Page 359 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

344 Environmental Applications of Nanomaterials

Eq. 10 is a restatement of Fick’s law, where the concentration gradient
is approximated by the difference in concentration across the mem-
brane divided by the thickness of the membrane, and the diffusion
coefficient, D, is equal to kT/f. Similarly, when a difference in pressure
is the sole driving force for moving a fluid through a network of pores
and this force is balanced by a resistance force that is proportional to
fluid velocity (F P 5 F resistance ), then we obtain an expression similar to
D’Arcy’s law:
V i P
v fluid 5 J fluid 5 (11)
f p d m
where f is the friction coefficient for the pores. In each of these cases,
p
the flux of one component in the system is shown to be proportional to
a driving force.
However, in many systems of practical interest, energy gradients and
the transport of multiple components do not act in isolation. The trans-
port of one component in the system may affect the transport of another.
One example is the flow of fluid through a membrane pore carrying a
charge that gives rise to a flux of current known as the streaming cur-
rent. Desalination by reverse osmosis is a second example. In RO desali-
nation, a pressure gradient is applied to transport water across a
semipermeable membrane that rejects much of the salts in solution.
Calculation of the flux of either salt or water requires that gradients in
both pressure and concentration be considered. Phenomena such as
these are examples of coupled transport.
Irreversible thermodynamics addresses coupled transport in a gen-
eralization of expressions such as Fick’s law that describes the flux of
a component of a system as being proportional to a driving force. We gen-
eralize the relationship for flux as a function of driving force by consid-
ering the flux of any one component, i, as a function of a vector of driving
forces, X. A Taylor series expansion from the point at which the driving
forces disappear (X 0 at thermodynamic equilibrium where the fluxes
will also be equal to zero) to X yields the following result:
2
'J i # 1 ' J j
3
J sX d 5 J s0d 1 ` X 1 : X X 1 OsX d (12)
k
j
i
i
j
j 'X X50 j,k 2! 'X 'X k
j
j
If the system is near equilibrium, then we can ignore the higher order
terms and truncate the Taylor’s series after the linear terms. This
implies that the partial derivatives evaluated at X 0 can be treated
as constants. These constants are referred to as phenomenological coef-
ficients, which relate the flux of component i to driving force j.
The result is the second postulate of irreversible thermodynamics,
referred to as the Onsager principle, which states that for small deviations