# Spherical Droplet Deposition—Mechanistic Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{4}) [18,19]. Such a number of water molecules correspond to the volume of a spherical drop 2.14 × 10

^{−9}μL, i.e., a droplet diameter 80 nm. In addition, simulations are conducted by treating such a small droplet as a two-dimensional object [19] a priori assuming that the contact angle meets Young’s equation, i.e., it does not depend on its shape. Often, the simulation also ends when Rayleigh instabilities appear. All of these simplifications do not ensure that the drop of liquid has reached a state of equilibrium of forces.

## 2. Theory

#### 2.1. Model Development

- In the vicinity of the state of equilibrium of forces acting on the droplet, it takes the shape of a sphere segment, and the fluid velocity and its changes become so small that the inertial forces of the moving fluid and its impact on the surface between fluids due to the so-called dynamic pressure become negligible.
- During droplet deposition on the substrate, the temperature of any of the system components does not change.
- The liquid forming the droplet does not change its volume—the entire system under consideration is an isochoric system. As a result of the assumed isothermal nature of the system, it is possible to avoid the need to consider the issue of droplet evaporation [14]. On the other hand, the assumption of complete insolubility of the components of both fluid phases allows to ignore the influence of the Marangoni effect.

#### 2.2. Model Solution

## 3. Results and Discussion

#### 3.1. Young’s Solution

#### 3.2. Improved Young’s Solution

#### 3.3. Influence of Adhesion Force on Droplet Deposition

## 4. Remarks on the Model and its Experimental Verification

^{3}], ${R}_{\pi}=3.82\times {10}^{-4}$ [m], ${\sigma}_{CI}=0.075$ [N/m], the value $\Delta =0.038$ was calculated. However, since even in the case of such small droplets their shape slightly differed from the spherical one, the value of $\Delta $ should be lower.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Adopted coordinate system and designations of the geometrical dimensions of the drop being a sphere segment. The symbol marked $\theta $ shows the direction of the axis of an angular variable in a cylindrical coordinate system, perpendicular to the drawing plane.

**Figure 2.**The dependence of the parameter $D=\frac{{F}_{L}}{{\sigma}_{CI}{R}_{\pi}}$ on the contact angle and several values of the parameter $B=\frac{{\sigma}_{FI}}{{\sigma}_{CI}}$. The gray area marks solutions to the unstable balance of forces.

**Figure 3.**The dependence of the parameter $B=\frac{{\sigma}_{FI}}{{\sigma}_{CI}}$ on the contact angle and several values of the parameter $D=\frac{{F}_{L}}{{\sigma}_{CI}{R}_{\pi}}$. The gray area marks solutions to the unstable balance of forces.

**Figure 4.**Dependence of the $E=\frac{\epsilon {R}_{\pi}}{{\sigma}_{CI}}$ parameter on the contact angle for several values of the $B=\frac{{\sigma}_{FI}}{{\sigma}_{CI}}$ parameter—the equilibrium of forces acting on the droplet. The gray area marks solutions to the unstable equilibrium of forces.

**Figure 5.**Dependence of the $B=\frac{{\sigma}_{FI}}{{\sigma}_{CI}}$ parameter on the contact angle for several values of the $E=\frac{\epsilon {R}_{\pi}}{{\sigma}_{CI}}$ parameter—the equilibrium of forces acting on the droplet. The gray area marks solutions to the unstable equilibrium of forces.

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Michalski, J.A.; Jakiela, S.
Spherical Droplet Deposition—Mechanistic Model. *Coatings* **2021**, *11*, 248.
https://doi.org/10.3390/coatings11020248

**AMA Style**

Michalski JA, Jakiela S.
Spherical Droplet Deposition—Mechanistic Model. *Coatings*. 2021; 11(2):248.
https://doi.org/10.3390/coatings11020248

**Chicago/Turabian Style**

Michalski, Jacek A., and Slawomir Jakiela.
2021. "Spherical Droplet Deposition—Mechanistic Model" *Coatings* 11, no. 2: 248.
https://doi.org/10.3390/coatings11020248