Biomedical Engineering Reference
In-Depth Information
membrane proteins acting as transporters or forming pores. Among the latter
are aquaporins forming pores permeable to water, ion channels for perme-
ation of selected ions, or pores formed by certain antibiotics, such as nys-
tatin. These pores have hydrophilic internal walls and are filled with water.
They all differ in their dimensions and specialize in fluxing water and/or
various nonelectrolytic and electrolytic solutes (Hejnowicz 1996; Elmoazzen
et al. 2002; Zeuthen and MacAulay 2002). With respect to their transport
properties we divide porous membranes into homogeneous and inhomoge-
neous. A homogeneous porous membrane has all pores of identical sizes, while
pores of an inhomogeneous porous membrane have different geometry and
sizes.
8.3.1 Homogeneous and Inhomogeneous Porous Membranes
Examples of homogeneous porous membranes are so-called nucleopore mem-
branes. They are manufactured by bombarding very thin solid foils with high-
velocity particles. If one type of particle is used, we obtain a membrane with
cylindrical pores of the same diameter. Depending on the size of the particles
used in manufacturing, the membrane may be permeable to water and some
solutes. If such a membrane has pores of radius r larger than the radius of the
water molecule r w , then it is permeable to water. When the size of a given
solute molecule r s 1 is larger than the pore radius r , the membrane is imperme-
able, and the KK formalism has a reflection coecient σ = 1 (a semipermeable
membrane). For a solute with molecular radius r w <r s 2 <r, the membrane
has a reflection coecient 0 <σ< 1 (a selective membrane), and for a solute
with r s 3 <r w, the coecient is σ = 0 (a permeable membrane).
A membrane with pores differing in their dimensions is called inhomoge-
neous. Such membranes usually have irregular pore geometry, and they differ
locally in their transport properties. For simplicity, however, we assume here
that all pores have cylindrical geometry and are perpendicular to the mem-
brane surface. An example is a nucleopore membrane obtained with a beam of
different particles. In real membranes the pores are randomly distributed, but
for modeling purposes we can order them according to their radii. We assume
that the smallest pores are larger than the dimensions of a water molecule
( r w <r 1 ), and for a solute with molecular radius r s we have
r 1 <r 2 <
···
<r s <
···
<r N
(8.27)
that is, out of N pores in the membrane a certain number, n a, of suciently
small pores is impermeable to the solute, while the remaining n b = N
n a
pores are permeable. To describe the degree of selectivity of individual pores
we introduce a coecient σ p . We postulate that a single pore with pore radius
r p can have the coecient σ p = 1 or 0 expressing the fact that it is either per-
meable to the solute molecules (if r s <r p ) or not. By similarity with the KK
formalism, we will call it the reflection coecient but it needs to be understood
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