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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(it 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 2 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. 3 (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<2d region of the computational domain. 4 (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 5 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 ( 103 kgm 1s 1 , 103 kgm 3 , D 1012 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. 6 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) 7 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). 8 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. 9 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. 10 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. 11 References Abboud H, El Chami F, Sayah T (2011) A priori and a posteriori estimates for three-dimensional Stokes equations with nonstandard boundary conditions. Numerical Methods for Partial Differential Equations, doi: 10.1002/num.20676, http://hal.archives-ouvertes.fr/hal-00442468 Crews N, Darabi J, Voglewede P, Guo F, Bayoumi A (2007) An analysis of interdigitated electrode geometry for dielectrophoretic particle transport in micro-fluidics. 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