Membrane-Based Nanotechnology and Drug Delivery - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 143

6.4.3 Theoretical Understanding of the Nanoparticle Diffusion
Through Ion Pores/Channels in Membranes

We consider two simple cases in order to better understand the problem outlined
above. We first assume that the solutes (nanoparticles) can cross through the porous
membrane with cylindrical pores that are perpendicular to the membrane surface,
due to the influence of a pressure gradient across the membrane. The other possibility
is that the solutes diffuse through the porous membrane, due to a gradient of solute
concentrations across the membrane.
In the case of nanoparticle diffusion due to the influence of a pressure gradient, a
very simple formula, the so-called Poiseuille formula, can be applied, which states
that

J v = L P P (6.7)

Here J v is defined as the total volume of fluid crossing the membrane per second and
per unit membrane surface area. L P is the hydraulic permeability of the membrane,
and is expressed in the units of (cm/s)/atmosphere, which can be derived from the
known values of fluid viscosity, pore density, cross-sectional area, and membrane
thickness. P is the pressure gradient across the membrane. The generalized form

of J is as follows:

v

flow per pore
channel

J v = × pore number density on membrane surface

Here, in the case of nanoparticle diffusion, we assume that the number of nanopar-
ticles crossing the membrane per second and per unit membrane surface area J np , P
changes in proportion to the value of J .

In another case of nanoparticle diffusion due to the influence of nanoparticle
concentration gradient across the membrane, a simple formula called Fick's law can
help to analytically describe the diffusion mechanism. For a gradient of nanoparticle
concentration

v

c np , the nanoparticle diffusion through the pores can be expressed
by the following formula

J np , c = η m c np (6.8)

Here, J np , c is the number of nanoparticles crossing a unit membrane area per second,
and η m is the membrane permeability.
Equations 6.7 and 6.8 describe situations where the nanoparticles pass through
cylindrical pores. However, in the proposed novel nanotechnology application for
nanoparticle transport through a lipid-lined toroidal pore (see Fig. 6.11 ), the parti-
cle flow is expected to take place through a different class of pores. This accounts
for mainly two important aspects which are: (i) that membrane thickness vanishes
at the opening of the pore so that the time to cross the membrane by a nanopar-
ticle is almost zero and (ii) that the pore cross-section changes back-and-forth
with time. These dynamical aspects are very important and require a totally novel