An extended view for acoustofluidic particle manipulation: Scenarios for actuation modes and device resonance phenomenon for bulk-acoustic-wave devices

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M. Bülent Özer and Barbaros Çetin

Citation: The Journal of the Acoustical Society of America 149, 2802 (2021); doi: 10.1121/10.0004778 View online: https://doi.org/10.1121/10.0004778

View Table of Contents: https://asa.scitation.org/toc/jas/149/4 Published by the Acoustical Society of America

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An extended view for acoustofluidic particle manipulation:

Scenarios for actuation modes and device resonance phenomenon for bulk-acoustic-wave devices

a)

M. B€ulentOzer€ 1,b)and BarbarosC¸etin2,c)

1Mechanical Engineering Department, Middle East Technical University, Ankara, 06800, Turkey

2Microfluidics and Lab-on-a-chip Research Group, Department of Mechanical Engineering, Bilkent University, Ankara, 06800, Turkey

ABSTRACT:

For the manipulation of microparticles, ultrasonic devices, which employ acoustophoretic forces, have become an essential tool. There exists a widely used analytical expression in the literature which does not account for the effect of the geometry and acoustic properties of the chip material to calculate the acoustophoretic force and resonance frequencies. In this study, we propose an analytical relationship that includes the effect of the chip material on the resonance frequencies of an acoustophoretic chip. Similar to the analytical equation in the literature, this approach also assumes plane wave propagation. The relationship is simplified to a form which introduces a correction term to the acoustophoretic force equation for the presence of the chip material. The proposed equations reveal that the effect of the chip material on the resonance frequency is significant—and is called the device resonance—for acoustically soft materials. The relationship between the actuation modes of the piezoelectric actuator(s) and position of the nodal lines inside the channel are discussed. Finite element simulations are performed to verify the proposed equations. Simulations showed that even if some of the assumptions in the derivations are removed, the general conclusions about the motion of the microparticles are still valid.VC 2021 Acoustical Society of America.

https://doi.org/10.1121/10.0004778

(Received 25 November 2020; revised 26 March 2021; accepted 28 March 2021; published online 23 April 2021)

[Editor: Max Denis] Pages: 2802–2812

I. INTRODUCTION

When inclusions are immersed in a fluid and placed in an acoustic field, they experience a steady force as a result of the interaction of the incident and scattered acoustic waves, which is called the acoustic radiation force (ARF).

The ARF has been used in many scientific and industrial applications such as acoustic levitation (Forestiet al., 2013), acoustic tweezers (Wu, 1991), the measurement of the inten- sity of the ultrasonic transducers of medical equipment, elasticity imaging of fluids and biological tissues (Sarvazyan, 2010), the measurement of the canonical momentum and spin angular momentum densities of acous- tic fields (Toftulet al., 2019), among many others. With the developments of micro/nanofabrication techniques, fol- lowed by the rapid development of the microfluidic technol- ogies, ARF has attracted even more attention in the last two decades as an efficient way of label-free and noncontact manipulation of bio-particles (Connacher et al., 2018; Wu et al., 2019), especially for clinical applications which require the processing of a vast amount of samples (due to

its potential for high-throughput applications; Cetin et al., 2014).

For all acoustofluidic platforms, the ARF is used to manipulate the bio-particles’ motion within the microfluidic channels. For the bulk acoustic wave (BAW) devices, a standing acoustic wave is built up within a liquid-filled microchannel by a single piezoelectric actuator or two pie- zoelectric actuators placed either in transversal or layered configurations against the acoustofluidic chip. The operation of the BAW devices with high performance depends on the material of the entire device because a high acoustic imped- ance ratio is beneficial between the chip material and the liq- uid filling the microchannel. In addition, the actuation mode [i.e., the placement and number of the piezoelectric trans- ducer (Cetinet al., 2016) together with the actuation polar- ity] is also critical for the BAW devices. The issue regarding the actuation mode has been overlooked in the literature because the numerical models generally do not include piezoelectric actuators and employ much simpler acoustic wave modes. As a result, as far as we know, there is no clear explanation with the derivations in the literature about the geometrical requirements of the channel and acoustofluidic chip, as well as how actuation needs to be applied to move particles/cells to certain desired locations within a microchannel. The issue is somewhat understood for widely used chip materials like silicon and glass, but a full-understanding and explanation with an analytical model

a)This paper is part of a special issue on Theory and Applications of Acoustofluidics.

b)Electcronic mail: ozerb@metu.edu.tr, ORCID: 0000-0002-0380-5125.

c)Also at: UNAM—National Nanotechnology Research Center and Institute of Materials Science and Nanotechnology, Bilkent University, Ankara, 06800, Turkey. ORCID: 0000-0001-9824-4000.

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on the effect of the chip material and actuation modes are still missing. There is a general understanding in the litera- ture that to reach a resonant condition, the channel width should be half of the wavelength of the acoustic waves of the fluid inside the channel (Adams et al., 2012; Lenshof et al., 2012a,b; Petersson et al., 2007). Currently, this approach does not address the importance of the chip mate- rial and chip geometry whatsoever. Interestingly, there are also some examples available in the literature where half- wavelength requirement is not satisfied, yet particles/cells were successfully focused and/or separated. For instance, Mueller et al. (2013) reported successful focusing of red blood cells in a polystyrene chip where the channel width was 430 lm at a frequency of 1.015 MHz, which does not correspond to a half-wavelength. In the same study, they also reported reaching resonance conditions in a channel with a 530 lm width at 0.92 MHz, which, again, does not coincide with the accepted norm of the half-wavelength width of the channel. Furthermore, in the acoustofluidics lit- erature on the BAW devices, there is a wide variety of choices for the excitation methods while using piezoelectric materials as actuators. There are some unexplained ques- tions such as why a symmetrically placed piezoelectric actu- ator placed under or over the microchannel does not move the particles to the center of a microchannel, but an asym- metrically placed piezoelectric actuator can move the par- ticles to the center of a microchannel. Recently, in the work of Gautam et al. (2018), different locations for the piezo- electric actuator were tested, and the results were reported for three different actuator locations. In the first two config- urations, the piezoelectric actuator was centered with respect to the microchannel and placed at the top of the chip in one configuration and at the bottom of the chip in the other configuration. These two configurations did not result in a strong acoustic field that moved the particles toward the center of the microchip. However, when they moved the actuator to a location which was asymmetric compared to the chip and microchannel (the piezoelectric actuator was placed to one side of the microchannel parallel to the micro- channel), a resonance condition was reached and the cells were moved to the center of the channel. This indicates that the symmetric configurations of a single piezoelectric actua- tor with respect to the channel do not result in the required movement of the particles toward the center. So far, to the best of our knowledge, a rigorous analysis of why certain configurations of actuators lead to the motion of the particles to the center of the channel (along the channel width direc- tion) and some configurations do not is not available in the literature. For a better understanding of the acoustic particle manipulation within a polymeric device, and only in a very recent study, Moiseyenko and Bruus (2019) demonstrated that there is an alternative to the conventional approach of standing half-wave resonances in the fluid domain for poly- mer systems, which they defined as a whole-system reso- nance (which we will refer to as device resonance in this study). Furthermore, they also clearly showed how the actu- ation mode of the actuator affects the operation.

In the present study, we aim to provide a simple analyti- cal setting that captures the physics of the presence of the interactions of the walls of a microfluidic channel with the chip material by assuming only plane waves are propagating within the acoustofluidic chip. Through this analytical deri- vation, the two important questions are answered. (i) How do the actuation modes (i.e., placement, number, and polar- ity arrangement) of the piezoelectric actuators affect the acoustofluidic particle manipulation? (ii) What is the reso- nance condition for the acoustofluidic chip when the geome- try beyond the microchannel is considered? Specifically, the answer to the latter question reveals the device resonance phenomenon, which suggests an alternative operating fre- quency to the classical resonance frequency based on the channel width.

II. ANALYTICAL MODEL FOR PLANE WAVES

The acoustic domains employed in the model are shown in Fig.1(A). The blue regions shown in Fig.1represent the chip material, and the red region in the middle represents the fluidic channel. For our analytical model, the following assumptions are introduced: (i) acoustic attenuation is not accounted for, (ii) the system is assumed to be infinitely long along the length direction and no acoustic reflections occur along that direction, (iii) the acoustofluidic chip is excited from outer boundaries only, and (iv) if the wall on the left side of Fig.1moves with the harmonic amplitude ofUw, the wall on the right side moves with an amplitude of K UwwhereK is a real number ranging between1  K  1.

In Fig.1, it can be seen that the incident plane waves with an acoustic pressure of pin1 and acoustic speed of uin1 originate from the left wall and propagate in the right direc- tion (þx-direction). The reflected plane waves with an acoustic pressure ofpr1and acoustic speed ofur1 are moving toward the left (-x-direction). With a similar convention, the incident and reflected waves with acoustic pressures ofpin3 andpr3and acoustic speeds ofuin3 andur3, respectively, prop- agate at the other side of the chip. Similarly, the incident and reflected waves with acoustic pressures of pin2 and pr2 and acoustic speeds ofuin2 andur2, respectively, travel within the microfluidic region (i.e., the red region in Fig.1). The channel width is ‘f and the width of the chip material on each side is ‘, which gives a total width ofð2‘ þ ‘fÞ. Based on this convention, the incident and reflected acoustic waves can be expressed as follows:

uinmðx; tÞ ¼ Uinmejðxtkm; urmðx; tÞ ¼ Urmejðxtþkm; pinmðx; tÞ ¼ Pinmejðxtkm; prmðx; tÞ ¼ Prmejðxtþkm;

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where the subscript m can be either 1, 2, or 3, referring to Fig.1, and the wave numberk is

km¼ k for m¼ 1; 3;

kf for m¼ 2:



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The acoustic velocity and pressure within the microflui- dic channel can be obtained by solvingU2in andUr2, respec- tively, from a system of equations, which is composed of the matching boundary conditions at x¼ ð‘ þ ‘f=2Þ;

x¼ ‘ þ ‘f=2; x¼ ‘f=2; and x¼ ‘f=2, and the equality of the acoustic pressure on the boundaries at x¼ ‘f=2 and x¼ þ‘f=2 (the details of this derivation are given in the supplementary material1). The ARF can be determined via the gradient of the acoustic radiation energy as (Gorkov, 1962)

Frad ¼ rUrad; (3)

where

Urad¼4p 3 a3 f1

2qfc2f hp2ðx; tÞi 3f2

4 qfhu2ðx; tÞi

" #

; (4)

f1ð~jÞ ¼ 1  ~j; f2ð~qÞ ¼2ð~q 1Þ

2~qþ 1 ; (5)

where ~j ¼ ðqpc2pÞ=ðqfc2fÞ and ~q¼ qp=qf, a is the particle radius, qfis density of the fluid,cf is the speed of sound of the fluid, qpis the density of the particle,cpis the speed of sound of the particle,f1is the monopole coefficient, andf2is the dipole coefficient.

The acoustic velocity and pressure within the microflui- dic channel can be written as (the details of the derivation of the acoustic velocity and pressure are given in the supple- mentary material1)

uðxÞ Uw

¼ð1 þ KÞ

2Cin cosðkfxÞ ð1  KÞ

2Cout sinðkfxÞ; (6a) pðxÞ

jZfUw

¼ð1 þ KÞ

2Cin sinðkfxÞ þð1  KÞ

2Cout cosðkfxÞ: (6b) Substituting Eqs. (6a) and (6b) into Eq. (3), the ARF can be obtained as

Frad¼ pa3qfkfU2wU Cð1Þchipsinð2kfxþ /Þ; (7a) /¼ arctan Cð2Þchip=Cð1Þchip

 

; (7b)

Cð1Þchip¼ 1þ K Cin

 2

 1 K Cout

 2

; (7c)

Cð2Þchip¼2ð1  K2Þ CoutCin

; (7d)

Cin¼ cos ðk‘Þ cos ðkff=2Þ  ^Z sinðk‘Þ sin ðkff=2Þ;

(7e) Cout¼ cos ðk‘Þ sin ðkff=2Þ þ ^Z sinðk‘Þ cos ðkff=2Þ;

(7f) Uð~j; ~qÞ ¼ 1

12f1ð~jÞ þ1

8f2ð~qÞ; (7g)

where Cð1Þchip and Cð2Þchip are the correction terms, which take into account the presence of the chip material, and Uð~j; ~qÞ is the acoustophoretic contrast factor. Equation(7a) repre- sents the ARF generated within the channel when one wall is moving with an amplitude ofUw and the other channel wall withK Uw.

When the wall velocities are in-phase (i.e., piezoelectric transducers are actuated out-of-phase,K¼ 1), the ARF becomes

Frad¼ pa3qfkfU2w f1

3þf2

2

 

sinð2kf

C2in : (8)

Alternatively, if the wall velocities are out-of-phase (i.e., piezoelectric transducers are actuated in-phase, K ¼ 1), the ARF becomes

Frad¼ pa3qfkfUw2 f1

3þf2

2

 

sinð2kf

C2out : (9)

Note that Eqs.(8) and(9) are very similar to the con- ventional expression in the acoustofluidics literature given

FIG. 1. (Color online) Schematics of the (A) analytical model and (B) two-dimensional (2D) computational domain.

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inBruus, [2012, Eq. (30)], except for the terms Cinand Cout, which takes into account the presence of the chip material.

These terms can be regarded as correction terms due to the existence of the chip material. As shown in Fig.1, the par- ticles that are in the right half of the channel have positive x-values and the particles that are in the left half of the chan- nel have negative x-values. Keeping in mind the positive sign convention and assuming the positive contrast factor, one can see that the forcing expression given in Eq. (8) results in particles moving toward the center of the channel (hence, the name of the correction term is Cin), and the forc- ing expression given in Eq. (9) results in particles moving toward the walls of the channel (therefore, the variable name is chosen as Cout).

The resonance condition can be achieved when Coutor Cingoes to zero. Using Eq.(7e), when Cin! 0,

tan 2pf ‘ cchip

 

tan pf ‘f

cf

 

¼1

Z^¼qchipcchip

qfcf

; (10)

where qchip andcchip are the density and speed of sound of the chip material, respectively. The frequency values, which satisfy Eq.(10), will create the resonance condition for the ARF and, hence, move particles efficiently toward the chan- nel walls. Using Eq.(7f), when Cout! 0,

cot 2pf ‘ cchip

 

tan pf ‘f

cf

 

¼  ^Z ¼  qfcf

qchipcchip

: (11)

Similarly, the frequency values, which satisfy the above expression, will create a resonance condition for the ARF and, thus, move particles toward the channel walls.

The analytical formulations derived in this section can be used to calculate the acoustic pressures inside a micro- channel. To verify the analytical predictions, a finite ele- ment model is developed and solved using COMSOL Multiphysics (Stockholm, Sweden). The geometrical param- eters are defined in Fig.1(B). The velocity boundary condi- tions are assigned at the top and bottom boundaries as shown in Fig.1(B), the fixed (i.e., acoustically hard) boundary conditions are implemented on the left and right boundaries.

All domains are modeled as acoustic domains. The geometry shown in Fig. 1(B) is two-dimensional (2D), but the given boundary conditions result in acoustic waves that change only with x-direction. Therefore, the 2D geometry can be used to verify the analytic formulations derived under the plane wave assumption.

In the numerical simulations, we are also interested in the movement of the microparticles and where these par- ticles are focused. Therefore, the motions of the polystyrene particles with a 10 lm diameter are simulated. Although it is a second-order effect, together with the inviscid flow approximation, which is valid if the particle radius is much greater than the viscous boundary layer thickness (i.e., a d), the ARF can be calculated based on the time- average of the first-order quantities as in Bruus (2012).

When the particle size is small compared to the channel

dimensions, the Rayleigh limit (i.e., single scattering events) is valid and any particle-wall and particle-particle acoustic interactions are ignored (Baasch and Dual, 2018). The ARF can be obtained from Eqs.(3)–(5)where the acoustic pres- sure and acoustic velocities are for incident waves pin and vin, respectively. Once the first-order pressure is obtained, the trajectory of the particles can be obtained by employing the point-particle approach (i.e., Lagrangian tracking method;Buyukkocaket al., 2014;Cetinet al., 2017;S¸ahin et al., 2019). Newton’s second law of motion can be written as

mp

dvp

dt ¼ Fradþ Fdrag; (12)

wherempis the mass of the particle, vpis the particle veloc- ity, and Fdragis the drag force, which is modeled as Stoke’s drag force.

III. RESULTS AND VERIFICATION OF THE ANALYTICAL MODEL

To verify our analytical model, we consider several cases for an acrylic acoustophoretic chip filled with water.

The magnitudes of the ARF, the values of Cin, Cout, and Cin Cout are obtained as a function of the actuation fre- quency for different K values. For the verification of our model, the results for the 2D COMSOL model are also pre- sented and the equilibrium distributions of the particles are also simulated. Following these results, the device reso- nance phenomenon is discussed on the basis of the effect of the acoustic impedance of the chip material on the resonant frequency; therefore, the comparison of the acrylic and sili- con carbide (SiC) chips are also presented. The geometric parameters and material properties used in the simulations are summarized in Tables S1 and S2, respectively, in the supplementary material.1

A. Scenarios for actuation modes 1. Same wall velocity amplitude

Figure 2 shows the results for the cases in which the wall velocities are in-phase (K ¼ 1, walls moving in the same direction with the same amplitude) and out-of-phase (K¼ 1, walls moving in the opposite directions with the same amplitude). Both the ARF and Cin Cout terms are illustrated as a function of the actuation frequency. The solid yellow lines shown in Fig.2are obtained from Eq. (8) for Fig.2(A)and from Eq.(9) for Fig. 2(B), respectively. The circle markers on the ARF figure are the results obtained by the finite element simulations with COMSOL. As seen from Fig.2, the analytical formula and COMSOL results are in excellent agreement. The values of Cinand Coutat different frequencies are calculated from the analytical expressions given in Eqs.(7e)and(7f). The multiplication of the Cinand Cout terms appears in the denominator of the Cð1Þchip term in Eq.(7c), and Cð1Þchip appears in the expression for the ARF in Eq. (7a). Therefore, if the Cin Cout term is equal to zero, then the ARF tends to go to infinity, creating a resonance

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condition inside the channel. When the wall velocities are in-phase (K ¼ 1), the ARF distribution along the channel width direction is such that microparticles are focused to the midplane of the channel for all frequencies. As seen in Fig.

2(A), the frequencies where the ARF is maximum clearly correspond to the frequencies where Cin Cout is approach- ing zero, which is due to Cin¼ 0. This leads to a resonance condition referring to Eq.(8)for this specific case. Actually, these resonance frequencies correspond to the frequencies that satisfy the condition given in Eq.(10). It is also possible to observe that the in-phase motion of the walls leads to par- ticle motion toward the center of the channel at all frequen- cies [the final locations of the particles are shown by the white circles in Fig.2(A)]. Although the particles intend to move to the center of the channel for all frequencies, in practice, the particle motion may be observable only around the resonant frequencies (noting that in these case studies, damping is not included). Similarly, for the out-of-phase (K¼ 1) cases, the high amplitude ARF occurs at the fre- quency where the Cin Cout term approaches zero, but this time Cout¼ 0 leads to a resonance condition, referring to Eq. (9). The corresponding frequency for this case can be calculated using Eq. (11). As illustrated in Fig. 2(B), the ouf-of-phase velocity of the walls (K¼ 1) results in parti- cle motion toward the channel walls for all frequencies.

2. Different wall velocity

Figure3shows the results for the cases where the upper and lower wall velocities are not equal to each other. This situation can be realized by providingK values that are not

equal to K ¼ 1 or K ¼ 1. In simulations, the different amplitudes are realized by assigning different wall velocity amplitudes as the wall boundary conditions. Figure 3(A) shows the case where one wall velocity amplitude is half of the other wall with an in-phase motion of the walls (K

¼ 0.5). Figure3(B)demonstrates the same condition except the out-of-phase motion of the walls (K¼ 0:5). As in the previous cases, there is excellent agreement between the analytical derivation and the COMSOL results. Examining the results given in Figs. 2 and 3 reveals some common themes as well as some differences. One observation is that the frequencies where resonances in the ARF occur forK

¼ 0.5 and K ¼ 0:5 are exactly the same frequencies as shown in Figs.3(A)and3(B), respectively. Therefore, if the wall velocities do not have the same amplitudes, then the resonance frequencies occur at the resonance frequencies of the same wall velocity amplitude motion cases.

One common difference among these cases is that for the cases shown in Fig.2, the particles move either toward the middle of the channel or toward the channel walls; how- ever, for the cases shown in Fig.3, it can be observed that the particles move toward the middle of the channel, the channel walls, as well as the regions above or below the mid-plane of the channel. This shift can be quantified by cal- culating the corresponding x-value using the analytical expressions given in Eq.(7a). Actually, the only frequency which focuses the particles exactly at the mid-plane or the walls of the channel occurs at the resonance frequencies.

This issue can be clearly understood when Eqs. (7c) and (7d)are substituted in Eq.(7b), which can be simplified to

FIG. 2. (Color online) Acoustic pressure distribution within the chip material and channel and magnitude of the ARF and Cin Coutas a function of the fre- quency. (A)K¼ 1.0 and (B) K ¼ –1.0. The corresponding particle equilibrium positions are also included (blue and red correspond to negative and positive acoustic pressures, respectively).

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/¼ arctan 2Cin Coutð1  K2Þ C2outð1 þ KÞ2 C2inð1  KÞ2

" #

: (13)

Keeping in mind that the resonance frequencies occur where the term Cin Coutis equal to zero, the phase angle (/) needs to be zero or p for any K at resonance frequencies. At this point, we can conclude from Eq. (7a)that the particles are collected at the mid-plane or at the channel walls at the reso- nance frequencies where the phase angle is equal to zero or p, which can clearly be observed from Fig. 3. For non- resonance frequencies, if the absolute value ofK is not equal to one, then the phase angle (/) is nonzero and a shift occurs at the location where the particles are collected (observe particle accumulations at 500 and 900 kHz).

B. Device resonance phenomenon

Acoustic impedance of the chip material is an important parameter that affects the resonance frequency values.

There is a general agreement in the literature that to reach a resonant condition, the separation channel should be half of the wavelength of the acoustic waves of the fluid inside the channel. Actually, the studies which relate the resonance condition solely on the channel width use acoustically hard materials as the chip material (such as silicon or glass). This issue can be understood under the light of the expression given in Eq.(7e). In this expression, the second term has ^Z as the multiplier, which approaches zero for hard materials (because the acoustic impedance of the chip material is sig- nificantly larger than the acoustic impedance of water, i.e., Z^! 0). If the contribution of the second term can be neglected and Cinbecomes close to zero, then it can be writ- ten as

Cin¼ cos ðk‘Þ cos ðkff=2Þ ¼ cos ðk‘Þ cos ðp=2Þ  0:

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As noted above, if the channel width (‘f) is equal to half of the wavelength of the acoustic wave, then Cinapproximates to zero regardless of the chip width (‘). Therefore, the acoustofluidic devices made of hard materials experience a resonance frequency at a value when half of the acoustic wavelength is equal to the channel width (which is a func- tion of only ‘f and, therefore, can be identified as thechan- nel resonance). On the other hand, this is not the case for acoustofluidic devices made of acoustically soft materials.

For instance, Fig.2(A)shows that the resonance frequencies occurred at 557.5 kHz and 842.5 kHz, neither of which cor- responds to a frequency where the channel width is equal to the half-wavelength at these frequencies. One can under- stand the reason for this situation by again examining Eq. (7e). When Cin¼ 0, the resonance condition is met according to Eq. (8). Once the expressions for the half- wavelength for the channel width (kf ¼ 2p=kf; ‘f ¼ kf=2 where kfis the wavelength of the acoustic waves in the fluid) is substituted into the Cinexpression [Eq.(7e)], the follow- ing expression can be written:

Cin¼  ^Z sinðk‘Þ: (15)

Clearly, Eq.(15)shows that Cinis not zero; hence, the resonance frequency is not equal to the frequency at which the channel width is equal to the half-wavelength of the acoustic waves in water. Interestingly, the acoustofluidic device with a channel width ‘f ¼ 1:0 mm (which is equal to the half-wavelength at 700 kHz) can achieve a resonant fre- quency by forcing Cinto be equal to zero with an appropri- ate choice of the channel width. This appropriate width is equal to the half-wavelength inside the chip material at 700 kHz. For example, the width of the channel can be cho- sen as equal to half of the wavelength of the acoustic waves, which corresponds to ‘f ¼ kf=2¼ 1:07 mm at 700 kHz.

Thus, if the width of the chip material is chosen as

‘¼ kchip=2¼ 1:93 mm instead of ‘ ¼ 3:0 mm (the width of

FIG. 3. (Color online) Magnitude of the ARF and Cin Coutas a function of the frequency. (A)K¼ 0.5 and (B) K ¼ –0.5. The corresponding particle equi- librium positions are also included (blue and red correspond to negative and positive acoustic pressures, respectively).

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the chip in the previous case study), the resonance is equal to the channel resonance (i.e., frequency of 700 kHz). This shift of the resonance frequency to 700 kHz for the acousti- cally soft material (acrylic) can be clearly seen in Fig.4(A).

Therefore, it can be concluded that for the acoustofluidic chips made of acoustically hard materials, the resonance fre- quency depends strongly on the channel width (‘f) and can be called the channel resonance; however, for acoustofluidic chips made of acoustically soft materials, the resonance fre- quencies significantly depend on the channel width (‘f) and the width of the chip material (‘) and can be called the device resonance rather than the channel resonance. To fur- ther clarify this issue, the sensitivity of the resonant fre- quency on the width of the chip material is demonstrated in Figs.4(B)and4(C). It can be clearly seen that the sensitivity of the resonance frequency to the width of the chip material is high for the soft material, and the sensitivity of the reso- nance frequency on the width of the chip material is low for the hard material. Similar to previous cases, the analytical results are in excellent agreement with the simulations.

Further investigation of Eq.(7e)also reveals that in the case of a soft material, for a certain actuation frequency and channel width, it is possible to obtain the resonance through an adjustment of the width of the chip. For instance, if an actuation frequency of 700 kHz is targeted as the resonance frequency, even if an arbitrary channel width is selected (say ‘f ¼ 0:6 mm), which is not equal to the half- wavelength, using Eq.(7e), an appropriate width of the chip can be calculated as ‘¼ 4:51 mm to achieve the resonance

frequency in the ARF at 700 kHz. This resonance occurs at 700 kHz due to an appropriate choice of chip width, which can be defined as thedevice resonance. This is successfully illustrated in Fig.4(C). Hence, a device resonance can also be achieved for a width of the channel which is a non- integer multiple of the half-wavelength when microfluidic chips made of polymers are employed. This discussion explains how, in the literature, resonant conditions were achieved in polymer chip materials with channel widths that were not integer multiples of the half-wavelength (Dow et al., 2018;Lissandrello et al., 2018;Muelleret al., 2013;

Savageet al., 2017;Silvaet al., 2017).

IV. 2D/3D NUMERICAL MODELS

In Sec.III, the analytical derivations are verified with numerical simulations consistent with the assumptions of the analytical derivation. In this section, to understand the practical acoustophoresis problem better, the assumptions are relaxed and the effects are discussed through 2D and three-dimensional (3D) numerical simulations. In the 2D simulations, first, the damping assumption is removed.

Then, the acoustically hard boundary conditions of the acoustic domains are replaced with the acoustic impedance (of air) at the boundaries, which results in the violation of the plane wave assumption. Finally, piezoelectric actuators are included in the 2D model and the effect of the presence of the piezoelectric actuator is investigated. For the case with the piezoelectric actuators, the acoustic domains have

FIG. 4. (Color online) Magnitude of the ARF as a function of the frequency with 625% variation in the width of the chip material. (A) Soft (acrylic) and (B) hard (SiC) material. (C) Demonstration of the device resonance behavior.

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the acoustic impedance of air as the boundary conditions, and the structural domains (piezoelectric material) have the free boundary conditions. In the one-dimensional (1D) verification studies, the chip walls at the two sides of the microchannel move either in equal amplitudes in-phase or out-of-phase with respect to each other. Additionally, the cases in which the two walls move in different amplitudes are also considered. The described wall motions can be achieved rather easily by assigning the velocity boundary conditions on the walls of the chip material. Achieving these boundary movements is not as straightforward when piezo- electric actuators are employed for actuation because the chip wall motion is provided by the piezoelectric actuator rather than direct application of the boundary conditions on the walls of the chip material. The details of the 2D/3D numerical models and the implemented boundary conditions are given in the supplementary material.1

V. RESULTS FOR 2D NUMERICAL MODEL

Figure 5shows the cases of K¼ 1.0 and K ¼ 0.5. In Fig.5, the yellow curve corresponds to the case of the fre- quency response of the mean ARF amplitude in the case

with a fixed boundary condition along with a damped acous- tic domain (i.e., the same boundary condition as that of the 1D numerical model; see Fig. S1A in the supplementary material1). Due to the fact that damping is included, the results of the analytical derivation, which exclude damping, is not given in Fig.5. For the case whenK¼ 1.0, the inclu- sion of damping does not shift the resonance frequencies significantly from the values predicted by the analytical model. However, when the boundary conditions are changed from acoustically fixed boundary conditions to air imped- ance boundary conditions, the planar wave assumption is violated. As a result of the existence of nonplanar waves, we can see additional resonance peaks between the initial reso- nance peaks at 557.5 and 842.5 kHz. It can be seen clearly that these resonance peaks are due to resonances occurring along the length of the chip. It can be plainly observed from the equilibrium positions of the particles in Fig.5that there is an increasing number of nodal points (along the length of the channel) as the longitudinal resonance frequency values increase. However, the particle accumulations near the reso- nance frequencies at 562.5 and 847.5 kHz, which are pre- dicted by the analytical model, have no ARF nodal lines along the length direction of the channel. At these two

FIG. 5. (Color online) Magnitude of the ARF as a function of the frequency and the corresponding particle equilibrium positions for the 2D numerical model without the piezoelectric actuator (PZT). (A)K¼ 1.0 (in-phase channel wall motion) and (B) K ¼ 0.5. In the particle equilibrium figures, blue and red corre- spond to negative and positive acoustic pressures, respectively.

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frequencies, the acoustophoretic forces are almost constant along the length of the chip. For the case with K¼ 0.5, it can be seen in the damped case [solid line in Fig.5(B)] that there exists a resonance peak (near 690 kHz) that belongs to the out-of-phase wall motion resonance (K ¼ –1.0) of the 1D case. As expected, particles are not collected at the cen- terline of the channel for that resonance frequency (692.5 kHz). For the case of K ¼ 0.5, the other resonance peaks are at similar locations as the K¼ 1.0 case but with lower resonance peak values.

Figure 6shows the cases in which piezoelectric actua- tors are employed for the actuation of the acoustophoretic system. The existence of the piezoelectric actuator moved two modes, predicted by the analytical model at 557.5 kHz and 842.5 kHz, only slightly. As seen in Figs. 6(A) and 6(B), there are several resonances around 700 kHz. These resonances are due to the resonance behavior of the actua- tors (the thickness mode along the poling direction is near 700 kHz) and complex vibration patterns imposed on the chip (compared to the uniform wall motion in the previous cases). For this case, the ARF amplitude is the highest at 692.5 kHz, which is believed to be caused by the

piezoelectric resonance. Similar to the previous cases, all of the particles are collected around the center of the channel for case whenK¼ 1.0, whereas this is not the case when K

¼ 0.5. Because of damping within the chip material, the presence ofnonplanar waves, and the nonuniform excitation of the chip walls due to the piezoelectric actuators, the reso- nance frequencies predicted by the analytical model are no longer accurate. On the other hand, the equilibrium positions of the particles predicted by the analytical model and 2D numerical models are quite similar for different actuation modes (K ¼ 1.0 and K ¼ 0.5); therefore, the analytical approach can be used to understand the behavior of the par- ticles in the chip for different actuation modes.

VI. RESULTS FOR THE 3D NUMERICAL MODEL The results of the 3D simulations are given for an acrylic chip in Fig.7. The simulations are performed with voltage values that lead to in-phase motion (K ¼ 1.0) and out-of-phase motion (K¼ –1.0) of the chip material on both sides of the channel, as well as different amplitude vibra- tions of the chip materials for K ¼ 0.2. The detailed

FIG. 6. (Color online) Magnitude of the ARF as a function of the frequency and corresponding particle equilibrium positions for the 2D numerical model with the PZT. (A)K¼ 1.0 (in-phase channel wall motion) and (B) K ¼ 0.5. In the particle equilibrium figures, blue and red correspond to negative and posi- tive acoustic pressures, respectively.

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explanations of how the voltages are set to conduce the required chip material vibrations are explained in the supplementary material (please see Fig. S2).1The 3D simu- lation studies show that, similar to 1D and 2D cases, the in- phase motion of the chip walls (K¼ 1.0) leads to the ARF distribution, which moves the particles toward the center of the channel and out-of-phase (K ¼ –1.0) leads to particle motion toward the channel walls. Again, similarly, when the chip walls have vibrations that are not equal in magnitude, the frequency response has the resonance frequencies of the in-phase and out-of-phase configurations.Figure 7(C)illus- trates that for the acrylic chip, the channel resonance of 937.5 kHz is not an ARF resonance. This was also the case in the previous 1D and 2D cases for the acrylic chip material.

VII. CONCLUDING REMARKS

In this study, implementing similar assumptions, the well-known ARF expression is extended to include the effect of the chip material on the acoustophoretic force and resonance behavior inside the channel. A major contribution of this study is the introduction of a modified ARF expres- sion, which reveals the effect of the chip material and

actuation modes on the resonance frequencies and the result- ing equilibrium position of the manipulated particles.

Through analytical and numerical calculations, it is demonstrated that the final locations of the particles for a given frequency are determined by the value and sign ofK (which is the ratio of the outer chip wall velocities of each sidewall along the channel width direction). The effects of the acoustic properties of the chip are also discussed. It is shown with the analytical expressions that when the acousti- cally hard materials are used, one of the ARF resonances is expected to be at the channel resonance (i.e., near the fre- quency where the acoustic half of the acoustic wavelength is equal to the channel width). However, for acoustically soft materials (like polymers), the ARF resonance is at the device resonance, which is significantly affected by the chip width. In addition, these findings suggest that for acousti- cally soft materials, the traditional practice of matching the piezoelectric actuator resonance to the channel resonance is not a good strategy because the ARF resonance will not be at the channel resonance.

Following our discussions based on our analytical model, we also developed a numerical model to implement realistic boundary conditions and the effect of the presence of piezoelectric transducers, and we systematically remove

FIG. 7. (Color online) 3D numerical model for an acrylic chip. The displacement field are (A) K ¼ 1.0 (out-of-phase PZT voltage) and (B) K ¼ –1.0 (in-phase PZT voltage).

Magnitude of the ARF as a function of the frequency and corresponding parti- cle equilibrium positions are (C) K

¼ 1.0 (out-of-phase PZT voltage), (D) K¼ 0.2 (uneven PZT voltage), and (E) K ¼ –1.0 (in-phase PZT voltage). In the particle equilibrium figures, blue- green and red-yellow correspond to negative and positive acoustic pres- sures, respectively.

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the underlying assumptions of our analytical model through case studies and discuss how these conclusions can be extended for real-life acoustophoretic applications. It is observed that the fundamental observations are still valid when 3D numerical models are considered. In the 3D mod- els, the chip material is modeled as a structural domain rather than an acoustic domain. Two different piezoelectric actuators are employed from the two sides of the chip mate- rial. When the two actuators are fed with the same amplitude in-phase voltage, which corresponds to K¼ –1.0, our deri- vations showed that the particles cannot move toward the center of the channel but rather move toward the channel walls. This clearly explains why a single simple piezoelec- tric actuator placed symmetrically on top of a microchannel (which correspond toK¼ –1.0 because the chip material on two sides of the channel moves with opposite phases) cannot be used to move the particles toward the center of the chan- nel. That is why, in practice, the piezoelectric actuator is always shifted with respect to the center of the microchan- nel, which creates the case withK6¼ –1.0.

We believe that the extended ARF formulation, which accounts for the acoustophoretic chip material [Eq.(7a)], is an essential contribution to the acoustofluidic field. We also believe that in light of these discussions, the fundamental and/or practical aspects of the BAW based acoustofluidic particle manipulation can be better understood for the fur- ther development of more efficient acoustofluidic platforms for specific applications. As one of our future research direc- tions, we will investigate the applicability of the present results to a system where out-of-plane bending waves domi- nate the acoustophoretic response.

ACKNOWLEDGMENTS

This study is financially supported by the Turkish Scientific and Technological Research Council under Grant No. 115M684. B.C¸. would like to acknowledge funding from the Turkish Academy of Sciences through the Outstanding Young Scientist Program (T €UBA-GEB_IP) and The Science Academy, the Turkey Distinguished Young Scientist Award (BAGEP).

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