Kinetics of Droplet Wetting Mode Transitions on Grooved Surfaces

Article pubs.acs.org/Langmuir

Kinetics of Droplet Wetting Mode Transitions on Grooved Surfaces: Forward Flux Sampling Azar Shahraz,† Ali Borhan,† and Kristen A. Fichthorn*,‡ †

Department of Chemical Engineering and ‡Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States S Supporting Information *

ABSTRACT: The wetting configuration of a liquid droplet on a rough or physically patterned surface is typically characterized by either the Cassie wetting mode, in which the droplet resides on top of the roughness, or the Wenzel mode, in which the droplet penetrates into the roughness. For a fixed surface topology and droplet size, one of these modes corresponds to the global free-energy minimum. However, the other state is often metastable and long-lived due to a free-energy barrier that hinders the transition between the two wetting states. Metastable wetting states have been observed experimentally, and we also observe them in molecular dynamics (MD) simulations of a droplet on a grooved surface. Using forward flux sampling, we study the kinetics of the Cassie to Wenzel and Wenzel to Cassie transitions for two-dimensional droplets on periodically grooved substrates. The globalminimum wetting states that emerge from our nanoscale MD approach are consistent with those predicted by a macroscopic model based on free energy minimization. We find that the free-energy barriers for these transitions depend on the droplet size and surface topology. A committor analysis indicates that the transition-state ensemble consists of droplets that are on the verge of initiating/breaking contact with the substrate at the bottom of the grooves.

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INTRODUCTION The ability to control the wetting of liquid drops on solid surfaces by varying the surface roughness or by applying external stimuli (e.g., temperature gradients, droplet compression or impact onto a solid surface, vibration, or electric voltage) is a subject of significant interest for a wide range of applications.1−7 The configuration of a drop on a rough or physically patterned surface is typically described by either the Wenzel8 model or the Cassie−Baxter model.9 The Wenzel model refers to the homogeneous wetting regime, in which the drop penetrates into the surface cavities and completely wets the substrate, while in the Cassie−Baxter model the droplet rests on top of the roughness with air pockets trapped underneath, leading to inhomogeneous or composite wetting (see Figure 1). While we expect either the Cassie or the Wenzel mode to be the global-minimum free-energy configuration for a droplet on a given surface, experimental studies provide evidence for wetting-mode multiplicity,10−15 such that the observed wetting state on a given surface depends on the history of the system.10 The energetically favored configuration may not be observable because of the large energy barrier between a metastable state and the most stable state.11−13,16−22 Because metastable wetting states can be long-lived, it is important to be able to predict the free-energy barriers and transition rates between Cassie and Wenzel wetting states (i.e., kC→W and kW→C in Figure 1) for a surface with a given morphology. This knowledge will indicate the lifetimes of droplets and the stability of the wetting states, © XXXX American Chemical Society

Figure 1. Sketch of the Cassie (C) and Wenzel (W) wetting states. The rate constants for the C → W and W → C wetting-state transitions are represented by kC→W and kW→C, respectively.

which should be a factor in designing a surface with controlled wettability.23,24 The magnitude of the free-energy barrier for the Cassie− Wenzel transition will determine the robustness of the surface wetting properties in practical applications.25 For example, in electrowetting applications,26 an appropriate surface topology is required that allows for a controlled, barrierless, and reversible switching between wetting states, whereas in the design of superhydrophobic surfaces topographies with high energy Received: September 8, 2014 Revised: November 25, 2014

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dx.doi.org/10.1021/la5035917 | Langmuir XXXX, XXX, XXX−XXX

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preparing the liquid drops has been described previously.41 The droplet radii studied here are R0 = (12 ± 2)σff, (17 ± 2)σff, and (23 ± 2)σff, corresponding to 3196, 7189, and 12 433 atoms, respectively. Here, σff is the length parameter in the truncated LJ potential that is used to describe pairwise interactions uff between fluid (f) atoms i and j separated by a distance of rij. This has the form

barriers for the Cassie−Wenzel transition are preferred to stabilize the Cassie state.23 While it is known that the magnitude of Cassie−Wenzel free-energy barrier is a critical factor in the optimal design of patterned surfaces, the understanding of the phenomena that affect the barrier is limited. Several theoretical studies based on continuum equations16,23,27−34 have been performed to elucidate wetting-state transitions and free-energy barriers between the Cassie and Wenzel states. These studies are based on a thermodynamic analysis of a hypothetical pathway from the Cassie to the Wenzel state. Thus, they can neither explain nor precisely determine the configuration of the droplet at the transitionstate ensemble, which is important in predicting the lifetime of a droplet in a given wetting state. As reported by Susarrey-Arce and coworkers,35 these models can be insufficient to explain experimental observations. The actual details of the transition from composite to wetted contact are not well understood.16,17 Recently, approaches based on atomistic-level simulation have been employed to study the rates and free-energy barriers for wetting-state transitions.36−39 Such approaches employ fewer assumptions regarding the transition mechanism than continuum approaches, they can provide specific values for transition rates, and they are able to accommodate a detailed chemical description of the droplet and the substrate. In our recent work,40,41 we examined the wetting of a droplet on a grooved surface using molecular dynamics (MD). We observed metastable wetting states in our MD simulations, analogous to those observed experimentally.11−15 In this paper, we use MD simulations in conjunction with forward flux sampling (FFS)42 to examine the rates and free-energy barriers for these transitions, as well as their dependence on the morphology of a periodically-grooved surface. These simulations provide a detailed description of how the wetting-state transition depends on the morphology of the surface. This knowledge will be useful in efforts to design surfaces with the optimal morphology for desired wetting properties.

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⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ ⎪ ⎢⎜ σff ⎟ ⎪ 4εff ⎢⎜ ⎟ − ⎜⎜ ff ⎟⎟ ⎥⎥, if rij ≤ 3.8σff u ff (rij) = ⎨ ⎣⎝ rij ⎠ ⎝ rij ⎠ ⎦ ⎪ ⎪ 0, otherwise ⎩

(1)

where εff is the energy parameter and rc = 3.8 σff is the cutoff distance. We fix the energy parameter at a value of εff/kB = 120 K, where kB is the Boltzmann constant, and use σff and εff as the length and energy scales, respectively, to make the equations dimensionless. Fluid−solid interactions are described by a LJ-based patterned-surface potential.43,44 The parameters for this potential are set to σfs = 0.921σff, σss = 1.2σfs/√2, and ρs = √2/σss3. These values lead to an equilibrium contact angle of θe = 126° ± 5° on a smooth substrate. We used the Nosé−Hoover thermostat.45,46 to maintain a constant temperature of T = 0.7εff/kB. Other details of these simulations were described previously.41 In a previous study, we probed the wetting states of this system using MD simulations and continuum free-energy calculations.40,41 The continuum free-energy calculations unambiguously indicated the global-minimum wetting states, allowing us to obtain wetting phase diagrams, such as that shown in Figure S1, Supporting Information. However, metastable states occurred in the MD simulations, especially near the phase boundaries, such that the observed wetting mode in these simulations depended on the initial configuration of the droplet. We selected five different droplet configurations from among the many for which we observed wetting-mode multiplicity. These configurations are designated by the points on the wetting phase diagram shown in Figure S1, Supporting Information. At each of these points, we observed the Cassie mode if we started our simulations in the Cassie mode, and we observed the Wenzel mode if we started our simulations in the Wenzel mode. We then characterized the wettingmode transitions (C → W and W → C, as shown in Figure 1) using FFS. FFS. The Cassie−Wenzel transition is a rare event that is difficult to observe over the times that can be probed in typical MD simulations. To characterize this relatively slow transition, we use FFS.42 The basic idea behind FFS is to use a series of interfaces to partition the phase space between neighboring stable and metastable states along an order parameter λ and employ a systematic, ratchet-like scheme to drive the system from the initial configuration to the final configuration. The initial state (A) is defined by λ < λA = λ0, the final state (B) by λ > λB = λn, and the intermediate interfaces by {λ1,...,λi,...,λn−1}. The rate constant kA→B for transitions from A to B can be calculated from the average total flux from A to B. This average can be expressed as the product of the flux of trajectories Φ̅ A,0 that leave state A and reach λ0 and the probability P(λn|λ0) that a trajectory crossing λ0 from A will reach λn without returning to A,42 namely,

SIMULATION METHODS

Model System. We performed MD simulations using the constant number, volume, and temperature (NVT) ensemble to study the wetting of an infinitely long (periodic), cylindrical Lennard−Jones (LJ) liquid droplet on a periodic, rectangular-grooved solid surface. As shown in Figure 2, the grooved surface morphology is characterized by groove width G, step height H, and step width W. These lengths are made dimensionless with R0 (the cylindrical-equivalent droplet radius), and the resulting dimensionless variables are designated by overbars (i.e., H̅ = H/R0, W̅ = W/R0, and G̅ = G/R0). Our procedure for

kA → B = Φ̅A,0P(λn|λ 0)

(2)

From among the various techniques that have been proposed to carry out FFS simulations,42 we chose to use the constrained branched growth (CBG) method proposed by Velez-Vega et al.47 Details of the CBG method are described in the Supporting Information. The order parameter (λ) is defined as

λ=

ρG ρG,max

(3)

where ρG is the fluid density in the groove and ρG,max is the fluid density in the completely filled groove. The number of required orderparameter interfaces to partition the phase space ranges from 4 for the fastest transition to 16 for the slowest transition, depending on the

Figure 2. Morphological parameters of the periodically grooved surface considered in this study: groove width G, step height H, and step width W. B


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both decrease as H̅ increases. However, the C → W transition rate decreases by 17 orders of magnitude with increasing H̅ , whereas the W → C transition rate only decreases by a factor of 100. To gain insight into this trend, we calculated the committor function for the three different surface morphologies. In Figure 4, we present the committor functions PB(λ) for the three surface topologies with varying H̅ . The points represent the average values of all committor functions PB(λ) obtained for all configurations at each interface (λ). The error bars indicate the standard deviation. Using the average value of PB at λ, the interpolation is carried out using piecewise cubic Hermite interpolating polynomials within the MATLAB curvefitting toolbox. The dashed lines corresponding to PB(λ*) = 0.5 identify the transition-state ensemble for each of the surface geometries. We determined λ* through the interpolation shown in Figure 4, and performed simulations at λ* to obtain the snapshots of the droplet. The snapshots in Figure 4 are selected from among many configurations obtained from simulations at λ*. Apparently, λ* increases as H̅ increases. As shown in the insets to Figure 4, the transition state for each geometry is mostly composed of droplet configurations that are on the verge of initiating or breaking contact with the substrate at the bottom of the grooves. Both experimental10 and previous theoretical studies37 support our finding that subsequent to initial contact of the droplet with the bottom of the groove, the C → W transition can occur easily. Also, we find that once the droplet detaches from the bottom surface, the W → C transition proceeds quickly. This trend is consistent with the dependence of the transition rate on the step height shown in Figure 3. Namely, because the transition state is associated with a meniscus configuration near the bottom of the groove, the W → C transition is not significantly affected by step height. However, to undergo the C → W transition, the droplet must progress to the bottom of the groove, which implies a greater sensitivity to the step height. Using the method proposed by Valeriani and coworkers,51 we calculated the free energy ΔG as a function of λ for surface geometries with different values of H̅ and fixed values of W̅ and G̅ . Figure S3 in the Supporting Information shows these freeenergy pathways. The value of λ = λts, at which the free energy goes through a maximum, indicates the location of the transition state (ts). These are shown in Table 1. Comparing λts in Table 1 to the values of λ* obtained from committor analysis in Figure 4, we find good agreement. We obtained the free-energy barrier for the W → C transition from the difference between the free energy at the transition state and that in the Wenzel state (λ → 1). Similarly, the free-energy barrier for the C → W transition was found from the freeenergy difference between the transition state and the Cassie state (λ → 0). These values are shown in Table 1. The equilibrium constant for the C ↔ W transition is given by

geometry of the surface and the size of the droplet. Also, FFS requires the system dynamics to be stochastic to generate uncorrelated paths from a given configuration at interface i. Since the Nosé−Hoover thermostat is completely deterministic, only one trajectory is produced for a given system configuration. To deal with this issue, we reassign velocities to each starting conformation at λi by selecting them randomly from the Maxwell-Boltzmann distribution at the temperature of interest, as suggested by Velez-Vega et al.48 To characterize the mechanisms behind the wetting-state transitions, we performed a committor analysis. The committor function PB(x) is defined as the probability that a given configuration x will end up in final state B before returning back to initial state A. PB(x), which is related to the ideal reaction coordinates of the system, represents the tendency of configuration x along the transition path to relax toward the final state under the system’s intrinsic dynamics.49 The committor function increases from 0 at the initial state to 1 at the final state. Configurations with PB = 0.5 correspond to the transitionstate ensemble of the system. A suitable order parameter can parametrize the committor function in such a way that a sharp change is seen around PB = 0.5. Here, we apply the procedure suggested by Borrero and Escobdeo to calculate PB(x), which is clearly described in ref 50. Finally, we apply the method proposed by Valeriani and coworkers51 to characterize the free energy ΔG as a function of λ. We compute ΔG/kBT for different surface topographies and different droplet sizes in order to obtain the energy barrier and its relationship to surface geometry and droplet size. Details of this method are given in the Supporting Information.

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RESULTS AND DISCUSSION To quantify the effect of the step height H on the C → W and W → C transitions, we tested three surface geometries with different values of H̅ (but fixed values of W̅ and G̅ ) near the C1−W1 boundary for a droplet with N ≈ 3000 atoms (R0 ≈ 12σff). The rate constants for these geometries are shown in Figure 3. This figure shows that kC→W is larger than kW→C for

Figure 3. Effect of step height H̅ on the C → W and W → C rate constants for a step width of W̅ = 0.32 ± 0.02 and a groove width of G̅ = 0.72 ± 0.06. The inset depicts the points in Figure S1, in the Supporting Information, that were studied, and the error bars reflect the uncertainty in determining the droplet radius R0 from MD simulations.

Ke =

the smallest value of H̅ , where the Wenzel state is predicted to be more stable by a thermodynamic model40 (see the inset). For the larger values of H̅ , kW→C is larger than kC→W, which is consistent (within uncertainty) with the model prediction that the Cassie state is more stable (see the inset). In Figure 3, we also see that the rates for the C → W and W → C transitions

⎛ −ΔGC → W ⎞ exp( −ΔG W → ts /kBT ) = exp⎜ ⎟ exp( −ΔGC → ts /kBT ) ⎝ kBT ⎠

(4)

From Table 1, we see that the Wenzel state is more stable for the smallest step height and the Cassie state is favored for the two larger heights. This is consistent with the values of the rate constants in Figure 3, as well as with the predictions of the mathematical model (cf., Figure S1, Supporting Information). Comparing the C → W and the W → C free-energy barriers, C


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Figure 4. Committor probability (PB) versus the order parameter (λ) for surface patterns with W̅ = 0.32 ± 0.02, G̅ = 0.72 ± 0.06, and three different values of H̅ : (a) 0.40 ± 0.03, (b) 0.56 ± 0.045, and (c) 0.72 ± 0.06. The dashed lines indicate the values of λ* at which PB = 1/2, corresponding to the transition-state ensemble. The insets show snapshots of droplet configurations from the transition-state ensemble for each surface geometry.

Table 1. Values of the Free-Energy Barriers for the W → C and C → W Transition along with λts for a Periodically Grooved Surface with Step Width W̅ = 0.32 ± 0.02, Groove Width G̅ = 0.72 ± 0.06, and Various Step Heights H̅

λts

ΔGW→ts/kBT

ΔGC→ts/kBT

H̅ = 0.40 ± 0.03 H̅ = 0.56 ± 0.05 H̅ = 0.72 ± 0.06

0.55 0.72 0.82

6.52 7.45 11.10

3.52 20.20 42.63

We note that Savoy and Escobedo38 used FFS to study the Cassie−Wenzel transition for droplets composed of LJ tetramers on post and nail-shaped pillars with θe = 105°. Data extracted from Figures 3 and 4 of their paper indicate that the C → W free-energy barrier, ΔGC→ts/kBT, increases about five-fold (from 8.5 to 38.3) when the height of the posts increases by 70% (from 9σ to 15σ), whereas the corresponding W → C free-energy barrier, ΔGW→ts/kBT, increases only 50% from 12.1 to 18.7. This is qualitatively similar to the trends observed here (with θe = 126°); Namely, ΔGC→ts/kBT increases 12-fold (from 3.5 to 42.6) when the step height increases by 80% from (0.40R to 0.72R), whereas the corresponding ΔGW→ts/kBT increases only 70% (from 6.5 to 11.1). In both studies, the C → W and W → C free-energy barriers increase

we find that they both increase in H̅ ; however, the C → W transition is more sensitive to changes with H̅ than the W → C transition. This is consistent with the location of the transitionstate ensemble near the bottom of the groove, as we found in the committor analysis (cf., Figure 4).

Figure 5. Meniscus configuration within the groove for Cassie-to-Wenzel transition mechanisms proposed by Patankar:28 (a) sag mechanism and (b) depinning mechanism. δ is the maximum depth of the meniscus below the contact line and α is the contact angle of the meniscus with the side walls of the groove, which can vary between θe and the advancing contact angle θa. D


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sufficiently large for the contact angle of the transition-state meniscus in the sag mechanism to reach the advancing contact angle θa, the contact lines are eventually depinned from the edges. At that point, eq 7 for the sag mechanism requires

with increasing step height, with the effect of step height being much more pronounced for ΔGC→ts/kBT. Some mechanisms for the Cassie-to-Wenzel transition have been proposed.28,29 Here we focus on two mechanisms suggested by Patankar:28 One is the “sag mechanism” (Figure 5a), in which the meniscus remains pinned to the edges of the steps as the liquid penetrates into the groove, and the other is the “depinning mechanism” (Figure 5b), in which the meniscus detaches from the step corners and moves downward to reach the substrate at the bottom of the groove.28 Requiring the local contact angle α of the liquid on the side walls of the groove to be θe,16 the maximum depth δc (below the contact line) for a constant curvature meniscus within the groove is given by δc =

1 G(tan θe − sec θe) 2

⎛ G ⎞ ⎟ θa = cot−1⎜ − ⎝ 4H ⎠

Applying this condition to the transition state in Figure 4a (i.e., assuming depinning occurs for H > 0.40 when G = 0.72) yields θa ≃ 156°. Once the contact lines become depinned, they move into the groove with contact angle θa until the meniscus makes contact with the substrate at the bottom of the groove at transition state. For this configuration (see Figure 5b), λ* can be written as

(5)

λ* = 1 −

Achieving the transition-state configuration via the sag mechanism requires δc ≥ H, which means that H 1 ≤ (tan θe − sec θe) G 2

G cot α 4

(6)

(7)

H 1 ≤ − cot α G 4

H̅ = 0.40 ± 0.03 H̅ =0.56 ± 0.05 H̅ = 0.72 ± 0.06

ρG,max

=

(8)

Am HG

λ* from eq 11

λ* from PB(x) (Figure 4)

λts from ΔG (FigureS3)

0.67 0.76 0.82

0.62 0.77 0.84

0.55 0.72 0.82

Figure 6 shows the effect of the groove width G on the C → W and W → C transition rates for three selected topographies around the C−W boundary shown in the inset. As depicted in Figure 6, kW→C is larger than kC→W for the two smallest values of G̅ , where the Cassie state is predicted (within uncertainty) to be more stable by a thermodynamic model40 (see the inset). For the largest value of G̅ , kC→W is larger than kW→C, which is consistent with the model prediction that the Wenzel state is more stable (see the inset). Figure 6 also shows that the C → W transition rates increase by 15 orders of magnitude and the W → C transition rates decrease by 9 orders of magnitude as G̅ increases. Comparing these results to those reported in Figure 3, we see that W → C transition rates are more sensitive to changes in the groove width than to changes in the step height. From eq 5, we see that the width of the groove affects the shape of the meniscus within the groove, which can also affect the transition state and the transition rate. The average λ* values for the transition states corresponding to the transition rates in Figure 6 are identified by committor analysis and depicted in Figure 7. Here, we see that, for fixed H̅ , λ* decreases with increasing G̅ , indicating that as the groove widens, a lower liquid fraction is required in the groove to reach

For the sag mechanism, λ* can be written as ρG*

(11)

Table 2. Comparison of λ* Values Calculated from eq 11 with the Corresponding Values Obtained from FFS Simulations for a Periodically Grooved Surface with Step Width W̅ = 0.32 ± 0.02, groove width G̅ = 0.72 ± 0.06, and Various Step Heights

where the contact angle α of the pinned meniscus on the sides of the groove can have any value between θe and the advancing contact angle θa. Achieving the transition-state configuration via the sag mechanism now requires

λ* =

1 ⎛ δp ⎞ 1 ⎛⎜ G ⎞⎟ cot α ⎜ ⎟=1+ 3⎝H⎠ 12 ⎝ H ⎠

where Am remains constant as the meniscus moves into a groove of fixed width G. For θa > π/2, the second term on the right-hand side of eq 11 is negative. Therefore, for fixed G, eq 11 predicts larger values of λ* with increasing H for the depinning mechanism, consistent with the observed effect of H on the transition state in MD simulations with θe = 126° ± 5° (see Figure 4). As shown in Table 2, there is close agreement between the values of λ* calculated from eq 11 and the corresponding values obtained from the FFS simulations for fixed G̅ and W̅ .

must be satisfied. However, eq 6 is only satisfied for sufficiently shallow grooves (e.g., those with (H/G) ≤ 0.16 for θe = 126°), and is not valid for the three geometries probed in Figure 3. We note that Luo et al.29 observed experimentally that the proposed sag transition states in Figure 5 might actually be stable intermediate states for droplets on patterned surfaces with shallow grooves. Studies of droplets on such surface morphologies would be an interesting topic for future work. Relaxing the assumption of a constant curvature meniscus within the groove, and considering the meniscus to have a parabolic shape which more closely resembles the transitionstate meniscus profiles shown in Figure 4, the maximum parabolic meniscus depth δp below the contact line is given by δp = −

(10)

(9)

where ρG* is the density in the groove at transition state and Am is the liquid area below the contact line within the groove. For a pinned parabolic meniscus at transition state, this expression reduces to λ* = 2/3. For the FFS results shown in Figure 4a, both the transition-state meniscus configuration and its corresponding λ* value of 0.62 are consistent with the above model for the sag mechanism. For transition-state configurations with larger values of λ* (such as those shown in Figure 4b and c), we must consider that the “depinning mechanism” (Figure 5b) might be responsible for the wetting transition. As the height of the groove increases for fixed G, the sag mechanism for the transition-state configuration requires larger contact angles at the pinned contact lines while maintaining a constant value of λ* = 2/3. When the groove height becomes E


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Table 3. Comparison of the λ* Values Calculated from eq 11 with the Corresponding Values Obtained from FFS Simulations for a Periodically Grooved Surface with Step Width W̅ = 0.32 ± 0.02, Step Height H̅ = 0.56 ± 0.04, and Various Groove Widths G̅ G̅

λ* from eq 11

λ* from PB(x) (Figure 7)

λts from ΔG (FigureS4)

0.56 ± 0.04 0.72 ± 0.06 0.88 ± 0.07

0.81 0.76 0.70

0.86 0.77 0.65

0.84 0.72 0.61

Table 4. Values of the Free-Energy Barriers for the W → C and C → W Transition along with λts for a Periodically Grooved Surface with Step Width W̅ = 0.32 ± 0.02, Step Height H̅ = 0.56 ± 0.04, and Various Groove Widths G̅

Figure 6. Effect of scaled groove width G̅ on the C → W and W → C transition rate constants for a periodically grooved surface with step width W̅ = 0.32 ± 0.02 and step height H̅ = 0.56 ± 0.04. The inset depicts the points in Figure S1 in the Supporting Information that were studied and the error bars reflect the uncertainty in determining the droplet radius R0 from MD simulations.

G̅

λts

ΔGW→ts/kBT

ΔGC→ts/kBT

0.56 ± 0.04 0.72 ± 0.06 0.88 ± 0.07

0.84 0.72 0.61

0.39 7.45 23.88

38.3 20.20 5.92

transitions. The free-energy barrier for the C → W transition decreases with increasing groove width, while the free-energy barrier for the W → C transition increases with increasing groove width. Quantitatively, ΔGC→ts/kBT decreases by about 85% (from 38.30 to 5.92) and ΔGW→ts/kBT increases more than 60 times (from 0.39 to 23.88) with an approximately 60% increase (of ∼0.32) in G̅ . We showed previously40,41 that the equilibrium wetting mode and its corresponding contact angle remain invariant for changes in droplet size for a fixed scaled surface geometry.

the transition state from the Cassie mode. As shown in Table 3, the theoretical values of λ* obtained from eq 11 compare well with the corresponding values generated by FFS simulations for fixed H̅ and W̅ . The free-energy profiles for different groove widths G̅ with fixed W̅ and H̅ are plotted in Figure S4 and the free-energy barriers and transition-state locations from those calculations are given in Table 4. As shown in this table, the groove width has a considerable effect on both the C → W and W → C

Figure 7. Committor probability (PB) versus the order parameter (λ) for surface patterns with W̅ = 0.32 ± 0.02, H̅ = 0.56 ± 0.04, and three different G̅ values: (a) 0.56 ± 0.04, (b) 0.72 ± 0.06, and (c) 0.88 ± 0.07. The dashed lines indicate the values of λ* at which PB = 1/2, corresponding to the transition-state ensemble. The insets show snapshots of droplet configurations from the transition-state ensemble for each surface geometry. F


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surface geometry is constant, the nonscaled groove width, step height, and step width all linearly increase with an increase in the drop radius. The change in the nonscaled groove width affects the shape of the meniscus within the groove, as shown qualitatively in Figure 5b and quantitatively by eq 7. Comparing Figure 8 to Figures 3 and 6, we see that increases in G and H act in concert to magnify the change in the Wenzel−Cassie transition rate, while the increase in G moderates the effect of an increase in step height on the Cassie-to-Wenzel transition rate. Committor analysis of the wetting transition for different droplet sizes (Figure 9) shows that the critical order parameter λ* does not depend on the size of the droplet. The theoretical values of λ* predicted by eq 11, are also independent of droplet size because G/H is fixed for constant G̅ and H̅ . The theoretical value of λ* corresponding to the fixed scaled geometrical parameters in Figure 9 is λ* = 0.62 ± 0.06, which agrees well with the FFS values shown in Figure 9. The effect of droplet size on the free-energy profile for the fixed scaled geometry in Figure 9 is shown in Figure S5 and the free-energy barriers and transition states from these calculations are given in Table 5. As shown in this table, the energy barriers

However, the invariance of the wetting mode and contact angle does not necessarily extend to the kinetic behavior of the droplet. To understand how the kinetics is affected by the droplet size, we used FFS to study three droplet sizes for a fixed scaled surface geometry characterized by H̅ = 0.4 ± 0.03, W̅ = 0.32 ± 0.02, and G̅ = 0.72 ± 0.06. The droplet radii are R0 = (12 ± 2)σff, (17 ± 2)σff, and (23 ± 2)σff, corresponding to 3196, 7189, and 12 433 atoms, respectively. The response of the rate constant to changes in droplet size is shown in Figure 8, where we see that the C → W and W →

Table 5. Values of the Free-Energy Barriers for the W → C and C → W Transition along with λts for a Periodically Grooved Surface with Step Width W̅ = 0.32 ± 0.02, Step Height H̅ = 0.40 ± 0.03, Groove Width G̅ = 0.72 ± 0.06, and Various Droplet sizes R0

Figure 8. Effect of droplet size on transition-rate constants for surfaces with the same scaled topography (H̅ = 0.40 ± 0.03, W̅ = 0.32 ± 0.020, and G̅ = 0.72 ± 0.06).

C transition rates decrease by 9 and 14 orders of magnitude, respectively, as the droplet size increases. Although the scaled

R0 (σff)

λts

ΔGW→ts/kBT

ΔGC→ts/kBT

12.0 ± 2.0 17.0 ± 2.0 23.0 ± 2.0

0.55 0.64 0.63

6.82 22.31 41.36

3.86 14.74 24.75

Figure 9. Committor probability (PB) versus the order parameter (λ) for three droplet sizes with fixed values of the scaled geometrical parameters (H̅ = 0.4 ± 0.03, G̅ = 0.72 ± 0.06, and W̅ = 0.32 ± 0.02): (a) R0 = (12 ± 2)σff, (b) R0 = (17 ± 2)σff, and (c) R0 = (23 ± 2)σff. The insets show snapshots of droplet configurations from the transition-state ensemble for each drop size. G


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for both the C → W and W → C transitions increase with an increase in the size of the droplet. There is about a sixfold increase in both ΔGC→ts/kBT (from 3.86 to 24.75) and ΔGW→ts/kBT (from 6.82 to 41.36) as the droplet size increases by almost a factor of two (from 12σff to 23σff). Although the droplet configurations, characterized by the wetting mode and contact angle,40,41 as well as the value of λ* (cf. Figure 9) are invariant to changes in the droplet size for a fixed scaled surface geometry, the droplet kinetics depend on its size.

(7) Liu, K.; Yaob, X.; Jiang, L. Recent developments in bio-inspired special wettability. Chem. Soc. Rev. 2010, 39, 3240−3255. (8) Wenzel, R. N. Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 1936, 28, 988−994. (9) Cassie, A. B. D.; Baxter, S. Wettability of porous surfaces. Trans. Faraday Soc. 1944, 40, 546−551. (10) Forsberg, P.; Nikolajeff, F.; Karlsson, M. Cassie−Wenzel and Wenzel-Cassie transitions on immersed superhydrophobic surfaces under hydrostatic pressure. Soft Matter 2011, 7, 104−109. (11) He, B.; Patankar, N. A.; Lee, J. Multiple equilibrium droplet shapes and design criterion for rough hydrophobic surfaces. Langmuir 2003, 19, 4999−5003. (12) Liu, G.; Fu, L.; Rode, A. V.; Craig, V. S. J. Water droplet motion control on superhydrophobic surfaces: Exploiting the Wenzel-toCassie transition. Langmuir 2011, 27, 2595−2600. (13) Lafuma, A.; Quere, D. Superhydrophobic states. Nat. Mater. 2003, 2, 457−460. (14) Quéré, D. Wetting and roughness. Annu. Rev. Mater. Res. 2008, 38, 71−99. (15) Dorrer, C.; Rühe, J. Condensation and wetting transitions on microstructured ultrahydrophobic surfaces. Langmuir 2007, 23, 3820− 3824. (16) Patankar, N. A. Transition between superhydrophobic states on rough surfaces. Langmuir 2004, 20, 7097−7102. (17) Barbieri, L.; Wagner, E.; Hoffmann, P. Water wetting transition parameters of perfluorinated substrates with periodically distributed flat-top microscale obstacles. Langmuir 2007, 23, 1723−1734. (18) DeConinck, J.; Dunlop, F.; Huillet, T. Metastable wetting. J. Stat. Mech. 2011, P06013. (19) Ran, C.; Ding, G.; Liu, W.; Deng, Y.; Hou, W. Wetting on nanoporous alumina surface: Transition between Wenzel and Cassie states controlled by surface structure. Langmuir 2008, 24, 9952−9955. (20) Bahadur, V.; Garimella, S. V. Preventing the Cassie−Wenzel transition using surfaces with noncommunicating roughness elements. Langmuir 2009, 28, 48115−4820. (21) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Resonance Cassie−Wenzel wetting transition for horizontally vibrated drops deposited on a rough surface. Langmuir 2007, 23, 12217− 12221. (22) Nosonovsky, M.; Bhushan, B. Biomimetic superhydrophobic surfaces: Multiscale approach. Nano Lett. 2007, 9, 2633−2637. (23) Whyman, G.; Bormashenko, E. How to make the Cassie wetting state stable? Langmuir 2011, 27, 8171−8176. (24) Zheng, Q. S.; Yu, Y.; Zhao, Z. H. Effects of hydraulic pressure on the stability and transition of wetting modes of superhydrophobic surfaces. Langmuir 2005, 21, 12207−12212. (25) David, R.; Neumann, W. Energy barriers between the Cassie and Wenzel states on random, superhydrophobic surfaces. Colloids Surf., A 2013, 425, 51−58. (26) Vrancken, R. J.; Kusumaatmaja, H.; Hermans, K.; Prenen, A. M.; Pierre-Louis, O.; Bastiaansen, C. W. M.; Broer, D. J. Fully reversible transition from Wenzel to Cassie−Baxter states on corrugated superhydrophobic surfaces. Langmuir 2010, 26, 3335−3341. (27) Bormashenko, E.; Musin, A.; Whyman, G.; Zinigrad, M. Wetting transitions and depinning of the triple line. Langmuir 2012, 28, 3460− 3464. (28) Patankar, N. A. Consolidation of hydrophobic transition criteria by using an approximate energy minimization approach. Langmuir 2010, 26, 8941−8945. (29) Luo, C.; Xiang, M.; Heng, X. A stable intermediate wetting state after a water drop contacts the bottom of a microchannel or is placed on a single corner. Langmuir 2012, 28, 9554−9561. (30) Ishino, C.; Okumura, L.; Quéré, D. Wetting transitions on rough surfaces. Europhys. Lett. 2004, 68, 419−425. (31) Sheng, Y. J.; Jiang, S.; Tsao, H. K. Effects of geometrical characteristics of surface roughness on droplet wetting. J. Chem. Phys. 2007, 127, 1−7.

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CONCLUSION We studied the kinetics cylindrical LJ droplets on periodicallygrooved surfaces using FFS. We investigated the effect of surface topology and droplet size on the Cassie-to-Wenzel and Wenzel-to-Cassie transition-rate constants and free-energy barriers. The results show that the C → W transition rate is more sensitive to step height than groove width, while the W → C transition rate is affected by both groove width and height. A committor analysis indicates that the transition-state ensemble consists of droplets that are on the verge of initiating or breaking contact with the substrate at the bottom of the grooves. The critical order parameter corresponding to the transition-state ensamble, λ*, depends on the surface geometry but not on the droplet size. We also obtained a free-energy profile for each FFS simulation, and found good agreement between the relative stability of the wetting mode obtained by FFS and the predictions of a thermodynamic model based on free energy minimization.

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ASSOCIATED CONTENT

S Supporting Information *

Wetting phase diagram for the systems studied, free-energy profiles for wetting-mode transitions, and details of the computational methods employed. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fi[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Grant CBET 0730987 from the National Science Foundation. REFERENCES

(1) Anastasiadis, S. H. Development of functional polymer surfaces with controlled wettability. Langmuir 2013, 29, 9277−9290. (2) Liu, X.; Liang, Y.; Zhou, F.; Liua, W. Extreme wettability and tunable adhesion: Biomimicking beyond nature? Soft Matter 2012, 8, 2070−2086. (3) Bhushan, B.; Jung, Y. C. Natural and biomimetic artififial surfaces for superhydrophobicity, self-cleaning, low adhesion, and drag reduction. Prog. Mater. Sci. 2011, 56, 1−108. (4) Guoa, Z.; Liuc, W.; Sua, B. L. Superhydrophobic surfaces: From natural to biomimetic to functional. J. Colloid Interface Sci. 2011, 353, 335−355. (5) Yan, Y. Y.; Gao, N.; Barthlott, W. Mimicking natural superhydrophobic surfaces and grasping the wetting process: A review on recent progress in preparing superhydrophobic surfaces. Adv. Colloid Interface Sci. 2011, 169, 80−105. (6) Liu, M.; Jiang, L. Switchable adhesion on liquid/solid interfaces. Adv. Funct. Mater. 2010, 20, 3753−3764. H


Langmuir

Article

(32) Chatain, D.; Lewis, D.; Baland, J. P.; Carter, W. C. Numerical analysis of the shapes and energies of droplets on micropatterned substrates. Langmuir 2006, 22, 4237−4243. (33) Li, W.; Amirfazli, A. A thermodynamic approach for determining the contact angle hysteresis for superhydrophobic surfaces. J. Colloid Interface Sci. 2005, 292, 195−201. (34) Li, W.; Amirfazli, A. Microtextured superhydrophobic surfaces: A thermodynamic analysis. Adv. Colloid Interface Sci. 2007, 132, 51− 68. (35) Susarrey-Arce, A.; Marín, Á . G.; Nair, H.; Lefferts, L.; Gardeniers, J. G. E. Absence of an evaporation-driven wetting transition on omniphobic surfaces. Soft Matter 2012, 8, 9765−9770. (36) Koishi, T.; Yasuoka, K.; Fujikawa, S.; Ebisuzaki, T.; Zeng, X. C. Coexistence and transition between Cassie and Wenzel state on pillared hydrophobic surface. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 8435−8440. (37) Savoy, E. S.; Escobedo, F. A. Molecular simulations of wetting of a rough surface by an oily fluid: Effect of topology, chemistry, and droplet size on wetting transition rates. Langmuir 2012, 28, 3412− 3419. (38) Savoy, E. S.; Escobedo, F. A. Simulation study of free-energy barriers in the wetting transition of an oily fluid on a rough surface with reentrant geometry. Langmuir 2012, 28, 16080−16090. (39) Giacomello, A.; Meloni, S.; Chinappi, M.; Casciola, C. M. Cassie-Baxter and Wenzel states on a nanostructured surface: Phase diagram, metastabilities, and transition mechanism by atomistic free energy calculations. Langmuir 2012, 28, 10764−10772. (40) Shahraz, A.; Borhan, A.; Fichthorn, K. A. A theory for the morphological dependence of wetting on a physically-patterned solid surface. Langmuir 2012, 28, 14227−14237. (41) Shahraz, A.; Borhan, A.; Fichthorn, K. A. Wetting on physically patterned solid surfaces: The relevance of molecular dynamics simulations to macroscopic systems. Langmuir 2013, 29, 11632− 11639. (42) Allen, R. J.; Valeriani, C.; Wolde, P. R. Forward flux sampling for rare event simulations. J. Phys.: Condens. Matter 2009, 21, 463102. (43) Wu, H.; Borhan, A.; Fichthorn, K. A. Coarse-grained interaction of a fluid with a physically patterned solid surface: Application to manodroplet wetting. J. Low Temp. Phys. 2009, 157, 277−295. (44) Wu, H.; Borhan, A.; Fichthorn, K. A. Interaction of fluids with physically patterned solid surfaces. J. Chem. Phys. 2010, 133, 054704. (45) Nosé, S. Constant temperature molecular dynamics methods. Prog. Theor. Phys. Suppl. 1991, 103, 1−46. (46) Hoover, W. G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 1985, 31, 1695−1697. (47) Velez-Vega, C.; Borrero, E. E.; Escobedo, F. A. Kinetics and mechanism of the unfolding native-to-loop transition of trp-cage in explicit solvent via optimized forward flux sampling simulations. J. Chem. Phys. 2010, 133, 105103. (48) Velez-Vega, C.; Borrero, E.; Escobedo, F. A. Kinetics and reaction coordinate for the isomerization of alanine dipeptide by a forward flux sampling protocol. J. Chem. Phys. 2009, 130, 225101. (49) Escobedo, F. A.; Borrero, E. E.; Araque, J. C. Transition path sampling and forward flux sampling: Applications to biological systems. J. Phys.: Condens. Matter 2009, 21, 333101. (50) Borrero, E. E.; Escobedo, F. A. Reaction coordinates and transition pathways of rare events via forward flux sampling. J. Chem. Phys. 2007, 127, 164101. (51) Valeriani, C.; Allen, R. J.; Morelli, M. J.; Frenkel, D.; Wolde, P. R. Computing stationary distributions in equilibrium and nonequilibrium systems with forward flux sampling. J. Chem. Phys. 2007, 127, 114109.

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Kinetics of Droplet Wetting Mode Transitions on Grooved Surfaces

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