Article pubs.acs.org/Langmuir
Microdroplet Evaporation with a Forced Pinned Contact Line Kevin Gleason and Shawn A. Putnam* Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, Florida 32816, United States S Supporting Information *
ABSTRACT: Experimental and numerical investigations of water microdroplet evaporation on heated, laser patterned polymer substrates are reported. The study is focused on both (i) controlling a droplet’s contact line dynamics during evaporation to identifying how the contact line influences evaporative heat transfer and (ii) validating numerical simulations with experimental data. Droplets are formed on the polymer surface using a bottom-up methodology, where a computer-controlled syringe pump feeds water through a 200 μm diameter fluid channel within the heated polymer substrate. This methodology facilitates precise control of the droplet’s growth rate, size, and inlet temperature. In addition to this microchannel supply line, the substrate surfaces are laser patterned with a moatlike trench around the fluid-channel outlet, adding additional control of the droplet’s contact line motion, area, and contact angle. In comparison to evaporation on a nonpatterned polymer surface, the laser patterned trench increases contact line pinning time by ∼60% of the droplet’s lifetime. Numerical simulations of diffusion controlled evaporation are compared the experimental data with a pinned contact line. These diffusion based simulations consistently over predict the droplet’s evaporation rate. In efforts to improve this model, a temperature distribution along the droplet’s liquid−vapor interface is imposed to account for the concentration distribution of saturated vapor along the interface, which yields improved predictions within 2−4% of the experimental data throughout the droplet’s lifetime on heated substrates.
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INTRODUCTION Microdroplet evaporation has been an important field of research in the last 2 decades due to its relevance in numerous technologies/applications including spray cooling,1−3 combustion,4−6 inkjet printing,7,8 and metrology.9−11 Recent research on microdroplets has in large part focused on understanding/ controlling the dynamics of a droplet’s contact line during evaporation.3,8,12−23 The forces associated with the pinning, depinning, and motion of a droplet’s contact line have been correlated to the substrates surface energy,13 imperfections (roughness),19 and wettability.3,17,24 Surfactants can be used to pin/lock the contact line during evaporation, although surfactants have additional effects on the evaporation rate.14,25−27 Mollaret et al.17 reported pinning forces increase with temperature on high energy surfaces (aluminum), while low energy surfaces (PTFE) showed little temperature dependence on pinning forces. Putnam et al.3 and Mollaret et al.17 both discussed pinning forces correlated to the wettability of the substrate. Hydrophilic surfaces require the greatest depinning force, which decreases as the surface becomes more hydrophobic.3,24 This helps explain why superhydrophobic surfaces typically facilitate a continuously depinned mode of contact line motion during evaporation,12,21,28 although not exclusively.29 In general, four different modes of contact line behavior are used to describe droplet evaporation: a constant contact angle (CCA) mode, a constant contact radius (CCR) mode, a stick−slip mode, and a mixed mode where both radius and contact angle decrease simultaneously.30 Recent work31,32 discusses a “master” model dependent on both the surface © 2014 American Chemical Society
wettability and droplet’s receding contact angle for predicting CCA, CCR, and stick−slip modes of evaporation. A better understanding of the contact line pinning/depinning transitions allows further validation of numerical models. Deegan et al.26 reported a model providing the solution of a nonuniform local evaporation flux, depending on the droplet’s contact angle (from here on out, termed Deegan’s model). This diffusion-limited model provides the framework for later developed models. Hu and Larson33 reported an approximate analytic solution, which stems from Deegan’s model, providing the evaporation rate for wetted droplets (θ ≤ 90) experiencing a CCR mode of evaporation. Popov34 used the work by Deegan et al. to provide the total rate of mass loss valid for all contact angles (from here on out, termed Popov’s model). Dash and Garimella28 introduced a correction factor to Popov’s model to predict droplet evaporation on a superhydrophobic substrate dominated by a CCA mode of evaporation. The correction factor is to reduce the evaporation rate caused by (i) the droplet’s thermal resistance and (ii) the evaporation cooling effect during evaporation. This introduces concerns for the validity of Popov’s model over the full range of droplet contact angles. Reports of a temperature distribution forming as the droplet evaporates led to discussions regarding the resulting effect on evaporation rate. While small, nonheated water droplets show small temperature gradients.16 large temperature gradients (as Received: May 8, 2014 Revised: July 14, 2014 Published: August 7, 2014 10548
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Figure 1. (a) A surface view and (d) a zoomed cut view of a laser patterned polymer substrate with a pattern radius R ≈ 500 μm is shown (fluid-flow channel d ≈ 200 μm, dp ≈ 100 μm, δp ≈ 50 μm). The effectiveness of the pattern can be seen when comparing droplets seen in parts b and c of similar volumes. A nonpatterned polymer substrate is represented in part b, which demonstrates a naturally formed droplet with the equilibrium contact angle according to Youngs’ equation (R0 ≅ 450 μm, θ0 ≅ 72.6°, V0 ≅ 123 nL). A patterned substrate is represented in part c, which provides a method for forcing the contact line to be fixed and a contact angle θ > θE (R0 ≅ 276 μm, θ0 ≅ 122, V0 ≅ 122 nL). The droplet profile during evaporation of (b) a naturally pinned droplet (dashed lines) and (c) a force pinned droplet (solid lines) with surface temperatures of Ts = 50 °C and Ts = 65 °C are shown in parts e and f, respectively. The error bars in parts e and f represent the largest deviation in the droplet profile data for at least 10 data sets/experiments.
reported with methanol and acetone16 or heated water droplets35) result in more influential effects on the evaporation rate.16,35,36 Additionally, this cooling effect becomes more influential for hydrophobic and superhydrophobic surfaces due to the increase in thermal resistance.3,9,28 Attempts at experimentally measuring this induced temperature distribution along the liquid−vapor interface have shown interesting results. Large interfacial temperature jumps/discontinuities have been reported,37,38 which increase with increasing substrate temperature. Thus, numerical simulations are more common for determining the temperature distribution.28,39−41 Briones et al.39,40 simulated an ∼9 °C temperature suppression at the apex of a wetted droplet on a high temperature surface (100 °C). Dash and Garimella reported a ∼7 °C temperature difference (apex to contact line) for droplets evaporating at room temperature with high contact angles (θ ≈ 160 °C), hence, their introduction of a correction factor to reduce the predicted
evaporation rate from Popov’s model which assumes a constant surface temperature. This work focuses on two aspects of droplet evaporation: (i) experimentally controlling the contact line dynamics during evaporation and (ii) proposing a modification to the models reported by Deegan et al.25,26 and Popov34 for providing a more accurate numerical solution by imposing a temperature distribution along the liquid−vapor interface, which is validated through experiments. To do this, a method of substrate fabrication is discussed, creating a circular “moatlike” trench which refrains further advancement of the contact line. This method of substrate fabrication is useful for controlling the surface area and contact line during evaporation, increasing the pinning time, and facilitates systematic experimental studies of contact angle dependence on droplet evaporation. The experimental data is compared to the numerical models, which is found to not accuracy represent the data. A modification to the numerical model is proposed which 10549
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measured equilibrium contact angle for the polymer substrate is θE = 72° ± 3°, with advancing and receding contact angles of θa = 85° ± 2° and θr = 61° ± 3°, respectively. Alternatively, the patterned substrate restricts motion of the contact line such that the droplet’s initial contact angle is dependent on the infused volume (experiencing a fixed contact line). A static contact angle within the pattern region can be achieved such that θ > θE. The pinning forces caused by the micropattern trench forces the droplet’s contact radius to remain fixed within the laser patterned region for 48° ≲ θ ≲ 120° (in comparison to 61° ≲ θ ≲ 85° on a nonpatterned substrate). For θ ≳ 120°, pinning forces are not enough to keep the droplet within the patterned region (i.e., depinning from the patterned section). Thus, the maximum observed contact angle before depinning on the patterned substrate is θmax ≈ 1.5θE, allowing the formation of both wetted and nonwetted droplets while maintaining a constant contact line length and area. The nonflat surface trench may have effects on the evaporation rate if the trench size is comparable to the mean free path (i.e., Knudsen number, Kn ≈ 1).43 To verify the patterned trench does not cause a molecular level influence on the evaporation rate, the Knudsen number is calculated for a 50 μm channel (i.e., the pattern depth δp, the smallest length scale of the trench). The atomic mean path of vapor molecules is on the order of 10 nm.44,45 The calculated Knudsen number for the pattern trench (Kn ≈ 0.0001, which is ≪1) verifies the application of continuum theory remains valid. For evaporation rate studies, patterned and nonpatterned substrates are heated to elevated temperatures of 50 and 65 °C. Data recordings are started once the syringe pump has been stopped (i.e., once the droplet is fully formed). The measured droplet parameters (R, θ, h, and V) are nondimensionalized with respect to the initial conditions (R0, θ0, h0, V0, respectively) and plotted over the droplet’s lifetime (t/τE), shown in Figure 1e,f. The image analysis software is terminated when the droplet becomes significantly small (due to increased error in the edge-finding software for droplet dimensions (i.e., R and h) comparable to 5 times the imaging resolution). As shown in Figure 1, the laser patterned substrate is effective in controlling the contact line (forcing to remain pinned). The results are best demonstrated by the significant increase in CCR mode of evaporation between the two substrate cases (Figure 1e,f). Droplet evaporation on a nonpatterned substrate (represented by dashed lines in Figure 1e,f) show a contact line depinning time of t ≈ 0.2τE (this short pinning time is caused by the energy barriers associated with contact angle hysteresis46). Comparatively, the contact line depinning time on the patterned substrate is 60% longer at t ≈ 0.8τE. While the contact line depinning times for the patterned and nonpatterned substrates differ from each other, each substrate (patterned or nonpatterned) shows identical time periods associated with a CCR mode of evaporation. There is a significant deviation between the droplets evaporating on patterned and nonpatterned substrates for t ≳ 0.2τE (once the naturally pinned droplet depins). A CCA mode of evaporation may be approximated during 0.2τE ≲ t ≲ 0.5τE for naturally pinned droplets, whereas the forced pinned droplets avoid this mode of evaporation. By maintaining a CCR mode of evaporation, the laser pattern substrates demonstrate simplified contact line dynamics. Numerical Simulations. Popov’s model34 (eq 1) transformed from Deegan’s model25,26 (eq 2) are the two diffusion-
considers the temperature distribution along the liquid−vapor interface. The proposed modification facilitates accurate predictions of water droplet evaporation within a wide range of substrate temperatures and initial contact angles.
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EXPERIMENTAL SETUP
For controlling the dynamics of a droplet’s contact line during evaporation, micropatterned acrylic polymer substrates are used. A femtosecond laser (Ti:sapphire, 140 fs at 80 MHz, 795 nm) is used to fabricate a circular trench that acts as a barrier to restrict further advancement of the droplet’s contact line, keeping the contact line fixed/pinned as the droplet evaporates (see Figure 1). Preparing the substrate for patterning, the surface is darkened using a black Sharpie permanent marker and black Expo dry erase marker, sequentially. This order of layers assists heat absorption by the transparent acrylic substrate for a direct solid−vapor transition (ablation), while reducing/eliminating annealing within the immediate surrounding. The prepared substrate is placed on an assembled 6 degree of freedom sample stage for positioning and rotating the substrate during the ablation process. The result produces a pattern spot diameter of dp ≈ 100μm on the surface, with a depth of δp ≈ 50μm (see Figure 1a,d). Water microdroplets are formed within the patterned region using a nontraditional method of droplet formation. A microchannel is created within the substrate, exiting at the substrate surface (centered about the circular trench laser pattern). A syringe pump controls the flow rate through the microchannel (0.2 μm filtered distilled water) to form droplets. This bottom-up methodology is advantageous by (i) gaining full control of the droplet’s initial volume, (ii) introduces the ability to increase a droplet’s volume after the droplet is formed, and (iii) avoids impingement dynamics commonly experienced using ink jet dispensers. Once the droplet is fully formed, the syringe pump is turned off to allow the droplet to evaporate. A custom image analysis LabVIEW program facilitates data collection.3,9 The image analysis program continuously captures images from the CCD camera (at 4 fps) and analyzes the droplet through a greyscale edge-detection technique. The image analysis/acquisition program measures the droplet’s contact radius (R), contact angle (θ), and apex height (h) by incorporating a spherical cap fit to the edge contour. The volume (V) of the droplet is then calculated through a spherical cap relation. The proposed method for droplet formation does have a minor influence on the evaporation rate. For example, once the syringe pump is stopped, there is still fluid flow into the droplet due to either (i) the pressure buildup within the fluid microchannel leading to the droplet (for θ < 90°) or (ii) capillary action by the droplet (for θ > 90°). In either case, the measured evaporation rate is slightly slower than other reported values (see Figure 8 in ref 3). This error is minimized by evaluating numerical predictions using the experimentally measured droplet profile (i.e., R and θ) during evaporation, thus simulating an equivalent evaporation rate. Nevertheless, the reported data provides an upper limit prediction to the droplet lifetime.
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RESULTS Evaporation with Constant Contact Radius. The droplet profile is measured during evaporation on both (i) a nonpatterned acrylic polymer substrate (Figure 1b) and (ii) a laser patterned acrylic polymer substrate (Figure 1c). The laser patterned substrate classifies a forced pinned contact line due to the step edge “moatlike” trench (Figure 1a,d) that confines the droplet within the patterned region. Naturally pinned is termed to droplets evaporating on a nonpatterned substrate, where contact line pinning (CCR mode of evaporation) is experienced due to unavoidable contact angle hysteresis and surface imperfections/inhomogeneities. The droplet equilibrium contact angle on a nonpatterned substrate is predicted by Youngs’ Equation.42 Therefore, the contact radius is dependent on the infused volume. The 10550
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Figure 2. Droplet evaporation on a (a) nonpatterned and (b) patterned polymer substrates at an elevated surface temperatures of Ts = 50 °C. Experimental data: (a) Ts = 50 °C, R0 ≅ 460.9 μm, θ0 ≅ 62.9°, V0 ≅ 103.3 nL, h0 ≅ 278.8 μm, τE ≅ 38.25 s; (b) Ts = 50 °C, R0 ≅ 469.2 μm, θ0 ≅ 121.0°, V0 ≅ 583.6 nL, h0 ≅ 828μm, τE ≅ 157.5 s.
limited models used for comparing the evaporation rate measured on both patterned and nonpatterned substrates. ⎡ sin θ dm = −πRD(cs − c∞)⎢ ⎣ 1 + cos θ dt ∞ 1 + cosh 2θτ ⎤ tanh[(π − θ)τ ] dτ ⎥ +4 ⎦ 0 sinh 2πτ
∫
J (α ) =
∫0
∞
D(cs − c∞) ⎡ 1 ⎢ sin θ + ⎣2 R
resulting in a temperature distribution along the liquid−vapor interface.28,39,40 Correction to Numerical Model. Dash and Garimella28 presented a simple correction factor to reduce the predicted evaporation rate by Popov’s model for superhydrophobic surfaces (the temperature distribution for a hydrophobic surface was not found to influence the evaporation rate significantly). Our proposed modification is a correction to both Popov’s model and Deegan’s model based on the temperature distribution along the liquid−vapor interface (shown in eqs 3 and 4, respectively).
(1)
2 (coshα + cosθ)3/2 ×
⎡ sin θ dm = −πRD(cs(α , θ ) − c∞)⎢ ⎣ 1 + cos θ dt ∞ 1 + cosh 2θτ ⎤ tanh[(π − θ)τ ] dτ ⎥ +4 ⎦ 0 sinh 2πτ
⎤ cosh(θτ ) tanh[(π − θ)τ ]P −1/2 + iτ(cosh α)τ dτ ⎥ ⎦ cosh(πτ )
∫
(2)
The numerical approach implemented for solving eqs 1 and 2 are provided in the Supporting Information. Figure 2 shows comparisons between Popov’s prediction (dotted black) and the measured droplet volume (solid red). The experimental data is used to provide the time evolution of the contact angle for evaluating eq 1. Numerical solutions are evaluated for both (i) a pinned contact line (dashed lines) and (ii) a depinning contact line (dotted lines). Only the pinned numerical solutions are reported for the laser patterned substrate. As shown in Figure 2a, the pinned model deviates significantly near the end of the droplet’s lifetime for the nonpatterned substrate. In this case, the model assumes that the contact line (where a majority of the evaporation takes place9) is larger than that observed experimentally. This result verifies our expectation of the increased role of contact line evaporation. The depinned model predicts a lower droplet lifetime and is a better representation of the evaporation on a nonpatterned substrate. The pinned contact line numerical solutions (Figure 2b) also provide reasonable predictions for evaporation on patterned substrates. In all cases, Popov’s model over predicts the evaporation rate. The error is best explained by Popov’s model assuming a constant surface vapor concentration (i.e., constant temperature) along the entire liquid−vapor interface. Because the fluid channel is located within the heated substrate we assume that the initial bulk fluid temperature is equal to the substrate temperature. Once the infuse rate is terminated, it is expected that bulk temperature will decrease during evaporation,16
J (α ) = +
(3)
D(cs(α , θ ) − c∞) ⎡ 1 ⎢ sin θ ⎣2 R
2 (cosh α + cos θ)3/2 ×
⎤ − θ )τ ]P −1/2 + iτ(cosh α)τ dτ ⎥ ⎦
∫0
∞
cosh(θτ ) tanh[(π cosh(πτ ) (4)
A temperature distribution imposes a concentration gradient along the interface (i.e., variable cs), thereby reducing the evaporation flux. Our modification changes the surface concentration (cs) to a variable surface concentration along the liquid−vapor interface (cs(α,θ)) dependent on the temperature distribution. Figure 3 represents the effect of contact angle (θ) and substrate surface temperature (Ts) on the interfacial temperature distribution. The temperature gradient increases with increasing contact angle, as expected.28 In this study, measurements of the temperature distribution are not practical. Thus, numerical results provided by Briones et al. (Figure 4 in ref 39 and Figure 8 in ref 40) and the Supporting Information accompanying Dash and Garimella28 are interpolated for determining the temperature distribution (details provided in the Supporting Information). Comparisons for Deegan’s Model. Demonstrating the difference between Deegan’s model (a constant surface concentration) and the modified model (a variable surface 10551
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using Deegan’s model (eq 2), and the right side is the solution of the modified model (eq 4). The vector plots exploit the difference between the two numerical solutions for each contact angle. A major improvement with the modified model is based on the change of curvature of the evaporation flux from the apex (α = 0, r/R = 0) to the contact line (α → ∞, r/R = 1) for contact angles θ ≥ 90°. The solution of Deegan’s model shows that the largest evaporation flux for hydrophobic droplets (θ > 90°) is at the apex. This disagrees with the results/conclusions provided by many authors on contact line dependence on evaporation rate.3,9,12−23,39,40 The modified model maintains a high evaporation flux near the contact line and a minimum at the apex regardless of contact angle (hence maintaining a contact line dependence). Comparisons for Popov’s Model. To further evaluate the corrected model, Popov’s model is modified and compared to experimental data for a forced pinned contact line (Figure 5). Our hypothesis of the decreasing temperature gradient causing a lower evaporation rate has been confirmed theoretically from the previous analysis on evaluating the improvements of Deegan’s model (Figure 4). Three interpolation models (detailed in the Supporting Information) are compared to Popovs model and shown in Figure 5: (i) using data gathered from Briones et al.;39,40 (ii) using data gathered from Dash and Garimella;28 and (iii) combining data gathered from all references. The measured evolution of the contact angle is used to provide the time dependent vapor concentration along the liquid−vapor interface for each comparison (neglecting any internal/external convection). Each model provides a better representation of the experimental data, where the “combined model” demonstrates the best fit (presumable to the range of contact angles, substrate temperatures, and largest number of interpolated data points).
Figure 3. Contour plots of temperature distribution (right contour axis) and corresponding vapor concentration (left contour axis) along the liquid−vapor interface on an evaporating droplet on a heated substrate of Ts = 50 °C and Ts = 65 °C using the surface fit produced through interpolation of numerical results by Briones et al.39,40 The thickness of the liquid−vapor interface is magnified to provide a better visual of the distribution.
concentration), the solution of the evaporation flux for both models are compared at various contact angles (see Figure 4). In each figure, the left side represents the solution evaluated
Figure 4. Magnitude and vector plot comparisons of the evaporation flux, J, obtained using Deegan’s model and the modified model (i.e., constant cs versus a cs distribution) for hydrophilic and hydrophobic droplets. 10552
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Figure 5. Plots comparing experimental data (solid red lines), numerical solutions obtained from Popov’s model assuming constant temperature along interface (dashed black lines), and the solution from the proposed modification to Popov’s model with a variable vapor surface concentration (dot-dashed lines). Experiments were performed on patterned substrate at (a) Ts = 50 °C and (b) Ts = 65 °C. The relative errors are shown in parts c and d. Experimental data: (a) R0 ≅ 455.4 μm, θ0 ≅ 117.4°, V0 ≅ 459.4 nL, h0 ≅ 745.2 μm, τE ≅ 133.0 s; (b) R0 ≅ 394.7 μm, θ0 ≅ 106.6°, V0 ≅ 205.7 nL, h0 ≅ 527.2 μm, τE ≅ 24.5 s.
evolution of the droplet’s volume remains within 2−4% of the experimental data. Sophisticated volume of fluid (VOF)39,40 and other high fidelity models47 can replace the interpolated models, incorporating internal flow circulation (e.g, Marangoni flow) and vapor transport through natural convection. While simulations including additional modes of transport remain within 2−3% of experimental data,47 the remaining error can be caused due to unknown phenomenon. For example, the recent discussions on hydrothermal waves (HTWs)48−51 have come to attention with their resulting influence on the evaporation rate.48 HTWs represent this idea that unknown, unrepresented phenomenon will yield less than perfect predictions. However, HTWs have only been observed in large volatile droplets49 (R ≳ 2 mm) or evaporating films;51 therefore, HTWs are not expected to contribute to the evaporation of small water microdroplets (R ≲ 500μm) unless these droplets have evaporated into thin-film “pancakes”. Alternatively, the additional 2−4% in error may also be due to a decrease in substrate surface temperature because the water droplets studied are evaporating on a low thermal conductivity polymer substrate.16 However, this study only measured water droplet evaporation on polymer acrylic substrates, thus the effect of a patterned
Table 1 provides the error analysis for all four models at each elevated surface temperature. Table 1. Error for Each of the Four Models Analyzed for Predicting the Droplet’s Lifetimea error in ṁ lv interpolated model
Ts = 50 °C
Ts = 65 °C
Popov’s model34 Briones et al.39,40 Dash and Garimella28 combined model
9.70% 7.44% 2.40% 1.84%
10.54% 7.43% 3.69% 2.83%
a The linearized error is the error in slope of the linear fitted curve of each curve provided in Figure 5.
Our combined model uses both (i) the interpolated data from Briones et al.39,40 (providing the evolution of the contact angle and the corresponding temperature distribution during evaporation) and (ii) the interpolated data from Dash and Garimella28 (providing data of the temperature distribution for θ > 90). This model increases the accuracy in predicting the droplet’s lifetime by ∼10% (compared to Popov’s model with a constant surface concentration). As a result, the predicted 10553
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(4) Arcoumanis, C.; Bae, C.; Crookes, R.; Kinoshita, E. The Potential of Di-Methyl Ether (DME) as an Alternative Fuel for CompressionIgnition Engines: A Review. Fuel 2008, 87, 1014−1030. (5) Hartranft, T. J.; Settles, G. S. Sheet Atomization of NonNewtonian Liquids. Atomization Spray 2003, 13, 191. (6) Putnam, S. A.; Byrd, L. W.; Briones, A. M.; Hanchak, M. S.; Ervin, J. S.; Jones, J. G. Role of Entrapped Vapor Bubbles During Microdroplet Evaporation. Appl. Phys. Lett. 2012, 101, 071602. (7) Calvert, P. Inkjet Printing for Materials and Devices. Chem. Mater. 2001, 13, 3299−3305. (8) Li, G.; Flores, S. M.; Vavilala, C.; Schmittel, M.; Graf, K. Evaporation Dynamics of Microdroplets on Self-Assembled Monolayers of Dialkyl Disulfides. Langmuir 2009, 25, 13438−13447. (9) Putnam, S. A.; Briones, A. M.; Ervin, J. S.; Hanchak, M. S.; Byrd, L. W.; Jones, J. G. Interfacial Heat Transfer During Microdroplet Evaporation on a Laser Heated Surface. Int. J. Heat Mass Transfer 2012, 55, 6307−6320. (10) Verkouteren, R. M.; Verkouteren, J. R. Inkjet Metrology II: Resolved Effects of Ejection Frequency, Fluidic Pressure, and Droplet Number on Reproducible Drop-on-Demand Dispensing. Langmuir 2011, 27, 9644−9653. (11) Whiteman, C. D. Mountain Meteorology: Fundamentals and Applications; Oxford University Press: New York, 2000. (12) Anantharaju, N.; Panchagnula, M.; Neti, S. Evaporating Drops on Patterned Surfaces: Transition from Pinned to Moving Triple Line. J. Colloid Interface Sci. 2009, 337, 176−182. (13) Bormashenko, E.; Musin, A.; Zinigrad, M. Evaporation of Droplets on Strongly and Weakly Pinning Surfaces and Dynamics of the Triple Line. Colloids Surf., A 2011, 385, 235−240. (14) Chandra, S.; di Marzo, M.; Qiao, Y.; Tartarini, P. Effect of Liquid-Solid Contact Angle on Droplet Evaporation. Fire Safety J. 1996, 27, 141−158. (15) Crafton, E. F.; Black, W. Heat Transfer and Evaporation Rates of Small Liquid Droplets on Heated Horizontal Surfaces. Int. J. Heat Mass Transfer 2004, 47, 1187−1200. (16) David, S.; Sefiane, K.; Tadrist, L. Experimental Investigation of the Effect of Thermal Properties of the Substrate in the Wetting and Evaporation of Sessile Drops. Colloids Surf., A 2007, 298, 108−114. (17) Mollaret, R.; Sefiane, K.; Christy, J. R.; Veyret, D. Experimental and Numerical Investigation of the Evaporation into Air of a Drop on a Heated Surface. Chem. Eng. Res. Des. 2004, 82, 471−480. (18) Nguyen, T. A.; Nguyen, A. V.; Hampton, M. A.; Xu, Z. P.; Huang, L.; Rudolph, V. Theoretical and Experimental Analysis of Droplet Evaporation on Solid Surfaces. Chem. Eng. Sci. 2012, 69, 522− 529. (19) Pittoni, P. G.; Chang, C.-C.; Yu, T.-S.; Lin, S.-Y. Evaporation of Water Drops on Polymer Surfaces: Pinning, Depinning and Dynamics of the Triple Line. Colloids Surf., A 2013, 432, 89−98. (20) Saada, M. A.; Chikh, S.; Tadrist, L. Evaporation of a Sessile Drop with Pinned or Receding Contact Line on a Substrate with Different Thermophysical Properties. Int. J. Heat Mass Transfer 2013, 58, 197−208. (21) Shin, D. H.; Lee, S. H.; Jung, J.-Y.; Yoo, J. Y. Evaporating Characteristics of Sessile Droplet on Hydrophobic and Hydrophilic Surfaces. Microelectron. Eng. 2009, 86, 1350−1353. (22) Starov, V.; Sefiane, K. On Evaporation Rate and Interfacial Temperature of Volatile Sessile Drops. Colloids Surf., A 2009, 333, 170−174. (23) Yu, Y.-S.; Wang, Z.; Zhao, Y.-P. Experimental and Theoretical Investigations of Evaporation of Sessile Water Droplet on Hydrophobic Surfaces. J. Colloid Interface Sci. 2012, 365, 254−259. (24) Blake, T.; Coninck, J. D. The Inuence of Solid-Liquid Interactions on Dynamic Wetting. Adv. Colloid Interface Sci. 2002, 96, 21−36. (25) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary Flow as the Cause of Ring Stains from Dried Liquid Drops. Nature 1997, 389, 827829.
substrate’s thermal properties on droplet evaporation cannot be evaluated.
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CONCLUSIONS Experimental and numerical investigations of water microdroplet evaporation are presented. A laser patterned substrate is fabricated for controlling the dynamics of the contact line during droplet evaporation, where a moatlike trench pins the contact line and forces the droplet to undergo a time-extended constant contact radius (CCR) mode of evaporation. This methodology also facilitates (i) precise control of droplet formation, size, and wetting dynamics, (ii) droplet heating prior to formation, and (iii) future steady-state evaporation studies on heated surfaces with a variety of different nonequilibrium contact angles (including θ > θa) for a given solid−liquid-vapor system (e.g., steady-state studies of both wetted and nonwetted droplets on a single substrate). Numerical solutions provided by Deegan et al.25,26 and Popov34 are used for comparisons to experimental data, which are unable to accurately represent the experimental data. The implementation of constant vapor concentration in both models is assumed to be a key source of error because the droplet’s surface/interface temperature is not constant (i.e., the apex temperature of the droplet should be less than the substrate temperature during evaporation22,28,39,40). In result, both Deegan’s model and Popov’s model are modified by considering the temperature distribution along the liquid− vapor interface (interpolated using results reported by Briones et al.39,40 and Dash and Garimella28). Evaluation of the modified Deegan model results provides a solution that matches the widely accepted concept of contact line dependence on droplet evaporation. In comparison to Popov’s model, the modified Popov model shows an increase in droplet lifetime, becoming better representative of the experimental data (to within 2−4%). On the basis of these results, the incorporation of more sophisticated phenomenon like HTWs and substrate cooling are expected to have minor effects during evaporation with submillimeter sized water droplets, yet may explain the remaining 2−4% errors in prediction.
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ASSOCIATED CONTENT
S Supporting Information *
Numerical evaluation for evaporation flux and determination of the interfacial temperature distribution. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Briones, A. M.; Ervin, J. S.; Putnam, S. A.; Byrd, L. W.; Gschwender, L. Micrometer-Sized Water Droplet Impingement Dynamics and Evaporation on a Flat Dry Surface. Langmuir 2010, 26, 13272−13286. (2) Deng, W.; Gomez, A. Electrospray Cooling for Microelectronics. Int. J. Heat Mass Transfer 2011, 54, 2270−2275. (3) Putnam, S. A.; Briones, A. M.; Byrd, L. W.; Ervin, J. S.; Hanchak, M. S.; White, A.; Jones, J. G. Microdroplet Evaporation on Superheated Surfaces. Int. J. Heat Mass Transfer 2012, 55, 5793−5807. 10554
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dx.doi.org/10.1021/la501770g | Langmuir 2014, 30, 10548−10555
