A practical extension of hydrodynamic theory of porous transport for hydrophilic solutes

The equations for transport of hydrophilic solutes through aqueous pores provide a fundamental basis for examining capillary-tissue exchange and water and solute flux through transmembrane channels, but the theory remains incomplete for ratios, alpha, of sphere diameters to pore diameters greater than 0.4. Values for permeabilities, P, and reflection coefficients, sigma, from Lewellen, working with Lightfoot et al., at alpha = 0.5 and 0.95, were combined with earlier values for alpha < 0.4, and the physically required values at alpha = 1.0, to provide accurate expressions over the whole range of 0 < alpha < 1.

The "data" were the long-accepted theory for alpha < 0.2 and the computational results from Lewellen and Lightfoot et al. on hard spheres (of 5 different alpha's) moving by convection and diffusion through a tight cylindrical pore, accounting for molecular exclusion, viscous forces, pressure drop, torque and rotation of spheres off the center line (averaging across all accessible radial positions), and the asymptotic values at alpha = 1.0. Coefficients for frictional hindrance to diffusion, F(alpha), and drag, G(alpha), and functions for sigma(alpha) and P(alpha), were represented by power law functions and the parameters optimized to give best fits to the combined "data."

The reflection coefficient sigma = {1 - [1 - (1 - phi)2]G'(alpha)} + 2alpha2 phi F'(alpha), and the relative permeability P/Pmax = phi F '(alpha)[1 + 9alpha5.5 x (1.0 - alpha5)0.02], where phi is the partition coefficient or volume fraction of the pore available to solute. The new expression for the diffusive hindrance is F'(alpha) = (1 - alpha2)(3/2) phi/[1 + 0.2 x alpha2 x (1 - alpha2)16], and for the drag factor is G'(alpha) = (1 - 2alpha(2)/3 - 0.20217 alpha5)/(1 - 0.75851 alpha5) - 0.0431[1 - (1 - alpha10)]. All of these converge monotonically to the correct limits at alpha = 1.

These are the first expressions providing hydrodynamically based estimates of sigma(alpha) and P(alpha) over 0 < alpha < 1 They should be accurate to within 1-2%.