Superhydrophobic Surface Curvature Dependence of Internal

Subscriber access provided by Iowa State University | Library

Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

Superhydrophobic surface curvature dependence of internal advection dynamics within sessile droplets Purbarun Dhar, Gargi Khurana, Harikrishnan Anilakkad Raman, and Vivek Jaiswal Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03932 • Publication Date (Web): 15 Jan 2019 Downloaded from http://pubs.acs.org on January 17, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Superhydrophobic surface curvature dependence of internal advection dynamics within sessile droplets

Purbarun Dhar1, *, Gargi Khurana1, Harikrishnan Anilakkad Raman2 and Vivek Jaiswal1

1

Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar–140001, India

2Department

of Mechanical Engineering, Indian Institute of Technology Madras, Chennai–600036, India

* Corresponding author E–mail: [email protected] Phone: +91–1881–24–2173

Abstract Sessile droplets seated on superhydrophobic surfaces are known to exhibit internal circulation patterns. The present article experimentally demonstrates and theoretically confirms for the very first time that the nature and velocity of the internal circulation in sessile droplets seated on superhydrophobic surfaces is strongly governed by the curvature of the surface and its 1 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

directionality. Sessile droplets were rested on concave and convex superhydrophobic surfaces, both with one curvature (cylindrical) and two curvatures (spherical) and varying droplet diameter to curve diameters were studied. Particle Image Velocimetry (PIV) was employed for flow visualization and quantification. It was observed that that increasing convexity of the surface leads to deterioration in the velocity of advection within the droplet, whereas increasing concavity of the surface augments the velocity of circulation. A scaling model based on the effective curvature modulated change in wettability has been put forward to predict the phenomenon, but it was found to be weak in deducing the circulation velocities. Consequently, potential flow theory is appealed to and the curvatures are approximated as equivalent wedges, with the rested droplet engulfing the wedge partly. Based on the curvature of the surface, the equivalent included wedge angle is deduced. Flow theory over wedged structures is employed to deduce the changes in the internal velocity in the presence of curved surfaces. The spatiotemporally averaged experimental velocities are found to conform to predictions from the proposed model and good agreement between the theoretical predictions and experimental observations is achieved. The present findings may have strong implications in thermofluidics transport phenomena or multiphase transport processes at the interfacial and/or microscale.

Keywords: Superhydrophobicity, droplets, curvature; potential flow, internal circulation, flow visualization

2 ACS Paragon Plus Environment

Page 2 of 25

Page 3 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Introduction Fluid dynamics, thermal and species transport in droplets has been an area of tremendous academic interest, since droplets are essential working components in several scientific and engineering applications. It is thereby of major importance that transport phenomena of droplets be understood for design and development of such applications and processes. Droplets form the backbone of a wide gamut of engineering and technological applications and utilities. Typical examples involve internal combustion engines1, 2 (where the fuel is sprayed into the combustion chamber and the spray consists of a population of microscale droplets), microfluidic biological manipulation3, 4 (such as lab-on-a-chip devices which are used for sorting fatty emulsions and globules for testing), and spray coating5 (largely used in industrial processes for painting, polishing and for erosion and corrosion resistance). Other major areas of application are in drug administering6 (such as in nebulizers involving liquid medicines and in medical sprays), in printing technology7, 8 (the whole process of printing is dependent on the proper deposition and drying of microscale ink droplets), in agriculture9,

10

(where crop spraying with fertilizers,

pesticides, fumigants, etc. is common practice, and in moisture controlled greenhouses), cooling technologies11(such as cooling towers, desert coolers, in HVAC systems, humidifiers, etc.) and so forth. Droplets are broadly classified into the pendent (suspended) and sessile (seated). Hence understanding the fluid dynamics, heat and mass transfer in droplets is of essential importance for thermo-hydrodynamic optimization and economization of such systems.

Droplets have been categorized based on their configuration, and the two major modes are the pendent drop and the sessile drop. As the name suggests, the typical pendent drop


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 25

suspends freely from a needle or a surface in the form of a pendent or a bulb by virtue of its surface tension. Since the interaction with the adjoining surface is minimal, pendent drops are studied due to their close similarity to freely suspended droplets which are in general encountered in applications. On the contrary, sessile droplets are seated on a given surface and the equilibrium of the interfacial tensions between the solid, liquid and ambient gas phase determines the shape of the seated droplet. Compared to the pendent droplet, the sessile counterpart poses a more complex phenomenon as its shape and size is determined by the equilibrium of three phases (liquid, surrounding gas and solid surface). Additionally, sessile droplets are studied widely to understand wetting dynamics, spreading behavior, heat and mass transfer characteristics in utility systems where droplets and surfaces interact. Consequently, thermofluidic behavior in sessile droplets has received vast attention among academic researchers. Typically droplet impact and spreading events12-14, thermal transport in droplets15 and mass transfer to and from droplets16, has received extensive attention among academicians. Major areas of studies have been impact dynamics and coalescence behavior of sessile droplets, wettability modulated spreading and heat transfer in such droplets, and evaporation kinetics of sessile droplets under different conditions. The fluid dynamics and thermal transport behavior in sessile droplets are strongly governed by the wettability of the surface, and work in this direction, ranging from superhydrophobicity (SH)

17, 18

to superhydrophilicity19 have received

immense focus in the last few years. Self-cleaning surfaces, water repellent coatings, oleophobic surfaces, anti-fog and anti-frost surfaces, fog harvesting, etc. are typical examples of research avenues in the area of wettability modulation.


Page 5 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Among the more interesting phenomena in sessile droplets, typically in case of those resting on SH surfaces, is the presence of prominent internal fluid circulation or advection patterns20-22. This is brought about by the Marangoni stresses generated across the droplet interface due to proximity to the surface molecules which have aversion towards water molecules. The stresses generated, along with mass continuity principle, leads to advective motion within an otherwise stationary droplet. As time evolves and the droplet evaporates, the thermal gradients generated within the droplet further aids the Marangoni circulation, leading to increased circulation velocities. In the event of hydrophilic surfaces, the circulation is very weak and only confined to regions close to the three phase contact line. Interestingly, pendent droplets also exhibit internal circulation when thermal or solutal Marangoni advection is forcefully imposed23,

24,

either by the presence of solvated ions, or due to externally imposed thermal

conditions, or due to evaporation. There have been several studies involving simple and complex fluids on superhydrophobic surfaces 25-28. The role of internal circulation has been shown to be a governing phenomenon behind several mechanisms, such as evaporation behavior, colloidal particle deposition patterns, etc. The motivation behind the present study derives clue from reports in literature that the curvature of a surface essentially modulates the effective wettability of a droplet29,

30.

It has been shown that the effective contact angle can be varied by changing the

curvature of the surface a droplet rests on. Consequently, the modulated wettability would thus be expected to govern the strength and pattern of internal circulation in sessile droplets seated on curved SH surfaces. The present article describes a novel phenomenon corroborating this theory and experimentally and analytically illustrates the curvature dependent modulation of internal circulation dynamics on SH surfaces. Flow visualization employing Particle Image Velocimetry has been conducted on sessile droplets seated on variant curved SH surfaces, and the internal


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

velocity field has been mapped. A theoretical formalism using potential flows over wedged structures has been put forward and good agreement between the theoretical velocities and experimental observations have been noted.

Materials and methodologies The study involved a customized experimental setup. The typical arrangement of the experimental setup has been illustrated in Figure 1. It consists of a digitized droplet dispensing mechanism, where a precision syringe pump (Holmarc Optomechatronics Ltd. India) is used to pump the fluid though a flexible, polyurethane tubing (Unitech, India). The syringe pump is capable of dispensing accurate to ± 0.5μl. The tubing is connected to a dispensing needle (a flat end 21 gauge needle, Unitech, India) which generates the droplet. The syringe and the tubing are initially primed with water to eliminate any bubbles and provide uninterrupted flow during droplet generation. Sessile droplets of average volume ~15 μl are carefully dispensed onto the superhydrophobic (SH) surfaces, ensuring that the droplet does not roll off the curved surface with low wettability. The droplet volume is chosen such that the final droplet diameter is below the capillary length scale for water (to ensure the role of gravity on the flow dynamics is negligible) and the droplet is sufficiently large to obtain proper imaging for the PIV. The surfaces used are of three different types, a) cylindrical rods (of polished stainless steel) (such rods promote single positive curvature surfaces), b) spheres (of polished stainless steel) (spherical balls promote dual positive curvature), and c) concave semi-cylindrical grooves (of stainless steel, and manufactured employing an automated wire-cut micro electro discharge machining (EDM)) (such concave cylindrical grooves promote single negative curvature).


Page 6 of 25

Page 7 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

A control case has also been employed using a flat polished stainless steel (such a surface promotes zero curvature). Different cylinder, sphere and cylindrical groves of different diameters were fabricated to understand the role of curvature and its magnitude. All the mentioned shapes were rendered SH by spray coating with a commercial SH twin component infusion (Rust-oleum Never Wet, USA). The infusion consists of two liquids which are sprayed on to the surfaces, yielding superhydrophobic surfaces due to the chemical structure of the constituent liquids in the infusion. Deionized water droplets exhibits fairly weak internal circulation24 which is often intermittent and hence difficult to map using velocimetry. Accordingly, droplets of dilute solution (0.025 M) of aqueous NaCl solution (analytical grade, Sigma Aldrich, India) are employed. The presence of a minute amount of salt has been shown to enhance the internal circulation within droplets due to solutal Marangoni advection, without any appreciable changes in the values of viscosity or surface tension24. The static contact angle of the solution on flat SH surface is determined to be ~ 157 ± 1.5oand the roll-off angle as < 3o. Consequently, the SH surfaces were carefully positioned to ensure that the droplet rests on the surface as motionless and does not tend to roll off.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1: Schematic of the experimental setup used in the present work. 1) acrylic chamber to eliminate external convective disturbances 2) SH surface of particular curvature 3) (Light Emitting Diode) LED array controller 4) LED backlight with intensity control 5) syringe pump system attached to glass syringe and flexible tubing 6) computer for camera control and data acquisition 7) needle assembly to generate droplet from the tubing 8) laser with lens assembly to generate light sheet 9) laser controller and power source 10) (Charged Coupled Device) CCD camera with microscopic lens assembly and 11) digital thermometer and hygrometer.

The dilute salt solution is seeded with neutrally buoyant (with respect to water at 300 K), fluorescent polystyrene particles (~20 µm diameter, Cospheric LLC, USA) for flow visualization and quantification using Particle Image Velocimetry (PIV). A charge-coupled device (CCD) based monochromatic camera (Holmarc Optomechatronics Ltd. India) with a microscopic lens assembly and an intensity controllable light emitting diode (LED) array backlight (DPLED 5S, China) is used to focus the droplet image using an image acquisition software. The camera is operated at 120 pixels/mm resolution at 20 fps for the velocimetry. A continuous wave laser, of wavelength 532nm and 15mW peak power (Roithner GmbH, Germany) is used as the source of illuminationfor the PIV. A laser sheet of ~0.25 mm thickness is used to illuminate the droplet 8 ACS Paragon Plus Environment

Page 8 of 25

Page 9 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

interior using a plano-convex cylindrical lens. Major care has been taken to ensure that the approximate vertical midplane of the droplet is illuminated by the light sheet, to ensure reduction in the noise in the PIV images. Once the camera is focussed onto the droplet midplane, the LED backlight is extinguished and the laser is used as the illumiation source.The PIV data acquisition was initiated within 2-3 minutes of placing the droplet, thereby exterminating possibilities of evaporation driven internal advection in addition to the surface energy driven advection24, 23. The PIV images were recorded for a time frame of 90 s. the images were pre-processed with contrast enhancement algorithms and intensity capping was done to reduce the noise signals from stray pixels. For post-processing, a cross-correlation, four-pass algorithm with subsequent interrogation windows of 64, 32, 16 and 8 pixels was used to obtain high signal-to-noise ratio in the velocity quantification. The typical signal-to-noise ratio was estimated to be ~8-9 for the present PIV studies. The data was processed for typically 1200 images for each case in the open source code PIVlab. Finally, the time-averaged, mean spatial velocity contours and vectors were obtained from the set of 1200 processed images. From the time-averaged, mean spatial velocity contour at the vertical mid-plane of the droplet, the average advection velocity within the droplet was obtained by considering the area averaged velocity in the mid-plane.The complete experimental setup was housed in an acrylic chamber to eliminate convective disturbances from the ambient and maintain nearly constant ambient conditions during the short period of imaging. A digital thermometer and hygrometer was employed to monitor conditions 10 mm away from the droplet (using a sensing probe attached to the digitized meter). All experiments were performed at 30 ± 1 o C and relative humidity of 48 ± 2.5 %.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Results and discussions Observations and results This section discusses the main results obtained from the flow visualization studies and the possible mechanisms by which the curvature of the SH surface modifies the internal flow dynamics. Experimentally, the internal advection dynamics were studied for droplets rested on SH convex cylinders of radii 5, 3.5 and 2 mm, on convex spheres of radii 4.75, 3.5 and 2.35 mm, andon concave cylindrical grooves of radii 3, 2 and 1 mm were studied. As visualization is not practically feasible for droplets seated within hemispherical grooves, the concave cases have been restricted to cylindrical grooves only. The droplet was carefully rested axisymmetric on the curved surfaces and flow visualization was performed as per conditions discussed previously. The velocimetry analysis has been done typically for 1000 images. Each image is correlated with its immediate successor image, and the velocity contours for each time frame is determined. The average velocity for each time frame is also determined by considering the area weighted average velocity at the interrogation plane of the droplet. When plotted with respect to time, it is observed that the average velocity exhibits random fluctuations about a mean value. Consequently, the time averaged mean velocity contour is determined for the interrogation period and is considered as the typical average circulation behavior within the droplet during the velocimetry period. The spatially averaged velocity of the time averaged contour gives the magnitude of the mean velocity of circulation within the droplet during the interrogation period. Fig. 2 illustrates the time averaged spatial velocity contours and vector field (at the droplet midplane) for droplets seated on different convex SH curved systems compared to the flat SH case. It is observable that while the circulation pattern is more or less retained throughout the droplet, the overall strength of the mean circulation velocity, as well as the maximum velocity within the 10 ACS Paragon Plus Environment

Page 10 of 25

Page 11 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

droplet diminishes as a function of increasing convex curvature. Additionally, it is observed that the reduction in velocity is further augmented in case of spheres than the cylinders, which signifies that the magnitude of curvature as well as the nature of the directrix of the curve governs the internal circulation behavior.

The fact that spheres has two curvatures (the

generatrix as well as the directrix); causes them to promote greater resistance to the flow than the single curvature of cylinders (the directrix being a line parallel to the axis). A few additional velocity contours have been illustrated in the supporting information document a.

Figure 2: Velocity contours and vector plots for internal circulation in droplets on (a) flat (b) 5 mm convex cylinder (c) 2 mm convex cylinder (d) 4.75 mm concave sphere and (e) 2.35 mm concave sphere SH surfaces. The volume of the droplet is maintained constant for all cases.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The contours and velocity fields have been illustrated for concave cylindrical grooves in fig. 3. In these cases, an opposite trend is observed and augmentation in the average velocity of circulation is noted. Additionally, the velocity is noted to increase with increased concave curvature. It is noteworthy however, that for droplet radius equal to concave groove radius (illustrated in fig. 3 (c)) the velocities drastically reduce. While it may seem counter-intuitive when compared against the other cases, a closer introspection reveals physical consistency. In case the droplet diameter is the same as (or marginally larger) than the groove diameter, a large portion of the lower hemisphere of the droplet is completely in contact with the surface of the grooved region. This leads to the wall shear resistance to the internal flow features, which in turn exhibits severely weakened velocity. It is known from literature29 that the advection within droplets becomes prominent as the hydrophobicity of the surface increases. In case of curved interfaces, the effective contact angle at the three phase contact line changes due to the local curvature of the liquid-solid contact, thereby modifying the effective wettability of the surface. The effective contact angle on such curved surfaces can be deduced from image analysis and geometrical considerations, along the lines detailed and illustrated in Fig. 4. The effective contact angle for a droplet seated on a curved superhydrophobic (SH) surface is different from its static contact angle on a flat SH surface. This effective change in contact angle essentially signifies that the wettability on a curved surface is a function of the curvature29.As perfig 4, it may be considered that the static contact angle on a flat SH surface is θ. It is assumed that a droplet of the same volume is now rested onto a surface with a known curvature (inverse of radius of the curve, a) of the same SH material. The convex curvature has been treated as positive, whereas the concave curvature has been treated as negative and the sign needs to be incorporated in the curve radius a, which is aligned differently for different curvatures with respect to the droplet


Page 12 of 25

Page 13 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

center. If the frontal projected length of the contact diameter (c)is determined, the effective contact angle (θ’) can be mathematically expressed from geometrical considerations as29 𝜃′ = 𝜃 ― 𝑠𝑖𝑛 ―1(𝑐 𝑎)

(1)

Figure 3:Velocity contours and vector plots for internal circulation in droplets on superhydrophobic (a) flat (b) 3 mm concave cylindrical groove (c) 1 mm concave cylindrical groove. The droplet volume is maintained constant in all cases.

It may be readily observed (from fig. 4) that with increasing positive curvature (or convexity) or decreasing curve radius, the difference between the effective contact angle and true contact angle increases (on the equivalent flat surface), indicating higher effective wetting. From an intuitive standpoint also this mechanism is correct. A droplet sitting on a spherical cap engulfs a larger contact surface compared to its flatter counterparts, thereby making if a more wetted contact. On the contrary, increased concavity reduces the contact area of the droplet, leading to augmented

effective

contact

angle

and

improved

hydrophobicity.


The

deteriorated

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

hydrophobicity in convex SH surface signifies that it has shifted a little closer to the hydrophilic regime from the hydrophobic regime (on the equivalent flat case), which qualitatively explains the reduction in the magnitude of circulation velocity (droplets exhibit mostly stagnant interiors on hydrophilic surfaces12, 30)and the same reasoning explains augmented circulation velocity with increasing concavity. Thereby, it can be proposed that the ratio of contact angles with respect to the flat case could be a measure of the change in wettability (the ratio, θ’/θ will be > 1 for increasing concavity and < 1 for increasing convexity). Based on order of magnitude scaling, it could be argued that the ratio of average circulation velocity in curved surface case (Uc) to that of the flat case (Uf) will scale as (Uc/ Uf)≈(θ’/θ)n, where n is a real number (n>0). However, analysis shows that the typical scaling does not hold true and the experimental observations do not comply with the scaling over the whole curvature spectrum. Thereby, while the effective wettability ratio might be important and explains the phenomena qualitatively, it alone cannot predict the observations fully and additional mechanisms need to be probed.

Figure 4: Geometric considerations to determine effective contact angles (θ’) for curved surfaces on the superhydrophobic I) flat II) convex and III) concave cases. Based on the effective


Page 14 of 25

Page 15 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

wetted diameter (c) and the curvature of the surface (a-1), the effective contact angle at the three phase contact point can be deduced.

Mathematical analysis Consequently, it is imperative that an improved mathematical approach is employed to model the phenomena and deduce circulation velocity values as function of curvature, given that the circulation velocity on the flat counterpart is already known. A major component which is expected to modulate the flow within the drop would be the presence of the curvature, which the droplet engulfs partially while seated. This would establish itself as a partial bluff body which modulates the flow behavior within the drop, which is otherwise (on a flat surface) similar to a plane circulation flow. The presence of the hump or depression at the base of the rested droplet distorts or modulates the otherwise plane circulation flows. It may be seen from the experimental velocity fields that the flow is modulated to one where the streamlines near the base of the droplet typically try to follow the profile of the curved surface. Consequently, a feasible approach could be to model the flow behavior using potential flow theory over angular deformations (commonly called wedge flows), with the curvatures being approximated as an equivalent wedge. Fig. 5illustrates the method of approximation of a curved interface to an equivalent wedge with the included angle α (in radians).Based on the knowledge of the droplet contact radius when seated in equilibrium and the radius of the curved surface, a triangular geometry can be conceived (as shown in fig. 5) and the included angle α can be determined. For wedge flows, the complex flow potential ω can be expressed as a function of z (where z=reiθ represents the complex coordinate plane). Here r and θ represent the radial and angular components of a polar coordinate system with origin positioned at the center of the curvature


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

(Fig. 5).From potential flow theory of wedge flows, the flow potential is expressible in an argand plane as 𝜔(φ,ψ) = 𝑉𝑧(𝑟,𝜃)𝑚

(2)

Where the index m represents the nature of the wedge flow (m=π/α) and V is a real number constant.

Figure 5: Geometric considerations to determine effective wedge angle (α) of flow and engulfment angles (β1 and β2) for curvatures, (left) convex and (right) concave cases. α is the included angle of the wedge approximated curvature. β1 and β2 represent the left hand and right hand engulfing angle of the droplet with respect to the curvature center.

The complex flow potential can be further decomposed into the real and imaginary components, viz. the velocity potential (φ) and stream function (ψ), in the form𝜔 = 𝜑 + 𝑖𝜓. From the definition of the stream function in the polar coordinate system, the radial (Um,r) and angular (Um,θ) components of circulation velocity within the droplet mid-plane, while seated on a


Page 17 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

wedge-approximated-curvature (of included wedge angle m) can be expressed as per eqns. 3 and 4 respectively. 𝑈𝑚,𝑟(𝑟,𝜃) = 𝑚𝑉𝑟𝑚 ― 1cos (𝑚𝜃)

(3)

𝑈𝑚,𝜃(𝑟,𝜃) = ―𝑚𝑉𝑟𝑚 ― 1sin (𝑚𝜃)

(4)

Since the PIV studies have been performed at the vertical mid-plane of the droplet, the flow field has been evaluated at the mid-plane of the droplet and the flow field is thus expressed as a 2D field. To introduce the effect of curvature (with respect to the center of the curved surface), the variable r is non-dimensionalized (r*=r/d) with the distance between the centers of the droplet and the curved surface (where d=a+rd for convex surfaces and d=a–rd for concave surfaces, where rdis the radius of the droplet, which is essentially half of the droplet height). Upon introducing r*, eqns. 3 and 4 possess the dimension of velocity in the left hand, thereby forcing that the variable V is also a similar term. It is scaled here that the variable V is the spatially averaged circulation velocity for the flat SH case (Uf). In order to evaluate for the spatially averaged components of velocity, the expressions obtained from eqns. 3 and 4 are integrated with respect to radial direction (from the droplet-curvature interface to the apex of the droplet) and with respect to angular direction (from the left hand engulfing angle, β1, to the right hand engulfing angle, β2) (fig. 5).The expressions to be evaluated are as shown in eqns. 5 and 6.

𝑈𝑚,𝑟 𝑈𝑓

𝑈𝑚,𝜃 𝑈𝑓

𝑚

(

𝑎 + 2𝑟𝑑 ∗ 𝑚 ― 1

(

𝑎 + 2𝑟𝑑 ∗ 𝑚 ― 1

= 2𝑟𝑑(𝛽2 ― 𝛽1) ∫𝑎 ―𝑚

= 2𝑟𝑑(𝛽2 ― 𝛽1) ∫𝑎

𝑟

𝑟

)(

𝛽

)(

𝛽

)

(5)

)

(6)

dr ∗ ∫𝛽2cos (𝑚𝜃)𝑑𝜃 1

dr ∗ ∫𝛽2sin (𝑚𝜃)𝑑𝜃 1


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 25

Where, the overhead bars represent the spatial average value of that particular variable. Upon evaluating Eqns. 5 and 6, the expressions for the spatially averaged velocities for a convex system can be expressed as 𝑛

[

𝑈𝑚,𝑟

(𝜃′ 𝜃)

𝑈𝑓

2𝑚𝑟𝑑(𝛽2 ― 𝛽1)

| |

(𝜃′ 𝜃)

| |= 𝑈𝑚,𝜃 𝑈𝑓

𝑛

(𝑎 + 2𝑟𝑑)𝑚 ― 𝑎𝑚

[

= 2𝑚𝑟𝑑(𝛽2 ― 𝛽1)

(𝑎 + 𝑟𝑑)𝑚 ― 1

][𝑠𝑖𝑛(𝑚𝛽 ) ― 𝑠𝑖𝑛(𝑚𝛽 )]

(7)

][𝑐𝑜𝑠(𝑚𝛽 ) ― 𝑐𝑜𝑠(𝑚𝛽 )]

(8)

(𝑎 + 2𝑟𝑑)𝑚 ― 𝑎𝑚 (𝑎 + 𝑟𝑑)𝑚 ― 1

2

1

2

1

For a concave system, the geometry illustrated in fig. 5 is used to determine the integration limits. The limits of integration for the radial component will be –a to (2rd – a) while the radial span remains from β1 to β2. The integrals are evaluated in a similar manner as eqns. 5 and 6. In addition to the wedge flow features, the modulated contact angle due to the curved surfaces also play a minor role in governing the internal circulation, as discussed in the previous sections. To incorporate the effects due to the change in effective wettability, the scaled contact angle ratio is introduced into the final expressions in eqns. 5 and 6. The average radial and angular components of velocity are determined from eqns. 5 and 6 and the spatially averaged velocity at the central plane of the droplet is determined as𝑈𝑚 = 𝑈𝑚,𝑟2 + 𝑈𝑚,𝜃2.It is observed that for single curvature surfaces, the index n=1, while for dual curvatures; n=2provides consistent predictions of the circulation velocity. Thus, essentially, n represents the number of curvatures. The theoretically deduced Um are compared with respect to experimental spatiotemporally averaged velocities (at the droplet mid-plane) in Fig. 6 and good predictability is observed for the model. Since mid-plane visualization of the internal circulation is not feasible for doubly concave surfaces (hemispherical indents or depressions), the same has not been 18 ACS Paragon Plus Environment

Page 19 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

considered in the present study.The present theoretical analysis fails when the droplet radius is equal to (or less than) the curvature radius for concave systems. This is observed from fig. 3 c (as well as from fig. 6), where the velocity is largely diminished when the droplet diameter just surpasses the groove diameter. Physically, this is a consistent observation, as the walls of the groove touch the droplet from all sides near the base, which hinders the internal flow due to shear at the wall.

Figure 6: Experimental and theoretical values of the spatially averaged circulation velocities. The normalized curvature is the ratio of droplet radius to curved surface radius. Cyl and sph refer to cylinders (n=1) and spheres (n=2).The positive abscissa indicates the convex systems, and the negative direction represents the concave systems. Since proper velocimetry is not feasible for spherical depressions, only concave cylindrical grooves have been studied.


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Conclusions To infer, present the article discusses the phenomenon of curvature dependent behavior of internal circulation within sessile droplets on SH surfaces. Curved surfaces, both concave and convex, and of varying curvatures, were fabricated by precision electro-discharge machining process. For convex shapes, both cylinders and spheres were employed, whereas for concave shapes, only cylindrical grooves were used given the difficulties and errors associated with flow visualization in case of droplets seated within spherical depressions. All surfaces were rendered superhydrophobic by the use of a commercial coating. An experimental setup was custom made to dispense microliter droplets onto the said surfaces, with the fluid already seeded with fluorescent, neutrally buoyant microparticles. A laser light sheet is employed to illuminate the droplet mid-plane and the internal advection is recorded at 20 fps using a CCD camera. The flow visualization experiments show that the magnitude of the circulation velocity within the droplet augments for concave SH surfaces whereas for convex surfaces, the circulation velocity diminishes. It is further noted that surfaces with dual curvature (spherical surfaces) are more potent in modulating the flow dynamics compared to surfaces with single curvatures (cylinders). The spatially averaged velocity is observed to be function of the curvature and alteration of the effective wettability on curved surfaces is proposed as a mechanism for the change in circulation velocity. A mathematical model has been proposed to predict the effective spatio-temporally averaged circulation velocity based on potential flow theory. The curvatures are approximated as wedges and potential flow theory for flow past wedged bodies is used to deduce the flow velocities. The mathematical predictions are observed to be consistent with experimental observations and good agreement between the theoretical velocities and experimental velocities


Page 20 of 25

Page 21 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

are obtained. The present findings may have important implications in microscale thermofluidic transport phenomena involving droplets and discrete fluidic phases.

Acknowledgements The authors thank the staff of the Central Workshop, IIT Ropar, for their technical assistance in fabrication of the test sections. PD also thanks IIT Ropar for funding the present research (vide ISIRD grant IITRPR/Research/193 and Interdisciplinary grant IITRPR/Interdisp/CDT).

Conflicts of interest The authors declare no conflicts of interest with any individual or agencies. a Supporting

information

This document contains additional velocity contours and vector fields for PIV studies on different superhydrophobic surfaces of different curvatures.

References 1.

Abramzon, B., & Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations. International journal of heat and mass transfer, 32(9), 1605-1618.

2.

Sirignano, W. A. (1983). Fuel droplet vaporization and spray combustion theory. Progress in Energy and Combustion Science, 9(4), 291-322.

3.

Anna, S. L., Bontoux, N., & Stone, H. A. (2003). Formation of dispersions using “flow focusing” in microchannels. Applied physics letters, 82(3), 364-366.

4.

Thorsen, T., Roberts, R. W., Arnold, F. H., & Quake, S. R. (2001). Dynamic pattern formation in a vesicle-generating microfluidic device. Physical review letters, 86(18), 4163. 21 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

5.

Page 22 of 25

Ogihara, H., Xie, J., Okagaki, J., & Saji, T. (2012). Simple method for preparing superhydrophobic paper: spray-deposited hydrophobic silica nanoparticle coatings exhibit high water-repellency and transparency. Langmuir, 28(10), 4605-4608.

6.

Hess, J., Bo, H., Weber, R., Ortega, I., Barraud, C., van der Schoot, B., De Rooij, N.F. and De Heij, B. (2001). Liquid droplet spray device for an inhaler suitable for respiratory therapies. Microflow Engr SA, U.S. Patent 6,196,219.

7.

Minemawari, H., Yamada, T., Matsui, H., Tsutsumi, J.Y., Haas, S., Chiba, R., Kumai, R. and Hasegawa, T. (2011). Inkjet printing of single-crystal films. Nature, 475(7356), 364.

8.

Sirringhaus, H., Kawase, T., Friend, R. H., Shimoda, T., Inbasekaran, M., Wu, W., & Woo, E.

P.

(2000).

High-resolution

inkjet

printing

of

all-polymer

transistor

circuits. Science, 290(5499), 2123-2126. 9.

Reichard, D. L., Cooper, J. A., Bukovac, M. J., & Fox, R. D. (1998). Using a videographic system to assess spray droplet impaction and reflection from leaf and artificial surfaces. Pesticide Science, 53(4), 291-299.

10. Murakami, S. (2006). A proposal for a new forest canopy interception mechanism: Splash droplet evaporation. Journal of Hydrology, 319(1-4), 72-82. 11. Kim, J. (2007). Spray cooling heat transfer: the state of the art. International Journal of Heat and Fluid Flow, 28(4), 753-767. 12. Harikrishnan, A. R., Dhar, P., Gedupudi, S., & Das, S. K. (2017). Effect of interaction of nanoparticles

and

surfactants

on

the

spreading

dynamics

of

sessile

droplets. Langmuir, 33(43), 12180-12192. 13. Deng, T., Varanasi, K. K., Hsu, M., Bhate, N., Keimel, C., Stein, J., & Blohm, M. (2009). Nonwetting of impinging droplets on textured surfaces. Applied Physics Letters, 94(13), 133109. 14. Reyssat, M., Yeomans, J. M., & Quéré, D. (2007). Impalement of fakir drops. EPL (Europhysics Letters), 81(2), 26006. 15. Wachters, L. H. J., & Westerling, N. A. J. (1966). The heat transfer from a hot wall to impinging water drops in the spheroidal state. Chemical Engineering Science, 21(11), 10471056. 16. Picknett, R. G., & Bexon, R. (1977). The evaporation of sessile or pendant drops in still air. Journal of Colloid and Interface Science, 61(2), 336-350. 22 ACS Paragon Plus Environment

Page 23 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

17. Tsai, P., Pacheco, S., Pirat, C., Lefferts, L., & Lohse, D. (2009). Drop impact upon microand nanostructured superhydrophobic surfaces. Langmuir, 25(20), 12293-12298. 18. Ma, M., & Hill, R. M. (2006). Superhydrophobic surfaces. Current opinion in colloid & interface science, 11(4), 193-202. 19. Koch, K., & Barthlott, W. (2009). Superhydrophobic and superhydrophilic plant surfaces: an inspiration for biomimetic materials. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 367(1893), 1487-1509. 20. Ristenpart, W. D., Kim, P. G., Domingues, C., Wan, J., & Stone, H. A. (2007). Influence of substrate conductivity on circulation reversal in evaporating drops. Physical Review Letters, 99(23), 234502. 21. Harris, S. R., Lempert, W. R., Hersh, L., Burcham, C. L., Saville, D. A., Miles, R. B., Gee, K. & Haughland, R. P. (1996). Quantitative measurements of internal circulation in droplets using flow tagging velocimetry. AIAA journal, 34(3), 449-454. 22. Clift, R., Grace, J. R., & Weber, M. E. (2005). Bubbles, drops, and particles. Courier Corporation. 23. Mandal, D. K., & Bakshi, S. (2012). Internal circulation in a single droplet evaporating in a closed chamber. International Journal of Multiphase Flow, 42, 42-51. 24. Jaiswal, V., Harikrishnan, A. R., Khurana, G., & Dhar, P. (2018). Ionic solubility and solutal advection

governed

augmented

evaporation

kinetics

of

salt

solution

pendant

droplets. Physics of Fluids, 30(1), 012113. 25. Pan, Z., Dash, S., Weibel, J. A., & Garimella, S. V. (2013). Assessment of water droplet evaporation

mechanisms

on

hydrophobic

and

superhydrophobic

substrates. Langmuir, 29(51), 15831-15841. 26. Shaikeea, A. J. D., & Basu, S. (2016). Evaporating sessile droplet pair: Insights into contact line motion, flow transitions and emergence of universal vaporisation pattern. Applied Physics Letters, 108(24), 244102. 27. Dash, S., Chandramohan, A., Weibel, J. A., & Garimella, S. V. (2014). Buoyancy-induced on-the-spot mixing in droplets evaporating on nonwetting surfaces. Physical Review E, 90(6), 062407. 28. Shaikeea, A. J. D., & Basu, S. (2016). Insight into the evaporation dynamics of a pair of sessile droplets on a hydrophobic substrate. Langmuir, 32(5), 1309-1318. 23 ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

29. Extrand,

C.

W.,

&

Moon,

S.

I.

(2008).

Page 24 of 25

Contact

angles

on

spherical

surfaces. Langmuir, 24(17), 9470-9473. 30. Viswanadam, G., & Chase, G. G. (2012). Contact angles of drops on curved superhydrophobic surfaces. Journal of colloid and interface science, 367(1), 472-477.


Page 25 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Table of Contents Graphic 306x275mm (150 x 150 DPI)

ACS Paragon Plus Environment

Superhydrophobic Surface Curvature Dependence of Internal

Recommend Documents