Acoustic radiation force on a heated spherical particle in a fluid including scattering and microstreaming from a standing ultrasound wave.

Analytical expressions are derived for the time-averaged, quasisteady, acoustic radiation force on a heated, spherical, elastic, solid microparticle suspended in a fluid and located in an axisymmetric incident acoustic wave. The heating is assumed to be spherically symmetric, and the effects of particle vibrations, sound scattering, and acoustic microstreaming are included in the calculations of the acoustic radiation force. It is found that changes in the speed of sound of the fluid due to temperature gradients can significantly change the force on the particle, particularly through perturbations to the microstreaming pattern surrounding the particle. For some fluid-solid combinations, the effects of particle heating even reverse the direction of the force on the particle for a temperature increase at the particle surface as small as 1 K.

Acoustic radiation force on a spherical thermoviscous particle in a thermoviscous fluid including scattering and microstreaming.

We derive general analytical expressions for the time-averaged acoustic radiation force on a small spherical particle suspended in a fluid and located in an axisymmetric incident acoustic wave. We treat the cases of the particle being either an elastic solid or a fluid particle. The effects of particle vibrations, acoustic scattering, acoustic microstreaming, heat conduction, and temperature-dependent fluid viscosity are all included in the theory. Acoustic streaming inside the particle is also taken into account for the case of a fluid particle. No restrictions are placed on the widths of the viscous and thermal boundary layers relative to the particle radius. We compare the resulting acoustic radiation force with that obtained from previous theories in the literature, and we identify limits, where the theories agree, and specific cases of particle and fluid materials, where qualitative or significant quantitative deviations between the theories arise.

Theory and modeling of nonperturbative effects in thermoviscous acoustofluidics.

A theoretical model of thermal boundary layers and acoustic heating in microscale acoustofluidic devices is presented. Based on it, an iterative numerical model is developed that enables numerical simulation of nonlinear thermoviscous effects due to acoustic heating and thermal advection. Effective boundary conditions are derived and used to enable simulations in three dimensions. The theory shows how friction in the viscous boundary layers causes local heating of the acoustofluidic device. The resulting temperature field spawns thermoacoustic bulk streaming that dominates the traditional boundary-driven Rayleigh streaming at relatively high acoustic energy densities. The model enables simulations of microscale acoustofluidics with high acoustic energy densities and streaming velocities in a range beyond the reach of perturbation theory, and is relevant for design and fabrication of high-throughput acoustofluidic devices.

Transition from Boundary-Driven to Bulk-Driven Acoustic Streaming Due to Nonlinear Thermoviscous Effects at High Acoustic Energy Densities.

Acoustic streaming at high acoustic energy densities E_{ac} is studied in a microfluidic channel. It is demonstrated theoretically, numerically, and experimentally with good agreement that frictional heating can alter the streaming pattern qualitatively at high E_{ac} above 400 J/m^{3}. The study shows how as a function of increasing E_{ac} at fixed frequency, the traditional boundary-driven four streaming rolls created at a half-wave standing-wave resonance transition into two large streaming rolls. This nonlinear transition occurs because friction heats up the fluid resulting in a temperature gradient, which spawns an acoustic body force in the bulk that drives thermoacoustic streaming.

Constant-Power versus Constant-Voltage Actuation in Frequency Sweeps for Acoustofluidic Applications.

Supplying a piezoelectric transducer with constant voltage or constant power during a frequency sweep can lead to different results in the determination of the acoustofluidic resonance frequencies, which are observed when studying the acoustophoretic displacements and velocities of particles suspended in a liquid-filled microchannel. In this work, three cases are considered: (1) Constant input voltage into the power amplifier, (2) constant voltage across the piezoelectric transducer, and (3) constant average power dissipation in the transducer. For each case, the measured and the simulated responses are compared, and good agreement is obtained. It is shown that Case 1, the simplest and most frequently used approach, is largely affected by the impedance of the used amplifier and wiring, so it is therefore not suitable for a reproducible characterization of the intrinsic properties of the acoustofluidic device. Case 2 strongly favors resonances at frequencies yielding the lowest impedance of the piezoelectric transducer, so small details in the acoustic response at frequencies far from the transducer resonance can easily be missed. Case 3 provides the most reliable approach, revealing both the resonant frequency, where the power-efficiency is the highest, as well as other secondary resonances across the spectrum.

Numerical study of bulk acoustofluidic devices driven by thin-film transducers and whole-system resonance modes.

In bulk acoustofluidic devices, acoustic resonance modes for fluid and microparticle handling are traditionally excited by bulk piezoelectric (PZE) transducers. In this work, it is demonstrated by numerical simulations in three dimensions that integrated PZE thin-film transducers, constituting less than 0.1% of the bulk device, work equally well. The simulations are performed using a well-tested and experimentally validated numerical model. A water-filled straight channel embedded in a mm-sized bulk glass chip with a 1- µm-thick thin-film transducer made of Al0.6Sc0.4N is presented as a proof-of-concept example. The acoustic energy, radiation force, and microparticle focusing times are computed and shown to be comparable to those of a conventional bulk silicon-glass device actuated by a bulk lead-zirconate-titanate transducer. The ability of thin-film transducers to create the desired acoustofluidic effects in bulk acoustofluidic devices relies on three physical aspects: the in-plane-expansion of the thin-film transducer under the applied orthogonal electric field, the acoustic whole-system resonance of the device, and the high Q-factor of the elastic solid, constituting the bulk part of the device. Consequently, the thin-film device is remarkably insensitive to the Q-factor and resonance properties of the thin-film transducer.

Fast Microscale Acoustic Streaming Driven by a Temperature-Gradient-Induced Nondissipative Acoustic Body Force.

We study acoustic streaming in liquids driven by a nondissipative acoustic body force created by light-induced temperature gradients. This thermoacoustic streaming produces a velocity amplitude nearly 100 times higher than the boundary-driven Rayleigh streaming and the Rayleigh-Bénard convection at a temperature gradient of 10 K/mm in the channel. The Rayleigh streaming is altered by the acoustic body force at a temperature gradient of only 0.5 K/mm. The thermoacoustic streaming allows for modular flow control and enhanced heat transfer at the microscale. Our study provides the groundwork for studying microscale acoustic streaming coupled with temperature fields.

Theory of pressure acoustics with thermoviscous boundary layers and streaming in elastic cavities.

We present an effective thermoviscous theory of acoustofluidics including pressure acoustics, thermoviscous boundary layers, and streaming for fluids embedded in elastic cavities. By including thermal fields, we thus extend the effective viscous theory by Bach and Bruus [J. Acoust. Soc. Am. 144, 766 (2018)]. The acoustic temperature field and the thermoviscous boundary layers are incorporated analytically as effective boundary conditions and time-averaged body forces on the thermoacoustic bulk fields. Because it avoids resolving the thin boundary layers, the effective model allows for numerical simulation of both thermoviscous acoustic and time-averaged fields in three-dimensional models of acoustofluidic systems. We show how the acoustic streaming depends strongly on steady and oscillating thermal fields through the temperature dependency of the material parameters, in particular the viscosity and the compressibility, affecting both the boundary conditions and spawning additional body forces in the bulk. We also show how even small steady temperature gradients ( ∼1 K/mm) induce gradients in compressibility and density that may result in very high streaming velocities ( ∼1 mm/s) for moderate acoustic energy densities ( ∼100 J/m3).

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.

Theory and simulation of electroosmotic suppression of acoustic streaming.

Acoustic handling of nanoparticles in resonating acoustofluidic devices is often impeded by the presence of acoustic streaming. For micrometer-sized acoustic chambers, this acoustic streaming is typically driven by viscous shear in the thin acoustic boundary layer near the fluid-solid interface. Alternating current (ac) electroosmosis is another boundary-driven streaming phenomenon routinely used in microfluidic devices for the handling of particle suspensions in electrolytes. Here, we study how streaming can be suppressed by combining ultrasound acoustics and ac electroosmosis. Based on a theoretical analysis of the electrokinetic problem, we are able to compute numerically a form of the electrical potential at the fluid-solid interface, which is suitable for suppressing the typical acoustic streaming pattern associated with a standing acoustic half-wave. In the linear regime, we even derive an analytical expression for the electroosmotic slip velocity at the fluid-solid interface and use this as a guiding principle for developing models in the experimentally more relevant nonlinear regime that occurs at elevated driving voltages. We present simulation results for an acoustofluidic device, showing how implementing a suitable ac electroosmosis results in a suppression of the resulting electroacoustic streaming in the bulk of the device by 2 orders of magnitude.

Acoustophoresis in polymer-based microfluidic devices: Modeling and experimental validation.

A finite-element model is presented for numerical simulation in three dimensions of acoustophoresis of suspended microparticles in a microchannel embedded in a polymer chip and driven by an attached piezoelectric transducer at MHz frequencies. In accordance with the recently introduced principle of whole-system ultrasound resonances, an optimal resonance mode is identified that is related to an acoustic resonance of the combined transducer-chip-channel system and not to the conventional pressure half-wave resonance of the microchannel. The acoustophoretic action in the microchannel is of comparable quality and strength to conventional silicon-glass or pure glass devices. The numerical predictions are validated by acoustic focusing experiments on 5-µm-diameter polystyrene particles suspended inside a microchannel, which was milled into a polymethylmethacrylate chip. The system was driven anti-symmetrically by a piezoelectric transducer, driven by a 30-V peak-to-peak alternating voltage in the range from 0.5 to 2.5 MHz, leading to acoustic energy densities of 13 J/m3 and particle focusing times of 6.6 s.

Acoustic trapping based on surface displacement of resonance modes.

Acoustic trapping is a promising technique for aligning particles in two-dimensional arrays, as well as for dynamic manipulation of particles individually or in groups. The actuating principles used in current systems rely on either cavity modes in enclosures or complex arrangements for phase control. Therefore, available systems either require high power inputs and costly peripheral equipment or sacrifice flexibility. This work presents a different concept for acoustic trapping of particles and cells that enables dynamically defined trapping patterns inside a simple and inexpensive setup. Here, dynamic operation and dexterous trapping are realized through the use of a modified piezoelectric transducer in direct contact with the liquid sample. Physical modeling shows how the transducer induces an acoustic force potential where the conventional trapping in the axial direction is supplemented by surface displacement dependent lateral trapping. The lateral field is a horizontal array of pronounced potential minima with frequency-dependent locations. The resulting system enables dynamic arraying of levitated trapping sites at low power and can be manufactured at ultra-low cost, operated using low-cost electronics, and assembled in less than 5 min. We demonstrate dynamic patterning of particles and biological cells and exemplify potential uses of the technique for cell-based sample preparation and cell culture.

Toward optimal acoustophoretic microparticle manipulation by exploiting asymmetry.

The performance of a micro-acousto-fluidic device designed for microparticle trapping is simulated using a three-dimensional (3D) numerical model. It is demonstrated by numerical simulations that geometrically asymmetric architecture and actuation can increase the acoustic radiation forces in a liquid-filled cavity by almost 2 orders of magnitude when setting up a standing pressure half wave in a microfluidic chamber. Similarly, experiments with silicon-glass devices show a noticeable improvement in acoustophoresis of 20-µm silica beads in water when asymmetric devices are used. Microparticle acoustophoresis has an extensive array of applications in applied science fields ranging from life sciences to 3D printing. A more efficient and powerful particle manipulation system can boost the overall effectiveness of an acoustofluidic device. The numerical simulations are developed in the COMSOL Multiphysics® software package (COMSOL AB, Stockholm, Sweden). By monitoring the modes and magnitudes of simulated acoustophoretic fields in a relatively wide range of ultrasonic frequencies, a map of device performance is obtained. 3D resonant acoustophoretic fields are identified to quantify the improved performance of the chips with an asymmetric layout. Four different device designs are analyzed experimentally, and particle tracking experimental data qualitatively supports the numerical results.

Particle-size-dependent acoustophoretic motion and depletion of micro- and nano-particles at long timescales.

We present three-dimensional measurements of particle-size-dependent acoustophoretic motion of microparticles with diameters from 4.8 µm down to 0.5 µm suspended in either homogeneous or inhomogeneous fluids inside a glass-silicon microchannel and exposed to a standing ultrasound wave. To study the crossover from radiation force dominated to streaming dominated motion as the particle size is decreased, we extend previous studies to long timescales, where the particles smaller than the crossover size move over distances comparable to the channel width. We observe a particle-size-dependent particle depletion at late times for the particles smaller than the crossover size. The mechanisms behind this depletion in homogeneous fluids are rationalized by numerical simulations which take the Brownian motion into account. Experimentally, the particle trajectories in inhomogeneous fluids show focusing in the bulk of the microchannel at early times, even for the particles below the critical size, which clearly demonstrates the potential to manipulate submicrometer particles.

Suppression of Acoustic Streaming in Shape-Optimized Channels.

Acoustic streaming is an ubiquitous phenomenon resulting from time-averaged nonlinear dynamics in oscillating fluids. In this theoretical study, we show that acoustic streaming can be suppressed by two orders of magnitude in major regions of a fluid by optimizing the shape of its confining walls. Remarkably, the acoustic pressure is not suppressed in this shape-optimized cavity, and neither is the acoustic radiation force on suspended particles. This basic insight may lead to applications, such as acoustophoretic handling of nm-sized particles, which is otherwise impaired by the streaming.

Theory of acoustic trapping of microparticles in capillary tubes.

We present a semianalytical theory for the acoustic fields and particle-trapping forces in a viscous fluid inside a capillary tube with arbitrary cross section and ultrasound actuation at the walls. We find that the acoustic fields vary axially on a length scale proportional to the square root of the quality factor of the two-dimensional (2D) cross-section resonance mode. This axial variation is determined analytically based on the numerical solution to the eigenvalue problem in the 2D cross section. The analysis is developed in two steps: First, we generalize a recently published expression for the 2D standing-wave resonance modes in a rectangular cross section to arbitrary shapes, including the viscous boundary layer. Second, based on these 2D modes, we derive analytical expressions in three dimensions for the acoustic pressure, the acoustic radiation and trapping force, as well as the acoustic energy flux density. We validate the theory by comparison to three-dimensional numerical simulations.

Microparticle Acoustophoresis in Aluminum-Based Acoustofluidic Devices with PDMS Covers.

We present a numerical model for the recently introduced simple and inexpensive micromachined aluminum devices with a polydimethylsiloxane (PDMS) cover for microparticle acoustophoresis. We validate the model experimentally for a basic design, where a microchannel is milled into the surface of an aluminum substrate, sealed with a PDMS cover, and driven at MHz frequencies by a piezoelectric lead-zirconate-titanate (PZT) transducer. Both experimentally and numerically we find that the soft PDMS cover suppresses the Rayleigh streaming rolls in the bulk. However, due to the low transverse speed of sound in PDMS, such devices are prone to exhibit acoustic streaming vortices in the corners with a relatively large velocity. We predict numerically that in devices, where the microchannel is milled all the way through the aluminum substrate and sealed with a PDMS cover on both the top and bottom, the Rayleigh streaming is suppressed in the bulk thus enabling focusing of sub-micrometer-sized particles.

Bulk-driven acoustic streaming at resonance in closed microcavities.

Bulk-driven acoustic (Eckart) streaming is the steady flow resulting from the time-averaged acoustic energy flux density in the bulk of a viscous fluid. In simple cases, like the one-dimensional single standing-wave resonance, this energy flux is negligible, and therefore the bulk-driven streaming is often ignored relative to the boundary-driven (Rayleigh) streaming in the analysis of resonating acoustofluidic devices with length scales comparable to the acoustic wavelength. However, in closed acoustic microcavities with viscous dissipation, two overlapping resonances may be excited at the same frequency as a double mode. In contrast to single modes, the double modes can support a steady rotating acoustic energy flux density and thus a corresponding rotating bulk-driven acoustic streaming. We derive analytical solutions for the double modes in a rectangular-box-shaped cavity including the viscous boundary layers, and use them to map out possible rotating patterns of bulk-driven acoustic streaming. Remarkably, the rotating bulk-driven streaming may be excited by a nonrotating actuation, and we determine the optimal geometry that maximizes this excitation. In the optimal geometry, we finally simulate a horizontal 2×2, 4×4, and 6×6 streaming-roll pattern in a shallow square cavity. We find that the high-frequency 6×6 streaming-roll pattern is dominated by the bulk-driven streaming as opposed to the low-frequency 2×2 streaming pattern, which is dominated by the boundary-driven streaming.

Theory of pressure acoustics with viscous boundary layers and streaming in curved elastic cavities.

The acoustic fields and streaming in a confined fluid depend strongly on the viscous boundary layer forming near the wall. The width of this layer is typically much smaller than the bulk length scale set by the geometry or the acoustic wavelength, which makes direct numerical simulations challenging. Based on this separation in length scales, the classical theory of pressure acoustics is extended by deriving a boundary condition for the acoustic pressure that takes viscous boundary-layer effects fully into account. Using the same length-scale separation for the steady second-order streaming, and combining it with time-averaged short-range products of first-order fields, the usual limiting-velocity theory is replaced with an analytical slip-velocity condition on the long-range streaming field at the wall. The derived boundary conditions are valid for oscillating cavities of arbitrary shape and wall motion, as long as both the wall curvature and displacement amplitude are sufficiently small. Finally, the theory is validated by comparison with direct numerical simulation in two examples of two-dimensional water-filled cavities: The well-studied rectangular cavity with prescribed wall actuation, and a more generic elliptical cavity embedded in an externally actuated rectangular elastic glass block.

Acoustic Streaming and Its Suppression in Inhomogeneous Fluids.

We present a theoretical and experimental study of boundary-driven acoustic streaming in an inhomogeneous fluid with variations in density and compressibility. In a homogeneous fluid this streaming results from dissipation in the boundary layers (Rayleigh streaming). We show that in an inhomogeneous fluid, an additional nondissipative force density acts on the fluid to stabilize particular inhomogeneity configurations, which markedly alters and even suppresses the streaming flows. Our theoretical and numerical analysis of the phenomenon is supported by ultrasound experiments performed with inhomogeneous aqueous iodixanol solutions in a glass-silicon microchip.