Active micromixers - Micromixers Fundamentals Design and Fabrication - page 261

Rhodamine B was used as the dye for visualization of the mixing effect. With an applied voltage of 0 to
900 V and a distance between the two electrodes of 450 m m, a field strength ranging from 0 to

10 6 V/cm was used. The mixing channel was fabricated in PDMS. The channel depth and width
are 36.5

2

10 6 V/
cm, which is one order of magnitude higher than that required for the configurations depicted in Figs.
7.15 (a) and (b).

m m and 78 m m, respectively. EHD instabilities occurred at a field strength of about 1

7.4 DIELECTROPHORETIC DISTURBANCE
The background of DC dielectrophoresis was discussed in Section 2.6.3. Since a DC electric field can
only trap dielectric particles and does not induce instability, mixing application require the stirring
motion of particles induced by AC dielectrophoretic forces. Considering an AC electric field E el ( t ) ¼

E el exp(- iut ) with the driving frequency u and a spherical dielectric particle with a radius of r p , the
dielectric force acting on the particle is:

m

1
2 <

E el

f DEP ð t Þ¼ ð u Þ $ V (7.36)

represents the real component of a complex number and E el is the complex conjugate of E el .
Due to the frequency dependence of permittivities of both the fluid and the particle, the dipole moment
is also frequency dependent:

where <

3 p ð u Þ 3 f ð u Þ

4 p3 f r p m ¼ E el (7.37)

3 p ð u Þ 2 3 f ð u Þ

where 3 f and 3 p are the complex permittivities of the fluid and the particle. The complex permittivity is
defined as:

i s el
u

3 ð u Þ¼ 3 (7.38)

where s el is the conductivity of the dielectric medium. The term with the complex permittivities is
called the Clausius-Mossotti factor:

3 p ðuÞ3 f ðuÞ
3 p ð

K ð u Þ¼

Þ :

(7.39)

u Þ 2 3 f ð u

Integrating (6.36) over time results in the time-averaged DEP force:

3 p ð u Þ 3 f ð u Þ

2 p3 f r p E rms f DEP ¼

Þ V

(7.40)

3 p ð u Þþ 2 3 f ð u

p is the root-mean-square electric field. The Clausius-Mossotti factor can take
any value between -0.5 and 1. That means, the DEP force can change its direction at a critical driving
frequency u c . This critical frequency can be determined by setting the DEP force or the real part of
K ( u ) to zero:

where E rms ¼ E el =

s

ð

s el ; f s el ;p Þð s el ; p þ 2 s el ; f Þ

u c ¼ : (7.41)

ð 3 p 3 f Þð 3 p þ

2 3 f Þ