# Optimization of Ultrasonic Powder Coatings on the Surface of Treated Materials

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

_{0}, a ball of the coating material standing motionless against the wall, and the wall itself, iinto which the ball of the coating material must be pressed to a depth h ≡ h(t) depending on time t. The optimal value of this depth of h

_{opt}is determined by the requirements of the strength of the applied coating, its thickness, esthetic features, etc. Its value can be expressed in terms of the dimensionless coefficient κ, specified by the above requirements, with the help of the following formula:

_{opt}= κR

_{h}U = F

^{2}ξ

_{m0}

^{2}/2.

_{m0}is the amplitude of ultrasonic exposure on the walls of the resonant chamber that accelerates the working bodies.

_{t}h)

^{2}/2, and the potential energy (3). By virtue of the law of conservation of energy using (3) and (4), one can obtain the following:

_{t}h turns to zero, there is obviously a maximum convergence of the ball to the surface, when there should be h = h

_{opt}. Using (1), we can rewrite Equation (5) at this point in time in the following form:

_{opt}is the optimal value of the amplitude of ultrasonic exposure, at which the powder particles of the coating material are pressed to the required depth into the treated surface.

_{opt}the formula describing the dependence of the optimal amplitude of ultrasonic exposure on the parameters of the material of the treated surface and the properties of the nanopowder used:

_{opt}, regardless of the magnitude of this mass. Therefore, a working body with a higher mass acquires a higher value of kinetic energy. Additionally, since the value of this energy is fixed to drive a particle of nanopowder to a given depth, then with an increase in mass, a decrease in the speed ωξ

_{opt}is required, and, therefore, a lower amplitude value ξ

_{opt}.

_{opt}of the amplitude increases with increasing microhardness of the treated surface. It corresponds to simple physical considerations that the harder the surface, the more difficult it is to embed nanoparticles into it, and a more vigorous external influence is required. It is also clear that the greater the radius of nanoparticles, the greater the amplitude of ultrasound is needed to introduce them to a set depth. As the frequency increases, the speed increases, with which the working body flies away from the wall of the resonance chamber. Therefore, with increasing frequency, the value of the optimal amplitude decreases.

_{0}

^{2}v(t)/l

^{3}≡ q v(t) is the frequency of ball collisions.

_{s}= ωξ

_{m0}exp(−q l)

_{opt}working bodies for a resonance chamber, the free volume in which, for the span of the working bodies, is approximately equal to the value of l

^{3}, is followed from the evident inequality:

_{opt}≤ l

^{2}/R

_{0}

^{2}

_{0}nanoparticles, the total cross-sectional area of which is S at the level of depth into the treated surface. It corresponds to the case when all nanoparticles lie next to each other without intersecting, which, of course, is unlikely. Therefore, in reality, the number of embedded nanoparticles should be taken twice as much. In fact, this number corresponds to the number of impacts of the working bodies necessary for the qualitative coating of the treated surface with a material in the form of nanopowder.

_{0}R

^{2}= 2S, that is N

_{0}= 2S/π R

^{2}

_{0}working bodies flying at a distance l to the surface of the processed product from the walls of the resonant chamber, which accelerate the working bodies. The distance that a working body will travel between two consecutive collisions is obviously equal to the value of 2l. Moving with a speed of about the value of ωξ

_{m0}, the working body will spend the process time t

_{s}equal to the following:

_{0}nanoparticles will be embedded in the surface. Additionally, it will take the time t

_{opt}to drive N

_{0}nanoparticles:

_{0}= N

_{opt}, we have the following, with the help of (12) and (8):

## 4. Discussion

## 5. Conclusions

- The obtained ratios show that the size of powder particles lies in the range from a few units to ten nanometers in the case when the frequency of ultrasound is in the operating range from 18 to 25 kHz, and the amplitude of the displacements of the walls of the resonance chamber does not exceed 15 μm. This means that for the operating parameters commonly used in standard ultrasonic equipment, the optimal size of coating particles is in the nanoscale range of values.
- The coating technology under consideration allows one to work with a surface of a very complex shape. It is necessary to take into account the fact that for high-quality processing of recesses, it is necessary to use working bodies, the size of which is less than the radius of curvature of the surface of this part of the product.
- The above expressions for the optimal values of the number of working bodies and the time of surface treatment give only approximate guidelines in the establishment of the technological process. Their exact values are determined in practice from the conditionsused toobtain the required quality of coverage.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Coating on the surface of a product: 1—treated surface; 2—working bodies in the form of steel balls; 3—resonance chamber; 4—coating material in powder form and 5—connection to the transducer.

**Figure 2.**Scheme depicting coating on a treated surface by pressing the impact force of a working body into it, which is a powder in the form of a hard ball from the coating material: R—ball radius of coating material; a—residual trace radius of implementation; h—depth of residual trace of implementation; and F—the force used to insert the ball.

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**MDPI and ACS Style**

Gradov, O.M.
Optimization of Ultrasonic Powder Coatings on the Surface of Treated Materials. *Appl. Sci.* **2023**, *13*, 6034.
https://doi.org/10.3390/app13106034

**AMA Style**

Gradov OM.
Optimization of Ultrasonic Powder Coatings on the Surface of Treated Materials. *Applied Sciences*. 2023; 13(10):6034.
https://doi.org/10.3390/app13106034

**Chicago/Turabian Style**

Gradov, Oleg M.
2023. "Optimization of Ultrasonic Powder Coatings on the Surface of Treated Materials" *Applied Sciences* 13, no. 10: 6034.
https://doi.org/10.3390/app13106034