Arising from an interplay between capillary, acoustic and intermolecular forces, surface acoustic waves (SAWs) are observed to drive a unique and curious double flow reversal in the spreading of thin films. With a thickness at or less than the submicrometre viscous penetration depth, the film is seen to advance along the SAW propagation direction, and self-similarly over time t1/4 in the inertial limit. At intermediate film thicknesses, beyond one-fourth the sound wavelength λℓ in the liquid, the spreading direction reverses, and the film propagates against the direction of the SAW propagation. The film reverses yet again, once its depth is further increased beyond one SAW wavelength. An unstable thickness region, between λℓ/8 and λℓ/4, exists from which regions of the film either rapidly grow in thickness to exceed λℓ/4 and move against the SAW propagation, consistent with the intermediate thickness films, whereas other regions decrease in thickness below λℓ/8 to conserve mass and move along the SAW propagation direction, consistent with the thin submicrometre films.
Useful for many microfluidic applications [1,2], high-frequency sound waves ((10–103 MHz)) strongly interact with fluids at the micro- to nanoscale and underpin a diverse range of complex and nonlinear phenomena, from colloidal patterning , chaotic advection  and interfacial jetting , to self-similar droplet generation  and fingering instabilities . Rayleigh surface acoustic waves (SAWs) are used in a majority of these studies for their microscale wavelengths, monolithic device construction, efficient acoustic energy carriage and effective coupling of the energy into a fluid placed upon the device. Here, SAWs were found to induce rapid spreading of distinct, submicrometre to millimetre-thick oil films in an arrangement as shown in figure 1, with the film exhibiting three thickness ranges and associated reversal of flow as illustrated in figure 2. The flow was found to reverse between each of these film thickness regimes, resulting in a complex, thickness-dependent flow in which the film is observed to spread along, opposite and again along the SAW propagation direction depending upon its thickness.
While the thicker pair of films and a single associated flow reversal was observed in an earlier study , none of the remaining phenomena has been reported. The ability to form a submicrometre film, and—between the thinner pair of these three film thickness ranges—a second flow reversal and an associated unstable thickness region have now been discovered and found to lead to the formation of sharply defined film protrusions and depressions that bizarrely move in different directions depending on their particular thicknesses.
The behaviour is shown through analysis to be governed by a scale-dependent competition between capillary, acoustic attenuation-driven, acoustic radiation-driven and intermolecular forces. In past work, no theoretical prediction was made for the critical thicknesses at which the flow reversals occur, hc1 and hc2, but predictions for these values and an explanation of the mechanisms responsible for both flow reversals, the instabilities and the thin, intermediate and thicker film behaviours are all provided here for the first time.
The paper is organized as follows. In §2, the procedure used to form and spread the thin oil films using the SAW is explained. Then, the behaviour of the film over the three thickness regimes is briefly explored in §3, from its thinnest condition, through its first critical transition to the intermediate thickness, and through the second transition to the thickest regime. This is followed by an analysis of the film as it transitions between the thickest and intermediate regimes in §4, an analysis of the unique behaviour of the thinnest film in §5 and the peculiar instability associated with the transition from this thinnest film to the intermediate regime in §6.
2. Experimental procedure
The SAW was generated using a continuous sinusoidal electrical input using a signal generator (SML01, Rhode & Schwarz, North Ryde, NSW, Australia) and amplifier (10W1000C, Amplifier Research, Souderton, PA) into a simple interdigital electrode composed of 42 straight electrode finger pairs formed from 175 nm Au atop 5 nm Cr. This was patterned via standard liftoff ultraviolet lithography with 50, 30 and 20 μm wide fingers and separated by gaps of similar width, leading to resonance frequencies of 19.5, 32.7 and 48.5 MHz, respectively, for the formation of Rayleigh SAW on 127.68° Y -rotated cut, X-propagating single crystal lithium niobate (LN, Roditi Ltd., London, UK). For these frequencies, the SAW wavelength λSAW is 204, 122 and 82 μm, respectively, and the sound wavelength in the fluid λSAW is 73, 44 and 29 μm, respectively. To prevent stray SAW radiation and edge reflections, we used silicone gel (μ-gel, Geltec Ltd., Tokyo, Japan) along the device's edges. The input electric power was measured using a digital oscilloscope (TDS4012B, Tektronix, Beaverton, OR) with appropriate current and voltage probes. A laser Doppler vibrometer (LDV, UHF 120, Polytec GmBH, Waldbronn, Germany) was used to measure the vibration velocity U perpendicular to the LN surface to confirm the existence of the Rayleigh SAW at the appropriate resonance frequencies during the experiments.
As shown in figure 1a an oil strip was deposited on the LN substrate by carefully dipping a 0.5 mm wide polyethylene strip in a silicone oil (Sigma-Aldrich Pty. Ltd., North Ryde, NSW, Australia) reservoir, placing it in a clamp fastened to a translation stage, and ‘writing’ the thin oil film across the LN substrate—as a simple version of dip-pen lithography —in a controlled and repeatable manner with a desired thickness dependent upon the translation speed of the polyethylene strip. The oil film width was approximately 0.5 mm for the ultrathin films and slightly wider (0.75 mm) owing to the slower writing speed for thicker films, although the slight variation in width was not observed to affect the film dynamics.
Silicone oil was used to prevent evaporation and to facilitate accurate placement and thickness, although other fluids with the requisite wettability to form thin films upon LN exhibit similar behaviour. Upon application of the SAW, the film spreads, but the spreading behaviour strongly depends on the film thickness, whether as set initially or as it develops over time.
The motion of the oil film was captured using a USB video camera (AM4023 Dino-Eye, AnMo Electronics Corp., Taipei, Taiwan) at 30 fps attached to an upright epifluorescence stereomicroscope (BXFM, Olympus, Tokyo, Japan). The film's thickness was determined using fringes formed by the incident light propagating through the film and transparent LN substrate, and filtered by a 550±10 nm optical notch filter (FB550–10, ThorLabs, Newton, NJ): adjacent fringes , for example those visible in figures 2a(i) and 2b(i), are separated by a vertical (y-axis) distance of approximately Δh∼225 nm.
3. Behaviour of the surface acoustic waves-driven film
Exceptionally thin films having a thickness at or less than the viscous penetration length, , as exemplified in figure 2a, are seen to spread along the SAW propagation direction (figures 2a(ii,iii)); f=ω/2π is the SAW frequency and μ and ρ are the fluid viscosity and density, respectively.
Regions of the film with thicknesses between one-eighth and one-fourth the sound wavelength in the fluid (λℓ/8<h<λℓ/4; figure 2b) are unstable (figure 2b(ii)). Portions of these regions quickly grow in thickness beyond λℓ/4 to form stable ‘steppes’ surrounded by the remaining fluid which thins to less than λℓ/8 as dictated by mass conservation (figure 2b(iii)). The thinner regions (h<λℓ/8) move along the SAW propagation direction in a manner similar to the thin films of figure 2a, but the steppes move in opposition to the SAW, representing the first reversal. The steppe motion corresponds to the motion of thicker, stable films with thicknesses in figure 2c. Such films can nevertheless increase further in thickness  such that , as shown by the small steppe region in figure 2d(iii). Films possessing this greater thickness will once again move along the direction of the SAW—the second reversal.
4. Critical transition thickness hc2 between thick and intermediate films and the associated flow reversal
(a) Thick films
Turning now to an analysis of the phenomenon, we consider films with thicknesses greater than the sound wavelength, h≫λℓ, i.e. films whose behaviour is depicted in figure 2d. The Rayleigh SAW propagating on the substrate along the x-axis is composed of longitudinal and transverse motions of similar amplitude along the surface of the substrate, y=0, possessing particle velocities given by  4.1 where U, k and α are the amplitude of the SAW's vibration velocity, its wavenumber, and its attenuation coefficient in the substrate, respectively. When it encounters the fluid body above the substrate, the SAW leaks its energy into the fluid to generate sound waves that propagate in the fluid at a small angle with respect to the normal to the solid–fluid interface, known as the Rayleigh angle, and with the particle velocities  4.2 thus resulting in the attenuation of the transverse component of the SAW.
An upper bound for the effective body force invoked by gradients in the sound field along the SAW propagation axis in the fluid  can be determined from an upper bound for the rate of sound attenuation along the film. We estimate this upper bound for the sound attenuation rate by neglecting reflections of sound off the free surface of the film, so that the sound field in (4.2) is attenuated strictly owing to the SAW attenuation, a common approach , because its attenuation length (measured in millimetres for most fluids) is three orders of magnitude smaller than the attenuation length of sound in the film (in the range of metres) .
To illustrate this we note that, should we incorporate continuous reflections of sound off the free surface of the film, resulting in sound propagation along the film owing to the initial leakage of sound at the Rayleigh angle and propagation of the sound in the fluid as an acoustic waveguide between the free interface and solid substrate, the rate of attenuation of sound intensity in the film will be reduced. The spatial attenuation of the SAW, quantified by α in (4.1), results in the corresponding reduction in sound intensity along the film in (4.2). This reduction in intensity will then be balanced by the propagation of sound along the film and result in a reduced attenuation of sound intensity, producing a smaller effective attenuation coefficient in (4.2) compared with α. If we choose to neglect sound reflections, then we can thus safely ignore the attenuation of sound in the fluid owing to viscous and thermal effects, because the attenuation occurs at such long length scales compared with the attenuation of the SAW, the dominant effect defining our upper bound for the sound's attenuation length. The upper limit for the corresponding body force along the film at long times is then 4.3 where denotes time averaging of the quantity in the angled brackets, is the unit vector along x and ξ≡[(cSAW/cf)2−1]1/2 in which cSAW and cf are the phase velocity of the wave in the solid and the fluid, respectively.
This body force generates flow—Eckart streaming—along the direction of the SAW [13,15] whose volume flux for a slender film geometry can be approximated by seeking a solution to the streamfunction ψE defined by (ux, uy)≡(∂yψE,−∂xψE) that satisfies the no-slip boundary condition at the substrate–fluid boundary, 4.4 a tangential stress-free condition at the free surface of the film, which can be approximated by 4.5 and the equation governing conservation of momentum, subjected to the thin film approximation, 4.6 Capillary contributions are ignored in the analysis above for Eckart streaming within a relatively flat film to facilitate its comparison with Rayleigh streaming, responsible in a significant part for flow in a thinner form of the film as explored in §4b. Solving the above by integrating (4.6), and using the boundary conditions imposed by (4.4) and (4.5), results in the following volume flux (per unit film width): 4.7 along the direction of the SAW propagation, consistent with the motion of the steppe in figure 2d(iii). While such Eckart flows usually dominate large sessile drop translation experiments, in which the drops are observed to move along the SAW propagation direction [15,16], we have shown that these flows are also dominant in the film in figure 2d(iii) and previously in the solitary wave-like ridges in , as long as the thickness of these features exceed the sound wavelength (h≫λℓ).
(b) Intermediate thickness films
Intermediate thickness films (; figure 2c), however, propagate in opposition to the SAW. The thicknesses are quantized to specific values  as dictated by the acoustic radiation pressure upon the film interface , discussed in §6. In our previous work, we have shown that such counterpropagating films arise owing to a drift flow induced at the outer edge of the viscous boundary layer—Rayleigh streaming—which has a corresponding volume flux per unit width of the film : 4.8 This directional volume flux, formed by the propagating SAW, was previously explained in detail ; the flow morphology in the film is also notably different to the classical vortices formed by Rayleigh streaming from a standing wave. Equating the two volume fluxes associated with Eckart streaming dominant in thick (h≫λℓ) films and Rayleigh streaming dominant in intermediate thickness () films specified by (4.7) and (4.8), respectively, then gives an order-of-magnitude estimate for the critical thickness at which the second flow reversal occurs 4.9 Substitution of the appropriate physical properties into (4.9) gives hc2≈1–1.5λℓ, in good order-of-magnitude agreement with the experimental transition hc2∼2.8λℓ∼λSAW previously observed for different experimental parameter sets .
5. Submicrometre thin films
When the film is far thinner, on the order of or below the viscous penetration length (; figure 2a), however, and therefore too thin to sustain sound waves [19,20], the film moves in response to a convective drift flow induced by the SAW along the substrate (i.e. (4.1)), subjected to the following conditions enforcing tangential stress continuity at the solid–fluid interface y=0 and the normal stress discontinuity at the free surface y=h: 5.1 and 5.2 respectively, with p, γ and Π being the hydrodynamic pressure, surface tension, and disjoining pressure, respectively.
Conservation of the local mass and momentum within the thin film is governed by the classical periodic boundary layer flow equation [19,20] 5.3 where ψ is a streamfunction satisfying (ux,uy)≡(∂yψ,−∂xψ). This may be solved to determine the film flow in the limit of small SAW attenuation (α→0) subjected to the boundary conditions stipulated by (4.1), (5.1) and (5.2). Omitting linear periodic solutions of the flow field that become negligible as by time averaging the general solution over the SAW period ω−1 using a tedious mathematical procedure documented elsewhere , the re-dimensionalized leading order, steady volume flux ψ|y=h(x)−ψ|y=0 may then be written as 5.4 in which the first term represents the volume flux, driven directly by the drift flow for small h(x) and the second term represents contributions to the fluid transport in the film from capillary, acoustic radiation and intermolecular forces, respectively. The dynamics of the film spreading can then be obtained by imposing integral mass conservation through the kinematic condition: 5.5
The dynamics of the film is then governed by a balance between the drift flow and the capillary and molecular forces in (5.4), reducing the film evolution equation given by (5.4) and (5.5) to 5.6 with a form functionally similar to those describing inertia-dominant films [21,22], suggesting that the thinnest films represented by the behaviour shown in figure 2a spread in the direction given by the average volume flux term in (5.4): along the SAW propagation direction.
The evolution of the film profile according to a numerical solution of (5.4) and (5.5) using the method of lines  and imposing the arbitrary initial condition 5.7 and no-flux boundary conditions 5.8 for a fully wetting film is depicted in figure 3. We note that the assumption of a fully wetting film is reasonable given the small contact angle (<20°) of silicone oil, for which the disjoining pressure can be described by Π=A/6πh3. The film spreading dynamics was insensitive to the specific choice of the Hamaker constant A=10−19 to 10−20 J. In the numerical scheme, fourth-order-centred differences were used for discretization of the spatial derivatives and Gear's method was used for integration in time. The numerical results shown in figure 3b,c were evaluated over 4500 evenly spaced positions within a spatial domain of length 600 k−1, where k−1=λSAW/2π, using the parameters associated with the diamonds in figure 3a with ρ=1000 kg m−3, γ=0.021 N m−1 and cSAW=3980 m s−1. Other choices of physical parameters provided only quantitatively different results over short timescales; the results were qualitatively similar throughout and were, regardless, quantitatively similar as .
Both the experimental data and numerical prediction for the film spreading velocity are observed in figure 3a to decay to a constant, non-zero value as . More specifically, the film is seen to spread in a self-similar manner, advancing as t1/4, as indicated by both the collapse in the film evolution profile in figure 3b through rescaling h and x according to the similarity transformation h∼t−1/4 and xf∼t1/4 (figure 3c), and by the constant asymptotic exponent, n, in xf∼tn, to which the experimentally measured film velocity data in figure 3a decays as .
The capillary and intermolecular forces in the second term in (5.4), ignored in the similarity scaling, appear to only influence the shape of the spreading film, not its dynamics. Only the first term in (5.6) was retained in the derivation of the similarity scaling and plotted in figure 3; this term is associated with the drift flow. The good agreement in the prediction of the velocity of the advancing film front between the experimental and numerical results therefore suggests that the t1/4 spreading law, and hence the spreading dynamics is governed by this boundary layer drift.
6. The critical transition thickness hc1 between thin and intermediate films, and the associated thickness instability and flow reversal
We finally turn our attention to the instability apparent in experiments on films with thicknesses λℓ/8<h<λℓ/4 while exposed to the SAW (figure 2b(ii)), corresponding to region (ii) in the phase plot given in figure 4a. The instability appears to be driven when the acoustic radiation pressure imparted on the free surface of the film, forcing it upwards such that the film thickness is increased to an extent that it can no longer be balanced by restoring capillary forces. This is because while radiation pressure is negligible in a sufficiently thin film (), causing it to spread solely owing to the drift flow along the direction of the SAW as governed by (5.4) and (5.5), an increase in the film thickness into the range λℓ/8<h<λℓ/4 (figure 2b) leads to significant acoustic radiation pressure on the film, overwhelming the restoring capillary forces, in turn, driving the observed film thickness instability. We now consider a film which is sufficiently thick, so that acoustic radiation pressure effects are significant, while excluding the relatively weak contributions arising from the disjoining pressure. At the same time, this film is presumed to be sufficiently thin (h≪λℓ) that Eckart streaming is weak.
The acoustic radiation pressure upon the free surface of the film, pr, essentially arises from the reflection of the sound waves propagating in the fluid from the large acoustic impedance change at the fluid–air interface . If the SAW attenuation is assumed negligible, for the sake of simplicity, the acoustic radiation pressure at the free surface of the film owing to the SAW can be written for small film thickness as  6.1 where B and A are the Fox and Wallace coefficients, ρa is the air density and kℓ=2π/λℓ is the wavenumber in the fluid. It is then apparent that pr approaches a minimum at h=(1+2j)λℓ/8 but becomes singular when h→2jλℓ/8, where j=0,1,2,…: the acoustic radiation pressure acts to define the thickness of the film at discrete intervals . Analysis similar to the previous section then gives the film evolution equation 6.2
We now consider a linear stability analysis to obtain a theoretical prediction for this observed unstable film thickness range and hence the critical film thickness hc1 at which the transition occurs. Briefly, we linearize the film evolution equation given by (6.2) far from the contact line. Introducing the periodic disturbance 6.3 to the film thickness in (6.2), in which δ/h0≪1 and , where h0, C, n and σ are the base state film thickness, an arbitrary constant, the disturbance wavenumber and the corresponding growth rate of the film thickness, respectively. Omitting complex and small contributions then simplifies (6.2) to the linear equation 6.4 that translates to 6.5 from which we obtain the real disturbance growth rate coefficient 6.6 indicating that capillary forces act to stabilize the film (σ<1), whereas the radiation pressure pr acts to destabilize it (σ>1) when σ increases with increasing h.
Substituting for pr in (6.6) using (6.1) then reveals that the film is stable only over specific thicknesses in the range 2jλℓ/8<h<(1+2j)λℓ/8 ( j=0,1,2,…). This behaviour is represented, in part, by the solid horizontal lines in figure 4a, which denote the range of stable film thicknesses found in the experiment. These primarily reside in region (i) when h<λℓ/8, consistent with the range predicted above with j=0 and corresponding to the films whose behaviour is exemplified by figure 2a. In region (ii) between the predicted j=0 and j=1 stable regions, as illustrated by the shaded region in figure 4b, however, the film tends to rapidly form thicker plateaus under the action of the acoustic radiation pressure, scavenging fluid from adjacent regions to reduce the thickness of those regions, and altogether remaining within the stable thickness ranges. This is seen by the truncation of the solid horizontal lines by the open squares in figure 4a, with the curved arrows in the plot indicating the jump in the film thickness to the points indicated by the filled squares in the next stable region, region (iii) in figure 4a corresponding to the next predicted stable region j=1, associated with the formation of the steppes.
In reality, the experimental data denoted by the solid horizontal lines in figure 4a indicate that the film remains stable even into region (ii), the region of instability predicted by linear stability theory. This is probably as a result of the omission of azimuthal γ/h curvature effects from the capillary stress term in (5.2) in the linear stability analysis, as demonstrated through a dominant balance between the azimuthal curvature and the time-averaged Reynolds stress ρU2 that gives rise to the acoustic radiation pressure. Imposing volume conservation of the film hL∼1 in the resultant relationship h∼LWe−1, where L is the characteristic film length scale and We ≡ρU2L/γ is an acoustic Weber number, explains the h∼1/U scaling of the experimental data in figure 5 for the threshold film thickness hc1 beyond which the film becomes unstable in region (ii) of figure 4.
For sufficiently thin films () that cannot sustain the propagation of acoustic waves into the liquid, the acoustic radiation pressure acting on the interface is negligible. The convective drift generated by this dominant viscous periodic flow described by (5.6) (without the far right term, i.e. the capillary term) is then the primary mechanism that drives the spreading of these thin films in the SAW propagation direction. Films of intermediate thicknesses (), however, permit the leakage of the acoustic energy from the substrate to produce sound waves in the liquid, and are therefore influenced by an additional stress mechanism—the acoustic radiation pressure upon the film's interface—giving rise to stable and unstable regions within a film depending on its thickness. This mechanism, in addition to the drift flow, imposes a volume flux that causes the film to spread with constant velocity in opposition to the drift flow, and therefore in opposition to the SAW. For thick films and sessile drops whose characteristic thickness is sufficiently large to permit viscous attenuation of the sound wave as it propagates in the liquid, long-range vortical Eckart streaming dominates and drives the fluid film to translate in the direction of the SAW propagation. The broader implications of the theoretical predictions become apparent when, for a single film possessing thicknesses across the entire range from near the edges of the film to nearer the centre, the film moves with, against and again with the SAW, respectively, all the while avoiding the unstable regions predicted by the stability analysis through the formation of steppes of fluid.
The thin films, possessing a thickness of less than 1 μm, have a velocity of order 1 mm s−1 in the direction of the SAW. The intermediate thickness films, with thickness of approximately 20 μm, typically have velocity approximately 1 mm s−1 opposing the SAW propagation. Eckart-driven streaming typical of thick films have a velocity of order 100 mm s−1 in the direction of the SAW. If the film width is 5 mm as in our experiments, this roughly produces a volumetric flow rate of approximately 0.01, approximately 0.1 and approximately 100 mm3 s−1, respectively, for the thin, intermediate and thick films, is one to two orders of magnitude larger than typical microfluidic flows actuated by other means, including electrokinetics and magnetohydrodynamics. Driving these films via megahertz acoustics with 1 W order miniature, battery-powered circuits offers substantial versatility in that the film width is defined by the interdigital transducer's aperture, therefore, films of greater or smaller widths can also be easily achieved by changing the aperture or the nature of the transducer's construction, incorporating tapered or stepped structures.
The mechanisms driving the film spreading phenomena and the double reversal rely upon a balance between the inertial forces driving the SAW-induced convective drift flow, intermolecular forces at the edges of the film, the capillary forces present on the free surface of the film and, for cases where the film is beyond the threshold thickness hc1, the acoustic radiation pressure impinging on the free surface that acts to destabilize the film when it lies within a specific thickness range. Beyond the second critical transition hc2, when the film further increases in thickness beyond the SAW wavelength in the fluid, the attenuation then becomes sufficient to generate a body force and concomitantly, the Eckart streaming that dominates flow in the direction of the SAW propagation.
This work was supported by a CSIRO Flagship Grant on Sensors Systems for Analysis of Aquatic Environments and Australian Research Council grants nos. DP120100013 and DP120100835. L.Y.Y. is grateful for funding from the Australian Research Council for an Australian Research Fellowship under Discovery Grant Project no. DP0985253. J.R.F. is grateful to the Melbourne Centre for Nanofabrication for a Senior Tech Fellowship and to RMIT University for a Vice-Chancellor's Senior Research Fellowship.
This work was performed in part at the Melbourne Centre for Nanofabrication, partly supported by the Australian National Fabrication Facility as an initiative of the Commonwealth of Australia and the Victorian Government.
- Received November 15, 2013.
- Accepted May 1, 2014.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.