# Numerical analysis of nanoparticle behavior in a microfluidic ```Numerical analysis of particle behavior under dielectrophoresis
1. Problem formulation
All the numerical simulations were performed using a finite element code, FreeFEM++
(Abboud et al. 2011, Lin et al. 2007, Hecht et al. 2012).
For the validation of the code we referred to a particular case in which our problem admits
an analytical solution. This is the case of a DEP chamber with electrodes placed on both the bottom
and top surfaces as illustrated in Fig. 1.
Fig. 1 Typical DEP device: micro-channel with interdigitated top and bottom electrodes.
If a harmonic electric potential V  V0 exp(it  qx) is imposed at the boundaries y   d , where V0
is the amplitude of the electric potential, q 
2

is the wave number, with  the wavelength of the
traveling wave, by solving the Laplace equation V  0 assuming that Ly   Lx , Lz  , and the
uncharged sidewalls, one can obtain the analytical solution of the problem (Shklyaev, Straube
2008):
V (r )  V0 exp(iqx) cosh qy / cosh qd ,
(14)
and the final expression of the dielectrophoretic force becomes:
F  r   F0 F ,
(15)
where F0  3 mV02 q 3k R / 2 cosh 2 (b / 2) is a parameter related to the intensity of the force field,
corresponding to term F0 mentioned above, F   k  cosh by,sinh by,0  corresponds to the term F 
in equation (13a), k  is the ratio of the imaginary, ki , and real, kr , parts of the Clausius-Mossotti
factor given by equation (5), and b  2qd is the so called dimensionless wave number. The first
component of term F  represents the dimensionless twDEP force, while the second component is
the dimensionless DEP force.
The numerical computation was performed for the case d  100μm and   400μm , which
leads to a value b   for the dimensionless wave number. The comparison between the analytical
solution, represented by the components of the F vector, and the numerical solutions, specifically
FDEP / F0 DEP and
FtwDEP / F0twDEP , is presented in Fig. 2 and reveals an excellent agreement
(Neculae et al., 2012).
Fig. 2 Dimensionless DEP and twDEP forces as a function of the vertical position in the channel in
Fig. 1; numerical (full lines) and analytical (squares and circles) are shown for comparison.
After the analytical validation of our numerical method, we can move on to analyzing the
behavior of a more realistic DEP device for nanoparticle manipulation. Fig. 3 presents a typical
DEP array of rectangular shaped electrodes, where virtual slowing down of the electromagnetic
waves is realized by using arrays of electrodes excited with phase-shifted AC signals (Crews et al.
2007). We denote by d1 the electrode width, d2 the distance between two adjacent electrodes, w the
electrodes height, h the height and L the length of dielectrophoretic chamber, respectively.
Fig. 3 Schematic of the DEP patterning chamber with interdigitated bar electrodes at the
bottom surface used for dielectrophoretic separation and traveling wave dielectrophoresis.
Next in this section we discuss two important aspects. First, we show how the DEP forces
are distributed in a typical DEP device and the influence of the device parameters on this
distribution. Also, we discuss here as well the projected trajectories of a nanoparticle in such a
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design. Second, we look at the more general question of determining the distribution of a
concentration of nanoparticles under the influence of DEP.
2. Dielectrophoresis forces and particle trajectories
For the computation of the dielectrophoretic force for an array of interdigitated microelectrodes, we solved the Laplace equations (2) for the real and imaginary components of the
electric potential, together with the associated boundary conditions. Due to the symmetry of the
problem and considering the electrodes long compared to their width, the problem can be treated as
two-dimensional. The computational domain and the boundary conditions can be assumed as
shown in Fig. 4, where the particular case d1=d2=d= 50μm and h=10d was considered. The height
was chosen so as to allow for the simplified boundary conditions which required that h  d .The
vertical lines mark the period over which the system is repeated (the basic unit cell). Each electrode
was assigned its corresponding value for the real and imaginary parts of the potential phasor.
Fig. 4 Schematic representation of the computational domain with boundary conditions for the real
part (VR ) and imaginary part (VI ) of the electric potential. The solid lines indicate the “unit cell”.

The results obtained for the magnitudes of the vectors  VR  VI
2
2

and
   VR  VI  , proportional to the dimensionless DEP and twDEP forces given by equations
(10), (11), are presented in Figs. 5a,b, The area shown corresponds to the region with y  100μm ,
the magnitude of the DEP forces above this level being negligible.
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(a)
(b)
Fig. 5 Calculated values for the magnitudes of the dimensionless DEP and twDEP forces,
plotted on logarithmic scale: (a) FDEP / F0 DEP and (b) FtwDEP / F0twDEP .
One observes a periodic variation for both vectors; the magnitudes of the computed quantities
increase to a maximum value at the electrode edge and diminish rapidly with distance in the
vertical direction.
Fig. 6 shows a more refined numerical analysis regarding the longitudinal variation of the
dimensionless DEP force (along the x-axis) at different distances y from the substrate, in which
either the electrode height is neglected, Fig. 6(a), or when rectangular electrodes with d1  d2  d
and w  0.1d are assumed, Fig. 6(b). The computation of the force profile reveals a highly
inhomogeneous distribution. The details presented in Fig. 6 shows that both DEP and twDEP forces
vary rapidly in the vicinity of the electrode edge. For the chosen geometry, the amplitudes can
range over more than three orders of magnitude, with the highest values at the electrode’s limits. As
was previously discussed, this periodic distribution of the potential allows for an analytical solution
for the DEP and twDEP problems to be found. As we expected, the obtained similar behavior
confirms that dielectrophoretic forces are especially strong near the corners of the electrodes and
fall quickly with the distance. For this reason, in what follows, we will show only the y&lt;2d region
of the computational domain.
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(a)
(b)
Fig. 6 Logarithmic plot of the dimensionless DEP force as a function of the horizontal position
for different distances from the substrate and for electrodes with d1  d2  d and w =0 (a),
respectively w  0.1d (b).
Next, we analyze the vertical behavior of the DEP force for different electrode heights, w.
Figure 7 presents the vertical variation taken for x  0 (the electrode centre to the left of the
domain), x  0.25d (in the electrode body) x  0.5d (the electrode’s edge) and x  d (the middle of
the domain).
(a)
(b)
(c)
(d)
Fig. 7 Vertical variation of the DEP force for electrodes with rectangular shape ( d1  d 2  d ) and
different heights w at: (a) x  0 , (b) x  0.25d , (c) x  0.5d and (d) x  d
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As we can see, increasing the height of the electrodes leads to a translation of the DEP force, which
causes it to slightly increase but does not affect the shape of its variation. Thus, we can say that
slight variations in the geometry have little influence on the overall behavior of our device. This is
a useful trait when considering the experimental implementation of such a structure.
Before moving on to a detailed analysis of the behavior of a suspension of nanoparticles
subjected to DEP and twDEP forces, we first undertake a preliminary study of single particle
trajectories. The particle trajectories are calculated by integrating the velocity equation using the
appropriate expressions for the DEP and twDEP forces. We considered particles with characteristic
size
a=200nm
in
water
(   103 kgm 1s 1 ,   103 kgm 3 , D  1012 m 2s 1 ,  m
80 ).
The
characteristic length of the device is d  50μm . For a real part of the Clausius-Mossotti factor
k R  0.6 (roughly corresponding to latex particles in water), an amplitude of the electric potential
V0
1V and a traveling wave with   400μm , we obtain for the dimensionless parameter in
equation (13a) a typical value of Q  0.2 . The fluid flow in the microchannel is assumed to be
laminar and described by a Poiseuille profile with a typical value of 1 μm/s for the maximum flow
velocity.
a)
b)
c)
d)
Fig. 8 Calculated trajectories of particles with radius a=200nm in case of positive DEP (a), negative
DEP (b), positive twDEP (c) and negative twDEP (d). Ordinate shows vertical launch point of
particles.
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As we can see in Figs. 8, the trajectories are strongly influenced by the DEP force in the
vicinity the electrode. When the particle-electrode distance is large, the influence of the DEP force
is very weak. In the case for which particles enter in the y  100μm region of the computational
domain, there is practically no effect of the DEP force, and as such these results are not discussed
here. So it is important to use very thin devices in order to control the trajectories using electric
fields. This analysis reveals that the nanoparticles in suspension tend to concentrate near the
channel walls for positive DEP or near the channel center for negative DEP, as was expected. Also,
we can see that for positive DEP, the particles will cluster around the edges of the electrodes due to
the distribution of the DEP forces.
We can thus conclude that, in this type of setup, it become easy to locate different types of
nanoparticles, after separation, based on their physical properties such as shape or size. The
distribution of the field profiles, coupled with the projected nanoparticle trajectories also clearly
show that this device can be easily tuned to allow for different types of applications simply by
tuning the size of the electrodes or the electric potential applied. In addition, one observes that the
twDEP forces practically do not influence the trajectories along the array; therefore in the following
considerations only the contribution of the DEP force will be taken into account as in this case we
are interested mainly in the vertical movement of the nanoparticles. We do note however that our
results thus far show that twDEP is a good candidate for horizontal nanoparticle movement with
potential applications in nanoparticle transport and tracking.
3. Dielectrophoretic effects in nanoparticle suspensions
Taking into account the results presented above, we consider for the global computation of
the concentration field a simplified computational domain of a two row system (i.e. both bottom
and surface distributions of electrodes) array, as shown in Fig. 1, in which we neglect the height w
of the electrodes. We consider the same simulation parameters as in the previous calculations,
taking into account the new device geometry.
For the simulation of the behavior of a suspension subjected to dielectrophoretic forces, the
system of equations (13) for the force given by equation (15) is solved. The parameter Q can be
varied by modifying the applied signal. Thus, by increasing or decreasing the voltage, Q increases
or decreases corresponding to a square law. Also, by shifting the frequency of the input potential,
the sign of Q can be changed from positive to negative and vice-versa.
In what follows, we use for our numerical analysis values for Q ranging from -0.3 to 0.3 in order to
describe the repulsive (negative values of Q) and respectively attractive (positive values of Q)
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effects for different intensities of the external field (Lungu et al. 2010). The concentration profile
(Eq. 15) describes the accumulation of particles near the boundaries for Q  0 , or in the center
plane for Q  0 , which correspond to p-DEP or n-DEP, respectively. The results presented in Fig. 9
are obtained for a DEP device with h  200μm and L  2000μm . For simplicity we only show in
Fig. 9 a portion along the length of the device.
a)
b)
Fig. 9 Numerical results regarding the stationary concentration field for k  =1, b   . The
panels (a) and (b) correspond to positive DEP ( Q  0.2 ), respectively negative DEP ( Q  0.2 ).
The results reveal that the nanoparticles in suspension tend to concentrate on the channel walls
(positive DEP) or in the center of the channel (negative DEP) depending on their properties (nature,
size), by adjusting the applied voltage at the command electrodes.
Fig. 10 presents the results of a more refined analysis regarding the influence of the external
field and characteristic size on the particle concentration in the DEP channel.
a)
b)
Fig. 10 Calculated dimensionless concentration profiles  / 0 as a function of the vertical distance
from the bottom electrodes, for several values of Q (a) and particle sizes (b).
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The distances y  100μm and respectively y  200μm correspond to the symmetry axis and
the upper wall of the channel, respectively. The case Q  0 corresponds to the initial distribution of
the dimensionless concentration. The simulations show that in the case of a positive DEP
characterized by Q  0.1 , the concentration increases by 50% near the wall and diminishes by 30%
in the center of the channel. When Q  0.3 , compared to the initial value, the concentration is more
than three times larger near the wall and eight times smaller in the center of the channel. In the case
of a negative DEP, the effect is opposite: for Q  0.1 the concentration on the wall is 20% smaller
than the initial value, while the concentration in the center of the channel presents an augmentation
of 15%. When Q  0.3 , the diminishing of the concentration near the wall is 50% and the
increasing in the center of the channel is about 30%. One observes that for the same absolute values
of the intensities of the external field, Q , the effect of the positive DEP is stronger. The validity of
the computations was carefully checked by verifying that in all the considered cases mass
conservation is preserved. We observe a similar behavior of the concentration profile when varying
the characteristic size of the particles while keeping the voltage constant at V0  1V for both
positive and negative DEP. The results are shown in Fig. 10b. This is expected, as any variation of
the particle size, a , will directly influence the values for Q .
We also compared the behavior of nanoparticles of various sizes, specifically, particles with
a  100 nm and a  200 nm . The results are shown in Fig. 11. As expected, we see a strong
influence of the characteristic size of the particles on their dynamics when subjected to DEP forces.
At the exact same launch point, or in the same area of the velocity profile, different sized particles
have distinct trajectories.
a)
b)
Fig. 11 Compared trajectories for particles with radius a=100nm si a=200 nm, in case of positive
DEP (a) and negative DEP (b), respectively.
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This is an important result to consider as it clearly shows that our method scales very well
even at the nanometer level. Specific to applications, this shows that DEP particle sorting is, in
theory, possible at the nanoscale. One can easily envision a setup wherein, various types of
particles can be concentrated in different areas of the device based on their size. Such an effect
could have potential applications in both particles filtering and sorting.
Another set of numerical of simulations reveals the effect of twDEP force on the suspended
nanoparticles. The considered geometry corresponds to a typical DEP micro-chamber with parallel,
planar, interdigitated electrode array placed on both the bottom and top surfaces. Due to the
symmetry of the geometry, only the bottom part of the device is depicted in Fig. 12.
Fig. 12 Schematic of the DEP patterning chamber with interdigitating bar electrodes at bottom
surface. The bottom surfaces (shaded) is no conducting glass. Geometric variables include chamber
height (h), electrode spacing (l), and electrode width (d).
In our analysis we considered the case of negative dielectrophoresis, when the DEP force
(nDEP) will concentrate the suspended particles in the median plan of the device while the twDEP
force drags the particles towards one side of the device.
At a first step we calculate the components of the dielectrophoretic force inside the device.
The presented results were obtained in the particular case d1=d2=d=h= 50μm (Lungu et al., 2011).
For the computation of the dielectrophoretic force, we solved the dimensionless form of
Laplace’s equations for the real and imaginary components of the potential, together with the
associated boundary conditions. Due to the symmetry of the problem and considering the electrodes
long compared to their width, problem can be treated as two-dimensional in the xOz plane. The
computational domain and the boundary conditions can be assumed as in Fig. 4.
The dimensionless components of the twDEP force ( FtwDEP / F0twDEP ) are presented in Fig. 13.
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Fig. 13 Computed components of the dimensionless twDEP force. The twDEP force acts
practically only along the Ox axis.
The behavior of the twDEP is the main factor in the manipulation process: the force acts only along
the Ox axis and does not change sign, so in this case the particles are continuously driven to the
lateral wall. Obviously, the deterministic lateral transport is determined by the dielectric properties
of the particles and fluid, respectively, as described by transport equations.
CONCLUSIONS
The presented results show that these very thin devices are valuable tools for controlling the
trajectories using electric fields. By adjusting the amplitude or the frequency of the applied voltage
we can obtain am adequate control of the suspended particles, which is a very difficult task at the
nanometer scale. By using DEP forces, micro and nano particles can be concentrated in or removed
from distinct areas of a liquid within microfluidic systems. Due to the enormous potential for new
applications in various fields including, but not limited to, medicine, biology, environmental
sciences and engineering, a solid knowledge of the principles and properties of DEP-based devices
becomes an important aspect. We have shown that numerical tools can help us both understand the
physics and behaviour of dielectrophoresis as well as allow us to design new and more efficient
devices for nano-particle sorting and tracking. We firmly believe that more research in this area can
lead to further advancements, improved designs and new applications of both DEP and twDEP as a
tool for nano-scale research and engineering.
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feedback effect.
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