XIX Sympozjum Srodowiskowe PTZE - materialy.pdf

XIX Sympozjum PTZE, Worliny 2009
recently derived the time-averaged dielectrophoretic force for the general electric field case as
[1], [2]
( ) ( )( 0 0 0 )
3 2 2 2 2
F () t = 2πε mr ⎡RefCM ∇ Erms + Im⎡
fCM Ex ∇ ϕx + Ey ∇ ϕ y + Ez
∇ϕ⎤⎤
⎣ z
⎣ ⎦ (6)
⎦
where Erms is the rms value of the electric field strength. Ei0 and φi (i = x, y, z) are the
magnitude and phase, respectively, of the field components in the principal axis directions.
This expression can be used to investigate the forces arising from any form of applied field. It
contains two terms that allow an appreciation that there are two independent force
contributions to DEP motion. The first term correspond to the real component of the induced
dipole moment in the particle and to the spatial nonuniformity of the field magnitude. This
force directs the particle toward strong or weak field regions, depending upon whether is
positive or negative. This is the conventional cDEPterm. The second term corresponds to the
imaginary component of the induced dipole moment and to spatial nonuniformity of the field
phases ∇φx, ∇φy, ∇φz. Depending on the polarity of this force directs the particle toward
regions where the phases of the field components are larger or smaller in other words, against
or with the direction of travel of the electric field.
Another approach to dielectrophoretic force calculation employed by Sauer and Schloegl is
based on the Maxwell stress tensor formulation where the stress tensor T is integrated over
the surface particle [5, 6]:
�
FDEP () t = � ∫ ( T⋅n) dS
(7)
where n is the unit vector normal to the surface. This method is regarded as the most general
approach to computation of the field induced forces. The Maxwell stress tensor is given by
1 * 1 2
( ( ) ) Re( )
2 2 E
ε
� � ⎛ � ⎞
T= DE+ ED− E⋅ D U = ⎜EE− U⎟(8)
⎝ ⎠
where ε * = ε − σ/jω. Only real part of the medium permittivity appears in the stress tensor. For
the applied harmonic electric field the Maxwell stress tensor is given by
� 1 * * * 1
�
* *
T= Re( ε ) ( E+ E )( E+ E ) − ⎡( ) ( ) ⎤
4 2⎣
E+ E ⋅ E+ E
⎦
U =
1 � * * * 2 1 * ⎡⎛ * * 1
�
* * ⎞⎤
= Re( ε ) ⎡( EE + E E− E U) ⎤+
Re( ε ) ⎜ + − ( ⋅ + ⋅ ) ⎟
4 ⎣ ⎦ 4
⎢ EE E E E E E E U
2
⎥
⎣⎝ ⎠⎦
The first term in the above expression is time-averaged stress tensor the second term vanishes
under time averaging. The time averaged net DEP force on particle can be now written as [9]
1 * * * 2
DEP Re( m) ( m m m m Em ) dS
4
ε
�
F = � ∫ E E + E E − U ⋅n
(10)
In our case the field is described by set of following well known equations [8]:
( σ jωε) V 0
∇⋅ + ∇ = (11)
99
(9)