Dynamic Behavior of Water Droplets on Solid Surfaces with Pillar

Article pubs.acs.org/Langmuir

Dynamic Behavior of Water Droplets on Solid Surfaces with Pillar-Type Nanostructures Woog-Jin Jeong,† Man Yeong Ha,*,† Hyun Sik Yoon,‡ and Matthew Ambrosia† †

School of Mechanical Engineering, ‡Advanced Ship Engineering Research Center, Pusan National University, San 30, Jangjeon-dong, Geumjeong-gu, Busan 609-735, Korea ABSTRACT: In the present study, we investigated the static and dynamic behavior of water droplets on solid surfaces featuring pillar-type nanostructures by using molecular dynamics simulations. We carried out the computation in two stages. As a result of the first computational stage, an initial water cube reached an equilibrium state at which the water droplet showed different shapes depending on the height and the lateral and gap dimensions of the pillars. In the second computational stage, we applied a constant body force to the static water droplet obtained from the first computational stage and evaluated the dynamic behavior of the water droplet as it slid along the pillar-type surface. The dynamic behavior of the water droplet, which could be classified into three different groups, depended on the static state of the water droplet, the pillar characteristics (e.g., height and the lateral and gap dimensions of the pillars), and the magnitude of the applied body force. We obtained the advancing and receding contact angles and the corresponding contact angle hysteresis of the water droplets, which helped classify the water droplets into the three different groups.

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INTRODUCTION Hydrophobicity is the ability of a solid surface to repel water. Static hydrophobicity is reinforced by roughness and can be quantitatively evaluated by two different theories proposed by Wenzel1 and Cassie and Baxter.2 The Wenzel model, which consists of the relation developed by Wenzel, assumes that water fills up the space between the structures on the

stays on top of the structures on the solid surface, as shown in Figure 1, whose contact angle correlation is expressed as cos θCB = f (cos θ Y + 1) − 1

where f is the fractional area of the wetted solid surface and liquid interface and θCB is the contact angle of the droplet in the Cassie−Baxter state. Besides the Cassie−Baxter and Wenzel wetting regimes, studies have shown that mixed wetting is also quite common. Marmur3 shows cases between the Cassie−Baxter state and the Wenzel state where a portion of the water drop falls between the textured structures. These cases are very common when dealing with various textures. However, in this study, square pillars are used for the texture which tend to give either the Cassie−Baxter state or the Wenzel state in nearly all cases. It is generally accepted that the control of dynamic hydrophobicity is difficult in comparison to that of static hydrophobicity due to the lack of information on the important factors controlling dynamic hydrophobicity.4 The sliding angle and contact angle hysteresis, which is the difference between the receding and advancing contact angles, has been used as a factor for evaluating dynamic hydrophobicity.5−11 Recently, other studies on dynamic hydrophobicity using sliding acceleration or velocity have been actively pursued.4,12−15 There are many experimental and theoretical studies on the dynamic hydrophobicity of water droplets, and they have served to elucidate the behavior of the water droplet at the macroscopic level.1−21 However, there are few nanoscale studies about the

Figure 1. Schematic of roughness-filling (with water) according to the Wenzel and Cassie−Baxter models.

surface, as shown in Figure 1, whose contact angle correlation is expressed as

cos θ W = r cos θ Y (1) where θw is the apparent contact angle on a solid surface with a certain level of roughness, θY is the ideal contact angle called Young’s angle, and r is the roughness factor, which is defined as the ratio of the actual solid surface area to the projected solid surface area due to roughness. However, the Cassie−Baxter model, involving the relation developed by Cassie and Baxter, assumes that the water droplet © 2012 American Chemical Society

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Received: December 27, 2011 Revised: February 28, 2012 Published: March 2, 2012 5360

dx.doi.org/10.1021/la205106v | Langmuir 2012, 28, 5360−5371

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(noncovalent) spring between the outer i and k atoms, and it contributes to the potential when constant kub ≠ 0. Like the spring bond, rij = ∥rj⃗ − ri⃗ ∥ gives the distance between the pair of atoms, and rub is the equilibrium distance. The four-body torsion angle (also known as dihedral angle) potential describes the angular spring between the planes formed by the first three and last three atoms of a consecutively bonded (i, j, k, l)-quadruple of atoms

dynamic hydrophobicity of water droplets using molecular dynamics (MD) simulations. Although there is a substantial difference in scale between experimental studies and numerical studies using MD simulations on the static and dynamic behavior of water droplets on solid surfaces, the contact angles obtained from MD simulations on smooth and structured surfaces are generally in good agreement with those obtained from experimental results.22−24 However, some studies show that nanoscale results may have differences from the macroscale. Hirvi et al.25 compared the velocity applied to a droplet on a nanosized pillared surface to the distance it travels and discovered that under these conditions the droplet does not travel as far as predicted by classical mechanics. In this study, the static and dynamic behaviors of a water droplet on solid surfaces with pillar-type nanostructures were studied using molecular dynamics simulations. The main purpose of this study is to understand how the existence of pillars and their size affect the static and dynamic states of a water droplet. In order to achieve this goal, we carried out the present computation in two different stages. From the first computational stage results, we could observe the static equilibrium state of a water droplet formed on smooth and pillared solid surfaces. In the second computational stage, we applied a constant body force to the static water droplet obtained from the first computational stage and evaluated the dynamic behavior of the water droplet sliding on pillared solid surfaces by varying the height and the lateral and gap dimensions of the pillars and the magnitude of the applied body force.

⎧ ⎪ k[1 + cos(n φ + ϕ)] if Utors = ⎨ ⎪ 2 if ⎩ k(φ − ϕ)

Unonbonded = ULJ + Uelec

⎡⎛ R ⎞12 ⎛ R min ⎞6⎤ ij ⎥ ⎢⎜ minij ⎟ ⎟ ⎜ ULJ = εij⎢⎜ ⎟ −⎜ r ⎟ ⎥ r ij ij ⎠ ⎥⎦ ⎝ ⎠ ⎝ ⎢⎣

(3)

(5)

where rij = ∥rj⃗ − ri⃗ ∥ gives the distance between the atoms, r0 is the equilibrium distance, and k is the spring constant. The threebody angular bond potential describes the angular vibrational motion occurring between an (i, j, k)-triple of covalently bonded atoms Uangle = k θ(θ − θ0)2 + k ub(rik − rub)2

(9)

where rij = ∥rj⃗ − ri⃗ ∥ gives the distance between a pair of atoms, and qi and qj are the charges on the respective atoms. ε0 is the vacuum permittivity. Figure 2 shows the coordinate system, computational domain, and initial shape of the water cube used in the present computation. The number of water molecules used in the simulation was 5124. The water molecules were initially placed on a rectangular surface whose size was Lx × Lz = 255.24 Å × 185.43 Å. To minimize the number of molecules used per unit volume, we chose (0001) graphite, which has a hexagonally tabular structure, as the solid surface.28 In the MD simulations, the atoms of the solid surface were spatially fixed in both steps of the computation. Figure 3 shows the shape of the pillars and their arrangement on the solid surface. The lateral sizes of the quadrangular pillar installed on the flat surface, Px × Pz, were chosen to be 8.51 Å × 7.37 Å and 12.76 Å × 12.28 Å, which give the same f value in eq 2. The gap size between the pillars, Gx × Gz, was equal to the lateral

(4)

The two-body spring bond potential describes the harmonic vibrational motion between an (i, j)-pair of covalently bonded atoms Ubond = k(rij − r0)2

(8)

where rij = ∥rj⃗ − ri⃗ ∥ gives the distance between a pair of atoms. εij = (εi·εj)1/2 represents the characteristic energy, and Rminij = (Rmini + Rminj)/2 represents the characteristic length between a pair of atoms. The Lennard-Jones potential approaches 0 rapidly as rij increases, so it is usually truncated (smoothly shifted) to 0 past a cutoff radius. The electrostatic potential Uelec in eq 8 is repulsive for atomic charges with the same sign and attractive for atomic charges with opposite signs qiqj Uelec = 4πε0rij (10)

The first term on the right side of eq 3 describes the bonded potential energy that involves 2-, 3-, and 4-body interactions of covalently bonded atoms, and it can be expressed as Ubonded = Ubond + Uangle + Utors

(7)

The Lennard-Jones potential ULJ in eq 8 accounts for the weak dipole attraction between distant atoms and for the hard core repulsion as atoms come closer

SIMULATION DETAILS We used the NAMD simulation package to simulate the static and dynamic behavior of the water droplet on pillar-type solid surfaces. NAMD was developed by the Theoretical and Computational Biophysics Group (TCB) and the Parallel Programming Laboratory (PPL) at the University of Illinois in Urbana− Champaign. NAMD is a parallel, object-oriented molecular dynamics program designed for high-performance simulations of large biomolecular systems.26 The force field used was the CHARMM force field.27 Its functional form is

∑ Ubonded( r ⃗) + ∑ Unonbonded( r ⃗)

n=0

where φ is the angle in radians between the (i, j, k)-plane and the (i, j, k, l)-plane. For n > 0, ϕ is the phase shift angle and k is a multiplicative constant. For n = 0, ϕ acts as an equilibrium angle and the unit of k changes to potential/rad2. The last term on the right side of eq 3 describes the nonbonded potential energy that involves interactions between all (i, j)-pairs of atoms, usually excluding pairs of atoms already involved in the bonded term. Even when using fast evaluation methods, the cost of computing the nonbonded potentials is the main part of the cost in each time step of an MD simulation. The nonbonded potential energy is expressed as

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Utotal( r ⃗) =

n>0

(6)

where θ is the angle in radians between vectors rij = rj⃗ − ri⃗ and rkj = rj⃗ − rk⃗ , θ0 is the equilibrium angle, and kθ is the angle constant. The second term is the Urey−Bradley term used to describe a 5361


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integrated numerically using the velocity Verlet algorithm with a time step of 2.0 fs. The simulation was performed in two stages. The aim in the first computational stage was to reach a static equilibrium state from an initial water cube. To achieve this objective, the computation was carried out during 2.5 × 106 steps corresponding to 5 ns. The aim in the second computational stage was to evaluate the dynamic behavior of the water droplet sliding on a solid surface. In order to investigate the movement of the water droplet in the presence of an applied force, we applied constant body forces to the water droplet obtained from the static computation in the first stage along the x-direction for an additional 2.5 × 106 steps corresponding to an additional 5 ns, as shown in Figure 4. The body force applied to move the water droplet on

Figure 2. Coordinate system, computational domain, and initial shape of water cube.

Figure 3. Schematic view of the shape of the pillars and their arrangement on the pillar-patterned surface.

size of the pillars, Px × Pz. The height of the pillars was varied in the range from 1 for a graphite interlayer distance of 3.35 Å to 7 interlayer distances for 23.45 Å. The MD simulations were carried out at constant-volume constant-temperature conditions (NVT) at 298.15 K.26 The temperature was controlled using the velocity scaling method. A periodic boundary condition was applied for all three spatial dimensions; 255.24 Å × 255.00 Å × 185.43 Å in the x, y, and z directions, respectively. The x and z dimensions were chosen to fit the solid surface dimensions. The y dimension was chosen to be large enough to ensure that there were no interactions in the y-direction beyond the periodic boundary. The TIP3P water model was used in this study. The TIP3P model has a single Lennard-Jones center representing an oxygen atom together with three charges that have −0.834 kcal/mol for the O atom and +0.417 kcal/mol for two H atoms arranged in a triangle. The bond length between O and H and the bond angle between three atoms in each water molecule are established as 0.9572 Å and 104.52°, respectively. The long-range charge− charge interactions between the water molecules were calculated using the Ewald method.29 Atoms of the flat and pillared surfaces were simply assumed to be Lennard-Jones atoms with a characteristic energy and a characteristic length of εs = −0.0385 kcal/mol and R̅ min,s = Rmin,s/2, respectively, where Rmin,s/2 = 1.9924 Å. Newton’s equation of motion was

Figure 4. Snapshot of a moving water droplet on the pillared surface in the presence of the body force applied.

the solid surface in the x-direction was defined as F = exp(Cεs/ R̅ min,s), where εs and R̅ min,s represent the solid surface characteristic energy and the solid surface characteristic length, respectively. In this study, we applied five different forces to the water droplet by adopting five different C values, as shown in Table 1. Table 1. Constant Body Forces Applied force C (F = exp(Cεs/R̅ min,s))

kcal/mol

nN/mol

100 150 200 250 300

0.1448 0.0551 0.0210 0.0080 0.0030

6.0588 2.3056 0.8774 0.3339 0.1270

To measure the hydrophobicity of the water droplet on the pillared surface for different cases considered in this study, the 5362


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studies about the values of the static contact angle,32−34 the present computational results for the static contact angle on smooth, flat surface represent the measured value of Fowkes and Harkins32 well. Figure 5b−k shows snapshots of water droplets in static equilibrium and their density fields for P1 and P2 surfaces at different pillar heights from H1 to H5. The snapshots obtained for taller pillars H6 and H7 are nearly identical to those shown in Figure 5 for H5, and they have therefore not been shown for the sake of brevity. In all cases shown in Figure 5, the droplet is either in the Cassie−Baxter state or the Wenzel state with the exception of Figure 5k. In this case, a mixed wetting state can be seen where the complete droplet is not above the pillars, nor does the droplet fill all the space between the pillars. Table 3 shows the static state and contact angles of the stationary water droplet after it reached static equilibrium for all the cases shown in Figure 5. As shown in Figure 5 and Table 3, the static equilibrium state of the stationary water droplet depends on the height of the pillar, the lateral size of the quadrangular pillar (Px × Pz), and the gap size between the pillars (Gx × Gz). As the height of the pillar increases, the static contact angle of the water droplet increases and the water droplet changes its state from the Wenzel state to the Cassie−Baxter state. As the lateral size of the pillar and the space between the pillars decreases, the water droplet changes its state from the Wenzel state to the Cassie−Baxter state. This means that the hydrophobicity is enhanced as the size of the water droplet increases in comparison to the top surface of the pillar and the space between the pillars.35 It is also noteworthy to add that the three-phase contact line where the edge of the droplet contacts the surface can be influenced by the protrusion edge of a pillar on which the droplet sits.36 This may make the apparent contact angle larger and make the surface seem more hydrophobic. In this study, only the measured apparent contact angle will be the concern. The behavior of the water droplet on the P2H4 surface shows some interesting features, as can be seen in Figure 5i. The extent to which the water droplet penetrates the space between the pillars is relatively small. Thus, the static contact angle for the P2H4 surface is 134.8°, which is almost identical to that for the P1H4 surface, which is in the Cassie−Baxter state. Figure 6 shows the static water droplet contact angles as a function of the dimensionless pillar height Ĥ for the different lateral pillar sizes of P1 and P2. Here, Ĥ is the dimensionless pillar height defined as the pillar height divided by 3.35 Å, which corresponds to 1 graphite interlayer distance (Ĥ = H/3.35 Å). Therefore, Ĥ = 1 corresponds to H1 and Ĥ = 2 corresponds to H2 and so on. The Cassie−Baxter angle model, expressed in eq 2, is also plotted in Figure 6 as the baseline against which the present computational results for different pillar heights and lateral pillar size are compared. Note that the area fraction f in eq 2 for P1 is the same as that for P2, with a value of 0.25. Since the contact angle obtained from the Cassie− Baxter model in eq 2 does not depend on the pillar height, the static contact angle obtained from eq 2 has a constant value of 137.1°, as shown in Figure 6. If the pillar height is relatively small, the water droplet is in the Wenzel state and the values of the static contact angles of the water droplet are significantly lower than 137.1°. However, as the pillar height increases, the water droplet changes from the Wenzel state to the Cassie− Baxter state as the droplet becomes more hydrophobic. As a result, the contact angle of the water droplet approaches the value of 137.1°. The contact angle for the P1 surface in the

intrinsic contact angles of the water droplets on the surfaces were calculated first. The following computational approach was used to determine the water droplet shapes and contact angles. From the computational results, the x, y, and z coordinates of all the water molecules were obtained. The density ρ̅ of the water droplet was defined as the average number of water molecules in a given unit cubic cell with a side length of 3 Å. To make the density field smooth, the density was calculated using a smoothed distribution ρ̂ obtained from ρ̅ through several successive applications of a spatially weighted average over the surrounding grid points. In three dimensions, the elementary smoothing procedure around a given grid point (i, j, k) is expressed as follows by modifying the threedimensional smoothing procedures given by Bonometti et al.30 ρ̂i , j , k =

3 1 ρi̅ , j , k + ρ̅ + ρi̅ + 1, j , k + ρi̅ , j − 1, k 4 24 i − 1, j , k

(

)

+ ρi̅ , j + 1, k + ρi̅ , j , k − 1 + ρi̅ , j , k + 1

(12)

The density field was normalized so that its highest value was 1. To extract the water droplet contact angle from the density field, the water droplet periphery was defined as the circular line with a density value of 0.5 obtained by performing leastsquares fitting through the water droplet periphery.31 The contact angle was defined as the angle between a tangential line at the droplet periphery where it is in contact with the solid surface and a line from the droplet periphery where it is in contact with the solid surface to the droplet’s projected center of mass on the smooth flat surface or the top surface of the pillars.

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RESULTS AND DISCUSSIONS Static State. For the sake of simplicity and readability, a subscript notion was devised to distinguish the different cases where P, H, and F represent the lateral pillar size, pillar height, and force applied to the droplet, respectively. These are defined in Table 2. Figure 5 shows snapshots of the stationary water

Table 2. Subscript Notation for Lateral Pillar Size, Pillar Height, And Body Force Applied to the Dropletsa subscripts

lateral pillar size (P)

1 2 3 4 5 6 7

Px × Pz = 8.51 Å × 7.37 Å Px × Pz = 12.76 Å × 12.28 Å -

pillar height (H) 3.35 6.70 10.05 13.40 16.75 20.10 23.45

Å Å Å Å Å Å Å

force (F) C C C C C

= = = = =

300 250 200 150 100 -

a Example: The case where Px × Pz = 12.76 Å × 12.28 Å with a pillar height of 13.40 Å and a body force corresponding to C = 100 translates as P2H4F5).

droplets in the static equilibrium state and their density fields in the x−y plane for different values of lateral pillar size and pillar height; they were obtained in the first computational stage. Figure 5a shows a water droplet on the smooth, flat surface whose time-averaged static contact angle is 85.9°. Fowkes and Harkins32 carried out the experiment to measure the static contact angle on a graphite surface whose measured static contact angle varied between 85.3° and 85.9°. To account for the static contact angle hysteresis even on a flat surface, the advancing and receding contact angles were averaged over time. Although there seem to be some discrepancies among different 5363


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Figure 5. Snapshots and density profiles of the static water droplets on the smooth flat surface and pillared surface for different values of lateral pillar sizes and pillar heights: (a) smooth flat surface, (b) P1H1, (c) P2H1, (d) P1H2, (e) P2H2, (f) P1H3, (g) P2H3, (h) P1H4, (i) P2H4, (j) P1H5, (k) P2H5.

Dynamic State. To elucidate the dynamic state of the water droplets quantitatively, the advancing and receding contact angles of the moving droplets were obtained. The process used to obtain the advancing and receding contact angles is similar to that used by Hong et al.37 The characteristics of the moving water droplets on the pillar-type surfaces can generally be classified into one of three different categories. Figure 7 shows typical snapshots of moving water droplets belonging to these three groups. The first group corresponds to the case where the water droplet is continuously stretched as the receding portion of the droplet is moving at a slower velocity in the positive x-direction than the advancing portion of the droplet. This behavior results in considerable stretching of the water droplet during its movement in the x-direction, as shown in Figure 7a. Such a droplet is primarily observed when the static state of the water droplet in Figure 5 and Table 3 is in the Wenzel state and the magnitude of the applied constant body force is relatively

Table 3. Static State and Contact Angles for Different Values of the Dimensionless Pillar Height Ĥ and Px × Pz static state and contact angle P1

P2

pillar height (H)

state

angle [°]

state

angle [°]

0 1 2 3 4 5 6 7

Wenzel Wenzel Cassie Cassie Cassie Cassie Cassie

85.9 111.7 130.1 134.5 135.8 134.8 134.8 133.5

Wenzel Wenzel Wenzel Wenzel Cassie Cassie Cassie

85.9 95.8 99.6 116.8 134.8 138.5 136.6 136.7

Wenzel state is larger than that for the P2 surface because the hydrophobicity of the water droplet for the P1 surface is larger than that for the P2 surface. 5364


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droplet leads to the stretching of the water droplet. If the water droplet is stretched to the total computational domain length Lx, the values of all the variables at x = Lx become equal to those at x = 0 owing to the periodic boundary condition. As the water droplet continues to stretch, the water droplet is translated into a stream of water molecules, as shown on the right side of Figure 7a. The second group consists of water droplets that are considerably distorted during their motion along the solid surface in the presence of the applied body force, that are not stretched in a continuous manner in the x-direction, and that have a large contact angle hysteresis (Figure 7b). This group corresponds to the case where the static state of the water droplet in Figure 5 and Table 3 corresponds to the Wenzel state and the applied body force is relatively weak compared to that of the first group. The initial velocity difference between the top and the bottom of the water droplet caused by the resistance of the pillar-type surface leads to a distortion of the water droplet during its movement in the x-direction, causing the hemispherical droplet in the initial static state to change into a more asymmetric hemispherical droplet. However, because the constant body force applied is not strong enough to stretch the droplet continuously in the x-direction, we cannot observe the transition from the water droplet to the water stream observed in Figure 7a,

Figure 6. Static contact angle of the water droplet as a function of the dimensionless pillar height Ĥ for P1 and P2.

strong. Such a situation occurs because the pillars retard the movement of water molecules between the pillars but not the movement of water molecules above the pillars. The large velocity difference between the top and the bottom of the water

Figure 7. Snapshots of the dynamic shape of the water droplet moving along the pillar-type surface corresponding to the three categories for different values of P, H, and F. 5365


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Figure 8. Snapshots of the dynamic shape of the water droplet moving along the pillar-type surface corresponding to P1H2F5 and P1H2F3.

Figure 9. Snapshots of the dynamic shape of the water droplet moving along the pillar-type surface corresponding to P2H4F4 and P2H4F2.

but the water molecules maintain their droplet shape during their motion in the x-direction. The third group corresponds to the case where the water droplet maintains its initial shape during its movement along the pillar-type surface in the presence of an applied body force, as shown in Figure 7c. When the static state of the water droplet in Figure 5 and Table 3 corresponds to the Cassie− Baxter regime, the droplet belongs to this group. Although the advancing and receding contact angles of the water droplet depend on the magnitude of the applied body force, the droplet

generally maintains its spherical shape as it moves across the pillar-type surface. In these cases, the water droplet moves smoothly along the top surface of the pillars without penetrating into the space between the pillars. Figure 8 shows snapshots of moving water droplets for the P1H2 surface for forces F5 and F3. The case of P1H2F5 does not fall into one of the three groups presented in Figure 7. The static state of the water droplet for the P1H2 surface is the Wenzel state, as shown in Figure 5d and Table 3. However, the water droplet on the P1H2 surface changes its state from the 5366


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Wenzel state in the absence of a constant body force to the Cassie−Baxter state in the presence of the constant body force F5. This occurs because the water molecules jump to the top surface of the pillars from the gap between the pillars and move in the x-direction along the top surface of the pillars, as shown in Figure 8a. However, in cases F1−F4, the transition from the Wenzel state to the Cassie−Baxter state does not occur (see Figure 8b). As a result, the cases for the P1H2 surface with a force of F4 or less correspond to the classification shown in Figure 7a. Figure 9 shows snapshots of moving water droplets on the P2H4 surface for different forces (i.e., F4 and F2). The case of P2H4F4 also does not fall in the three different groups presented in Figure 7. The static state of the water droplet for the P2H4 surface is the Wenzel state in the absence of the applied body force, and a small number of water molecules penetrate into the gap between the pillars, as shown in Figure 5i and Table 3. If the relatively strong body force corresponding to F3 or stronger is applied to the water droplet, a small number of water molecules in the receding part of the water droplet are stretched in the x-direction and penetrate into the gap between the pillars, but the water molecules in the advancing part of the water droplet maintain their initial droplet shape during the droplet’s movement in the x-direction (see Figure 9a). However, in the case of P2H4F2, the dynamic state of the water droplet is the Wenzel state, as shown in Figure 9b. Figure 10 shows the advancing and receding contact angles of the moving water droplet as a function of time for the cases shown in Figure 7. Here, A-CA and R-CA represent the advancing and receding contact angles, respectively. In this study, we calculated the advancing and receding contact angles quantitatively every 5 ps using two circle fitting methods. Moreover, we expressed them as an exponential function of time: θAorR = exp[A · log t + B]

(13)

where A and B are constants and t is the time in nanoseconds. The difference between the advancing and receding contact angles of the water droplet for the first group shown in Figure 7a is very large, resulting in a large contact angle hysteresis, as shown in Figure 10a. If the dynamic behavior of the water droplet falls into the category of the second group, as shown in Figure 7b, the value of the receding contact angle for the second group becomes larger than that for the first group, resulting in a decrease in the difference between the advancing and receding water droplet contact angles. Therefore, the contact angle hysteresis for the second group is lower than that of the first group, as shown in Figure 10b. For the third group shown in Figure 7c, the values of the receding contact angles are much larger than those of the first and second groups, since the water droplets in the third group are much more hydrophobic than those in the first and second groups. The value of the advancing contact angle for the third group is also larger than that of the first and second groups. As a result, the difference in the advancing and receding contact angles for the third group is much smaller than that for the first and second groups, resulting in a smaller contact angle hysteresis for the third group (see Figure 10c). The advancing and receding contact angles of the water droplets at t = 5 ns were calculated using eq 4 while the water droplet moved along the pillar-type surface. These results are shown in Figure 11 as a function of the pillar height H for different values of applied forces F in order to evaluate the dynamic behavior of the water droplet on the pillar-type

Figure 10. Advancing and receding contact angles as a function of time corresponding to the three categories for different values of P, H, and F.

surface. If the solid surface is more hydrophobic, the contact angle hysteresis of the water droplet decreases. Figure 11a,b shows the advancing and receding contact angles of the water droplet as functions of the dimensionless pillar height Ĥ for the P1 and P2 surfaces in the presence of a relatively strong body force of F5. In the presence of the applied body force F5, the variation in the receding contact angles with respect to the pillar height is larger than the variation in the ascending contact angles because of the stretching of the water 5367


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Figure 11. Advancing and receding contact angles of the water droplet as a function of the dimensionless pillar height Ĥ for different body forces for P1 and P2.

droplet during its movement on the pillar-type surface. In the case of P1H1F5, as shown in Figure 11a, the water molecules change their shape from the shape of a water droplet in the Wenzel state to that of a water stream, as shown in Figure 7a. This transition is caused by the dynamic behavior of the water droplet, which corresponds to the first group. As a result, the contact angle hysteresis of the water droplet is 129.4° for the P1H1F5 case, which is larger than the hysteresis for any other P1 pillar height with a force of F5. If the pillar height increases to H2 for the P1 surface with a a force of F5, then the dynamic state of the water droplet changes to the Cassie−Baxter state from the initial Wenzel static state. This change is shown in Figure 8a. As a result, as the pillar height for the P1 surface for F5 increases from H1 to H2, the value of the contact angle hysteresis at H2 is smaller than that of H1. This occurs because, at H2, the receding contact angle of the water droplet increases while the advancing contact angle of the water droplet decreases slightly. If the pillar height is H3 or larger for the P1 surface for F5, both the static and dynamic states of the water droplet are in the Cassie−Baxter state, which corresponds to the third group (Figure 7c), and as a result, the contact angle hysteresis has an almost uniform value with nearly uniform advancing and receding contact angles. In the case of the P2 surface and F5, shown in Figure 11b, when the pillar height is H1, H2, and H3, the dynamic state of the water droplet corresponds to the first group in the Wenzel state, and as a result, the contact angle hysteresis has a large value in this region with the maximum value at H2. When the pillar height is H4 for the P2F5 cases, the value of the contact angle hysteresis becomes smaller than those for H1, H2, and H3

because the dynamic state of the water droplet is intermediate between the Wenzel and Cassie−Baxter states, as shown in Figure 9a. At H5, the dynamic state of the water droplet corresponds to the third group in the Cassie−Baxter state, and the contact angle hysteresis of the water droplet has a nearly uniform value. When the force is F5, the value of the contact angle hysteresis for the P1 surface is less than that for the P2 surface, implying that the P1 pillar-type surface is more hydrophobic than the P2 surface. When the P1 and P2 surfaces have an applied body force of F5, we can observe the dynamic states of the water droplets corresponding to only the first and third groups. Figure 11c,d shows the advancing and receding contact angles of the water droplet as a function of the dimensionless pillar height Ĥ for the P1 and P2 surfaces in the presence of a body force F4. The variation in the advancing and receding contact angles in the presence of a body force of F4 is generally similar to that of F5. However, as the applied body force decreases from F5 to F4, the dynamic behavior of the water droplet shown in Figure 8a disappears at F4 and the dynamic state of the water droplet on the P1H2 surface corresponds to the first group. In this case, the maximum value of the contact angle hysteresis for the P1F4 cases is found at H2. That is different from the P1F5 cases, which has the maximum value at H1. The value of the contact angle hysteresis of the water droplet decreases with decreasing applied body force from F5 to F4 for all pillar heights except for H2. In the cases of P1F4, the dynamic state of the water droplet corresponds to the first group at H1 and H2 and to the third group at H3−H5. For P2F4 cases, the dynamic state of the water droplet corresponds to the 5368


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first group at H1, H2, and H3 and to the third group at H4 and H5. Figure 11e,f shows the advancing and receding contact angles of the water droplet as a function of the dimensionless pillar height Ĥ for the P1 and P2 surfaces in the presence of a body force of F3. The distribution of the advancing and receding contact angles of the water droplet for the pillar height for F3 is also generally similar to the F4 and F5 cases, with the contact angle hysteresis decreasing slightly in the presence of a decreasing body force when the force decreases from F4 to F3. However, for P1H1 and P2H1 cases, the dynamic state of the water droplet in the presence of the body force at F3 corresponds to the second group, unlike that in the cases of F4 and F5. The contact angle hysteresis of the water droplet for the P2 surface in the presence of the body force F3 has the maximum value at Ĥ = 3. In the presence of the body force F3, the dynamic state of the water droplet has all three groups shown in Figure 7. For the P2H4 surface, the water droplet in the presence of the body force F3 shows the dynamic behavior of Figure 9a, and the receding contact angle of the water droplet becomes larger than that for H3. For P1F4 cases, the dynamic state of the water droplet corresponds to the second group at H1, the first group at H2, and the third group at H3−H5. For P2F4 cases, the dynamic state of the water droplet corresponds to the second group at H1, the first group at H2 and H3, and the third group at H5. Figure 11g−j shows the advancing and receding contact angles of the water droplet as a function of the dimensionless pillar height Ĥ for P1 and P2 in the presence of the body forces F2 and F1. The general distribution of the advancing and receding contact angles of the water droplet according to the pillar height for the body forces F1 and F2 is similar to that for the larger body forces. However, since the magnitude of the body forces F1 and F2 is much weaker than for F3−F5, the values of the contact angle hysteresis of the water droplet for F1 and F2 are much less than for F3−F5, especially when the pillar height is relatively small. For P2F2 cases, the receding contact angle for the water droplet has the minimum value at H3, whereas the contact angle hysteresis has the maximum value at H4. For the F1 and F2 cases, the dynamic state of the water droplet corresponding to the second group was not found because of the presence of a very weak body force compared to F3−F5. As a result, in the cases P1F1 and P1F2, the dynamic state of the water droplet corresponds to the second group at H1 and H2 and to the third group at H3 and larger. In the cases P2F1 and P2F2, the dynamic state of the water droplet corresponds to the second group at H1−H4 and the third group at H5. Figure 12 shows the difference between the cosine of the receding contact angle (cos θR) and the cosine of the advancing contact angle (cos θA) as a function of the dimensionless pillar height Ĥ for different forces. Figure 12a shows cos θR − cos θA for the P1 surface. If the pillar height is H3 or higher (Ĥ ≥ 3), since the dynamic state of the water droplet corresponds to the third group in the Cassie−Baxter state, cos θR − cos θA has relatively smaller and nearly uniform values compared to that for H1 and H2. It is evident that this pillar-type surface is hydrophobic and the contact angle hysteresis is small. As the body force applied to the water droplet decreases, the values of cos θR − cos θA also decrease. For H1 and H2 cases, since the dynamic state of the water droplet generally corresponds to groups other than the third group as mentioned in Figure 11, the values of cos θR − cos θA for H1 and H2 are much larger than those for taller pillars. When the applied body force is

Figure 12. Difference between the cosine of the receding contact angle and the cosine of the advancing contact angle as a function of the dimensionless pillar height Ĥ for different applied body forces for P1 and P2.

largest, F5, for the P1 surface, the maximum value of cos θR − cos θA is at H1. However, for the applied body forces F1−F4 for the P1 surface, the cos θR − cos θA value is maximum at the pillar height H2. As the applied body force increases at H1, the value of cos θR − cos θA also increases because of the increased stretching of the water droplet during its movement. Figure 12b shows cos θR − cos θA for the P2 surface. In the case of P2, the pillar height at which the dynamic state of the water droplet corresponds to the third group in the Cassie− Baxter region moves to the region of H5 and larger due to the increasing top area of the pillar and the increasing gap between the pillars. The general variation of cos θR − cos θA with respect to the pillar height at H5 and larger for different forces with P2 is similar to that at H3 and higher with P1, resulting in a decreasing value of cos θR − cos θA because of the decreasing body force. The dynamic state of the water droplet for the P2 surface corresponds to a different group when the pillar height is less than H5. The general variation of cos θR − cos θA with the pillar height less than H5 for different forces with P2 is also similar to that for P1 when the pillar height is less than H3. However, when the pillar height is less than H5 for the P2 surface, the pillar height at which the maximum value of cos 5369


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θR − cos θA depends on the applied body force. Those maxima are at H2 for F5, H2 for F4, H3 for F3, H4 for F2, and H3 for F1.

due to the increase in hydrophobicity with decreasing lateral and gap dimensions of the pillars.

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CONCLUSIONS The present study investigated the static and dynamic behavior of a water droplet on smooth and pillar-type solid surfaces using molecular dynamics simulations. We considered the effect of the height and the lateral and gap dimensions of the pillar and the body forces applied to the water droplet on the dynamic behavior of the water droplet during its movement along pillar-type surfaces. We carried out this computation in two different stages. From the first computational stage results, we obtained the static state of the water droplet on the surface after it reached the static equilibrium from an initial water cube. From the second computational stage, by applying a constant body force to the static water droplet obtained from the first computation stage, we could evaluate the dynamic behavior of the water droplet sliding on pillar-type solid surfaces. The static state of the water droplet is in the Wenzel or Cassie−Baxter state, depending on the height and the lateral and gap dimensions of the pillars. When the pillar height is small, the water droplet is in the Wenzel state with a relatively low static contact angle compared to that in the Cassie−Baxter state. As the pillar height increases, the water droplet changes its state from the Wenzel to the Cassie−Baxter state and the corresponding static contact angle increases. The pillar height at which the water droplet changes its state from the Wenzel to the Cassie−Baxter state for P2 is larger than that for P1, because P2 is less hydrophobic than P1. As the pillar height increases and the state of the water droplet changes from the Wenzel state to the Cassie−Baxter state, the static contact angle of the water droplet on the pillared surface approaches the angle predicted by the Cassie−Baxter equation due to increasing hydrophobicity. The dynamic state of the water droplet can be categorized into three different groups, depending on the height and the lateral and gap dimensions of the pillars and the magnitude of the applied body force. The water droplet in the first group is translated into a stream of water molecules due to the continuous stretching of the water droplet as the receding part of the water droplet moves at a slower velocity than the advancing part of the water droplet. This occurs when a relatively strong body force is applied to the static water droplet in the Wenzel state. The water droplet in the second group changes its shape from a hemispherical shape to an asymmetric hemispherical shape when a relatively weak body force is applied to the static water droplet in the Wenzel state. The water droplet in the third group maintains its initial shape well during its movement along a pillared surface with relatively high pillars when a constant body force is applied to the static water droplet in the Cassie−Baxter state. The contact angle hysteresis of the water droplet, corresponding to the difference between the advancing and receding contact angles, depends on the dynamic behavior of the water droplet in the presence of a body force. If the dynamic state of the water droplet corresponds to the third group, the contact angle hysteresis of the water droplet is small. However, if the dynamic state of the water droplet does not correspond to the third group, the contact angle hysteresis has a much larger value than that for the third group. The dimensionless pillar height at which the contact angle hysteresis has the maximum value depends on the lateral and gap dimensions of the pillars and the magnitude of the applied body force. The contact angle hysteresis for P2 is larger than that for P1 for all constant body forces

AUTHOR INFORMATION

Corresponding Author

*Telephone: +82-51-510-2440; Fax: +82-51-515-3101; E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by Leading Foreign Research Institute Recruitment Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (MEST) (No. K20703001798-11E0100-00310). This work was supported by the Korean Research Foundation (KRF) grant funded by the Korea government (MEST) (No. 20110027445). This work was also supported by PLSI supercomputing resources of the Korea Institute of Science and Technology Information.

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REFERENCES

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