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A Steady Method for the Analysis of Evaporation Dynamics Ahmet Alperen Gunay, Soumyadip Sett, Junho Oh, and Nenad Miljkovic Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02821 • Publication Date (Web): 26 Sep 2017 Downloaded from http://pubs.acs.org on September 28, 2017

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A Steady Method for the Analysis of Evaporation Dynamics A. Alperen Günay1, Soumyadip Sett1, Junho Oh1, Nenad Miljkovic1,2,3,*

1

Department of Mechanical Science and Engineering, University of Illinois at Urbana – Champaign, 1206 W Green St, Urbana, Illinois 61801 USA

2

Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, USA 3

International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

* Corresponding Author. E-mail address: [email protected]

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ABSTRACT Droplet evaporation is an important phenomenon governing many man-made and natural processes. Characterizing the rate of evaporation with high accuracy has attracted the attention of numerous scientists over the past century. Traditionally, researchers have studied evaporation by observing the change in the droplet size in a given time interval. However, the transient nature coupled with the significant mass-transfer governed gas-dynamics occurring at the droplet three-phase contact line make the classical method crude. Furthermore, the intricate balance played by the internal and external flows, evaporation kinetics, thermocapillarity, binary-mixture dynamics, curvature, and moving contact lines make the decoupling of these processes impossible with classical transient methods. Here, we present a method to measure the rate of evaporation of spatially and temporally steady droplets. By utilizing a piezoelectric dispenser to feed microscale droplets ( ≈ 9 µm) to a larger evaporating droplet at a prescribed frequency, we can both create variable-sized droplets on any surface, and study their evaporation rate by modulating the piezoelectric droplet addition frequency. Using our steady technique, we studied water evaporation of droplets having base radii ranging from 20 µm to 250 µm on surfaces of different functionalities (45˚ ≤ , ≤ 162˚, where , is the apparent advancing contact angle). We benchmarked our technique with the classical un-steady method showing an improvement of 140% in evaporation rate measurement accuracy. Our work not only characterizes the evaporation dynamics on functional surfaces, it also provides an experimental platform to finally enable the decoupling of the complex physics governing the ubiquitous droplet evaporation process. Keywords: Evaporation, Droplet, Hydrophobic, Hydrophilic, Contact Angle, Steady, Steady State, Superhydrophobic, Contact Line, Non-Condensable Gases.


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1. Introduction The evaporation of droplets is an important phenomenon for both natural and man-made processes such as the hydrological cycle1-4, oil-water separation5-8, liquid hydrocarbon combustion9-12, phase change heat transfer13-18, and power generation19-22. Thus, scientists have been studying droplet evaporation over the past century23-31. The main physical phenomenon characterizing evaporation is the rate of evaporation, which determines the amount of liquid mass that is converted into vapor in a fixed amount of time. Numerous studies have been performed in an attempt to more accurately characterize the rate of evaporation on functional surfaces15, 32-39. However, the developed techniques typically rely on detecting the non-steady size changes in a droplet for a given amount of time, which include transient effects governed by vapor dynamics outside the droplet, interfacial dynamics at the liquid-vapor interface, and liquid dynamics inside the droplet. In addition, droplet evaporation dynamics are made more complex by the intricate and length-scale-dependent balance played by the bulk flow inside the droplet40-44, evaporation kinetics at the liquid-vapor interface15,

45-46

, thermocapillary stability47-48, and binary-mixture dynamics49-51, making the

decoupling of these processes impossible with classical transient methods. Furthermore, the internal, interfacial, and external flows are complicated further by the presence of moving contact lines and high curvature regions, both of which govern evaporation52. In this study, we develop a steady method capable of decoupling droplet length scale from the evaporation measurement, to examine droplet evaporation dynamics for a range of droplet sizes (20 ≤ ≤ 250 m, where = sin , is the base radius) and apparent advancing contact angles (45˚ ≤ , ≤ 162˚). The steady technique involves the feeding of microscale droplets ( ≈ 9 µm, where is the radius of curvature of the droplet) to a larger evaporating droplet (≫9 µm) residing on a functional surface at a prescribed frequency. At the instant the droplet reaches the required size of interest, the microscale droplet injection rate is modulated 3 ACS Paragon Plus Environment

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in order to ensure that the evaporating droplet shape and size are constant. At this instant, the rate of liquid evaporation is balanced by the rate of liquid microdroplet deposition. To ensure stability of our technique, we monitor the droplet evaporation for long times (>120 seconds). We benchmarked our technique with classical non-steady methods of evaporation characterization on functional surfaces, showing significant improvement (≈ 140%) in evaporation rate measurement accuracy with our steady method stemming from the elimination of transient effects on a droplet evaporating in the steady state. Our results show that the rate of droplet evaporation in non-condensable gas (NCG) leaden environments increases linearly with increasing droplet base radius due to the linear dependence of contact line length with base radius. Furthermore, we show that the droplet evaporative flux decreases linearly for increasing contact angles at fixed droplet sizes for both hydrophilic and hydrophobic surfaces. The outcomes of this work not only show the effects of surface functionality, droplet length scale, and contact line dynamics on the droplet evaporation rate, but also establish a powerful characterization platform that can be used to decouple the intricate and length-scale-dependent balance played by internal and external flows, evaporation kinetics at the liquid-vapor interface, thermocapillary motion, and binary-mixture dynamics, and provide avenues for the development of novel, and versatile functional surface technologies for a wide range of applications utilizing droplet evaporation.

2. Experimental Methods 2.1. Experimental Procedure. To benchmark our steady method, and to study the effects of droplet size scale, base area, contact line length, and contact angle, we fabricated six different samples with distinct and wide ranging wetting characteristics, including a gold coated silicon wafer (Si-Au, Figure 1a, , = 45.1˚ ± 5.2˚), polished copper (Cu, Figure 1b, , = 85.0˚ ± 1.9˚), smooth silicon wafer coated with a fluorinated silane (TFTS) (Si-TFTS, Figure 1c, , = 110.6˚ ± 2.8˚), carbon nanotube mats coated with a 90 nm layer of P2i 4 ACS Paragon Plus Environment

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fluoropolymer (CNT1, Figure 1d, , = 130.6˚ ± 2.9˚), carbon nanotube mats coated with a 30 nm layer of P2i fluoropolymer (CNT2, Figure 1e, , = 141.3˚ ± 3.1˚), and HDTS coated copper oxide microstructures (CuO-HDTS, Figure 1f, , = 162.0˚ ± 2.9˚). Note, lower contact angles approaching superhydrophilicity were not studied here due to requirement for wide angle imaging techniques to capture the large base radius. To study droplet evaporation and to eliminate transient effects of moving interfaces, we interfaced a high speed camera (Fastcam SA2, Photron) with a frequency-controlled piezoelectric micro-goniometer (MCA-3, Kyowa Interface Science). All experiments were performed in ambient conditions ( = 25 ± 0.5 ℃, = 50% ± 5%, where is the relative humidity). To grow droplets, the piezoelectric dispenser was placed above our functional surfaces with a spacing ranging from 5 to 10 mm (ℎ), and turned on to dispense monodisperse microscale droplets (Figure 2). 2.2. Surface Fabrication. For CNT growth by chemical vapor deposition (CVD), a 20 nm thick Al2O3 diffusion barrier and a 5 nm thick film of Fe catalyst layer were deposited on silicon growth substrates using electron beam deposition. After purging in He/H2 atmosphere in a 2.54 cm quartz furnace for 15 min, the silicon substrate was heated to 750°C in a 2.54 cm quartz furnace and annealed for 3 min under a flow of H2 and He at 400 and 100 sccm, respectively. Randomly aligned CNTs were then grown by flowing C2H4 at 200 sccm for 1 min. These thermally grown CNT had characteristic diameter of ≈ 7 nm53. CuO nanostructured surfaces were created on commercially available copper (Cu) tabs (99.9% purity, 25 mm x 25 mm x 0.8 mm). Similar to silicon substrates, each Cu tab was thoroughly rinsed with acetone, ethanol, isopropyl alcohol (IPA), and deionized (DI) water. The cleaned tabs were then dipped in 2.0 M Hydrochloric acid solution for 2 min to remove native oxide film on the surface, rinsed with DI water and dried with nitrogen gas. 5 ACS Paragon Plus Environment

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Nanostructured CuO films were formed by immersing the cleaned tabs in hot (96 ± 3 ˚C) alkaline solution composed of NaClO2, NaOH, Na3PO4·12H2O, and DI water (3.75:5:10:100 wt%). This oxidation process leads to initial formation of a thin (≈300 nm) Cu2O layer which reoxidizes to form sharp, knife-like CuO oxide structures ( ≈ 1µm). 2.3. Surface Functionalization. Trichloroperfluorooctyl-silane (TFTS, CAS no. 78560-45-9, Sigma Aldrich) was deposited on silicon growth substrate by liquid phase deposition (LPD). Initially, the obtained silicon substrate was thoroughly cleaned by rinsing it with acetone, methanol, ethanol, isopropyl alcohol (IPA) and deionized (DI) water. After drying the silicon substrate with nitrogen gas, it was plasma cleaned for 10 minutes. For hydrophobic functionalization, a drop of TFTS was added to 500 mL hexane and the resulting solution was vigorously mixed before placing the clean silicon substrate inside the solution. The coated silicon substrate (referred to as Si-TFTS in the text) was removed from the solution after 24 hours and was rinsed with ethanol and DI water and dried in nitrogen gas. Goniometric measurements (MCA-3, Kyowa Interface Science Ltd.) of ≈ 100 nL droplets on TFTS coated silicon substrates showed advancing contact angle of = 110.6˚ ± 2.8˚. For hydrophilic functionalization, the plasma cleaned silicon substrates were sputter coated with gold having 10 nm thickness (referred to as Si-Au in the text). Goniometric measurements (MCA-3, Kyowa Interface Science Ltd.) of ≈ 100 nL droplets on Si-Au substrates showed advancing contact angle of , = 45.1˚ ± 5.2˚. Hexadecyltriethoxysilane (HDTS, CAS no. 16415-12-6, Sigma Aldrich) was deposited by liquid phase deposition on CuO nanostructured surfaces. Before deposition, the nanostructured CuO tab was dried in clean nitrogen gas. A solution containing a drop of HDTS in 500 mL of hexane was prepared and vigorously mixed, before placing the CuO tab in it. The coated Cu tab (referred to as CuO-HDTS in the text) was removed from the solution after 24 hours, rinsed with ethanol, DI water and dried with clean nitrogen gas before placing 6 ACS Paragon Plus Environment

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it in an atmospheric oven at 80 °C for 90 mins. Goniometric measurements (MCA-3, Kyowa Interface Science Ltd.) of ≈ 100 nL droplets on HDTS coated CuO substrates showed advancing contact angle of , = 162˚ ± 2.9˚. Plasma-enhanced vapor deposition was used for obtaining P2i hydrophobic coatings on the CNT surfaces. Within a vacuum chamber at room temperature and low pressure, the P2i fluoropolymer coating is introduced as vapor and ionized. By varying the time of deposition, two different thicknesses of P2i coating, 90 nm and 30 nm were obtained. Goniometric measurements (MCA-3, Kyowa Interface Science Ltd.) of ≈ 100 nL droplets on the thicker (90 nm, referred as CNT1 in text) and thinner (30 nm, referred as CNT2 in text) P2i coated sample showed advancing contact angles of , = 130.6˚ ± 2.9˚ and , = 141.3˚ ± 3.1˚, respectively. Note, the functionalized CNT samples were used extensively in condensation experiments prior to this study54, resulting in degradation and lower than expected advancing contact angle behavior. 2.4. Surface Characterization. Contact angle measurements of ≈100 nL droplets on all samples were performed using a microgoniometer (MCA-3, Kyowa Interface Science). Field emission scanning electron micrographs (QUANTA FEG 450 ESEM, FEI) were performed on all samples at an imaging voltage of 5.0 kV with 2.0 nm of spot size. 2.5. Experimental Setup. Rate of evaporation measurements were performed on a microgoniometer (MCA-3, Kyowa Interface Science) combined with a high speed camera (Fastcam SA2, Photron) (Figure 2). The samples were horizontally placed on a movable stage with 3D spatial control ability (X-Y-Z). Illumination was supplied by a LED source attached to the dispenser (AITECSYSTEM, TSPA22x8). Droplets were added on the sample through a 2D piezoelectric dispenser (X-Z) with voltage and frequency control. The experiments were carried out at 6 V, 5-120 Hz (), and a sample-dispenser spacing of 5-10 mm (ℎ). Imaging 7 ACS Paragon Plus Environment

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was performed at 7×, 12×, 18× and 24× magnification depending on the droplet size. Image acquisition was performed at 200 fps. All the experiments were performed under ambient conditions ( = 25 ± 0.5 ℃, = 50% ± 5%).

3. Results and Discussion 3.1. Steady Measurement Technique. To study droplet evaporation in ambient conditions, a high speed camera was interfaced with a frequency-controlled piezoelectric microgoniometer. The piezoelectric dispenser was placed 5 to 10 mm above the functional surfaces to grow droplets (Figure 2). The surface-dispenser spacing was chosen to have close control of the dispensed droplet trajectory, which is not possible at larger spacings due to the presence of random air currents which act to deflect microscale droplets. For the spacing range studied (5 to 10 mm), the evaporation dynamics of steady droplets were independent of dispenser to surface spacing. Due to the relatively small size of departing droplets (≈9 µm), the shape of droplets remained spherical during flight. This assumption is justified given that the Bond, Weber, and Capillary numbers are all much less than one (Bo = #$ % & /( ≪ 1, We = #- .& /( ≪ 1, Ca = - ./( ≪ 1, where % is the gravitational constant, . is the dispensing speed of droplets (≈ 1.5 cm/s), - is the vapor dynamic viscosity, #$ and #- are the liquid and vapor densities of water, respectively, and ( is the water liquid-vapor surface tension). For a detailed list of physical property values used in this study, please see Table S1 of the Supplementary Information. As the microscale droplets landed on the functional surface, they began to accumulate and grow to form a single larger droplet (Figure 3a). The camera focal plane was located at the growing droplet mid-plane for the desired optical zoom level. Subsequently, while the droplet was growing, contact angle measurements on the surface were performed in order to characterize functionality via the apparent advancing contact angle. Careful stroboscopic imaging of the dispensing droplets 5 mm beneath the piezoelectric dispenser confirmed the production of spherical and monodisperse droplets ( = 8.77 ± 0.56 8 ACS Paragon Plus Environment

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µm, Figure 2b) coming from the piezoelectric dispenser that were independent of both dispensing frequency (2 Hz < < 10 kHz) and driving voltage (3-9 V). After surface characterization and droplet formation, the frequency of deposition was modulated in order to find the dispensing frequency at which the stationary droplet size becomes spatially and temporally steady (Figure 3b). Once the droplet was steady, the frequency was noted, as well as the standard deviation from measurements made on multiple locations on the same sample. The frequency of deposition () that resulted in a steady droplet for a certain droplet diameter was then related to the corresponding rate of evaporation 12 = 1, where 1 is the mass of dispensed microscale droplets, 1 = 4/35 6 #$ . For each surface, multiple frequencies were probed in order to evaluate the droplet size and dependent evaporation dynamics. The piezoelectric head is capable of dispensing picoliter-scale droplets at frequencies of 2 – 10,000 Hz during measurement. The potential to introduce vibrations in the evaporating droplet by dispensing droplets for the piezoelectric head, which may cause the contact line to oscillate, was considered. The Ohnesorge number for a water droplet of the size observed during evaporation measurements (diameter 7 ≈ 50 µm) was Oh = $ /:#$ (7 ≈ 0.01. Therefore, viscous effects are neglected and a balance between inertial and surface tension forces is considered. Determining the droplet natural frequency for the first resonant mode in this case55-56: = ( [6(ℎA, B]/[#D A1 − cos , BA2 + cos , B] ≈ 6 kHz, where , is the apparent advancing contact angle of the droplet residing on the surface, and ℎA, B is a geometry factor57. The experimental data points were obtained with droplet dispensing frequencies of 5 – 120 Hz 20 µm, indicating that the steady approach is suitable for high accuracy characterization of droplet evaporation. It is important to note, due to the finite size of dispensing droplets ( ≈ 9 µm), the minimum recommended steady droplet size is ≈ 20 µm due to the added possibility of incoming dispensed droplets (at any frequency) affecting the local vapor diffusion dynamics surrounding the droplet residing on the surface. For the steady analysis of smaller droplets (< 20 µm), a piezoelectric dispenser having an ejection droplet size of ≪ 9 µm should be utilized to avoid coupling between the deposition and steady droplets. The steady technique of droplet evaporation characterization presents a few significant advantages when compared to classical transient based methods. The removal of moving liquid-vapor interfaces and three-phase contact lines during evaporation allows for the elimination of liquid-gas momentum exchange (i.e. motion of the liquid-vapor interface) during evaporation and the characterization of a true evaporation rate at a fixed geometry. The elimination of boundary motion is particularly important in evaporation processes where vapor diffusion is a limiting factor, such as that observed in NCGs58-59. Furthermore, the ubiquitous presence of contact angle hysteresis and pinning on surfaces of interest results in difficulty in obtaining consistent evaporation results at a fixed contact angle or droplet base radius. Instead, an amalgamation of data is obtained with varying droplet parameters which 10 ACS Paragon Plus Environment

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must be post processed in order to extract meaningful information. The steady technique on the other hand can probe the effects of droplet size, droplet base area, contact line length, and contact angle, with high precision, high accuracy, and elimination for the need to post-process data in order to obtain limited results. Lastly, the consistent addition of liquid at a constant enthalpy (temperature) from the piezoelectric dispenser ensures that the temperature profile inside the droplet remains steady and that self-cooling due to evaporation does not introduce secondary transient effects during the measurement27. 3.2. Droplet Radius. Figure 4a shows the droplet evaporation rate (12) as a function of the droplet radius of curvature (R) as determined by the steady technique. A linear trend was observed for all surfaces studied. Interestingly, the results show that the droplet contact angle did not play a large role in the rate of evaporation when the considered size parameter is droplet radius of curvature. Due to the inability to control for contact line length and interfacial area for the varying contact angles, we re-plotted our evaporation data as a function of droplet base radius as determined using the droplet radius and the contact angle information. Figure 4b shows the droplet evaporation rate as a function of droplet base radius, showing a linear dependence for each tested sample. However, the contact angle now plays a significant role. As the hydrophobicity of the sample increases (i.e. , increases), the rate of evaporation increases for the same droplet base radius. This was due to the increase in the exposed liquid-vapor surface area as the advancing contact angle increases, resulting in higher heat transfer with the environment60. It is important to note, however, that the rate of evaporation of the hydrophilic materials (Cu and Si-Au) are higher than Si-TFTS, which was hydrophobic, indicating that a balance between droplet conduction, contact line length, and liquid-vapor interfacial area govern the evaporation process. 3.3. Evaporative Flux. In order to have a better understanding of the droplet evaporation dynamics, the evaporative mass flux (1") was studied. The evaporative flux was evaluated by 11 ACS Paragon Plus Environment

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normalizing the rate of evaporation (12, Figure 4a, b) with the liquid-vapor surface area in order to include the contact angle effects in the analysis (Figure 5a). Examining Figure 5a, the effect of the advancing contact angle on the evaporation process can be understood as a balance between liquid-vapor interfacial area for vapor diffusion and efficient heat transfer through the droplet required to facilitate evaporation. The results show that the evaporative flux increases with decreasing contact angle for the same droplet base radius, indicating that efficient heat transfer between the surface and liquid-vapor interface is a key enabler for efficient vaporization. The thermal resistance due to heat conduction through the droplet can be approximated as61-63: IJKLM ≅

1 , 4O$ sin(, )

(1)

where O$ is the liquid water thermal conductivity. Assuming that vapor diffusion and evaporative cooling limitations are negligible compared to droplet conduction (large radius limit), the evaporative flux can be determined from 1" ~ ∆/IJKLM / & ~ 1/ ~ 1/ , in excellent agreement with the results of Figure 5a at larger length scales ( > 100 µm, for a log-log plot of Figure 5a, please see Figure S1 of the Supplementary Information). In addition, Eq. (1) shows that as the contact angle increases from hydrophilic to hydrophobic, the droplet conduction resistance increases and 1" decreases, as shown in Figure 5a. Note, Eq. (1) is a valid approximation for droplets residing on hydrophobic surfaces, and does not accurately predict thermal resistances of droplets on hydrophilic substrates, which continue to decrease as contact angle decreases, unlike that suggested by Eq. (1). 3.4. Vapor Diffusion. Droplet conduction may not be the only limit governing droplet evaporation. Indeed, vapor diffusion and evaporative cooling at the liquid-vapor interface have been shown to be a critical limit to evaporation in the presence of NCGs58-59 and at high heat fluxes64-65. In order to gain further insight, we compared the experimental results with the


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power law exponent model which can be utilized to explain vapor-diffusion controlled condensation and evaporation.66 In these experiments, due to the presences of NCGs, we expect vapor diffusion limitations to play a key role in droplet evaporation dynamics at low length scales where droplet conduction is not important ( < 100 µm) as well as at much larger length scales where droplet conduction affects vapor diffusion via the droplet temperature distribution, and the two couple to create non-uniform diffusive fields surrounding the evaporating droplets. At small droplet length scales ( < 100 µm) as the droplet evaporates, direct evaporation of water molecules from the liquid-vapor interface induces a 3D concentration gradient of water molecules inside the vapor surrounding the droplet. For a constant flux of water molecules leaving the water droplet surface, the concentration gradient should vary as 1/ &, resulting in a scaled droplet volumetric evaporation rate of: R 1 = 45 & ~ 45 & T & U 7V& S S

(2)

where, R is the volume of water droplet, 7V& is mutual diffusion coefficient of water in air. Rearranging Equation 2 reveals: 1" ~ (R/S)/ & ~ 1/ & ~ 1/ & , in excellent agreement with the results of Figure 5a at small length scales ( < 100 µm) (see Figure S.1). Note, the scaling analysis used to obtain Equation 2 is only an order-of-magnitude analysis and therefore is devoid of constants related to the droplet geometry. For larger evaporating droplets ( > 100 µm) which are conduction limited (Eq. 1), the majority of the evaporation occurs at the contact line as observed by the convergence of the evaporative fluxes at large base radii (Figure 5a). The resulting 2D concentration gradient varies as 1/. Thus, scaling analysis yields: R 1 = 45 & ~ 45 & T U 7V& S S 13 ACS Paragon Plus Environment

(3)

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Rearranging Eq. (3) reveals: 1" ~ (R/S)/ & ~ 1/ ~ 1/ , in excellent agreement with the results of Figure 5a at very large length scales ( > 200 µm, see Figure S.1). The power law results reinforce the critical importance that NCGs play in the dynamics of droplet evaporation. 3.5. Natural Convection. To investigate the effects of natural convection with the surroundings of the evaporating droplet, the Rayleigh number (Ra) of the droplet for the experimental conditions was evaluated. As the droplet size has an important effect on the Rayleigh number (~ 6 ), the Rayleigh number had a broad range for the droplets studied here, calculated as67: %X7 6 ∆ Ra = , YZ

(4)

where % is the gravitational constant, X is the thermal expansion coefficient, 7 is the diameter of the droplet, ∆ is the temperature difference between the ambient air and the evaporating droplet, Y is the thermal diffusivity and Z is the kinematic viscosity of air. Utilizing Eq. (4) yields Ra ≈ 0.08 for the largest droplet of interest ( ≈ 500 µm), revealing that the heat transport with the surroundings is governed by conduction since the maximum Rayleigh number was much smaller than the critical Rayleigh number for air (Ra ≪ RaJ = 1708)68-69. 3.6. Evaporative Cooling. Evaporative cooling of droplets has been shown to play an important role in the dynamics of droplet evaporation16-17. During evaporation, the absorption of latent heat results in the cooling of the droplet interface, which in turn reduces the local liquid temperature, vapor pressure, and hence evaporation rate. Evaporative cooling effects are especially more pronounced in high contact angle (> 90°) droplets due to larger liquidvapor interfacial area, leading to increased resistance to evaporation.


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For droplets undergoing evaporation, the relative effect of evaporative cooling compared to vapor diffusion can be characterized from the dimensionless Evaporative cooling number (Ec), defined as70: Ec =

ℎ\] 7- ^ , O$

(5)

where ℎ\] is the latent heat of evaporation of water, 7- is the diffusion coefficient of water vapor (subscript v) in air (subscript a), ^ = _`a ()/, and _`a () is the saturated vapor concentration of liquid water at temperature . For liquid water at room temperature (295 K) and pressure (101 kPa), Ec ≈ 0.11, below the critical value, EcJa = 1, implying that evaporative cooling had negligible influence on the droplet evaporation experiments conducted here. 3.7. Effect of Contact Angle. To further emphasize the scaling results and the delicate balance between vapor diffusion and droplet conduction, and to help visualize the effect of the contact angle, the evaporative flux data at constant base radii ( = 60 µm and = 100 µm) were compared (Figure 5b). The results show that droplets having lower contact angle are more sensitive to droplet size effects due to the low conduction resistance (Eq. 1), dominating 3D vapor diffusion resistance (Eq. 2) and mitigation of evaporative cooling due to the flat shape of the droplets16. For droplets having larger contact angles (> 90°), droplet size effects are less pronounced due to the conduction-limit enabling 2D vapor diffusion resistance (Eq. 3) which acts to govern evaporation by contact line length. The results indicate that the two regimes of evaporation governing behavior are delineated between hydrophilic and hydrophobic surfaces, in excellent agreement with numerical results obtained by previous studies16. 3.8. Comparison with Classical Transient Methods. In order to validate our method and to establish a comparison with the traditional derivative based technique, we performed 15 ACS Paragon Plus Environment

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additional experiments on the hydrophilic (Cu), hydrophobic (CNT1) and a superhydrophobic surfaces (CuO-HDTS) with the traditional method to obtain rates of evaporation. Briefly, the evaporation rate was calculated from image processing and using the rate approximation of (b/bS ≈ Δ/ΔS). Figure 6 shows the evaporation rate as a function of droplet base radius for the classical and steady approach. The transient method overestimates the rate of evaporation for each case due to the uncertainties associated with it such as variable contact angle during the process, pinning, and interfacial motion. In fact, pinning to the surface was observed during the experiments which led to a constant droplet base radius for most of the cases with varying contact angle; hence, an accurate calculation of the rate of evaporation cannot be performed due to this effect. Furthermore, the transient method experiments are conducted by turning the droplet flow off, and allowing evaporation to take place as the droplet recedes. Thus, the rate of evaporation is obtained for receding contact angle values. On the contrary, the steady method developed here utilizes continuous droplet flow and is hence characteristic of the droplet evaporation rate in the advancing state with little variability due to pinning. A secondary drawback of the classical transient method is need to measure time as an additional parameter in the calculations since transient effects are observed due to the changes in the size and shape, especially for a small droplets. The need for temporal and spatial information can therefore increase error in the calculation. However, for the steady technique, we eliminate the need for including time as a parameter and only match the frequency of deposition to the frequency of evaporation to get a higher fidelity estimation of the rate of evaporation. Finally, due to the changes in the droplet size, vapor diffusion from the droplet also induces errors in the classical transient based method. However, for droplet evaporating at steady state; vapor effects are steady as well since no driving forces exist to induce changes on the vapor dynamics in the vicinity of the droplet. Hence, for applications


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requiring steadiness of evaporating droplets, the newly developed steady method corrects the shortcomings of the traditional unsteady methods. The steady method developed here presents a unique platform to study the evaporation dynamics of droplets. In addition to higher accuracy, the approach allows for the study of a multitude of working fluids at length scales not previously possible using transient based methods. For example, low surface tension fluids such as ethanol, methanol, and toluene are notoriously difficult to dispense in small quantities on omniphobic surfaces. The use of a piezoelectric dispenser eliminates this problem by quantizing the dispensed fluid to a small volume and ‘building’ it on the surface independently of the dispenser. In the future, it would be interesting to use the steady approach to build an evaporation rate database for a multitude of fluids and conditions. Furthermore, the inexorable presence of contact angle hysteresis on any real surface introduces a range of contact angles a droplet can exist in for a single droplet base radius in the pinned state. The steady approach overcomes this limitation by enabling the study of droplet evaporation on a surface for the entire range of contact angles between the receding and advancing state without the need for crude approximations. Future studies should use the data presented here as a benchmark for 3D numerical simulations of droplet evaporation,

which classically use non-moving boundaries and the

quasi-steady

approximation71-72. Furthermore, the steady approach developed here has a plethora of other applications where the volume flux of a working fluid needs to be measured, whether it be permeation through a membrane73, or the Wickability of a superhydrophilic surface74-75. Indeed, the use of the steady approach to characterize Wickability of surfaces would eliminate the need for contact based methods which may introduce errors in the measurement. 4. Conclusions In this study, a steady evaporation method is developed for surfaces of different functionalizations (i.e. hydrophilic or hydrophobic). By utilizing a piezoelectric dispenser to 17 ACS Paragon Plus Environment

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feed microscale droplets ( ≈ 9 µm) to a larger evaporating droplet at a prescribed frequency, we studied water evaporation of droplets having base radii ranging from 20 µm to 250 µm on surfaces of different functionalities (45˚ ≤ , ≤ 162˚, where , is the apparent advancing contact angle). Utilizing our steady method we showed that the rate of evaporation has a linear dependence on the droplet base radius; and the evaporative flux has a linear dependence on contact angle for both hydrophobic and hydrophilic samples. We benchmarked our technique with the classical unsteady method showing significant improvement (≈ 140%) in evaporation rate measurement accuracy for droplets evaporating in steady state where the contact line and interfacial effects are negligible. Our work not only sheds light into the evaporation dynamics for various droplet sizes and contact angles on functional surfaces, it provides an experimental platform that enables the decoupling of the intricate and length-scale-dependent balance played by internal and external flows, evaporation kinetics, thermocapillary motion, and binary-mixture dynamics. ASSOCIATED CONTENT Supporting Information The supporting information is available free of charge on the ACS Publications website at DOI: Logarithmic plots of the evaporative flux, and a table of values of the parameters used in the calculation of non-dimensional numbers. Author Contributions A.A.G. and N. M. conceived the initial idea for this study. N. M. guided the work. A.A.G and S.S. fabricated, functionalized and characterized the samples and carried out the experiments. J.O. carried out SEM imaging. A.A.G, S.S. and J.O analyzed the data. All authors contributed to the writing of the manuscript and have given approval to the final version of this manuscript. 18 ACS Paragon Plus Environment

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Notes The authors declare no competing financial interest.

Acknowledgments The authors would like to thank Xiao Yan for his support in image analysis. The authors gratefully acknowledge funding support from the National Science Foundation under Award No. 1554249. N.M. gratefully acknowledges funding support from the International Institute for Carbon Neutral Energy Research (WPI-I2CNER), sponsored by the Japanese Ministry of Education, Culture, Sports, Science and Technology. SEM imaging was carried out at the Beckman Institute for Advanced Science and Technology, University of Illinois.


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Figures

Figure 1. Scanning electron microscopy images of (a) gold coated silicon wafer (Si-Au) (de = 45.1° ± 5.2°), (b) polished copper (Cu) (de = 85˚ ± 1.9˚), (c) TFTS coated silicon wafer (SiTFTS) (de = 110.6˚ ± 2.8˚), (d) 90 nm P2i Carbon Nanotubes (CNT1) (de,eff = 130.6˚ ± 2.9˚), (e) 30 nm P2i Carbon Nanotubes (CNT2) (de,eff = 141.3° ± 3.1°), and (f) HDTS Coated Copper Oxide (CuO-HDTS) (de,eff = 162° ± 2.9°).


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Figure 2. (a) Schematic of the experimental setup. The substrate is placed on top of the stage at ambient temperature, and droplets grown on the substrate. The piezoelectric dispenser is placed a distance g above the substrate, and forms small microdroplets (≈ 9 µm diameter) to feed the bigger droplet growing on the substrate. The droplet growth process is monitored by a high speed camera from the side. Images from the high speed camera of the (b) strobed dispensing microdroplets, and the growing droplet on the substrate (c) right before and (d) right after incoming droplet coalescence. The small droplet on top of the large droplet residing on the surface in figure (c) is the droplet dispensed by the piezoelectric dispenser.


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Figure 3. Side view time lapse images of droplet growth for frequency of microdroplet deposition of (a) h = 5 Hz, h = 10 Hz, and h = 15 Hz. (b) Droplet radius as a function of time for different frequency of deposition. It is seen that the droplet asymptotes a constant size as the deposition time increases due to the matching of rate of evaporation and rate of deposition at a certain droplet radius.


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Figure 4. Rate of evaporation as a function of the (a) droplet radius and (b) droplet base radius for all tested samples. The rate of evaporation varies linearly with increasing droplet size due to the linear dependence of contact line length with base radius.


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Figure 5. Droplet evaporative flux as a function of the (a) droplet base radius (ij ) and (b) droplet advancing contact angle (de). A cut-off exists at de = kl° due to the change in the cause of evaporation (for a hydrophilic sample, the heat transfer from the base governs evaporation; whereas evaporation is governed by vapor diffusion for a hydrophobic sample). For log-log plots of (a) with associated slopes, please see supplementary information, Figure S1.


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Figure 6. Droplet rate of evaporation characterized by the traditional transient method (solid symbols) and the steady method (hollow symbols). For all surfaces tested, the traditional method overestimates the rate of evaporation by approximately 140% due to the inner transient effects and the liquid-gas momentum exchange at the interface. The main sources of error included measurement error of droplet size for the traditional method, and error due to spread in the dispensed droplet size (≈ ±6%) for the steady method.


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Figure 1. Scanning electron microscopy images of (a) gold coated silicon wafer (Si-Au) (θ = 45.1° ± 5.2°), (b) polished copper (Cu) (θ = 85˚ ± 1.9˚), (c) TFTS coated silicon wafer (Si-TFTS) (θ = 110.6˚ ± 2.8˚), (d) 90 nm P2i Carbon Nanotubes (CNT1) (θ = 130.6˚ ± 2.9˚), (e) 30 nm P2i Carbon Nanotubes (CNT2) (θ = 141.3° ± 3.1°), and (f) HDTS Coated Copper Oxide (CuO-HDTS) (θ = 162° ± 2.9°). 437x192mm (96 x 96 DPI)

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Figure 2. (a) Schematic of the experimental setup. The substrate is placed on top of the stage at ambient temperature, and droplets grown on the substrate. The piezoelectric dispenser is placed a distance h above the substrate, and forms small microdroplets (≈ 9 µm diameter) to feed the bigger droplet growing on the substrate. The droplet growth process is monitored by a high speed camera from the side. Images from the high speed camera of the (b) strobed dispensing microdroplets, and the growing droplet on the substrate (c) right before and (d) right after incoming droplet coalescence. The small droplet on top of the large droplet residing on the surface in figure (c) is the droplet dispensed by the piezoelectric dispenser. 155x115mm (120 x 120 DPI)


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Figure 3. Side view time lapse images of droplet growth for frequency of microdroplet deposition of (a) f = 5 Hz, f = 10 Hz, and f = 15 Hz. (b) Droplet radius as a function of time for different frequency of deposition. It is seen that the droplet asymptotes a constant size as the deposition time increases due to the matching of rate of evaporation and rate of deposition at a certain droplet radius. 180x227mm (120 x 120 DPI)


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Figure 4. Rate of evaporation as a function of the (a) droplet radius and (b) droplet base radius for all tested samples. The rate of evaporation varies linearly with increasing droplet size due to the linear dependence of contact line length with base radius. 274x106mm (120 x 120 DPI)


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Figure 5. Droplet evaporative flux as a function of the (a) droplet base radius (R_b) and (b) droplet advancing contact angle (θ_a). A cut-off exists at θ_a=90° due to the change in the cause of evaporation (for a hydrophilic sample, the heat transfer from the base governs evaporation; whereas evaporation is governed by vapor diffusion for a hydrophobic sample). For log-log plots of (a) with associated slopes, please see supplementary information, Figure S1. 260x107mm (120 x 120 DPI)


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Figure 6. Droplet rate of evaporation characterized by the traditional transient method (solid symbols) and the steady method (hollow symbols). For all surfaces tested, the traditional method overestimates the rate of evaporation by approximately 140% due to the inner transient effects and the liquid-gas momentum exchange at the interface. The main sources of error included measurement error of droplet size for the traditional method, and error due to spread in the dispensed droplet size (≈ ±6%) for the steady method. 101x70mm (600 x 600 DPI)


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