^{1}, Shyam Sridhar

^{1}and Jeffrey R. Errington

^{1,a)}

### Abstract

We introduce Monte Carlo simulation methods for determining the wettingproperties of model systems at geometrically rough interfaces. The techniques described here enable one to calculate the macroscopic contact angle of a droplet that organizes in one of the three wetting states commonly observed for fluids at geometrically rough surfaces: the Cassie, Wenzel, and impregnation states. We adopt an interface potential approach in which the wettingproperties of a system are related to the surface density dependence of the surface excess free energy of a thin liquid film in contact with the substrate. We first describe challenges and inefficiencies encountered when implementing a direct version of this approach to compute the properties of fluids at rough surfaces. Next, we detail a series of convenient thermodynamic paths that enable one to obtain free energy information at relevant surface densities over a wide range of temperatures and substrate strengths in an efficient manner. We then show how this information is assembled to construct complete wetting diagrams at a temperature of interest. The strategy pursued within this work is general and is expected to be applicable to a wide range of molecular systems. To demonstrate the utility of the approach, we present results for a Lennard-Jones fluid in contact with a substrate containing rectangular-shaped grooves characterized by feature sizes of order ten fluid diameters. For this particular fluid-substrate combination, we find that the macroscopic theories of Cassie and Wenzel provide a reasonable description of simulation data.

We gratefully acknowledge the financial support of the National Science Foundation (Grant No. CBET-0828979). Computational resources were provided in part by the University at Buffalo Center for Computational Research and the Rensselaer Polytechnic Institute Computational Center for Nanotechnology Innovations.

I. INTRODUCTION

II. SIMULATION METHODS

A. The interface potential

B. Convenient thermodynamic paths

1. Canonical expanded ensemble

2. Grand canonical expanded ensemble

3. Groove maker simulation

C. Constructing wetting diagrams

D. Calculation of probability distributions

III. MODEL SYSTEM AND SIMULATION DETAILS

IV. RESULTS AND DISCUSSION

A. Flat substrate

B. Contoured substrate

V. CONCLUSION

### Key Topics

- Wetting
- 58.0
- Free energy
- 36.0
- Interfacial properties
- 30.0
- Fluid drops
- 19.0
- Free surface
- 15.0

## Figures

Wetting diagram for a fluid on a geometrically rough surface. The curves are based upon the macroscopic theories of Wenzel and Cassie (see Eq. (15)). Dashed portions of the curves represent metastable states.

Wetting diagram for a fluid on a geometrically rough surface. The curves are based upon the macroscopic theories of Wenzel and Cassie (see Eq. (15)). Dashed portions of the curves represent metastable states.

Illustrative example of an interface potential for a system within the partial wetting regime. *l* _{min} marks the position of the global minimum and *l* _{plat} marks the onset of the plateau region. The configuration images provide representative snapshots from a GC-TMMC simulation. The placement of the snapshots along the abscissa is coupled to the surface density of the system.

Illustrative example of an interface potential for a system within the partial wetting regime. *l* _{min} marks the position of the global minimum and *l* _{plat} marks the onset of the plateau region. The configuration images provide representative snapshots from a GC-TMMC simulation. The placement of the snapshots along the abscissa is coupled to the surface density of the system.

Interface potentials for a system in the vicinity of a Wenzel-impregnation transition. The potential curves are terminated after the global minimum region, i.e., the plateau region is omitted. The curves are generated with the “small” contoured system (the simulation cell contains just one period of the heterogeneity) at *T* = 0.7. Curves from top to bottom correspond to *α* = 0.50, 0.52, 0.54, and 0.56. The configuration images provide representative snapshots from a GC-TMMC simulation.

Interface potentials for a system in the vicinity of a Wenzel-impregnation transition. The potential curves are terminated after the global minimum region, i.e., the plateau region is omitted. The curves are generated with the “small” contoured system (the simulation cell contains just one period of the heterogeneity) at *T* = 0.7. Curves from top to bottom correspond to *α* = 0.50, 0.52, 0.54, and 0.56. The configuration images provide representative snapshots from a GC-TMMC simulation.

Schematic of the off-coexistence strategy for obtaining *V*(*l* _{cmin}). It is assumed that the free energy at (*α* _{H}, *ξ* _{b}) is known (e.g., from direct GC simulation).

Schematic of the off-coexistence strategy for obtaining *V*(*l* _{cmin}). It is assumed that the free energy at (*α* _{H}, *ξ* _{b}) is known (e.g., from direct GC simulation).

Geometrically rough surfaces. (a) Schematic of a substrate with a regular one-dimensional heterogeneity. (b) Snapshot of a substrate with *a* = 4, *λ* = 8, and *λ* _{s} = 4 unit cells.

Geometrically rough surfaces. (a) Schematic of a substrate with a regular one-dimensional heterogeneity. (b) Snapshot of a substrate with *a* = 4, *λ* = 8, and *λ* _{s} = 4 unit cells.

Interface potential for the flat substrate at *T* = 0.7. Blue and red curves trace the four paths used to determine the *α*-dependence of the spreading coefficient along the *T* = 0.7 isotherm. Green curves represent interface potentials obtained at various *α* for illustrative purposes. Data are generated with the small system.

Interface potential for the flat substrate at *T* = 0.7. Blue and red curves trace the four paths used to determine the *α*-dependence of the spreading coefficient along the *T* = 0.7 isotherm. Green curves represent interface potentials obtained at various *α* for illustrative purposes. Data are generated with the small system.

Evolution of the spreading coefficient (top) and contact angle (bottom) with substrate strength for the flat substrate. Within the top panel and right side of the bottom panel, the curves from bottom to top correspond to *T* = 0.6, 0.7, 0.8, 0.9, and 1.0. Symbols connected with dotted lines represent data obtained from temperature EE simulation. Filled symbols denote a reference point.

Evolution of the spreading coefficient (top) and contact angle (bottom) with substrate strength for the flat substrate. Within the top panel and right side of the bottom panel, the curves from bottom to top correspond to *T* = 0.6, 0.7, 0.8, 0.9, and 1.0. Symbols connected with dotted lines represent data obtained from temperature EE simulation. Filled symbols denote a reference point.

Interface potentials generated at *T* = 0.7 with *α* _{L} = 0.2 and *α* _{H} = 1.0. Solid and dashed lines represent the contoured and flat substrates, respectively.

Interface potentials generated at *T* = 0.7 with *α* _{L} = 0.2 and *α* _{H} = 1.0. Solid and dashed lines represent the contoured and flat substrates, respectively.

Evolution of *V*(*l* _{min}) and *V*(*l* _{plat}) with substrate strength at *T* = 0.7. The curves labeled min and plat represent *V*(*l* _{min}) and *V*(*l* _{plat}), respectively. The accompanying letter(s) in parenthesis denotes the wetting state(s) that the curve corresponds to: flat substrate (F), Cassie state (C), Wenzel state (W), and impregnation state (I). Solid and dashed lines represent the contoured and flat substrates, respectively. The off-coexistence path is marked by a dashed-dotted line. The line labeled GM denotes the connection between the plat (F) and plat (C) curves provided by a groove maker simulation. The arrow near the bottom of the figure indicates that the plat (W,I) curve continues to higher values of *α* and lower values of *V*(*l* _{plat}) than that shown in the figure.

Evolution of *V*(*l* _{min}) and *V*(*l* _{plat}) with substrate strength at *T* = 0.7. The curves labeled min and plat represent *V*(*l* _{min}) and *V*(*l* _{plat}), respectively. The accompanying letter(s) in parenthesis denotes the wetting state(s) that the curve corresponds to: flat substrate (F), Cassie state (C), Wenzel state (W), and impregnation state (I). Solid and dashed lines represent the contoured and flat substrates, respectively. The off-coexistence path is marked by a dashed-dotted line. The line labeled GM denotes the connection between the plat (F) and plat (C) curves provided by a groove maker simulation. The arrow near the bottom of the figure indicates that the plat (W,I) curve continues to higher values of *α* and lower values of *V*(*l* _{plat}) than that shown in the figure.

Snapshots from the simulations used to generate the data presented in Fig. 9: (a) min (F); (b) plat (F); (c) min (C,W); (d) plat (W,I); (e) min (I); (f) plat (C); (g) off (I); (h) GM. Blue, red, and green spheres represent fluid, substrate, and groove particles, respectively.

Snapshots from the simulations used to generate the data presented in Fig. 9: (a) min (F); (b) plat (F); (c) min (C,W); (d) plat (W,I); (e) min (I); (f) plat (C); (g) off (I); (h) GM. Blue, red, and green spheres represent fluid, substrate, and groove particles, respectively.

Evolution of the spreading coefficient (top) and contact angle (bottom) with substrate strength for the contoured substrate. Within the top panel and right side of the bottom panel, the curves from bottom to top correspond to *T* = 0.6, 0.7, 0.8, 0.9, and 1.0. Symbols connected with dotted lines represent data obtained from temperature EE simulation, with the filled symbols denoting a reference point. Shaded circles and squares mark the location of first-order Cassie-Wenzel and Wenzel-impregnation transitions, respectively. Metastable states are not included in the bottom panel.

Evolution of the spreading coefficient (top) and contact angle (bottom) with substrate strength for the contoured substrate. Within the top panel and right side of the bottom panel, the curves from bottom to top correspond to *T* = 0.6, 0.7, 0.8, 0.9, and 1.0. Symbols connected with dotted lines represent data obtained from temperature EE simulation, with the filled symbols denoting a reference point. Shaded circles and squares mark the location of first-order Cassie-Wenzel and Wenzel-impregnation transitions, respectively. Metastable states are not included in the bottom panel.

Wetting diagram for the contoured system studied here. Solid lines represent simulation data. On the right side of the figure, the curves from top to bottom correspond to *T* = 0.6, 0.7, 0.8, 0.9, and 1.0. The dashed line provides predictions from the macroscopic models of Wenzel and Cassie.

Wetting diagram for the contoured system studied here. Solid lines represent simulation data. On the right side of the figure, the curves from top to bottom correspond to *T* = 0.6, 0.7, 0.8, 0.9, and 1.0. The dashed line provides predictions from the macroscopic models of Wenzel and Cassie.

## Tables

Bulk fluid properties.

Bulk fluid properties.

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