Dynamic contact angles and mechanisms of motion of water droplets

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Dynamic contact angles and mechanisms of motion of water droplets moving on nano-pillared superhydrophobic surfaces: A molecular dynamics simulation study Hao Li, Tianyu Yan, Kristen A. Fichthorn, and Sirong Yu Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01324 • Publication Date (Web): 30 Jul 2018 Downloaded from http://pubs.acs.org on August 1, 2018

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Dynamic contact angles and mechanisms of motion of water droplets moving on nano-pillared superhydrophobic surfaces: A molecular dynamics simulation study Hao Li *,a,b, Tianyu Yanb, Kristen A. Fichthorn *,b, and Sirong Yuc a

School of Material Science and Engineering, Shandong University of Science and Technology, Qingdao, 266590, China

b

Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States c

College of Mechanical and Electronic Engineering, China University of Petroleum (East China), Qingdao 266580, China

Abstract: In this work, we investigate the dynamic advancing and receding contact angles, and the mechanisms of motion of water droplets moving across nano-pillared superhydrophobic

surfaces

using

molecular-dynamics

simulation.

We

obtain

equilibrium Cassie states of droplets on nano-pillared surfaces with different pillar heights, groove widths, and intrinsic contact angles. We quantitatively evaluate the dynamic advancing and receding contact angles along the advancing direction of an applied body force, and find that they depend on the roughness parameters and the applied body force in a predictable way. The maximum dynamic advancing contact angle is 180°, and the minimum dynamic advancing contact angle is close to the static contact angle. On the receding side, the maximum dynamic receding contact angle is as large as 180°, while the minimum dynamic receding contact angle is close to the intrinsic contact angle of smooth surface. Interestingly, water droplets exhibit a “rolling” mechanism as they move across the surface, which is confirmed by movies of interfacial water molecules, as well as droplet velocity profiles.

Keywords: Cassie wetting state, dynamic advancing contact angle, dynamic receding contact angle, droplet motion, nano-pillared surface, molecular dynamics simulation.

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Introduction The unique surface wettability of the lotus leaf [1, 2] has inspired numerous research groups to prepare superhydrophobic surfaces due to their many beneficial properties, including self-cleaning [3], drag reduction [4], and anti-icing [5]. It is well known that both the surface chemical composition and surface morphology play important roles in achieving superhydrophobicity [6]. Therefore, hydrophobic surfaces can achieve superhydrophobicity if they are decorated with micro- or nanometer length-scale structures. There are two possible wetting states of a water droplet on such surfaces: the Cassie wetting state (composite wetting state) [7] and the Wenzel wetting state (noncomposite wetting state) [8]. In the Cassie wetting state, the water droplet is lifted up by surface micro- or nanometer structures and air is trapped in the grooves among the structures. In contrast, in the Wenzel wetting state, the water droplet enters the grooves among the micro- or nanometer structures. In general, a water droplet adheres more strongly to structured surfaces in the Wenzel state than in the Cassie state [9, 10]. Therefore, in many practical applications, the Cassie state is preferable to the Wenzel state. In this work, we consider a water droplet that is in the Cassie state on a structured superhydrophobic surface. The hydrophobicity of a solid surface can be quantified as static hydrophobicity and dynamic hydrophobicity [11]. Generally, the advancing and receding contact angles measured by the sessile drop method are termed as the static advancing and receding contact angles, while the advancing and receding contact angles measured during droplet motion are defined as the dynamic advancing and receding contact angles [12]. It is generally accepted that it is more difficult to control dynamic hydrophobicity than static hydrophobicity because of the lack of information on important factors controlling dynamic hydrophobicity [13]. The effect of surface structure on the static contact angle, as well as on the static advancing and receding contact angles, has been studied comprehensively by many research groups [14-17]. However, knowledge about the dynamic advancing and receding contact angles of a moving water droplet and how these are affected by surface structure is still incomplete.

For the Cassie state, Kusumaatmaja and Yeomans used 2


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analytical and numerical approaches to minimize the free energy of a drop on two- and three-dimensional patterned surfaces [12].

They reported that the advancing contact

line can only move after the advancing contact angle reaches about 180°, and the receding contact angle can be equal to the intrinsic contact angle of the smooth surface [12]. Gao et al. [18] observed that the static advancing contact angle can reach 180°, and the static receding contact angle can also approach 180° by fabricating a second level of nanostructure on a surface with microposts. Schellenberger et al. [19] studied the apparent advancing and receding contact angles of a water droplet on micrometer-scale superhydrophobic surfaces immediately prior to droplet motion, and also found that the apparent advancing contact angle can be up to 180°. The sliding angle [11, 20], as well as the advancing and receding contact angles [21, 22], have been evaluated to characterize dynamic hydrophobicity. However, to our knowledge, there is no simulation studying the dynamic advancing and receding contact angles of water droplets on superhydrophobic surfaces during water droplet motion. In addition, an understanding of the mechanisms of droplet motion will lead to better control in applications. Several studies have mentioned that nanoscale surface structures can increase the effective intrinsic contact angle (θe, or the Young’s angle) of the microscale features [14, 23]. Therefore, it is important to understand hydrophobicity at the nanoscale to improve hydrophobicity of solid surfaces in practical applications. Molecular-dynamics (MD) simulations can be used to describe droplets on solid surfaces, especially at the nanoscale [24-30]. For example, several groups have studied the effect of surface structure on the wetting state and the static contact angle [24, 31] using MD simulations. MD simulations have also been used to quantify the kinetics of droplet-wetting transitions between the Cassie and Wenzel states [32, 33]. There are also MD simulations studying about small droplets evaporation and condensation on solid surfaces [34, 35]. In addition, MD simulations can be applied to investigate dynamic hydrophobicity [36-38] and droplet impact behavior on solid surfaces [39, 40]. A few studies have investigated the movement of a nanodroplet on a solid surface in the presence of a body force [34, 39, 41]. 3


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In this work, we study the dynamic advancing and receding contact angles of water droplets in the Cassie state on nano-pillared superhydrophobic surfaces by applying a body force to initially static droplets. The nano-pillared surfaces have various pillar heights, groove widths, and intrinsic contact angles. We also study the influence of the magnitude of applied body force on the dynamic advancing and receding contact angles. More importantly, we study the motion mechanism of the water droplets moving across nano-pillared surfaces with different roughness parameters, including pillar heights, groove widths, and intrinsic contact angles, to gain an understanding of key factors relevant to dynamic hydrophobicity. Simulation Methods The MD simulations are carried out using LAMMPS [42, 43] in the canonical ensemble, where we have a constant number of molecules, simulation volume, and temperature (298.16K). Periodic boundary conditions are applied in all three spatial dimensions. We set up the solid surface as a simple cubic crystal with a nearest-neighbor distance of 2.9095 Å, and use 6 atomic layers as the thickness of the flat surface. The rigid SPC/E model is employed to describe the structure and intermolecular interactions between water molecules [44]. A Lennard-Jones (LJ) potential is used to describe the water-surface interactions. The LJ distance parameter is set at σ = 2.858 Å. We obtained droplets with various intrinsic contact angles (θe) by varying the LJ energy parameter (εsurf) across a set of values including 0.31, 0.27, 0.23, 0.19 and 0.15 kcal/mol. The long-range cutoffs for the LJ and Coulombic terms are both 10 Å. The long-range electrostatic interactions were handled by using the Ewald summation method. A water droplet (36,000 water molecules) with an initially cubic shape is placed above the solid surface. The time step is 1.0 fs, and the simulation equilibrium was achieved with a total time of 4.0 ns when the droplet achieved a virtually constant shape. Actually, a total time of 3.0 ns is sufficient to achieve equilibration, according to our simulation results. Here, we use a three-dimensional simulation model to demonstrate the dynamic advancing and receding contact angles of water droplets moving on nano-pillared superhydrophobic surfaces.

Figure 1 depicts various views of a water droplet on a flat 4


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surface [Figure 1 (a) and (b)] and on a pillared surface with a pillar length and width of 4 atomic layers, a groove width of 4 atomic layers, and a pillar height of 6 atomic layers [Figure 1(c)].

Figure 1. Views of an equilibrated water droplet on a smooth surface with εsurf = 0.27 kcal/mol: (a) side view, and (b) top view; (c) Three-dimensional simulation model of an equilibrated water droplet on a pillared surface with εsurf = 0.27 kcal/mol.

To evaluate the static contact angle, a series of density profiles was obtained in the x-y plane perpendicular to the surface [cf., Fig. 1(a)] at a value of z corresponding to the center of the droplet in the z direction [cf., Fig. 1(b)]. We divided the x-y plane into a series of boxes, with lengths of 4.0 Å in the x and y directions in a slice. We obtained the local water density in each box as an average over the final 0.5 ns of a 4.0-ns trajectory. After obtaining the density contour of a static droplet on the flat surface, as shown in Figure 2(a), we identified the points where the local density is equal or lower than half that of bulk water at these conditions [28]. The static contact angle of the water droplet, as shown in Figure 2(b), is then defined as the angle between the tangential line of the fit droplet surface and the horizontal line representing the liquid-solid interface.

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Figure 2. (a) The density contour (ρ) of a static droplet on the flat surface; (b) Measurement of the intrinsic contact angle θe: the red curve is a fit to the droplet surface, the blue line denotes the tangential line of the droplet surface, and the black line represents the flat surface.

To study nanodroplet motion on the nano-pillared surfaces, a body force is applied to the droplet after equilibration [36]. We apply a body force of 1×10-10 N in the negative x direction (e.g., see Fig. 2) to the water molecules, so that each molecule experiences a constant acceleration. As a result, the water droplet moves in the direction of the force. We save the trajectories of all water molecules every 4000 time steps while we carry out the MD simulation. To verify our results and improve the statistics, we obtained dynamic advancing and receding angles based on the average of 2 trajectories, which are similar to the behavior of the macroscopic droplet in the experiment using the tilting plate method [3, 5]. For measurement of the dynamic advancing and receding contact angles, just as Schellenberger et al. reported [19], the dynamic advancing contact angle is the temporary angle obtained by extrapolating the fit of the advancing contact line to the line formed by the top faces of the pillared surface, while the dynamic receding contact angle is the temporary angle obtained by extrapolating the fit of the receding contact line to the line formed by the top faces of the pillared surface. Gao et al. distinguished “sliding” and “rolling” mechanisms for drops to move across surfaces [48]. They discussed that “sliding” occurs when near-surface water molecules in the bulk of the drop exchange with water molecules at the liquid-solid interface, in a tank-tread fashion. “Rolling” occurs when water molecules at the liquid-solid interface exchange with molecules at the liquid-vapor interface, such that interfacial water molecules rotate as the drop advances. To distinguish between these 6


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two mechanisms, we use two different measures. In one measure, we track the motion of water molecules initially at the liquid-solid interface between the water droplet and the nano-pillared surface (see Figure S1 in the Supporting Information). If the droplet “slides” we should observe these interfacial molecules moving up into the droplet bulk as the droplet moves. If the droplet “rolls”, we should see these molecules moving around the liquid-vapor interface as the droplet moves. We also evaluate the velocity fields of the droplet. To do this, we divide the x-y plane into a series of boxes, with lengths of 4.0 Å in the x and y directions. The magnitude and direction of the local velocity v in each box was obtained as an average (including the z direction) over a 0.1-ns trajectory.

Such an average, over relatively

short times and limited spatial extents, was necessary to eliminate the random, diffusive component of the velocity so the convective component could be resolved. We also obtained the velocity relative to the center of mass (vrel) in each box by subtracting the droplet’s center-of-mass velocity (vCOM) from v, such that vrel = v - vCOM. Thus, we obtain several measures to characterize the motion of water droplets on the nano-pillared surfaces: The intrinsic contact angles (θe) of equilibrated water droplets on smooth surfaces; The static contact angles (θs) of equilibrated water droplets on the nano-pillared surfaces; The dynamic advancing contact angle (θa) between the temporary advancing contact line and the nano-pillared surface in the direction of motion; The dynamic receding contact angle (θr) between the temporary receding contact line and the nano-pillared surface in the direction of motion and; The droplet velocity profiles v and vrel.

Results and Discussion The value of the LJ energy parameter (εsurf) for the interaction between water molecules and atoms in the smooth solid surface influences the value of θe [45]. Therefore, to achieve control over this quantity, we first compute θe of water droplets on the smooth surfaces with various εsurf to determine the range for hydrophobic surfaces. The results of these studies for droplets at a constant temperature (T = 298.16 K) and a fixed water-surface distance parameter (σ = 2.858 Å) are shown in Figure S2 in the 7


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Supporting Information. These results confirm that the contact angle decreases with increasing εsurf. Thus, we can tune the hydrophobicity θe of the smooth surfaces by varying εsurf.

Effect of nanopillar height To study the influence of nanopillar height on θa and θr, we first study the effect of nanopillar height on θs. The pillar widths and grooves are both 4 atomic layers, and θe of a water droplet on the smooth surface is 111.23° with εsurf = 0.27 kcal/mol. Heights of 6, 7, 8, and 9 atomic layers correspond to θs of 141.1°, 142.48°, 142.5° and 142.79°, respectively (see Figure S3 in the Supporting Information). The reported values of θs (and their errors) are averages over four to five values after equilibrium is achieved in the simulations. Within the analyzed error, we find that θs is independent of the pillar height, because the contact area between the water droplets and the nano-pillared surfaces does not change, in good agreement with Cassie’s theory [7]. We proceed to study the effect of nanopillar height on θa and θr. Snapshots of an advancing droplet on a surface with a nanopillar height of 9 atomic layers are shown in Figure 3(a).

Here, we observe that the advancing contact line is initially pinned at the

edge of a nanopillar and that θa increases continuously from its initial value of 143.13° (close to θs) to nearly 180°. Subsequently, the droplet proceeds to the neighboring pillar and θa decreases to 140.53°, close to θs. The advancing contact line on the neighboring pillar moves with θa ≈ θs until it pins at the edge of this pillar and the cycle continues. Figure 3(b) shows a summary of the events that occur as a water droplet moves on the nano-pillared surface. Overall, there are two stages in which θa changes. In the first stage, θa increases continuously from θs to about 180° as the droplet deforms with the advancing contact line pinned at the edge of the pillar. Subsequently, the advancing contact line detaches from the edge of the pillar and moves to the neighboring pillar, resulting in a large jump of θa from 180° to around θs. In the second stage, θa remains at a value around θs as the advancing contact line moves across the pillar until it reaches the edge, where it is pinned again. There is a jump of θa from about 180° in the first stage to θs in the second stage. As this occurs, the advancing contact line moves and the 8


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process recommences to form a cycle, as shown in Figure 3(c). Thus, the lowest value of θa is close to θs, while the largest value of θa reaches 180°.

Figure 3. Advancing contact line of a water droplet on a nano-pillared surface with a height of 9 atomic layers. (a) Side view of a water droplet at various times during motion, with an indication of θa. (b) Schematic diagram of a side view of the advancing contact line of the droplet. (c) Dynamic advancing contact angles (θa) measured from MD simulation trajectories plotted versus time.

A maximum θa of about 180° was also observed by Schellenberger et al. [19] in their experimental studies of incipient contact angles of water droplets prior to their rolling off micro-pillared surfaces (Figure 2 of ref. [19]). There are other similarities between our work and theirs. We see a relatively slow increase of θa from θs to 180°, then a rapid plunge of θa back to θs, as Schellenberger et al. saw [19]. However, we find that θa stays at the minimum near θs for a longer time interval than it spends at 180°, whereas the time periods at θs and 180° seem nearly equal in the study of Schellenberger et al. [19]. This may occur because the MD results are resolved over ps time scales, while the time interval is seconds in the experimental study and changes 9


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over a shorter time may be difficult to resolve. Figure 4(a) shows the receding contact line of a water droplet on the nano-pillared surface. Initially, the local θr is 141.53°, close to θs. As the receding contact line continues to retreat, θr decreases to 126.35° and, as the droplet moves across the tops of the pillars, θr further decreases to 113.78° − a value close to θe (=111.23°). This is because the receding contact line has a similar configuration to a water droplet on a smooth surface [46]. Subsequently, the receding contact line detaches from the edge of the pillar and suspends above the gap between pillars, increasing θr to around 180°. This is because the tail of the droplet forms a surface of zero curvature as the receding contact line pins on the edge of a pillar [47]. Since the receding contact line is pinned to the neighboring pillar, θr decreases as the water droplet advances. When θr decreases to a value of 143.15°, close to θs, the rear contact line retreats and the cycle begins again. Figure 4(b) shows a schematic diagram of a side view of the receding contact line. We find that the motion of the receding contact line from pillar to pillar has two steps – similar to the advancing contact line. In the first step, θr decreases from θs to a value close to θe as the contact line advances from one edge of the pillar to the other. Then, the receding contact line detaches from the edge of the pillar and is suspended above the inter-pillar gap, resulting in an increase of θr to around 180°. In the second step, θr decreases from about 180° to θs as the tail of the drop moves from one pillar to the next. Thus, the receding contact line advances and the process recommences to form a cycle, as shown in Figure 4(c). The trends of θr in Figure 4(c) here seem to be qualitative agreement with the work of Schellenberger et al. [19], in that the receding contact angles in both studies exhibit a rapid jump followed by a relatively slow decline as the drop moves across the pillars. However, the range of values that they find is significantly different than the range that we find here. We find that the maximum in θr is close to 180° here, while it is significantly lower in Schellenberger’s study, where it is apparently lower than 180° [19]. Also, we find that the minimum in θr is close to θe here, while it is smaller than θe in Schellenberger’s study [19]. The origins of this discrepancy are unclear. Kusumaatmaja and Yeomans [12] studied two-dimensional water droplets in the Cassie 10


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state on pillared surfaces using static energy minimization, and also found that θr can be equal to θe. Further work is needed to resolve the theoretical-experimental discrepancy.

Figure 4. Motion of the receding contact line of a water droplet on a nano-pillared surface with a pillar height of 9 atomic layers. (a) Side view near the receding contact line during motion. (b) Schematic diagram of side view near the receding contact line. (c) Dynamic receding contact angles (θr) measured from MD simulation trajectories plotted versus time.

To view the synchronization of the advancing and receding contact lines, Video S1 in the Supporting Information shows a water droplet moving on the nano-pillared surface in Figures 3 and 4. We also plot the advancing and receding contact angles as a function of time on the same graph in Figure 5. Initially, θa increases to 180°, as the advancing contact line moves from one pillar to the next. Subsequently, θr decreases to a value close to θe, resulting in motion of the receding contact line from one pillar to the next. Thus, motion of the advancing side of the droplet is followed by motion of the receding side of the droplet. The advancing and receding contact angles of a water droplet advancing on surfaces for all pillar heights studied are shown in Figure S4 and Figure S5, respectively, of the Supporting Information. We find the trends of the advancing and receding processes are the same when water droplets move on nano-pillared surfaces with different pillar heights. 11


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Figure 5. Dynamic advancing (θa) and receding contact angles (θr) versus time, in a composite of Figs. 3 and 4.

As discussed above, Gao et al. distinguished “sliding” and “rolling” mechanisms for drops to move across surfaces [48]. Here, we were able to distinguish these mechanisms. In Figure 6 and Video S2 in the Supporting Information, we show a series of side views of water molecules initially at the liquid-solid interface between the water droplet and the nano-pillared surface with a pillar height of 9 atomic layers. While there is some diffusion of interfacial water molecules into the droplet bulk (which is to be expected), we clearly observe that water molecules at the liquid-solid interface rotate to the liquid-vapor interface as the drop advances. We note that in recent experimental studies with inverted laser scanning confocal microscopy, Schellenberger et al. also observed rolling motion of drops in the Cassie mode as they progressed down a tilted, pillared surface. Figure 7 shows two velocity fields of water molecules in the x-y plane perpendicular to the nano-pillared surface with a pillar height of 9 layers. In Figure 7(a), we see that the velocity is the highest for the water molecules at the liquid-vapor interface furthest from the solid surface. The velocity decreases on approaching the substrate and is nearly zero at the liquid-solid interface, which indicates that the drop exhibits the “no-slip” boundary condition. As shown in Figure 7(b), the velocity profile 12


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relative to the center of mass indicates that the droplet consists of two regions: A core that exhibits almost pure translation (with vrel ≈ 0), surrounded by a rotating shell with a velocity magnitude that increases on approaching the liquid-vapor interface. Thus, the relative velocity profile indicates the rolling mechanism for these droplets.

Figure 6. Side view of the motion of water molecules initially at the liquid-solid interface between a water droplet and a nano-pillared surface with a pillar height of 9 atomic layers.

Figure 7. Velocity fields of a water droplet in x-y plane perpendicular to a nano-pillared surface with a pillar height of 9 atomic layers: (a) the velocity field (v) and (b) the velocity field relative to the droplet center-of-mass velocity (vrel).

Thus, our studies of the effect of nanopillar height on θa and θr reveal that three key angles are involved in droplet motion: θs, which is the minimum angle in motion of the advancing contact line (cf., Figure 3), θe, which is the minimum angle in motion of the receding contact line (cf., Figure 4), and 180◦, which is the maximum angle in motion of both the advancing aand receding contact lines. We also learned that the droplets exhibit rolling motion as they move across the surface. By varying the surface characteristics, we can change θs and θe to learn their influence on θa and θr and confirm the generality of the trends shown in Figures 3-7. 13


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Effect of groove width By varying the groove width of the nano-pillared surface, we can change θs and verify that the minimum advancing contact angle is in fact θs. Thus, we studied droplets on surfaces with various groove widths. In these studies, the nanopillar width and height are 4 and 8 atomic layers, respectively, and θe is 129.50° with εsurf = 0.19 kcal/mol. θs for an equilibrated water droplet as a function of groove width is presented in Figure S6 in the Supporting Information. Here, we see that θs is sensitive to the groove width. Groove widths of 2, 4, 6, and 8 atomic layers correspond to θs of 145.60°, 152.75°, 158.38° and 160.71°, respectively. Thus, θs increases with increasing groove width. As the groove width increases, the contact area between the drop and the pillared surface decreases, according to Cassie’s theory [7], which leads to the increase of θs. Figure 8 shows θa on nano-pillared surfaces with different groove widths (G) measured from MD simulation trajectories as a function of time. Details of this process for a side view near the advancing contact line are shown in Figure S7 in the Supporting Information. We find that the trends in Figure 8 are the same as those in Figure 3. For all G studied, we find that θa increases from θs to about 180° because the advancing contact line is pinned at the edge of the pillar, and then jumps to θs as the advancing contact line moves from one pillar to the next. Subsequently, θa remains at θs as the advancing contact line moves across the neighboring pillar until it pins at the edge of this pillar, and the cycle continues. Thus, we reconfirm that the minimum θa is close to θs, while the largest θa is around 180° when a water droplet moves on a nano-pillared surface. The receding contact angles of a water droplet advancing on surfaces for all groove widths studied are shown in Figure S8 of the Supporting Information. Here, we note that there is an initial restructuring period for the drop with G = 8, in which the drop maintains a constant angle between θs and θe before deceasing to θe. However, subsequent to this initial period, we find that Figure S8 confirms the results that we find for the dynamic receding contact angles as a function of groove height in Figures 4 and S5: Initially, θr decreases from θs to a value close to θe as the contact line advances from 14


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one edge of the pillar to the other. Subsequently, θr decreases to θs as the tail of the drop moves from one pillar to the next. Thus, the receding contact line advances and the cycle recommences.

Figure 8. Dynamic advancing contact angles (θa) from MD simulation trajectories plotted versus time on nano-pillared surfaces with different groove widths (G).

To confirm the “rolling” mechanism of water droplets on nano-pillared surfaces with different groove widths, Figure S9 and Video S3 in the Supporting Information show water molecules initially at the liquid-solid interface between the water droplet and a nano-pillared surface with a groove width of 6 atomic layers. We find that these water molecules rotate to the liquid-vapor interface as the drop advances. Figure S10 in the Supporting Information shows the velocity fields of water molecules in the x-y plane perpendicular to a nano-pillared surface with a groove width of 6 atomic layers, which confirm that water droplets move on the nano-pillared surfaces via the “rolling” mechanism.

Effect of intrinsic contact angle By varying θe, we can verify the trend in Figure 4 that the minimum receding contact angle is θe. Thus, we studied the effect of θe on θr.. In these studies, the pillar width is 4 atomic layers, the pillar height is 6 atomic layers, and the groove width is 4 15


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atomic layers. Figure S11 in the Supporting Information shows θs on the nano-pillared surface as a function of θe. θe of 111.23°, 120.5°, 129.5°, and 137.5° correspond to θs of 141.1°, 149.7°, 153.19°, and 157.9°, respectively. Thus, θs of water droplets on the nano-pillared surfaces increases with increasing θe, consistent with Cassie’s theory [7]. Figure 9 shows θr as a function of time from MD trajectories on nano-pillared surfaces with various θe. In addition, a time series of detailed side views near the receding contact line is shown in Figure S12 in the Supporting Information. In Figure 9, we see that with θe1 (120.5°) and θe2 (129.5°), θr decreases from θs to a value close to θe as the receding contact line moves across the top of the pillar, then it jumps to a value around 180° as the receding contact line detaches from the pillar and moves to the neighboring pillar. Subsequently, θr decreases to a value close to θs, resulting in the water droplets’ advancing in the next cycle. For θe3 (137.5°), the contact line is initially pinned to the inner edges of the nanopillars [see Figure S12(b) in the Supporting Information], and θr is close to θe when the water droplet begins to advance. Therefore, it first jumps to a value around 180°, and then decreases to a value close to θs, resulting in the water droplets’ advancing in the cycle that is the same as for θe = 120.5° and 129.5°. Thus, though details of the substrate microstructure can influence the drop configuration initially, the minimum θr is close to θe, whereas the maximum θr is also around 180° for a moving drop.

Figure 9. Dynamic receding contact angles (θr) on the nano-pillared surface with different θe 16


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measured from MD simulation trajectories plotted versus time.

Figures S13, S14 and Video S4 in the Supporting Information show that water molecules in this system also move via the rolling mechanism.

Effect of applied body force In this study, we use a body force as a means to facilitate droplet motion and it is important to establish if there is any dependence of the motion on this force. To study the effect of the magnitude of the applied body force on θa and θr, three different body forces (0.5×10-10 N, 1.0×10-10 N and 3.0×10-10 N) were applied to equilibrated water droplets. We save the trajectories for the position of all water molecules with the applied body force of 0.5×10-10 N, 1.0×10-10 N and 3.0×10-10 N every 6000, 4000 and 3000 time steps, respectively, while we carry out the MD simulations. To improve the statistics, we obtained θa and θr based on the average of two trajectories. The nano-pillared surface for these simulations has a height of 9 atomic layers, the groove and width are both 4 atomic layers, and θe = 111.23° with εsurf = 0.27 kcal/mol. The values of θa and θr resulting from a body force of 1.0×10-10 N are illustrated in Figures 3 and 4, respectively. Figures S15 and S16 in the Supporting Information show θa and θr with applied body forces of 0.5×10-10 N and 3.0×10-10 N, respectively. Videos S5 and S6 in the Supporting Information show the water droplet moving on the nano-pillared surface with applied body forces of 0.5×10-10 N and 3.0×10-10 N, respectively. We find that both θa and θr exhibit the same trends for all three applied body forces. The lowest θa is close to θs, while the largest θa is around 180°. The lowest θr is close to θe, whereas the largest θr is around 180°. It is evident that the magnitude of the body force sets the time scale for droplet motion, as the smallest applied body force (0.5×10-10 N) induces motion over longer time scales than the largest applied body force (3.0×10-10 N). In addition, we find that a smaller body force than 0.5×10-10 N could not induce motion of the water droplet on the time scale of our simulations, while a body force exceeding 3.0×10-10 N results in deformation of the droplet. Therefore, in the range of the applied body force that we can probe in our simulations, the magnitude of 17


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the applied body force has no obvious effect on the trends in θa and θr observed here. We also studied the motion mechanism of water molecules at the interface between water droplet and a nano-pillared surface with an applied body force of 3.0×10-10 N. Figure S17, S18 and Video S7 in the Supporting Information show that water molecules initially at the interface rotate as the drop advances and the water molecules rotate around the center of the droplet, verifying that the water droplet moves on the nano-pillared surface via the “rolling” mechanism. This indicates that the magnitude of the applied body force has no effect on the motion mechanism of water molecules.

Conclusions In this study, we quantified the dynamic advancing and receding contact angles of water droplets in the Cassie wetting mode moving on nano-pillared superhydrophobic surface with different parameters, including pillar height, groove width, intrinsic contact angle, and applied force using MD simulations. We find that the dynamic advancing and receding contact angles of the water droplets do not depend on the roughness parameters within the range studied. The maximum dynamic advancing contact angle is around 180°, and the minimum dynamic advancing contact angle is close to the static contact angle. On the receding side, the maximum dynamic receding contact angle is also close to 180°, while the minimum dynamic receding contact angle is close to the intrinsic contact angle of the droplet on a smooth surface made of the same material. We also study the motion mechanism of water droplets on the nano-pillared surfaces. The water molecules at the liquid-solid interface exchange with those at the liquid-vapor interface and rotate as the droplet moves, in a “rolling” mechanism. We find that the magnitude of the applied body force has no effect on the change trend of dynamic advancing and receding contact angles, and the motion mechanism of water droplets on the nano-pillared surface. Interestingly, the trends we observe here for nano-sized drops over ps times agree with those seen experimentally, in studies of mm-scale water droplets rolling down pillared surfaces over second time scales [19]. In trying to compare the macroscopic experiments to the nanoscopic simulations here, we expect that the experiments and 18


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simulations could give similar results if the appropriate dimensionless numbers are similar. For example, one of us showed in previous work [31, 49] that in the low Bond-number limit, the static contact angles of drops on two-dimensional patterned surfaces in MD simulations were the same as those from a continuum analysis if the roughness parameters (groove width, pillar height, and pillar width) are scaled by the droplet radius. However, the relevant experiments are not performed in the limit of low Bond number because gravity induces the motion of the drops in experiment [19]. We note the Bond number (‫ܴ݃ߩ = ݋ܤ‬଴ଶ /ߛ ) depends on gravitational force and drop size and here the body force plays an analogous role to gravity. Since the density and liquid-vapor interfacial tension of SPC/E water are comparable to those in experiment, we have ଶ ଶ ‫݋ܤ‬௡௔௡௢ /‫݋ܤ‬௠௔௖௥௢ = ݃௡௔௡௢ ܴ଴,௡௔௡௢ /ܴ݃଴,௠௔௖௥௢ ,

(1)

where gnano is the gravitational acceleration (force) that must be applied to a nanoscopic drop with a radius of R0,nano to induce a macroscopic result. Here, ݃௡௔௡௢ = ‫ܨ‬௡௔௡௢ /݉, where Fnano is the body force in MD simulations and m is the mass of water. For example, gnano = 3 × 1015 m/s2 for an applied body force of 1×10-10 N. For our droplet radii of 6 nm and typical experimental radii of 2 mm in ref. 19, we have ‫݋ܤ‬௡௔௡௢ /

‫݋ܤ‬௠௔௖௥௢ = 1.39 × 10ସ , which is far from one. To have a Bond number that is comparable to experiment, we would need to apply a body force that is four orders of magnitude lower than we use here. Moreover, in the experimental scenario of a tilted plate, the force on the droplet will have components normal and tangential to the surface, instead of just the tangential component that we consider here. Yet the trends we see here still agree with experiment. Since we find that the observed trends are independent of the applied body force, in a range that is amenable to MD simulation, we conjecture that we would still observe these trends using a smaller body force – though the time scale for droplet motion would be exceed the MD time scale. Future studies establishing the relationship between nanoscale MD simulations and macroscopic experiments would be fruitful.

Supporting Information. 19


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Static contact angle (θs), dynamic advancing contact angles (θa) and dynamic receding contact angles (θr) as a function of different parameters, side view near the advancing and receding contact line with different parameter, side view of the motion process and velocity fields of water molecules with different parameters.

Acknowledgements Financial support to the first author (Hao Li) from the China Scholarship Council (CSC) is fully acknowledged.

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Dynamic contact angles and mechanisms of motion of water droplets

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