medium contains one or more solutes, an osmotic pressure may be produced in the fluid if the osmolarity of the interstitial fluid is non-uniform, or if it is different from that of the external bathing solution surrounding the porous medium. In general, since the osmolarity of the interstitial fluid may vary over time in transient problems, the analysis of such swelling effects may be addressed using, for example, the biphasic-solute material model described in Section Biphasic-Solute Material. However, if we are only interested in the steady-state response for such types of materials, when solvent and solute fluxes have subsided, the analysis may be simplified considerably.
The Cauchy stress tensor for a mixture of a porous solid and interstitial fluid is given by
where \(p\) is he fluid pressure and \(\boldsymbol{\sigma}^{e}\) is the stress in the solid matrix resulting from solid strain. When steady-state conditions are achieved, the fluid pressure \(p\) results exclusively from osmotic effects and ambient conditions (i.e., it does not depend on the loading history). Thus, in analogy to eq.(2.8-11), \(p=\tilde{p}+R\theta\Phi c\) where \(\tilde{p}\) is the mechanical pressure resulting from ambient conditions and \(R\theta\Phi c\) is the osmotic pressure resulting from the osmolarity \(c\) of the solution.
The osmotic pressure \(p\) may produce swelling of the solid matrix, which is opposed by the solid matrix stress. This becomes more apparent when considering, for example, the case of a traction-free body. The traction is given by \(\mathbf{t}=\boldsymbol{\sigma}\cdot\mathbf{n}\), where \(\mathbf{n}\) is the unit outward normal to the boundary. When \(\mathbf{t}=\mathbf{0}\), the relation of eq.\eqref{eq141} produces \(p=\mathbf{n}\cdot\boldsymbol{\sigma}^{e}\cdot\mathbf{n}\), clearly showing that the osmotic pressure \(p\) is balanced by the swelling solid matrix.
The interstitial osmolarity (number of moles of solute per volume of interstitial fluid) may be related to the solute and solid content according to
where \(c_{r}\) is the number of moles of solute per volume of the mixture in the reference configuration, \(\varphi_{r}^{s}\) is the volume fraction of the solid in the reference configuration, and \(J=\det\mathbf{F}\) is the volume ratio of the porous solid matrix. Neither \(c_{r}\) nor \(\varphi_{r}^{s}\) depend on the solid matrix deformation, thus eq.\eqref{eq142} provides the explicit dependence of \(c\) on \(J\). This relation shows that the osmolarity of the interstitial fluid is dependent on the relative change in volume of the solid matrix with deformation. Effectively, under equilibrium swelling conditions, the term \(-p\mathbf{I}\) in eq.\eqref{eq141} represents an elastic stress and may be treated in this manner when analyzing equilibrium swelling conditions.
Since \(p\) also depends on the osmotic coefficient, if we assume that \(\Phi\) depends on the solid strain at most via a dependence on \(J\), we may thus state generically that \(p=p\left(J\right)\) under equilibrium swelling. It follows that the elasticity tensor for \(\boldsymbol{\sigma}\) is
where \(\boldsymbol{\mathcal{C}}^{e}\) is the elasticity tensor of \(\boldsymbol{\sigma}^{e}\).
Perfect Osmometer¶
Consider a porous medium with an interstitial fluid that consists of a solvent and one or more solutes, whose boundary is permeable to the solvent but not to the solutes (e.g., a biological cell). Since solutes are trapped within such a medium, \(c_{r}\) is a constant in this type of problem. Since the boundary is permeable to the solvent, \(\tilde{p}\) must be continuous across the boundary. Assuming ideal physicochemical conditions, \(\Phi=1\), and zero ambient pressure, this continuity requirement implies that \(p=R\theta\left(c-c^{\ast}\right)\), where \(c^{\ast}\) is the osmolarity of the external environment. Using eq.\eqref{eq142}, it follows that
The reference configuration (the stress-free configuration of the solid) is achieved when \(J=1\) and \(p=0\), from which it follows that \(c_{r}=\left(1-\varphi_{r}^{s}\right)c_{0}^{\ast}\), where \(c_{0}^{\ast}\) is the value of \(c^{\ast}\) in the reference state. Therefore eq.\eqref{eq144} may also be written as
and this expression may be substituted into eq.\eqref{eq143} to evaluate the corresponding elasticity tensor.
A perfect osmometer is a porous material whose interstitial fluid behaves ideally and whose solid matrix exhibits negligible resistance to swelling (\(\boldsymbol{\sigma}^{e}\approx\mathbf{0})\). In that case \(p=0\) and eq.\eqref{eq145} may be rearranged to yield
This equation is known as the Boyle-van't Hoff relation for a perfect osmometer. It predicts that variations in the relative volume of such as medium with changes in external osmolarity \(c^{\ast}\) is an affine function of \(c_{0}^{\ast}/c^{\ast}\), with the intercept at the origin representing the solid volume fraction and the slope representing the fluid volume fraction, in the reference configuration.
FEBio implements the relation of eq.\eqref{eq145} for the purpose of modeling equilibrium swelling even when solid matrix stresses are not negligible. The name “perfect osmometer” is adopted for this model because it reproduces the Boyle-van't Hoff response in the special case when \(\boldsymbol{\sigma}^{e}=\mathbf{0}\).
Cell Growth¶
The growth of cells requires the active uptake of soluble mass to provide the building blocks for various intracellular structures, such as the cytoskeleton or chromosomes, and growth contributes to the osmolarity of the intracellular space. The resulting mechano-chemical gradient drives solvent into the cell as well, contributing to its volumetric growth.
Cell growth may be modeled using the “perfect osmometer” framework by simply increasing the mass of the intracellular solid matrix and membrane-impermeant solute. This is achieved by using eq.\eqref{eq144} to model the osmotic pressure and allowing the parameters \(\varphi_{r}^{s}\) and \(c_{r}\) (normally constant) to increase over time as a result of growth. Since cell growth is often accompanied by cell division, and since daughter cells typically achieve the same solid and solute content as their parent, it may be convenient to assume that \(\varphi_{r}^{s}\) and \(c_{r}\) increase proportionally, though this is not an obligatory relationship. To ensure that the initial configuration is a stress-free reference configuration, let \(c_{r}=\left(1-\varphi_{r}^{s}\right)c^{\ast}\) in the initial state prior to growth.
Donnan Equilibrium Swelling¶
Consider a porous medium whose solid matrix holds a fixed electrical charge and whose interstitial fluid consists of a solvent and two monovalent counter-ions (such as Na\(^{\mathrm{+}}\) and Cl\(^{\mathrm{-}})\). The boundaries of the medium are permeable to the solvent and ions. The fixed charge density is denoted by \(c^{F}\); it is a measure of the number of fixed charges per volume of the interstitial fluid in the current configuration. This charge density may be either negative or positive, thereby producing an imbalance in the concentration of anions and cations in the interstitial fluid. To determine the osmolarity of the interstitial fluid, it is necessary to equate the mechano-chemical potential of the solvent and the mechano-electrochemical potential of the ions between the porous medium and its surrounding bath. When assuming ideal physicochemical behavior, the interstitial osmolarity (resulting from the interstitial ions) is given by
where \(c^{\ast}\) is the salt concentration in the bath. Alternatively, we note that the osmolarity of the bath is \(\bar{c}^{\ast}=2c^{\ast}\). Though this expression may be equated with eq.\eqref{eq142}, the resulting value of \(c_{r}\) is not constant in this case, since ions may transport into or out of the pore space; therefore that relation is not useful here.
However, since the number of charges fixed to the solid matrix is invariant, we may manipulate eq.\eqref{eq142} to produce a relation between the fixed charge density in the current configuration, \(c^{F}\), and the corresponding value in the reference configuration,\(c_{r}^{F}\),
Now the osmotic pressure resulting from the difference in osmolarity between the porous medium and its surrounding bath is given by
This expression may be substituted into eq.\eqref{eq143} to evaluate the corresponding elasticity tensor.
When the osmotic pressure results from an imbalance in osmolarity produced by a fixed charge density, it is called a Donnan osmotic pressure. The analysis associated with this relation is called Donnan equilibrium.