manner, as the electro-migration of the ions (toward/away from the wall) is dynamically
balanced by the thermal motions in the diffuse layer. The concentration of ions in the diffuse
layer follows the Maxwell-Boltzmann distribution
ScxpRT
RT ), (1-1)
where 0,, is the electric potential in the EDL induced by the surface charge, co is the
concentration far from the surface where 4~, approaches zero, F is the Faraday constant, R is
the gas constant, Tis the temperature of the fluid, and z is the valence number of the ions15. For
a solution with a symmetric electrolyte, the charge density becomes
p,= -2Fzco sinh zF..f
SRT ) .(1-2)
Coupling Equation 1-2 with the Poisson's equation
V20..r = PI
s (1-3)
where e is the permittivity of fluid, it yields
2 2Fzc sinh zF'4 .
V 2 sinhSnh
S -RT ) (1-4)
The above equation can be simplified from Debye-Hiickel approximation,16 which
considers that the potential energy of the ions is small in comparison to their thermal energy in
the EDL. As a result, the hyperbolic sine term is approximated by the first perturbation only,
S2Fzco zFAf
Se RT (1-5)
Hence, the resulting expressions for electric potential and charge density are
S=-pot Xp Y-
S) (1-6)
and