Page 360 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Membrane Processes 345

from equilibrium, the fluxes of materials across a membrane, J , can be
i
expressed as linear combinations of all the pertinent driving forces, X j
[1, 2]. Thus, for the case of several driving forces, the flux of component
i, J is calculated as:
i
(13)
J i 5 L ij X j
i
where the L , are the phenomenological coefficients.
ij
Consider the case of transport across a membrane involving two molec-
ular species (i and j) such as might occur in a diffusion dialysis sepa-
ration. In this case, the two conjugate driving forces (X and X ) are
j
i
simply the gradient of the chemical potential for each of these species
across the membrane. In the absence of gradients in pressure or elec-
trical potential, and for dilute solutions, these driving forces are
proportional to the concentration gradients for each species. The phe-
nomenological coefficients for i j account for the possibility that the
flux of i (J ) may affect the flux of j (J ), that is to say that these fluxes
i
j
are coupled.
In the case of our two molecules transported by diffusion:
J L X L X j
i
i
ij
ii
J L X L X j
j
jj
ji
i
The matrix of phenomenological coefficients will be a square matrix
since the flux of each species “i” is described as a linear sum of the gra-
dients of the chemical potentials of all other species “j.” We can begin
to better appreciate the meaning of the phenomenological coefficients
by rewriting Eq. 13 so that terms with j i are removed from the
summation:
J 5 L X 1 L X k (14)
i
ii
ik
i
k2i
The first term in Eq. 14 describes the flux of component i due to a driv-
ing force such as a concentration gradient in i. In the case of Fick’s law,
L is proportional to the diffusion coefficient. For the case of D’Arcy’s law,
ii
L 11 can be interpreted as the membrane hydraulic permeability. The
second term in Eq. 14 describes the impact that the transport of other
components may have on the flux of component i. The phenomenologi-
cal coefficients in the summation are referred to as coupling coefficients
since they relate the impact of a driving force for one component (and
hence the transport of that component) on the transport of a second com-
ponent. The Onsager reciprocal relationship further states that cou-
pling effects are symmetric and thus:
L 5 L ki (15)
ik