Article pubs.acs.org/Langmuir
Electrowetting of Partially Wetting Thin Nanofluid Films Monojit Chakraborty, Rahul Chatterjee, Udita Uday Ghosh, and Sunando DasGupta* Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721302, India ABSTRACT: It is observed that the presence of negatively charged, suspended nanoparticles significantly changes the electric-field-induced spreading and contact line dynamics of partially wetting liquid films. Image-analyzing interferometry is used to accurately measure the meniscus profile, including the spatial change in the meniscus curvature. The nanoparticlecontaining meniscus exhibits enhanced spreading with an increase in the particle size and weight fraction. The instantaneous contact line velocities are measured using video microscopy and a frame-by-frame analysis of the extracted images. The effects of electric field polarity reversal on the flow toward the contact line are explored as well. The movement of the meniscus is analyzed taking into account the capillary forces and Maxwell-stress-induced flows. An analytical model based on the Young−Laplace equation is used to analyze the electric-field-induced contact line motion, and the modelpredicted velocities are compared to the experiments. upon external perturbations, e.g., heat input.20 The resulting heat-transfer coefficient has also been theoretically estimated at the three-phase contact line region.21 The use of optical techniques, such as image-analyzing interferometry coupled with ellipsometry, has greatly improved the measurement capability of the ultrathin film thicknesses and the subsequent estimation of the shape-dependent intermolecular stress field.22,23 The effects of forced cycles of evaporation and condensation brought about by externally induced thermal perturbations on the film meniscus have also been investigated.24 The concepts of velocity slip as well as temperature jump at the solid−liquid interface have been introduced to accurately model evaporation of thin films.25 Molecular dynamics has also been used to examine nanoscale evaporating meniscus.26 Ma et al. has provided a comprehensive theoretical analysis27 outlining a solution of the thin-film profile, interfacial temperature, and heat-flux distribution. The concept of disjoining pressure coupled with the Kelvin−Clapeyron model has also been shown to capture the non-isothermal effects, as in the case of constrained vapor bubble (CVB) systems.28 It has been established that evaporation from the thin-film region of the extended meniscus29 can be enhanced by the introduction of nanoparticles,30,31 owing to the ordering of nanoparticles near the contact line, resulting in an excess pressure known as the structural disjoining pressure.32,33 The EWOD technique coupled with nanosuspensions also lead to suppression of the coffee stain effect, attributed to the EWOD [direct current (DC) potential]-generated electrophoretic forces 34 and internal flows [alternating current (AC)
1. INTRODUCTION Miniaturization has brought in new systems collectively known as microelectromechanical systems (MEMS) and, more recently, “lab-on-a-chip devices”.1−3 These dynamic fluid flow microsystems rely on manipulation of small quantities of fluid volume through channels or as individual droplets by controlling surface properties, precisely the surface energies. Electrowetting on dielectric (EWOD)4 achieves droplet manipulation through the use of an externally applied electric field to bring about a reduction in the effective solid−liquid interfacial tension, leading to an increase in wetting. Various other forms of energy, such as optical,5 acoustic,6 and thermal,7 can also be used to trigger and manipulate droplet motion. However, the use of electrical energy offers the advantages of availability of standard fabrication processes, ease of operation, control,8 and integration. Potential applications of these miniaturized devices range from biomedical diagnostics,9 disruption of human serum albumin (HSA) fibrils,10 variable focus lenses,11 microchip cooling12−14 and its enhancement,15 reflective display devices,16 etc. The state of research in droplet EWOD is presented in an excellent review paper by Mugele and Baret.4 However, there is a dearth in literature concerning EWOD of the extended meniscus of partially wetting thin films, which will have specific applications in microcooling devices related to cooling of hot spots. Bhaumik et al. demonstrated EWOD of partially wetting thin films under equilibrium and non-equilibrium conditions.17 In a subsequent work,18 EWOD of evaporating meniscus of thin films of partially wetting liquids were experimentally studied, highlighting appreciable mass flux enhancement. The physics of an extended evaporating meniscus was first analyzed by Potash and Wayner19, establishing that thin film evaporation is controlled by fluid flow, resulting from the change in meniscus curvature, which, in turn, strongly depends © 2015 American Chemical Society
Received: December 6, 2014 Revised: March 18, 2015 Published: March 19, 2015 4160
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Figure 1. Schematic of the experimental setup (reproduced with permission from ref 17, with minor modifications).
on flow toward the contact line can be analyzed and are the motivations for the present study.
frequency]35 that overcome the evaporation-based advection flows. Moreover, thermal conductivity and wettability of nanofluids are functions of the particle concentration,36 which may be modulated as per process requirements. Magnetic immunoassays have also been demonstrated using suspensions containing submicron magnetic particles on digital microfludic (DMF) platforms, which significantly lowered sample volume requirement.37,38 These thermophysical properties make this class of fluids an ideal choice for miniaturized systems besides their already existing wide range of applications, e.g., as coolant in chillers, solar water heaters,39 heat pipes,40 and fuel cells.41 The effect of particle addition on spreading during EWOD has been examined42 using submicron particle-laden sessile droplets. Besides an observed decrease in the contact angle with an increase in the particle concentration, the actuation voltages are lowered because of the altered effective permittivity of the system. The addition of nanoparticles showed an increase in electric-field-induced wetting behavior43 and absence of contact angle saturation.44 It was postulated that nanoparticle adsorption at the solid−liquid interface modified the interfacial energy.43 These studies point to the likely benefits associated with the usage of nanofluids in EWODbased platforms. The combined use of electrowetting and nanoparticle-laden films may provide an attractive strategy for enhancement of microscale transport processes occurring at the three-phase contact line region. It would thus be imperative to probe, both experimentally and theoretically, the changes in the wetting behavior, shape change of the meniscus, and contact line dynamics of partially wetting nanofluid films subjected to a small electric field. These can be accurately measured using non-obtrusive optical techniques and lead to the appropriate resolution of the interfacial force field capturing the underlying physics. Additionally, the interplay of the polarity of the applied electric field and the nanoparticle charge vis-à-vis their effects
2. EXPERIMENTAL SECTION 2.1. Experimental Setup and Materials Used. The schematic of the experimental setup is presented in Figure 1, which is similar to the one used in our previous experiments,17 consisting of a stainless-steel base and a top cover enclosing a silicon wafer (substrate) between Teflon gaskets and O-rings for sealing and isolating the liquid film. However, the major difference is the presence of negatively charged nanoparticles in varying concentrations and size in the partially wetting liquid film. The cell has provisions for entry of the working liquid and a platinum wire. The wire serves as an electrode, with the plate under the Si wafer serving as the other electrode, as portrayed in Figure 1. Cleaning plays a vital role in all of these experiments in terms of the quality and reproducibility of the data, and special care has been taken to standardize the cleaning procedure. Silicon wafers are prone to contamination, given its inherent high surface energy. The cleaning protocol followed herein comprises of dipping the wafers in piranha solution (30% dilute H2O2 and 98% pure H2SO4 in the ratio of 1:1), followed by repeated deionized water rinse and subsequent drying in a jet of pure nitrogen. It is to be emphasized that the entire cleaning and the assembly of the experimental cell parts are performed in a Class 100 laminar flow hood (MFD-V-W-2400, Micro Flow Devices India Private Limited) to reduce the chance of contamination and presence of dust particles, which can significantly alter the contact line dynamics because of contact line pinning and disturb the local film thickness in an arbitrary way. The working liquid consists of negatively charged nanoparticles (carboxylate-modified polystyrene beads commercially obtained from Sigma-Aldrich) suspension dispersed in an aqueous solution of 0.1 M sodium chloride and sodium dodecyl sulfate (SDS) [0.1 times its critical micelle concentration (cmc)]. Prior to the experiment, the suspension is sonicated in an ultrasonicator for 10 min. The addition of the surfactant (SDS) lowers the liquid interfacial tension (measured to be 33 mN/m using pendant drop method in a goniometer, 290-G1 Ramehart, Germany) and enhances the effect of applied electric fields.45 It also allows for easy viewing of the interferometric fringes for the magnifications used herein. On the other hand, the addition of 4161
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The flow of the current is not detected during the experiments because the potential difference is kept well below 12 V (the experimentally obtained breakdown voltage for the experimental surface). The entire movement of the film is recorded and used later to extract images for subsequent analysis of the advancing film at different instants of time as well as at its final equilibrium state. To investigate the effect of charged nanoparticles during electrowetting of a partially wetting liquid meniscus, a nanofluid comprising of negatively charged nanoparticles with a nominal diameter of 30 nm is added to the electrolyte−surfactant solution with a particle weight fraction of 0.05% (w/w). The particle concentration can also be expressed (as per information provided by the supplier) in number of particles per milliliter (N) as
nanoparticles in the concentration range used in this study (maximum of 0.5%, w/w) does not alter the surface tension of SDS-laden water in an appreciable manner, as confirmed by repeated measurements. However, the presence of nanoparticle influences processes near the contact line region46 and causes interfacial deformations of the meniscus (discussed in detail in section 3). The liquid, introduced through a microtube, forms a liquid pool with an extended curved meniscus, and the system is allowed to reach equilibrium before the application of the electric field. The entire setup is tilted at an angle of about 15° to create an extended, curved liquid meniscus. The movement of the liquid film is continuously viewed through the top glass cover using a LEICA DM-LM microscope at 20× magnification and a charge-coupled device (CCD) camera (LEICA DFC290) attached to the microscope. Each pixel of the digitized image represents the average reflectivity of a region of 0.29 μm diameter for the magnification and camera settings used in these experiments. Monochromatic light of wavelength (λ = 543.5 nm) is used throughout the experiments. Interferometric fringes are readily observed because of the constructive and destructive interference of the reflected light from the curved liquid meniscus and the solid silicon surface. The reflectivity profile of the bare substrate is measured and used as a reference for subsequent calculations. The dynamics of the meniscus as a function of the applied voltages are measured by capturing the video of the advancing meniscus between the equilibrium states using a CCD camera. Xilisoft HD Video Converter (version 5.1) is used to extract images at regular time intervals, enabling frame-by-frame analysis of the advancing meniscus. The velocity of the contact line is measured using a central difference scheme.17 Gray values of the digitized images are used to evaluate the film thickness profiles, its slope, and curvature profiles.17,24 2.2. Image Analysis. Image-analyzing interferometry22 is used to measure the meniscus thickness profile and the curvature using methodologies described earlier.17,23,24 The images of the meniscus showing the interference fringes are analyzed using Image-Pro Plus software (version 6.0). To relate the local film thickness with the local light intensity, each image is digitized into 640 (horizontal) × 480 (vertical) pixels, which act as individual light sensors, and is assigned a gray value between 0 (black) and 255 (white). From the line profile analysis of each image, a plot of the pixel gray value (G) versus pixel position, x, is extracted. Using the interpolated peak/valley envelopes and analyzing the relative reflectivity, G̅ , of each pixel with respect to the envelopes, a film thickness at every pixel is determined.22 The basic relation that is used to connect the film thickness with the gray values and other parameters, such as λ, the wavelength of the monochromatic light, nl, refractive index of the liquid, and β and κ, which are functions of the refractive index of solid, liquid, and vapor, is provided in eq 1.
δ(x) =
⎡ β + κ(1 − 2G̅ (x)) ⎤ λ cos−1⎢ ⎥ ⎣ β(2G̅ (x) − 1 − κ ) ⎦ 4πnl
N=
(6 × 1010)S PL πPS d3
(2)
where S is the percentage of solids (w/w), d is the diameter (μm), PS is the density of the bulk polymer (g/mL) of 1.05 g/ mL (polystyrene), and PL is the density of latex (g/mL) of 1.005 g/mL. The nanoparticles are carboxylate-modified, implying that their surfaces carry a net negative charge,44 as discussed in the Experimental Section. The nanofluid meniscus is subjected to the same applied voltage (3 V), and the contact line motion during electrowetting is monitored and compared to that of the same liquid without the nanoparticles. Whenever a wetting or a partially wetting liquid comes in contact with a solid substrate, an extended meniscus is formed. This extended meniscus is traditionally divided into three zones, namely, an adsorbed flat thin film, where intermolecular forces predominate over surface forces, a capillary meniscus at the thicker end having a constant curvature governed principally by surface tension forces, and a region bridging these two, known as the transition zone, where both forces are active. The effects of an external force field can be broadly studied by examining the changes in each of these individual zones as reactions to the induced stress. The augmented Young−Laplace equation is normally used to express the pressure jump across the liquid−vapor interface at equilibrium as Pl − Pv = −σK −
(1)
∏
(3)
where
The detailed methodology to obtain the spatial variation of the thickness profile from the measured gray values at every pixel is available in the literature17,23,24 and is not reiterated herein. The errors associated with measuring the thickness in the capillary and transition zone are estimated to be ±0.01 μm and about ±10% for the adsorbed region. The curvature varies from a value equal to zero at the adsorbed end to a constant value at the thicker end (capillary part) of the meniscus, along with the possibility of a maximum between them. The sensitivity of the film thickness (especially in the adsorbed region) and the importance of the shape of the profile as an indicator of the intermolecular force field have been discussed in detail in a number of related publications.23,47
K (curvature) =
d2δ /dx 2 dδ /dx + [1 + (dδ /dx)2 ]3/2 x[1 + (dδ /dx)2 ]1/2
≈
d2δ (because the slope is quite small herein) dx 2 (4a)
∏
(disjoining pressure) =
−B δn
(4b)
In these equations, δ represents the film thickness, σ is the surface tension, and B is the modified Hamaker constant (B < 0 for completely wetting systems). In the limit of very thin films of pure simple fluid (non-retarded region), n = 3 and B = A/6π, in which A is the classical Hamaker constant, while in the thick film region (retarded region), n = 4 and B is a retarded dispersion constant.48 For the range of adsorbed films encountered in this study, the use of the retarded dispersion constant is more appropriate. The additional stress because of
3. RESULTS AND DISCUSSION 3.1. Effect of Negatively Charged Particle Addition on Contact Line Dynamics. An extended meniscus of an aqueous solution of 0.1 M sodium chloride and SDS (0.1 times its cmc) is subjected to a potential difference of 3 V, which causes immediate spreading of the partially wetting film. 4162
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Langmuir the applied electric field has a non-vanishing component, known as Maxwell stress, which is instrumental in the motion of the contact line during spreading. Thus, during electrowetting, the effective pressure jump near the adsorbed thin film region comprises of the disjoining pressure, attributed to the intermolecular forces, and the electrostatic pressure because of Maxwell stress. Equation 3 can be written for a point in the capillary meniscus and for another point in the transition region, and because, at equilibrium, the effective liquid pressure (Pl) is same everywhere, the following relation is obtained σ
d2δ B − 4 = σK ∞ 2 δ dx
(5)
where K∞ is the constant curvature at the capillary end of the meniscus. It has been experimentally observed that electricfield-induced spreading is associated with a reduction in the curvature at the capillary end of the meniscus but the adsorbed layer thickness remains unaffected.17 Since the effect of Maxwell stress is localized (over a distance ∼20 Å), it can be treated as a point force.49,50 The net effect of the electric field can be visualized as a horizontal point force pulling the three phase contact line along the adsorbed layer without altering its thickness.17 The above mechanism is distinct from that of normal spreading, where the advancing front is accompanied by an increase in the thickness of the adsorbed layer. The effect of Maxwell stress is thus implicitly accounted for in the model equation through the change in the values of K∞. The addition of negatively charged nanoparticles attracts counterions in the electrolyte solution, and a localized cloud of positive counterions forms around the surface of the negatively charged particles. Additional experiments are performed for an in-depth analysis, and the results are discussed in detail in section 3.3.1. The migration of these anions is governed by the polarity of the electrode, and because the platinum wire is connected to the negative end of the DC source (configuration 1, Figure 2), they drift toward the contact line in the relatively
Figure 3. Particle addition effect on an electrowetted (applied voltage of 3 V) thin film [A, without particle; B, with 30 nm, 0.05% (w/w) particle] (configuration I). A ±5−8% error is associated with each experimental value.
velocities (the peak of the contact line velocity is found to be almost doubled in the presence of nanoparticles). To summarize, the alteration in the meniscus shape is primarily governed by this electric-field-induced Maxwell stress and the presence of nanoparticles near the three-phase contact line.46 These complex effects are difficult to explicitly incorporate in a model. However, these effects are manifested through a change in the shape of the liquid meniscus, e.g., the value of the constant curvature at the capillary end of the meniscus and the maximum curvature near the three-phase contact line. This change in meniscus curvature has been measured experimentally. Thus, the effects of Maxwell stress and nanoparticles are implicitly accounted for in the model equation. 3.2. Effect of Interchanging Circuit Polarity. In configuration II (Figure 4), the polarity of the system has
Figure 2. Schematic of the advancement of nanofluid films (configuration I, favorable). The figure is not drawn to scale.
Figure 4. Schematic of the advancement of nanofluid films (configuration II). The figure is not drawn to scale.
thinner part of the meniscus. In this process of migration, the anions drag the adjacent fluid along with it, generating an additional flow, which augments the already existing fluid flow generated by the Maxwell stress. Thus, the combined effects of these two phenomena result in an enhanced flow toward the contact line region with an associated increase in the contact line velocity. The contact line velocities are measured using a frame-by-frame analysis of the recorded contact line motion and are presented in Figure 3, clearly showing the effect of the negatively charged nanoparticles in amplifying the contact line
been reversed such that the platinum wire is now connected to the positive terminal of the DC source while keeping the nanofluid properties and electric field parameters unaltered. The goal is to evaluate the contact line dynamics when the free (excess) negative ions in the electrolyte are attracted toward the electrode in the capillary part of the meniscus. The motion of the anions toward the capillary part of the meniscus (as schematically presented in Figure 4) drags the adjacent fluid layer along with them, creating a flow in a direction opposite the Maxwell-stress-induced flow toward the contact line. 4163
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Langmuir The reduced flow toward the contact line (in comparison to the case depicted in Figure 3, configuration I) has resulted in slower movement of the contact line, as displayed in Figure 5.
experiments, reported in section 3.3.1, corroborate this hypothesis. 3.3. Effect of Particle Characteristics on Contact Line Dynamics. 3.3.1. Effect of the Particle Size Variation. Electric-field-induced spreading of dilute [0.35% (w/w) particle concentration] suspensions containing three different size particles, namely, 30, 48, and 100 nm, are studied at a constant applied voltage of 3 V. The variation of the contact line velocities with the particle size is depicted in Figure 6.
Figure 5. Polarity reversal effect (applied voltage of 3 V) [A, without particle; B, with 30 nm, 0.05% (w/w) particle in configuration II; and C, with 30 nm, 0.05% (w/w) particle in configuration I]. A ±5−8% error is associated with each experimental value.
Figure 6. Relative comparison of contact line velocity to the variation of particle size with a fixed particle loading of 0.35 wt % (A, without particle; B, 30 nm particle; C, 48 nm particle; and D, 100 nm particle). A ±5−8% error is associated with each experimental value.
Additionally, a relative comparison of the time-averaged contact line motion is presented in the inset of Figure 5. It is to be noted here that the contact line velocity for the configuration II case is still larger than that of the control experiments, i.e., the electric-field-induced spreading of the partially wetting film in the absence of nanoparticles. The well-established concepts of spreading and adhesion of simple liquids do not apply to fluids containing particles of nanometer dimensions.30 It has been proposed that the solid-like ordering of suspended spheres occurs in the confined three-phase contact region at the edge of the spreading fluid, which tends to become more disordered and fluid-like toward the bulk phase. The pressure arising from such colloidal ordering in the confined region also enhances the spreading behavior of nanofluids. Chakraborty et al.42 performed experiments to measure the variations in the contact angles of water droplets containing homogeneously dispersed submicron particles under the influence of external electric potential in EWOD configuration. They have shown that the presence of the submicron particles increases the effective permittivity [about 20% increase at 210 V when 53 nm, 0.005% (v/v) particles are added to a sessile water drop], resulting in larger variation in the contact angle, as compared to a pure water droplet subjected to the same voltage. This may explain the enhanced spreading and, hence, the higher contact line velocity in the presence of nanoparticles, as compared to EWOD of partially wetting liquid films without any nanoparticles under identical electrowetting conditions. Moreover, the particles used herein differ only in the diameter but carry the same charge. If the imposed polarity at the solid−liquid interface is the same as that of the particles, it results in repulsion of the particles, and they start moving toward the contact line with higher velocity, dragging the liquid along with it. A reduction in the contact line velocity would result if the polarity is opposite that of the particles, because the resulting flow will be opposite to the Maxwell stress directed flow toward the contact line. The results of additional
Introduction of electrolyte and the addition of charged particles in an electrolyte suspension results in the dissociation of the particle surface functional group as well as the electrolyte.44 The dissociated counterions become attached to the particles that are responsible for the decreased availability of free ions in the solution. Nanoparticles interact with their local environment, resulting in significant variations in the distribution of ions around them.51 The surfaces of the nanoparticles are screened by the counterions, which, in turn, results in variation in the concentrations of ions near the nanpoparticle surfaces. Measuring the exact number and concentration distribution of ions around the nanoparticles is an involved excercise51 and is beyond the scope of the present work. However, the decrease in the availability of free ions in the solution is macroscopically manifested by a decrease in the conductivity (measured using Multi-Parameter TESTR 35 Series, Eutech Instruments) and ζ potential (measured using Malvern NanoZS, Germany). A decrease in ζ potential, in the presence of electrolyte, is observed (Table 1). It is intuitive that a larger number of counterions will be attached to the particle with an increase in the particle diameter [while keeping the weight fraction of the particle constant at 0.35% (w/w)] to neutralize the surface potential of the charged particle. Thus, the availability of the free pair of ions in the solution decreases, resulting in a decrease of the measured values of conductivity with an increase in the particle diameter, as shown in Table 2. The meniscus thickness profiles of the partially wetting films in the presence of particles are evaluated using the methodology described in section 2.2 and presented in Figure 7. It can be clearly seen from these profiles that the film becomes steeper with the introduction of nanoparticles as well as with the 4164
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Langmuir Table 1. Variation of the ζ Potential of the Particle with Size ζ potential (mV) particle size (nm)
(particle + 0.1 cmc SDS) solution
(particle + 0.1 cmc SDS + 0.1 M NaCl) solution
30 48 100
−43.5 −48.6 −54.5
−40.8 −43.4 −49.7
field of 3 V are also evaluated, and they clearly show a maximum (Figure 8) near the contact line region (Kmax) and a constant curvature at the capillary meniscus region (denoted by K∞), which is consistent with results reported in the literature.17,23 To delineate the effect of nanoparticles only on meniscus deformation, experiments are performed in the absence of an electric field. It is apparent from Figure 8a that the introduction of particles affects the thin-film curvature even in the absence of an electric field, whereas the results presented in Figure 8b clearly indicate the additional increases in the curvatures (Kmax and K∞) when an electric field is imposed.17 The addition of particles to a sessile droplet can alter the contact angle, generally showing an increasing trend with an increasing particle size.52 It has also been reported that a very small increase in the slope of the extended meniscus (indicative of a change in the apparent contact angle) may result in significant increases in the curvature of the capillary meniscus region.20,24,25,53 Thus, even a microscopic change in the meniscus shape near the contact line region, because of the presence of nanoparticles, is sufficient to alter the curvature of the meniscus at the capillary end and is measured accurately. The curvature difference is principally responsible for the flow from the capillary meniscus region toward the contact line during electrowetting or evaporation. Consistent with the experimental observation of the increase in curvature difference with the increase in the particle size, the contact line velocities show an increasing trend with an increase in the particle size (Figure 6). 3.3.2. Theoretical Model. It has been shown experimentally that the sizes of the nanoparticles can alter the slope and curvature of the film thickness profiles. The net changes in curvature or, more specifically, the differences in curvature at the capillary end of the meniscus and the transition region govern the flow of liquid toward the contact line region, which, in turn, dictates the velocity of the contact line. A control
Table 2. Conductivity Variation of Nanofluid Solution with Particle Size 0.1 cmc SDS + 0.1 M NaCl + particle
conductivity (mS)
no particle 30 nm 48 nm 100 nm
4.03 3.82 3.77 3.62
Figure 7. Particle size effect on the spatial profile of electrowetted meniscus (A, without particle; B, with 48 nm particle; and C, with 100 nm particle). Only three particle sizes are shown, to enhance readability.
increase in their sizes. However, as reported by Bhaumik et al.17 and in section 3.1, the adsorbed film thickness remains nearly constant (increases marginally from ∼50 to ∼66 nm when 100 nm particles are added to the solution), implying that the Maxwell stress does not influence the flat adsorbed layer. The curvature profiles of the thin film under the constant electric
Figure 8. Variation in the values of the curvatures at the transition (Kmax) and the capillary meniscus region (K∞) for different particle sizes, with and without the application of an electric field (3 V potential difference is applied to the system): (a) at 0 V and (b) at 3 V. 4165
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considering the small slopes of the thickness profiles encountered in the experiments reported here.
volume approach is used herein to correlate the shapedependent curvature with the pressure in the liquid. The governing equation for liquid flow in a slightly tapered thin film can be modeled through the use of lubrication approximation and is given as21 dPl d2u =μ 2 dx dy
(6)
Integrating the above equation with the appropriate boundary conditions (no slip at the wall and no shear at the interface), the instantaneous as well as average interfacial velocity (Uint) can be obtained as Uint =
1 μδ
Uint = −
∫0
δ
⎞ ⎛ dPl ⎞⎛ y 2 ⎜ ⎟⎜ − δy⎟dy ⎝ d x ⎠⎝ 2 ⎠
δ 2 ⎛ dPl ⎞ ⎜ ⎟ 3μ ⎝ dx ⎠
(7)
where δ is the average film thickness over the control volume of a finite length (ΔL). The control volume is selected starting from the point of maximum curvature (Kmax) in the transition region to the point in the capillary meniscus beyond which the curvature assumes a constant value (K∞). The modified expression for Uint can be written as
Uint = −
δ 2 ⎛ Pl 2 − Pl1 ⎞ ⎜ ⎟ 3μ ⎝ ΔL ⎠
Figure 9. Comparison of the experimentally obtained averaged interfacial velocity to theoretical predictions (applied voltage of 3 V; A, experimentally obtained averaged interfacial velocity; B, interfacial velocity calculated using eq 10).
(8)
where subscript 1 stands for the onset of a constant curvature region (K∞) and subscript 2 stands for the point where curvature reaches its maximum (Kmax). The effect of the Maxwell stress and the electrostatic pressure [given as P = −(1/ 2)εE2] are implicitly accounted for in the measured values of the curvature.17,49 The augmented Young−Laplace equation (eq 3) can be used to express the pressure jump across the liquid−vapor interface,24,54 where the first term on the right-hand side is the disjoining pressure and −σK is the capillary pressure, as described before. It is appropriate to take the retarded form of the disjoining pressure24 in the transition region and can be expressed as Π = −B/δ4 (eq 4b), where B is the retarded dispersion constant, as discussed earlier. An order of magnitude analysis of eq 5 at the two extreme locations, namely, the capillary and the transition region, reveals that σk varies from ∼58 to ∼84.1 N/m2, whereas Π varies from 3.7 × 10−2 to 1.2 × 10−4 N/m2 (as the particle size increases from 0 to 100 nm). The retarded dispersion constant B is taken to be −1.37 × 10−29 N m2.17 Therefore, the pressure drop along the selected control volume can be expressed solely in terms of the capillary pressure using the Young−Laplace equation as Pl 2 − Pl1 = −σ(K max − K∞)
Figure 10. Contact line velocity enhancement with particle (30 nm) loading (applied voltage of 3 V) [A, without particle; B, 0.05% (w/w); C, 0.1% (w/w); D, 0.35% (w/w); and E, 0.5% (w/w) particle concentration]. A ±5−8% error is associated with each experimental value.
A comparison between the velocities calculated using eq 10 to the average velocity obtained from the frame-by-frame analysis of the advancing meniscus (averaged over time for a specific particle size) as functions of the nanoparticle diameter is depicted in Figure 9. The close match, with an average error of ∼7.1%, between these two sets indicates that the electrowetting of a partially wetting liquid film can be analyzed successfully using the Young−Laplace equation. 3.3.3. Effect of the Particle Weight Fraction Variation. The weight fraction of the particles in the suspension has been varied from 0.05, 0.1, 0.35, and 0.5% (w/w) at a constant particle diameter (30 nm) and applied voltage (3 V). The choices of the particle weight percent are made keeping in mind that precipitation is observed for the particle concentration of about 1 wt % and above. Experimental observations under a bright-field microscope also confirmed that the particles beyond
(9)
Substitution of eq 9 into eq 8, the contact line velocity can be expressed as Uint =
δ 2 ⎛ σ(K max − K∞) ⎞ ⎜ ⎟ ⎠ ΔL 3μ ⎝
(10)
Therefore, the contact line velocity can be obtained from the evaluated values of the maximum and constant curvatures of the advancing film, with δ representing the thickness of the meniscus at ΔL/2. This is a reasonable approximation 4166
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1 wt % may not remain well dispersed for the given experimental conditions. An increase in the particle concentration brings about an increase in the concentration of anions, and these excess anions get repelled from the negatively charged platinum wire acting as a cathode (configuration I). The migration of the anions and the associated induced bulk flow of the liquid augment the contact line velocity with an increase in the particle concentration (Figure 10). Additionally, as mentioned before, an increase in the particle concentration leads to an increase in the electrical permittivity of the composite droplet, resulting in a decrease in the contact angle, as predicted by the traditional Young−Lippmann equation, and this effect is more pronounced at lower values of applied potentials,42 which is the case herein. Thus, the combined action of the repelling of similar charges (configuration I) and the effect of the increase in the particle concentration resulted in a higher contact line velocity during electrowetting of nanoparticle-laden partially wetting liquids and is consistent with the basic physics of the phenomena.
4. CONCLUSION Electrowetting-induced spreading and contact line dynamics of a partially wetting, surfactant-laden water meniscus containing negatively charged nanoparticles of varying sizes and concentrations have been studied. Image-analyzing interferometry is used to accurately measure the meniscus profile, whereas frame-by-frame analysis of the captured images of the spreading meniscus is used to evaluate the contact line velocities as functions of the applied electric field, its polarity, nanoparticle size, and concentrations. The results indicate significant increases in the electric-field-induced spreading and the contact line velocity (an increase in peak velocity from 1.5 to about 3.75 μm/s) with the addition of negatively charged nanoparticles at an applied potential of 3 V. Upon reversal of the polarity, the contact line velocity of the meniscus decreases (peak velocity decreases to 2.1 μm/s) but is still greater than that obtained for the case with no nanoparticles. An expression for the contact line velocity has been obtained following a control volume approach, taking into account the capillary pressure gradients and the effects of Maxwell stress. The predicted velocities are successfully compared (within an average deviation of 7.1%) to the experimental results.
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AUTHOR INFORMATION
Corresponding Author
*Telephone: +91-3222-283922. E-mail:
[email protected]. ernet.in and/or
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support provided by the Indian Institute of Technology Kharagpur [Sanction Letter no.: IIT/SRIC/ATDC/CEM/2013-14/118, dated 19.12.2013].
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Langmuir
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