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Axisymmetric and Non-Axisymmetric Oscillations of Sessile Compound Droplets in Open Digital Microfluidic Platform SHUBHI BANSAL, and Prosenjit Sen Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02042 • Publication Date (Web): 18 Sep 2017 Downloaded from http://pubs.acs.org on September 24, 2017
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Axisymmetric and Non-Axisymmetric Oscillations of Sessile Compound Droplets in Open Digital Microfluidic Platform Shubhi Bansal and Prosenjit Sen* Centre for Nano Science and Engineering (CeNSE) Indian Institute of Science, Bangalore Karnataka, India
Abstract: Manipulating droplets of biological fluids in Electrowetting on Dielectric (EWOD) based digital microfluidic platform is a significant challenge due to biofouling and surface contamination. This problem is often addressed by operating in an oil environment. We study an alternate configuration of sessile compound droplets having an aqueous core surrounded by a smaller oil shell. In contrast to conventional EWOD, open digital microfluidic platform enabled by the coreshell configuration will allow electrical, mechanical or optical probes to get unrestricted access to the droplet and thus enabling highly flexible and dynamically reconfigurable lab-on-chip systems. Understanding droplet oscillations is essential as they are known to enhance mixing. To our knowledge, this is the first study of axisymmetric and non-axisymmetric oscillations of compound droplets actuated using EWOD. Modes shapes for both axisymmetric and non-axisymmetric oscillations were studied and explained. Enhancement in axisymmetric oscillation of the core by decreasing the shell volume was obtained experimentally and modeled theoretically. Smaller shell volumes reduce the damping losses and hence allowed appearance of non-axisymmetric modes over a larger range of operating parameters. The oscillation frequency regime for obtaining prominent non-axisymmetric oscillations for different shell volumes was identified. Compound droplets provide a mechanism to reduce biofouling, sample contamination, and evaporation. We demonstrate axisymmetric and non-axisymmetric oscillations of compound droplets with biological core of red blood cells, providing crucial first steps for promoting applications like rapid efficient assays, mixing of biological fluids and fluidic photonics on hysteresis free surfaces.
Keywords: compound droplet; sessile core; non-axisymmetric oscillations; shell volume; mode amplitude; electrowetting.
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Introduction One of the basic functions required for enabling bio-chemical applications on miniaturized lab-onchip platforms is mixing of various liquids. Rapid assays are made possible by improving the efficiency of mixing, which is often a very slow process in micro scale devices due to low Reynolds number. For laminar flows, rate of mixing is limited by diffusion through the interface. In order to speed up mixing in microfluidic devices several strategies have been employed
[1]
which often
requires complicated designs and additional mechanisms. Electrowetting on Dielectric (EWOD) based droplet platforms allow controlling each droplet independently for digital microfluidics
[2, 3]
,
and accessing the interfacial oscillation modes through actuation at specific frequencies provides an efficient technique for reducing both time and chip-space required for mixing [4,5]. Mixing in droplets actuated by electrowetting is benefited from surface flows that lead to an increase in the interface length between the mixing liquids. By moving the droplet in perpendicular paths mixing rate can be enhanced significantly
[6]
. By harnessing the interface oscillation modes further speed-up of the
mixing phenomena can be achieved. Rapid mixing using EWOD and associated non-laminar flows during droplet oscillations have been studied in past
[7,8]
. When voltage is applied, the droplet
spreads symmetrically about the central axis (normal to the substrate) due to symmetric field along the contact line. However, at higher actuation voltages the droplet loses the axisymmetric shape and non-axisymmetric modes are actuated because of parametric instability. For coverless open droplet configuration, enhancement in mixing of single phase sessile droplets by using non-axisymmetric oscillations has been demonstrated previously
[9]
.Interfacial oscillations in single droplet invoking
non-axisymmetric modes have been shown to achieve mixing in shorter times when compared to axisymmetric oscillations [9].
In EWOD based platforms actuation of biological samples or colloidal reagents are often limited by phenomena that degrade the surface such as protein fouling, surface charging etc. Performing the microfluidic operations in oil reduces surface degradation and allows efficient handling of biological samples [10]. Silicone oil has been used extensively as an lubricant [11,12] and surrounding medium [4] to prevent surface biofouling and to reduce hysteresis by eliminating surface pinning. However, the presence of surrounding medium affects the interfacial dynamics of the droplets. We have studied the impact of the encapsulating oil volume on the dynamics of compound droplets. Compound droplet oscillations in liquid bath
[13,14]
and on fibers
[15]
have been studied in past in order to
understand their resonance frequencies [16,17], stability [18,19] and several other characteristics [20,21, 22]. A class of compound drops that are sessile on a planar solid surface has been explored recently [23,24] where the different configurations based on the droplet size or volume ratio have been investigated.
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Movement of discrete compound drops on chip sample core
[26]
[25]
and with an oil-carrier engulfed biological
using electrowetting has also been reported. To the best of our knowledge, this is
the first report of studying the impact of shell volume on the oscillation dynamics of the sessile core of a compound droplet oscillated by electrowetting. Here, we report about both the axisymmetric and non-axisymmetric mode oscillations of compound droplets actuated using electrowetting. We have studied the shape oscillations and associated mode patterns in concentric compound drops of water surrounded by oil shell using AC electrowetting. The volume of the surrounding oil affects the behavior of the oscillating core droplet due to the contact line hysteresis and viscous forces which damps the actuated interface of the core droplet. As the shell volume increases, the resonance frequency shows slow reduction for axisymmetric oscillations. We present an extended model for determining the resonance frequency of the compound droplets. We determined shell volume and frequency regime for achieving non-axisymmetric oscillations in compound droplets. The parametric instability regime for obtaining non-axisymmetric modes of compound droplet was observed to be dependent on the shell volume. There have been numerous studies on the oscillations and behavior of single droplet [9,27,28,29] but compound droplet dynamics has never been investigated.
Finally, the oscillations of a sessile core consisting of red blood cells (RBC) suspended in phosphate buffered saline (PBS) solution were studied for different oil volumes.
These controlled non-
axisymmetric oscillations of colloidal samples in presence of an optimized shell volume are useful for enhancing speed of reactions and biological assays on lab-on-chip platforms. We demonstrate this by mixing a dye with a RBC populated PBS solution. In comparison to mixing by pure diffusion a significant reduction in mixing time is obtained for non-axisymmetric oscillations. Sessile compound droplets can also find use in fluidic displays, electronic paper
[30]
and other opto-fluidic
[31, 32]
applications.
Materials and Methods Device Fabrication Devices were fabricated on Borofloat glass wafers having 150 nm of sputter deposited transparent conducting ITO (indium tin oxide) to make the actuation electrodes. Use of ITO for bottom electrode allows bottom view imaging of the droplet. SU8 2002, used as a dielectric, was spin coated at 1000 rpm for 30 s to get a thickness of 2.6 ± 0.4 μm which was measured using Dektak-XT Surface profiler. The topmost hydrophobic layer was spin coated using Teflon® (AF-2400) solution (DuPont) at 4000
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rpm for 40 s and hard baked at 180 °C. The thickness of the Teflon layer was measured to be 170 ± 30 nm using Dektak-XT Surface profiler.
Experimental Setup Experiments were carried out with the compound droplets formed with two immiscible liquids – DI water as core droplet and silicone oil (viscosity-1 cSt) as the shell droplet. Firstly the oil droplet was placed on the substrate and then a fixed volume of water was added. Due to the Laplace pressure gradient caused by the oil curvature, the water core would have a stable position at the center of oil shell
[33]
as the volume ratio satisfied the relation
/
≥ 0.05 for our experiments. After
stabilization, the compound droplet was grounded by inserting a thin wire (diameter~100
) from
the top. The droplet was actuated by applying voltage to the ITO base actuation electrode. A National Instruments multifunctional card (NI-DAQ 6363) was used to generate the AC actuation voltage which was then amplified by a gain of 50 using a high voltage amplifier (Trek 2205). A Photron SA4 Fastcam was used to capture the bottom view of the droplet oscillations at a frame rate of 2000 fps. A 5 V step signal generated by NI-DAQ was used to trigger the high-speed camera for synchronized image acquisition. The schematic of the compound droplet and experimental setup is shown in Figure 1(A).
Compound Droplet Radius and Volume Measurements A DI water droplet of fixed volume (8 μl) was used as the core and silicone oil droplet was used as the shell with varying volumes ranging between 3.6 μl to 25.4 μl. The droplets were dispensed using a micro-pipette leading to fairly accurate volumes (standard deviation of ±0.5 ) of the dispensed water droplets. However, low surface tension of oil (17.4 mN/m) led to additional intake and incomplete dispensing as the oil liked to coat and remain adhered to the tip. Hence, the volumes of the dispensed oil droplets were determined by fitting Laplace equation to the shape of the oil droplets using a MATLAB code. The base radius of the droplets was measured using ImageJ from the side-view and bottom-view images of the compound droplets with different shell volumes. In Figure 2, the normalized total base radius of the compound droplets has been plotted with respect to oil volumes normalized by the volume of the water droplet.
Mode Amplitude Calculation The recorded videos were analyzed and images were extracted using MATLAB. The extracted images were analyzed using ImageJ software to study the mode amplitude of axisymmetric oscillations. As discussed later, the core droplet was completely engulfed in oil and hence the water-substrate-oil
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contact line was not present. The water droplet however formed a relatively flat interface with the oil near the substrate as depicted in Figure 1(B). Since the images were captured from bottom view, we could clearly see the flat interface of the water droplet. Base radius ( ) of the flat interface was measured at different instants of an oscillation cycle. For axisymmetric modes the contact line shape is not perfectly circular due to the biasing wire and non-uniform surface properties which vary from device to device. Variation from the circular shape is however minimal and we measure the base diameter by selecting points on the perimeter. The axisymmetric mode amplitude was calculated as the maximum change in radius (
,
= max ( ) − min ( )) over one cycle. For each experiment,
the data was averaged for 3 actuation cycles. A software was developed to calculate the mode amplitudes for the non-axisymmetric oscillations from the captured videos. For non-axisymmetric oscillations, the captured videos for each shell-volume and different actuation frequencies were analyzed by extracting the time varying mode amplitudes (with components components define the core interface shape ( , )= where
,
,
( ) +
(t)
(t)). These
( ) which is expanded using the Fourier series [9] as { (t) cos(
)) +
(t)sin ( )}
is the axisymmetric mode amplitude of the water core. The mode amplitude for the
cores non-axisymmetric mode 2 was calculated from the time varying coefficients as +
1
,
( )=
. At each frequency, the mode amplitude was averaged for 2-4 actuation cycles. The
variation within different cycles for a given experiment on the same device was usually minimal. At least three videos were taken at each frequency on different devices to determine the standard deviation in calculated amplitudes. Most of the standard deviation in the measured data is attributed to variations in surface quality from one device to another. We calculated the mode amplitudes for mode numbers 0 and 2 only. Different non-axisymmetric modes have been studied in literature [34, 35, 36] but we have not investigated the higher modes because the amplitude was found to be maximum for mode 2 only. However, the next higher resonance mode was explored for both axisymmetric and non-axisymmetric oscillations to prove the reduction in mode amplitude at higher resonances. For this, 5.9 μl shell volume was used (data is available in Supplementary Information section (E)). For axisymmetric oscillations the next higher mode was found to have less mode amplitude (~0.11) than the second mode (~0.45). Similar reduction in the mode amplitude was also observed for the higher non-axisymmetric oscillation. The measured modal amplitudes were normalized with respect to the radius of the unactuated droplet for each experiment.
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Results and Interpretation 1. Sessile Compound Droplet Configuration Shape of Compound Droplets Compound drops comprising of four-phases namely air, solid-substrate, immiscible core and shell liquids can have different configurations based on interfacial tensions of the two liquids[24], different core-shell size ratio and substrate wettability
[23, 37]
. For a compound sessile-droplet three different
three-phase contact lines are possible, namely, core-shell-air, shell-substrate-air and core-substrateshell. These three phase contact lines can be formed only when the interfacial energies of the various interfaces favors retention of all the interfaces and formation of the contact line, which can be calculated through the Neumann’s triangle law
[38]
and the triangle inequality
[24]
. In the current
study using water core and silicone oil shell in air, the spreading coefficient is positive, i.e. + [40]
, where
(35 mN/m
[39]
) is the surface tension of oil-water interface,
) is the surface tension of the air-oil interface and
>
(17.4 mN/m
is the surface tension of the air-water (72
mN/m) interface. The positive spreading coefficient favors the formation of a sessile compound droplet where the water core is cloaked by the silicone oil [37] to minimize the interfacial energy and hence the three phase contact line for the oil-air-water interfaces is not formed.
In the current scenario where the oil-air-water triple line is absent, four different configurations of the compound droplet on partially wettable substrates are possible
[41,23]
as shown in Figure 3.
Interfacial energies and densities of the liquids determine the stable configuration. The possible configurations are as follows: (i) sessile core with sessile shell capping happens when the density of core is greater or equivalent to the density of shell, and the interfacial energies prefer the formation of core-shell-substrate contact line with negative spreading coefficient. The spreading coefficient of the base (water-oil-substrate) interface is given as S
=
−
−
, where
and
are
the surface tensions of water-substrate and oil-substrate interfaces respectively; (ii) sessile core cloaked with a layer of shell happens when the core density is greater than the shell and the spreading coefficient (S
) is positive
[24,42]
. This leads to formation of a thin film of shell which
separates the core from the substrate; (iii) history dependent configuration is observed when the negative spreading coefficient favors that the core remains attached to the solid, whereas the higher density of shell favors that the engulfed core float to the top. In this case the order in which the droplets are deposited on the substrate will decide the final configuration; (iv) completely engulfed core is formed when spreading coefficient (S
) is positive, and the shell density is more than the
core density due to which the buoyancy force positions the core near apex of the sessile drop. Although the interfacial energies and relative densities play a major role in determining the
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compound droplet configuration, but its overall shape is also dependent on the shell volume as compared to the core volume. Using a value of 20 mN/m (
), the interfacial energies of Teflon®-oil (
[43]
for the Teflon®-air interfacial energy
= 5 ± 0.5 mN/m) and Teflon®-water (
= 53 ± 3
mN/m) interfaces were calculated using Young’s equation with measured contact angles of water on Teflon as 117° ± 3° and oil on Teflon as 30° ± 3°. From the calculated interfacial energies, we observe >
the inequality (
+
) to hold true. This implies that a thin film of silicone oil will lubricate
and stay between the water droplet and the PTFE substrate. Hence, the sessile compound droplet will be in configuration (ii). The experimental images of side view of compound droplet for different shell volumes are shown in Figure 4. It is important to mention that the base radius ( ) of the flattened water droplet was measured from the bottom view images and was found to be ~1.56 mm with calculated apparent contact angle of ~95°.
Thin Oil Film Formation and its Stability In the current compound droplet configuration, two thin oil films are formed. The first one is between water and Teflon at bottom having a thickness as shown in Figure 1. The second one is between air and water at the top having thickness as shown in Figure 1 (B). The overall shape of the interfaces considering the disjoining pressures and in absence of the actuating electric field is determined by the following set of equations: ∆ ∆
=
−
(ℎ − ) +
=
−
(ℎ − ) −
6
6
2
3
where ℎ is the height of the compound droplet as depicted in Figure 1(B), is the coordinate axis, is the curvature of the respective interfaces, oil shell and
,
is the density of water core,
is the density of the
are the Hamaker constants which were calculated using standard equations
available in various texts
[44,45,46]
. In the calculation, a typical value of 3 × 1015 was used for the
electronic absorption frequency. The values of dielectric constants and refractive indices used for calculating the Hamaker constant are given in the Supplementary Information Section (A). The calculated Hamaker constants were ~10-19 and ~10-20 for the air-oil-water interface and water-oilTeflon interface respectively. Positive values of the calculated Hamaker constants indicate that the disjoining pressure stabilizes the thin film
[47]
. Realizing that the cloaking oil film follows the same
curvature as the core droplet at the top, its equilibrium thickness ( ) can be calculated approximately from equation 3 as [44]:
=
12
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is the radius of the compound drop at the top (as depicted in Figure 1B) which was
estimated from the captured images to be ~1.56 mm. The top film thickness was calculated to be ~62 nm. Similarly, the thickness of the oil film between water and substrate can be estimated as: =
6 (2
+
ℎ
5
)
The bottom film thickness was estimated to be ~21 nm. The stability of the bottom film especially in the presence of the electrostatic pressure, which tends to destabilize the film has been studied earlier [48]. The stability of the bottom film can be determined by balancing the surface energies, van der Waals interaction energies and electrostatic energy due to the applied voltage. For a perturbation of wavelength (λ), the film becomes unstable when the applied voltage (V) is larger than critical value, given as (see Supplementary Information Section (B) for details) ≥ where
(
) (2 / ) + /2 (1 + ) /
is the dielectric constant of the solid dielectric (SU-8) having a thickness
6
and
is the
dielectric constant of the oil. This relationship is plotted in Figure S.1 (Supplementary Information Section (B.1)). As the film thickness increased, the voltage required for disruption of the thin film decreased for the same perturbation wavelength. We also observed that for a given film thickness, there is a minimum voltage below which the film remains stable for all wavelengths. For film thickness of 21 nm, the film destabilizes above 98 V and the voltages used in our experiments (53 Vrms and 88 Vrms) are below this threshold voltage, implying, we have a stable oil film entrapment beneath the water drop.
2. Axisymmetric Oscillations of Sessile Compound Droplet Axisymmetric Mode Shapes At low actuation voltages of 53 Vrms the triple phase contact line of the compound droplet remained mostly circular and we captured axisymmetric oscillations of the core using high speed camera at 2000 fps (images are shown in Figure 5). We performed an actuation voltage frequency sweep from 10 Hz - 46 Hz with a step of 3 Hz to determine the frequency response. If not otherwise stated we used applied actuation voltage frequencies (represented by
) for all graphs and discussions.
Mechanical response frequencies (represented by ) are twice the voltage frequencies as actuation force is proportional to ∝
. In the range of frequencies studied in this work, we were able to
capture the first resonance mode for the compound droplets. Axisymmetric mode shapes were also studied by capturing side-view videos of the core oscillations in a 8.6 μl shell. In Figure 6 a), we have plotted the calculated profiles of single sessile droplets with moving contact lines for axisymmetric mode 2 and mode 3. The droplet profile was calculated by solving spherical harmonic functions using
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MATLAB. Single droplet undergoing mode 2 of oscillations shows in-phase behavior i.e. height and contact angle of sessile drop increase and decrease simultaneously while during mode 3 as the height increases, contact angle decreases and vice-versa. The profile for the oil shell has been drawn approximately on the periphery of sessile droplet in the plots to explain oil contact angle variation in case of compound drop due to volume conservation (when oil contact line is pinned). The arrows show the direction of interface motion. In Figure 6b), the experimental images of compound droplet with 8 μl core surrounded by 8.6 μl silicone oil shell are shown during the axisymmetric oscillations with 53 Vrms actuation voltage. These images clearly depict that at the first axisymmetric oscillation mode 2 (i.e., at 25 Hz), the height of compound droplet increased as the contact angle of oil shell decreased. This observed phase relationship between the compound droplet height and contact angle is in contradiction with the case of a single phase sessile droplet [49,3] where for first mode, as height increases, contact angle also increases. This apparent contradiction is however easily explained by taking into consideration the fact that the oil contact line remains stationary while the core interface moves. As the water (core) interface moves in and up, the oil (shell) interface moves in because the oil contact line is pinned due to strong hysteresis. Similarly, as the height of the core droplet decreased for mode 2, the oil interface moved out leading to an increase in the contact angle. In comparison, at 50 Hz, mode 3 of axisymmetric oscillation, the height of compound droplet and contact angle of oil shell increased simultaneously and vice-versa as shown by overlapped image with marked edges (Figure 6b right) Frequency Response for different Shell Volumes The amplitude of the core oscillations was measured from the bottom view videos and normalized with respect to its initial radius and has been plotted for each oil volume in Figure 7. To account for the sparse frequency sampling and standard deviation in the measured data, the resonance frequencies (both damped and undamped), and maximum amplitudes were obtained by fitting in MATLAB the measured response curves with the response of a one degree of freedom spring-massdamper model (see Supplementary Information Section (D.1) for details) given by Γ( ) =
where
(
−
) + (2
)
7
is a constant related to the actuation force, is the damping ratio related to the losses, and
is the undamped natural frequency. For the smallest oil volume of 3.6 µl the resonance frequency extracted from the fit of measured response was at an actuation voltage frequency of 25.6 Hz. As the oil volume increased from 3.6 µl to 25.4 µl, the resonance frequency was found to decrease from 25.6 Hz to 21.4 Hz, as can be observed in Figure 8.
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This shift in the resonance frequencies for increasing oil volumes is explained by extending the semiempirical model proposed by Noblin et. al.
[27]
which has successfully predicted the single phase
sessile droplet oscillations in multiple studies. In that model droplet oscillations were modeled as shallow capillary waves with resonance frequencies (Ω ) given by Ω =
tanh ( ℎ)/ , where
=
2 / is the wave number, is surface tension, ℎ is the average height of the droplet and
is
density. The semi-empirical model relates the wavelength ( ) with the perimeter ( ) of the liquid air interface (as shown in Figure 4A) of the droplet profile as
= , where
depends on mode
number and the state of contact (pinned or free). The model predicts a resonance frequency (
) for
the 8 µl single phase sessile droplets as ~25 Hz (actuation voltage frequency) which has been experimentally confirmed in our previous study [9]. In the limit of very small oil volumes, where most of the oil exists in the cloaking film, the equation presented above can be used directly by considering the thin oil layer cloaking the water to be a part of the core droplet interface. When the core interface is cloaked by a thin layer of oil, the effective interfacial tension of the compound thin film is given by [50] =
+
+
8
4
The contribution of the third term in Equation 8 is negligible, and the calculated value of
is 52.4
mN/m. For the opposite limit where the droplet is completely submerged in a large volume of oil, we extended the model to account for the oil medium (see Supplementary Information Section (C.1)) and we obtainedtanh( ℎ ) 9 + tanh( ℎ ) is density of oil. To cover the intermediate oil volumes, we Ω = where
is density of water and
propose an empirical approach where the surface tension term in the numerator and the density term in the denominator is scaled linearly between the two extreme cases. For compound droplets, the resonance frequency is then given bytanh( ℎ )
Ω ∝ where effective surface tension is given as
10
=
+ (1 − ℎ /ℎ)
. Here ℎ is the ridge height
where the oil shell transitions from bulk to thin film as shown in Figure 1. This approach is justified by the fact that the core droplet interface consists of two sections, one with oil-water interfacial tension
and the other consist of the thin oil film cloaking over the core droplet with water-oil-air
interfacial tension
. Similarly, the effective density is approximated as
=
+
(ℎ /ℎ ) tanh ( ℎ). This simplification is justified as only a part of the core interface (up to the ridge height) sees the increased inertial effect due to the shell.
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The model presented above is used to explain the shift in resonance frequency in the measured responses as seen in Figure 8. For calculating the resonance frequencies from the model, we used ridge height (ℎ ) values measured from the side view images of the compound droplet. Since this is an empirical approach a fitting parameter with a value of 1.15 was used to obtain a good match with the experimental data. For the case of small oil volumes, the presented model simplifies to the model presented by Noblin et. al. [51] with
as the effective surface tension for the droplet. Using
the fitting parameter obtained during fit, we calculate the resonance frequency (actuation voltage frequency) for the limit of very small oil volumes (i.e. ℎ → 0) as 28.5 Hz. In the opposite limit where the water core is completely engulfed in oil (i.e. ℎ = ℎ), our model predicts the resonance at an actuation frequency of ~17.4 Hz. In order to explore the limiting conditions, we performed additional experiments. For ℎ → 0, we used an oil volume of ~1.7 ± 0.3μl, such that it just cloaks the water core. Further reducing the oil volume had challenges in repeatable dispensing of volumes less than 1μl. In this case, the observed damped resonance frequency was ~28 Hz which is comparable to 28.5 Hz predicted by the model. Droplet oscillations were also performed for a larger oil volume of ~37 ± 3μl, such that ℎ = ℎ, i.e. the water core is fully enclosed by the oil drop. The measured resonance frequency in this case was ~20 Hz.
Effect of Shell Losses As seen in Figure 8 (left), damping causes the measured resonance frequencies to shift to lower values. Calculation of exact contribution of damping losses is complicated and hence we extract the damping ratio ( ) from the fits to our experimental results as tabulated in Supplementary Information Section (D.2). With damping ratio value less than 1 the compound droplet oscillations are underdamped. An increase in the losses with increase in oil volume was observed as seen through the reduction of Q factor values. This leads to decrease in the peak amplitudes at resonance frequencies with increase in oil volume as seen in Figure 8 (right). We distinguish three regions in the compound droplet where the viscous losses are taking place. First dissipation component is due to loss in bulk core (water) droplet. Similarly, another component of equal order of magnitude will be due to bulk loss in the oil shell. The third component is associated with losses in the draining of the thin oil film entrapped between the substrate and the advancing water interface [48].
The oscillations of shell contact line were largely damped due to surface pinning and hysteresis. As shown in Figure 5, the displacement of shell contact line that of the core base radius(
>
was significantly less when compared to
). The amplitude for the oil shell contact line motion was
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measured to be 8 times smaller than the amplitude of the core for 3.6 µl shell volume. As the oil volume increased, the amplitude of the oil triple line motion decreased and it was nearly pinned to the surface for larger oil volumes (as seen in Figure 5 for the case of 11.4 μl and 25.4 μl shell volumes). The weak response of the shell (oil-air-substrate) contact line is primarily attributed to the large hysteresis 17° with advancing angle 45° ± 3° and receding angle 26° ± 3°. The actuated oilwater interface drives fluid flow both in the core and shell. This induced oil flow leads to the motion of the oil-air interface. As seen in the derivation given in Supplementary Information Section (C.1), oil velocity normal to the interface decays exponentially with increase in distance from the interface (
,
=(
/
)∝
). With increasing oil volume, the distance between the oil-air interface
and the water-oil interface increases, leading to a reduction in the displacement of the oil-air interface. Reduced amplitude of oil-air interface in addition to the strong hysteresis resulted in ) from ~0.07
change of oil displacement (
) ~0.04
for 3.6 µl oil shell volume to (
11.4 µl oil shell volume and for larger oil shell volume (25.4 µl) it become negligible i.e.,
for
~0
implying that for higher oil volumes the oil shell transitioned from moving contact line oscillations to pinned contact-line oscillations.
3. Non-Axisymmetric Oscillations of Sessile Compound Droplet Non-Axisymmetric Mode Shapes Upon actuation with higher voltages (88Vrms), the non-axisymmetric modes were excited. The nonaxisymmetric oscillations were found to occur at half the mechanical excitation frequency i.e. at half the oscillation frequency of axisymmetric modes. Thus, these modes were subharmonic in nature. In our previous paper
[9]
, we have modelled these non-axisymmetric modes for single droplets as
parametrically coupled to the axisymmetric modes where the resonant frequency of the nonaxisymmetric modes was shown to be a function of time. These non-axisymmetric oscillations are driven by the energy transfer from the axisymmetric modes, in resemblance with a classical parametric oscillator system. As the parametric coupling between the two modes remain unchanged in presence of an oil shell, the non-axisymmetric mode in our case mostly arises from the parametric coupling with the axisymmetric mode. The dynamics of the core undergoing non-axisymmetric mode 2 oscillation is given bÿ
,
+
̇
,
+ Ω∗
,
+ ℋ
̇
,
=0
11
where is the viscous damping coefficient for the non-axisymmetric modes, Ω∗ is the resonant frequency of the non-axisymmetric mode, ℋ is the hysteresis loss term and sign of interface velocity ̇
,
̇
,
returns the
. Using similar arguments as above, the non-axisymmetric resonant
frequencies can be given by Ω∗ ∝
/(
∗ ),
where
∗
is the equatorial perimeter of the
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Langmuir
compound droplet as observed in the top or bottom view. The periodic variation in the droplets radius (
,
) due to the driven axisymmetric modes leads to a periodic variation in the resonant
frequency of the non-axisymmetric modes, which is stated as Ω∗ = Ω∗
where
1−
3 12
, the displacement of core radius, is modulated with twice the actuation voltage
frequency, thereby giving the expression for time modulated resonance frequency as Ω∗ = Ω∗ [1 + cos(2
)]. Here
is the amplitude of modulation, which is proportional to ∝ 3
/ .
This leads to a form of Mathieu equation resembling to a classical parametric oscillator system. The solution of this equation provides regions of instabilities in ( , ) space where the non-axisymmetric modes are able to grow due parametric coupling with the axisymmetric modes. Therefore, the nonaxisymmetric modes appear only within a specific range of voltage and frequencies.
The non-axisymmetric oscillations of the compound droplet were studied for actuation voltage of 88 Vrms by varying the actuation voltage frequencies between 10 Hz – 40 Hz. The non-axisymmetric modes were predominantly observed to appear between 20 Hz and 30 Hz. The droplet undergoing non-axisymmetric oscillations contains two mode components [9]: i) mode 0: having a axisymmetric shape pattern for droplet equator; and ii) non-axisymmetric mode 2: non-axisymmetric shape pattern along the droplet equator with lobe formation
[52]
degenerate modes corresponding to the sectoral harmonics
. These non-axisymmetric modes are
[6]
in spherical harmonic functions. For
non-axisymmetric modes at low shell volumes, the core gets constrained by the shell because the shell interface does not deform as much as the core interface. This interaction between the two interfaces restricts the expansion of the core leading to formation of bulges in the core at the lobe tips during the maximum stretch position, as seen in Figure 9 for 3.6 μl shell volume. At higher oil volumes, the interaction of the water-oil core interface with the oil-air shell interface is absent and we observe unconstrained normal lobe patterns which are similar to the case of sessile droplets without a shell.
Similar to axisymmetric modes discussed above, the oil boundary also showed significantly smaller oscillations. The oil triple line displacements decreased with increasing oil volumes. Significantly larger displacements were observed for 3.6 μl shell volume than for 25.4 μl shell volume. This is attributed to the reduced coupling (as explained for axisymmetric modes) between the interfaces as the distance between them increased. Another interesting aspect is that the oil triple line fails to completely recover from the displacements from the previous cycle and hence the oil shape
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Langmuir
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resembles that of a higher mode (mode 4). This is due to the high contact angle hysteresis and low surface tension of the oil-air interface both of which hinder the recovery of the oil-air interface.
Effect of Shell Volume on Frequency Regime and Mode Amplitude The normalized non-axisymmetric oscillation amplitudes for different shell volumes have been plotted in Figure 10 (left). The amplitudes of mode 2 were found to decrease with increase in shell volumes at a particular frequency and prominent modes were observed within 20 Hz - 30 Hz frequency range. By changing shell volumes, we observed that frequency range for obtaining nonaxisymmetric modes also shifted. The frequency range for obtaining significant non-axisymmetric oscillations has been plotted against different shell volumes in Figure 10 (Right). Non-axisymmetric modes were observed to appear inside the hatched shaded region. The non-axisymmetric modes for smaller shell volumes (3.6-5.9 μl) were observed between 25 Hz and 30 Hz actuation. At 8.6 μl shell volume (which is equal to the core volume), the non-axisymmetric modes were observed for widest range of frequency i.e., between 20 Hz and 30 Hz. For 11.4 μl, non-axisymmetric modes were observed at a lower frequency range of 20-25 Hz. This frequency range corresponds to parametric resonance near Ω∗ and is given by Ω∗ 2
1−
5 24
