Asymmetric Spreading of a Drop upon Impact onto a Surface

May 15, 2017 - (7-9) Studying the surface motion (i.e., tangential velocity) effect is also helpful in understanding the systems like drop impact onto...
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Asymmetric Spreading of a Drop upon Impact onto a Surface H. Almohammadi and A. Amirfazli* Department of Mechanical Engineering, York University, Toronto, ON M3J 1P3, Canada S Supporting Information *

ABSTRACT: Study of the spreading of an impacting drop onto a surface has gained importance recently due to applications in printing, coating, and icing. Limited studies are conducted to understand asymmetric spreading of a drop seen upon drop impact onto a moving surface; there is no relation to describe such spreading. Here, we experimentally studied the spreading of a drop over a moving surface; such study also provides insights for systems where a drop impacts at an angle relative to a surface, i.e., drop has both normal and tangential velocities relative to the surface. We developed a model that for the first time allows prediction of time evolution for the asymmetric shape of the lamella during spreading. The developed model is demonstrated to be valid for a range of liquids and surface wettabilities as well as drop and surface velocities, making this study a comprehensive examination of the topic. We also found out how surface wettability can affect the recoil of the drop after spreading and explained the role of contact angle hysteresis and receding contact angle in delaying the recoil process.

■

INTRODUCTION

through modeling investigate the effect of tangential velocity on spreading of a drop upon impact onto a surface. When drop impacts onto a stationary surface (Figure 1a), due to high kinetic energy of the drop, it flattens symmetrically as a radially expanding liquid film. This process (called spreading) ends once the drop kinetic energy has been converted into surface energy or dissipated by viscous forces. The spreading ends when maximum spreading diameter is reached at a certain time, tmax (Figure 1a). The maximum spreading diameter is expressed as maximum spread factor: ξmax = Dmax/D0, the ratio of the maximum spreading diameter, Dmax, to the initial drop diameter, D0. In the literature, many studies have been developed to predict ξmax. The ξmax was found to be dependent on nondimensional numbers including the following: the normal Reynolds number (Ren = ρVnD0/μ, the ratio of inertia to the viscous forces, where ρ is the liquid density, Vn is the drop normal velocity, and μ is the liquid dynamic viscosity), and the normal Weber number (Wen = ρVn2D0/σ, the ratio of inertia to surface forces, where σ is the liquid surface tension), and surface wettability.22−30 However, it was found by Clanet et al.27 that, for low viscous liquids, the ξmax is independent from

Understanding the spreading of a liquid drop on a surface, i.e., the amount and way of wetting of the surface by drop impact, is of fundamental importance for natural and industrial processes.1−9 Many studies are conducted to understand the most simple and basic format of drop impact, i.e., drop spreading over a stationary surface.10−14 Recently, drop impact for complex drop/surfaces is studied to gain practical insights or achieve desired results like reducing the drop interaction time with surface or controlling the drop deposition.4,15−20 However, drop spreading on a moving surface as one of the most common and practical configurations is rarely studied. Drop spreading over moving surface can be seen in many agricultural and industrial instances including printing (e.g., inkjet printing1,2), spraying (e.g., pesticide,3,4 turbine blades,5 and rail industry6), and icing (aircraft and wind turbine).7−9 Studying the surface motion (i.e., tangential velocity) effect is also helpful in understanding the systems like drop impact onto an inclined surface or drop arriving at an angle onto a surface.21 Thus, an even wider range of applications can be listed. The practical importance and lack of comprehensive understanding of tangential velocity effect on drop impact are also highlighted as a future work in a review paper last year in the context of drop impact onto a surface.11 Here, we experimentally and © 2017 American Chemical Society

Received: March 1, 2017 Revised: May 1, 2017 Published: May 15, 2017 5957

DOI: 10.1021/acs.langmuir.7b00704 Langmuir 2017, 33, 5957−5964

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Figure 1. Overhead views of the water drop spreading over (a) stationary and moving (b) hydrophilic and (c) hydrophobic surfaces. Drop velocity is Vn = 1.43 m/s, and surface velocities are (a) Vs = 0 m/s, (b) Vs = 8.0 m/s, and (b) Vs = 8.0 m/s. White cross denotes the drop impact point on the surface. Surface moves from right to left.

Figure 2. Overhead views of a water drop spreading (results from Figure 1) in the Lagrangian frame of reference; a zoomed view is provided in the insets. Drop impacts on (a) stationary and moving (b) hydrophilic and (c) hydrophobic surfaces. Drop velocity is Vn = 1.43 m/s, and surface velocities are (a) Vs = 0 m/s, (b) Vs = 8.0 m/s, and (c) Vs = 8.0 m/s. White crosses denote the impact point on a surface, which are aligned along the vertical dashed line. White lines show the superimposed outlines of lamellae from consecutive frames shown. Arrows show the lamella motion relative to the surface.

Ren. A few studies also have been done analytically to predict the time evaluation of spread factor ξ(t) (=D(t)/D) for drop impact onto a stationary surface.31−34 There are relations to describe spreading for a symmetric lamella (i.e., drop spreading on a stationary surface). Previously, we found that when a drop impacts onto a moving surface, it spreads asymmetrically (i.e., has an elongated outline in the direction of surface movement; see Figure 1).35 This is also seen in the literature for drop impact onto inclined surfaces and other similar systems,36−39 but there is no relation to describe or explain such asymmetric spreading. Note that there are some conditions where drop splashes upon impact onto a moving surface, but we are not concerned with these conditions in this paper. Spreading phase for the drop impact onto a moving surface is defined as the period from impact until reaching the maximum width for the lamella (Wmax, see Figure 1). The corresponding time to maximum width is defined as the tmax.

There are works in the literature and our past study that provide some qualitative initial ideas about spreading, but a comprehensive understanding of how drop and surface velocities affect the lamella shape (both qualitatively and quantitatively) is missing,35−43 also how surface wettability and liquid viscosity change the lamella outline. Put together, the question of how the outline of the spreading liquid film evolves over time for systems with tangential velocity for the impact is not known. The answers to these questions are the basic information to comprehensively understand how tangential velocity affects the spreading of a drop upon impact onto a surface. To answer the above questions, here we provide novel hypotheses for the spreading behavior of a drop over a moving surface. Using the hypotheses, the effect of the normal and tangential velocities and surface wettability will be discussed. The experimental observation will be reported to verify our 5958

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models in the literature.31−34 However, due to the complexity of the exiting correlations where they should be solved numerically, here for internal consistency we use our experimental data for r(t). As such, we provide relations to calculate r(t) for both hydrophilic and hydrophobic surfaces as in the following (see the Supporting Information for details)

hypotheses. Finally, a model will be developed to mathematically predict not only the maximum spreading of lamella but also the time evolution of the lamella. The model prediction is validated with the experimental data.

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METHODS

Surfaces. Two different surfaces (hydrophilic and hydrophobic) were used in the experiments. A stainless steel surface with advancing and receding contact angles of 89 ± 1° and 34 ± 2°, respectively, was considered as a hydrophilic surface. To fabricate a hydrophobic surface, a solution of FC-75 and Teflon (5:1 (v/v) FC-75, 3-M/Teflon AF, Dupont) was spray coated over the cleaned (with DI water and acetone) stainless steel, where the advancing and receding contact angles were measured to be 123 ± 1°and 109 ± 1°, respectively. Liquids. The glycerol/water mixtures with 0, 24, and 42% glycerol (w/w) and the viscosities of 1.0, 2.0, and 4.1 mPa.s, respectively, were used as the working fluids.44,45 The viscosity of the liquids is considered in the range of μ/ρ ≲ 4 cSt which is referred to as a low viscous liquid in the context of drop impact literature.46−48 The surface tension and density of the liquids are considered approximately the same, as for the highest concentration of glycerol, the surface tension is decreased from 71.7 to 69.2 mN/m and density is increased from 998.2 to 1104.7 kg/m3.44,45 Note that surface tension and density of the liquids are taken from ref 44 (Tables 3 and 35), and viscosity of the liquid is read from ref 45 (Table 6). Impact Experiments. All of the experiments were performed at room temperature (20 °C). Drops with diameter of D0 = 2.5 ± 0.1 mm were generated from a syringe-needle system. A wide range of normal drop velocities from 0.5 to 3.4 m/s was studied. The tested surfaces were mounted on the side of a rotating large wheel with diameter of D = 56.9 cm. Linear velocities in the range of 0−17 m/s were generated by rotating the wheel at various rotational velocities. The deviation of the surface from the horizon in the frame of study is found to be 0.53%, which confirms that one can consider the target surface as a flat one. The experiments were recorded by high speed phantom cameras from both side and overhead, respectively, at frame rates of 5 000 and 10 000 fps.

Hydrophilic

(2)

Hydrophobic

ξ(t )/ξmax = 1 − exp(ct d)

(3)

where t(ms) is the given time, and a, b, c, and d are fitting coefficients. The values of the coefficients are found to be a = −0.535, b = 0.850, c = −1.352, and d = 0.815. The value of ξmax can be calculated through correlations from literature.22−30 Upon drop impact onto a moving hydrophilic surface (Figure 2b), a drop spreads in all directions at the very initial time (0 < t ≪ tmax in Figure 2b) of impact. The presence of the lamella around the impact point confirms this issue; see the zoomed view in Figure 2b. Then, the lamella stops spreading in one side of the drop (downstream in Figure 2b), while it continues spreading at the other side (upstream in Figure 2b) until the maximum width is reached. Note that the upstream and downstream are defined for a given time, and the line of maximum width at any given time defines the boundary between upstream and downstream. Arrest of the lamella at downstream can be seen from superimposed outlines of lamella in Figure 2b (see white lines). Using a surface with a lower wettability (i.e., hydrophobic, see Figure 2c) changes the behavior of the lamella. The lamella starts recoiling downstream (see superimposed outlines of lamella in Figure 2c). To sum up, in the case of a drop spreading on a stationary surface, the lamella outline is symmetric. However, a drop spreads asymmetrically over a surface when the surface is moving (i.e., there is a tangential velocity). The question is can the lamella outlines (circles in Figure 2a) for the drop spreading over stationary surface be used to provide any insights for the outline of a spreading drop over a moving surface? If yes, then how do lamellae propagate when surface is moving? Hypothesis Development. The asymmetric spreading of the drop over a moving surface can be deconstructed as being due to radial spreading of a drop impacting onto a stationary surface and the linear motion of the surface. As such, we propose that spreading of the drop over a moving surface is a superposition of the two aforementioned components. Accordingly, we developed the following two hypotheses to describe how a lamella propagates when a surface is moving. First, the lamella propagation relation (i.e., r(t) in eq 1) is not affected by the motion of the surface; i.e., for a given condition and time, the radius of each circle is the same for a moving and a stationary surface. Second, the source where the lamella propagates around it (i.e., center point of each circle) moves with the velocity of the surface. That is to say, the lamella propagation (r(t) in eq 1) is affected by the governing parameters of a drop impact onto a stationary surface, and the location of the source (bulk of the drop), where the lamella propagates around it, is affected by its relative velocity with respect to the surface. Based on the developed hypotheses, lamella propagation in the case of moving the surface is presented in Figure 3.

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RESULTS AND DISCUSSION Drop Spreading. Figures 2a−c show the relative motion of the lamella with respect to the surface for spreading of a drop over stationary and moving surfaces. To do so, the Lagrangian frame of reference is taken to observe the results of drop spreading in Figures 1a−c. In fact, Figures 2a−c are shifted images in space (by the surface velocity times the corresponding time interval) from that of Figures 1a−c, respectively. In the case of drop impact onto a stationary surface (Figure 2a), drop spreads radially outward until it reaches the maximum diameter. The superposed outlines of the lamella at different times (0 < t < tmax) clearly show the radial spreading of the drop (see Figure 2a). The diameter of the lamella is D(t), and drop impact point is at the center of all lamella outlines (circles). One can take either drop impact point or the drop apex as the center point for outline of the lamella; these two points for the drop impact on a stationary surface are always overlapped.35 Mathematically, one can write the equation of propagation of the lamella for spreading of drops on a stationary surface as in the following (X−Y coordinate origin is located at drop impact point, see Figure 2a) y 2 + x 2 = r(t )2

ξ(t )/ξmax = 1 − exp(a(t /Wen−1/4)b )

(1)

where r(t) (=D(t)/2) is the radius of the lamella at time t. The radius of the lamella, depending on impact, liquid, and surface parameters, differs and can be calculated through the existing 5959

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source (see eq 2, where surface velocity changes the Vs × t). Figure 4b shows schematically the lamella propagation for various surface velocities at the same normal drop impact velocity. Increase of surface velocity leads to a larger displacement of the source, while the lamella propagation is the same at a given time (e.g., consider first or second circle in each case). Taken together, drop normal velocity affects the lamella propagation behavior, while any changes in motion of the surface relative to the drop (i.e., tangential velocity) result in changes in displacement of the source (distance between centers of two successive circles). As discussed, during drop spreading over a moving surface, the lamella can experience the following: expansion, pinning, or recoiling (see Figure 2). The expansion of the lamella is related to the lamella propagation. For the case of pinning, one can think of lamella as remaining frozen after propagation stops. Following this concept, the recoiling of the lamella can be explained by the shrinking of the lamella after propagation. As such, Figures 4c and 4d show schematically the sequence of circles representing the outline of a drop spreading over moving hydrophilic and hydrophobic surfaces, respectively. In the case of a hydrophilic surface, the propagated lamella (circles) remains frozen, while the lamella propagates upstream. For a hydrophobic surface, the propagated lamella (circles) shrinks after propagation stops. As it is clear, one can observe smaller lamella (circles) downstream for the hydrophobic case in comparison to that of the hydrophilic one. Hypotheses Verification. To evaluate the developed hypotheses, several experiments in a wide range of normal drop impact velocities and surface (or tangential) velocities were performed. Drops of liquids with different viscosities and both hydrophilic and hydrophobic surfaces were considered. Starting with the first hypothesis, i.e., lamella propagation (r(t) in eq 4) is the same as that of stationary one, it was observed that the upstream of the lamella always has a semicircle shape (Figure 5a). The radius of the circles on moving surface was compared with that of stationary ones; see Figure 5b. Note that error bars (representing one standard deviation) are smaller to the symbol sizes in Figure 5, which shows high repeatability of data. As it is apparent, at a given time the results are similar for both stationary and moving surfaces. Note that for all of the tested liquids (low viscous liquids), surfaces (both hydrophilic and hydrophobic), and surface velocities, we found the same observation (see Supporting Information for more experimental data). As such, we can make a conclusion that the first hypothesis is supported by the empirical data. Next we test whether the source (i.e., center of circles) is moving with the surface velocity or not (second hypothesis). The experimental observations reveal that the apex point (see Figure 5a) of the drop always moves with the surface velocity relative to the drop impact point on a surface (see Figure 5c and ref 35 as well). However, the question is whether the apex is the center of the circle (outline of the lamella) or not? To verify, we considered the center point of the circle with respect to the drop apex (Figure 5a). We realized that there is a very small shifting for the center relative to the drop apex (Figure 5a). The possible reason for shifting can be the shear transfer from surface to the liquid film. In the end, our second hypothesis is correct by adding a shifting factor as in the following

Figure 3. (a) Schematic view of the drop lamella outlines on a stationary surface at different times. X−Y coordinate origin is located at the drop impact point; see the gray + . (b) The lamella propagation behavior based on the developed two hypotheses (eq 4); the circles from (a) are displaced with the surface velocity but keeping the same radius. Colored crosses show the center point of the displaced circles; each pair is highlighted with the same color. Color plot online.

Using the hypotheses, the following will be the equation which can be used to predict how the lamella propagates when the surface is moving y 2 + (x − Vs × t )2 = r(t )2

(4)

where Vs × t is the displacement of the circle’s center point, i.e., source (which moves with the surface velocity) and r(t) is the lamella propagation relation which is the same as that of a stationary surface (see first hypothesis). Based on a proposed relation for the propagation of the lamella (i.e., eq 2), any changes in drop velocity will result in a change for r(t). As it is understood, increase of the drop impact velocity leads to an increase of the r(t) at a given time.31−34 As such, Figure 4a shows schematically how lamella propagates for

Figure 4. Schematic view for (a) drop impact velocity effect and (b) surface velocity effect on how circles evolve over time. Black cross refers to the drop impact point (as the reference). Colored crosses show the center point of the circles; each pair is highlighted with the same color. (c) Shows schematically the arrangement of the circles over a hydrophilic surface, where the circles remain frozen after propagation. (d) Circle sequence over a hydrophobic surface; the circles experience recoiling after propagation over a hydrophobic surface.

various drop impact velocities (low, medium, and high) at the same surface velocity. Increase of the drop normal velocity results in larger lamella propagation (circles with higher radii), while displacements of the source (centers of circles) are the same at a given time (e.g., consider the first circle in each case). Surface motion only affects the displacement of the lamella

Xdisp = Vs × t − C 5960

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Figure 5. (a) Selected snapshots of water drop impact (Vn = 1.43 m/s) onto a moving surface (Vs = 5.9 m/s). White crosses refer to the drop impact point on a surface. Red cross shows the circle (lamella outline) center point. Drop apex is denoted with a triangle. (b) Comparison of the lamella circular radius with that of stationary one (Vn = 2.01 m/s). (c) The apex location relative to drop impact point for various relative motion velocities; apex moves away from drop impact point with surface velocity (Vn = 2.01 m/s).

⎧ 0 tgiven ≤ (t + Δtadv → rec) ⎪ Δt = ⎨ ⎪ ⎩ tgiven − (t + Δtadv → rec) tgiven > (t + Δtadv → rec)

where the value of the C (shifting factor) depends on drop impact, liquid, and surface parameters. The value of the C is found to be very small (C/D(t) = 6 ± 3.5%), and it is provided through an empirical correlation in the Supporting Information. Model Development. Having verified the hypotheses, we develop a relation to predict the lamella behavior at any given time for when impact involves both normal and tangential velocities as in the following: y 2 + (x − Xdisp(t ))2 = (r(t ) − Vr(Δt ))2

where Δtadv→rec is the residence time that liquid−solid contact line remains pinned as the liquid front stops spreading and starts to recoil. This time increases as the contact angle hysteresis (CAH) increases.30 Equation 8 means that there is recoiling, if tgiven > (t + Δtadv→rec); otherwise, there is no recoiling for lamella (Δt will be equal to zero). For the hydrophobic surface studied here, Δtadv→rec is considered to be zero as contact angle hysteresis is small (CAH = 14 ± 1°). For the hydrophilic, the recoiling is negligible (i.e., contact line seems pinned, see Figure 2b). The reason is explained by high contact hysteresis and low receding angle. The former leads to higher Δt (liquid spends more time to change angle from advancing for spreading to receding for recoiling), and the latter leads to a very low recoiling velocity, i.e., Vr (Δt) ≈ 0 (in eq 6). Figure 6 schematically shows the sequence of lamella outlines (circles) that are drawn based on eq 6 for drop impact onto a moving surface. As each circle is a snapshot in time, the outline

0 < t < tgiven (6)

The developed model can be used for the drop spreading phase (i.e., the period from impact until reaching the maximum width for the lamella); the spreading phase takes place in 2−4 ms depending on the drop velocity (for details regarding the tmax, see ref 35). In this equation, Vr is the recoiling velocity for the lamella downstream and Δt is the time interval in which recoiling takes place. In addition, the r (t) − Vr (Δt) term describes the nonlinear behavior of the lamella outlines (circles), where it is achieved by combining the propagation (eq 4) and after propagation (remaining frozen or receding, see Figures 4c and 4d) behaviors of the lamella. The recoil velocity for the liquid film can be expressed as49 Vr =

2γ(1 − cos θrec)/ρh

(8)

(7)

where h is the thickness of the liquid film. The thickness of the lamella can be calculated by having the volume of the liquid film (i.e., how much of the drop is delivered to surface) divided by the area which is covered by the liquid film (see below). In addition, θrec is the receding contact angle of the surface. The time at which recoiling happens can be calculated as the following

Figure 6. (a) Modeling of lamella behavior. Black cross refers to the drop impact point (as a reference). Colored crosses show the center point of the circles; each pair is highlighted with the same color. (b) Tangent line to circles’ sequence which envelops all of the circles. 5961

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the experimental results of drop spreading over another type of the surface, i.e., polystyrene coated surface with advancing contact angle of 94 ± 2° and receding contact angle of 71 ± 1° (see the Supporting Information). This shows that our modeling approach is of a general nature as it is able to predict the spreading of liquids over surfaces of various wettabilities. Although here we focused on the effect of surface motion on drop spreading, the tangential velocity also can be seen in other common systems including drop impact onto an inclined surface or drop arriving at an angle onto a surface. These systems can be seen in many applications from printing, coating, and cooling of the surfaces to windshield raindrops and icing. Practically, depending on the application, it is needed to have controlled wetted area, i.e., minimum, desired, or maximum area. Based on the findings in this paper, one is able to design a system with the knowledge of how the tangential velocity affects drop spreading, and as a result, how much area is being wetted by a drop at any given time (both qualitatively and quantitatively). The developed model (eq 6) is capable of predicting not only the maximum wetted area and its outline but also the time evolution of the wetted area. The general nature of our analysis is another advantage, as it can be used to explain the spreading for drop impact where tangential velocity is zero (surface is stationary). Practically, in applications (like printing, coating, and cooling of hot surfaces) where several drop generators are used to wet the surface, it is of importance to know what is the efficient frequency of drop generation and how the arrangement of the drop generators should be. Here, it is shown that increase in normal drop velocity results in stronger lamella propagation (lamella outlines with higher r(t) in eq 6), which leads to the lamella with larger width. And, an increase in surface velocity makes the source of the lamella propagation to displace more (higher Xdisp(t) in eq 6) and results in an increase in the length of the wetted area. This implies that to have the highest efficiency in wetting the area; if one uses a low tangential velocity, it is advised to use high frequency for drop generation with a wider space interval between drop generator tips. For high tangential velocity, a closer space interval for drop generator tips with low frequency of drop generation is advised. For low viscous liquids which are used in this study, the liquid viscosity does not have significant effect on the spreading behavior of the drop (see eq 6 where liquid viscosity affects only through the shifting factor (C) which is found to be very small). This is consistent with the study of Clanet et al.,27 where for low viscous liquids the viscous dissipation is ignored and the maximum spread factor described as a function of Wen. The wettability of the surface affects the wetted area over the surface (by affecting (r(t) − Vr(Δt) in eq 6), where there is a smaller wetted area for hydrophobic surfaces. Interestingly, the lamella analysis is also helpful in terms of reducing drop interaction time with a surface which is important in applications like icing. For instance, it has been shown that having a microridge on a nonwetting surface can decrease the drop contact time.17,18 The faster recoiling over the ridge (due to lower lamella thickness, see eq 7) results in splitting the lamella, and hence the recoiling distance will be reduced by roughly half before rebounding. But when the surface is moving, we showed that the spreading of the drop will be asymmetric; as such, to reduce the contact time one has to pay attention to the orientation of the microridge. Orienting the

of the lamella over a moving surface will be the line which envelops all of the propagated lamella (i.e., the curve which is tangent to all of the circles). Mathematically, the tangent to circles can be found by solving eq 6 and its derivative together50 (see Figure 6b): d 2 [y + (x − Xdisp(t ))2 = (r(t ) − Vr(Δt ))2 ] dt

(9)

The resultant solution is noted as f tangent (x, y). Having the equation of the line which envelops all of the propagated lamella (circles), one can calculate the area which is covered by the liquid film at any given time as the following: A=

∫ ftangent (x , y)

(10)

As such, the thickness of the lamella (assuming being uniform, this simplification is common in the context of drop impact) can be calculated by knowing the area (eq 10), so to be used for calculation in eq 7. The model provided in eq 6 can predict the shape of the lamella (or wetted area) for different drop and surface velocities, liquid properties, and surface wettabilities. Changes in drop velocity result in different values for r(t), and the value of the Xdisp(t) differs for different surface velocities. The liquid properties affect the values of r(t) and recoiling velocity, where recoiling velocity is also a function of the surface wettability. Figure 7 shows the comparison between the model predictions

Figure 7. Overhead view of drops spreading on a moving (a, b, and c) hydrophilic and (d) hydrophobic surfaces. Drop and surface velocities and liquids are (a) Vn = 0.7 m/s, Vs = 4.17 m/s, and water; (b) Vn = 0.7 m/s, Vs= 3.69 m/s, and glycerol/water mixture (μ = 2.0 mPa·s); (c) Vn = 2.01 m/s, Vs = 8.11 m/s, and water; (d) Vn = 2.01 m/s, Vs = 7.62 m/s, and water. The top of the each panel shows the experimental results and the bottom shows the model prediction (eq 6). The second panel in each case (i.e., a, b, c, or d) refers to the maximum spreading (i.e., at tmax). Red cross denotes the drop impact point on the surface.

and our experimental observations for different cases with various drop and surface velocities, liquid viscosities, and surface wettabilities at different times. As it is seen, our model predicts the outline of the lamella very well. The agreement between experimental results and the model confirms the appropriateness of our hypotheses and the discussion regarding the lamella propagation (see Figure 4a−d). Note that we also observed a good agreement between the model predictions and 5962

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microridge in the direction of the surface movement will have the best benefit for contact time reduction, compared to any other direction, as the spreading length for the lamella is the longest in the direction of the surface motion (see Figure 5a). Therefore, if a ridge is oriented in the direction of surface motion, one can split the lamella along its length and most effectively reduce the time for the lamella to recoil.

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CONCLUSION In the present study, through an experimental study, the asymmetric nature of a drop spreading over a moving surface is studied. The developed model for the first time is able to predict accurately the asymmetric shape of the lamella. The model can predict the time evolution of the lamella for different drop and surface velocities, liquid properties, and surface wettabilities. It is found that an increase in normal drop velocity leads to an increase in the lamella width, while an increase in surface velocity results in a lamella with larger length. The role of wettability (i.e., contact angle hysteresis and receding contact angle) in recoil process of the lamella is discussed. At a given time, a smaller wetted area was observed for hydrophobic surfaces compared to hydrophilic ones. For low viscous liquids used in this study, the liquid viscosity does not have significant effect on the outline of the lamella. It was also discussed that our findings help with better understanding of the similar systems like oblique drop impact onto a surface and drop impact onto an inclined 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.7b00704. Detailed information about the relations of r(t), experimental results of the drop spreading, shifting factor (C), and example for polystyrene coated surfaces (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +1-416-736-5901. E-mail: [email protected]. ORCID

A. Amirfazli: 0000-0002-8391-0493 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work of Mr. Mohammad Al Ramahi in the design of the experimental setup as well as funding from Discovery and Discovery Accelerator Supplement Grants through NSERC are acknowledged.

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REFERENCES

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DOI: 10.1021/acs.langmuir.7b00704 Langmuir 2017, 33, 5957−5964