ACOUSTOFLUIDIC DEVICE

Abstract

The invention concerns a novel acoustofluidic device to separate acoustically active particles from fluids comprising a novel device arrangement for improved acoustic pressure and particle velocity; and a method of separating particles from a fluid comprising use of same.

Claims

1. An acoustofluidic device comprising: at least one interdigitated transducer (IDT) deposited on the surface of a piezoelectric substrate; and functionally coupled therewith at least one channel having a first end and second end forming a fluid flow path, wherein said at least one channel is positioned adjacent said at least one IDT and comprises a first sidewall; a second sidewall; a floor and an acoustic wave source defining a roof of the at least one channel.

2. The device according to claim 1 comprising at least a pair of interdigitated transducers (IDTs) deposited on the surface of a piezoelectric substrate to form at least one standing surface acoustic wave (SSAW) transducer wherein the at least one channel is positioned between said at least one pair of IDTs.

3. The device according to claim 1, wherein said piezoelectric substrate is: polyvinylidene difluoride (PVDF), gallium nitride (GaN), Aluminum nitride (AlN), silicon carbide (SiC), aluminum gallium nitride (AlGaN), Langasite (La3Ga5SiO.sub.14), gallium orthophosphate (GaPO.sub.4), a lithium niobate (LiNbO.sub.3), lithium tantalate (LiTaO.sub.3), barium titanate (BaTiO.sub.3), lead zirconate titanate (Pb[Zr.sub.xTi.sub.1-x]O.sub.3 with 0≤x≤1), potassium niobate (KNbO.sub.3), sodium tungstate (Na2WO.sub.3), or zinc oxide (ZnO).

4. (canceled)

5. The device according to claim 1, wherein a longitudinal axis of said at least one channel is substantially orthogonal with respect to said at least one IDT or is provided at an angle with respect to said at least one IDT.

6. (canceled)

7. (canceled)

8. The device according to claim 1, wherein said at least one channel floor and/or sidewalls is manufactured from: polycarbonates or polymethyl methacrylates, polyphenylsulfone (PPS), glass, silicone, ceramic, elastomers, thermoset polyester (TPE), poly-methyl methacrylate (PMMA), polystyrene (PS), polycarbonate (PC), poly-ethylene glycol diacrylate (PEGDA), teflons, polyurethane (PU), paper, hydrogels, pyrex or polydimethyl siloxane (PDMS).

9. (canceled)

10. The device according to claim 1, wherein said acoustic wave source is provided as a further (a) at least one interdigitated transducer (IDT) deposited on the surface of a piezoelectric substrate, (b) a standing surface acoustic wave (SSAW) transducer, or both.

11. (canceled)

12. The device according to claim 10, wherein said at least one IDT deposited on the surface of a piezoelectric substrate and said acoustic wave source are configured such that, in use, a phase difference of between about Δφ=π/2 and Δφ=3π/2 exists between the acoustic wave(s) originating in said piezoelectric substrate and the acoustic wave(s) originating in said roof of the at least one channel.

13. The device according to claim 12 wherein, in use, a phase difference of between about Δφ=π exists between the acoustic wave(s) originating in said piezoelectric substrate and the acoustic wave(s) originating in said roof of the at least one channel.

14. The device according to claim 1, wherein said acoustic wave source is provided as a bulk acoustic wave (BAW) piezoelectric transducer producing bulk acoustic waves (BAWs).

15. The device according to claim 14 wherein said BAW piezoelectric transducer is a piezoelectric ceramic.

16. The device according to claim 15 wherein said piezoelectric transducer is PZT or LiNbO.sub.3.

17. The device according to claim 1, wherein said at least one channel has a width to height ratio of between about 10:1 and 1:1.

18. The device according to claim 1, wherein said at least one channel has a width between about 10-1000 μm and a height between about 1-250 μm including every 1 μm therebetween.

19. The device according to claim 1, wherein the at least one channel comprises at least one inlet configured to introduce a fluid into a proximal end portion of the at least one channel and/or at least one outlet which is located at a downstream portion of the at least one channel positioned substantially along the longitudinal axis of the at least one channel.

20. The device according to claim 19 wherein the at least one inlet and/or the at least one outlet is branched to permit separation of particles into different flow streams.

21. The device according to claim 1, wherein said device comprises a plurality of channels in fluid communication with one another.

22. The device according to claim 21 wherein each of the plurality of channels is functionally coupled with at least one IDT deposited on the surface of a piezoelectric substrate or a standing surface acoustic wave (SSAW) transducer such that each channel can separate different particles with respect to one another according to a standing wave generated for each respective channel.

23. The device according to claim 1, wherein the at least one IDT or a standing surface acoustic wave (SSAW) transducer and/or acoustic wave source can generate a resonance frequency, or a mean resonance frequency, of between about 100 KHz to 1000 MHz.

24. (canceled)

25. A method for separating a mixture of acoustically active particles comprising: suspending a mixture of acoustically active particles in a liquid flow stream; and flowing said liquid flow stream through the at least one channel of the device of claim 1, thereby separating the mixture of acoustically active particles.

26. An apparatus, comprising: a signal generator configured to couple to an acoustofluidic device having (i) at least one interdigitated transducer (IDT) deposited on the surface of a piezoelectric substrate and (ii) at least one channel coupled to the IDT, wherein the at least one channel has a first end and second end forming a fluid flow path and wherein the at least one channel is positioned adjacent the at least one IDT and includes a first sidewall, a second sidewall, a floor, and an acoustic wave source defining a roof of the at least one channel; wherein the signal generator is configured to produce a radio frequency (RF) voltage for the at least one IDT that controls a distribution of an acoustic pressure field within the at least one channel.

Description

[0042] The Invention will now be described by way of example only with reference to the Examples below and to the following Figures wherein:

[0043] FIG. 1. A partial side sectional view of the acoustofluidic device according to the invention;

[0044] FIG. 2. Cross sectional side views of the channels of the state of the art (a-b) and according to the invention (c-d). (a) A typical acoustofluidic structure consisting of a PDMS channel and a SSAW transducer (SAW-PDMS). (b) A hybrid acoustofluidic resonator employing a glass slide as the reflector positioned at the top of the PDMS channel (SAW-Glass). (c) An acoustofluidic configuration equipped by two SSAW transducers as the top and bottom plates (SAW-SAW). (d) An integrated acoustofluidic configuration consisting a top BAW transducer and a bottom SAW transducer (BAW-SAW). (e) The computational domain of the model, the boundary Γ.sub.t, Γ.sub.b, and Γ.sub.s are modelled as the top, bottom and side walls, respectively;

[0045] FIG. 3. Colour plots of the first-order acoustic pressure p.sub.1, the first-order velocity field v.sub.1 and the time-averaged second-order velocity {circumflex over (v)}.sub.2 in the SAW-PDMS and SAW-Glass channels. (a) The maximum pressure in the SAW-PDMS and SAW-Glass is 13.4 kPa and 33.6 kPa, respectively. (b) The amplitude of the first-order velocity in the SAW-PDMS and SAW-Glass is 5.42 mm/s and 37.6 mm/s, respectively. (c) The maximum second-order velocity in the SAW-PDMS and SAW-Glass is 0.65 μm/s and 12.2 μm/s, respectively.

[0046] FIG. 4. Particle trajectories and velocities in the SAW-PDMS and SAW-Glass configurations. (a) Particle size is 1 μm, the maximum velocity is 0.55 μm/s in SAW-PDMS and 5.89 μm/s in SAW-Glass. (b) Particle size is 5 μm, the maximum velocity is 0.65 μm/s in SAW-PDMS and 10.7 μm/s in SAW-Glass. (c) Particle size is 10 μm, the maximum velocity is 10.4 μm/s in SAW-PDMS and 40.8 μm/s in SAW-Glass;

[0047] FIG. 5. Colour plots of the first-order acoustic pressure p.sub.1, the first-order velocity field v.sub.1 and the time-averaged second-order velocity {circumflex over (v)}.sub.2 in the SAW-SAW channel. The left panel shows the phase difference Δφ=0 while the right panel shows the phase different Δφ=π. (a) The maximum pressure is 14.2 kPa and 224 kPa, respectively. (b) The amplitude of the first-order velocity is 2.0 mm/s and 70.6 mm/s, respectively. (c) The maximum second-order velocity is 0.88 μm/s and 41.5 μm/s, respectively;

[0048] FIG. 6. Plots of the maximum first-order acoustic pressure p.sub.1 (a) and the acoustic pressure distribution (b) for phase difference Δφ between 0 and 2π in the SAW-SAW configuration;

[0049] FIG. 7. Colour plots of the first-order acoustic pressure p.sub.1 for the BAW-SAW configuration when the amplitude of the vibration of the BAW is (a) the same as the SAW transducer, and (b) 10 times higher than the SAW transducer;

[0050] FIG. 8. Colour plots of the first-order acoustic pressure p.sub.1, the first-order velocity field v.sub.1 and the time-averaged second-order velocity {circumflex over (v)}.sub.2 in the BAW-SAW configuration with the channel dimension of 450 μm×120 μm. (a) The maximum pressure is 373 kPa. (b) The maximum first-order velocity is 295 mm/s. (c) The maximum time-averaged second-order velocity is 161 μm/s. (d) The maximum pressure achieves 3,200 kPa when the BAW amplitude is ten times higher than the SAW;

[0051] FIG. 9. Particle trajectories and velocities in the SAW-SAW (phase difference Δφ=π) and −SAW (u.sub.r=10u.sub.0) configurations. (a) Particle size is 1 μm, the maximum velocity is 17.5 μm/s in SAW-SAW and 80.1 μm/s in BAW-SAW. (b) Particle size is 5 μm, the maximum velocity is 131 μm/s in SAW-SAW and 847 μm/s in BAW-SAW. (c) Particle size is 10 μm, the maximum velocity is 573 μm/s in SAW-SAW and 3310 μm/s in BAW-SAW;

[0052] FIG. 10. Cross sectional side views of additional exemplary acoustofluidic devices according to the invention:

[0053] (A) An acoustofluidic configuration comprising PDMS channel sidewalls, a piezoelectric substrate top plate and a piezoelectric substrate bottom plate. A single IDT is deposited on the surface of each of the top and bottom plates, wherein the IDTs of the top and bottom plates are positioned adjacent to and on the same side of the channel sidewalls.

[0054] (B) An acoustofluidic configuration comprising PDMS channel sidewalls, a piezoelectric substrate top plate and a piezoelectric substrate bottom plate. A single IDT is deposited on the surface of each of the top and bottom plates, wherein the IDTs of the top and bottom plates are positioned adjacent to and on the opposite side of the channel sidewalls.

[0055] (C) An acoustofluidic configuration comprising PDMS channel sidewalls, a piezoelectric substrate top plate and a piezoelectric substrate bottom plate. A single IDT is deposited on the surface of the top plate, and a pair of IDTs forming a SSAW transducer are deposited on the surface of the bottom plate.

[0056] (D) An acoustofluidic configuration comprising PDMS channel sidewalls, a top BAW transducer and a piezoelectric bottom plate. A single IDT is deposited on the surface of the bottom plate, adjacent to the channel sidewalls.

[0057] (E) An acoustofluidic configuration comprising PZT (or any other piezoelectric substrate) sidewalls, a top BAW transducer and a piezoelectric bottom plate. A single IDT is deposited on the surface of the bottom plate, adjacent to the channel sidewalls. Alternatively, a pair of IDTs forming a SSAW transducer may be deposited on the surface of the bottom plate.

[0058] Table 1. Parameters used in numerical analysis at T=25° C.

[0059] Referring to the figures and, firstly, to FIG. 1 there is shown a partial side sectional diagrammatic view of an exemplar acoustofluidic device [1] according to an embodiment of the invention. It can be seen that a channel [2] is bonded to the surface of a piezoelectric substrate [3a] between ideally, although not exclusively, a pair of interdigitated transducers (IDTs) [3b] forming a SSAW transducer [3]. In this particular arrangement, at least one pair of IDTs [3b] is provided to generate at least one SSAW transducer [3]. As will be appreciated, alternative arrangements are envisaged such as where a single IDT [3b] or more than one pair of IDTs [3b] are provided to generate one of more SSAWs. The IDTs are routine in the art and comprise electrodes deposited on the surface of a piezoelectric substrate to form a periodic structure and, upon application of RF voltage, convert electric signals to surface acoustic waves (SAW) by generating periodically distributed mechanical forces via a piezoelectric effect causing the substrate to expand and contract. In the case of the present arrangement, when an RF voltage is applied to the pair of IDTs [3b] this generates two series of identical SAWs within the piezoelectric substrate that propagate in counter directions towards one another wherein they interfere to generate a SSAW field of periodic pressure nodes and antinodes in the space in between said pair of IDTs [3b] i.e. where the channel [2] is located, thereby exposing a flow stream in the channel to a standing acoustic wave field to affect acoustic fluid relocation of acoustically active particles contained therein.

[0060] As is shown in FIG. 1, the channel is configured such that the longitudinal axis of same is substantially orthogonal with respect to the IDTs [3b] such that acoustic waves generated by the IDTs [3b] are substantially transverse thus exposing any particles present in the fluid to lateral pressure force permitting generation of fluid flow paths that allow separation of particles according to shape and/or size.

[0061] As will be appreciated, to separate particles in a fluid it is required to provide fluid flow path through said channel [2] through which a fluid containing particles to be separated can flow. In FIG. 1, this is achieved by providing at least one fluid inlet [4] configured to introduce a fluid into a proximal end portion of the channel and at least one outlet [5] located at a downstream portion of the channel positioned substantially along the longitudinal axis of the channel. As will be appreciated by those skilled in the art, in this embodiment, a fluid can be introduced through the first inlet [4] and flowed through the at least one channel [2] and out of the fluid outlet [5]. Through exposure to the standing acoustic wave generated by the SSAW transducer [3] and/or acoustic wave source (best seen in FIG. 2), particles present in the fluid can be separated into different specific outlets [5] of the channel according to particle shape and/or size. As is shown, in a preferred arrangement the channel [2] comprises at least two outlets [5], although more or less outlets can be envisaged according to the particles to be separated. Preferably, as shown, said inlet(s) [4] and/or outlet(s) [5] are branched to permit separation of particles into different flow streams, although equally they may be unbranched. Further, the device may comprise a pump (not shown) to control flow rate of fluid through the inlet(s)/channel(s)/outlet(s). Also, by providing multiple inlets it is possible to introduce multiple fluids into the channel at any one time and so, in this arrangement, separate multiple particles from multiple fluid sources. In a preferred arrangement, said inlet(s) [4] and/or outlets(s) [5] are provided with tubing to permit flow of a fluid into the inlet(s) [4a] and/or out of the outlet(s) [5b].

[0062] According to the invention, as shown, a single channel [2] is provided to allow a single staged particle separation stage. However, in alternative arrangements it is envisaged that multiple stages of separation can be achieved. For example, a single channel [2] positioned between multiple staggered pairs of IDTs [3b] may be provided wherein each pair of IDTs [3b] produces a separate SSAW thereby providing differing fields of separation along the flow path. Alternatively, the device [1] comprises a plurality of channels [2] (not shown) in fluid communication with one another. Preferably, the channels are connected in series, so that each channel shares a connection with at least another channel, wherein the outlet of a first channel forms the inlet for the second, and so on. More preferably, each channel is connected via tubing. More preferably still each channel comprises a SSAW transducer such that each channel can separate different particles with respect to one another according to the standing wave generated for each respective channel. In this arrangement multi-stage particle separation can be achieved.

[0063] Turning to the channel, referring to FIG. 2, channel cross-sections of the art (FIGS. 2a and b) and according to the invention (FIGS. 2c and d) are shown as a side sectional diagrammatic views. In both arrangements (c and d), the channel comprises a first sidewall [2a] and second sidewall [2b], a floor [2c], and an acoustic waves source [7] defining the roof [2d]. In the embodiments shown, said channel floor [2c] is configured to functionally couple with the at least one SSAW transducer [3] such that the SSAW from the IDTs [3b] is propagated across same. As will be appreciated, in this preferred embodiment said floor [2c] of the channel [2] is provided by the piezoelectric substrate [3a] wherein the walls [2a/2b] of the channel [2] are bonded to the surface of the substrate [3a] to provide a channel. Alternatively (not shown), said channel floor and/or walls can be made from any suitable material that can house a fluid mixture and permits coupling of the acoustic wave energy from the piezoelectric substrate to the fluid in the channel. As is known by those skilled in the art, silicon, glass, or metal materials are commonly used for acoustophoresis because the rigid channel walls provide a near ideal acoustic boundary against the sample fluid, enhancing the required standing wave resonance. Such suitable materials for channel floor and/or wall include, but are not limited to, medical grade plastics, but most ideally, said channel walls and/or floor are PDMS.

[0064] With reference to the channel roof [2d], an acoustic wave source [7] refers to any means for generating acoustic wave energy wherein said acoustic wave energy is transmitted into the fluid of the channel [2]. In this way it is believed that by providing a roof [2d] defined by an acoustic wave source [7] one can further generate a vertical pressure field in the channel, in addition to the longitudinal pressure field induced by the orthogonal IDTs, producing another standing wave thereby increasing the channel pressure field. Further, the vertical channel pressure field also permits manipulation of particles in the vertical direction of the channel in addition to laterally.

[0065] In a first embodiment, as shown in FIG. 2c, the acoustic wave source [7] is provided as a further SSAW transducer [7a] thus forming two SSAW transducers (i.e. as the channel floor [3] and roof [7a]), termed SAW-SAW. Respectively, each SSAW transducer [3],[7a] can be independently controlled such that the acoustic waves generated by same can be carefully controlled to manipulate particle flow in both vertical and lateral planes thus providing for an extra level of particle flow separation. Preferably, to achieve higher resolution of particle flow and thus separation, it has been found that fine tuning the phase difference between each respective SSAW transducer one can generate symmetrical pressure anti-nodes to form good particle trajectories.

[0066] Alternatively, in a second embodiment as shown in FIG. 2d, the acoustic wave source is provided as a bulk acoustic piezoelectric transducer [7b] producing bulk acoustic waves (BAWs). In this arrangement (called BAW-SAW) the channel floor [2c] is provided by the SSAW transducer of the substrate and the roof [2d] is defined by a bulk acoustic (BAW) piezoelectric transducer [7b]. It has been found that in this particular arrangement one can also achieve stronger pressure gradients in the channel of several orders of magnitude greater than conventional arrangements known in the art. RF signals from a source (not shown) drive both the BAW [7b] and SSAW [3] transducers to produce a combined acoustic energy in the channel [2]. Changing the input voltage of the BAW [7b] and SSAW [3] transducers can vary the integrated acoustic field.

[0067] Therefore, as will be appreciated, in all embodiments the SSAW transducers [3] and/or acoustic wave source [7a, 7b] can be operated in phase with each other, or operated out of phase with each other depending on the configuration. Each SSAW transducer [3] and/or acoustic wave source [7a, 7b] of the present disclosure may have individual electrical attachments (e.g. electrodes), so that each SSAW transducer and/or acoustic wave source can be individually controlled for frequency and power. Configuration allows for not only the generation of a multi-dimensional acoustic standing wave, but also improved control of the acoustic standing wave. In this way, it is possible to drive individual transducers with arbitrary phasing and/or different or variable frequencies and/or in various out-of-phase modes.

[0068] As will be appreciated, the channel [2] takes the form predominantly of a longitudinal channel whose dimensions (height and width) can vary according to the nature of the fluid to be flowed therethrough, the number of particles to be separated, or respective number of inlet and outlet channels, for example. It has been found that where the channel dimensions are proportionally greater in width than height, maximum pressure field can be achieved across the entire cross-section of the channel allowing more careful particle manipulation. Ideally, said channel has a width between about 400-650 μm and a height between about 100-150 μm, although variations outside these ranges are possible and within the spirit of the invention.

[0069] The flow dynamics and particle separation of the devices according to the invention are described in the following examples.

[0070] Methods

[0071] When acoustic wave is applied to a suspension of particles, the scattering of the wave on the particles will exert an acoustic radiation force that can be utilised to manipulate the particles. To understand how the design of the acoustofluidic devices affects distribution patterns of the particles in a channel, we performed numerical simulations for different design scenarios. In all cases, the channel length is significantly longer than the height and the width, and the acoustic waves are perpendicular to the longitudinal direction. Thus, the fluid flow and particle movement in the channel are investigated as two-dimensional problems.

[0072] A. Governing Equations for Fluid Flow

[0073] For very dilute suspensions, influences of the particles on the bulk fluid flow can be neglected as long as the particle size is significantly smaller than the dimension of the channel and the acoustic wavelength. Thus, the governing equations for the bulk fluid flow are,

[00001]$\begin{array}{cc}\frac{\partial \hat{\rho }}{\partial t}=-\nabla .Math.\left(\hat{\rho }\hat{\upsilon }\right)& \left(1\right)\\ \hat{\rho }\frac{\partial \hat{\upsilon }}{\partial t}=-\nabla \hat{p}-\hat{\rho }\left(\hat{\upsilon }.Math.\nabla \right)\hat{\upsilon }+\eta {\nabla }^2\hat{\upsilon }+\mathrm{\beta \eta }\nabla \left(\nabla .Math.\hat{v}\right)& \left(2\right)\\ \beta =\frac{{\eta }_b}{\eta }+\frac{1}{3}& \left(3\right)\end{array}$

[0074] where {circumflex over (ρ)} is the fluid density, the bold letter {circumflex over (v)} is the vector of fluid velocity, {circumflex over (p)} is the fluid pressure, η and η.sub.b are the shear viscosity and bulk viscosity, respectively.

[0075] In our devices, there is no fluid flow before application of acoustic waves. As a result, the fluid density and pressure, ρ.sub.0 and p.sub.0, are uniform and time-independent. When acoustic waves propagate through the fluid, they cause small perturbations in the density, pressure, and velocity fields, which can be expressed as,

{circumflex over (p)}=p.sub.0+{circumflex over (p)}.sub.1+{circumflex over (p)}.sub.2+ . . .   (4)

{circumflex over (ρ)}=ρ.sub.0+{circumflex over (ρ)}.sub.1+{circumflex over (p)}.sub.2+ . . .   (5)

{circumflex over (v)}={circumflex over (v)}.sub.1+{circumflex over (v)}.sub.2+ . . .   (6)

[0076] where the subscripts 1 and 2 indicate the first and the second order terms, respectively. Higher order terms are neglected in the simulations. Additionally, we assume that {circumflex over (p)}.sub.1 is proportional to {circumflex over (ρ)}.sub.1,

{circumflex over (p)}.sub.1=c.sub.0.sup.2{circumflex over (ρ)}.sub.1  (7)

[0077] where c.sub.0 is a constant and approximately equal to the speed of sound in the fluid. Substituting Eq. (4) through (7) into Eq. (1) and Eq. (2) yields the continuity and momentum equations for the first- and second-order terms:

[00002]$\begin{array}{cc}\frac{\partial {\hat{\rho }}_1}{\partial t}=-{\rho }_0\nabla .Math.{\hat{\nu }}_1& \left(8\right)\\ {\rho }_0\frac{\partial {\hat{v}}_1}{\partial t}=-c_0^2\nabla {\hat{\rho }}_1+\eta {\nabla }^2{\hat{\nu }}_1+\beta \eta \nabla \left(\nabla .Math.{\hat{\nu }}_1\right)& \left(9\right)\\ \frac{\partial {\hat{\rho }}_2}{\partial t}=-{\rho }_0\nabla .Math.{\hat{\nu }}_2-\nabla .Math.\left({\hat{\rho }}_1{\hat{\nu }}_1\right)& \left(10\right)\\ {\rho }_0\frac{\partial {\hat{v}}_2}{\partial t}=-\nabla {\hat{p}}_2+\eta {\nabla }^2{\hat{\nu }}_2+\beta \eta \nabla \left(\nabla .Math.{\hat{\nu }}_2\right)-{\hat{\rho }}_1\frac{\partial {\hat{v}}_1}{\partial t}-{\rho }_0\left({\hat{\nu }}_1.Math.\nabla \right){\hat{\nu }}_1& \left(11\right)\end{array}$

[0078] For periodic perturbations, the time average of the Eq. (10) and (11) become

[00003]$\begin{array}{cc}{\rho }_0\nabla .Math..Math.{\hat{\nu }}_2.Math.=-\nabla .Math..Math.{\hat{\rho }}_1{\hat{\nu }}_1.Math.& \left(12\right)\\ \eta {\nabla }^2.Math.{\hat{\nu }}_2.Math.+\beta \eta \nabla \left(\nabla .Math..Math.{\hat{\nu }}_2.Math.\right)-\nabla .Math.{\hat{p}}_2.Math.=.Math.{\hat{\rho }}_1\frac{\partial {\hat{v}}_1}{\partial t}.Math.+{\rho }_0.Math.\left({\hat{\nu }}_1.Math.\nabla \right){\hat{\nu }}_1.Math.& \left(13\right)\end{array}$

[0079] where X denotes the temporal average of X over an oscillation period. To solve the first order equations, Eq. (8) and Eq. (9) are first combined to obtain the governing equation for {circumflex over (p)}.sub.1.

[00004]$\begin{array}{cc}\frac{{\partial }^2{\hat{p}}_1}{\partial t^2}=c_0^2\left[1+\frac{\left(1+\beta \right)\eta }{{\rho }_0{c}_0^2}\frac{\partial }{\partial t}\right]{\nabla }^2{\hat{p}}_1& \left(14\right)\end{array}$

[0080] In the study, we assume the first-order fields of the density, pressure, and velocity to be harmonic time dependence, i.e.

{circumflex over (ρ)}.sub.1(r,t)=ρ.sub.1(r)e.sup.iωt  (15)

{circumflex over (p)}.sub.1(r,t)=e.sup.iωt  (16)

{circumflex over (v)}.sub.1(r,t)=v.sub.1(r)e.sup.iωt  (17)

[0081] where ω=2πf, which is the angular frequency, and f is the wave frequency. Substituting Eq. (15) through (17) into Eq. (9) and (14) yields,

[00005]$\begin{array}{cc}i\omega {\rho }_0{\nu }_1=-\nabla p_1+\eta {\nabla }^2{\nu }_1+\beta \eta \nabla \left(\nabla .Math.{\nu }_1\right)& \left(18\right)\\ \left[1+\frac{i\omega \left(1+\beta \right)\eta }{{\rho }_0{c}_0^2}\right]{\nabla }^2{p}_1+\frac{{\omega }^2}{c_0^2}{p}_1=0& \left(19\right)\end{array}$

[0082] Eq. (19) can be solved with specific boundary conditions (see descriptions below) to obtain p.sub.1, which can be substituted into Eq. (18) to determine the first order velocity, v.sub.1(r). In the current study, however, we determined v.sub.1(r) using an approximate method. Based on the Helmholtz decomposition theorem, a vector field can be separated into two terms: irrotational and solenoidal. Previous studies have shown that the second term for v.sub.1 is negligible in the bulk fluid. It is important only within the boundary layer around a solid surface. Thus, we assumed v.sub.1 to be irrotational in the bulk fluid, which can be expressed as the gradient of a velocity potential, v.sub.1=∇ϕ.sub.1. Substituting this relationship into Eq. (18) yields,

[00006]$\begin{array}{cc}{v}_1=-\frac{\nabla p_1}{i\omega {\rho }_0}\left[1+\frac{i\omega \left(1+\beta \right)\eta }{{\rho }_0{c}_0^2}\right]& \left(20\right)\end{array}$

[0083] This is the equation used to calculate v.sub.1 after solving the governing equation for p.sub.1 (i.e., Eq. (19)).

[0084] B. Governing Equation for Acoustophoretic Trajectories of Particles

[0085] Once the first order acoustic pressure p.sub.1 and velocity v.sub.1 are obtained, we can determine the time-averaged acoustic radiation force F.sup.rad on a spherical particle, which leads to the net movement of the particles besides local oscillation. F.sup.rad is the sum of the second-order pressure and the first-order momentum flux integrated over the particle surface,

F.sup.rad=.sub.∂Ωn.Math.[{circumflex over (p)}.sub.2I+ρ.sub.0({circumflex over (v)}.sub.1{circumflex over (v)}.sub.1)]dA  (21)

[0086] where ∂Ω is a fixed surface in the bulk fluid around the particle. If the particle radius is much smaller than the wave length, an analytical expression of the force has been derived by Settnes and Bruus,

[00007]$\begin{array}{cc}{F}^{rad}=-{\mathrm{\pi a}}^3\left[\frac{2{\kappa }_0}{3}\mathrm{Re}\left(f_1^{*}{p}_1^{*}\nabla p_1\right)-{\rho }_0\mathrm{Re}\left(f_2^{*}{\nu }_1^{*}\nabla {\nu }_1\right)\right]& \left(22\right)\end{array}$

[0087] where the asterisk denotes the complex conjugate of the quantity, a is the particle radius, and κ.sub.0 is the isentropic compressibility of the fluid defined as

[00008]$\begin{array}{cc}{\kappa }_0=-\frac{1}{V}{\left(\frac{\partial V}{\partial \hat{p}}\right)}_s=\frac{1}{\hat{\rho }}{\left(\frac{\partial \hat{\rho }}{\partial \hat{p}}\right)}_s& \left(23\right)\end{array}$

[0088] where V and s are the volume and entropy of the fluid. After neglecting second and higher order terms, κ.sub.0=1/(ρ.sub.0c.sub.0.sup.2). The scattering coefficients f.sub.1 and f.sub.2 are calculated by

[00009]$\begin{array}{cc}{f}_1=1-\frac{{\kappa }_p}{{\kappa }_0}& \left(24\right)\\ f_2=\frac{2\left(1-\gamma \right)\left({\rho }_p-{\rho }_0\right)}{2{\rho }_p+{\rho }_0\left(1-3\gamma \right)}& \left(25\right)\\ \gamma =-\frac{3}{2}\left[1+i\left(1+\frac{\delta }{a}\right)\right]\frac{\delta }{a}& \left(26\right)\\ \delta =\sqrt{\frac{2\eta }{\omega {\rho }_0}}& \left(27\right)\end{array}$

[0089] where ρ.sub.p and κ.sub.p is the mass density and compressibility of the particle, respectively. δ is called the viscous penetration depth, which characterises the boundary layer thickness.

[0090] Apart from F.sup.rad, particles also experience drag force from the viscous fluid due to the relative movement of the particle with respective to the fluid. Since the time-averaged streaming velocity is {circumflex over (v)}.sub.2, the time-averaged drag force is,

F.sup.drag=6πηa({circumflex over (v)}.sub.2−v.sub.p)  (28)

[0091] where a is the particle radius and v.sub.p is the particle velocity vector. Applying the Newton's second law of motion to the particle yields,

[00010]$\begin{array}{cc}{m}_p\frac{{\mathrm{dv}}_p}{\mathrm{dt}}=F^{\mathrm{rad}}+F^{\mathrm{drag}}& \left(29\right)\end{array}$

[0092] where m.sub.p is the mass of the particle. In most experimental setup, the particle acceleration time is much shorter than the time scale of experimental observation. As a result, we can neglect the acceleration term in Eq. (29) to obtain an expression for v.sub.p:

[00011]$\begin{array}{cc}{\nu }_p=.Math.{\hat{\nu }}_2.Math.+\frac{F^{rad}}{6\mathrm{\pi \eta }a}& \left(30\right)\end{array}$

[0093] Once {circumflex over (v)}.sub.2 is determined by numerically solving Eq. (12) and Eq. (13) with the boundary conditions described below, Eq. (30) can be used to calculate the particle velocity.

[0094] C. Model Configurations and Boundary Conditions

[0095] C1. Model Configurations

[0096] The four different model configurations considered in this study are shown in FIG. 2(a-d). The typical SSAW transducer made by patterning a pair of interdigital electrodes on LiNbO.sub.3 bonded with a PDMS channel is given in FIG. 2(a). Operating under the same RF signal, two SAWs generated by the IDTs counter-propagate to produce a standing SAW (SSAW) within the channel. Stable acoustic pressure gradients are formed in the water flowing in the channel which exerts acoustic radiation force and streaming drag force on the particles inside the channel. We call this typical acoustofluidic structure as SAW-PDMS. FIG. 2(b) shows the model of the novel structure of an acoustofluidic chip.sup.29, 30 for high throughput CTC separation, which employs a glass slide as an acoustic reflector attached on the top of the channel, namely hybrid PDMS-glass resonator (we call it SAW-Glass). The reflector prevents the acoustic energy loss caused by the PDMS absorption on the top.

[0097] To further increase the acoustic energy pressure in the channel for enhanced manipulation of particles, we developed two new models of acoustofluidic structures as shown in FIG. 2(c) and FIG. 2(d). In FIG. 2(c), the top wall of the PDMS channel is replaced by a second SSAW transducer which configures into a sandwich SSAW transducer spacing by PDMS walls, we call this structure as SAW-SAW, The SAW-SAW model can provide stronger pressure gradients in the channel by replacing the passive glass top with the active SSAW transducer. By tuning the phase difference between the top and bottom SSAW transducers, the acoustic pressure distribution can be controlled. The model shown in FIG. 2(d) is a variance of the sandwich structure by replacing the top wall with a BAW transducer producing acoustic wave into the water, namely BAW-SAW. The channel is constructed by two PDMS walls supporting the BAW transducer. RF signals are driving both the BAW and SSAW transducers to produce a combined acoustic energy in the channel. Changing the input voltage of the BAW and SSAW transducers can vary the integrated acoustic field. FIG. 2(e) shows the computational domain of the model, where the top (Γ.sub.t) and bottom (Γ.sub.b) boundaries are modelled differently using the displacement boundary condition in the SAW-SAW and BAW-SAW, and impedance boundary conditions are used to model the two PDMS side walls (Γ.sub.s). All the parameters and values given in Table 1 are used in the numerical analysis.

[0098] C2. Boundary Conditions

[0099] To solve the first order pressure field, we employ the impendence or lossy-wall boundary condition at the water-PDMS interface, due to partial absorption of the acoustic energy by PDMS:

[00012]$\begin{array}{cc}n.Math.\nabla p_1=i\frac{\omega {\rho }_0}{{\rho }_m{c}_m}{p}_1& \left(31\right)\end{array}$

[0100] where ρ.sub.m and c.sub.m are the mass density of PDMS and the speed of sound in PDMS, respectively. n is the normal vector of the solid boundary surface. The same lossy-wall boundary condition also applies to the water-glass interface shown in FIG. 2(b),

[00013]$\begin{array}{cc}n.Math.\nabla p_1=i\frac{\omega {\rho }_0}{{\rho }_g{c}_g}{p}_1& \left(32\right)\end{array}$

[0101] where ρ.sub.g and c.sub.g are the mass density of glass and the speed of sound in glass, respectively. To derive the boundary condition at the water-LiNbO.sub.3 interface, we considered the LiNbO.sub.3 substrate to be actuated by the SSAW, and ignored the wave decay along the propagation path in the substrate because of the short path length. Thus, the displacement and the velocity of the substrate in the z direction at the interface are,

û=u.sub.0{e.sup.i(ωt-k.sup.s.sup.y)+e.sup.i[ωt-k.sup.s.sup.(w.sup.0.sup.-y)]}   (33)

{circumflex over (v)}=iωu.sub.0{e.sup.i(ωt-k.sup.s.sup.y)+e.sup.i[ωt-k.sup.s.sup.(w.sup.0.sup.-y)]}   (34)

[0102] where u.sub.0, t, k.sub.s, y and w.sub.0 denote displacement amplitude, time, wave number, location on y-axis and channel width, respectively. The continuity of the displacement in the z direction requires the z component of the velocity to be continuous. Using Eq. (20), the boundary condition for p.sub.1 at the water-LiNbO.sub.3 interface is,

[00014]$\begin{array}{cc}n.Math.\nabla p_1=\frac{{\omega }^2{u}_0{\rho }_0\left[e^{-i{k}_{s}y}+e^{-i{k}_s\left(w_0-y\right)}\right]}{\left[1+\frac{i\omega \left(1+\beta \right)\eta }{{\rho }_0{c}_0^2}\right]}\left(e_z.Math.n\right)& \left(35\right)\end{array}$

[0103] where e.sub.z is the unit vector in the z direction. In the design of the SAW-SAW configuration, the same attenuation boundary condition as that shown in Eq. (35) applies to both the top and the bottom boundaries.

[0104] For the water-PZT interface, we simulated the design in which PZT vibrated only in the z direction. Thus, the displacement and the velocity of the substrate at this interface are,

û=u.sub.Te.sup.iωt   (36)

{circumflex over (v)}=iωu.sub.Te.sup.iωt   (37)

[0105] where u.sub.T denotes the maximum displacement amplitude of the PZT surface which is controlled by the applying RF voltage. Again, the continuity of the displacement in the normal direction at the interface requires the normal velocity to be continuous. Using Eq. (20), the boundary condition for p.sub.1 at the water-PZT interface is,

[00015]$\begin{array}{cc}n.Math.\nabla p_1=\frac{{\omega }^2{u}_T{\rho }_0}{\left[1+\frac{i\omega \left(1+\beta \right)\eta }{{\rho }_0{c}_0^2}\right]}\left(e_z.Math.n\right)& \left(38\right)\end{array}$

[0106] In the design with the BAW-SAW configuration, Eq. (38) and Eq. (34) apply to the top and the bottom boundaries respectively.

[0107] D. Numerical Simulations

[0108] Computational mesh with maximum element size length d.sub.b at the domain boundary and 10 d.sub.b in the bulk of the domain is reasonable to capture the physics of the model. We use an illustrative mesh with d.sub.b=20δ, where δ is the viscous penetration depth defined in Eq. (27). For verifying the correctness of this solution, an investigation of the mesh-convergence is required. We compare a series of meshes with decreasing mesh element size length and define a relative convergence function C(g) for a solution g with respect to a reference solution g.sub.ref taken to be the solution for the smallest value of d.sub.b.

[00016]$\begin{array}{cc}C\left(g\right)=\sqrt{\frac{\int {\left(g-g_{\mathrm{ref}}\right)}^{2}dydz}{\int {\left(g_{\mathrm{ref}}\right)}^{2}dydz}}& \left(39\right)\end{array}$

[0109] where we use a reference solution g.sub.ref with d.sub.b=0.4δ, which resulted in 2.2×10.sup.5 elements.

[0110] To simulate the flow and particle distribution patterns, we first solve Eq. (19) to determine the first order pressure field p.sub.1 and use it to calculate the velocity field v.sub.1 with Eq. (20). The results are substituted into Eq. (12) and Eq. (13) to solve for the time averaged, second order velocity field {circumflex over (v)}.sub.2. Finally, the particle velocity and trajcetories are calculated with Eq. (30).

[0111] Results

[0112] Acoustofluidic Field and Particle Trajectories in the SAW-PDMS and SAW-Glass

[0113] FIG. 3(a) shows the first-order pressure gradient p.sub.1 inside the SAW-PDMS and SAW-Glass channels. The maximum pressure is 33.6 kPa in the SAW-Glass structure which has a similar pressure distribution reported by Wu's work.sup.29, where the pressure anti-nodes located near the four corners of the channel. Comparing with the maximum pressure of 13.4 kPa in the SAW-PDMS, the higher acoustic pressure in the SAW-Glass channel is achieved attributing to the reflected acoustic energy at the water-glass interface.

[0114] FIG. 3(b) shows the first-order velocity v.sub.1 inside the SAW-PDMS and SAW-Glass configurations, the amplitude of the actuation velocity is less than the amplitude of the first-order velocity |v.sub.1| in both SAW-PDMS (5.42 mm/s) and SAW-Glass (37.6 mm/s) structures. The glass reflector produces 89% reflection at the water-glass interface allowing the acoustic wave to travel back to the channel, which results approximately 7-fold greater first-order velocity comparing to that in the SAW-PDMS configuration.

[0115] The time-averaged second-order velocities v.sub.2 in the SAW-PDMS and SAW-Glass are given in FIG. 3(c), with the maximum velocity of 0.65 μm/s and 12.2 μm/s, respectively.

[0116] Acoustofluidic Field and Particle Trajectories in the SAW-SAW

[0117] In the SAW-Glass channel, the reflected wave from the water-glass interface interacts with the leaky wave in the water produced by the bottom SSAW transducer to produce a pseudo-standing wave (PSW) on the z direction. The PSW can be further improved and controlled by using another SSAW or BAW transducer such as PZT to replace the glass positioned on the top of the channel (FIG. 2(c) and FIG. 2(d)). In the SAW-SAW configuration, the phase difference, Δφ, between the two SSAWs generated on the top and the bottom SSAW devices can be controlled by a signal generator, which can enhance the acoustic energy within the channel and control the distribution of the pressure field. The first-order acoustic pressure p.sub.1, the first-order velocity field v.sub.1 and the time-averaged second-order velocity {circumflex over (v)}.sub.2 in the SAW-SAW are given in FIG. 5, where the left panel and right panel show the results when Δφ=0 and Δφ=π, respectively.

[0118] Varying the phase difference Δφ between the top and bottom SSAW transducers can redistribute the pressure gradients and alter the pressure amplitude in the channel, due to the phase shift results in an interchange in the position of the nodes and the anti-nodes. By sweeping the Δφ, the maximum acoustic pressure and pressure gradients are shown in FIG. 6(a). The largest pressure of 232 kPa is obtained at Δφ=5π/6 and Δφ=7π/6 with four pressure anti-nodes (FIG. 6(b)), and the smallest pressure of 14.2 kPa is noted at Δφ=0 (FIG. 5(a)). The four pressure anti-nodes presented at Δφ=5π/6 or 7π/6 are not entirely symmetrical, which could not produce useful particle trajectories for particle separation. The maximum acoustic pressure at Δφ=π (red arrow in FIG. 6) is slightly lower but producing four symmetrical pressure anti-nodes (FIG. 5(a), right), which can form good particle trajectories (FIG. 9).

[0119] Acoustofluidic Field and Particle Trajectories in the BAW-SAW

[0120] By replacing the top SSAW transducer by the BAW transducer, such as PZT, as shown in FIG. 2(d), the hybrid device achieved even larger acoustic pressure and allowed more powerful particle manipulation. To allow high throughput particle manipulation by the primary acoustic radiation force to drive particles towards a belt like pressure node in the z direction, it is desired to have two pressure anti-nodes formed at the central top and bottom areas in the channel. For the same channel geometry used in the above simulations, i.e. 600 μm (W)×125 μm (H), when the BAW and the SSAW transducers produce the same vertical vibration amplitude, i.e. u.sub.T=u.sub.0, the pressure anti-nodes are asymmetrical as shown in FIG. 7(a). We increased the amplitude of the BAW to 10 times as the amplitude of the bottom SSAW transducer, i.e. u.sub.T=10u.sub.0, as shown in FIG. 7(b), the two pressure anti-nodes are formed.

[0121] The particle trajectories in the BAW-SAW configuration are simulated in FIG. 9 right panel, which can be compared with that in the SAW-SAW configuration shown in FIG. 9 left panel. Both configurations struggle to concentrate 1-μm particles due to the strong drag force dominating over the radiation force. Particles sized 5 and 10 μm are effectively driven by both the SAW-SAW and the BAW-SAW, the latter configuration displays much faster transition velocities achieving 847 μm/s and 3,310 μm/s for 5 and 10 μm, respectively.

CONCLUSIONS

[0122] A comprehensive comparison amongst traditional SAW-PDMS, hybrid SAW-Glass, novel SAW-SAW and BAW-SAW structures have been presented in the study. The model of the SAW-SAW transducers has notably increased the 10-μm particle velocity to 573 μm/s, comparing to the velocity of 10.4 μm/s in the state-of-the-art hybrid SAW-Glass configuration. The active acoustic generation by the top SAW transducer instead of the passive glass reflection employs the same actuation boundary condition as the bottom IDT used in most of acoustofluidic devices. The BAW-SAW transducer whose piezoelectric material (e.g. PZT) can produce much greater vibration to incorporate with the bottom SSAW transducer allowing significantly stronger acoustic resonance generated in the channel. The 10-μm particles migrate at the maximum velocity of 3,310 μm/s when the BAW actuation amplitude is 10 times larger than the SAW actuation. The future work is to manufacture the SAW-SAW and BAW-SAW acoustofluidic chips to verify the model system and numerical analysis.

TABLE-US-00001 TABLE 1 Water Density ρ.sub.f 997 kg/m.sup.3 Speed of sound c.sub.0 1497 m/s Shear viscosity η 0.890 mPa s Bulk viscosity η.sub.b 2.47 mPa s Compressibility κ.sub.0 448 T/Pa Lithium niobite (LiNbO.sub.3) Speed of sound c.sub.sub 3994 m/s Poly-dimethylsiloxane (PDMS, 10:1) Density ρ.sub.wall 920 kg/m.sup.3 Speed of sound c.sub.wall 1076.5 m/s Attenuation coeff. (6.65 MHz) 31 dB/cm Polystyrene Density ρ.sub.p 1050 kg/m.sup.3 Speed of sound c.sub.p 2350 m/s Poisson’s ratio σ.sub.p 0.35 Compressibility κ.sub.p 249 T/pa Acoustic actuation parameters Wavelength λ 600 nm Forcing frequency ƒ 6.65 MHz Displacement amplitude μ.sub.0 0.1 nm Displacement decay coefficient Cd 116 m.sup.−1 Acoustic impedance PDSM Z.sub.PDMS 0.98 MPa .Math. s/m Water Z.sub.water 1.49 MPa .Math. s/m Glass Z.sub.glass 12.0 MPa .Math. s/m

REFERENCES

[0123] 29. M. Wu, P. H. Huang, R. Zhang, Z. Mao, C. Chen, G. Kemeny, P. Li, A. V. Lee, R. Gyanchandani, A. J. Armstrong, M. Dao, S. Suresh and T. J. Huang, Small (Weinheim an der Bergstrasse, Germany), 2018, 14, e1801131. [0124] 30. M. Wu, K. Chen, S. Yang, Z. Wang, P. H. Huang, J. Mai, Z. Y. Li and T. J. Huang, Lab on a chip, 2018, 18, 3003-3010.