Lab on a Chip PAPER. Numerical study of acoustophoretic motion of particles in a PDMS microchannel driven by surface acoustic waves. 1.

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1 PAPER View Article Online View Journal View Issue Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. Cite this: Lab Chi, 5,5, 7 Received 6th February 5, Acceted 5th May 5 DOI:.39/c5lc3a Introduction The emergence of lab-on-a-chi technologies has sarked a renewed interest in microfluidics. One of the requirements for the success of lab-on-a-chi systems is to recisely maniulate fluids and articles immersed in them at microscales., Here, surface acoustic wave (SAW) based systems, recently reviewed in ref. 3 5 have shown great otential in recent years. SAW-based systems rely on iezoelectric actuation of surface acoustic waves in a solid substrate. These waves roagate along the substrate surface and, as they encounter a fluid interface, they radiate acoustic energy into the fluid. This drives acoustic streaming in the fluid itself as well as the motion of the immersed articles. The articles exerience rimarily two a Deartment of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 68, USA. junhuang@su.edu b Institute of Fluid Mechanics and Aerodynamics, Bundeswehr University Munich, Neubiberg, Germany c Center for Neural Engineering, The Pennsylvania State University, University Park, PA 68, USA. costanzo@engr.su.edu d Deartment of Bioengineering, The Pennsylvania State University, University Park, PA 68, USA Electronic sulementary information (ESI) available. See DOI:.39/ c5lc3a Numerical study of acoustohoretic motion of articles in a PDMS microchannel driven by surface acoustic waves Nitesh Nama, a Rune Barnkob, b Zhangming Mao, a Christian J. Kähler, b Francesco Costanzo* ac and Tony Jun Huang* ad We resent a numerical study of the acoustohoretic motion of articles susended in a liquid-filled PDMS microchannel on a lithium niobate substrate acoustically driven by surface acoustic waves. We emloy a erturbation aroach where the flow variables are divided into first- and second-order fields. We use imedance boundary conditions to model the PDMS microchannel walls and we model the acoustic actuation by a dislacement function from the literature based on a numerical study of iezoelectric actuation. Consistent with the tye of actuation, the obtained first-order field is a horizontal standing wave that travels vertically from the actuated wall towards the uer PDMS wall. This is in contrast to what is observed in bulk acoustic wave devices. The first-order fields drive the acoustic streaming, as well as the time-averaged acoustic radiation force acting on susended articles. We analyze the motion of susended articles driven by the acoustic streaming drag and the radiation force. We examine a range of article diameters to demonstrate the transition from streaming-drag-dominated acoustohoresis to radiation-force-dominated acoustohoresis. Finally, as an alication of our numerical model, we demonstrate the caability to tune the osition of the vertical ressure node along the channel width by tuning the hase difference between two incoming surface acoustic waves. forces, the acoustic radiation force arising from the scattering of sound waves on the articles and the Stokes drag force from the induced acoustic streaming. However, while bulk acoustic wave (BAW) based systems have been heavily studied, 6 9 the theoretical and numerical work on SAW-driven systems is rather limited and so is the full understanding of the underlying hysics. For examle, little investigation has been made on the mechanisms underlying the vertical focusing of articles in oly-dimethylsiloxane (PDMS) channels, the mechanism of cell cell interactions in a SAW device, the effect of using PDMS channels as oosed to silicon walls, the recise bulk acoustic fields and associated acoustic streaming, and the critical article size for the transition between radiation-dominated and streaming-dominated acoustohoresis. The latter has been extensively studied within BAW-driven systems, 9,,3 but it is yet to be examined in SAW-driven systems. One of the rimary reasons for the lack of a detailed theoretical understanding of the hysical rocesses involved in SAW devices is the difficulty in the identification of recise boundary conditions. From a numerical viewoint, the difference between BAW systems and SAW systems is limited to the differences in actuation and wall conditions, while the governing equations remain the same. While SAW-based systems with free boundaries in form of drolets have been 7 Lab Chi, 5, 5, 7 79 This journal is The Royal Society of Chemistry 5

2 Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. heavily studied numerically, 4 6 SAW-driven systems with closed boundaries have received less attention. Using hardwall boundary conditions, few studies have been reorted for the acoustic streaming in a closed SAW-driven system. 7,8 However, while BAW systems utilize walls that are often made of hard material like glass or silicon making hard-wall boundary conditions aroiate, SAW-based systems often utilize soft materials such as PDMS leading to significant radiative energy losses. In this work, we emloy imedance boundary conditions to model the PDMS walls of a tyical SAW-based device to setu a numerical model for investigating the acoustohoretic motion in SAW devices. In line with the work by Muller et al. 9 we emloy erturbation theory and use the solution of the first-order equations to calculate the timeaveraged solutions, such as the acoustic streaming induced in the liquid and the acoustic radiation force acting on susended articles. These are then used to determine the article trajectories and to study the transition of dominance on articles' motion between the two forces. The numerical method and the results resented in this work will be helful in roviding a better understanding of the hysics in SAW-driven devices as well as to allow for future otimization and reliable control of SAW-based microfluidic devices, such as those emloyed in ref. 9.. Governing equations The mass and momentum balance laws governing the motion of a linear viscous comressible fluid are,3 and v t, () v v v v b v, () t 3 where ρ is the mass density, v is the fluid velocity, is the fluid ressure, and μ and μ b are the shear viscosity and bulk viscosity, resectively. Here, the fields ρ,, and v are understood to be in Eulerian form, 3 i.e., functions of time t and satial osition r within a fixed volume. Furthermore, in order to describe the fluid motion, we need a constitutive relation linking the ressure and density. We assume a linear relation between and ρ: = c ρ, (3) where c is the seed of sound in the fluid at rest. Combining eqn () (3) with aroriate boundary conditions, the system is fully determined. Nonetheless, the above-mentioned nonlinear system of equations is numerically challenging to solve via a direct numerical simulation due to the widely searated time scales (characteristic oscillation eriods vs. characteristic times dictated by the streaming seed). 4 For examle, a tyical SAW device is oerated at frequencies in the range of MHz, while the streaming fields are characterized by time scales of the order of tenth of seconds to several minutes. Therefore we neglect the transient build-u of the acoustic fields and in this work we only consider time-harmonic forcing. However, because of viscous dissiation, the resonse of the fluid to a harmonic forcing is, in general, not harmonic. The fluid resonse can be understood to be comrised of two comonents: (i) a eriodic comonent with eriod equal to the forcing eriod, and (ii) a remainder that can be viewed as being steady. It is this second comonent which is generally referred to as the streaming motion. 4 Following our recent model, 5 we emloy Nyborg's erturbation technique 6 in which fluid velocity, density, and ressure are assumed to have the following form v = v + ε + ε +O(ε 3 )+, (4a) 3 O, (4b) 3 O, (4c) where ε is a non-dimensional small arameter. Following Köster, 7,7 we define ε as the ratio between the amlitude of the dislacement of the boundary in contact with the iezoelectrically driven substrate (i.e., the amlitude of the boundary excitation) and a characteristic length. We take the zeroth order velocity field v to be equal to zero thus assuming the absence of an underlying net flow along the microchannel. Letting v v,,, (5) v v, substituting eqn (4a), (4b) and (4c) into eqn () (3), and setting the sum of all the terms of order one in ε to zero, the following roblem, referred to as the first-order roblem, is obtained v (6) t v v b v. (7) t 3 Reeating the above rocedure for the terms of order two in ε, and averaging the resulting equations over a eriod of oscillation, the following set of equations, referred to as the second-order roblem, is obtained t v v, (8) This journal is The Royal Society of Chemistry 5 Lab Chi, 5,5,

3 Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. v v v v t t v b v, 3 where A denotes the time average of the quantity A over a full oscillation time eriod. As ointed out by Stuart, 8 inertial terms in eqn (9) can be significant and must be retained in the formulation. Also, to fully account for viscous attenuation of the acoustic wave, both within and without the boundary layer, the last term in eqn (9) associated with the bulk viscosity must also be retained. 3. Numerical model 3. Model system and comutational domain A tyical standing SAW (SSAW) device for article maniulation consists of a PDMS channel bonded on a iezoelectric substrate (Fig. (a)). The device emloys a air of metallic interdigitated transducers (IDTs) sitting on the surface of the iezoelectric substrate. A SSAW is roduced via the suerosition of two counter-roagating traveling SAWs generated on the surface of the iezoelectric substrate by alying a harmonic electric signal to the IDTs. The full hysical system is governed by couling of elastic, electromagnetic, and hydrodynamic effects, which makes numerical modeling challenging. 7,7 Thus, we simlify the system by modeling the PDMS walls of the channel using imedance boundary conditions limiting this study to cases with PDMS walls of thickness mm or greater, while the effect of iezoelectric substrate is modeled using a dislacement function at the substrate boundary. As a result our comutational domain Ω shown in Fig. (b) consists of a rectangular microchannel of (9) width w = 6 μm and height h = 5 μm, where the boundaries subject to the imedance condition are denoted by Γ i, while the actuated boundary is denoted by Γ d. The boundary conditions are discussed in detail in section 3.. In this work, we analyse the case where the iezoelectric substrate is made of lithium niobate actuated with a surface wave of wavelength λ = 6 μm and frequency f = 6.65 MHz, and where the channel is filled with water. The values for all the relevant roerties of the iezoelectric substrate and water, as well as the tyical oerational arameters used in our numerical model, are listed in Table. 3. Boundary conditions As the objective of this work is to study the fluid and article motion inside the microfluidic channel shown in Fig., we simlify the system considerably by modeling the effect of iezoelectric substrate via a dislacement boundary condition while the PDMS walls are modeled using imedance boundary conditions. The tye of waves usually considered in SAW devices are the so-called Rayleigh waves. The amlitude of these waves decay exonentially with the deth into the substrate, thereby confining most of the energy to the surface. 37 The two wave motions in the y and z direction are known to be 9 out of hase in time, thereby resulting in ellitical dislacements. Based on these considerations, it is ossible to find dislacement functions for waves which roagate along the y direction and decay exonentially in both y and z direction. Taking these considerations into account, Gantner et al. 38,39 analyzed numerically the Rayleigh waves in iezoelectric substrates in great detail. We use the dislacement results from their analysis, also used by Köster, 7,7 to describe the dislacement rofile due to a traveling SAW which takes the form C y yw d uy t, y 6. u e sin t, C y d uz t, yue cos y w t, () Fig. (a) Cross-sectional sketch of the SAW-driven device consisting of a lithium niobate substrate and liquid-filled PDMS channel (width w = 6 μm and height h = 5 μm). The substrate is acoustically actuated via two sets of interdigitated transducers (IDTs). Note, the figure is not drawn to scale and that the PDMS channel walls are considered to be of thickness mm or greater. (b) Sketch of the comutational domain is Ω with imedance boundaries Γ i and Dirichlet actuation boundary Γ d. where u y and u z are the dislacements along the y and z direction, resectively, C d is the decay coefficient, and ω = πf is the angular frequency. The value of C d emloyed by Köster 7,7 (86 m ) is aroriate for a SAW device loaded with an infinite layer of water at frequencies in the range of MHz. Recently, Vanneste and Bühler 4 investigated streaming atterns using an attenuation coefficient of 8 m for a frequency of aroximately 5 MHz. Using the leaky SAW disersion relation emloyed by Vanneste and Bühler 4 for a frequency of 6.65 MHz, we get an attenuation coefficient of 6 m. However, we note that the disersion relation emloyed by Vanneste and Bühler 4 is valid for a SAW roagating under an infinitely thick layer of water. A finite thickness might further reduce the 7 Lab Chi, 5,5, 7 79 This journal is The Royal Society of Chemistry 5

4 Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. Table Material arameters at T = 5 C Water Density 9 ρ 997 kg m 3 Seed of sound 9 c 497 m s Shear viscosity 9 μ.89 mpa s Bulk viscosity 3 μ b.47 mpa s Comressibility a κ 448 TPa Lithium niobate (LiNbO 3 ) Seed of sound 3 c sub 3994 m s Poly-dimethylsiloxane (PDMS, : ) Density 3 ρ wall 9 kg m 3 Seed of sound 33 c wall 76.5 m s Attenuation coeff. (6.65 MHz) b33 3 db cm Polystyrene Density 9 ρ 5 kg m 3 Seed of sound 34 (at C) c 35 m s Poisson's ratio 35 σ.35 Comressibility c κ 49 TPa Acoustic actuation arameters Wavelength (set by IDTs) λ 6 μm Forcing frequency f 6.65 MHz Dislacement amlitude μ. nm Dislacement decay coefficient C d 6 m a Calculated as κ =/IJρ c ). b Calculated via ower law fit to data by 3 Tsou et al. 33 c Calculated as c from Landau and Lifshitz. 36 attenuation coefficient. Noting this, for comarison uroses we considered a case with C d = (see ESI Fig. ). We observed that decreasing the value of C d from 6 m to zero does not change the solution significantly. With this in mind, we have used an attenuation coefficient of 6 m for all the results resented in this article. Using eqn (), we construct the standing SAW dislacement rofile over Γ d by suerimosing the dislacement rofile of two SAWs traveling in oosite directions with a hase difference of Δϕ: The dislacement function is lotted in Fig.. This dislacement function is then differentiated with resect to time to obtain the first-order velocity that we imose over Γ d (Fig. (b)): u v ty ty,,, on d. () t For the boundary condition on the channel walls, marked as Γ i in Fig. (b), we use the so-called imedance or lossywall boundary condition given as 4 n c i, on i, (3) m m where i is the imaginary unit, and ρ m and c m are the mass density and the seed of sound of the wall material, resectively. Note that this boundary condition is very different from the hard-wall condition, n v =, used to model silicon or glass walls in BAW systems as the imedance boundary condition allows a non-zero first-order wall velocity. Futhermore, with this boundary condition, the model assumes all transmitted wave energy to be absorbed in the PDMS, i.e. no reflected waves, from a otential PDMS/air interface, are allowed to re-enter the water channel. The model therefore only alies to cases, where the PDMS walls are thick enough to attenuate waves transmitted from the channel. In commonly-used PDMS ( : ), the attenuation coefficient for frequencies of 5 MHz and 7 MHz are.3 db cm to db cm, resectively, which translate to decay coefficients of 49 m and 773 m. 33 For the secific frequency of 6.65 MHz used in this work, the attenuation coefficient is close to 3 db cm corresonding to a decay coefficient of 74 m. Therefore, if the channel walls are thicker than mm, only a ex( 74 m. m) =.58 fraction of the transmitted waves at the water/pdms interface will reach the PDMS/air interface and come back again. This corresonds to an absortion of more than 94% and thus the assumtion of total absortion of acoustic waves in the PDMS walls is reasonable for channel walls thicker than mm. For higher actuation frequencies commonly used in SAW devices (tens of MHz) the attenuation coefficient C y y w uy t y u 6. t e d sin w y sin t, C y yw uz t yue d cos w y cos t). t () Fig. Plot of standing SAW dislacement vectors along the interface of the channel and the iezoelectric substrate at z = at (a) t =, (b) t = π/6ω, (c) t = π/3ω, (d) t = π/ω, (e) t =π/3ω, and (f) t =5π/6ω. The dislacement function is obtained by suerimosing two incoming traveling SAWs from the left and the right direction, see eqn (). This journal is The Royal Society of Chemistry 5 Lab Chi, 5,5,

5 Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. increases further (more than double at MHz) thus making the assumtion of total absortion even more reasonable. For the second-order roblem, Bradley 4 offered a careful analysis of the boundary conditions to be satisfied on the moving surfaces. Secifically, the no-sli boundary condition needs to be satisfied on the deformed ositions of the moving surfaces, and not on the initial rest ositions. However, the dislacement amlitude in SAW devices is usually in subnanometer range, thus it is ossible to neglect the minute difference between the initial and the deformed ositions. Noting this, we emloy the zero-velocity boundary condition on all the boundaries, similar to those used by Muller et al.: 9 v =, onγ i Γ d. (4) 3.3 Single-article acoustohoretic trajectories In order to be able to redict acoustohoretic article trajectories of a tyical microfluidic exeriment using olystyrene tracer articles, we imlement a article tracking strategy. Such trajectories will also establish the foundation for exerimental validation of our numerical model by using threedimensional article tracking velocimetry. 9,43 We consider articles susended in a susension dilute enough to neglect article article interactions, hydrodynamic as well as acoustic. The tracking strategy is redicated on the determination of the acoustic radiation force due to the scattering of waves on the article as described by Settnes and Bruus. 44 Considering a article of radius a much smaller than the wavelength λ, mass density ρ, and comressibility κ, the radiation force takes the form: 44 rad F a 3 * * * * Re f f v v Re, 3 (5) where κ =/IJρ c ) is the comressibility of the fluid, ReijA] denotes the real art of quantity A, the asterisk denotes the comlex conjugate of the quantity, and the coefficients f and f are given by with f and f, (6) 3 3 i,,. a (7) Note that we use the general exression for the radiation force without a riori assumtion of whether we deal with traveling or standing waves. In addition to the radiation force, a bead is assumed to be subject to a drag force roortional to v v, which is the velocity of the bead relative to the streaming velocity. When wall effects are negligible, the drag force is estimated via the simle formula F drag = 6πμa( v v ). The motion of the bead is then redicted via the alication of Newton's second law m dv d t F F (8) rad drag where m is the mass of the bead. In many acoustofluidics roblems the inertia of the bead can be neglected since the characterstic time of acceleration is small in comarison to the time scale of the motion of the articles. 45 Doing so, eqn (8) can be solved for v rad v F v 6a (9) For steady flows, we can identify the bead trajectories with the streamlines of the velocity field v in eqn (9). 3.4 Numerical scheme For the first-order roblem we seek solutions of the following form v(r, t) =v(r)ex( iωt), (r, t) =(r)ex( iωt), (a) (b) where vijr) is a vector-valued function of sace while IJr) is a scalar function of sace. For the second-order roblem, we seek steady solutions. We also note that the second-order roblem has ure Dirichlet boundary conditions on all sides and hence does not admit a unique solution unless we assign an additional ressure constraint. However, since the radiation force used here is comletely deendent only on the first-order fields, we do not use the second-order ressure in any of our calculations. Combining information from the first- and the second-order solutions, it is then ossible to estimate the mean trajectory of articles in the flow. All the solutions discussed later are for two-dimensional roblems. The numerical solution was obtained via the finite element software COMSOL Multihysics For both the first- and second-order roblems we used Q Q elements for velocity and ressure, resectively, where Q and Q denote triangular elements suorting Lagrange olynomials of order one and two, resectively. 4. Results and discussions 4.. Mesh convergence analysis To cature the hysics inside the boundary layers near the walls, we use a comutational mesh with a maximum element size near the boundary, d b while the maximum element size in the bulk of the domain was set to μm. Fig. 3(a) shows an illustrative mesh with d b =3δ, where δ is the viscous boundary layer thickness given by eqn (7). To check for mesh convergence, we investigate the behavior of the variables solved for on a series of meshes generated by rogressively decreasing the mesh element size, d b. We 74 Lab Chi, 5, 5, 7 79 This journal is The Royal Society of Chemistry 5

6 Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. define a relative convergence function CIJg) for a solution g with resect to a reference solution g ref obtained on the finest mesh as g gref dy dz Cg ( ), g dy dz ref () where we use a reference solution g ref obtained for d b =.δ with aroximately elements. The results of the mesh convergence analysis are shown in Fig. 3(b) where the convergence function C is lotted as a function of δ/d b.as the value of d b reaches.3δ all the variables have reached sufficient convergence and throughout the rest of the work we use a mesh with d b =.3δ. 4. Wall imedance swee To study the effect of the wall material of the microfluidic channel, we erform a series of simulations with increasing value of wall imedance while the fluid imedance is ket constant. For each value of increasing imedance we comare the solution g to the solution g ref obtained using hardwall boundaries (i.e. n v =atγ i ) by calculating the convergence function CIJg) in eqn (). The first-order ressure field from the hard-wall solution is shown in Fig. 4(a) and features a resonance with no traveling waves as tyically observed in BAW systems. The convergence function CIJg) for the firstorder ressure and velocity fields is lotted in Fig. 4(b) and it is seen that as the wall imedance increases, the solutions converge to the hard-wall solution. The values of the convergence function C for imedance values for those of glass (z gl =.3 7 kg m s ) and silicon (z si =. 7 kg m s ) were around.45 and.3, resectively, while C for PDMS is around. Thus, to a reasonable aroximation, hard-wall boundary conditions can be used for BAW systems using tyically silicon or glass walls, while it is an inaccurate condition for SAW systems using PDMS walls. This is in good agreement with the fact that PDMS, having an acoustic imedance similar to water, absorbs most of the incident waves with little reflections while silicon and glass, having very different acoustic imedances from water, reflect most of the incident waves leading to the building u of resonances inside the microchannel Acoustic fields Having identified the roer mesh refinement level in section 4. and that imedance boundary conditions are aroriate for modeling PDMS walls in section 4., we investigate the acoustic fields that are set u inside the channel. In Fig. 5 we show the first-order ressure field, first-order velocity field v, and the second-order velocity field v, where the lotted colors indicate the field magnitude (from blue minimum to red maximum) and the black arrows indicate the field vectors. For the first-order ressure field and first-order velocity field v in Fig. 5(a) and (b), resectively, we observe a clear horizontal standing wave along y, but as indicated by the uwards-ointing magenta arrows, we observe that the first-order fields are traveling waves moving from the bottom wall towards the to wall along the z direction. The first-order ressure amlitude a is observed to be.9 kpa as oosed to 7.5 kpa observed when using the hard-wall boundary conditions as shown in Fig. 4(a). This difference in ressure amlitude can be attributed to the fact that a resonance is set u in the channel when using the hard-wall boundary condition, leading to an increased ressure amlitude. Furthermore, we notice that the first-order velocity amlitude v a is observed to be 5.3 mm s. This amlitude is greater than the actual velocity amlitude v a = ωu = 4.7 mm s imosed via the actuation function described in eqn (), which indicates that the traveling wave is not comletely transmitted through the PDMS walls and thus reflections occur from the channel walls due to the small but non-zero imedance mismatch between the PDMS and water. Fig. 5(c) shows the second-order velocity field v, in which four streaming vortices are observed along the y direction with a maximum velocity of.47 μm s close to the bottom wall. Fig. 5(d) shows a zoomed version of the secondorder velocity near the bottom boundary and we note that no streaming rolls are observed within the viscous boundary layer of width /. m, which were observed numerically by Muller et al. 9 for a BAW system. The difference between the model by Muller et al. and our model, is that we use imedance boundary conditions instead of hardwall conditions, which allow the first-order velocity to have a sli-velocity thus minimizing the velocity gradients near the walls. Furthermore, in contrast to the work by Muller and co-workers, we actuate the bottom wall from where the streaming is driven. 4.4 Particle trajectories Based on the acoustic fields described in the former section and the theory described in section 3.3 (see the radiation force field in ESI Fig. ), we calculate the velocities and trajectories of olystyrene articles of diameters ranging from μm toμm. The trajectories are lotted in Fig. 6 for 43 articles with uniformly-distributed initial ositions as shown in Fig. 6(a). The anels IJb) IJf) show the trajectories of (b) μm articles during s, (c)5 μm articles during s, (d) μm articles during 6 s, (e) 5 μm articles during 6 s, and (f) μm articles during 4 s. For each article trajectory the colors denote the article velocity ranging from zero (blue) to its maximum (red), while the colored disks show the articles' final ositions and velocities. For the μm and 5 μm articles in anel (b) and (c), resectively, we clearly see that their motion is governed by the viscous drag from the acoustic streaming as lotted in Fig. 5(c). The articles are carried around in the four horizontal streaming rolls and the maximum velocities of around.5 μm s are very close to the maximum streaming velocity of.47 μm s.as This journal is The Royal Society of Chemistry 5 Lab Chi, 5,5,

7 View Article Online Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. the article size increases, the acoustic radiation force becomes influential and for the motion of the μm articles in anel (d) we observe that far from the strong streaming at the bottom wall, the acoustic radiation force ushes the articles out of the streaming vortices towards the to wall. This is even more evident for the 5 μm articles in anel (e), where the radiation force contribution is IJ5/).5 times larger, which is seen by an almost comlete vanishing of the vortex motion as well as a maximum article velocity of.43 μm s. In anel (f) for the μm articles, the acoustic streaming attern has comletely vanished and the acoustic radiation force is fully dominating the motion. We note that the radiation force carries the articles outwards from (±w/4, h/) towards the channel walls consequently bringing the articles to the standing ressure nodes at y equal to w/,, and w/. Furthermore, we notice that the μm articles obtain velocities 4.5/.43.7 times larger than those of the 5 μm articles, which is close to the exected ratio of IJ/5) =.8 if both the articles were fully dominated by the radiation force, see eqn (5) and (9). For the investigated system, the critical article size for the transition between streaming-dominated and radiation-dominated motion is around μm deending on the article z-osition as the acoustic streaming is strongest at the bottom wall. If we comare this to the BAW system studied by Muller et al.9,47 and Barnkob et al. where they found a transition diameter around μm, it is clear that the acoustic streaming has a larger influence in the system studied in this work. However, tyically SAW systems are driven at higher frequencies, which will increase the effect from the radiation force Fig. 3 Mesh convergence analysis. (a) An illustrative comutational mesh with 478 triangular elements obtained with maximum element size near the boundary, db = 3δ, while the maximum element size in the bulk of the domain was set to μm. (b) Semi-logarithmic lot of the relative convergence arameter C, as given in eqn (), for decreasing mesh element size near the boundaries, db. 76 Lab Chi, 5, 5, 7 79 due to its linear deendence on the actuation frequency, eqn (5). Finally, note that we have not taken into account the enhanced viscous drag force due to the resence of the channel walls, which would decrease the radiation force contribution for the large articles and for articles close to the channel walls Phase sweeing As an alication of our numerical model, we investigate the effect of the hase difference between the two incoming SAWs by changing their relative hase Δϕ in eqn (). Fig. 7 shows the lots of the first-order ressure fields for varying hase difference Δϕ. We see that the osition of the ressure node along the y direction can be tuned by changing the hase difference between the two incoming SAWs. For a hase difference of π/, the shift in the hase difference is λ/8. In other words, a hase difference of π results in an interchange in the osition of the nodes and the antinodes. This is in agreement with the results obtained by Meng et al.,49 where they used the tuning of the ressure node to transort single cells or multile bubbles. This rincile has recently been utilized by Li et al.5 to study heterotyic cell cell interaction by sequentially atterning different tyes of cells at different ositions inside the microfluidic channel. Fig. 7 also oints to the fact that a minor shift in actuation does not affect the solution in a drastic manner, indicating the robustness of the solution with resect to minor erturbations in the alied actuation. Fig. 4 Imedance convergence analysis. (a) First-order ressure field when using hard-wall conditions n v = at Γi boundaries [color lot ranging from 7.5 kpa (blue) to 7.5 kpa (red)]. (b) Semilogarithmic lot of the relative convergence arameter C, as given in eqn (), as a function of the wall imedance zwall. The solution with hard-wall boundary conditions in anel (a) was chosen as the reference solution. As the imedance of the walls increases, the solution with imedance boundary conditions converges to the solution with the hard-wall boundary conditions. This journal is The Royal Society of Chemistry 5

8 View Article Online Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. Fig. 5 Color lots of the first-order ressure and velocity v fields as well as the time-averaged second-order velocity v. The first-order fields oscillate in time with a standing wave along y and a travelling wave along z indicated by the uwards-ointing magenta arrows. (a) Oscillating first-order ressure field [colors ranging from.9 kpa (blue) to.9 kpa (red)]. (b) Oscillating first-order velocity field v [magnitude shown as colors ranging from zero (blue) to 5.3 mms (red), vectors shown as black arrows]. (c) Time-averaged second-order velocity field v [magnitude shown as colors ranging from zero (blue) to.47 μm s (red), vectors shown as black arrows]. (d) Zoom of the time-averaged second-order velocity field v in (c) in a slab of.3 μm height from the bottom wall [magnitude shown as colors ranging from zero (blue) to.7 μm s (red), vectors shown as black arrows]. 5. Conclusions We have successfully used a finite element scheme to model the acoustohoretic motion of articles inside an isentroic comressible liquid surrounded by PDMS walls. The system is acoustically actuated via two counter-roagating surface acoustic waves that form a standing wave in a iezoelectric material interfacing the liquid channel. Our model emloys an actuation condition from the literature based on iezoelectric simulations as well as imedance boundary conditions to model the PDMS channel walls. Our model results in significantly different acoustic fields from those observed in bulk acoustic wave devices. Firstly, the first-order acoustic fields are travelling in the vertical direction away from the actuated boundary, while the horizontal standing wave feature remains. This results in a time-averaged second-order velocity field (the so-called acoustic streaming) driven by roducts of first-order fields with the characteristics of four horizontal streaming rolls er wavelength, each of which decay vertically from the actuated boundary. In contrast to reorted bulk acoustic wave cases, we do not observe any acoustic streaming rolls inside the viscous boundary, which we attribute to the differences in actuation condition as well as differences in the established first-order fields. This journal is The Royal Society of Chemistry 5 Fig. 6 Particle trajectories with article velocities as colors from blue minimum to red maximum and colored disks denoting the final ositions within the observation time. (a) Starting osition of 43 articles distributed uniformly within the microchannel. The anels IJb) IJf) show the trajectories of (b) μm articles during s, (c) 5 μm articles during s, (d) μm articles during 6 s, (e) 5 μm articles during 6 s, and (f) μm articles during 4 s. The motion of the smaller articles is dominated by the viscous drag force from the acoustic streaming, while the larger articles are ushed to the ressure nodes by the acoustic radiation force. The motion of the articles is governed by the viscous drag force from the acoustic streaming as well as the direct acoustic radiation force due to scattering of sound waves on the articles. For our secific model arameters (6 μm wavelength, 6.65 MHz actuation frequency, and olystyrene articles susended in water), we obtain an aroximate critical article size of μm for which the article motion goes from being streaming-drag dominated to being radiation dominated. The critical article size is only aroximate due to the acoustic streaming decaying strongly along the height of the channel. The next imortant ste is to obtain direct exerimental validation of our numerical results by use of 3D astigmatism article tracking velocimetry caable of determining the three-dimensional three-comonent article trajectories.43 Lab Chi, 5, 5,

9 View Article Online Published on May 5. Downloaded by Pennsylvania State University on /6/5 5:36:4. Fig. 7 Color lots of the first-order ressure field for different values of hase difference ϕ, (a) ϕ = π/, (b) ϕ =, (c) ϕ = π/, and (d) ϕ = π, as in eqn (3), between the two incoming traveling waves. The osition of the ressure node along the y direction can be tuned by changing the value of ϕ. The ressure node moves by a distance of λ/8 for each hase difference of π/. Such exerimental validation would ave the road for further enhancements of our numerical model to include wallenhancement effects of the viscous drag force as well as the inclusion of the heat-transfer equation in the governing equations in order to account for temerature effects as recently studied by Muller and Bruus.5 Acknowledgements This work was suorted by National Institutes of Health (R GM48-A, R33EB9785-), National Science Foundation (CBET-4386 and IIP-34644), the Penn State Center for Nanoscale Science (MRSEC) under grant DMR844, and the German Research Foundation (DFG), under the individual grants rogram KA 88/6-. References S.-C. S. Lin, X. Mao and T. J. Huang, Lab Chi,,, 766. J. Shi, H. Huang, Z. Stratton, Y. Huang and T. J. Huang, Lab Chi, 9, 9, M. Gedge and M. Hill, Lab Chi,,, X. Ding, P. Li, S.-C. S. Lin, Z. S. Stratton, N. Nama, F. Guo, D. Slotcavage, X. Mao, J. Shi, F. Costanzo and T. J. Huang, Lab Chi, 3, 3, L. Y. Yeo and J. R. Friend, Annu. Rev. Fluid Mech., 4, 46, A. Neild, S. Oberti, A. Haake and J. Dual, Ultrasonics, 6, 44, Lab Chi, 5, 5, J. Dual, P. Hahn, I. Leibacher, D. Möller, T. Schwarz and J. Wang, Lab Chi,,, J. Lei, P. Glynne-Jones and M. Hill, Lab Chi, 3, 3, P. B. Muller, R. Barnkob, M. J. H. Jensen and H. Bruus, Lab Chi,,, J. Shi, S. Yazdi, S.-C. S. Lin, X. Ding, I.-K. Chiang, K. Shar and T. J. Huang, Lab Chi,,, F. Guo, P. Li, J. B. French, Z. Mao, H. Zhao, S. Li, N. Nama, J. R. Fick, S. J. Benkovic and T. J. Huang, Proc. Natl. Acad. Sci. U. S. A., 5,, R. Barnkob, P. Augustsson, T. Laurell and H. Bruus, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.,, 86, J. Lei, M. Hill and P. Glynne-Jones, Lab Chi, 4, 4, L. Y. Yeo and J. R. Friend, Biomicrofluidics, 9, 3,. 5 R. Shilton, M. K. Tan, L. Y. Yeo and J. R. Friend, J. Al. Phys., 8, 4, M. Alghane, B. Chen, Y. Fu, Y. Li, M. Desmulliez, M. Mohammed and A. Walton, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.,, 86, D. Köster, SIAM J. Sci. Comut., 7, 9, H. Antil, R. Glowinski, R. H. Hoe, C. Linsenmann, T.-W. Pan and A. Wixforth, J. Comut. Math,, 8, X. Ding, Z. Peng, S.-C. S. Lin, M. Geri, S. Li, P. Li, Y. Chen, M. Dao, S. Suresh and T. J. Huang, Proc. Natl. Acad. Sci. U. S. A., 4,, X. Ding, S.-C. S. Lin, B. Kiraly, H. Yue, S. Li, I.-K. Chiang, J. Shi, S. J. Benkovic and T. J. Huang, Proc. Natl. Acad. Sci. U. S. A.,, 9, 5 9. P. Li, Z. Mao, Z. Peng, L. Zhou, Y. Chen, P.-H. Huang, C. I. Truica, J. J. Drabick, W. S. El-Deiry, M. Dao, S. Suresh and T. J. Huang, Proc. Natl. Acad. Sci. U. S. A., 5,, L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, nd edn, 993, vol. 6, Course of Theoretical Physics. 3 M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, New York,. 4 T. Frommelt, D. Gogel, M. Kostur, P. Talkner, P. Hanggi and A. Wixforth, IEEE Trans. Sonics Ultrason., 8, 55, N. Nama, P.-H. Huang, T. J. Huang and F. Costanzo, Lab Chi, 4, 4, W. L. Nyborg, in Nonlinear Acoustics, ed. M. F. Hamilton and D. T. Blackstock, Academic Press, San Diego, CA, 998, D. Köster, Ph.D. thesis, Universität Augsburg, Germany, 6. 8 J. Stuart, J. Fluid Mech., 966, 4, CRCnetBASE Product, CRC Handbook of Chemistry and Physics, Taylor and Francis Grou, 9th edn,. 3 M. J. Holmes, N. G. Parker and M. J. W. Povey, J. Phys.: Conf. Ser.,, 69,. 3 S. Sankaranarayanan and V. R. Bhethanabotla, IEEE Trans. Sonics Ultrason., 9, 56, This journal is The Royal Society of Chemistry 5

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