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

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. General Remarks

#### 2.2. Approximation of the Field Gradient

#### 2.3. DEP Force in the Classical Dipole Approximation

#### 2.4. Charging Work for External and Suspension Media Boxes

#### 2.5. DEP Force Approximation by a Capacitor-Charging Cycle

#### 2.6. Electrorotation (ROT) Torque

^{−1}) has been dropped from Equations (31) and (32), so that the units of force in Equations (10) and (27) are changing to units of torque.

## 3. Modelling Results and Discussion

#### 3.1. Model Parameters

- I
- ${\sigma}_{i}$ = 0.01 S/m, with ${\epsilon}_{i}$ = 800, and
- II
- ${\sigma}_{i}$ = 1 S/m with ${\epsilon}_{i}$ = 8.

#### 3.2. Clausius-Mossotti Factor

#### 3.3. Conductivity and Dissipation

#### 3.4. Dispersion Relation, Active, Reactive, and Apparent Power

#### 3.5. Dissipation and Charging Work in the Box Chain

^{3}, which corresponds to the external conductivity of 0.1 S/m at a field strength of 1 V/m. Without the object, the dissipation along the box chain increases with the square of the field strength (Table 1; Figure 6A, left ordinate). The charge work shows the same field strength dependence (Equations (15) and (16)). It is shown on the right ordinate as “apparent relative permittivity”. This parameter assigns an apparent permittivity to each box, so that charging each box with 1 V/m requires the same work as charging the box with the actual field strength in the inhomogeneous field (Table 1). Accordingly, the “apparent relative permittivity” in the first box is 80, the actual permittivity of the suspension medium (cf. left ordinate in Figure 4). For each box, the dissipation and apparent permittivity are plotted for three cases, in the absence and presence of the low (0.01 S/m) or high (1 S/m) conductive sphere. In the calculations, the low frequency limit (Equation (30)) of Equation (36) was used as an example. In the presence of the spheres, the dissipation decreases or increases as indicated for box 6. In DEP, the spheres travel through the box chain against (negative DEP) or in (positive DEP) the direction of the field gradient, depending on their conductivity, i.e., their polarizability relative to that of the suspension medium. The dissipation differences induced by the presence of the spheres increase in the direction of the field gradient, resulting in decreasing and increasing force magnitudes along the DEP trajectories for negative and positive DEP, respectively.

#### 3.6. DEP Force in the Box Chain

## 4. General Discussion

#### 4.1. Higher Precision for the DEP Force?

#### 4.2. DEP as Conditioned Polarization Process

#### 4.3. Relations to the Law of Maximum Entropy Production (LMEP)

## 5. Conclusions and Outlook

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**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).

Box$\mathit{i}$ | Normalized Field${\mathit{E}}_{\mathit{i}}/{\mathit{E}}_{\mathbf{0}}$ | Normalized Field Squared${\mathit{E}}_{\mathit{i}}^{\mathbf{2}}/{\mathit{E}}_{\mathbf{0}}^{\mathbf{2}}$ | Transition from Box$\mathit{i}\to \mathit{i}+\mathbf{1}$ | Squared Normalized Mean Field at Box Interfaces${{}^{\left(\frac{{\mathit{E}}_{\mathit{i}+\mathbf{1}}+{\mathit{E}}_{\mathit{i}}}{\mathbf{2}{\mathit{E}}_{\mathbf{0}}}\right)}}^{\mathbf{2}}$ |

1 | 1 | 1 | ||

2 | 1.1 | 1.21 | $1\to 2$ | 1.1025 |

3 | 1.21 | 1.4641 | $2\to 3$ | 1.3340 |

4 | 1.331 | 1.7716 | $3\to 4$ | 1.6142 |

5 | 1.4641 | 2.1436 | $4\to 5$ | 1.9531 |

6 | 1.6105 | 2.5937 | $5\to 6$ | 2.3633 |

7 | 1.7716 | 3.1384 | $6\to 7$ | 2.8596 |

8 | 1.9487 | 3.7975 | $7\to 8$ | 3.4601 |

9 | 2.1436 | 4.5950 | $8\to 9$ | 4.1867 |

10 | 2.3579 | 5.5599 | $9\to 10$ | 5.0660 |

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**MDPI and ACS Style**

Gimsa, J.
Active, Reactive, and Apparent Power in Dielectrophoresis: Force Corrections from the Capacitive Charging Work on Suspensions Described by Maxwell-Wagner’s Mixing Equation. *Micromachines* **2021**, *12*, 738.
https://doi.org/10.3390/mi12070738

**AMA Style**

Gimsa J.
Active, Reactive, and Apparent Power in Dielectrophoresis: Force Corrections from the Capacitive Charging Work on Suspensions Described by Maxwell-Wagner’s Mixing Equation. *Micromachines*. 2021; 12(7):738.
https://doi.org/10.3390/mi12070738

**Chicago/Turabian Style**

Gimsa, Jan.
2021. "Active, Reactive, and Apparent Power in Dielectrophoresis: Force Corrections from the Capacitive Charging Work on Suspensions Described by Maxwell-Wagner’s Mixing Equation" *Micromachines* 12, no. 7: 738.
https://doi.org/10.3390/mi12070738