Radiation Torque Research Articles

Numerical calculation of acoustic radiation force and torque on non-spherical particles in Bessel beams

Manipulation of biological cells using acoustic radiation force has drawn a lot of attention in recent years. The force and torque acting on cells are usually estimated from analytical and semi-analytical solutions derived for simple shapes, such as spheres and ellipsoids, typically in an axisymmetric configuration. Since biological cells come in various shapes and sizes and they may have an arbitrary orientation in a microfluidic channel, there is a need for a more versatile and robust numerical model for evaluating the acoustic radiation force and torque. Motivated by this, a three-dimensional boundary element model is developed for calculating radiation force and torque on particles of arbitrary shapes and sizes subjected to arbitrary acoustic waves. The first order acoustic field is solved by using the boundary element method. The second-order, time-averaged tractions are then obtained from the first order field. Subsequently, the resultant radiation force and torque are calculated by integrating the tractions over the surface of a fictitious sphere that encapsulates the particle. The force and torque on non-spherical particles subjected to acoustic Bessel beams are obtained using this numerical model. The effects of the beam cone angle and particle orientation on the radiation force and torque are investigated.

Acoustic radiation and viscous torque for micromanipulation controlled rotation of particles in fluid cavities

Our investigations at ETH Zurich aimed at the theoretical analysis of the acoustic torque and its experimental realization of a controlled rotation of spherical and non-spherical particles by ultrasound. Ultrasonic manipulation of particles provides a contactless handling method for particles suspended in a fluid by acoustic streaming and radiation forces. In addition to the translation of particles in all three spatial directions, particles like functional beads, cells, clumps of cells, fibers, etc., can be rotated. Various methods for the rotation of non-spherical particles were developed with the acoustic radiation torque. The necessary varying pressure field, where the orientation of the nodal pressure lines was controlled by two orthogonal standing waves, was achieved by a modulation of one single parameter over time, e.g., amplitude, phase, and frequency. Stable particle and fiber rotations up to 40 rpm were reached. The rotation can be performed continuously or in a stepwise fashion. Moreover, the rotation of spherical objects was realized by the viscous torque. This torque is formed by acoustic streaming, due to two orthogonal standing waves shifted in phase at the same excitation frequency and amplitude. Angular rotations up to 1200 rpm for spherical 35.5 μm copolymer particles were reached.

Acoustic forces acting on particles and fluids in microscale acoustofluidics

Acoustofluidics relying on acoustic forces to handle fluids and particles in microfluidic systems has emerged as a useful tool for characterizing, focusing, separating, and sorting cells based on their acousto-mechanical properties. Here, we present recent advances in the theoretical understanding of acoustic forces on particles and fluids. In particular, we address the effects of thermoviscous boundary layers on acoustic scattering off sub-micron particles or droplets. Re-examining the far-field method of calculating acoustic radiation forces and torques, we show that exact non-perturbative expressions can be derived regardless of boundary layer dissipation. The necessary condition for moving the surface of integration from the particle surface to the far field, is the time-periodicity of the system rather than negligible boundary-layer dissipation. In the long-wavelength limit, this approach leads to particularly simple expressions for the force and torque acting on a particle in a thermoviscous fluid. ...

Radiation stress, momentum, angular momentum, and power relationships for axisymmetric objects in acoustic vortex and coaxial wavefields

Starting with P. Westervelt’s formulation of the radiation stress tensor and G. Maidanik’s application to angular momentum in the 1950s, some recent theorems concerning acoustic radiation forces, torques, angular momentum, and power flow will be illustrated. Applications include acoustic vortex wavefields, Bessel beams, and other invariant beams. While derivations are simplified for objects surrounded by inviscid fluids [L. Zhang and P. L. Marston, Phys. Rev. E. 84, 035601 (2011); L. Zhang and P. L. Marston, Phys. Rev. E. 84, 065601 (2011); L. Zhang and P. L. Marston, J. Acoust. Soc. Am. 131, EL329–EL335 (2012)], for spheres in slightly viscous liquids limiting approximations are known at long wavelengths that recover results derivable by other approaches [L. Zhang and P. L. Marston, J. Acoust. Soc. Am. 136, 2917–2921 (2014); P. L. Marston, POMA 19, 045005 (2013)]. Theorems obtained are helpful for relating negative radiation forces to the asymmetry of the scattering pattern for spheres in Bessel beams. The relation with extinction theorems is noted [L. Zhang and P. L. Marston, Bio. Opt. Express 4, 1610–1617 (2013); (E) 4, 2988 (2013)] as well as momentum and angular momentum radiated by sources. [Work supported in part by ONR.]

Vortex beams and radiation torque for kidney stone management

Our team previously developed an instrument to reposition kidney stones with acoustic radiation force. In a clinical trial, the technology was used to transcutaneously facilitate passage of small stones and to relieve pain by dislodging obstructing large stones. Acoustic trapping and manipulation of kidney stones in water has recently been investigated using both single element and sector arrays in the range of 0.3–1.5 MHz. Experimental holographic reconstruction of the transducer surface velocity confirmed the proper operation of each transducer. Human stones approximately 5 mm, as well as glass and aluminum beads, were placed on a flat tissue phantom in a water bath. During exposure, stones were drawn to the beam axis, and then controllably translated along the surface in any direction transverse to the beam. The phase between sector elements could be used to control the vortex size, as well as rate and direction of rotation of the trapped object. The trapping effect was disrupted at increased transducer output, possibly by generation of acoustic streaming. In conclusion, a method was tested for transverse acoustic trapping of kidney stones with vortex beams. [This work was supported by RBBR 14-02-00426, NIH NIDDK DK43881, DK104854, and DK092197, and NSBRI through NASA NCC 9-58.]

Extended optical theorem for scalar monochromatic acoustical beams of arbitrary wavefront in cylindrical coordinates

One of the fundamental theorems in (optical, acoustical, quantum, gravitational) wave scattering is the optical theorem for plane waves, which relates the extinction cross-section to the forward scattering complex amplitude function. In this analysis, the optical theorem is extended for the case of 3D-beams of arbitrary character in a cylindrical coordinates system for any angle of incidence and any scattering angle. Generalized analytical expressions for the extinction, absorption, scattering cross-sections and efficiency factors are derived in the framework of the scalar resonance scattering theory for an object of arbitrary shape. The analysis reveals the presence of an interference scattering cross-section term, which describes interference between the diffracted or specularly reflected inelastic (Franz) waves with the resonance elastic waves. Moreover, an alternate expression for the extinction cross-section, which relates the resonance cross-section with the scattering cross-section for an impenetrable object, is obtained, suggesting an improved method for particle characterization. Cross-section expressions are also derived for known acoustical wavefronts centered on the object, defined as the on-axis case. The extended optical theorem in cylindrical coordinates can be applied to evaluate the extinction efficiency from any object of arbitrary geometry placed on or off the axis of the incident beam. Applications in acoustics, optics, and quantum mechanics should benefit from this analysis in the context of wave scattering theory and other phenomena closely connected to it, such as the multiple scattering by many particles, as well as the radiation force and torque.

Acoustophoresis of disk-shaped microparticles: A numerical and experimental study of acoustic radiation forces and torques.

Disk-shaped microparticles experience an acoustic radiation force and torque in an ultrasonic standing wave. Hence, they are translated by the acoustic field, an effect called acoustophoresis, and rotated. The torque effect is also known from the "Rayleigh disk" which is described in literature for sound intensity measurements. In this paper, inviscid numerical simulations of acoustic radiation forces and torques for disks with radius ≪ wavelength in water are developed in good agreement with former analytical solutions, and the dependence on disk geometry, density, and orientation is discussed. Experiments with alumina disks (diameter 7.5 μm), suspended in an aqueous liquid in a silicon microchannel, confirm the theoretical results qualitatively at the microscale and ultrasonic frequencies around 2 MHz. These results can potentially be applied for the synthesis of disk-reinforced composite materials. The insights are also relevant for the acoustic handling of various disk-shaped particles, such as red blood cells.

Acoustic Interaction Forces and Torques Acting on Suspended Spheres in an Ideal Fluid.

In this paper, the acoustic interaction forces and torques exerted by an arbitrary time-harmonic wave on a set of N objects suspended in an inviscid fluid are theoretically analyzed. We utilize the partial-wave expansion method with translational addition theorem and re-expansion of multipole series to solve the related multiple scattering problem. We show that the acoustic interaction force and torque can be obtained using the farfield radiation force and torque formulas. To exemplify the method, we calculate the interaction forces exerted by an external traveling and standing plane wave on an arrangement of two and three olive-oil droplets in water. The droplets' radii are comparable to the wavelength (i.e., Mie scattering regime). The results show that the acoustic interaction forces present an oscillatory spatial distribution which follows the pattern formed by interference between the external and rescattered waves. In addition, acoustic interaction torques arise on the absorbing droplets whenever a nonsymmetric wavefront is formed by the external and rescattered waves' interference.

Induced radiation force and torque on a viscoelastic particle in an ideal fluid

The interaction of an acoustic wave with a suspended particle may produce radiation force and torque on the particle through linear and angular momentum transfer. Theoretical analysis of the radiation force exerted on a rigid, compressible fluid, elastic solid, and layered spherical particle is abundantly found in the literature. Nevertheless, less attention has been devoted to cases involving a homogeneous viscoelastic particle, even though viscoelastic materials are ubiquitous. Here, the radiation force and torque exerted on this kind of particle is studied in detail using the fractional Kelvin-Voigt viscoelastic model. Analytical expressions are obtained in the monopole-dipole approximation to a small particle in the long-wave limit. Considering a traveling plane wave interacting with a high-density polyethylene (HDPE) particle, we found that the axial radiation force is negative, i.e., the force and the wave propagation are in opposite directions. For a first-order Bessel vortex beam, a full 3D tractor beam develops on the HDPE particle placed in the beam's axis. Furthermore, negative axial radiation torque also appears on the particle, i.e., the torque and the beam's angular momentum are contrariwise. In addition, possible implications of this method to the advancement of acoustophoretic techniques will be outlined.

Acoustics of finite Bessel tractor beams

Surprises and counterintuitive effects occurring in physical phenomena rank high on the list of things that make science discoveries a source of perpetual excitement. Some examples illustrated here include, (i) the effects of attracting (i.e., pulling) a spherical particle placed in the field of a Bessel acoustical beam of progressive waves back towards the radiator’s surface [Mitri, “Single Bessel tractor-beam tweezers,” Wave Motion 51, no. 6, pp. 986–993, 2014], (ii) suppressing some of the sphere’s resonances [Mitri, “Near-field acoustic resonance scattering of a finite Bessel beam by an elastic sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 61, no. 4, pp. 696–704, 2014], (iii) and the emergence of a “negative radiation torque,” meaning that an incoming high-order Bessel vortex beam with a positive topological charge (known also as the order of the beam) would rotate a viscoelastic sphere in the opposite sense of the beam handedness [Silva et al., “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett., 97, art. no. 54003, 2012]. Results and numerical predictions illustrate the analysis and extensions to other geometries, such as rigid oblate and prolate spheroids [Mitri, “Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid,” http://arxiv.org/abs/1505.06754v3, ibid. "Acoustic radiation force on oblate and prolate spheroids in Bessel beams,” Wave Motion, 57, pp. 231–238, 2015] are also mentioned.

Influence of the external torques in the angle between the spin axis and the Sun direction for spin stabilized satellite

The goal of this paper is the study of the influence of the environmental torques in the angle between the spin axis and the Sun direction (solar aspect angle) for spin stabilized satellite. The theory uses a cylindrical satellite in an illumined orbit, considering the gravity gradient, aerodynamic, solar radiation, residual magnetic and eddy current torques. The mathematic model for each torque is shown. The dynamic equations are represented in a reference system fixed in the satellite and described by spin velocity and the right ascension and declination angles of the spin axis. An analytical solution for the spin velocity and the attitude angles is used to study the behavior of the solar aspect angle. The theory is applied for the real data of the Brazilian Satellite of Data Collection - SCD1 and SCD2. Two approaches are presented. The results agree with the real satellite behavior for specific time simulation. Then the theory has consistency and can be applied to predict the behavior of the solar aspect angle.

Magnitude of Solar Radiation Torque in the Transition Region from the Umbra to the Dark Shadow of the Earth

The analysis of solar radiation pressure force and its influence on the motion of artificial satellites has been developed by researchers. Accurate models to describe the influence of the Earth's shadow on the torque and force due to solar radiation pressure have been presented. In this work the solar radiation torque (SRT) and its influence on the attitude of an artificial satellite are taken into account by the introduction of the Earth's shadow function in the equations of motion. This function assumes a unitary value when the satellite is in the fully illuminated region of its orbit, and the value zero for the full shade region. The main objective of this study is to analyze the magnitude of SRT using the equations described by quaternions during a 35 day period and to compare the results with the satellite transition through the shadow region and the time interval in this region. The duration and transition through the shadow region were obtained using the software "Shadow Conditions of Earth Satellites". The formulation is applied to the Brazilian Data Collection Satellites SCD1 and SCD2, and the torque model is presented in terms of the satellite attitude quaternion, distance of the satellite to the Sun, orbital elements, right ascension and declination of the Sun.

Computational study of radiation torque on arbitrary shaped particles with MLFMA.

The surface integral equation (SIE) method is used for the computational study of radiation torque on arbitrarily shaped homogeneous particles. The Multilevel Fast Multipole Algorithm (MLFMA) is employed to reduce memory requirements and improve the capability of SIE. The resultant matrix equations are solved iteratively to obtain equivalent electric and magnetic currents. Then, radiation torque is computed using the vector flux of the pseudotensor over a spherical surface tightly enclosing the particle. We use, therefore, the analytical electromagnetic field expression for incident waves in the near region, instead of the far-field approximation. This avoids the error which may be caused when describing the incident beam. The numerical results of three kinds of non-spherical particles are presented to illustrate the validity and capability of the developed method. It is shown that our method can be applied to predict, in the rigorous sense, the torque from a beam of any shape on a particle of complex configuration with a size parameter as large as 650. The radiation torques on large ellipsoids are exemplified to show the performance of the method and to study the influence that different aspect ratios have on the results. Then, the code is used for the calculation of radiation torque on objects of complex shape including a biconcave cell-like particle and a motor with a non-smooth surface.

Generalization of the optical theorem for monochromatic electromagnetic beams of arbitrary wavefront in cylindrical coordinates

The optical theorem constitutes of the fundamental theorems in optical, acoustical, quantum, and gravitational wave scattering, which relates the extinction cross-section to the forward scattering complex amplitude function of plane waves. In this analysis, a generalized formalism is presented for beams of arbitrary character in cylindrical coordinates without restriction to the plane wave case of the angles of incidence and scattering. Based on the partial-wave series expansion method of cylindrical multipole, analytical expressions for the extinction, absorption, scattering cross-sections and efficiency factors are derived for an object of arbitrary shape. An “interference scattering” term arises in the cross-section (or efficiency), which describes the mutual interference between the diffracted or specularly reflected waves. Examples for plane waves and 2D scalar quasi-Gaussian focused beams are also considered, which illustrate the theory. The generalized optical theorem in cylindrical coordinates can be applied to evaluate the extinction efficiency from any object of arbitrary geometry placed on or off the axis of the incident beam. Applications in the context of wave scattering theory by a single particle or multiple particles would benefit from the results of the present study, in addition to other phenomena such as the radiation force and torque.

Numerical Calculation of Acoustic Radiation Force and Torque Acting on Rigid Non-spherical Particles

Numerical Calculation of Acoustic Radiation Force and Torque Acting on Rigid Non-spherical Particles

Radiation torque, streaming, and viscous absorption due to orthogonal waves in slightly viscous fluids

Torque generated by orthogonal waves in slightly viscous fluids is analyzed to explore its connection with the near-boundary viscous absorption and streaming [Zhang and Marston, J. Acoust. Soc. Am. 131, 2917–2921 (2014)]. The analysis is based on a viscous correction and beam superposition of vortex fields. The analysis reports an approximation for the torque on a compressible small solid sphere and the viscous power absorption [Eqs. (15) and (21) in Zhang and Marston]. The torque expression in the dense sphere limit recovers prior expressions using near-boundary flow and streaming analyses: [Busse and Wang, J. Acoust. Soc. Am. 69, 1634–1638 (1981)] and [Lee and Wang, J. Acoust. Soc. Am. 85, 1081–1088 (1989)]. Our analysis extends prior results to give the torque on small compressible solid spheres of finite mass. The torque does not depend explicitly on the compressibility of the sphere, but does depend on the density of the sphere. Our analysis also gives the proportional relationship between the torque and the near-boundary viscous absorption. The results are generalized to cases of standing orthogonal waves and acoustic vortex fields, and other small axisymmetric obstacles such as circular cylinders and disks. [Work supported by the 2013-14 F. V. Hunt Postdoctoral Fellowship (Zhang) and by ONR (Marston).]

Experimental and numerical acoustofluidics in bulk acoustic wave devices at ETH Zurich

Ultrasonic fluid cavity resonances in acoustofluidic micro-devices can be exploited to miniaturize important operations for the handling of beads, cells, droplets, and other particles. With a growing number of experimentally tested unit operations, acoustofluidics holds increasing promise for emerging applications in bio- and microtechnology on lab-on-a-chip systems. We provide an overview of our research activities during the last years with a focus on the latest experimental setups and advances in the numerical simulation. Specifically, we present micro-devices with impedance matched cavity walls that allow a more flexible device design. Further, we show devices for the handling of fluid droplets and report on a method for the direct measurement of the acoustic radiation force on micro-particles. Due to the rapidly growing computational capabilities, numerical simulation has become a valuable tool in acoustofluidics research. We present a numerical model that accurately mimics the boundary layer damping inside the fluid cavity, allowing to make predictions of the attainable acoustic amplitudes and radiation forces. Furthermore, we demonstrate how numerical optimization of the device geometry is used to design new devices in an automatic fashion. Finally, we show how the radiation forces and torques can be deduced from the simulated acoustic fields to compute trajectories of complex shaped particles.

Examples of viscous phenomena relevant to second-order responses to ultrasound

It has long been realized that viscosity strongly affects the response of drops and bubbles to the radiation pressure of modulated ultrasound. See for example an analysis of the viscous damping of bubble and drop shape oscillations and related experiments [T. J. Asaki and P. L. Marston, J. Fluid Mech. 300, 149–167 (1995)]. There has been some recent interest in the importance of the viscosity of the fluid surrounding objects of interest when considering other second-order responses. These include the following quantities or responses associated with scattering by spheres and other objects: radiation force of progressive plane waves and Bessel beams [P. L. Marston, Proc. Meet. Acoust. 19, 045005 (2013)]; power extinction of acoustic beams [L. Zhang and P. L. Marston, Bio. Opt. Express 4, 1610–1617 (2013); (E) 4, 2988 (2013)]; and the radiation torque of vortex beams and orthogonal standing waves [L. Zhang and P. L. Marston, J. Acoust. Soc. Am. 131, 2917–2921 (2014)]. The underlying assumptions and limitations of some current and prior approaches will be noted along with some applications to acoustophoresis. [Work supported by ONR (Marston) and by the 2013-14 F. V. Hunt Postdoctoral Fellowship (Zhang).]

Optical theorem for two-dimensional (2D) scalar monochromatic acoustical beams in cylindrical coordinates

The optical theorem for plane waves is recognized as one of the fundamental theorems in optical, acoustical and quantum wave scattering theory as it relates the extinction cross-section to the forward scattering complex amplitude function. Here, the optical theorem is extended and generalized in a cylindrical coordinates system for the case of 2D beams of arbitrary character as opposed to plane waves of infinite extent. The case of scalar monochromatic acoustical wavefronts is considered, and generalized analytical expressions for the extinction, absorption and scattering cross-sections are derived and extended in the framework of the scalar resonance scattering theory. The analysis reveals the presence of an interference scattering cross-section term describing the interaction between the diffracted Franz waves with the resonance elastic waves. The extended optical theorem in cylindrical coordinates is applicable to any object of arbitrary geometry in 2D located arbitrarily in the beam’s path. Related investigations in optics, acoustics and quantum mechanics will benefit from this analysis in the context of wave scattering theory and other phenomena closely connected to it, such as the multiple scattering by a cloud of particles, as well as the resulting radiation force and torque.

Discoid and Asymmetrical Gyroless Micro-Satellite Off-Modulation Attitude Control with Kalman Filter

The gyroless attitude control method and kalman algorithm procedures presented in this paper are applicable to asymmetrical microsatellites of any shape with large mass variation and without angular rate sensors. The attitude sensors include a three-axis magnetometer, a horizon sensor, and a coarse sun sensor (CSS), which together serve as an analytic platform and help in ensuring the stability of attitude controls. The attitude control problem of microsatellites has become a focal point because microsatellite fabrication processes are short, costless, and can be flexibly used for various purposes. The center of mass of the microsatellite can be offset because of fuel consumption during propulsion, irrespective of the existence of interference from the external orbital environment, such as gravity gradient torque and solar radiation torque. For a microsatellite with a discoid and asymmetrical shape, attitude control is difficult. One of the solutions to overcome the difficulty is to design a robust controller that assists the attitude pointing of the satellite to satisfy requirements in the presence of internal parameter perturbations and external disturbances. The robust nonlinear state feedback used in the design of the propulsion mode attitude control for FORMOSAT-3 was applied in this study, and the feasibility of the controller was cross-validated through time and frequency domain stability analyses. The time-domain performance indexes (e.g., rise time, maximum overshoot, and stabilization time) of the designed state feedback gain were consistent with a robust stability margin of the stable performance index in the frequency domain. Furthermore, to reduce the weight and manufacturing cost of the satellite, an extended Kalman filter algorithm was used to obtain the gyroless satellite attitude rate. Other sensors, such as the CSS and earth horizon sensor were adopted to help sense satellite attitude controls.