Capillary wave motion excited by high frequency surface ...users.monash.edu/~lyeo/Dr_Leslie_Yeo/SAW_Atomisation_files/Tan_… · Capillary wave motion excited by high frequency surface

Capillary wave motion excited by high frequency surface acoustic wavesMing K. Tan,1 James R. Friend,1 Omar K. Matar,2 and Leslie Y. Yeo1,a�

1Department of Mechanical and Aerospace Engineering, Micro/Nanophysics Research Laboratory,Monash University, Clayton, Victoria 3800, Australiaand The Melbourne Centre for Nanofabrication, Clayton, Victoria 3800, Australia2Department of Chemical Engineering, Imperial College, London SW7 2AZ, United Kingdom

�Received 24 May 2010; accepted 29 September 2010; published online 15 November 2010�

This paper presents a numerical and experimental study of capillary wave motion excited by highfrequency surface acoustic waves �SAWs�. The objective of this study is to provide insight into thedynamic behavior of the fluid free surface and its dependence on the excitation amplitude. Atwo-dimensional numerical model that couples the motion of the piezoelectric substrate to a thinliquid layer atop the substrate is constructed. A perturbation method, in the limit of small-amplitudeacoustic waves, is used to decompose the equations governing fluid motion to resolve the widelydiffering time scales associated with the high frequency excitation. While this model focuses on thefree surface dynamics in the low-amplitude flow regime, the experimental study focuses on thehigh-amplitude flow regime. Transformation of time series data from both experiments andsimulations into the frequency domain reveals that, in the low-amplitude regime, a fundamentalresonant frequency and a superharmonic frequency are found in the frequency spectra. The formeris found to be identical to that of the applied SAW, and the free surface displacement magnitude iscomparable to that of the substrate displacement. Our numerical results also confirm previousspeculation that the separation distance between two displacement antinodal points on the freesurface is �St��SAW /2 for a film and �St��f /2 for a drop, where �SAW and �f denote the SAWwavelength and the acoustic wavelength in the fluid, respectively. Finally, in the high-amplituderegime, strong nonlinearities shift the acoustic energy to a lower frequency than that of the SAW;this low-frequency broadband response, quite contrary to the subharmonic half-frequency capillarywave excitation predicted by the classical linear or weakly nonlinear Faraday theories, is supportedby a scaling analysis of the momentum equations. © 2010 American Institute of Physics.�doi:10.1063/1.3505044�

I. INTRODUCTION

The actuation and manipulation of liquids in microde-vices using surface acoustic waves �SAWs� offer many ad-vantages over conventional methods. SAW devices havetraditionally been utilized for a variety of applications1 in-cluding microcentrifugation,2,3 micromixing,3,4 mico-pumping,5,6 drop spreading and transport,7–9 micro-reactors,10–12 jet microejectors,13 and microatomizers.14–16

These varied applications arise because the behavior of asessile drop on a SAW device can be dramatically changedby altering the displacement amplitude of the SAW, which istypically on the order of ��10−9–10−8 m. At low displace-ment amplitude, the drop vibrates atop the substrate �see Fig.1�a��. However, at high displacement amplitude �above acritical Weber number�, the drop elongates rapidly into aslender jet.13 In between these two limits, the entire droptranslates along the direction of the SAW propagation. Inter-estingly, when the drop spreads into a film of thickness com-parable to or smaller than the acoustic wavelength, the vibra-tion immediately changes form, resulting in rapidatomization,14–16 which arises from the violent destabiliza-tion of capillary waves at the free surface.16 SAW atomiza-tion has been used in many bioengineering processes includ-

ing protein chip fabrication,17 generation of aerosolscontaining proteins18 or drugs,19 formation of periodic poly-mer patterns on a substrate,20 and the extraction of liquid,protein molecules, and yeast cells from paper throughatomization.21 With the advent of SAW atomization basedprocesses, the understanding of the underlying physics,which gives rise to the formation of capillary waves and theirsubsequent breakup to produce atomized droplets, is of criti-cal importance in order to fully explore the potential of thistechnique.

There has been a longstanding history of investigationsinvolving vertical, pistonlike excitation of free surfaces. Thisdates back to the experiments of Michael Faraday22 who ob-served a subharmonic response: the resultant capillary wavesoscillate at one-half the excitation frequency. Faraday’s studyhas since led to a body of work which has employedMathieu-type equations in modeling the free surfaceresponse.23–25 The possibility of a harmonic free surface re-sponse was captured by the analysis of Kumar andTuckerman26,27 and later observed by Müller et al.28 for liq-uid films thinner than the viscous boundary layer.

Nevertheless, such simple harmonic and subharmonicFaraday responses were noticeably absent in the very firstexperiments measuring the capillary wave undulation drivenby the SAW wherein the vibration source is in the form of aa�Electronic mail: [email protected].

PHYSICS OF FLUIDS 22, 112112 �2010�

1070-6631/2010/22�11�/112112/22/$30.00 © 2010 American Institute of Physics22, 112112-1

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

http://dx.doi.org/10.1063/1.3505044

http://dx.doi.org/10.1063/1.3505044

surface �Rayleigh� wave.16 While a weak harmonic responsewas observed, the free surface also oscillated, quite incred-ibly, at frequencies some three to five orders of magnitudelower than the excitation frequency of 107 Hz and, further-more, assuming a broadband nature which spanned two or-ders of magnitude from 102 to 104 Hz.16 This indicates thatclassical Faraday-wave theory may be capable of providingonly a partial explanation of the observed SAW-induced phe-nomena.

Further, Keolian et al.29 found that period-doubling se-quences far beyond the subharmonic frequencies f−→ f0 /2to f0 /4, f0 /12, . . . f0 /35 were present in their Faraday-likeexperiment; the f0 /35 appeared due to an additional degreeof freedom—the width of the fluid parcel—which also ex-hibited an influence in the period-doubling and altered thecontinuity of the overall period-doubling sequence. As theexcitation amplitude increases, the physical appearance ofadditional capillary wave modes is known to occur. Since theacceleration induced by the SAW is several orders of mag-nitude larger than other methods in the literature and theexcitation frequency itself is also several orders of magnitudehigher than what is used by other investigators in the litera-ture, it is conceivable that Qi et al.16 saw an extended period-doubling of the excitation at fSAW to frequencies of102–104 Hz. Moreover, the two-dimensional and stronglycurved fluid surface permits many more capillary wavemodes due to three-wave resonant interaction that is perhapsbest explained by Chen and Viñals30 and Rucklidge31 whoshowed the formation of patterns similar to capillary wavesappearing on temporal scales remarkably different than thatof the excitation signal.

With our system, beyond the SAW excitation, there isalso the factor of the finite size of the drop. In general, themotion at the free surface occurs over both macroscopic andmicroscopic length scales.16 In the former case, the wave-

length of the free surface waves is comparable to the dropgeometry, �c�Rd, whereas in the latter, the wavelength ofthe free surface waves is much smaller than the drop dimen-sion, �c�Rd. Hereafter, the former is defined as apparentdeformations �Fig. 1�a��, and the latter is defined as surfacedeformations �Fig. 1�b��. Depending on the disturbance oractuation mechanism, many different types of surface defor-mations can be formed �see Fig. 2�. For a sufficiently deepliquid, the fluid elements on the free surface may follow anelliptical orbit32 with characteristics similar to the SAW trav-eling on a piezoelectric substrate �see Fig. 2�a��.Longuet-Higgins33 investigated the instabilities of deep wa-ter irrotational gravity waves as a result of superharmonicand subharmonic34 resonances. In addition to deep waterwaves, interfacial longitudinal waves35,36 or interfacial shearwaves37 can be generated by placing an oscillator that vi-brates along the x- or y-directions at the free surface, asshown in Figs. 2�b� and 2�c�, respectively. The interfaciallongitudinal wave motion is associated with the elastic �com-pression� modulus of the free surface,35,36 while the interfa-cial shear wave motion is associated with the shear modulusof the free surface.37 We note that some aspects of the Fara-day instability can be understood through the framework of aRayleigh–Taylor38 analogy39–41 although this is beyond thescope of the present work.

Whether driven by pistonlike vibration or the more com-plex vibration in the form of Rayleigh waves in the case ofthe SAW, the spectrum of vibrational modes present on thefree surface is known to vary with a change in the displace-ment amplitude of the vibrating substrate.42 More perplex-ing, the typical resonant response of a fluid interface withdiscrete resonance frequencies can give way to a broadband,chaotic response—turbulent capillary waves. Driving a fluid

Droplet

Substrate

TransducerDroplet

Droplet

TransducerTransducer

SAW SAW SAW SAW

(a)

(b)

SAW

FIG. 1. �Color online� Schematic and experimental images showing capil-lary wave motion across multiple length scales: �a� apparent deformation��c�Rd� and �b� surface deformation ��c�Rd�, induced by the SAW. Theexperimental images in �a� were captured at 500 frames/s and indicate adrop vibration frequency around 100 Hz, whereas images in �b� were cap-tured at 5000 frames/s. Note that the drop in �b� spreads into a thin film.Such spreading dynamics, however, is beyond the scope of the presentstudy.

Liquid

Stationary substrate

xy

Deep water waves

Liquid

* Oscillator

xx

yyz z*Oscillator

motion*Oscillator

motion

Interfacial

shear waves

Interfacial

longitudinal

waves

(a)

(b) (c)

(d)

Vibrating substrate

xy

Capillary

waves

FIG. 2. �Color online� Common types of free surface deformations: �a� deepwater waves, �b� interfacial longitudinal waves �Ref. 36�, �c� interfacialshear waves, and �d� capillary waves. The free surface waves shown in�a�–�c� are frequently generated by introducing disturbances at the free sur-face using an oscillator or through the motion of a fluid layer atop theliquid-liquid interface. The free surface wave shown in �d� is generatedthrough the excitation induced by a substrate vibrating with a pistonlikemotion.

112112-2 Tan et al. Phys. Fluids 22, 112112 �2010�


interface at low-frequency �102 Hz� and high displacementamplitude, Brazhnikov et al.43 have experimentally demon-strated that the behavior of capillary waves departs from lin-earity and enters into a “turbulent” capillary wave regime,exhibiting broadband power spectra in the superharmonicfrequency range �f+�102–103 Hz�; some choose to de-scribe this as acoustic turbulence.44 With SAW excitation,broadband response of this sort does appear at frequenciesfar less than the excitation frequency, yet the broadband re-sponse appears at frequencies one to two orders of magni-tude greater than the fundamental resonance frequency of thedrop itself.16 In one sense, the turbulent capillary wave re-sponse is entirely consistent with past researchers’ results asthe turbulence cascade appears across a broad frequencyrange above the drop’s fundamental resonance frequency. Onthe other hand, the mechanism of multiple period-doublingcascades to such low-frequencies from the actual SAW exci-tation frequency seems to be unique to the SAW system.Given that previous studies considering broadband capillarywave excitation due to a monofrequency source have onlyfocused on the fundamental resonant frequency f0, however,there is no evidence of the existence of superharmonic fre-quencies observed on a liquid free surface to the best of ourknowledge. Therefore, one of the aims of this work is toshow that the superharmonic frequencies do indeed exist forthe case of high-frequency SAW excitation.

Due to the difficulties in capturing the flow field beneaththe free surface and measuring the complex dynamics of thefree surface at fine temporal and spatial scales, experimentalinvestigations are extremely challenging. As a result, numeri-cal analyses have typically been used to conduct these stud-ies. To date, numerical studies by Qi et al.,16 Schindleret al.,45 and Köster46 have attempted to elucidate the mecha-nisms that give rise to the vibration of the free surface andthe internal flow field in a drop driven by the SAW. In Qi’smodel, the lubrication approximation for a thin, incompress-ible liquid film was employed to construct a model govern-ing the destabilization of the free surface leading toward at-omization, which was solved numerically using an adaptiveform of the method of lines with fourth-order centered dif-ferences for spatial discretization and an implicit Runge–Kutta scheme for time integration. To decouple the fluid-solid interaction, an analytical expression to describe apressure disturbance imposed on the free surface that re-sembles the SAW, albeit with a decay in amplitude due to theenergy absorbed by the fluid, was employed. This decou-pling, however, omits important physics underlying how thefluid vibrates in relation to the substrate excitation.

Unlike Qi’s model, the numerical studies conducted bySchindler and Köster focused on the flow field within thedrop bounded by its free surface and the solid substrate.Schindler’s numerical results, obtained via finite elementboundary parametrization of the free surface using Galerkindiscretization, successfully showed the apparent deformationof the free surface driven by SAWs and concluded that theapparent deformation of the drop was caused by the pressureforce; the viscous force was far too weak to result in a sub-stantial deformation. Nevertheless, Schindler invoked a num-ber of questionable assumptions, including incompressibility,

steady-state flow, and a fixed body force on the fluid from theSAW, ignoring the effect of the fluid on the substrate and thedetermination of the actual force from the SAW on the fluidas it is generated: as a decaying, distributed body force de-pendent on the shape of the drop of fluid sitting on the sub-strate. On the other hand, Köster’s approach generally fol-lows the conventional method by using a perturbationexpansion of the equations of motion for the substrate andfluid: equations at the first-order govern the propagation oflinear acoustic waves and equations at the second-order �af-ter time averaging� govern SAW streaming.47–50 Although hewas unable to resolve the surface deformation on the lengthscale of the capillary wave, Köster, using an arbitraryLagrangian–Eulerian approach with second-order Taylor–Hood finite element spatial discretization and a backwardEuler or Crank–Nicholson scheme for time integration, man-aged to capture the apparent deformation of the drop in theform of an asymmetric cone when excited from beneath by atraveling SAW, consistent with experimental observations.The complementary approach of Dong et al.,51 in which thethree-dimensional Navier–Stokes equations are solved usinga volume-of-fluid method to simulate the apparent deforma-tion of a drop subjected to lateral vibrations, demonstratesthe formation of capillary waves on the drop surface. Thiswork, which was carried out in order to analyze the effect ofvibration frequency on the difference in the drop contactangles that determine the velocity of its motion along thesubstrate, did not account for compressibility effects and wasrestricted to very low vibration frequencies �of order102 Hz�. More recently, Brunet et al.52 conducted a numeri-cal study of SAW generated fluid motion in a drop withnondeformable free surface and found a “chaotic” behaviorof the acoustic field within the drop as a result of multiplewave-reflection from the free surface. Brunet et al.’s finiteelement numerical model was based on a linear acoustic as-sumption and used a nondeformable stress-free surface con-dition to examine the acoustic field in a large hemisphericaldrop, i.e., Rd�10�f. Due to the nondeformable free surface,their model, however, was unable to provide analysis of thesurface deformation. The present work is therefore motivatedby the lack of a comprehensive investigation of the dynamicsof the free surface motion driven by SAW.

The rest of this paper is organized as follows. First, weconstruct a simple numerical model to verify the existence ofhigh frequency capillary waves �fc� fSAW�, which has yet tobe captured theoretically, and also to verify the existence ofsuperharmonic frequencies �f+�2f0 ,4f0 ,6f0 , . . .�, which, tothe best of our knowledge, has never been shown either theo-retically or experimentally. The mathematical formulationand the numerical scheme are discussed in Sec. II. Using thismodel, we examine the flow field within a film excited by alow-amplitude SAW. In Sec. III, we report on experimentsperformed to investigate the relationship between the capil-lary frequency and the SAW amplitude, paying particularattention to the transition from low- to high-amplitude SAW.While laser Doppler vibrometry measurements of the capil-lary wave frequency under SAW excitation have been brieflymeasured in our previous work,16 the experiments carried outhere provide a more rigorous investigation of the frequency

112112-3 Capillary wave motion excited by high frequency Phys. Fluids 22, 112112 �2010�


spectra as a function of the excitation amplitude over abroader frequency range. Finally, in Sec. IV, we examine theacoustic field in a drop and provide a case study of the align-ment of microparticles on the free surface of a hemisphericaldrop driven by an extremely low-amplitude standing-SAW��1 nm�. The prime objective of this case study is to pro-vide a qualitative verification of the consistency between thenumerical results and experimental observations.

II. NUMERICAL SIMULATION

A schematic of the flow configuration is shown in Fig. 3.The model, which will be derived below, consists of the lin-ear piezoelectric coupling equations,53 a set of first-orderacoustic equations,49 and a set of second-order acousticstreaming equations50 to accommodate the time-averagedfluid flow arising from the SAW irradiation, viscous effects,and the effect of attenuation of the SAW propagation arisingfrom the fluid present on the substrate. We adopt a similarnumerical approach reported by Qi et al.,16 with a few im-portant improvements: �1� we omit the lubrication approxi-mation, and instead use a regular perturbation expansion todecompose the governing equations such that they can beresolved separately at small and large time scales; �2� wefully capture the fluid-structural coupling by completelysolving for the equation of motion in both the solid and fluiddomains. The variables across the solid-fluid interface aresolved simultaneously both spatially and temporally to forma fully coupled field, allowing complex fluid-solid interac-tions to be studied.6 Below, we first introduce the governingequations underpinning the motion of the piezoelectric sub-strate and the equations that describe the acoustic streamingfield. We then transform the coordinate system of the fluiddomain to allow the free surface to deform in the physicalspace, while remaining as a fixed-grid in the computationaldomain. The numerical procedure used to carry out the com-putations, and the initial and boundary conditions imposed,are then discussed.

A. Governing equations

The time-domain constitutive equations governing themotion of a piezoelectric substrate are given by53–55

�Di

�t= eikl

�Skl

�t+ �ik

S �Ek

�t, �1�

�Tij

�t= cijkl

E �Skl

�t− ekij

�Ek

�t, �2�

where eikl are the piezoelectric stress coefficients, �ikS are the

dielectric coefficients at constant strain S, Tij are the stresscomponents, and cijkl

E are the elastic stiffness coefficients at aconstant electric displacement D or field E. Note that the firstterm of the electric field displacement equation, Eq. �1�,arises from the relation between the electric displacementand electric field �Di=�ikEk�, while the first term of the stressequation, Eq. �2�, arises from Hooke’s law �Tij =cijklSkl�.Equations �1� and �2� take into account the piezoelectric cou-pling between the electrical and mechanical variablesthrough the piezoelectric stress tensor eikl. The tensor eikl issymmetric with respect to the last two indices k and l, i.e.,eikl=eilk, because of the symmetric strain Skl=Slk. For an or-thogonal Cartesian coordinate system i , j ,k , l� �1,2 ,3.

In a piezoelectric substrate, the elastic field, described byEqs. �1� and �2�, and the electromagnetic field, as defined byMaxwell’s equations, are coupled. The distribution of thesefields can be determined by simultaneously solving thecoupled equations. However, the velocity of the electromag-netic wave is 105 times faster than the elastic wave. There-fore, the mechanical dynamics and its effect on the electro-magnetic field generated through the piezoelectric couplingare considerably slower than the electromagnetic radiation,so much so that we can treat the electric field as being qua-sistatic. This allows us to simplify Eq. �1� based on the qua-sistatic assumption �D /�t=��H�0, where H is the mag-netic field vector. Together with the infinitesimal strain-displacement relationship,

�Skl

�t=

1

2 �2�k

�xl � t+

�2�l

�xk � t� . �3�

Equations �1� and �2� are then simplified into the followingform:

�Ek

�t= −

eikl

�ikS �1

2 �vk

�xl+

�vl

�xk� , �4�

�Tij

�t= cijkl

E �1

2 �vk

�xl+

�vl

�xk� − ekij

�Ek

�t, �5�

where � is the particle displacement and v is the velocity.Following this, Newton’s second law of motion facilitatesclosure of Eqs. �4� and �5� to describe the basic motion of thepiezoelectric solid,

s�v j

�t=

�Tij

�xi, �6�

where s is the crystal mass density. Materialproperties—eikl, �ik

S , and cijklE —need to be transformed to the

Transducer

Absorber

Absorber

Absorber

Stress free

surface

Fluid

Solid

SAWSAW

x1x3

h(x,t)

H

Absorber

L

x

y

Full coupling

between

solid and fluid

A

L1 L2 L3 L4

FIG. 3. �Color online� A two-dimensional schematic of the computationaldomain used to model the surface deformation and to analyze the flow fielddriven by the SAW. A liquid film of undisturbed thickness H is placed ontop of the piezoelectric substrate. At the solid-fluid interface, the sectiondefined by �L2−L3� represents the region of full coupling between the fluidand solid, whereas the sections defined by �L1−L2� and �L3−L4� are tran-sition regions. Therefore, the solid-fluid interactions are restricted within thesection defined by L2−L3 to reduce wall effects.



correct orientation. We use the bond transformationprinciple53 to transform these material constants from thecrystal’s principal axes to the 127.68° Y-rotated-about-X-cut,X-propagating configuration. The transformed material prop-erties are listed in Appendix A.

In the fluid medium, the equations describing Newtonianfluid motion are those pertaining to mass and momentumconservation,47,49

�f

�t+ � · �fu� = 0, �7�

f�u

�t+ f�u · ��u = − �p + �2u + � � � · u , �8�

where u is the fluid velocity, f is its density, is its pressureand �= �B+ /3�, in which B is the bulk viscosity and the shear viscosity. A thermodynamic equation of state isused to complete the above set of equations,48,56

p − p0 = c02�f − f0� +

1

2� �c2

��

s

�f − f0�2

+�p

�s�s − s0� + ¯ , �9�

where c is the sound speed and s is the entropy. In the above,the subscript “0” refers to equilibrium properties associatedwith the ambient conditions, i.e., u0=0 m /s, p0=101 kPa,and f0=998 �water� kg /m3.

Assuming infinitesimally small amplitude waves, a per-turbation expansion can then be used to linearize the equa-tions. We begin by examining the dependent variables for thefluid. The perturbation expansion is carried out in powers ofthe small parameter �=U1 /c0�1, where U1 is the character-istic acoustic particle velocity. Expansions of the fluid veloc-ity, pressure, and density field then take the form47,49,50,57

u = u0 + �u1 + �2u2 + O��3� , �10�

p = p0 + �p1 + �2p2 + O��3� , �11�

f = f0 + �f1 + �2f2 + O��3� , �12�

where the subscript “1” refers to the first-order approxima-tions and the subscript “2” refers to the second-order field.Substitution of Eqs. �10�–�12� into Eqs. �8� and �7� subse-quently yields the following first-order approximations forthe mass and momentum conservation equations:47,49,50

�f1

�t+ f0�� · u1� = 0 �13�

and

f0�u1

�t= − �p1 + �2u1 + � � � · u1, �14�

respectively. From Eq. �9�, the first-order approximation ofthe equation of state for an adiabatic process49 �s=s0� is

p1 = c02f1. �15�

At the next order, the fluid motion comprises a superpo-sition of the steady-state and harmonic time-oscillatingflows. The second-order approximation to the continuity �Eq.�7�� and the momentum �Eq. �8�� equations are47,50

�f2

�t+ � · �f0u2� + � · �f1u1� = 0, �16�

f0�u2

�t+ f1

�u1

�t+ f0�u1 · ��u1

= − �p2 + �2u2 + � � � · u2. �17�

Rewriting Eq. �9�,

p − p0 = A�f − f0

f0+

1

2

B

Af − f0

f0�2 , �18�

the second-order approximation of the equation of state thenreads

p2 =c0

2

f0�1

2

B

A1

2 + f02 . �19�

In the above, A and B are the adiabatic bulk elastic mod-ulus and nonlinear modulus given by A=f0c0

2 and B=f0

2 ��c2 /�f�s, respectively.Subsequent time-averaging ��a�=1 /T�0

Ta�t�dt, whereT=nf−1 and n=1,2 ,3 , . . .� all the terms in Eqs. �16�, �17�,and �19� produces the following equations:

2

�t+ f0�� · u2� + � · �f1u1� = 0, �20�

f0�u2

�t+ �f1

�u1

�t� + f0��u1 · ��u1�

= − �p2 + �2u2 + � � � · u2, �21�

p2 =1

2

c02

f0�B

A�1

2� + 2f02 . �22�

To accommodate the deformation of the free surface aty=0 �see Fig. 3�, we apply the following transformation:

� = x, =y − h�x,t�H − h�x,t�

, �23�

where H is the undisturbed thickness and h is the free sur-face displacement from its undisturbed location y=0. We ex-pand h�x , t� via the following ansatz:

h = h0 + �h1 + �2h2 + O��3� , �24�

where h0=0 and h1=h2=0 at t=0. Substituting Eq. �24� intoEq. �23� and eliminating terms of order O��3� and higher, weobtain



= 0 +�

H y

H− 1�h1 +

�2

H− h2 +y

Hh2 −

h12

H+

y

H2h12� ,

�25�

where 0�y /H. Equations �23� and �25� are then substitutedinto Eqs. �B1�–�B6� in Appendix B, from which we obtainthe following:

�

�x=

�

��− �

�

H�

�

�h1

��+ O��2� , �26�

�2

�x2 =�2

��2 − �2 �

H�2

� � �

�h1

��− �

�

H�

�

�2h1

��2 + O��2� ,

�27�

�

�y=

1

H�

� + �

1

H2h1�

� + O��2� , �28�

�2

�y2 =1

H2

�2

� 2 + �1

H32h1�2

� 2 + O��2� , �29�

�2

�x � y=

1

H�2

�� + �

h1

H2

�2

�� − �

1

H2

�2

� 2

�h1

��

+ �1

H2

�

�

�h1

��+ �

1

H2

�

��

�h1

��+ �

0

H2

�2

� 2

�h1

��

+ O��2� , �30�

�

�t=

�

��− �

�

H�

�

�h1

��+ O��2� , �31�

where �=1− 0. Finally, we combine Eqs. �26�–�31� withEqs. �13� and �14� to arrive at

�f1

��+ f0

�u1x

��+

f0

H�u1

y

� = 0, �32�

f0�u1

x

��= −

�p1

��+

1

H2�2u1

x

� 2 + ��2u1

x

��2 + �1

H�2u1

y

�� ,

�33�

f0�u1

y

��= −

1

H�p1

� +

�2u1y

��2 + �1

H2

�2u1y

� 2 + �1

H�2u1

x

�� ,

�34�

and Eqs. �21� and �20� to give

�f2

��= − f0

�u2x

��−

f0

H�u2

y

� − Favg

mass, �35�

f0�u2

x

��= −

�p2

��+

H2

�2u2x

� 2 + ��2u2

x

��2 +�

H�2u2

y

�� + Favg

m−x,

�36�

f0�u2

y

��= −

1

H�p2

� +

�2u2y

��2 +�

H2

�2u2y

� 2 +�

H�2u2

x

��

+ Favgm−y , �37�

where

Favgmass = − f0 1

H−

0

H �� h1

��

�u1x

� � + �u1

x �f1

��

− 1

H−

0

H �� h1

��

�f1

� � +

1

H�u1y �f1

� �

+ f01

H2�h1�u1

y

� � +

1

H�f1�u1

y

� � + �f1

�u1x

�� ,

�38�

Favgm−x = − �1

�u1x

�� −

f0

H �u1y �u1

x

� � +

�

H2�h1�2u1

y

��

+ f0 �

H� �h1

��

�u1x

� � − f0�u1

x �u1x

��

+ �

H� �h1

��

�p1

� � − �

�

H� �2h1

��2

�u1x

� �

+1

H2�� h1

��

�u1y

� � +

2

H3�h1�2u1

x

� 2�+

1

H2�� h1

��

�u1y

�� − �

�

H2� �h1

��

�2u1y

� 2�− 2�

�

H� �h1

��

�2u1x

�� , �39�

Favgm−y = − �1

�u1y

�� −

f0

H �u1y �u1

y

� � − f0�u1

x �u1y

��

+ f0 �

H� �h1

��

�u1y

� � − 2

�

H� �h1

��

�2u1y

��

−1

H2�h1�p1

� � +

1

H2�� h1

��

�u1x

� �

− �

H� �2h1

��2

�u1y

� � − �

�

H2� �h1

��

�2u1x

� 2�+

2

H3��h1�2u1

y

� 2� +1

H2�� h1

��

�u1x

��

+1

H2��h1�2u1

x

�� , �40�

in which �=B+ �4 /3�; the superscript for the velocitydenotes its direction.

B. Boundary conditions

When SAW is coupled with air, a stress-free boundarycondition is assumed on the surface of the piezoelectric sub-strate. As such, the mechanical boundary conditions at eachpoint on the surface of the substrate are T33=T31=T13=T32



=T23=0 at x3=0 �see Fig. 3�. The left, right, and bottom ofthe substrate are surrounded by a split-field perfectlymatched layer �PML� to minimize wave reflection fromthe boundaries.58 Within the PML, the wave amplitude de-cays quadratically with a loss function represented byDi=Dmax��xi−x0� /�PML�2,6,58 where Dmax is an empiricalconstant, �xi−x0� indicates the location of the nodes withinthe PML layer, and �PML is the thickness of the PML. Toimplement the split-field PML for the two-dimensionalanalysis in x1- and x3-space, Eq. �6� is rewritten to separatethe derivative into the x1- and x3-directions,59

�

�t+ D1�v1

� =1

s

�T1

�x1, �

�t+ D3�v1

� =1

s

�T5

�x3, �41�

�

�t+ D3�v3

� =1

s

�T3

�x3, �

�t+ D3�v3

� =1

s

�T5

�x1, �42�

where v1=v1� +v1

� and v3=v3� +v3

�; the superscripts � and �

refer to the parallel and perpendicular directions of the asso-ciated velocity component, respectively. Abbreviated sub-scripts are used for the stress field, i.e., T1=T11, T3=T33, andT5=T13=T31. To generate the SAW, a sinusoidal electric po-tential,

� = �p−p sin�2�xi/�SAW�sin��t� , �43�

where �p−p is the peak-to-peak voltage, is applied on thesurface of the piezoelectric substrate.

At the air-liquid interface, y=0, we impose the followingtangential and normal stress conditions, respectively, givenby

2 �ux

�x−

�uy

�y� �h

�x+ �uy

�x+

�ux

�y��1 − �h

�x�2 = 0, �44�

p = �c�c + � � · u

+

2� �ux

�x �h

�x�2

+ �uy

�x+

�ux

�y� �h

�x+

�uy

�y

�h

�x�2

+ 1

, �45�

where �c denotes surface tension and �=B− �2 /3�. Onthe other hand, the surface curvature �c is given by

�c =�2h

�x2�1 + �h

�x�2 −3/2

. �46�

In addition to the above, the kinematic boundary condition isalso imposed,

�h

�t+ ux�h

�x= uy . �47�

By substituting Eqs. �10�, �11�, �24�, and �26�–�31� into Eqs.�44�, �45�, and �47�, we obtain the boundary conditions aty=0 for the first-order flow field by selecting terms of orderO��,

�u1y

��+

1

H�u1

x

� = 0, �48�

p1 = �� u1x

��+

1

H�u1

y

� +

2

H�u1

y

� + �c

�2h1

��2 , �49�

�h1

��= u1

y , �50�

and, the boundary conditions at y=0 for the second-orderflow field by selecting terms of order O��2�,

�u2y

��= −

1

H�u2

x

� − 2� �h1

��

�u1x

�� −

1

H2�h1�u1

x

� �

+3

H� �h1

��

�u1y

� � , �51�

p2 = �c�2h2

��2 +2

H�u2

y

� + �� u2

x

��+

1

H�u2

y

�

+2

H � �h1

��

�u1x

� � +

2

H2�h1�u1

y

� �

− �1

H� �h1

��

�u1x

� � +

1

H2��h1�u1

y

� �

+ 2� �h1

��

�u1y

�� , �52�

�h2

��= u2

y − �u1x �h1

�� . �53�

A no-slip boundary condition is imposed on the left andright walls x= �0,L�: u1

y =u1x =0 and u2

y =u2x =0. In addition,

the no-slip condition implies the height fluctuation shouldalso be zero on the walls: h1=0 and h2=0 at x= �0,L�. At thesolid-fluid interface y=H and between x=L1 and L4 �seeFig. 3�, the first-order velocity and pressure of the fluid arecoupled with the solid substrate through continuity of veloc-ity, u1

x =v1, u1y =v3, and stress conservation, �x=L1

x=L4�33

=�x=L1

x=L4T33 and �x=L1

x=L4�13=�x=L1

x=L4T13. As for the second-orderfield, we employ the velocity boundary condition,50

u2�y=H = − 12 � � ��1

f � u1� , �54�

where �1f is the displacement of the fluid element. This con-

dition arises as a result of the elliptical motion of the solidsubstrate �acoustic radiator� as the SAW propagates along thesubstrate surface. In addition, the left and right ends in thefluid domain of the first-order field are surrounded by thesplit-field PML to minimize wave reflections. This imple-mentation is similar to that for the solid. Equation �32� cantherefore be rewritten as

�

�t+ D1�1

� = − f0�u1

x

��, �

�t+ D3�1

� =f0

H�u1

y

� , �55�

where 1=1� +1

�.



C. Numerical method

The necessary equations to be solved for the piezoelec-tric solid involve the time-domain constitutive piezoelectricequations, i.e., Eqs. �4� and �5�, Newton’s second law ofmotion, i.e., Eq. �6�, and the PML boundary condition, i.e.,Eqs. �41� and �42�. In addition, Eq. �43� is required to gen-erate the electric field in the substrate. For the acoustic fieldin the fluid, the appropriate equations are continuity, i.e., Eq.�32�, momentum conservation equations �33� and �34�, andthe equation of state, i.e., Eq. �15�. On the free surface of thefluid, the equations for the normal stress jump, i.e., Eq. �48�,tangential stress continuity, i.e., Eq. �49�, and the kinematicboundary condition, i.e., Eq. �50�, are solved simultaneouslyusing the finite-difference time-domain method.55,60 Theequations of motion for the solid are discretized using asecond-order centered difference scheme and the variablesare distributed on a staggered-spatiotemporal-grid system,where stress and electric fields are located at the same nodesand velocities occupy the others. However, due to the diffi-culties in computing the fluid variables at the free surface ona spatially staggered grid, variables for the fluid are allocatedon a spatially collocated but temporally staggered grid sys-tem. These equations are also discretized using a second-order centered difference scheme. The stress and electricfields of the solid, and the pressure and density fields of thefluid, are allocated at the same time frame at �n /2��t, wheren is a positive integer and �t is the time step. Similarly, thevelocity fields for the solid and fluid are allocated at the sametime frame at n�t. To ensure stability of the numerics, thelargest Courant number of the solid or fluid domain C=c�t /�x,61 where c is the phase velocity of the wave and �xis the grid spacing, is selected �approximately 0.02� to deter-mine the maximum allowable time step �t. Since we selectidentical �t for both the solid and fluid, the calculations canbe carried out synchronously. In other words, the equationsfor solving the solid variables at time �n /2��t are solved inparallel with the equations for solving the fluid variables attime �n /2��t.

In each time step, the terms in the � � brackets are com-puted and summed over one sinusoidal cycle �i.e., to numeri-cally approximate the time integration of each term in thebrackets�. To complete the time averaging, these terms arethen divided by the number of steps to complete the cycleand subsequently substituted back into Eqs. �38�–�40� andEqs. �51�–�53�. After one cycle is completed, we temporaryhalt the computation of the acoustic fields in the solid andfluid, and proceed to calculate the acoustic streaming field.The streaming field involves solving the equations governingcontinuity, i.e., Eq. �35�, momentum conservation, i.e., Eqs.�36� and �37�, and the equation of state, i.e., Eq. �19�. Inaddition, the tangential stress balance at the free surface, i.e.,Eq. �51�, the normal stress jump at the free surface, i.e., Eq.�52�, the kinematic boundary condition at the free surface,i.e., Eq. �53�, and the velocity boundary condition at thesolid-fluid interface, i.e., Eq. �54�, are also required. The nu-merical scheme used to solve the second-order field is iden-tical to the method used to solve the first-order field. In ourmodel, the dimension of the solid domain is 4.8 mm long

�x1-axis� and 1.6 mm deep �x3-axis�, and the dimension ofthe fluid domain is 2 mm long �x-axis� and 0.4 mm deep�y-axis�. The spacing between nodes in the solid and fluiddomain are 4 and 2 m, respectively. However, in the fluiddomain and at regions adjacent to the solid substrate, adenser grid is required to capture the boundary layerthickness49 �v= � /2�ff�1/2�100 nm generated by 20MHz acoustic waves. Therefore, the node spacing along they-direction in this region has been reduced to 20 nm in orderto provide approximately five computational points withinthe boundary layer.

D. Results

Figure 4 shows the results from a discrete Fourier trans-form �DFT� of the free surface displacement at point A �Fig.

3� from a time to frequency domain, i.e., h�t�→ h��, where

h�t�=h1�t�+�h2�t�. The h spectra therefore constitute qua-dratic terms of order O�� smaller than their linear counter-parts, representing weak nonlinearities. With the presence ofnonlinear terms in the equations, the solution may constituteharmonic waves62 with frequencies 0, 2�, 3�, 4� , . . . Thenumber of points for each data set is 53 900 and the sam-pling rate is 523 MHz. Therefore, the upper bound on thehighest frequency the recorded data can capture is approxi-mately 261 MHz �Nyquist frequency� and the lower bound isapproximately 10 kHz. The lowest frequency is limited bythe total sampling time, approximately 103 s. We examinethe effects of varying the surface displacement magnitude ofthe substrate, ��0.3, 1.2, and 2.5 nm, in Figs. 4�a��i�,4�a��ii�, 4�b��i�, 4�b��ii�, and 4�c��i�, 4�c��ii�, respectively,and of varying the initial film thickness, H=400 and600 m, in Figs. 4�b��i�, 4�b��ii� and 4�d��i�, 4�d��ii�, respec-tively.

The results in Fig. 4 clearly show that the resonant fre-quency f0 of the free surface is 20 MHz, identical to theapplied SAW frequency and consistent with the previouslyreported findings of Li et al.63 No distinct subharmonic cap-illary frequency at fc= f0 /2�10 MHz can be identified,which is in agreement with the Qi et al.’s experimentalfindings,16 and quite contrary to the original Faradayexperiment22 and the classical theories that have sincefollowed.23–25 Within the frequency band of 1–10 MHz, nospecific pattern of the frequency spectra can be identified�Figs. 4�a��i�, 4�b��i�, 4�c��i�, and 4�d��i��. In addition, thebroadband low-frequency vibrations �fc�1 MHz� f0�, firstobserved by Qi et al.,16 are absent in our numerical results.This is anticipated as the excitation of the low-frequencyvibration is associated with the high-amplitude SAW and faststreaming regime, i.e., ReA�1, where ReA is the acousticReynolds number, given by

ReA =fu1�f

2��. �56�

The present model, which is based on successive approxima-tions, is, however, limited to the low-amplitude SAW andslow streaming regime, i.e., ReA�1, since �h2�h1 is a nec-essary requirement of the regular perturbation expansion. We



note here that the phrase “successive approximations” is of-ten used in the acoustic streaming literature instead of per-turbation expansion to describe the analysis technique.

Nonetheless, the results do show the existence of weaksuperharmonic resonant frequencies at f1

+�40 MHz�2f0,f2

+�80 MHz�4f0, f3+�120 MHz�6f0, f4

+�160 MHz�8f0, and f5

+�200 MHz�10f0 �see Fig. 4�. The free sur-

face displacements h at these superharmonic frequencies areat least two orders of magnitude smaller than the resonantfrequency f0. The value of �h� at f1–6

+ decreases, following apower law in the frequency domain, as shown by the solidline in Fig. 4. A power law relationship between the peakamplitude at each resonance and the frequency may be ex-

pressed by h�f�=Cf−q, where C and q are constants. The es-timated exponent q for the curves in Figs. 4�a�–4�d� is ap-

proximately 4.9, 3.5, 3.1, and 3.5, respectively. For the

power law analysis, an increase in q �larger h− f slope� isassociated with a lower content of harmonics at higher fre-quencies, resulting in a purer monofrequency sinusoidal re-sponse at the excitation frequency f0, whereas a smaller q

�smaller h− f slope� represents higher amplitudes of the har-monics and, hence, a weaker monofrequency sinusoidal re-sponse. Increasing the SAW amplitude can increase the mag-

nitude of h at the superharmonic frequencies, which thenresults in smaller q. As the height of the liquid film increases,more sideband frequencies appear, showing an increase ofthe content of harmonics at higher frequencies. Therefore,within ReA�1, the shape of the deformed surface becomesslightly less sinusoidal by increasing the magnitude of the

Frequency (MHz)

|ξ| ~ 0.3 nmH ~ 400 µm

(a-ii)

|ξ| ~ 1.2 nmH ~ 400 µm

(b-ii)

|ξ| ~ 1.2 nmH ~ 600 µm

(d-ii)

10 60 110 160

(nm)

(nm)

(nm)

(nm)

|ξ| ~ 2.5 nmH ~ 400 µm

(c-ii)

100

10-1

10-2

10-3

10-4

10-5

10-6

10

100

10-1

10-2

10-3

10-4

10-5

10-6

(nm)

Frequency (MHz)

(nm)

(nm)

(nm)

(a-i)

(b-i)

(d-i)

(c-i)

10-7

10-8

10

100

10-1

10-2

10-3

10-4

10-5

10-6

10

100

10-1

10-2

10-3

10-4

10-5

10-6

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

0 5 10 15 20

Frequency (MHz)0 5 10 15 20

10

100

10-1

10-2

10-3

10-4

10-5

10-6


10

100

10-1

10-2

10-3

10-4

10-5

10-6


10

100

10-1

10-2

10-3

10-4

10-5

10-6

|ξ| ~ 0.3 nmH ~ 400 µm

|ξ| ~ 1.2 nmH ~ 400 µm

|ξ| ~ 1.2 nmH ~ 600 µm

|ξ| ~ 2.5 nmH ~ 400 µm

Frequency (MHz)10 60 110 160



FIG. 4. �Color online� Numerical results from the DFT of h at point A �see Fig. 3�. Three substrate displacement magnitudes ��—��a��i� and �a��ii�� 0.3 nm,��b��i� and �b��ii� and �d��i� and �d��ii�� 1.2 nm, and, ��b��i� and �b��ii�� 2.5 nm—and two different initial thicknesses for the liquid film—��a��i� and �a��ii�,�b��i� and �b��ii�, and �c��i� and �c��ii�� 400 m and ��d��i� and �d��ii�� 600 m—are selected for the analysis. In this figure, the frequency spectra for eachsubstrate displacement magnitude �� are separated for clarity. Six resonant frequencies are identifiable �marked by the solid-dots�. The solid line represents a

regression fit of the data, which are based on the power law h�f�=Cf−q with the regression coefficients q of 0.79, 0.9, 0.94, and 0.9 for �a��ii�, �b��ii�,�c��ii�, and �d��i�, respectively. The acoustic Reynolds number is approximately ReA�0.15 for ��0.3 nm, ReA�0.5 for ��1.2 nm, and ReA�1 for��2.5 nm.



SAW and/or the height of the liquid film. Also, as discussedin Sec. I, the presence of superharmonics and period-halvingbehavior is always associated with a likewise subharmonic,period-doubling sequence. We, therefore, expect the subhar-monic frequencies, from f0 to f0 /10, to be present althoughthey are not discernible in the results presented in Figs.4�a��i�, 4�b��i�, 4�c��i�, and 4�d��i�; this is due to limitationsof the model, which are highlighted below.

Figure 5�a��i� shows the computed velocity field of thesolid and the fluid film of height H�400 m after t=2000f−1 s, as a consequence of 20 MHz SAW excitation.The displacement magnitude of the substrate is ��1.2 nm excited by a 10 Vp−p potential applied to the sub-strate. Figure 5�a��ii� shows details of the decomposed in-stantaneous acoustic velocity field u1, and Fig. 5�a��iii�shows the total fluid velocity field u. Within the fluid do-main, the amplitude of the acoustic particle velocity u1

�O�10−1� m /s and the amplitude of the second order ve-locity �u2�O�10−6� m /s, indicating that the flow is in the

low-amplitude SAW regime �ReA�1�. The numerical re-sults, therefore, remain valid after 2000 cycles since the con-dition of u1��u2 is still satisfied.

The velocity profiles along the line A-A in Fig. 5�a��i�are shown in Fig. 5�b�. On the free surface, the wavelengthalong the x-direction is approximately �c�228 m and,within the bulk of the fluid, the wavelength along they-direction is approximately 80 m �see Figs. 5�b��i� and5�b��ii��. By comparing the wavelength in the y-direction tothe acoustic wavelength in the fluid, �f�75 m for a 20MHz acoustic wave, the results suggest that the compres-sional acoustic waves are propagating in the fluid predomi-nantly along the y-direction. The waves arriving at the air-liquid interface are then reflected back into the liquid due tothe acoustic impedance mismatch of the air��103 kg m−2 s� and water ��106 kg m−2 s�. Similarly,waves arriving at the solid-fluid interface are partially re-flected into the fluid. The superposition of these reflected andincident waves forms strong acoustic standing waves in the

Lithium niobate substrate

x1x3

xy

SAWSAW

Transducer

t ~ 2000 / f

2 �SAW

0

6

6

h (nm)

Instantaneous

velocity magnitude (m/s) 0.3 0.60

401

400.8

400.6

400.4

400.2

0.05 0.09 0.13

350

300

250

200

150

100

50

0

-0.15 -0.05 0.05 0.15

0

350

300

250

200

150

100

50

400-0.2 -0.1 0 0.1 0.2

Free surface Free surface

Velocity, ux (m/s)

Substrate

Velocity, uy (m/s)

Heigh

t(µm)

Heigh

t(µm)

Heigh

t(µm)

Heigh

t(µm)

LineA-A

(b)

(i) (ii)

(iii)(iv)

(a)

�v~ 0.1 µm

2 �SAW

0

0.2

0.4

0.6

0.8

1-0.05 0 0.05 0.1 0.15

Depth

(mm)

v1 velocity (m/s)

0

0.2

0.4

0.6

0.8

1-0.1 -0.05 0 0.05 0.1 0.15

0.115 0.117

Depth

(mm)

v3 velocity (m/s)

(v) (vi)

(iii)

(ii)

(i)

Velocity antinodal point

400

400

401

400.8

400.6

400.4

400.2

400

�SAW~�SAW/2

Substrate

0

6

6

h1(nm)

�R

FIG. 5. �Color online� �a��i� Computed acoustic velocity field �color density plot� in the solid and in the fluid after t=2000fSAW−1 , i.e., after 2000 sinusoidal

cycles each of period fSAW−1 seconds. At the solid-fluid interface, the transmission of acoustic waves into the fluid begins at a distance away from the left wall,

i.e., interactions between solid and fluid only within L1−L4 as shown in Fig. 3. �a��ii� Enlarged view of the fluid domain showing the instantaneous acousticvelocity field u1 and the computed instantaneous displacement h1. The Rayleigh angle �R is approximately 23°, which is very close to the theoretical predictionof �R�sin−1�c /cs��22°, where c=1485 m /s is the sound speed in the fluid and cs=3965 m /s is the sound speed in the substrate. �a��iii� Magnitude of thevelocity field u in the fluid and the instantaneous displacement of the free surface h, showing the formation of capillary waves on the liquid/gas interface. Theinstantaneous displacement of the free surface has been exaggerated for clarity. The profiles of h1 and h are almost identical, suggesting that the contributionof �h2 is negligible. However, adjacent to the free surface, the contribution of �u2 augments the velocity of the standing wave and hence facilitates thecollection of particles in this region, consistent with the results reported by Li et al. �Ref. 63�. The distance between two adjacent acoustic velocity antinodalpoints is approximately 114 m��SAW /2. The velocity profiles along line A-A are shown in �b�: ��i� and �ii�� instantaneous ux and uy velocity amplitude inthe bulk of the fluid, ��iii� and �iv�� instantaneous ux and uy velocity amplitude at the region adjacent to the substrate surface, and ��v� and �vi�� instantaneousv1 and v3 velocity amplitudes in the substrate.



y-direction; the mechanism is similar to Bragg resonanceresulting from the interaction of the free surface waves onwater and the �stationary� undulating seabed as observed byHeathershaw.64 In order to illustrate the interference ofacoustic waves more clearly, we use a 20 MHz pulse modeexcitation �one cycle on and four cycles off of a sinusoidalsignal� and extract only the first-order results at t=12fSAW

−1 ,13fSAW

−1 , 14fSAW−1 , 15fSAW

−1 , and 16fSAW−1 seconds, as shown in

Figs. 6�a�–6�e�, respectively.Curiously, the wavelength in the x-direction �c more

closely matches the SAW wavelength, �SAW�200 m. Thisindicates that the compressional wave that propagates in they-direction is strongly coupled with the SAW on the sub-strate. This results in a phase-locking phenomenon65 inwhich the phase of the u1

y velocity at the air-liquid interfaceis in-phase with the v3 velocity at the solid-fluid interface.The process is more clearly illustrated in Fig. 6, showing thesuperposed acoustic waves forming a moving “grid-structure,” which then appears to assist the phase-lockingmechanism. Moreover, the result obtained for the H�600 m film �not shown� reveals that �c is increased fur-ther to approximately 270 m, suggesting the �c is a func-tion of the film height; that is, �c increases with increasing

film height. We attribute this to be caused by the dispersionof acoustic waves and the weakening of the phase-lockingmechanism, whereby the effect should be strongest for H��SAW. Next to the substrate surface, a thin viscous bound-ary layer of thickness �v�0.1 m is generated, as shown inFigs. 5�b��iii� and 5�b��iv�. The velocities of the piezoelectricsubstrate along line A-A are also shown in Figs. 5�b��v� and5�b��vi�. The displacement of the substrate as the SAWpropagates across it is mostly confined within a SAW wave-length from the free surface, i.e., from x3=0 to x3=�SAW.

At the free surface of the fluid, acoustic velocity antin-odal points are formed as shown in Fig. 5�a��iii�. We notehere that from linear acoustic theory, a displacement nodalpoint is equivalent to an antinode in the velocity due to the90° phase difference between the displacement and velocityfield. The instantaneous displacement of the free surface istherefore at an equilibrium state �h=0� at displacement nodalpoints or at velocity antinodal points �Fig. 5�a��iii��. The for-mation of a standing wave on the free surface here is consis-tent with Li et al.’s experimental results,63 wherein the clus-tering of nanoparticles on the free surface of a slenderdroplet �L�H� along lines specified by the displacement

(a)t ~ 12/f

SAW

SAW

SAW

SAW

SAW

(b)

(c)

(d)

t ~ 13/fSAW

t ~ 14/fSAW

t ~ 15/fSAW

SAW

(e)t ~ 16/f

SAW

100 µm

100 µm

100 µm

100 µm

100 µm

0.23 0.460Instantaneous acoustic velocity magnitude (m/s)

FIG. 6. �Color online� Numerical results showing thepropagation/reflection of acoustic waves into a flat liq-uid film �H�L� at t= �a� 12fSAW

−1 , �b� 13fSAW−1 , �c�

14fSAW−1 , �d� 15fSAW

−1 , and �e� 16fSAW−1 s, under a 20 MHz

pulse mode excitation �one cycle on and four cycles offof a sinusoidal signal�. The sequential images show theformation of strong acoustic standing waves in they-direction and a moving grid-structure within the liq-uid film. The instantaneous displacement h1 of the freesurface has been exaggerated for clarity.



antinodal locations is reported. The interval between two ad-jacent particle cluster lines is approximately 100 m��SAW /2, which compares favorably with the fluid velocityfield in Fig. 5�a��iii�.

Further evidence for the formation of standing waves isobtained from Alvarez et al.’s experimental analysis20 inwhich they report the spreading of a polymer drop atop aSAW device into a thin film �L�H�. Breakup occurs acrossthe entire free surface due to the depletion and rupture of thefilm at the acoustic velocity antinodal positions. Due to atransverse instability across a two-dimensional free surface,the velocity antinodal points form a hexagonal closed-packedlattice66 and with a longitudinal pitch spacing equivalent to�SAW /2.20 As a consequence, the polymer spots appear in anoffset close-packed pattern. We note here that the distance of�SAW /2 on the free surface and the phase-locking phenom-enon requires L�H; that is, the free surface is relatively flatand parallel to the substrate, and with a small film height�H��SAW�.

In summary, the numerical model above, based on suc-cessive approximations, is not capable of capturing thebroadband low-frequency vibration �fc� f0� driven byhigh-amplitude SAW excitation �ReA�1�, observedexperimentally.16 The displacement amplitude of the free sur-face in the experiments is far larger than the substrate dis-placement, i.e., h��. However, the numerical results �Fig. 4�show the computed values of h and � to be of the same orderof magnitude, h��. This is primarily due to the criteria ofthe method of successive approximations in which h1��h2

and h1�� as a result of the linearization of the first-orderacoustic field. This can be more clearly demonstrated by Eq.�50� in which the linearized kinematic equation shows thatthe time-derivative of h1 varies linearly with u1

y; since theacoustic field in the fluid is linear, we expect �v��u1� foracoustic waves in a bounded medium with negligible acous-tic absorption coefficient �. We also note that the h�� re-striction causes the suppression of apparent deformations,which requires h�H �see Figs. 1 and 10�a��iv��. The succes-sive approximation method also limits the second-order fieldto slow streaming and hence is not capable of capturing thefast-moving streaming. Furthermore, the two-dimensionalmodel only allows a limited interaction of wave vectors �ofcapillary waves� as a consequence of the reduced dimensionof the free surface; the free surface is modeled as a single-line in a two-dimensional model. The importance of usingmore than one-dimension for the free surface is clearly dem-onstrated in the experiments in which capillary wave patternsare formed—the transition between patterns on a two-dimensional free surface is associated with the interactionof wave vectors, thereby generating subharmonic/superharmonic waves.67,68 The interaction between wavevectors could also result in the cascade of subharmonic fre-quencies and give rise to the broadband low-frequency phe-nomenon observed. Therefore, the high-amplitude vibrationof the free surface �with nonrestricted degree of freedoms� inthe low-frequency band will be analyzed using experimentaltechniques in Sec. III.

III. EXPERIMENTS

A. Experimental setup

The experimental setup is shown in Fig. 7. The interdigi-tal transducer �IDT� was fabricated on 500 m thick,127.68°-rotated Y-cut X-propagating single-crystal lithiumniobate �128-YX-LN� substrates �Roditi Ltd., London, UK�,using a standard UV photolithography process. We useddual-layer metallization for the IDT; a titanium layer ofthickness of 4 nm is first deposited followed by an aluminumlayer of thickness of 250 nm. The design was based on aconventional bidirectional IDT scheme69,70 and consists of30 electrode finger pairs, each of width �SAW /4�200 m,to generate a 20 MHz SAW.

To contain the fluid, a rectangular area of approximately10 mm in length L, 5 mm in width W, and 5 mm in heightH, was placed on the substrate using a damping material��-gel, Geltec Ltd., Yokohama, Japan�, as shown in Fig. 7�a�.This material absorbs the SAW energy, thus minimizingwave reflection. The liquid was dispensed until the thicknessof the film reached �1 mm. A scanning laser Doppler vibro-meter �LDV� �MSA-400, Polytec PI, Waldbrunn, Germany�was used to record the vertical �perpendicular to the sub-strate� displacement of the fluid free surface at a samplingrate of 52 MHz; 220 data points were recorded for eachmeasurement.71 The Nyquist frequency is 26 MHz �half ofthe sampling frequency�, which is the highest frequency therecorded data can accurately represent. The lowest frequencyis determined by the total sampling time, approximately0.02 s. The lowest frequency is therefore 50 Hz. Thus, fre-quency data outside the band from 50 Hz to 26 MHz arediscarded. The recorded time-domain data were subsequently

Transducer

(a) Absorber

Water

Isometric view

Transducer

Sectional side view

LaserAbsorber

128-YX LN Substrate

Point A Water

(b)

W

LH

Scanning

LDV

FIG. 7. �Color online� Schematic illustrating the experimental setup em-ployed to measure the capillary wave vibration. �a� To minimize wave re-flections, the water was contained within an absorber except on the sidefrom which the SAW approached. �b� A sectional side view shows the lo-cations where the vertical displacement was measured using the scanningLDV.



transformed into the frequency domain using a fast Fouriertransform. For all measurements, water was mixed withblack food coloring �Queen Fine Foods Pty. Ltd., Queen-sland,Australia� to enhance the reflection of the laser beam fromthe free surface and to reduce the reflected light from thesubstrate to ensure that the time-domain displacement data ofthe capillary waves is independent of the time-domain dis-placement data of the vibrating substrate. At each inputpower, the vertical displacement amplitude of the substrate atpoint A �see Fig. 7�b�� was measured. Four different magni-tudes were selected—1.5, 2, 2.4, and 2.8 nm—for the studyof the transition from different flow regimes. We note herethat no SAW atomization occurred for this range of displace-ment magnitudes.

B. Results and discussion

Figure 8 shows the frequency spectrum of the free sur-face vibration. Only a single peak at 20 MHz is observed forlow magnitude SAW excitation, i.e., ��1.5 and 2.0 nm. Asshown by the solid line, the resonances represented by thedata in Figs. 8�a��ii� and 8�b��ii� are Lorentzian and may beexpressed by L��=�−1� / ��−�0�2+�2�, where � specifiesthe width and �0 is the central frequency, indicating the phe-nomena at the current excitation amplitude is linear. By in-creasing the SAW magnitude to ��2.4 and 2.8 nm, theamplitude of the 20 MHz vibration peak of the free surfacedecreases significantly, as shown in Figs. 8�c��ii� and 8�d��ii�,respectively, and the Lorentzian function no longer fits thedata. This suggests that the system’s response begins to de-

Frequency (MHz)

|ξ| ~ 1.5 nm

Frequency (MHz)

(nm)

|ξ| ~ 2.0 nm

|ξ| ~ 2.4 nm

|ξ| ~ 2.8 nm

19.97 19.98 19.99 20 20.01 20.02 20.03

19.97 19.98 19.99 20 20.01 20.02 20.03

(nm)

(nm)

(nm)

(nm)

(a-i)

(b-i)

(d-i)

(c-i)

10

100

10-1

10-2

10-3

10-4

(nm)

10

100

10-1

10-2

10-3

10-4

(nm)

(nm)

(a-ii)

(b-ii)

(c-ii)

(d-ii)

Frequency (MHz)19.97 19.98 19.99 20 20.01 20.02 20.03

10

100

10-1

10-2

10-3

Frequency (MHz)19.97 19.98 19.99 20 20.01 20.02 20.03

10

100

10-1

10-2

10-3

Frequency (MHz)

10

100

10-1

10-2

10-3

100

10-1

10-2

10-3

10-4

Frequency (MHz)10

010

-110

-210

-310

-4

10

100

10-1

10-2

10-3

Frequency (MHz)10

010

-110

-210

-310

-4

10

100

10-1

10-2

10-3

10-4

10

100

10-1

10-2

10-3

10-4

Frequency (MHz)10

010

-110

-210

-310

-4

|ξ| ~ 1.5 nm

|ξ| ~ 2.0 nm

|ξ|~ 2.4 nm

|ξ|~ 2.8 nm

FIG. 8. �Color online� Capillary wave frequency spectra over a range of vertical substrate displacement magnitudes measured using the LDV at low kilohertzorder frequencies—�a��i� ��1.5 nm, �b��i� 2.0 nm, �c��i� 2.4 nm, and, �d��i� 2.8 nm—and high megahertz order frequencies—�a��ii� ��1.5 nm and

hpeak�2.9 nm, �b��ii� ��2.0 nm and hpeak�1.8 nm, �c��ii� ��2.4 nm and hpeak�1.2 nm, and �d��ii� ��2.8 nm and hpeak�0.3 nm—when driven bya 20 MHz SAW. In this figure, the frequency spectra for each substrate displacement magnitude �� are separated for clarity. The frequency spectra between0.1 kHz and 1 MHz show that the free surface vibration spectra broadens considerably as the SAW magnitude increases from �� a��i� 1.5 to �d��i� 2.8 nm.On the other hand, the high frequency measurements show a single peak at 20 MHz, which is the excitation frequency, and the peak becomes increasinglyweak as the SAW magnitude increases from �� a��i�1.5 nm to �d��ii� 2.8 nm. The solid line is the Lorentzian function used to fit to the data.



part from a linear one with excitation magnitude. Overall, asthe SAW magnitude increases from 1.5 to 2.8 nm, the vibra-tion amplitude of the fluid free surface at 20 MHz decreasesfrom 2.9 to 0.3 nm. Also, under higher amplitude SAW ex-citation, the low-frequency broadband vibration of the freesurface begins to emerges, as shown in Figs. 8�a��i�, 8�b��i�,8�c��i�, and 8�d��i�. The frequency band shifts from 10–50kHz downward to 700–900 Hz when the SAW magnitude isincreased from ��1.5 to 2.8 nm, suggesting that the energyappearing at the excitation frequency fSAW=20 MHz is be-ing drawn into the far lower frequency modes via an ex-tended subharmonic period “doubling” sequence.

Due to the limitation of the measurement system and therequirement to maintain the spectra resolution, the spectrumof the first superharmonic frequency for the 20 MHz SAW�f1

+�40 MHz� exceeds the upper bound of the frequencylimit �26 MHz�. As such, the existence of the first superhar-monic frequency for 20 MHz SAW cannot be verified basedon the current experimental setup. However, this problemcan be easily resolved by using an 8.78 MHz SAW devicein lieu of the 20 MHz SAW device. For the case offSAW=8.78 MHz, we expect f0�8.78 MHz and f1

+

�17.56 MHz; the first superharmonic frequency is thereforebelow the upper bound of the measurable frequency range

�26 MHz�. Figure 9 shows the frequency spectra of the freesurface vibration driven by an 8.78 MHz SAW; Figs. 9�a�and 9�b� ��1.5 nm, Figs. 9�c� and 9�d� ��2.0 nm, andFigs. 9�e� and 9�f� ��2.5 nm. As expected, the resultsshow the existence of the first superharmonic frequencyf1

+�17.56 MHz.Finally, we speculate that the transition to the regime

dominated by subharmonic frequencies, i.e., h�f−�� h�f0� asshown in Figs. 8�d��i� and 8�d��ii�, is a consequence of thedeparture from the infinitely small amplitude wave regimeand the occurrence of the subharmonic cascade. Using Eq.�56�, for 20 MHz acoustic waves, the acoustic Reynoldsnumber for ��1.5, 2, 2.4, and 2.8 nm, is approximatelyReA�0.7, 1, 1.2, and 1.4, respectively. However, ReA doesnot consider the physical dimension of the liquid film, andtherefore it is essential to obtain a different nondimensionalparameter that provides a better representation of the signifi-cance of the change in the wave magnitude �� and the di-mension of the film H. Nondimensionalizing the momentumequation in Eq. �8�, assuming the flow to be steady, using the

following transformation: �x ,d�=L�H /�v��x , d�, =f0,and, u2=U2�H /�v�u2, together with Eq. �10�, and selectingterms of order O��2� give

(b)

(d)

8.5 8.6 8.7 8.8 8.9 9.0Frequency (MHz)

10

100

10-1

10-2

10-3

10

100

10-1

10-2

10-3

17.0 17.2 17.4 17.6 17.8 18.0Frequency (MHz)

(nm)

(nm)

|ξ| ~ 2.0 nm(c)

(a)

|ξ| ~ 2.0 nm~ 8.78 MHz

~ 17.56 MHz

8.5 8.6 8.7 8.8 8.9 9.0Frequency (MHz)

10

100

10-1

10-2

10-3

10

100

10-1

10-2

10-3

17.0 17.2 17.4 17.6 17.8 18.0Frequency (MHz)

(nm)

(nm)

|ξ| ~ 1.5 nm |ξ| ~ 1.5 nm~ 8.78 MHz

~ 17.56 MHz

(f)

8.5 8.6 8.7 8.8 8.9 9.0Frequency (MHz)

10

100

10-1

10-2

10-3

10

100

10-1

10-2

10-3

17.0 17.2 17.4 17.6 17.8 18.0Frequency (MHz)

(nm)

(nm)

|ξ| ~ 2.5 nm(e)

|ξ| ~ 2.5 nm~ 8.78 MHz

~ 17.56 MHz

FIG. 9. �Color online� Capillary wave frequency spectra for two different vertical substrate displacement magnitudes measured using the LDV at highmegahertz order frequencies—��a� and �b�� 1.5 nm, ��c� and �d�� 2.0 nm, and ��e� and �f�� 2.5 nm—when driven by an 8.78 MHz SAW. Theresults show the existence of the first superharmonic frequency f1

+�17.56 MHz�2f0, where f0=8.78 MHz.



��u2 · ��u2 = �2

Lf0U2��2u2 + � +

1

3� � � · u2 ,

�57�

where the tilde decoration represents dimensionless quanti-ties, L is the characteristic length, �=B /, and U2 is thecharacteristic streaming velocity. The ratio H /�v captures theway changes in the film height contribute to nonlinear ef-fects, such as the appearance of additional harmonics ob-served in Fig. 4�d��ii�. We note that this is an important pa-rameter in Liobashevski et al.’s study of low-frequency�20–80 Hz� Faraday instabilities. Bifurcation to a subhar-monic global pattern �the spatiotemporal behavior of the freesurface� is observed for large H /�v as the acceleration of thefluid layer is increased, whereas propagating solitary struc-tures are observed for small H /�v.72

A parameter that captures the onset of fast streaming isdefined as �=�2�H /�v�2. This is similar to the nonlinear“Reynolds number” ReNL=2MA

2 R2 /�v2 introduced by Men-

guy and Gilbert,73 for acoustic streaming in a resonator ofradius R and acoustic Mach number MA=U1 /c0=�. We fur-ther assume that U1�hf0 and, therefore, ��hf0 /c0, where his the vertical surface displacement of the free surface. In the20 MHz SAW experiments, ��108 s−1, �v�10−7 m, c0

�1450 m /s, and H�10−3 m. Using the recorded time se-ries data, we found that the maximum instantaneous dis-placement amplitudes for ��1.5, 2.0, 2.4, and 2.8 nm werehmax�3, 6, 17, 60 nm, respectively. Therefore, for the caseswhere h�hmax�3, 6, 17, and 60 nm, ��0.09, 0.36, 2.9, and36, respectively, clearly suggesting that the nonlinearitybuilds quickly as the SAW magnitude increases from 2.4 to2.8 nm. This is consistent with our speculation that as theamplitude increases beyond a critical level, the energy shiftsfrom the 20 MHz peak to the subharmonic frequencies andsubsequently gives rise to the broadening of low-frequencyspectra via a subharmonic cascade from the fundamentalresonance mode of the free surface. For the 8.78 MHz SAWexperiments, ��1 for all cases shown in Fig. 9, and there-fore no flow transition is observed. We note here that thefourth-order nonlinear term �u2 ·��u2 is usually neglected inthe classical approach.47,74,75 However, Lighthill76 pointedout that it is this term that should be retained when the orderof magnitude of the viscous term becomes comparable to theinertia term.

IV. DROP

The interesting phenomenon in which �c��SAW ob-served when L�H, i.e., a film, warrants further investiga-tion, at least to verify that �c��SAW when L�H, i.e., adrop. Here, we perform a numerical simulation of a dropsubjected to traveling-SAW excitation with the following pa-rameter values: fSAW=20 MHz, H=1 mm, and, L=2 mm.The height H in Eq. �23� has been redefined as H�x�=−2L−1x2+2x. In place of the transformation for the case ofa film in Eq. �23�, we apply the following for the case of adrop:

� = x, =y − ��h1�x,t� + �2h2�x,t��

H�x� − ��h1�x,t� + �2h2�x,t��, �58�

where h1�x , t� and h2�x , t� are the first and second order sur-face displacements previously defined in the perturbation ex-pansion of Eq. �24�. Substituting Eq. �58� into Eqs.�B1�–�B6�, we obtain

�

�x=

�

��−

0

H�H��

�

� + O�� , �59�

�2

�x2 =�2

��2 −2 0

H�H��

�2

�� +

2 02

H2 �H��

�2 �2

� 2

− 0

H�2H��2

�

� +

2 0

H2 �H��

�2 �

� + O�� , �60�

�

�y=

1

H�

� + O�� , �61�

�2

�y2 =1

H2

�2

� 2 + O�� , �62�

�2

�x � y=

1

H�2

�� −

0

H2

�H��

�2

� 2 −1

H2

�H��

�

�

−1

H2

�H��

�

��+ O�� , �63�

�

��=

�

�t+ O�� , �64�

where 0=y /H. Next, we combine Eqs. �59�–�64� with Eqs.�13� and �14� to arrive at

�f1

��+

f0

H�u1

y

� + f0

�u1x

��− f0

0

H�H��

�u1x

� = 0, �65�

f0�u1

x

��=

0

H�H��

�p1

� +

2 0

H2 � �H��

�2�u1x

� −

�p1

��

− 0

H�

�2H��2

�u1x

� −

�

H2

�H��

�u1y

� +

H2

�2u1x

� 2

+ 0

2

H2� �H��

�2�2u1x

� 2 − 0

2

H2��H��

�2u1y

� 2

− ��H��

�u1y

��−

2 0

H�

�H��

�2u1x

�� +

�

H�2u1

y

��

+ ��2u1

x

��2 , �66�



f0�u1

y

��= −

1

H�p1

� −

�

H2

�H��

�u1x

� +

2 0

H2 �H��

�2�u1y

�

− 0

H

�2H��2

�u1y

� −

0

H2��H��

�2u1x

� 2 +�

H2

�2u1y

� 2

+ 0

2

H2 �H��

�2�2u1y

� 2 −�

H2

�H��

�u1x

��+

�

H�2u1

x

��

−2 0

H

�H��

�2u1y

�� +

�2u1y

��2 . �67�

The boundary conditions at the free surface and solid-fluidinterface remain unchanged. Under low excitation power, thedrop is observed in all previous experiments to be essentiallypinned at the contact line. Therefore, at the contact line, the

following boundary conditions are prescribed: h1=0, u1x =0,

and, u1y =0 at x=0 and x=L. The results for the film in Figs.

5�a��ii� and 5�a��iii� show that the contribution of the second-order field to the surface deformation profile h is insignifi-cant. As such, the present drop model only involves solvingthe first-order equations.

As shown in Fig. 10, the numerical results for drops withL�H show that an acoustic standing wave is formed pre-dominantly along the radiation angle �R�22° and that thedistance between two pressure nodal points on the free sur-face is approximately �f /2. In order to illustrate the interfer-ence of acoustic waves within a drop more clearly, we use a20 MHz pulse mode excitation �one cycle on and four cyclesoff of a sinusoidal signal� and extract the first-order results att=8fSAW

−1 , 9fSAW−1 , 10fSAW

−1 , 11fSAW−1 , 12fSAW

−1 , 13fSAW−1 , 14fSAW

−1 ,

(b-i) (b-ii)

(d-i)

t ~ 10/fSAW t ~ 10/f

SAW

t ~ 50/fSAW

125 µm 125 µm

125 µm

Substrate

x

xSubstrate

x

125 µmh1

(d-ii)

h1

Traveling

SAW Substrate

(c-i)

t ~ 30/fSAW

125 µm

Substrate

xSubstrate

x

h1

(c-ii)~ �

f

h1

1.9 nm

x

t ~ 50/fSAW

t ~ 30/fSAW

~ �f

0 680-680

Instantaneous acoustic

pressure amplitude (kPa)0.16 0.320


velocity magnitude (m/s)

~ �f

~ �f

22o

22o

Traveling

SAWTraveling

SAW

Traveling

SAWTraveling

SAW

22o

22o

(a-i) (a-ii)

t ~ 7/fSAW t ~ 7/f

SAW

125 µm 125 µm

Droplet

H

L

xTraveling

SAW Substrate

x

SubstrateTraveling

SAW

h1

h1

FIG. 10. �Color online� Numerical results showing the transmission of acoustic waves into a sessile drop �H�L� at t= ��a��i� and �a��ii�� 7fSAW−1 , ��b��i� and

�b��ii�� 10fSAW−1 , ��c��i� and �c��ii�� 30fSAW

−1 , and ��d��i� and �d��ii�� 50fSAW−1 s, demonstrating the formation of an acoustic standing wave in the drop along the

Rayleigh angle �R�22°. The magnitude of the acoustic particle velocity field �u1� is shown in �a��i�, �b��i�, �c��i�, and �d��i�, and the amplitude of the acousticpressure amplitude p1 is shown in �a��ii�, �b��ii�, �c��ii�, and �d��ii�. The instantaneous displacement h1 of the free surface has been exaggerated for clarity. Theheight and diameter of the drop is represented by H and L, respectively. Thus, for L�H, �c��f, whereas for L�H, �c��SAW �see Fig. 5�.



and 15fSAW−1 seconds, as shown in Figs. 11�a�–11�h�, respec-

tively. The results show that the interaction of acoustic waveswithin a drop is far more complex compared to that in thecase of a film �Fig. 6�; further, the grid-structure formed bythe superposed waves is absent, suggesting no phase-lockingmechanism is possible and therefore �c��SAW. The dropmodel presented here, however, is essentially valid for low-amplitude SAW �ReA�1� and therefore unable to capturethe apparent deformation ��c�Rd� of the free surface. None-theless, capturing the formation of the acoustic standingwave along the Rayleigh angle �R�22° generated by a trav-eling SAW is consistent with the previously reported experi-mental observation in which a rapid microjet along the Ray-leigh angle is generated by a high intensity traveling SAW.13

Furthermore, the results also show that polygonlike struc-tures are formed at the top-right corner of the drop due to the

superposition of the incident and reflected acoustic waves�Fig. 10�, consistent with Brunet et al.’s52 numerical results.

Finally, we provide a direct qualitative comparison be-tween experimental and numerical results by presenting acase study that analyzes the alignment of microparticles on ahemispherical sessile drop. A consequence of standing capil-lary waves is the assembly of hydrophobic particles alongantinodal displacement lines.77 To reduce the effect of acous-tic streaming on the particles, the amplitude of SAW needs tobe sufficiently low ��1 nm�, such that the net flow of thefluid can be neglected. In the experiments, the substrate wascoated with a layer of hydrophobic material �Teflon AF,DuPont Corp., Wilmington, Delaware�, and therefore thedrop shape is almost hemispherical with a contact angle ofapproximately 118°. The motion of 6 m fluorescent poly-styrene �hydrophobic� particles �Duke Scientific, Palo Alto,

0.13 0.260Instantaneous acoustic velocity magnitude (m/s)

(a) (b)

(c)

t ~ 8/fSAW

125 µm 125 µm

125 µm 125 µm

(d)

SAW

(e)

125 µm 125 µm

(f)

SAW

(g)

125 µm 125 µm

(h)

t ~ 9/fSAW

t ~ 10/fSAW t ~ 11/f

SAW

t ~ 12/fSAW t ~ 13/f

SAW

t ~ 14/fSAW

t ~ 15/fSAW

SAW SAW

SAW SAW

SAW SAW

FIG. 11. �Color online� Numerical results showing the propagation/reflection of acoustic waves in a sessile drop �H�L� at t= �a� 8fSAW−1 , �b� 9fSAW

−1 , �c�10fSAW

−1 , �d� 11fSAW−1 , �e� 12fSAW

−1 , �f� 13fSAW−1 , �g� 14fSAW

−1 , and �h� 15fSAW−1 s, under a 20 MHz pulse mode excitation �one cycle on and four cycles off of a

sinusoidal signal�. The results show the complex interaction between the incident and reflected waves within the drop, and therefore no phase-lockingmechanism arises. In addition, the motion of the free surface under pulse mode excitation is irregular and no specific �c can be identified. The instantaneousdisplacement h1 of the free surface has been exaggerated for clarity.



CA� suspended in the drop was observed under fluorescencemicroscopy �BXFM stereomicroscope and iSpeed camera,Olympus, Japan� at 500 frames/s. Figure 12�a� illustrates theexperimental setup. The excitation frequency was 30 MHzand hence �SAW�130 m and �f�49 m. Two focusingsingle-phase unidirectional transducers were used to generatehigh intensity focusing standing SAWs. We note that thistype of SAW device has been used to generate slender SAWmicrojet in previous studies.13 Under extremely low excita-tion power ��1 nm�, the particles are observed to align ontop of a 3� droplet �H�L� with the distance �St betweentwo adjacent alignment lines varying in the range between�St=32�4 m; therefore, �f /2��St��f, as shown in Figs.12�b�–12�d�. With only a slight increase in the amplitude ofSAW, the aligned particles are observed to be dispersed andare seen to move along the streamlines �or to accumulate atthe apex of the drop� as a consequence of the buildup of slowstreaming, as shown in Fig. 12�d�.

It is then possible to use the numerical model developedabove to verify the formation of acoustic standing wavesand, hence, to verify the �St separation along the free surfacefor a drop of radius Rd=500 m under 30 MHz standingSAW excitation. Since the amplitude of SAW is extremelylow ��1 nm and hence ReA�1�, it is therefore possible toneglect the O��2� contributions—only O�� contributions

are considered. Figure 13 shows the numerical results of theacoustic velocity magnitude �u1� and the acoustic pressure p1

field in the drop after t=15fSAW−1 and 30fSAW

−1 s. Near the dropapex, the distance �St between two adjacent displacementantinodal points is within the limits of �f /2��St��f, con-sistent with the experimental results. Therefore, it appearsthat for H�L, the distance between two adjacent displace-ment antinodes lines is associated with the acoustic wave-length of the liquid.

V. CONCLUSIONS

A combination of both numerical and experimental stud-ies has been presented to capture the salient features of thefree surface dynamics of a liquid vibrating under high fre-quency surface acoustic wave excitation. A two-dimensionalnumerical model was constructed that couples the motion ofthe piezoelectric substrate and a thin liquid layer atop thesubstrate from both the solid to the fluid and vice versa. Forthe solid domain, the time-domain constitutive equations thatdescribe the motion of the piezoelectric substrate were em-ployed. For the fluid domain, the assumption of infinitesi-mally small amplitude acoustic waves was employed, whichthen permitted the use of successive approximations to de-compose the equations governing the fluid motion such that

Fluorescence

microscope

Side viewSAW SAW

DropletTransducer Transducer

Focal plane

(a)

�f/2 < ~32 µm < �

f

Top view

(b) (c)

�f/2 < ~33 µm < �

f

Top view

Focusing

SPUDT

Focusing

SPUDT

Focal

point

Teflon

coated area

Top view (no droplet)

(d) (e)

�f/2 < ~32 µm < �

f

Top view

Top view

FIG. 12. �Color online� �a� Schematic of the experi-mental setup used to visualize the collection of 6 mparticles on a 3 � drop �H�L� due to the vibrationof its free surface using 30 MHz standing SAWs gen-erated by two focusing SPUDTs. The images—capturedwith �b� 5�, �c� 10�, and �d� 50� microscope lenses—show that the microparticles are collected at the top ofthe drop �focal plane�. The distance �St between twoadjacent collection lines is 32�4 m, therefore fallingwithin the limits of �f /2��St��f, where �f�49 mfor 30 MHz acoustic waves in water. �e� The alignedparticles are dispersed and move along the streamlines�or accumulate at the apex of the drop� when the am-plitude of SAW is increased.



it can be resolved at the two widely varying time scalesrelevant to the problem. Due to this method of successiveapproximation, the numerical model is, however, limited tothe study of free surface dynamics in the low-amplitude flowregime �ReA�1�. We thus augment the study with experi-ments to capture the free surface dynamics in the high-amplitude flow regime �ReA�1�. Together, the combinednumerical and experimental studies therefore offer a basisupon which the free surface dynamics driven by surfaceacoustic waves across the entire range of excitation ampli-tudes from low ReA to high ReA can be analyzed.

In both the simulations and experiments, a time seriesdata of the free surface deformation h were captured. Theresults revealed that the flow regime had a critical influenceon the dynamics of the free surface. For the low-amplitudeflow regime �ReA�1�, only a single peak was found in thefrequency spectra. The resonant frequency of the free sur-face, in this case, is identical to the frequency of the SAW�fc� fSAW�, consistent with previous experimental results.The omission of strong nonlinear terms as well as the smallfree surface assumption ��h2�h1� required by the numericalmodel therefore cannot capture the behavior of the large am-plitude flow regime �ReA�1�, where strong nonlinear effects

become dominant, resulting in a shift of acoustic energyfrom a single peak at fSAW to a low-frequency broadbandspectra. This phenomenon is associated with the emergenceof fast streaming. A simple scaling analysis of the leadingorder terms in the momentum equation further shows theimportance of such strong nonlinearity arising from thehigher order terms, which is postulated to be the reason forthe low-frequency broadband vibration. A nonlinear param-eter is derived to define the onset of such strong nonlineareffects; i.e., when ��1.

The results of the numerical simulation, however, areable to provide insight into the �SAW /2 separation distancebetween two adjacent displacement nodal points on the freesurface reported in Li et al.63 and Alvarez et al.;20 the formerobserved that particles suspended on the free surface collectat displacement nodal lines, whereas the latter reported theformation of polymer spot patterns, also along displacementnodal lines. Thus, despite the simplicity of our numericalmodel, the results are able to capture these salient featuresobserved in experiments, suggesting that they occur as a re-sult of phase-locking of the vertical component of the veloci-ties; that is, the phases of uy at y=0 and at y=H along thex-axis are closely matched. The numerical results further re-

0 530-530


pressure amplitude (kPa)0.15 0.30


velocity magnitude (m/s)

(b-i)

t ~ 15/fSAW

125 µm

Droplet

SAW

(b-ii)h1

SAW

Standing SAW

(c-ii)

t ~ 30/fSAW

125 µm

SAW

(c-ii)

t ~ 30/fSAW

h1

SAW

Standing SAW

�f/2 < ~33 µm < �

f

t ~ 15/fSAW

Standing SAW

Standing SAW

(a-i)

t ~ 10/fSAW

125 µm

Droplet

SAW

(a-ii)h1

SAW

Standing SAW

t ~ 10/fSAW

Standing SAW

FIG. 13. �Color online� Numerical results showing the transmission of 30 MHz standing SAWs into a sessile drop �H�L� at t= ��a��i� and �a��ii�� 10fSAW−1 ,

��b��i� and �b��ii�� 15fSAW−1 , and ��c��i� and �c��ii�� 30fSAW

−1 s; �a��i�, �b��i�, and �c��i� are the instantaneous acoustic velocity magnitude �u1� plots, whereas �a��ii�,�b��ii�, and �c��ii� are the instantaneous acoustic pressure amplitude p1 plots. The results suggest that on the top of the drop, the distance between two adjacentdisplacement antinodal lines ��St� is �f /2��St�33 m��f, consistent with the experimental results in Figs. 12�b�–12�d�.



veal that the separation distance depends on the geometry ofthe system: for a film where L�H, the separation distanceapproaches �SAW /2, whereas for a drop where L�H, theseparation distance approaches �f /2.

The close agreement between the predictions afforded byour simple model and the experimental results lends confi-dence to the numerical model for the prediction of the flowfield under low excitation amplitudes. Good agreement wasobtained between experiments and numerics, in that, �1� thefundamental resonant frequency of the free surface is identi-cal to the excitation frequency, i.e., f0= fSAW, �2� the separa-tion distance between two adjacent pressure nodal points onthe free surface of a film �L�H� is approximately �SAW /2excited by a traveling-SAW, and �3� the separation distancebetween two adjacent displacement antinodal points on thefree surface of a hemispherical drop �L�H� is within therange of �f /2 excited by a standing SAW. Finally, the exis-tence of superharmonic frequency f1

+�2f0 has also beenshown. The present results offer guidance for the numericalstudy of SAW atomization, which involves the use of largeexcitation powers to drive the transition from the slowstreaming regime �ReA�1� into the fast streaming regime�ReA�1�, in anticipation of a more complex numericalscheme that spans both regimes and fully satisfies the largeamplitude capillary wave vibrations that proceed towardpinch-off and droplet breakup.

APPENDIX A: TRANSFORMED MATERIALPROPERTIES

The transformed piezoelectric constant ekij, dielectricconstant �ik

S , and elastic stiffness constant cijklE for 128° YX

lithium niobate �LiNbO3, LN� are53

�e� = � 0 0 0 0 4.46 0.4

− 1.77 4.46 − 1.59 0.12 0 0

1.68 − 2.67 2.4 0.59 0 0� C

m2 , �A1�

�� = �44 0 0

0 37.9 − 7.81

0 − 7.81 34� F

m, �A2�

�c� = �20.3 7.23 6.02 1.07 0 0

7.23 19.4 9.06 0.89 0 0

6.02 9.06 22.03 0.81 0 0

1.07 0.89 0.81 7.49 0 0

0 0 0 0 5.63 − 0.44

0 0 0 0 − 0.44 7.6

�1010 N

m2 .

�A3�

APPENDIX B: COORDINATE TRANSFORMATION

The following general transformation of the partial de-rivatives from a physical space �x ,y , t� to a computationalspace �� , ,�� for a two-dimensional unsteady flow is used:61

�

�x=

�

��

�x� +

�

� �

�x� , �B1�

�2

�x2 =�

�� 2�

�x2� +�

� �2

�x2 � +�2

��2 ��

�x�2

+�2

� 2 �

�x�2

+ 2�2

� � � �

�x� ��

�x� , �B2�

�

�y=

�

��

�y� +

�

� �

�y� , �B3�

�2

�y2 =�

�� 2�

�y2� +�

� �2

�y2� +�2

��2 ��

�y�2

+�2

� 2 �

�y�2

+ 2�2

� � � �

�y� ��

�y� , �B4�

�2

�x � y=

�

�� 2�

�x � y� +

�

� �2

�x � y� +

�2

��2 ��

�x� ��

�y�

+�2

� 2 �

�x� �

�y� +

�2

��

�x� ��

�y�

+ ��

�x� �

�y� , �B5�

�

�t=

�

��

�t� +

�

� �

�t� +

�

��

�t� . �B6�

1L. Y. Yeo and J. R. Friend, “Ultrafast microfluidics using surface acousticwaves,” Biomicrofluidics 3, 012002 �2009�.

2H. Li, J. R. Friend, and L. Y. Yeo, “Surface acoustic wave concentration ofparticle and bioparticle suspensions,” Biomed. Microdevices 9, 647�2007�.

3R. Shilton, M. K. Tan, L. Y. Yeo, and J. R. Friend, “Particle concentrationand mixing in microdrops driven by focused surface acoustic waves,” J.Appl. Phys. 104, 014910 �2008�.

4W. K. Tseng, J. L. Lin, W. C. Sung, S. H. Chen, and G. B. Lee, “Activemicro-mixers using surface acoustic waves on Y-cut 128° LiNbO3,” J.Micromech. Microeng. 16, 539 �2006�.

5M. Cecchini, S. Girardo, D. Pisignano, R. Cingolani, and F. Beltram,“Acoustic-counterflow microfluidics by surface acoustic waves,” Appl.Phys. Lett. 92, 104103 �2008�.

6M. K. Tan, L. Y. Yeo, and J. R. Friend, “Rapid microchannel fluid actua-tion and particle manipulation induced by surface acoustic waves,” Euro-phys. Lett. 87, 47003 �2009�.

7G. McHale, M. K. Banerjee, and M. I. Newton, “Surface acoustic waveresonances in the spreading of viscous fluids,” Phys. Rev. B 59, 8262�1999�.

8A. Wixforth, C. Strobl, C. Gauer, A. Toegl, J. Scriba, and Z. v. Guttenberg,“Acoustic manipulation of small droplets,” Anal. Bioanal. Chem. 379,982 �2004�.

9M. K. Tan, J. R. Friend, and L. Y. Yeo, “Microparticle collection andconcentration via a miniature surface acoustic wave device,” Lab Chip 7,618 �2007�.

10N. Shimizu, J. Kondoh, Y. Matsui, and S. Shiokawa, “Localized heatingeffects of liquid based on saw streaming,” Proceedings of the 2004 IEEEUltrasonics Symposium �IEEE, Los Alamitos, CA, 2004�, Vol. 3, pp.2235–2238.

11J. Kondoh, N. Shimizu, Y. Matsui, M. Sugimoto, and S. Shiokawa, “De-velopment of temperature-control system for liquid droplet using surfaceacoustic wave devices,” Sens. Actuators, A 149, 292 �2009�.



http://dx.doi.org/10.1063/1.3056040

http://dx.doi.org/10.1007/s10544-007-9058-2

http://dx.doi.org/10.1063/1.2951467

http://dx.doi.org/10.1063/1.2951467

http://dx.doi.org/10.1088/0960-1317/16/3/009

http://dx.doi.org/10.1088/0960-1317/16/3/009

http://dx.doi.org/10.1063/1.2889951

http://dx.doi.org/10.1063/1.2889951

http://dx.doi.org/10.1209/0295-5075/87/47003

http://dx.doi.org/10.1209/0295-5075/87/47003

http://dx.doi.org/10.1103/PhysRevB.59.8262

http://dx.doi.org/10.1007/s00216-004-2693-z

http://dx.doi.org/10.1039/b618044b

http://dx.doi.org/10.1016/j.sna.2008.11.007

12K. Kulkarni, J. Friend, L. Yeo, and P. Perlmutter, “Surface acoustic wavesas an energy source for drop scale synthetic chemistry,” Lab Chip 9, 754�2009�.

13M. K. Tan, J. R. Friend, and L. Y. Yeo, “Interfacial jetting phenomenainduced by focused surface vibrations,” Phys. Rev. Lett. 103, 024501�2009�.

14M. Kurosawa, T. Watanabe, A. Futami, and T. Higuchi, “Surface acousticwave atomizer,” Sens. Actuators, A 50, 69 �1995�.

15K. Chono, N. Shimizu, Y. Matsui, J. Kondoh, and S. Shiokawa, “Devel-opment of novel atomization system based on SAW streaming,” Jpn. J.Appl. Phys. 43, 2987 �2004�.

16A. Qi, L. Y. Yeo, and J. R. Friend, “Interfacial destabilization and atomi-zation driven by surface acoustic waves,” Phys. Fluids 20, 074103 �2008�.

17J. W. Kim, Y. Yamagata, M. Takasaki, B. H. Lee, H. Ohmori, and T.Higuchi, “A device for fabricating protein chips by using surface acousticwave atomizer and electrostatic deposition,” Sens. Actuators B 107, 535�2005�.

18M. Alvarez, J. Friend, and L. Y. Yeo, “Rapid generation of protein aero-sols and nanoparticles via surface acoustic wave atomization,” Nanotech-nology 19, 455103 �2008�.

19A. Qi, J. R. Friend, L. Y. Yeo, D. A. V. Morton, M. P. McIntosh, and L.Spiccia, “Miniature inhalation therapy platform using surface acousticwave microfluidic atomization,” Lab Chip 9, 2184 �2009�.

20M. Alvarez, J. Friend, and L. Y. Yeo, “Surface vibration induced spatialordering of periodic polymer patterns on a substrate,” Langmuir 24,10629 �2008�.

21A. Qi, L. Y. Yeo, J. R. Friend, and J. Ho, “The extraction of liquid, proteinmolecules and yeast cells from paper through surface acoustic wave at-omization,” Lab Chip 10, 470 �2010�.

22M. Faraday, “On a peculiar class of acoustical figures; and on certainforms assumed by groups of particles upon vibrating elastic surfaces,”Philos. Trans. R. Soc. London 121, 299 �1831�.

23T. B. Benjamin and F. Ursell, “The stability of the plane free surface of aliquid in vertical periodic motion,” Proc. R. Soc. London, Ser. A 225, 505�1954�.

24W. Eisenmenger, “Dynamic properties of the surface tension of water andaqueous solutions of surface active agents with standing capillary waves inthe frequency range from 10 Kc/S to 1.5 Mc/S,” Acustica 9, 327 �1959�.

25J. Miles and D. Henderson, “Parametrically forced surface waves,” Annu.Rev. Fluid Mech. 22, 143 �1990�.

26K. Kumar and L. S. Tuckerman, “Parametric instability of the interfacebetween two fluids,” J. Fluid Mech. 279, 49 �1994�.

27K. Kumar, “Linear theory of Faraday instability in viscous liquids,” Proc.R. Soc. London, Ser. A 452, 1113 �1996�.

28H. W. Müller, H. Wittmer, C. Wagner, J. Albers, and K. Knorr, “Analyticstability theory for Faraday waves and the observation of the harmonicsurface response,” Phys. Rev. Lett. 78, 2357 �1997�.

29R. Keolian, L. A. Turkevich, S. J. Putterman, and I. Rudnick, “Subhar-monic sequences in the Faraday experiment: departures from period dou-bling,” Phys. Rev. Lett. 47, 1133 �1981�.

30P. Chen and J. Viñals, “Amplitude equation and pattern selection in Fara-day waves,” Phys. Rev. E 60, 559 �1999�.

31A. M. Rucklidge, “Pattern formation in large domains,” Philos. Trans. R.Soc. London, Ser. A 361, 2649 �2003�.

32J. Lighthill, Waves in Fluids �Cambridge University, Cambridge, 1978�.33M. S. Longuet-Higgins, “The instabilities of gravity waves of finite am-

plitude in deep water: I. Superharmonics,” Proc. R. Soc. London, Ser. A360, 471 �1978�.

34M. S. Longuet-Higgins, “The instabilities of gravity waves of finite am-plitude in deep water: I. Subharmonics,” Proc. R. Soc. London, Ser. A360, 489 �1978�.

35J. Lucassen, “Longitudinal capillary waves. Part 1—Theory,” Trans. Far-aday Soc. 64, 2221 �1968�.

36J. Lucassen, “Longitudinal capillary waves. Part 2—Experiment,” Trans.Faraday Soc. 64, 2230 �1968�.

37J. A. De Feijter, “The propagation of surface shear waves: 1. Theory,” J.Colloid Interface Sci. 69, 375 �1979�.

38D. Sharp, “An overview of Rayleigh-Taylor instability,” Physica D 12, 3�1984�.

39O. Lioubashevski, J. Fineberg, and L. S. Tuckerman, “Scaling of the tran-sition to parametrically driven surface waves in highly dissipative sys-tems,” Phys. Rev. E 55, R3832 �1997�.

40E. Cerda and E. Tirapegui, “Faraday’s instability in viscous fluid,” J. FluidMech. 368, 195 �1998�.

41S. Kumar, “Mechanism for the Faraday instability in viscous liquids,”Phys. Rev. E 62, 1416 �2000�.

42A. J. James, B. Vukasinovic, M. K. Smith, and A. Glezer, “Vibration-induced drop atomization and bursting,” J. Fluid Mech. 476, 1 �2003�.

43M. Y. Brazhnikov, G. V. Kolmakov, A. A. Levchenko, and L. P. Mezhov-Deglin, “Observation of capillary turbulence on the water surface in awide range of frequencies,” Europhys. Lett. 58, 510 �2002�.

44K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics�Cambridge University, Cambridge, 1998�.

45M. Schindler, P. Talker, and P. Hänggi, “Computing stationary free-surfaceshapes in microfluidics,” Phys. Fluids 18, 103303 �2006�.

46D. Köster, “Numerical simulation of acoustic streaming on surface acous-tic wave-driven biochips,” SIAM J. Sci. Comput. �USA� 29, 2352 �2007�.

47W. L. Nyborg, in Acoustic Streaming, edited by W. P. Mason and R. N.Thurston �Academic, San Diego, CA, 1965�, Chap. 11, pp. 265–329.

48L. D. Rozenberg, High-Intensity Ultrasonic Fields �Plenum, New York,1971�.

49P. M. Morse and K. U. Ingard, Theoretical Acoustics �McGraw-Hill, NewYork, 1968�.

50C. E. Bradley, “Acoustic streaming field structure: The influence of theradiator,” J. Acoust. Soc. Am. 100, 1399 �1996�.

51L. Dong, A. Chaudhury, and M. K. Chaudhury, “Lateral vibration of awater drop and its motion on a vibrating surface,” Eur. Phys. J. E 21, 231�2006�.

52P. Brunet, M. Baudoin, O. B. Matar, and F. Zoueshtiagh, “Droplet dis-placements and oscillations induced by ultrasonic surface acoustic waves:A qualitative study,” Phys. Rev. E 81, 036315 �2010�.

53B. A. Auld, Acoustic Fields and Waves in Solids �Wiley, New York, 1973�,Vol. 1.

54H. F. Tiersten, “Wave propagation in an infinite piezoelectric plate,” J.Acoust. Soc. Am. 35, 234 �1963�.

55F. Chagla and P. M. Smith, “Finite difference time domain methods forpiezoelectric crystals,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control53, 1895 �2006�.

56R. T. Beyer, in Nonlinear Acoustics: The Parameter B/A, edited by M. F.Hamilton and D. T. Blackstock �Academic, New York, 1998�, Chap. 2, pp.25–40.

57F. V. Hunt, “Notes on the exact equations governing the propagation ofsound in fluids,” J. Acoust. Soc. Am. 27, 1019 �1955�.

58C. T. Schröder and W. R. Scott, “A finite-difference model to study theelastic wave interaction with buried land mines,” IEEE Trans. Geosci.Remote Sens. 38, 1505 �2000�.

59F. Collino and C. Tsogko, “Application of the perfectly matched absorbinglayer model to the linear elastodynamic problem in anisotropic heteroge-neous media,” Geophysics 66, 294 �2001�.

60D. Botteldooren, “Vorticity and entropy boundary conditions for acousticalfinite-difference time-domain simulations,” J. Acoust. Soc. Am. 102, 170�1997�.

61J. D. Anderson, Computational Fluid Dynamics: The Basics with Appli-cation �McGraw-Hill, New York, 1995�, p. 261.

62O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of NonlinearAcoustics �Consultants Bureau, New York, 1977�.

63H. Li, J. R. Friend, and L. Y. Yeo, “Microfluidic colloidal island formationand erasure induced by surface acoustic wave radiation,” Phys. Rev. Lett.101, 084502 �2008�.

64A. D. Heathershaw, “Seabed-wave resonance and sand bar growth,” Na-ture �London� 296, 343 �1982�.

65C. Sarasola, F. J. Torrealdea, A. d’Anjou, A. Moujahid, and M. Graña,“Energy balance in feedback synchronization of chaotic systems,” Phys.Rev. E 69, 011606 �2004�.

66D. Binks, M. T. Westra, and W. van de Water, “Effect of depth on thepattern formation of Faraday waves,” Phys. Rev. Lett. 79, 5010 �1997�.

67B. Christiansen and M. T. Alstrøm Levinsen, “Ordered capillary-wavestates: quasicrystals, hexagons, and radial waves,” Phys. Rev. Lett. 68,2157 �1992�.

68H. Arbell and J. Fineberg, “Pattern formation in two-frequency parametricwaves,” Phys. Rev. E 65, 036224 �2002�.

69C. K. Campbell, Surface Acoustic Wave Devices �Academic, London,1998�.

70K. Y. Hashimoto, Surface Acoustic Wave Devices in Telecommunications�Springer-Verlag, Berlin, 2000�.

71J. Friend and L. Yeo, “Using laser Doppler vibrometry to measure capil-lary surface waves on fluid-fluid interfaces,” Biomicrofluidics 4, 026501�2010�.



http://dx.doi.org/10.1039/b819217k

http://dx.doi.org/10.1103/PhysRevLett.103.024501

http://dx.doi.org/10.1016/0924-4247(96)80086-0

http://dx.doi.org/10.1143/JJAP.43.2987

http://dx.doi.org/10.1143/JJAP.43.2987

http://dx.doi.org/10.1063/1.2953537

http://dx.doi.org/10.1016/j.snb.2004.11.012

http://dx.doi.org/10.1088/0957-4484/19/45/455103

http://dx.doi.org/10.1088/0957-4484/19/45/455103

http://dx.doi.org/10.1039/b903575c

http://dx.doi.org/10.1021/la802255b

http://dx.doi.org/10.1039/b915833b

http://dx.doi.org/10.1098/rstl.1831.0018

http://dx.doi.org/10.1098/rspa.1954.0218

http://dx.doi.org/10.1146/annurev.fl.22.010190.001043

http://dx.doi.org/10.1146/annurev.fl.22.010190.001043

http://dx.doi.org/10.1017/S0022112094003812





http://dx.doi.org/10.1103/PhysRevE.60.559

http://dx.doi.org/10.1098/rsta.2003.1267

http://dx.doi.org/10.1098/rsta.2003.1267



http://dx.doi.org/10.1039/tf9686402221

http://dx.doi.org/10.1039/tf9686402221

http://dx.doi.org/10.1039/tf9686402230

http://dx.doi.org/10.1039/tf9686402230

http://dx.doi.org/10.1016/0021-9797(79)90127-9

http://dx.doi.org/10.1016/0021-9797(79)90127-9

http://dx.doi.org/10.1016/0167-2789(84)90510-4

http://dx.doi.org/10.1103/PhysRevE.55.R3832

http://dx.doi.org/10.1017/S0022112098001578

http://dx.doi.org/10.1017/S0022112098001578


http://dx.doi.org/10.1209/epl/i2002-00425-9

http://dx.doi.org/10.1063/1.2361291

http://dx.doi.org/10.1137/060676623

http://dx.doi.org/10.1121/1.415987

http://dx.doi.org/10.1140/epje/i2006-10063-7


http://dx.doi.org/10.1121/1.1918438

http://dx.doi.org/10.1121/1.1918438

http://dx.doi.org/10.1109/TUFFC.2006.122

http://dx.doi.org/10.1121/1.1908106

http://dx.doi.org/10.1109/36.851950

http://dx.doi.org/10.1109/36.851950

http://dx.doi.org/10.1190/1.1444908

http://dx.doi.org/10.1121/1.419780


http://dx.doi.org/10.1038/296343a0

http://dx.doi.org/10.1038/296343a0






http://dx.doi.org/10.1063/1.3353329

72O. Lioubashevski, H. Arbell, and J. Fineberg, “Dissipative solitary statesin driven surface waves,” Phys. Rev. Lett. 76, 3959 �1996�.

73L. Menguy and J. Gilbert, “Non-linear acoustic streaming accompanying aplane stationary wave in a guide,” Acustica 86, 249 �2000�.

74J. W. S. Rayleigh, Theory of Sound �Dover, Toronto, 1945�, Vol. 2.

75P. J. Westervelt, “The theory of steady rotational flow generated by asound field,” J. Acoust. Soc. Am. 25, 60 �1953�.

76S. J. Lighthill, “Acoustic streaming,” J. Sound Vib. 61, 391 �1978�.77G. Falkovich, A. Weinberg, P. Denissenko, and S. Lukaschuk, “Floater

clustering in a standing wave,” Nature �London� 435, 1045 �2005�.




http://dx.doi.org/10.1121/1.1907009

http://dx.doi.org/10.1016/0022-460X(78)90388-7

http://dx.doi.org/10.1038/4351045a

Documents

Capillary wave motion excited by high frequency surface ...users.monash.edu/~lyeo/Dr_Leslie_Yeo/SAW_Atomisation_files/Tan_… · Capillary wave motion excited by high frequency surface