pubs.acs.org/Langmuir © 2010 American Chemical Society
Molecular Simulation Study of Anisotropic Wetting Eric M. Grzelak,† Vincent K. Shen,‡ and Jeffrey R. Errington*,† †
Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200, and ‡Chemical and Biochemical Reference Data Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8320 Received December 12, 2009. Revised Manuscript Received February 3, 2010 We study anisotropic wetting in systems governed by Lennard-Jones interactions. Molecular simulation is used to obtain the macroscopic contact angle a fluid adopts on face-centered-, body-centered-, and simple-cubic lattices with the (100), (110), or (111) face in contact with the fluid. Several amorphous substrates are also examined. Substrates are modeled as a static collection of particles. For a given set of calculations, the atomistic density of the substrate and the particle-particle interactions (surface-fluid and fluid-fluid) remain fixed. These constraints enable us to focus on the extent to which substrate structure influences the contact angle. Three substrate-fluid interaction strengths are considered, which provide wetting conditions that span from near-dry to near-wet. Our results indicate that the manner in which particles are organized within the substrate significantly influences the contact angle. For strong substrates (near-wet case), a change in the substrate structure can change the cosine of the contact angle by as much as 0.5. We also examine how well certain structural and energetic features of the substrate-fluid system serve as suitable metrics for predicting the variation of the contact angle with substrate topography. Three parameters are considered: the density of atoms within the crystalline plane closest to the fluid, a measure of the effective strength of the substrate-fluid interaction, and the roughness of the solid-liquid interface. The effective strength of the substrate potential shows the strongest correlation with the contact angle. This energy-based parameter is defined in a general manner and therefore could serve as a useful tool for describing the anisotropic wetting of solids. In contrast, the metrics based on planar density and interface roughness are found to correlate with contact angle data relatively weakly.
I. Introduction Within numerous industrial applications and natural phenomena, a fluid is brought into contact with a crystalline solid. In such cases, the molecular-level topography and underlying atomistic organization that the solid presents to the fluid influences the wetting characteristics the system displays. Given that the various faces of a crystalline solid possess unique topographies, one should generally expect interfacial properties to change upon variation of the crystalline face. Indeed, experimental studies indicate that wetting anisotropy is observed for a wide range of *To whom correspondence should be addressed. E-mail: jerring@ buffalo.edu. (1) Nikolopoulos, P.; Agathopoulos, S.; Angelopoulos, G. N.; Naoumidis, A.; Grubmeier, H. Wettability and Interfacial Energies in Sic-Liquid Metal Systems. J. Mater. Sci. 1992, 27 (1), 139-145. (2) Heng, J. Y. Y.; Bismarck, A.; Lee, A. F.; Wilson, K.; Williams, D. R. Anisotropic Surface Energetics and Wettability of Macroscopic Form I Paracetamol Crystals. Langmuir 2006, 22 (6), 2760-2769. (3) Heng, J. Y. Y.; Bismarck, A.; Lee, A. F.; Wilson, K.; Williams, D. R. Anisotropic Surface Chemistry of Aspirin Crystals. J. Pharm. Sci. 2007, 96 (8), 2134-2144. (4) Heng, J. Y. Y.; Bismarck, A.; Williams, D. R. Anisotropic Surface Chemistry of Crystalline Pharmaceutical Solids. AAPS PharmSciTech 2006, 7 (4), 84. (5) Naidich, Y. V.; Grigorenko, N. F.; Perevertailo, V. M. Interphase and Capillary Phenomena in Crystal Growth and Melting Processes. J. Cryst. Growth 1981, 53 (2), 261-272. (6) Chatain, D. Anisotropy of Wetting. Annu. Rev. Mater. Res. 2008, 38, 45-70. (7) Chatain, D.; Metois, J. J. A New Procedure for the Determination of the Free-Energies of Solid Fluid Interfaces from the Anisotropy of Wetting of a Melt on Its Solid. Surf. Sci. 1993, 291 (1-2), 1-13. (8) Rao, G.; Zhang, D. B.; Wynblatt, P. A Determination of Interfacial Energy and Interfacial Composition in Cu-Pb and Cu-Pb-X Alloys by Solid-State Wetting Measurements. Acta Metall. Mater. 1993, 41, (11), 3331-3340. (9) Shen, P.; Fujii, H.; Nogi, K. Effect of Substrate Crystallographic Orientation on Wettability and Adhesion in Several Representative Systems. J. Mater. Process. Technol. 2004, 155-56, 1256-1260. (10) Shi, Z.; Lowekamp, J. B.; Wynblatt, P. Energy of the Pb{111} Parallel to Al{111} Interface. Metall. Mater. Trans. A 2002, 33 (4), 1003-1007.
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crystalline solids.1-15 While the existence of wetting anisotropy has been recognized for some time, our ability to predict the evolution of interfacial properties upon variation of the substrate geometry remains lacking. In this work, we use molecular simulation to determine the wetting properties of several crystalfluid systems and subsequently examine the extent to which various characteristics of the substrate and fluid can be used to describe the observed trends. Our study is motivated in part by the important role that the anisotropic wetting of crystalline solids plays in the development of numerous products and technologies. Pharmaceutical crystals, such as aspirin, are known to display substantially different behaviors when the crystalline face exposed to the fluid is changed.2-4 As a result, these therapeutics are often closely examined to determine the relative stability, adhesion, and release rates associated with different faces of the crystal. The wetting of a ceramic by a molten metal, which is featured prominently within several applications, including the fabrication of catalysts, electronic packaging, and composites, is also known to be sensitive to the crystal face in contact with the fluid. A number of studies focused on the wetting of R-alumina by liquid metals such as (11) Shi, Z.; Wynblatt, P. A Study of the Pb/Al(100) Interfacial Energy. Metall. Mater. Trans. A 2002, 33 (8), 2569-2572. (12) Nogi, K.; Okada, Y.; Ogino, K.; Iwamoto, N. Wettability of Diamond by Liquid Pure Metals. Mater. Trans., JIM 1994, 35 (3), 156-160. (13) Dezellus, O.; Eustathopoulos, N. The Role of van der Waals Interactions on Wetting and Adhesion in Metal Carbon Systems. Scr. Mater. 1999, 40 (11), 1283-1288. (14) Shen, P.; Fujii, H.; Matsumoto, T.; Nogi, K. The Influence of Surface Structure on Wetting of R-Al2O3 by Aluminum in a Reduced Atmosphere. Acta Mater. 2003, 51 (16), 4897-4906. (15) Shen, P.; Fujii, H.; Matsumoto, T.; Nogi, K. Surface Orientation and Wetting Phenomena in Si/R-Alumina System at 1723 K. J. Am. Ceram. Soc. 2005, 88 (4), 912-917.
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aluminum, gold, and copper have demonstrated the significant influence anisotropy has on the droplet contact angle.14,15 Previous studies addressing anisotropic wetting have focused almost exclusively on molecular crystals. For these systems, there are two effects that contribute to the change in wetting behavior upon variation of the crystalline face: (1) surface chemistry, associated with variation in the population of atom or functional group identities within the crystalline face, and (2) surface structure, including the topography of the face and the manner in which the underlying substrate atoms are organized. In general, these two effects are difficult to decouple. Those that have probed the relative contributions generally conclude that surface chemistry plays the dominant role in establishing the wetting properties of a system.2-5,9 In this work, we focus on the extent to which substrate structure influences the interfacial properties of a system. We isolate this effect by restricting our attention to monatomic substrates. The anisotropic wetting of monatomic crystals has been examined in relatively few experimental studies.5,8-10,12,13,16 These investigations have revealed that the identity of the crystal face in contact with the fluid can have a substantial impact on the interfacial properties. For example, Naidich et al. studied the anisotropic wetting of germanium in contact with its own melt.5 For the conditions examined, the fluid adopted contact angles of 30°, 17°, and 9° at the (111), (110), and (100) faces of an fcc crystal, respectively. While attempts have been made to identify a substrate characteristic that provides a means to predict the change in contact angle upon variation of the crystalline face, no metric has proven to be a robust indicator of the shift in interfacial properties. Some have suggested9 that the contact angle correlates well with the density of the first layer of the crystalline face in contact with the fluid, with higher first-layer densities associated with lower contact angles. However, a number of counterexamples have been identified (including the germanium case mentioned above) that weaken this claim. Finally, we note that in a number of the experimental studies6 the authors cited the possibility of influences other than substrate structure (e.g., surface reactions, impurities, etc.) that could have contributed significantly to the reported contact angles. Molecular simulation is an attractive tool to probe the influence of substrate structure on wetting behavior. Within a computer simulation, one has complete control over the structural characteristics of the substrate and of the molecular-level interactions within a system. By adopting such an approach, one can rigorously relate microscopic information regarding a system to macroscopically observed properties. Here, we use molecular simulation to determine the wetting properties of a fluid at several model surfaces that differ only in the manner in which the substrate atoms are organized. More specifically, we look at the interfacial properties of an atomistic fluid governed by LennardJones interactions in contact with the (111), (110), and (100) faces of face-centered-cubic (fcc), body-centered-cubic (bcc), and simple-cubic (sc) lattices. Several amorphous surfaces are also considered. We consider three substrate-fluid interaction strengths that produce fcc (100) contact angles of 38°, 91°, and 132°. These substrate strengths were selected to produce wetting conditions broadly distributed over the partial wetting regime. In what follows, we refer to these conditions as strong (or near-wet), moderate, and weak (or near-dry), respectively. Overall, we find that the arrangement of the substrate particles (16) Shi, W.; Zhao, X. C.; Johnson, J. K. Phase Transitions of Adsorbed Fluids Computed from Multiple-Histogram Reweighting. Mol. Phys. 2002, 100 (13), 2139-2150.
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has a significant influence on the contact angle the fluid adopts. After examining the interfacial properties of the model surfaces, we consider the ability of certain metrics to describe the variation in contact angle with substrate structure. We first consider the efficacy of the atomic density of the first crystalline layer. As mentioned above, this quantity has been suggested as a predictive quantity with respect to wetting anisotropy. We find that this metric provides a useful tool for relatively weak substrates only. We then look at a parameter that provides a measure of the effective strength of the substrate-fluid interaction. The topography of a substrate places restrictions on the manner in which fluid particles organize near its surface. As a result, some substrates facilitate a stronger net interaction with the fluid than others. Given that the contact angle is known to be sensitive to the strength of the substrate potential, it is reasonable to suspect that a quantity that provides a measure of the effective potential will correlate well with the contact angle. Below we show that such an energy-based metric provides a reasonable description of the variation in contact angle with substrate structure. Inspired by the Wenzel model, we also consider the predictive ability of the roughness of the substrate-fluid interface. The Wenzel model indicates that the cosine of the contact angle on a structured surface scales with a so-called roughness factor, which is defined as the ratio of the true interfacial area of the solid-liquid region to that of a simple planar projection of the interfacial region. Within the context of the current study, each face of the various crystalline lattices considered presents a unique topography to the fluid. If the ideas behind the Wenzel model persist down to the atomistic scale, then the roughness should serve as a suitable metric for describing the variation in the contact angle with substrate structure. To complete this analysis, we first introduce two techniques for quantifying the substrate roughness. The first of these approaches utilizes a Voronoi tessellation of particles in the vicinity of the solid-liquid interface, and the second relies upon an analysis of the potential energy field exerted by the substrate on a fluid particle. Our results suggest that there is relatively little connection between the observed contact angle and these two measures of surface roughness. The analysis described above requires a computational approach to determine the macroscopic contact angle with a relatively high degree of precision. We have found free-energybased simulation techniques to be particularly useful in this regard.17-19 Specifically, we employ an approach in which one obtains the spreading coefficient via measurement of the surface excess free energy as a function of surface density. The qualitative features that this curve adopts under different wetting scenarios, as well as the quantitative link between it and various interfacial properties, have been described in a number of theoretical studies.20-23 In this work, we utilize grand canonical transition (17) Grzelak, E. M.; Errington, J. R. Computation of Interfacial Properties via Grand Canonical Transition Matrix Monte Carlo Simulation. J. Chem. Phys. 2008, 128 (1), 014710. (18) MacDowell, L. G.; Muller, M. Observation of Autophobic Dewetting on Polymer Brushes from Computer Simulation. J. Phys.: Condens. Matter 2005, 17 (45), S3523-S3528. (19) MacDowell, L. G.; Muller, M. Adsorption of Polymers on a Brush: Tuning the Order of the Wetting Phase Transition. J. Chem. Phys. 2006, 124 (8), 084907. (20) Degennes, P. G. Wetting - Statics and Dynamics. Rev. Mod. Phys. 1985, 57 (3), 827-863. (21) Dietrich, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1988; Vol. 12, p 1. (22) Indekeu, J. O. Line Tension near the Wetting Transition - Results from an Interface Displacement Model. Phys. A 1992, 183 (4), 439-461. (23) Indekeu, J. O. Line Tension at Wetting. Int. J. Mod. Phys. B 1994, 8 (3), 309-345.
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matrix Monte Carlo simulation17,24-27 to obtain the aforementioned free energy curve and associated spreading coefficient for the substrate-fluid systems outlined above. The paper is organized as follows. In the following section, we describe the model examined in this work. We then introduce the simulation tools employed here and provide details related to our calculations. Next, we present our simulation results for the contact angle and discuss the ability of select metrics to describe the trends observed. Finally, the salient conclusions are reviewed.
II. Model System We examine the properties of model systems consisting of a monatomic fluid that interacts with itself and substrate particles according to a truncated and shifted Lennard-Jones potential. The energy of interaction uij between any two particles separated by a distance r is given by ( uij ðrÞ ¼
LJ c uLJ for r < rcij ij rij -uij rij 0 for rgrcij
ð1Þ
with
Figure 1. Particle number probability distribution collected with the fcc (100) substrate.
III. Computational Approach " 6 # σij 12 σij LJ uij ðrÞ ¼ 4εij r r
ð2Þ
where εij and σij are energy and size parameters, respectively. The fluid-fluid (ff) and substrate-fluid (sf) interactions are cut off at distances of rcff =2.5σff and rcsf =5.0σff. The characteristic length scale of the sf interaction relative to that for the ff interaction is set at σsf =1.1σff. This particle size ratio is consistent with a system containing fluid argon on solid xenon.28 We work with substrate strengths of εsf=0.25εff, 0.45εff, and 0.65εff, which produce neardry (weak), moderate, and near-wet (strong) wetting scenarios, respectively. In what follows, all quantities are made dimensionless using εff and σff as characteristic energy and length scales, respectively. Each substrate considered consists of a static collection of particles at density Fs=0.58. For the crystalline systems examined, the specific configuration of atoms depends on both the choice of lattice and the Miller index of the face exposed to the fluid. We consider fcc, bcc, and sc lattices with the (100), (110), or (111) face directed toward the fluid. This selection yields nine surfaces. In addition to the crystalline substrates, three amorphous configurations are examined. These substrates are constructed by first performing a bulk canonical simulation of the LennardJones fluid at a density of Fs =0.58 and temperature of T=0.9. At these conditions, the system remains in a liquid state. During the simulation, equilibrated configurations are stored at regular intervals. Three of these configurations are randomly selected with one of the faces of each simulation cell taken as the surface. (24) Errington, J. R. Direct Calculation of Liquid-Vapor Phase Equilibria from Transition Matrix Monte Carlo Simulation. J. Chem. Phys. 2003, 118 (22), 9915-9925. (25) Errington, J. R. Prewetting Transitions for a Model Argon on Solid Carbon Dioxide System. Langmuir 2004, 20 (9), 3798-3804. (26) Sellers, M. S.; Errington, J. R. Influence of Substrate Strength on Wetting Behavior. J. Phys. Chem. C 2008, 112 (33), 12905-12913. (27) Errington, J. R. Evaluating Surface Tension Using Grand-Canonical Transition-Matrix Monte Carlo Simulation and Finite-Size Scaling. Phys. Rev. E 2003, 67 (1), 012102. (28) Tang, J. Z.; Harris, J. G. Fluid Wetting on Molecularly Rough Surfaces. J. Chem. Phys. 1995, 103 (18), 8201-8208.
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Contact Angle Measurement. We employ the “one adsorbing surface” technique outlined in ref 17 to compute the contact angle. Using this approach, one obtains the spreading coefficient via calculation of the surface excess free energy as a function of surface density at conditions that correspond to bulk liquid-vapor saturation. From a statistical mechanics perspective, this free energy curve is directly related to the probability of observing a specified surface density at the bulk saturation activity for the temperature of interest. Given that the surface density is trivially related to the fluid particle number N within a grand canonical simulation, it follows that the spreading coefficient can be extracted from the particle number probability distribution Π(N). An example of such a curve for one of the systems studied here is presented in Figure 1. For partial wetting conditions, a vapor peak emerges at low particle number followed by a plateau region at relatively high particle number. The free energy of the vapor peak is consistent with a system containing a solid-vapor interface, while the free energy of the plateau region is associated with a system containing a solid-liquid and liquid-vapor interface. The spreading coefficient s is defined as the difference between these two free energies, and therefore, one can extract this quantity directly from a Π(N) distribution as illustrated in Figure 1, βsA ¼ ln ΠðNplateau Þ -ln ΠðNvapor Þ
ð3Þ
where β is the inverse temperature and A is the interfacial area of the substrate within the simulation box. The contact angle θ follows from the relation cosðθÞ ¼ 1 þ s=γlv ð4Þ where γlv is the liquid-vapor interfacial tension. Grand canonical transition matrix Monte Carlo simulation (GC-TMMC) is used to compute the necessary interfacial particle number probability distributions. As mentioned above, these calculations are completed at conditions that correspond to bulk liquid-vapor saturation. Therefore, we first use a bulk version of the GC-TMMC method24 to determine the coexistence activity at the temperature of interest and subsequently utilize this value within interfacial simulations. Evaluation of the contact angle via eq 4 also requires knowledge of the surface tension γlv. We again use an independent simulation to obtain this quantity. Specifically, we employ an expanded ensemble area sampling scheme29 to compute γlv. (29) Errington, J. R.; Kofke, D. A. Calculation of Surface Tension via Area Sampling. J. Chem. Phys. 2007, 127 (17), 174709.
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Simulation Details. All calculations were performed at a temperature of T = 0.7. Bulk GC-TMMC simulations were conducted in a cubic cell with a volume of V=512. Other aspects of the simulations are similar to those described in ref 24. The key quantity extracted from these simulations was the particle number probability distribution Π(N ) for the bulk system. Histogram reweighting techniques30 were used to shift this distribution to an activity ξ=Λ-3 exp(βμ), where μ is the chemical potential and Λ is the thermal de Broglie wavelength, that satisfied the conditions for phase coexistence. This procedure provided a saturated activity of ξ=6.5953 10-3 at T=0.7. The liquid-vapor surface tension was calculated using an expanded ensemble areasampling approach with details identical to those described in ref 29. With this technique, we found γ1v=0.584(22). Interfacial GC-TMMC simulations were performed in a rectangular parallelepiped box with periodic boundary conditions applied in the x and y directions. The system was closed at the two ends of the nonperiodic z direction by an atomistically defined substrate and hard wall of cross-sectional area A separated by a distance of H = 80. The lateral dimensions of the substrate depended upon the crystalline lattice and face selected. More specifically, the box dimensions were adjusted to a multiple of the natural periodicity of the lattice for the substrate density specified above. The cross-sectional area for each substrate was within 5% of A=115, with the box length in a given direction no smaller than L = 10.18. We found that sampling a particle number range between 0 and 1500 adequately characterized the surface free energy of the system. Our previous work17 suggests that finite-size effects are small for systems of the type examined here. For crystalline lattices, substrate atoms were placed such that the first layer of particle centers sat at z = 0 and all subsequent layers adopted positions with z < 0. For the amorphous substrates, we first constructed a surface defined by βΦ(z;xi,yj) = 100, where Φ(z;xi,yj) is the z-dependent potential energy of a test particle at fixed (xi,yj). The substrate was then shifted such that the lowest point in the repulsive contour sat at z=0. This allowed for fluid particles to penetrate into the repulsive wall while always occupying positions with z > 0. We also used the aforementioned surface to ensure that all fluid particles occupied positions “above” the surface during grand canonical simulations. In other words, we prevented nonphysical situations in which fluid particles occupy holes in the amorphous substrate. Finally, we note that the location of the dividing surface has no impact on the contact angle.17 In principle, substrates can be placed at any elevation within the simulation box. Statistical uncertainties were determined by performing four independent sets of simulations. The standard deviation of the results from the four simulation sets was taken as an estimate of the statistical uncertainty.
IV. Results and Discussion Contact Angle. An illustrative example of a particle number probability distribution obtained for a system within the partial wetting regime is provided in Figure 1. The local minimum in this curve between the vapor peak and liquid-slab plateau region indicates that a free energy barrier separates these two regions. The presence of this barrier signifies first-order wetting behavior.20-23 With an increase in substrate strength, the system tends toward the wetting transition. At this point, the values of Π(N) corresponding to the vapor peak and liquid-slab plateau are equal (s = 0), and a local minimum in Π(N) separates the two regions. At conditions slightly beyond the wetting point (complete wetting regime), the value of Π(N) within the liquid-slab plateau region exceeds that of the vapor peak. Again, a free energy barrier separates the two regions, and therefore, one can differentiate the (30) Ferrenberg, A. M.; Swendsen, R. H. New Monte-Carlo Technique for Studying Phase-Transitions. Phys. Rev. Lett. 1988, 61 (23), 2635-2638.
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Article Table 1. Values for the Cosine of the Contact Angle for Strong, Moderate, and Weak Substratesa substrate
strong
moderate
weak
amorph 1 0.263(4) -0.579(4) amorph 2 1.106(4) 0.232(3) -0.592(4) amorph 3 1.103(4) 0.203(4) -0.607(5) bcc 100 0.687(2) -0.086(3) -0.694(2) bcc 110 0.801(5) 0.021(2) -0.638(3) bcc 111 0.785(3) -0.031(4) -0.698(2) fcc 100 0.788(4) -0.010(3) -0.671(3) fcc 110 0.797(3) -0.031(3) -0.702(2) fcc 111 0.835(3) 0.041(2) -0.626(3) sc 100 1.048(3) 0.158(3) -0.587(3) sc 110 0.868(3) 0.018(3) -0.640(5) sc 111 0.578(2) -0.140(2) -0.703(2) a The uncertainties indicated do not account for the uncertainty in the liquid-vapor interfacial tension.
vapor peak and plateau region. This construction allows us to deduce a spreading coefficient (and the associated contact angle) at conditions slightly into the complete wetting regime. In this case, the positive spreading coefficient indicates a metastable wetting scenario. The shape for the Π(N ) curve just described also serves as a marker for the existence of a prewetting transition, which one expects to find over a limited range of conditions within the complete wetting regime for systems that exhibit first-order wetting.20-23 In fact, the Π(N) curve can be used to locate directly the prewetting saturation point.25,26 At conditions far enough removed from the wetting point, the free energy barrier separating the vapor peak and liquid-slab plateau vanishes and spreading coefficients can no longer be obtained. Finally, we note that the nature of the wetting transition (first-order versus continuous) depends sensitively on a number of factors, including the temperature and the range of the substrate-fluid and fluid-fluid interactions.31,32 Table 1 and Figure 2 provide the cosine of the contact angle for each of the substrates considered here. The cases for which cos(θ) > 1 correspond to the metastable wetting scenarios described immediately above. The data presented in Figure 2 clearly indicate that the manner in which the substrate atoms are organized influences the contact angle a fluid adopts. Among the crystalline systems, the sc lattice exhibits the strongest anisotropy, with the bcc and fcc lattices showing a smaller variation of contact angles. For the highest substrate-fluid interaction strength, the spreads in cos(θ) for the sc, bcc, and fcc lattices are 0.47, 0.11, and 0.05, respectively. Based on these observations, one could argue that the role substrate structure plays in wetting anisotropy is just as significant as surface chemistry. Our calculations also indicate that the effect of surface topography diminishes as the strength of the substrate is reduced. As is discussed below, this trend can be rationalized by considering the general form of the relationship between substrate strength and cos(θ). Our results indicate that the rank order of the contact angle with substrate identity remains relatively constant with substrate strength. While there are a few exceptions (e.g., bcc (100) and bcc (111) switch rank with substrate strength), the position of each substrate generally remains fixed. This observation suggests that one may be able to identify a single structural characteristic of the substrate that provides a qualitative guide for the evolution of the contact angle with substrate identity. However, it is clear that structural information alone will not provide one with the ability (31) Parry, A. O.; Rascon, C. The Trouble with Critical Wetting. J. Low Temp. Phys. 2009, 157 (3-4), 149-173. (32) Gatica, S. M.; Cole, M. W. To Wet or Not to Wet: That Is the Question. J. Low Temp. Phys. 2009, 157 (3-4), 111-136.
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Figure 2. Evolution of the cosine of the contact angle with substrate identity and strength of interaction. The three panels from top to bottom correspond to the strong, moderate, and weak interaction strengths. The amorphous, bcc, fcc, and sc substrates span from left to right. A legend for the face identity is provided at the top of the figure.
to make quantitative predictions, as the magnitude of the change in contact angle with substrate identity varies with substrate strength. In what follows below, we examine the extent to which various metrics correlate with the contact angle. Planar Density. The density of the first layer of atoms in contact with the fluid has been suggested9 as a potential metric for describing the variation in contact angle with substrate identity. Specifically, it is expected that the contact angle will decrease with increasing first-layer density. The physical rationale for this assertion can be understood by considering the manner in which the effective strength of the substrate varies with first-layer density. As the fraction of substrate particles within close proximity of the fluid is increased, one generally finds that the overall substrate-fluid interaction becomes stronger. Since it is wellestablished that the cosine of the contact angle increases with increasing substrate strength,17,26 one would generally expect cos(θ) to increase with increasing first-layer density as well. Finally, we note that the distance between the first and second layers of substrate atoms within a crystal has also been mentioned as a potentially useful correlating parameter.9 In the case of perfect crystalline lattices, one can show that the density of the first layer of substrate atoms is linearly related to the distance between the first and second layers. Therefore, we simply focus on the first-layer density here. Figure 3 provides the relationship between contact angle and first-layer density Ffirst for the three substrate strengths examined here. We observe a positive correlation between the two quantities; an increase in Ffirst generally leads to an increase in cos(θ). The quality of the correlation improves as the strength of the substrate decreases. To quantitatively probe this issue, we performed a linear fit of the data for each of the substrate strengths and subsequently examined the correlation coefficient R2. As expected, the correlation coefficient increases from R2 =0.35 to 0.77 upon variation of the substrate strength from the strong to weak case. It appears that a gross structural characteristic of the substrate, the first-layer density, provides a reasonable correlation for the weak case but fails upon increasing the substrate strength. 8278 DOI: 10.1021/la9046897
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Figure 3. Relationship between cosine of the contact angle and density of the first plane of substrate particles in contact with the fluid. Amorphous substrates are not included in this analysis. Symbol notation is the same as that used in Figures 2 and 4. Thin horizontal lines separate the strong, moderate, and weak cases.
This analysis suggests that the local fluid structure plays a greater role for stronger surfaces. This notion is supported by considering the relative extent to which the microscopic topography of strong and weak surfaces influences fluid structure. Markers of fluid organization, such as the z-dependent density profile, indicate that the fluid is more likely to preferentially occupy select positions when in the vicinity of a stronger substrate (e.g., stronger peaks in the density profile). Given that weak surfaces provide relatively little influence on the neighboring fluid, an aggregate substrate property is more likely to succeed as a predictive tool for the contact angle. Substrate Strength. We now turn our attention to a more direct measure of the substrate-fluid interaction strength. Given that the relationship between contact angle and substrate strength is well-established, we anticipate that a metric that provides a reasonable description of the effective strength of the substratefluid interaction will serve as a useful predictive tool for the contact angle. Such a parameter should incorporate the atomatom interaction potential between substrate and fluid particles as well as information regarding the topography of the surface. One can imagine a number of metrics that would satisfy this criterion. In this work, we consider the average energy Eeff over a surface defined by minima in the aggregate substrate-fluid interaction potential, Eeff ¼ -
Nj Ni X 1 X Φmin ðxi , yj Þ Ni Nj i ¼1 j ¼1
ð5Þ
where the average is computed over a Ni Nj grid of lateral positions and Φmin(xi,yj) is the minimum value of the aggregate substrate-particle potential energy at fixed (xi,yj). To identify such a surface, we first construct a uniform grid of lateral positions with a spacing of Δx = Δy = 0.01. For a given grid point (xi,yj), we minimize the substrate-particle potential energy with respect to z. These minima are averaged to yield an effective substrate strength. Figure 4 shows the relationship between contact angle and effective substrate strength. In general, we observe a decent Langmuir 2010, 26(11), 8274–8281
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Within the geometric approach, we estimate the interfacial area from information gathered from a Voronoi tessellation of atomistic configurations containing a liquid slab in contact with the substrate. For a given configuration of N particles, the Voronoi tessellation, or the dual of the Delaunay tessellation, tiles space, dividing it into regions vi such that all points within it are closer to particle i than any other particle. That is, dðr, ri Þ < dðr, rj Þ for 1 < j ¼ 6 i
