#### 2.1. Quasi-Spherical Catom as a Module

As pointed previously, the cube shaped catom has several advantages but the sharp edges do not allow an ease of movement. To overcome this limitation, a spherical catom would be the theoretical best structure allowing an ease of movement with the minimal amount of energy. This also allows to mimic nature and the atoms that compose matter. However, a spherical catom is not convenient for latching because of the point-point contact between the two spheres. To tackle this limitation, a quasi-spherical shape concept was adopted in Reference [

20] within the B3PM project. In this paper we use the quasi-spherical shape for a catom and proposes to design the actuation that will permit to this to latch and to move.

The suggested quasi-spherical catom has a face-centered cubic (FCC) lattice structure as depicted in

Figure 2a. In this configuration each catom can have up to twelve neighbors catoms. Each neighbor catom can be locked onto the dedicated latching surface, called connector, of the initial catom by the proposed electrostatic actuation. The other surfaces (non-connectors) serve as rolling surfaces during the displacement. The length of all the surfaces, connectors and non-connectors, are designed such that a catom in movement can always reach another catom latched and connected on the initial catom. Even if this quasi-spherical shape offers several advantages, we propose a modified wuasi-spherical shaper as illustrated in

Figure 2b. In fact, the latching force that will be obtained from the electrostatic actuation is proportional to the connector surface, hence the quasi-spherical shape of

Figure 2b will provide better taching strength than that of

Figure 2a. In the new suggestion, the connectors surfaces for a given volume in order to ensure the required latching force while rolling and passing surfaces (the non-connectors surfaces) based on triangle empty surfaces are still possible. For the rest of the study, a catom of a diameter of 2 cm is considered with connector’s surface of 8 mm × 8 mm.

#### 2.2. Principle of Electrostatic Latching

We present here the principle and model of the electrostatic actuation that will be behind the connectors surfaces and that serve first as latching mechanism. The idea is to distribute electrodes on each connector surface. Hence, when applying an electrical potential to the electrodes on one surface of a catom, and when another potential is applied to the electrodes on the surface of the other catom, the potentials difference will create an attraction force between the two surfaces. Thus, a latching of the two catoms is obtained. The advantages of such electrostatic actuation are—(i) suitable for miniaturization, (ii) easy to implement and (iii) can be used for both latching and displacement actuation and l (iv) low energy consumption. This latter advantage is of particular interest since the theoretical energy consumption is zero in steady state and non zero in transient state, which is very fast. In this paper, we investigate only the latching capability of the catoms. The idea consists in assessing the provided force according to the applied voltage and the geometrical features of the electrodes on each connectors surfaces. In particular, we will study and derive the required force to maintain two catoms attached to each others.

Figure 3 illustrates the principle of the electrostatic actuation where two electrodes surfaces facing each other are electrically charged. Considering this configuration as a parallel capacitor, it is easy to derive the attractive force

F between them using the principle of virtual work:

where

W_{e} is the stored electrical energy in the capacitor. In the case of parallel capacitor the following expression of the force along the vertical axis can be derived.

where

${\epsilon}_{0}$ and

${\epsilon}_{r}$ are respectively the dielectric permittivity of the void and the relative permittivity of the insulation material,

A is the area of a connector surface,

${x}_{ins}$ is the width of the insulation,

${x}_{air}$ is the distance of the air gap separating the two plates and

U is the voltage or potentials difference between the two electrodes.

Having the relationship that links the applied voltage

U and the electrostatic force

F as described in Equation (

2), it is easy to derive the required condition to latch two catoms to each other. For this, let us consider the four configurations presented in

Figure 4. They represent the possible positions between two catoms in latching condition and based on a FCC latice. Among these positions, two are of interest: Case A and Case B. Case A is the critical case where the entire weight of one catom has to be overcame by the connection, whilst case B represents the worst case where the moment to be overcame is maximal. Cases C and D represent less force and less moment than the two first cases, therefore they will not be considered here.

First let us consider case A. The minimum force to maintain the catom is straightforward from the Newton’s first law.

where

m is the weight of the catom and

g is the gravity. By combining Equations (

2) and (

3), considering that there is an air gap between the two electrodes, the condition in term of voltage that guarantees the latching is obtained:

Let us now consider case B. Still using Equations (

2) and (

3), the required moment around point I (Case B in

Figure 4) to maintain the catom latched is derived:

where

c is the connector side and

r the external radius of the quasi-spherical shape that forms the catom. Combining Equations (

2) and (

3) leads to the second condition in term of voltage required to make sure that the catom will not roll down when latched:

From the two conditions in Equations (

4) and (

6), the final condition to be ensured in order to guarantee both case A and case B, and consequently case C and case D, is:

#### 2.3. Simulations

To perform the simulations, numerical values for the further realization are used. The material to be used is the VISIJET M3 Crystal material, a plastic used for 3D printing. From this material and using Autodesk Inventor 2019 CAD software, we estimate the weight of the catom to be 500 mg. From this weight and using Equation (

3), the force has to be greater than

F = 5.10 mN in order to lift a catom in vertical position (case A). To derive the necessary conditions in term of applied voltage and distance between electrodes that can provide this amount of force, we start by simulating the general governing equation of electrostatic actuation as described in Equation (

1). To this aim, some hypothesis are made: (i)

${\epsilon}_{r}$ is taken equal to 1 as we assume that only air insulates the two electrodes, and (ii) the electrodes are perfectly planar. The resulting force versus the applied voltage and the gap between electrodes is given in

Figure 5.

As expected from Equation (

2),

Figure 5 clearly reveals that the distance between the electrodes has an important role regarding the generated force: the further distance between two electrodes is, the less generated force will be. In other words, few hundreds of nanometers of additional distance will drastically decrease the generated force. In counterpart, reducing the electrodes gap in order to increase the amount of force for a given voltage has a limitation. Indeed, one has to take into account the dielectric strength (breakdown voltage) of the material isolating the electrodes. When subjected to a voltage higher than its dielectric strength, the material breaks and electrical arcs will appear between the two electrodes and will reduce the generated force to zero. Furthermore when the material is broken, the electrodes can not be used anymore. As an example, let us consider a gap between

${x}_{air}$ two electrodes equals to 10 μm. Doing so, it is easy to determine the required force to lift the catom respecting the two previous conditions. Considering as example a voltage of 65.8 V, one can obtain from the intersection of the two curves illustrated in

Figure 6: the force generated by the weight of the catom and the electrostatic force attracting each other the electrodes separated by a distance of 10 μm. For example, the air has a dielectric strength of 3.0 Mv/m. Hence, a voltage of 30 V can cause breakage of the air between the two electrodes distanced of 10 μm. Thus, another insulation material has to be used or added to separate the electrodes in the suggested electrostatic actuators. In this regard, SiO

_{2} (silica) material is of particular interest since it has a dielectric strength of 40 MV/m that requires 390 V of voltage to reach the breakage [

21]. In addition, the integration and deposition of this material to the electrodes is standard using clean room facilities and thus a good flatness of the final electrodes will be ensured. With this SiO

_{2} layer on each electrode, the final gap between two electrodes is:

$x={x}_{air}+{x}_{ins}$.