Active, Reactive, and Apparent Power in Dielectrophoresis: Force Corrections from the Capacitive Charging Work on Suspensions Described by Maxwell-Wagner's Mixing Equation

Abstract

A new expression for the dielectrophoresis (DEP) force is derived from the electrical work in a charge-cycle model that allows the field-free transition of a single object between the centers of two adjacent cubic volumes in an inhomogeneous field. The charging work for the capacities of the volumes is calculated in the absence and in the presence of the object using the external permittivity and Maxwell-Wagner's mixing equation, respectively. The model provides additional terms for the Clausius-Mossotti factor, which vanish for the mathematical boundary transition toward zero volume fraction, but which can be interesting for narrow microfluidic systems. The comparison with the classical solution provides a new perspective on the notorious problem of electrostatic modeling of AC electrokinetic effects in lossy media and gives insight into the relationships between active, reactive, and apparent power in DEP force generation. DEP moves more highly polarizable media to locations with a higher field, making a DEP-related increase in the overall polarizability of suspensions intuitive. Calculations of the passage of single objects through a chain of cubic volumes show increased overall effective polarizability in the system for both positive and negative DEP. Therefore, it is proposed that DEP be considered a conditioned polarization mechanism, even if it is slow with respect to the field oscillation. The DEP-induced changes in permittivity and conductivity describe the increase in the overall energy dissipation in the DEP systems consistent with the law of maximum entropy production. Thermodynamics can help explain DEP accumulation of small objects below the limits of Brownian motion.

Keywords: 2D and 3D modelling; DEP force in narrow volumes; DEP trajectories; DEP-induced polarizability increase; Rayleigh’s dissipation function; capacitor charge cycle; conditioned polarization; law of maximum entropy production (LMEP); lossy dispersive materials; micro-fluidic volumes; thermodynamics.

Figure 1

Diagram illustrating the consecutive positions of a spherical object during DEP movement in the inhomogeneous field of a 2D or 3D radial setup. The squares i and i + 1 represent cuboid boxes with $x$ by $x$ geometry $\left(y=x\right)$ in the sheet plane. The field gradient points in the radial direction. Without any limitation in generality for 3D models of microscopic systems, perpendicular to the sheet plane, a depth of $z=x$ or of $z=$ 1 m can be assumed, as is common in 2D models. The distance of travel between the box centers is $\Delta x=x$.

Figure 2

Real and imaginary parts of the Clausius-Mossotti factors of two homogeneous spherical objects in aqueous medium (${\sigma }_e$ = 0.01 S/m, ${\epsilon }_i$ = 800) plotted over frequency (A) and in the complex plain (B) according to the identical Equations (7) or (29). Dashed lines: ${\sigma }_i$ = 0.01 S/m, ${\sigma }_i$ = 800; Full lines: ${\sigma }_i$ = 1 S/m, ${\epsilon }_i$ = 8). The low and high frequency plateaus (Equations (30)) are clearly visible.

Figure 3

Frequency (A) and complex plots (B) of DEP-force and ROT-torque spectra of the two spheres of Figure 2 according to Equations (27) and (32) compared with the classical model (dotted lines; Equations (4), (7), (28) and (31)). All spectra were calculated for field strengths of 1 V/m. To obtain corresponding numerical values of forces and moments, the DEP forces were calculated for $\gamma =1 {\mathrm{m}}^{-1}$.

Figure 4

Relationships of the real parts of the relative permittivity and conductivity (Equations (26) and (36)) of the suspension containing two different homogeneous spheres (Dashed lines: ${\sigma }_i$ = 0.01 S/m, ${\epsilon }_i$ = 800; solid lines: ${\sigma }_i$ = 1 S/m, ${\epsilon }_i$ = 8) to the induced DEP force (Equation (27)). The DEP forces are identical to those in Figure 3 when using the relative permittivity and conductivity of the suspension medium (80, 0.1 S/m) as a reference for zero DEP force. The dotted lines show the classical DEP force (Equation (10)).

Figure 5

Component plots of the real parts of the relative permittivity ((A); Equation (18)) and conductivity ((B); Equation (35)) of two suspensions containing a single homogeneous sphere (Dashed lines: ${\sigma }_i$ = 0.01 S/m, ${\epsilon }_i$ = 800; solid lines: ${\sigma }_i$ = 1 S/m, ${\epsilon }_i$ = 8). The apparent permittivity and conductivity are the sums of their active and reactive components.

Figure 6

Illustration of DEP-induced changes in field-normalized dissipation in a box chain subjected to an inhomogeneous low-frequency field according to the values of Table 1. (A): In the absence of the object, the dissipation in the direction of the field gradient increases with the square of the field strength (circles, gray columns). In the presence of a high (triangles, ${\sigma }_i$ = 1 S/m) or low polarizable sphere (squares, ${\sigma }_i$ = 0.01 S/m), the active dissipation in the box is increased or decreased, respectively. The work of charging has the same field strength dependence (Equations (15) and (16)). It is plotted as “apparent relative permittivity” above the right ordinate (see text). (B): Dependence of the sum of dissipation in the box chain system on the positions of the single objects. Arrows denote DEP “trajectories”. The dissipations in the chain system (dashed horizontal lines) correspond to effective relative polarizabilities (right ordinate), which are proportional to the charge work of the whole chain system. Effective relative polarizabilities of one correspond to the average dissipation throughout the chain, for an even distribution of the 10 starting positions for the model spheres (dashed horizontal lines).

Figure 7

Low frequency plateaus of the DEP forces calculated with Equation (30) for the model spheres with ${\sigma }_i$ = 0.01 S/m (squares) and ${\sigma }_i$ = 1 S/m (triangles) compared with the classical model of Equation (10) (dotted lines). A field inhomogeneity of $\gamma =2500 {\mathrm{m}}^{-1}$ and normalized field strengths at the box interfaces were assumed (Figure 1, Table 1).