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Droplet Oscillation as an Arbitrary Waveform Generator Kyle Yu, Jinlong Yang, and Yi Y. Zuo Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01059 • Publication Date (Web): 30 May 2018 Downloaded from http://pubs.acs.org on May 30, 2018
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Droplet Oscillation as an Arbitrary Waveform Generator Kyle Yu, Jinlong Yang, and Yi Y. Zuo * Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, HI, 96822, USA *Corresponding author Mailing address: 2540 Dole St, Holmes Hall 302, Honolulu, HI, 96822, USA. Phone: 808-9569650; Fax: 808-956-2373; E-mail:
[email protected].
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Abstract Oscillating droplets and bubbles have been developed into a novel experimental platform for a wide range of analytical and biological applications, such as digital microfluidics, thin film, biophysical simulation, and interfacial rheology. A central effort of developing any droplet-based experimental platform is to increase the effectiveness and accuracy of droplet oscillations. Here, we developed a novel system of droplet-based arbitrary waveform generator (AWG) for feedback controlling single droplet oscillations. This AWG was developed through closed-loop axisymmetric drop shape analysis (CL-ADSA) and based on the hardware of constrained drop surfactometry (CDS). We have demonstrated the capacity of this AWG in oscillating the volume and surface area of a millimeter-sized droplet to follow four representative waveforms, sine, triangle, square, and sawtooth. The capacity of oscillating the surface area of a droplet across the frequency spectrum makes the AWG an ideal tool for studying interfacial rheology. The AWG was used to determine the surface dilational modulus of a commonly studied nonionic surfactant, dodecyldimethylphosphine oxide. The droplet-based AWG developed in this study is expected to achieve accuracy, versatility, and applicability in a wide range of research areas, such as thin film and interfacial rheology.
Keywords: Droplet oscillation; Arbitrary waveform generator; Axisymmetric drop shape analysis; Constrained drop surfactometry; Interfacial rheology
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Introduction Oscillating droplets and bubbles have been developed into a novel experimental platform for a wide range of analytical and biological applications, such as digital microfluidics,1 signal encryption,2 thin film,3 smart materials,4 biosensors,5 biophysical simulations,6 and interfacial reactions and rheology.7 Popularity of the oscillating droplet/bubble platform is multifactorial.8 First, the droplet platform offers a large surface-area-to-volume ratio which increases the rate of surface reaction and interfacial assembly. Second, the small sample consumption facilitates the study of expensive chemicals and scarce biological fluids. Third, the miniaturized droplet platform allows precise environmental control, thereby permitting versatile studies across the droplet surface. A central research effort of developing any droplet-based experimental platform is increasing the effectiveness and accuracy of droplet oscillations.9 Effective control of droplet oscillation is not trivial and often requires sophisticated mechanisms. Most existing droplet oscillation methods are through preprogrammed fluid actuators, such as motorized syringe pumps or piezoelectric micropumps.10 However, these methods directly control only drop volume but not its surface area. For micron-sized droplets with small Bond numbers, the surface area can be approximated from the volume as the droplets can be considered as spheres.11-13 However, when oscillating millimeter-sized droplets with sufficiently large Bond numbers, surface area cannot be approximated from volume due to droplet deformations.14-16 Precise control of surface area oscillations is an essential requirement for many analytical studies of interfacial phenomena, for instance, interfacial rheology, which studies the viscoelastic properties of thin-film materials assembled at the liquid-fluid interface.10, 17 Understanding the dilational rheological behaviors of thin-film materials made of surfactants, proteins, lipids,
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polymers, and colloidal particles is important for studying monolayers, foams, emulsions, and other soft matters.7, 17-18 The surface dilational modulus of a monolayer can be measured by oscillating the surface area with a harmonic motion, i.e., sinusoidal oscillations with a constant amplitude and a constant frequency, and then comparing to the resultant surface tension response.7 A number of techniques have been developed to determine the surface dilational modulus, such as oscillating barriers,19 capillary waves,20 and most importantly, oscillating droplets and bubbles.7, 9, 21-23 However, to the best of our knowledge, none of these techniques are capable of directly controlling the harmonic oscillation of the surface area of droplets and bubbles. Here, we developed a novel arbitrary waveform generator (AWG) capable of oscillating the surface area, as well as the volume, of a millimeter-sized droplet. This AWG is based on closed-loop axisymmetric drop shape analysis (CL-ADSA), recently developed in our laboratory,15 in company with an experimental methodology called the constrained drop surfactometry (CDS).24 CL-ADSA is a feedback control system which allows the direct manipulation of the volume, surface area, and surface tension of a droplet by real-time analysis and control.15 To demonstrate the feasibility of this AWG, we first showed its capacity in oscillating the volume or surface area of a water droplet to four representative waveforms, i.e., sine, triangle, square, and sawtooth waveforms. Subsequently, we applied the droplet-based AWG to study the interfacial dilational rheology of a commonly studied surfactant, dodecyldimethylphosphine oxide (C12DMPO).25 Our studies demonstrated that this AWG is capable of precisely controlling the oscillation of volume or surface area of a droplet to follow theoretical waveforms at various frequencies and amplitudes. Therefore, the AWG is ideal for
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studying surface phenomena such as interfacial rheology, where harmonic oscillation of the surface area is a central requirement.
Experimental Section Constrained Drop Surfactometry (CDS). The CDS is a new generation of droplet-based experimental platform, developed in our laboratory, for studying interfacial phenomena. Detailed description of the CDS can be found elsewhere.24 Briefly, as shown in Figure 1c, the CDS used a 3-mm sessile drop as the reaction vessel. The air-water or liquid-liquid interface of the sessile drop was monitored for molecular self-assembly and interfacial reaction. The sessile drop was “constrained” on a carefully machined droplet pedestal using a sharp knife-edge to prevent film leakage even at very low surface or interfacial tensions. Droplet oscillation was performed by a motorized syringe with a displacement resolution of 0.1 µm, equivalent to a volume resolution of approximately 4 nL.15 Physical properties of the droplet, including its volume, surface area, and surface tension, can be simultaneously determined from the shape of the droplet using axisymmetric drop shape analysis (ADSA).26 Development of the Arbitrary Waveform Generator (AWG). As shown in Figure 1a, an arbitrary waveform was first discretized into segments with time intervals as small as 0.2 s, bottlenecked by the frequency of the servo motor. Within each discretized interval, as shown in Figure 1b, the volume or surface area of the droplet was feedback controlled through CLADSA.15 CL-ADSA controlled the physical properties of the droplet, including its volume, surface area, or surface tension, through proportional-integral-derivative (PID) control, described in Eqs. 1-3,
= + +
(Eq. 1)
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= + + = +
(Eq. 2)
+
(Eq. 3)
where kp is the proportional gain, ki is the integral gain, kd is the derivative gain, Ts is the sampling time, e is the error, and u is the command signal, expressed as a function of t in the time domain, as a function of s in the Laplace domain, and as a function of z in the Z domain. To minimize oscillatory transients, a fuzzy control component was added in parallel to the PID control. In addition, for periodic waveforms, the PID gains and the fuzzy sets can be automatically tuned from previous periods to mitigate errors for upcoming periods, thus achieving adaptive control. Subsequently, as shown in Figure 1d, CL-ADSA controlled the oscillation of the droplet within the targeted interval, as a part of completing the waveform. To prove the feasibility of the droplet-based AWG, we controlled the oscillation of either volume or surface area of a water droplet (Millipore, Billerica, MA) to follow four representative waveforms, i.e., sine, triangle, square, and sawtooth waveforms, as described in Eqs. 4-7, & sin*+ + ,A"#$% = A + A & × +* + , − A./#0$12% = 3 + 4A
(Eq. 4)
7
=
82+ + , + :-−18 7 :
& × sgnBsin*+ + ,-C A"?@0/% = 3 + A
& × + H + , − *+ + , -I A"0DEFFEG = 3 + A 7
(Eq. 5) (Eq. 6) (Eq. 7)
where 3 is the reference value, 3J is the amplitude, ω is the angular frequency, and , is the
phase shift. All four parameters of 3 , 3J, +, and , can be controlled. These four waveforms represent the most fundamental periodic waveform functions. Without loss of generality, combinations of these four waveforms can produce any arbitrary waveform.27
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Measurement of the surface dilational modulus. The surface dilational modulus can be expressed as a complex property with the real part expressing surface elasticity and the imaginary part expressing surface viscosity.7 For elastic behavior, the surface stress is proportional to the relative surface area perturbation. For viscous behavior, the surface stress is proportional to the rate of the relative surface area perturbation.28 A general expression for the surface dilational modulus is shown in Eq. 8, N=
P O QR/QT
= N + U2V+W = NX + UN
(Eq. 8)
where N is the complex surface dilational modulus, YZ is amplitude of surface tension variations,
3 is the reference surface area, 3J is the amplitude of the surface area oscillation, N is the
dilational elasticity, + is the angular frequency of the harmonic surface area oscillations, η is the dilational viscosity, NX = N is the real part of the dilational modulus, and N = 2V+W is the
imaginary part of the dilational modulus. To demonstrate the capacity of the AWG in determining the surface dilational modulus, we studied a nonionic surfactant, dodecyldimethylphosphine oxide (C12DMPO, Sigma-Aldrich) at 0.22 mM, which is below the critical micelle concentration (CMC) of this surfactant (i.e., 0.3 mM).29 This surfactant concentration was selected to ensure a diffusion-controlled adsorption process and comparison with literature values.25 All measurements of the surface dilational modulus were performed at room temperature and were initiated only after adsorption equilibrium at the drop surface. The surface area of the surfactant film was first oscillated using the AWG to undergo an ideal harmonic perturbation, as described in Eq. 9, 3 = 3 + 3Jsin2V+
(Eq. 9)
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where A is the surface area, A0 is the initial reference surface area, 3J is the amplitude of surface
area oscillations, and + is the frequency. Then, the resulting harmonic response of the surface tension was determined and extracted from the regression described in Eq. 10, Y = Y + YZsin2V+ + ,
(Eq. 10)
where Y is the surface tension, γ is the equilibrium reference surface tension, YZ is the amplitude of surface tension variations, and ϕ is the phase shift between the surface area perturbations and the surface tension response. &, and + were controlled using the AWG, and Y, YZ and ϕ During the experiments, A0, A were extracted through a Levenberg–Marquardt curve fit (OriginPro, Northampton, MA) of the measured surface tension response. The complex surface dilational modulus, and the elastic and viscous moduli of the surfactant film were determined with Eqs. 11-13. |N| = 3
P O QR
(Eq. 11)
P O
(Eq. 12)
P O
(Eq. 13)
|NX | = 3 R cos, Q
|N | = 3 R sin, Q
Results and Discussion Droplet-based AWG. Figure 2 demonstrates the capacity of the AWG in oscillating the volume and surface area of a water droplet in four representative waveforms, i.e., sine, triangle, square, and sawtooth. In the volume control shown in Figure 2a, the droplet was oscillated around a reference volume of 15 µl, with an amplitude of 3 µl (i.e., 20% of the reference volume), and a frequency of 0.05 Hz (i.e., a period of 20 s). In the surface area control shown in Figure 2b, the water droplet was oscillated around a reference area of 0.25 cm2, with an amplitude of 0.05 cm2 (i.e., 20% of the reference area), and a frequency of 0.05 Hz. For comparison, the theoretical 8 ACS Paragon Plus Environment
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waveforms, as described by Eqs. 4-7, are also shown in Figure 2 in red lines. It can be seen that relatively large deviations only occur during the square waveform control where the AWG has difficulties following the instantaneous jumps in volume and surface area. Except for square waveforms, both the controlled volume and surface area, at the studied amplitude and frequency, match the theoretical waveforms very closely. Videos S1-4 in the Supporting Information (SI) demonstrate the dynamic process of the AWG for controlling the surface area of the water droplet oscillated in sine, triangle, square and sawtooth waveforms, respectively. It is not unexpected that the accuracy of the AWG depends on the amplitude and frequency of the controlled waveform. Figures S1-4 in SI show the waveform dependence on amplitude and frequency. Figure 3 summarizes the accuracy of the AWG using the coefficient of determination (R2), evaluated with the Levenberg–Marquardt curve fitting, as a statistically significant quantitative measure of the regression. For both volume and area control, we studied two amplitudes, i.e., 10% and 20% of the reference value, and six frequencies, i.e., 0.01 Hz, 0.025 Hz, 0.05 Hz, 0.1 Hz, 0.2 Hz, and 0.5 Hz. Several conclusions can be drawn from Figure 3. First, the R2 value is less sensitive to the amplitude in comparison with the frequency. Second, the R2 value generally decreases with increasing frequency. Third, different waveforms show different sensitivities to the increase in frequency. Compared to sine and triangle waveforms, the square and sawtooth waveforms are more susceptible to increasing frequency. Given a R2 value of 0.9 to represent an acceptable waveform tracking, both sine and triangle waveforms maintain a satisfactory accuracy even at the highest frequency tested, i.e., 0.5 Hz. However, the square and sawtooth waveforms only maintain a satisfactory accuracy at a frequency of 0.1 Hz and below. Particularly, it is found that accuracy of the sawtooth waveform decays rapidly when the frequency is increased above 0.1 Hz.
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This is mainly due to the lack of odd symmetry in comparison with the other three tested waveforms. Taking the AWG as a free-standing technique for droplet oscillation and fluid actuation, it is therefore safe to conclude that its limiting frequency for generating any arbitrary waveform is between 0.1 and 0.5 Hz. This frequency limitation is close to the theoretical maximum frequency at which any drop/bubble shape analysis technique is still valid. This maximum frequency is limited by deviations of the drop/bubble profile from the classical Laplacian shape that is determined by the balance between the surface tension forces and gravity. Leser et al. estimated this theoretical limitation of ADSA to be 1 Hz, above which the drop profile is significantly deviated from the Laplacian shape, thus failing ADSA calculation.30 When oscillated with higher frequencies, the drop/bubble profiles are affected by two additional hydrodynamic forces, i.e., the inertial forces and the viscous forces. Freer et al. studied the effects of these two forces, and concluded the viscous effect is only negligible when the Capillary number (Ca) is less than 0.002.31 Karbaschi et al. also estimated that the hydrodynamic effect on drop shape analysis is only negligible when the Reynolds number (Re) is less than 21.32 It should be noted that all these reported limitations in the oscillation frequency, Capillary number, and Reynolds number were determined for aqueous fluids (i.e., viscosity close to 1 cP). These threshold limit values were found to be significantly lower for viscous fluids.30-32 Table 1 summarizes the dimensionless parameters relevant to the AWG at the angular frequency of 0.1 Hz. Both the Capillary number and the Weber number are much smaller than 1. The Reynolds number, being a ratio of the Weber number to the Capillary number, is in the order of magnitude of 10. All these dimensionless parameters confirm the validity of the droplet-based AWG. Thus, the drop profile is only determined by the balance between the surface tension
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forces and the gravity, as predicted by the Bond number or a newly defined variation of the Bond number called the Neumann number.26 Significantly higher frequency of surface oscillation can be achieved by the oscillating bubble method with a piezoelectric actuator under the assumption of a spherical bubble shape.25 Surface dilational modulus. After proving the feasibility of the AWG, we engaged the AWG in studying the interfacial rheology of a widely studied surfactant film, i.e., 0.22 mM C12DMPO. Figure 4a demonstrates a typical experiment of measuring the surface dilational modulus of the surfactant film. The surface area of the droplet was harmonically oscillated in the sine waveform around a reference value of 0.2 cm2, with an amplitude of 10% of the reference value and a frequency of 0.1 Hz. The theoretical sine waveform is shown in the red line for comparison. The surface tension response to the harmonic oscillation of the surface area was measured. The Levenberg–Marquardt regression of the surface tension response is shown in the blue line. Upon slow harmonic area oscillations (
