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Forced spreading behavior of droplets undergoing low frequency vibration
Colloids and Surfaces A: Physicochem. Eng. Aspects 393 (2012) 144–152
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Forced spreading behavior of droplets undergoing low frequency vibration James David Whitehill, Adrian Neild ∗ , Mark Howard Stokes Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia
a r t i c l e
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Article history: Received 15 September 2011 Received in revised form 14 November 2011 Accepted 16 November 2011 Available online 25 November 2011 Keywords: Droplets Contact angle Vibration Spreading
a b s t r a c t The profile of a droplet sitting on a flat surface is related to the contact angle, when the surface is vibrated the droplet oscillations result in a fluctuation in the contact angle over time. Depending on the amplitude of vibration this can be accompanied with an oscillation of the position of the contact line, this motion occurs when the instantaneous contact angle exceeds the advancing angle or falls below the receding angle. In this work we demonstrate that a net spreading can occur over multiple cycles. The result is that the final contact angle, after the cessation of vibration, is much reduced from the initial value. This has applications in the uniform filling of wells, such as in a microplate. The degree of forced spreading is shown to depend on the mode of oscillation, rather than purely the amplitude of vibration, with cases being demonstrated whereby an increase in amplitude decreases the degree of spreading due to an axisymmetric oscillation changing into a ‘spinning’ mode. Four regions of behavior are identified as a function of vibrational amplitude. In the lowest amplitude state no spreading is observed. The second region exhibits modest spreading resulting from axisymmetric oscillation. Upon further increase in amplitude, a spinning mode occurs in which no spreading results. Finally, the fourth region is characterized by a complex oscillation in which large scale spreading can be achieved. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The reduction in scale of fluidic based chemical and biological processes offers significant analytical and sensitivity improvements. Further benefits are a reduction in reagent usage, increased automation and reduced manufacture costs [1], these factors motivate interest in “lab-on-a-chip” or micro total analysis systems (TAS) [2]. Most of these microfluidic systems involve enclosed fluidic channels, these can be fabricated in a range of ways including etching in silicon and sealed with glass [3], hot embossed in plastics [4], or molding in PDMS [5]. One requirement, which can arise when using such enclosed microfluidic systems is that the systems must be completely filled. This allows samples can be handled without failure of the pumping mechanisms [6]. Hence when attention is turned to very small sample volumes, they either need to be located in an immiscible buffer solution [7], or it is necessary to switch from using enclosed volumes to the use of droplets deposited on plane surfaces [8]. In this switch, some of the key technological building blocks required remain the same, including the manipulation of suspended matter [9,10], sample mixing [11] and fluid motion [12] (whether pumping or droplet movement).
∗ Corresponding author. Tel.: +61 3 9905 3545, fax: +61 3 9905 1825. E-mail address: [email protected] (A. Neild). 0927-7757/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2011.11.015
A range of actuation methods have been applied to droplets to induce mixing. These include electrowetting in which the contact angle is varied repeatedly causing droplet deformation [13–15], thermal currents [16] and ultrasonic actuation where acoustic streaming is generated at high frequencies (typically in the MHz range) [17]. Indeed ultrasound has been used in microfluidic systems to achieve mixing [18] and particle manipulation [19,20]. However, less work has been conducted using lower frequency excitation (referring to 100 s Hz range) despite the potential advantages this offers. These include simpler instrumentation and the ability to actuate simultaneously and almost identically over a much larger area than achievable by ultrasound (due to the longer wavelengths). Low frequency fluid oscillations [21] and direct vibration [22] have been shown to cause fluid mixing in enclosed microfluidic channels and streaming in droplets [23]. These vibrations have been used for particle manipulation within droplets [24] and such manipulation has been predicted in channels [25]. Droplets have been moved on flat surfaces by asymmetric lateral [26,27] or oblique vibration [28], up an inclined surface with vertical vibration [29,30] and across a surface by a combination of both [31]. The mechanisms behind these methods of obtaining droplet movement relate to the fluctuation of the contact angle and the associated hysteresis. Hysteresis is the difference in advancing and receding contact angles on real surfaces. A droplet placed on a real surface exhibit many metastable energy states, which cause a range of contact angles to occur [32]. This is due to an array of
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thermodynamic equilibrium states, which a droplet of a certain volume can make. Each of these metastable states is a local global energy minimum for the drop with the most stable equilibrium contact angle being at the lowest global energy state. In between these metastable states are finite energy barriers. These energy barriers must be overcome for the drop to move to the next metastable state [33].For example, when a puddle (a larger volume than a droplet where gravity plays a significant role in the shape of the fluid interface) is deposited on a surface the contact angle will be between the advancing and receding angle depending on the manner of deposition. Any subsequent movement of the contact line is prevented by the metastable energy state the droplet is in. However, upon vibration of the surface in the vertical direction, the contact line de-pins negating the effect holding the drop in a metastable energy barrier. The result is that the final contact angle will be a balance of only the surface tensions present, meaning that the hysteresis of the drop has effectively been removed [34]. This stick slip behavior has been well documented in previous studies [35,36]. For example, droplet vibration at resonant frequencies causes distinct surface wave behavior [37], and if the amplitude is sufficient de-pinning of the contact line can occur resulting in its’ oscillatory motion [38,39]. Previously the authors demonstrated the spreading of droplets under strong vibration [40], with an emphasis on applications involving spreading of fluid to evenly fill wells and coalescing of droplets. In this work, the behavior of an unconstrained droplet under a range of vibration conditions, on and off resonance, is carefully examined. The parameters, which affect the ability of the droplet to spread are also investigated. At resonance an increase in vibration amplitude leads to the contact line de-pinning and oscillating. Under higher amplitudes still the resonance mode of the droplet breaks down, due to stronger parametric instabilities, which can cause the droplet to spread. In contrast at a non-resonant frequency the behavior of the droplet spreading is more complex. There is a lower and upper amplitude region, which exhibits spreading, whilst a middle region displays no significant spreading. The droplet vibrations are studied with a high speed camera to distinguish the modes of oscillation in each of these regions displaying different behavior.
2. Experimental set-up The experimental apparatus, depicted in Fig. 1, consists of a circular droplet (30 l) deposited on to the surface of a ‘StarFrost’ coated slide (ProSciTech, Thuringowa, Australia, model: G312Si-W). Deionised (DI) water droplets were colored with a low concentration of food dye to create a stronger contrast with the background. The slide was arranged horizontally and vibrated in the vertical direction, with minimum off-axis vibration, by an electromagnetic shaker (LDS, model V201) driven by a signal generator (Stanford Research SDR 345) via a power amplifier (LDS, model PA 25E). It was necessary to capture three different aspects of the droplet’s behavior; this required three different camera orientations. The first camera was positioned directly above the droplet, in order to obtain the changes in the droplet shape. The second CCD camera and a digital SLR camera (Canon, PowerShot SX1 IS) viewed the contact angle of the drop and were arranged slightly above the horizontal plane of the slide. These cameras were positioned at right angles to each other in order to view the contact angle at four separate locations around the drop. This allowed an average contact angle to be calculated, regardless of the non-axisymmetric nature of the spreading. Low speed video image recordings were made using two CCD cameras (Hitachi, KP-D20AU) coupled with a magnification lens
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(InfiniVar Video Microscope, Infinity Photo-Optical Company). The images were recorded at 25 frames/s directly onto a standard DVD recorder. Images obtained by playback from the DVD were transferred to a PC via a frame grabber driven by imaging software (Alliance Vision, Vision Stage). High speed video recordings were made using a Fastec Imaging Troubleshooter (TS1000ME) camera at 1000 frames/s. These were registered directly onto the internal hard drive of the camera, producing an uncompressed video file, which was then transferred to the PC. In both cases lighting was provide by a cold source gooseneck lamp (Olympus, LG-PS2). 3. Contact angle parameters In this work the effect of low frequency vibration on unconstrained droplets is under investigation. The term unconstrained is used to indicate that the droplets are placed on nominally flat surfaces, with the shape of the droplet being defined by contact angle. This is the angle at which the air fluid interface meets the solid surface, it can be found by way of a force balance between the various surface tensions (fluid/gas, fluid/solid, and gas/solid). However, the actual angle usually takes a range of values (exhibiting hysteresis) from the lowest, the receding angle, to the highest the advancing angle [41]. This range can be linked to the energy required to move the contact line from its current metastable state. In other words the force balance must be overstepped by a certain amount so that the contact line actually moves, whether receding as fluid is extracted or advancing when fluid is added. The technique used to measure the hysteresis of the droplet on the differing surfaces was tilting-plate goniometry (TPG) [42]. This method measures two contact angles of a drop that is rotated with respect to the horizontal. As the droplet is tilted, the steeper gradient causes the uppermost and lowermost parts of the droplet interface to approach minimum and maximum contact angles, respectively. These boundaries are measured just prior to the droplet sliding down the slope. These maximum and minimum values can correspond to the advancing and receding angles of the drop [43]. The advancing and receding angle pairs were measured to be 57◦ /19◦ for the ‘StarFrost’ coated glass slides used. These measurements were taken using the same low concentration dyed water to provide a consistency with other experimental data sets. It should be noted that the values of advancing and receding contact angles TPG provides can be highly dependent of the placement of the drop on the surface [44] Confirmation of the advancing and receding contact angles were provided by the captive-drop goniometry (CDG) method. We can estimate the most stable equilibrium contact angle by assuming the energy required to advance or recede the contact line is equal, discussions of the accuracy of such an estimate have been previous presented [34,39,45]. The force (per unit length) required to move the contact line could be expressed as [45]: fc+ = (cosms − cosa ), fc− = (cosr − cosms )
(1)
where the superscripts + and − indicate advancing and receding respectively, is the surface tension and ms , a , r are the most stable equilibrium, advancing and receding contact angles respectively. Hence the most stable contact angles can be estimated as cosms ≈
(cosr + cosa ) 2
(2)
This method is known as the average cosine approximation. Alternately the most stable contact angle can be estimated using an arithmetic mean approach [46–49]. This method is simply the average of the sum of the advancing and receding contact angles
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Fig. 1. Experimental setup.
measured. The most stable equilibrium contact angle be expressed as ms ≈
(r + a ) 2
(3)
For the ‘StarFrost’ slides the average cosine and arithmetic mean approximations yield different results, with the most stable equilibrium contact angle approximations of 42◦ and 38◦ , respectively. 4. Characterization of behavior starting at resonance The movement of the contact line when a droplet is excited at resonance has previously been studied. Noblin et al. [39] were able to observe the oscillatory motion of the contact line. In their work the accelerations of excitation were below 1 g and the contact line oscillation was observed to match the excitation frequency. For the oscillation of the contact line to occur, a certain value of amplitude of excitation needed to be exceeded. In lower acceleration cases, the contact angle fluctuation would remain within the hysteresis bounds hence the contact line remained pinned. The droplet behavior would change under higher accelerations as the hysteresis bounds can be overcome. These conditions would allow the contact line to de-pin and as a result an oscillation of the contact line was clearly demonstrated [39,50]. A series of experiments were performed with the actuation commencing at the resonant frequency, in order to create a comparison to these previous studies and an extension into higher amplitude regimes. Resonant frequencies of these unconstrained drops can easily be affected by the boundary conditions. To account for this, prior to an experiment being conducted, a low acceleration actuation was applied to the droplet in order to determine the resonant frequency. This test was performed within the pinned region as to not manipulate the droplet and confirmed at what frequency a resonant mode was forming. The results of resonant actuation are shown in Fig. 2. An acceleration of 2.8 g (row a) causes some fluctuation in the contact angle but no line movement is present. However, contact line oscillation is present under increased excitation (5.9 g) as seen in row b. For both acceleration conditions, the change in location of the contact line over one time period is presented in Fig. 2(c). In both scenarios the mode shape are a similar axisymmetric shape and the behavior
agrees with that described previously [36]. Furthermore, in both cases the oscillation is linear, with the frequency of fluid motion matching that of the excitation frequency. Under an even higher oscillating acceleration of 10.2 g the conditions for resonance start to break down. Whilst initially the frequency has been determined as resonant for the droplet, a significantly different non-axisymmetric mode shape forms. In this scenario considerable spreading can be observed across the images presented in Fig. 3 (images taken 18.6 cycles apart). In contrast to the contact line oscillation, for spreading to occur the time averaged displacement of the contact line must be non-zero, hence it is a non-linear process occurring over multiple actuation cycles. The fluid appears to rotate in the plane of the solid surface. This rotation occurred at 47.6 revolutions per second, which is approximately half the excitation frequency. Examination of the images shows the motion of the contact line on one side of the drop advancing whilst the other side is in the receding phase (due to the contact angle values). Clearly for spreading to occur over a time period any segment on the droplet’s contact line must spend more time within the advancing region than the receding region. Clearly it is possible for spreading to be caused when sufficient oscillation amplitudes are used at the resonant frequency. However, it is worth noting that as soon as contact line oscillation occurs the resonant frequency will alter [38]. This effect becomes much more dramatic when the shape of the droplet alters due to spreading; hence what commences as a resonant frequency will not remain so as spreading ensues. It is for this reason that attention will now be turned to investigating the behavior of droplets at an arbitrary frequency. 5. Characterization of behavior starting at non-resonance Experiments have been performed on 30 l droplets at 200 Hz over a range of amplitudes. The initial (×) and final (+) contact angles are plotted in Fig. 4. Due to differences, which occur when the droplet is deposited there is a range of initial angles, however, what is more important is the resulting final contact angles. The spreading characteristics fall into four different regions (labeled in the figure) based on the final contact angle. In between these distinct regions are transition zones in which the droplet spreading behavior does not conform to these regions.
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Fig. 2. A sequence of high speed images, is shown in (a) and (b), of a 30 l droplet excited by a resonant 93 Hz frequency over an actuation cycle. The contact angle remaining pinned at an acceleration of 2.8 g (row a), whilst de-pinning and oscillating at 5.9 g (row b). An image analysis (c) of the two previous accelerations, 2.8 g (䊉) and 5.9 g (), showing the change in droplet diameter in pixels over one actuation cycle.
The spreading can be considered a non-linear effect in so much as a net spread occurs over a period of many cycles. An analogy would be the alignment of particles by acoustic radiation force. The ultrasonic oscillation may be in the MHz range but a steady alignment is achieved over a long period [20]. This gives rise to two time frames within the experiments that of an oscillation and that of the spreading. We will present comparative data in each of these time frames for the different regions prior to examining each region separately. Non-linearity is well known to exist within oscillating droplets. Miles and Henderson [51] discussed the non-linear motion of
Fig. 3. High speed images, 18.6 cycles apart, of a 30 l droplet spreading when influenced by a large acceleration (10.27 g) at resonant frequency of 93 Hz. The droplet mode shapes, before the start of actuation (a), after transition into an oscillatory motion whilst still remaining relatively symmetric (b) and oscillating in a non-symmetric way (c–f).
droplets by way of an analogy with a vertically driven simple pendulum. This non-linearity exists when the actuation frequency is within a certain parameter of resonance, with this parameter being in part actuation amplitude dependant. However, this non-linearity results in sub-harmonics rather than time average effects. Firstly we examine the timescale of the spreading over numerous cycles (>440) as shown in Fig. 5. The behavior of the spreading varies greatly from region to region. In region 2 the change in the wetted diameter is approximately the same for both data sets plotted. However, the lower acceleration (4 g) requires over 60 cycles for the hysteresis bounds to be overcome and for spreading to start occurring. Whereas the higher acceleration region 2 data (5.6 g) shows evidence that the contact line de-pins after approximately 15 cycles. In region 3 the change in wetted diameter is relatively low which corresponds to the minimal change in contact angle data shown in Fig. 4. The highest acceleration data shows a large amount of spreading until finally becoming more stable after approximately 220 cycles. To show a comparison of region behavior over the timescale of a single cycle, Fig. 6 shows a series of images of the oscillating droplets for each region identified. A significant variation in modes is seen as drive amplitude is increased. The images for region 1 clearly repeat every actuation cycle. Whilst for region 2 the fluid oscillation require two full actuation cycles to complete, though due to the spreading there are slight differences (best seen from the bright reflected light) between the t/T = 0 and t/T = 2 images. Region 3 exhibits a fluid cycle which takes 2 T to complete, here the first and last images shown are identical, due to the lack of spreading. Finally in region 4 the motion is highly energetic with a complex mode shape. The transition between these oscillation modes is a much better indicator of the spreading characteristics than the fluid motion frequency. Fig. 7 presents a series of side profile images of the oscillating drops. Again significant differences are seen across the regions. Region 1 (not shown) and region 2 are axisymmetric, differing
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Fig. 4. A plot of the different regions of spreading behavior. In region 1, the droplet has insufficient excitation energy to allow the droplet to de-pin and spread. Region 2, the contact line de-pins, the contact angle relaxes and the droplet spreads. Region 3, the droplet forms spatiotemporal oscillating wave forms the result of which is that minimal spreading occurs. Region 4, the droplet wave forms move into chaotic spatiotemporal regime that causes the droplet to impact with the substrate and a large amount of spreading occurs.
predominately in fluctuation amplitude. The profile from droplets at actuation amplitude corresponding to region 3 becomes asymmetric, and finally a very complex and irregular profile characterizes region 4. In the following subsections the behavior in each of these regions are discussed, with emphasis on those in which significant spreading is present. 5.1. Region 1 No spreading of droplets actuated by acceleration of less than 1.75 g is observed. Within this region the contact line does not
de-pin at any stage through the oscillation cycle. The contact angle remains within the bounds of the advancing and receding angles, due to insufficient energy being supplied to the system to overcome these bounds, consequently no spreading is observed. The oscillation shape can be clearly identified as repeating once per actuation time period, indicating linear behavior (Fig. 6a). In the first transition zone between region 1 and 2, the droplet spreads very slightly. In this area, the droplet has sufficient energy to create waveforms that will de-pin the droplet. Although once the droplet has spread slightly the contact angle fluctuations no longer cause the upper angle in a cycle to continue to exceed the advancing angle, so spreading stops short of that seen in region 2.
Fig. 5. A plot of the growth in the diameter of a droplet over a few seconds at different accelerations. Low acceleration excitation (region 2) 4 g and 5.6 g, and * respectively. The lowest acceleration, 4 g, does not initially spread until the droplet has sufficient energy to overcome its current global energy barrier. The oscillatory spatiotemporal droplet regime (region 3) exhibits minimal spreading at an acceleration of 8.7 g (triangles). Under even higher acceleration (14.7 g) the droplet moves into a chaotic spatiotemporal behavior (region 4) with significant spreading (diamonds).
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Fig. 6. A sequence of high speed camera images showing a droplet over time periods from above, showing the behavior at different accelerations. The top row shows linear waves at an acceleration 1.45 g (row a). The next two rows, at 5.2 g (row b) and 8 g (row c), are in the non-linear behavior with repetition every second cycle. The last row (d) is spatiotemporal regimes chaotic motion (acceleration of 14.7 g).
5.2. Region 2 When the acceleration is increased further (2.5–6 g) the final contact angle is observed to change. In region 2 the profile of the oscillating droplet remains axisymmetric (Fig. 7column a), with a large vertical displacement at the center of the droplet. The fluid oscillation cycle in this region is complete in ether one or two cycles (in Fig. 5 an example requiring two cycles is shown). This demonstrates that the boundary of the regions does no coincide with a shift from linear to non-linear oscillation; rather it becomes clear that the degree of oscillation in contact angle is the determining feature of spreading behavior. Despite variations in the initial contact angle due to alterations in the manual deposition of the fluid, the final angle is seen to be tightly defined at close to 38◦ . The consistency in the final contact
angle (after the cessation of vibration) occurs as the contact line relaxes to the most stable contact angle, conforming to the mean average contact angle approximation of 38◦ rather than the average cosine most stable equilibrium angle estimate (42◦ ). The droplet expanded in an approximately radial manner. Andrieu et al. [34] have previously demonstrated that the contact lines of puddles relax under vibration resulting in the most stable contact angle. This driven sinusoidal system causes the drop to behave as a damped harmonic oscillator. The droplet has sufficient energy to overcome the pinning of the contact line, causing the droplet to behave in a stick-slip manner, and unlike the previous transition region the energy supplied to the droplet is sufficient to allow the most stable equilibrium contact angle state to be reached. The stick-slip manner of contact lien movement can be further demonstrated by examining the contact angle variation over the
Fig. 7. High speed camera images showing the side view of a droplets behavior over a cycle. This sequence of images shows the contact angle variation when influenced by different acceleration. The left column demonstrates the axisymmetric mode shapes with the acceleration of 5.2 g (region 2). Column b, 8 g, illustrates the droplet profile during region 3. It should be noted the contact angle of the drop does not change greatly. The spatiotemporal regime exhibits the droplets chaotic motion when influenced by an acceleration of 14.7 g (region 4). This shows the large range of contact angles, and results in a large amount of spreading.
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Fig. 8. (a) The behavior of the contact angle and diameter growth of a droplet has been plotted for the 91 Hz example over the first 18 cycles of excitation at an acceleration of 3.9 g. This plot shows that advancing and receding contact angle (dots) and the contact line behavior (triangles). The contact line spends more time in the advancing segment of the cycle resulting in net droplet spreading over each cycle. The image sequence over half a cycle (b–g) shows the change in droplet profile and contact angle.
oscillation cycle, as is shown in Fig. 8. This experiment was performed at 91 Hz and 3.9 g, a frequency close to but not at resonance. At this lower frequency the profile of the droplet was less complex and the contact angles are clearer to measure as the gradient of the profile changes less rapidly than at higher frequencies. The spreading, observed in Fig. 8, was of a nature comparable to region 2 in Fig. 4. It can be seen from the profile images that a considerable range of contact angles occur over a cycle. When plotted against time (for the first 18 cycles of excitation) it becomes apparent that the angle exceeds the advancing angle for more time per period than it exceeds the receding angle. When either of these values are passed a shift in the contact line occurs, hence a net movement outwards results. Over a longer time period the diameter of the droplet will settle to a steady value (as was shown in Fig. 5). This occurs when the contact angle becomes equal to the average of the receding and advancing angles. As under such a condition, the sinusoidal oscillation of the contact angle would result in equal portions of the cycle occurring below the receding angle and above the advancing angle, hence the cessation of spreading.
5.3. Region 3 In region 3 (7.5–10.5 g) the drop experiences even higher accelerations, however, surprisingly the droplet behavior greatly changes and spreading almost ceases. In this region the axisymmetric behavior (seen previously) of the drop breaks down and the parametric instability of the wave form grows. These instabilities form droplet oscillation modes that appear to spin. As a consequence of this mode change the advancing and receding contact angles are not exceeded often in this region, as can be seen from the side profiles of the droplet shown in Fig. 7(column b). Even though a large motion of fluid is present over the cycle, the contact
angles do not vary significantly. This causes minimal spreading, as the majority of the time is spent within the stick region.
5.4. Region 4 The final section occurs when accelerations of between 12 and 17 g are applied to the droplet. In this region a large degree of spreading is observed. Here, the droplet oscillation mode moves into a spatiotemporal chaotic motion (Fig. 7column c). The spreading that occurs becomes fairly random in nature, and the resulting contact line cannot be considered as being circular. The relaxation of the contact line, which accounted for behavior demonstrated in region 2 cannot be applied to this chaotic system. In comparison to region 2, the final contact angles in region 4 are significantly lower at around 29◦ . Under slightly higher amplitudes it has been shown previously that spreading can create a final contact angle below the receding angle [40], clearly far removed from the most stable equilibrium angle. This behavior occurs within the pre-ejection state of the droplet. It is only upon further increase in forcing acceleration that daughter droplets will be ejected. During the first 70 cycles the wetted diameter changes greatly (as seen in Fig. 5). When examining the start of the oscillation in the region at 14.7 g (Fig. 9), the initial droplet behavior is quite extreme. This figure shows a series of images taken from the start of oscillation, the images are clearly axisymmetric and as such more similar to those of region 2 shown in Fig. 8. However, the fluid motion is much more severe with a large droplet of fluid nearly ejected. This almost detached daughter droplet will ‘impact’ with the substrate and resultantly cause a very large contact angle change. Similar behavior was found by Pasandideh-Fard et al. [52] after modelling droplet ‘impacts’ on a solid surface. The resultant downward motion of the almost detached daughter droplet will cause
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Fig. 9. Demonstrating the effect on the droplet under high amplitude excitation during the very early stages of excitation, with this sequence taken 3 ms apart. The droplet is excited at an acceleration of 14.7 g, this figure showing the droplet almost ejecting part of itself, before crashing back down into the surface of the substrate. This impact causes significant spreading and after a few cycles the droplet will transition into the chaotic regime.
a high amount of spreading. Once the droplet has spread (after numerous ‘impact’ cycles) the mode shape in Fig. 9 becomes damped out of the system, and this ‘impacting’ effect no longer affect the spreading of the drop. The droplet behavior will transform into the fluctuating motion and further large spreading previously shown to characterize this region. It should be noted that all data presented is from experiments in which no fluid ejection was observed in the videos taken or by examination of the substrate for daughter droplets.
5.5. Spreading over a range of parameters It should not be thought that the spreading can only occur due to gravitation potential energy promoting a spreading droplet when the contact line detaches. To confirm this, the slide was inverted; the ensuing spreading is presented in Fig. 10. It can be seen that the inverted droplet of 30 l, spreads extensively when excited with an acceleration of 9.5 g. With the spreading occurring regardless of the new orientation, it is clear that gravity is not a key driver in the mechanism. The region cut-off accelerations are, unsurprisingly, affected by the change in parameters. For a non-inverted drop these conditions result in a ‘spinning’ droplet (region 3) exhibiting little spreading. Whereas the inverted experiment found that the droplet behavior was in the transition area between region 3 and 4. Furthermore these non-resonance spreading characteristics have also been observed across different droplet diameters. A degree of spreading after 200 actuation cycles is demonstrated for droplets in Fig. 11. A range of droplet sizes, from 1.5 l to 30 l,
Fig. 10. An upside down droplet of 30 l, excited with a frequency of 200 Hz at 119.1 m amplitude. The droplet exhibits similar spreading to the normal orientation drop, demonstrating that this effect is not gravity dependant.
Fig. 11. The spreading before (left column) and one second after excitation (right column) of different sized droplets. The conditions that the droplets are constant for all experiments the same amplitude and frequency were used (139.9 m at 200 Hz). The droplet volumes of 1.5 l (row a), 3 l (row b), 7 l (row c), 15 l (row d) and 30 l (row e) are used in this experiment. The red lines are superimposed over the images to indicate the original droplet diameter and the relative spreading.
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Acknowledgement The authors are grateful that this work was supported by an Australian Research Council Discovery grant: DP110104010. References
Fig. 12. Three sets of images are shown of droplets (30 m) on different surface exposed to the same vibration (within region 4) until spreading has ceased. The surfaces are uncoated lime soda glass slides (a–b) and silicon coated glass slides (c–d).
excited at a non-resonant frequency with an acceleration of 11.2 g. The cut-off between the different regions differs with volume, so though fixed excitation conditions (the same accelerations and frequency) are used, the droplets exhibit characteristics consistent with different spreading regions. The smaller droplets (rows a–c) have symmetric mode shapes and display spreading characteristics of region 2. The larger drops, rows d and e, behave with a non-symmetrical waveform. In row d, the 15 l droplet can be seen to display spreading characteristics of region 3. Whilst the largest droplet 30 l demonstrating traits of the transition region between region 3 and 4. The degree of spreading differs across the droplets (the red lines indicating the original wetted diameter of the droplet), this is due to the mode shapes that can form within the droplet length scale. Finally Fig. 12 shows two sets of images showing droplets on different surface exposed to the same vibration (11.2 g) until spreading has ceased. The surfaces are silicon covered glass slides and uncoated lime soda slides. The respective most stable equilibrium contact angle for the silicon and lime-soda glass slides were approximated as 84◦ and 29◦ , respectively. Again spreading is clearly observable across all surfaces presented. 6. Conclusion The nature of spreading which occurs in droplets when exposed to strong low frequency vibration has been investigated. The spreading has been linked to the change in contact angle, which occurs during the oscillating cycle. This short term effect can cause a longer term spreading if the advancing angle is exceeded more prevalently than the receding angle. The way in which the contact angle oscillates is related to the oscillation of the fluid. Hence an increase in off resonance vibration amplitude can cause a decrease in spreading as an axisymmetric mode becomes a ‘spinning’ mode. Further increase causes a more complex mode resulting in extensive spreading. Hence the link between amplitude and degree of spreading is via oscillation mode. In addition, the spreading behavior described has been shown to occur under different conditions including changed slide orientation and differing droplet sizes.
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Forced spreading behavior of droplets undergoing low frequency vibration James David Whitehill, Adrian Neild ∗ , Mark Howard Stokes Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia
a r t i c l e
i n f o
Article history: Received 15 September 2011 Received in revised form 14 November 2011 Accepted 16 November 2011 Available online 25 November 2011 Keywords: Droplets Contact angle Vibration Spreading
a b s t r a c t The profile of a droplet sitting on a flat surface is related to the contact angle, when the surface is vibrated the droplet oscillations result in a fluctuation in the contact angle over time. Depending on the amplitude of vibration this can be accompanied with an oscillation of the position of the contact line, this motion occurs when the instantaneous contact angle exceeds the advancing angle or falls below the receding angle. In this work we demonstrate that a net spreading can occur over multiple cycles. The result is that the final contact angle, after the cessation of vibration, is much reduced from the initial value. This has applications in the uniform filling of wells, such as in a microplate. The degree of forced spreading is shown to depend on the mode of oscillation, rather than purely the amplitude of vibration, with cases being demonstrated whereby an increase in amplitude decreases the degree of spreading due to an axisymmetric oscillation changing into a ‘spinning’ mode. Four regions of behavior are identified as a function of vibrational amplitude. In the lowest amplitude state no spreading is observed. The second region exhibits modest spreading resulting from axisymmetric oscillation. Upon further increase in amplitude, a spinning mode occurs in which no spreading results. Finally, the fourth region is characterized by a complex oscillation in which large scale spreading can be achieved. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The reduction in scale of fluidic based chemical and biological processes offers significant analytical and sensitivity improvements. Further benefits are a reduction in reagent usage, increased automation and reduced manufacture costs [1], these factors motivate interest in “lab-on-a-chip” or micro total analysis systems (TAS) [2]. Most of these microfluidic systems involve enclosed fluidic channels, these can be fabricated in a range of ways including etching in silicon and sealed with glass [3], hot embossed in plastics [4], or molding in PDMS [5]. One requirement, which can arise when using such enclosed microfluidic systems is that the systems must be completely filled. This allows samples can be handled without failure of the pumping mechanisms [6]. Hence when attention is turned to very small sample volumes, they either need to be located in an immiscible buffer solution [7], or it is necessary to switch from using enclosed volumes to the use of droplets deposited on plane surfaces [8]. In this switch, some of the key technological building blocks required remain the same, including the manipulation of suspended matter [9,10], sample mixing [11] and fluid motion [12] (whether pumping or droplet movement).
∗ Corresponding author. Tel.: +61 3 9905 3545, fax: +61 3 9905 1825. E-mail address: [email protected] (A. Neild). 0927-7757/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2011.11.015
A range of actuation methods have been applied to droplets to induce mixing. These include electrowetting in which the contact angle is varied repeatedly causing droplet deformation [13–15], thermal currents [16] and ultrasonic actuation where acoustic streaming is generated at high frequencies (typically in the MHz range) [17]. Indeed ultrasound has been used in microfluidic systems to achieve mixing [18] and particle manipulation [19,20]. However, less work has been conducted using lower frequency excitation (referring to 100 s Hz range) despite the potential advantages this offers. These include simpler instrumentation and the ability to actuate simultaneously and almost identically over a much larger area than achievable by ultrasound (due to the longer wavelengths). Low frequency fluid oscillations [21] and direct vibration [22] have been shown to cause fluid mixing in enclosed microfluidic channels and streaming in droplets [23]. These vibrations have been used for particle manipulation within droplets [24] and such manipulation has been predicted in channels [25]. Droplets have been moved on flat surfaces by asymmetric lateral [26,27] or oblique vibration [28], up an inclined surface with vertical vibration [29,30] and across a surface by a combination of both [31]. The mechanisms behind these methods of obtaining droplet movement relate to the fluctuation of the contact angle and the associated hysteresis. Hysteresis is the difference in advancing and receding contact angles on real surfaces. A droplet placed on a real surface exhibit many metastable energy states, which cause a range of contact angles to occur [32]. This is due to an array of
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thermodynamic equilibrium states, which a droplet of a certain volume can make. Each of these metastable states is a local global energy minimum for the drop with the most stable equilibrium contact angle being at the lowest global energy state. In between these metastable states are finite energy barriers. These energy barriers must be overcome for the drop to move to the next metastable state [33].For example, when a puddle (a larger volume than a droplet where gravity plays a significant role in the shape of the fluid interface) is deposited on a surface the contact angle will be between the advancing and receding angle depending on the manner of deposition. Any subsequent movement of the contact line is prevented by the metastable energy state the droplet is in. However, upon vibration of the surface in the vertical direction, the contact line de-pins negating the effect holding the drop in a metastable energy barrier. The result is that the final contact angle will be a balance of only the surface tensions present, meaning that the hysteresis of the drop has effectively been removed [34]. This stick slip behavior has been well documented in previous studies [35,36]. For example, droplet vibration at resonant frequencies causes distinct surface wave behavior [37], and if the amplitude is sufficient de-pinning of the contact line can occur resulting in its’ oscillatory motion [38,39]. Previously the authors demonstrated the spreading of droplets under strong vibration [40], with an emphasis on applications involving spreading of fluid to evenly fill wells and coalescing of droplets. In this work, the behavior of an unconstrained droplet under a range of vibration conditions, on and off resonance, is carefully examined. The parameters, which affect the ability of the droplet to spread are also investigated. At resonance an increase in vibration amplitude leads to the contact line de-pinning and oscillating. Under higher amplitudes still the resonance mode of the droplet breaks down, due to stronger parametric instabilities, which can cause the droplet to spread. In contrast at a non-resonant frequency the behavior of the droplet spreading is more complex. There is a lower and upper amplitude region, which exhibits spreading, whilst a middle region displays no significant spreading. The droplet vibrations are studied with a high speed camera to distinguish the modes of oscillation in each of these regions displaying different behavior.
2. Experimental set-up The experimental apparatus, depicted in Fig. 1, consists of a circular droplet (30 l) deposited on to the surface of a ‘StarFrost’ coated slide (ProSciTech, Thuringowa, Australia, model: G312Si-W). Deionised (DI) water droplets were colored with a low concentration of food dye to create a stronger contrast with the background. The slide was arranged horizontally and vibrated in the vertical direction, with minimum off-axis vibration, by an electromagnetic shaker (LDS, model V201) driven by a signal generator (Stanford Research SDR 345) via a power amplifier (LDS, model PA 25E). It was necessary to capture three different aspects of the droplet’s behavior; this required three different camera orientations. The first camera was positioned directly above the droplet, in order to obtain the changes in the droplet shape. The second CCD camera and a digital SLR camera (Canon, PowerShot SX1 IS) viewed the contact angle of the drop and were arranged slightly above the horizontal plane of the slide. These cameras were positioned at right angles to each other in order to view the contact angle at four separate locations around the drop. This allowed an average contact angle to be calculated, regardless of the non-axisymmetric nature of the spreading. Low speed video image recordings were made using two CCD cameras (Hitachi, KP-D20AU) coupled with a magnification lens
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(InfiniVar Video Microscope, Infinity Photo-Optical Company). The images were recorded at 25 frames/s directly onto a standard DVD recorder. Images obtained by playback from the DVD were transferred to a PC via a frame grabber driven by imaging software (Alliance Vision, Vision Stage). High speed video recordings were made using a Fastec Imaging Troubleshooter (TS1000ME) camera at 1000 frames/s. These were registered directly onto the internal hard drive of the camera, producing an uncompressed video file, which was then transferred to the PC. In both cases lighting was provide by a cold source gooseneck lamp (Olympus, LG-PS2). 3. Contact angle parameters In this work the effect of low frequency vibration on unconstrained droplets is under investigation. The term unconstrained is used to indicate that the droplets are placed on nominally flat surfaces, with the shape of the droplet being defined by contact angle. This is the angle at which the air fluid interface meets the solid surface, it can be found by way of a force balance between the various surface tensions (fluid/gas, fluid/solid, and gas/solid). However, the actual angle usually takes a range of values (exhibiting hysteresis) from the lowest, the receding angle, to the highest the advancing angle [41]. This range can be linked to the energy required to move the contact line from its current metastable state. In other words the force balance must be overstepped by a certain amount so that the contact line actually moves, whether receding as fluid is extracted or advancing when fluid is added. The technique used to measure the hysteresis of the droplet on the differing surfaces was tilting-plate goniometry (TPG) [42]. This method measures two contact angles of a drop that is rotated with respect to the horizontal. As the droplet is tilted, the steeper gradient causes the uppermost and lowermost parts of the droplet interface to approach minimum and maximum contact angles, respectively. These boundaries are measured just prior to the droplet sliding down the slope. These maximum and minimum values can correspond to the advancing and receding angles of the drop [43]. The advancing and receding angle pairs were measured to be 57◦ /19◦ for the ‘StarFrost’ coated glass slides used. These measurements were taken using the same low concentration dyed water to provide a consistency with other experimental data sets. It should be noted that the values of advancing and receding contact angles TPG provides can be highly dependent of the placement of the drop on the surface [44] Confirmation of the advancing and receding contact angles were provided by the captive-drop goniometry (CDG) method. We can estimate the most stable equilibrium contact angle by assuming the energy required to advance or recede the contact line is equal, discussions of the accuracy of such an estimate have been previous presented [34,39,45]. The force (per unit length) required to move the contact line could be expressed as [45]: fc+ = (cosms − cosa ), fc− = (cosr − cosms )
(1)
where the superscripts + and − indicate advancing and receding respectively, is the surface tension and ms , a , r are the most stable equilibrium, advancing and receding contact angles respectively. Hence the most stable contact angles can be estimated as cosms ≈
(cosr + cosa ) 2
(2)
This method is known as the average cosine approximation. Alternately the most stable contact angle can be estimated using an arithmetic mean approach [46–49]. This method is simply the average of the sum of the advancing and receding contact angles
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Fig. 1. Experimental setup.
measured. The most stable equilibrium contact angle be expressed as ms ≈
(r + a ) 2
(3)
For the ‘StarFrost’ slides the average cosine and arithmetic mean approximations yield different results, with the most stable equilibrium contact angle approximations of 42◦ and 38◦ , respectively. 4. Characterization of behavior starting at resonance The movement of the contact line when a droplet is excited at resonance has previously been studied. Noblin et al. [39] were able to observe the oscillatory motion of the contact line. In their work the accelerations of excitation were below 1 g and the contact line oscillation was observed to match the excitation frequency. For the oscillation of the contact line to occur, a certain value of amplitude of excitation needed to be exceeded. In lower acceleration cases, the contact angle fluctuation would remain within the hysteresis bounds hence the contact line remained pinned. The droplet behavior would change under higher accelerations as the hysteresis bounds can be overcome. These conditions would allow the contact line to de-pin and as a result an oscillation of the contact line was clearly demonstrated [39,50]. A series of experiments were performed with the actuation commencing at the resonant frequency, in order to create a comparison to these previous studies and an extension into higher amplitude regimes. Resonant frequencies of these unconstrained drops can easily be affected by the boundary conditions. To account for this, prior to an experiment being conducted, a low acceleration actuation was applied to the droplet in order to determine the resonant frequency. This test was performed within the pinned region as to not manipulate the droplet and confirmed at what frequency a resonant mode was forming. The results of resonant actuation are shown in Fig. 2. An acceleration of 2.8 g (row a) causes some fluctuation in the contact angle but no line movement is present. However, contact line oscillation is present under increased excitation (5.9 g) as seen in row b. For both acceleration conditions, the change in location of the contact line over one time period is presented in Fig. 2(c). In both scenarios the mode shape are a similar axisymmetric shape and the behavior
agrees with that described previously [36]. Furthermore, in both cases the oscillation is linear, with the frequency of fluid motion matching that of the excitation frequency. Under an even higher oscillating acceleration of 10.2 g the conditions for resonance start to break down. Whilst initially the frequency has been determined as resonant for the droplet, a significantly different non-axisymmetric mode shape forms. In this scenario considerable spreading can be observed across the images presented in Fig. 3 (images taken 18.6 cycles apart). In contrast to the contact line oscillation, for spreading to occur the time averaged displacement of the contact line must be non-zero, hence it is a non-linear process occurring over multiple actuation cycles. The fluid appears to rotate in the plane of the solid surface. This rotation occurred at 47.6 revolutions per second, which is approximately half the excitation frequency. Examination of the images shows the motion of the contact line on one side of the drop advancing whilst the other side is in the receding phase (due to the contact angle values). Clearly for spreading to occur over a time period any segment on the droplet’s contact line must spend more time within the advancing region than the receding region. Clearly it is possible for spreading to be caused when sufficient oscillation amplitudes are used at the resonant frequency. However, it is worth noting that as soon as contact line oscillation occurs the resonant frequency will alter [38]. This effect becomes much more dramatic when the shape of the droplet alters due to spreading; hence what commences as a resonant frequency will not remain so as spreading ensues. It is for this reason that attention will now be turned to investigating the behavior of droplets at an arbitrary frequency. 5. Characterization of behavior starting at non-resonance Experiments have been performed on 30 l droplets at 200 Hz over a range of amplitudes. The initial (×) and final (+) contact angles are plotted in Fig. 4. Due to differences, which occur when the droplet is deposited there is a range of initial angles, however, what is more important is the resulting final contact angles. The spreading characteristics fall into four different regions (labeled in the figure) based on the final contact angle. In between these distinct regions are transition zones in which the droplet spreading behavior does not conform to these regions.
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Fig. 2. A sequence of high speed images, is shown in (a) and (b), of a 30 l droplet excited by a resonant 93 Hz frequency over an actuation cycle. The contact angle remaining pinned at an acceleration of 2.8 g (row a), whilst de-pinning and oscillating at 5.9 g (row b). An image analysis (c) of the two previous accelerations, 2.8 g (䊉) and 5.9 g (), showing the change in droplet diameter in pixels over one actuation cycle.
The spreading can be considered a non-linear effect in so much as a net spread occurs over a period of many cycles. An analogy would be the alignment of particles by acoustic radiation force. The ultrasonic oscillation may be in the MHz range but a steady alignment is achieved over a long period [20]. This gives rise to two time frames within the experiments that of an oscillation and that of the spreading. We will present comparative data in each of these time frames for the different regions prior to examining each region separately. Non-linearity is well known to exist within oscillating droplets. Miles and Henderson [51] discussed the non-linear motion of
Fig. 3. High speed images, 18.6 cycles apart, of a 30 l droplet spreading when influenced by a large acceleration (10.27 g) at resonant frequency of 93 Hz. The droplet mode shapes, before the start of actuation (a), after transition into an oscillatory motion whilst still remaining relatively symmetric (b) and oscillating in a non-symmetric way (c–f).
droplets by way of an analogy with a vertically driven simple pendulum. This non-linearity exists when the actuation frequency is within a certain parameter of resonance, with this parameter being in part actuation amplitude dependant. However, this non-linearity results in sub-harmonics rather than time average effects. Firstly we examine the timescale of the spreading over numerous cycles (>440) as shown in Fig. 5. The behavior of the spreading varies greatly from region to region. In region 2 the change in the wetted diameter is approximately the same for both data sets plotted. However, the lower acceleration (4 g) requires over 60 cycles for the hysteresis bounds to be overcome and for spreading to start occurring. Whereas the higher acceleration region 2 data (5.6 g) shows evidence that the contact line de-pins after approximately 15 cycles. In region 3 the change in wetted diameter is relatively low which corresponds to the minimal change in contact angle data shown in Fig. 4. The highest acceleration data shows a large amount of spreading until finally becoming more stable after approximately 220 cycles. To show a comparison of region behavior over the timescale of a single cycle, Fig. 6 shows a series of images of the oscillating droplets for each region identified. A significant variation in modes is seen as drive amplitude is increased. The images for region 1 clearly repeat every actuation cycle. Whilst for region 2 the fluid oscillation require two full actuation cycles to complete, though due to the spreading there are slight differences (best seen from the bright reflected light) between the t/T = 0 and t/T = 2 images. Region 3 exhibits a fluid cycle which takes 2 T to complete, here the first and last images shown are identical, due to the lack of spreading. Finally in region 4 the motion is highly energetic with a complex mode shape. The transition between these oscillation modes is a much better indicator of the spreading characteristics than the fluid motion frequency. Fig. 7 presents a series of side profile images of the oscillating drops. Again significant differences are seen across the regions. Region 1 (not shown) and region 2 are axisymmetric, differing
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Fig. 4. A plot of the different regions of spreading behavior. In region 1, the droplet has insufficient excitation energy to allow the droplet to de-pin and spread. Region 2, the contact line de-pins, the contact angle relaxes and the droplet spreads. Region 3, the droplet forms spatiotemporal oscillating wave forms the result of which is that minimal spreading occurs. Region 4, the droplet wave forms move into chaotic spatiotemporal regime that causes the droplet to impact with the substrate and a large amount of spreading occurs.
predominately in fluctuation amplitude. The profile from droplets at actuation amplitude corresponding to region 3 becomes asymmetric, and finally a very complex and irregular profile characterizes region 4. In the following subsections the behavior in each of these regions are discussed, with emphasis on those in which significant spreading is present. 5.1. Region 1 No spreading of droplets actuated by acceleration of less than 1.75 g is observed. Within this region the contact line does not
de-pin at any stage through the oscillation cycle. The contact angle remains within the bounds of the advancing and receding angles, due to insufficient energy being supplied to the system to overcome these bounds, consequently no spreading is observed. The oscillation shape can be clearly identified as repeating once per actuation time period, indicating linear behavior (Fig. 6a). In the first transition zone between region 1 and 2, the droplet spreads very slightly. In this area, the droplet has sufficient energy to create waveforms that will de-pin the droplet. Although once the droplet has spread slightly the contact angle fluctuations no longer cause the upper angle in a cycle to continue to exceed the advancing angle, so spreading stops short of that seen in region 2.
Fig. 5. A plot of the growth in the diameter of a droplet over a few seconds at different accelerations. Low acceleration excitation (region 2) 4 g and 5.6 g, and * respectively. The lowest acceleration, 4 g, does not initially spread until the droplet has sufficient energy to overcome its current global energy barrier. The oscillatory spatiotemporal droplet regime (region 3) exhibits minimal spreading at an acceleration of 8.7 g (triangles). Under even higher acceleration (14.7 g) the droplet moves into a chaotic spatiotemporal behavior (region 4) with significant spreading (diamonds).
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Fig. 6. A sequence of high speed camera images showing a droplet over time periods from above, showing the behavior at different accelerations. The top row shows linear waves at an acceleration 1.45 g (row a). The next two rows, at 5.2 g (row b) and 8 g (row c), are in the non-linear behavior with repetition every second cycle. The last row (d) is spatiotemporal regimes chaotic motion (acceleration of 14.7 g).
5.2. Region 2 When the acceleration is increased further (2.5–6 g) the final contact angle is observed to change. In region 2 the profile of the oscillating droplet remains axisymmetric (Fig. 7column a), with a large vertical displacement at the center of the droplet. The fluid oscillation cycle in this region is complete in ether one or two cycles (in Fig. 5 an example requiring two cycles is shown). This demonstrates that the boundary of the regions does no coincide with a shift from linear to non-linear oscillation; rather it becomes clear that the degree of oscillation in contact angle is the determining feature of spreading behavior. Despite variations in the initial contact angle due to alterations in the manual deposition of the fluid, the final angle is seen to be tightly defined at close to 38◦ . The consistency in the final contact
angle (after the cessation of vibration) occurs as the contact line relaxes to the most stable contact angle, conforming to the mean average contact angle approximation of 38◦ rather than the average cosine most stable equilibrium angle estimate (42◦ ). The droplet expanded in an approximately radial manner. Andrieu et al. [34] have previously demonstrated that the contact lines of puddles relax under vibration resulting in the most stable contact angle. This driven sinusoidal system causes the drop to behave as a damped harmonic oscillator. The droplet has sufficient energy to overcome the pinning of the contact line, causing the droplet to behave in a stick-slip manner, and unlike the previous transition region the energy supplied to the droplet is sufficient to allow the most stable equilibrium contact angle state to be reached. The stick-slip manner of contact lien movement can be further demonstrated by examining the contact angle variation over the
Fig. 7. High speed camera images showing the side view of a droplets behavior over a cycle. This sequence of images shows the contact angle variation when influenced by different acceleration. The left column demonstrates the axisymmetric mode shapes with the acceleration of 5.2 g (region 2). Column b, 8 g, illustrates the droplet profile during region 3. It should be noted the contact angle of the drop does not change greatly. The spatiotemporal regime exhibits the droplets chaotic motion when influenced by an acceleration of 14.7 g (region 4). This shows the large range of contact angles, and results in a large amount of spreading.
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Fig. 8. (a) The behavior of the contact angle and diameter growth of a droplet has been plotted for the 91 Hz example over the first 18 cycles of excitation at an acceleration of 3.9 g. This plot shows that advancing and receding contact angle (dots) and the contact line behavior (triangles). The contact line spends more time in the advancing segment of the cycle resulting in net droplet spreading over each cycle. The image sequence over half a cycle (b–g) shows the change in droplet profile and contact angle.
oscillation cycle, as is shown in Fig. 8. This experiment was performed at 91 Hz and 3.9 g, a frequency close to but not at resonance. At this lower frequency the profile of the droplet was less complex and the contact angles are clearer to measure as the gradient of the profile changes less rapidly than at higher frequencies. The spreading, observed in Fig. 8, was of a nature comparable to region 2 in Fig. 4. It can be seen from the profile images that a considerable range of contact angles occur over a cycle. When plotted against time (for the first 18 cycles of excitation) it becomes apparent that the angle exceeds the advancing angle for more time per period than it exceeds the receding angle. When either of these values are passed a shift in the contact line occurs, hence a net movement outwards results. Over a longer time period the diameter of the droplet will settle to a steady value (as was shown in Fig. 5). This occurs when the contact angle becomes equal to the average of the receding and advancing angles. As under such a condition, the sinusoidal oscillation of the contact angle would result in equal portions of the cycle occurring below the receding angle and above the advancing angle, hence the cessation of spreading.
5.3. Region 3 In region 3 (7.5–10.5 g) the drop experiences even higher accelerations, however, surprisingly the droplet behavior greatly changes and spreading almost ceases. In this region the axisymmetric behavior (seen previously) of the drop breaks down and the parametric instability of the wave form grows. These instabilities form droplet oscillation modes that appear to spin. As a consequence of this mode change the advancing and receding contact angles are not exceeded often in this region, as can be seen from the side profiles of the droplet shown in Fig. 7(column b). Even though a large motion of fluid is present over the cycle, the contact
angles do not vary significantly. This causes minimal spreading, as the majority of the time is spent within the stick region.
5.4. Region 4 The final section occurs when accelerations of between 12 and 17 g are applied to the droplet. In this region a large degree of spreading is observed. Here, the droplet oscillation mode moves into a spatiotemporal chaotic motion (Fig. 7column c). The spreading that occurs becomes fairly random in nature, and the resulting contact line cannot be considered as being circular. The relaxation of the contact line, which accounted for behavior demonstrated in region 2 cannot be applied to this chaotic system. In comparison to region 2, the final contact angles in region 4 are significantly lower at around 29◦ . Under slightly higher amplitudes it has been shown previously that spreading can create a final contact angle below the receding angle [40], clearly far removed from the most stable equilibrium angle. This behavior occurs within the pre-ejection state of the droplet. It is only upon further increase in forcing acceleration that daughter droplets will be ejected. During the first 70 cycles the wetted diameter changes greatly (as seen in Fig. 5). When examining the start of the oscillation in the region at 14.7 g (Fig. 9), the initial droplet behavior is quite extreme. This figure shows a series of images taken from the start of oscillation, the images are clearly axisymmetric and as such more similar to those of region 2 shown in Fig. 8. However, the fluid motion is much more severe with a large droplet of fluid nearly ejected. This almost detached daughter droplet will ‘impact’ with the substrate and resultantly cause a very large contact angle change. Similar behavior was found by Pasandideh-Fard et al. [52] after modelling droplet ‘impacts’ on a solid surface. The resultant downward motion of the almost detached daughter droplet will cause
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Fig. 9. Demonstrating the effect on the droplet under high amplitude excitation during the very early stages of excitation, with this sequence taken 3 ms apart. The droplet is excited at an acceleration of 14.7 g, this figure showing the droplet almost ejecting part of itself, before crashing back down into the surface of the substrate. This impact causes significant spreading and after a few cycles the droplet will transition into the chaotic regime.
a high amount of spreading. Once the droplet has spread (after numerous ‘impact’ cycles) the mode shape in Fig. 9 becomes damped out of the system, and this ‘impacting’ effect no longer affect the spreading of the drop. The droplet behavior will transform into the fluctuating motion and further large spreading previously shown to characterize this region. It should be noted that all data presented is from experiments in which no fluid ejection was observed in the videos taken or by examination of the substrate for daughter droplets.
5.5. Spreading over a range of parameters It should not be thought that the spreading can only occur due to gravitation potential energy promoting a spreading droplet when the contact line detaches. To confirm this, the slide was inverted; the ensuing spreading is presented in Fig. 10. It can be seen that the inverted droplet of 30 l, spreads extensively when excited with an acceleration of 9.5 g. With the spreading occurring regardless of the new orientation, it is clear that gravity is not a key driver in the mechanism. The region cut-off accelerations are, unsurprisingly, affected by the change in parameters. For a non-inverted drop these conditions result in a ‘spinning’ droplet (region 3) exhibiting little spreading. Whereas the inverted experiment found that the droplet behavior was in the transition area between region 3 and 4. Furthermore these non-resonance spreading characteristics have also been observed across different droplet diameters. A degree of spreading after 200 actuation cycles is demonstrated for droplets in Fig. 11. A range of droplet sizes, from 1.5 l to 30 l,
Fig. 10. An upside down droplet of 30 l, excited with a frequency of 200 Hz at 119.1 m amplitude. The droplet exhibits similar spreading to the normal orientation drop, demonstrating that this effect is not gravity dependant.
Fig. 11. The spreading before (left column) and one second after excitation (right column) of different sized droplets. The conditions that the droplets are constant for all experiments the same amplitude and frequency were used (139.9 m at 200 Hz). The droplet volumes of 1.5 l (row a), 3 l (row b), 7 l (row c), 15 l (row d) and 30 l (row e) are used in this experiment. The red lines are superimposed over the images to indicate the original droplet diameter and the relative spreading.
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Acknowledgement The authors are grateful that this work was supported by an Australian Research Council Discovery grant: DP110104010. References
Fig. 12. Three sets of images are shown of droplets (30 m) on different surface exposed to the same vibration (within region 4) until spreading has ceased. The surfaces are uncoated lime soda glass slides (a–b) and silicon coated glass slides (c–d).
excited at a non-resonant frequency with an acceleration of 11.2 g. The cut-off between the different regions differs with volume, so though fixed excitation conditions (the same accelerations and frequency) are used, the droplets exhibit characteristics consistent with different spreading regions. The smaller droplets (rows a–c) have symmetric mode shapes and display spreading characteristics of region 2. The larger drops, rows d and e, behave with a non-symmetrical waveform. In row d, the 15 l droplet can be seen to display spreading characteristics of region 3. Whilst the largest droplet 30 l demonstrating traits of the transition region between region 3 and 4. The degree of spreading differs across the droplets (the red lines indicating the original wetted diameter of the droplet), this is due to the mode shapes that can form within the droplet length scale. Finally Fig. 12 shows two sets of images showing droplets on different surface exposed to the same vibration (11.2 g) until spreading has ceased. The surfaces are silicon covered glass slides and uncoated lime soda slides. The respective most stable equilibrium contact angle for the silicon and lime-soda glass slides were approximated as 84◦ and 29◦ , respectively. Again spreading is clearly observable across all surfaces presented. 6. Conclusion The nature of spreading which occurs in droplets when exposed to strong low frequency vibration has been investigated. The spreading has been linked to the change in contact angle, which occurs during the oscillating cycle. This short term effect can cause a longer term spreading if the advancing angle is exceeded more prevalently than the receding angle. The way in which the contact angle oscillates is related to the oscillation of the fluid. Hence an increase in off resonance vibration amplitude can cause a decrease in spreading as an axisymmetric mode becomes a ‘spinning’ mode. Further increase causes a more complex mode resulting in extensive spreading. Hence the link between amplitude and degree of spreading is via oscillation mode. In addition, the spreading behavior described has been shown to occur under different conditions including changed slide orientation and differing droplet sizes.
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