Two-Dimensional Particle Assembly Based on the Synchronized Evolution of Centrosymmetric Off-Axis Acoustic Vortexes

Ning Ding; Gepu Guo; Juan Tu; Dong Zhang; Qingyu Ma

doi:10.1016/j.eng.2024.01.032

Engineering ›› 2025, Vol. 47 ›› Issue (4) :149 -161. DOI: 10.1016/j.eng.2024.01.032

Research Biomedical Engineering—Article

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Two-Dimensional Particle Assembly Based on the Synchronized Evolution of Centrosymmetric Off-Axis Acoustic Vortexes

Ning Ding ^a
, Gepu Guo ^a^,^c^,^*
, Juan Tu ^b
, Dong Zhang ^b
, Qingyu Ma ^a^,^c^,^*

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Abstract

Acoustic-vortex (AV) tweezers ensure stable particle trapping at a zero-pressure center, while particle assembly between two vortex cores is still prevented by the high-potential barrier. Although a one-dimensional low-pressure attractive path of particle assembly can be constructed by the interference between two independent cylindrical Bessel beams, it remains challenging to create two-dimensional (2D) neighboring vortexes using a source array in practical applications. In this paper, a three-step phase-reversal strategy of 2D particle assembly based on the synchronized evolution of a centrosymmetric array of M off-axis acoustic vortexes (OA-AVs) with a preset radial offset is proposed based on a ring array of planar sources. By introducing initial vortex phase differences of −2π/M and +2π/M to the vortex array, low-pressure patterns of an M-sided regular polygon and M-branched star are formed by connecting the vortex cores and the field center before and after the tangent state of adjacent OA-AVs. Center-oriented particle assembly is finally realized by a central AV constructed by coincident in-phase OA-AVs. The capability of particle manipulation in the lateral and radial directions is demonstrated by low-pressure patterns with acoustic radiation forces pointing to the field center during a synchronized central approach. The field evolution is certified by experimental field measurements for OA-AVs with different vortex numbers, initial vortex phase differences, and radial offsets using a ring array of 16 planar sources. The feasibility of particle assembly in two dimensions is also verified by the accurate manipulation of four particles using the low-pressure patterns of a four-sided polygon, a four-branched star, and a central AV in experiments. The three-step strategy paves a new way for 2D particle assembly based on the synchronized evolution of centrosymmetric OA-AVs using a simplified single-sided source array, exhibiting excellent potential for the precise navigation and manipulation of cells and particles in biomedical applications.

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Keywords

Centrosymmetric array of off-axis acoustic vortexes / Phase-reversal strategy / Initial phase difference / Particle assembly / Single-sided ring array

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Ning Ding, Gepu Guo, Juan Tu, Dong Zhang, Qingyu Ma. Two-Dimensional Particle Assembly Based on the Synchronized Evolution of Centrosymmetric Off-Axis Acoustic Vortexes. Engineering, 2025, 47(4): 149-161 DOI:10.1016/j.eng.2024.01.032

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1. Introduction

In 1986, Ashkin et al. [1] discovered that micron-sized particles and neutral atoms could be moved under the action of an optical radiation force. Since then, contactless optical tweezers have been developed to manipulate particles and cells with good selectivity and accuracy [2], [3], [4]. Although optical tweezers have been widely applied in biology, their practical application in the human body still suffers from the fundamental constraints of limited penetration depth and possible thermal effect [5], [6]. To meet the demand of biomedical applications, the concept of acoustic tweezers was proposed by Wu [7] in 1991. The acoustic manipulations of trapping [8], [9], [10], separating [11], [12], levitating [13], and mixing [14] particles or cells have been accomplished using the acoustic radiation force (ARF) of sound beams in the past decades. Thanks to their capability for the noninvasive manipulation of objects of different materials and sizes with good biocompatibility, acoustic tweezers hold great significance in the fields of material science and biomedical engineering. As typical acoustic tweezers for collective manipulation, the phase-controllable nodes of standing waves have been demonstrated to be able to trap objects in two [15], [16], [17] or three dimensions [18], [19], [20] and to perform particle sorting [21], [22] and patterning [23], [24], [25]. However, the movement and assembly of adjacent particles are still prevented by nodes and antinodes, and the manipulation flexibility is limited by the double-sided transducers or reflectors on both sides of objects. Meanwhile, for the selective trapping of multiple particles, more dexterous acoustic tweezers have been constructed using travelling waves, such as a single-beam or holographic framework [26], [27] launched from a single-sided transducer array.

An acoustic-vortex (AV) beam generated by specially designed phase modulation has helical wave fronts around its axis with a phase singularity at the center, carrying orbital angular momentum [28], [29]. The central zero-pressure surrounded by a high-pressure ring can serve as a potential well to trap [20], levitate [30], [31], move [16], [18], and rotate [32] objects. Moreover, to realize stable three-dimensional (3D) trapping, acoustic focusing has been applied to create pushing and pulling forces toward the focus along the axial direction. As a form of AV tweezers, the manipulation capability can be adjusted by regulating the acoustic field with the modulation of both amplitude and phase for each source element. To manipulate multiple targets individually and simultaneously, several technologies [26], [33], [34], [35], [36] have been developed to build on-axis or off-axis AVs, including a holographic framework using a single-sided array [26] or a double-sided arrangement of two opposed arrays [33], an anomalous reflection based on the multi-bit coding of acoustic metasurfaces [34], off-axis acoustic vortex (OA-AV) generated by two coaxial vortex beams [35], and self-navigated 3D AV tweezers based on the time-reversal method [36].

The positions of composite optical vortices formed by two interfering parallel light beams are determined by their relative phase, amplitude, or distance [37]. Cheng and Lü [38], [39] reported that the motion, creation, and annihilation of composite coherence vortices could be accomplished by varying the off-axis distance, the coherence parameter, and the propagation distance. Hence, the number, shape, and position of vortex potential wells can be adjusted by the interference of acoustic beams, which can be applied to move multiple trapped targets. In 2019, Gong and Baudoin [40] studied the synchronized evolution of two parallel Bessel beams and demonstrated that an attractive low-pressure path between the cores of two neighboring tangent in-phase AVs could be created by means of lateral destructive interference, making particle movement and assembly possible. However, the constructed one-dimensional (1D) low-pressure path for particle movement cannot be applied to two-dimensional (2D) cases with more AVs. Moreover, the ideal Bessel beam can only be constructed by the precise control of an infinite source array [41], which is still difficult to build accurately in practical applications [42], [43]. In addition, the synchronized approach of two parallel Bessel beams is influenced by source overlap and field blocking. Hence, tilted source arrays should be employed to generate a composite acoustic field via obliquely incident AV beams. In contrast, the intersection of two inclined beams varies with the incident angle, and the shape and position of the interference region should be studied in three dimensions. In 2021, by introducing additional phase delays to a traditional ring array of 16 planar transducers, directional OA-AV beams [44] with an approximate linear trajectory passing through a preassigned point were constructed to route acoustic packets along a predictable path. The successful realization of OA-AV beams based on a ring array laid a theoretical and experimental foundation for the construction of multiple AVs. Kotlyar et al. [45] studied the superposition of four identical parallel single-ringed Laguerre–Gaussian beams and concluded that the helical phase formed a series of high-intensity potential barriers outside the singularity in the center of the composite field, which made it difficult to break through the barriers and accomplish particle assembly in the process of gradually approaching.

In the current study, we propose a three-step phase-reversal strategy for particle assembly based on the synchronized evolution of centrosymmetric OA-AVs. By applying multiplexing technology [46] to a ring array of N planar transducers, a centrosymmetric array of M OA-AVs with a preset radial offset is constructed in the transverse plane. Then, initial vortex phase differences of −2π/M and +2π/M are introduced to the vortex array to form the low-pressure patterns of an M-sided polygon and M-branched star by paths connecting the vortex cores and the field center before and after the tangent states of adjacent OA-AVs. The capability of 2D particle assembly in the lateral and radial directions is demonstrated by low-pressure patterns with the corresponding ARFs pointing to the field center. The central accumulation of trapped particles is finally accomplished by coincident in-phase OA-AVs. The synchronized evolution of centrosymmetric OA-AVs based on the three-step phase-reversal strategy is demonstrated by experimental field measurements generated by a ring array of 16 planar sources, and the feasibility of particle assembly is demonstrated by the accurate manipulation of four particles using a low-pressure four-sided polygon, four-branched star, and central AV constructed by four OA-AVs with corresponding radial offsets. The proposed strategy provides a new way of 2D particle assembly based on the synchronized evolution of centrosymmetric OA-AVs formed by a simplified ring array; it may also be applied to 3D applications with a non-axisymmetric field distribution. Thus, the proposed strategy holds great potential for the precise navigation and manipulation of cells and particles in biomedical applications.

2. Principle and method

In the sketch map shown in Fig. 1(a), a single-sided ring array (radius R) of N planar transducers (center frequency f) with the spatial angle difference of Δφ = 2π/N is used as the source to construct a centrosymmetric array of multiple OA-AV beams in free space. For the nth source S_n, the center is located at (R, φ_n, 0) with the spatial angle of φ_n = (n − 1)Δφ for n = 1, 2, ..., N in cylindrical coordinates. To generate a traditional on-axis AV beam with a topological charge (TC) of l, the initial phase of S_n is set to ϕ_n = 2π(n − 1)l/N. Hence, the acoustic pressure at (r, φ, z) generated by the nth source, p_n, is as follows:

(1)

p_{n} (r, φ, z) = \int_{0}^{2 π} \int_{0}^{a} u_{0} \frac{i k_{0} ρ_{0} c_{0}}{2 π R_{n}} \exp (i k_{0} R_{n}) \exp (i ϕ_{n}) r_{n}^{'} d r_{n}^{'} d φ_{n}^{'}

where (r, φ, z) is the coordinate of the observation position in free space in cylindrical coordinates, and

(r_{n}^{'}, φ_{n}^{'}, 0)

denotes the cylindrical coordinate of the surface element dS_n in the nth source; i is the imaginary unit; a is the radius of the planar sources; u₀ is the surface particle velocity; k₀ = ω/c₀ is the wave number, in which ω is the angular frequency; and ρ₀ and c₀ are the density and sound speed in the medium, respectively.

R_{n} = {[{(R \cos φ_{n} + r_{n}^{'} \cos φ_{n}^{'} - r \cos φ)}^{2} + {(R \sin φ_{n} + r_{n}^{'} \sin φ_{n}^{'} - r \sin φ)}^{2} + z^{2}]}^{1 / 2}

is the distance between

(r_{n}^{'}, φ_{n}^{'}, 0)

and (r, φ, z). Therefore, the acoustic pressure of the traditional on-axis AV beam, p, can be obtained as follows:

(2)

p (r, φ, z) = \sum_{n = 1}^{N} \int_{0}^{2 π} \int_{0}^{a} u_{0} \frac{i k_{0} ρ_{0} c_{0}}{2 π R_{n}} \exp (i k_{0} R_{n}) \exp (i ϕ_{n}) r_{n}^{'} d r_{n}^{'} d φ_{n}^{'}

In the transverse plane at z = z₀, the vortex core P₀ of the on-axis AV is located at the field center (r = 0), and the distance between P₀ and S_n is simplified to

R_{0} = {(R^{2} + z_{0}^{2})}^{1 / 2}

Also, as illustrated in Fig. 1(a), the centrosymmetric array of M OA-AVs (l = 1, anti-clockwise phase rotation) in the transverse plane is positioned around the field center P₀. P_m is the mth OA-AV centered at (r_m, φ_m, z_m) with the radial offset r_m (distance between the mth vortex core and P₀) and the initial source phase φ_m = 2π(m − 1)/M for m = 1, 2, ..., M. To produce the mth OA-AV beam, an additional phase shift of the nth source determined by the distance variation ΔR_nm = R_nm − R₀ is introduced, where the distance between (r_m, φ_m, z₀) and S_n is

(3)

R_{nm} = {[{(r_{m} \cos φ_{m} - R \cos φ_{n})}^{2} + {(r_{m} \sin φ_{m} - R \sin φ_{n})}^{2} + z_{0}^{2}]}^{1 / 2}

Hence, the initial source phase of S_n is revised to

ϕ_{nm}^{'} = ϕ_{n} + k_{0} Δ R_{nm}

[44]. Furthermore, considering the same r_m for the centrosymmetric OA-AV array, the acoustic pressure of the composite field is determined as follows:

(4)

p (r, φ, z) = \sum_{n = 1}^{N} \int_{0}^{2 π} \int_{0}^{a} u_{0} \frac{i k_{0} ρ_{0} c_{0}}{2 π R_{n}} \exp (i k_{0} R_{n}) \sum_{m = 1}^{M} \exp [i (ϕ_{n} + k_{0} Δ R_{nm})] r_{n}^{'} d r_{n}^{'} d φ_{n}^{'}

Cross-sectional phase maps of centrosymmetric OA-AV arrays for M = 2 and 4 are illustrated in Fig. 1(b); as shown in the figure, the phase at the field center produced by the in-phase OA-AVs varies with M. A low-pressure line can be formed by the opposite phases between the cores of OA-AVs p₁ and p₂ for M = 2 [40]. Furthermore, the synchronized approach of two OA-AVs toward the field center with a decreasing r_m does not affect the phase pattern, while maintaining a low-pressure line between the vortex cores. Influenced by the spatial positions of the in-phase OA-AVs for M > 2, a phase singularity may be created at the field center by the circular phase evolution of 2π. Taking M = 4 as an example, the phases of 0, π/2, π, and −π/2 are produced by four OA-AVs at the field center with a total phase shift of 2π, and a central AV can be created by the phase spiral during the synchronized approach of the four OA-AVs. Therefore, a new central AV with l = 1 forms by the centrosymmetric array of the in-phase OA-AVs for M ≥ 3, and the particle motion from the vortex cores to the field center is still hindered by the surrounding high-pressure barrier. Thus, to prevent the formation of the central AV, the initial phases of the centrosymmetric OA-AVs should be adjusted to destroy the phase singularity at the field center. By introducing the initial vortex phase β_m to the mth OA-AV, the pressure is revised to

(5)

p (r, φ, z) = \sum_{n = 1}^{N} \int_{0}^{2 π} \int_{0}^{a} u_{0} \frac{i k_{0} ρ_{0} c_{0}}{2 π R_{n}} \exp (i k_{0} R_{n}) \sum_{m = 1}^{M} \exp [i (ϕ_{n} + k_{0} Δ R_{nm} + β_{m})] r_{n}^{'} d r_{n}^{'} d φ_{n}^{'}

For the centrosymmetric array of M OA-AVs with a fixed radial offset r_m, the vortex phase difference β = β_m₊₁ − β_m is a constant, with the positive and negative polarities representing the phase lead and lag between adjacent OA-AVs. The radial offset and the rotation direction of each OA-AV are determined by the additional phase shift and the polarity of the TC, respectively. The vortex phase difference of the centrosymmetric array determines the low-pressure distribution and further influences the effect of particle motion. Moreover, the unique low-pressure patterns of an M-sided regular polygon and M-branched star can be constructed by means of the appropriate selection of the vortex phase difference and radial offset.

As is well known, if the size of the particles is much smaller than the wavelength of the acoustic waves, the Gor’kov potential [47] of particles in an ideal fluid can be described by the following:

(6)

U = π a_{p}^{3} (f_{0} \frac{{|p|}^{2}}{3 ρ_{0} c_{0}^{2}} - f_{1} \frac{ρ_{0} {|v|}^{2}}{2})

where v is the particle velocity; a_p is the radius of the spherical particles;

f_{0} = 1 - (ρ_{0} c_{0}^{2}) / (ρ_{p} c_{p}^{2})

and

f_{1} = 2 (ρ_{p} - ρ_{0}) / (2 ρ_{p} + ρ_{0})

, with ρ_p and c_p respectively representing the density and sound speed in the particle material. In our previous study, Zhou et al. [32] calculated the acoustic gradient force (AGF) exerted on particles by the negative gradient of the Gor’kov potential and demonstrated that the radial AGF in the focal plane was determined by the acoustic pressure and the vortex radius of the AV. Millimeter-scale particles were successfully trapped in a rotational manner by the focused AV tweezers, with a trapping force of 10^–9 N.

Therefore, during the synchronized central approach of M OA-AVs, particles trapped at the vortex cores can move along low-pressure paths under the action of the exerted AGFs, forming a controllable 2D pattern of an M-sided polygon and realizing central accumulation through an M-branched star. However, assembling particles from the OA-AV cores to the field center still requires reasonable optimization of the radial offset and the vortex phase difference of the centrosymmetric array.

3. Numerical studies

It has been reported that the high-pressure annulus of an OA-AV deteriorates with an increase in the radial offset due to the greater distance difference between the vortex core and the sources, while the phase spiral remains basically unchanged [44]. To improve the manipulation accuracy of the AV tweezers, centrosymmetric arrays of multiple OA-AVs with a large radial offset were simulated based on a single-sided ring array of planar sources. Next, a synchronized approach toward the central axis was conducted by decreasing the radial offset for all the OA-AVs. The acoustic field formed by the OA-AVs was synthesized for M = 2, 3, and 4, and the corresponding characteristics of particle manipulation were analyzed by the exerted AGF in the transverse plane. An optimized annular source model with a continuous phase spiral of l = 1 was employed to improve the precision of field simulation, with the following parameters: R = 30 mm, ring width w = 5 mm, c₀ = 1500 m∙s⁻¹, and f = 500 kHz. The source pressure in water was calibrated to p₀ = 90 kPa with a surface particle velocity of u₀ = 60 mm∙s⁻¹. By defining r₀ as the vortex radius between the core and the first pressure peak for the OA-AVs, the radial offset ratio of δ = r_m/r₀ was defined to describe the relative distance between the field center and the vortex cores of adjacent OA-AVs.

With the parameters listed above, the vortex radius of the OA-AVs with l = 1 was about 6.0 mm in the transverse plane at z = 200 mm. Cross-sectional pressure maps of centrosymmetric arrays of in-phase OA-AVs (β = 0) with M = 2 and δ = 2, M = 3 and δ = 4, and M = 4 and δ = 5 were simulated and are presented in Fig. 2(a). As shown in the figure, M OA-AVs with obvious zero-pressure cores and high-pressure annuli are distributed evenly around the field center with radial offsets of 2r₀, 4r₀, and 5r₀ at the increased peak pressures of about 115.0, 117.9, and 121.3 kPa for M = 2, 3, and 4, respectively. Meanwhile, as shown in Fig. 2(b), the phase evolution of 2π around the vortex core in r < 12 mm (distance from the vortex center to the first pressure valley) demonstrates the formation of M independent OA-AVs. Moreover, the axial pressure profiles in the xoz plane shown in Fig. 2(c) indicate that each OA-AV beam can pass through the preassigned point P_m with a fixed radial offset r_m. Due to the spatial angle difference of 2π/3 for M = 3, as shown in Fig. 2(a-ii), only the high-pressure annulus of the OA-AV p₁ centered on the +x axis is observed in Fig. 2(c-ii). The vortex node V at z = 40 mm produced by the zero-pressure radiation between the main and side lobes of the annular transducer [48], as shown in Fig. 2(c), can be adjusted by changing the radiation pattern of the planar sources.

Then, the synchronized evolution of four in-phase OA-AVs was simulated to study the capability for particle assembly. It is known that a complex singularity distribution can be generated by noncoaxial interference between the outer pressure rings of tangent vortex beams [49], and the positions of these singular points can be calculated by p(r, φ, z) = 0. Cross-sectional distributions of the pressure and zero-contours of the real and imaginary parts of the complex pressure are plotted in Fig. 3 for different radial offset ratios. As shown in Fig. 3(a-i) for δ = 2.5, four OA-AVs with a central AV are distributed on the ±x and ±y axes, where the dotted circles and arrows indicate the positions and movement directions of the OA-AVs, respectively. Corresponding to the distributions of the zero-contour in Fig. 3(b-i), several phase singularities (marked with ×) are clearly displayed at the intersections of the real (blue) and imaginary (red) lines of complex pressure; the positions of the singularities coincide well with those of the zero-pressure points in Fig. 3(a-i). During the central approach of the OA-AVs, the low-pressure regions deviate from the corresponding vortex cores [40] due to the gradually enhanced interference. Furthermore, the phase at the field center does not vary with the radial offset ratio, and the central singularity always exists in the composite field. When the OA-AVs are tangential to each other at δ = 1.4, an approximate quadrilateral high-pressure barrier with a distinct phase spiral around the center (as indicated by the inset phase map) is formed by the central AV, as shown in Fig. 3(a-ii), which prevents trapped particles from moving toward the field center. Further decreasing the radial offset ratio enhances the continuity of the high-pressure barrier and finally results in the formation of an annular distribution caused by the central AV when δ = 0, demonstrating that the central accumulation of vortex-captured particles cannot be realized by the synchronized central approach of in-phase OA-AVs.

To destroy the generation condition of the central AV, initial vortex phase differences of β = –π/2 and π/2 were introduced to the centrosymmetric array in order to construct different low-pressure patterns (Fig. 4). The phases at the field center produced by the OA-AVs are the same for β = −π/2, as illustrated in Fig. 4(a-i). Hence, besides the OA-AVs distributed independently on the ±x and ±y axes, a high-pressure center with an approximate low-pressure annulus is created for δ = 2.5 in Fig. 4(a-ii). Moreover, four low-pressure lines connecting the vortex cores and the apexes of the low-pressure annulus along the ±x and ±y axes form special routes of particle assembly. As δ decreases, the vortex cores move toward the low-pressure annulus and produce an approximately regular quadrilateral pattern when the four OA-AVs are in the tangent state at δ = 1.4. Meanwhile, for the centrosymmetric array with a reversed vortex phase difference of β = π/2, the phases at the field center produced by adjacent OA-AVs are opposite, as shown in Fig. 4(b-i). Four high-pressure regions located on the ±x and ±y axes between the vortex cores and the field center are created, as shown in Fig. 4(b-ii) for δ = 2.5, which prevent particle movement toward the field center. However, it is worth noting that the phase maps in the insets show circular phase shifts of 0 and 4π for β = −π/2 and π/2, indicating the creation of central AVs with l = 0 and 2, respectively. Moreover, four low-pressure lines with obvious phase singularities can be created by the opposite phases of adjacent OA-AVs. By further decreasing δ to the tangent state, a four-branched star is constructed by the low-pressure lines between the vortex cores and the field center in Fig. 4(b-iii), which provides a feasible solution for the central accumulation of trapped particles. In addition, for the special case of δ = 0, cross-sectional pressure distributions of the centrosymmetric array with β = −π/2 and π/2 (Figs. 4(a-iv) and (b-iv)) are completely counteracted by four coincident OA-AVs with a helical phase shift of 2π, resulting in a null field of pressure.

Therefore, based on the central approach of four OA-AVs, a three-step phase-reversal strategy for synchronized evolution is developed to realize 2D particle assembly from vortex cores to the field center. A vortex phase difference of β = −π/2 is used first to construct a low-pressure pattern of a four-sided polygon (quadrilateral) with a large radial offset ratio to accomplish big-radius particle assembly. Then, at the tangent state of the OA-AVs during the synchronized central approach, a reversed vortex phase difference of β = π/2 is employed to construct a low-pressure pattern of a four-branched star in order to move the particles toward the field center. Finally, the stable central accumulation of trapped particles is accomplished by the central AV formed by the superposition of coincident OA-AVs at the field center with β = 0.

The curve in Fig. 5 shows the radial distance distribution of the pressure valley (potential well) along the x direction with respect to the radial offset ratio, where the blue and red lines represent the positions of the vortex core and the pressure valley, and the initial vortex phase differences of the OA-AV array are indicated by different colors. The figure shows that, during the synchronized central approach of four OA-AVs with β = −π/2, the pressure valley moves toward the field center with a distance smaller than the radial offset and stabilizes at the low-pressure side of the quadrilateral at x = 7.5 mm after small position fluctuations. Then, by introducing β = π/2 to the OA-AV array in the tangent state at δ = 1.4, the pressure valley located at x = 0 mm is prevented from changing with the further decreased radial offset, ensuring unidirectional particle motion toward the field center. Finally, a strengthened central AV is formed by the four coincident OA-AVs with β = 0 at δ = 0.

To demonstrate that the three-step phase-reversal strategy of the synchronized evolution of OA-AVs is capable of particle assembly, spherical polyethylene particles with radius a_p = λ/10, density ρ_p = 918 kg∙m⁻³, and sound speed c_p = 1900 m∙s⁻¹ were employed to calculate the AGF. Cross-sectional distributions of the Gor’kov potential for four OA-AVs with β = –π/2, π/2, and 0 were simulated, as plotted in Fig. 6(a), where the corresponding AGFs exerted on particles are indicated by white arrows. As shown in the figure, all the AGFs are pointing to the low-potential sides of the four-sided polygon from both inside and outside for β = −π/2, while they are pointing to the field center along the low-potential branches of the four-branched star for β = π/2.

Next, the corresponding AGF distributions along the x direction were simulated, as illustrated in Fig. 6(b), for different radial offset ratios, where the green arrow represents the trapping force (positive AGF) toward the field center. The maximum AGF marked by the black dotted line and the zero-AGF denoting the static equilibrium position are used to analyze the direction of particle movement. As δ decreases, the maximum AGFs are pointing to the field center at x = 15.9, 10.6, and 9.8 mm with amplitudes of 2.14 × 10⁻¹², 6.60 × 10⁻¹², and 1.28 × 10⁻¹¹ N for δ = 2.2, 1.7, and 1.4, respectively. Driven by the positive AGF, the trapped particles move toward the field center from x = 9.5 mm and stabilize on the low-potential sides at x = 7.5 mm. After the vortex phase reversal of β = π/2 at δ = 1.4, the positions of the maximum AGF and the static equilibrium continuously move inward. The maximum AGFs are 1.87 × 10⁻¹² and 6.94 × 10⁻¹² N at x = 6.8 mm for δ = 1.4 and 1.1, respectively. The particles captured at the vortex cores are pushed by the positive AGF (i.e., the trapping force) and move toward the static equilibrium position at x = 0 mm. For the special case of δ = 0, an initial phase difference of β = 0 is applied to form a central AV with a strengthened maximum AGF of 6.77 × 10⁻¹¹ N at x = 3.0 mm inside the peak-pressure annulus. In this way, 2D particle assembly based on the synchronized central approach of four OA-AVs can be accomplished by the three-step phase-reversal strategy; the specific process of field evolution is shown in Movie S1 in Appendix A.

To demonstrate the generality of the proposed three-step phase-reversal strategy for particle assembly, field evolutions of a centrosymmetric array of OA-AVs in the transverse plane at z = 200 mm were simulated for M = 3 and 5 with β = ±2π/3 and ±2π/5, respectively. The phase maps and cross-sectional pressure distributions at the tangent state are plotted in Figs. 7(a) and (b), respectively, where the critical radial offset ratios for the phase reversal are at δ = 1.15 and 1.7 for M = 3 and 5. The insets of phase distribution in the central region show l = 0 for β = −2π/3 and −2π/5, and l = −1 and 2 for M = 3 and 5 with β = 2π/3 and 2π/5, respectively. As expected, low-pressure 2D patterns of a three-/five-sided polygon and three-/five-branched star around the field center are constructed by the centrosymmetric array of three or five OA-AVs with opposite vortex phase differences, further validating the three-step strategy of field evolution.

The radial pressure distributions along the +x axis at five radial offset ratios during the central approach are then extracted to verify the feasibility of particle assembly; these are plotted in Figs. 7(c) and (d) for M = 3 and 5, respectively, where solid dots of different colors on the x-axis represent the vortex cores with the corresponding radial offset ratios. For M = 3, as shown in Fig. 7(c), the position of the pressure valley moves from x = 8.3 to 6.8 mm as δ decreases from 2.2 to 1.15. The low-pressure pattern evolves from a trilateral into a three-branched star before and after the phase reversal at δ = 1.15. At the same time, two pressure valleys are formed at x = 0 and 6.8 mm, as shown in Fig. 7(c). The outer pressure valley of the vortex core at x = 6.8 mm gradually moves forward with a decreasing δ and reaches x = 4.5 mm when δ = 0.8. The final stable particle trapping is realized by the central AV at δ = 0 with β = 0. Similarly, the pressure valley approaches the field center for five OA-AVs, as shown in Fig. 7(d). The pressure valleys are positioned at x = 13.6, 7.5, 0, 0, and 0 mm for the arrays with δ = 2.2 and β = −2π/5, δ = 1.7 and β = −2π/5, δ = 1.7 and β = 2π/5, δ = 1.2 and β = 2π/5, and δ = 0 and β = 0, respectively. The obvious advance of the pressure valley during the central approach demonstrates the possibility of successful particle assembly. Detailed field evolutions of centrosymmetric arrays of three and five OA-AVs are provided as Movies S2 and S3 in Appendix A.

Influenced by the initial vortex phase and the spatial position, the phase at the field center produced by the mth OA-AV is 2πm/M; hence, a new central AV with l = 1 can be constructed by M in-phase OA-AVs with β = 0, preventing the particles from moving from the vortex cores toward the field center. To destroy the high-pressure barrier of the central AV, β = −2π/M and 2π/M are introduced to the vortex array of M OA-AVs. The phase maps and cross-sectional phase distributions in the tangent state are presented in Fig. S1 in Appendix A for M OA-AVs where M = 3, 4, 5, and 6. For β = −2π/M, as shown in Figs. S1(a) and (b), the phase at the field center produced by the mth OA-AV is 2πm/M − 2πm/M = 0, and a central AV of l = 0 with a high-pressure center can be generated to trap particles on the low-pressure M-sided polygon. Meanwhile, for β = 2π/M, as shown in Figs. S1(c) and (d), the phase at the field center produced by the mth OA-AV is 2πm/M + 2πm/M = 4πm/M, resulting in a circular phase shift of 4π for the M OA-AVs. Hence, a new central AV with l = 2 in a large vortex radius is formed for M > 3, whereas a small central AV with l = −1 is produced by the phases of 0, 4π/3, and 8π/3 for M = 3. The central AV can be used to move trapped particles from the OA-AV cores to the field center through the low-pressure branches of the M-branched star. In this way, the high-pressure barrier of the central AV is destroyed by the reversed initial vortex phase differences of −2π/M and 2π/M for the centrosymmetric array of M OA-AVs before and after the tangent state during the synchronized central approach. By adjusting the vortex number and the radial offset ratio of the OA-AV array, the shape and size of the low-pressure pattern can be optimized in two dimensions; accurate particle movement toward the field center can be accomplished through the low-pressure sides and branches, finally fulfilling particle assembly at the field center by means of the strengthened central AV of coincident in-phase OA-AVs.

By introducing an additional phase shift to the optimized annular source model, a centrosymmetric array of arbitrary OA-AVs can be created in theory, which is applicable to the construction of the low-pressure patterns of an M-sided polygon and M-branched star. However, it is difficult to realize a continuous phase spiral for the annular source in the experiment. Hence, a discrete annular model should be employed based on the ring array of N transducers. With ring arrays of 16 (experimental transducer) and ∞ sources, cross-sectional pressure maps are simulated for four OA-AVs at different radial offsets, as shown in Fig. S2 in Appendix A. It is clear that the shapes and positions of the four OA-AVs are basically consistent with each other. However, the uniformity and continuity of the high-pressure annulus of the four OA-AVs constructed by the ring array of 16 sources obviously decrease with more significant interference. In addition, limited by the Nyquist theorem [50], the maximum number of OA-AVs generated by a ring array of N sources is M = N/2. Therefore, a centrosymmetric array of more OA-AVs with more consistent high-pressure annuli can be constructed by a ring array of more sources.

4. Experimental demonstration

A block diagram of the experimental system is shown in Fig. 1(a). Sixteen planar piston transducers with a = 2.5 mm and f = 500 kHz were fixed uniformly around a circumference (R = 30 mm) with a space angle of Δφ = 2π/16, manufactured via 3D printing using acrylonitrile butadiene styrene (ABS) resin. The phase difference of adjacent sources was set to 2π/16 to build an OA-AV beam with l = 1. The entire system was immersed in distilled water with c₀ = 1500 m∙s⁻¹. The particle velocity on the transducer surface was calibrated to u₀ = 60 mm∙s⁻¹ by means of a laser vibrometer (OFV-503, Polytec Company, Germany). Controlled by the computer through a WiFi module, 16 phased square waveforms at f = 500 kHz were sent out from a field-programmable gate array (FPGA) (EP4CE6E22C8N, Altera Corporation, USA). After band-pass filtering and power amplification, 16 sinusoidal signals with controllable amplitudes and phases were sent out to drive the ring array to create OA-AV beams. Controlled by a motion controller (Newport ESP301, Newport Corporation, USA), a needle hydrophone fixed on two-stage stepper motors (Newport M-ILS250, Newport Corporation) was used as the receiver to conduct 2D measurements in the xy plane. The acoustic pressure at each position was collected by a digital oscilloscope (Agilent DSO9064A, Agilent Corporation, USA) and saved in the computer for field reconstruction.

To construct the mth OA-AV beam with a TC of l, the phase-coded approach [44], [46] was applied to the nth source to radiate the acoustic signal

S_{n} = A \cos [ω t + 2 π (n - 1) l / N + k_{0} Δ R_{nm}]

, where A and k₀ΔR_nm are the pressure amplitude and the additional phase determined by the radial offset, respectively. Hence, for a composite field of M OA-AV beams with a vortex phase difference β, the acoustic signal of the nth source can be synthesized as follows:

(7)

S_{n} = \sum_{m = 1}^{M} A \cos [ω t + 2 π (n - 1) l / N + k_{0} Δ R_{nm} + β_{m}]

This can be simplified to

S_{n} = A_{n} \cos (ω t + Φ_{n})

, meaning that the acoustic field can be formed by accurate adjustments of amplitude (A_n) and phase (Φ_n) for each source.

With an experimental ring array of 16 planar transducers, composite acoustic fields of centrosymmetric arrays of in-phase OA-AVs with M = 2 and δ = 2, M = 3 and δ = 4, and M = 4 and δ = 5 were constructed in water and scanned at a motion step of 1.0 mm. The measured cross-sectional distributions of pressure and phase at z = 200 mm, as shown in Fig. 8, agree well with the corresponding maps in Fig. 2. The actual positions of the OA-AVs coincide well with the preassigned coordinate with r₀ = 6.0 mm. With an increase in M, the inconsistency and asymmetry of the high-pressure annuli are more obvious due to the strengthened energy dispersion and phase aliasing of the OA-AVs, as shown in Fig. 8(a), while the helical phase distributions are still well maintained, as shown in Fig. 8(b).

Next, the three-step phase-reversal strategy was applied to a centrosymmetric array of four OA-AVs in order to construct composite fields with different radial offset ratios. The corresponding cross-sections of pressure and phase at z = 200 mm are plotted in Fig. 9. The maps of the initial state with δ = 2.5 and β = −π/2 (Figs. 9(a-i) and (b-i)) show a low-pressure four-sided polygon with its apexes connecting the OA-AV cores on the x and y axes at a radial distance of 15 mm. By decreasing δ to 1.4, a low-pressure four-sided polygon (an approximate ring) with a high-pressure center is formed by the tangent OA-AVs, as shown in Fig. 9(a-ii), and no phase spiral can be observed inside the four-sided polygon in Fig. 9(b-ii). By reversing the vortex phase difference to β = π/2, a low-pressure four-branched star is created with the zero-pressure core located at the field center (Fig. 9(a-iii)), which is also demonstrated by the circular phase shift of 4π (central AV with l = 2) in Fig. 9(b-iii). In addition, for the coincident OA-AVs with δ = 0 and β = 0, the generation of a central AV with l = 1 is proved by the higher-pressure annulus with the corresponding phase spiral shown in Figs. 9(a-iv) and (b-iv). These experimental results demonstrate the successful field evolution of a centrosymmetric array of multiple OA-AVs.

To verify the feasibility of particle assembly from off-axis to on-axis based on the three-step phase-reversal strategy, an experimental system (Fig. 10(a)) was established to conduct 2D particle assembly based on the synchronized evolution of OA-AVs. The transducer array was placed on the bottom to radiate OA-AV beams upward with a calibrated source pressure of p₀ = 90 kPa. The distance from the transducer array to the water surface was adjusted to 200 mm. Polyethylene particles with a diameter of d = 0.6–1.2 mm floated on the water surface and were manipulated by the OA-AVs. Their motions were captured from above by a digital camera (MVC3000F, Microview, China) and saved in a computer for further analysis. The particle assembly based on the three-step phase-reversal strategy was performed under the three conditions of δ = 1.4 and β = −π/2, δ = 1.4 and β = π/2, and δ = 0 and β = 0 using four OA-AVs. The experimental videos are provided as Movies S4–S6 in Appendix A, and the corresponding pressure maps are provided as backgrounds for reference. Under the action of potential wells, four floating particles (d = 1.2 mm) are captured independently by four OA-AVs. For δ = 1.4 and β = −π/2, as shown in Movie S4, four particles are trapped by the low-pressure four-sided polygon with slight floats, realizing stable particle capture in Fig. 10(b-i). Then, by introducing β = π/2 to four OA-AVs, four particles move to the field center through the low-pressure four-branched star, and form a big cluster in Fig. 10(b-ii). In contrast, due to the weak rotational capacity of the large-scaled central AV with l = 2, the particle cluster basically does not rotate, as shown in Movie S5. Despite the obvious movements caused by environmental disturbances, the cluster is still restricted inside the low-pressure pattern by four surrounding high-pressure regions, and finally stabilizes at the lowest pressure center. Driven by the strengthened central AV with l = 1 constructed by β = 0 at δ = 0, the trapped particle cluster spins quickly at the field center, as shown in Movie S6 and Fig. 10(b-iii). These favorable results demonstrate the feasibility of the three-step phase-reversal strategy for 2D particle assembly based on the synchronized evolution of multiple OA-AVs.

5. Discussion

In our previous research [44] on constructing inclined OA-AV beams, the maximum radial offset was demonstrated to be determined by the parameters of the source array, the directivity of the transducers, the axial distance of the acoustic beams, and so on. For a centrosymmetric array of multiple OA-AVs with a larger radial offset, more serious field distortions in the pressure annulus and phase spiral are produced by the introduced pressure attenuation and phase lag, which are created by the distance variations between the vortex cores and the sources. Our research [44] showed that the maximum radial offset ratio of the OA-AVs generated by the experimental system is about δ = 5. Although OA-AVs can form outside this range, the distribution in both amplitude and phase are significantly distorted, as shown in Fig. S3 in Appendix A. Hence, in practical particle-assembly applications, the maximum radial offset ratio of the OA-AVs should be optimized by the radius of the ring array, the transmission distance, the vortex radius, and the array number.

In 2022, Wang et al. [51] presented a theoretical study on the generation of vortex states based on the coherent superposition of waves radiated from a discrete ring array of limited sources. Based on the Jacobi–Anger expansion (in terms of Bessel functions), the researchers provided an analytical conclusion of the least source number of

N_{\min} = 2 |l| + 1

for a vortex with a TC of l. Later, the topology and polarization of optical vortex fields radiated from atomic phased arrays were also investigated [52], and the least number for the generation of vector vortices with a given TC was demonstrated. In 2013, Yang et al. [53] derived the maximum TC that could be created by a circular array of N sources, namely,

|l_{\max}| = Fix [(N - 1) / 2]

, where Fix(x) rounds x toward 0, and the least source number of N_min = 3 is reached for an AV with l = 1. Thus, to construct a centrosymmetric array of M OA-AVs with l = 1, the least source number of N_min ≥ 2M should be selected in the current study for M > 1. As shown in Fig. S1, by introducing the vortex phase differences of β = −2π/M and 2π/M to a centrosymmetric array of M OA-AVs, a new central AV with l = 0 can be generated by an M-sided polygon, and that with l = −1 (M = 3) or 2 (M > 3) can be generated by an M-branched star. Thus, the three-step phase-reversal strategy is only applicable to a vortex array with M ≥ 3 for realizing successful particle assembly in two dimensions. However, for the special case of M = 2, the synchronized evolution shown in Movie S7 in Appendix A demonstrates that a 1D low-pressure path between the cores of two in-phase OA-AVs can be constructed to move particles toward the field center, accomplishing perfect central accumulation without reversing β. Moreover, the numerical and experimental cross-sectional maps of pressure and phase for M = 2, as shown in Fig. S4 in Appendix A, are similar to distributions of two approaching Bessel beams and provide a further validation of the conclusion drawn by Gong and Baudoin [40]. It should be noted that it is difficult to create ideal Bessel beams in experiments, especially when two source arrays are partially or completely overlapped. In contrast, a centrosymmetric array of multiple OA-AVs can be constructed perfectly using a single-sided ring array, as in the current study. Compared with 1D particle assembly using two approaching parallel Bessel beams, the synchronized evolution of the centrosymmetric array based on the three-step phase-reversal strategy can construct the controllable low-pressure patterns of an M-sided polygon and M-branched star, which are more universal and applicable to controllable 2D particle assembly. Furthermore, since the OA-AV beam can be used to flexibly route the trajectory passing through a preassigned position, the proposed strategy is also applicable to obstacle-avoidant particle assembly in three dimensions in a complicated biological environment.

It is known that, with an increase in the TC, the acoustic pressure of AV tweezers decreases accordingly with a larger vortex radius. For p₀ = 90 kPa, the maximum AGFs of four OA-AVs with l = 1, 2, 3, and 4 were calculated to be 1.40 × 10⁻¹¹, 4.50 × 10⁻¹², 1.76 × 10⁻¹², and 7.00 × 10⁻¹³ N at the peak pressures of 121.3, 118.0, 111.0, and 94.0 kPa, respectively, resulting in a reduced capability for particle trapping. Moreover, it was reported that the phase singularities of two off-axis high-order Laguerre–Gauss optical vortexes could be separated by the interference [54] and split into multiple first-order vortexes, which was demonstrated by a simulation and experimental results for vortexes of different orders. For this reason, field simulations were conducted for a centrosymmetric array of four OA-AVs with l = 2, 3, and 4; the corresponding cross-sectional maps of pressure and phase are plotted in Fig. S5 in Appendix A. The splitting of phase singularities poses difficulties for precise field regulation and particle manipulation. Therefore, only the centrosymmetric array of OA-AVs with l = 1 is feasible in the current study.

In previous studies, particles of different materials and sizes have been successfully manipulated by several AV tweezers with an ARF from 10⁻⁶ to 10⁻¹² N [15], [55], [56], [57], [58], [59], [60]. In the current study, for a source pressure of 90 kPa, cross-sectional pressure maps of the pressure and Gor’kov potential and the radial AGF distributions for a centrosymmetric array of four OA-AVs generated by continuous and 16-element annular transducer models were simulated and are illustrated in Fig. S6 in Appendix A. The maximum AGFs are about 1.01 × 10⁻¹¹ and 2.46 × 10⁻¹² N for the continuous and 16-element ring array, respectively. Although the AGF of the OA-AV array is obvious lower than that of focused ones, it is still strong enough to manipulate drug and bio-particles at the nanometer and micrometer scales. Moreover, based on the calculation of initial phases for each source, centrosymmetric OA-AVs can be constructed at the preset positions by means of multi-ring arrays, as shown in Fig. S7 in Appendix A. The low-pressure four-sided polygon and four-branched star shown in Fig. S8 in Appendix A, generated by a triple-ring array, show good agreement with those in Figs. 4(a-iii) and (b-iii) for the single-ring array. Because of the strengthened interference of more sources in multi-ring arrays, the manipulation capability of the OA-AVs can be improved by a more independent vortex distribution with a higher peak pressure, a smaller vortex radius, and a more uniform phase spiral. Therefore, the three-step phase-reversal strategy is applicable to multi-ring arrays and may enable more precise and powerful applications of particle assembly.

In addition, although centrosymmetric arrays of multiple OA-AVs were experimentally generated by discrete phase modulation of a ring array of 16 transducers, obvious distortions of pressure annuli and phase spirals—which are more serious for a larger M—were still observed. Moreover, the asymmetry and inhomogeneity of the cross-sectional maps resulted from the inaccuracy of the experimental system, including the inconsistency of the source array and the driving circuit, misalignment of the hydrophone and the acoustic field, and inaccurate control of the amplitude and phase. The experimental error can be improved by enhancing the phase resolution of the driving signals, improving the uniformity of the transducers, increasing the accuracy of the scanning measurement, and so on.

6. Conclusions

A three-step phase-reversal strategy for 2D particle assembly was proposed based on the synchronized central approach of a centrosymmetric array of multiple OA-AV beams using a simplified ring array of sources. Construction of the low-pressure patterns of an M-sided regular polygon, an M-branched star, and a strengthened central AV with vortex phase differences of −2π/M, +2π/M, and 0 was demonstrated through numerical simulations and experimental measurements. The feasibility of particle assembly was also verified by the accurate manipulation of four particles at the tangent and coincident states of four OA-AVs using a ring array of 16 planar transducers. It was demonstrated that, based on the shape and size of the low-pressure patterns determined by the radial offset of the OA-AVs, accurate particle assembly can be predesigned by analyses of the zero-pressure and zero-AGF. Superior to the 1D approach of two parallel Bessel beams, the 2D assembly and central accumulation of particles can be accomplished by the specific design of low-pressure patterns through the synchronized evolution of multiple OA-AVs. In addition, by setting radial offsets for OA-AV beams independently, the 2D low-pressure pattern can be expanded to a 3D pattern with a non-axisymmetric field distribution, indicating that the proposed strategy holds excellent potential for application in particle capture and assembly in complex biological environments.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was funded by the National Nature Science Foundation of China (11934009, 12174198, and 12227808), the Natural Science Foundation of Jiangsu Province, China (BE2022814), the Universal Technology for Primary and Secondary Schools, the National Research Institute for Teaching Materials, and the Qing Lan Project of Jiangsu Province, China.

The authors would like to thank Jianchun Cheng (Nanjing University, Nanjing, China) for his valuable comments on the construction of acoustic-vortex beams.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.01.032.

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