We describe the concept of a dielectrophoretic nanoparticle injector and its use in a plasmonic/photonic-based nanoparticle manipulation system. Particle motion is achieved by generating an electrostatic, non-uniform field between two tilted plates and applying the corresponding dielectrophoretic force to net-neutral nanoparticles. We investigate the dependence the dielectrophoretic force has on the plate angle of the charged plates as well as their separation distance, dielectric filler material, and exit interface membrane. Our results indicate an increasing average and maximum dielectrophoretic force attainable in the axial direction with corresponding decreasing plate angle and gap distance. The model also predicts larger field variation and deviation from the average with smaller plate angle and gap distance. Lastly, we conclude that the nanoparticles must be suspended in a dielectric medium with permittivity greater than their own permittivity so that their net motion is outward through the exit interface membrane and into the manipulator system.
The manipulation of micro/nanoparticles of solid or aqueous material by way of gradient electromagnetic fields is used extensively in the fields of photonics and microfluidics. One such action mechanism is that of the dielectrophoretic force. Dielectrophoresis is well known and has been utilized in the manipulation of liquid microflows and pico/nanoliter droplets for siphoning,1 separation and mixing2 in chemical and biological experiments, and transport applications.3 Liquid tendrils have been guided from a droplet along wall-less straight4 and curved5 virtual microchannels by way of the dielectrophoretic force enabling enhanced flexibility for the aforementioned transport techniques. Additional nanoparticle manipulation schemes make use of plasmon generated gradient force fields which have been studied for the acceleration of net-neutral nanoparticles via dielectrophoresis with applications in nanosatellite propulsion systems.6–8 We study a dielectrophoretic tilted plate geometry that enables variable injection of nanoparticles or microliter quantities of liquids into manipulation/acceleration schemes such as those mentioned above and can double as a mass storage reservoir when injection is inactive.
II. DIELECTROPHORETIC INJECTOR
A. Theory of dielectrophoresis
Dielectrophoresis occurs when a net-neutral particle is placed in a non-uniform electric field. The electric field polarizes the particle and the polarized particle then feels a force due to the gradient in the magnitude of the field (Figure 1). The direction of the force depends on the difference between the permittivity of the particle and that of the surrounding medium. The dielectrophoretic (DEP) force is utilized in a variety of research fields but most commonly in microfluidics and biomedical applications.9 Its effectiveness in these areas is due, in part, to its ability to separate particles according to their polarizability and/or size. We desire to make use of its ability to precisely control the motion and flow of a concentration of net-neutral nanoparticles.
Research has shown that dielectrophoresis can be used to continuously pump particle-laden microfluidic flows through virtual (wall-less) channels using microstructured electrodes in a variety of configurations.3 Research has also shown that dielectrophoresis can filter particles from a stream of gas; expanding the usability of the DEP mechanism.10 Further progress in this field has demonstrated that particulate matter can be separated by use of dielectrophoresis in a vacuum environment, known as vacuum dielectrophoresis.11 Vacuum dielectrophoresis eliminates certain interactions due to the particles moving in a medium/fluid, making the DEP force the only interaction with the particles in the plane perpendicular to the force of gravity.
The DEP-induced motion depends on the dielectric properties of the particles and the surrounding medium. Specifically, it depends on the effective polarization of the suspended particles. If the polarizability of a net-neutral nanoparticle is greater than the polarizability of the surrounding medium, then the nanoparticle will be pushed toward the stronger region of the electric field (pDEP) and vice-versa (nDEP) if the medium has higher polarizability. Equations (1) and (2) define the DEP force acting on a particle.
is the radius of the particle. is the permittivity of the surrounding medium in which the particles are suspended. is the Clausius-Mossotti factor, defined in Equation (2), that relates the permittivity of the particle and medium and is positive when the particle permittivity is greater than the medium permittivity. is the electric field. From these equations we see that the DEP force is proportional to the cube of the radius of the suspended particles as well as the gradient of the magnitude of the electric field.
B. Dielectrophoretic injector design
The DEP force can be used to inject nanoparticles into a photonic particle manipulator. As described above, the DEP force is active in the presence of a non-uniform electric field. It acts on net-neutral particles along the direction of the gradient of the non-uniform field. Therefore, to harness the DEP force and use it to propel nanoparticles into particle manipulating platforms, we must design an electric field that is non-uniform and whose gradient tends to lie along a single direction. We investigate here a wedge-shaped prism, whose 2-D cross section is a simple tilted plate capacitor. This geometry creates a steady, non-uniform electric field when supplied with a DC voltage and the electric field can be easily solved analytically in 2-dimensions using the following equation derived from Coulomb’s Law for the electric field due to a distributed charge.
For 2-dimensions, let the charge density where Q is the charge on the plate and L is its length. With this reduction and the definitions in Equations (4), (5) and (6) that give the location of a test point and shape of the surfaces, we can define the total electric field between the surfaces as the sum of the electric fields produced by each surface, Equation (7).
The tilted plate capacitor cross-section is shown in Figure 3, where the red lines show the silhouette of the injector, and the design is such that the particles will start in the injector (injector doubles as a storage tank) and then exit to the manipulation platform on the right. The upper and lower surfaces of the injector (red lines in the image) are electrically separated and a potential difference is maintained across them in order to produce the desired electric field. A dielectric, rectangular guide-sleeve (solid blue lines) is inserted along the axis of the injector between the charged surfaces with separation distance equal to the opening width, , of the injector exit. The dielectric guide sleeve keeps the particles away from the conducting plates where, in close proximity to the plates, the gradient of the electric field pointing toward the plate would act to trap them in pDEP.
Summing the electric fields due to the upper and lower plates allows the conversion of Equation (3) to Equation (7) with , the electric field produced by the lower plate, defined by Equation (8) and , the electric field produced by the upper plate, defined in Equation (9).
The electric field produced by this setup is plotted in Figure 4(a) for plate angle and Figure 4(b) for . For validation, the case is compared to the electric field produced by a parallel plate capacitor, which assumes infinite electrodes, . is the electric field, the electric potential difference between the two electrodes, and the separation distance between the two capacitor electrodes. With and , . Let and for the analytical model solution with finite electrodes. The minor difference in plate separation values is an intentional offset for the analytical solution because the electric field derived from Coulomb’s law is proportional to , which means that as approaches , the distance to the second plate approaches zero and the electric field contribution goes to infinity. The electric field produced by the plates in the analytical solution, at the point where the field is max, is . The percent error between the finite plate analytical and infinite plate parallel capacitor solutions is . The electric field distribution in Figure 4(b) for the plate angle is steady and non-uniform, increasing in strength from left to right. The solid, black, diagonal lines represent the edges, or silhouette, of the wedge-shaped injector while the short vector lines indicate the electric field produced between the electrodes. Along the centerline (), one can see that the electric field lines increase in strength with increasing y-position. These properties are plotted in Figure 5. We have disregarded the configuration of the electric field outside of the particle injection structure because it has no effect on the motion of the particles and is assumed to be shielded.
In Figure 5, contours of the electric field along the centerline and above and below the centerline are plotted versus the distance along the axis of the injector. Ez and Ey are the electric field components in the z- and y-directions, respectively. The dotted red line in Figure 5 shows that the y-component of the electric field along the axis of the injector is zero for the whole axis. This is expected because as the electric field lines curve from one electrode to the other, they are perpendicular to the centerline axis of the injector at the centerline. The off-axis y-components of the electric field contour show that there is a transition region where the electric field reverses direction inside the injector (yellow dotted line and black dot-dash line at ). This phenomenon poses a potential problem because the gradient of the electric field also changes sign (observe the slope of the yellow dotted line and black dot-dashed line for ) which could indicate a trapping region for the nanoparticles if pDEP is utilized. The transition region is also visible in the slope of the off-axis contours of the z-component of the electric field (blue solid, red dot-dot, and cyan dash lines for ).
As stated previously, the DEP force is calculated from the gradient of the magnitude of the electric field. Figure 6 contains contour plots of the gradient in the (a) y-direction and (b) z-direction (the signed natural logarithm is used to create a higher contrast visual of the data) and Figure 6(c) is a line contour of the characteristic force along the axis of the injector. The characteristic force is the DEP force divided by the cubed radius of the nanoparticles such that Equation (1) becomes Equation (10) with units of . , the dielectric permittivity of the surrounding medium, is set to , the permittivity of free space. , the dielectric permittivity of the particles, .
The force in the y- and z-directions at three values of are plotted in Figure 6(c). for all three values begins positive and then goes negative near . Studying the y-gradient of the electric field magnitude indicates that this behavior is expected because following the gradient from left to right in Figure 6(a), along the axis, one sees that the gradient begins positive, increases, then decreases and goes negative. The force is proportional to this profile as shown in Equation (10). Contrary to , behaves differently for each value. When the force in the z-direction is also zero along the entire axis because the z-gradient of the electric field magnitude crosses an inflection point in this location. When the z-gradient is positive which shows the electric field increasing toward the electrode as the dependence indicates it should. Increasing position effectively brings the electrode closer to the line which also follows the dependence and we see a corresponding increase in . When , mirrors the behavior of when . The force is negative, pointing along the gradient directed towards the lower electrode.
C. Parametric analysis of the dielectrophoretic injector
A parametric analysis was performed to develop a more comprehensive understanding of the dependence that the DEP force has on the plate angle, θ, of the charged plates as well as their separation distance, . In Figure 7(a) the largest force magnitude produced in the y-direction (positive or negative) within the guide sleeve is plotted for three electrode plate separation distances and various plate angles. This value is determined for each plate angle by calculating the maximum of the absolute value of the force in the y-direction then re-introducing the sign of the force value so as not to lose information regarding which direction the maximum force acts. Figure 7(b) indicates the average force in the y-direction, calculated as the statistical mean of within the entirety of the guide sleeve area, and (c) shows the associated standard deviation from that statistical average.
Addressing the separation distance, , the maximum y-force magnitude plotted in Figure 7(a) increases as decreases. The discontinuity at occurs when the positive values of the gradient of the electric field become stronger in magnitude than the negative gradient values, as explained later in this section. From Figure 6(a), we expect the magnitude of the force to be greatest where the electric field gradient is strongest, such as in the region that is between and . In this region the gradient is negative which means the force will also point along the negative y-direction as seen in Figure 7(a) for plate angles less than . For plate angles greater than Figure 7(a) indicates that the max force is positive in the y-direction. This can be explained by analyzing the field region of the example contour plot shown in Figure 8(a), for plate angle , near the electrode and at the narrow end of the particle injector where an increase in the strength of the positive gradient of the field occurs (annotated in Figure 8(a)).
The positive gradient values in the Figure 8(a) contour increase by when compared to the positive gradient values in the plot in Figure 6(a). (The light green contour represents values 15-20 and the yellow contour 20-25 versus the previous plot of light green 10-15 and yellow 15-20 ). This makes the largest positive gradient greater than the smallest negative one such that the maximum force magnitude in the y-direction for plate angles greater than is positive. The piece-wise step from negative to positive max force values, though interesting, is not the important information from Figure 7(a). Converting the data from Figure 7(a) into the simple, unsigned magnitude of the maximum y-force versus plate angle and plotting it in Figure 8(b) indicates that, as the plate angle increases, the gradient, and thus the force in the y-direction, decreases in strength. The above analysis of the maximum y-force magnitude indicates that small plate angle and narrow plate separation distance are preferable in order to generate strong DEP force fields.
Analysis of the average y-force plotted in Figure 7(b) indicates the same conclusion. As the plate angle increases, the regions where the electric field gradient is strongest decrease in size which can be seen by comparing Figure 6(a) to Figure 8(a). The strong positive gradient region between and -2 in Figure 6(a) decreases in size to a region between and in Figure 8(a) while simultaneously maintaining strength at . This decrease in region size of the strong gradient brings down the overall average of the electric field gradient and subsequently the force. We expect the average y-force to be zero, as seen in Figure 7(b), when the plate angle is zero because any fringe effects at the ends of the charged plates will be equal and opposite when the injector structure is mirrored across the vertical line . The structure plotted in Figure 4(a) exemplifies these mirrored fringe effects where it can be seen that the electric field points outward at both ends of the bottom plate and inward at both ends of the upper plate.
Analysis of the y-force deviation from the statistical average plotted in Figure 7(b) indicates that choosing a larger plate separation distance and larger plate angle will produce a more consistent force field in the y-direction, although, an overall weaker force.
D. Finite element analysis
The analytical solution is used to study particle dynamics within the tilted plate injector. COMSOL Multiphysics numerical models were also developed in order to study more complex electrode geometries and electric field structures. The initial geometry used for the numerical models is composed of the tilted, charged plates and the dielectric guide sleeve that restricts the motion of the nanoparticles. Figure 9 shows a contour plot of the magnitude of the electric field between the two tilted plates. The field structure supports our understanding of the analytically obtained results in Figure 4(b) wherein the field strength increases towards the narrow end of the injector and near the electrodes. The electric field magnitude at location.
The perforated exit/inlet membrane that guides the injection of the nanoparticles, indicated in Figure 3, was also included in COMSOL models to determine its affect on the electric field structure. Our design calls for a metallic membrane that acts as a floating potential and shields part of the dielectric guide sleeve from the applied electric field at the location where the electric field gradient changes direction as indicated in Figure 5. A concern we have with this membrane is that the electric field will reduce to zero too quickly at the location of the perforations and the resultant gradient will create a strong DEP force acting against the motion of the nanoparticles. Figure 10 shows the force field contours for the tilted plate injector with a perforated metallic membrane acting as a floating electrical ground. The perforations in the membrane are perpendicular to the axis of the injector and act as a gate through which the nanoparticles pass. From this contour plot, we see that the presence of the metallic membrane does create force acting against the motion of the particles (the red/orange/yellow bulge in the center of the plot, focused at the perforated membrane).
The DEP force depends on the relative polarizability of the nanoparticles to the surrounding medium which is vacuum, as stated earlier. This means that the dielectric constant of the nanoparticles is greater than the dielectric constant of the medium, vacuum, and the DEP force takes on the sign of the gradient of the electric field magnitude. If the medium had a dielectric constant that was greater than that of the nanoparticles then the DEP force would be the negative of the sign of the gradient in the electric field magnitude. Therefore, a possible solution to the strong negating force issue that the perforated membrane poses is to suspend the nanoparticles in a liquid such as water () rather than vacuum (). The resulting force field contour of the tilted plate injector that is filled with water and has a dielectric guide sleeve and a metallic perforated membrane is plotted in Figure 11. This contour shows that the repulsive hump seen in Figure 10 now becomes an accelerating ramp that assists the motion of the nanoparticles.
We have investigated a dielectrophoretic nanoparticle injection mechanism that can couple with a photonic acceleration/manipulation platform. The injector consists of tilted plates that are electrically isolated and charged to maintain a steady, nonuniform electric field across a vacuum or liquid-filled gap. We have analytically and numerically modeled the electric field and dielectrophoretic force within the space between the tilted plates. Our results indicate that an injector with small plate angle, , and narrow plate separation distance, , will produce stronger DEP force fields than an injector with large plate angle, , and wide separation distance, . This selection will provide a maximum y-force magnitude of and an average y-force of . We also conclude that choosing small and will increase the amount by which the fields vary and deviate from the average within the guide sleeve such that the standard deviation of the y-force is . Finally, we conclude that the nanoparticles must be suspended in a medium with dielectric constant greater than that of the nanoparticles so that the metallic membrane acting as a floating potential and gate will aid their motion and not hinder it.
The authors would like to thank the Air Force Office of Scientific Research for partially supporting this work through grant FA9550-14-1-0230 with Dr. Mitat Birkan as program monitor. The authors also thank the NASA Innovative Advanced Concepts program for supporting this work through grant NNX16AL26G. J. N. Maser would like to thank Missouri University of Science and Technology for sponsoring his graduate program through the Chancellor’s Fellowship.