Identifying Differences and Similarities in Static and Dynamic Contact

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Identifying Differences and Similarities in Static and Dynamic Contact Angles between Nanoscale and Microscale Textured Surfaces Using Molecular Dynamics Simulations Mitchell Ross Slovin, and Michael R. Shirts Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b00842 • Publication Date (Web): 25 Jun 2015 Downloaded from http://pubs.acs.org on July 8, 2015

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Identifying Differences and Similarities in Static and Dynamic Contact Angles between Nanoscale and Microscale Textured Surfaces Using Molecular Dynamics Simulations Mitchell R. Slovin and Michael R. Shirts∗ Department of Chemical Engineering, University of Virginia, Charlottesville, VA, 22904 E-mail: [email protected]

Abstract We quantify some of the effects of patterned nanoscale surface texture on static contact angles, dynamic contact angles, and dynamic contact angle hysteresis using molecular dynamics simulations of a moving Lennard-Jones droplet in contact with a solid surface. We observe static contact angles that change with the introduction of surface texture in a manner consistent with theoretical and experimental expectations. However, we find that the introduction of nanoscale surface texture at the length scale of 5-10 times the fluid particle size does not affect dynamic contact angle hysteresis even though it changes both the advancing and receding contact angles significantly. This result differs significantly from microscale experimental results where dynamic contact angle hysteresis decreases with the addition of surface texture due to a reduction in the receding contact angle. Instead, we find that molecular-kinetic theory, previously applied only to nonpatterned surfaces, accurately describes dynamic contact angle ∗

To whom correspondence should be addressed

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and dynamic contact angle hysteresis behavior as a function of terminal fluid velocity. Therefore, at length scales of tens of nanometers, the kinetic phenomena such as contact line pinning observed at larger scales becomes insignificant in comparison to the effects of molecular fluctuations for moving droplets, even though the static properties are essentially scale invariant. These findings may have implications for the design of highly hierarchical structures with particular wetting properties. We also find that quantitatively determining the trends observed in this paper requires the careful selection of system and analysis parameters in order to achieve sufficient accuracy and precision in calculated contact angles. Therefore, we provide a detailed description of our two-surface, circular-fit approach to calculating static and dynamic contact angles on surfaces with nanoscale texturing.

Introduction Superhydrophobic surfaces possess many desirable characteristics including low friction coefficients, low roll-off angles, resistance to ice formation, self-cleaning properties, and antiwetting properties that are ideal for a wide variety of important applications. 1 For example, synthetic microstructured and nanostructured surfaces, including superhydrophobic surfaces, have become increasingly important for many biomedical applications including the control of protein uptake, cell and tissue interactions, and bacterial attachment. 2 Superhydrophobicity results from high liquid-solid surface tension relative to both vaporsolid and vapor-liquid surface tensions in addition to certain surface texture characteristics and is typically quantified by the contact angles and contact angle hysteresis of fluid droplets on a surface. 3,4 The contact angle is angle formed by a vapor-liquid phase interface with respect to a solid surface at the point of intersection. 5 When water is the fluid of interest, systems with contact angles less than 90◦ are typically referred to as hydrophilic while systems with contact angles greater 90◦ are typically referred to as hydrophobic. 3 Superhydrophobic surfaces have extremely large contact angles, typically in excess of 150◦ for water, and very 2


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low contact angle hysteresis. 3,6 When a body force is applied to either the fluid or the solid surface, resulting in motion of the contact line of the droplet relative to the surface, both the solid-liquid surface tension and the vapor-liquid phase interface area of the droplet increase with respect to static conditions. 7 As a result, advancing and receding dynamic contact angles differ from each other and from their corresponding static contact angles. 7 Contact angle hysteresis is defined as the difference between the larger, advancing and smaller, receding contact angles, where both angles are determined with respect to a contact line in the direction of droplet motion. 3,7,8 The static contact angle is always an intermediate value between the advancing and receding contact angles. 9 Droplets with low contact angle hysteresis will roll off a surface more easily than those with larger contact angle hysteresis, with all other factors constant. There is an important distinction between dynamic contact angle hysteresis, which involves the contact angle during movement of the contact line, and static contact angle hysteresis, where the droplet is changing in size and/or shape but the contact line remains pinned, with the droplet remaining in a metastable state while its overall shape changes. 10 There is some confusion in terminology, with dynamic contact angle hysteresis occasionally being used to refer to both types of hysteresis. This study focuses only on the dynamic contact angle hysteresis involving moving contact lines. Dynamic contact angle hysteresis is frequently measured experimentally using a moving Wilhelmy plate or moving droplet methods. Static contact angle hysteresis, on the other hand, is frequently measured experimentally using the sessile drop or tilted plate methods. At higher droplet velocities, the dynamic contact angle hysteresis is the relevant quantity. 8 Extensive theory, including variants of the Young, Cassie-Baxter, and Wenzel theories, relates equilibrium contact angles for fluids on solid and chemically-homogeneous surfaces to physical properties, such as surface tension. 4 These models can also predict contact angles from surface tension and surface roughness measures very accurately under more specific circumstances following certain modifications. 11 However, these models are not capable of

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accurately determining contact angles under many physically important situations. For example, the Cassie-Baxter model fails to accurately predict contact angles for a variety of pillar geometries. 12 Additionally, the accuracy of the Wenzel model decreases for LennardJones fluids on surfaces with a roughness periodicity less than 10 nm. 13 Extremely superhydrophobic surfaces can be created by imitating the hierarchical texturing found in natural surfaces such as lotus leaves or bird feathers. 6,14,15 Hierarchical texture involves texture present at multiple length scales, such as both µm and nm scale texture features. 6,14,16–18 Hierarchical texture enables fluids to minimize their surface free energy by maintaining a geometry suspended on the surface texture present at multiple length scales, resulting in very large macroscopic contact angles and low contact angle hysteresis. A range of experimental techniques can be used to accurately and precisely build surfaces and measure contact angles via a number of techniques under a wide variety of microscale situations. 5,9,19 Researchers have previously determined the sensitivity of both static and dynamic contact angle hysteresis to a variety of surface texture features. 20,21 Both static and dynamic contact angle hysteresis increase as the solid area fraction of a particular surface increases. 20 Researchers explained this observation as an increase in contact line depinning per unit of area when the solid area fraction increases, resulting in an increase in static contact angle hysteresis 20 or increased work of adhesion for dynamic contact angles. 21–23 This macroscopic static pinning behavior is maintained for surfaces with nanostructures as small as approximately 200 nm. 24 Other experimentalists found that pinning plays an important role in the behavior of dynamic contact angles. 25 However, experimental techniques are limited by the minimum droplet size they can analyze, and often require significant amounts of time and effort to study the effect of surface modification in a systematic way. These constraints make it difficult to determine the influence of molecular and nanoscale properties on macroscopic fluid behavior at interfaces, particularly at the sub-100 nm scale. Molecular dynamics (MD) simulations can help investigate these regimes. Physical properties such as surface tension and contact angles obtained with atomistic

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MD simulations for static droplets are generally consistent with experimentally observed values at larger scales. 26 Researchers using MD simulations of Lennard-Jones liquid and water droplets have determined that the contact angles of liquids on a smooth, solid surface decrease with increasing temperature and solid-fluid potential parameter. 27,28 Atomistic MD simulations have also shown that increasing temperature, decreasing droplet size, and decreasing surface roughness increases wettability for nanoscale water droplets on randomly textured gold substrates. 29 Other researchers used atomistic MD simulations of water droplets on smooth and randomly textured surfaces to conclude the influence of roughness height and droplet size on contact angles depends on the characteristic surface energy. 30 Researchers using MD simulations determined nanodroplet contact angles are strongly dependent on the root-mean-square surface roughness amplitude of randomly textured nanoscale surfaces but not the surface fractal dimension. 31 Others studying the wetting of cylindrical droplets on physically textured surfaces have found that nanoscale MD simulations are consistent with static properties described by macroscale models. 32 A few researchers have performed MD simulations in order to determine dynamic contact angles and dynamic contact angle hysteresis on relatively smooth, solid surfaces. 3,33–35 For atomistically rough surfaces no dependence of contact angle hysteresis on roughness was found, though roughness levels were below that required to create Cassie-Baxter states. 35 Smaller contact angle hysteresis was seen on flexible surfaces than than on rigid surfaces. 36 Continuum modeling of moving droplets 37–39 also provides insight on the dependence of contact angle hysteresis on surface properties, but cannot reach the smallest scales where atomic fluctuations may be relevant. There is a still shortage of information about the behavior of the dynamic contact angles on surfaces with features at the nanoscale, particularly at the lower end of the scale, at tens of nanometers. It is frequently dynamic contact angles that are the most relevant descriptors for wetting behavior, and not the static contact angle. Additionally, for the design of hierarchical materials with specific wetting properties, understanding the behavior

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of dynamic contact angles at all length scales is required. Additionally, despite the number of studies involving contact angle calculations using MD simulations, there has been little study of the sensitivity of measured contact angles to various analysis techniques and parameter selections, either for static or dynamic. Most of such simulations have examined problems that require only moderate levels of precision or accuracy. However, both precision and accuracy are of particular importance when determining contact angle hysteresis, as it involves taking the difference between two relatively similar contact angles, each with non-canceling statistical error. Therefore, in this study, we examine the dependence of measured contact angles on various regression functional forms and analysis parameters to identify which approaches yield the highest statistical efficiency and the least bias. We also carefully optimize the parameters associated with calculating contact angles and construct a two-surface system with effective and efficient contact angle calculations in mind. We then apply this optimized contact angle calculation procedure to quantify the effects of nanoscale physical surface texture on static contact angles under differing levels of fluidsurface interaction. We also examine dynamic contact angles and their hysteresis for differing applied body forces on both smooth and textured surfaces. In both cases, we compare our observations to theoretical predictions and experimental behavior at larger scales, as well as simulations where available.

Methodology System Configuration. We use the the Lennard-Jones potential function, ULJ (r) = 4ǫ[( σr )12 − ( σr )6 ], to model a fluid on a solid surface. 40 We space the innermost atoms of two parallel, rigid walls a distance d apart and use multiple layers of atoms, positioned in a primitive cubic arrangement, to construct their surfaces, as shown in Fig. 1. We include a minimum of 5 complete layers of atoms in each wall, regardless of physical texturing, so

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that wall atoms are always present beyond the maximum cutoff radius of the outermost fluid atoms. We fix these wall atoms in place and position each wall atom one lattice spacing a from its closest neighbor. We define the length y and width z of the walls to be the longer and shorter dimensions in the plane of the walls respectively. We define the height dimension x of the system to be perpendicular to both walls. Fig. 1 and 2 show the geometry of this system. The two walls contain liquid atoms in a cylindrical arrangement spanning the system’s periodic boundaries in the z dimension, as shown in Fig. 1. Researchers commonly use cylindrical droplets in MD simulations to calculate contact angles. 31,32 A derivation in Section S2 of the Supporting Information shows that the circular functional form we use to fit the vapor-liquid interface of the fluid and calculate contact angles does not depend on the specific shape of the fluid. Finally, we can remove any effect of line tension by contact angles of our cylindrical droplet as the contact line has zero curvature. 32 We initially position fluid atoms in a body-centered cubic arrangement, occupying half of the box volume, and space each fluid atom 1.15σF luid from its closest neighbor, a value similar to previous simulations analyzing closely related systems. 28 This spacing is slightly √ greater than the 6 2σF luid energy minimum, resulting in a small contraction of the fluid during equilibration. We incorporate patterned physical texture by constructing pillars protruding from the base atom layers of each wall while still maintaining a consistent spacing between the innermost atoms of each wall, as shown in Fig. 1. In this study, we use vertically oriented cylindrical pillars arranged in a square lattice to model surface texture, as shown in Fig. 1. This geometry is similar to the quadrangular pillars and rectangular posts used in similar MD studies but is expected to reduce liquid shearing when examining dynamic contact angles and their hysteresis because fewer edges and corners are present. 7,37,41 Additionally, µm-scale cylindrical pillar arrays are commonly generated in order to create superhydrophobic and superoleophobic surfaces. 12,20 We use a pillar height and diameter of 5a and a pillar center

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spacing of 8a, roughly consistent in size with the pillars from other simulation studies. 41

Figure 1: We use cylindrical pillars, with dimensions defined in terms of the wall lattice parameter a, to model patterned physical surface texture. The liquid forms a horizontal aligned droplet due to the system geometry and periodic boundaries. We use two walls, as opposed to a single wall, in order to increase computational efficiency, regression accuracy, and the ease of handling of dynamic angle calculations. Since a normal force is not required to maintain liquid-wall contact when two walls are used, liquid shearing is reduced and motion of the fluid center of mass in the height dimension of the system becomes less of a concern. 33,34 Additional contact data, resulting from the presence of a second wall, enables us to gather information from two contact surfaces per unit of simulation time. Further, contact between the liquid and each wall along the entire system width generates substantially more contact data for a given system size than a droplet on a single surface, which may only contact the surface over a small fraction of its projected area. Droplet symmetry, due to the presence of two walls, enables us to use lower bias circular regressions to calculate contact angles for both static and dynamic contact angles. This droplet symmetry is particular important with calculating dynamic contact angles, as a spherical droplet’s contact line will have a different orientation with respect to the direction of motion at each point. System size must be carefully considered when attempting to accurately analyze contact angles and contact angle hysteresis on the nanoscale. An excessively small system may contain an insufficient number of fluid atoms to allow for well-defined contact angles. However, a system that is excessively large will result in unnecessary computational expenses, which limit statistical precision. Previous simulation studies found the number of fluid atoms in a droplet does not have a significant effect on contact angles between 16, 000 and 320, 000 8


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atoms, 36 while another study observed significant changes in contact angles where fewer than 8, 000 water molecules were present. 29 Our preliminary simulations, varying only the system size with a constant ratio of system dimensions, showed that a system with x = 25a, y = 70a, and z = 35a is large enough that contact angles are essentially independent of the droplet size. Our resulting system contains 46, 800 fluid and 24, 500 wall atoms with 5 base wall layers, with texturing adding additional wall atoms. Periodic boundary conditions are of particular importance for this system, as their presence allows the liquid to remain stable in a non-spherical geometry. As a result, the liquid forms a horizontal, cylindrical formation that spans the entire width of the system with coherent shape. Liquids minimize their surface tension, and therefore free energy, by forming a sphere if they are not constrained by other factors. 7 In this periodic system, there is no curvature of the contact line as would be observed in a spherical droplet, simplifying the analysis. 33 One must be careful of the ratio of system dimensions and periodic boundary conditions to obtain a single droplet. The system must not be too wide or else the liquid splits into multiple droplets. The presence of a second wall does not significantly change observed contact angles because molecular fluctuations due to the surface only extend a relatively small distance into the fluid, approximately 5 σW all above each surface. Fig. S3 of the Supporting Information shows that observed contact angles do not depend significantly on the wall spacing as long as the liquid height-to-length ratio is greater than approximately 0.5. Below this value, the molecular effects of the interface extend through approximately 50% of fluid and confinement effects become a concern. For these simulations, we maintain the spacing large enough so that the contact angle becomes independent of separation distance. We use a liquid heightto-length ratio of approximately 0.7 for all subsequent production simulations in order to minimize confinement effects and prevent separation of the droplet from either wall. Simulation Details. We obtained the parameters for fluid and wall atoms from previous simulations of an argon fluid on a platinum surface. 28 We used σF luid = 0.3045 nm for each

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Lennard-Jones fluid atom, an ǫF luid = 1.0 kJ/mol, and a mass of 39.75 g/mol. For each wall atom, we choose σW all = 0.2705 nm and an ǫW all that varies and is defined in terms √ luid of reduced units of ǫW ǫall−F , where ǫW all−F luid = ǫW all ǫF luid . The mass of wall atoms is F luid irrelevant due to fixed wall atom positions. The temperature was controlled using Langevin dynamics at 85 K (0.7 in Lennard-Jones reduced units). This is an intermediate temperature within the range where meaningful contact angles were observed in simulations analyzing the effects of temperature on similar systems in detail. 28 We chose a for the primitive cubic wall atom lattice equal to the experimentally determined 0.39236 nm lattice spacing for platinum (1.45045σW all ). 42 We note that the platinum lattice is actually a face-centered cubic lattice but a primitive cubic lattice allows for more precise pillar construction. Our objective is not to accurately model a platinum surface but to calculate trends in contact angles and the effects of surface texture on these contact angles. The density difference between wall lattice structures does have an impact on resulting contact angles. As shown in Fig. S4 and S5 of the Supporting Information, observed contact angles increased significantly with a for small a since a decrease in wall atom spacing increases the cohesive energy. When a approached the value used in these simulations, contact angles became approximately independent of a. We used the GROMACS MD package (version 4.5.6) for all simulations. 43 We first minimized the energy of the system before performing a 1 ns canonical ensemble equilibration simulation. In most cases, 1 ns was sufficient to reach equilibrated droplets, though we required between 2 and 4 ns of additional equilibration time for systems with small contact angles, where reduced ǫW all−F luid was greater than 0.7. We then obtained production data from 5 ns MD simulations with a 0.01 ps time step. We computed the autocorrelation function of the timeseries of the vapor-liquid interface voxel densities, allowing us to estimate the correlation time of the interface motion. This autocorrelation time was approximately 0.1 ps, meaning that approximately one in ten time steps consist of independent fluid configurations, and we therefore sampled configurations every 0.1 ps.

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We performed simulations to determine static contact angles using the leapfrog Verlet integrator with the velocity rescaling thermostat of Bussi et al. 44 However, we used the stochastic dynamics implementation of Langevin dynamics as both an integrator and thermostat for dynamic contact angle simulations. We specified a coupling time scale, τt = 0.1, for static contact angle simulations and τt = 10 for dynamic contact angle simulations. We used a larger τt for dynamic contact angle simulations to minimize the effects of the thermostat on the large-scale motion 45 while still removing heat generated from friction and the body force. We used a switched potential energy cutoff between 3 and 3.5 σF luid . Previous studies found that a cutoff radius of approximately 3 σF luid does not significantly alter contact angle results compared to longer cutoffs. 28 To observe dynamic contact angle hysteresis, we applied a constant body force to all fluid atoms during both the equilibration and production simulations. The droplet eventually reached a terminal center-of-mass velocity when the applied force acting to accelerate the fluid came into balance with the frictional forces resulting from contact with the wall surfaces and the random forces of the stochastic integrator. We provide an overview of the contact angle calculation process here, with all items being discussed in more technical detail in Supporting Information Section S2. Contact Angle Calculation Procedure 1. We center the fluid atoms of every sampled configuration. This procedure requires multiple steps to account for the droplet crossing the boundary conditions. These steps are detailed in Supporting Information Section S1. 2. We divide the volume between the system walls into a large number of small voxels with a side length equal to some fraction of σF luid , and take advantage of the symmetry of the system to create a 2D density distribution. We then calculate the average number density at each pixel, obtaining results like those shown in Fig. 2. 3. We use a threshold average density to differentiate vapor and liquid regions of the fluid. The calculation of this value is discussed in detail in Supporting Information Section S1. 11


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number of subsamples are roughly normally distributed, as shown Supporting Information Section S1. 6. For circular regressions, we report the single advancing and single receding contact angles obtained from the cumulative density data. We average these values to obtain the reported contact angle in the case of contact angle measurements in static droplets. For linear regressions, we average the two advancing contact angles (top and bottom surfaces) from the cumulative density data to obtain the reported advancing contact angle, and similarly average the two receding contact angles. For static contact angle calculations, we again average both the advancing and receding contact angles. 7. Finally, we determine the average velocity of the fluid center of mass. We can then report dynamic contact angles and dynamic contact angle hysteresis as a function of a physically meaningful quantity. This also allows us to confirm that the fluid has reached a terminal center-of-mass velocity at the start of each production simulation. We first record the fluid center-of-mass position in the system length dimension for every sampled configuration prior to fluid centering. We then determine the velocity of the fluid center of mass for every sampled configuration by dividing the change in center-of-mass position from the previous sampled configuration by the elapsed time between the adjacent configuration samples. We average segments of velocity values, where the center-of-mass position is increasing linearly with time, over no less than 1 ns of production simulation to determine reported average velocity and uncertainty measures. We include a 95% confidence interval for all reported velocities but this is typically not visible on our plot scales. Contact Angle Calculation Parameter Optimization. We optimized the fraction, f , of interface regressed, length of contiguous subsamples used estimating the error, pixel and voxel size, and simulation length self-consistently by trial and error for the case of linear regression, and present the final results of that process. The interface fraction, f , must be small to minimize bias for linear regressions. However, a sufficiently large fraction is required to obtain precise results by including many phase 13


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and 3h show that 5 ns of production simulation provides sufficiently high levels of both accuracy and precision necessary to clearly distinguish the contact angle features analyzed in this study. Observed contact angles did not change substantially over explored production simulation times while the standard error in the mean decreased with the square root of simulation time, as predicted by the central limit theorem. Circular Regressions. Researchers have previously used circular functional forms to calculate contact angles for MD simulations. 3,30,32,33 Using two wall surfaces to contain the fluid, as shown in Fig. 2, results in a symmetry of the liquid with respect to the height, x, of the system. This symmetry enables each phase interface to be fit with a half-circle. Eq. 1, derived in Section S2 of the Supporting Information, describes the vapor-liquid phase d interface location and results in a circular section with a radius 2s and center (x, y) = √ d 1 − s2 ), where d is the diameter of the liquid droplet, h is the distance from (d/2, h + 2s

the midpoint of the droplet area in contact with wall surface, and s is related to the contact angle, θc , via s = cos(θc ).

f (x) = h +



d  1 − s2 − 2s √

s

1− s−

2sx d

2

 

(1)

Note that this fit assumes the effects of gravity are negligible. This assumption is reasonable for nanoscale simulations as the Bond number associated with even our smallest applied acceleration is 1.0 × 1010 times the Bond number associated with gravitational influences on the system. Due to the extraordinarily large ratio of surface area to volume for nanoscale droplets compared to macroscale droplets, the radius of the droplet used in our simulations would need to increase by a factor of 4,000, up to the micrometer scale, before the effects of a gravitational force would equal 0.1% of those due to the smallest applied acceleration. As detailed in Section S2, d in Eq. 1 is the distance between the fluid-wall planes of interaction. However, there is some ambiguity as to the exact value of d for structured surface, as the exact height of the liquid density can vary from point to point on the surface.

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that, for an equilibrated system, uncertainty, but not accuracy, depends on the simulation time. Therefore, we assume the optimal number of configuration samples per contiguous subsample, the pixel size, and the production simulation time are all independent of the regression type and use the optimal linear regression values for circular regressions. However, we optimized f for circular regressions by varying only this parameter while holding all other parameters constant at their optimal values. As we show in Fig. 4a and 4b, the lowest error estimates are obtained by regressing the largest fraction of the fluid interface possible (f = 1) while the contact angle is largely independent of f as long as f is greater than 0.4.

Results and Discussion

Figure 5: Smooth (top) and textured (bottom) system configurations corresponding to reduced ǫW all−F luid of 0.4 (left), 0.65 (middle-left), 0.7 (middle-right), and 1.0 (right) show decreasing contact angles with increasing ǫW all−F luid values. Static Contact Angle Results. We quantified the effects of surface texture on static contact angles using the high accuracy circular fit procedure discussed above. We show a selection of the simulations used for this analysis in Fig. 5. Fig. 6 shows static contact angles on both smooth and textured surfaces as a function of reduced ǫW all−F luid . We use circular regression fits to the phase interface as Fig. 4 indicates that the accuracy of linear fits is questionable. However, we found that linear regressions typically result in slightly greater

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Figure 7: Dynamic contact angle hysteresis on a smooth (top) and textured (bottom) surface increases when we increase the terminal fluid center-of-mass velocity from approximately 0 (left) to approximately 25 (right) m/s via the application of a constant body force. The Bond number, Bo =

ρaR2 , γ

which compares surface tension to body forces, is between

0.58 and 1.74 for our nanoscale fluid droplets. This Bo is calculated from ρ = 30.4 argon atoms per nm3 , the range of applied accelerations mentioned above, and R = 7.67 nm. We use a vapor-liquid γ of 2.045 × 10−2 N/m from tabulated Lennard-Jones fluid property data for the ǫ and σ used in these simulations, a cutoff radius equal to 2.5 σ, and the 0.70 reduced temperature used in these simulations. 46 In comparison, experimentalists observed Bond numbers of 0.15 and 0.30 for 1.5 mm radius aqueous zinc chloride droplets on slightly inclined and textured superhydrophobic surfaces, resulting in a characteristic velocity of 9.7 cm/s. 47 Therefore, our large nanoscale accelerations are consistent with moderately fast macroscopic dynamic contact angle experiments. The terminal fluid velocity is linearly related to the applied acceleration for both smooth and textured surfaces within the range examined in this study. This linear relationship is shown in Supporting Information Fig. S8 and is consistent with other Langevin dynamics simulations, which found an approximately linear relationship between the kinetic-friction coefficient of oil fluid layers contained between parallel walls, and sliding velocity at the relatively low velocities examined in this study. 48 These previous kinetic-friction coefficient results were empirically determined but consistent with theory, experiments, and other simulations. 48 Additionally, we find an approximately linear relationship between dynamic con-

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in circular fits near 180◦ . In contrast to static contact angles, where nanoscale behavior is similar to microscale behavior, there appear to be fundamental differences between nanoscale and microscale dynamic contact angle behavior. In our simulations, there is no statistical difference in dynamic contact angle hysteresis for fluids in the Cassie-Baxter regime as a function of velocity between smooth surfaces and surfaces with nanoscale texturing, as shown in Fig. 9c. This consistency of contact angle hysteresis between smooth and textured surfaces has been observed before on surfaces with atomistic level roughness at levels smaller than that required to create droplets suspended in the Cassie state. 35 However, this finding differs significantly from experimental measurements of dynamic contact angle hysteresis conducted with moving droplets on surfaces with microscale cylindrical pillars, where researchers have generally found that surface texture plays a crucial role in dynamic contact angle hysteresis. 21,22,25 . These measurements show that dynamic contact angle hysteresis increases with solid area fraction independent of pillar size. 20–22 Adhesion at the receding contact line increases dynamic contact angle hysteresis for microscopic systems 21,22 , leading to slip-stick transitions. 25 Instead, the droplets in our study move more or less uniformly while suspended over the pillars. In contrast to the pinning transitions observed in microscale experiments, 21,22,25 the thermal fluctuations that are assumed to control the kinetics of motion on much smoother surfaces also appear to dominate here, 35 even though the droplets are clearly in a Cassie-Baxter state with relatively large surface structures. Dynamic contact angle pinning at this length scale appears to behave in a different manner than static contact angle pinning, which does occur at the nanoscale. Experimentalists have determined that static contact angle pinning is preserved for nanostructures as small as 200 nm. 24 Further, researchers have used simulations to observe pinning of static contact angles on the atomistic length scale and concluded that theoretical models explaining this behavior are consistent with experiments utilizing small drops in addition to trends in the

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Cassie and Wenzel models. 32,53,54 While these researchers have observed static pinning at the nanoscale via MD simulations, they did not analyze the fluid dynamics of rapidly moving droplets. 32,53,54 The droplet kinetics in our simulations are well described by the atomic level molecularkinetic theory of Blake. This independence of dynamic contact angle hysteresis on pillared nanoscale texturing agrees with an equivalent form of molecular-kinetic theory, U = S )−cos(θD )) 2κ0 λ sinh( γ(cos(θ2nk ). 52 In this equation, n is the number of adsorption sites per unit BT

area of the solid surface and W = γ(cos(θS ) − cos(θD )) is the work required to displace a unit length of the wetting line. 52 All of the variables in molecular-kinetic theory remain approximately the same with the introduction of surface texture on the nanoscale assuming the droplet stays in a Cassie state. The spacing between sites λ should not change, because although nanoscale pillars change the area density of surface sites, there are no additional potential barriers to overcome when the contact line jumps between pillars. Similarly, κ0 should remain largely unchanged as the depth of the surface potential wells remains the same for surface adsorption sites, except possibly those at the edge of the surface texture features. 52 Clearly, kB , T , and γ do not depend on the particular surface features, and although surface texture changes θS , this mainly affects the intercept. There are two places where the area density of surface sites must be accounted for but these contributions cancel out for this system. By adding surface texture, we reduce the W required to move the three-phase interface per length of fluid, since we reduce the number of potential barriers that must be overcome per unit length. The reduction factor is rW = ℓT exture /ℓSmooth , where ℓSmooth and ℓT exture are the lengths of the three-phase interface along the width of the system for smooth and textured surfaces respectively. The addition of surface texture pillars also scales n by a multiplicative constant since the number of adsorption sites per unit area of the solid surface decreases by rn = AT exture /ASmooth , where ASmooth and AT exture are the solid area fractions for smooth and textured surfaces respectively. For this

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particular texture pattern, rW and rn , which we calculate in Section S9 of the Supporting Information, are approximately equal and cancel each other. Therefore, the equation reduces to the previous equation for θD because n ∝ λ−2 . Regressing of our data to this model yields κ0 for both advancing and receding contact angles on both smooth and textured surfaces. In all cases κ0 is between 0.9 × 1011 and 2.4 × 1011 s−1 for circular regression contact angles, with 95% confidence bounds approximately 0.1 to 0.2 × 1011 s−1 in all cases. Values of κ0 for rough surfaces are, within noise, 2 times those of smooth surfaces, while in both cases, advancing contact angles have κ0 larger than receding contact angles by between 40% and 50%. As can be seen from the plots, despite any quantitative differences in κ0 , the hysteresis predicted by molecular-kinetic theory is, within noise, the same for both smooth and textured surfaces. In particular, Fig. 9c and Fig. 9d show that molecular-kinetic theory is consistent with the sharp deviation from linearity near 20 m/s. Our finding that dynamic contact angle hysteresis, in the Cassie-Baxter regime, does not depend on nanoscale surface texture features for a regularly pillared surface is also consistent with previous MD simulations of water droplets on similar surfaces. 34 In these simulations, dynamic contact angle hysteresis is essentially unchanged for various pillar heights, surface area fractions, and applied body forces as long as the droplet remains suspended on the pillars in the Cassie-Baxter regime. 34 However, these researchers also found that surface texture significantly changes dynamic contact angle hysteresis and impedes droplet motion compared to smooth surfaces when the droplet is in the Wenzel regime. 34 These observations concerning the Wenzel regime exhibit significant shearing of the fluid and are of less interest for the design of superhydrophobic surfaces. Dynamic contact angle hysteresis calculated from linear regression techniques leads to the incorrect conclusion that the addition of surface texture results in a decrease in dynamic contact angle hysteresis, which researchers might naively believe since it agrees so well with macroscopic results. 21,22 This numerical biasing effect is due to the contact angle-dependent

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bias of using linear regressions to fit to a circular function. According to Fig. 9d, inaccuracies in linear regression dynamic contact angle hysteresis can be attributed to the fact that bias, equal to the difference between the circular and linear regression contact angles, is strongly and linearly dependent on the contact angle but not the terminal fluid velocity, whether the contact angle is advancing or receding, or the presence of surface texture. This linear regression bias also accounts for the inaccuracy of Fig. 9a compared to Fig. 9b. Correct understanding of these behaviors required a well-validated procedure with minimal bias, such as one we present here. Computational Efficiency. A detailed analysis of the computational resources required to run and analyze these simulations is included in Fig. S10 of the Supporting Information.

Conclusions We found that a patterning of surface texture at the nanoscale increases static contact angles greater than 90◦ and decreases static contact angles less than 90◦ , a result consistent with contact angle theory. However, we did not observe a change in dynamic contact angle hysteresis with the introduction of surface texture for fluids in the Cassie-Baxter regime, a result that differs from microscale experimental 20–22,25 and continuum computational results 37–39 results where static contact angle hysteresis was sensitive to the surface texture solid area fraction and texture played a critical role in influencing the pinning behavior of dynamic receding contact angles. Instead, we found that molecular-kinetic theory models accurately describe both dynamic contact angles and dynamic contact angle hysteresis under the conditions examined in these simulations. Our study suggests that droplet dynamics can change considerable at low nanoscale length scales even if static properties are independent of scale. Thermal fluctuations appear to govern contact line kinetics at this low nanometer length scale instead of pinning effects on the receding contact angle observed on the microscale. In contrast, researchers have

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used both experiments and MD simulations to observe similar static pinning effects at the nanometer length scale to those observed at the microscale. 24,32,53,54 This lack of scale invariance for dynamic contact angle behavior may have significant consequences for the design of hierarchical omniphobic surfaces with texture features at the tens of nanometer level. We also found that the way in which simulations are analyzed can significantly affect the quality of the results, especially for calculating dynamic contact angle hysteresis, which involves the difference between two often-similar contact angles. We described a detailed and effective procedure for calculating both the static and dynamic contact angles of atomistic droplets on arbitrarily textured nanoscale surfaces using MD simulations. We presented a number of approaches to improve the accuracy and precision of these contact angle measurements as well as optimized parameters for these calculations. Significantly, we found that the use of circular regressions, as opposed to linear fits, is necessary to quantitatively and in some cases even qualitatively test hypotheses of wetting behavior. The details of this contact angle calculation approach may enable others to more effectively and efficiently calculate contact angles via atomistic MD simulations. We have made the code used for this analysis available on the GitHub site https://github.com/shirtsgroup/droplet-analysis.

Acknowledgments We acknowledge the University of Virginia Harrison Undergraduate Research Award and the ARCS Foundation Undergraduate Scholar Award for providing funding to support this project. We thank Eric Loth, Yong Yeong, and Athanasios Milionis for helpful discussions concerning the experimental details of dynamic contact angles in addition to Joe Basconi and Levi Naden for constructive feedback.

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Supporting Information Derivations, expanded discussion of technical issues and additional figures are included as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

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Identifying Differences and Similarities in Static and Dynamic Contact

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