A water droplet-cleaning of a dusty hydrophobic surface: influence of dust layer thickness on droplet dynamics

Effect of dust layer thickness on the droplet behavior is examined and droplet the fluid spreading rate on dust surface is determined. Wicking conditions for dust layer is assessed and the dust cleaning process by rolling droplet is evaluated.

Hydrophobic surface and dust particles

Figure 1a depicts SEM images of functionalized nano-silica particles coated surface while Fig. 1b shows atomic force microscopy of the surface line scan. The coated surface resembles closely formed nano-size silica units with a nominal size of about 30 nm. Small size porous appearance on the surface (Fig. 1a) is related to agglomeration of nano-silica units, i.e. modifier silane gives rise to the side reactions during condensation on silica particles, which trigger silica particles agglomeration22. Agglomerated particles do not form a thick coating layer on the surface. It can be noted that once the particles agglomerate in the coating the coating thickness is expected to increase, which generates a texture profile while altering the wetting state locally. However, from atomic force microscope line scan, the average roughness of the coated surface is about 65 nm (Fig. 1b). Hence, the coating produced has fine-size thickness and the agglomerated particles enable to form air-gaps in the coating (Fig. 1a). In addition, some rippling behavior is noted along the surface line scan, which represents the clustered particles on the surface (Fig. 1b). The roughness parameter of the coating surface is determined using surface topology obtained from the atomic force microscope. The roughness parameter, which corresponds to the ratio of pillars area over the projected area23 and it is about 0.49 for coating surface. The state of the wetting of the coating surface is evaluated adopting the contact angle method20. In addition, advancing and receding angles of the droplet are also measured and contact angle hysteresis is assessed. Figure 1c depicts droplet image on the coating surface. The contact angle of the coating surface is about 150° ± 2° and hysteresis is about 2° ± 1°. The measurement contact angle and hysteresis is repeated five times at leach location on the surface while incorporating the technique used in the early work20. This arrangements ensures the repeatability of the contact angle data. Droplet rolls on the coating surface rather than sliding, i.e. pinning of droplet due to adhesion under the effect of surface tension is significantly small. The free energy of the coated surface is determined via contact angle measurements24,25 and it is estimated about 35.51 mJ/m2. On the other hand, the dust particles collected are analyzed using scanning electron microscope (SEM), energy dispersive spectroscopy (EDS) and X-ray diffraction (XRD). Figure 2a depicts SEM image of the dust particles while Fig. 2b shows dust particle size distribution. The particles have various sizes and shapes (Fig. 2a,b). Small particles cluster and attaches to large particles (Fig. 2a). The shape of particles can be assessed through introducing a shape factor parameter4. The parameter is determined from the estimation of cross-sectional area and dust perimeter (({R}_{Shape}= frac{4pi A}{{P}^{2}}), here P represents the dust perimeter and A is the cross-sectional area). The shape factor resembles the circularity of a dust particle. SEM micrographs of individual dust particle are used determining the dust particle perimeter and the corresponding dust particle cross-section. However, no strong correlation is observed between the dust size and the shape factor. Nevertheless, the value of shape factor improves with increasing dust size. Hence, it is close to 1 for dust with size ≤ 2 μm, and the median of shape factor becomes 3 for dust with size ≥ 5 μm. Table 1 gives EDS dust data (in wt%). Dust contains Na, K, Si, S, Ca, Fe, O, Mg, and Cl. The existence of Na, K, and Cl in the dust reveals that dust contain salt compounds. Chlorine concentration is higher for large size dust (> 1.2 µm) and Cl concentration does not satisfy the stoichiometric ratio for NaCl; hence, NaCl is present in the dust with nonstoichiometric concentration. X-ray photoelectron spectroscopy (XPS) is carried out towards assessing Na, K, and Cl binding energies. The findings reveal that chlorine covers inorganic chloride, i.e. Cl2p3/2 peak appears at 198.9 eV (3a) as consistent with early studies26,27. The data obtained from CasaXPS indicate that the concentration ratio (in terms of the mass base) of potassium over chlorine is about 1.62, which is similar to that obtained from the energy dispersive spectroscopy data (Table 1). Moreover, sodium data demonstrate that the XPS peak occurs at 1,072.8 eV for Na1s. Hence, CasaXPS shows that the ratio of sodium concentration over chlorine (in terms of mass percentage) is 2.67, which is similar to that of the energy dispersive spectroscopy data (Table 1). To evaluate the dust compounds, X-ray diffractogram is carried out. Figure 2c depicts X-ray diffractogram of the dust collected, which possess same to the data reported in the early work5. Here, the peaks of Na and K are because of salt while sulfur is because of calcium, such as in anhydrite or gypsum components (CaSO4). However, Fe is because of hematite (Fe2O3) in the dust particle. To assess the surface energy of dust, the contact angle technique is adopted, as presented in the early studies24,25. It is should be noted that the Washburn technique can be used to assess the surface free energy of porous-like structures, such as dust layer28. The Washburn technique28 is accommodated measuring the contact angle of three-different fluids (water, glycerol, and ethylene glycol). A glass-tube of 3 mm diameter is for the assessment of the selected liquid contact angle, which are drawn-up in the tube under the capillary force. The mass increase in the tube and the time for this mass increase is related through the Washburn technique, which is: (frac{{Delta m}^{2}}{Delta t}=frac{c.{rho }^{2}gamma costheta }{mu }), where Δm is the mass gain, Δt is the time corresponding to mass gain (flow time), c is the capillary constant of the dust, ρ is the fluid density, θ is the contact angle, µ is the fluid viscosity. The capillary constant of dust is evaluated using n-hexane as a liquid, which results in zero contact angle (θ = 0). Hence, the variation of mass gain square (({Delta m}^{2})) with corresponding time ((Delta t)) provides the contact angle of the fluid. The capillary constant for dust particles is estimated as 5.82 × 10–16–6.54 × 10–16 m−5. However, the variation of particle size influences the capillary constant29. Hence, the variation of capillary constant may be associated with dust particle sizes and shapes, which are different. Moreover, to validate the contact angle measurements, dust pellets are prepared in line with early work29. In the process, dust particles are lightly compressed to form pellet-like samples. Water, glycerol, and ethylene glycol are used in the experiments. The contact angle determined from Washburn technique slightly differs from that of measured from the dust pellet surface, i.e., the contact angle measured on the pellet surface for water is about 38.2° while the Washburn technique results in 37.4°. Nevertheless, the difference is small. Table 2 gives the Lifshitz-van der Walls components and electron-donor parameters used in the surface free energy assessments24,25. The surface free energy measured for the dust pellets is about 112.2 ± 5.2 mJ/m2. The experiments are repeated twelve times to secure the accurate measurement of the surface free energy of the dust pellets.

Figure 1

(a) SEM micrograph of coated surface, (b) AFM line scan on coated surface, and (c) droplet contact angle on coated surface.

Figure 2

(a) SEM micrograph of dust particles. Small particles adhere on large particle and small particles form a cluster, (b) size distribution of dust particles, and (c) X-ray diffractogram of dust particles.

Table 1 Elemental composition of dust particles (wt%).
Table 2 Lifshitz-van der Walls components and electron-donor parameters used in the simulation24,25.

Droplet liquid wets dust particles during rolling on the dusty hydrophobic surface. This forms a liquid wrapping layer on the dust surface and gradually infusing liquid, which is droplet fluid (water), encapsulates the dust surface. This phenomenon is defined as the liquid cloaking30. The droplet liquid spreading on the dust surface satisfies the hemi-wicking criteria31, which yields: (S={gamma }_{s}-{gamma }_{L}-{gamma }_{s-L}), here, γs represents dust surface free energy, γL is surface tension of droplet liquid, and γs-L resembles the interfacial tension at droplet liquid and the dust interface. The interfacial tension at droplet interface can be devised as:({gamma }_{s-L}={{gamma }_{s}-frac{{gamma }_{L}}{r}costheta }_{w}), 32, here θw corresponds to contact angle of a droplet fluid on the pellet surface and r represents the roughness parameter of the pellet surface. 3-dimensional imaging microscope is used to determine the roughness parameter, which is found to be 0.61. The contact angle θw is measured as 37° ± 4°. Therefore, ({gamma }_{s-L}=18.23) mJ/m2 and inserting in spreading rate (S) formula, it yields S = 22.27 mJ/m2, i.e. S > 0. Hence, the droplet fluid (water) spreads over the dust surface. Experiments are carried out to determine the water cloaking velocity on the dust surface. In this case, high speed video recording using the micro-lens was utilized to monitor ridges of water on dust particle surface. The tracking program is used to monitor and temporal variation of the height of water ridges on dust surface via uploading the high speed video recorded. Figure 3a shows the cloaking velocity with time for a single dust particle. The cloaking velocity sharply decays with time and shows almost quasi-steady behavior with progressing time until the particle is totally cloaked. The variation resembles the exponential decay in the form of ~ C.e-mt, where C is a constant and m is the parameter which is about 0.25 s−1. The exponential decay of the cloaking velocity is also reported in the early work33 and it occurs because of the force balance among the gravity, surface tension, interfacial tension, and shear resistance34. Energy dissipated due to shear resistance during the cloaking is related to the Ohnesorge number ((Oh={mu }_{o}/sqrt{{rho }_{o}a{gamma }_{L}})); here, ({gamma }_{L}) is the surface tension and a is particle size35. Incorporating the dust particle size ≥ 1.2 µm, the Ohnesorge number becomes much less than one (Oh ~ 0.057). Hence, the dissipation force due share rate becomes small during the cloaking. On the other hand, the dust particle has open porous structure, which can cause droplet water penetration into the dust particle via porous sites. Figure 3b shows computerized tomography (CT) nano-scan of a dust particle. The size of the dust particle is about 25 µm and it is difficult to measure the pore size less than 0.5 µm from nano-CT scan image (Fig. 3b). This is because of the resolution of nano-CT scan, which is 0.5 µm. However, SEM micrographs are analyzed to assess the average pores size. SEM micrograph of a typical large size dust particle is shown in Fig. 3c while Fig. 3d depicts SEM micrograph of magnified large dust particle surface. The pore size various on dust surface and the averaged pore size is estimated at about 450 nm. The open porous sites covers about 35% of the dust particle. To evaluate liquid (water) penetration into the dust particles via porous sites, experiments are carried out to determine mass gain by the dust particle with time. Since, the transition time of droplet wetting diameter on the dusty hydrophobic surface and cloaking time of droplet fluid are small, which is in the order of fraction of a second, experimental duration for the weight gain of the dust particle is extended in the order of 10 times of the cloaking and the transition times. Figure 4 shows the percentage of the mass gain of the dust particle with time. The mass gain of the dust particle is determined from the ratio of difference between final and initial masses of the dust particle over the initial mass of the dust particle. The initial and final mass of the dust particle is measured using sensitive scale (Thomas Scientific). Since the liquid film is formed on the dust particles after complete cloaking, the final mass of the dust particle is measured for the periods after the cloaking period. The initial mass of the particle is measured onset of complete cloaking. Moreover, the percentage of the mass gain of the dust particle ((frac{({m}_{cl}-{m}_{dr})}{{m}_{dr}}), here mcl is the particle mass after cloaking and mdr is the mass of dry particle)) during the cloaking is also measured and it is about 0.1728, i.e. almost 17.28% of the mass of the dry dust is increased during cloaking period. In line with Fig. 4, as cloaking period increases further, the mass gain of the particle remains small, which in the about 0.1% after the duration 140 s inside water after the cloaking period. Hence, infusion of the droplet fluid through the porous sites of the dust particle is extremely small. Consequently, the mass gain of the dust particle during the transition time of the droplet wetting area on the dusty hydrophobic surface is almost limited with the mass of the liquid covering the dust surface due to cloaking. Moreover, the dust layer is considered to have porous structure and the wetted height of the liquid in the dust layer is estimated experimentally. In this case, 3 mm diameter the dust columns with 12 mm height are prepared from the dust particles. High speed recoding system is used to monitor the wetted front in the dust column while the column is located on the liquid (water) film. Infusion of the liquid into porous dust column is governed by the capillary pressure and the pressure drop due to fluid weight and porous resistance along the column height. The capillary pressure can be estimated from Laplace Young Equation36 as: (Delta {P}_{Cap}=frac{2{gamma }_{L}}{{R}_{eff}}), here, γL represents surface tension of fluid and Reff is the effective capillary radius. The pressure drop (pressure loss) can be associated with the friction due to pores structure and gravitational influence (hydrostatic) during the liquid infusion. The pressure loss ((Delta {P}_{loss})) can be formulated incorporating the friction and hydrostatic influences, which yields36: (Delta {P}_{loss}=frac{mu varepsilon }{K}hfrac{dh}{dt}+rho gh), here µ is the fluid viscosity, ε is the porosity, K is permeability, h is the wetting fluid height, ρ is the fluid density, and g is the gravity. SEM micrograph of the cross-section of the dust column (Fig. 5) is used to estimate the porosity. The porosity is determined as the ratio of area covered by pores over the cross-sectional area, which is estimated at about 0.18. Several dust columns are produced from the dust particles to assess the porosity variation. Porosity varies depending on the distribution of the dust size and the shapes in the column prepared. This gives rise to the variation of porosity within 25% in the dust columns. The permeability (K) of the dust columns are measured incorporating permeability meter and K is estimated to be about 4 × 10–16 m2. The tests are repeated ten times using different dust columns prepared and findings revealed that the data for permeability (K) changes with 20%. This is attributed to different size and shapes of the dust particles in the dust column. Moreover, after considering the inertial force of the infusing fluid in the dust column is negligibly small as compared to capillary force36. The force balance between the force generated due to pressure drop and the force due to capillary pressure should be satisfied. This consideration yields the condition that the capillary force is same order of the force for pressure drop during the fluid infusion, i.e. (pi {R}_{eff}^{2}Delta {P}_{Cap}=2pi {R}_{eff}{L}_{eff}{Delta P}_{loss}), where Leff is the effective capillary length. The arrangement of the force balance leads to the equation for wetting fluid height, which becomes:

Figure 3

(a) Cloaking velocity of water on a dust particle, (b) CT scan of a large dust particle, (c) SEM micrograph of a large dust particle, and (d) SEM micrograph of magnified large particle surface.

Figure 4

Percentage of weight gain of a dust particle in water after cloaking is completed during 150 s.

Figure 5

SEM micrograph of dust column cross-section. Large porosity is observed on micrograph.

$$frac{mu varepsilon }{K}hfrac{dh}{dt}+rho gh=frac{{gamma }_{L}}{{L}_{eff}}$$


The effective capillary length is evaluated using the SEM micrograph (Fig. 5); however, the capillary length varies along the different locations of the dust column because of the variation of the dust size, dust shape, and the dust orientation in the dust column. Hence the approximate effective capillary length is considered as same as the column length (0.012 m). The solution of Eq. (1) yields37:

$$h=frac{C left[Wleft(-expleft[frac{-({B}^{2}t+AC}{AC}right]right)+1right]}{B}$$


The Lambert (W) function has the series expansion, i.e.:

$$Wleft(xright)= sum_{n-1}^{infty }frac{{(-1)}^{n-1} {n}^{n-2}}{(n-1)!} {x}^{n}$$


The Lambert function takes the form: (Wleft(xright)=x-{x}^{2}+ {frac{3}{2}x}^{3}-{frac{8}{3}x}^{4}+{frac{125}{24}x}^{5}-{frac{54}{5}x}^{6}+{frac{16807}{720}x}^{7}+dots ). The coefficients in Eq. (2) are: (A= frac{mu varepsilon }{k}), (B=rho g), and (C=frac{{gamma }_{L}}{{L}_{eff}}) (Eq. 1). Incorporating the fluid viscosity, porosity, permeability, surface tension of fluid, and the effective capillary length, Eq. (2) can be solved and height (h) variation with time can be obtained. Figure 6 shows the wetting height (h) with time obtained from the experiment. The predicted wetting height is also included in Fig. 6. The wetting front of the liquid reaches 2.54 mm height in 0.04 s. Hence, the time for the droplet liquid infusing into the dust layer of 150 µm thickness on the hydrophobic surface becomes about 1 ms, which is less than the transition time of wetted length of the rolling droplet on the dusty surface (0.02 s). Consequently, the droplet fluid fully infuses and wets the dust layer on the hydrophobic surface during its rotational transition.

Figure 6

Wetting liquid height along dust column with time due to experiment and predictions of Eq. (1).

Droplet motion on dusty surface

Figure 7 depicts translational velocity of the rolling droplet on the dusty surface for different dust thicknesses, 1° inclination angle of surface, and various droplet volumes. Translational velocity attains lower values for the dusty surface as compared to that corresponding to the clean hydrophobic surface with same wetting state. This is because of the resistance created between the droplet and the dusty surface, and dissolution of some dust compounds in droplet fluid. Dust compounds, such as KCl and NaCl, can dissolve in droplet fluid (water) causing increased surface tension3. Hence, experiments are carried out to determine change of droplet surface tension due to picking up of the dust particles. Surface tension of the droplet fluid increases from 0.072 to 0.119 mJ/m2 after mixing with the dust particles during rolling. Hence surface tension increase becomes almost 4%, which in turn enhances the pinning force (({F}_{pin}cong frac{24}{{pi }^{3}}{gamma }_{L}D{phi }_{s}left(cos{theta }_{R}-cos{theta }_{A}right)), where D is droplet wetting diameter, ϕs is solid fraction, θR and θA are receding and advancing angles of droplet4). However, the gravitational force (({F}_{grav}=mgsindelta ), where m is droplet mass, g is gravitational acceleration, and δ is inclination angle of surface) remains larger than the pinning force4. Moreover, as the dust layer thickness increases, frictional and adhesion forces (causing pinning) lower the droplet velocity significantly, which is more pronounced for the large volume droplet (60 µL). This may be because of: (i) small inclination angle results in small gravitational force for rolling, and (ii) increasing droplet volume enhances wetting radius on the surface so that adhesion force increases. The droplet velocity reduces zero for 20 µL as the dust thickness becomes 150 µm. This is related to the inertial force acting on the droplet under the gravity, which becomes small as droplet mass reduces. Hence, the interfacial friction and adhesion force overcomes the inertial force generated under the gravitational pull. In addition, droplet velocity shows decreasing trend as the droplet volume and the dust thickness increase simultaneously. The balancing force for the droplet rolling is influenced by the increased wetting diameter and the dust thickness, i.e. enlarging wetting diameter due to droplet volume increase enhances the adhesion force; in addition, increasing the dust layer thickness boosts the frictional force. Figures 8 and 9 show droplet translational velocity with distance on the surfaces with dust presence for different droplet sizes and two inclination angles (5° and 10°). Increasing the dust thickness reduces the droplet translational velocity. In addition, increasing droplet volume lowers the translational velocity. This is attributed to large wetting diameter with increasing droplet size, which enhances the frictional and adhesion forces on the surface. As comparing Figs. 8 and 9, increasing inclination angle increases the droplet translational velocity because of the gravitational influence. This becomes more apparent for the small dust thicknesses. Hence, the droplet cleaning of the dusty hydrophobic surfaces is faster as the dust thickness becomes small. Figure 10a–c show high speed recording data for 40 µL droplet images on the dusty surface for various dust thicknesses. The droplet during its transition picks up the dust particles for all the dust thicknesses. As the dust thickness increases, amount of dust picked up by the droplet increases and visual transparency of the rolling droplet reduces. Large amount of dust mixing with droplet fluid occurs for the case of large dust thicknesses. This alters the droplet mass and increase the gravitational force acting on droplet; however, dissolution of some dust compounds (alkaline metal compounds) increases the liquid surface tension. Hence, the pinning force on the surface increases while creating adverse influence on the droplet translational velocity. Moreover, as the dust thickness increases, the ability of droplet picking up dust from the surface reduces. This gives rise to the dust residues remaining on the droplet path. The reduced ability of droplet picking up dust particles is related to (i) the amount of particles cloaked by the droplet fluid, during its transition, is limited and as the dust thickness increases, particles only cloaked by the droplet fluid are picked up by the rolling droplet, and (ii) short transition duration of the droplet wetted area on the dusty surface limits the cloaking and spreading of the droplet fluid on the surface of the dust particles. Figure 11a shows SEM micrograph of the dust residues on the surface. In addition, Fig. 11b depicts optical image of 20 µL droplet terminating on the thick dusty surface (150 µm thick) for the inclination angle of 1°. The dust residues have sharp edges and the residues with sharp edges can poke on the surface and they become hard to pick up by the rolling droplet. In the case of Fig. 11b, dusts are present on the droplet path and the point of droplet termination (end of transition on the surface), the droplet picks up almost all dust. The Bond number measures the importance of gravitational force over the surface tension force and it can be expressed as (Bo=frac{Delta rho g{l}^{2}}{gamma }), here Δρ is the density variation in the droplet, g is gravitational acceleration, l is the droplet characteristics diameter, and γ is the surface tension. For small volume droplets (≤ 40 µL), the Bond number remains less than unity. For mall volume droplets, the wetting diameter on the surface is about 2.1–3.2 mm. However, for large volume droplet (60 µL), the Bond number is about 3.7, which is greater than unity and the wetting diameter on the surface is about 4.1 mm due to large puddling of the droplet. Hence, increasing Bond number causes large area of cleaning on the surface. However, mixing of the droplet fluid with dust particles slightly alters the surface tension, i.e. surface tension increases from 0.072 N/m to 0.119 N/m during rolling. This is because of dissolution of alkaline (Na, K) and alkaline earth (Ca) metals in the droplet fluid3. In addition, mixture of the droplet fluid and dust alters the droplet density from 1,000 kg/m3 to 1,070 kg/m3 during droplet rolling over the length scale of 5 mm on the dusty surface for dust layer thickness of 100 µm and droplet volume of 40 µL. This changes the Bond number slightly because change of density and surface tension as the droplet rolls on the dusty surface. Moreover, the change of surface tension is about 4% while the density increase is 7%. Hence, Bond number increases slightly as the droplet rolls over the surface. This is more pronounced with increasing droplet volume, i.e. amount of dust particles picked up increases.

Figure 7

Translational velocity of droplet obtained from experiment on 1° inclined surface for various droplet volumes and dust layer thicknesses.

Figure 8

Translational velocity of droplet obtained from experiment on 5° inclined surface for various droplet volumes and dust layer thicknesses.

Figure 9

Translational velocity of droplet obtained from experiment on 10° inclined surface for various droplet volumes and dust layer thicknesses.

Figure 10

(a) Optical image of 20 µL droplet on dusty hydrophobic surface. Dust thickness is 50 µm. (b) Optical image of 20 µL droplet on dusty hydrophobic surface. Dust thickness is 100 µm. (c) Optical image of 20 µL droplet on dusty hydrophobic surface. Dust thickness is 150 µm.

Figure 11

(a) SEM micrograph of dust residues on the coated surface. Dust particle has sharp edges, and (b) optical image of 20 µL droplet on 150 µm dust layer with inclination angle of 1° (droplet terminates on dusty surface). Circle shows dust residues on inclined dusty hydrophobic surface.

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