Imaging the position-dependent 3D force on microbeads subjected to acoustic radiation forces and streaming

Acoustofluidics 24: theory and experimental measurements of acoustic interaction force

The motion of small objects in acoustophoresis depends on the acoustic radiation force and torque. These are nonlinear phenomena originating from wave scattering, and consist of primary and secondary components. The primary radiation force is the force acting on an object due to the incident field, in the absence of other objects. The secondary component, known as acoustic interaction force, accounts for the interaction among objects, and contributes to the clustering patterns of objects, as commonly observed in experiments. In this tutorial, the theory of acoustic interaction forces is presented using the force potential and partial-wave expansion approaches, and the distinguishing features of these forces such as rotational coupling and non-reciprocity are described. Theoretical results are compared to experimental measurements of interaction forces using a glass micro-capillary setup to explain the practical challenges. Finally, the phenomenon of clustering patterns induced by the close-range interaction of objects is demonstrated to point out the considerations about multiple collision and the predicted clustering patterns entirely due to the interaction force. Understanding the principles of acoustic interaction enables us to develop novel acoustofluidic applications beyond the typical processing of large populations of particles and with focus on the controlled manipulation of small clusters.

Influence of particle shape and material on the acoustic radiation force and microstreaming in a standing wave

In view of its influence on the acoustic radiation force, we investigate the microstreaming around a small solid elastic particle in an ultrasonic standing wave in dependence of its material properties and shape. The configuration is axisymmetric, making it accessible to numerical methods, such as the finite element method. The results reveal a transition from viscous scattering- to microstreaming-dominated acoustic radiation force that depends on the particle density. When a deviation of the particle shape from a sphere becomes smaller than the viscous boundary layer thickness, we show that the influence of the shape on the viscous contributions to the acoustic radiation force diminishes, allowing the use of theoretical models for a spherical particle. However, extreme asymmetric shape perturbations, such as crowns with sharp edges, can give rise to noticeable viscous contributions for a dense particle that is larger than the viscous boundary layer thickness. We also introduce a hybrid analytical model for the acoustic radiation force on a spherical particle that accounts for the microstreaming and particle compressibility and shows a good agreement with numerical simulations for an arbitrary particle size and density.

Measuring the effects of a pulsed excitation on the buildup of acoustic streaming and the acoustic radiation force utilizing an optical tweezer

Pulsed excitations of piezoelectric transducers affect during the buildup the force contributions from acoustic streaming (AS) and the acoustic radiation force (ARF) to the total force in a standing pressure wave differently. We find with an optical tweezer as measuring instrument that during the first 120 000 excitation periods and across different pulsing frequencies, the AS-induced displacement is on average less than 20% of its nonpulsed value for a duty cycle of 50%, whereas the ARF-induced displacement is around 50%. These findings show that a pulsed excitation can be a tool for reducing AS compared to the ARF.

Dynamic measurement of the acoustic streaming time constant utilizing an optical tweezer

The combination of a bulk acoustic wave device and an optical trap allows for studying the buildup time of the respective acoustic forces. In particular, we are interested in the time it takes to build up the acoustic radiation force and acoustic streaming. For that, we measure the trajectory of a spherical particle in an acoustic field over time. The shape of the trajectory is determined by the acoustic radiation force and by acoustic streaming, both acting on different time scales. For that, we utilize the high temporal resolution (Δt=0.8μs) of an optical trapping setup. With our experimental parameters the acoustic radiation force on the particle and the acoustic streaming field theoretically have characteristic buildup times of 1.4μs and 1.44ms, respectively. By choosing a resonance mode and a measurement position where the acoustic radiation force and acoustic streaming induced viscous drag force act in orthogonal directions, we can measure the evolution of these effects separately. Our results show that the particle is accelerated nearly instantaneously by the acoustic radiation force to a constant velocity, whereas the acceleration phase to a constant velocity by the acoustic streaming field takes significantly longer. We find that the acceleration to a constant velocity induced by streaming takes in average about 17 500 excitation periods (≈4.4ms) longer to develop than the one induced by the acoustic radiation force. This duration is about four times larger than the so-called momentum diffusion time which is used to estimate the streaming buildup. In addition, this rather large difference in time can explain why a pulsed acoustic excitation can indeed prevent acoustic streaming as it has been shown in some previous experiments.

Rapid measurement of the local pressure amplitude in microchannel acoustophoresis using motile cells

Acoustic microfluidics (or acoustofluidics) provides a non-contact and label-free means to manipulate and interrogate bioparticles. Owing to their biocompatibility and precision, acoustofluidic approaches have enabled innovations in various areas of biomedical research. Future breakthroughs will rely on the translation of these techniques from academic labs to clinical and industrial settings. Here, accurate characterization and standardization of device performance are crucial. Versatile, rapid, and widely accessible performance quantification is needed. We propose a field quantification method using motile Chlamydomonas reinhardtii algae cells. We previously reported qualitative mapping of acoustic fields using living microswimmers as active probes. In the present study, we extend our approach to achieve the challenging quantitative in situ measurement of the acoustic energy density. C. reinhardtii cells continuously swim in an imposed force field and dynamically redistribute as the field changes. This behavior allows accurate and complete, real-time performance monitoring, which can be easily applied and adopted within the acoustofluidics and broader microfluidics research communities. Additionally, the approach relies only on standard bright-field microscopy to assess the field under numerous conditions within minutes. We benchmark the method against conventional passive-particle tracking, achieving agreement within 1% for field strengths from 0 to 70 J m^-3 (from 0 to ∼0.8 MPa).

Optical manipulation: advances for biophotonics in the 21st century

SIGNIFICANCE

Optical trapping is a technique capable of applying minute forces that has been applied to studies spanning single molecules up to microorganisms.

AIM

The goal of this perspective is to highlight some of the main advances in the last decade in this field that are pertinent for a biomedical audience.

APPROACH

First, the direct determination of forces in optical tweezers and the combination of optical and acoustic traps, which allows studies across different length scales, are discussed. Then, a review of the progress made in the direct trapping of both single-molecules, and even single-viruses, and single cells with optical forces is outlined. Lastly, future directions for this methodology in biophotonics are discussed.

RESULTS

In the 21st century, optical manipulation has expanded its unique capabilities, enabling not only a more detailed study of single molecules and single cells but also of more complex living systems, giving us further insights into important biological activities.

CONCLUSIONS

Optical forces have played a large role in the biomedical landscape leading to exceptional new biological breakthroughs. The continuous advances in the world of optical trapping will certainly lead to further exploitation, including exciting in-vivo experiments.

Numerical study of the coupling layer between transducer and chip in acoustofluidic devices

By numerical simulation in two and three dimensions, the coupling layer between the transducer and microfluidic chip in ultrasound acoustofluidic devices is studied. The model includes the transducer with electrodes, microfluidic chip with a liquid-filled microchannel, and coupling layer between the transducer and chip. Two commonly used coupling materials, solid epoxy glue and viscous glycerol, as well as two commonly used device types, glass capillary tubes and silicon-glass chips, are considered. It is studied how acoustic resonances in ideal devices without a coupling layer are either sustained or attenuated as a coupling layer of increasing thickness is inserted. A simple criterion based on the phase of the acoustic wave for whether a given zero-layer resonance is sustained or attenuated by the addition of a coupling layer is established. Finally, by controlling the thickness and the material, it is shown that the coupling layer can be used as a design component for optimal and robust acoustofluidic resonances.

Controlled orientation and sustained rotation of biological samples in a sono-optical microfluidic device

In cell biology, recently developed technologies for studying suspended cell clusters, such as organoids or cancer spheroids, hold great promise relative to traditional 2D cell cultures. There is, however, growing awareness that sample confinement, such as fixation on a surface or embedding in a gel, has substantial impact on cell clusters. This creates a need for contact-less tools for 3D manipulation and inspection. This work addresses this demand by presenting a reconfigurable, hybrid sono-optical system for contact-free 3D manipulation and imaging, which is suitable for biological samples up to a few hundreds of micrometers in liquid suspension. In our sono-optical device, three independently addressable MHz transducers, an optically transparent top-transducer for levitation and two side-transducers, provide ultrasound excitation from three orthogonal directions. Steerable holographic optical tweezers give us an additional means of manipulation of the acoustically trapped specimen with high spatial resolution. We demonstrate how to control the reorientation or the spinning of complex samples, for instance for 3D visual inspection or for volumetric reconstruction. Whether continuous rotation or transient reorientation takes place depends on the strength of the acoustic radiation torque, arising from pressure gradients, compared to the acoustic viscous torque, arising from the shear forces at the viscous boundary layer around the particle. Based on numerical simulations and experimental insights, we develop a strategy to achieve a desired alignment or continuous rotation around a chosen axis, by tuning the relative strengths of the transducers and thus adjusting the relative contributions of viscous and radiation torques. The approach is widely applicable, as we discuss in several generic examples, with limitations dictated by size and shape asymmetry of the samples.

Motile cells as probes for characterizing acoustofluidic devices

Acoustic microfluidics has emerged as a versatile solution for particle manipulation in medicine and biology. However, current technologies are largely confined to specialized research laboratories. The translation of acoustofluidics from research to clinical and industrial settings requires improved consistency and repeatability across different platforms. Performance comparisons will require straightforward experimental assessment tools that are not yet available. We introduce a method for characterizing acoustofluidic devices in real-time by exploiting the capacity of swimming microorganisms to respond to changes in their environment. The unicellular alga, Chlamydomonas reinhardtii, is used as an active probe to visualize the evolving acoustic pressure field within microfluidic channels and chambers. In contrast to more familiar mammalian cells, C. reinhardtii are simple to prepare and maintain, and exhibit a relatively uniform size distribution that more closely resembles calibration particles; however, unlike passive particles, these motile cells naturally fill complex chamber geometries and redistribute when the acoustic field changes or is turned off. In this way, C. reinhardtii cells offer greater flexibility than conventional polymer or glass calibration beads for in situ determination of device operating characteristics. To illustrate the technique, the varying spatial density and distribution of swimming cells are correlated to the acoustic potential to automatically locate device resonances within a specified frequency range. Peaks in the correlation coefficient of successive images not only identify the resonant frequencies for various geometries, but the peak shape can be related to the relative strength of the resonances. Qualitative mapping of the acoustic field strength with increasing voltage amplitude is also shown. Thus, we demonstrate that dynamically responsive C. reinhardtii enable real-time measurement and continuous monitoring of acoustofluidic device performance.

Acoustic Compressibility of Caenorhabditis elegans

The acoustic compressibility of Caenorhabditis elegans is a necessary parameter for further understanding the underlying physics of acoustic manipulation techniques of this widely used model organism in biological sciences. In this work, numerical simulations were combined with experimental trajectory velocimetry of L1 C. elegans larvae to estimate the acoustic compressibility of C. elegans. A method based on bulk acoustic wave acoustophoresis was used for trajectory velocimetry experiments in a microfluidic channel. The model-based data analysis took into account the different sizes and shapes of L1 C. elegans larvae (255 ± 26 μm in length and 15 ± 2 μm in diameter). Moreover, the top and bottom walls of the microfluidic channel were considered in the hydrodynamic drag coefficient calculations, for both the C. elegans and the calibration particles. The hydrodynamic interaction between the specimen and the channel walls was further minimized by acoustically levitating the C. elegans and the particles to the middle of the measurement channel. Our data suggest an acoustic compressibility κ_Ce of 430 TPa^-1 with an uncertainty range of ±20 TPa^-1 for C. elegans, a much lower value than what was previously reported for adult C. elegans using static methods. Our estimated compressibility is consistent with the relative volume fraction of lipids and proteins that would mainly make up for the body of C. elegans. This work is a departing point for practical engineering and design criteria for integrated acoustofluidic devices for biological applications.

Acoustic streaming induced by two orthogonal ultrasound standing waves in a microfluidic channel

A mathematical model is derived for acoustic streaming in a microfluidic channel confined between a solid wall and a rigid reflector. Acoustic streaming is produced by two orthogonal ultrasound standing waves of the same frequency that are created by two pairs of counter-propagating leaky surface waves induced in the solid wall. The magnitudes and phases of the standing waves are assumed to be different. Full analytical solutions are found for the equations of acoustic streaming. The obtained solutions are used in numerical simulations to reveal the structure of the acoustic streaming. It is shown that the interaction of two standing waves leads to the appearance of a cross term in the equations of acoustic streaming. If the phase lag between the standing waves is nonzero, the cross term brings about circular vortices with rotation axes perpendicular to the solid wall of the channel. The vortices make fluid particles rotate and move alternately up and down between the solid wall and the reflector. The obtained results are of immediate interest for acoustomicrofluidic applications such as the ultrasonic micromixing of fluids and the manipulation of microparticles.

Retrieving acoustic energy densities and local pressure amplitudes in microfluidics by holographic time-lapse imaging

The development of techniques able to characterize and map the pressure field is crucial for the widespread use of acoustofluidic devices in biotechnology and lab-on-a-chip platforms. In fact, acoustofluidic devices are powerful tools for driving precise manipulation of microparticles and cells in microfluidics in non-contact modality. Here, we report a full and accurate characterization of the movement of particles subjected to acoustophoresis in a microfluidic environment by holographic imaging. The particle displacement along the direction of the ultrasound wave propagation, coinciding with the optical axis, is observed and investigated. Two resonance frequencies are explored, varying for each the amplitude of the applied signal. The trajectories of individual tracers, accomplished by holographic measurements, are fitted with the theoretical model thus allowing the retrieval of the acoustic energy densities and pressure amplitudes through full holographic analysis. The absence of prior calibration, being independent of the object shape and the possibility of implementing automatic analysis make the use of holography very appealing for applications in devices for biotechnologies.

Acoustofluidic particle dynamics: Beyond the Rayleigh limit

In this work a numerical model to calculate the trajectories of multiple acoustically and hydrodynamically interacting spherical particles is presented. The acoustic forces are calculated by solving the fully coupled three-dimensional scattering problem using finite element software. The method is not restricted to single re-scattering events, mono- and dipole radiation, and long wavelengths with respect to the particle diameter, thus expanding current models. High frequency surface acoustic waves have been used in the one cell per well technology to focus individual cells in a two-dimensional wave-field. Sometimes the cells started forming clumps and it was not possible to focus on individual cells. Due to a lack of existing theory, this could not be fully investigated. Here, the authors use the full dynamic simulations to identify limiting factors of the one-cell-per-well technology. At first, the authors demonstrate good agreement of the numerical model with analytical results in the Rayleigh limiting case. A frequency dependent stability exchange between the pressure and velocity was then demonstrated. The numerical formulation presented in this work is relatively general and can be used for a multitude of different high frequency applications. It is a powerful tool in the analysis of microscale acoustofluidic devices and processes.

Digital holography as 3D tracking tool for assessing acoustophoretic particle manipulation

The integration of digital holography (DH) imaging and the acoustic manipulation of micro-particles in a microfluidic environment is investigated. The ability of DH to provide efficient 3D tracking of particles inside a microfluidic channel is exploited to measure the position of multiple objects moving under the effect of stationary ultrasound pressure fields. The axial displacement provides a direct verification of the numerically computed positions of the standing wave's node, while the particles' transversal movement highlights the presence of nodes in the planar direction. Moreover, DH is used to follow the aggregation dynamics of trapped spheres in such nodes by using aggregation rate metrics.

Acoustic streaming in a microfluidic channel with a reflector: Case of a standing wave generated by two counterpropagating leaky surface waves

A theory is developed for the modeling of acoustic streaming in a microfluidic channel confined between an elastic solid wall and a rigid reflector. A situation is studied where the acoustic streaming is produced by two leaky surface waves that propagate towards each other in the solid wall and thus form a combined standing wave in the fluid. Full analytical solutions are found for both the linear acoustic field and the field of the acoustic streaming. A dispersion equation is derived that allows one to calculate the wave speed in the system under study. The obtained solutions are used to consider particular numerical examples and to reveal the structure of the acoustic streaming. It is shown that two systems of vortices are established along the boundaries of the microfluidic channel.

Huygens-Fresnel Acoustic Interference and the Development of Robust Time-Averaged Patterns from Traveling Surface Acoustic Waves

Periodic pattern generation using time-averaged acoustic forces conventionally requires the intersection of counterpropagating wave fields, where suspended micro-objects in a microfluidic system collect along force potential minimizing nodal or antinodal lines. Whereas this effect typically requires either multiple transducer elements or whole channel resonance, we report the generation of scalable periodic patterning positions without either of these conditions. A single propagating surface acoustic wave interacts with the proximal channel wall to produce a knife-edge effect according to the Huygens-Fresnel principle, where these cylindrically propagating waves interfere with classical wave fronts emanating from the substrate. We simulate these conditions and describe a model that accurately predicts the lateral spacing of these positions in a robust and novel approach to acoustic patterning.

Multibody dynamics in acoustophoresis

Determining the trajectories of multiple acoustically and hydrodynamically interacting as well as colliding particles is one of the challenges in numerical acoustophoresis. Although the acoustic forces between multiple small spherical particles can be obtained analytically, previous research did not address the particle-particle contacts in a rigorous way. This article extends existing methods by presenting an algorithm on displacement level which models the hard contacts using set-valued force laws, hence allowing for the first time the computation of a first approximation of complete trajectories of multiple hydrodynamically and acoustically interacting particles. This work uses a semi-analytical method to determine the acoustic forces, which is accurate up to the dipole contributions of the multipole expansion. The hydrodynamic interactions are modeled using the resistance and mobility functions of the Stokes' flow. In previous experimental work particles have been reported to interact acoustically, ultimately forming stacked lines near the pressure nodes of a standing wave. This phenomenon is examined experimentally and numerically, the simulation shows good agreement with the experimental results. To demonstrate the capabilities of the method, the rotation of a particle clump in two orthogonal waves is simulated. The presented method allows further insight in self-assembly applications and acoustic particle manipulation.