## Abstract

The ability to manipulate dense micrometre-scale objects in fluids is of interest to biosciences with a view to improving analysis techniques and enabling tissue engineering. A method of trapping micrometre-scale particles and manipulating them on a two-dimensional plane is proposed and demonstrated. Phase-controlled counter-propagating waves are used to generate ultrasonic standing waves with arbitrary nodal positions. The acoustic radiation force drives dense particles to pressure nodes. It is shown analytically that a series of point-like traps can be produced in a two-dimensional plane using two orthogonal pairs of counter-propagating waves. These traps can be manipulated by appropriate adjustment of the relative phases. Four 5 MHz transducers (designed to minimize reflection) are used as sources of counter-propagating waves in a water-filled cavity. Polystyrene beads of 10 μm diameter are trapped and manipulated. The relationship between trapped particle positions and the relative phases of the four transducers is measured and shown to agree with analytically derived expressions. The force available is measured by determining the response to a sudden change in field and found to be 30 pN, for a 30 V_{pp} input, which is in agreement with the predictions of models of the system. A scalable fabrication approach to producing devices is demonstrated.

## 1. Introduction

The force applied to micrometre-scale objects as a result of interaction with ultrasonic waves offers opportunities for application in the biosciences. Standing waves, generated in resonant cavities, have attracted the most interest, as they produce greater forces; however, the dexterity of these devices is limited by the strong correlation of the acoustic field shape to the device geometry. In this paper, a method of using counter-propagating waves to generate standing-wave patterns is developed. This allows pressure nodes, which act to trap dense particles, to be moved by varying the phase relationships between exciting transducers. The work builds on previous work, which used a pair of transducers for one-dimensional manipulation (Courtney *et al.* 2010), by using two opposing pairs of transducers, positioned orthogonally, in order to trap and manipulate particles in two dimensions.

Although practical applications are only just being explored, the acoustic radiation force has been of academic interest for many years: Kundt (1868) demonstrated the effect of using cork dust in a glass tube in the nineteenth century, and King (1934) provided analytical solutions for the force on an incompressible sphere in an inviscid fluid under the influence of standing or travelling waves. This analysis relied on the particle diameter being substantially smaller than the incident wavelength. Further development of this analysis produced solutions for compressible spheres (Yosioka & Kawasima 1955; Gor'kov 1962), still in inviscid fluids. More extensive calculations, to include viscosity and heat conductivity, have been performed (Doinikov 1994, 1997; Mitri 2009). However, the effects of these factors are only appreciable where travelling waves are involved (Doinikov 1996), rather than standing waves; so for most devices developed to date, the results due to Gor'kov (1962) and Yosioka & Kawasima (1955) are suitable for calculating the forces involved.

Trapping particles with megahertz-frequency standing waves has been used to agglomerate cells allowing filtering of cells from fluids by sedimentation or enhancement of bead-based assays (Coakley 1997; Coakley *et al.* 2000). Ultrasonic standing waves have been used to guide particles suspended in a fluid undergoing laminar flow in devices with split inlets and/or outlets. This allows the division of particles from fluid, for concentration, filtration (Yasuda *et al.* 1995) or the sorting of particles (Johnson & Feke 1995). These approaches have been developed, primarily in the area of micro-scale devices, and the field of manipulating micro-particles (and in particular cells) is reviewed by Laurell *et al.* (2007) and appears in a general review of separation techniques using continuous flow by Pamme (2007). In addition to applications in the biosciences, there has been interest in using the acoustic radiation force in the production of acoustic metamaterials. Saito *et al.* (1998) demonstrated the construction of a structured composite material consisting of 10 μm diameter acrylic spheres trapped in layers in a epoxy solution that was then cured. The same authors extended this approach by using two pairs of transducers arranged orthogonally to produce a grid pattern (Saito *et al.* 1999). Pairs of transducers were used to improve field uniformity and not, as in the current work, to introduce a manipulation capability. Mitri *et al.* (2011) demonstrated the construction of such a material by using opposing transducers to trap layers of diamond nanoparticles in liquid epoxy which were fixed in position as the epoxy cured.

A recent development in the field of micro-particle manipulation is the use of surface acoustic waves (SAWs) generated using interdigital transducers (IDTs) on lithium niobate substrates to generate acoustic fields in fluid-filled cavities and trap particles. Wood *et al.* (2008) used opposing IDTs to trap 0.5–2 μm diameter particles in lines in a 25 μm deep channel. Trapping of particles in a grid-pattern using SAWs has been demonstrated using two pairs of IDTs arranged orthogonally (Wood *et al.* 2009) and using two IDTs at right angles, but without opposing IDTs (Shi *et al.* 2009). By varying the excitation frequency, Wood *et al.* (2008, 2009) showed particle manipulation over distances of 5 μm.

The earlier-mentioned techniques have almost exclusively concentrated on acoustic fields defined by the geometry of the system, which drive particles to particular points in the fluid cavity. However, there has also been interest in increasing versatility by using devices that can generate variable fields to allow a more generalized manipulation of particles. There are currently four approaches to manipulating particles: mode switching, ‘acoustical tweezers’, linear arrays and counter-propagating waves. Trinh *et al.* (1986) used frequency switching between resonant standing waves to levitate and manoeuvre, in one dimension, hollow quartz spheres in air. Work by Min *et al.* (1992) with Styrofoam spheres used a similar approach, but added the concept of trapping in a node of a particular mode, switching the excitation off to allow the particle to drop slightly before pulsing the excitation at that mode to provide a momentum kick, thus allowing the particle to be trapped in another mode by applying an excitation at a different frequency at an appropriate time. Another recent variation on this approach uses switching between two modes (a half wavelength and a quarter wavelength, respectively) to force particles to an equilibrium position between the nodes of the two modes (Glynne-Jones *et al.* 2010).

Focused ultrasonic beams have been used in a manner analogous to optical tweezers (Wu & Du 1990; Wu 1991) by trapping latex particles with a diameter of 270 μm using 3.5 MHz ultrasonic fields. Initially, these ‘acoustical tweezers’ used opposing focused transducers to produce a trap at their, common, focal point (Wu 1991). More recent theoretical (Lee *et al.* 2005; Lee & Shung 2006) and experimental (Lee *et al.* 2009) work has demonstrated the feasibility of using a single focused transducer working at a high frequency (30 MHz) to trap particles much larger than a wavelength in diameter against a surface. Particle manipulation has been achieved by moving the transducer and hence the focal position. An approach that could be used for manipulation in a similar manner was proposed and demonstrated by Yamakoshi & Noguchi (1998): two curved piezoelectric plates were placed next to each other with a common centre of curvature and driven out of phase resulting in a field with a node along the axis perpendicular to the centre of the pair of plates and anti-nodes either side, allowing 30 μm-diameter polystyrene spheres to be trapped against a fluid flow perpendicular to the node. Recent theoretical work suggests that the versatility of acoustic tweezers may be improved by the use of Bessel beams (Marston 2006; Mitri 2008).

The third method of manoeuvring particles using the acoustic radiation field uses a linear array of transducers placed opposite to a reflector. Simultaneously exciting three adjacent array elements allow a standing wave to be generated above those elements, which is sufficient to levitate the target particles (80 μm alumina powder) and leads to particles trapped at the nodes of the field over the active elements. By switching elements, it is possible to smoothly move the region of activity and hence move the trapped particles along the array (Kozuka *et al.* 1996). This technique was further improved by using focused transducers to improve the trapping and adding frequency switching to allow some control perpendicular to the trap positions (Kozuka *et al.* 1998*a*). Recently, work has been done on moving from large (tens of millimetre) devices to microfluidic devices, with 10 μm diameter polystyrene beads being successfully manipulated in a 300 μm channel, using a linear array (Démoré *et al.* 2011).

The fourth approach to particle manipulation, which is exploited in this paper, involves generating a standing wave as the sum of two independently generated travelling waves. Using such a standing wave (rather than one generated by resonance) allows the nodal positions of the field to be changed by varying the relative phase of the two sources. The difficulty is generating opposing travelling waves without generating resonant standing waves. The first implementation of this method worked by using transducers in a large water tank and inclining the transducers slightly (15^{°}) so that their beams intersected, but there was no reflection from the front faces of the opposing transducer (Kozuka *et al.* 1998*b*). Haake & Dual (2002) showed analytically that the position of the pressure nodes in a fluid between two piezoelectric plates could be varied as a function of the relative phase between the voltages applied to the plates. However, they indicated that controlled positioning would not be possible at certain points owing to strong reductions in pressure amplitude. An alternative implementation of this principle was used by the current authors to manipulate 10 μm diameter polystyrene particles in one dimension by using transducers acoustically matched to the fluid with an absorbing backing to prevent reflections leading to resonant modes: it was demonstrated that the particles could be trapped and then moved arbitrary distances (Courtney *et al.* 2010). This approach removed the need to work in a large chamber (as would be required for the approach of Kozuka *et al.* (1998*b*)) and is extended to motion in two dimensions in this paper.

The use of counter-propagating waves for particle manipulation has the advantage of allowing dexterous manipulation, without being limited by fixed nodal positions, with a monotonic relationship between relative phase and particle position. The absence of resonances in the system allows for rapid changes in the field even in chambers that are tens of wavelengths in size and avoids potential problems with the trapping of large numbers of particles leading to a change in the resonant frequency of the device (as observed by Kwiatkowski & Marston (1998)). Operation in a narrow frequency band allows transducer optimization, taking advantage of the high *Q*-values of piezoceramics. The use of matched transducers, positioned around the periphery of the fluid chamber, allows the use of relatively small chambers (rather than large tanks), while allowing access from above for handling, observation or optical tweezing.

The paper starts with an analytical section deriving the relationship between phase and nodal position for a pair of counter-propagating waves and the shapes of traps produced by two orthogonal pairs of counter-propagating devices (§3). This is followed by a description of the device designed to demonstrate the approach (§4). Two models are then described, a one-dimensional electro-acoustic model used to optimize the design of the transducers required and a two-dimensional finite-element (FE) model used to predict the field shape and pressure amplitude in the device (§5). The experimental results section demonstrates the manipulation of particles by the device, and the relationship between phase and particle position (§6). This section includes the estimation of the force applied to a particle by measurement of the response to a sudden shift in the applied field. Finally, a scalable fabrication method is proposed and results showing its viability are presented (§7).

## 2. Radiation force

The acoustic radiation force acting on a small compressible sphere in an inviscid fluid was first calculated by Yosioka & Kawasima (1955), and developed into a more convenient potential function by Gor'kov (1962). Gor'kov writes the force, *F*, in terms of a potential *U*, such that
2.1The potential for a spherical particle of radius, *a*, density *ρ*_{0} and bulk modulus *K*_{0} in a fluid of density *ρ*, bulk modulus *K* and bulk sound velocity is given by
2.2where is the mean square pressure of the incident wave at the particle position and the mean-squared velocity. The calculation is valid for particles with radius substantially smaller than one wavelength in the fluid (*a*≪*λ*) and for arbitrary fields, provided that they are not purely travelling waves (Gor'kov 1962). The factors *f*_{1} and *f*_{2} depend on the relative densities and sound speeds of the spheres and fluids, and are given by
2.3

## 3. Counter-propagating wave method

The method of trapping and manipulating particles used in this paper relies on the principle that a standing wave can be generated as a superposition of two counter-propagating waves and that, by varying the phase difference between the counter-propagating waves, it is possible to move the nodal positions of the resultant standing wave. In order to generate the two waves, opposing transducers, matched to prevent reflection at the front surface and backed to absorb energy, are used. A one-dimensional propagation analysis is used to demonstrate the principle and calculate the relationship between position, acoustic pressure and phase.

### (a) Principle of particle control in one dimension

Consider a one-dimensional linear system consisting of two opposing transducers, one positioned at *x*=−*x*_{0} and another at *x*=*x*_{0}. If each transducer has a reflection coefficient *R*, and secondary reflections can be ignored, then the resultant field, *P*_{1}, in the region −*x*_{0}≤*x*≤*x*_{0} owing to the transducer at −*x*_{0} can be written as the superposition of the wave travelling away from the activated transducer, which has amplitude, *P*_{0}, and the reflected part, amplitude *P*_{0}*R*:
3.1The field for the transducer at +*x*_{0} is
3.2Dropping the time dependence term (*e*^{−iωt}), and introducing an arbitrary phase difference of Δ*ϕ* between the two excitations, the total field when both transducers are excited is
3.3

Figure 1*a* shows the behaviour of the pressure field as a function of the phase change for *R*=0.2 and *x*_{0}=3*λ*; note that the effect of the non-zero reflection coefficient is to introduce a deviation from the linear relation between position of the nodes and phase and introduce a variation in the anti-node amplitude with phase. Both these can be evaluated from equation (3.3). The locus of the minimum value of *P*(*x*), calculated from equation (3.3), varies as
3.4Note that this has a similar form to the zero-pressure position predicted for such a system with only single transducer activated (see Haake & Dual 2002, eqn (18)); however, unlike the single transducer case, in the two transducer system, the nodal positions depend on the relative phase and do not vary in time unless the relative phase is changed. For the case where the reflection coefficient is non-zero, the anti-node has a maximum value when Δ*ϕ*=0 of ; however, the amplitude drops to for Δ*ϕ*=*π* (i.e. when trapping the particles at a distance *λ*/2 from the Δ*ϕ*=0 trap). So there is a reduction of up to a ratio of depending on the preferred trapping position. In principle, the effect of reflection could be removed by ensuring that , which is the case when the transducer separation, 2*x*_{0}, is an odd number of quarter wavelengths. However, this is difficult in practice, where there are many wavelengths across the fluid chamber, and minimizing reflection is the best way of ensuring consistent traps and good positional control.

### (b) Principle of particle control in two dimensions

In order to elucidate how a trap can be created in two dimensions using two pairs of transducers, consider an idealized case where there is no reflection and each transducer produces a plane travelling wave. In the absence of reflection, the transducer position is unimportant and so the effect of transducer position is included in the phase delays. Omitting the harmonic time dependence, the contribution from each transducer can be written as
3.5Each contribution has the same amplitude, *P*_{0}, but the phase delay applied to each, *ϕ*_{n}, can be varied independently. The total pressure can be written as the sum of these contributions:
3.6Defining
3.7equation (3.6) can be rewritten:
3.8

The aim is to be able to trap particles at pressure nodes, which can then be moved by changing the signals applied to the transducers. To this end it is necessary to calculate the positions and shapes of pressure nodes, which can most easily be achieved by calculating the modulus squared of the pressure in equation (3.8)
3.9If *ϕ*_{x}=*ϕ*_{y} (as is the situation when *ϕ*_{x}=*ϕ*_{y}=0 for example), then pressure nodes occur where *y*=(2*n*−1)(*λ*/2)−Δ*ϕ*_{x}/2*k*−Δ*ϕ*_{y}/2*k*−*x* or *y*=(2*n*−1)*λ*/2+Δ*ϕ*_{x}/2*k*−Δ*ϕ*_{y}/2*k*+*x*. Although this indicates that there is a node pattern that can be shifted in the *x*- and *y*-directions by varying Δ*ϕ*_{x} and Δ*ϕ*_{y}, the lack of any local trapping (the nodes form a grid of intersecting lines) limits the usefulness of this pattern. This pressure field can be seen in figure 1*b*. More useful is the pattern, shown in figure 1*c*, formed if the following relationship is obeyed:
3.10In that case, the third term of equation (3.9) is zero and pressure nodes occur only where both and . This leads to a regular grid of traps (regions of zero pressure) at
3.11where *n*_{x} and *n*_{y} are independent integers. The phase relationship in equation (3.10) and the resulting relationships between nodal positions and relative phases in equation (3.11) provide the basis for a method of manipulating particles in two dimensions. The following section describes the design of a practical device for realizing this.

## 4. Device design

In order to obtain manipulation in two dimensions, two pairs of matched transducers are required, along with a single transducer, positioned on the base of the device in order to hold particles against gravity. The four manipulation transducers are designed to be acoustically matched to water (the fluid in the chamber) and include an absorbing backing. These features minimize the reflection of the incoming acoustic energy in order to reduce the standing waves produced by the reflections and ensure that the result of driving a single transducer is wave travelling away from the transducer. As demonstrated in §3, counter-propagating travelling waves can be used to produce a standing wave whose nodal positions (which act as particle traps) are controlled by the phase of the excitation signals.

Matching layer performance is strongly dependent on frequency and so the transducers are designed for a specific frequency. In the frequency range in which existing analytical solutions are applicable (for particles much smaller than a wavelength), the acoustic radiation force increases linearly with frequency (Yosioka & Kawasima 1955). Higher frequencies require higher tolerances in manufacture and so a compromise between manufacturing complexity and force is sought, and 5 MHz is selected as the operation frequency.

The device can be considered as three orthogonally arranged systems; the levitation system (arranged vertically), which lifts the particles and holds them in horizontal planes, and then two manipulation stages, which use the counter-propagating wave method to trap the particles in a grid of traps that can then be manipulated by altering the phases of the signals applied to the transducers.

The levitation stage consists of a straightforward resonant system consisting of a 15×15 *mm* piezoceramic plate of thickness 3 mm positioned 2 mm from a 1 mm thick microscope slide, which acts as a reflector. Application of a 5 MHz signal produces a standing wave with 13 nodes in the water-filled cavity between the piezoelectric plate front face and the reflector. The acoustic radiation force traps particles in layers at the nodes, allowing manipulation in the *x*–*y* plane. The levitation transducer extends many (approx. 50) wavelengths in both the *x*- and *y*-directions and so the field is uniform (or at least very slowly varying in those directions), leading to forces (from the levitation field) directed only along the *z*-axis.

The four manipulation transducers are designed to minimize reflection in order to produce controllable fields; they consist of 2×15×1.3 *mm* thick piezoceramic plates with an alumina-loaded epoxy matching layer applied to the front face and a tungsten-loaded epoxy absorbing layer applied to the back face.

The matching layer is designed to minimize reflection by adding a layer with a thickness that is an odd-integer multiple of one quarter of the wavelength in the matching layer and an acoustic impedance of:
4.1where *z*_{water}=1.5 MRayl is the acoustic impedance of water and *z*_{T} is the acoustic impedance of the transducer material. For the NCE51 material used (a soft-doped lead zirconate titanate (PZT) material: Noliac group), the acoustic impedance is 35 MRayl and so equation (4.1) gives *z*_{m}=7.2 MRayl. This would require a very large concentration of alumina (in excess of the 30% by volume used by Wang *et al.* (2001)); in practice, a more conservative approach was taken with 10 per cent alumina by volume used, resulting in an acoustic impedance of 3.6 MRayl (Wang *et al.* 2001). In the following section, it is demonstrated—using a one-dimensional acousto electric model—that this leads to a reduction in reflection that is still sufficient to allow particle manipulation.

The aim of the backing layer is to ensure that all acoustic energy entering the transducer is absorbed. The absorption of tungsten-doped epoxy peaks at 7.5 per cent tungsten by volume; however the acoustic impedance increases with increasing tungsten content (Wang *et al.* 2001). A compromise value of 10 per cent tungsten by volume was used, giving an absorption of 38 dB mm^{−1} (at 30 MHz) and an acoustic impedance of 5.6 MRayl (Wang *et al.* 2001). The backing layer was 9 mm thick. Assuming that the attenuation varies linearly with frequency, as the tungsten particle radius (5 μm) is much smaller than one wavelength, then 99 per cent of the acoustic energy should be absorbed in a single, 18 mm, roundtrip through the backing layer.

## 5. Modelling device behaviour

The analysis in §3 described the behaviour of particles in a somewhat idealized manipulation system, but more involved models are required to assess the actual behaviour of the system shown in figure 2. Even when considering the system as a one-dimensional system, the reflection coefficient, *R*, and acoustic pressure amplitude need to be included to predict the behaviour of the system. To this end, a one-dimensional electro-acoustic model was used to predict the behaviour of the particle trapping system. This, relatively straightforward, model allowed the study of design parameters (in particular, the thickness of the component layers of the transducers) and prediction of the particle positions and trap strengths. The other major consideration in the assessment of the device design is the effect of the finite nature of the transducers on the resultant field, particularly, in order to evaluate the region over which the particles can be effectively controlled and the pressure amplitude generated. This is addressed using a two-dimensional FE model in §5*b*.

### (a) One-dimensional electro-acoustic model

The design of the manipulation system was analysed by a one-dimensional electro-acoustic model, an approach similar to that previously used by Hill (2003) for the design of resonant devices. The system, including the fluid-filled cavity, is modelled as a series of layers, infinite in the directions perpendicular to the wave propagation. Linear wave propagation is assumed through passive materials, and an acousto-electric solution is used for the electrically active piezoelectric plates (Bui *et al.* 1977). The general approach is the same as the Krimholtz, Leedom and Matthaei equivalent circuit approach (Leedom *et al.* 1971), and a description of the method used here is given in Wilcox *et al.* (1998). The one-dimensional model assumes that the dimensions perpendicular to the propagation direction are large compared with the wavelength. The system is 2 mm in the *z*-direction, which is 6.7 wavelengths at 5 MHz. This suggests that this one-dimensional model will give only approximate results and a three-dimensional model would be required to fully model the device. However, the one-dimensional model offers clear physical insights and the ability to perform rapid parameter studies to aid the design process.

Each layer is defined in terms of its thickness, *d*, cross-sectional area, *A*, acoustic impedance, *Z*, and *Q*-factor, *Q*, with piezoelectric layers also having a permittivity, *ϵ* and piezoelectric constant, *h*_{xx}. The piezoelectric coefficient is obtained from more readily available parameters by
5.1where *k*_{t} is the piezoelectric coupling coefficient, and the open circuit stiffness in the direction of propagation (*x*).

The arrangement of the layers used to model the manipulation stage is shown in figure 3 and the material parameters are shown in table 1.

Using the one-dimensional model to evaluate the reflection coefficient (as the ratio of waves propagating in the positive and negative *x*-directions in the fluid chamber under continuous excitation) allows the potential performance of the device to be assessed as a function of the material thicknesses and compositions.

Figure 4 shows the reflection coefficient as a function of excitation frequency for the system described in figure 3 and table 1, but with the thickness of the matching layers on both transducers varied. It can be seen that there are two sources of reductions in the reflection coefficient: resonances either in the transducers or in the matching layer. For this system, the frequencies at which the former occur are mainly dependent on the transducer thickness, and are only weakly dependent on the matching layer thickness (and occur even where there is no matching layer). The matching layers behave as expected with reductions in reflection where the matching layer thickness is equal to an odd integer number of quarter wavelengths. Reflection can best be reduced by making these two frequencies coincident. For this reason, the transducer thickness (1.33 mm) was chosen to provide a resonance at 5 MHz, and a matching layer thickness of 0.4 mm (3/4 *λ* at 5 MHz) was chosen to minimize reflection at 5 MHz.

The other key consideration is the pressure amplitude available. Figure 5 shows the pressure amplitude in the fluid cavity according to the one-dimensional electro-acoustic model of the device with 0.4 mm matching layers. For each frequency, the model is solved for an excitation of 30 V_{pp} and the resulting pressure peak in the cavity calculated. In addition to many peaks owing to cavity resonances, there are large peaks at 1.4 MHz and 5 MHz corresponding to resonances of the transducers. The excitation voltage is 30 V_{pp}, as used experimentally, and the peak pressure at 5 MHz is 300 kPa.

### (b) Two-dimensional finite-element model

The simplicity of the one-dimensional model makes it well suited to studying the effect of varying experimental parameters, but it does not give the true field shape generated by orthogonal pairs of finite-sized transducers. In particular, the positions of the nodes and the region over which particles can be trapped and manipulated is of interest.

#### (i) The model

A commercial FE package (PZFlex, Weidlinger Associates Inc.) was used to model the device described in §4 in two dimensions with the direction perpendicular to the manipulation plane treated as plane strain. Square elements were used with 15 elements per wavelength and the model solved with an explicit time stepping solver. The geometry of the model is shown in figure 6. A symmetrical boundary condition is applied along the *x*-axis to halve the size of the model. The remaining model boundaries have unconstrained boundary conditions. Two virtual experiments were undertaken. In the first, a pulse of three cycles (with a top-hat envelope) at 5 MHz central frequency was applied, and the incident and reflected pressure from a transducer was used to evaluate the reflection coefficient of the transducer. In the second experiment, a continuous excitation at 5 MHz was applied until a steady state was achieved in order to evaluate the field shape of the device in operation. In each case, the excitation was applied as a voltage across one of the transducers. The response to excitation of four transducers is calculated as a superposition of the solution for a single transducer, rotated appropriately.

### (c) Model results

Figure 7*a* shows the time domain signal for the pressure at the centre of the model system after a three-cycle burst at 5 MHz centre frequency has been applied across one of the transducers. Region (1) corresponds to the pulse passing the centre of the system directly from the input, and region (2) encompasses the first reflection from the opposing transducer. Figure 7*b* shows the reflection coefficient calculated from the Fourier transforms of the two pulses (solid line) and is shown compared with the reflection coefficient calculated using the one-dimensional model.

Although the two-dimensional model indicates a higher reflection coefficient than would be expected from the one-dimensional model, the value is still low enough to expect manipulation to be possible, and the optimal operation frequency is unchanged at 5 MHz. The FE model runs in the time domain, but running it with continuous sinusoidal excitation until a steady state is reached allows continuous single frequency excitation to be modelled effectively. Once a steady state is reached, a fast Fourier transform is performed on the time response at each spatial point and the response at 5 MHz extracted. The pressure field, in terms of a complex amplitude field, in the fluid-filled cavity resulting from continuous excitation at 5 MHz was calculated by this method and the response to simultaneous excitation by all four transducers was calculated as the sum of the calculated field rotated appropriately and with an independently variable phase applied to each contribution. Figure 8*a* shows the pressure amplitude in the central 15×15 *mm* region of the device (the region over the levitation stage), and expanded images of the central 0.6×0.6 *mm* region are shown in figure 8*b*,*c*. There is a consistent pressure across most of the region of interest (the area over the levitation transducer), with some amplitude reduction and distortion near the sides. The behaviour away from the edges is as predicted by equation (3.8). The application of phases such that Δ*ϕ*_{x}=0, *ϕ*_{x}=0, Δ*ϕ*_{y}=0 and *ϕ*_{y}=0 (figure 8*a*,*b*) gives a series of intersecting diagonal minima, making trapping difficult. In figure 8*c*, the phase condition given in equation (3.10) is used and a regular grid of nodes is produced.

The pressure amplitudes in figure 8 are given per volt of excitation. For the excitation used experimentally (30 V_{pp}), the maximum pressure in the central region when Δ*ϕ*_{x}=0, *ϕ*_{x}=0, Δ*ϕ*_{y}=0 and *ϕ*_{y}=0 (shown in figure 8*a*,*b*) is predicted to be 450 kPa. When Δ*ϕ*_{x}=0, *ϕ*_{x}=0, Δ*ϕ*_{y}=0 and *ϕ*_{y}=*π*/2, the maximum pressure is 340 kPa. As can be seen in figure 8*c*, these peaks occur on the diagonals of the traps. Of particular interest is the variation of pressure along the *x*- and *y*-directions passing through the traps, as this determines the force available for manipulation in those directions. In the central region, this pressure varies sinusoidally with an amplitude of 225 kPa. Using this pressure, equations (2.1)–(2.3) and the material parameters in table 2 can be used to predict a maximum force on 10 μm diameter polystyrene spheres of 35 pN.

## 6. Experimental results

Each of the five transducers was attached to a signal generator. The resonant nature of the levitation stage ensured that the signal direct from the generator (10 V_{pp} at 5 MHz) was sufficient to trap 10 μm diameter polystyrene spheres against gravity. Without signal applied to the manipulation transducers, the particles were randomly distributed across the region over the levitation transducer and appeared to move freely in the *x*–*y* plane, suggesting that any lateral forces resulting from the levitation stage are small. The four manipulation transducers (arranged in opposing pairs) require additional power amplification to provide voltages of 30 V_{pp}. The manipulation transducers were excited at 5 MHz in order to minimize reflection (see §5) and each pair was synchronized to allow the relative phase of the excitations of the opposing transducers to be adjusted. The levitation stage is not synchronized with the manipulation transducers.

### (a) Trapping and manipulation

Figure 9 shows the distribution of particles in a small region of the device, near the centre, for two operating conditions. In figure 9*a*, the transducer phases are adjusted to conform to equation (3.10) and trapping is achieved at a grid of points, as expected from the analytical result in equation (3.8). Figure 9*b* shows the effect of running all four transducers in phase and, although poor trapping makes getting a good image difficult, particles can be seen to line up along diagonals as predicted analytically in §3*b* (see figure 1*b* for comparison) and by the FE model in §5*c* (figure 8*b*).

Figure 10 shows the manipulation of agglomerates of particles in a single trap; the image is a composite of five images, with *π*/2 changes in Δ*ϕ*_{x} and Δ*ϕ*_{y} used to shift the trap position by *λ*/8=38 μ*m*.

### (b) Device characterization

In order to quantify the behaviour of the trap, a series of images were produced by varying the phase of the particle over a range −2*π*≤Δ*ϕ*_{x}≤2*π* in 36 steps and then the same for −2*π*≤Δ*ϕ*_{y}≤2*π*. In each case, the condition given in equation (3.10) is obeyed throughout. The images were used to determine the positions of the particle agglomerates as a function of phase using image tracking software (Crocker & Grier 1996). Figure 11 shows the relationship between position and phase for an agglomerate of particles when the relative phase of each pair of transducers is varied over a range of −2*π*≤Δ*ϕ*≤2*π*. In each case, the theoretical variation of position calculated from equation (3.4) is plotted, assuming a reflection coefficient of 0.06 as predicted by the one-dimensional electro-acoustic model in §5. The same equation is evaluated using an value determined with a least squares fit between the data and equation (3.4). For the variation of Δ*ϕ*_{x} (i.e. the data in figure 11*a*,*b*), the data appear to follow the expected result well and is extracted from the least squares fit. Where Δ*ϕ*_{y} is varied (in figure 11*c*,*d*), the measured position again follows equation (3.4), but the value of calculated is 0.33, higher than suggested by the one-dimensional model. This suggests that one pair of transducers is less well matched than the other, but that the matching is still sufficient to allow manipulation of particles.

### (c) Force calibration

By changing, in a single step, Δ*ϕ*_{x} by *π* the nodal position of the pressure field can be moved by *λ*/4. Observing the response of a particle to this change in field allows the amount of force to be evaluated. In a sinusoidal pressure field, the force on the particle varies sinusoidally and the equation of motion, assuming a pressure node at *x*=0, can be written as
6.1where *F*_{0} is the maximum force (occurring at *x*=*λ*/8=38 μ*m*), *r* is the radius of the polystyrene sphere and *μ* is the dynamic viscosity of the fluid. Note that, although the viscosity can be neglected in the calculation of the acoustic radiation force, which results from the scattering of the acoustic wave, it can have a substantial effect on the motion of the particle in response to that force. If the system is overdamped, then there is an analytical solution for the particle position:
6.2where *X*_{t=0} is the initial position of the particle.

A node with a single trapped particle was selected and the response of the particle to a change of Δ*ϕ*_{x} by *π* was recorded with a high-speed camera at 92 fps and the particle position extracted using feature tracking software (Crocker & Grier 1996). The change in position along the *x*-axis is shown in figure 12. Also shown is the trajectory given by equation (6.2) with *F*_{0} evaluated by minimizing the r.m.s.e. between the analytical response and the data. This approach indicates that in practice the maximum force generated is 30 pN, which is comparable to the value of 35 pN predicted using the FE model in §5.

## 7. Scalable implementation

Although it is possible to vary the field shape using four transducers, the variation is limited to moving between the field shapes shown in figure 1. Additional transducers would allow more flexible field control; however, the fabrication approach of the device described previously makes construction of such prototypes time-consuming. To alleviate this, a scalable approach, allowing more rapid prototype development and arbitrary numbers of transducers, has been investigated. The approach uses a flexible printed circuit board formed into a polygon to attach matched and backed transducers. To demonstrate that this is a viable approach, it is shown that this method can be used to reproduce the technique of using four transducers to trap and manipulate polystyrene beads.

### (a) Device description

The transducers with alumina-doped matching layers are bonded to a flexible printed circuit board, or ‘flex circuit’ (Flexible dynamics Ltd, UK), and formed into an octagon. The octagonal flex circuit is first sandwiched between two Plexiglas plates to create a sealed unit. Then this sealed octagon is mounted on a rigid PCB to allow simple connection to each transducer element. Finally, an alumina-doped absorbing backing is moulded to the back of the transducers. The device is shown schematically in figure 13*a*; the depth of the fluid chamber is 10 mm. The piezoceramic plates were 5×5 *mm* with a thickness of 0.5 mm. The acoustic matching layer and absorbing backing have been designed to acoustically match the transducers to water and absorb the unwanted reflections, as in the previous device.

The flex circuit was a ribbon of 10 mm width and 80 mm length, each face of the octagon was 10 mm long. The eight transducers have a common ground. Four transducers (arranged in two orthogonal pairs) were each driven using a 4 MHz sine wave of 8 V_{pp}. Synchronization between channels was achieved using two arbitrary waveform generators providing four outputs each allowing independent control of the amplitude, phase and frequency. The signals from the waveform generators were amplified by high-speed buffers, BUF634T. The system is controlled by a virtual control panel developed in Labview that allows real-time voltage, frequency and phase control. No levitation transducer is used for these tests; the rate of sedimentation of the beads is slow enough to allow trapping and manipulation without one. Adding a levitation transducer in the same configuration as before would be straightforward.

### (b) Key results

Four transducers, positioned as two orthogonal pairs in a manner analogous to the device described in §4, were excited simultaneously and as a result, 10 μm diameter polystyrene beads were trapped on a grid with a nodal spacing of *λ*/2, as shown in figure 13*b*. When the phase difference between opposing transducers is changed by *π*, the particles move with a distance of 94 μm (*λ*/4). This is demonstrated in figure 13*c*, which shows an overlay of pictures of the particles when the phase difference is 0 (in black) and when the phase difference is *π* (in grey).

## 8. Conclusions

A method of trapping and manipulating particles in two dimensions in fluid has been described, modelled and demonstrated. Pairs of opposing transducers, sufficiently well acoustically matched to the fluid and with an absorbing backing, allow a standing wave to be generated with nodal positions determined by the relative phases of the transducers. It was shown analytically that imperfect matching leads to a divergence from the linear relation between phase difference and position, and introduces a variation in pressure amplitude with phase. The relationship between the phases of orthogonal pairs of transducers affects both the position and shape of nodes: the best approach for localized trapping and manipulation was derived. By considering each pair of transducers in terms of their phase difference and the arithmetic mean of their phases, it is possible to trap particles at points by fixing the relationship between the mean phases of the two pairs, and then vary the trap position in each axis by adjusting the phase differences of the respective pairs. A manipulation device was designed and the effect of varying experimental parameters assessed using a one-dimensional electro-acoustic model. A two-dimensional FE model was used to predict the shape and amplitude of the field in the device. A device consisting of a resonant levitation stage to hold particles against gravity and two pairs of acoustically matched transducers was constructed. Particle trapping and manipulation was demonstrated and behaviour was found to match that predicted analytically. The available force was determined by measuring the response of individual particles to a sudden change in the applied field. The force was found to be 30 pN for a 30 V_{pp} applied signal, and this is in reasonable agreement with the value of 35 pN predicted from the FE model. An implementation using a flexible circuit material allowing the easy assembly of a variety of devices was demonstrated.

The ability demonstrated, to trap and freely manipulate particles in two dimensions in an enclosed device represents, a significant step in the application of the acoustic radiation force, with potential implications for the biosciences.

## Acknowledgements

This work was performed as part of the Sonotweezers project, funded by the EPSRC (grant no. EP/G012067/1). The authors are grateful to their colleagues at the Institute for Medical Sciences and Technology (University of Dundee), for their aid in fabricating the transducers and to Weidlinger Associates for their assistance with the FE modelling.

- Received April 29, 2011.
- Accepted September 1, 2011.

- This journal is © 2011 The Royal Society