pubs.acs.org/Langmuir © 2010 American Chemical Society
Rationalization of the Behavior of Solid-Liquid Surface Free Energy of Water in Cassie and Wenzel Wetting States on Rugged Solid Surfaces at the Nanometer Scale Frederic Leroy* and Florian M€uller-Plathe Eduard-Zintl-Institut f€ ur Anorganische und Physikalische Chemie and Center of Smart Interfaces, Technische Universit€ at Darmstadt, Petersenstrasse 22, 64287 Darmstadt, Germany Received October 5, 2010. Revised Manuscript Received November 29, 2010 The present work aims to contribute to the understanding at a molecular level of the origin of the hydrophobic nature of surfaces exhibiting roughness at the nanometer scale. Graphite-based smooth and model surfaces whose roughness dimension stretches from a few angstroms to a few nanometers were used in order to generate Cassie and Wenzel wetting states of water. The corresponding solid-liquid surface free energies were computed by means of molecular dynamics simulations. The solid-liquid surface free energy of water-smooth graphite was found to be -12.7 ( 3.3 mJ/m2, which is in reasonable agreement with a value estimated from experiments and fully consistent with the features of the employed model. All the rugged surfaces yielded higher surface free energy. In both Cassie and Wenzel states, the maximum variation of the surface free energy with respect to the smooth surface was observed to represent up to 50% of the water model surface tension. The solid-liquid surface free energy of Cassie states could be well predicted from the Cassie-Baxter equation where the surface free energies replace contact angles. The origin of the hydrophobic nature of surfaces yielding Cassie states was therefore found to be the reduction of the number of interactions between water and the solid surface where atomic defects were implemented. Wenzel’s theory was found to fail to predict even qualitatively the variation of the solid-liquid surface free energy with respect to the roughness pattern. While graphite was found to be slightly hydrophilic, Wenzel states were found to be dominated by an unfavorable effect that overcame the favorable enthalpic effect induced by the implementation of roughness. From the quantitative point of view, the solid-liquid surface free energy of Wenzel states was found to vary linearly with the roughness contour length.
1. Introduction Recent works have shown that experimental techniques have progressed in such a way that surfaces can now be prepared with hierarchical implementation of roughness from the nanoscale to the microscale, yielding surfaces of extreme hydrophobic character.1-3 This observation partly relies on the fact that surfaces exhibiting nanoscopic roughness have particular properties. For instance, Joly et al. have experimentally shown that nanoscale roughness of hydrophobic surfaces strongly modifies surface slippage properties.4 In a theoretical study illustrated by experimental results, Wong and Ho showed that surfaces with nanoscopic patterns may possess the ability to behave like ultrahydrophobic materials.5 Current trends in the preparation of surfaces which allow typical roughness dimension to be reduced down to the nanometer scale are therefore expected to produce materials which amplify the wetting properties of materials that already exhibit micrometer scale roughness. In that context, the characterization at the nanometer scale of interfaces where water interacts with solid surfaces has become an important topic of investigation. The nanometer length domain is naturally accessible to molecular simulation methods. These methods have yielded studies where the water structure and dynamics close to solid surfaces *Corresponding author. E-mail:
[email protected].
(1) Kwon, Y.; Patankar, N.; Choi, J.; Lee, J. Langmuir 2009, 25, 6129. (2) Xia, D. Y.; He, X.; Jiang, Y. B.; Lopez, G. P.; Brueck, S. R. J. Langmuir 2010, 26, 2700. (3) Cho, K. L.; Liaw, I. I.; Wu, A. H. F.; Lamb, R. N. J. Phys. Chem. C 2010, 114, 11228. (4) Joly, L.; Ybert, C.; Bocquet, L. Phys. Rev. Lett. 2006, 96, 046101. (5) Wong, T. S.; Ho, C. M. Langmuir 2009, 25, 12851.
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have been reported. Properties such as water mass density profiles or the individual behavior of water molecules such as their orientation at the interface have been characterized.6-24 Some simulation studies even attempted to establish a direct connection between the shape of nanosized liquid droplets and some microscopic interfacial properties of fluids.7,8,18,25 For example, Giovambattista et al. have found in a molecular dynamics (MD) study that the contact angle of a water droplet on a homogeneous surface of variable polarity was directly related to the water molecules’ orientation in the immediate vicinity of the surface.18 (6) Li, L. W.; Bedrov, D.; Smith, G. D. J. Phys. Chem. B 2006, 110, 10509. (7) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 15181. (8) Trudeau, T. G.; Jena, K. C.; Hore, D. K. J. Phys. Chem. C 2009, 113, 20002. (9) Gordillo, M. C.; Martı´ , J. J. Phys. Chem. B 2010, 114, 4583. (10) Park, J. H.; Aluru, N. R. J. Phys. Chem. C 2010, 114, 2595. (11) Lee, C. Y.; McCammon, J. A.; Rossky, P. J. J. Chem. Phys. 1984, 80, 4448. (12) Gordillo, M. C.; Martı´ , J. J. Chem. Phys. 2002, 117, 3425. (13) Grigera, J. R.; Kalko, S. G.; Fischbarg, J. Langmuir 1996, 12, 154. (14) Pal, S.; Weiss, H.; Keller, H.; M€uller-Plathe, F. Phys. Chem. Chem. Phys. 2005, 7, 3191. (15) Pal, S.; M€uller-Plathe, F. J. Phys. Chem. B 2005, 109, 6405. (16) Janecek, J.; Netz, R. R. Langmuir 2007, 23, 8417. (17) Sendner, C.; Horinek, D.; Bocquet, L.; Netz, R. R. Langmuir 2009, 25, 10768. (18) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. J. Phys. Chem. B 2007, 111, 9581. (19) Godawat, R.; Jamadagni, S. N.; Garde, S. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 15119. (20) Gordillo, M. C.; Martı´ , J. Phys. Rev. B 2008, 78, 075432. (21) Hua, L.; Zangi, R.; Berne, B. J. J. Phys. Chem. C 2009, 113, 5244. (22) Mittal, J.; Hummer, G. Faraday Discuss. 2010, 146, 341. (23) Wang, J. W.; Kalinichev, A. G.; Kirkpatrick, R. J. J. Phys. Chem. C 2009, 113, 11077. (24) Argyris, D.; Cole, D. R.; Striolo, A. J. Phys. Chem. C 2009, 113, 19591. (25) Ohler, B.; Langel, W. J. Phys. Chem. C 2009, 113, 10189.
Published on Web 12/13/2010
DOI: 10.1021/la104018k
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As for rugged surfaces, Daub et al. predicted in a MD simulation study also that nanoscopic roughness would yield drastic changes in the surface wetting properties.26 It is shown in that work that when roughness is implemented at the scale of a few atomic diameters, both static and dynamic behavior of water droplets on different rugged substrates are strongly affected by minute changes in the surfaces topography. When we put these simulation results in the perspective of the experimental facts described above, it appears that molecular simulations have an important role to play in the characterization of interfacial wetting properties of water on surfaces with roughness organized in nanoscopic patterns. The aim of the present work is to contribute to the understanding at the molecular scale of the origin of the hydrophobic nature of nanostructured surfaces. In the context of static wetting, the contact angle of a sessile droplet on a partially wet surface is often employed as an easy experimental way to characterize the interaction strength between water and solid substrates. Certain theories aim to predict the contact angle behavior of a sessile droplet on chemically heterogeneous or rugged surfaces: Cassie and Baxter,27 Israelachvili and Gee,28 and Wenzel29,30 derived equations in order to connect the actual contact angle of droplets on heterogeneous surfaces to its value on smooth surfaces using factors quantifying the surface roughness or the chemical heterogeneity. Cassie-Baxter and Wenzel equations will be recalled later in the text. For now, it can be noted that these two equations were derived disregarding the ratio between the different length scales existing in the system, while Israelachvili and Gee proposed an equation in order to better describe the situation where the scale of the heterogeneity domain approaches that of molecular dimensions. The aim of Israelachvili and Gee was to take into account the influence of molecular properties such as the molecular polarizability or dipole moment. It must be noted that the theories quoted above have been derived on the basis of interfacial energetic arguments. Such an approach suggests that the shape of a droplet on a heterogeneous surface is driven by the interfacial surface free energies of the system. However, a recent discussion in connection with this topic has emphasized the importance of the different length scales in the system.31-37 When rugged and/or chemically heterogeneous surfaces are considered, it has been shown that if the wavelength of the surface heterogeneity is much smaller than the length-scale of the contact line, the droplet shape actually depends on the interfacial free energies in the system. On the other hand, when the droplet size and the surface heterogeneities occur on comparable length scales, the actual droplet shape is driven by the local equilibrium condition at the three-phase contact line which can be expressed as a force balance, while the global equilibrium condition determines the most stable apparent contact angle.32 Moreover, the roughness pattern topography can also play an (26) Daub, C. D.; Wang, J.; Kudesia, S.; Bratko, D.; Luzar, A. Faraday Discuss. 2010, 146, 67. (27) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (28) Israelachvili, J. N.; Gee, M. L. Langmuir 1989, 5, 288. (29) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (30) Wenzel, R. N. J. Phys. Colloid Chem. 1949, 53, 1466. (31) Gao, L. C.; McCarthy, T. J. Langmuir 2007, 23, 3762. (32) Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277. (33) McHale, G. Langmuir 2007, 23, 8200. (34) Panchagnula, M. V.; Vedantam, S. Langmuir 2007, 23, 13242. (35) Nosonovsky, M. Langmuir 2007, 23, 9919. (36) Gao, L. C.; McCarthy, T. J. Langmuir 2009, 25, 14105. (37) Bormashenko, E. Langmuir 2009, 25, 10451. (38) Halverson, J. D.; Maldarelli, C.; Couzis, A.; Koplik, J. Soft Matter 2010, 6, 1297. (39) Kusumaatmaja, H.; Vrancken, R. J.; Bastiaansen, C. W. M.; Yeomans, J. M. Langmuir 2008, 24, 7299.
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important role and yield the droplet to have an asymmetric shape.38,39 When simulations deal with nanosized droplets, it is moreover important to note that the droplet radius has a noticeable influence on the contact angle.40 In fact, the line tension should also be taken into account.41 It should also be noted that the definition of the contact angle of such small droplets may be fuzzy because their shape fluctuations are large and the density distribution at the contact line may not be homogeneous. Instead, quantities like the surface free energies or surface tensions have a clear definition independent of the system size. Before we recall the equation of Cassie and Baxter and that of Wenzel in detail, it must be stressed that these equations are applicable only when the surface heterogeneity dimension and the droplet size have dissimilar length scales. Cassie and Baxter derived an equation in order to predict the contact angle θCB of a macroscopic droplet on a heterogeneous surface made of two components. It reads cos θCB ¼ fA cos θA þ fB cos θB
ð1Þ
The parameters fA and fB in eq 1 are the surface fraction of component A and B while θA and θB are the intrinsic contact angles on smooth surfaces A and B, respectively. If the surface is rugged, it is assumed that the liquid wets the surface in such a way that it does not penetrate the roughness pattern. There exist then macroscopic liquid-air interfaces in the system and eq 1 takes the form cos θCB ¼ f cos θ þ ð1 - f Þ
ð2Þ
Note that the constraint fA þ fB = 1 was used and fA = f. Equations 1 and 2 show that a given surface can be rendered less wettable when chemical or roughness heterogeneity is implemented. Wenzel attempted to compute the contact angle θW of a droplet on a rugged surface from its value θ on a smooth one. His equation takes the form cos θW ¼ r cos θ
ð3Þ
The parameter r in eq 3 is the ratio between the total contact area between the liquid and the solid and the projected solid surface area. Therefore, r is always larger than 1. Equation 3 predicts that either the hydrophobic or hydrophilic character of a smooth surface is amplified by making it rugged. It is assumed in Wenzel equation that the liquid completely wets the surface roughness pattern. Let us now consider a millimeter-sized droplet resting on a rugged surface exhibiting a pattern of roughness at the micrometer scale. Furthermore, those elements of micrometer size are characterized by nanometer scale roughness. Since the droplet size always differs from the roughness length scale by several orders of magnitude, both Cassie-Baxter and Wenzel equations are expected to hold. Furthermore, independent contact angle equations can be employed to describe the wetting character of each scale of roughness. In more detail, the equations can first be used to describe the hypothetical contact angle of a droplet on the nanometer rugged surface. The result can next be implemented in the equation corresponding to the micrometer level. Finally, it appears that a nanorugged surface which is hydrophobic may amplify the hydrophobic character of a surface having millimeter scale roughness. Note that both θCB in eq 1 and θW in eq 3 depend (40) Werder, T.; Walther, J. H.; Jaffe, R. L.; Halicioglu, T.; Koumoutsakos, P. J. Phys. Chem. B 2003, 107, 1345. (41) Amirfazli, A.; Kwok, D. Y.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1998, 205, 1.
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on the contact angle on the nanorugged surface. Since the droplet dimension is much larger than that of roughness, that angle θ* can be obtained from Young’s equation: cos θ ¼
γsv - γsl γlv
ð4Þ
θ* in eq 4 corresponds to θ in eqs 2 and 3; γ is the surface free energy. The subscripts lv, sv, and sl refer to the liquid-vapor, solid-vapor, and solid-liquid component, respectively. In practical cases, the solid-vapor component may safely be neglected. We conclude that cos θ* in eq 4 can be evaluated from the ratio -γsl/γlv. It can be observed that the solid-liquid surface free energy γsl which is not directly accessible experimentally becomes the key quantity of the contact angle prediction process outlined above. The goal of the present work is to accomplish a further step with respect to that issue. We proceed through MD computations on systems containing water in the vicinity of model rugged graphite-based surfaces. The roughness dimension stretches from a few angstroms to a few nanometers. We base our study on surface free energy calculations that we use to assess the original thermodynamic approaches of Cassie and Baxter and Wenzel at the scale of the modeled systems. We discuss the validity of a Cassie-Baxter-type equation for surface free energy. We probe Wenzel’s concept and show that it does not hold at least when the hole in which water adsorbs is not deeper than the size of a water molecule. The simulation methodology is explained in the next section. Results are reported and discussed on the basis of a molecular description in the subsequent section, while conclusions close the article.
2. Methodology We have employed the phantom-wall algorithm42 recently developed in our group in order to compute the solid-liquid surface free energy of water-graphite systems having different roughness patterns. In the phantom-wall method, thermodynamic integration is employed in order to turn the solid-liquid interface of interest into a reference one where the structured solid surface has been transformed in a flat, unstructured wall named the phantom wall. Under the condition that calculations are carried out with a constant number of particles N, constant pressure normal to the interface PN, constant temperature T, and constant cross-sectional area of the simulation cell A (NPNTA ensemble), these calculations provide the difference in solidliquid surface free energy Δγsl between the actual system surface free energy γsl and that of the phantom wall-water interface γslref. The superscript “ref” is used to show that the phantom wallwater interface becomes the reference system. γslref is thus a common value to calculations where different surfaces in contact with water are transformed in the same phantom wall. It means that γslref has to be further determined in order to compute the absolute surface free energy γsl. We will show that γsl can be indirectly obtained in the particular present study without additional calculation. This extends the ability of the phantom-wall algorithm to yield absolute solid-liquid surface free energies. Although the derivation of the algorithm has been detailed and illustrated in two recent publications,42,43 we recall here its main features in order to ease the understanding of our simulations. In the present phantom-wall calculations, the periodic condition was employed at the boundaries of the simulation cell. The solid was arranged in a slab geometry with two outer surfaces. Thus, two (42) Leroy, F.; dos Santos, D.; M€uller-Plathe, F. Macromol. Rapid Commun. 2009, 30, 864. (43) Leroy, F.; M€uller-Plathe, F. J. Chem. Phys. 2010, 133, 044110.
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phantom walls had to be used. In such a configuration, the walls are initially located within the solid at a position such that they do not interact with the liquid. (Note that they do not interact with the solid either.) They are progressively displaced in order to begin to interact with the liquid. They consequently lift it away from the solid. At intermediate position, the liquid interacts with both the walls and the solid. The phantom walls are further moved until they reach a position where the liquid interacts only with them. If the transformation from the solid surface to the phantom wall is carried out reversibly, the general formalism of thermodynamic integration can be used in order to extract the Gibbs free energy change ΔG between the initial and the final states: Z DHðλÞ ð5Þ ΔG ¼ 01 dλ Dλ N , PN , T , A H is the Hamiltonian of the system that is made dependent on a parameter λ that quantifies the position of the system along the transformation path. The difference in Gibbs interfacial excess free energy between the actual system and the reference ΔGS = GS - GSref can then be obtained from the Gibbs free energy change ΔG and the system’s volume change term PNΔV=PN(Vref - V): ΔGS ¼ - ΔG þ PN ΔV
ð6Þ
where V and Vref are the volumes of the actual system and of the reference, respectively. When it is normalized by the total simulation cell cross-sectional area 2 A, the interfacial excess free energy change ΔGS becomes the surface free energy change Δγsl. For that purpose, the simulation cell cross-sectional area must remain constant during the transformation induced by the phantom walls. The Hamiltonian is made dependent on the parameter λ via the potential energy. More precisely, the phantom walls are attached to one atom of the solid central layer with harmonic springs of which the equilibrium distance Z0 directly depends on λ: Z0 ðλÞ ¼ λðZ 0 ðλ ¼ 1Þ - Z 0 ðλ ¼ 0ÞÞ þ Z 0 ðλ ¼ 0Þ
ð7Þ
The potential of interaction between the phantom walls and the solid is harmonic: UðZ, λÞ ¼
kW ½Z - Z0 ðλÞ2 2
ð8Þ
where Z is the instantaneous position of the phantom walls along the z direction and kW is the spring force constant. (Note that the z direction was chosen to be perpendicular to the interface.) It is straightforward to obtain an operational form of eq 5: Z Z 0 ðλ ¼ 0Þ ð9Þ ΔG ¼ Z0 ðλ ¼ 1Þ dZ 0 ÆF1 - F2 æ where F1 and F2 are the harmonic forces acting between the solid and each phantom wall. In practice, the system is prepared in different states corresponding to different values of Z0(λ). The average of the total bonding force between the walls and the solid in eq 9 is computed before the integration is performed numerically. We send the interested reader to our previous article43 for more details about technical aspects of the procedure.
3. Models and Computational Details We carried out molecular dynamics simulations of water in the vicinity of graphite-based surfaces. The graphite sample was arranged in a slab geometry with two outer surfaces, and the simulation cell was arranged with periodic boundary conditions. The liquid phase contained 2790 water molecules, and the DOI: 10.1021/la104018k
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graphite layer contained 2688 carbon atoms arranged in seven layers. The interlayer distance was 0.34 nm. Water molecules were modeled with the SPC/E model, and the water-carbon interactions were computed from the model of Werder et al.40 The internal dynamics of graphite were modeled with the force field of Bedrov and Smith.44 Water molecules on opposite sides of the graphite layer could not interact due to its thickness. The crosssectional area of the simulation box was 3.408 2.951 nm2 and was kept the same for all the systems. The electrostatic interactions were computed through the reaction field (RF) methodology with a RF dielectric constant set to infinity. A cutoff distance of 1.375 nm for water-water and water-graphite interactions was employed. No long-range correction to pressure was applied. The two phantom walls were represented by two interaction centers having a planar symmetry parallel to the outer solid surfaces. The interaction cutoff between water and the phantom walls was 0.558 25 nm. The phantom walls were given the mass and the Lennard-Jones interaction parameters of carbon but did not bear any electrostatic charge. They interacted through a 12-6 Lennard-Jones potential with water molecules but not with the solid atoms at all. The corresponding force was shifted so that it canceled at the cutoff. The force constant kW of the spring attaching the phantom walls to one atom of the central layer of the solid was 50 000 kJ mol-1. The pressure and the temperature were 101.3 kPa and 298 K, respectively. They were controlled by means of Berendsen’s barostat and thermostat45 with coupling constants of 0.2 and 2 ps-1, respectively. A 1 fs time-step was used. The force wall curve in the phantom-wall computation was obtained using between 18 and 26 intermediate points which were spaced by 0.05, 0.1, or 0.2 nm. The value of Z0 typically spanned the range between 0.4 and 2.1 nm. The calculation of each of those points lasted 1 ns, and the last 500 ps was used to produce results based on data collected every picosecond. A modified version of the parallel OpenMP-YASP simulation package46 was run using eight processors of our group computing clusters and external machines (see Acknowledgments). In addition to the computation dealing with water in contact with the smooth graphite surface, calculations were carried out where some selected carbon atoms of graphite were made transparent to water. This was achieved by canceling their interaction with water molecules even though they still existed in the simulation cell. By this way, the atomic structure of graphite was maintained, and all the simulations were performed with the same number of atoms. The corresponding lack of carbon atoms in a real system would yield an imbalance in the electronic structure and a modification of the solid local atomic structure, at least. However, our purpose is to present a study about the effect of roughness at the atomic scale with clear geometrical arrangements having strong sharpness. Therefore, our study should be taken as conducted on model systems, although the conclusions it yields are claimed to be more general.
4. Results 4.1. Cassie Wetting States. In a first set of calculations, groups of atoms in a hexagonal configuration have been removed from the top carbon layer of graphite. It should be noted that there are in fact two such top carbon layers since the solid has two outer surfaces. The configurations of the top layer have been represented in Figure 1. It must be remembered that five other full (44) Bedrov, D.; Smith, G. D. Langmuir 2006, 22, 6189. (45) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Dinola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (46) Tarmyshov, K. B.; M€uller-Plathe, F. J. Chem. Inf. Model. 2005, 45, 1943.
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Figure 1. Atomic structure of the top layer of the graphite structures used to study the Cassie wetting states. R is the surface fraction of carbon atoms interacting with water. Transparent atoms to water have not been represented.
carbon layers exist underneath, even though they are not represented in the figure. The well-known honeycomb structure of graphite can be recognized in the sketch S which corresponds to the top carbon layer of smooth graphite. We have considered structures where atoms transparent to water have been arranged in linear grooves, namely C1, C2, C3, and C4, and a structure referred to as C5 where transparent atoms were arranged in holes. We introduced a parameter R which corresponds to the fraction of carbon atoms on the top layer with which water molecules had nonzero interaction. We performed simulations at R= 0.5, 0.75, 0.875, and 1. We have seen from the water mass density profiles over the simulation cell (not shown) that the roughness of the surfaces represented in Figure 1 yields Cassie states where water can penetrate neither the grooves nor the holes. It should be observed that the size of a unit of six transparent carbon atoms that has been used to implement the mentioned structures is larger than the size of a water molecule. The projected area of a water molecule can be approximated to 0.080 nm2 based on the Lennard-Jones distance parameter of the SPC/E model. The area of the unit of six carbon atoms corresponds roughly to 0.157 nm2. We conclude that the adsorption of water molecules in the holes or in the grooves is not limited by steric hindrance, a priori. On a smooth graphite surface, the equivalence between solid-liquid surface free energy and solid-liquid surface tension holds. Furthermore, in the case of water on poorly hydrophilic or hydrophobic substrates it is accepted that the solid-vapor surface free energy has a minor contribution to the contact angle because of water low vapor pressure in normal condition. Moreover, the force field we have employed to model carbon-water interactions was developed to reproduce a contact angle of 86.40 Combining those elements with Young’s equation formulated with surface free energies (eq 4) allows us to anticipate that the liquid-vapor surface free energy is significantly larger than the solid-liquid surface free energy, as will be quantitatively shown later. It can be concluded that the attraction exerted on water by the carbon atoms underneath the top layer where transparent atoms were implemented is not strong enough to overcome the work required to adsorb a water molecule in the holes or grooves. Although water molecules cannot completely adsorb in those spaces, mass density profiles show that hydrogen atoms can be found at positions corresponding roughly to the position of the transparent carbon atoms; i.e., the water surface in the vicinity of rough surfaces is slightly distorted in comparison to its shape on smooth graphite. Simulations at pressure up to 101.3 102 kPa were Langmuir 2011, 27(2), 637–645
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Table 1. Difference in Solid-Liquid Surface Free Energy Δγsl between the Actual System Surface Free Energy γsl of the Systems in Cassie and Wenzel Wetting States and That of the Phantom Wall-Water Interface system
Δγls/(mJ/m2)
γls/(mJ/m2)
system
Δγls/(mJ/m2)
γls/(mJ/m2)
C1 C2 C3 C4 C5 S
-62.7 ( 1.7 -54.1 ( 1.4 -38.3 ( 2.0 -53.6 ( 1.2 -53.8 ( 1.9 -70.0 ( 1.7
-5.4 ( 5.4 3.2 ( 5.4 19.0 ( 5.7 3.7 ( 4.9 3.5 ( 5.6 -12.7 ( 3.3
W1 W2 W3 W4 W5 W6
-63.5 ( 1.6 -62.2 ( 1.7 -60.1 ( 1.6 -56.7 ( 1.4 -49.2 ( 1.4 -29.5 ( 1.4
-6.2 ( 5.3 -4.9 ( 5.4 -2.8 ( 5.3 0.6 ( 5.1 8.1 ( 5.1 27.8 ( 5.1
carried out in order to test whether such a pressure condition could yield wetting states, referred to as Wenzel states, where water adsorbs in the holes of the structure. We found no evidence of such wetting states and therefore consider that the present Cassie states are stable. Note that a more comprehensive study of transitions between Cassie and Wenzel states has been reported by Koishi et al. in the context of pillared hydrophobic surfaces.47 The surface free energy difference Δγsl obtained from phantomwall calculations on the Cassie states has been computed for each system. The results are reported in Table 1. It can be observed that the smooth graphite surface leads to the free energy difference having the largest amplitude, though negative. Since the phantom walls were only weakly attractive, the water-wall surface free energy is expected to be positive. We can thus conclude that water-smooth graphite has the lowest surface free energy of all the systems we have studied. In what follows we quantitatively comment the results that were found concerning the influence of the pattern of transparent atoms to the surface free energy at constant R. When R=0.75, the arrangement of transparent atoms has a very weak influence on the solid-liquid surface free energy. It can be noted from Table 1 that Δγsl is comparable for the systems C2, C4, and C5. This means that the interfacial thermodynamics of all the corresponding Cassie states are equivalent. In order to probe whether the periodicity of the arrangement of atoms in the patterns had an influence, we have generated five different solid surfaces where the position of transparent atoms in the top layer of graphite have been randomly chosen with the constraint that R = 0.75. All the wetting states corresponded to Cassie states. Their average Δγsl value was -51.05 mJ/m2 with a very narrow distribution quantified by a standard deviation of 0.13 mJ/m2, while the statistical error on each individual Δγsl was 1.4 mJ/m2. Consequently, the surfaces at R = 0.75 where atoms have been randomly made transparent are roughly 5% more hydrophobic than those which exhibit organized patterns. The mass density profiles have shown that the distance of least approach of water to graphite is slightly larger in the case of randomly distributed transparent atoms than in the case of organized systems with R=0.75. When transparent atoms were randomly distributed, the generated holes always had a smaller size than in the case of the organized patterns. Therefore, there was a steric hindrance that prevented water for penetrating the holes; i.e., water could not approach graphite as close as when it was in contact with organized surfaces at R=0.75. We will show later that the main contribution to the interaction energy between water and graphite arises from interaction with the first two carbon layers. We conclude that a decrease in the interaction energy as well as in the surface free energy finds its origin in the reduced number of interactions between water and those carbon layers. At this point in the discussion, we suggest that the surface free energy of a Cassie state is dominated by its enthalpy contribution. (47) Koishi, T.; Yasuoka, K.; Fujikawa, S.; Ebisuzaki, T.; Zeng, X. C. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 8435.
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Figure 2. Variation of Δγsl with respect to the density of carbon atoms R. Circles correspond to simulations where atoms of the first layer of graphite have been made transparent to water. Squares correspond to simulations where carbon atoms of the first two layers have been made transparent. Triangles correspond to simulations where carbon atoms have been removed in all the layers of graphite within the cutoff distance of water-carbon interaction. This corresponds to five layers. The symbols have been slightly shifted either to higher or to lower values of R for clarity. The horizontal dashed line corresponds to the zero of the absolute scale of solid-liquid surface free energy γsl. The other dashed lines correspond to the result of linear regressions.
In other words, surfaces which have the same density of missing atoms arranged with comparable homogeneity can approximately be identified with a single surface of which the interaction with water has been modulated by a factor depending on the fraction of missing atoms. We aim to show that the solid-liquid surface free energy linearly depends on the density of transparent atoms R. For this purpose, we have plotted the variations of Δγsl against R in Figure 2 for the systems with linear grooves (C1, C2, C3, and S) only. The variation could be fitted to a linear relation with a regression coefficient of 0.9997. We now demonstrate that this result can be understood through a linear combination similar to that of the Cassie-Baxter equation. The Cassie-Baxter equation has been derived in order to predict the contact angle of a macroscopic droplet on a heterogeneous surface made of two components. Substituting the cosines in eq 2 by their definition in Young’s equation (eq 4) yields γsv ðf Þ - γsl ðf Þ ¼ f γsv, A þ ð1 - f Þγsv, B - ðf γsl, A þ ð1 - f Þγsl, B Þ ð10Þ where γ(f) is the surface free energy of a rough interface. The subscripts A and B refer to two different homogeneous smooth surfaces. The solid-vapor and the solid-liquid contributions can be separated in eq 10. In order to analyze our results, we wrote the linear combination for the solid-liquid contribution only. Note that a similar linear combination could be written for the solid-vapor contribution due to the structure of Young’s equation where both the solid-liquid and the solid-vapor contributions are independent. Moreover, we assumed that the surface DOI: 10.1021/la104018k
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fraction f could be identified with the fraction of nontransparent carbon atoms R. For that purpose, we associated with each transparent atom the area of its projection in the graphite top layer plane. When a group of atoms is made transparent, the projected areas of the individual atoms are not all equivalent. Atoms in the center of such a group have an effective area larger than those at the boundaries. We assumed that the group of atoms which were made transparent was always sufficiently large to neglect the effect of the boundaries. Therefore, we approximated the total projected area of transparent atoms to be equal to the sum of the individual projected areas. This allowed us to identify the surface fraction f with the atom fraction R. In that context eq 10 yields ½ - n
þ ð1 - RÞγsl γsl ðRÞ ¼ Rγgraphite sl
ð11Þ
is the solid-liquid surface free energy of the where γgraphite sl smooth graphite-water interface. γ[-n] sl is the solid-liquid surface free energy of a smooth graphite-water interface where n carbon layers starting from the top layer have been removed, but where water molecules cannot go further than the position they cannot overcome when they are interacting with all the carbon layers of graphite. The validity of eq 11 has already been proven by Schneemilch et al. under the condition that the interaction between the liquid and the substrate is not too strong.48,49 This condition is obviously fulfilled when water interacts with hydrophobic or slightly hydrophilic substrates like graphite. Introducing the solid-liquid surface free energy of the reference system of the phantom-wall transformation yields ½ - n
graphite þ ð1 - RÞγsl γsl ðRÞ - γref sl ¼ Rγsl
- γref sl
ð12Þ
- γref sl Þ
ð13Þ
which finally leads to ½ - n
- γsl Δγsl ðRÞ ¼ Rðγgraphite sl
½ - n
Þ þ ðγsl
Equation 13 shows the linear relation between Δγsl and R. It contains three unknowns while only two parameters can be extracted from a linear regression. In fact, it is possible to build a graphite sample such that the number of layers where atoms have been removed is larger than the cutoff of water-carbon interaction. Considering the interlayer distance in graphitic materials and the interaction cutoff, we have found that this corresponded to five carbon layers. In that case, the surface free is γ[-5] energy γ[-n] sl sl , and it corresponds to the surface free energy of water interacting with vacuum. We have previously stated that there was no steric hindrance for water to penetrate the grooves. Moreover, we have mentioned that the water surface could be slightly deformed, which allowed it to relax the additional stress due to the confinement fixed by the artificial boundary corresponding to the virtual graphite surface. Finally, we conclude that is identical to the liquid-vapor surface free energy γlv. That γ[-5] sl quantity can be easily computed from simulations through the so-called virial route.50,51 We have computed its value and found 59.5 ( 2.4 mJ/m2. This is consistent with the value of 61.2 mJ/m2 assigned to the SPC/E model52 even though it slightly (48) Schneemilch, M.; Quirke, N.; Henderson, J. R. J. Chem. Phys. 2003, 118, 816. (49) Schneemilch, M.; Quirke, N.; Henderson, J. R. J. Chem. Phys. 2004, 120, 2901. (50) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1949, 17, 338. (51) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Dover: Mineola, NY, 2002. (52) Vega, C.; de Miguel, E. J. Chem. Phys. 2007, 126, 154707.
642 DOI: 10.1021/la104018k
underestimates it. This is due to the use of the reaction field method in combination with an infinite reaction field dielectric constant, though with a large cutoff. The effect of such an approximation lowers the strength of the electrostatic cohesive interaction. We found that the mass density of water far from the interface was 998 kg/m3, in agreement with SPC/E model properties in the bulk. We carried out simulations where R was varied but also where some atoms were removed in all the carbon layers so as to produce vertical grooves of apparently infinite depth. Additionally, calculations were carried out where the groove depth reached the second and third carbon layers. The variations of Δγsl with respect to R are reported in Figure 2. In all cases, the variation could be fitted to the linear relation in eq 13, where was dependent on the layer parameter n. The variation of γ[-n] sl Δγsl with respect to the grooves depth was well fitted to an exponential of n (not shown here) which showed that the first two carbon layers of graphite yield the main part of the interaction between graphite and water. The case where the grooves were graphite from infinitely deep enabled the computation of γref sl and γsl ref the fitting parameters in eq 13. They yielded γsl =57.3 ( 3.7 mJ/m2 and γgraphite =-12.7 ( 3.3 mJ/m2. It must be noted that the value sl ref of γsl can be used in any further study where the phantom walls of the present work are employed. It can also be observed that γref sl is comparable to the value of SPC/E water surface tension. This is consistent with the work of Huang et al. where the surface tension of water in the vicinity of large hydrophobic substrates was found to be identical to the liquid-vapor surface tension.53 Therefore, if purely repulsive phantom walls were used, the value of γref sl could safely be approximated by the water model surface tension without additional calculation. The value of γref sl obtained in the present study was further used to obtain absolute values of the surface free energies reported in Table 1. The horizontal dashed line in Figure 2 shows the origin of the γsl absolute scale. We can note that the systems S and C1 have a negative surface free energy, while systems C2-C5 have a positive surface free energy. If one considers that the solid-vapor surface free energy negligibly contributes to the contact angle of water on graphite, the use of Young’s equation with the surface free energies that we obtained yields a value of the contact angle of θ ≈ 77.2. We carried out a simulation of a cylindrical water droplet on smooth graphite and found a value of the contact angle of 76. The cylindrical shape was used in order to avoid the influence of the line tension on the contact angle. The agreement between the two independent calculations of the contact angle validates all of the quantitative approach that has been derived above in order to quantify the variations of the solid-liquid surface free energy in Cassie wetting states. If we neglect again the solid-vapor surface free energy contribution to the contact angle, experimental values for the contact angle (86) and water surface ≈ -5 mJ/m2. Although our tension (72 mN/m) yield γgraphite ls underestimates the guessed experimental one, it value for γgraphite sl is consistent with the fact that the model employed also underestimated the liquid-vapor surface tension. In summary, even though the precise arrangement of atomic defects in a surface structure that yield Cassie wetting states has an influence on the value of the solid-liquid surface free energy, that quantity can approximately be inferred from the surface density of defects and the value of the surface free energy of the smooth surface, as stated by the Cassie-Baxter theory. Furthermore, our results show that a smooth surface which includes atomic defects yields a solid-liquid surface free energy higher than a perfectly (53) Huang, D. M.; Geissler, P. L.; Chandler, D. J. Phys. Chem. B 2001, 105, 6704.
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Figure 3. Atomic structure of the top layer of the graphite structures used to study the Wenzel wetting states.
smooth surface because of the reduction of the liquid-surface average interaction strength. 4.2. Wenzel Wetting States. We have prepared systems where transparent carbon atoms of the top layer were arranged such that water could penetrate the empty spaces. Such wetting states are referred to as Wenzel states. The corresponding graphite surfaces are sketched in Figure 3 where the surfaces W1 and W4 are the inverse images of W2 and W5, respectively. Surface W3 exhibits a large domain of transparent atoms, while W6 contains pillars. The solid-liquid surface free energy change Δγsl was computed for each system. The absolute values of γsl were then obtained using γref sl and are reported in Table 1. The calculations show that all the rugged surfaces are less hydrophilic than the smooth graphite. At the macroscopic scale, where water can be considered as a continuum, Wenzel theory predicts that inverse image surfaces interacting with water would yield the same surface free energy because they have identical water-surface contact area. Our results show that such reasoning no longer holds at the molecular scale. Grzelak and Errington have recently made the same observation in the context of Lennard-Jones systems.54 It is well accepted that even a solute of small size represents a perturbation to the water hydrogen bond (HB) network. The dimension of the protrusions in the systems W2, W5, and W6 is sufficient to induce such a perturbation. We aim to understand whether the change in surface free energy with respect to the roughness pattern directly depends on the alteration of the HB network arising from the surface roughness. Therefore, we have undertaken a detailed water-water HB spatial distribution analysis. The following energetic criterion was employed for that purpose:55 Two water molecules were considered to form a hydrogen bond when their interaction energy was lower than -9.2 kJ/mol. This criterion yielded the average number of hydrogen bonds NHB per water molecule to be 3.6 in the bulk water. We simultaneously analyzed the water mass density profile of each system over the simulation cell in the direction perpendicular to the surface. The profiles of the systems S and W1-W6 are reported in Figure 4. Those profiles show that water is organized in approximately two layers in the vicinity of the smooth graphite surface (see the black curve in Figure 4). The two layers, referred to as L1 and L2, extend within the intervals 1.15-1.55 and 1.55-1.85 nm, respectively. Such a layered structure is conserved (54) Grzelak, E. M.; Errington, J. R. Langmuir 2010, 26, 13297. (55) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926.
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Figure 4. Water mass density profiles over the simulation cell along the z direction in the solid center-of-mass reference frame. Top: results for the systems S (black solid line), W1 (red solid line), W2 (red dashed line), W4 (blue solid line), and W5 (blue dashed line). Bottom: results for the system S (black solid line), W3 (black circles), and W6 (dashed solid line). Only the interval where water is between 0.5 and 3 nm away from the solid center of mass has been represented.
when some carbon atoms are made transparent to water. Moreover, an additional layer, referred to as L0, appears in the interval 0.9-1.15 nm, as is visible in Figure 4. In other words, water molecules adsorbed into the holes of the structures W1-W6 form a first layer (L0), while the second layer (L1) lies on top of the top carbon layer. It was found that NHB in L1 is similar in all the systems having a rugged surface and is equal to the value in L1 on the smooth surface. Furthermore, no difference in NHB was noticed in all the systems at distances further than L2 where NHB was found to be equal to its value in bulk water. This means that the perturbation to the HB network generated by the roughness is very localized and occurs in the vicinity of the contact surface. Other authors have recently reported a similar result.8,22 The difference in the distribution of NHB between the systems mainly occurs in the adsorbed layer L0. We have reported in Figure 5 the spatial distribution of NHB in that layer for the systems W1-W6. It can be observed that the value of NHB was mostly affected only in the immediate neighborhood of the steps in the rugged structures. This means that the maximum perturbation to the HB network occurs close to the contour of the holes or the protrusions in structures W1-W6, as is shown in Figure 5. A more precise inspection of the NHB spatial distribution in W1, W2, W3, and W5 shows that NHB is close to its value in bulk water when water molecules are located in between regions where NHB is minimized (center of the hole in W1, for example). In the case of W4 and W6, there is space for only one water molecule to adsorb in a given hole at a time; therefore, NHB is maximized at positions where water preferentially adsorbs, i.e., close to the contour. In fact, the mass density spatial distribution in L0 (not shown here) reveals that water molecules in that layer are preferentially located close to the steps in the roughness pattern such that they can DOI: 10.1021/la104018k
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Figure 6. Variation of the solid-liquid surface free energy with respect to the roughness contour length of the systems in the Wenzel wetting states.
Figure 5. Average number of hydrogen bonds NHB per water molecule in the first water layer of the systems W1-W6. The first water layer was defined as the layer of water parallel to the solid external surface within the interval 0.9-1.15 nm (see Figure 4). The xy plane was chosen to be parallel to the solid external surfaces. The color in the plot refers to the average number of hydrogen bonds per water molecules NHB. The colored scale under the graphs indicates the intensity of the local value of NHB. The plots were obtained from a grid of 128 128 bins.
maximize contacts with the carbon atoms. This is consistent with the observation we have made about the favorable nature of the water-graphite interaction. When we consider together the information arising from the HB and mass density analysis above, we can describe the arrangement of water in the layer L0 as follows: On the one hand, water molecules prefer to adsorb close to the steps of the roughness pattern because they can maximize the number of simultaneous contacts with carbon atoms. On the other hand, such an arrangement represents an unfavorable perturbation of the local hydrogen-bonding network that counterbalances the gain of enthalpy due to multiple contacts between water and carbon atoms. Note that the possible temporary immobilization of water molecules induced by multiple contacts with carbon atoms may also contribute as an unfavorable entropy contribution. It can be observed that the effect is maximized in the case of W6 where the increased number of simultaneous contacts between water molecules and atoms belonging to two carbon layers is the largest. We conclude that the increase in surface free energy with respect to the smooth graphite surface is closely related to the contour of the roughness pattern. We define this roughness contour as the line following the border between the regions where NHB is zero and where it is not. In most cases, such a line appears as the purple line around the motifs in Figure 5. (Note that a definition using a similar border in the water mass density profile in L0 would yield the same result.) We have evaluated the total length of each contour and plotted the solid-liquid surface 644 DOI: 10.1021/la104018k
free energy of the Wenzel systems with respect to this length in Figure 6. The result we obtained is remarkable since it shows that the solid-liquid surface free energy increases with the linear extent of the perturbation introduced by roughness. Wenzel macroscopic theory predicts that either the hydrophilic or the hydrophobic character of a smooth surface is amplified by making it rugged, assuming that the liquid wets the entire surface available. The underlying reasoning applied to the present water-graphite system is the following: When roughness is introduced at constant cross-sectional area, the free energy is expected to decrease because of the enhanced contact surface area between water and graphite. In such a situation, the surface free energy would decrease under the influence of roughness. In a recent publication, we have studied the variation of the surface free energy of pure Lennard-Jones systems in Wenzel wetting states on surfaces having linear grooves whose depth corresponded to a single atom dimension. The behavior of the solid-liquid surface free energy of those systems followed the trend suggested by Wenzel’s theory.43 The result in Figure 6 shows that the trend in systems where water interacts with rugged graphite surfaces is opposite. The gain of enthalpy obtained from the enhancement of contacts between water molecules and carbon atoms due to the surface roughness is overcome by an effect to which the perturbation of the HB network along the roughness contour contributes. We conclude that Wenzel theory does not apply here because of a molecular feature of water, i.e., its ability to form an extended hydrogen-bonding network. It may be asked what would happen if the surface was strongly hydrophobic, in contrast to the present graphite case. In order to answer this question, we discuss further MD simulations results about water in contact with smooth and rugged surfaces of a hydrophobic n-eicosane crystal layer that were obtained in our group.56,57 The roughness was implemented by means of a hole of which the depth was 0.5 nm and the side-walls surface area of ∼4 nm2. An inverse image of that structure was implemented by means of a protrusion. It can be seen from the mass density distribution in Figure 3 of the article of Pal et al.56 that water molecules avoid the roughness contour of the hole and protrusion structures. This is expected because of the hydrophobic character of the surface. Water molecules tend to avoid the confined space of the hole because occupying it would imply to have an increased number of unfavorable contacts with the hydrophobic crystal. This yields an increase of enthalpy which is the origin of the observed difference in excess free energy between the protrusion (56) Pal, S.; Roccatano, D.; Weiss, H.; Keller, H.; M€uller-Plathe, F. ChemPhysChem 2005, 6, 1641. (57) M€uller-Plathe, F.; Pal, S.; Weiss, H.; Keller, H. Soft Mater. 2005, 3, 21.
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and the hole structure. Note that the confining effect of the hole represents a possible perturbation of the HB network that would come in addition to the mentioned unfavorable enthalpic effect. According to the arguments we have developed above, the hole structure should yield a higher surface free energy than the protrusion. This is what was observed. We conclude that the molecular mechanism that yields the enhancement of the hydrophobic character of rugged surfaces may be different depending on the intrinsic nature of the smooth surface. Nevertheless, none of the situations can be described by the theory of Wenzel, mainly because of the molecular dimension of the perturbation. This observation is consistent with the recent work of Grzelak and Errington, who showed that Wenzel’s approach is no longer adapted to describe wetting states where the roughness length scale is approximately less than 20 molecular diameters.54 The question about what would happen if the roughness depth of the graphite surfaces was extended much deeper than one single carbon layer remains. According to Wenzel theory, the system should converge toward a state having a surface free energy lower than that of the smooth interface. In that case the gain of enthalpy generated by roughness should be the largest contribution to the free energy change. The influence of the roughness wavelength also has an important impact on the solid-liquid surface free energy variation. The question of whether a crossover to the prediction of Wenzel theory would be observed by tuning the ratio between the roughness depth and wavelength should also find an answer in a future publication.
5. Conclusion We have carried out molecular dynamics simulations in order to determine the solid-liquid surface free energy of systems where water is in contact with smooth as well as rugged graphite-based surfaces in both Cassie and Wenzel wetting states at the nanometer scale. For that purpose, we used the phantom-wall method recently developed in our group. We have found that the simulated rugged surfaces were always less hydrophilic (more hydrophobic) than the smooth graphite surface. We have tested the Cassie-Baxter theory and confirmed that a Cassie-Baxter equation can be employed in order to estimate the solid-liquid surface free energy of Cassie states. The results show that the hydrophobic character of the rugged surfaces arises from the reduced intensity of the interaction between water molecules and the graphite layers where atoms are missing. The intensity of the observed surface free energy increase is driven by the density of atomic defects rather than their precise arrangement, as predicted by the Cassie-Baxter theory. The computations allowed us to estimate the value of the solid-liquid surface free energy of the reference interface employed in the phantom-wall calculations. We found that this value was close to the water liquid-vapor surface tension in agreement with general results about the interfacial thermodynamics of water in interaction with hydrophobic solutes. This represents an improvement of the
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method, since the result can be used without additional computation in any further phantom-wall study where liquid water is studied. We used the above-mentioned reference value to compute the absolute surface free energy of all the systems in the present study. The water-smooth graphite solid-liquid surface free energy was found to be -12.7 ( 3.3 mJ/m2, in relatively good agreement with a value deduced from experiments and fully consistent with the simplification employed to compute the electrostatic interactions. The theory of Wenzel was found to qualitatively and quantitatively fail to describe systems where water wets a surface having a roughness pattern not deeper than one molecular diameter. We have shown that the solid-liquid surface free energy variation was linearly dependent on the roughness contour length along which the water molecules can experience simultaneous contact with carbon atoms. The roughness contour corresponds also to the location where the hydrogen bonding network feels the most intense perturbation. We suggest that the perturbation of the hydrogen bonding network is the most important contribution to the enhancement of the excess surface free energy and is at the origin of the hydrophobic character of hydrophilic rugged surfaces at the scale of a few nanometers. It was observed that the solid-liquid surface free energies of the Cassie and Wenzel wetting states are of the same order of magnitude and vary in the range between -10 and 30 mJ/m2. The effect is strong since it represents up to 50% of the liquid-vapor surface free energy. For example, if patterns with short wavelength roughness were implemented in a smooth surface, they would yield a change of contact angle from ∼90 on the smooth surface to 110-120 on the rugged one, assuming that the surface is wet by a large water droplet in a Wenzel state. More generally, surfaces having comparable densities of atomic defects in the top layer of graphite lead to comparable solid-liquid surface free energies in the two different wetting states, unless the roughness contour is very large. From the experimental point of view, even though roughness patterns like those of the Cassie states (C1-C5) may not be implemented, the results concerning the systems leading to Wenzel states (W1-W6) suggest that it is possible to tune the hydrophobic/hydrophilic character of a solid surface by controlling the roughness contour’s length and the depth of the roughness pattern. Acknowledgment. This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) and the Center of Smart Interfaces (CSI) of the TU Darmstadt. We are grateful to the high performance computing center of the TU Darmstadt for allocating computation time and to the John von Neumann Institute for Computing in J€ulich for allocating computation time on the machine JUROPA. We thank Aoife Fogarty for inspiring discussion about the roughness contour length calculation algorithm, for her critical reading of the manuscript and for her help in its preparation. We thank Michael B€ohm for his critical reading of the manuscript.
DOI: 10.1021/la104018k
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