Particles suspended in a medium between two electrodes under tension are polarized by the ensuing electric field. Charges inside the particles accumulate near the particle-medium interfaces and attract charges inside the medium. Charges also accumulate at the interfaces on the medium side. These accumulations of electric charges form what are called electric double layers [6] .

The polarizabilities of both medium and particles determine the relative quantity of charges inside and outside the particles. As negative and positive charges attract, and effectively cancel each other, electric dipoles are created. These relative differences determine the orientation of the dipoles [6] .

A net dipole inside an electric field is subjected to forces resulting from the attraction of opposite charges and the repulsion of identical charges. In the case of a symmetrical electric field generated by electrodes of identical sizes and shapes, for example, the forces cancel each other and no resulting force occurs. If the electric field is inhomogeneous, when electrodes of different sizes are used for example, a resulting force is applied on each particle. This force can be expressed by [24] :

where is the permittivity of the medium, is the radius of the particle, is the Clausius-Mossotti factor and is the electric field. In a DC electric field, the Clausius-Mossoti factor is given by [25]

where and are the conductivity of each particle and the medium, respectively. This factor, which varies from −0.5 to 1, determines if the particles are attracted or repulsed by regions of high electric field intensity [26] . The Clausius-Mossotti factor also affects the value of the dielectrophoretic force. By carefully selecting the medium conductivity, can be tailored to obtain the required force, the absolute maxima being bounded by the extreme values of −0.5 and 1.

Particles being acted upon by dielectrophoresis accelerate inside the medium. The acceleration is dictated by the dielectrophoretic force and by the drag. As the velocity of the particles increase, so does the drag. When those two forces become equal, the particles reach their terminal velocity. Usually, the particles reach their terminal velocities inside the medium in a very short time (in the order of 10⁻⁴ s for 10 μm particles) [19] . The acceleration phase can then be neglected and velocity can be used instead of force to provide a clearer understanding of the particle behavior. The velocity of particles subjected to DEP is given by [27]

where, a measure of the impact of the electric field gradient on the particle velocity, is called the dielectrophoretic mobility. It is given by [27]

where is the fluid viscosity.

Figure 1(a) shows the design of the 3D separator that consisted of one inlet reservoir, two outlet reservoirs and a channel connecting the inlet to the outlet. In order to separate a bimodal particle dispersion, the channel and output reservoirs were filled with a conductive aqueous sodium chloride solution. The input reservoir was filled with the particle dispersion. Three electrodes made of 250 µm diameter gold wire, were immersed in the reservoirs. A DC voltage was then applied between the inlet and outlet as shown in Figure 1(a).

The electric field distribution for a voltage of 900 VDC is shown in Figure 1(b). The field values were obtained using COMSOL Multiphysics (Comsol, Burlington, MA, USA). Under the influence of the electric field, the particles were subjected to tangential forces resulting from electrically-induced phenomena called electroosmosis and electrokinesis and the resulting particle velocities are noted. Figure 1(c) shows the separator from the side with the three particle arrangement during the separation experiments. First, the particles entered the channel leading to the first helix in a random arrangement as shown in Figure 1(d). Upon entering the first helix, the particles were subjected to radial dielectrophoretic forces resulting in velocities noted and given by equation (3). These velocities were toward the outside wall of the channel and their magnitude depended on the size of the particles. The larger particles had a larger radial velocity than the smaller particles. Due to the sufficient length of the first helix, both types of particles reached the outside wall and were aligned when exiting the first helix, as shown in Figure 1(e). This first step was crucial in order to separate the particles because their random positions would nullify the effect of the difference in rendering separation nearly impossible.

After the alignment, the particles entered the second helix which was coiled in the opposite direction and in which the outside wall became the inside wall. The electric field gradient pushed the particles with the same respective velocities toward the outside wall. The difference in velocities separated the particles until they reached

the end of the second helix. That length was shorter than that of the first helix to insure that the particles did not reach the outside wall but were properly positioned with respect to the bifurcation shown in Figure 1(f). The particles were then collected in the outlet reservoirs.

The microchannels were fabricated using the direct-write fabrication method [28] . The sacrificial ink, which was made of 70 wt% of petroleum jelly (Unilever, Toronto, Canada) and 30 wt% microcrystalline wax (Strahl & Pitsch, West Babylon, NY, USA), was extruded through a steel nozzle by the piston.

Fugitive ink was deposited on rotating epoxy 1.2 mm diameter spindles with the direct-write method to create the helices, as shown on Figure 2(a). This diameter of spindle, which determined the curvature radius, was chosen because of ease of fabrication. These spindles were rotated using a precision electric motor (VT-80, Pi Micos GmbH, Eschbach, Germany). For each helix, a straight section was deposited first to facilitate the connection of the channels. The helix was deposited on the rotating spindle and followed by another straight section. The first helix had 6 coils and the second had 4. These numbers of coils were determined by analysis of the results of previous prototypes. They were both made using a 100 μm nozzle and a pressure of 550 kPa. This nozzle size was chosen because of its availability.

The two epoxy spindles were glued upright on a glass slide. The inlet and outlet channels were deposited and connected to the lower part of the helices as shown on Figure 2(b). The bifurcation was completed by depositing its second branch. The reservoirs were fabricated by depositing circular walls 1.5 mm high and with a diameter of 5 mm which were subsequently filled with fugitive ink to prevent resin infiltration during the encapsulation step. The entire apparatus was encapsulated with a two-part liquid epoxy (Epon 862/Epikure 3274, Momentive, Colombus, OH, USA) and left to cure as illustrated in Figure 2(c). The top section of the epoxy was then carefully cut using a precision saw (Isomet Low Speed Saw, Buehler Canada, Whitby, ON, Canada) to prevent the introduction of foreign particles that could clog the channels. The two exposed filaments were connected with a filament of fugitive ink deposited with a 100 µm nozzle thus completing the connection of all the filaments as shown on Figure 2(d). This last filament was encapsulated with epoxy and left to cure. The glass slide was removed to expose the filaments. The encapsulated model was heated in boiling water to melt the ink that was then removed by applying vacuum to one end of the channel. Any remaining ink residue was dissolved and removed by circulating hexane through the channel. The remaining epoxy block had empty channels and reservoirs and constituted a finished and functional separator.

The particle suspension was prepared by mixing 4 and 10 µm polystyrene microbeads (Fluka Analytical, Buchs, Switzerland) with an aqueous solution of 100 ppm of sodium chloride. The solution was also used without particles to fill the channel and the outlet reservoirs. The final suspension had a conductivity measured of 210 µS/cm and contained approximately 5 × 10⁶ particles per mL of each type for a total of 10⁷ particles per mL. The conductivities of the particles were 10 µS/cm and 4 µS/cm for the 4 µm and 10 µm particles, respectively (these conductivities were calculated using the surface conductance recommended by Ermolina and Morgan [29] ). The Clausius-Mossotti factor of both particle types were computed as −0.465 and −0.486 for the 4 µm and 10 µm particles, respectively, from Equation (2). These values are very close to the theoretical minimum of the Clausius-Mossotti factor of −0.5, which means that, during separation experiments, the particles were pushed toward the regions of low electric field value with forces very close to the maxima allowed by a negative Clausius- Mossotti factor. The specific gravity of polystyrene being 1.05, the particles tended to settle. Manual stirring was required before the particle introduction in the inlet reservoir.

The outlet reservoirs and channel were filled with the NaCl solution using a syringe. The inlet reservoir was then filled with approximately 30 µL of particle suspension using a pipette. The gold electrodes were immersed inside the reservoirs. The voltage was supplied by a 1000 VDC electrophoresis power supply (FB1000Q, Cole- Parmer, Montréal, Canada). The inlet reservoir received the positive potential and the outlet reservoirs were grounded. The reservoirs were carefully filled to the same level to avoid the interference of a pressure induced flow.

Particle motion was captured using an Evolution VF Camera (MediaCybernetics, Rockville, MD, USA) and an SZX12 stereomicroscope (Olympus, Richmond Hill, Ontario, Canada). Close-up fluorescent optical images were taken using an Evolution VF camera and a BX61 microscope (Olympus). Pictures were treated with Image Pro 7 (MediaCybernetics). Close-up images were assembled from several images with different focus heights to extend the depth of field.

Figure 3(a) shows the completed 3D separator having a total microchannel length of 65 mm and a planar footprint is 2.6 cm², which is a 35% reduction compared to a reported spiral-shaped separator [8] . The effective separation length, which was comprised of the two helices and their connection, was 42 mm. The curvature radius of the helices at the center of the channels, which takes into account the channel roughness, was 0.650 ± 0.004 mm. The channel in the helical part of the separator, which is shown in Figure 3(b), had a diameter of 111 ± 6 µm. The three straight channel parts were fabricated with the same procedure. Their width should have been similar. Two of the three were similar with a width of 147 ± 4 µm. However, the top channel (shown in Figure 3(c)) had a width of 215 ± 15 µm. This difference can be seen in Figure 3(c) where the angled part is smooth and the horizontal part is wider and more jagged. The junction between the different parts of the channel, such as those visible at the bottom of Figure 3(b) and the top of Figure 3(d), had a diameter of 216 ± 54 µm.

The overall dimensions and shape of the designed separator were respected. The curvature radius was relatively constant in the device and the channel dimensions had variations of less than 7%. Despite the overall smoothness, one part of the output channel had some visible roughness that induced local electric field gradients and could potentially have hindered the separation. The difference between the total length and the effective length did not contribute to the separation but reduced the available voltage in the helices. The connections between the different parts of the channel had diameters that were nearly twice that as large as the average diameter of the helices, which also created local electric field distortions. Finally, the fabrication process is complex and heavily reliant on manual operations.

Separation experiments have been performed with the method described in Section 2.5. Voltages between 800 and 1000 VDC were tested. The highest separation efficiency was obtained using a voltage of 900 VDC.

Figure 4(a) shows the particles before separation. The particles accumulated in both output reservoirs after their separation using a voltage of 900 VDC. Figure 4(b) shows a still image from the video captured of the bifurcation during a test using a voltage of 900 VDC. The 4 and 10 µm particles can be seen being separated and entering both branches of the bifurcation which lead them to the output reservoirs. Figure 4(c) and Figure 4(e) show representative images of the separated particles in each of the reservoirs after the test. Figure 4(d) shows the separation efficiency calculated by counting the proportion of each type of particle that reached the expected reservoir. After their transit through the separator, 94.0% of the 4 µm particles reached the reservoir at the end of the top channel and 92.5% of the 10 µm particle reached the reservoir at the end of the bottom channel. The particles were tracked and counted from recorded videos of the separation experiments.

Difficulties were encountered during the separation experiments. The results were difficult to reproduce because valid tests were challenging to obtain. The fine adjustment of the level of solution in each reservoir was especially difficult to make. An uneven level of solution led to pressure-induced flow which deviated the particles and prevented efficient separation. The settling of the particles during the short initial period between the introduction of the particle in the input reservoir and the beginning of the separation experiments was also a problem observed. Since the 10 µm particles settled more rapidly than the 4 µm particles, the smaller particles were more numerous altogether during the experiments even though the same number of each type of particle was introduced in the input reservoir.

Reported 2D separators have efficiencies varying between 90% and 98% [7] [8] . These results are comparable to the results obtained in the experiments with the helical-shaped separator. The fabrication process generated geometrical flaws, such as channel roughness and increased cross-section in the connections, which most probably hindered the separation. Considering these shortcomings, the results, and how they compare with other devices, are promising.