Active micromixers - Micromixers Fundamentals Design and Fabrication - page 268

FIGURE 7.20

Model of control of zeta potential: (a) parallel shielding electrode; (b) shielding electrode with an arbitrary shape.

where 3 r is the relative permittivity of the device material, 3 0 is the dielectric constant of vacuum, d is
the gap between the channel wall and the electrode. The initial voltage V i corresponds to the case of
zero control voltage V c ¼ 0, where the resulting charge density is equal the initial charge density at the

channel wall. The charge density of the capacitor formed at the electric double layer in the micro-
channel is:

sinh zez
2 k B T

4 n ze

N

r el ; d ¼ (7.54)

k

where n is the ion concentration in the channel, e is the elementary charge, z is the ionic valence, k is

the Debye parameter, k B is the Boltzmann constant, and T is the absolute temperature. Setting
r el,0 ¼

N

r el,d , the zeta potential can be expressed explicitly as a function of the control voltage:

arsinh 3 r 3 0 k

2 k B T
ze

ð V c V i Þ

z ¼ : (7.55)

4 n N zed

For a transient process, the charging time of the capacitors need to be considered. Since the
capacitance of the double layer is much smaller than the capacitance formed by the shielding electrode
and the channel wall. The time constant is determined by the control capacitance and its resistance only:

r Ohm
A

3 r 3 0 A
d ¼

3 r 3 0 r Ohm
d

s ¼

RC ¼ (7.56)

where r Ohm is the specific resistance of the device material. The transient function of the zeta potential
is:

arsinh 3 r 3 0 k

2 k B T
ze

ð V c V i Þ

z ð t Þ¼ exp ð t = s Þ : (7.57)

4 n N zed