Wetting Behavior of Water near Nonpolar Surfaces - The Journal of

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Wetting Behavior of Water near Nonpolar Surfaces Vaibhaw Kumar and Jeffrey R. Errington* Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260, United States S Supporting Information *

ABSTRACT: We use molecular simulation to study the wetting behavior of water near flat nonpolar surfaces. The interface potential approach is used to capture the evolution of various interfacial properties over a broad range of temperature and substrate strength. Three model substrates are considered: an atomistically detailed face-centered cubic (FCC) lattice, a graphite lattice, and a structureless wall. We first examine the evolution of the contact angle with substrate strength for conditions ranging from near drying to complete wetting at select temperatures. A notable characteristic of all systems is the presence of a surface strength at which the contact angle is independent of temperature. We also identify an analogous point in which the quantity γlv cos θ is temperature invariant. We discuss the relationship between this point and the excess entropy of the solid−liquid interface. We next consider the temperature dependence of interfacial properties. We find the contact angle to decrease with temperature at relatively strong surfaces and increase with the temperature at relatively weak surfaces. The work of adhesion is found to be a useful quantity for describing the interfacial properties of water. The enthalpic and the entropic contributions to the work of adhesion are obtained from its temperature dependence. These properties are found to be directly related to the affinity of water for a solid surface and therefore serve as useful measures of the hydrophobicity of a surface. Finally, we examine the effect of surface strength and temperature on the density depletion associated with water at hydrophobic surfaces. We study various metrics that quantify the density and compressibility of water in the vicinity of hydrophobic surfaces. Comparisons are drawn between the behavior of water and simple nonpolar fluids at solvophobic surfaces.

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simulation techniques developed in our group14−17 to explore the evolution of select interfacial properties over a wide range of conditions that span from the drying to wetting regimes and from ambient to near-critical temperatures. Three model surfaces are considered: graphite, a face-centered cubic (FCC) lattice, and a structureless Lennard-Jones 9−3 substrate. We use these data to explore several issues. We begin by examining the substrate strength dependence of the contact angle at select temperatures. Previous studies focused on nonpolar fluids point to a substrate strength at which the contact angle is approximately independent of temperature.17−20 Interestingly, we find a similar point for each of the three systems considered here. We also observe an analogous temperature independence in the quantity γlv cos θ, where θ is the contact angle and γlv is the liquid−vapor surface tension, which occurs at a relatively high substrate strength. We discuss the ramification for these points in terms of interfacial thermodynamic properties, such as the excess energy and entropy. We also briefly examine the impact of local atomistic substrate structure on interfacial properties. Following the seminal work of Rossky and co-workers,21 several simulation studies have considered metrics related to the local structure of

INTRODUCTION Interfaces involving water are commonplace in our daily experiences, in biological assemblies, and within industrial processes. We regularly find water interacting with solid surfaces, including water droplets falling onto the auto windshield during rain, daily washing activities, and watering plants. Aqueous interfaces play a crucial role in several biological phenomena, such as protein folding and cellular transport.1−4 Industrial processes utilize water as a common operating fluid, and the way water behaves near a solid surface influences important process parameters, such as heat transfer rates5 and hydrodynamic slip.6 Also, knowledge of the interfacial properties of water is important for the design and fabrication of superhydrophobic surfaces. Therefore, there is considerable motivation to better understand how water behaves near a solid surface. It is well-known that the wetting properties of a fluid depend upon (1) the surface chemistry, which dictates the strength of the solid−fluid interaction, and (2) the surface structure, including the nanoscale topography and the atomistic-scale organization of atoms within the substrate. The effect of the surface−fluid interaction strength on interfacial properties, such as the contact angle and solid−liquid interfacial tension, has been examined in several computer simulation studies.6−13 Within many of these studies a narrow band of temperatures and/or substrate strengths is examined. Here, we employ © XXXX American Chemical Society

Received: August 23, 2013 Revised: October 3, 2013

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dx.doi.org/10.1021/jp4084647 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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The local structure of water near a surface is of considerable interest. One property that has garnered significant attention is the so-called depletion layer thickness, which provides a measure of the extent to which water retreats from a hydrophobic surface. This quantity has been studied using analytical theories,30,31 experiments,32−35 and atomistic simulations.36−38 As discussed by Garde and co-workers,29,39 molecular level roughness and the chemistry of real surfaces prevent the depletion layer thickness from becoming a universal characteristic of hydrophobicity. Surprisingly, several outstanding questions related to the origin of depletion layer remain.8,40 Is the depletion layer unique to water, or is it a common aspect of all solvophobic surfaces? How does the depletion layer evolve with temperature? Answers to these questions will help provide a better understanding of the relationship between the local interfacial structure of water and macroscopic observables. In this paper, we present the evolution of the depletion layer thickness with the contact angle for water at select temperatures with the three model surfaces considered here. We also address the question of whether the density deficit is “special” to water by comparing results for water with those for a Lennard-Jones system. The paper is organized as follows. In the following section we describe the models examined in this work and the molecular simulation methods used to compute the interfacial properties of interest. Next, we present our simulation results for the wetting properties of water near three nonpolar surfaces and discuss several issues related to the wetting behavior of water near these surfaces.

water, such as the hydrogen bonded network, molecular orientation, and density, near solid surfaces (see ref 8 and references therein). In contrast, the effect of substrate structure on macroscopic observables, such as contact angle, has been examined in relatively few studies. We find that the substrate strength dependencies of the contact angle for the three surfaces are nearly identical when the substrate strength is appropriately normalized. The most significant deviations are at stronger surface strengths, wherein the local substrate structure plays a greater role. There is considerable interest22 in how the contact angle of water varies with temperature. Exploring this issue via experiment has proven difficult.23 Garcia et al. (experiments and theory),24 Shi and Dhir9 and Dutta et al.25 (nanodroplet molecular dynamics (MD) simulations), and Taherian et al.13 (phantom wall MD simulations) found the contact angle of water to decrease with increasing temperature. On the other hand, Giovambattista et al.26 and Zangi and Berne27 (nanodroplet MD simulations) reported that the contact angle of water at a hydrophobic surface is insensitive to temperature. Recently, utilizing the interface potential approach, we examined the variation in the contact angle with temperature for the Lennard-Jones fluid17,19 and water.16 We observed that the evolution of contact angle with temperature depends on the surface strength. For a relatively strong surface, the contact angle decreases with temperature, while it increases with temperature at a relatively weak surface. In this paper, we further characterize these trends with the three model systems considered here. The contact angle serves as a useful metric to gauge the hydrophobicity of a surface. However, it is not always possible to use the contact angle to measure this affinity. As discussed by Granick and Bae,28 in certain situations, such as water near protein surface or inside nanochannels, it is difficult to establish a contact angle, and one should work with some other suitable metrics. Several interesting studies have now addressed this issue. Garde and co-workers29 indicate that the local density of water is generally a poor indicator of the degree of hydrophobicity of a surface. They suggest that one should instead focus on the density fluctuations near a surface to gauge hydrophobicity. More specifically, they indicate that the probability of observing a cavity near a surface can serve as a useful metric. This probability is found to be relatively large for hydrophobic surfaces and small for hydrophilic surfaces. In another study, Striolo and co-workers6 examined whether hydrodynamic slip provides a macroscopic signature of water− substrate affinity. In this paper, we discuss the role the work of adhesion Wadh can play in describing the wettability of a surface. This quantity is defined as the reversible work done in creating an interface between the solid and liquid phases. Recently, Leroy and coworkers13 showed that Wadh evolves almost linearly with temperature. They used thermodynamic arguments to write Wadh in terms of its enthalpic and entropic components and subsequently used these properties to rationalize the difference in the contact angle of water at graphite and graphene surfaces. Here, we examine the temperature dependence of Wadh at several surface strengths. This analysis provides insight into the prospects for using Wadh to quantify the hydrophobicity of a surface. More specifically, we find that the enthalpic and entropic components of Wadh serve as robust macroscopic signatures of hydrophobicity.

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MOLECULAR SIMULATIONS Molecular Models. We work with a modified version of extended simple point charge (SPC/E) model41 for water introduced by Werder et al.7 For this model, the Lennard-Jones interactions are truncated (simple truncation) at a separation distance of 10 Å, and the electrostatic interactions are described by a shifted-force Coulomb potential42 with a cutoff distance of 10 Å. We study three different model surfaces. The first surface is the structureless Lennard-Jones 9−3 wall, with the energy of interaction usf between the substrate and the oxygen atom of a water molecule separated by a distance z given by ⎡⎛ 2 ⎞⎛ σ ⎞ 9 ⎛ σ ⎞ 3 ⎤ usf (z) = εsf ⎢⎜ ⎟⎜ sf ⎟ − ⎜ sf ⎟ ⎥ ⎝z⎠⎦ ⎣⎝ 15 ⎠⎝ z ⎠

(1)

with σsf = 3.5 Å and variable εsf. The second model consists of an atomistically detailed surface with the atoms occupying the positions of a σ = 0.018 25 Å−3 FCC lattice and the (100) plane exposed to the fluid. Surface atoms interact with the oxygen atom of water molecules according to a truncated and shifted Lennard-Jones 12−6 potential LJ LJ c c ⎧ ⎪ usf (r ) − usf (rsf ) r < rsf usf (r ) = ⎨ ⎪ r > rsfc ⎩0

(2)

with ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ usfLJ(r ) = 4εsf ⎢⎜ sf ⎟ − ⎜ sf ⎟ ⎥ ⎝r ⎠⎦ ⎣⎝ r ⎠

(3)

with σsf = 3.48 Å and variable εsf. The cutoff distance is taken to be rcsf = 15.8 Å. The third model is the graphite system studied B



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by Werder et al.7 Substrate atoms are fixed at their respective positions as described by Werder et al.7 These atoms interact with the oxygen atom of water molecules via a truncated Lennard-Jones 12−6 potential LJ c ⎧ ⎪ usf (r ) r < rsf usf (r ) = ⎨ ⎪ r > rsfc ⎩0

surface models examined in this study. For the FCC model, interfacial simulations are completed with A = 3491 Å2 and H = 100 or 150 Å. Direct spreading and drying mode grand canonical (GC) simulations are performed at εsf = 2.08 and 0.91 kJ/mol, respectively, at T = 500 K. Spreading mode temperature expanded ensemble (TE) simulations focused on the plateau region are completed at εsf = 2.08 kJ/mol. In order to obtain the spreading vapor free energies, direct GC simulations are performed at T = 300 and 400 K. Drying mode TE simulations focused on the plateau and liquid peak are completed at εsf = 0.91 kJ/mol. A set of surface strength expanded ensemble (SE) simulations are performed at T = 300, 400, and 500 K using both the spreading and drying approaches. For the graphite model, interfacial simulations are completed with A = 1170.6 Å2 and H = 120 Å. Direct spreading and drying mode GC simulations are performed at εsf = 0.57 and 0.25 kJ/ mol, respectively, at T = 500 K. Spreading and drying mode TE simulations are completed at εsf = 0.25 kJ/mol. We also perform spreading and drying mode SE simulations at T = 300, 400, and 500 K. For the 9−3 model, interfacial simulations are completed with A = 1600 Å2 and H = 100 Å. Direct spreading and drying mode GC simulations are performed at εsf = 16.63 kJ/mol and εsf = 3.33 kJ/mol, respectively, at T = 500 K. Spreading and/or drying mode TE simulations are completed at εsf = 0.831, 1.66, 2.49, 3.33, 4.16, 4.99, and 5.82 kJ/mol. We also perform spreading and drying mode SE simulations at T = 300, 400, and 500 K. Statistical uncertainties are determined by performing four independent sets of simulations. The standard deviations of the results from the four simulation sets are taken as an estimate of the statistical uncertainty.

(4)

with σsf = 3.19 Å and variable εsf. The cutoff distance is taken to be rcsf = 10 Å. For each of the two atomistically detailed surfaces considered, the depth of the substrate extends beyond the range of the surface−fluid interaction (defined by rcsf). We refer to the three models described above as the “9−3”, “FCC”, and “graphite” models. For each model, we explore a range of εsf over which the system evolves from near drying to complete wetting conditions along a given isotherm. Within the discussion that follows below, we compare behaviors exhibited by the water systems to those of a simple nonpolar fluid in the vicinity of a nonpolar surface. The simple system is modeled as a truncated Lennard-Jones fluid that interacts with a structureless surface via the Lennard-Jones 9−3 potential provided in eq 1. The manner in which various interfacial properties of this system evolve with temperature and substrate strength is detailed in a previous publication from our group.17 Within the earlier report we referred to this model as the “homogeneous” system. Here, we use these data to calculate various properties relevant to the discussion below. Methods. In this study, we employ the interface potential approach43−46 to compute the interfacial properties of water at nonpolar surfaces. Detailed information related to this approach is available in previous papers from our group.14−17 In a recent study,16 we outline the application of this method to the 9−3 water model described above. The approach is based upon the interface potential, which provides the surface excess free energy associated with the growth of a fluid film near a surface. We work with two variants of the interface potential. The first is the spreading potential, which focuses on the growth of a liquid film near a surface in a mother vapor phase.43−46 The qualitative and quantitative features of the spreading potential under different wetting scenarios have been discussed in detail by us14−17,19,47−52 and others.53,54 Recently, we developed another variant of the interface potential approach, known as the drying method.17 The drying potential focuses on the growth of a vapor film near a surface in a mother liquid phase. The spreading potential is better suited for systems characterized by moderate to strong surface−fluid interaction strength, while the drying potential is more relevant for systems with weak to moderate strength substrates. Both potentials are appropriate for deducing the wetting properties of systems within the partial wetting regime. We couple the interface potential approach with various expanded ensemble (EE) schemes, which allow us to capture the evolution of a system’s free energy over a path of interest. For example, we have employed EE techniques to understand how components of the interface potential evolve with temperature,16,17,19,49 surface strength,16,17,19 and surface topology.19 Collectively, these schemes allow one to obtain the evolution of important interfacial properties, such as the spreading coefficient, drying coefficient, contact angle, and liquid−vapor surface tension, over a wide range of conditions. Simulation Details. We detailed the bulk saturation properties of the water model studied here in a previous report.16 Here we provide the simulation details for the three

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RESULTS AND DISCUSSION Figure 1 provides the evolution of the contact angle with surface strength at select temperatures for the three model systems. First, we note the good agreement between our results for the graphite system and those from the study of Werder et al.7 Within Figure 1, we provide the results Werder et al.7 obtain by extrapolating cos θ data acquired from simulations of nanodroplets of various sizes to the infinite-system limit. For example, with εsf = 0.3135 kJ/mol at T = 300 K, they obtain cos θ = −0.431, −0.329, and −0.304 for nanodroplets of size N = 1000, 4000, and 17 576, respectively, resulting in an infinitesystem value of cos θ = −0.237. Using the interface potential approach, we obtain cos θ = −0.22 (5) at these conditions. It is encouraging to see that the interface potential and nanodroplet techniques agree in this limit. We now consider the general features of the curves provided in Figure 1. As is expected,7−9 cos θ monotonically increases with increasing substrate strength. The approach employed here14−17,19 provides pseudocontinuous curves for cos θ (εsf), which allows one to appreciate how the shape of these curves evolves with temperature and substrate strength. The results indicate that cos θ is near-linear in εsf for relatively strong surfaces, while exhibiting noticeable curvature for relatively weak surfaces. It is also clear that this trend becomes more pronounced with increasing temperature. The near-linear behavior of cos θ for relatively high εsf can be justified using mean-field arguments, where cos θ can be shown to be linearly dependent on the interaction energy.8 In contrast, for surfaces characterized by low εsf, these arguments do not remain valid. C



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Figure 1. Substrate strength dependence of cos θ. Data for the FCC, graphite, and “9−3” wall models are provided in the top, middle, and bottom panels, respectively. Red, blue, and green curves represent temperatures of T = 300, 400, and 500 K, respectively. Maroon circles in the middle panel correspond to data from the study of Werder et al.7 Uncertainties are provided at select conditions.

Figure 2. Substrate strength dependence of γlv cos θ. The curves are formatted in the same manner as for Figure 1.

γlv cos θ = γsv − γsl with εsf. The intersection point for γlv cos θ (εsf) is located at a strong surface strength εγinter relative to lv cos θ that for cos θ (εsf). In fact, at T = 500 K, the surface strengths identified are above the wetting point for the three model surfaces. Interfacial thermodynamics reveals that d(γsv − γsl)/ dT = 0 at the substrate strength at which the γlv cos θ (εsf) isotherms meet. At relatively low T, γsv ≈ 0. More specifically, we obtain γsv = −1.0 × 10−3, −0.04, and −0.4 mN/m at T = 300, 400, and 500 K, respectively, for the 9−3 system with εsf = 4.16 kJ/mol and the dividing surface located at z = 0. For comparison, we note that γlv = 54.5, 35.1, and 15.3 mN/m at these respective temperatures. Therefore, the intersection point in γlv cos θ (εsf) corresponds to the surface strength at which dγsl/dT is zero. Hence, one can relate this point to the εsf at which the excess entropy (excess with respect to the bulk fluid) of the solid−liquid interface Sex sl = −dγsl/dT changes sign. Based upon the temperature dependence of the data provided in inter Figure 2, it is clear that Sex sl is negative for εsf exceeding εγlv cos θ, ex while for lower εsf, Ssl is positive. From a physical perspective, the decrease in excess entropy with increasing substrate strength is related to the increase in the structural order of water molecules in the vicinity of strong surfaces. As the substrate strength increases, interfacial water is progressively more constrained (e.g., fluid layering, organization of the hydrogen bond network), resulting in a decrease in the excess entropy. Finally, we note that the intersection of γlv cos θ (εsf) isotherms does not appear to be a common aspect of all systems. These isotherms meet over a relatively broad range of εsf for the Lennard-Jones system17 (see Figure S1 in the Supporting Information). We now consider the effect of the atomistic-level substrate structure on the contact angle of water. Figure 3 provides cos θ for the three model surfaces as a function of scaled surface

Entropic effects and enhanced density fluctuations near the hydrophobic surface render analysis of the relation between cos θ and εsf difficult.8 The near-linear behavior of cos θ (εsf) near the wetting point indicates that the systems exhibit first-order wetting transitions, while near the drying point, enhanced curvature in cos θ (εsf) signals that the systems exhibit continuous drying transitions.55,56 The shape of the interface potentials (not shown) that we collect for these systems supports this point. Spreading interface potentials (see Figure 1 of ref 17 for an example from a Lennard-Jones system) show a free energy barrier associated with the creation of a thick liquid film at a solid surface, whereas the analogous drying interface potential (see Figure 1 of ref 16) does not show such a barrier. One of the more striking features of the cos θ (εsf) curves is the manner in which isotherms appear to intersect at a common point. This intersection point is observed for all of the model surfaces. At this surface strength εinter cos θ, the contact angle is temperature invariant. Above this surface strength, the contact angle decreases with increasing temperature, while below this surface strength it shows the opposite trend. The variation in cos θ with temperature is rather pronounced for stronger surfaces and is relatively weak for lower substrate strengths. This behavior is not unique to water, as a similar intersection point in cos θ (εsf) isotherms has been reported for the Lennard-Jones fluid.17,19 The cos θ (εsf) isotherms for the water and Lennard-Jones systems intersect at approximately the same value of cos θ ≈ −0.5. The intersection points for cos θ (εsf) isotherms encouraged us to look for similar trends in other interfacial properties. Interestingly, we find an analogous intersection point in the quantity γlv cos θ. Within Figure 2 we present the evolution of D



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Figure 4. Temperature dependence of the work of adhesion Wadh for the 9−3 water model. The curves from bottom to top correspond to surface strengths of εsf = 0.831, 1.66, 2.49, 3.33, 4.16, 4.99, and 5.82 kJ/mol. Uncertainties are provided at select conditions.

reported by Taherian et al.13 after studying water at graphite and graphene model surfaces over a relatively narrow temperature range. We previously examined the temperature dependence of cos θ for this model system16 and found this quantity to be a nonlinear function of temperature. The fact that Wadh shows linear temperature dependence over a wide range of temperature and substrate strength can be utilized by researchers to estimate cos θ at a given temperature via linear interpolation/extrapolation of Wadh. The Lennard-Jones system shows similar trends for Wadh (see Figure S2 in the Supporting Information). The higher precision of the Wadh(T) data for the Lennard-Jones system enables us to detect a slight degree of systematic curvature, with the high-εsf curves concave and the low-εsf curves convex. The work of adhesion can be written in terms of energetic and entropic contributions, with Wadh = ΔUadh − TΔSadh. Hence, the slope and the intercept of Wadh(T) are related to the entropy ΔSadh and the energy ΔUadh associated with the adhesion process, respectively.13,58 Formally, the energy term depends on the working ensemble. In the isothermal−isobaric ensemble, the energy term corresponds to ΔUadh + PΔVadh, while in the grand canonical ensemble it corresponds to ΔUadh + μΔΓadh, where ΔVadh and ΔΓadh are changes in the volume and the adsorption during the adhesion process, respectively. For systems in the partial wetting regime, one can assume these to be negligible relative to ΔUadh.13,58 From a thermodynamics ex ex perspective, one has ΔXadh = (Xex sv + Xlv ) − Xsl , where X is S or 13,58 U. If we again take the contribution of the solid−vapor ex interface to be negligible, then ΔXadh ≈ Xex lv − Xsl ; i.e., the change in a property due to the adhesion process is given by the difference in the excess property associated with the liquid− vapor and solid−liquid interfaces. Therefore, at sufficiently low temperature, ΔXadh isotherms convey the evolution of Xex sl with a given parameter of interest. In this work, we aim to capture the evolution of ΔSadh and ΔUadh with substrate strength at relatively low temperature. We estimate these quantities by performing linear curve fits with Wadh(T) data over the temperature range T = 250−400 K. Figure 5 provides the evolution of TΔSadh, ΔUadh, and Wadh with surface strength at T = 300 K for the 9−3 system. All of

Figure 3. Scaled substrate strength dependence of cos θ. Data for T = 300, 400, and 500 K are provided in the bottom, middle, and top panels, respectively. Red, blue, and green curves represent the 9−3, FCC, and graphite models, respectively. Uncertainties are provided at select conditions.

strength at temperatures of T = 300, 400, and 500 K. More specifically, the surface strengths are scaled by the surface strength εosf at which cos θ = 0. This analysis is motivated by a recent paper57 in which Machlin reports that the contact angle of water at nonpolar surfaces is almost independent of the substrate structure. We notice a similar trend here. At a given temperature, cos θ curves for the three surfaces approximately collapse to a common curve. The agreement is more apparent at relatively weak surface strengths in comparison to stronger surface strengths. This trend can be justified by again considering the manner in which water organizes at the surface. For relatively strong surfaces, the density profile of interfacial water is highly structured, indicating that water molecules occupy preferential positions relative to the location of the substrate atoms. In contrast, for weaker surfaces, water retreats from the interface, resulting in a diffuse density profile. It follows that interfacial properties are more likely to be influenced by the atomistic-scale contours of the surface at high substrate strength. Here, we see this effect reflected in the enhanced differentiation of cos θ curves at high εsf. Finally, we note that chemical and physical surface heterogeneities are known to have a significant impact on the wetting properties of a system.39 The surfaces studied here are relatively uniform. We suspect that systems with heterogeneous surfaces will show deviations from the trend captured in Figure 3. We now focus our attention toward the effect of temperature on the interfacial properties of water. Figure 4 provides the evolution of the work of adhesion Wadh = γlv(1 + cos θ) with temperature at select surface strengths for the 9−3 system. Note that the substrate strength dependence of this quantity is provided in Figure 2, as one simply needs to shift each isotherm by γlv to obtain Wadh. We observe Wadh to be a near-linear function of temperature, with the slope of the curves decreasing as the surface strength decreases. A similar observation was E



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consistent with those displayed by the water system. At relatively low temperature, we find ΔSadh to be a slightly concave function of substrate strength and ΔUadh and Wadh to be convex functions. An example of these curves at a dimensionless temperature of T = 0.65 is provided in Figure S3 of the Supporting Information. The results for these two systems suggest that the observed qualitative behaviors for the excess thermodynamic properties associated with a solid−liquid interface may be general in nature. Godawat et al.29 have examined the excess chemical potential associated with inserting a hard sphere particle in the solid− liquid interfacial region. They show that this quantity increases with cos θ in a near-linear manner, thereby providing a robust microscopic measure of hydrophobicity. They associate this metric with the probability of observing a particle-sized cavity in the interfacial region. One can also think of this quantity as the work required to create a cavity within this region. As the size of the test particle becomes very large, one expects this microscopically based metric to behave in the same manner as the macroscopically based work of adhesion Wadh. As noted above, the latter quantity is defined as the work required to insert a thick vapor film at a solid−liquid interface. At low temperature, the low-density saturated vapor is negligibly different from a cavity. It follows that one can consider the excess entropy and excess energy associated with inserting a cavity in the solid−liquid interfacial region to be microscopic versions of ΔSadh and ΔUadh, respectively. The results generated here suggest that the excess chemical potential, entropy, and energy of cavity formation could all serve as useful microscopic measures of hydrophobicity. This point has also been appreciated by Taherian et al.13,58 That being said, the excess entropy may prove most useful, as the shape of ΔSadh (εsf) suggests that this quantity changes relatively rapidly upon variation of the surface strength at low εsf. We now examine select metrics that describe fluid density in the vicinity of the solid−liquid interface. These metrics are computed from data collected during substrate strength expanded ensemble (SE) simulations.16,17,19 More specifically, within the “drying” interface potential approach17,51 that we employ, we consider the properties of a liquid confined within a nonperiodic rectangular parallelepiped simulation box constrained at one end by the substrate of interest (e.g., graphite surface) and at the opposing end by a “sticky wall”.17,51 Within a grand canonical SE simulation, the system visits subensembles differentiated by substrate strength (for the substrate of interest

Figure 5. Surface strength dependence of adhesion properties for the 9−3 water model at T = 300 K. The energy ΔUadh, entropy ΔSadh, and work Wadh of adhesion are represented by blue circles, red triangles, and green squares, respectively. Solid lines result from second-order polynomial curve fits applied to the data.

these metrics are positive quantities that increase monotonically with surface strength. The energetic term ΔUadh is near-linear, with mildly convex character, which is not surprising, as one can establish ΔUadh ∝ εsf for conditions away from the drying point from mean-field arguments.8 In contrast, ΔSadh is a slightly concave function, with this quantity changing relatively rapidly at low εsf. It follows that Wadh is convex. Taherian et al. recently studied the surface−fluid contribution to adhesion properties.58 Interestingly, they find that this contribution exhibits the same qualitative features as the total (surface−fluid and fluid−fluid contributions) adhesion properties examined here. We find that ΔSadh and ΔUadh approach zero as εsf approaches zero. This result implies that as the surface strength decreases; i.e., as the surface becomes more hydrophobic, the water−solid interface behaves much like a liquid−vapor interface. This result is not new, as it is well-known that the extended hydrophobic interface resembles a liquid−vapor interface.59 The analysis performed here indicates that the quantities ΔSadh, ΔUadh, and Wadh are macroscopic manifestations of the manner in which water behaves near extended hydrophobic surfaces and can serve as quantitative measures of hydrophobicity. An analysis of data for the Lennard-Jones system reveals that ΔSadh, ΔUadh, and Wadh exhibit trends that are qualitatively

Figure 6. Evolution of the joint probability distribution P(N;εsf) with particle number and surface strength for the fluid near “9−3” wall. Plots from left to right represent T = 500, 400, and 300 K. F



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only) with the temperature and activity set to bulk saturation conditions. Given that the substrate strength only changes along this path, such simulations allow one to track how various thermodynamic and structural properties evolve with this surface characteristic. We first consider the particle number probability distribution at select substrate strengths P(N,εsf). It is this distribution that provides information related to the interfacial density and the intensity of interfacial density fluctuations. This surface is obtained by normalizing the particle number probability distributions collected within each subensemble. Examples of this surface for the 9−3 system at temperatures of T = 300, 400, and 500 K are provided in Figure 6. Two trends clearly emerge. First, as εsf decreases, the peak position in P(N;εsf) decreases to smaller values of N, indicating that the fluid is progressively depleted from the interfacial region as εsf decreases. Second, as εsf becomes small, the width of the P(N;εsf) distributions increases, indicating that density fluctuations increase as the surface becomes relatively hydrophobic. Finally, we note that the εsf at which the P(N;εsf) distributions noticeably broaden appears to shift to higher εsf with increasing temperature. The peak positions within P(N;εsf) are directly related to the excess adsorption at the solid−liquid interface Γsl. From grand canonical simulation data, one can evaluate this quantity from the ensemble average of the particle number ⟨N⟩ or the density profile ρ(z) Γsl =

1 (⟨N ⟩ − ρl V ) = A

∫0

Figure 7. Evolution of solid−liquid excess adsorption Γsl and depletion thickness dl with cos θ. Data for T = 300, 400, and 500 K are provided in the bottom, middle, and top panels, respectively. Red, blue, and green curves represent the 9−3, FCC, and graphite models, respectively. Uncertainties are provided at select conditions.

∞

[ρ(z) − ρl ] dz

(5)

where ρl is the bulk saturated liquid density and V is the volume of the simulation box. Note that for the simulations completed here, one must remove contributions from the sticky wall when employing the ensemble average approach, a correction that is relatively straightforward to evaluate.14 Enumerating the volume V requires one to (arbitrarily) specify the location of the dividing surface that separates the solid and liquid phase. The location of this surface also establishes the origin of the integration scale in eq 5. Here, the dividing surface is located using the “Boltzmann factor criteria”, which provides the effective hard-core length scale of a specified interaction.60−62 To determine the surface−fluid hard-core diameter σHS sf , we employ a procedure similar to that discussed in ref 14. Although σHS sf strictly varies with T and εsf, we observe relatively little change in the diameter with these parameters. Therefore, HS we use a single value of σHS sf for a given system, with σsf = 2.5861, 2.6315, and 2.8164 for the FCC, 9−3, and graphite systems, respectively. Finally, we note that we have computed Γsl using both definitions provided in eq 5 and have obtained consistent results. Before examining the temperature and substrate strength dependence of the excess adsorption, we note that this quantity is related to the so-called depletion layer thickness dl, another metric commonly used to quantify the extent to which water retreats from a hydrophobic surface. This distance is obtained from the density profile at the solid−liquid interface: dl =

∫0

∞⎡

⎢1 − ⎢⎣

ρ (z ) ⎤ ⎥ dz ρl ⎥⎦

In all cases, Γsl monotonically decreases with decreasing cos θ, signaling that the fluid progressively retreats from the surface as the substrate strength decreases. The variation in Γsl with substrate strength is relatively modest at moderate values of cos θ. From a quantitative perspective, dl is smaller than one molecular diameter at T = 300 K for all conditions examined. Interestingly, there is relatively little change in fluid adsorption upon variation of the microscopic details of the substrate, as is reflected by similar values of Γsl for the three model systems. These observations are consistent with the results from room temperature experimental studies, which point to depletion layer thicknesses less than one molecular diameter for water near large hydrophobic surfaces.40 Finally, we note that modifying the temperature impacts the response of the fluid adsorption to substrate strength, with the variation in Γsl with cos θ increasing with increasing temperature. For example, as cos θ shifts from 0.2 to −0.8, the depletion layer thickness increases by approximately 3.5 Å at 500 K and just 1.5 Å at 300 K. The behavior of the adsorption in the vicinity of the drying point is rather interesting. From a practical perspective, these conditions are not experimentally realizable, as the largest contact angle reported for water on a flat homogeneous surface at room temperature is 120°.63 Nonetheless, the results are of general interest from a theoretical perspective and may prove useful in understanding the manner in which water organizes near geometrically rough nanostructured substrates, where contact angles approaching 180° have been observed.63 As a system approaches the drying point, Γsl decreases rapidly. For systems associated with continuous drying transitions, Γsl diverges as the drying point is approached.64 The response of Γsl to variation in cos θ presented in Figure 7 is consistent with this expectation. In terms of the drying interface potential (see,

(6)

Comparison of eqs 5 and 6 reveals Γsl = −ρldl. Figure 7 provides the substrate strength dependence of Γsl and dl for the three model systems at temperatures of T = 300, 400, and 500 K. We use cos θ to describe the substrate strength. G



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dividing surface. The metric is defined such that χsl > 0 when the interfacial density fluctuations exceed those for a system containing a bulk saturated liquid. Figure 8 provides the substrate strength dependence of χsl for the three model systems at temperatures of T = 300, 400, and

for example, Figure 1 of ref 16), this divergence can be understood by considering how the free energy basin associated with a vapor film in contact with the surface evolves with substrate strength. As a system approaches the drying point, this basin shifts to a location characteristic of thicker vapor films, becomes shallower, and broadens. This progression of the underlying free energy surface is consistent with increased depletion layer thicknesses and an increase in the excess solid− liquid entropy as the surface weakens. The presence of a depletion layer for systems consisting of a nonpolar fluid at a solvophobic surface has been reported in earlier studies.36−38 Here, we compare the depletion layers associated with water and the simple Lennard-Jones system. Figure S4 of the Supporting Information provides depletion layer thickness data for the LJ system at reduced temperatures T/Tc = 0.505, 0.673, and 0.841 as well as for the water at 9−3 wall system studied here at T/Tc = 0.5, 0.667, and 0.833. We observe similar trends in the depletion layer thickness for both systems. The evolution of dl with both substrate strength and temperature is qualitatively consistent. The results indicate that the emergence of a depletion layer for a fluid at a solvophobic surface is not unique to associating fluids like water. The compressibility of water in the vicinity of a hydrophobic surface has gathered considerable attention. Giovambattista et al. report that water confined between hydrophobic plates shows enhanced density fluctuations relative to the bulk, leading them to characterize hydrophobic interfaces as relatively “soft”.65 The Garde group has explored the relationship between interfacial compressibility and hydrophobicity for a wide range of systems.29,39,66,67 For example, they have examined the compressibility of the solvation shell surrounding hydrophobic solutes,66 density fluctuations in the vicinity of self-assembled monolayers,29 and the compressibility of water in the vicinity of proteins.67 In all cases, they find a strong link between microscopic measures of interfacial compressibility and hydrophobicity. Here, we consider how a macroscopic measure for interfacial compressibility varies with temperature and substrate strength. More specifically, we examine the metric

Figure 8. Evolution of compressibility χ with surface strength. Data for T = 300, 400, and 500 K are provided in the bottom, middle, and top panels, respectively. Red, blue, and green circles represent the 9−3, FCC, and graphite models, respectively. Uncertainties are provided at select conditions.

500 K. Note that when considering the substrate strength dependence of χsl at a given temperature, a change in the bulk properties (e.g., ρl, kT) leads to a constant shift in the χsl curve. Within Figure 8 we include uncertainties related to ⟨(δN)2⟩ only. Analogous data for the Lennard-Jones system at the three reduced temperatures noted above are provided in Figure S5 of the Supporting Information. While the trend is difficult to detect in some cases, it appears that χsl increases slightly with decreasing εsf for moderate to strong surfaces. In contrast, at the higher temperatures of T = 400 and 500 K, χsl clearly increases appreciably with decreasing εsf for cos θ ≲ −0.5. The poor precision of the T = 300 K data makes it difficult to draw any definitive conclusions at this temperature. The results for χsl are consistent with the P(N;εsf) distributions displayed in Figure 6, for which one does not observe an appreciable broadening of the particle number distribution until the substrate becomes relatively weak. We again find qualitatively similar trends for the Lennard-Jones and water systems (Supporting Information). Collectively, the results suggest that the compressibility serves as a sensitive metric for gauging the hydrophobic nature of an interface for relatively weak substrates only. This finding is perhaps a bit surprising, as microscopically based measures of compressibility examined by the Garde group39,67 appear to be effective measures of hydrophobicity over a broader range of substrate strength. That

⎛ ∂Γ ⎞ 1 ⎛ ∂ 2Ωex ⎞ χ =− ⎜ =⎜ ⎟ 2 ⎟ A ⎝ ∂μ ⎠T , V , A ⎝ ∂μ ⎠T , V , A 1 = [β(δN )2 − ρl2 VκT] A

(7)

where Ωex is the excess grand potential, μ is the chemical potential, δN = N − ⟨N⟩, and kT is the bulk isothermal compressibility. The property is related to fluctuations in the interfacial adsorption and therefore is generally applicable to a wide range of systems. We compute χ via analysis of the P(N;εsf) distributions collected within the expanded ensemble simulations described above. Recall that these calculations are completed with a simulation box in which the fluid is confined at one end by the substrate of interest and by a “sticky wall” at the other end. To eliminate the contribution of the sticky wall, we perform an independent simulation with a system containing sticky walls at both ends of the box and apply eq 7 to obtain χtotal = 2χsticky, where χsticky is the contribution associated with a single sticky wall. We then apply eq 7 with data obtained from the expanded ensemble simulation to calculate χtotal = χsl + χsticky for each εsf examined, where χsl is the contribution from the substrate of interest. As is the case for Γsl and dl, the value of χsl is dependent upon the location of the H



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Institute Computational Center for Nanotechnology Innovations.

being said, the compressibility changes abruptly with variation in the strength of the underlying surface−fluid interaction strength for very solvophobic surfaces. Such a metric may prove useful in the analysis of geometrically rough solvophobic surfaces for which cos θ < −0.5 conditions are relatively common.

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CONCLUSIONS We have used molecular simulation to determine how various interfacial properties evolve with temperature and substrate strength for systems containing water in the vicinity of nonpolar flat surfaces. Three molecular models were utilized, including an atomistically detailed surface with the FCC (100) plane exposed to the fluid, a graphite surface, and a structureless 9−3 wall. We first examined the effect of surface interaction strength and surface topology on the contact angle of water. We found that the properties cos θ and γlv cos θ show temperature invariance at relatively low and high substrate strength, respectively. We also found the substrate strength dependence of the contact angle of water to be relatively insensitive to the atomistic-level structure of the surface, particularly for substrates of weak to moderate strength. We next considered the temperature dependence of various interfacial properties of water. We found that the contact angle decreases with temperature for relatively strong surfaces and for relatively weak surfaces the contact angle increases with temperature. The work of adhesion proved to be a near-linear function of temperature over a broad range of substrate strength. We determined the entropic and the enthalpic components of the work of adhesion and found that these properties provide robust measures of the degree of hydrophobicity of a surface. We also investigated the density and compressibility of interfacial water. In general, we find that both of these metrics are correlated with substrate strength, with the sensitivity of the metrics to variation in substrate strength enhanced significantly for very hydrophobic interfaces. Interestingly, we found that the depletion layer thickness and interfacial compressibility associated with water and the Lennard-Jones fluid near a solvophobic surface show similar qualitative trends. A natural extension of this work is to investigate the interfacial behavior of water near polar surfaces. We plan to study such systems in the near future.

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ASSOCIATED CONTENT

* Supporting Information S

Information related to the wetting behavior of a Lennard-Jones system. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: jerring@buffalo.edu (J.R.E.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support of the National Science Foundation (CHE-1012356). Computational resources were provided in part by the University at Buffalo Center for Computational Research and the Rensselaer Polytechnic I



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