Research on a Focused Acoustic Vortex that Can Be Used to Capture Tiny Underwater Objects

Abstract

The energy of a focused acoustic field is quite concentrated, and the ability of an acoustic vortex formed by a concave focusing transducer array to capture objects in a flowing medium remains to be investigated. In this paper, the focused pressure distributions generated by an acoustic lens and a concave focused transducer array are firstly simulated, and the analyzed results show that the focusing effect of the latter is significantly better than that of the former. The acoustic gradient force and orbital angular momentum density distributions of the focused transducer array were investigated. A focused acoustic vortex tiny object capture system was built by simulating the hydrothermal column that forms in the seafloor hydrothermal zone. It was discovered that the forces affecting microorganisms and other small objects primarily consist of acoustic gradient force, viscous force, and additional mass force. The non-destructive capture of tiny seafloor objects was accomplished by adjusting the focused acoustic vortex’s propagation direction and the transducer array’s emitted power, thereby enabling more potential applications in ocean equipment.

1. Introduction

At present, a focused acoustic field can be produced by means of (1) phased arrays [1], (2) concave spherical focused transducers [2], (3) acoustic metamaterials [3,4], (4) phononic crystals [5], and (5) structural phase modulation [6]. The idea here is that acoustic focusing technology could be used to form focused acoustic vortices (FAVs), thereby realizing more precise and stable object manipulation with strengthened acoustic gradient force (AGF).

As we know, the vortex formed by a planar transducer as the vibration source has dispersed energy in the near field, the directivity of the ultrasonic transducer is strong, and the axial distance of the vortex formed by the main lobe is relatively far from the vibration source [7,8]. Therefore, less energy is actually involved in manipulating an object in the near field. Focused acoustic fields can gather energy in a smaller area to improve the utilization efficiency of acoustic energy, and these focused acoustic fields are mainly emitted by concave focused transducers. Lim et al. [9] established an experimental setup of micropipette aspiration and ultrahigh-frequency focused transducers and quantitatively measured the magnitude of acoustic trapping force. The measured trapping forces were in the range of nanonewtons, and the trap stiffness was in the range of nanonewtons per micrometer. When the focusing transducer is moved in a certain direction, the beads with radius a = λ trapped in the near field move along the same direction as if they were fixed on the acoustic axis [10]. In an experiment using a concave focusing transducer to trap an object [11], the motion process of polystyrene spheres in the near field demonstrated that the maximum capture distance of a single-beam focused ultrasound is closely related to the operating frequency of the transducer. When the transducer is operated at the center frequency, the capture distance of the focused acoustic beam is maximum, and the maximum distance increases with the increase in the transducer amplitude. Takatori et al. [12] used focused acoustic tweezers to manipulate the particles, and these particles were clustered together when the acoustic tweezers acted; in the paper, the Brownian motion of the particles was analyzed, as well as the motion of the particles in the presence of the repulsive force between the particles and the acoustic tweezers’ entrapment force. Hongyu Sun et al. proposed a composite acoustic lens for underwater 3D focusing [13]. Jun Zhang et al. simulated and measured the concave spherical acoustic lens transducer using the ray method [14].

Although these focused single-beam acoustic fields achieve the capture of particles of specific sizes, the maximum acoustic intensity at the focal point tends to deflect tiny objects away from the acoustic axis, and axial repulsive forces due to backward scattering make single-beam acoustic fields difficult to achieve for a wide range of sizes. Zhixiong Gong et al. theoretically demonstrated that synchronized spherical vortexes can capture or assemble particles in three dimensions [15]. The emergence of focused acoustic vortices solves the problem of manipulating tiny objects out of focus by allowing the vortex field to both push and pull the polystyrene spheres to move in the axial direction and to precisely control the position of the spheres in the radial direction [16]. The central potential well of the focusing vortex can firmly “grab” the small object, effectively suppressing the off-target effect.

Stone crushing employs an ideal focused waveform that results from focused acoustic waves utilizing an acoustic lens [17], and the acoustic lens designed based on the genetic algorithm improves the efficiency of acoustic field calculation [18]. The focusing effect of a conventional acoustic lens is achieved by bending materials into specific shapes, that is, changing the propagation path of acoustic waves, and thus the wave aberration of an acoustic lens with a defined geometry is fixed. The rapid development of phononic crystals [19,20] and acoustic materials [21,22] has made acoustic lenses more promising for applications, where the refractive index of the acoustic lens is adjusted by controlling the dimensions of the acoustic metamaterials’ cellular inclusions and the focal point of the acoustic lens can be controlled by pre-compressing a two-dimensional ball-and-chain array, as this will change the wave velocity inside it [23]. Alternatively, the focusing characteristics of the acoustic lens can be improved by a modified triangular acoustic crystal plate. The simplicity of operation and low experimental cost are the main advantages of the focused acoustic field formed by acoustic lenses compared to concave transducers. However, it is precisely due to the presence of the acoustic lens that a significant amount of acoustic energy is lost.

The study of marine microorganisms offers significant research value and importance for industrial fields, pharmaceutical applications, genetic engineering, and environmental protection. Due to the complexity and variability of the marine environment, its unique high salt, high pressure, low temperature (or occasionally high temperature), low light, and other extreme conditions impose stringent technical requirements on the collection equipment. Therefore, the development of high-quality marine equipment is a necessary way to achieve high-value use of marine resources. At present, the in situ enrichment of marine microorganisms is achieved by using deep-water pumps to suck up large quantities of seawater through filter membranes. A method has been developed to capture marine bacterial strains, replacing the conventional two-step collection method [24]. However, those methods may cause some extrusion or damage to the microorganisms, and the microorganisms will even lose their vital signs.

Ultrasound has the advantages of good directionality, strong penetration ability, and easy access to more concentrated acoustic energy and is widely used in medicine, industry, agriculture, and other fields. The technology of acoustic tweezers driven by ultrasound has the advantages of being non-invasive, radiation-free, biocompatible, and able to manipulate objects of different sizes and types, showing unique advantages in biological particle manipulation, which can realize non-invasive and non-polluting enrichment sampling of microbes and other tiny objects in the ocean.

1.1. Principle and Method

In this study, an N-element sector transducer array with a radius a is adopted to generate FAV beams, as shown in Figure 1. In the source plane, N-sector concave piston transducers are distributed uniformly to form a complete concave surface. The spatial angle of each sector source is $\Delta \phi =2\pi /N$. The initial phase function of the nth sector is expressed by ${\varphi }_n=2\pi l{\displaystyle \frac{\left(n-1\right)}{N}}$, where l is the topological charge. Assuming that the surface vibration is perpendicular to the source plane at a uniform strength, the particle velocity at each point on the nth sector transducer can be described by $A=A_0\exp \left[j\left(\omega t-{\varphi }_n\right)\right]$, where A₀ is the particle velocity amplitude and $\omega$ is the angular frequency.

By dividing the sector surface into infinitely small portions (or infinitely infinitesimal elements), each vibration element can be treated as a unit source. Thus, the acoustic pressure at the observation point S_n produced by the nth sector source can be calculated by

$$d{p}_n\left(r,\varphi ,z\right)={\displaystyle \frac{jk{\rho }_0{c}_0}{2\pi R_n}}{A}_{0}d{S}_n\exp \left[j\left(\omega t_n-k{R}_n-{\phi }_n\right)\right],$$

(1)

where $d{S}_n=r_{n}d{r}_{n}d{\phi }_n$ is the area of the unit source in cylindrical coordinates, $R_n=\sqrt{{\left[r\cos \varphi -r_n\cos {\varphi }_n\right]}^2+{\left[r\sin \varphi -r_n\sin {\varphi }_n\right]}^2+z^2}$ is the transmission distance from source $\left(r_n,{\phi }_n,0\right)$ to observation position $\left(r,\phi ,z\right)$, $t_n=R_n/c_0$ is the corresponding transmission time, ${\rho }_0$ and $c_0$ are the density and acoustic speed of the medium, and $k=\omega /c_0$ is the wave number. The acoustic pressure at the observation point generated by the sector transducer array can be achieved as [25]

$$p\left(r,\phi ,z\right)={\displaystyle \sum _{n=1}^N\frac{jk{\rho }_0{c}_0}{2\pi }{A}_0}{\displaystyle {\int }_0^a{\displaystyle {\int }_{{\phi }_n}^{{\phi }_n+\Delta \phi }\frac{{\mathrm{e}}^{j(\omega t_n-k{R}_n-{\varphi }_n)}}{R_n}}}{r}_{n}d{r}_{n}d{\phi }_n,$$

(2)

The relationship between the acoustic pressure and the particle velocity can be described by the motion equation $\rho {\displaystyle \frac{\partial v}{\partial t}}=-\nabla p$, $\nabla =\widehat{r}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{1}{r}}\widehat{\phi }{\displaystyle \frac{\partial }{\partial \phi }}+\widehat{z}{\displaystyle \frac{\partial }{\partial z}}$ is the gradient operator in cylindrical coordinates, and $\widehat{r}$, $\widehat{\varphi }$, and $\widehat{z}$ are the unit vectors along the respective directions. Then, the particle velocity ${\overrightarrow{v}}_{\perp }$ in the transverse plane can be calculated by ${\displaystyle \frac{{\nabla }_{\perp }p}{\left(i\omega \rho \right)}}$, where ${\nabla }_{\perp }=\nabla -\widehat{z}{\displaystyle \frac{\partial }{\partial z}}$. Hence, the transverse particle velocity can be calculated by ${\overrightarrow{v}}_{\perp }={\displaystyle \frac{\left(\widehat{r}\frac{\partial p}{\partial r}+\widehat{\phi }\frac{1}{r}\frac{\partial p}{\partial \phi }\right)}{\left(j\omega {\rho }_0\right)}}$. In object manipulation, the capability of particle trapping and rotating is often described by the exerted acoustic gradient force as

$$F_{\perp }=-\nabla U_{\perp }=-\pi a_p^3\left\{\left[1-{\displaystyle \frac{c^2\rho }{c_p^2{\rho }_p}}\right]{\displaystyle \frac{\nabla {|p|}^2}{3\rho c^2}}-\left[{\displaystyle \frac{2\left({\rho }_p-\rho \right)}{2{\rho }_p+\rho }}\right]{\displaystyle \frac{\rho \nabla {\left|{\overrightarrow{v}}_{\perp }\right|}^2}{2}}\right\},$$

(3)

where $U_{\perp }$ is the Gor’kov potential [26] in the transverse plane and $a_p$, ${\rho }_p$, and $c_p$ are the radius, density, and acoustic speed of the particle, respectively.

1.2. Numerical Studies

In the top-view schematic, the sector angles of the eight focused sector transducers are all π/4, as shown in Figure 2(a1), and the radius of the sector is 42 mm, as shown in Figure 2(a2). In the planar schematic diagram, the red part indicates the transducer, and the curvature radius of the focused sector transducer is 50 mm, the height is 22.8 mm, and the width is 84 mm, as shown in Figure 2(b1). The planar sector transducer is closely attached to the concave acoustic lens, and the acoustic lens, made of ABS resin, has a curvature radius of 50 mm, a height of 22.8 mm, and a width of 84 mm, as shown in Figure 2(a2,b2).

Figure 3(a1) displays the cross-sectional acoustic pressure distribution produced by the focused sector transducer array. The resulting vortex field has a topological charge of 1, the axial distance is 45 mm, and the sound-source vibration frequency is 100 kHz. The higher pressure is collected in a 10 mm radius circle, where acoustic energy is also concentrated. The pressure in the vortex center is zero, and the potential well’s central region is sufficiently small to trap the object there. In the region of higher pressure, the phase distribution is regular, smoothly transitioning from −π to π, as shown in Figure 3(a2). Conversely, in areas of lower pressure, the cross-sectional phase does not display a gradual variation spanning from 0 to 2 π. As depicted in the axial pressure distribution in Figure 3(a3), the concentration of higher pressure in a narrower region clearly indicates that the energy of the vortex field, produced by the focused sector transducer array, is more intensely focused. While the acoustic field generated by the planar sector transducer array and acoustic lens does exhibit a focusing effect, it is notably less pronounced compared to the focused acoustic field produced by the focused sector transducer array. As seen in Figure 3(b1), the cross-section at the axial position of z = 79 mm coincides with the location of the maximum pressure within the vortex acoustic field. The phase distribution of the cross-section is more regular, as shown in Figure 3(b2). However, from the axial pressure distribution, while the acoustic lens achieves the focusing effect, the distribution of the higher pressure is relatively dispersed, and the pressure inside the acoustic lens medium is higher, as shown in Figure 3(b3). The focusing ability is obviously weaker than that of the focused sector transducer array. Owing to its superior focusing effect, the vortex field generated by the focused sector transducer array is primarily employed for the capture of tiny objects in intricate marine environments.

The normalized radial pressure distributions generated by the focused transducer array and the combination of the planar transducer array and the acoustic lens are shown in Figure 4a. The maximum pressure generated by the former is 4.65 times that of the latter, and the vortex radii are 6.1 mm and 11.4 mm, respectively. The axial locations are at z = 45 mm and 79 mm, respectively.

The normalized radial acoustic gradient force distributions generated by the focused transducer array and the combination of the planar transducer array and acoustic lens are shown in Figure 4b. At distances of 2.8 and 6.5 mm from the vortex center at z = 45 mm and z = 79 mm, respectively, the force generated by the focused transducer array is 35 times larger than that generated by the combination. In Figure 4b, positive values represent the force’s direction moving from left to right, while negative values indicate the opposite direction, from right to left. This enables the focused transducer array to circularly maneuver an object towards the vortex center with a radius of 6.5 mm. In terms of acoustic gradient force, the focused transducer array exhibits a remarkable enhancement in its ability to capture the object. During the simulation, the controlled object within the acoustic field was a polyethylene bead with a radius of 2 mm. For the focused transducer, the vortex field generated an acoustic gradient force with a magnitude of 10⁻⁸ N at an amplitude of 30 μm.

The distributions of axial acoustic pressure and acoustic gradient force generated by the focused transducer array are shown in Figure 5. The white arrows in Figure 5a–c indicate the magnitude and direction of the acoustic gradient force for topological charges l = 1, 2, and 3, respectively. As the topological charge of the acoustic vortex increases, so does its vortex radius, resulting in a more dispersed acoustic energy and a corresponding decrease in the acoustic gradient force directed towards the vortex center. Specifically, the acoustic gradient force at topological charge l = 1 is 2.9 and 3.2 times larger than that at topological charges l = 2 and l = 3, respectively.

The distribution of the acoustic pressure generated by the focused transducer array is shown in Figure 6a, and the magnitude and direction of the acoustic gradient force are indicated by the white arrows in the figure. The position with the maximum pressure (radial distance r = 6.5 mm, which is also the direction’s cut-off point of the acoustic gradient force) is identified as the critical point, and the object approaches (or moves away from) the vortex center under the action of the acoustic gradient force.

The cross-sectional distribution of the acoustic gradient force is shown in Figure 6b, where the higher of the acoustic gradient forces shows an annular distribution, with relatively small forces near the vortex center within the annulus and very small forces outside it. The white arrows indicate the magnitude and direction of the acoustic gradient force.

As depicted in Figure 6c, the distribution of the orbital angular momentum density within the focused vortex is displayed. Within this distribution, the rotational moment originating from the vortex field is imparted to the controlled object through the transfer of orbital angular momentum, subsequently inducing its rotation. The black arrows indicate the direction and magnitude of the velocity vector. Regions with large orbital angular momentum correspond to high velocities, while velocities near the potential well are small. The direction of the velocity vector is deflected counterclockwise. The small object moves towards the low-pressure position under the action of the acoustic gradient force. Finally, it is bound to the vortex center. At the time of simulation, the axial position of the cross-section is z = 45 mm, and the topological charge l = 1.

In addition to acoustic gradient force, there are also additional mass forces, viscous forces, gravity, buoyancy, Saffman lift, Magnus lift, Basset forces, and other forces acting on the tiny particles in seawater. Notably, the density of the particle material is nearly identical to the density of seawater, leading to a neutralization of the gravity and buoyancy forces. The lift force generated by the rotation of the spherical particles (i.e., Magnus lift) is relatively small. When the particle size is very small, the Saffman lift is neglected. According to Melling’s description of the motion of solid particles, the Basset force can also be neglected [27]. In the direction perpendicular to the axis of acoustic propagation, the forces acting on the particles are primarily composed of the acoustic gradient force, viscous force, and additional mass force. If the acoustic gradient force surpasses the combined magnitude of the viscous force and the additional mass force, the particles will be drawn towards and captured in the vortex center. The forces acting on the particles are shown in Figure 7, and the direction of the water flow is negative along the x-axis. A hydrothermal column is formed by the seafloor hydrothermal area where microbial communities are more enriched, and the hydrothermal fluid that erupts from the hydrothermal vents flows upward. The flow rate of the hydrothermal fluid from the vents is generally about 1 m/s, significantly faster than the velocity of the seafloor bottom current, which is generally about 0.011 m/s. Compared to the surrounding seawater, the flow of the hydrothermal fluid from the vents remains relatively stationary. The area near the microbial community forms a plume movement with a slower flow rate.

The viscous drag at low particle Reynolds numbers satisfies Stokes’ law. Therefore, the viscous drag can be expressed as $F_D=-3\pi \mu d_{p}V$, where μ is the hydrodynamic viscosity (Pa·s) and $V=u_f-u_p$ is the relative flow rate of particles and fluid (m/s). When a spherical particle is accelerated in an ideal static fluid, it drives the fluid near the particle to perform the acceleration together, which is equivalent to adding an additional force to the particle. This force is called the additional mass force, and its expression is shown as $F_{vm}=k_m{\displaystyle \frac{\pi d_p^3}{6}}{\rho }_f{\displaystyle \frac{dV}{dt}}$, where d_p is the radius of the particle, ${\rho }_f$ the density of the fluid, and k_m is an empirical constant with a value of 0.5. Under the influence of the orbital angular momentum, the spherical particle rotates and thus generates a Magnus lift force, which can be expressed as $F_m={\displaystyle \frac{1}{8}}{d}_p^3\pi {\rho }_f{\omega }_{p}V$, where ${\omega }_p$ is the particle rotational speed. The direction of the force is inward and perpendicular to the xoz plane. The seawater also generates a drag moment $\tau =-8d_p^3\pi \mu {\omega }_p$ opposite to the direction of rotation of the object, causing the rotational speed of the object to decrease under its effect.

The density ${\rho }_f$ of seawater is determined by its temperature and salinity and is calculated as 1.025 × 10³ kg/m³. Other parameters used in the calculation are shown in

Table 1. Value of parameters.

Name	Density of Seawater	Particle Radius	Hydrodynamic Viscosity	Particle Rotation	Fluid Velocity
Symbol	${\rho }_f$	dp	μ	${\omega }_p$	V
Value	1.025 × 103 kg/m3	1 × 10−6 m	1.025 × 103 Pa·s	13.2 rad/s	1 m/s

. In addition, the calculation results are shown in

Table 2. Calculations of forces.

Name	Acoustic Radiation Force	Additional Mass Force	Viscous Drag	Magnus Lift	Rotational Torque
Symbol	F	Fvm	FD	Fm	$\tau$
Value	1 × 10−8 N	0.27 × 10−15 N	9.66 × 10−15 N	5.3 × 10−15 N	3.9 × 10−19 N·m

.

2. Experimental Studies

To verify the characteristics of the FAVs generated by a focus sector transducer array, an experimental system was established for manipulating polyethylene particles in both radial and axial directions. An 8-channel driving circuit was designed to excite the 8 concave sector transducers. Controlled by a computer, a field programmable gate array (FPGA) 8 was used to generate 8 phase-controlled square waves at a frequency of 100 kHz. The configuration of the sector transducer array is the same as shown in Figure 2. The transducer array was placed in the water. Polystyrene particles, with a diameter of 2 mm, were utilized to analyze the trapping and rotating characteristics of object manipulation, based on videos recorded by a camera.

When the topological charge l is set to 1, the peak pressure occurs precisely at the location where the vortex radius measures 6.5 mm, as depicted in Figure 8a. Notably, the radius of the red circle also corresponds to this value of 6.5 mm. At the initial time t = 0 s, the camera starts recording while the vibration source remains active, the orbital angular momentum within the acoustic field is disregarded, and the beads are displaced towards both sides of the red curve due to the radial acoustic gradient force. As indicated by the red circle, the bubbles, which are produced by the dissolved gas in the water, coalesce into clusters and are dispersed evenly around the vortex radius. Progressing to t = 0.5 s, the particles have already deviated significantly from the position of circle, as visible in Figure 8b. At t = 3 s, the beads have retreated from the region of maximum pressure. Subsequently, upon deactivation of the vibration source, the bubbles dissipate, and the small balls move freely, as illustrated in Figure 8c. This experiment confirms the existence of radial trapping force and the trapping range of the experimental system. It should be noted that the proper measurement method for the radial acoustic gradient force is still under study.

The z-axis is perpendicular to the horizontal plane, and the forces acting on the bead at the vortex center include its gravity, the buoyancy f, and the axial radiation force F_Z of the vortex. In the axial direction, the bead remains static, as shown in Figure 9a, and the three forces are in equilibrium; that is, mg = f + F_Z, where g is the acceleration of gravity. At this time, the bead rotates in the vortex center, and the vertical distance to the sound source is s. When the sound source is turned off, the small ball reaches the equilibrium state in the water after a period of time; that is, mg = f₁, as shown in Figure 9b. At this point, the distance of the bead from the sound source is s₁, and s₁ − s = 0.52 mm. Therefore, in the absence of other external forces, the axial radiation force can be expressed as F_Z = f₁ − f. Due to the small diameter of the bead, the change in the volume of water discharged by the bead before and after the vortex field action can be approximated as the volume of the cylinder. Plugging the data into the formula for axial radiation force approximation gives F_z = 65.3 μN. This experiment proves that, under the influence of axial acoustic radiation force, an FAV can effectively transport the captured object into a collector. Of course, the radial capture force and axial thrust of the FAV need to be perfectly coordinated in order to accomplish this task.

3. Discussion

Based on the stronger FAV of a concave transducer array, numerical simulations indicate that the strengthened AGF can be produced through the effective concentration of acoustic waves. Although the distribution of beads in Figure 8a shows generally good agreement with the numerical simulations, obvious differences are still observed. These differences may be attributed to inconsistent bead sizes and variations in the elements of the sector transducer. Under the premise of neglecting the surface tension of water, the experimental results for the axial acoustic radiation force also constitute an approximation.

When the fluid flow velocity above the object increases while the fluid flow velocity below decreases, this results in a pressure difference between the fluid above and below the object. This pressure difference produces a lift force perpendicular to the direction of the fluid flow velocity, a phenomenon known as the Magnus effect. The resulting lift is also referred to as Magnus lift. As introduced in previous sections, the orbital angular momentum carried by the acoustic vortex causes the object to rotate, subsequently accelerating its radial movement speed and enhancing capture efficiency.

Also, the position of the separation point above the sphere is pushed further back within the sphere, creating an asymmetrical flow pattern of the fluid around it. When the surface of the ball is somewhat rough, it increases the lift of the fluid against the rotating bead. As the object rotates, the fluid exerts a lift force in a direction perpendicular to the flow velocity. The magnitude of the Magnus lift force is related to the speed of motion of the object, the speed of rotation, the density of the fluid, the size of the object, and the fluid’s viscosity. Therefore, calculating this force is not very straightforward, and many problems in fluid mechanics rely on experimental data to accurately estimate the Magnus lift force on an object in a given condition. The Jukowski theory of circulation is often used to calculate the lift force on a rotating object.

When objects are in motion and undergo rotation within a fluid medium, such as that induced by an acoustic vortex, the trajectory of those objects is altered. Consequently, it is imperative to conduct further extensive research to establish a robust theory of object manipulation utilizing acoustic vortices, ensuring their effective application in the lossless exploration of marine environments.

4. Conclusions

The complex environment of the seafloor hydrothermal zone makes it difficult to capture tiny objects in flowing seawater. Based on a comparison of two systems, the superiority of FAVs was verified by the clearly focused fields observed in the focal plane. In this paper, an FAV capture system equipped with a concave focused transducer array was utilized, employing focused acoustic energy and strengthened AGFs. During the experiment, polyethylene beads were selected as the controlled objects for simulation. The FAV could radially capture them in the vortex center and then move them to the collector using the axial thrust generated by the focused acoustic field. The favorable results demonstrate that the FAVs generated by the concave sector transducer array are experimentally applicable for more accurate object manipulation with improved flexibility. Additionally, FAVs can serve as an efficient means of capturing tiny marine objects, improving survival rates through non-contact and non-invasive methods and thereby enabling broader potential applications in marine exploration and ocean equipment.