Dynamic Behavior of Water Droplet Impact on Microtextured Surfaces

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Dynamic behavior of water droplet impacting microtextured surfaces: the effect of geometrical parameters on anisotropic wetting and the maximum spreading diameter Xiying Li, liqun Mao, and Xuehu Ma Langmuir, Just Accepted Manuscript • DOI: 10.1021/la304567s • Publication Date (Web): 25 Dec 2012 Downloaded from http://pubs.acs.org on December 25, 2012

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Dynamic behavior of water droplet impacting micro-textured surfaces: the effect of geometrical parameters on anisotropic wetting and the maximum spreading diameter Xiying Li1,*, Liqun Mao1, Xuehu Ma*2 *Corresponding author: E-mail address: [email protected] *Corresponding author: E-mail address: [email protected] Phone: (86)-378-3881589, Fax: (86)-378-3881589. 1

School of Chemistry and Chemical Engineering, Henan University, Kaifeng, 475001, China 2

Institute of Chemical Engineering, Dalian University of Technology, Dalian 116012,

China

ACS Paragon Plus Environment

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Abstract: Textured silicon surfaces decorated by the square arrays of pillars with adjustable

pitch

were

fabricated.

The

wetting

behavior,

especially

for

direction-dependent water contact angles on the textured silicon surfaces after silanization, was investigated by incorporating the contact line fraction into a modified Wenzel model. Also, the effect of geometrical parameters on the anisotropic wetting behavior of water was examined with respect to water droplet impacting textured surface. Moreover, the maximum spreading factor was studied theoretically in terms of energy conservation allowing for surface topography and viscous friction of the liquid flowing among the arrays of the posts. Theoretical models were found to be in good agreement with experimental data.


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1. Introduction Controlling the wettability of solid materials is a fundamental issue in surface engineering, due to its importance and applications for manipulating adhesion, printing, cleaning, painting, and flocculation.1,2 From a theoretical point of view, the Young equation, formulated around 200 years ago, remains as the basic equation in the regime of wetting. According to the Young equation, the contact angle of a drop resting on an ideally flat surface can be determined from γSV, γSL, and γ, which represent different surface tensions (solid/vapor, solid/liquid, and liquid/vapor, respectively) involved in the system. The balance of these forces leads to the classical Young equation: cos θ Y =

γ SV − γ SL , γ

(1)

where θ Y is the Young contact angle or intrinsic contact angle. As a matter of fact, the apparent contact angle in practice is influenced by the complex interaction of surface chemistry and topography. Two existing models, referred to as Wenzel3 and Cassie-Baxter4 models, can account for the apparent contact angle on a rough surface. According to Wenzel’s theory, the liquid on a solid surface thoroughly follows the roughness of the surface and thus leads to homogeneous wetting. From the perspective of surface energy under the condition of thermodynamic equilibrium state, there is a linear relationship that accounts for the apparent contact angle and the intrinsic contact angle on a rough surface. Thus, the roughness factor r is defined as the ratio of actual solid surface area in contact with water over the projected one and can be introduced to compute the apparent contact angle in combination with the


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Young contact angle in the Wenzel state:

cos θ ∗ = r cos θY ,

(2)

where θ ∗ is the apparent contact angle. According to the Wenzel model, surface roughness reinforces original hydrophilic/hydrophobic nature, since the value of r is constantly larger than 1. Apart from homogeneous wetting (Wenzel model), heterogeneous wetting (Cassie-Baxter model) defines the other wetting state where a liquid droplet stays on a composite surface. For Cassie-Baxter model, the weighted effect of each acting phase corresponding to its own fraction is responsible for the apparent contact angle. Accordingly, a liquid droplet residing on a composite surface (here specially referring to the composite surface made of air and solid) displays a contact angle as follows: cos θ ∗ = −1 + φs (cos θ Y + 1) ,

(3)

where φs refers to the fraction of solid in contact with the liquid. Air entrapment below the liquid droplet can remarkably increase the contact angle up to 150D or more and simultaneously maintain very low contact angle hysteresis, namely achieving

a

superhydrophobic

state.5,6

However,

Extrand,7

MacCarthy,8-10

Bormashenko11,12 and Anantharaju13 have recently argued that apparent contact angles are determined by the three phase contact line rather than the contact area. The Cassie-Baxter theory, based on a minimization of the global Gibbs free energy of the system, cannot account for the anisotropic wetting phenomenon essentially dependent on local conditions where triple phase line is located.12-17 If the roughness geometry is on the whole random, the profile of a sessile drop remains almost spherical and the


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apparent contact angle of the drop on the rough surface is nearly uniform along the contact line. However, if the roughness geometry is not isotropic, the apparent contact angle may no longer be uniform along the contact line. In this case, the modified Cassie-Baxter model is qualified to express the apparent contact angles on a rough surface by incorporating the line fraction of triple phase contact line14 or differential parameter.15 As a result, geometrical parameters in conjunction with surface chemistry can govern anisotropic wetting and therefore control the dynamic behavior of liquid brought into contact with micro-textured surface.16-25 Zipping wetting is currently attracting more attention due to its potential applications in microfluidic system. For example, Courbin et al.16 recently discussed the wetting behavior on topographically patterned but chemically homogeneous surfaces and investigated the mechanism of shape selection in the phase of liquid imbibition within the interstices. Furthermore, they argue that wetting shape selection can be manipulated by adjusting surface topography or adopting liquids with different equilibrium contact angles. Similarly, when superhydrophobic state breaks down, the wetting front propagates in a stepwise manner and the interstices among pillars are impregnated with water through a zipping mechanism.20-22 Under this condition, the square-shaped wetted area eventually emerges on a micro-structured surface decorated by the square arrays of pillars. Moreover, some researchers have revealed that polygonal wetting also occurs on micro-structured surfaces after liquid droplet impact.23,24 Nevertheless, the intrinsically complex nature of the liquid-solid interaction makes it difficult to elucidate anisotropic wetting behavior, and further


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work is urgently needed to acquire more insights into the phenomena. Therefore, in this paper we examined anisotropic wetting in light of the influence of contact line fraction on the apparent contact angle. Furthermore, we also investigated the shape selection of wetted area and the dynamics in relation to liquid droplet impacting the micro-structured surfaces. 2. Experimental section

Textured surfaces with the square arrays of pillars were fabricated by photolithography combined with reactive ion etching. The method was invented by Laemer and Schilp and is also known as time-multiplexed alternating process or, more commonly, as the Bosch method.26 At first, with the assistance of the lithographic technique (BP212 positive photoresist), a pattern of circular shapes with square arrangement was transferred from a photomask onto a silicon wafer with an oxide layer of 1μm which is achieved by a thermal treatment in the presence of oxide gas. Thereafter, the oxide layer, which was free after development, was etched by reactive gas of C4F8. Then photoresist unexposed was dissolved in its benign solvent and isotropic etching with reactive gas of SF6 was subsequently applied to etch Si during which the remaining oxide layer acted as the mask. A three-dimensional (3D) structure with the square arrays of micro-posts was eventually fabricated. All the textured surfaces adopted were provided by Micro Electro Mechanical Systems (MEMS) of Dalian University of Technology. The height of the micro-posts was regulated by controlling the etching time and the height of the micro-posts was set at 20 µm. A patterned surface topography was designed by independently adjusting the


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P

2R

H

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L

50µm Figure 1. SEM images and schematic pictures of the square arrays of micro-posts. The definition of characteristic length for the micro-posts is given in the left picture.

pillars’ pitch. The as-prepared textured surfaces are denoted as TP with superscript P referring to pitch. The pillars have a diameter of 10 µm and contain pitches of different sizes (20, 40, 60, and 100 µm). Figure 1 shows the representative scanning electron microscopic (SEM) images of the patterned surfaces, where the directions along the lattice axis and diagonal line are defined to discuss anisotropic wetting in subsequent sections. These patterned surfaces turned out to be hydrophobic after being immersed in 1wt% solution of octadecyltrichlorosilane (OTS; Alfa Aesar) at room temperature for five minutes. The equilibrium contact angles against water droplet with a volume of 4µl on the textured surfaces were measured with a goniometer (OCA20, Dataphysics Co., Germany). Five repeat measurements were conducted at different locations of the same textured surface where a water droplet was placed randomly. Experimental results of contact angles show good repeatability, with relative deviation of ±2o . The equilibrium contact angle on a flat silane-terminated silicon wafer was also measured ( 103 ± 2D ), and it was assumed to be


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equal to the Young angle described in Eq. 1-3.27 The scenarios of the water droplet impinging events were recorded by a high-speed camera (CCD; Photron, Fastcam Apx-Rs equipped with a long distance microscope Hirox, OL-35, Japan). CCD works at 10,000 frames per second at a resolution of 512 × 512 pixels. An image processing software (Image-Pro Plus) was used to measure the contact diameter after a liquid droplet collided with textured surfaces. In our experiments, the spatial resolution is determined to have an order of 10 μm per pixel. Each impinging event was repeated under the same impact condition at least six times. The systematic error of length measurement for water droplet impacting all textured surfaces is about 0.08 mm. All the experiments were implemented at room temperature 25DC .

a

b

146°

160°

c

d

108°

104°

Figure 2. Apparent contact angles on different textured surfaces in the lattice axis direction. Here a, b, c and d represent the textured surfaces T20, T40, T60 and T100, respectively.


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3. Results and discussion 3.1 Discussion on contact angles in context of contact line fraction Figure 2 shows the images of water droplets (volume 4 µL) gently depositing on the

textured surfaces. It is obvious that the water droplets resting on T20 and T40 are in the Cassie wetting regime, showing contact angles of 146o and 160o respectively, conforming to 148o and 164o calculated by the Cassie model. Furthermore, the light passing through the interstices on the T40 surface, as shown in Figure 2, also confirms that the water droplet lies in the Cassie state. However, the wetting states on T60 and T100 are in the Wenzel regime and the corresponding contact angles in the lattice axis 160o

(a)

(b)

114o

(c)

108o

Figure 3. Wetting regime is apt to undergo a transition triggered by external actuation. The images represent wetting transition from Cassie regime (a) to Wenzel regime (b and c) on the surface with the pitch of 40 µm. Here (b) and (c) in Wenzel regime correspond to the contact angles in the lattice axis direction and diagonal direction, respectively.


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direction are 108o and 104o respectively, conforming to 105o and 104o calculated by the Wenzel model. The Cassie state on T20 and T40 is metastable and tends to undergo a transition from the Cassie to the Wenzel regime, provided that the static liquid droplets are exposed to vibrating, squeezing and impacting motions.5,28-33 Once the transition of wetting state occurs on the T40 surface, the contact angles become direction-dependent, illustrating the difference of water contact angles along lattice axis and diagonal line (Figure 3). As has been discussed elsewhere,7-13 the contact angle is determined by a triple phase line instead of the contact area. Likewise, we examined the anisotropic wetting behavior of water on the micro-structured surfaces decorated by the square arrays of pillars by substituting the contact area ratio in the Wenzel equation with the contact line fraction ratio. Neglecting the local torsion of

Table 1. Comparison of the measured contact angles and the predicted ones

θL

Sample* 20

T T40 T60 T100

θD

θ∗

Measured

Predicted

Measured

Predicted

128±2° 114±2° 108±2° 104±2°

132° 117° 112° 108°

118±2° 108±2° 105±2° 101±2°

123° 112° 109° 107°

125° 108° 105° 104°

θL and θD represent the contact angles in lattice axis and diagonal directions while resting droplets remain homogeneous wetting state. θ* denotes the apparent contact angle stemming from Wenzel equation, cos θ = r cos θ Y , where r , defined as the ratio of the actual area ∗

contacting with liquid to the projected area, refers to roughness factor. The predicted contact angles are derived from the modified Wenzel equation, cos θ = r ′ cos θ Y , where the roughness ∗

factor is replaced by the ratio of the real length of the triple line to the projected area. Along the lattice axis, rL′ = 1 + 2 H P ; along the diagonal direction, rD′ = 1 + 2 H *The superscript indicates the pitch value.


2P .

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the contact line, we can simply rewrite the roughness factors corresponding to the lattice axis direction and diagonal line direction denoted in Figure 1 as rL′ = 1 + 2 H P and rD′ = 1 + 2 H

2 P , respectively.

Thus, the apparent contact angles can

be calculated according to the modified Wenzel equation incorporating the contact line fractions. Table 1 shows the resultant apparent water contact angles calculated and the relevant experimental ones. From Table 1, there is a noticeable discrepancy between the theoretical contact angles estimated from the above-mentioned modified Wenzel equation and our experimental results. This could be explained two ways. First, the above-mentioned modified Wenzel equation does not take into account the torsion of the contact line due to its pinning on micropillars; and second, the contact angle hysteresis is ubiquitous on the actual surface, which inevitably leads to errors for contact angle measurement (the apparent contact angle can take any value of an angle between receding and advancing angles34). As discussed by Lafuma and Quéré,5 there is considerable contact angle hysteresis (up to about 100o) for a Wenzel regime on a rough surface. According to previous works,31,35,36 there are also 2~15o of difference between an experimental contact angle and one computed from the Wenzel equation. We do not intend here to determine which model is better, but we argue that the direction-dependent contact angles can be explained by introducing line fractions in the lattice axis and diagonal directions, as discussed by Courbin et al.16 More recently, Papadopoulos et al.,37 also examined a similar phenomenon using a laser scanning confocal microscopy (LSCM).


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(a)

(b)

(c)

(d)

0.1ms

2

4.5

11.5

16

Figure 4. Sequential images of water droplet impacting onto patterned hydrophobic surfaces with the pitch of (a) 20, (b) 40, (c) 60 and (d) 100 µm. The scale is 2 mm; and We = 14.71. The small background specks are unrelated to droplet impacting events.

3.2 Dynamics of water droplets impacting on textured surfaces

The dynamic behavior of the water droplets impacting on the textured surfaces was experimentally investigated. As shown in Figure 4, as the water droplet comes into contact with the textured surface, the spherical droplet changes its shape from truncated cap to pancake, while liquid film rapidly spreads outward under the action of kinetic energy. At the initial stage of spreading, a pyramidal structure with the capillary waves,38 propagating from the bottom to the top of the droplet, emerges from the liquid/gas interface. In the phase of the droplet spreading outside, the kinetic


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Figure 5. Temporal evolution of the contact diameter along the lattice axis direction and diagonal after water droplet impacting onto different micro-patterned surfaces. The subscripts L and D represent lattice axis and diagonal directions, respectively. Here We = 14.71; and error bars are of the order of the symbol sizes.

energy is mainly converted to surface energy and partly dissipated by viscous friction, and therefore the propagating velocity of the contact line becomes slower and eventually quiescent. Thereafter, triple line temporarily remains quiescent subject to pinning forces and then recedes at a low speed driven by surface tension and capillary force. After the liquid droplet impact, the maximum wetted diameter becomes longer with increasing the pitch between adjacent posts. Relevant quantitative information is given in Figure 5. It should be noted that the maximum wetted diameters for water droplets in collision with the T40 and T60 surfaces become direction-dependent. Namely, the wetted length in the diagonal direction is slightly larger than that in the


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(a)

(b)

(c)

(d)

Figure 6. Different shape selections of water droplet impacting different textured surfaces. The shape selection depends on the topographic features arising from different pitch of patterned surfaces. Resultant wetted shapes include (a) octagon, (b) rounded octagon, (c) irregular circle and (d) circle, corresponding to the pitch of 20, 40, 60 and 100 µm, respectively. The length scale is all 2 mm except for 0.5 mm in (a); We = 14.71.

lattice axis direction. After the droplet impingement, we also examined the wetted footprints when the bulk of the liquid was carefully removed with a slim thread. The top view of the wetted profiles is demonstrated in Figure 6. As shown in Figure 6, the anisotropic wetting definitely occurs on the T20 and T40 surfaces. Also, the wetting shapes undergo a successive transition from octagon to rounded octagon and irregular


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Liquid

(a)

P

dx

L

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(b)

(c)

(d)

Figure 7. Schematic pictures of liquid fronts passing through the interstices between the posts. (a) The liquid front just contacts the outmost rims of the posts and advances ahead. (b) The liquid front approaches the nearest region between two adjacent posts. (c) The liquid front passes the nearest region and propagates forward. (d) The liquid front moves outside to the right side of the two posts.

circle, and eventually to circle. Many researchers have discussed the similar phenomena.16-25 Some researchers suggested that anisotropic wetting can be explained by a “zipping” mechanism in which the dynamics depend on the balance of interfacial energy and viscous dissipation.20-22 The wetting phenomena observed in our experiments, however, differ from those reported previously, since the textured surfaces employed herein are decorated by circular posts rather than square pillars as adopted by Sbragaglia’s group.20-22 Therefore the dynamic of the liquid front advancing among the micro-posts should be different from what has been reported by Sbragaglia’s group.20-22 Pointing to such differences, we take geometrical parameters into consideration to address the critical contact angle at the specified location which


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is responsible for anisotropic wetting. Figure 7 illustrates the water droplet passing through the crevice between two adjacent micro-posts. When the descending droplet contacts the hydrophobic micro-textured surfaces, the liquid-air interface tends to preferentially contact the bottom of the cavities in the diagonal direction within a single cell of the array, under the action of dynamic pressure or water hammer.30-33, 39 Thereafter, the liquid front continuously advances horizontally between the micro-posts. As for scenario (a) in Figure 7, the liquid front moves ahead and contacts the outermost sides of two adjacent posts. With the assumption that the liquid fronts move forward with a small displacement dx P , the interfacial energy changes with the liquid-gas interface and the replacement of the solid-gas interface by solid-liquid interface. The total energy conversion is therefore established in combination with simple knowledge of trigonometric function:

(

)

(

dEs = −γ Pdx +2 H 2 Rdx − γ cos θ Y Pdx +2 H 2 Rdx

).

(4)

The limit dEs = 0 features a critical contact angle θ c , below which the liquid front can spontaneously spread ahead horizontally. The critical contact angle can be expressed as:

(

cos θ c = − Pdx + 2 H 2 Rdx

) ( Pdx + 2 H

)

2 Rdx = −1 .

(5)

Namely, θ c = 180D , which means that all liquids can spontaneously spread forward within crevice just as they contact the outermost rims of the posts. Similarly, as to scenario (b) in Figure 7 (the liquid front approaches the nearest region between two adjacent posts), the total energy gained can be obtained when the liquid front moves forward with a small distance dx:


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(

)

(

)

dEs = −γ L+ ( dx ) R dx − γ cos θ Y L + ( dx ) R dx − 2γ cos θ Y Hdx − γ H ( dx ) R . 2

2

2

(6)

When the terms with a higher order of dx are neglected, the critical contact angle θ c is therefore derived from the limit dEs=0 referring to equation 6: cos θ c = − L ( L + 2 H ) .

(7)

As a result, while the liquid front moves from the outmost sides to the shortest region between two adjacent posts, the critical contact angles are within the scope of ⎡⎣ cos -1 ( − L ( L + 2 H ) ) ,180D ⎤⎦ , where cos

-1

represents the inverse function of cosine.

Similarly, the critical contact angles corresponding to scenarios (c) and (d) in Figure 7 are also derived as cos θ c = − L ( L + 2 H ) and cos θ c = 1 , respectively. Thus, as the

liquid front passes through the nearest region and reaches the rims outside downstream, the critical contact angles cover a range of ⎡⎣ 0, cos -1 ( − L ( L + 2 H ) ) ⎤⎦ . It is certain that the aforementioned analysis is addressed from the perspective of thermodynamic equilibrium. However, the dynamic mechanism pertaining to liquid droplet impacting textured surfaces is substantially complex and still remains obscure, because the trajectory of the moving triple line is directly related to the interaction of inertial, capillary, viscous and wetting hysteresis forces. In this regard, there is still no convincing theory to resolve the dynamic of the moving contact line with regard to droplet impacting.40 In fact, the contact line will also move forward when the dynamic contact angle gets up to advancing contact angle under dynamic pressure, regardless of the restriction from the critical contact angle. In this respect, the critical contact angle, which is derived according to interfacial energy conversion, is only applicable to the quasi-equilibrium state. For example, it works well only when the front velocity


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within the posts is very slow as validated by Courbin et al. and Sbragaglia et al.16, 20-22 Our experimental results (see Figure 6) demonstrate that a polygonal wetting shape emerges on the T20 due to the fact that its critical contact angle at the nearest region (102o) is roughly equal to corresponding equilibrium contact angle on flat silanized silicon substrate (103 ± 2o), which is consistent with Sbragaglia’s arguments.20-22 As for the droplet impacting textured surfaces, it is therefore reasonable to suppose that polygonal wetting is not only related to the critical contact angle but also to the movement of the contact line, which might be retarded by capillary and viscous forces at the nearest region between adjacent posts. The arguments herein about the critical contact angle have potential applications in tailoring wetting patterns16,41 and capturing heterogeneous nucleation of gas microbubbles42 under the condition that the geometrical parameters on textured surfaces are smartly designed. Moreover, as shown in Figure 6, the length of the wetted profile on T20 surface after impact (about 1.2 mm) is considerably shorter than its maximum wetted diameter (about 3.85 mm, see Figure 5), which implies that the impacting water droplet lies in the coexistence of the Wenzel state and the Cassie state, under the influence of water hammer and capillary force.39 After the liquid droplet reaches its maximum spread, the receding velocity is high, since the pinning force that arose from the pillars’ top in the Cassie state is greatly reduced compared to that in Wenzel state. However, when receding contact line reaches the regime in the Wenzel state, the contact diameter remains unchanged for a short while, as shown in Figure 5; before dynamic contact angle decreases to receding contact angle the contact line always remains stationary.


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Finally, the bulk droplet detaches from the textured surface while the liquid thread linking the bulk droplet to the textured surface pinches off under inertial effect, therefore leaving behind a wetted patch as shown in Figure 6. In the case of the coexistence of the Wenzel and the Cassie states, the contact time between the liquid droplet and T20 surface can be estimated by the method reported in our previous work.43 The pinch-off liquid thread retards the receding motion and thus leads to a time-delay for the droplet to detach from the textured surface. Extra time arising from the pinch-off of liquid thread, calculated via τ = ( ρ a 30 γ ) , is 1.52 ms (here a 0 12

refers to the radius of the liquid thread).44 Similarly, the contact time on superhydrophobic surface with the same solid fraction as that of T20 is also calculated as 18.77 ms and the total contact time on T20 is therefore 20.29 ms,43 which is very close to our experimental result (20.60 ms, as shown in Figure 5). In this regard, the coexistence of the Wenzel and the Cassie states for the impingement on T20 can also be validated indirectly. Such coexistence of the Wenzel and the Cassie states during droplet impacting has also recently been noticed and discussed by Reyssat et al.24,30 Furthermore, as for the water droplet impacting the textured surfaces, the octagonal wetted footprint in our experiments is different from the square counterpart reported by Sbragaglia et al.;20-22 the reason for anisotropic wetting herein might be that the liquid front propagation in interstices was triggered by dynamic pressure for droplet impacting, intrinsically different from that dominated by the spontaneous breakdown of Cassie state as discussed by Sbragaglia et al.20-22 Also, Figure 8 shows the two-dimensional schematic diagram signifying a liquid front advancing through the


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ΔD

ΔL

2R

L

Figure 8. Schematic diagram of the profiles of the triple line distorted in lattice axis direction and diagonal direction for the square arrays of the micro-posts.

spacing between the posts. Assuming a slight distortion at the liquid-air interface, the balance of capillary forces and dynamic pressure can be established as follows: γ

ΔL 2

L

≈γ

ΔD

⎡ 2L + 2 ⎣

(

)

2 −1 R⎤

2

= pd ,

(8)

⎦

where Δ D and Δ L refer to the slight distortions along the lattice axis and the diagonal direction, respectively. According to equation 8, it is easy to get Δ D > Δ L . Furthermore, the distances needed for the liquid front to advance and reach the next post are L + R and

(

) (

2 2 L − 1−

)

2 2 R,

corresponding to the lengths in the lattice

axis and diagonal direction respectively. Thus, the liquid fronts in the diagonal direction preferably start to move and touch the next posts forward at the same dynamic pressure, eventually leading to octagonal wetted footprints instead of the


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square-shaped counterparts, reflecting the underlying square arrays of the micro-posts. It should be noted that Cassie impregnating state cannot occur for impacting water droplets in both spreading and receding stages. Unlike the water droplets vibrating on hydrophobic textured surface,35 where the sustaining and constant driving force gives rise to wetting transition, the driving force herein, which originates instead from the dynamic pressure for water droplet impact, is attenuated as the process proceeds. As a result, the liquid front is unable to spontaneously advance ahead among the arrays of the posts. In addition, with respect to the receding phase of water droplet impact, the Cassie impregnating state would arise only if the liquid film retracts from the patchwork of the solid and liquid itself. Under this condition, the inertial force must be sufficient to surpass the co-action of wetting hysteresis force and viscous force,2 which is described as:

ρ bω 2 A = where ρ , γ

μUC π D + γ D ( cos θ r − cos θ ∗ ) , θr

(9)

and μ refer to liquid density, surface tension and viscosity;

b = (1 6 ) π D03 and it represents the volume of the liquid droplet with a diameter of D0

(D0 = 2.7 ± 0.02 mm in our experiments); D is the contact diameter and is regarded as the maximum spreading diameter as receding phase starts; U is receding velocity (equal to 0.1 m/s in our experiment); A denotes oscillating amplitude; ω is oscillating frequency with the form of ω = ( 4 π ) γ ρ D03 , corresponding to first-order vibration frequency of freely oscillating liquid drop;45,46 θ r signifies receding contact angle in Cassie impregnating state and has the form of


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cos θ r = 1 − φs (1 − cos θ r , f ) ( θ r , f is the receding contact angle on a flat solid surface, and θ r , f = 87 ± 2D , as determined in our experiments); θ ∗ represents the apparent contact angle for Wenzel state and cos θ ∗ = r cos θ Y ( θ Y is Young contact angle on a flat solid surface, and θY = 103 ± 2D in our experiment); and C is constant with an order of 10.2 According to order of dimensional analysis with respect to equation 9, the threshold oscillating amplitudes, which are necessary to induce Cassie impregnating state on the textured surfaces of T40, T60, and T100 during receding phase, are calculated to be 8.7, 10.0, and 13.1 mm, respectively. However, in our experiments the maximum oscillating amplitude after droplet impacting is less than 5 mm, which means that Cassie impregnating state is inaccessible in receding stage. In fact, referring to the wetted footprints after impacting (see Figure 6), the characteristic length of the wetted area is always smaller than the maximum spreading length and nearly equal to that of the eventual contact area between the solid and the liquid droplet, which also means that Cassie impregnating state is unable to happen during liquid impacting. It is known that the accurate prediction of the maximum spreading diameter is crucial in achieving a fundamental knowledge of the dynamic features on the liquid droplet impingement. Many researchers, by taking into account the viewpoint of energy conservation, solving mass and momentum equations, and conducting experiments, have established theoretical models to estimate the maximum spreading factor.47-58 In our previous research, we have also investigated the effect of surface topography on the receding dynamics and contact time between impacting droplet and textured


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surfaces. Furthermore, we have revealed that the remaining liquid film influences the receding contact angle and modifies the retraction dynamics, therefore responding to the dependence of the contact time between liquid droplets and superhydrophobic surfaces on geometrical parameters.43 Unfortunately, until now, the dynamics, especially for the maximum spreading factor for droplet impacting textured surfaces, still remain unclear in that most models cannot incorporate the changes of surface energies associated with the geometrical parameters and viscous dissipation of the liquid flowing within the arrays of the micro-posts. To overcome this drawback, we have established a theoretical model allowing for geometrical parameters and viscous dissipation arising from liquid flowing within the micro-posts. Several key dimensionless parameters are essential for dealing with the impact dynamics of a liquid droplet, mainly including Weber number (We), Reynolds number (Re), and capillary number (Ca), which can be described as: We = ρV0 D0 γ , Re = ρV0 D0 μ , Ca = We Re = μV0 γ , 2

(10)

where D0 and V0 represent initial diameter and impacting velocity of the liquid droplet, respectively. In our experiments, the maximum impacting velocity is set at 0.99±0.01m/s, corresponding to We = 36.75, below which the splashing fails to emerge after impact. Just before collision with the solid substrates, the total energy of the impacting droplet includes kinetic energy ( EK,0 ) and surface energy ( ES,0 ), which are given as: EK,0 =

1 12

πρ D03V02 ,

Es,0 = π D02γ .


(11) (12)

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After impact, the kinetic energy of the impinging droplet is stored as the surface energy during its deformation and liquid lamella moves toward to peripheral rim. At the maximum spreading with the kinetic energy being negligible, two parts of energy, referred to as surface energy and viscous dissipation, are involved in the spreading phase. The surface energy of the liquid film with an approximate cylindrical shape can be expressed as: Es,1 =

1 4

2

D03

3

Dmax

2 π Dmax γ (1 − r cos θY ) + π

γ,

(13)

where r = 1 + 2π RH P 2 , and it defines roughness factor representing the ratio of the actual contact area to the projected one. Considering cos θ ∗ = r cos θY (the Wenzel law applies), we can rewrite equation 13 as: 1 2 D3 2 Es,1 = π Dmax γ (1 − cos θ ∗ ) + π 0 γ . 4 3 Dmax

(14)

As to the liquid flowing on the textured surface in Wenzel state, the viscous dissipation energy can be divided into two parts: one comes from the interaction between the liquid and the pillars’ top, and another comes from liquid flow among the arrays of the micro-posts. According to the modified expression of Pasandideh-Fard,49 taking into account the fraction of the pillars’ top together with the characteristic time D0 V0

during which impacting droplet spreads up to the maximum diameter, the

relevant viscous dissipation energy can be described as:

W1 =

π 2 μV0 Dmax φs Re , 8

(15)

where φs = πR 2 P 2 , and it represents the area fraction of the pillars top. Additionally, the viscous dissipation arising from liquid flowing among the arrays of the


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micro-posts can be further divided into two parts: one corresponds to the friction between the moving liquid and the bottom of the textured surface, another takes place against the micro-posts. According to the argument of Ishino et al.,59 the viscous force from the bottom is expressed as: V D2 FV = πμ , H 2

(16)

and corresponding dissipative energy with a little displacement dx is expressed as: ⎛ V D2 ⎞ dW2 = ⎜ πμ ⎟ ( dx ) . H 2 ⎠ ⎝

Here,

we

V = DmaxV0 D0

replace

transient

propagating

velocity

(17) with

averaged

velocity

to circumvent the difficult task in depicting the dynamics of the liquid

while flowing upon the textured surface. After integration over the distance Dmax 2 , we have the dissipative energy as: 4 Dmax V0 1 . W2 = πμ 12 D0 H

(18)

Similarly, as suggested by Ishino et al.,59 the dissipative energy due to the viscous friction between the liquid and the ensemble of the posts has the following form: 4 π 2 μV0 Dmax H W = . 6 ⎡⎣ln ( P R ) − 1.31⎤⎦ D0 P 2 ' 2

(19)

Of course, although both parts of dissipative energy should coexist, their weighted contributions are dependent on geometrical parameters. Omitting the influence of the coefficients in equations 18 and 19, we have the ratio of the two parts of dissipative energy as below: W2 W2' ~ P 2 H 2 .


(20)

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Figure 9. Comparison between the maximum experimental spreading factors and the theoretically computed ones. The experiments were implemented at different Weber numbers including 11.02, 14.71, 22.06, 29.44 and 36.75; error bars are of the order of the symbol sizes.

For short posts (H < P), the dominant dissipative energy arises from the bottom of the textured surface. In contrast, for long posts (H > P), the viscous friction between the liquid and the forest of the posts is dominant. In terms of larger pitch as compared with the posts, considering the energy conservation condition EK,0 + Es,0 = Es,1 + W1 + W2 and combining equations 11, 12, 14, 15, and 18, we finally achieve the equation accounting for the maximum spreading coefficient: Ca

D0 4 ⎛ 1 ⎞ β + 3 ⎜1 + Ca Reφs − cos θ ∗ ⎟ β 2 + 8 β − (12 + We ) = 0 , H ⎝ 2 ⎠

(21)

where the maximum spreading coefficient is described as: β = Dmax D0 . Likewise, in the case of longer posts as compared with the pitch, corresponding equation can also


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be established by incorporating equation 19 as below: ⎛ 1 ⎞ β 4 + 3 ⎜ 1 + Ca Reφs − cos θ ∗ ⎟ β 2 + 8 β − (12 + We ) = 0 . ⎝ 2 ⎠ [ (ln ( P R ) − 1.31] P 2πCaD0 H

2

(22)

In Figure 9, the comparison of our experimental data and theoretical results is demonstrated. Both equation 21 and equation 22 are employed to deal with the maximum spreading factors on T40 (Figure 9), because the height of the pillars (20µm) is comparable to the length of the pitch (40 µm). The present model can be well used to evaluate the maximum spreading factor associated with droplet impacting textured surfaces. As of present, we are still unable to obtain the expression for the limit when the pitch is close to the height of pillars because the complex interactions between the impacting droplet and the textured surface still remain unclear. We believe that the discrepancy between our experimental results and those theoretically simulated from our model is mainly attributed to the adoption of equations 18 and 19 with the assumptions of H P and H P , respectively.59,60 After all, the viscous dissipation associated with liquid flow in interstices on the textured surface actually comprises two parts as discussed previously, and therefore the two parts of the energy dissipation simultaneously come into effect in that the resultant velocity gradients essentially depend on the interaction of two flow fields, i.e., the liquid flow on the bottom and the one around the cylindrical micro-pillars. Comprehensive knowledge about viscous dissipation with respect to this flow pattern is still deficient due to its complex dynamics.59,60 This may also partly account for the deviations between our experimental results and theoretical ones. Meanwhile, equations 18 and 19 are only suitable for low Reynolds number flow, and thus at an increased Weber number the


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Figure 10. Comparison between the maximum experimental spreading factors and the theoretical ones computed from Pasandideh-Fard et al. model49, Mao et al. model50, Ukiwe and Kwok model55 and present model. The experiments were implemented at different Weber numbers including 11.02, 14.71, 22.06, 29.44 and 36.75; error bars are of the order of the symbol sizes.

deviation between the simulated and experimental results becomes larger (as shown in Figure 9). In addition, a comparison between our model and the preceding theoretical

models (referring to Figure 10; the models employed are given in Table 2) shows that the present model is competent in estimating the maximum spreading factor of water droplet impact on the textured surfaces. To sum up, we have theoretically analyzed the maximum spreading factor in light of energy conservation. We have acquired good agreement between the experimental and theoretical results by taking into account the surface topography and viscous friction


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Table 2. Representative theoretical models for maximum spreading factor Correlations

β=

Pasandideh-Fard et al model49

We+12 3 (1-cosθ a ) + 4 We

⎡ 1 1-cosθ + 0.35 We ) Y ⎢⎣ 4 (

( We+12 ) β Ca

References

D0 H

Re

⎤ ⎥⎦

3

⎛ We + 1 ⎞ β + 2 = 0 ⎟ 3 ⎝ 12 ⎠

Re β − ⎜

3 = 8 + β ⎡⎣3 (1-cosθ Y ) + 4 We

⎛ ⎝

1

⎞ ⎠

Re ⎤⎦

β + 3 ⎜ 1 + Ca Reφs − cos θ ⎟ β + 8 β − (12 + We ) = 0 4

∗

2

2πCaD0 H

[ (ln ( P R ) − 1.31] P

2

(

β + 3 1+ 4

1 2

2

Ca Reφs − cos θ

∗

)

β + 8 β − (12 + We ) = 0 2

Mao et al model50 Ukiwe and model55 Present equation 21 Present equation 22

Kwok model

model

for liquid flowing within the forest of the posts. However, the present model is still inapplicable to the coexistence of Wenzel state and Cassie state (i.e. for T20 surface) during spreading period because it is infeasible to track the gas-liquid interface in-situ40,58 within the interstices, and thus the conversion of surface energy remains unresolved. In this respect, the present model may be improved provided that the dynamics of the liquid flow within the arrays of the micropillars is resolved thoroughly. Conclusion: In summary, the anisotropic wetting behavior of water droplet on

textured surfaces has been studied according to the measurement of apparent contact angle and the observation of droplet impingement. The direction-dependent contact angles in Wenzel state have been elucidated according to the modified Wenzel equation, which involves the substitution of the contact area with the triple line fraction accounting for roughness factor. The anisotropic wetting phenomenon of the


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water droplet impinging the textured surface has been addressed in light of thermal equilibrium state and the effect of dynamic pressure as well. This offers the possibility in tailoring wetting patterns by smartly devising geometrical parameters on textured surfaces, as wetting state undergoes a transition from Cassie to Wenzel regime. Furthermore, the maximum spreading factor is derived from the perspective of energy conservation and the resultant theoretical results are in good agreement with experimental results when the geometrical parameters and viscous friction are taken into account.

Acknowledgements

We are grateful to the anonymous reviewers who provided invaluable suggestions that contributed to improving the quality of the manuscript. We also thank the National Natural Science Foundation of China (grant No. 21104016) and Natural Science Foundation of Henan University (grant No. 2010YBZR041) for financial support. X.Y. Li also thanks L.G. Yu for his contribution in revising the manuscript.

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3.

Wenzel, R. N. Resistance of Solid Surfaces to Wetting by Water. Ind. Eng. Chem. 1936, 28 (8), 988-994.


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Dynamic Behavior of Water Droplet Impact on Microtextured Surfaces

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