diff-Albro-iso¶

Module: solute

Category: materialprop

Type string: "diff-Albro-iso"

Parameters¶

Name	Description	Default	Units
`free_diff`	free diffusivity	1	[L^2/t]
`cdinv`	cdinv	0	[L^3/n]
`alphad`	alphad	0	[]

Description¶

The material type for a porosity and concentration-dependent diffusivity which is isotropic is diff-Albro-iso.

This material uses a porosity and concentration-dependent diffusivity tensor that remains isotropic with deformation:

\[ \mathbf{d}=d_{0}\exp\left(-\alpha_{D}\frac{1-\varphi^{w}}{\varphi^{w}}-\frac{c}{c_{D}}\right)\mathbf{I}, \]

where \(c_{D}^{-1}=1/c_{D}\) and the porosity \(\varphi^{w}\) varies with deformation according to the kinematic constraint

\[ \varphi^{w}=1-\frac{\varphi_{r}^{s}}{J}\,. \]

\(J\) is the Jacobian of the deformation, i.e. \(J=\det\mathbf{F}\) where \(\mathbf{F}\) is the deformation gradient and \(\varphi_{r}^{s}\) is the referential solid volume fraction. Here, \(c\) represents the actual concentration of the solute whose diffusivity is given by \(\mathbf{d}\). This constitutive relation is based on the experimental findings reported by Albro et al. ¹.

Example:

<solute sol="1">
  <diffusivity type="diff-Albro-iso">
    <free_diff>123e-6</free_diff>
    <alphad>18</alphad>
    <cdinv>0.0625</cdinv>
  </diffusivity>
  <solubility type="solub-const">
    <solub>1</solub>
  </solubility>
</solute>

Albro, Michael B, Rajan, Vikram, Li, Roland, Hung, Clark T, and Ateshian, Gerard A, "Characterization of the Concentration-Dependence of Solute Diffusivity and Partitioning in a Model Dextran-Agarose Transport Sy…", Cell Mol Bioeng 2, 3 (2009), pp. 295-305. ↩