Dielectrophoresis from the System's Point of View: A Tale of Inhomogeneous Object Polarization, Mirror Charges, High Repelling and Snap-to-Surface Forces and Complex Trajectories Featuring Bifurcation Points and Watersheds

Abstract

Microscopic objects change the apparent permittivity and conductivity of aqueous systems and thus their overall polarizability. In inhomogeneous fields, dielectrophoresis (DEP) increases the overall polarizability of the system by moving more highly polarizable objects or media to locations with a higher field. The DEP force is usually calculated from the object’s point of view using the interaction of the object’s induced dipole or multipole moments with the inducing field. Recently, we were able to derive the DEP force from the work required to charge suspension volumes with a single object moving in an inhomogeneous field. The capacitance of the volumes was described using Maxwell−Wagner’s mixing equation. Here, we generalize this system’s-point-of-view approach describing the overall polarizability of the whole DEP system as a function of the position of the object with a numerical “conductance field”. As an example, we consider high- and low conductive 200 µm 2D spheres in a square 1 × 1 mm chamber with plain-versus-pointed electrode configuration. For given starting points, the trajectories of the sphere and the corresponding DEP forces were calculated from the conductance gradients. The model describes watersheds; saddle points; attractive and repulsive forces in front of the pointed electrode, increased by factors >600 compared to forces in the chamber volume where the classical dipole approach remains applicable; and DEP motions with and against the field gradient under “positive DEP” conditions. We believe that our approach can explain experimental findings such as the accumulation of viruses and proteins, where the dipole approach cannot account for sufficiently high holding forces to defeat Brownian motion.

Keywords: DEP trajectory; LMEP; LOC; MatLab® model; force spectroscopy; microfluidics; protein dielectrophoresis; system’s perspective; virus trapping; μTAS.

Figure 1

200 µm 2D sphere approximated with a diameter of 39 voxels in the horizontal and vertical directions. The central voxel is marked in white. Linear rows of 11 voxels form the horizontal and vertical edges.

Figure 2

(A) Potential and current line distributions in the 1 × 1-mm² chamber without the sphere energized with 1 V at the pointed electrode (center right) versus 0 V at the plain electrode (vertical gray bar on the left). At the symmetry line, the dipole range is marked. (B) Potential (dashed) and field strength (full) along the symmetry line of the chamber (500 µm ≤ x ≤ 500 µm, y = 0 µm). Vertical lines mark the limits of the chamber volume. The curve was enlarged by multiplication with a factor of 10 to show the field behavior in the dipole region more clearly. (C): Field gradient along the symmetry line.

Figure 3

Potential and current line distributions for different positions of the 1.0 S sphere in 0.1 S medium, in front of the plain electrode (A), on the watershed (B), in a largely homogeneous field region (C), and at the pointed electrode (D). The conductances are (A): 35.744 mS, (B): 35.563 mS, (C): 35.647 mS, and (D): 83.912 mS. The basic sheet conductance $L_{Basic}^{2D}$ of 34.908 mS without a sphere corresponds to a cell constant of $k^{2D}=0.34908$.

Figure 4

Potential and current line distributions for different positions of the 0.1 S sphere in the 1.0 S medium, at the plain electrode (A), in a largely homogeneous field region (B,C), and in front of the pointed electrode (D). The conductances are (A): 343.29 mS, (B): 342.28 mS, (C): 340.10 mS, and (D): 60.682 mS. The basic sheet conductance $L_{Basic}^{2D}$ of 348.97 mS without sphere corresponds to a cell constant of $k^{2D}=0.34897$.

Figure 5

Single 200 µm, 2D sphere of 1.0 S (reddish circles in (A)) in the chamber of Figure 2 with 0.1 S medium. The mean conductance is ${\overline{L}}^{2D}=35.739 \mathrm{mS}$. (A) Conductance field plot with trajectories (a–g). A watershed (bent white line) separates the two caption areas of the stable endpoints E₁ and E₂. E₃ is an instable saddle point in the middle of the watershed. (B) Sheet conductance along the trajectories. Basic and mean conductance are marked. The system’s sheet conductance increases steadily along each trajectory, reaching moderate and high peak values at the endpoints E₂ and E₁, respectively. Trajectories b, c, d, and e end at E₂. Trajectories a, f, and g end at E₁, reaching ${\overline{L}}^{2D}$ by coincidence (insert). (C) Normalized DEP forces calculated with Equation (16) from the conductance values in (B).

Figure 6

Single 200 µm 2D sphere of 0.1 S (reddish circles in (A)) in the chamber of Figure 2 with 1.0 S medium. The mean conductance is ${\overline{L}}^{2D}=340.14 \mathrm{mS}$. (A) Conductance field plot with trajectories (a–j). Two watersheds (bent white lines) and one symmetry line (trajectory f) separate four catchment areas with the four stable endpoints (E₄, E₁, E₃, and E₅). E₆, and E₇ are instable saddle points. E₂ is an instable minimum at the end of the symmetry line. Trajectories close to f, such as j, are diverted to E₁ or E₃. (B) Sheet conductance along the trajectories. Basic and mean conductance are marked. Trajectories b and e end at E₄; trajectories a, c, and j at E₁; trajectory f at E₂; trajectories d and g at E₃; trajectories i and h at E₅. (C) Normalized DEP force calculated with Equation (16). Forces along trajectories parallel to the plain electrode toward the endpoints E₁, E₂ and E₃ are very low. The constant forces observed at the start of trajectories e, d, and f (Figure 6C, left insert) may be due to the reversal of the effect discussed above.

Figure 7

Absolute value of the quotient of the normalized DEP forces acting on the 1.0 S and 0.1 S spheres plotted along the symmetry line of the chambers (trajectories d and f of Figure 5 and trajectory f of Figure 6).

Figure 8

DEP behavior in the dipole range marked in Figure 2. (A) Field strength and field gradient along the symmetry line for 1 V potential difference at the electrodes. The vertical auxiliary line at x = 186 µm is perpendicular at a field gradient of 1 V/m² (field strength of 0.5602 V/m). Horizontal auxiliary lines run out to the respective ordinates from the intersections with field strength and field gradient plots. (B) DEP of the sphere increases the overall conductance of the chamber. Top: 1.0 S sphere, 0.1 S medium (short, dashed line, left ordinate, positive DEP), bottom: 0.1 S sphere 1.0 S medium (long dashed line, right ordinate, negative DEP). At x = 186 µm, the conductances are 36.02 mS and 337.6 mS. The common baseline corresponds to the basic conductances of the two setups. (C) At x = 186 µm, the normalized forces from Equation (16) are 0.1211 (1.0 S sphere, 0.1 S medium) and 0.1234 (0.1 S sphere, 1.0 S medium).