Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 65

where q surf , u ep , and r p are the surface charge, the particle velocity and the radius of the particle. The
charge density on the particle surface is:
q surf
A surf ¼

r¼r p ¼ 1

3 d

d r 3z

r p

r p
l D

þ : (2.175)

If the radius of the particle is much larger than the Debye length ( r p / l D [ 1), the surface charge

can be calculated as follows:

1

3z
r p

r p
l D

4 pr p q surf ¼ þ ¼ 4 p3r p z: (2.176)

Combining (2.174) and (2.176) results in the velocity of the particle:

q surf E el
6 pmr p ¼

3zE el
m :

2
3

u ep ¼ (2.177)

At a constant dynamic viscosity m and a constant zeta potential z , the electrophoretic velocity is
proportional to the field strength E el . The proportional factor is called the electrophoretic mobility:

u ep
E el ¼

2 3z
3 m :

u ep ¼ (2.178)

If the particle radius is much smaller than the Debye length ( r p / l D << 1), the electrophoretic

mobility approaches the electroosmotic mobility on a flat wall:

u ep
E el ¼

3z
m :

u ep ¼ (2.179)

The electrophoretic mobility of a particle can generally be formulated as follows:

2 3z
3 m $

u ep ¼ C ð r p = l D Þ;

(2.180)

where the correction factor C is a function of the ratio r p / l D :

r p = l D 2
16

5 r p = l D 3
48

r p = l D 2 4
96

r p = l D 5
96

C ¼ þ þ

1

" r p = #

exp

(2.181)

l D 4
8

r p =

l D 6
96

Z r p

r p
l D <

exp ð x Þ

ð r p Þ

d x

:

for 1

x

N

r p
l D >

3
2

9
2 r p =l D þ

75 330

C ¼

2

for 1 :

3

2 ð r p = l D Þ ð r p = l D Þ

2.6.3 Dielectrophoresis
Dielectrophoresis (DEP) is the motion of a dielectric particle in a dielectric fluid. Because the particle
is charge neutral, dielectric force is caused by the inhomogeneity of the electric field. Assuming