Coexistence and transition between Cassie and Wenzel state on pillared hydrophobic surface
 ^{a}Department of Applied Physics, University of Fukui, 391 Bunkyo, Fukui 9108507, Japan;
 ^{b}Innovative Nanopatterning Research Laboratory and
 ^{d}Ebisuzaki Computational Astrophysics Laboratory, RIKEN, Wako, Saitama 3510198, Japan;
 ^{c}Department of Mechanical Engineering, Keio University, Yokohama 2238522, Japan; and
 ^{e}Department of Chemistry and Center for Materials and Nanoscience, University of Nebraska, Lincoln, NE 68588
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Edited by Benjamin Widom, Cornell University, Ithaca, NY, and approved April 3, 2009 (received for review February 24, 2009)
Abstract
Water droplets on rugged hydrophobic surfaces typically exhibit one of the following two states: (i) the Wenzel state [Wenzel RN (1936) Ind Eng Chem 28:988–994] in which water droplets are in full contact with the rugged surface (referred as the wetted contact) or (ii) the Cassie state [Cassie, ABD, Baxter S (1944) Trans Faraday Soc 40:546–551] in which water droplets are in contact with peaks of the rugged surface as well as the “air pockets” trapped between surface grooves (the composite contact). Here, we show largescale molecular dynamics simulation of transition between Wenzel state and Cassie state of water droplets on a periodic nanopillared hydrophobic surface. Physical conditions that can strongly affect the transition include the height of nanopillars, the spacing between pillars, the intrinsic contact angle, and the impinging velocity of water nanodroplet (“raining” simulation). There exists a critical pillar height beyond which water droplets on the pillared surface can be either in the Wenzel state or in the Cassie state, depending on their initial location. The freeenergy barrier separating the Wenzel and Cassie state was computed on the basis of a statisticalmechanics method and kinetic raining simulation. The barrier ranges from a few tenths of k_{B}T_{0} (where k_{B} is the Boltzmann constant, and T_{0} is the ambient temperature) for a rugged surface at the critical pillar height to ≈8 k_{B}T_{0} for the surface with pillar height greater than the length scale of water droplets. For a highly rugged surface, the barrier from the WenzeltoCassie state is much higher than from CassietoWenzel state. Hence, once a droplet is trapped deeply inside the grooves, it would be much harder to relocate on top of high pillars.
 freeenergy barrier
 molecular dynamics simulation
 nanodrop raining experiment
 WenzeltoCassie state transition
It is well known that microtextured or nanotextured hydrophobic surfaces can become superhydrophobic (1–39). In fact, nature provides first examples of superhydrophobic surfaces, such as lotus leaves and water striders' nonwetting legs (40–42). Synthetic microtextured surface structures like the lotus leaves have been fabricated to achieve high water repellency such that on these surfaces, water droplets are typically in the Cassie state (43) rather than the Wenzel state (44). In general, water droplets adhere more strongly to the textured surface in the Wenzel state than in the Cassie state, causing stronger contactangle hysteresis. Hence, in many practical applications such as selfcleaning surfaces (6, 17), the Cassie state is preferred over the Wenzel state. It is also known that as the degree of surface roughness increases, the Cassie state becomes increasingly favorable compared with the Wenzel state. Hence, at certain degree of roughness, the Wenzel state and Cassie state can become more or less equally favorable and may even coexist on the same surface. From a statisticalmechanics point of view, the 2 states can coexist when they are separated by a high freeenergy barrier by which one state is still metastable (freeenergy local minimum), and the other is thermodynamically stable (freeenergy global minimum). In this article, we present computer simulation evidence of coexisting Wenzel/Cassie state (or the bistable state) for water droplets on pillared hydrophobic surface. We have studied 4 conditions that affect the transition between the Wenzel and Cassie state: (i) The height of nanopillars, (ii) the spacing between pillars, (iii) the impinging velocity of water nanodroplet (“raining” effect), and (iv) the intrinsic contact angle of a droplet on the flat surface of the same material. For 2 special cases (when droplets are at the bistable state), we have computed freeenergy barrier separating the Wenzel and Cassie state. This is a quantitative computation, at the molecular level, of the freeenergy barrier between Wenzel and Cassie state of water nanodroplets.
The simulation system consists of a solid hydrophobic surface, either flat or rugged, and a nanodroplet of water. The flat surface assumes the structure of a (0001) graphite surface with hexagonally arranged atoms. In the molecular dynamics (MD) simulation (see Materials and Methods), atoms of the solid surface were fixed. The rugged surface is an artificial pillared surface. Quadrangular pillars with lateral size of 12.3 × 12.8 Å were arranged with spacing of 12.3/12.8 Å between them in the x/y direction. A larger spacing of 14.8/14.9 Å between pillars was also examined. This nearsquarelattice pillar arrangement has been used previously by Lundgren et al. (11, 32). The height of nanopillars is adjustable, ranging from 2 graphite–interlayer distance (6.7 Å) to 30 interlayer distance (100.4 Å). For MD simulations with a larger water droplet, the initial configuration of water droplet is a 18 × 18 × 18 (5,832) molecules cube with a side length of 55.9 Å. This length scale was selected based on liquid water density of 0.997 g/cm^{3} at 298 K. The lateral size of the solid surface is 167.4 × 168.4 Å, ≈3 times the length of the water cube. The length of the simulation cell (280 Å) is 5 times the length of the water cube. For MD simulations with a smaller water droplet, the initial configuration of the water droplet was a 12 × 12 × 12 (1,728) molecules cube; the lateral size of the solid surface is 110.8 × 110.8 Å, ≈3 times the length of the smaller water cube.
Results and Discussion
To measure wettability of the model hydrophobic surfaces, we first computed the intrinsic contact angle of a water nanodroplet on the flat surfaces (11, 32, 45). To this end, we used the following computational approach to determine the surface locus of a water nanodroplet. We divided the entire simulation cell into many cubic meshes, each with length scale of 5 Å. The average local water density in each cubic mesh was recorded. Having obtained the local water density, we can identify the spatial points where the local density is half of the bulk water. The locus of these points gives rise to the surface of the droplet. The contact angle is defined as the angle between a tangential line of the droplet surface (described by a fitting circle) at any 3phase contact point and the line in the flat surface. Note that both lines must be in the same plane through the center of the droplet. Snapshots of the water nanodroplet are shown in Fig. 1 A and B and supporting information (SI) Fig. S1. Depending on the interaction parameter between water molecule and atom of the solid surface (see Materials and Methods), the computed contact angles range from 92° to 127° (see Fig. 1C and Fig. S1). Hereafter, we will mainly focus on 2 weakly hydrophobic surfaces whose contact angles are ≈92° and ≈99°, respectively. Note that the contact angle of a graphite surface is ≈85°.
Next, we performed MD simulations of the larger water nanodroplet on pillared surfaces to examine texture effects on the shape and location of the droplet. Two initial locations for the larger water cube were considered, one on top of the pillars (Fig. 2Ai–Di) and another at the bottom of the grooves (Fig. 2 Ei–Hi). Here, the height of identical pillars ranges from 2 to 5 graphite–interlayer distance. Totally, 8 systems were studied, each equilibrated for 1 ns. Snapshots of the equilibrated droplet at t = 1.0 ns for these systems are shown in Fig. 2 Aii–Dii, and Eii–Hii. When the water cube is initially placed at the bottom of the grooves, the final state of the droplet is always the Wenzel state (Fig. 2 Eii–Hii), regardless of the height of pillars. However, when the water cube is initially placed on top of the pillars, the final state is sensitive to the height of the pillars. When the height is 2 or 3 graphite–interlayer distance, the droplet still prefers the Wenzel state (Fig. 2 Aii and Bii). But when the height is >4 graphite–interlayer distance (13.4 Å), the droplet adopts the Cassie state (here, the “air nanopockets” are essentially nanosized cavities between nanopillars). In other words, at low pillar height, the Wenzel state is the only stable state for the droplet. But when the height is beyond a critical value, e.g., 13.4 Å, the Cassie state is metastable, separated from the stable Wenzel state by a freeenergy barrier (18, 23, 37). As such, coexistence of Wenzel and Cassie state for water droplets is possible, depending on the initial location of the droplets (18, 19, 23, 35–37).
At high pillar height, the transition from the metastable Cassie state to the stable Wenzel state becomes more difficult due to an increasingly higher freeenergy barrier. To gain more insights into relative stability of the Cassie and Wenzel states, it is important to have quantitative values of the freeenergy barrier that separates the Wenzel and Cassie state, as schematically shown in Fig. 3(23, 37). To this end, we used 2 computer simulation methods: 1 kinetic and 1 equilibrium. The kinetic method is basically a computer experiment of nanodroplet “raining,” namely, a water droplet impinging the pillared surface. This raining simulation mimics macroscopic raining experiments (recently reported for designing micropatterned superhydrophobic surfaces) (28, 29). Besides the design of nanopatterned surfaces, the raining simulation allows quantitative evaluation of the freeenergy barrier through preparation of water droplets with exactly the same number of molecules and impurityfree (perfectly dry) surfaces. Specifically, a water droplet was initially located 60 Å from the top of pillars (Fig. 4A) and was equilibrated there for 200 ps at T_{0} = 298 K, where T_{0} =
The downward velocity v_{d} was carefully chosen to attain sufficient number of statistical events for both the Cassie and Wenzel states (Fig. 4 B and C; also see Movie S1 and Movie S2). If v_{d} is too high, the water droplet can easily go over the freeenergy barrier and reaches the Wenzel state. If v_{d} is too low, the water droplet favors the Cassie state. In the first raining experiment, we performed 926 totally independent MD simulations and recorded the number of events for the droplet in the Wenzel or Cassie state. Nine selected downward velocities and the number of MD runs for each given v_{d} are listed in Table 1. The probability for the droplet in the Wenzel state P_{w} was computed (for each given v_{d}) and shown in Table 1. The probability P_{w} can be fitted to an exponential equation where P_{0} is a preexponential factor, ΔG_{cw} is defined as the freeenergy barrier from the Cassie to Wenzel state, and e_{k} is thekinetic energy of the center of mass of the droplet (per molecule), given by In Fig. 5, a fit to numerical probabilities P_{w} vs. e_{k} (filled circles) to the natural logarithm of Eq. 1 gives rise to a freeenergy barrier of ΔG_{cw} = 0.334 kJ/mol ≈ 0.135 k_{B}T_{0}. This barrier corresponds to the condition that the height of pillars is at the critical value of 13.4 Å. In the second computer experiment, we used the smaller water droplet (with 1,728 water molecules) but otherwise the same pillared surface. After 816 independent MD simulations (102 MD runs for each given downward v_{d}), we obtained the freeenergy barrier of ΔG_{cw} = 0.328 kJ/mol ≈ 0.132 k_{B}T_{0}, very close to the barrier of ΔG_{cw} = 0.334 kJ/mol. These results indicate that the freeenergy barrier is not very sensitive, to some extent, to the size of water droplet. In the third computer experiment, we still used the smaller water droplet but a larger spacing between pillars, that is 14.8/14.9 Å in the x/y direction. Totally, 810 independent MD simulations were carried out. The obtained freeenergy barrier is ΔG_{cw} = 0.00286 kJ/mol, considerably lower than the barrier of ΔG_{cw} = 0.328 kJ/mol. This indicates that the spacing between pillars has a profound effect on the stability of the metastable Cassie state. In the fourth computer experiment, we used the same simulation system as in the second computer experiment but the pillared surface has a larger intrinsic contact angle of 99°. Totally, 773 independent MD simulations were carried out. The numerical probabilities P_{w} and kinetic energy of the droplet e_{k} (open circles) are shown in Fig. 5. After fitting the data to Eq. 1, we obtained a freeenergy barrier of ΔG_{cw} = 1.41 kJ/mol ≈ 0.57 k_{B}T_{0}. This barrier is appreciably higher than ΔG_{cw} = 0.328 kJ/mol, indicating that the intrinsic contact angle can also strongly affect the stability of the Cassie state.
Finally, we computed the freeenergy barrier for a system with very high pillar height whose length scale is greater than length scale of the water droplet. In this case, the raining experiment is not effective to compute the freeenergy barrier because the water droplet would overwhelmingly favor the Cassie state. We therefore invoke a statisticalmechanics method to evaluate the freeenergy barrier. We still used the larger droplet (with 5,832 water molecules) and the pillared surface with an intrinsic contact angle of 92°, but the height of the pillars amounts to 100.4 Å. The water cube was placed at 2 initial positions: (i) an upper position but not fully embedded by the pillars or (ii) a lower position where the entire water cube is embedded in the groove region, as shown in Fig. 6 A and D. The system was then relaxed to achieve equilibration. Interestingly, the water cube at the upper position ascended upward and turned to a sphericallike droplet on top of the high pillars (Fig. 6 B and C and Movie S3]. We found that as long as a small portion of water cube is above the top of the pillars, the water cube always ascends to the top of the pillars. On the other hand, the water cube at the lower position tends to descend toward the bottom of the grooves (Fig. 6 E and F). We performed 210 independent MD simulations, 105 each for the droplet ascending and descending. The freeenergy barrier ΔG_{l} can be computed by taking an integration of the positiondependent z component of the total force, f_{z}(z), acting on entire water droplet, that is, where z_{0} is the initial position of the center of mass of the droplet. The trajectory of averaged z coordinate of the center of mass of the droplet is shown in Fig. 7A. The integration of f_{z}(z) is plotted in Fig. 7B. The minimum at the upper equilibrium position (z ≈ 127.4 Å) gives rise to a freeenergy barrier of ΔG_{cw} = 4.83 kJ/mol ≈ 1.9 k_{B}T_{0}. This barrier is an order of magnitude higher than that (0.334 kJ/mol) in the case of lowpillar surface, indicating the strong effect of the pillar height to the stability of the metastable Cassie state. On the other hand, the minimum at the lower equilibrium position (z ≈ 28 Å) gives rise to a freeenergy barrier of ΔG_{wc} = 18.6 kJ/mol ≈ 7.5 k_{B}T_{0} (from the WenzeltoCassie state), indicating that once a droplet is trapped in the groove region, it would be much harder for it to reach to the top of high pillars because the Wenzel state is the thermodynamically stable state.
Conclusion
We present a simulation evidence of coexisting Wenzel/Cassie state for water droplets on a pillared hydrophobic surface. There generally exists a critical pillar height beyond which water droplets on pillared hydrophobic surfaces can be in the bistable Wenzel/Cassie state, depending on the initial location of the droplets. More importantly, the freeenergy barrier separating the Wenzel and Cassie states is computed on the molecular level, based on both raining experiments and a statisticalmechanics method. Typically, the barrier ranges from a few tenths of k_{B}T_{0} for a pillared surface at the critical pillar height to ≈8 k_{B}T_{0} for a pillared surface with pillar height greater than the size of the water droplet. Knowledge on the dependence of the freeenergy barrier to the pillar height, the spacing between pillars, and the intrinsic contact angle will benefit the future design of nanopatterned hydrophobic surfaces and practical applications in nanofluidics.
Materials and Methods
The MD simulation was carried out at a constantvolume and constanttemperature (298 K) condition. The temperature was controlled by using the velocity scaling method. The periodic boundary condition was applied in all 3 spatial dimensions. The z dimension of the simulation cell (perpendicular to the pillared surface) is sufficiently large to avoid direct interaction between the water droplet and periodic images of pillared surfaces. A rigidbody model of water, the SPC/E (46) model, was used. The potential function of the SPC/E model includes 2 terms, a Coulomb term and a Lennard–Jones (LJ) term. The longrange charge–charge interaction between water molecules was calculated by using the Ewald method. Atoms of the flat and pillared surfaces were simply assumed to be LJ particles whose size and energy parameters are σ = 3.4 Å and ε = 0.2325 kJ/mol, respectively (47). The hydrophobicity of the solid surfaces can be adjusted by multiplying a parameter (in the range of 0.5–1) and ε (see Fig. S1). The time integration for the translational and rotational motion was undertaken by using the velocity Verlet method and timereversible algorithm (48). The MD time step was set at 2.0 fs. In the initial MD simulation, translational motion of water molecules was not involved for 2.0 ps so that only orientational degrees of freedom of water molecules were relaxed.
Because of large number of simulations and a few very large system sizes involved in some simulations, we used a specialpurpose computer “MDGRAPE3” (49–51) to perform the MD simulations. The MDGRAPE3 chips handle 3 force calculations: (i) the realspace part of the longrange charge–charge interaction, (ii) the van der Waals interaction, and (iii) the reciprocalspace part of longrange charge–charge interaction. Other computationally lessintensive calculations are handled by a host computer, which include updating particle positions and evaluating temperature. The specialpurpose computer contains 1 MDGRAPE3 board that consists of 12 MDGRAPE3 chips. The peak performance of a MDGRAPE3 board at 250 MHz is 2.16 TFLOPS. We used 2 specialpurpose computers for the MD simulations; one for the real part of the Ewald calculation and another for the reciprocalspace part of Ewald calculation.
Acknowledgments
We are grateful for valuable discussions with Prof. Peter Rossky. This work was supported by the Japan Science and Technology Corporation, the Ministry of Education, Culture, Sport and Technology in Japan, and the GrantinAid (KAKENHI) for Young Scientists (B). X.C.Z. was supported by U.S. Department of Energy Grant DEFG0204ER46164, National Science Foundation Grant CHEM0427746, and by the Nebraska Research Initiative.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: xczeng{at}phase2.unl.edu

Author contributions: T.K., K.Y., S.F., and X.C.Z. designed research; T.K., K.Y., and X.C.Z. performed research; T.K., K.Y., S.F., and T.E. contributed new reagents/analytic tools; T.K., K.Y., and X.C.Z. analyzed data; and T.K. and X.C.Z. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0902027106/DCSupplemental.
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