# Foundation of the Manipulation Technology for Tiny Objects Based on the Control of the Heterogeneity of Electric Fields

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Apparatus and Methods

**v**is the translation velocity,

**F**is the sum of the electrostatic force exerted on the sphere,

**F**

_{d}is the frictional force exerted by the glass plate at the contact point,

**Ω**is the rotational angle velocity of the sphere,

**T**is the torque created by the electrostatic forces, R is the radius of the sphere,

**k**is the unit vector of vertical upward, and I is the momentum of inertia of the sphere. The force that transfers objects is the translational for, hence

**F**should be determined. Equation (5) can be deduced from Equations (1)–(4)

**F**was determined by neglecting the second term of Equation (5). This analysis was repeated at more than 20 initial positions to cover a broad area.

- (1)
- To determine whether the transfer from one initial holding point to an adjacent point is possible.
- (2)
- To investigate the distribution of the electrostatic force when only one of the two electrodes was supplied with high voltage.

#### 2.2. Methods for Numerical Simulation

**n**indicates the normal vector of the interfaces. V was determined for each different position of the glass sphere by 0.5 mm. Subsequently, the net force exerted on the glass sphere at every position is determined by integrating the Maxwell stress tensor at sphere surface S, as described by Equation (9) [12]

_{i}is the i-component of the force exerted on the glass sphere and Te

_{ij}is the ij-component of the Maxwell stress tensor. Theoretically,

**F**is the sum of the forces exerted on 2n-th multipole

**F**

^{(n)}(n = 1, 2, …), as described by Equation (10).

**F**

^{(n)}is the dielectrophoretic force and it is represented by Equation (11) using an effective scalar constant K

^{(n)}if the electric field is irrotational and the materials are isotropic.

**F**, a dc voltage, V

_{e}, of 4.243 kV, which is the rms of the 6-kV sinusoidal wave, is given for the upper electrode, as indicated by Equation (12)

_{f}, of the upper electrode adjacent to the high-voltage electrode is determined using the condition that no true charge exists inside a closed cylindrical surface S

_{c}, which includes the electrode with the floating potential

_{f}was obtained as 0.684 times V

_{e}, that is, 2.903 kV. This value is commonly used for all cases with the glass sphere.

## 3. Results and Discussion

#### 3.1. Case of an Isolated Cross Point

_{x}, F

_{y}) were determined by averaging the forces within the 0.5 mm square whose center is at the grid points. Because the glass sphere right under the upper electrode cannot be observed, only the forces at locations that are more than 0.5 mm away from the edges of the upper electrode are displayed. Forces other than at x = ±1 mm head to the cross point and decrease in magnitude with increasing distance from the cross point. In contrast, forces at x = ±1 mm are significantly larger, and their directions are more parallel to the x-axis and almost constant in magnitude with increasing distance from the cross point in the range of y from 0 to ±4 mm.

_{x,}at each y ranging from 0 to 5 mm obtained in the experiment. Significantly large forces at x = ±1 mm range from 10 to 20 µN, which are comparable to the weight of the sphere, 22.5 µN, and show no correlation with the y values in the range of y from 0 to ±4 mm. However, at remote locations ($\left|\mathrm{x}\right|$ ≥ 1.5 mm), the magnitude of F

_{x}clearly decreases with increasing y and $\left|\mathrm{x}\right|$.

_{x}in the experiment and simulation, respectively. The values in Figure 9a are four times than those shown in Figure 7. Although the experimental values are not stable, their magnitudes are of the same order as those in the simulation in the range of $\left|x\right|$ ≥ 1.5 mm, which suggests that dielectrophoretic forces are dominant. The forces at the locations within 1.5 mm from x = 0 were significantly larger compared to those obtained by the simulation. This difference can be explained by the dielectric barrier discharge that can occur between the glass sphere and edges of the upper electrode. Ions of both polarities are divided to adhere to the surfaces of the glass sphere and glass plate, resulting in an attractive Coulombic force. Such forces head to the upper electrode and not to the cross point. In addition, these attractive forces may generate torque because the charges on the insulator surface do not necessarily spread uniformly over a short period. Therefore, there may be a difference between the true forces and those obtained using Equation (5).

_{z}, exerted on the glass sphere obtained from the simulation. F

_{z}was downward at all the locations. This suggests that the assumption that the sphere does not slip but rolls on the glass plate is reasonable, considering that the friction coefficient between the glasses ranges from 0.9 to 1.0 [21].

#### 3.2. Case of Two Adjacent Cross Points

^{2}created by the large difference in the potential of the high-voltage and grounded electrodes. However, the forces were less than 15 µN even at x = −1 mm in the simulation result, which indicates the influence of the discharge at these locations in the experiment. In the region of x < −1.5 mm, comparison is impossible because the forces do not exceed the static friction force and they could not be estimated in the experiment. In the region of x > 0 mm, both the forces in the experiment and the simulation coincide well, except at several locations at x = 1 mm where discharge might occur.

## 4. Conclusions

- (a)
- Applying a high voltage between two perpendicularly facing electrodes creates forces toward the cross point. A comparison with the numerical simulation indicated that dielectrophoretic forces were dominant among the forces observed in the experiment at locations sufficiently remote from the cross point.
- (b)
- Large forces are generated near the electrodes to which high voltages were applied. These forces could be attributed to the Coulombic force that works between the charges created by the dielectric barrier discharge.
- (c)
- The glass sphere is successfully transferred from one holding point to an adjacent point.
- (d)
- Two cases, one of which the electrode of the initial holding point was grounded and in the other of which that electrode was floated, were compared. For grounding, large forces are generated immediately before the destination point. This was caused by the large field gradient between the two electrodes. For floating, forces toward the destination were created more uniformly between the initial and destination points. In the latter case, the glass sphere was transferred faster and more smoothly. Therefore, the floating condition was a better manipulation device.
- (e)
- There are at least two problems to be solved in the future work: The manipulation of multiple objects with diverse sizes and shapes, and the development of the countermeasures against the discharge that disturbs transportation by dielectrophoretic force.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Manipulation device tested: (

**a**) Top and side views of a set of electrode plates of single-cross-point type; (

**b**) Top view of an electrode plate of dual-cross-point type.

**Figure 5.**Calculation region (dimensions are all in mm): (

**a**) Top and side views of the region for single-cross-point type; (

**b**) Top view of the region for dual-cross-point type.

**Figure 8.**Comparison between the experimental and the numerical simulation on electrostatic forces exerted on the glass sphere: (

**a**) Experimental (four times magnified version of Figure 6 with arrows at x = ±1 mm hidden); (

**b**) Simulation.

**Figure 9.**Comparison between the experimental and the numerical simulation on x-component of the electrostatic forces: (

**a**) Experimental (four times magnified version of Figure 7); (

**b**) Simulation.

**Figure 10.**Magnitude of the electric field on the glass sphere: (

**a**) Definition of the position where electric field strengths are indicated; (

**b**) The magnitude of electric field on the sphere when the sphere center is at x = 1 mm and y = 0–3 mm.

**Figure 12.**Transfer from an initial holding point to the adjacent cross point (the upper electrode of the initial point is grounded).

**Figure 13.**Transfer from an initial holding point to the adjacent cross point (the upper electrode of the initial point is floating).

**Figure 14.**Comparison of experimentally obtained electrostatic forces between the cases where the upper electrode of the initial point is grounded and floating: (

**a**) The case of grounded; (

**b**) The case of floating.

**Figure 15.**Comparison of electrostatic forces between the experimental and the simulation for the cases where the upper electrode of the initial point is grounded: (

**a**) Experimental (four times magnified version of Figure 14a with arrows at x = −1 mm hidden); (

**b**) Simulation.

**Figure 16.**Comparison of electrostatic forces between the experimental and the simulation for the cases where the upper electrode of the initial point is floating: (

**a**) Experiment (four times magnified version of Figure 14b with arrows at x = −1 mm hidden); (

**b**) Simulation.

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**MDPI and ACS Style**

Shirota, I.; Yoshida, K.
Foundation of the Manipulation Technology for Tiny Objects Based on the Control of the Heterogeneity of Electric Fields. *Energies* **2022**, *15*, 4513.
https://doi.org/10.3390/en15134513

**AMA Style**

Shirota I, Yoshida K.
Foundation of the Manipulation Technology for Tiny Objects Based on the Control of the Heterogeneity of Electric Fields. *Energies*. 2022; 15(13):4513.
https://doi.org/10.3390/en15134513

**Chicago/Turabian Style**

Shirota, Isao, and Keiichiro Yoshida.
2022. "Foundation of the Manipulation Technology for Tiny Objects Based on the Control of the Heterogeneity of Electric Fields" *Energies* 15, no. 13: 4513.
https://doi.org/10.3390/en15134513