Metastable Sessile Nanodroplets on Nanopatterned Surfaces - The

Mar 19, 2012 - Department of Chemistry, Virginia Commonwealth University, ... the contact angle θ to interfacial free energies between solid and liqu...
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Metastable Sessile Nanodroplets on Nanopatterned Surfaces John A. Ritchie, Jamileh Seyed Yazdi,† Dusan Bratko,* and Alenka Luzar* Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284-2006, United States ABSTRACT: Small aqueous droplets on homogeneous surfaces surrounded by a reservoir of vapor are inherently unstable. Depending on the humidity, they keep evaporating and ultimately disappear or grow until they fully wet the surface under supersaturation. We are considering a system departing from this common picture. For nanoscale droplets sitting above hydrophilic patches on a heterogeneous surface, there can exist a range of supersaturated pressures at which the droplets maintain a stable volume, determined by the pertinent contact angle and the size of the patches. The region under the droplet perimeter controls the drop’s curvature. Vapor pressure rises along with increased curvature as soon as the drop extends into the hydrophobic area. The drop size may therefore remain stable when its base just covers the hydrophilic patch. The finite range of water−substrate interactions, however, blurs the boundaries between surface regions with different hydrophilicities; hence, the nanodrop contact angle varies with the patch size in a gradual manner. We use molecular simulations to examine this dependence on model surfaces with either chemical or topological heterogeneities. For both types of heterogeneities, our results show the contact angle of a nanodroplet can be predicted by the local Cassie−Baxter mixing relation applied to the area within the interaction range from the drop’s perimeter, which, in turn, enables predictions for drop condensation and saturated vapor pressure above partially wetted nanopatterned hydrophilic/hydrophobic surfaces.

1. INTRODUCTION Understanding the nanoscale wetting behavior is vital to a range of disciplines from biology to atmospheric chemistry, geoscience, and engineering. Modifications to surfaces imparting specific chemical and physical properties are important in the development of new technologies, materials design, and nanofluidics.1 Materials with nanosized surface heterogeneities2 can show wetting properties qualitatively different from both the homogeneous surfaces and the ones with heterogeneities of macroscopic length scale. A nanopatterned surface can support sessile droplets in equilibrium with supersaturated vapor above the substrate. This contrasts the behavior of droplets on uniform surfaces, which will either grow until they coalesce and form a contiguous aqueous phase or evaporate until extinction, depending on whether the actual vapor pressure in the system is above or below the equilibrium vapor pressure corresponding to the initial size of the droplets. In either case, the change will take place at accelerated pace because the droplets’ vapor pressure increases with decreasing radius of curvature of the nanodrop.3,4 If a nanodroplet sits above a hydrophilic patch on a heterogeneous surface, however, the condensation can stop when the drop barely outgrows the size of the underlying patch. When this happens, the drop acquires a higher contact angle θ, increased curvature, and higher vapor pressure. Similarly, evaporation can cease if the perimeter of a shrinking droplet, initially exceeding the patch size, crosses from the hydrophobic region onto hydrophilic area of a patch. In principle, only the region under the droplet perimeter controls the contact angle,5−9 which in turn determines the drops curvature and hence the vapor pressure of the liquid in the drop. While there exist different conventions defining hydrophobicity, consistent with previous works,2,10−12 here we term surfaces with acute © 2012 American Chemical Society

contact angles hydrophilic and those where the angle is obtuse hydrophobic. According to macroscopic thermodynamics, Young’s equation relates the contact angle θ to interfacial free energies between solid and liquid, γSL, solid/vapor, γSV, and liquid/vapor, γLV ≡ γ13 γ − γSL cos θ = SV γ (1) Contact angle, along with the volume of the drop, determines the radius of curvature of the drop, Rc,9 the excess chemical potential Δμ(Rc), and associated vapor pressure P(Rc)3,4,14 Δμ(R c) ∼ V̅ ΔPL = V̅

P(R c) 2γ = kBT ln Rc Po

(2) 3

in analogy to Kelvin equation for spherical drops. Above, kBT is thermal energy and V̅ partial molar volume of water. ΔPL is the Laplace pressure inside the droplet, and P(Rc)/Po is the relative increase in the saturated vapor pressure above the drop compared to the vapor pressure of a macroscopic liquid phase, Po. At fixed volume of a sessile drop, Rc decreases with contact angle,9 making chemical potential a monotonically increasing function of θ. According to eq 2, any significant dependence of P on contact angle is limited to nanosized sessile droplets, the effect being essentially negligible for macroscopic Rc. Cassie and Baxter generalized Young’s equation, eq 1, to composite surfaces whose cosine of contact angle is presumed to represent the area average of cos θi of individual components covering fractional areas f i15 Received: January 5, 2012 Revised: March 12, 2012 Published: March 19, 2012 8634

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reach a metastable equilibrium with a supersaturated vapor phase.

(3)

The Cassie−Baxter (CB) equation implicitly presumes any heterogeneities to occur on length scales small compared to the size of the drop (macroscopic drops); thus, there is no dependence on the drop location.2,6 Recent studies5−9 address scenarios where the drops and surface heterogeneities are of comparable length scales. These works highlight the importance of surface properties under the drop perimeter, which determine the force balance underlying the Young equation. To extend the approximate validity of the CB equation to nanoscale drops and nanoscale surface texture, eq 3 has to be replaced by a local CB equation where only the region covered by the three-phase contact line is considered in determining f1 and f 2.7−9 At the nanoscale, the situation is complicated further by the somewhat arbitrary definition of the perimeter region, which is blurred by both the finite range of water−substrate interactions and fluctuations of the nanodrop shape. Hysteresis associated with pinning/depinning processes16 can sometimes contribute to deviations from the Cassie−Baxter prediction.16,17 In this work, we use molecular simulations to examine generalizations of conventional surface thermodynamics to small length scale systems, relevant to nanofluidics and the design of surface-patterned nanomaterials. We explore the differences that inevitably separate macroscopic and nanoscale systems, as the continuum picture holds only approximately at the nano- and molecular levels. In experiments, surface heterogeneities comparable to the size of the droplets have so far been studied only on surfaces with macroscopic drops and surface patches.5 Molecular simulations provide an ideal framework for studies of nanoscale systems which are not accessible to laboratory measurements. To probe the role of tiny heterogeneities, comparable to the size of the nanodroplets on the surface, we consider model graphene-like surfaces with hydrophobic and hydrophilic domains. The origin of spatially varying hydrophilicity is either an inhomogeneous chemical composition or varied surface roughness.18 Chemical heterogeneity is introduced by considering patches of two types of atoms characterized by different interactions with water19 to render the surface strongly hydrophilic or hydrophobic. In parallel calculations on a chemically homogeneous surface, we produce hydrophilicity contrast by varying the degree of local surface roughness. Our computer experiments test how contact angle of a nanodrop depends on water−substrate interaction beneath the drop’s core and under its perimeter. We examine the variation of the contact angle upon increasing the size of a hydrophilic patch beneath the nanodrop as it approaches, and eventually extends beyond, the droplet’s three-phase contact line. We further determine the role of the finite range of water/ surface attraction on contact angle change, which we observe over a window of patch diameters. Within this window, the patch size deviates from the droplet’s base by less than the range of interaction. We find essentially identical crossover behaviors for both types of heterogeneities: chemical and topological. If a droplet on a hydrophilic patch is growing due to vapor condensation, the increase in its curvature and chemical potential, associated with extension onto the hydrophobic surrounding, can preclude further condensation. An evaporating droplet, on the other hand, can stabilize as it shrinks to the size of the patch. We show that, in contrast to droplets on homogeneous surfaces,4 at appropriate vapor pressure sessile nanodrops on a nanopatterned surface can

2. MODELS AND METHODS Molecular dynamics simulations are carried out by the LAMMPS20 simulation package in NVT ensemble at temperature 300 K maintained by a Nose-Hoover thermostat with 100 fs time constant. Because of vapor/liquid coexistence, the average pressure in the system corresponds to the saturated vapor pressure above the drop at given T. Verlet integrator is used with simulation time step 1 fs. Long-range electrostatic interactions are treated by particle−particle−particle mesh solver (pppm) with a real-space cutoff of 14 Å and precision tolerance of 10−5. Lennard-Jones atom−atom interactions are truncated at 14 Å. Periodically replicated simulation box is a rectangular prism, with box edges Lx = 117.9 Å, Ly = 119.1 Å, and Lz = 300.0 Å. To speed up the calculation, the graphitic substrate layers are held frozen in space during the simulation and the SHAKE algorithm21 is used to maintain the internal geometry of the water molecules. Water droplets containing 2000 molecules are placed on a single-layer substrate with two types of atoms in the case of chemical heterogeneity or a corrugated surface made from three layers of identical atoms in the case of topological heterogeneity. Drop size effects on θ have been analyzed systematically in previous studies,18,19,22,23 and we pick the above size because it has been found sufficient to secure a fair agreement with Young contact angles extrapolated to infinite drop size19 or calculated for extended surfaces by pressure tensor24 and thermodynamic integration2 techniques. From reported size dependencies,23,19 deviation in the contact angle close to 10° can be expected for 500 water molecule nanodrops. In fact, the first evidence that nanodrops can inform us about approximate macroscopic behavior has been obtained in a system as small as 90 water molecules.25 Using bigger drops with N approaching O(104), on the other hand, is of lesser interest to us as the sensitivity of chemical potential and associated vapor pressure, eq 2, declines with growing size. First, we consider patches of size small compared to the base of the droplet above the patch. We determine the drop’s contact angle for different combinations of surface type of the patch and its surroundings. We then proceed by performing consecutive simulation runs where we gradually increase the size of a hydrophilic patch under the drop, while the rest of the surface is hydrophobic. In these runs, we compare two different scenarios. In the first one, we achieve the hydrophilicity contrast through stronger water/substrate attraction in the patch region. In the second one, we introduce increased surface roughness to render the surface outside the patch strongly hydrophobic. Water molecules are represented by the extended simple point charge (SPC/E) model.26−28 This model captures qualitatively and often quantitatively a large number of properties of water, including interfacial water. Most importantly, it reproduces well the surface tension of water,28−32 an essential property for our study. Although both structure and thermodynamics become more sensitive to the shortcomings of the force field in nonuniform systems,33 the change in H bonding within an interfacial water layer, the main contribution to water’s surface tension34 is described well by a number of two-body water potentials, including SPC/E, as well as by a simple mean-field model34,35 when compared to experimental findings.36 8635

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Table 1. Lennard-Jones Parameters Used in the Simulations surface type

εCC (kcal/mol)

σCC (Å)

εOO (kcal/mol)

σOO (Å)

εCO (kcal/mol)

σCO (Å)

chemical heterogeneity: hydrophobic chemical heterogeneity: hydrophilic topological heterogeneity

0.023 0.092 0.092

3.214 3.214 3.214

0.155 0.155 0.155

3.165 3.165 3.165

0.060 0.120 0.120

3.190 3.190 3.190

We choose water/substrate atom interactions to tune surface hydrophilicity following Werder, Jaffe, and co-workers.19,37 Lennard-Jones interactions corresponding to our surfaces are consistent with previous studies, 18,19,22,38 with minimal modifications necessary to generate two coexisting surface types. Parameters we use are listed in Table 1. Water−graphene interaction parameters are calculated using geometric average rules incorporated in LAMMPS, εCO = (εCCεOO)1/2 and σCO = (σCCσOO)1/2, where σCO, σCC, and σOO are the carbon−oxygen, carbon−carbon, and oxygen−oxygen separation distance at minimum potential and εCO, εCC, and εOO are the LennardJones minimum potential energies.19 For given molecular sizes, contact distance σCO = 3.19 Å used here is essentially indistinguishable from the value 3.1895 Å obtained by the arithmetic (Berthelot) mixing rule. 2.1. Chemical Heterogeneity. We considered three distinct surface types. Each of the surfaces comprised 5376 atoms positioned on a hexagonal lattice mimicking the structure of graphene. Hydrophilic and hydrophobic surfaces were characterized by water/substrate Lennard-Jones parameters from Table 1, and we obtained a third, “intermediate” type of surface, by mixing both atomic species with equal representations. We designed heterogeneous surfaces with background of one surface type (hydrophilic, hydrophobic, or intermediate) and a central region or patch of another type. We also considered pure hydrophobic and pure hydrophilic surfaces and a sequence of mixed hydrophobic−hydrophilic surfaces with incrementally increased hydrophilic patches on a hydrophobic surface. In all cases, the droplet was initially located above the center of the patch. In the first series of runs, we considered small patches of size of approximately one-fifth the contact surface area of the water drop. We performed contact angle calculations for all possible combinations of surface types of the patch and its surroundings. In each run, the drop was equilibrated above the selected patch for 200 ps, followed by about 400 ps of production. If necessary, the drop was translated to bring its center above the center of the patch before production. While the droplets remain localized on hydrophilic patches,39 in the case of hydrophobic patches, during the course of the simulation, the water drop occasionally moved from over the patch. Only simulation trajectories where the drop’s center of mass continuously resides within 4 Å of the patch center were used in calculating the contact angle. To investigate the effects of patch size and range of interaction between the substrate and the water molecules at the three-phase contact line on contact angle, we created surfaces with a circular hydrophilic patch beneath the drop, surrounded by a hydrophobic surface. The patch radii rp used were 16.3, 17.4, 19.7, 20.9, 25, 30, 35, 40, and 45 Å. Again, only trajectories in which the drop remained atop the patch were used. Figure 1 illustrates typical situations considered in these calculations. Results are presented in the following sections. 2.2. Topological Heterogeneity. We created a corrugated surface with asperities made from two layers of atoms protruding from the graphene-like bottom layer comprised of

Figure 1. Snapshots of the water droplet above a chemical heterogeneity for a sequence of increasing radii of hydrophilic patch rp in the interval from 16 to 45 Å. Drop’s geometry becomes sensitive to patch size when the latter approaches the size of droplet’s base.

5376 atoms. The asperities added another 2668 atoms. The positions of pillar atoms were the same as positions of carbon atoms of graphite in the second and third layers at given lattice positions, with successive heights 3.35 Å (layer thickness of graphite) and 6.7 Å, while we leave all other atom sites in these two layers unoccupied to form grooves between pillars. The distance between asperities in x and y directions are Δx = 4.912 Å and Δy = 4.254 Å. These distances are sufficiently small to prevent penetration of water into the grooves.18 We grow a circular patch in the middle of this corrugated surface by replacing the pillars and grooves by fully occupied second and third graphite layers. We began with the patch radius of 10 Å and increased it to 55 Å in 5 Å increments. Lennard-Jones parameters were the same as used for the homogeneous hydrophilic surface with monolayer contact angle ∼67°. As demonstrated in previous works,18,40,41 the subnanoscale corrugations, reducing the atom density in the top surface layers, render the surface more hydrophobic, a behavior departing from predictions42 of Wenzel equation.43 In the absence of the patches, this surface is very hydrophobic (θ ∼ 135°). The hydrophilic extreme is the case with infinite radius of the patch, rp (θ ∼ 57°), meaning that the “patch” extends over the entire surface comprising 16 128 atoms. Our simulations were initiated by putting a water droplet on these surfaces above the center of the patch. We equilibrated the system and then used configurations sampled during 1−3 ns trajectories to measure the contact angle. Longer sampling times were required at high contact angles as thermal motion caused occasional bounces of the droplet, increasing statistical fluctuations. Figure 2 shows snapshots of three surfaces with 8636

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to the overall droplet’s contour and the horizontal plane at the height σCO (3.19 Å) above the graphene surface. In the case of topological heterogeneity, however, the contact angle was measured at the height of 9.89 Å to account for the additional two layers of carbon atoms in the pillars. We note that by using the global drop’s contour,45,50 the analysis differs from the one of Ingebrigtsen and Toxvaerd,47 which concerns the differences between the macroscopic (Young) prediction and the actual droplet contact angle measured in the first few layers above the substrate. Even so, the latter study47 revealed considerable deviations for large Lennard-Jones droplets only on strongly attractive (lyophilic) surfaces with Young contact angle 55°). For water, our comparisons between thermodynamic contact angles for planar geometry2,24 and their microscopic analogue for simulated nanodrops2,38 (size ∼2000 molecules) revealed an agreement within ±5°. Also, when we removed the cutoff in the Coulombic part19 of water−water interactions, simulation at different drop sizes indicates rather low line tension of aqueous nanodroplets τ ∼ 2 × 10−11 N,18,22,51 below experimental value of ∼7 × 10−11 N measured52,53 with micrometer sessile drops. This trend of τ with reducing the drop size is consistent with theoretical expectations.54 Surface prewetting55−57 extending beyond the droplet base is not observed on simulated weakly hydrophilic to hydrophobic substrates studied in this or previous2,19,22,38 related works.

Figure 2. Surfaces with different roughness patterns: hydrophilic patch radius rp = 0 (hydrophobic surface), rp = 30 Å, and rp = ∞ (top row from left to right). Snapshots of water droplet on corresponding surfaces (bottom row).

different patch radii (top) as well as water droplets on these surfaces (bottom). 2.3. Contact Angle Calculation. The microscopic analogue of contact angle of water droplets25,44 has been calculated according to procedures from previous works.18,22,38 We adapted the technique of de Ruijter et al.45 which basically entails fitting the cross section of the droplet to a truncated circle (Figure 3). The equimolar dividing surface of the droplet

3. RESULTS AND DISCUSSION 3.1. Small Patches. We first present the results from contact angle calculations for nanodroplets, situated above a smaller patch of ∼25 Å across, with the background of another surface type. In these cases, the drop’s base is consistently bigger than the patch. Results collected in Table 2 show that, Table 2. Contact Angles of a Drop above a Chemical Heterogeneity

Figure 3. Drop profiles on three different surfaces, hydrophilic (homogeneous surface, patch radius rp → ∞), hydrophobic (rp = 0), and on an intermediate-size patch (rp = 30 Å). Hydrophilic/ hydrophobic heterogeneity reflects nonuniform topological patterns on a chemically homogeneous substrate. Black solid lines are fitted to the simulated data. Dashed line represents the plane, where the contact angles were measured. R is the distance from the main axis of the drop, and the height is measured from the bottom layer.

surface−water interaction

is defined as the surface where the average density of the water drop decreases by 50% from the density of liquid water. The contact angle is calculated from the best circular fit of a line drawn along the equimolar dividing surface of the drop’s density profile. In order to alleviate the impact of possible distortions in the perturbed surface layer,45−47 the procedure fits the droplet’s shape far from the surface, while ignoring the immediate proximity of the substrate.45,48,49 Contact angles determined by this approach, the so-called microscopic analogue of the macroscopic contact angle,25 are known to converge to the thermodynamic (Young) value based on macroscopic interfacial free energies much faster than the real microcontact angle measured at the position of the three-phase contact line.50 The position of the liquid/vapor dividing surface is found by sectioning the water drop into horizontal layers. Each layer is divided into radial bins where the density profile is measured. From this profile the equimolar dividing line is calculated. On chemically heterogeneous but smooth surfaces, contact angle is measured as the angle between the circular fit

θ/deg

under drop’s core

under perimeter

115 115 114 92 91 91 65 68 66

hydrophobic intermediate hydrophilic hydrophobic intermediate hydrophilic hydrophobic intermediate hydrophilic

hydrophobic hydrophobic hydrophobic intermediate intermediate intermediate hydrophilic hydrophilic hydrophilic

for comparatively small surface patches, the macroscopic picture still applies; i.e., the contact angle is controlled exclusively by surface properties near the drop’s perimeter, independently of the surface character under the core of the drop. 3.2. Contact Angle Variation with Patch Size. Figure 4 (top graph) illustrates the dependence of the simulated contact angle of a sessile nanodrop of constant volume on the radius of underlying hydrophilic patch of circular shape. Although the substrate properties change abruptly at the border of the patch, both substrates with chemical and those with topological heterogeneities show a gradual transition from the contact angle characteristic of pure hydrophobic surface when the patch 8637

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→ ∞ (Δγ ∼ −34 mN m−1). The difference in the extent of hydrogen bonding can be attributed to density depletion of water molecules in the interfacial layer next to the corrugated surface. 3.4. Varied Patch Size and Local Cassie−Baxter Equation. To analyze contact angle as a function of the size of the patch requires the knowledge about substrate properties near the drop’s perimeter. Because of the finite range of water/ substrate interactions and perpetual droplet shape fluctuations, the apparent character of the surface affecting the droplet near the three-phase coexistence line changes in a gradual rather than abrupt fashion. This gradual change can be incorporated in the local form of Cassie−Baxter equation, in which we use the average fractions of hydrophilic and hydrophobic coverage, f1 and f 2 = 1 − f1, on the area within the range of interaction from the drop’s perimeter. To estimate f1 and f 2 in the proximity of the perimeter, we calculate the area of the hydrophilic patch within the range of interaction from the drop’s perimeter (Api) and the net area of the range of interaction (Ari) as follows: A ri = π(rd + ri)2 − π(rd − ri)2

(4)

A p i = πrp2 − π(rd − ri)2

(5)

Above, rd is the average radius of the drop’s base (perimeter radius) obtained from the simulation and rp is the radius of the patch. The parameter ri can be regarded as the range of water/ substrate attraction, approximated by a step function potential. Apparent surface composition under the perimeter can then be described by f1 = 1

Figure 4. (a, top) Simulated contact angles of a 2000 molecule nanodrop for topological (blue, circles) and chemical (red, diamonds) heterogeneity vs the radius of the patch, rp. Error bars are within the symbol sizes. Lines are guide to the eye. (b, middle; c, bottom) Predictions of local Cassie−Baxter equation with composition f given by eq 6 for topological (red circles) and chemical (blue circles) heterogeneity. Lines connect simulated points as shown in the top graph. Triangles correspond to predictions of local Cassie−Baxter equation with effective interaction range ri reduced (open triangle up) or increased (open triangle down) by 1 Å to illustrate sensitivity to selected value ri.

f1 =

A pi A ri

f1 = 0

if rp > rd + ri if rd − ri < rp < rd + ri if rp < rd − ri

(6)

f1 is the fractional area of hydrophilic surface overlapping the area of the range of interaction (see Figure 5), and f 2 = 1 − f1 is the fractional area of the hydrophobic surface type. Local values of f1 and f 2 can be substituted into eq 3 to calculate Cassie− Baxter contact angles. In view of eq 6 (see also Figure 5), the width of the transition window in which the contact angle crosses over from the background to patch value is expected to approximately equal the sum 2ri + Δrd. This width corresponds to twice the interaction range ri, increased by the change in the perimeter radius of the drop, Δrd, which accompanies the growth of hydrophilic patch from zero to infinite size. As we show in Figures 4b,c, if we use a plausible empirical range, ri = 5 Å (1.57σ), our generalization of the local Cassie− Baxter equation (local composition f based on eqs 4−6) predicts contact angles (solid circles) in good agreement with the simulation for entire data sets. The above value of ri is within the usual 1.5−2.0σ range of the representative step function potential approximating van der Waals attraction.59 To estimate the sensitivity to the fitted value of ri, Figures 4b,c include results of the local Cassie−Baxter equation for ri increased or reduced by ±1 Å (triangles down and up, respectively). While the good performance of the mean-field model behind the local CB equation is not surprising at macroscopic scale,8,16,60 our results for nanodrops manifest effective cancellation of shape fluctuations consistently

is small (compared to the base of the drop) to the much lower angle corresponding to pure hydrophilic surface for bigger patches. 3.3. Interfacial Hydrogen Bonds and Surface Roughness. According to the geometric definition of hydrogen bond in a simulation,58 a pair of water molecules are considered to share a bond when oxygen−oxygen distance rOO < 3.5 Å, the oxygen−hydrogen distance rOH < 2.45 Å, and the angle between rOO and rOH vectors is less than 30°. We monitored interfacial hydrogen bonds involving at least one molecule within σCO (3.19 Å) from the nearest substrate atom on topologically distinct surfaces for the two extreme cases: pure hydrophilic (100% coverage) and pure hydrophobic (25% coverage in top two graphitic layers of the substrate). We found the number of interfacial hydrogen bonds per molecule 3.1 ± 0.1 and 2.6 ± 0.3, in good agreement with previous results18 and consistent with the difference in observed wetting free energies Δγ = γ cos θ (eq 1) of the two surfaces. The two surface coverages (25% and 100%) correspond to the small and large patch limits in Figure 4, rp = 0 (Δγ ∼ 45 mN m−1) and rp 8638

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Figure 6. Comparison of radius of the perimeter obtained from molecular simulation (Solid circles) and predicted values (solid squares) from eqs 7 and 8 for (a) chemical and (b) topological heterogeneity. Lines are guides to the eye. Error bars are within the symbol sizes.

Figure 5. Parameters used to calculate local form of the Cassie−Baxter equation.

observed in nanodroplet simulations. Importantly, identical values of ri work with chemical and topological heterogeneities. 3.5. Nanodroplet Geometry. Because of their nanoscale dimensions, the average shape of small droplets we consider may deviate from the ideal truncated sphere with contact angle predicted by Young equation (eq 1). The differences between the static and dynamic contact angles, and contact angle hysteresis, important with macroscopic drops,61 are typically insignificant for nanodroplets. Recent systematic analyses62−64 and previous works18 show Young equation is obeyed surprisingly well for O(103) molecule drops, with small deviation observed on very hydrophilic surfaces. Calculations for semi-infinite cylindrical drops65 free of line tension effects66 render contact angles within a few degrees from those from aqueous nanodroplet contact angle calculation. In the present work, we examined the sphericity of nanodrops on varied surfaces. In Figures 6a,b we compare the long-time averages of drops’ perimeter radii from simulations with those predicted presuming ideal truncated sphere shape of the drops. In the latter case, the drop’s geometry follows the equation9 ⎛ 3V ⎞1/3 R c = ⎜ ⎟ , rd = R c sin θ ⎝ πβ ⎠

between geometric and thermodynamic contact angles of droplets in the literature.62,64 3.6. Droplet Stability in an Open System. Our observations together with affirmed reliability2,63,64 of nanoscopic contact angle simulations show that continuum representations provide surprisingly good descriptions of sessile nanodrops. The notion is reinforced by Figures 6a,b showing that predicted values for the radius of the perimeter of the drop agree well with simulated values. These results imply that eqs 7 and 8 also provide reliable predictions for the drop’s curvature radius Rc, which controls the vapor pressure of water in the drop (eq 2). As such, a self-consistent solution of eqs 2−8 yields analytic estimates for the sessile droplet curvature and vapor pressure as a function of drop’s volume in a practically relevant scenario, where the patch size is fixed while the droplet mass can change through condensation or evaporation, depending on the humidity in its surroundings. In Figure 7, we present the solution of these equations for vapor pressure as

(7)

V ∼ V̅ H2ONH2O is the volume of the drop (V̅ H2O ∼ 30 Å ), and β(θ) is a function of contact angle defined as9 3

β(θ ) = 2 − 3 cos θ + cos3 θ = (1 − cos θ )2 (2 + cos θ ) (8)

The predicted overall increase in the perimeter radius, Δrd, was 14.5 and 21.5 Å (simulated values 16 ± 1 and 20 ± 1 Å) for chemical and topological heterogeneities. The good agreement between simulated radii and spherical-drop predictions, shown in Figure 6, indicates negligible droplet distortion when surfaces are predominantly hydrophobic. Increasing the size of the hydrophilic (chemical or topological) heterogeneity results in small deviations of the simulated droplet shape from perfect sphere. This observation is consistent with comparisons

Figure 7. Ratio between vapor pressures of the sessile nanodroplet with N water molecules, P, and bulk vapor pressure, Po. The droplet is located above a circular hydrophilic patch of radius rp = 3 nm and Young contact angle 57° (top solid) or 20° (bottom solid curve), surrounded by hydrophobic surface with Young contact angle 135°. Dashed curves correspond to homogeneous hydrophilic surfaces. All curves converge to unity in the macroscopic limit. 8639

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drop’s perimeter. This holds for both topological and chemical surface heterogeneities. Our results are used in discussing implications of surface heterogeneities on saturated vapor pressure of water above the droplet. In equilibrium, the droplet curvature alone defines the chemical potential inside a drop and hence the saturated vapor pressure, which is therefore indirectly related to the nature of the substrate. The vapor pressure above the drop is generally higher than the equilibrium vapor pressure above the bulk liquid; hence, the drops are generally unstable in the absence of vapor supersaturation. Under supersaturated vapor, droplets on a uniform surface can show two behaviors. Depending on the initial difference between the chemical potential inside the drop and in the vapor, evaporation (shrinking) or condensation (droplet growth) will proceed at constant contact angle. On a patterned surface, however, the chemical potential inside the droplet can vary in a nonmonotonic way. When, due to the droplet growth, or shrinking, the droplet perimeter crosses the border of the underlying hydrophilic patch, the drop’s contact angle, curvature, and water chemical potential will be affected. If, due to condensation, the droplet base outgrows the size of the hydrophilic patch, the curvature and chemical potential will increase, thus slowing and sometimes preventing further droplet growth. In the opposite scenario, the perimeter of a shrinking droplet can move over from a hydropobic region to the area covered by the hydrophilic patch. In response, the curvature will be reduced and the chemical potential will decrease. Again, the droplet shape can reach a metastable equilibrium as chemical potentials in the drop and vapor converge. At appropriate humidity and realistic surface conditions, the depth of the local free energy minimum can significantly exceed the thermal energy, securing nanodroplet survival over experimentally relevant times of observation. This picture is general and applies to all evaporating liquids. For water, analogous considerations can play a role in the surprising longevity of sessile nanobubbles68−70 on submerged interfaces, an isomorphic problem we will address in a separate study.

a function of the droplets size (measured in the number of water molecules N) for a situation, where the drop is located atop a hydrophilic patch of radius 30 Å. When the droplet is small compared to the size of the patch, its radius of curvature, Rc, increases approximately as ∼N1/3, leading to monotonic decrease of vapor pressure (eq 2) with N. When the drop base extends to within the range of interaction ri from the patch border, the drop resists to spread onto the more hydrophobic region, rather increasing its volume by elevating the contact angle and concomitant curvature. As a consequence, the drop’s vapor pressure passes through a minimum and begins to increase with N until reaching the maximum when the drop base ultimately outgrows the patch. In the examples shown in Figure 7 the maximum P(Rc) is reached after more than doubling the droplet’s mass and number of molecules N. The nonmonotonic size dependence of P on the size (N) is qualitatively different from the case of a sessile drop on a homogeneous surface, where P is a monotonically decreasing function of N (dashed curves in Figure 7). While drops on a homogeneous surface are inherently unstable, sessile nanodrops atop a hydrophilic patch, with drop sizes N below the maxima in curves P(N) can reach a metastable equilibrium with supersaturated vapor when the environment vapor pressure falls between the two extrema on the pressure vs size curve. The equilibrium size N will be approached by evaporation, or condensation, and will remain stable because of the locally positive slope ∂P/∂N. Note that the source of the drop’s stability in our open system (fixed μ and variable N) differs from the balance between adhesion and cohesion forces at essentially constant N, highlighted67 in a recent analysis of wetting on a homogeneous mineral. The height of the grand-potential barrier, ΔΩ*, which the droplet has to overcome before outgrowing the patch, depends on the hydrophilicity contrast of the surface pattern. With the barrier height of the order O(N*Δμ*), and Δμ defined in eq 2, the two examples illustrated in Figure 7 support barriers of O(10) kBT and O(102) kBT at the lower and higher patch hydrophilicities, respectively. These estimates suggest that, at high hydrophilicity contrast, and appropriate (supersaturated) humidity, tiny droplets commensurate with nanoscale surface patches can withstand thermal fluctuations over macroscopic observation times.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Present Address †

Department of Physics, Faculty of Science, Vali-e-Asr University of Rafsanjan, Iran.

4. CONCLUDING REMARKS We examine the relation between contact angle of an aqueous nanodrop and surface water interaction energy at the perimeter and beneath the drop. We simulate droplets atop a surface heterogeneity on a patterned substrate with hydrophobic and hydrophilic domains. The microscopic analogue of the contact angle is extracted from simulation trajectory data. We show the contact angle of the nanodrop is exclusively related to the surface interaction energy in the region adjacent to its perimeter. We test the role of finite range of substrate−water interaction when the area of a circular hydrophilic patch beneath the drop’s core is incrementally expanded until the contact angle converges to that on the pure hydrophilic surface. We identify a range of interaction corresponding to a considerable reduction in contact angle when plotting contact angle as a function of patch size. We demonstrate the observed contact angle dependence on the size of the patch can be predicted by the Cassie−Baxter mixing relation, limited to the area within the water/substrate interaction range from the

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Jihang Wang for his initial efforts. This work was supported by the National Science Foundation (CHE0718724) in the early stage and by the U.S. Department of Energy, Office of Basic Science (DE-SC 0004406), in the late stage. The simulations were conducted with support from the XSEDE/National Center for Supercomputing Applications (NCSA) and the National Energy Research Scientific Computing Center (NERSC).

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NOTE ADDED IN PROOF Situations analogous to those discussed in paragraph 3.6 have been observed experimentally.71 We thank Jarek Drelich for bringing his work to our attention.

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