The Pennsylvania State University The Graduate School Department of Chemical Engineering WETTING CHARACTERISTICS OF PHYSICALLY-PATTERNED

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1 The Pennsylvania State University The Graduate School Department of Chemical Engineering WETTING CHARACTERISTICS OF PHYSICALLY-PATTERNED SOLID SURFACES A Dissertation in Chemical Engineering by Azar Shahraz c 2013 Azar Shahraz Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2013

2 The dissertation of Azar Shahraz was reviewed and approved by the following: Kristen A. Fichthorn Professor of Chemical Engineering Professor of Physics Dissertation Co-Adviser, Co-Chair of Committee Ali Borhan Professor of Chemical Engineering Dissertation Co-Adviser, Co-Chair of Committee Richard Hogg Professor of Mineral Processing and Geo-Environmental Engineering Themis Matsoukas Professor of Chemical Engineering Andrew L. Zydney Professor of Chemical Engineering Head of the Department of Chemical Engineering Signatures are on file in the Graduate School.

3 Abstract This work aims to clarify the effect of surface topology on the kinetic and equilibrium behavior of a droplet deposited on a grooved surface. Specifically the wetting behavior of a droplet on a grooved surface was studied in two different length scales: nano-scale simulation using molecular dynamics (MD) and macroscale modeling by minimizing expressions for the free energies of various droplet wetting modes. The main advantage of our proposed model is that it accounts for pinning/de-pinning of the contact line at step edges, a feature that is not captured by the Cassie and Wenzel models. We also accounted for the effects of gravity (via the Bond number) on various wetting configurations that can occur. Using freeenergy minimization, we constructed phase diagrams depicting the dependence of the wetting modes (including the number of surface grooves involved in the wetting configuration) and their corresponding contact angles on the geometrical parameters characterizing the patterned surface. In the limit of vanishing Bond number, the predicted wetting modes and contact angles become independent of drop size if the geometrical parameters are scaled with drop radius. To investigate the relevance of nanoscale simulations to wetting on macroscopic patterned surfaces, we employed MD to simulate the wetting of Lennard-Jones cylindrical droplets on surfaces patterned with grooves. Using a systematic surface parameter study, we found that, similar to macro-scale, the radius of the droplet is an appropriate length scale at nanoscale where the gravity does not play a significant role. Drop-size-independent wetting phase diagrams were constructed representing the possible equilibrium wetting configurations of a droplet as a function of surface geometry. Unlike macroscopic model, MD simulations exhibit wetting-mode multiplicity in the regions neighboring wetting-mode boundaries predicted by the model. This is likely due to a large free-energy barrier separating wetting states. Using forward flux sampling, we quantified the rate of theses wetting transitions iii

4 and the corresponding energy-barriers. Our results show that surface topology play a significant role in determining the kinetic and equilibrium wetting characteristics of the surfaces. iv

5 Table of Contents List of Figures List of Tables Acknowledgments vii xii xiii Chapter 1 Introduction Background Wetting models Macro-Scale (Continuum-Level) Modeling Nano-Scale (Atomistic-Level) Modeling Hybrid (Atomistic-Continuum) Modeling Kinetics of Droplet Wetting-Mode Transitions Summary of Chapters Chapter 2 Mathematical Modeling of Wetting on Physically-Patterned Solid Surfaces Introduction Model Formulation Results and Discussion Contact angle discontinuity Equilibrium wetting states Comparison to Cassie and Wenzel models Influence of gravity Comparison to experiment v

6 2.5 Summary Chapter 3 Molecular Dynamics Simulation of a Droplet On Nanopatterned Solid Surfaces Introduction Basics of Molecular Dynamics Boundary Condition Alternative Ensembles The relevance of molecular-dynamics simulations to macroscopic systems Simulation Details Results and Discussion Summary Chapter 4 Kinetics of Droplet Wetting-mode Transitions on Grooved Surfaces: Forward Flux Sampling Introduction Forward Flux Sampling FFS via CBG scheme The Committor Analysis Free Energy Calculation Simulation Details Results and Discussion Summary Chapter 5 Conclusions and future work Conclusion Recommendations for Future Work Appendix A Potentials for a grooved surface 87 Bibliography 89 vi

7 List of Figures 1.1 SEM images of different kinds of hierarchical structures in plants: (a) The lotus (Nelumbo nucifera), (b) Euphorbia myrsinites, and (c-e) Salvinia oblongifolia [1] Experimental observation of Cassie-Wenzel coexistence [61] View of the model grooved surface in the direction parallel to the grooves. The surface topology is characterized by groove width G, groove height H, and a step width W Various wetting modes considered in this study. θ is the apparent contact angle, θ e is the equilibrium contact angle for the droplet on a flat surface made of the same material as the grooved surface, and q is the fraction of the height of the groove that is filled with liquid Schematic of a droplet in the Cassie state with one groove underneath it. G and H are the width and height of the groove, respectively, R is the radius of curvature of the droplet, θ is the equilibrium contact angle, and z 0 is the elevation of the droplet center of mass relative to the substrate surface Two scenarios for contact-line pinning at step edges: (a) The droplet periphery resides over grooves and (b) the droplet periphery resides over steps The dimensionless free energy of a drop with R 0 =0.25 mm, Bo= , and θ e = 126 as a function of the number of grooves n beneath the drop on a surface with H = Ḡ = W = The apparent contact angle as a function of Ḡ for constant H (= 0.3) and W (= 0.5) for Bo = 0. Discontinuities in the contact angle result from pinning/depinning transitions, as well as from wettingmode transitions Cross-section of the wetting phase diagram for two different hydrophobic surfaces, θ e = 126 (a) and θ e = 115 (b), with fixed W = 0.2 and Bo = vii

8 2.8 Cross-section of the wetting phase diagram for two different hydrophobic surfaces, θ e = 126 (a) and θ e = 115 (b), with fixed W = 0.5 and Bo = Contact-angle isocontours with θ e = 126 based on the Wenzel and Cassie models for two step widths: (a) Wenzel model, W = 0.2 (b) Cassie model, W = 0.2 (c) Wenzel model, W = 0.5 (d) Cassie model, W = The apparent contact angle as a function of the ratio of the basal length of the drop profile d 0, to the groove width G for different drop volumes on a hydrophobic substrate (pphex); (a) 0.4 nl, (b) 12 nl, (c) 40 nl and, (d) 4.5 µl. Predictions of the model are compared to experimental data [89] Illustration of periodic boundary condition Side view (a) and top-down view (b) of the periodically-grooved surface characterized by groove width, G, groove height, H, and step width, W The equilibrium contact angle on a flat surface θ e as a function of the relative energy parameter ɛ r. The error bar represents the uncertainty in the computed contact angles. The surface is designated as hydrophobic for θ e 90 and hydrophilic for θ e < Various wetting modes observed in this study The effect of droplet size on wetting mode and apparent contact angle for surfaces with the same scaled topography (Ḡ = H = 0.45 ± 0.03, and W = 0.32 ± 0.02): (a) R 0 = (50 ± 2) σ ff, (b) R 0 = (25 ± 2)σ ff, and (c) R 0 = (12 ± 2)σ ff The effect of groove width Ḡ on apparent contact angle for a step width of W = 0.32 ± 0.03 and step heights (a) H = 0.10 ± 0.01 and, (b) H = 0.40 ± The solid lines represent predictions of the mathematical model in ref. [96] and the symbols correspond to MD simulations. The vertical error bar depicts the representative uncertainty in the MD contact angles and the horizontal error bars indicate the uncertainty in the MD value of R The effect of step width W on apparent contact angle for a step height of H = 0.64 ± 0.05 and groove widths of (a) Ḡ = 0.16 ± 0.01, and (b) Ḡ = 0.64 ± The solid lines represent predictions of the mathematical model in ref. [96] and the symbols correspond to MD simulations. The vertical error bar depicts the representative uncertainty in the MD contact angles and the horizontal error bars indicate the uncertainty in the MD value of R viii

9 3.8 The effect of step height H on apparent contact angle for a step width of W = 0.32 ± 0.03 and groove widths (a) Ḡ = 0.16 ± 0.01, and (b) Ḡ = 0.45 ± The solid lines represent predictions of the mathematical model in ref. [96] and the symbols correspond to MD simulations. The vertical error bar depicts the representative uncertainty in the MD contact angles and the horizontal error bars indicate the uncertainty in the MD value of R The effect of initial droplet configuration (i) on the final wetting configuration (f)realized in MD simulation for surfaces with W = 0.32 ± 0.03 and H = 0.32 ± 0.03, Ḡ = 0.40 ± 0.03 (A and B), and H = 0.8 ± 0.06, Ḡ = 0.48 ± 0.04 (C and D) The effect of initial droplet location (i) on final observed wetting configuration (f) for surfaces with pattern comparable to drop size. (For A, B and C, H = 0.64 ± 0.05, Ḡ = 1.12 ± 0.09, W = 0.8 ± 0.06) Cross-section of the wetting phase diagram for a hydrophobic surface with θ e = 126 and W = 0.16 ± Data points from the simulations are represented by symbols and colored regions indicate the predicted wetting regimes based on the simulations. Error bars have been shown for only a few points to improve visual clarity. However, the uncertainty in both H and Ḡ ranged from 0.01 to Phase boundaries between the most stable wetting modes predicted by the macroscopic model [96] are shown by dashed lines, and regions I and II correspond to the model predictions for Wenzel and Cassie modes, respectively Cross-section of the wetting phase diagram for a hydrophobic surface with θ e = 126 and W = 0.8 ± Data points from the simulations are represented by symbols and colored regions indicate the predicted wetting regimes based on the simulations. Error bars have been shown for only a few points to improve visual clarity. However, the uncertainty in both H and Ḡ ranged from 0.01 to Phase boundaries between the most stable wetting modes predicted by the mathematical model [96] are shown by dashed lines, and regions I, II, and III correspond to the model predictions for Wenzel, Cassie, and Epitaxial Cassie domains, respectively Schematic illustration of CBG method. Branched transition paths all are generated by firing a constant M i trial runs at each interface, λ i, shown by vertical black line. Successful partial paths are depicted by solid green lines and unsuccessful trajectories are shown by dashed gray lines ix

10 4.2 Sketch of different scenarios for a trajectory in transition region between A and B. Trajectories 1 and 2 initiate from A. Trajectory 1 ultimately reaches B, while trajectory 2 returns back to A after visiting other interfaces. Trajectories 3 and 4 start from B. Trajectory 3 goes to A, while trajectory 4 goes all the way back to B The location of different surface geometries in the theoretical phase diagrams for which wetting-mode multiplicity has been observed in MD simulations. W = 0.32 ± 0.02 and error bars represent the uncertainty in calculating the radius of the droplet in MD simulations, which in turn affects the uncertainty in computing scaled geometric parameters of the surface A schematic for C W and W C transitions. The corresponding rate constants are represented by k C W and k W C, respectively Example of (a) C W and (b) W C transition paths for a droplet with N 3000 and a surface with H = 0.4 ± 0.03, Ḡ = 0.72 ± 0.06, and W = 0.32 ± Committor probability P B as a function of order parameter, λ for a surface with H = 0.4 ± 0.03, Ḡ = 0.72 ± 0.06, and W = 0.32 ± The vertical dashed lines display interfaces that partition phase space. The intersect between the curve and the horizontal dash line of P B = 0.5 define the critical size λ, which corresponds to the transition state The effect of step height H on C W and W C transition-rate constants for a step width of W = 0.32 ± 0.02 and groove width of Ḡ = 0.72 ± Two Cassie-to-Wenzel wetting mechanisms: (a) sag, and (b) depinning The committor probability vs.λ for three geometries with W = 0.32 ± 0.02, Ḡ = 0.72 ± 0.06, and three different H values (a) 0.4 ± 0.03, (b) 0.56 ± 0.045, and (c) 0.72 ± Schematic view of transition state for de-pinning meniscus The effect of groove width Ḡ on C W and W C transition rate constant for a step width of W = 0.32 ± 0.02 and step height of Ḡ = 0.56 ± The committor probability vs. λ for three geometries with W = 0.32 ± 0.02, H = 0.56 ± 0.04, and three different Ḡ values (a) 0.56 ± 0.04, (b) 0.72 ± 0.06, and (c) 0.89 ± x

11 4.13 The effect of droplet size on transition-rate constant with the same scaled topography ( H = 0.4 ± 0.03, W = 0.32 ± 0.020, and Ḡ = 0.72 ± 0.06) The committor probability vs. λ for a fixed geometry ( H = 0.4±0.03, Ḡ = 0.72±0.06, and W = 0.32±0.02), and three droplet sizes, R 0 : (a) (12 ± 2)σ ff, (b) (17 ± 2)σ ff, and (c) (25 ± 2)σ ff The free energy profile vs. λ for three geometries with W = 0.32 ± 0.02, Ḡ = 0.72 ± 0.06, and three different H values (a) 0.4 ± 0.03, (b) 0.56 ± 0.045, and (c) 0.72 ± The free energy profile vs. λ for three geometries with W = 0.32 ± 0.02, H = 0.56 ± 0.04, and three different Ḡ values (a) 0.56 ± 0.04, (b) 0.72 ± 0.06, and (c) 0.89 ± The free-energy profile vs λ for a fixed geometry ( H = 0.4 ± 0.03, Ḡ = 0.72 ± 0.06, and W = 0.32 ± 0.02), and three droplet sizes, R xi

12 List of Tables 2.1 The effect of the Bond number on the wetting mode and apparent contact angle θ of a droplet (with θ e = 126 ) for different scaled surface topologies The effect of gravity on the wetting state and contact angle θ of a droplet on a grooved surface Commonly used ensembles in statistical mechanics. Each ensemble is defined by a sub-set from the following thermodynamic variables: energy E, temperature T, pressure P, volume V, number of particles N,and chemical potential µ A comparison of the theoretical λ (Eq. 4.17) and those generated by FFS simulations for a step width of W = 0.32 ± 0.02 and step gap of Ḡ = 0.72 ± A comparison of the theoretical λ (Eq. 4.17) and those of FFS simulations for a step width of W = 0.32 ± 0.02 and step height of H = 0.56 ± xii

13 Acknowledgments I would like to express my deepest appreciation to my thesis advisers, Prof. Kristen A. Fichthorn and Prof. Ali Borhan, who have supported me throughout my graduate study at Penn State with their guidance, enthusiasm, and immense knowledge. Without their help and insightful advice, this work would not have been completed. I would like to thank my other thesis committee members, Prof. Richard Hogg, and Prof. Themis Matsoukas, for their helpful suggestions and feedback. I thank the High Performance Computing (HPC) Group at Penn State for providing the computational resources required for this work. I would like to cease this opportunity to thank all my friends and lab-mates for helpful discussions, providing very friendly environment, and unforgettable coffee breaks. Finally I would like to extend my deepest gratitude to my dear mom and dad, Mehrangiz Nikahi and Majid Shahraz, my beloved brother, Saeid, and my kind sister, Farzaneh, who, as always, have provided unwavering support and love. This research is supported by the National Science Foundation, Grant CBET xiii

14 Chapter 1 Introduction 1.1 Background Wetting is defined as the fundamental process of liquid interaction at solid gas interfaces [1]. A common quantity used to characterize the wettability of surfaces is the contact angle (CA), the angle at which the liquid-vapor interface meets the solid-liquid interface. A low CA value indicates that the liquid spreads, while a high CA indicates poor wetting. In this case, the liquid droplet does not wet the surface. Instead, it beads up into a spherical cap that can roll off the surface when the surface is tilted to a specific angle, called the tilt angle (TA). Another important quantity in measuring surface wettability is hysteresis, which is defined as the difference between advancing (maximum) and and receding (minimum) contact angles just before the wetting line starts to move [100]. The ability to create surfaces with controlled wettability is important for a wide variety of applications. Superhydrophobic surfaces, which possess a high (> 150 ) contact angle for water droplets and a low (< 10 ) contact-angle hysteresis, have a number of beneficial properties, including water repellency, self cleaning, low

15 2 drag, and antifouling characteristics [3, 4]. Recently, particular attention has been paid to find a systematic approach of preparation of superhydrophobic surfaces in submicrometer scale, which have applications in the advancing field of nanotechnology including, micro/nanoelectromechanical systems (MEMS/NEMS), biosensing, lab-on-a-chip systems, microfluidics and microreactors [5, 6]. An archetypical, biologically-inspired surface for superhydrophobicity is the Lotus leaf. Superior water repellency of the lotus leaf was first reported by Barthlott and Neinhuis in 1997 [7]. They investigated the surface structures of more than 200 water-repellent plant species, and found that much of the superior water repellency of these plant surfaces derives from the hierarchical surface structures in conjunction with the waxy surface of their leaves. Figure 1.1 shows scanning electron microscope (SEM) images of various types of hierarchical structures in plants. The idea that the surface roughness can induce superhydrophobicity has inspired many studies aimed at synthesizing rough or patterned surfaces with superhydrophobic properties [3, 4, 8, 9, 10, 11, 12, 13, 14, 15] and quantifying the effect of roughness on superhydrophobicity [16, 17, 18, 19, 20, 21, 22, 23, 25, 42]. 1.2 Wetting models There are three main approaches commonly followed by theoretical researchers to address the wetting phenomena over a specific length and time scale. The first approach describes this phenomenon in macro-scale (continuum level modeling) which can be classified as either dynamic or static modeling [26]. The second approach offers an explanation for the wetting process from the molecular point of view by employing atmomistic simulation predominately based on molecular

16 3 Figure 1.1: SEM images of different kinds of hierarchical structures in plants: (a) The lotus (Nelumbo nucifera), (b) Euphorbia myrsinites, and (c-e) Salvinia oblongifolia [1]. dynamics (atomistic-level simulation). The third approach is based on a multiscale hybrid scheme, in which the continuum and atomistic subdomains are dynamically coupled together to use the most efficient description in each length scale. Each of these three models will be discussed further in following sections Macro-Scale (Continuum-Level) Modeling The study of dynamic wetting at macro-scale involves the modeling of the moving contact line withing the framework of conventional fluid mechanics [27, 28, 29, 30]. This subject continues to be very challenging because the stress singularity appears in the vicinity of the moving contact line by imposing the common no-slip boundary condition, i.e., zero fluid velocity relative to the solid at the fluid-solid

17 4 interface. Over the years there have been numerous attempts aiming to resolve the incompatibility of this boundary condition with the moving contact line [32]. The most usual hypothesis made to circumvent this difficulty is to relax the no-slip condition in the vicinity of the moving contact line and to replace it with a slip boundary condition, which specifies a slip velocity proportional to the local shear stress exerted on the solid. The constant of proportionality is usually referred to as the slip coefficient [33]. The problem becomes even more complicated in the case of patterned surfaces, and in spite of major advances, the validity of the proposed continuum theories for these patterned surfaces has not been proved yet. The principle of static or thermodynamic modeling of wetting usually relies on the minimization of the Gibbs free energy of the system. In the static condition, the contact angle is a direct result of the force balance acting on the droplet [34]. A simplified model based on the force and energy balance was first developed by Young [35] in 1805, in which he defines the equilibrium contact angle on ideal surfaces θ e in terms of liquid vapor surface tension, γ lv, solid liquid surface tension, γ sl, and solid vapor surface tension, γ sv : γ lv cos θ e = γ sv γ sl (1.1) The term ideal in this context, refers to an smooth, rigid, chemically homogeneous, insoluble, and nonreactive surface [26]. However, an ideal surface is very rarely encountered in practice [36] and most solid surfaces are rough and heterogeneous to some extent. Numerous modifications of Young s equation have been proposed to extend its applicability to real surfaces [37, 38]. Wetting on rough surfaces is typically classified by one of two models. In the Wenzel, or non-composite, model [39], liquid fills all the surface asperities beneath the droplet. According to Wenzel

18 5 equation, the apparent contact angle formed by a liquid wetting a rough surface, θ W, is given by, cos θ W = r cos θ e (1.2) where r is the roughness factor and defined as the ratio between the actual wettable area and geometric projected area; θ e is Young s angle on the flat surface. This equation indicates that the surface roughness enhances both hydrophobicity of hydrophobic surfaces and hydrophilicity of hydrophilic ones. Another possible wetting state is described by the Cassie-Baxter, or composite, model [40], in which the droplet is lifted up by surface roughness, with air pockets in the asperities underneath it, leading to the heterogeneous wetting on the surface. The contact angle of a Cassie drop, θ C is related to the fraction of solid in contact with the liquid, φ s and θ e by Cassie-Baxter equation: cos θ C = 1 + φ s (1 + cos θ e ) (1.3) The validity and applicability of the two classical Wenzel and Cassie models have been questioned several times [41]. Experimental studies have shown that these models can only describe contact angles for a limited range of surface morphologies [42, 43, 44, 45]. Gao and McCarthy argued that the Wenzel and Cassie equations are not directly relevant to water repellency and that events at the contact line not over the liquid-solid interfacial area control the contact angle [21]. Additionally, other wetting modes that are not predicted by the Cassie and Wenzel models can occur. These include a Mixed mode, in which liquid partially fills the surface asperities [47, 48, 46] and, as we will discuss in Chapter 2, the Epitaxial Cassie mode, an experimentally observed wetting state in which a

19 6 droplet rests on top of a single asperity [50, 91] Nano-Scale (Atomistic-Level) Modeling As mentioned above, most of the theoretical continuum models of three-phase contact line are based on assumptions that cannot be easily verified, and in spite of major advances in experimental techniques, it is difficult to get a clear picture from experimental studies of the contact zone at the microscopic level. Molecular dynamics (MD) can be used to shed some light and provide detailed, microscopic-level information to understand the physics underlying many classical wetting experiments. Substantial progress in applying MD to the study of wetting phenomena has been achieved. Examples include the spreading of a droplet over both homogeneous [51, 52] and heterogeneous surfaces [53, 54, 46], electro-wetting at the nano-scale, [55] forced wetting [56], and reactive wetting processes [57]. More details about MD simulations of a droplet on physically-patterned surfaces are presented in Chapter Hybrid (Atomistic-Continuum) Modeling Although MD simulation is a powerful tool for accurately describing the behavior of the moving contact line, it is extremely expensive in terms of CPU time and memory to implement for droplets larger than a few tens of nanometers for longer than few nanoseconds. In the hybrid approach, MD simulation is applied to the contact line region, where the continuum assumptions break down and the Navier- Stokes equations are used for the main body of the drop where the continuum assumptions hold true. These two regions are coupled together in the overlap domain [58]. The choice of an appropriate coupling method and the imposition of

20 7 boundary conditions on the MD region are two central issues in developing hybrid methods [59, 60]. The application of a hybrid MD-continuum algorithm is out of the scope of this thesis and is left for future studies. 1.3 Kinetics of Droplet Wetting-Mode Transitions The term wetting transition is used by different researchers to mean different things. In a number of studies [62, 63, 64], the wetting transition refers to structureinduced transition in which the observed or the most stable wetting state of the droplet changes due to change in surface morphology. Other researchers use this term to talk about passing from one wetting mode associated with local minimum in energy to the global-minimum wetting state on a given surface. Experimental observations provide evidence for the existence of wetting mode multiplicity [16, 17, 20, 25]. Droplets can remain in local minima over experimental time scales due to free-energy barriers that hinder attainment of the global-minimum wetting state. The energy required for overcoming an intervening barrier is provided by external stimuli such as temperature, compression, vibration, impact or electric voltage [62, 69]. An example of these experimental observation is presented in Figure 1.2 [61]. This Figure shows two millimetric water drops deposited on the same microtextured surface with different impact velocities. The right drop, which induced a Wenzel state falls from a few centimeters, wheres the drop on the left, which remains in Cassie state, is carefully deposited on the surface without any impact velocity. In addition to experimental works, considerable amount of theoretical studies

21 8 Figure 1.2: Experimental observation of Cassie-Wenzel coexistence [61]. based on continuum [65, 66, 67, 68, 69, 70, 71, 72, 80, 81] have been performed to elucidate wetting transitions and free-energy barriers between the Cassie and Wenzel states. These studies are based on thermodynamic analysis of a hypothetical pathway from Cassie to Wenzel and can neither explain nor precisely determine the position of the transitions. As it is reported by Susarrey-Arce and co-workers [73], these energy-based arguments are insufficient to explain the experimental observations. The actual details of the transition from composite to wetted contact are not well understood [19, 66]. Recently, different approaches to study the kinetically hindered energy barrier based on atomistic-level simulation are reported in few publications [75, 76, 77, 78].

22 9 1.4 Summary of Chapters This dissertation focuses on the surface topography and seeks a systematic procedure to develop a design methodology for constructing surfaces with the desired wettability in the macro-scale and nano-scale. As described in Chapter 2, a continuum-level model, based on free-energy minimization, is developed to predict the most stable wetting state of a two-dimensional droplet on a grooved surface. Unlike the Cassie and Wenzel models, our model accounts for pinning of the contact line as well as the the effect of gravity. Chapter 3 is devoted to the study of the wetting phenomena on nano-grooved surfaces using MD simulations. The dependence of the equilibrium wetting state on the size of the spreading droplet is examined to explore the appropriateness of using drop radius as a length scale to characterize equilibrium wetting states in different geometries. In chapter 3 we aimed to clarify whether large-scale MD simulations could correctly capture the hydrodynamic behavior relevant to experiments and applications involving macroscopic droplets. We also demonstrate how MD simulations can indicate wetting-mode multiplicity for various droplet configurations. We implement a forward flux sampling (FFS) approach to study the kinetics of wetting mode transitions in Chapter 4. FFS allows us to compute the rate constants and free energy barriers associated with rare events by using a series of interfaces that partition the phase space between meta-stable and stable states. Our studies show that metastable wetting states can exist over human time scales, and that the free-energy barrier for transitions between these states depends on topological parameters characterizing the grooved surface.

23 Chapter 2 Mathematical Modeling of Wetting on Physically-Patterned Solid Surfaces 2.1 Introduction In the design of superhydrophobic surfaces, it is important to be able to predict the wetting mode, contact angle, and contact-angle hysteresis that will occur for a given surface morphology. Among the possible local minimum wetting states for a given surface morphology, there exists a global minimum and it is important to be able to predict this state. Many prior theoretical efforts to predict wetting freeenergy minima have been based on the Wenzel or Cassie-Baxter models [67, 82, 83] and/or incorporate the assumption that the drop is large compared to the surface roughness [80, 81]. Detailed numerical simulations have been performed to solve for the droplet wetting mode and shape on various patterned surfaces [71, 72]. However, despite these studies, basic questions persist in efforts to design surfaces

24 11 Figure 2.1: View of the model grooved surface in the direction parallel to the grooves. The surface topology is characterized by groove width G, groove height H, and a step width W. with optimal superhydrophobicity. In this study, we present a model that predicts the global-minimum wetting modes of a two-dimensional droplet on a grooved surface. Unlike the Cassie and Wenzel models, our model accounts for pinning of the contact line, which is important for achieving superhydrophobicity. We also account for the effects of gravity on droplet wetting via the Bond number. As we will discuss below, our model reproduces, both qualitatively and quantitatively, many experimentally observed features of droplet wetting on patterned surfaces. 2.2 Model Formulation We consider a cylindrical liquid droplet on a patterned solid surface characterized by the periodic, rectangular grooves shown in Fig The grooves have width G and height H, and are separated by steps of width W. To simplify the problem, we consider the droplet to be of infinite length along the cylindrical axis, which

25 12 Figure 2.2: Various wetting modes considered in this study. θ is the apparent contact angle, θ e is the equilibrium contact angle for the droplet on a flat surface made of the same material as the grooved surface, and q is the fraction of the height of the groove that is filled with liquid. is aligned with the grooves, so that the problem becomes two-dimensional. We also assume that the drop has a constant-curvature profile in the cross section perpendicular to the grooves (i.e., the drop profile is a portion of a circle). The assumption of constant-curvature profile is valid provided the drop is sufficiently small that its shape is not affected by gravity [90]. We consider four different wetting modes, which are shown in Fig These include the Cassie and Wenzel modes, as well as the Epitaxial Cassie state, an experimentally observed [50, 91] wetting mode in which a droplet resides completely on top of a single step. We also consider the possibility of a Mixed mode, which is defined as a wetting configuration in which the droplet partially penetrates and forms a meniscus within the grooves. In each of the wetting modes in Fig. 3.4, the free-energy difference per unit length ( E) between a system consisting of a droplet on the patterned surface and a surface without a droplet is given by E = A lv σ lv + A sl (σ sl σ sv ) + F g, (2.1)

26 13 where the first two terms represent interfacial energies and the third term F g reflects the effect of gravity. In Eq. 2.1, A lv is the area per axial length of the liquid-vapor interface, A sl is the area per axial length of the solid-liquid interface, and σ lv, σ sl, and σ sv denote the liquid-vapor, solid-liquid, and solid-vapor interfacial energies, respectively. The expressions for A lv, A sl, and F g depend on the wetting mode. To illustrate our approach, we consider a droplet that covers one surface groove in the Cassie mode. A schematic for this wetting mode is shown in Fig Figure 2.3: Schematic of a droplet in the Cassie state with one groove underneath it. G and H are the width and height of the groove, respectively, R is the radius of curvature of the droplet, θ is the equilibrium contact angle, and z 0 is the elevation of the droplet center of mass relative to the substrate surface. For the droplet in Fig. 2.3, the liquid-vapor interfacial area (per unit axial length), A lv, is the sum of the area (per unit axial length) of the cylindrical cap resting on top of the pillars and that of the flat interface over the groove. This is given by A lv = 2Rθ + G, (2.2) where R is the radius of curvature of the cylindrical cap (see Fig. 2.3). The

27 14 liquid-solid interfacial area per unit axial length, A sl, can be written as A sl = 2R sin θ G. (2.3) The gravitational energy contribution per unit axial length, F g, is given by F g = πρgr 0 2 z 0, (2.4) where z 0 is the elevation of the center of mass of the droplet relative to the surface of the substrate, ρ is the liquid density, g is the gravitational acceleration, and R 0 is the cylindrical-equivalent radius of the drop. By substituting Eqs into Eq. 2.1 and recognizing that the static equilibrium contact angle on the analogous flat surface, θ e, is related to the interfacial energies through Young s equation, i.e., σ lv cos θ e = σ sv σ sl, the free energy difference per unit axial length for the configuration shown in Fig. 2.1 can be rewritten as E = [2R(θ cos θ e sin θ) + G(1 + cosθ e )] σ lv + πρgr 0 2 z 0, (2.5) where z 0 is given by z 0 = H R cos θ + 2R3 sin 3 θ 3πR 0 2, (2.6) and the radius of curvature R can be related to drop size R 0 and contact angle θ according to ( ) 1/2 π R = R 0. (2.7) θ sin θ cos θ Specific expressions for the free-energies of the wetting configurations in Fig. 3.4 are presented below. In what follows, all lengths and energies (per unit axial length)

28 15 are made dimensionless with R 0 and σ lv R 0, respectively, and all dimensionless variables are designated by overbars (i.e., H = H/R0, W = W/R0, Ḡ = G/R 0, R = R/R 0, and E = E/σ lv R 0 ). In addition, the subscripts W, C, M, and EC refer to the Wenzel, Cassie, Mixed, and Epitaxial Cassie modes, respectively. Cassie Mode E C = 2 R C (θ cos θ e sin θ) + nḡ(1 + cos θ e) [ ] 2 + Bo 3 R C 3 sin 3 θ + π( H R C cos θ), (2.8) where n is the number of grooves underneath the droplet, Bo = ρgr2 o σ lv is the Bond number, and R C, the dimensionless radius of curvature of the cylindrical portion of the drop that lies above the steps, is given by ( ) 1/2 π R C =. (2.9) θ sin θ cos θ For n = 1, the dimensionless free-energy difference in Eq. 2.8 is the same as that for our example in Fig. 2.3 and Eq Wenzel Mode E W = 2 R W (θ cos θ e sin θ) 2n H cos θ e [ 2 + Bo 3 R W 3 sin 3 θ + (π nḡ H)( H R W cos θ) + 1 ] 2 nḡ H 2, (2.10) where R W = ( π n Ḡ H ) 1/2. (2.11) θ sin θ cos θ

29 16 Mixed Mode E M = 2 R M (θ cos θ e sin θ) + nḡ [(π/2 θ e) sec θ e + cos θ e ] 2nq H cos θ e [ ] 2 + Bo 3 R 3 M sin 3 θ + R M( 2 H R M cos θ)(θ sin θ cos θ) + n [ {sec 4 BoḠ2 2 θ e (θ e + sin θ e cos θ e π/2) (1 q) H 1 ]} 2Ḡ tan θ e ] + nboḡ [(1 q/2)q H 2 112Ḡ2, (2.12) where π n R M = 4 Ḡ2 [(θ e π/2) tan 2 θ e + tan θ e + θ e π/2] nqḡ H θ sin θ cos θ 1/2, (2.13) and q is the fraction of the groove that is filled with liquid. For this configuration, the contact angle formed by the meniscus within the groove at the solid wall is θ e. Epitaxial Cassie Mode [ ] E EC = 2 R 2 EC (θ cos θ e sin θ)+bo 3 R EC 3 sin 3 θ + π( H R EC cos θ), (2.14) where ( ) 1/2 π R EC =. (2.15) θ sin θ cos θ The apparent contact angle for each wetting mode is obtained by minimizing the corresponding energy expression with respect to θ, subject to the constraint that the contact line be located on top of a step and not over a groove. When the contact line is not pinned at the edge of a step (i.e., it is free to move), minimizing the free-energy expressions for different wetting modes of a droplet covering n

30 17 grooves yields the following expression for the contact angle θ f j, where j refers to the wetting mode, which could be C, W, M, or EC: 2(cos θ f j cos θ e)(θ f j cos θf j sin θf j )(θf j sin θf j cos θf j ) + Bo R 2 j(θ f j sin θf j cos θf j )( 2 sin3 θ f j + θf j sin2 θ f j cos θf j + θf2 j sin θ f j ) = 0. (2.16) Here, Rj (θ f j ) is the dimensionless radius of curvature of the cylindrical cap that lies above the steps, given by Eqs. 2.9, 2.11, 2.13, and 2.15 for the different modes. In the limit Bo 0, which occurs for sufficiently small drops, Eq simplifies to (cos θ f j cos θ e)(θ f j cos θf j sin θf j )(θf j sin θf j cos θf j ) = 0, (2.17) whose only non-trivial solution in the range [0, π] is θ f j = θ e. Thus, for Bo = 0, if the contact line resides on top of a step without being pinned at the step edges, the droplet will maintain the same contact angle as that on a flat surface, regardless of the wetting mode. Figure 2.4: Two scenarios for contact-line pinning at step edges: (a) The droplet periphery resides over grooves and (b) the droplet periphery resides over steps. In the general case, Eq can be solved numerically to find θ f j for a given droplet configuration. Frequently, the solution to this equation leads to the

31 18 situation where the contact line resides above a groove, in which case we consider that it has to be pinned at a step edge. There are two different scenarios for contact-line pinning and these are shown in Fig In the first scenario shown in Fig. 2.4(a), which can occur for n 1, the basal length of the drop (with contact angle θ f j from Eq. 2.16) is smaller than (n 1) W + nḡ. In this case, the minimum free energy will be achieved when the contact line is pinned at a step edge, with the droplet periphery over grooves so that the drop covers n grooves and (n 1) steps. The resulting contact angle θ pg j is then obtained from the solution of 2 R j sin θ pg j = (n 1) W + nḡ, (2.18) where R j (θ pg j ) is given by Eqs. 2.9, 2.11, and 2.13 for the different modes. Similarly, if the resulting basal length of the drop (with contact angle θ f j from Eq. 2.16) exceeds (n + 1) W + nḡ, the minimum free energy will be achieved when the contact line is pinned at a step edge, with the droplet periphery over steps so that the droplet covers (n + 1) steps and n grooves [cf., Fig. 2.4(b)]. The resulting contact angle θ ps j is obtained from the solution of 2 R j sin θ ps j = (n + 1) W + nḡ, (2.19) where R j (θ ps j ) is given by Eqs. 2.9, 2.11, 2.13, and 2.15 for the different modes. We note that this scenario also holds for n 0, so the Epitaxial Cassie mode (n = 0) falls under this case. For a given surface topology, the equilibrium contact angle is the one corresponding to the minimum free energy among all possible wetting modes and number of grooves involved in the wetting configuration. We find the global

32 19 Figure 2.5: The dimensionless free energy of a drop with R 0 =0.25 mm, Bo= , and θ e = 126 as a function of the number of grooves n beneath the drop on a surface with H = Ḡ = W = minimum in free energy using the approach illustrated in Fig. 2.5, wherein the minimum free energies are plotted for different configurations of a water droplet with R 0 = 0.25 mm and θ e = 126 on a fixed surface topology with H = Ḡ = W = To obtain each point on Fig. 2.5, we used a bisection algorithm to solve Eq for the contact angle that minimizes the free energy. If in the resulting configuration, the contact line lies above a groove, we use Eq or Eq. 2.19, as discussed above, to find the contact angle that yields the minimum free energy for a pinned configuration. The results in this figure show that the Cassie mode with n = 12 represents the global free-energy-minimum configuration in this case. We note that no points are shown for the Mixed mode in Fig This is because, for every case considered in our study we found that the Mixed mode was never the minimum-energy configuration. For Bo = 0, we found that the

33 20 Mixed mode cannot represent a global minimum in free energy for any value of 0 < q < 1 since, for fixed surface topology, E M does not have a local minimum, i.e. E M q 0 for 0 < q < 1. We also did not observe the Mixed mode in any of the corresponding MD simulations that we will discuss in Chapter 3. The Mixed mode has been observed in MD [47, 46] and lattice Boltzmann simulations [48] of droplets on surfaces patterned with square pillars. Apparently, this wetting mode is sensitive to details of the surface geometry. Thus, we exclude the Mixed mode in our discussion below. 2.3 Results and Discussion Below, we present various predictions of our model. In most of the discussion below, we focus on the limit of vanishing Bond number, for which the effects of gravity are unimportant. The small Bond-number limit is of interest for many applications, including electrowetting, microfluidic devices, and Lab-on-a-Chip systems [84, 85, 86]. The zero Bond-number limit is also appropriate for comparing our results to those from classical MD simulations. We presented such a comparison in Chapter 3. We include a discussion on the influence of gravity (finite Bo) on droplet configurations and wetting modes. We also include a comparison of our model predictions to experiment, for experiments in which the drop sizes span a range including both small and finite Bond numbers Contact angle discontinuity One of the interesting features of surface roughness is that it introduces discontinuities in the apparent contact angle as the droplet size or surface topology is varied in a systematic way. These discontinuities are due to both wetting-mode

34 21 Figure 2.6: The apparent contact angle as a function of Ḡ for constant H (= 0.3) and W (= 0.5) for Bo = 0. Discontinuities in the contact angle result from pinning/depinning transitions, as well as from wetting-mode transitions. transitions and pinning/de-pinning transitions of the contact line. To illustrate this point, we consider the case of Bo = 0, for which gravity does not play a significant role. The apparent contact angle for the wetting mode with the minimum free energy is plotted as a function of Ḡ in Fig. 2.6, for fixed values of W = 0.5 and H = 0.3. Each equilibrium wetting state in Fig. 2.6 is identified by a letter indicating the wetting mode, followed by a number denoting the number of grooves beneath the droplet. The W 1 configuration, in which the droplet is in the Wenzel state with one liquid-filled groove beneath it, occurs for the largest values of Ḡ. For Ḡ > 0.695, the contact line is de-pinned in the W 1 state, so that θ = θ e = 126. As the grooves become narrower, the contact lines move toward the step edges with constant apparent contact angle until contact-line pinning occurs

35 22 Table 2.1: The effect of the Bond number on the wetting mode and apparent contact angle θ of a droplet (with θ e = 126 ) for different scaled surface topologies. G = W = H = 0.1 G = 0.15 W = 0.1 G = W = H = 0.01 H = 0.05 Bo θ Wetting θ Wetting θ Wetting mode mode mode C W C C W C C W C C W C C W C C W C C W C C W C C W C 83 at Ḡ = While the contact lines remain pinned, the apparent contact angle increases with decreasing groove width, because it is energetically unfavorable for the drop to reduce its contact angle by jumping across a groove. As the groove width decreases further, a discontinuity in the apparent contact angle eventually appears due to a wetting-mode change from W 1 to C 1, followed by a much larger second discontinuity associated with the transition from C 1 to C 2. The apparent contact angle is reduced sharply to θ e when it becomes energetically favorable for the droplet to cover two grooves. In the C 2 state, contact-line pinning occurs again as Ḡ is reduced below 0.127, and the apparent contact angle increases with decreasing groove width Equilibrium wetting states One interesting feature of our model in the limit of small Bond numbers is that the apparent contact angle and the equilibrium wetting state of a droplet are not affected by droplet size when the geometrical parameters of the surface are scaled with R 0. This is apparent from inspection of Eqs , and is also illustrated in

36 23 Table 2.1 which lists wetting modes and contact angles for droplets as a function of the Bond number. The entries in Table 2.1 can be interpreted as a series of experiments on different surfaces in which the drop size is increased along with the geometrical parameters describing the surface topology, such that Ḡ, H, and W remain constant. These results show that the contact angle and wetting mode remain constant and independent of drop size for sufficiently small Bond numbers (drop sizes). As the Bond number (drop size) increases beyond a threshold that depends on surface geometry, the influence of gravity becomes significant and the contact angles and wetting modes become dependent on drop size. As we shall elaborate in next chapter, the drop-size independence of the wetting mode/contact angle for scaled surface geometries indicates that nanoscale MD simulations can be used to predict wetting behavior for droplet sizes and surface topologies beyond the nanometer scale (for which this technique is typically used). To explore the full range of wetting behavior predicted by our model, we construct wetting phase diagrams illustrating droplet wetting modes and apparent contact angles as a function of H, W, and Ḡ in the small Bond-number limit. Figures 2.7 and 2.8 show cross-sections of the wetting phase diagram for two different values of W on hydrophobic surfaces with θe = 126 and θ e = 115. These phase diagrams show that the Wenzel mode is energetically favorable for large groove widths and small step heights. In these surface geometries, the energetically unfavorable interaction of the droplet with the groove walls is minimized. Conversely, the Cassie mode is favored for narrower and/or taller grooves. The contact angle is independent of groove height in all the Cassie configurations. The Epitaxial Cassie (EC) mode becomes the lowest-energy state when the dimensions of the groove pattern become comparable to drop size. This occurs with θ e = 126 for wide steps (relative to the droplet s basal length)

37 24 (a) (b) Figure 2.7: Cross-section of the wetting phase diagram for two different hydrophobic surfaces, θ e = 126 (a) and θ e = 115 (b), with fixed W = 0.2 and Bo = 0.

38 25 (a) (b) Figure 2.8: Cross-section of the wetting phase diagram for two different hydrophobic surfaces, θ e = 126 (a) and θ e = 115 (b), with fixed W = 0.5 and Bo = 0.

39 26 separated by sufficiently high and wide grooves that inhibit the droplet from bridging a single groove. The phase diagrams for the two values of θ e are qualitatively similar although with decreasing θ e, the boundary separating the Wenzel and Cassie modes shifts to lower Ḡ. This shift can be understood as resulting from the more favorable liquid-solid interaction with lower θ e, which creates less of an energetic penalty for liquid to reside within the grooves. Within the Wenzel and Cassie regions, we see a wider range of wetting configurations with larger values of n (e.g., W 4, W 5, and C 6 ) with decreasing θ e. This shift to larger n can be understood in terms of the increasing basal area of the drop with lower θ e. The boundaries of the Epitaxial Cassie region are shifted to larger values of W as the value of θe is reduced. As a result, the Epitaxial Cassie region disappears for W = 0.5 over the range of parameters shown in Fig. 2.8(b), but it can be recovered by increasing the step width to W 0.9. Defining a superhydrophobic surface as one for which the contact angle is larger than 150, the results in Fig. 2.7 show that for W = 0.2, most of the topological region corresponding to the C 1 wetting mode, as well as portions of the W 1 and C 2 domains, are superhydrophobic. For wider steps, the EC mode is superhydrophobic in Fig. 2.8(a), and the contact angles associated with this mode are in the same range as those for water on the Lotus leaf [7, 87]. In all of these wetting modes (and especially in the EC mode), the droplet rests on a relatively small number of steps, so its size is comparable to the length scale of the surface pattern. This finding is in line with results from the MD simulation study of Yang, Tartaglino, and Persson, who probed liquid droplets in contact with self-affine fractal surfaces, and found that long-wavelength roughness plays an important role in determining the contact angle [88]. Our results may provide an indication of why multiple-scale

40 27 roughness that reaches droplet size scales has been observed in conjunction with superhydrophobicity [4, 7, 8]. Although most of the contact angles reported in Figs. 2.7 and 2.8 are greater than θ e, there are regions in the phase diagram [e.g., portions of the W 2 region in Fig. 2.7(a) and the W 1 region in Fig. 2.7(b)] where the contact angle is less than θ e. These regions tend to be characterized by wide grooves relative to the droplet size. Here, the grooves have a sufficiently large volume-to-wall-area ratio that a significant amount of liquid can reside in them. For the W 1 wetting mode [cf., Fig. 2.7(b)], this reduces the volume of the drop that resides above the groove to the point that there is not sufficient liquid to produce a cylindrical cap with a contact angle of θ e. A similar argument applies for the W 2 configuration, where we see [in Figs. 2.7(a) and 2.8(b)] that droplets with θ < θ e lie close to the W 2 W 1 boundary. Here, the decrease in the liquid-vapor interfacial energy in going from a W 1 configuration to a W 2 configuration more than compensates for the increase in the solid-liquid interfacial energy, thereby resulting in a lower energy for the W 2 configuration. The resulting W 2 configuration has a lower contact angle than θ e because the volume of the cylindrical cap above the grooves is not sufficient to have θ e Comparison to Cassie and Wenzel models Contact-angle isocontours based on the Cassie and Wenzel models are presented in Fig. 2.9 for θ e = 126 and two step widths of W = 0.2 and W = 0.5. These correspond to the wetting phase diagrams of our model shown in Figs. 2.7(a) and 2.8(a). In the Cassie model (Eq. 1.3), φ s = 1 G/(W + G), and for Wenzel model (Eq. 1.2), r = 1 + 2H/(W + G) for the grooved geometry presented in Fig. 2.1.

41 28 Figure 2.9: Contact-angle isocontours with θ e = 126 based on the Wenzel and Cassie models for two step widths: (a) Wenzel model, W = 0.2 (b) Cassie model, W = 0.2 (c) Wenzel model, W = 0.5 (d) Cassie model, W = 0.5. As shown in Fig. 2.9, the Wenzel model is only applicable over a small portion of the Ḡ H domain, which roughly coincides with the region where we observe the Wenzel wetting mode in our model. However, for large Ḡ, our model predicts the Wenzel mode for a larger range of H than is seen for the Wenzel model. Equation 1.2 yields contact angles that increase monotonically with H for a fixed value of Ḡ, and predicts superhydrophobic contact angles ( > 150 ). In contrast, our

42 29 model shows only a weak dependence of the contact angle on H for a fixed Ḡ unless a wetting-mode boundary is crossed, in which case the contact angle varies discontinuously with H due to contact-line pinning. The Wenzel contact angles from our model do not tend to fall within the superhydrophobic range. The Cassie model predicts that the contact angle is independent of groove height and that it increases monotonically with increasing groove width. Although our model also shows that the contact angle is independent of groove height, it exhibits discontinuities associated with contact-line pinning as the wetting modes change (e.g., from C 1 to C 2 ). The Cassie model does not predict the Epitaxial Cassie mode, which we find for wide steps, as shown in Fig. 2.8(a). Compared to the Cassie and Wenzel models, the main advantages of our model are that it provides a detailed description of wetting modes (e.g., we distinguish W 1 from W 2 ) and it accounts for pinning/de-pinning of the contact line at step edges. Such effects are not taken into account in the Cassie and Wenzel models. 2.4 Influence of gravity We now turn our attention to the influence of gravity, which becomes increasingly important as the Bond number increases, as was shown in Table 2.1. In Table 2.2, we compare the predicted wetting modes and contact angles of water droplets (ρ = 997 kg/m 3 and σ lv = N/m) of different sizes to those for the zero-bond-number (g = 0) counterparts. A comparison of the results for Bo = 0 to those for nonzero Bond number indicates that droplets with radii as large as several hundred microns can fall within the small Bond-number limit. The effect of gravity becomes significant when the droplet radius exceeds a threshold value that depends on the surface geometry. We find that the threshold radius decreases

43 30 Table 2.2: The effect of gravity on the wetting state and contact angle θ of a droplet on a grooved surface. G = W = H = 0.1 (mm) G = W = H = 0.01 (mm) R o θ Wetting θ Wetting θ Wetting θ Wetting (mm) mode mode mode mode (g = 0) (g = 0) W W C C EC 161 EC 142 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C 94 with decreasing size of the surface features. For example, the threshold radius decreases from ± mm for G = H = W = 0.1 mm to ± mm for G = H = W = 0.01 mm. Once the threshold radius is reached, the droplet assumes an extended surface configuration, in which it covers an increasing number of surface grooves and possesses decreasingly smaller apparent contact angles as its size increases. As the Bond number increases further, we expect the interplay between gravity and interfacial energy to lead to a flattening of the droplet profile, such that the droplet no longer has a constant-curvature cross-section. [90] Such deformation effects are not included in our model Comparison to experiment To further confirm the validity of our theoretical analysis, we compared our model predictions to available experimental data for droplet wetting on grooved solid surfaces fabricated from a relatively hydrophobic material (pphex, with θ e = 100 ), with a fixed groove height of 3 µm and equally-sized groove and step widths of 5,

44 31 Figure 2.10: The apparent contact angle as a function of the ratio of the basal length of the drop profile d 0, to the groove width G for different drop volumes on a hydrophobic substrate (pphex); (a) 0.4 nl, (b) 12 nl, (c) 40 nl and, (d) 4.5 µl. Predictions of the model are compared to experimental data [89]. 10, 25 and 100 µm [89]. In this study, wetting is anisotropic and apparent contact angles were measured from optical images obtained in the directions parallel and perpendicular to the grooves. Since our theoretical model is two-dimensional and assumes the droplet to be infinitely long in the direction parallel to the grooves, we compare our predictions to experimental contact-angle measurements made from a parallel view of the droplet. Figure 2.10 presents a comparison of the measured contact angles for differentsized water droplets on pphex to the predictions of our model for the Cassie and Wenzel configurations on a surface characterized by the same values of Ḡ, H, W,

45 32 and θ e as in the experiments. For this comparison, we use the basal length of the drop profile in the parallel view, d 0, to scale the geometrical parameters of the surface, as was done in the experiments. According to our model, the Wenzel mode is the lowest-energy state for all of the cases shown in Fig For small droplets [Figs. 2.10(a) and (b)], the experimental contact angles agree well with those predicted for the Wenzel mode for small values of d 0 /G, and those predicted for the Cassie mode for large values of d 0 /G. Although the Wenzel mode is energetically preferred, it is possible that the Cassie mode is metastable and long-lived in the experiments. Even though the Cassie mode is not the lowest-energy state, our model can predict the correct contact angle especially if the Bond number is small. With increasing drop size [Fig. 2.10(c) and (d)], the experimentally measured contact angles become increasingly larger than those predicted by the model. This is likely because our assumption of a constantcurvature drop profile is no longer valid for these drops. The Bond numbers associated with the experiments range from for the smallest (0.4 nl) drops to about 0.14 for the largest (4.5 µl) drops. As can be seen in Table 2.1, gravity has a non-negligible effect on the contact angles predicted by our model at Bond numbers as small as Although the model predictions presented in Fig correspond to non-zero values of the Bond number, we do not account for the effect of gravity-induced shape deformation on the equilibrium drop configuration. Thus, our model under-predicts contact angles for large drops with significant Bond numbers.

46 Summary We presented a mathematical model based on free-energy minimization to predict the equilibrium wetting state of a two-dimensional liquid droplet on a periodicallygrooved solid surface. The relationship between the resulting contact angle and the geometry of the surface has been established for different droplet configurations and the effect of drop size on the equilibrium contact angle has been described in terms of the Bond number. For small droplets (characterized by Bond numbers much smaller than unity), the relationship between contact angle and drop size is non-monotonic. However, if the geometrical parameters characterizing the surface are scaled with the drop radius, the contact angle becomes independent of the scaled geometry. We used our model to construct phase diagrams depicting the dependence of the wetting configurations and their corresponding contact angles on geometrical parameters describing the surface. The resulting wetting phase diagrams and contact-angle isocontours can serve as guidelines for designing surfaces with desired wettability. Interestingly, we find the largest contact angles when the size of the drop is comparable to the length scales characterizing the surface topology. While the contact angles predicted by our model exhibit some of the qualitative trends in the Cassie and Wenzel models, significant quantitative differences occur. Compared to the Cassie and Wenzel models, the main advantages of our model are (i) it provides a more detailed description of droplet wetting configurations and; (ii) it accounts for pinning/de-pinning of the contact line at step edges. Analysis of our detailed model indicates that contact-line pinning is essential for achieving large contact angles. Our contact-angle predictions are also in good agreement with experimentally-measured contact angles of small drops, for which gravity-induced

47 interface deformation is negligible. 34

48 Chapter 3 Molecular Dynamics Simulation of a Droplet On Nanopatterned Solid Surfaces 3.1 Introduction Molecular dynamics (MD) simulations can be used to describe many aspects of droplet wetting on structured surfaces. For example, several MD studies have probed the influence of surface patterning on droplet wetting modes and contact angles [92, 93, 47, 94, 46, 95, 96, 97]. Recent studies have demonstrated that MD simulations can describe contact-angle hysteresis [98, 99, 100]. A clear strength of these simulations is that they do not rely on continuum assumptions, they are able to resolve the influence of atomic-scale phenomena on wetting, and, provided that sufficiently large systems can be probed, they can resolve the interplay of molecular and continuum phenomena. For example, the studies of Weijs et al. and Ritos et al. demonstrate that nanoscale phenomena can play an important

49 36 role for bubbles [101] on superhydrophobic surfaces and in advancing and receding contact angles of nanoscale droplets on solid surfaces [102]. However, a concern has been the limited range of length and time scales that can be covered in a typical MD simulation, as these do not extend far beyond the nanoscale and they fall short of the length and time scales relevant to experimental studies. Recent studies have demonstrated that the time limitation can be overcome in special simulations that focus on the kinetics of wetting transitions [75, 76, 77, 78, 79]. The recent work of Giacomello et al. [79] indicates that their continuum model can describe many aspects of the Cassie-to-Wenzel transition although a precise continuum description of the free-energy barrier remains elusive. Despite this progress, many aspects of the relationship between nanoscale MD simulations and macroscale experiments remain unresolved. In this chapter, after a brief overview of MD methodology, we address the relevance of nanoscale MD simulations to the wetting of liquid droplets on macroscopic patterned surfaces in experimental systems. We demonstrate that the wetting modes and contact angles of droplets on grooved solid surfaces are independent of droplet size if the dimensions of the pattern are scaled by the droplet radius. Our results are consistent with a macroscopic model [96] for the free energies of various droplet wetting modes in the limit of a negligible gravitational effect. Contact angles resulting from our MD simulations are also in excellent agreement with those predicted by the macroscopic model. Further, MD simulations exhibit metastable wetting states, similar to experiments. Thus, our studies indicate that MD simulations can provide considerable insight into macroscopic experimental studies of droplet wetting.

50 Basics of Molecular Dynamics Molecular dynamics is a deterministic approach to investigate the macroscopic properties of systems by following the evolution at molecular scale. This microscopic information is converted to macroscopic quantities such as pressure, energy, heat capacities, etc., by using statistical mechanics tools mainly based on the ergodic hypothesis: the statistical ensemble averages are equal to the time averages of the system. The classical MD simulations boils down to numerically solving Newton s equations of motion for N particles which build up the system. Newton s equation states that the acceleration of a particle i is proportional to the force acting on the particle i, F i = m i r i (3.1) The force can be expressed as the gradient of the potential function, F i = i U. In the absence of external forces, the potential can be represented in the simplest case as a sum of pairwise interactions: U = N N u(r ij ) (3.2) i=1 j>i where r ij = r i r j, and the condition j > i avoids double counting of pair (i, j). The best known of these potentials, originally proposed for liquid argon, is the Lennard-Jones (LJ) potential [103, 104]. For a pair of atoms i and j located at r i and r j the potential energy is [ ( ) 12 ( ) ] 6 σ σ 4ɛ if r ij r c u(r ij ) = r ij r ij 0 otherwise, (3.3)

51 38 The parameter ɛ is the depth of the potential well and governs the strength of the interaction, and σ is the distance at which the inter-particle potential is zero. The interaction potential is usually cut off at some large enough distance, r c for computational reasons. For d dimensional system, Eq. 3.1 and Eq. 3.2 result in a set of d N second-order differential equations, and a total of d N degrees of freedom. This set of equations are discretized in time and then integrated with a finitedifference algorithm for given initial positions and velocities of particles. A variety of numerical algorithms have been developed for integrating the equations of motion. However just few of them are computationally efficient and have achieved widespread use. Some popular algorithms are central difference (Verlet, leapfrog, velocity Verlet, Beeman algorithm)[105, 106], predictor-corrector [107], and symplectic integrators [108] Boundary Condition In macroscopic systems, only a small fraction of atoms are close enough to a wall; however, in a typical MD system a considerable fraction of atoms are adjacent to walls. Therefore, to correctly capture the properties of bulk fluid atoms, the finite-size effects should be minimized. One possible approach is replicating the simulation region and using periodic boundary conditions, as shown schematically in Fig The central simulation region is surrounded by identical copies of itself; therefore, the simulated system is pseudo-infinite. The image particle moves in exactly the same direction as the particle in the central cell. Implementing periodic boundary conditions has two consequences. First, as a particle leaves the central simulation region, it immediately reenters the central region through the

52 39 Figure 3.1: Illustration of periodic boundary condition. opposite side. Second,a wraparound effect occurs, wherein particles lying within a distance r c of a boundary interact with the atoms near the opposite boundary. This effect must be taken into account in both the integration of the equations of motion and the interaction computations [109] Alternative Ensembles In order to connect macroscopic properties of the system to its microscopic behavior, time independent statistical averages are often introduced. On the macroscopic scale, the thermodynamic state of the system is usually defined by a set of parameters, such as the temperature T, the pressure P, and the number of particles N. Other thermodynamic properties are derived by using equations of state. In microscopic systems, mechanical or microscopic state is defined by atomic positions q and momenta p, also referred to as coordinates in a multidimensional

53 40 Table 3.1: Commonly used ensembles in statistical mechanics. Each ensemble is defined by a sub-set from the following thermodynamic variables: energy E, temperature T, pressure P, volume V, number of particles N,and chemical potential µ. Ensemble Fixed quantities Comments Microcanonical N, V, E This corresponds to an isolated system. Canonical N, V, T Temperature is kept constant by using a thermostat. Isobaric-Isothermal N, P, T Used when the density of the system is unknown. Grand Canonical µ, V, T The system is open - Used where the chemical composition of the system is unknown. space or phase space. An ensemble is a collection of all possible systems that have different microscopic states but identical macroscopic or thermodynamic states. Statistical ensembles are characterized by thermodynamic variables such as energy E, temperature T, pressure P, volume V, number of particles N, and chemical potential µ, and imply the system variables that are regulated or conserved. Commonly used ensembles are listed in Table The relevance of molecular-dynamics simulations to macroscopic systems Simulation Details We consider the wetting of a periodically grooved solid surface by an infinitely long (periodic) cylindrical Lennard-Jones (LJ) liquid droplet. The grooves are characterized by width G and height H, and are separated by steps of width W, as shown in Fig The surface is periodic in the y direction and infinite in x direction (see Fig. 3.2). The cylindrical droplet is infinite along its long axis (the x

54 41 Figure 3.2: Side view (a) and top-down view (b) of the periodically-grooved surface characterized by groove width, G, groove height, H, and step width, W. axis in Fig. 3.2), which allows us to use periodic boundary conditions to simulate a larger droplet for a fixed number of atoms [51]. Moreover, since the curvature of the contact line of a cylindrical droplet is zero, the contact angle of a cylindrical droplet is not affected by line tension [110]. The number of atoms used to simulate the droplets ranged from 3,196 to 28,550. To describe pair-wise interactions between fluid (f) atoms, we use a truncated LJ potential (Eq. 3.3) with r c = 3.8. σ ff, where σ ff is the length parameter. The energy parameter is fixed at a value of ɛ ff /k B = 120 K, where k B is the Boltzmann constant. σ ff and ɛ ff are used as the length and energy scales, respectively, to make the equations dimensionless. We also use a LJ potential to describe the interaction of fluid-phase atoms with the solid surface (s). For patterned surfaces, the sum of pairwise LJ interactions can be coarse-grained to enhance the computational efficiency. Here, we use our recently developed patterned-surface potential (PSP) [111] (see Appendix) for describing fluid-solid interactions. The parameters for this potential are set to σ fs = 0.921σ ff, σ ss = 1.2σ fs / 2, and ρ s = 2/σ 3 ss, with the value of ɛ fs selected so as to achieve a desired equilibrium contact angle for the droplet on a flat substrate, as described below. To simulate the fluid-phase atoms, we used MD in the canonical ensemble,

55 42 with a constant number, volume, and temperature. We integrated the equations of motion using a fourth-order predictor-corrector algorithm with a time step of 4 fs, and used the Nosé-Hoover thermostat [112, 113] to maintain a temperature of T = 0.7ɛ ff /k B. To generate a cylindrical droplet of radius R, we placed 16R 2 atoms at fcc lattice sites in half of an 8 2R 2R rectangular prism, and allowed the system to equilibrate at a temperature of T = 2ɛ ff /k B for about 1 ns. Subsequently, the system was cooled to T = 0.7ɛ ff /k B in a rectangular prism with dimensions of 8 7R 7R to form a condensed cylindrical droplet. For a Cassie initial configuration, the resulting droplet was cut into halves and the upper half was placed over the solid surface such that the center of the mass of the drop was centered over a groove, with the lower-most atom was initially a distance of 0.8σ ff above the steps to avoid strong repulsion from the solid surface. To achieve an Epitaxial Cassie initial configuration, the center of mass of the droplet was centered over a step. For simulations with a Wenzel initial configuration, the solid-liquid interaction energy was initially increased to allow the liquid to fill the grooves and subsequently reduced to its intended value for the simulation. MD simulations with different initial droplet configurations were performed for different surface geometries, and the apparent contact angle θ was monitored as a function of time. Since the observed drop profiles had constant curvature, we determined the contact angle by measuring the apex height and basal length of the best-fit circular interface for the droplet profile [93]. The interface position was determined by calculating the local density of the fluid and recognizing that the maximum variation in local density occurs at the interface [114]. The system was considered to be equilibrated when the contact angle as a function of time reached a plateau value. The total simulation time for achieving equilibrium was 5-15 ns, depending on the system size.

56 43 Figure 3.3: The equilibrium contact angle on a flat surface θ e as a function of the relative energy parameter ɛ r. The error bar represents the uncertainty in the computed contact angles. The surface is designated as hydrophobic for θ e 90 and hydrophilic for θ e < 90. The affinity of the liquid for the solid surface was varied by changing the relative liquid-solid interaction energy ɛ r, which is defined as ɛ r = ɛ fs /ɛ ff. For different relative liquid-solid interaction energies, the corresponding contact angle θ e on a smooth substrate was used to characterize the surface as either hydrophobic or hydrophilic. For sufficiently small ɛ r, the surface is hydrophobic, with θ e 90, as shown in Fig Figure 3.3 indicates that the surface is hydrophobic when ɛ r < 0.5 and hydrophilic when ɛ r 0.5. For all of the simulations in this study, we used ɛ r = 0.25, corresponding to an equilibrium contact angle of θ e = 126 ± 5 on a smooth substrate. As we will discuss below, a thermodynamic analysis of our system [96] indicates that the equilibrium contact angle is the key variable that expresses the identities of the liquid and solid-substrate material when the effects

57 44 of gravity are negligible. 3.4 Results and Discussion Figure 3.4: Various wetting modes observed in this study. Figure 3.4 shows the different wetting modes observed in this study. These include the Cassie, Wenzel, and Epitaxial Cassie modes. In a previous chapter, we showed in the limit Bo 0, the effect of gravity is negligible and the dimensionless free energy per unit length becomes independent of drop size. Indeed, this implies that a nanoscale MD simulation can predict macroscale wetting configurations, as long as the droplet is sufficiently large that its interfacial energies exhibit continuum-like behavior. Figure 3.5: The effect of droplet size on wetting mode and apparent contact angle for surfaces with the same scaled topography (Ḡ = H = 0.45 ± 0.03, and W = 0.32±0.02): (a) R 0 = (50±2) σ ff, (b) R 0 = (25±2)σ ff, and (c) R 0 = (12±2)σ ff.

58 45 To test the ramifications of this prediction, we ran a series of MD simulations. In the first set of simulations, we studied droplets of different sizes on surfaces with the same scaled surface topography. The droplet radii from MD are R 0 = (12 ± 2) σ ff, (25±2)σ ff and (50±2)σ ff, corresponding to 3196, 12,433, and 28,550 atoms, respectively, and the surface topography is characterized by H = 0.45 ± 0.03, Ḡ = 0.45 ± 0.03, and W = 0.32 ± The results are shown in Fig. 3.5, and demonstrate that the droplet is in the Cassie mode in all three cases. The contact angles of the droplets in Figs. 3.5(a), (b), and (c) are 140 ± 5, 141 ± 5, and 142 ± 5, respectively. Using n = 1 and θ e = 126 (the same values as in the MD simulations) in Eq. 2.8 for the same scaled surface parameters, the corresponding model prediction for the contact angle in all three cases is θ = 139, which agrees well with the MD results. In addition to exhibiting a droplet-size invariance of the wetting mode and contact angle for a fixed scaled surface topography, the MD simulations also show the same wetting-mode transitions as the model [96]. We performed a series of simulations in which Ḡ, H, and W were varied, and we recorded the wetting mode and contact angle for each configuration. The results are shown in Figs , where the effect of groove width, step width, and step height on contact angle are compared to those predicted by the model for the same wetting configuration. The effect of the groove width on contact angle and wetting mode is shown in Fig. 3.6 for two different values of the step height ( H = 0.1±0.01 and H = 0.4±0.03) with a fixed step width of W = 0.32 ± There is excellent agreement between the wetting modes and contact angles predicted by the macroscopic model and those obtained from MD simulations for the corresponding lowest-energy wetting modes. Figure 3.6 also shows the role of surface roughness in producing discontinuities in the apparent contact angle as the surface topography or drop size changes. For

59 46 (a) (b) Figure 3.6: The effect of groove width Ḡ on apparent contact angle for a step width of W = 0.32 ± 0.03 and step heights (a) H = 0.10 ± 0.01 and, (b) H = 0.40 ± The solid lines represent predictions of the mathematical model in ref. [96] and the symbols correspond to MD simulations. The vertical error bar depicts the representative uncertainty in the MD contact angles and the horizontal error bars indicate the uncertainty in the MD value of R 0.

60 47 the widest gap in Fig. 3.6(a), the droplet remains in the W 1 configuration as its contact angle increases with decreasing gap width the contact line is pinned at the step edges until it transitions to the W 2 mode at Ḡ = 0.55 ± 0.04, where the contact line is unpinned and the contact angle is abruptly reduced to θ = θ e. As Ḡ is further reduced, the contact line remains unpinned, and moves toward the outer step edges with nearly constant apparent contact angle until it becomes pinned again at Ḡ = 0.39±0.03 and the contact angle begins to increase in the W 2 mode. A small jump in the contact angle is observed in the transition from W 2 to C 2 as the liquid that resided within the gap in the Wenzel mode is pushed above the surface roughness in the Cassie mode. The contact line remains pinned in the C 2 mode as the gap is further reduced, until it becomes energetically favorable for the drop to jump across a third groove to enter the C 3 mode. We note that the contact line remains pinned in the jump from C 2 to C 3 (as apposed to the jump from W 1 to W 2 ). For higher steps, as shown in Fig. 3.6(b), we observe a sequence of such pinned wetting-mode transitions. Comparing Fig. 3.6(b) to Fig. 3.6(a), we note that the W 2 wetting mode becomes energetically unfavorable for higher steps. Figure 3.7 illustrates how the apparent contact angle and wetting mode of the droplet are affected by the step width W, for two different values of the groove width (Ḡ = 0.16 ±0.01 and Ḡ = 0.64 ±0.05) with a fixed step height of H = 0.64 ± For the widest step in Fig. 3.7(a), the droplet is in the C 1 state, with an unpinned contact line and θ = θ e. As the step width is reduced, the contact line becomes pinned to the step edges and the contact angle increases. After a transition to the unpinned C 2 state, reducing the step width causes the droplet to go through a series of wetting-mode transitions in which the contact line remains completely pinned. For the wider groove in Fig. 3.7(b), the droplet begins in the pinned Epitaxial Cassie (EC) state, and the contact line remains pinned as

61 48 Figure 3.7: The effect of step width W on apparent contact angle for a step height of H = 0.64±0.05 and groove widths of (a) Ḡ = 0.16±0.01, and (b) Ḡ = 0.64±0.05. The solid lines represent predictions of the mathematical model in ref. [96] and the symbols correspond to MD simulations. The vertical error bar depicts the representative uncertainty in the MD contact angles and the horizontal error bars indicate the uncertainty in the MD value of R 0.

62 49 the droplet undergoes a series of wetting-mode transitions with decreasing step width. Figure 3.8 shows the dependence of the contact angle and wetting mode on the step height H for two groove widths (Ḡ = 0.16 ± 0.01 and Ḡ = 0.45 ± 0.04) with a fixed step width of W = 0.32 ± The contact angle and wetting mode are essentially independent of step height for the smaller gap [Fig. 3.8(a)]. For the larger gap in Fig. 3.8(b), there is a wetting-mode change from W 1 to C 1 as the step height increases. However, within the uncertainty in determining apparent contact angles from the MD simulations, the contact angle remains nearly the same. This is consistent with our macroscopic model [96], as well as with previous MD simulations [95] and experiments [115]. Thus, we find excellent agreement between the wetting modes and contact angles predicted by MD simulations and our macroscopic model over a wide range of surface-feature and droplet sizes, for which the effect of gravity is negligible. This indicates that if the macroscopic system and the nanoscale MD simulation possess the same feature-to-droplet ratio sizes, then the MD simulation can predict the macroscopic wetting mode and contact angle. Going beyond this agreement, we demonstrate how MD simulations can indicate wetting-mode multiplicity due to the existence of free-energy barriers between various droplet configurations. To show how the final observed configuration of a droplet is influenced by its initial configuration, we considered the same droplet in three different initial configurations, as shown in the top panels of Figs. 3.9 and In the first of these, which can be seen in Figs. 3.9 A(i), 3.9C(i), and 3.10A(i), the drop is initially in the Cassie mode with its center of mass over a groove. In a second initial configuration, the drop is in the Wenzel mode, as shown in Figs. 3.9B(i), 3.9D(i), and 3.10B(i). Finally, in a third initial condition, the droplet is in the

63 50 Figure 3.8: The effect of step height H on apparent contact angle for a step width of W = 0.32±0.03 and groove widths (a) Ḡ = 0.16±0.01, and (b) Ḡ = 0.45±0.04. The solid lines represent predictions of the mathematical model in ref. [96] and the symbols correspond to MD simulations. The vertical error bar depicts the representative uncertainty in the MD contact angles and the horizontal error bars indicate the uncertainty in the MD value of R 0.

64 51 Cassie mode with its center of mass over a step, as shown in Fig. 3.10C(i). The final configuration of the droplet after 10 ns of MD simulation starting with each of these initial states is shown in Figs. 3.9A(f)-D(f) and 3.10A(f)-C(f). Figure 3.9: The effect of initial droplet configuration (i) on the final wetting configuration (f)realized in MD simulation for surfaces with W = 0.32 ± 0.03 and H = 0.32 ± 0.03, Ḡ = 0.40 ± 0.03 (A and B), and H = 0.8 ± 0.06, Ḡ = 0.48 ± 0.04 (C and D). We find that wetting-mode multiplicity is strongly dependent on the topography of the surface. For narrow steps ( W = 0.32 ± 0.03), the observed wetting mode is unique and independent of the initial configuration of the droplet when H = 0.40±0.03 and Ḡ = 0.48±0.04. An example of this wetting-mode uniqueness is shown in panels A and B in Fig. 3.9, where the droplet s final configuration is C 1 regardless of whether it is initially in a Cassie or Wenzel configuration. However, as the groove width increases to Ḡ = 0.48 ± 0.04 and the step height increases to H = 0.40 ± 0.03, wetting-mode multiplicity appears, as shown by panels C and D in Fig Here, when the droplet is initially in the Cassie mode [Fig. 3.9C(i)], its final configuration is C 1 [Fig.3.9C(f)], whereas the Wenzel initial configuration [Fig. 3.9D(i)] leads to a W 1 final configuration [Fig. 3.9D(f)].

65 52 Figure 3.10: The effect of initial droplet location (i) on final observed wetting configuration (f) for surfaces with pattern comparable to drop size. (For A, B and C, H = 0.64 ± 0.05, Ḡ = 1.12 ± 0.09, W = 0.8 ± 0.06). Figure 3.10 shows an example in which both the Wenzel and Epitaxial Cassie states are observed for the same surface topography with different initial conditions. When the initial configuration of the droplet is either a Cassie state, with the center of mass over a groove, or a Wenzel state (panels A and B in Fig. 3.10), the final configuration of the drop is the W 1 mode. For a Cassie initial configuration with the center of mass over a step, the drop achieves an Epitaxial Cassie final configuration [Fig C(f)]. When the final wetting state depends on the initial position of the droplet, there is a free-energy barrier separating the metastable state(s) from the global-minimum free-energy state. Based on the results here, we cannot comment on the magnitude of the free-energy barrier for a given transition in a macroscopic system. Theoretical methods exist to quantify these barriers within MD simulations [117, 118, 125] and this would be discussed in next chapter.

66 53 However, without applying these special techniques, canonical MD simulations can indicate the morphological trends that are likely to lead to wetting-state multiplicity in a macroscopic system. We can combine our observations of the wetting modes and contact angles to construct a phase diagram of possible wetting modes as a function of the scaled geometrical parameters describing the surface. Our computational results lead to a three-dimensional phase diagram in terms of H, Ḡ, and W, covering parameter ranges of [0.08, 1.2], [0.08, 0.8], and [0.16, 0.8], respectively. The increment sizes for H, Ḡ, and W used in the search were 0.08, 0.04, and 0.16, respectively. For each set of surface parameters, we considered three different initial droplet configurations (Cassie with the droplet center of mass over a groove, Cassie with the center of mass over a step, and Wenzel). Two different constant- W cross sections of the wetting phase diagram on a hydrophobic surface with θ e = 126 ± 5 are shown in Figs and To construct these cross sections, we considered fixed values of H and W with a given initial condition and we searched for the critical value of Ḡ at which a wetting-mode transition occurred. To compare the MD phase diagrams to those obtained from the macroscopic model, phase boundaries between the most stable wetting modes predicted by the macroscopic model are shown by dashed lines in Figs and Regions I, II and III correspond to model predictions for the Wenzel, Cassie, and Epitaxial Cassie wetting modes, respectively. Figures 3.11 and 3.12 show that the Cassie state becomes favorable for narrow grooves and tall steps. Conversely, the Wenzel state is observed for wide grooves and short steps. The Epitaxial Cassie (EC) mode appears when the dimensions of the surface pattern become comparable to the size of the droplet (i.e., for wide and tall steps separated by sufficiently wide grooves). These figures also show that the

67 54 Figure 3.11: Cross-section of the wetting phase diagram for a hydrophobic surface with θ e = 126 and W = 0.16 ± Data points from the simulations are represented by symbols and colored regions indicate the predicted wetting regimes based on the simulations. Error bars have been shown for only a few points to improve visual clarity. However, the uncertainty in both H and Ḡ ranged from 0.01 to Phase boundaries between the most stable wetting modes predicted by the macroscopic model [96] are shown by dashed lines, and regions I and II correspond to the model predictions for Wenzel and Cassie modes, respectively. most stable wetting mode predicted by the macroscopic model is always among the wetting modes observed in MD simulations. However, MD simulations exhibit wetting-mode multiplicity in the regions neighboring wetting-mode boundaries predicted by the model. For the narrow steps ( W = 0.16 ± 0.01) in Fig. 3.11, only two wetting-mode domains (Cassie and Wenzel) with a single two-phase boundary are predicted by the model. The MD simulations also result in a dual-mode (Cassie- Wenzel) region in the vicinity of the predicted Cassie-Wenzel boundary. For the wider ( W = 0.80 ±0.06) steps in Fig. 3.12, a more complex wetting behavior is observed in MD simulations, with both dual-mode and triple-mode regions being

68 55 Figure 3.12: Cross-section of the wetting phase diagram for a hydrophobic surface with θ e = 126 and W = 0.8 ± Data points from the simulations are represented by symbols and colored regions indicate the predicted wetting regimes based on the simulations. Error bars have been shown for only a few points to improve visual clarity. However, the uncertainty in both H and Ḡ ranged from 0.01 to Phase boundaries between the most stable wetting modes predicted by the mathematical model [96] are shown by dashed lines, and regions I, II, and III correspond to the model predictions for Wenzel, Cassie, and Epitaxial Cassie domains, respectively. realized. This is due to the presence of a third wetting mode (i.e., Epitaxial Cassie) for wide steps. In this case, the macroscopic model predicts three twophase boundaries that meet at the triple point (Ḡ, H) = (0.44±0.03, 0.23±0.02) in Fig The triple mode (Epitaxial Cassie-Wenzel-Cassie) region resulting from MD simulations is quite large, encompassing nearly the entire model-predicted Epitaxial Cassie domain for H < 0.80, and indicating the presence of (presumably) metastable Cassie states within the model-predicted Wenzel region around the Wenzel-Epitaxial Cassie boundary. This is likely due to a large free-energy barrier

69 56 for the transition from the Cassie to the Epitaxial Cassie state when the groove width is comparable to the drop size. Nevertheless, there is good agreement between the model and MD simulation results in all three regions in the sense that the most stable mode predicted by the model is always among the modes realized by MD simulations. 3.5 Summary We investigated the wetting behavior of a LJ cylindrical droplet on a periodicallygrooved surface by using MD simulations. A systematic study on the effect of surface topographical parameters revealed that the drop size is an appropriate length scale for characterizing wetting states and their corresponding contact angles, implying that nanoscale MD simulations can be used to predict macroscale wetting behavior. This observation is consistent with our derived [96] thermodynamic model for droplet wetting in the limit of a negligible effect of gravity. Good agreement was found between contact angles obtained in MD simulations and those predicted by our macroscopic model [96]. We constructed drop-size-independent wetting phase diagrams representing possible equilibrium wetting configurations of a droplet as a function of surface topography. Here, we found that the most stable wetting mode predicted by the macroscopic model is always among the wetting modes observed in MD simulations. However, a droplet can achieve multiple wetting modes for a given surface morphology, each corresponding to a stable or metastable state. Metastability is especially prevalent near wetting-mode boundaries. The drop can remain trapped in metastable states over relatively long time scales, due to the presence of free-energy barriers impeding wetting-mode transitions. A similar phenomenon has been observed experimentally. Thus, our studies

70 57 indicate that suitably applied MD simulations can yield insight into the largelength-scale behavior of droplets on patterned surfaces. Such simulations can play a role in the design of patterned surfaces for optimal wettability.

71 Chapter 4 Kinetics of Droplet Wetting-mode Transitions on Grooved Surfaces: Forward Flux Sampling 4.1 Introduction As shown in Chapter 3, multiple minima of the free energy of a droplet on a structured surface promote multiplicity of wetting configurations. The energetically favored configuration may not be observable because of the large energy barrier between metastable and stable states. Predicting the rate of the wetting-state transition and the existing energy barrier to this transition will help to quantify the relative stability of wetting-state energy minima, which stability must be a factor in any proposed criteria for designing surfaces with a specific wettability. Although extensive effort, both theoretical and experimental, has been made to understand the droplet equilibrium wetting states, less attention has been paid to the kinetics of the wetting transition.

72 59 Using forward flux sampling, we study the effect of surface topology and droplet size on both Cassie to Wenzel (C W) and Wenzel to Cassie (W C) transition rates and mechanisms. The global-minimum wetting states that emerge from our nanoscale MD approach are compared to those that had been predicted by a macroscopic model for the free energy. A committor analysis indicates that the transition-state ensemble consists of droplets that are on the verge of initiating or breaking contact with the substrate below the grooves. 4.2 Forward Flux Sampling Many systems exhibit multiple local minima that correspond to stable or metastable states with switching events occur infrequently. These systems are difficult to simulate because there is little chance of observing these events in a typical simulation time. Different algorithms have been proposed to simulate rare events in various contexts [118, 119]. One class of these algorithms, known as forward flux sampling (FFS), uses trajectories developed using molecular dynamics [120, 121]. The basic idea of FFS is to use a series of interfaces to partition the phase space between stable and metastable states along an order parameter λ and employ a systematic, ratchet-like scheme to drive the system from the initial configuration to the final configuration. The initial state (A) is defined by λ < λ A = λ 0, the final state (B) by λ > λ B = λ n, and the intermediate interfaces by {λ i, λ i+1,..., λ n 1 }. It is required that λ i+1 > λ i for all i. FFS starts by performing a long MD run in the initial state region in order to store N 0 configurations that correspond to reaching the first interface, λ 0. The rate constant k A B for transitions from A to B can be calculated from the total average flux from A to B. This average can be expressed as the product of a flux of trajectories, Φ A,0, that leave state A and reach λ 0, and

73 60 the probability P (λ n λ 0 ) that a trajectory crossing λ 0 from A will reach λ n without returning to A [118, 122]: k A B = Φ A,0 h A P (λ n λ 0 ) (4.1) where h A is the average of a history-dependent function that is either 1 if a trajectory was more recently in A than B, or 0 otherwise, and P (λ n λ 0 ) is equal to n 1 P (λ n λ 0 ) = P (λ i+1 λ i ) (4.2) i=0 where P (λ i+1 λ i ) is the probability that a trajectory that has crossed interface i, coming from A, will subsequently reach λ i+1 before returning to A. Various techniques have been proposed in the framework of FFS including direct FFS, branched growth (BG), constrained branched growth (CBG), random branched growth (RBG), and the Rosenbluth-like method. These methods differ in the protocol used for selecting configurations, firing trajectories, and computing the probability P (λ n λ 0 )[118]. Here we apply CBG, which is most the robust of these methods and, unlike conventional BG algorithms, does not lead to an exponential increase in number of runs [76, 123] FFS via CBG scheme In the conventional BG method, one configuration is randomly selected from stored configuration N 0 at λ 0 to generate a branching tree of paths to connect subsequent interfaces λ i, 0 i n. This method becomes computationally taxing when the number of trial runs per interface increases rapidly due to highly branched transition paths generated by harvesting partial paths. CBG is a modified version

74 61 of BG, proposed by Velez-Vega et al.[123], aimed at avoiding rapid proliferation of branched paths as one approaches state λ n. This algorithm proceeds as follows (see Fig. 4.1): (I) Evaluate Φ A,0 h A by carrying out a simulation in the initial state. (II) Select one configuration from the set of configurations collected at λ 0, and fire M i trial runs, which are continued until λ 1 or λ 0 is reached. Store the N 0 s configurations of successful trajectories that reach λ 1 from λ 0. (III) L i (L i N (i 1) s ) configurations randomly selected from N 0 s states as the starting points to initiate M i trail runs to λ 2 or back to λ 0. The N 1 s successful configurations are stored. (IV) Repeat step (III) over all successive interfaces until λ n is reached, or until no trials are successful at a given interface. For each interface i store N (i 1) s successful trajectories coming from interface i 1.This will generate a branched tree of paths, all with the starting point λ 0. In this method, M i is constant for 0 < i n. k j i trajectories are lunched from each L i point. The value of L i is selected such that, L i = M i k min, for N (i 1) s > M i k min (4.3) N (i 1) s for N (i 1) s M i k min where k min is the minimum number of shots per point, and k j i M i L i. P (λ n λ 0 ) is expressed as, n 1 P (λ n λ 0 ) = P (λ i+1 λ i ) = i=0 n 1 i=0 N i s M i (4.4)

75 62 Figure 4.1: Schematic illustration of CBG method. Branched transition paths all are generated by firing a constant M i trial runs at each interface, λ i, shown by vertical black line. Successful partial paths are depicted by solid green lines and unsuccessful trajectories are shown by dashed gray lines The Committor Analysis The committor function P B (x) is defined as the probability that a given configuration x will end up in final state B before returning back to initial state A. In fact, P B (x), which is in some way related to the ideal reaction coordinates of the system, represents the tendency of the configuration x along the transition path to relax toward the final state under the system s intrinsic dynamics [124]. The committor function increases from 0 at the initial state to 1 at the final state. Configurations with P B = 0.5 correspond to the transition state of the system and help us to understand the mechanism behind the transition. A suitable order parameter can parameterize the committor function in such a way that a sharp change is seen around P B = 0.5. The committor analysis requires the information about the number of successful trial runs to λ i+1 from λ i reached as well as the details of partial

76 63 path connectivity [125]. There are many committor analysis methods proposed in the literature [124]. Here, we apply the procedure suggested by Borrero and Escobdeo [125] for committor probability analysis of the BG method. According to their formulation, the committor probability of point j at λ i, P i Bj is given by P i Bj = P i j (λ i+1 λ i ) N i j m=1 Nj i P i+1 Bm (4.5) where P i j (λ i+1 λ i ) is the probability that the point j at λ i will reach λ i+1 before initial state A and Nj i is the number of points reaching λ i+1 from point j at interface N j i / i. Nj i represents the average P i+1 Bj of all points at λ i+1 that connect with m=1 P i+1 Bm that configuration j at λ i. Equation 4.5 explains a recursive algorithm. Starting from P n Bj = 1, the committor probability of each point at n 1 interface is simply given by P n 1 Bj = P n 1 j (λ n λ n 1 ). Then P n 1 Bj values are used to calculate P n 2 Bj for each point at n 2 interface all the way back to A Free Energy Calculation The method used here to calculate the free energy profile was proposed by Valeriani and coworkers in 2007 [122]. The key idea of their approach is introducing the order parameter q, which is not required to be the same as λ. The free energy profile is given by G = k B T ln [ρ(q)], where ρ(q) is a stationary distribution and ρ(q)dq, which determines the probability of observing q in the range [q, q + dq], is given by ρ(q) = ψ A (q) + ψ B (q) (4.6)

77 64 Figure 4.2: Sketch of different scenarios for a trajectory in transition region between A and B. Trajectories 1 and 2 initiate from A. Trajectory 1 ultimately reaches B, while trajectory 2 returns back to A after visiting other interfaces. Trajectories 3 and 4 start from B. Trajectory 3 goes to A, while trajectory 4 goes all the way back to B. where ψ A (q) is the contribution from region A to the probability distribution of trajectories obtained by A B transition data, and, similarly, ψ B (q) is the contribution from region B to trajectories obtained from B A transition data. It is important to stress that both successful trajectories that ultimately reach B from A (or A from B) and unsuccessful paths that return to where they started contribute to the computation of probability density ρ(q) (see Fig 4.2). ψ A (q) is given by ψ A (q) = p A Φ A,0 τ + (q; λ 0 ), (4.7) where p A is the probability that the system is in region A and is calculated by using A B and B A transition rates, p A = k B A/k A B 1 + k B A /k A B (4.8)

78 65 where τ + (q; λ 0 ) determines the average time spent at the order parameter q by a trajectory coming from A and is given by n 1 i 1 τ + (q; λ 0 ) = π + (q; λ 0 ) + π + (q; λ i ) P (λ j+1 λ j ) (4.9) i=1 j=0 where π + (q; λ 0 ) is the average time spent at the order parameter q by a trajectory fired from λ i that goes to λ i+1 or returns back to λ 0, and is calculated as π + (q; λ 0 ) = N q qm i (4.10) where N q is the number of times that during the total M i trial runs the order parameter of the system has a value between q and q + dq. ψ B is calculated by a similar algorithm, taking into account the following relation: p A k A B = p B k B A (4.11) We apply the above procedure to compute G/k B T for different surface topographies and different droplet sizes in order to obtain the energy barrier and its relationship to surface geometry and droplet size. 4.3 Simulation Details We combine the CBG method with the MD simulation framework, which is fully described in Chapter 2, to study C W and W C transitions in the region neighboring wetting-mode boundaries (see figures 3.11 and 3.12). The above description of FFS indicates that the implementation of any FFS method, including CBG, needs the system dynamics to be stochastic to generate uncorrelated paths

79 66 from a given configuration at interface i. Since the MD algorithm with Nosé-Hoover thermostat is completely deterministic, only one trajectory is produced for a given system configuration. One way to deal with this issue is to reassign velocities to the starting conformation at λ i [120] such that initial velocities for M i distinct trajectories at λ i can be generated by using a Maxwell-Boltzmann distribution at the temperature of interest (T = 0.7ɛ ff /k B ). The minimum number of trial runs per point at λ i, k min is 4 and the total number of trajectories fired from interface i, M i is 60. The number of required interfaces to partition phase space ranges from 4 for the fastest transition to 16 for the slowest transition, depending on the geometry of the surface and the size of the droplet. Five different surface geometries in the region neighboring wetting-mode boundaries (Fig. 4.3) are examined to calculate the rate constants and energy barriers of C W and W C transitions. For each of these configurations either Cassie or Wenzel is reported as the global-minimum energy state by our proposed mathematical model. However, starting with different initial locations of the droplet in MD simulation allows us to observe both Cassie and Wenzel. C W and W C transitions and corresponding rate constants are schematically shown in Fig The order parameter (λ) is defined as λ = Liquid density in the gap The density of the completely filled gap (4.12) 4.4 Results and Discussion Figure 4.5 shows the sequence of droplet configurations at each interface as it passes from Cassie at λ 0 to Wenzel at λ 4 (Fig. 4.5(a)) and from Wenzel at λ 6 to

80 67 Figure 4.3: The location of different surface geometries in the theoretical phase diagrams for which wetting-mode multiplicity has been observed in MD simulations. W = 0.32±0.02 and error bars represent the uncertainty in calculating the radius of the droplet in MD simulations, which in turn affects the uncertainty in computing scaled geometric parameters of the surface. Cassie at λ 0 (Fig. 4.5(b)) for a sample surface with H = 0.4±0.03, Ḡ = 0.72±0.06 and W = 0.32 ± The uncertainty in reporting the scaled surface topology parameter relates to the uncertainty in calculating the radius of the droplet with MD simulations.

81 68 Figure 4.4: A schematic for C W and W C transitions. The corresponding rate constants are represented by k C W and k W C, respectively. (a) (b)

82 69 Figure 4.6: Committor probability P B as a function of order parameter, λ for a surface with H = 0.4 ± 0.03, Ḡ = 0.72 ± 0.06, and W = 0.32 ± The vertical dashed lines display interfaces that partition phase space. The intersect between the curve and the horizontal dash line of P B = 0.5 define the critical size λ, which corresponds to the transition state. In this geometry Wenzel is predicted as the most stable wetting mode by the mathematical model. Therefore it is expected that the C W transition is faster than the W C transition. Comparing k C W (= s 1 ) and k W C (= s 1 ) corroborates the stability of the Wenzel state. In order to investigate the mechanism behind the wetting transition, we compute the committor probability as a function of λ for the wetting transition depicted in Fig The result of such calculation is shown in Fig The horizontal dashed line corresponding to P B (λ ) = 0.5 identifies the transition state ensemble. The transition state is mostly composed of droplet configurations that are on the verge of initiating or breaking contact with the substrate below the

83 70 grooves. Both experimental and theoretical studies support our finding that once the droplet first contacts the roughness groove the transition from the Cassie regime to the Wenzel regime can easily take place [76, 116]. Compared to the interest in the C W transition, less effort has been made to study the W C transition. However, quantifying this phenomenon is essential in many applications, including dewetting. Investigating the transition state of the W C transition, we find that once the droplet detaches from the surface substrate the W C transition proceeds quickly. To quantify the effect of the height of steps on C W and W C transition rates, we test three geometries near the C 1 W 1 boundary with constant W and Ḡ and different H. Corresponding transition-rate constants are shown in Fig This figure shows that the rate of transition in both C W and W C directions decreases as H increases. However, the C W transition rate decreases by 17 orders of magnitude for an increase in H, whereas the W C transition rate only decreases 100 times. This can be explained by the fact that reaching a detached point for a Wenzel drop does not depend on groove depth, whereas a Cassie drop has to pass all the way from the top to the bottom of steps to form the transition state. There are two proposed mechanisms for the Cassie-to-Wenzel transition. One is the sag mechanism (Fig. 4.8(a)), in which the meniscus remains pinned on the outer edges of the steps. The other is a depinning mechanism (Fig. 4.8(b)), in which the meniscus starts to detach from the step corners and moves downward to reach the bottom of the grooves and wet the surface??. In the suggested sag-transition mechanism, the local contact angle of the meniscus at the edges (α in Fig. 4.8) should be θ e to satisfy the Gibbs Equilibrium Condition [66]. Denoting δ as the maximum thickness of the meniscus,

84 71 Figure 4.7: The effect of step height H on C W and W C transition-rate constants for a step width of W = 0.32±0.02 and groove width of Ḡ = 0.72±0.06. Figure 4.8: Two Cassie-to-Wenzel wetting mechanisms: (a) sag, and (b) depinning. δ = 1 2 G(tan θ e sec θ e ) (4.13) where R is the radius of the curvature of the droplet (Eq. 2.7), considering the fact that H δ for sag transition, the following geometry-dictated condition should be

85 72 Figure 4.9: The committor probability vs.λ for three geometries with W = 0.32 ± 0.02, Ḡ = 0.72 ± 0.06, and three different H values (a) 0.4 ± 0.03, (b) 0.56 ± 0.045, and (c) 0.72 ± satisfied for the meniscus pinning, and H Ḡ 1 2 (tan θ e sec θ e ) (4.14) since the above condition is not met for the three examined geometries shown in Fig. 4.7, the depinning mechanism might be responsible for the wetting transition. To verify this statement, we present the committor functions for these typologies to identify the transition-state ensemble as well as the topographical dependency of the critical order parameter λ in Fig 4.9. This figure shows that when H increases a higher fraction of liquid is required to reach transition state, meaning that λ increases as H increases. It can be shown mathematically that

86 73 Figure 4.10: Schematic view of transition state for de-pinning meniscus. λ = ρ G ρ G,max = N = A m + (H δ)g ρ G,max V G HG = constant δ H (4.15) where N is the number of liquid atoms in the gap at transition state and V G is the volume of the gap, A m is the area of the meniscus and is constant for constant G, provided that the contact angle of the meniscus and side walls of the grooves conform to the local value of the Young CA, θ e (see figure 4.10), and is given by A m = θ e π 2 + sin θ e cos θ e 4 cos 2 θ e G 2 (4.16) Equation 4.15 justify the observed change of λ with the step height. By substituting Eq and Eq into Eq. 4.15, Eq can be rewritten as λ = (ξ 1 ξ 2 ) G H + 1 (4.17) where ξ 1 and ξ 2 are θ e -dependent coefficients as expressed below. ξ 1 = θ e π 2 + sin θ e cos θ e 4 cos 2 θ e and ξ 2 = tan θ e sec θ e (4.18) Very close agreement is observed in Table 4.1 between the values of λ obtained from Eq for constant Ḡ and W with those of FFS simulations. Figure 4.11