Three dimensional acoustic tweezers with vortex streaming

The study addresses the challenge of achieving robust three-dimensional acoustic trapping with a single beam and minimal hardware complexity. Conventional acoustic tweezers rely either on standing-wave radiation forces, which require multiple transducers and lack selectivity, or on traveling-wave vortices, which provide selective in-plane trapping but struggle to generate strong axial forces without strict constraints on particle properties. Additionally, nonlinear acoustic streaming often destabilizes radiation-force-based traps at practical amplitudes. The authors propose a hybrid approach that purposefully harnesses Eckart streaming generated by a focused acoustic vortex to provide upward drag for levitation along the axial (z) direction, while radiation forces maintain lateral (x–y) confinement. The purpose is to realize a simple, single-transducer 3D acoustic tweezer that can stably manipulate a wider range of particle sizes, shapes, and materials.

Prior work on acoustic tweezers splits into standing-wave and traveling-wave modalities. Standing-wave systems form node/antinode lattices using counter-propagating waves and excel at parallel manipulation but lack selectivity and typically require multi-transducer setups surrounding the trapping region (e.g., SSAW and bulk acoustophoretic devices). Traveling-wave tweezers engineer single-beam phase fronts to create pressure nodes, including beams imparting orbital angular momentum (acoustic vortices) via holographic transducers and metasurfaces. While focused vortices increase selectivity, they inherently produce an axial node line rather than a point, limiting 3D trapping. A notable 3D single-beam trap using a dipolar mode on-axis was demonstrated by Baresch et al., but it imposes tight constraints on particle properties and yields much weaker axial than lateral forces. Moreover, nonlinear streaming often appears at the amplitudes needed for heavier particles, undermining radiation-force-based traps. Separate lines of work exploit acoustic streaming (e.g., with oscillating bubbles or structures) for indirect manipulation, but these approaches often provide limited spatial resolution and are typically 2D or require complex control. This context motivates integrating controlled streaming with vortex radiation forces for practical 3D trapping.

Device design and lens synthesis: A single circular PZT transducer (38 mm diameter, 4.1 mm thick, 500 kHz resonance) was paired with a passive PDMS holographic lens to generate a focused acoustic vortex. The required 2D phase at the source plane is the superposition of a Fresnel focusing phase and a simple vortex phase (following BFGS-optimized prescriptions from prior work), assuming uniform amplitude due to low PDMS attenuation and water-like impedance. The PDMS thickness map is computed from the phase delay using the wavelength contrast between PDMS and water, yielding a 3D lens profile with Fermat–Archimedes spiral-like contours. Fabrication used standard PDMS molding (Ecoflex 00-30, 1:1 by weight, degassed, cured at 120 °C for 1 h) from a stereolithographically printed negative mold, then bonded to the PZT. Acoustic field simulation: Frequency-domain finite element simulations (Code Aster via Salome-MECA) solved the weak-form Helmholtz equation in a 3D waveguide. The bottom plane (z=0) imposed the analytically computed source-phase boundary, side walls were hard, and the top boundary was anechoic. The model produced intensity and phase distributions across x–y focal planes and x–z sections, revealing a ring-shaped intensity profile with central spiral phase indicative of orbital angular momentum and a central axial node line suitable for in-plane trapping. Acoustic streaming simulation: Streaming was modeled in OpenFOAM in three stages: (i) time-domain compressible acoustic propagation (sonicLiquidFoam) to obtain acoustic velocity and density, (ii) time-averaging of the nonlinear term to compute an effective body force F = −ρ0 ⟨u·∇u⟩ driving streaming, and (iii) incompressible steady-state flow solution (icoFoam) with the added body force. Water dynamic viscosity was set to 0.0044 Pa·s (estimated from the particle’s free-fall via Stokes’ law). The incident acoustic pressure was ~35 kPa. The simulation predicted converging upward streaming near the focal region with a surrounding toroidal vortex and reduced velocity along the axis, yielding a localized axial velocity gradient for levitation. Acoustic field measurements: Experiments were conducted in a 40-gallon water tank. A 500 kHz Gaussian-modulated pulse from a function generator (RIGOL DG4102) was amplified (ENI 2100L) to drive the PZT+PDMS lens. A hydrophone (ONDA HNR-0500) on a 3D stage scanned x–y and x–z planes; signals were digitized (AlazarTech ATS 9440, 125 MS/s), averaged over 1024 repetitions, and Fourier-transformed to extract amplitude and phase at 500 kHz. Measured fields agreed with simulations, showing a strong focused ring intensity and spiral central phase and a z-axis node line. Streaming field measurements (PIV): Polyamide seeding particles (density 1.03 g/cm³, mean diameter 60 µm) were dispersed in deionized water. A 532 nm laser line generated a fan-shaped light sheet aligned with x–z. Video captured in slow motion was processed in MATLAB PIVlab using an ensemble correlation approach over 2000 frames. The measured streaming map showed upward flow converging to a maximum near the focal region and diverging above, with streamlines forming a vortex around the axial node. Trapping demonstrations and force/stiffness estimation: A 1.5 mm diameter cellulose acetate sphere (density 1.3 g/cm³, bulk modulus 4.8 GPa) was trapped at z ≈ 42 mm with RMS 42 V drive. The levitation force was computed as weight minus buoyancy (≈ 5.2 µN). Semi-analytical radiation force estimates used angular spectrum analysis to obtain fields and a Gor’kov-potential-based model to compute lateral and axial components; the axial radiation force near the trap was ~0.02 µN and downward, indicating levitation is provided entirely by streaming drag. Trap stiffness was estimated from force gradients: lateral k ≈ 0.52 mN/m; axial k ≈ 0.064 mN/m (from a smooth fit to noisy PIV-derived velocity gradients). Additional tests demonstrated trapping of cylindrical particles (1.3 mm diameter × 1.3 mm height, ~6 mg total weight) with an estimated levitation force of 41.8 µN, stable trapping under up to ~21° tilt, and 3D path following by mechanically scanning the source.

- A single-transducer, passive-lens focused acoustic vortex can combine radiation forces for lateral confinement with Eckart streaming drag for axial levitation, yielding a true 3D acoustic trap in fluids. - The experimentally inferred levitation force via streaming is about three orders of magnitude larger than previously reported single-beam 3D radiation-force traps at comparable frequency. - Demonstration: A 1.5 mm cellulose acetate sphere (ρ ≈ 1.3 g/cm³) was levitated at z ≈ 42 mm using 42 V RMS at 500 kHz; levitation force ≈ 5.2 µN. The axial radiation force at the trap position was ≈ 0.02 µN and downward, confirming levitation is dominated by streaming. - Streaming velocity along z measured by PIV agrees in trend with simulations, peaking near the focal region. The particle’s free-fall speed in water (≈ 83 mm/s) closely matches the streaming velocity magnitude at the levitation height. - Trap stiffnesses: lateral k ≈ 0.52 mN/m; axial k ≈ 0.064 mN/m. - Trapping of cylindrical particles (1.3 mm diameter × 1.3 mm height) was achieved; estimated levitation force ≈ 41.8 µN. The trap remained stable under up to ~21° device tilt. - Simulations and measurements confirm a focused ring intensity and central spiral phase (vortex) with a localized streaming vortex that creates an axial velocity gradient for levitation while maintaining in-plane trapping.

The proposed hybrid approach directly addresses the difficulty of generating strong axial forces in single-beam vortex tweezers. By exploiting the naturally arising Eckart streaming near a focused ultrasonic focal spot, the device provides robust upward drag for levitation while using radiation forces for lateral confinement. This eliminates the need to finely tailor particle resonant modes (e.g., dipolar on-axis modes) and reduces sensitivity to particle material and shape, enabling manipulation of heavier and larger particles than radiation-force-only designs. The substantial increase in available levitation force (orders of magnitude larger) expands the operational regime and simplifies hardware, as the system requires only a single transducer and a passive lens. The observed agreement between simulated and measured acoustic and streaming fields supports the mechanism: a converging upward flow with a surrounding vortex concentrates drag near the focal region to balance gravity. These results are particularly relevant for applications where streaming is unavoidable (e.g., higher frequencies for microscale objects), turning a usual limitation into a functional advantage. The ability to scan the trap in 3D demonstrates practical controllability for contactless manipulation tasks.

This work introduces and validates a single-beam, hybrid 3D acoustic tweezer that integrates focused acoustic vortex radiation forces for lateral trapping with controlled streaming drag for axial levitation. The device, implemented with a simple PDMS holographic lens on a single PZT transducer at 500 kHz, delivers levitation forces about three orders of magnitude larger than prior single-beam 3D radiation traps at similar frequency and stably manipulates millimeter-scale particles irrespective of shape. Extensive simulations and measurements confirm both the acoustic and streaming field structures and their roles in trapping. Future directions include improving axial stiffness and resolution, extending operation to smaller particles via higher frequencies while leveraging streaming, developing more accurate coupled nonlinear-acoustic–fluid–structure models for predictive force calibration, and exploring pulsed or duty-cycled excitation to tune streaming forces. The approach opens avenues for versatile, low-complexity manipulation in fluidic and biomedical contexts where controlled flow around trapped objects is advantageous.

- Axial trapping stiffness is lower than radial stiffness due to inertia sustaining streaming velocities, a common limitation for single-sided 3D tweezers. - The trapped object size must remain smaller than the acoustic/streaming vortex; large scatterers can distort the flow and degrade trapping. - For very small particles, streaming may dominate dynamics and affect in-plane stability; higher operating frequencies are suggested to improve control. - The trap relies on balancing upward streaming drag against gravity and is primarily designed for upward orientation (limited performance when inverted). - Nonlinear streaming leads to a nonlinear relationship between applied acoustic amplitude and levitation force; precise characterization requires complex coupling of nonlinear acoustics, fluid mechanics, and fluid–structure interaction. In practice, adapting to various particles currently requires empirical amplitude tuning; alternative tuning (e.g., pulsed signals with varying duty cycles) may help.