Bulk acoustofluidic devices driven by thin-film transducers and whole-system resonance modes
BBulk acoustofluidic devices driven by thin-filmtransducers and whole-system resonance modes
Andr´e G. Steckel a) and Henrik Bruus b Department of Physics, Technical University of Denmark,DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark (Dated: 29 January 2021)
In acoustofluidics, acoustic resonance modes for fluid and microparticle handling are tradi-tionally excited by bulk piezoelectric transducers. In this work, we demonstrate by numericalsimulations in three dimensions (3D) that integrated piezoelectric thin-film transducers con-stituting less than 0.1% of the device work equally well. The simulations are done using awell-tested and experimentally validated numerical model. Our proof-of-concept example isa water-filled straight channel embedded in a mm-sized glass chip with a 1- µ m thick thin-filmtransducer made of Al . Sc . N . We compute the acoustic energy, streaming, and radiationforce, and show that it is comparable to that of a conventional silicon-glass device actuatedby a bulk PZT transducer. The ability of the thin-film transducer to create the desiredacoustofluidic effects in bulk acoustofluidic devices rely on three physical aspects: The in-plane-expansion of the thin-film transducer under the orthogonal applied electric field, theacoustic whole-system resonance of the device, and the high Q-factor of the elastic solidconstituting the bulk part of the device. Consequently, the thin-film device is surprisinglyinsensitive to the Q-factor and resonance properties of the thin-film transducer. © [http://dx.doi.org(DOI number)][XYZ] Pages: 1–13 I. INTRODUCTION
An increasing number of microscale ultrasoundacoustofluidic devices are used for applications withinclinical diagnostics, biology, and forensic sciences.
Ex-amples include but are not limited to rapid sepsis diag-nostics by detection of bacteria in blood, enrichment ofprostate cancer cells in blood, high-throughput cytome-try and multiple-cell handling, cell synchronization, single-cell patterning and manipulation, and size-independent sorting of cells. Furthermore, acoustoflu-idics has been used for massively parallel force mi-croscopy on biomolecules, acoustic tweezing, and non-contact microfluidic trapping and particleenrichment Most applications rely on one of two basic methodsfor exciting the ultrasound field. One method is based onsurface acoustic waves, excited by interdigitated metal-lic electrodes positioned on the surface of a piezoelec-tric (PZE) substrate. The other method relies on bulkacoustic waves excited locally in liquid-filled acoustic mi-crochannels or microcavities defined in acoustically hardmaterials by an attached bulk transducer, or in cavi-ties with a thin silicon-membrane lid driven by a lead-zirconate-titanate (PZT) thin film. Recently, the bulk-acoustic-wave method have beenextended the concept of whole-system ultrasound res- a) [email protected] b) [email protected] onances (WSUR), where the resonant acoustic wavesare defined by the whole system, and not just themicrocavities. In this paper we extend the WSURmethod by substituting the large bulk PZE transducerby a tiny PZE thin-film transducer integrated on the sur-face of the device and constituting less than 0.1% v/v ofthe resulting device.Integrated thin-film PZE transducers have beenused extensively for actuating electromechanical sys-tems, often made of aluminum nitride (AlN). Thin-filmtransducers made of AlN are structurally and chemi-cally stable, they have a low dielectric and mechan-ical loss, they are compatible with standard silicon-based CMOS microfabrication techniques. Academicapplications of AlN thin-film transducers include RFfilters, suspended microchannel resonators, contourmode resonators, switches, and accelerometers. AlN thin-film have been deposited on substrates of sap-phire, crystal quartz, fused silica, and silicon, andon 30- µ m-thick Si membranes. Commercially, AlN-sputtered thin films are used in thin-film bulk-waveacoustic resonator filters. However, hitherto thin-filmtransducers have not yet been applied in MHz bulkacoustofluidic devices.In this paper, based on a well-tested and experi-mentally validated numerical model, we demon-strate by three dimensional (3D) numerical simulationsthat glass chips with integrated PZE thin-film transduc-ers constituting less than 0.1% v/v of the system, formdevices with an acoustofluidic response fully on par withthat obtained in a conventional silicon-glass device ac-
J. Acoust. Soc. Am. / 2 February 2021 1 a r X i v : . [ phy s i c s . f l u - dyn ] F e b uated by a bulk PZT transducer. This, perhaps sur-prising, result offers several advantages for the practicalapplication of thin-film transducers within acoustoflu-idics: Thin-film devices do not depend on resonanceproperties of the thin-film transducer itself, and thin-film devices can be fabricated by well-controlled repro-ducible microfabrication techniques subject to parallelmass-fabrication processes.The paper is organized as follows: In Section IIwe summarize the theory and numerical model usedthroughout the paper. In Section III we present a proof-of-concept example showing that a thin-film acoustoflu-idic device perform on par with a conventional bulk-transducer device. In Section IV we discuss the physicalprinciple of the thin-film transduction process, the ro-bustness of the device to material, thickness, and qualityfactor of the thin-film transducer, the role of the shapeof the thin-film-transducer electrodes, and the sensitivityof the device to shifts in the channel position away fromexact centering in the glass chip. Finally, in Section Vwe present our conclusions. II. THEORY
We use, and in this section briefly summarize, thetheory and numerical model developed by Skov et al. including the effective boundary layer theory by Bachand Bruus. This model is well-tested and experimen-tally validated, and although originally stated foracoustofluidic devices with bulk PZT transducers, it istrivially extended to describe other types of PZE trans-ducers, including thin-film transducers.
A. Governing equations for the time-harmonic fields
We consider a time-harmonic electric potential˜ ϕ ( r , t ), which excites the PZE transducer and inducesa displacement field ˜ u ( r , t ) in the solids as well as anacoustic pressure ˜ p ( r , t ) in the fluid channel and in thecoupling layer. All of these fields ˜ F ( r , t ) separates intoa complex-valued amplitude F ( r ) and a complex time-harmonic phase factor with frequency f ,˜ F ( r , t ) = F ( r ) e − i ωt , with ω = 2 πf. (1)The phase factor e − i ωt cancels out in the following lineargoverning equations, leaving just the amplitude fields.From first-order perturbation theory follows that theacoustic pressure p in the fluid is governed by theHelmholtz equation with damping coefficient Γ fl , ∇ p = − ω c (cid:0) fl (cid:1) p , with Γ fl = (cid:16) η fl + η bfl (cid:17) ωκ fl , (2)where c fl is the speed of sound, ρ fl is the density, κ fl =( ρ fl c ) − is the isentropic compressibility, and η fl and η bfl are the dynamic and bulk viscosity of the fluid, respec-tively, see Table I. The acoustic velocity v of the fluid TABLE I. Parameter values of water at 25 ◦ C used in thenumerical simulations. Parameter Value Parameter Value ρ fl
997 kg m − η bfl .
485 mPa s c fl − Γ fl . − fκ fl
448 TPa − η fl .
890 mPa s is proportional to the gradient of the pressure p , v = − i 1 − iΓ fl ωρ fl ∇ p (3)For the linear PZE transducer, whether made of AlN,Al x Sc x N, or PZT, the electrical potential ϕ inside thePZT, is governed by Gauss’s law for a linear, homoge-neous dielectric with a zero density of free charges, ∇ · D = ∇ · (cid:2) − (1 + iΓ ε ) ε · ∇ ϕ (cid:3) = 0 , (4)where D is the electric displacement field and ε the di-electric tensor. The governing equation for the mechani-cal displacement field u in a linear elastic solid (includ-ing the PZE) with density ρ sl , is Cauchy’s equation − (1 + iΓ sl ) ρ sl ω u = ∇ · σ sl , (5)In the PZE, the complete linear electromechanical cou-pling relating the stress and the electric displacement tothe strain and the electric field is given in Voigt notationas, σ xx σ yy σ zz σ yz σ xz σ xy D x D y D z = C C C − e C C C − e C C C − e c − e
00 0 0 0 c − e c e ε e ε e e e ε ∂ x u x ∂ y u y ∂ z u z ∂ y u z + ∂ z u y ∂ x u z + ∂ z u x ∂ x u y + ∂ y u x − ∂ x ϕ − ∂ y ϕ − ∂ z ϕ . (6)The remaining three components of the stress tensor aregiven by the symmetry relation σ ik = σ ki . Similarly, theCauchy equation (5) governs u in a purely elastic solid,but now the stress-strain relation (6) includes only thefirst six rows and first six columns, as D and ϕ do notcouple to σ sl and u . The parameter values are listed inTable II. B. Governing equations for the steady time-averaged fields
The non-linearity of the governing equation results inhigher order responses to the time-harmonic actuation.Here, we are only interested in the steady time-averagedsecond-order response and define F ( r ) = (cid:10) F ( r , t ) (cid:11) = ω π (cid:82) πω F ( r , t ) d t . A time-average of a product of twofirst-order fields is also a second-order term, written as ABLE II. Parameters of the solids at 25 ◦ C used in thenumerical simulation. For glass C = C − C . For allPZE used in this work C = C − C . Parameter Value Parameter Value
Thin-film aluminum nitride , AlN ρ sl − Γ sl C C C C C C e − . − e − .
39 C m − e − Γ ε . (cid:15) (cid:15) (cid:15) (cid:15) Thin-film aluminum scandium nitride , Al . Sc . N ρ sl − Γ sl C C C C C C e − .
65 C m − e − .
32 C m − e − Γ ε . (cid:15) (cid:15) (cid:15) (cid:15) Bulk and thin-film lead zirconium titanate , PZT ρ sl − Γ sl C
168 GPa C
123 GPa C
110 GPa C C C e − . − e .
86 C m − e − Γ ε . ε ε ε ε Glass, Schott D263 ρ sl − E s C C C sl c lo − c tr − Glass, Pyrex ρ sl − E s C C C sl c lo − c tr − Silicon substrate ρ sl − Γ sl C C C (cid:10) A B (cid:11) = Re (cid:2) A B ∗ (cid:3) , where the asterisk denote com-plex conjugation. The acoustic streaming v is such atime-averaged field. It is a steady-state, incompressibleStokes flow driven by the slip velocity stated in Sec-tion II C and the time-averaged acoustic dissipation bodyforce proportional to Γ fl , η fl ∇ v = ∇ p − Γ fl ω c Re (cid:104) p ∗ v (cid:105) , ∇ · v = 0 . (7) The time-averaged acoustic energy density in thefluid is given as the sum of the kinetic and compressionalenergy densities, E flac = 14 ρ fl (cid:12)(cid:12) v , fl (cid:12)(cid:12) + 14 κ fl (cid:12)(cid:12) p , fl (cid:12)(cid:12) . (8)The acoustic radiation force F rad acting on particlesin the fluid is the gradient of the potential U rad , spec-ified for particles with radius a , density ρ ps , and com-pressibility κ ps , suspended in a fluid with density ρ fl andcompressiblity κ fl , F rad = − ∇ U rad , (9a) U rad = πa (cid:16) f κ fl | p , fl | − f ρ fl | v , fl | (cid:17) , (9b) f = 1 − κ ps κ fl , f = 2( ρ ps − ρ fl )2 ρ ps + ρ fl , (9c)where, f and f is the so-called acoustic monopole anddipole scattering coefficient, respectively. C. Boundary conditions fluids, solids, and PZE
The boundary conditions of the fields on all bound-aries and interfaces of the model are specified as follows.On the surfaces facing the surrounding air, we assumezero stress on the solid and the PZE as well as zerofree surface charge density on the PZE. On the surfaceswith electrodes, the PZE has a specified ac-voltage am-plitude. On the internal surfaces between solid and PZE,the stress and displacement are continuous, and likewiseon the fluid-solid interface, but here for the latter inthe form of the effective boundary conditions derived byBach and Bruus. These effective boundary conditionsinclude the viscous boundary layer analytically, and thuswe avoid resolving these very shallow boundary layers nu-merically. The effective boundary conditions include thevelocity v sl = − i ω u of the solid (sl) and the complex-valued shear-wave number k s = (1 + i) δ − of the fluid(fl), where δ fl = (cid:112) η fl / ( ρ fl ω ) ≈ . µ m is the thicknessof the boundary layer.sl-fl: σ sl · n = − p n + i k s η fl ( v sl − v (cid:1) , (10a)sl-air: σ sl · n = , (10b)PZE-air: D · n = 0 , (10c)top elec.: ϕ = 0 , (10d)bot elec.: ϕ = ± ϕ . (10e)Further, at fluid-solid interfaces, the slip velocity v bc2 driving the streaming takes into account both the mo-tion of the surrounding elastic solid and the Reynolds J. Acoust. Soc. Am. / 2 February 2021 3 tress induced in viscous boundary layer in the fluid, v = v bc2 n · v bc2 = 0 , (11a)( − nn ) · v bc2 = − Re (cid:20)(cid:18) − i4 ω ∇ (cid:107) · v ∗ (cid:107) + i2 ω ∂ ⊥ v ∗ ⊥ (cid:19) v (cid:107) (cid:21) − ω ∇ (cid:107) (cid:12)(cid:12) v (cid:107) (cid:12)(cid:12) . (11b)Here, we have stated the special case of the slip velocity v bc2 , which is only valid near acoustic resonance, wherethe magnitude | v | of the acoustic velocity in the bulk ismuch larger than ω | u bc1 , sl | of the walls.Finally, we use the symmetry at the yz - and xz -planeto reduce the model to quarter size in the domain x > y > yz -plane x = 0 and anti-symmetry at the xz -plane y = 0, Symmetry, x = 0 : ∂ x ϕ = 0 , ∂ x p fl1 = 0 , ∂ x p fl2 = 0 , (12a) v sl1 ,x = 0 , ∂ x v sl1 ,y = 0 , ∂ x v sl1 ,z = 0 , (12b) v fl2 ,x = 0 , ∂ x v fl2 ,y = 0 , ∂ x v fl2 ,z = 0 , (12c)Anti-symmetry, y = 0 : ϕ = 0 , p fl1 = 0 , ∂ y p fl2 = 0 , (12d) v sl1 ,x = 0 , ∂ y v sl1 ,y = 0 , v sl1 ,z = 0 , (12e) v fl2 ,x = 0 , v fl2 ,y = 0 , ∂ y v fl2 ,z = 0 . (12f) III. RESULTS COMPARING A THIN-FILM AND A BULKTRANSDUCER DEVICEA. The two main model devices
As a proof of concept that a tiny thin-film trans-ducer is able to drive an acoustofluidic device as well asa conventional bulk transducer, we study the two devicesshown in Fig. 1(a,b), oriented along the x -, y - and z -axis, and both containing a water-filled microchannel oflength L wa = 35 mm, width W wa = 0 .
377 mm, and height H wa = 0 .
157 mm, chosen as a typical channel size usedin the literature and specifically studied experimentallyand theoretical in Ref. 47.The thin-film device consist of a rectangular glassblock of L gl = 40 mm, width W gl = 3 .
02 mm, andheight H gl = 1 . . Sc . N thin-film trans-ducer of length L thf = L gl , width W thf = L gl , and height H thf = 1 µ m, is deposited on the bottom surface of the xy -plane. The anti-symmetric voltage actuation is madepossible by splitting the bottom electrode in half by a40- µ m-wide gap along the x -axis. Here, the microchan-nel is centered both along the x -axis and the y -axis, butits top aligns with the center of the glass height H gl tomimic that the glass block consists of two glass slabs ofequal height bonded together, but with the microchanneletched into the top surface of the bottom slab. This spe-cific device is chosen, because in a recent study, we suc-cessfully modeled and experimentally validated a similar thin-film-glass-block device without the microchannel. Note that the fraction of the total volume occupied by thethin-film transducer is minute, V thf / V tot = 0 .
07% v/v.The bulk-transducer device has been studied ex-tensively both experimentally and numerically in theliterature.
It consists of a silicon substrate oflength L sl = 40 mm, width W sl = 2 .
52 mm, and height H si = 0 .
35 mm, into which the centered microchannelis etched, and onto which is bonded a Pyrex glass lid ofthe same length and width, but with the height H py =1 .
13 mm. This silicon-glass chip is placed off-center on anominal 2-MHz PZT-transducer of L pzt = 40 mm, width W pzt = 5 mm, and height H pzt = 1 mm, such that theright-most side walls align. In the actual device, thetransducer is glued to chip, but here we neglect the gluelayer and assume an ideal bonding instead. Note that thefraction of the total volume occupied by the bulk PZTtransducer is large, V pzt / V tot = 57% v/v.Using these dimensions and the material parameterslisted in Tables I and II we implement these two 3Ddevice models in the commercial finite-element softwareCOMSOL Multiphysics 5.4, closely following the imple-mentation method described in Ref. 34. All simulationsare run on a workstation with a 16-core processor Inteli9-7960X @ 3 .
70 GHz boost clock and with 128 GB ram.
B. Mode analysis of the two devices
First step in our analysis is to identify good acousticresonances in the two devices, which we actuate in a com-parable manner with a peak-to-peak voltage of ϕ = 1 V.In the thin-film device the voltage amplitude of the ac-voltage on the ”positive” (”negative”) half of the topelectrode is set to + ϕ ( − ϕ , 180 ◦ out of phase) rel-ative to the grounded bottom electrode. Similarly, thevoltage amplitude on the top electrode in the bulk-PZTdevice is set to + ϕ relative to the grounded bottomelectrode. The frequency of the actuation voltage is thenswept from 0.1 to 3.5 MHz, while monitoring the acousticenergy density E flac , Eq. (8), in the water. The frequencysteps in the sweep is adaptive, range from ∆ f = 16 kHzwhen the local curvature in E flac ( f ) is small (far from res-onance peaks) down to ∆ f = 0 .
03 kHz when it is large(near resonance peaks).As expected, the strongest resonance peak in E flac happens near the hard-ward standing half-wave reso-nance f = c fl W wa = 2 MHz. This main resonance islocated at f thf = 1 .
946 MHz with a maximum energydensity of E flac ( f thf ) = 72 . − for the thin-film device,and at f pzt = 1 .
927 MHz with a E flac ( f pzt ) = 21 . − .The amplitude p of the pressure and u of the dis-placement for these main resonance modes in the twodevices are shown in Fig. 1(b,c). One immediate conclu-sion is that the quality of the resulting resonant pressuremode in the two devices are comparable, a nearly perfectanti-symmetric wave across the channel with only weakvariations along the channel, and the pressure amplitudeof 1.23 MPa in the thin-film device 2.2 times the 0.55- [1 .
23 MPa] p [0 .
55 MPa] v [190 (cid:181) m / s] v [34 (cid:181) m / s] F rad [45 pN] F rad [9 pN](a)(b) E flac = 72 J m − f = 1 .
946 MHz S c h o t t D l a s s Al . Sc . N 35 mm
HHY . m m . m m (c)(d) f = 1 .
927 MHz E flac = 22 J m − P y r e x S i P Z T
35 mm . m m . m m . m m . m m . m m (e) (f) (g) (cid:181) m157 (cid:181) m Al . Sc . N PZT Al . Sc . N PZT Al . Sc . N PZT ? x/L gl FIG. 1. (a,b) A glass chip driven by a 1-um-thick Al . Sc . N-thin-film transducer (not visible) at resonance f = 1 .
946 MHzactuated with 1 Vpp. The color plots show the displacement field u from 0 (blue) to 15 nm (yellow) and p from − − f = 1 .
927 MHz actuatedwith 1 Vpp. The color plots show the displacement field u from 0 (blue) to 3 . p from −
550 kPa (blue) to+550 kPa(red). (d) Cross section plots for the bulk pressure field p of the Al . Sc . N driven system on the left and the PZTdriven system of the right, with the lengths at which the cross sections were taken in the device. (e) Cross sections of thestreaming velocity v of the Al . Sc . N driven system on the left and the PZT driven system of the right. (f) Cross sectionsof the radiation force F rad for suspended 5-um-diameter polystyrene particles of the Al . Sc . N driven system on the left andthe PZT driven system of the right.
MPa amplitude in the bulk-PZT device. Clearly, the tiny0.07% v/v thin-film transducer can deliver a fully com-parable, and perhaps even better, acoustic response inthe device in comparison with conventional large large57% v/v bulk PZT transducer.When inspecting the displacement field, it is seenthat the displacement field has a more regular mode witha larger 15-nm amplitude in the thin-film device com- pared to the more complex resonance mode with a 3.6-nmamplitude in the larger volume of the bulk-PZT device.Perhaps this is an indication of the regular mode beingmore efficient in transferring acoustic energy from thetransducer through the solid into the microchannel. InFig. 1(d) is shown the pressure in seven vertical chan-nel cross sections equally spaced along the channel fromits center to its end, showing the weak axial variations
J. Acoust. Soc. Am. / 2 February 2021 5 n p for both devices. The bulk-PZT device appearsmore constant along the channel, however, it is 2.2 timesweaker than p in the thin-film device, in which pres-sure wave nevertheless dies out towards the ends of thechannel. C. The acoustic radiation force and streaming at resonance
The acoustic modes p and u are the basic fields giv-ing rise to the steady time-averaged responses used forapplications in acoustofluidic devices, namely the acous-tic streaming v in the fluid and the radiation force F rad acting on suspended microparticles. In Fig. 1(e,f) theseresponses are shown in the same seven cross sections asused in Fig. 1(d). Being second-order repsonses, the im-provement factor of 2.2 in pressure becomes an factorof 2 . ≈ IV. PHYSICAL ASPECTS OF ACOUSTOFLUIDIC DE-VICES DRIVEN BY THIN-FILM TRANSDUCERS
In this section we use the numerical model to studyvarious physical aspect of acoustofluidic devices withthin-film transducers. The study includes the physicalprinciple of the thin-film transduction process, the ro-bustness of the device to material, thickness, quality fac-tor of the thin-film transducer, the role of the shape ofthe thin-film-transducer electrodes, and sensitivity of thedevice to shifts in the channel position away from exactcentering in the glass chip.
A. The physical mechanism of thin-film-actuated bulk acous-tic waves
The ability of the thin-film transducer to create thedesired acoustofluidic effects in a bulk acoustofluidic de-vice rely on three physical aspects of the system: Thein-plane-expansion of the the thin-film transducer underthe action of the orthogonal applied electric field, theacoustic whole-system resonance of the device, and thehigh Q-factor of the elastic solid constituting the bulkpart of the device.Traditional bulk PZE transducers typically work byexciting a standing half-wave in the mechanical displace-ment field along the polarization axis of the transducerby applying an electric field in the same direction, facil- z y
FIG. 2. Numerical simulation of displacement field u inthe Al . Sc . N thin-film device of Fig. 1(a), enhanced by afactor 7000 for clarity, of the whole-system resonance f thf =1 .
946 MHz. Vector plot of u (red vectors) and color plot ofits magnitude | u | from 0 (dark blue) to 15 nm (light yellow).The mode is excited by the Al . Sc . N thin-film transducerdriven by a 1-V pp AC-voltage at the frequency f = f thf =1 .
946 MHz. The in-plane strain ∂ y u ,y e y (magenta vectors)generated by the transducer on the transducer-glass interfaceshowing an expansion (contraction) on the left (right) sidecompatible with strain of whole-system resonance mode. itated by a large longitudinal PZE coefficient e . Thisthickness mode is fairly easy to compute and to designfor. Moreover, this mode is also relatively good at trans-ferring acoustical power into a device attached to thetransducer, by the large component along the surfacenormal of the induced displacement field. Conventionalbulk PZE transducers with a millimeter-thickness typi-cally have good resonances in the low MHz regime.In contrast, the half-wave transducer resonances ofthe thin-film transducers are pushed up into the GHzregime due to the micrometer-sized thickness, muchhigher than the low-MHz frequencies usually used inacoustofluidics. Therefore, the transduction studied inthis work is dominated by the transverse PZE coeffi-cient e . The large electric field in the order of MV/mthat results from applying, say, a potential difference of1 V pp across a 1- µ m-thick thin-film transducer, generatesa large strain that accumulates along the millimeter-sizedtransducer-glass interface, the first of the three physicalprerequisites. When this strain oscillates at a resonancefrequency of the glass block constituting bulk part of thedevice, the corresponding eigenmode of the glass block isexcited if it has a compatible strain pattern, the secondphysical prerequisite. We emphasize that this transduc-tion mechanism does not rely on exciting any resonancesin the thin-film transducer, but instead on exciting res-onances in the whole system, of which the transducer isonly a minute part.The thin-film transduction mechanism is exemplifiedin in Fig. 2 by the mode f thf the thin-film device Fig. 1(a).The in-plane strain ∂ y u ,y e y on the transducer-glass in-terface generated by the anti-symmetrically driven split-electrode thin-film transducer correspond to an expan-sion on the left side and a contraction on the right side. his strain-pattern is compatible with that of the whole-system resonance mode, which therefore is excited witha large 15-nm displacement amplitude. The resultinganti-symmetric oscillatory displacement field of the glassblock pushes on the water in the channel, which leads ananti-symmetric pressure wave p , Fig. 1(ed), with the de-sired acoustofluidic properties shown in Fig. 1(e-f). Thethird and last physical prerequisite is the high quality fac-tor of the whole system as an acoustic resonator. Becausethe thin-film transducer in our main example constitutesonly a 0.07% v/v, the quality factor is completely domi-nated by that of the glass block, which has a high value Q ∼ in a typical glass. Note that the pressure wavein the water-filled microchannel does not need to be aresonant standing half-wave, but if it is, its amplitudemay be enhanced further.
B. The robustness of the device to material, thickness, andquality factor of the thin-film transducer
The above-mentioned thin-film transduction methodimplies that the functionality of the thin-film acoustoflu-idic device is robust to changes in several characteristicproperties of the thin-film transducer, essentially becauseof its small volume fraction of the whole system. In thefollowing we study three of such properties, namely thematerial, thickness, and quality factor of the thin-filmtransducer.We study three types of PZE materials. One is thecommonly used and commercially available PZT havinglarge PZE coefficient e . One drawback of this materialis its lead contents, which is being out-phased for health-and environmental reasons, and another is the difficultyin fabricating the material with a sufficiently low dissi-pation. Whereas other materials have lower PZE coeffi-cients than PZT, they may be fabricated with higher pu-rity and less dissipation. The lead-free AlN is a choice forits simpler and more well-controlled high-quality deposit-ing on a variety of substrates, which allows for higherbreak-down voltages that almost make op for the lowerPZE coefficient. Al x Sc x N offers a PZE coefficient be-tween the values of PZT and AlN, with many of thesame advantages as AlN, but has a more complex fabri-cation process and a lower break-down voltage. In Fig. 3we show simulation results three different PZE materi-als, while keep all other quantities fixed in the model:AlN, Al . Sc . N , and PZT Pz26, maintaining this orderwhen referring to the numerical results in the following.In spite of the very different material parameters listedin Table II, the resulting acoustofluidic response of themain resonance f thf is nearly the same. Within 0.3%, theresonance frequencies are identical f thf = 1 . | p | = 0 .
35, 1.20, and 3.50 MPa, the displacement | u | = 3 .
5, 15, and 38 nm, and the acoustic energy den-sity E flac = 6 .
66, 72.1, and 669 J m − . Besides the obvi-ous difference in amplitude, the computed whole-systemresonance is nearly the same in all three cases, showing (a) 3040 (cid:181) mGlassSchott D263700 (cid:181) m157 (cid:181) m543 (cid:181) m1400 (cid:181) m 377 (cid:181) m1- (cid:181) m thin film1000 (cid:181) m 1000 (cid:181) m+0 . − . L g l = (cid:181) m . M H z | u | = . n m E fl a c = . J m − | p | = . M P a (c) Al . Sc . N . M H z | u | = n m E fl a c = . J m − | p | = . M P a (d) Pz26 . M H z | u | = n m E fl a c = J m − | p | = . M P a L w a = (cid:181) m FIG. 3. (a) Sketch of cross section of the thin-film drivensystem used the simulations shown on (b)-(d), where the twoelectrodes are actuated out of phase, with a gap of 40 µ mbetween them. The glass, of type Schott D263, devices are40000 µ m long and the water channel is only 35000 µ m, sothe last part the channel is closed off by the glass. (b) Simu-lation of a 1 µ m thick AlN thin-film, (c) simulation of a 1 µ mthick Al . Sc . N thin-film, and (d) simulation of a 1 µ m thickPz26 thin-film. The Pz26 parameters are for the purpose ofthe simulations assumed to have the same parameters as bulkPz26. The different films show that they all have the samemode, with a slight shift in frequency, and for 1 Vpp Pz26 hasthe largest resonce, followed by Al . Sc . N and then AlN, al-though the breakdown voltage is also different for the differentfilms. a nearly ideal anti-symmetric pressure wave in the mi-crochannel.The thin-film device is also insensitive to the qual-ity factor of the thin-film transducer, which in terms ofthe damping coefficient Γ sl in the Cauchy equation (5),is given by Q = sl . For two reasons, we expect a weakdependency on Q . The smallness of the transducer im-plies that the Q-factor of the system is completely domi-nated by that of the glass block, and since the transduc- J. Acoust. Soc. Am. / 2 February 2021 7 .938 1.942 1.946 1.950806040200 E fl , maxac Q thf Q thf E fl a c ( J m − ) Frequency f (MHz) FIG. 4. The acoustic energy density E flac ( f ) of a resonancepeak for the system shown in Fig. 1(a) for f = 1 .
938 to1 .
954 MHz for the thin-film Q-factor Q thf in the range from1000 to 5. The insert is a plot of the maximum E flac vs. Q thf . tion mechanism does not rely on resonance properties ofthe thin-film transducer, the strong Q -dependence usu-ally associated with resonant modes is absent. The sim-ulation results shown in Fig. 4 confirms our expectation.Here, the acoustic energy density E flac in the microchan-nel of the thin-film device Fig. 1(a) at the resonance f thf = 1 .
946 MHz is shown as a function of Q from theoriginal value of 1000 down to an appalling low value of 5.The resonant behavior reflected in E flac ( f ) is maintained,and the change in Q by a factor of 200 results in a dropof the peak value of E flac of 2, from 75 to 40 J m − .Finally, in Fig. 5 the simulation result shows that themain resonance mode f thf is maintained when changingthe thickness by a factor of 3 from H thf = 1 to 3 µ m,numbers typical for current AlN and Al x Sc x N thin-filmfabrication technology. The amplitude of the resonancepeak in E flac decreases from 6.9 to 6 . − from thinnestto the thickest film. C. Enhancing the acoustic response of the device by shapingthe electrodes of the thin-film transducer
Thin-film transducers are fabricated by standard mi-crofabrication deposition techniques, and this impliesseveral distinct advantages. The lateral shape of thetransducer or its electrode can be chosen freely by pho-tolithography techniques, the attachment of the trans-ducer to the glass chip is reproducible, stable and strong,and the less controlled use of glue known form standardbulk transducer technology is avoided. Commercially,microfabrication techniques open up for volume produc-tion with relatively cheap unit prices, a necessary prereq-uisite for widespread single-use applications in biotech-nology and medicine, where the cross-contamination aris-ing from multiple use of the same device is a no go.An illustrative example of how the shape of the metalelectrodes on the surface of the thin-film transducer may E fl a c ( J m − ) f (MHz) H t h f ( (cid:181) m ) (cid:181) m (cid:181) m (cid:181) m (cid:181) m FIG. 5. 3D simulations of E flac ( f ) in a Schott D263 glassdevice with an AlN thin-film transducer thickness H thf . Themain resonance peak in E flac is shifted from f thf = 1 .
977 to1 .
985 MHz as H thf is increased from 1 to 3 µ m, and the cor-responding maximum E flac is decreased from 6.9 to 6 . enhance the acoustic response, is shown in Fig. 6. Herewe show a pyrex-glass block of width W sl = 2 . H sl = 1 . W wa = 0 . H sl = 0 .
15 mm. A 2- µ m-thick AlN thin-film transducer is attached to the topsurface. Two electrode configurations are considered. InFig. 6(a), the top and bottom electrodes cover the entirethin-film surface, and therefore even resonance modescan be excited having F ( x, y, z ) = F ( x, − y, z ) for anyfield F . A strong whole-system resonance is located at f thf = 3 .
49 MHz, where the associated pressure wave ofmagnitude | p | = 24 kPa in the channel have two verticalnodal planes (gray) place symmetrically around the cen-ter plane y = 0, in contrast to the anti-symmetric modewith a single vertical nodal plane at the center y = 0shown in Fig. 1(d).In Fig. 6(b) the same device is shown, but now withthe middle half of the top electrode removed. Of course,given this minute change in the system, the same whole-system resonance mode is excited, but the diminished electrode coverage has lead to an increase in the pressureamplitude by a factor of 6 to 146 kPa. The explanation ofthis perhaps surprising result is found in the spatial formof the whole-system resonance mode. By inspection wesee that the displacement field at the glass-transducerinterface forms a wave with in-plane contractions andexpansions. The PZE coefficient e in the transducerimplies the presence of an electric field with a verticalcomponent that changes sign along the in-plane direc-tion. This tendency is counteracted by the fully coveringtop electrode, which imposed a unidirectional electricalfield. Consequently, by removing the central part of thetop electrode, this constraining boundary condition is re-laxed, while the remaining side parts of the electrodeare still capable of exciting the whole-system resonance + + + + + + + (a) AlN f = 3 . MHz | u | = 0 . nm p = 24 kPa (cid:181) m 400 (cid:181) m (cid:181) m (cid:181) m (cid:181) m(b) AlN f = 3 . MHz | u | = 0 . nm p = 146 kPa (cid:181) m 400 (cid:181) m (cid:181) m (cid:181) m (cid:181) m FIG. 6. (a) 2D simulations of 2- µ m-AlN thin-film trans-ducer actuated by a un-split top electrode for symmetric ac-tuation at 1 V pp , with a frequency f = 3 .
49 MHz that ac-tuates the standing second nodal standing wave in the hor-izontal direction in the liquid, with a pressure amplitude of p = ±
24 kPa. The device is 2800 µ m-wide and 1400 µ mthick, and the substrate material is Pyrex. (b) same dimen-sions, materials, and frequency as in (a) but with half of theelectrode cut away, which gives a maximum pressure ampli-tude of p = ±
146 kPa. mode. This example offers a glimpse of the opportuni-ties for design improvements by performing a shape opti-mization of the electrodes or perhaps the entire thin-filmtransducer.
D. Spatially regular modes in the thin-film device
Intuitively, the simplicity of the thin-film device con-sisting essentially of just a glass block should lead to sim-pler modes with regular spatial dependencies. As men-tioned above, the presence of a bulky PZT transducerleads to the excitation of whole-system resonance modeswith a more irregular wave pattern in the displacementfields. Also experimentally, this has been observed as hotspots in the pressure field along an otherwise perfectlyshaped rectangular microchannel. In Fig. 7 we show the lowest whole-system reso-nance modes in a thin-film device with a 1- µ m-thicksplit top-electrode AlN thin-film transducer mounted onthe bottom of a rectangular Schott D263 glass block oflength L sl = 45 mm, width W sl = 2 . (a) G l a s s , S c h o t t D . M H z . M H z . M H z . M H z . M H z . M H z (b)0.6 E flac (cid:0) J m (cid:1) f (MHz) 1.00 1.01 . J m3 . J m3 . J m3 . J m3 . J m3 . J m3 (c) 0.43 mm0.15 mm FIG. 7. Simulation of the low-frequency modes near 1 MHzin a thin-film device with a 1- µ m-thick split bottom-electrodeAlN thin-film transducer. The system is symmetric aroundthe vertical center plane at x = 0 and anti-symmetric aroundthe vertical center plane at y = 0. (a) The lowest six res-onance pressure modes n = 0 , , . . . , n nodal planesalong the x axis, and 1 nodal plane along y . (b) The acousticenergy density spectrum E flac ( f ) identifying the six resonancemodes. (c) The cross-sectional shape of the microchannel inthe vertical yz -plane. H sl = 1 . L wa = 40 mm, width (at its top) W wa = 0 . H wa = 0 .
15 mm. To mimic the shape ob-tained by isotropic etching in glass, the side walls aremodeled as quarter circles. The anti-symmetric actu-ation of the split top electrode combined with the geo-metrical symmetry dictates that the system is symmetricaround the vertical plane at x = 0 across the device andanti-symmetric around the vertical plane at y = 0 alongthe device. In Fig. 7(a), one immediately notice the spa-tial regularity of both the displacement field u and thepressure field p in the six displayed modes. Both fieldshave the required symmetry along the x -axis and anti-symmetry along the y -axis, and both fields exhibits onenodal plane along the transverse y direction and respec-tively 2 n nodal planes with n = 0 , , , . . . , x direction. In Fig. 7(b) is shown the spectrum E flac ( f ) in the frequency range from 0.952 to 1.015 MHz,which allows the identification of the six resonance fre-quencies f n, , , where the indices refer to the number ofnodal planes in each direction. Of the six modes, the n = 0-mode without nodes along the x -axis has an axialstructure that matches the x -independent voltage bound-ary condition better than the other modes, and indeedit has the highest energy density. As the number n of x -axis nodes increases, the corresponding mode exhibits J. Acoust. Soc. Am. / 2 February 2021 9 n increasing number of nodes, and thus an increasingmismatch with the x -independent voltage boundary con-dition. This explains the monotonically decreasing peakvalue of E flac for increasing values of n seen in Fig. 7(b). E. Device performance as a function of breaking the geomet-rical symmetry
As a final point, we discuss the consequences ofbreaking the perfect anti-symmetry of the thin-film de-vice imposed in Figs. 1, 2, 3, and 7. Using microfabri-cation techniques, many geometrical features can be de-fined with accuracies down between 1 and 10 µ m, how-ever it can be problematic to reach such accuracies whendicing up a full-sized wafer into the individual devices.For microelectronics this is not problematic, if the in-tegrated circuits is sufficiently removed from the edges.However, for acoustofluidic devices the whole substrateinfluences the whole-system resonances. For this reasonit is interesting to investigate the sensitivity of a givenacoustofluidic device given shifts in the position of themicrochannels relative to the edges of the substrate.In Fig. 8, we study the acoustic response to a shiftin the center axis of the microchannel in the thin-filmdevice of Fig. 1(a) from the ideal (anti-)symmetric posi-tion at y = 0 to 50 and 100 µ m. It is gratifying to seethat the whole system resonance mode is not degradedsignificantly. A contributing factor to this robustness isthat the water-filled channel only constitutes 1.2% of thetotal volume of the device. We notice that the main anti-symmetric form of the acoustic pressure is unaffected bythe shift, and the acoustic energy density E flac remainshigh, in the range from 55 to 70 J m − . However, as theshift increases, more pronounced axial inhomogeneitiesdevelop. For application in the stop-flow mode this couldimply a degradation in functionality, however as is well FIG. 8. (a) Acoustic resonances of the half system of samedimensions as in Fig. 1(a) for a frequency of f = 1 .
954 MHz,an acoustical energy of E ac = 70 Pa, a maximum pressureamplitude p = ± .
23 MPa, and a max displacement of u =15 nm. (b) Acoustic resonance at 1 .
954 MHz of the sameglass dimensions as in (a), but with the channel off-set by50 µ m, with an acoustical energy of E ac = 78 Pa, a maximumpressure amplitude p = ± .
70 MPa, and a max displacementof u = 22 nm. (c) Acoustic resonance at 1 .
955 MHz wherethe glass dimensions are unchanged but the channel is off-setwith 100 µ m and it has an acoustical energy of E ac = 55 Pa,a maximum pressure amplitude p = ± .
44 Pa, and a maxdisplacement of u = 29 nm. known experimentally in several acoustofluidic devices,in flow-through applications such axial inhomogeneitiesaverages out, and the device would work essentially with-out degradation. V. CONCLUSION
In this paper, based on a well-tested and experimen-tally validated numerical model, we have by 3Dnumerical simulations shown in Fig. 1 that glass chipswith integrated piezoelectric thin-film transducers con-stituting less than 0.1% v/v of the system, have anacoustofluidic response fully on par with that obtainedin a conventional silicon-glass device actuated by a bulklead-zirconate-titanate (PZT) transducer. In Section IV,we have shown that the ability of the thin-film transducerto create the desired acoustic effects in a bulk acoustoflu-idic device relies on three physical aspects of the system:The in-plane-expansion of the the thin-film transducerunder the action of the orthogonal applied electric field,the acoustic whole-system resonances of the device, andthe high Q-factor of the elastic solid constituting the bulkpart of the device.There are several advantages to the use of thin-filmtransducers. Among them are the low sensitivity of thethin-film device to the material, the thickness, and thequality factor of the thin-film transducer discussed in Sec-tion IV B. It is also an advantage that thin-film devicescan be produced by clean-room microfabrication tech-nique, similar to the ones employed in the fabrication ofsurface acoustic waves. An example of this was stud-ied in Section IV C, where we demonstrated an enhancedacoustofluidic response by shaping the electrodes of thethin-film transducer, in this case by removing roughlyhalf of the top-electrode coverage. Other advantages ofmicrofabrication techniques are that the thin-film trans-ducers are integrated in the devices in a much more re-producible manner compared to the conventional bulk-transducer technique involving the use of glue.In an application perspective, the use of thin-film transducers offers new possibilities in the field ofacoustofluidcs. The fact that the thin-film transducerconstitutes such a low volume fraction implies not onlythat the device is relatively insensitive to the quality ofthe thin film, but also that the core part of the acoustoflu-dic system, namely the microchannel, constitutes a rela-tively large part of the system and is thus easier to con-trol. Several of the thin-film transducers can be fabri-cated with a high breakdown voltages ( ∼
20 V / µ m) thatallows for relatively high acoustic energy densities andlower dissipation and heat production.We hope that this theoretical analysis will inspire ourexperimental colleagues in the field to investigate the newapplication aspects offered by the thin-film acoustofluidicdevices.
10 J. Acoust. Soc. Am. / 2 February 2021
CKNOWLEDGEMENTS
This work was supported by the
BioWings projectfunded by the European Union’s Horizon 2020
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