Direct 2D measurement of time-averaged forces and pressure amplitudes in acoustophoretic devices using optical trapping

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.

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.

Acoustofluidic medium exchange for preparation of electrocompetent bacteria using channel wall trapping

Comprehensive integration of process steps into a miniaturised version of synthetic biology workflows remains a crucial task in automating the design of biosystems. However, each of these process steps has specific demands with respect to the environmental conditions, including in particular the composition of the surrounding fluid, which makes integration cumbersome. As a case in point, transformation, i.e. reprogramming of bacteria by delivering exogenous genetic material (such as DNA) into the cytoplasm, is a key process in molecular engineering and modern biotechnology in general. Transformation is often performed by electroporation, i.e. creating pores in the membrane using electric shocks in a low conductivity environment. However, cell preparation for electroporation can be cumbersome as it requires the exchange of growth medium (high-conductivity) for low-conductivity medium, typically performed via multiple time-intensive centrifugation steps. To simplify and miniaturise this step, we developed an acoustofluidic device capable of trapping the bacterium Escherichia coli non-invasively for subsequent exchange of medium, which is challenging in acoustofluidic devices due to detrimental acoustic streaming effects. With an improved etching process, we were able to produce a thin wall between two microfluidic channels, which, upon excitation, can generate streaming fields that complement the acoustic radiation force and therefore can be utilised for trapping of bacteria. Our novel design robustly traps Escherichia coli at a flow rate of 10 μL min^-1 and has a cell recovery performance of 47 ± 3% after washing the trapped cells. To verify that the performance of the medium exchange device is sufficient, we tested the electrocompetence of the recovered cells in a standard transformation procedure and found a transformation efficiency of 8 × 10⁵ CFU per μg of plasmid DNA. Our device is a low-volume alternative to centrifugation-based methods and opens the door for miniaturisation of a plethora of microbiological and molecular engineering protocols.

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.

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.

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.

Acoustic force measurements on polymer-coated microbubbles in a microfluidic device

This work presents an acoustofluidic device for manipulating coated microbubbles, designed for the simultaneous use of optical and acoustical tweezers. A comprehensive characterization of the acoustic pressure in the device is presented, obtained by the synergic use of different techniques in the range of acoustic frequencies where visual observations showed aggregation of polymer-coated microbubbles. In absence of bubbles, the combined use of laser vibrometry and finite element modelling supported a non-invasive measurement of the acoustic pressure and an enhanced understanding of the system resonances. Calibrated holographic optical tweezers were used for direct measurements of the acoustic forces acting on an isolated microbubble, at low driving pressures, and to confirm the spatial distribution of the acoustic field. This allowed quantitative acoustic pressure measurements by particle tracking, using polystyrene beads, and an evaluation of the related uncertainties. This process facilitated the extension of tracking to microbubbles, which have a negative acoustophoretic contrast factor, allowing acoustic force measurements on bubbles at higher pressures than optical tweezers, highlighting four peaks in the acoustic response of the device. Results and methodologies are relevant to acoustofluidic applications requiring a precise characterization of the acoustic field and, in general, to biomedical applications with microbubbles or deformable particles.

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.

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

Acoustic particle manipulation in microfluidic channels is becoming a powerful tool in microfluidics to control micrometer sized objects in medical, chemical and biological applications. By creating a standing acoustic wave in the channel, the resulting pressure field can be employed to trap or sort particles. To design efficient and reproducible devices, it is important to characterize the pressure field throughout the volume of the microfluidic device. Here, we used an optically trapped particle as probe to measure the forces in all three dimensions. By moving the probe through the volume of the channel, we imaged spatial variations in the pressure field. In the direction of the standing wave this revealed a periodic energy landscape for 2 μm beads, resulting in an effective stiffness of 2.6 nN m(-1) for the acoustic trap. We found that multiple fabricated devices showed consistent pressure fields. Surprisingly, forces perpendicular to the direction of the standing wave reached values of up to 20% of the main-axis-values. To separate the direct acoustic force from secondary effects, we performed experiments with different bead sizes, which attributed some of the perpendicular forces to acoustic streaming. This method to image acoustically generated forces in 3D can be used to either minimize perpendicular forces or to employ them for specific applications in novel acoustofluidic designs.

Acoustic force mapping in a hybrid acoustic-optical micromanipulation device supporting high resolution optical imaging

Many applications in the life-sciences demand non-contact manipulation tools for forceful but nevertheless delicate handling of various types of sample. Moreover, the system should support high-resolution optical imaging. Here we present a hybrid acoustic/optical manipulation system which utilizes a transparent transducer, making it compatible with high-NA imaging in a microfluidic environment. The powerful acoustic trapping within a layered resonator, which is suitable for highly parallel particle handling, is complemented by the flexibility and selectivity of holographic optical tweezers, with the specimens being under high quality optical monitoring at all times. The dual acoustic/optical nature of the system lends itself to optically measure the exact acoustic force map, by means of direct force measurements on an optically trapped particle. For applications with (ultra-)high demand on the precision of the force measurements, the position of the objective used for the high-NA imaging may have significant influence on the acoustic force map in the probe chamber. We have characterized this influence experimentally and the findings were confirmed by model simulations. We show that it is possible to design the chamber and to choose the operating point in such a way as to avoid perturbations due to the objective lens. Moreover, we found that measuring the electrical impedance of the transducer provides an easy indicator for the acoustic resonances.

Imaging local acoustic pressure in microchannels

A method for determining the spatially resolved acoustic field inside a water-filled microchannel is presented. The acoustic field, both amplitude and phase, is determined by measuring the change of the index of refraction of the water due to local pressure using stroboscopic illumination. Pressure distributions are measured for the fundamental pressure resonance in the water and two higher harmonic modes. By combining measurement at a range of excitation frequencies, a frequency map of modes is made, from which the spectral line width and Q-factor of individual resonances can be obtained.

Viscous torque on spherical micro particles in two orthogonal acoustic standing wave fields

This paper reports the experimental results of the acoustic rotation of spherical micro particles because of two orthogonal standing waves. When the standing waves are excited at equal frequency but with a phase shift between two external voltage signals there is an acoustic streaming around the particles. This streaming is due to a time averaging of the acoustic wave field and produces a nonzero viscous torque on the particles, driving them to rotate. The work investigates the micro-particle rotation due to the viscous torque and predict the particle's steady state rotational velocity. The previous theoretical discussions [Nyborg, J. Acoust. Soc. Am. 85, 329-339 (1958); Lee and Wang, J. Acoust. Soc. Am. 85, 1081-1088 (1989)] of the viscous torque on a non-rotating sphere are expanded to allow free rotations. The analytical calculations provide a deeper understanding of the viscous torque and explain the experimental observations of rotating particles. A macroscopic experimental device is designed to provide the necessary boundary conditions for the viscous torque to rotate spherical particles. The experiments not only show good agreement with the analysis, but also demonstrate that the viscous torque due to acoustic streaming may dominate for the case of near-spherical particle dynamics.

Acoustic interaction forces between small particles in an ideal fluid

We present a theoretical expression for the acoustic interaction force between small spherical particles suspended in an ideal fluid exposed to an external acoustic wave. The acoustic interaction force is the part of the acoustic radiation force on one given particle involving the scattered waves from the other particles. The particles, either compressible liquid droplets or elastic microspheres, are considered to be much smaller than the acoustic wavelength. In this so-called Rayleigh limit, the acoustic interaction forces between the particles are well approximated by gradients of pair-interaction potentials with no restriction on the interparticle distance. The theory is applied to studies of the acoustic interaction force on a particle suspension in either standing or traveling plane waves. The results show aggregation regions along the wave propagation direction, while particles may attract or repel each other in the transverse direction. In addition, a mean-field approximation is developed to describe the acoustic interaction force in an emulsion of oil droplets in water.