# Electrocoalescence of Water Droplets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiment Technique

_{0}, where E

_{0}= 3 × 10

^{6}V × m

^{−1}, which is the breakdown value of the electric field strength in air under atmospheric conditions close to standard (this is a fixed value). Thus, the voltage between the electrodes was sawtooth with a frequency of 1–10 Hz and adjustable in amplitude. According to the foregoing, the electric circuit between the electrodes consisted of (a) a near-electrode insulator layer (1 mm thick), (b) a system of drops on the surface (20 mm long), and (c) an insulator layer (1 mm thick) at the second electrode. The reactive resistance of the insulator layers at the moments of the electric discharge (with a duration of fractions of a millisecond) was estimated to be many orders of magnitude lower than the active resistance of the system of drops on the surface. Therefore, it could be assumed that at the instants of discharges, the entire voltage was applied to the droplet system. The amplitude of the field strength can be estimated in order of magnitude by the formula E = U/d = E

_{0}× Δ/d, where d is the distance between the electrodes. In reality, the field between the cylindrical electrodes was inhomogeneous. Thus, at a distance of 3 mm from the electrode and at the middle of the distance between the electrodes, the field could differ by almost a factor of two. Therefore, the estimation of the electric field strength according to the indicated formula has a large systematic relative error of several tens of percent. This error is the same for all our experiments. The random relative error in determining the field strength was less than 10%. It depended on the errors in determining the quantities E

_{0}, d, and Δ. We believe that the random error in the value of E

_{0}is small (<10%), since the experiments were performed at atmospheric pressure and temperature close to the standard values; the random errors of the other two quantities also did not exceed 10%, since these quantities were determined by the same method as the droplet radii and interdroplet distances.

## 3. Experimental Results

^{2}= 0.93, where δ and r are measured in meters. Below, this function will be useful for analyzing the mechanism of droplet ordering on the surface. The trend of the distance between the centers of drops L is described by a function $L=1.35\times {r}^{0.92}$ with a confidence value of R

^{2}= 0.998. For droplet radii in the range r = [0.03, 2] mm, the distances between the centers of neighboring drops is, on average, 20% larger than their average diameter.

^{6}V × m

^{−1}. At a slightly higher field strength, separate pairs of droplets coalesced, located one after the other along the field strength lines. That is, coalescence occurred due to dipole–dipole attraction. With a further increase in tension, ensembles of three, five, or more drops coalesced. The drops in the ensembles were closely spaced around a single geometric center. In all cases, coalescence occurred in a fraction of a second. At even higher field strengths, the probability of coalescence decreased. The reason for this was that the drops became few, and they were at too great a distance from each other.

_{c}~r

^{−1/2}. The obtained dependence will be required further when discussing the mechanism of coalescence.

## 4. The Discussion of the Results

#### 4.1. Ordering of Drops on a Hydrophobic Surface

_{F}

_{max}at the points of the local maximum of the free energy of all plasma particles depending on the droplet charge Z. The position of the curve does not depend on the radius of the droplets (free energy takes into account the gas-kinetic and electrostatic energies of the particles.) Curve 1 is well approximated by curve 5, which can be obtained by numerical calculation from the condition that the electrostatic and kinetic energies of all plasma particles are equal. This dependence is described by the formula ${\delta}_{F\mathrm{max}}\approx {10}^{-5}\times {Z}^{1/3}$.

_{F}

_{min}at the points of local minimum of the plasma free energy depending on the droplet charge Z at droplet radii of 10

^{−5}, 10

^{−6}, and 10

^{−8}m, respectively. These curves are approximated by the empirical Dependence (6) described by the formula

#### 4.2. Mechanism of Electrocoalescence

^{−5}m (curve 2) can coalesce, passing to the lower limit of attraction, at charges over 300 and 10

^{4}, respectively. At lower charge values, coalescence is impossible. But, electrocoalescence is possible. Indeed, in an external electric field, water droplets are polarized and can acquire a large surface electric charge sufficient for interdroplet distances to fall into the region $\delta <{\delta}_{0}$ and coalescence to occur.

_{0}is the electrical constant. (In the calculations, we choose a hemispherical drop model, which is a rather rough approximation. As a result, one can hope to obtain theoretical estimates of the drop parameters only by an order of magnitude). We integrate (5) over the surface of a hemisphere of radius r and obtain the value of the charge induced by the field

_{c}at which coalescence begins.

#### 4.3. Experimental Estimation of the Droplet Charge on the Surface

^{−4}m, $\nabla E\approx \frac{E}{l}={10}^{8}$V × m

^{−2}, where E = 10

^{6}V × m

^{−1}—field strength in the experiment and l = 10

^{−2}m—half of the interelectrode distance. We use these data in Formula (11) and get Z = 9 × 10

^{4}. Let’s compare this value with the value calculated by Formula (3): Z = 10

^{12}× r

^{2}= 6 × 10

^{4}. We see that the values coincide in order of magnitude. Thus, the use of Formula (3) for an approximate estimate of the charge of droplets on a hydrophobic surface is admissible.

## 5. Conclusions

- An experimental setup has been created to study the spatial ordering and electrocoalescence of water droplets on a hydrophobic plastic surface. An electrostatic (electrophore) machine with a voltage of up to 60 kV was used as a source of high voltage.
- The dependences of the average distance between the surfaces of the nearest drops on their average radius are studied. It has been established for the first time that drops on a hydrophobic surface exhibit patterns of spatial arrangement characteristic of a drop cluster and water fog. That is, the system of drops on the surface can serve as a good physical model of both. This circumstance allowed us to hope to obtain useful experimental data on electrocoalescence in fog using a system of droplets on the surface.
- The dependences of the number of coalesced droplets on the surface on the strength of the external electric field were measured. The dependences of the field strength at which mass coalescence begins on the average droplet radius in the range of average radii of [0.03, 2] mm are studied. A new model of droplet electrocoalescence based on the state diagram of a droplet-ion plasma, numerically consistent with experimental data, is proposed. Since drops on a hydrophobic surface are a physical model of a droplet cluster and fog, we hope that the pattern of electrocoalescence established by us can be suitable for understanding the phenomenon of coalescence in fog and clouds. Further numerical calculations of the growth rate of droplets in clouds, taking into account electrocoalescence, can put an end to the problem of “bottleneck condensation-coalescence in the formation of raindrops.”

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Drops of water on the wall of a plastic bottle before applying high voltage (

**a**), after applying high voltage (

**b**).

**Figure 2.**Block diagram of the installation. 1—camera, 2—cylindrical electrode, 3—plastic container with water, 4—drops of water, 5—discharger, and 6—electrostatic machine.

**Figure 3.**Dependence of the average distance between the surfaces of neighboring drops δ (1) and the average distance between the centers of neighboring drops L (2) on their average radius r. The inset shows a fragment of a photograph of the droplet structure.

**Figure 4.**Dependence of the concentration of drops on the surface N on the strength of the external electric field E. (1)—the average radius of the drops is 0.06 mm, (2)—0.16 mm. (3) and (4) are approximating curves for (1) and (2), respectively.

**Figure 5.**Dependence of the critical electric field strength E

_{c}on the average droplet radius r. (1)—experimental values and (2)—theoretical curve.

**Figure 6.**Diagram of states of a drop-ion plasma. (1) is the distance between the surfaces of the nearest drops ${\delta}_{F\mathrm{max}}$ at the points of the local maximum of free energy depending on the charge of the drops Z. (2), (3), and (4) are the distance between the surfaces of the drops ${\delta}_{F\mathrm{min}}$ at the points of the local minimum of the free energy, the radii of the drops are 10

^{−5}, 10

^{−6}and 10

^{−8}m, respectively. (5) and (6) are approximating curves. (7)—experimental values for drops on a hydrophobic surface, (8)—the same for a drop cluster, and (9)—the same for water fog. (10) is the distance between the droplet surfaces at which coalescence occurs.

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**MDPI and ACS Style**

Shavlov, A.V.; Dzhumandzhi, V.A.; Yakovenko, E.S.
Electrocoalescence of Water Droplets. *Plasma* **2023**, *6*, 127-138.
https://doi.org/10.3390/plasma6010011

**AMA Style**

Shavlov AV, Dzhumandzhi VA, Yakovenko ES.
Electrocoalescence of Water Droplets. *Plasma*. 2023; 6(1):127-138.
https://doi.org/10.3390/plasma6010011

**Chicago/Turabian Style**

Shavlov, A. V., V. A. Dzhumandzhi, and E. S. Yakovenko.
2023. "Electrocoalescence of Water Droplets" *Plasma* 6, no. 1: 127-138.
https://doi.org/10.3390/plasma6010011