*2.3. Surface Morphology*

The reduction of the roughness of the finished products was carried out by two methods of ultrasonic processing—cavitation-abrasive finishing and vibratory tumbling in water. It should be noted that dry vibratory tumbling is forbidden in many countries by sanitary norms and rules of production due to destructive action of the ceramic dust on human health (silicosis) [63].

Effects of a mechanical nature exert the most significant influence on the processing in liquid during ultrasonic cavitation abrasive finishing: cavitation, variable sound pressure, radiation pressure, acoustic streams of various scales, sound capillary effect. The introduction of ultrasonic vibrations into a liquid is an effective way of complex-shaped part processing with the internal cavities [44–46,64–66]. The method's effectiveness is due to many specific effects arising in a liquid technological medium under the influence of vibrations [67]. An acoustic pressure arises when ultrasound passes through a liquid as follows:

$$P\_d = P\_A \sin 2\pi f t\_\prime \tag{1}$$

where *PA* is a maximal amplitude of acoustic pressure (Pa); *f* is oscillation frequency (Hz); *t* refers to the propagating (collapse) time (s) [68,69]:

$$t = 0.915 R\_{\text{max}} \left( \frac{\rho}{P\_m} \right)^{\frac{1}{2}} \left( 1 + \frac{P\_{v\chi}}{P\_m} \right) \tag{2}$$

where *Rmax* is the radius of the cavity at the start of the collapse (m); ρ is the medium density (kg·<sup>m</sup>3); *Pm* is the medium pressure at the time of collapse (Pa); or [70]

$$t = 0.915 R\_{\text{max}} \left( \frac{\rho}{P\_h} \right)^{\frac{1}{2}} \,\text{\,\, \,}\tag{3}$$

where *Ph* is the hydrostatic pressure surrounding the cavity (Pa). The maximum pressure developed in bubble *Pmax*:

$$P\_{\text{max}} = P\_{\text{v\%}} \left( \frac{P\_{\text{m}}(\text{y} - 1)}{P\_{\text{v\%}}} \right)^{\frac{\mathcal{V}}{(\mathcal{V} - 1)}},\tag{4}$$

where *Pvg* is pressure inside the cavity (of vapor-gas mixture) at maximum radius *Rmax* (in the bubble at its maximum size, pressure at the initial stage) (Pa); γ is the polytropic index (exponent) for the gas mixture that is equal to the adiabatic exponent for the adiabatic process following Poisson's equation:

$$P\_"V^k = \text{const} \tag{5}$$

where *V* is the volume (m3), and κ:

$$\kappa = \frac{\mathcal{C}\_p}{\mathcal{C}\_v} \tag{6}$$

where *Cp* and *Cv* are heat capacity of gas at constant pressure and at constant volume, correspondingly. If:

$$\gamma = \frac{\mathbb{C} - \mathbb{C}\_p}{\mathbb{C} - \mathbb{C}\_v} \tag{7}$$

and *C* is heat capacity of gas in the given process, then, for adiabatic process:

$$
\gamma = \kappa \tag{8}
$$

The polytropic exponent γ determines the gas state in the cavity and is in the range of 1–1.33. The amplitude can be presented by acoustic force *FA* and system stiffness *ka*:

$$A\_{\rm nl} = \frac{F\_A}{k\_a} \tag{9}$$

At the same time, the stiffness of the system is determined by its mass:

$$K\_a = 4\pi^2 \frac{m\_a}{T^2} \,\mathrm{\,\, \,}\tag{10}$$

where *T* is a period of natural oscillations [s]; or by cross-sectional area perpendicular to the line of the force application, Young's modulus and length of an element:

$$K\_{\rm d} = \frac{S\_A \cdot E}{L}.\tag{11}$$

Because of the periodic action of tensile and compressive forces in the liquid, cavitation occurs. It consists of discontinuities in the continuity of the liquid with the subsequent collapse of these cavities. A feature of cavitation in the processing of solids is transforming a relatively low energy density of the sound field into a high density of local impulse action when cavitation bubbles collapse. Thus, if we consider the process of bubble collapse to be adiabatic, then the pressure inside is determined by expression:

$$P\_m = P\_{\text{vy}} \left(\frac{R\_{\text{max}}}{R\_{\text{min}}}\right)^{3\gamma} \tag{12}$$

or

$$P\_m = P\_{\text{xy}} \left(\frac{R\_{\text{max}}}{R\_{\text{min}}}\right)^{3-4} \tag{13}$$

where *Rmax* is a maximum bubble radius at the initial stage of collapse (m); *Rmin* is a minimum bubble radius at the end of the collapse (mm). The maximum pressure is determined by the ratio *Rmax Rmin* . The results of classical studies [71–73] show that during the stretching period of the liquid *Rmax* exceeds the radius of the cavitation nucleus by 100–300 times, and the pressures *Pmax* can reach up to 107–1011 (Pa) at the stage of the collapse of the cavitation bubble, which causes plastic deformation of the solid surface and local destruction of the surface (erosion) when specific pressures are exceeded.

The second effect that determines the efficiency of ultrasonic liquid treatment is acoustic flows, the role of which is in the transfer and distribution of cavitation bubbles over the sound volume, which is especially important for the treatment of complex-profile surfaces. The nature of acoustic flows primarily depends on the mode of ultrasonic treatment, determined by the amplitude of oscillations of the end of the radiator *Sm*. Thus, large-scale acoustic flows are virtually absent at low-amplitude processing mode (*Sm* < 10–12 μm for water), and random sections of the sound volume are involved in cavitation. The transition to a high-amplitude mode (*Sm* > 10–12 μm) is abrupt and is explained by the strong absorption of acoustic energy during the development of the cavitation region at the end surface of the radiator [74], as a result of which directional hydrodynamic flows are formed, which carry out an active transfer of bubbles from the radiation surface to the treated surface. Formed flows lead to the formation of a stable cavitation area. The height of this area characterizes the depth of penetration of the cavitating liquid flow into the treated volume and depends on the amplitude of vibrations and the absorbing capacity of the process medium:

$$a = \frac{2\pi^2 f^2}{\rho c^3} \left(\frac{4}{3}\eta + \frac{(\gamma - 1)\theta}{\gamma \mathbb{C}\_v}\right) \tag{14}$$

where ρ is medium density (kg·cm<sup>−</sup>3); *c* is the speed of sound in a medium (m·s<sup>−</sup>1); η is the viscosity of a fluid (Pa·s); θ is a coefficient of thermal conductivity of a material (W·(m·K)−1); *Cv* is the molar heat capacity at constant volume (J·(K·mol)−1).

One of the methods for intensifying the solids' ultrasonic treatment is adding the abrasive powder to the working fluid—cavitation-abrasive finishing. The addition of insoluble abrasive particles to the sonicated liquid leads to a significant change in the processing. The presence of inhomogeneities in the technological liquid medium leads to a decrease in the liquid's cavitation strength and an increase in the number of cavitation centers, which increases the volume of the effective cavitation region. The mechanism of the effect of cavitation-abrasive finishing on the surface of the product is based as well on the micro-cutting action of abrasive particles, which acquire acceleration due to impulse transmission from shock waves' large-scale acoustic currents.

The ultrasonic cavitation-abrasive finishing was carried out on a half-wave magnetostrictive oscillatory system powered by a UZG 2.0/22 generator (JSK "Ultra-resonance", Yekaterinburg, Russia) (Figure 3a). An ultrasonic emitter of a rod three-half-wave magnetostrictive oscillatory system was immersed in water at a distance of 20 mm from the end face to the workpiece. The processing was carried out with the following parameters: the vibration frequency *f* = 21,000 Hz, the vibration amplitude of the end face of the emitter *Sm* = 20 μm, and the processing time *t* = 120 s. During processing, the ultrasonic generator was operated in the automatic frequency control mode to maintain resonance conditions.

**Figure 3.** Schemes of ultrasonic treatment: (**a**) cavitation-abrasive finishing, where (**1**) is a transducer, (**2**) is medium, (**3**) is a cavitation bubble, (**4**) is a part to be processed, *Va* is a feed of abrasive, *Sa* is a direction of induced oscillations; (**b**) vibration tumbling, where (**1**) is a tank with the parts, (**2**) is a transducer, (**3**) is an abrasive particle, *Va* is an abrasive movement speed, *Vp* is a part movement speed, *Fa* is an induced acoustic force, *Fa*' is a medium force response.

The sample was placed in a radiator, on the bottom of which a layer of Elbor-R abrasive cubic boron nitride (β-BN) powder (JSC SPC "Abrasives and Grinding", Saint-Petersburg, Russia) of 2 mm thick was poured. An axial channel was made in the radiator to ensure the supply of abrasive powder (Elbor-R) to the cavitation zone. The optimal processing parameters were determined based on preliminary studies.

The vibratory tumbling was carried out while moving products and abrasive grains relative to each other in a vibrating container of an 80 L ZHM-80A vibratory tumbler finishing machine planetary drum type (Shengxiang, Zhejiang, China); a filler SCT VFC 10 mm × 10 mm ceramics prism gray (tumbling body) (CFT, Moscow, Russia) made of ceramics with 30% of silicon abrasive with a smooth surface (Figure 3b). The processing was carried out in the following modes: a vibration frequency of 50 Hz, a filler weight of 50 kg, an operating time of 20.5 h, and an engine speed of 1440 rpm. The harmonic vibration amplitude was 6–7 mm.
