Trapping force on a finite-sized particle in a dielectrophoretic cage
P. Singh and N. Aubry
Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
Received 24 February 2005; published 5 July 2005
The point dipole PD model is routinely used for estimating the dielectrophoretic DEP force acting on a
particle placed in the nonuniform electric fields of dielectrophoresis devices, such as square cages. We show
that if the particle size is much smaller than the dielectrophoretic cage size, the PD model accurately approxi-
mates the actual DEP force, computed numerically using the Maxwell stress tensor method. However, when
the two sizes are comparable, the actual DEP force differs significantly in both magnitude and direction from
that given by the PD model.
DOI: 10.1103/PhysRevE.72.016602 PACS numbers: 41.20.Cv
In recent years, primarily due to improvements in the mi-
crofabrication techniques, many new applications of dielec-
trophoresis have been developed for the handling of micron
and nanosized particles. For example, dielectrophoresis has
been used for separating submicron-sized latex spheres and
viruses 1,2, separating DNA molecules and proteins 3,
characterizing and separating bacterial cells 4,5, removing
cancer cells from human blood 6–8, and trapping submi-
cron sized particles in cages with dimensions comparable to
the particles size 9,10. Our focus in this paper is on the
contact less trapping force acting on a particle placed in a
dielectrophoretic square 3D cage which in the past has been
estimated using the point dipole PD model 11–15. The
goal of the present study is to compute the actual force in
this configuration and determine the validity of the PD
model.
A simple 3D dielectrophoretic cage can be formed by
placing four electrodes in the four side walls of a square
shaped channel, as shown in Fig. 1. The voltages of the four
electrodes are selected such that the electric field magnitude
is locally minimum at the center of the domain where one
wishes to attract and hold the particle. This device is of prac-
tical interest as it provides a way to trap a particle in a con-
tact less fashion, at the center of the cage. The magnitude of
the electric field E in the xz plane for the case without the
particle is shown in Fig. 2a. Notice that the magnitude is
locally minimum at the cage center and increases with in-
creasing distance from the domain center.
Figure 2b shows that the electric field inside the cage is
nonuniform, and its gradient near the domain center is non-
zero, except at the center itself where it is zero. The lines of
the gradient of the electric field magnitude shown in Fig.
2b emanate approximately radially from the cage center
and end at the edges of the electrodes. If a particle is placed
in this domain and its dielectric constant is smaller than that
of the liquid, it will experience the so-called dielectro-
phoretic DEP force towards the center of the domain, i.e.,
in the direction opposite to the lines of the gradient of the
electric field magnitude. If the dielectric constant of the par-
ticle is greater than that of the liquid, the DEP force is in the
direction away from the center. This situation will not be
considered here, since it has limited practical applications as
in this case the particle will not be trapped at the center of
the cage in a contact less manner.
An estimate of the trapping force can be obtained using
the PD model which considers the particle as a point dipole
and thus assumes that the gradient of the electric field is
approximately constant over the particle. According to the
PD model, the dielectrophoretic force acting on a linearly
and homogeneously polarizable spherical particle placed in a
dc electric field is given by the expression F
DEP, PD
=4a
3
0
c
E · E, where a is the particle radius,
0
= 8.8542 10
-12
F / m is the permittivity of free space, E is
the electric field, and =
p
-
c
/
p
+2
c
is the Clausius-
Mossotti factor,
c
and
p
are the dielectric constants of the
liquid and particle. We will assume that the particles and
liquid are both perfect dielectrics. Our results are also appli-
cable to ac electric fields, provided the rms value of the
electric field is used, is replaced by the real part of the
complex frequency dependent Clausius-Mossotti factor and
the force is the time averaged force. We will also present
some results where the DEP force is estimated using the
quadrupole model 11,15.
Clearly, when the size of the particle being trapped is
comparable to the cage size, the PD model is expected to be
in error because the assumption made on the electric field
nonuniformity i.e., the nonuniformity is modest and its scale
is large compared to the particle size is no longer valid and
also because the presence of the particle modifies the overall
electric field distribution in the cage, as the distance between
the particle and the cage walls is comparable to the particle
size 10. To correct this error, the electric field must be
obtained by including the particle in the electric field prob-
lem, as was done in 10 for the two canonical cases of a
cylindrical particle in a cylindrical shell and a spherical par-
ticle in a spherical shell. It has been noted in the past that the
error in the DEP force due to the assumptions made on the
electric field nonuniformity can be reduced by incorporating
the quadrupole, and if needed, additional higher order terms
11,15. However, the error due to the modification of the
electric field cannot be fully corrected by simply adding the
higher order multipolar terms, as these terms are evaluated
using the electric field computed without the particle. We
make this point by presenting some results where the quad-
rupole terms are also included for estimating the DEP force.
It is worth noting that in the uniform electric field case, the
method of image can be used to determine the modified elec-
PHYSICAL REVIEW E 72, 016602 2005
1539-3755/2005/721/0166025/$23.00 ©2005 The American Physical Society 016602-1