Droplet Mobility on Hydrophobic Fibrous Coatings Comprising

Article Cite This: Langmuir 2018, 34, 12488−12499

pubs.acs.org/Langmuir

Droplet Mobility on Hydrophobic Fibrous Coatings Comprising Orthogonal Fibers M. Jamali,† H. Vahedi Tafreshi,*,† and B. Pourdeyhimi‡ †

Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, Virginia 23284-3015, United States ‡ The Nonwovens Institute, NC State University, Raleigh, North Carolina 27695-7103, United States

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S Supporting Information *

ABSTRACT: Water droplet mobility on a hydrophobic surface cannot be guaranteed even when the droplet exhibits a high contact angle (CA) with the surface. In fact, droplet mobility on a surface, especially a fibrous surface, has remained an unsolved empirical problem. This paper is a combined experimental−computational study focused on droplet mobility on a fibrous surface. Electrospun polystyrene (PS) coatings were used in this work for their ability to exhibit high CAs simultaneously with low droplet mobility. To simplify this otherwise complicated problem and better isolate droplet−fiber interactions, the orientation of the fibers in the coatings was limited to the x and y directions. As the earth gravity was not strong enough to mobilize small droplets on PS coatings, experiments were conducted using ferrofluid droplets, and a magnet was used to make them move on the surface. Experimentally validated numerical simulations were used to enhance our understanding of the forces acting on a droplet before moving on the surface. Effects of Young−Laplace CA and fiber−fiber spacing on droplet mobility were investigated. In particular, it was found that droplet mobility depends strongly on the balance of forces exerted on the droplet by the fibers on the receding and advancing sides.

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INTRODUCTION Hydrophobicity is proven to be the required attribute for a surface used in applications involving anti-icing (e.g., refs1−3), water-droplet separation (e.g., refs 4 and 5), drag reduction (e.g., refs6−8), fog harvesting (e.g., refs 9 and 10), and selfcleaning (e.g., refs11,15), among others. Self-cleaning is a desirable property that allows a surface to remain clean for a longer period of time.11−15 There are two main requirements for such a surface. The first requirement is to allow the liquid to bead up on the surface, and the second is to allow the droplet to move on the surface.11−15 Although there has been tremendous progress in creating a surface on which a droplet beads up, there has not been much progress in guaranteeing that this surface also promotes droplet mobility, especially if it is made of fibers.16−26 The focus of the work presented here is therefore to study droplet mobility on hydrophobic fibrous surfaces with low droplet mobility, the so-called rose petal effect.27−31 Fibrous coatings are usually made by depositing fibers on top of one another. Although a droplet can exhibit high apparent contact angles (ACAs) on such surfaces, its adhesion to the surface may be very unpredictable.22−31 The root cause of this problem is the empirical nature of the current surface manufacturing procedures in which a fibrous surface is first manufactured and then it is tested for droplet mobility. An ambitious but yet logical alternative to the traditional manufacturing approach would be to first design © 2018 American Chemical Society

and test the surface virtually and then manufacture it if the performance was acceptable. This futuristic approach obviously requires detailed information about the impact of the surface micro-scale morphology on the forces acting on a droplet. The study presented in this paper is therefore aimed at providing additional insight into the physics of droplet−fiber interactions specific to fibrous hydrophobic coatings. The remainder of this paper is organized as follows. We first discuss the challenges involved in determining droplet mobility on a fibrous surface and the approaches considered in prior studies. We then present a quick overview of our experimental and computational methods, which also includes a validation study custom-designed to examine the accuracy of the computational and experimental results against one another. At the end, we discuss the contribution of each individual fiber in contact with a droplet in the total force resisting against droplet mobility.

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ADVANCING AND RECEDING CAs ON A FIBROUS SURFACE Predicting droplet mobility on a rough surface, that is, the tendency of the droplet to move on the surface in response to Received: August 17, 2018 Revised: September 13, 2018 Published: September 18, 2018 12488

DOI: 10.1021/acs.langmuir.8b02810 Langmuir 2018, 34, 12488−12499

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an external force, is a challenge. This is because droplet mobility on a surface depends on many factors including, but not limited to, (1) area of contact between the droplet and the solid surface (wetted area, WA), (2) length of the three-phase air−water−solid contact line (CL), (3) three-dimensional (3D) shape and orientation of WA and CL with respect to droplet’s direction of motion, and (4) slope of the air−water interface (AWI) along the CL. In addition, droplet pinning to surface nonhomogeneities is another unresolved issue that further complicates this problem.32−34 For these reasons, it is almost impossible to accurately predict the force required to move a droplet on a surface via a first-principle theoretical approach. For the lack of a better option, designing a selfcleaning surface has remained an empirical problem, often approached via try-and-error and characterized in terms of the intellectually insignificant but easy-to-image largest and smallest ACAs along the perimeter of a droplet (loosely referred to as the advancing and receding ACAs, respectively).35−39 Because of these limitations, the force required to detach a droplet from a surface is often presented in terms of the difference between the advancing and receding CAs [i.e., CA hysteresis (CAH)] but scaled by yet another empirical factor 1 < k < 3.14 that is there to compensate for all what is not known about the actual forces acting on the droplet,40−47 that is, F = kwσ(cos θmax − cos θmin)

Article

METHODS

Droplet Mobility Experiment. Electrospun PS coatings are considered in this work for their superhydrophobicity and ease of manufacturing. In our electrospinning unit, a dc electric field of 5.5 kV was applied to the PS solution contained in a needle mounted on a syringe pump with an infusion rate of 2.5 μL/min. When the electric force on the liquid exceeds a critical limit, a jet of charged solution emerges from the tip of the needle toward a substrate (microscope slides from McMaster Carr) held at a distance of about 85 mm. To produce the PS solution, PS pellets were dissolved in a 70−30 wt % toluene−tetrahydrofuran mixture to obtain a solution with 25 wt % PS concentration (chosen based on previous experience with the same setup26). To align the fibers, the substrate was placed on an axially moving rotating drum with rotational and translational speeds of 1200 rpm and 1.5 cm/s, respectively. The orthogonally layered structures were made by rotating the substrate by a 90° angle after depositing each layer (see ref 26), and the average fiber−fiber spacing was varied by varying the spinning time per layer. Our group has recently developed a new method to measure the force required to detach a droplet from a surface or a fiber in a direction perpendicular to the surface using a ferrofluid droplet in a magnetic field.62−65 The same method is used here to measure the force required to move the droplet on the surface of the abovementioned PS coatings (stickiness of the hydrophobic PS coatings allowed the experiment to be conducted using the same setup with the coatings mounted vertically) as shown in Figure 1. This force was measured using a sensitive scale (Mettler Toledo XSE105DU with an accuracy of 0.01 mg). The scale was zeroed, and a magnetic force was vertically applied to the droplet by a nickel-plated axially magnetized cylindrical permanent magnet with a diameter of 22 mm and a length of 22 mm (K&J Magnetics). The magnetic force was increased incrementally by lowering the magnet (attached to a Mitutoyo electronic height gauge) toward the droplet, and the corresponding

(1)

In this equation, σ is the surface tension and w is an arbitrary “width” perpendicular to the direction of the motion assigned to the droplet. Being so empirical in nature, this equation can only be used when one takes the effort to experimentally measure the advancing and receding CAs for the desired droplet−surface system at hand (where one may rather measure the force directly!). Fortunately however, a theoretical approach can still be devised to predict the force needed to move a droplet on a rough hydrophobic surface, but only if droplet pinning is not an issue (e.g., when the surface asperities are not sharp).48−52 As mentioned earlier in the Introduction, the study presented in this paper is aimed at providing some analytical insight into the interactions between a droplet and its underlying hydrophobic fibrous surface. Fibrous coatings, for example, electrospun coatings, are cost-effective in terms of manufacturing, and they can conform to the geometry of the surface to which they are applied. Although droplet mobility on a fibrous coating is not very predictable, one can still assume that it may depend on fiber orientation or fiber−fiber spacing to some degrees. For instance, previous studies have shown that a droplet on a coating with unidirectional fibers may exhibit different ACAs in different directions.53−56 Therefore, one can potentially improve or control the adhesion force between a droplet and a fibrous coating by controlling the orientation of the fibers. The easiest way to produce a fibrous coating with directional fibers is to deposit parallel fibers in orthogonal layers. This helps to produce coatings with a reasonably controlled thickness or porosity.57−61 As will be discussed in the next section, we produce electrospun polystyrene (PS) coatings comprising orthogonally oriented layers of aligned fibers and use these coatings in studying the role of surface microstructure on droplet mobility. Our study is also accompanied by numerical simulations to provide additional insight into the physics of droplet mobility on a surface.

Figure 1. Schematic and actual image of the experimental setup developed for the study. 12489


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CA of about 120° for the coating with a flat surface, that is, YLCA. A goniometer was used to measure the droplet contact and roll-off angles for droplet volumes ranging from 5 to 30 μL. Experiments were repeated several times and reported as average values with standard deviations. Figure 2a−f shows a water droplet with a volume of 20 μL

readings on the scale digital display were made into a video (to capture the scale reading at the moment of droplet detachment). The ferrofluid used in the experiment (purchased from EMG508, Ferrotech, USA) was an aqueous suspension of Fe3O4 nanoparticles (contained 1% volumetric) with a mixture density of ρ = 1.05 g/cm3 at 25 °C. Note that the ferrofluid response to the magnetic field is attributed to the Fe3O4 nanoparticles, and this should not be confused with the Moses effect (see, e.g., refs66−69). Droplet motion over the coatings was recorded using a high-speed camera (Phantom Miro Lab 340 with) with a Tokina 100 mm F 2.8 D lens. Droplet Mobility Simulation. We have used the energy minimization method implemented in the Surface Evolver (SE) finite element code to predict the force needed to move a droplet on an idealized fibrous structure comprising orthogonally oriented layers of fibers. To start, we first obtained the equilibrium shape of the droplet on the surface (in the presence of gravity in the z-direction, normal to the plain of the coatings). This requires one to program additional equations in SE to correctly calculate the volume of the droplet in contact with the fibers. Assuming the fibers in the first and second layers to be in the x and y directions, respectively, the general formula of the droplet−fiber energy equation can be given as

∬A

E = σA aw − σ cos θ YL +

dA sw + sw

∬ ρf y dA y

∭ ρgz dVa + ∬ ρfx dAx (2)

where E, ρ, and θ are the total energy of the system, fluid density, and fluid Young−Laplace CA (YLCA) with the material of the fibers, respectively. The external forces per unit mass of the droplet in the x and y directions are denoted in eq 2 by f x and f y, respectively. While SE calculates the AWI Aaw by default, it needs additional instructions to compute the solid−water interfacial area Asw or the volume of wetted fibers Vs (needed to calculate the droplet volume Va correctly). To calculate Asw and Vs, the following equations were developed for the solid−water elemental area and wetted fiber elemental volume and added to the SE code.56,70 xy xz dA sw = − dy + dz 2 2 2 y + z2 y +z (3) YL

dVs = (− xz dy + xy dz)/2

(4)

To obtain the equilibrium droplet ACA (i.e., in the presence of gravity but in the absence of a horizontal force), we set the last two terms on the right-hand side (rhs) of eq 2 equal to zero (f x = f y = 0). With the 3-D droplet shape and ACAs obtained, we then change the gravity from g to g cos α and the horizontal force from zero to f y = g sin α (or f x = g sin α depending on the orientation of the fibers) to simulate the droplet shape on an inclined surface with the inclination angle of α. Equation 2 should again be modified when the last two terms on the rhs of eq 2 are not zero, that is, the solid−water elemental areas dAx and dAy in eq 2 should be defined as dA x =

x 2y x 2z dy + dz 4 4

(5)

dA y =

y2 z y2 x dx + dz 4 4

(6)

Figure 2. Water droplet with a volume of 20 μL is shown from top, longitudinal, and transverse views on 3-D printed parallel fibers in (a− c), respectively. The same droplet is shown in (d−f) on a two-layer coating of orthogonally oriented 3-D printed parallel fibers. Comparison between experimental and computational droplet shapes for different inclination angles. The coating in (g−i) is composed of 3D printed parallel fibers, but the one in (j−l) is made up of orthogonal fibers. on the printed parallel and orthogonal fibers (fiber diameter and spacing are given in the figure) from top, longitudinal, and transverse views. Figure 2g−l compares the droplet shape and ACAs with their computational counterparts for different inclination angles. It can be seen that the experimental and numerical results are in good general agreement with one another. The parameters contributing to the mismatch between the experimental and numerical results (5−15% on average) include the possible nonuniformity of the surface coating (possibly causing some variations in the YLCA of the fibers) and the imperfection in the geometry of the printed fibers (in terms of fiber

Validation. To examine the accuracy of our SE simulations using experimental data, parallel and orthogonal fibers with diameters of 362 and 467 μm and spacing distances of 898 and 865 μm were printed using a fifth Gen Makerbot Replicator 3-D printer and spray coated with Ultra-Ever Dry solution from Ultratech Company. Prior to coating the fibers, a layer of adhesive was applied to the fibers, and the fibers were left to dry for 30 min. The Ultra-Ever Dry coating was sprayed in a fine mist to maximize the uniformity of the coating, and the fibers were left to dry for a day before being used for the experiment. The coating was also applied on a glass slide to obtain a 12490


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Langmuir spacing, fiber cross-sectional shape, and fiber surface roughness). These factors were obviously not included in the simulations as the fibers were assumed to be smooth cylinders. The experiment− simulation comparison given in Figure 2 is only for droplet motion in the direction perpendicular to the fibers in the top layer. This is because there is almost no resistance to a droplet moving in a direction parallel to the fibers (i.e., a roll-off angle of about 5°). In addition to validating our simulation results with the experiment, we also compared them with the predictions of the Cassie−Baxter equation given as cos θapp ̅ =

ϑd d sin ϑ cos θ YL + −1 s s

but rather additional insight. In fact, the simulations have been conducted for idealized coatings in which the fibers were equally spaced and perfectly paralleled to one another in each layer. The fibers were also considered to be larger than the typical electrospun fibers to ease the otherwise prohibitively time-consuming calculations. Nevertheless, experimental and computational data are in good qualitative agreement despite their scale differences, as will be seen later in this section. Experimental Results. As mentioned earlier, the force required to move a droplet on the surface of our PS coatings was measured using a sensitive scale on which the coatings were placed in the vertical position. To start, a ferrofluid droplet was gently deposited on the coating in the horizontal position, and the coating was then rotated 90° and mounted on the scale. A continuously increasing magnetic force was then applied to the droplet by lowering the magnet while recoding droplet deformation using a high-speed camera. While studying droplet mobility on fibrous coatings, we limit our work to droplets that have not penetrated deep into the coatings to reach the underlying substrate. While we realize that there exists a series of partially wetted transition states between the fully dry Cassie and fully wetted Wenzel states,73,74 we refer to such droplets as the Cassie droplets to distinguish them from the droplets that have come into contact with the substrate (referred to here as the Wenzel droplets) for the sake of simplicity. Because it is difficult to visually determine whether a droplet has come into contact with the substrate during an experiment, we use the ACAs, sliding force, and droplet residue on the coating in judging the state of the droplet as shown in Figure 3. This figure reports the droplet sliding force and ACAs on electrospun PS coatings comprising two, three, and four orthogonal layers of parallel fibers (3 min electrospinning per layer). We have also included the CAH for comparison. It can be seen that the force needed to move the droplet over the two-layer coatings is almost two times higher than that needed to slide the droplet over the three-layer or four-layer coatings. Likewise, the CAH is greater on the twolayer coatings. This information indicates that ferrofluid droplets with a volume of 4 μL stay in the Cassie state on the three-layer or four-layer coatings but they transition to the Wenzel state when deposited on the two-layer coatings. This argument is also supported by the images obtained from droplet motion and the presence or the lack of a measureable droplet residue on the surface (see the droplet detachment videos in the Supporting Information). As can be seen in the bottom-right image in Figure 3, a Wenzel droplet breaks up into smaller droplets as it moves over the surface, the so-called pearling effect as explained in ref 75. Figure 4 shows the sliding force and CA for droplets with a volume in the range of 2−7 μL on coatings comprising three orthogonal layers of parallel PS fibers each spun for 5 min (the CAs will be later used for Figure 10). It can be seen that the sliding force per droplet mass is smaller for larger droplets. These results are in qualitative agreement with those reported in ref 44 for droplet retention on surfaces coated with octadecyl trimethylammonium and with many other such studies in the literature. From a design perspective, it is important to explore the importance of fiber−fiber spacing (inversely proportional to spinning time per layer) and layer configuration on droplet mobility. Figure 5 shows the sliding force and CAs for a droplet with a volume of 4 μL placed on PS coatings with different fiber−fiber spacings. The results given in Figure 5a

(7)

where ϑ represents the immersion angle (dependent on fiber diameter d, fiber spacing s, YLCA θYL, and droplet pressure p). The immersion angle can be calculated for simple geometries (e.g., a row of parallel fiber) using a force balance approach, as given in eq 8.71,72 p=−

2σ sin(θ YL + ϑ) (s − d sin ϑ)

(8)

The pressure in eq 8 can be approximated with the Laplace pressure for a small droplet on a hydrophobic surface in the absence of strong external forces (p = 2σrd−1, where rd is the radius of a spherical droplet having the same volume). One can approximate the equilibrium average ACA of a Cassie droplet using the Cassie−Baxter equation (eqs 7 and 8). We also calculated droplet roll-off angle α for completeness of the study with printed fibers. For a droplet on an inclined surface, the capillary force from the fibers should balance the component of the gravitational force along the surface, that is, Fx = mg sin α

(9)

where m, g, and α are the droplet mass, the gravitational acceleration, and the roll-off angle. Tables 1 and 2 present the roll-off angles obtained from goniometer measurement, from SE simulations, and from using eqs 1 and 9. When

Table 1. Comparison between the Experimental, Numerical, and Analytical CAs for a Droplet on 3-D Printed Parallel Fibers of Figure 2 ACA (deg) roll-off CA (deg) F/M (N/kg) ACA of hanging droplet (deg)

experiment

simulation

eqs 1 and 9

± ± ± ±

132 39 6.2 122

130 36 5.8

136 45 6.9 119

3 5 0.8 4

Table 2. Comparison between the Experimental, Numerical, and Analytical CAs for a Droplet on 3-D Printed Orthogonal Fibers of Figure 2 experiment ACA (deg) roll-off CA (deg) F/M (N/kg) ACA of hanging droplet (deg)

130 19 3.2 116

± ± ± ±

5 2 2 5

simulation

eqs 1 and 9

125 16 2.7 123

139 10 1.7

using eq 1, one could use θmax and θmin from either the simulations or the experiments. For w, we used the droplet base width from simulations (as the droplets were not imaged from behind during the experiment, see Figure 10 and its discussion).

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RESULTS AND DISCUSSION Our experimental and computational results are given in this section but in different subsections, as the simulations were not intended to provide one-on-one comparison with experiment 12491


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Figure 3. Droplet sliding force and critical CAs are given vs the number of electrospun layers. Example snapshots of droplet deformation and sliding are also shown for Wenzel and Cassie droplets with a volume of 4 μL. The droplet profile is extracted and overlaid for comparison.

are for coatings made of only one layer of fibers (mounted on a stand to avoid contact with the underlying substrate), whereas those shown in Figure 5b are obtained with three-layer coatings. It can be seen that the sliding force and CAH increase with decreasing fiber spacing. It is interesting to note that the sliding force and CAH values reported in Figure 5b are greater than their counterparts in Figure 5a. This indicates that for the droplet−fiber combinations considered in these experiments, the droplets penetrate deep into the coatings enough to come into contact with the fibers below the first layer, as was discussed in ref 56. Additional discussion about the role of individual fibers and the effects of fiber−fiber spacing is given in the last section (see Figures 8 and 9). Simulation Results. Unless otherwise stated, the fiber diameter and droplet volume considered for the computational component of this research are 10 μm and 0.5 μL, respectively. Simulations start by first obtaining the droplet’s equilibrium shape on horizontal coatings (with gravity normal to the plane of fibers). By mimicking the droplet on a vertical coating as in the experiment (see Figure 1), we then set the gravity to zero but apply an external force to the droplet in a direction parallel to the plain of coating. In simulating the sliding force, we incrementally increase the force on the droplet (with an arbitrary increment of Δgz = 0.5 N/kg) until no equilibrium shape and position can be obtained for the droplet. The largest body force at which an equilibrium shape is obtained (plus an increment of Δgz) is then taken as the force required to slide the droplet on the surface. Note that the simulation method used in this work cannot be used to model the dynamics of droplet sliding or the volume of the droplet residue left on the fibers after detachment. Figure 6a−c shows examples of overlaid droplet shapes on fiber coatings with different properties. The blue-colored droplets are under the influence of gravity only (i.e., horizontal equilibrium position), whereas the red-colored droplets represent droplets on a vertical coating (exposed to an external

Figure 4. Droplet sliding force and CAs are given vs the droplet volume. The coatings used for the experiments were composed of three orthogonal layers of electrospun parallel fibers with 5 min fiber spinning per layer. Example snapshots of droplet deformation are also shown for three different droplet volumes. 12492


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Figure 5. Droplet sliding force and CAs are given vs spinning time (inversely proportional to fiber-to-fiber spacing) for a droplet with a volume of 4 μL on one-layer coatings made of aligned electrospun fibers in (a) and orthogonally layered aligned fibers in (b). The insets show the droplet shape at the moment of detachment.

Figure 6. Droplet profile and bottom views are given in (a−c) for droplets on coatings with different fiber−fiber spacings. Effects of increasing horizontal body force (leading to droplet sliding) on CL and fiber WA (d) and CAH (e) are also given for coatings with three different fiber spacings but an identical YLCA of 85°.

body force parallel to the plain of fibers). The red-colored droplets are at their final state of equilibrium, that is, they are about to slide on the surface. The footprint of each droplet on the coatings (portion of the droplet in contact with the coating) is also included for each case in Figure 6 with matching colors. Fibers in the first (top) and second layers are shown with green and dark-red colors, respectively. The droplets shown in Figure 6a are in contact with only one layer

of fibers (s = 70 μm), whereas those shown in Figure 6b,c are in contact with both layers of fibers (s = 140 μm and s = 125 μm). The external force is normal to the green fibers in Figure 6a,b, but it is parallel to the green fibers in Figure 6c. Effects of the external body force on CL, fibers’ WA (Figure 6d), and CAH (Figure 6e) are also presented for quantitative comparison. It can be seen that increasing the body force increases the CAH but decreases CL and fibers WA, which 12493


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Figure 7. Droplet profile and bottom views are given in (a) through (d) for a droplet on coatings with different fiber spacings. Effects of fiber spacing on CL and WA (e) and CAH (f) and sliding force (g) are also given for comparison. Droplet volume and fiber diameter are 0.5 μL and 10 μm, respectively.

were expected conceptually (see, e.g., ref 76) but never quantified for fibrous surfaces. Figure 7a−d shows the effects of fiber spacing and YLCA on the droplet 3-D shape and footprint on the coatings in the presence (red-colored) and absence (blue-colored) of a parallel external force. Coating’s WA, droplet CL, CAH, and droplet sliding force (parallel to the surface but perpendicular to the fibers in the top layer) are calculated and given in Figure 7e−g as a function of fiber−fiber spacing. Our results indicate that the sliding force is lower on coatings with high fiber−fiber spacing (regardless of the number of layers or YLCA of the fibers). This is because a droplet on a coating with a greater fiber spacing is in contact with less number of fibers (smaller capillary force). It can also be seen that the sliding force is less on coatings composed of fibers with a higher YLCA. The simulations also revealed that the sliding force increases rapidly if the droplet comes into contact with the fibers in the second layer (i.e., mobility is less on coatings that allow the droplet to

penetrate into the structure). The results given in Figure 7 are in qualitative agreement with the work of ref 77 who measured the surface WA for a microfabricated polydimethylsiloxane surface (and the corresponding droplet CAHs) under different lateral adhesion forces. Qualitative agreement can also be seen between our experimental data (Figure 5) and the computational results (Figures 6 and 7) despite the differences in dimensions of the fibers and the droplets. It is worth noting that the sliding forces obtained experimentally for our electrospun coatings are larger than those obtained from the simulations. This is because the number of fibers in contact with a droplet deposited on an actual electrospun coating is more than that in the virtual coatings used in the simulations. When the external force is applied in a direction parallel to the fibers in the first layer (i.e., in the y-direction), the sliding forces are about an order of magnitude smaller than their counterparts in the x-direction (see Figure S1 in the Supporting Information). In fact, the sliding force is zero for 12494


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Langmuir droplets that do not reach the fibers in the second layer in this case (in the absence of viscous forces and CL pinning). It is also interesting to mention that sliding forces in the ydirection are less than g, which means that the droplets can roll off by simply tilting the surface (which was not the case when the external force was in the x-direction). This confirms that the droplet mobility over a fibrous surface strongly depends on the relative angle between the fibers in contact with the droplet (fibers near the top of the surface) and the direction of the external force (compare Figures 7 with S1). Force Balance Analysis for Droplet Sliding Force. An unanswered (or perhaps partially answered) question in the literature has been the relative contributions of the advancing and receding sides of a droplet in resisting against the droplet motion (see, e.g., ref 78). In this section, we use our numerical simulations to shed some more light on this issue and help better understand the role of individual fibers in resisting droplet mobility. Fiber-Level Force Calculation. With the numerical simulation data available, one can study how each individual fiber in contact with a droplet contributes to the force resisting against the droplet motion. Figure 8a shows a schematic drawing of the contact between a fiber and a droplet. The capillary forces act on the opposite sides of the fiber in a direction tangent to the AWI along the fiber axis (the ydirection here) with a slope that is equal to the local slope of the AWI. Pressure forces, caused by the elevated pressure inside the droplet, act normal to the local curvature of the solid surface on the fiber’s WA (shown in red in Figure 8a). The fiber resistance to droplet motion is the summation of the components of the capillary and pressure forces in the direction opposite to that of the droplet motion, that is, Fx =

∫CL σ cos β dx + ∫WA p dA

(10)

where β is the angle between the tangent to the AWI and the horizontal plane along the CL on the fiber and dA is the elemental WA of the fiber projected onto a vertical plane as shown in Figure 8a. Expanding eq 10 using geometrical information given in Figure 8a, we obtain Fx = FxL + FxR = −

Figure 8. Free body diagram is given in (a) for the forces acting on a fiber in contact with a droplet. (b) Simulated force components are given for a droplet with a volume of 30 μL on the 3-D printed fibers of Figure 2 (with a diameter of 362 μm and a fiber spacing of 898 μm).

∫CL σ cos β1 dy + ∫WA p dA1

between a droplet and its underlying fibers. As mentioned earlier, the net force exerted on each fiber is the resultant of the capillary (along the CL) and pressure (over the WA) forces on the left and right sides of the fiber. Table 3 compares these

∫CL σ cos β2 dy − ∫WA p dA 2

=σ

∫ (cos β1 − cos β2)dy + pr∫ (cos ϑ2 − cos ϑ1)dy

Table 3. Force Components for a Droplet with a Volume of 30 μL on 3-D Printed Parallel Fibers with a Diameter of 362 μm and a Fiber Spacing of 898 μm from Figure 2

(11)

where r is the fiber radius. The superscripts L and R refer to left and right sides of the fiber as can be seen in Figure 8a. Applying eq 11 to all fibers in contact with the droplet results in the total resistance of the coating to droplet motion. Using Figure 8a, the angles β1 and β2 can be written in terms of immersion angles on each side of the fiber (ϑ1 and ϑ2), that is, iπ y cos β1 = sinjjj + ϑ1 + θ YL zzz k2 {

i 3π y cos β2 = sinjjj − ϑ2 − θ YL zzz k 2 {

f = Fx/m

fLσ

fiber 1 (receding) fiber 2 (middle) fiber 3 (advancing) resultant force experiment

20 26 26

fRσ

f Lp

f Rp

f Lnet

f Rnet

f net

−25 −5 17 15 −26 −15 15 11 −30 −14 6 12 summation of all three fibers g sin α, α = 30

8 11 24

7 0 −12 −5 4.9

(12)

forces with one another for the case shown in Figure 8b. It is interesting to note that the fiber in the middle (fiber 2) does not contribute to the total force acting on the droplet (forces cancel each other because of geometrical symmetry). The fiber on the advancing side (fiber 3) makes the strongest resistance against the droplet motion, whereas the fiber on the receding

(13)

The simulations reported earlier in Figure 2i are considered here again in Figure 8b for their simplicity (there are only three fibers in contact with the droplet) to further analyze the forces 12495


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Langmuir side (fiber 1) tends to help the droplet to move. Note that the length of the CL is quite different on the receding and advancing sides of the droplet (CL is longer on the fiber on the advancing side as can be seen in the magnified images in Figure 8b). This observation may seem to contradict the commonly accepted conclusion that the receding end of a droplet plays the most important role in controlling droplet mobility.78 However, the conclusion was probably reached for droplets on surfaces that allow CL pinning (e.g., surfaces made of microfabricated sharp-edged posts). Coming back to the more complicated case of droplet on virtual coatings with orthogonal fibers, we present force per fiber for four different coatings in Figure 9. The bar chart of

Approximate Expressions for Droplet Sliding Force. An alternative approach to using eq 11 for sliding force prediction is to slice the droplet at a location slightly above the surface and to consider the balance of forces acting on the droplet as shown in Figure 10a (the distance between the

Figure 9. Forces acting on each individual fiber in contact with a droplet at the final state of equilibrium under an external horizontal force (a). Each bar represents a coating with a different fiber spacing, and the colors in each bar represent the force acting on the individual fibers in each coating (b).

Figure 10. Droplet slicing method on the capillary forces at the cross section is shown in (a). Comparison between droplet sliding forces obtained from numerical simulations, eqs 14, and 1 is given in (b). Comparison between predictions of eq 1 and our experimental data is given in (c).

fibers and the slicing plane is exaggerated for illustration). The new approach does not provide any fiber-level information (see Figure 9 and its discussion), but it is more practical as will be seen later in this section. Figure 10a also shows the capillary forces (black arrows) acting on the sliced droplet. The capillary forces are projected onto the slicing plane and are shown from below. The red arrows in this figure are the x-components of these planar forces. Note that the pressure forces acting on droplet cross section have no components in the x-direction. The sliding force can therefore be calculated as the xcomponent of the force obtained using eq 14, that is,

Figure 9a shows the contribution of each individual fiber in the total force against the droplet motion (the total force shown with black circles) for a few virtual fibrous coatings with different fiber−fiber spacings (from Figure 7). The bar segments in Figure 9a are colored differently for different fibers. Figure 9b shows the droplet footprints on coatings with different fiber−fiber spacing values of 50, 70, 125, and 170 μm. Note that the fibers shown in red provide no resistance against the droplet motion in the x-direction (they are parallel to the x-direction). It is interesting to note in Figure 9a that the fibers on the droplet’s receding and advancing sides play the main role in resisting against the droplet motion. In this figure, the positive forces help droplet motion, whereas the negative forces resist against it.

Fx = 12496

cr dl ∫CL σ cos θapp

(14) DOI: 10.1021/acs.langmuir.8b02810 Langmuir 2018, 34, 12488−12499

Article

Langmuir where θcrapp is the local ACA of the droplet (around the 3-D CL) at its final equilibrium state before moving. Figure 10b compares the sliding force from simulations (Figure 7) with the predictions of eq 14. To do so, we sliced the simulated droplets at a distance of 6d above the top surface of the fibers to extract θcrapp and CL data for eq 14. Good agreement can be seen between the actual simulation data (black squares) and those obtained from the above slicing method (green circles), as expected. As mentioned earlier in the Introduction, one can estimate the droplet sliding force on a surface using eq 1. This equation however requires droplet’s advancing and receding CAs as well as the width of the droplet’s footprint on the surface w as the input (assuming the empirical correction factor to be k = 1). While in principle droplet’s width right before sliding wcr should be used in eq 1, better agreement with the simulation results was observed when we used footprint’s width in the absence of the magnetic force w0 in the equation (see Figure 10b). The inset in Figure 10b shows the droplet footprint on the surface as a function of the in-plane body force. It can be seen that w decreases (though not monotonically) with an increase in the in-plane body force on the droplet. The experimental counterparts to the computational results shown in Figure 10b are given in Figure 10c for a droplet with a volume of 4 μL on a single layer of aligned electrospun fibers (the experiments reported in Figure 5a). Because the droplet footprint on the surface can be approximated as being circular in the absence of an in-plane body force, we used the droplet length in place of the droplet width when measuring w0. To measure wcr, the droplet was imaged from behind as it was pulled up by the magnet. The inset in Figure 10c shows an example of such images obtained under different in-plane body forces (overlaid on top of each other for comparison). It can again be seen that eq 1 tends to underestimate the sliding force when w = wcr is used in the equation for the footprint width. Equation 1 however provides reasonable predictions with w = w0. Regardless, we believe droplet footprint dimensions on the surface right before detachment (be it the width or other dimensions) are more logical parameters to use in predicting the sliding force. The reported inaccuracies in predictions obtained from eq 1 seem to be inherent to this oversimplified empirical equation (hence the empirical correction factor of 1 < k < 3.14 as recommended in the literature).

droplet (i.e., the droplet does not roll off by tilting the surface). The only exception to this is when the droplet is only in contact with the fibers in the top layer and those fibers are in the direction of the external body force (where there will be no significant resistance against the droplet motion if the fibers are smooth). Our results indicate that droplet mobility is generally higher when the spacing between the fibers is larger. However, this depends on whether or not the droplet is in contact with the first layer of fibers. Excessive spacing between the fibers can lead to droplet penetration into the coating (even as small as one fiber diameter deep) to result in a significant reduction in droplet mobility (e.g., see Figure 7). Our study quantified the effects of droplet volume (as well as fiber spacing or YLCA) on the force needed to roll the droplet on the surface. An in-depth analysis was presented for the effects of the external body force on droplet CL or fibers’ WA. By calculating the force exerted on a droplet placed on a fibrous surface, it was found that the fibers on the receding and advancing sides of the droplet play the most important roles in determining the force needed to roll the droplet on the surface.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b02810. Additional information about sliding force and magnetic field (PDF) Ferrofluid droplet with a volume of 4 μL moving on an electrospun PS mat with a fiber diameter of 0.5 μm at the Wenzel state (AVI) Ferrofluid droplet with a volume of 4 μL moving on an electrospun PS mat with a fiber diameter of 0.5 μm at the Cassie state (AVI)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 804-828-9936. Fax: 804827-7030. URL: http://www.people.vcu.edu/?htafreshi/. ORCID

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H. Vahedi Tafreshi: 0000-0003-0689-6621 Notes

CONCLUSIONS Droplet mobility on electrospun PS coatings is studied experimentally and computationally in this work. To simplify the otherwise very complicated problem, we limited the orientation of the fibers to the x and y directions. Depending on the spacing between the fibers (and of course fiber and droplet diameters), a droplet on an electrospun PS coating can be at the Cassie state, at the Wenzel state, or at a transition state in between these extreme states. It appeared from our experiments that the Cassie (or near-Cassie) droplets leave a much smaller residue on the surface after sliding compared to the Wenzel (or near-Wenzel) droplets. The Wenzel droplets have to break up into two or more volumes before they can roll on the surface. Our results also indicate that Cassie droplets seem to require a smaller body force to roll on the surface. In the work presented here, we focused mostly on the Cassie droplets. The force needed to move a droplet on the surface of an electrospun PS coating is usually greater than the weight of the

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Nonwoven Institute for the financial support. NOMENCLATURE A, area d, fiber diameter F, force E, energy k, proportional constant L, contact length r, fiber radius P, pressure inside a droplet s, fiber’s spacing wcr, droplet’s width right before sliding w0, droplet’s width in the absence of the magnetic force

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Langmuir ϑ, immersion angle α, roll-off angle θYL, Young−Laplace contact angle θapp, apparent contact angle θmax, maximum apparent contact angle θmin, minimum apparent contact angle θH, hysteresis contact angle β, force angle σ, surface tension

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Droplet Mobility on Hydrophobic Fibrous Coatings Comprising

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