Nanodrop on a nanorough solid surface: density functional theory considerations

Machine-Learned Free Energy Surfaces for Capillary Condensation and Evaporation in Mesopores

Using molecular simulations, we study the processes of capillary condensation and capillary evaporation in model mesopores. To determine the phase transition pathway, as well as the corresponding free energy profile, we carry out enhanced sampling molecular simulations using entropy as a reaction coordinate to map the onset of order during the condensation process and of disorder during the evaporation process. The structural analysis shows the role played by intermediate states, characterized by the onset of capillary liquid bridges and bubbles. We also analyze the dependence of the free energy barrier on the pore width. Furthermore, we propose a method to build a machine learning model for the prediction of the free energy surfaces underlying capillary phase transition processes in mesopores.

Mechanism of Heterogeneous Bubble Nucleation in Polymer Blend Foaming

A three-dimensional heterogeneous bubble nucleation model is constructed to provide a reasonable explanation at the molecular level for the foaming mechanism of polypropylene (PP) and polystyrene (PS) blends. CO₂ solubilities and supersaturation rations are quantitatively calculated to help interpret the contribution of each phase of the blend in the CO₂ dissolution stage. The spatial density profiles of polymer/CO₂ binary melt around different polymer chains are presented to give an intuitive perspective to the thermodynamic driving force. The predicted interfacial tension and contact angles of critical bubbles provide valid evidence to distinguish the wettability of CO₂ in different regions. The values of predicted free-energy barriers, critical radii, and nucleation number densities imply that bubbles that nucleate in the PP and PS blend interfacial region attached to the PS-rich phase achieve the smallest size and largest number density. The reliability of the theoretical model has been tested by partial available experimental data.

Nucleation of Capillary Bridges and Bubbles in Nanoconfined CO2

Using molecular simulation, we examine the capillary condensation and the capillary evaporation of CO₂ in cylindrical nanopores. More specifically, we employ the recently developed μV T-S method to determine the microscopic mechanism associated with these processes and the corresponding free energy profiles. We calculate the free energy barrier for capillary condensation and identify that the key step consists in the nucleation of a liquid bridge of a critical size. Similarly, the free energy maximum found for the capillary evaporation process is found to correspond to the nucleation of a vapor bubble of a critical size. In addition, we assess the impact of the strength of the wall-fluid on the height of the free energy barrier and on the critical size of liquid bridges (condensation process) and vapor bubbles (evaporation process). We observe that the height of the free energy barrier increases with the strength of the wall-fluid interactions. Finally, we build a theoretical model, based on capillary theory, to rationalize our findings. In particular, the simulation results reveal a linear scaling of the free energy barrier with the critical size, in excellent agreement with the theoretical predictions.

A heuristic approach for nanodrops on a smooth solid surface

A heuristic approach is developed to obtain a simple equation for the contact angle of a nanodrop on a smooth planar solid surface. First, nanodrops of various fluids in contact with various solid surfaces are considered on the basis of nonlocal density functional theory (DFT). Along with the traditional (apparent) contact angle, θ_a, which the drop profile makes with the solid surface, another one, θ_d, formed by the smooth part of the drop profile and the horizontal plane separating that part from the oscillatory part of the profile was examined. For each of the contact angles, a separate simple equation resembling the Young equation for the macroscopic drops but containing, instead of surface tensions, the microscopic parameters of intermolecular interactions, temperature, and average density of the fluid was hypothesized and the parameters of this equation were determined using the results of DFT calculations. It was shown that predictions of these equations coincide with the results provided by DFT.

Geometry-induced interface pinning at completely wet walls

We study complete wetting of solid walls that are patterned by parallel nanogrooves of depth D and width L with a periodicity of 2L. The wall is formed of a material which interacts with the fluid via a long-range potential and exhibits first-order wetting transition at temperature T_{w}, should the wall be planar. Using a nonlocal density functional theory we show that at a fixed temperature T>T_{w} the process of complete wetting depends sensitively on two microscopic length scales L_{c}^{+} and L_{c}^{-}. If the corrugation parameter L is greater than L_{c}^{+}, the process is continuous similar to complete wetting on a planar wall. For L_{c}^{-}<L<L_{c}^{+}, the complete wetting exhibits first-order depinning transition corresponding to an abrupt unbinding of the liquid-gas interface from the wall. Finally, for L<L_{c}^{-} the interface remains pinned at the wall even at bulk liquid-gas coexistence. This implies that nanomodification of substrate surfaces can always change their wetting character from hydrophilic into hydrophobic, in direct contrast to the macroscopic Wenzel law. The resulting surface phase diagram reveals a close analogy between the depinning and prewetting transitions including the nature of their critical points.

An analog to Bond number for pendant nanodrops

A new dimensionless number is introduced which characterizes the shape and stability of a pendant nanodrop.

Prediction of Contact Angles and Density Profiles of Sessile Droplets Using Classical Density Functional Theory Based on the PCP-SAFT Equation of State

This study demonstrates the capability of the density functional theory (DFT) formalism to predict contact angles and density profiles of model fluids and of real substances in good quantitative agreement with molecular simulations and experimental data. The DFT problem is written in cylindrical coordinates, and the solid-fluid interactions are defined as external potentials toward the fluid phase. Monte Carlo (MC) molecular simulations are conducted in order to assess the density profiles resulting from the Helmholtz energy functional used in the DFT formalism. Good quantitative agreement between DFT predictions and MC results for Lennard-Jones and ethane nanodroplets is observed, both for density profiles and for contact angles. That comparison suggests, first, that the Helmholtz energy functional proposed in a previous study [ Sauer , E. ; Gross , J. Ind. Eng. Chem. Res. 56 , 2017 , 4119 - 4135 ] is suitable for three-phase contact lines and, second, that Lagrange multipliers can be used to constrain the number of molecules, similar to a canonical ensemble. Experiments of sessile droplets on solid surfaces are performed to assess whether a real solid with its microscopic roughness can be described through a simple model potential. Comparison of DFT results to experimental data is done for a Teflon surface because Teflon can be regarded as a substrate exhibiting only attractive interactions of van der Waals type. It is shown that the real solid can be described as a perfectly planar solid with effective solvent-to-solid interactions, defined through a single adjustable parameter for the solid. Subsequent predictions for the contact angle of eight solvents, including polar components such as water, are found in very good agreement to experimental data using simple Berthelot-Lorentz combining rules. For the eight investigated solvents, we find mean absolute deviations of 3.77°.

Shape and Stability of a Pendant Nanodrop

The shape and stability of a pendant nanodrop attached to a smooth or rough solid surface are considered on the basis of the microscopic density functional theory in the presence of a strong external force normal to the solid surface. The drop profile, width, and height of the drop and the contact angle that a nanodrop makes with the solid surface are calculated and some interesting features of the drop shape and stability are identified. In particular, a linear relationship between the height of the drop and the contact angle, as well as one between the critical external force at which the drop loses its stability and the strength of the fluid-solid interaction, was found. In some cases, a fixed point on the drop profile was observed whose spacial location does not depend on the value of external force. It is also argued that the Bond number that is a characteristic of the pendant macroscopic drop is not applicable to nanodrops.

The mechanism of roughness-induced CO2 microbubble nucleation in polypropylene foaming

Within the framework of classical density functional theory, the thermodynamic driving forces for CO₂ microbubble nucleation have been quantitatively evaluated in the foaming of polypropylene containing amorphous and crystalline structures. After the addition of fluorinated polyhedral oligomeric silsesquioxane particles into the polypropylene matrix, we construct different composite surfaces with nanoscale roughness for bubble nucleation. Meanwhile, as the dissolved CO₂ molecules increase, the corresponding CO₂/PP binary melts can be formulated in the systems. Due to the roughness effect coupled with the weak interactions of particle-PP, PP chains in the binary melts are depleted from the surfaces, leading to a significant enhancement of osmotic pressure in depletion regions. During the foaming process, a large number of dissolved CO₂ molecules are squeezed into the regions, thus local supersaturations are dramatically improved, and the energy barriers for bubble nucleation are dramatically reduced. Moreover, when the nanocomposite surfaces display ordered nanoscale patterns, the energy barriers can be further reduced to their respective minimum values, and the bubble number densities reach their maximum. Accordingly, the bubble number densities can be enhanced by 4 or 5 orders of magnitude for bubbles nucleated on the crystalline or amorphous PP nanocomposite surface. In contrast, when the foaming pressure is increased from 15 to 20 MPa, the elevated bubble number density in the foaming PP matrix is less than one order of magnitude. As a result, the enhancement of local supersaturation induced by the controlled nanoscale roughness is much more effective than that of bulk supersaturation given by high pressure.

Thermal fluctuations of an interface near a contact line

The effect of thermal fluctuations near a contact line of a liquid interface partially wetting an impenetrable substrate is studied analytically and numerically. Promoting both the interface profile and the contact line position to random variables, we explore the equilibrium properties of the corresponding fluctuating contact line problem based on an interfacial Hamiltonian involving a "contact" binding potential. To facilitate an analytical treatment, we consider the case of a one-dimensional interface. The effective boundary condition at the contact line is determined by a dimensionless parameter that encodes the relative importance of thermal energy and substrate energy at the microscopic scale. We find that this parameter controls the transition from a partial wetting to a pseudopartial wetting state, the latter being characterized by a thin prewetting film of fixed thickness. In the partial wetting regime, instead, the profile typically approaches the substrate via an exponentially thinning prewetting film. We show that, independently of the physics at the microscopic scale, Young's angle is recovered sufficiently far from the substrate. The fluctuations of the interface and of the contact line give rise to an effective disjoining pressure, exponentially decreasing with height. Fluctuations therefore provide a regularization of the singular contact forces occurring in the corresponding deterministic problem.

Calculation of nanodrop profile from fluid density distribution

Two approaches are examined, which can be used to determine the drop profile from the fluid density distributions (FDDs) obtained on the basis of microscopic theories. For simplicity, only two-dimensional (cylindrical, or axisymmetrical) distributions are examined and it is assumed that the fluid is either in contact with a smooth solid or separated from the smooth solid by a lubricating liquid film. The first approach is based on the sharp-kink interface approximation in which the density of the liquid inside and the density of the vapor outside the drop are constant with the exception of the surface layer of the drop where the density is different from the above ones. In this case, the drop profile was calculated by minimizing the total potential energy of the system. The second approach is based on a nonuniform FDD obtained either by the density functional theory or molecular dynamics simulations. To determine the drop profile from such an FDD, which does not contain sharp interfaces, three procedures can be used. In the first two procedures, P1 and P2, the one-dimensional FDDs along straight lines which are parallel to the surface of the solid are extracted from the two-dimensional FDD. Each of those one-dimensional FDDs has a vapor-liquid interface at which the fluid density changes from vapor-like to liquid-like values. Procedure P1 uses the locations of the equimolar dividing surfaces for the one-dimensional FDDs as points of the drop profile. Procedure P2 is based on the assumption that the fluid density is constant on the surface of the drop, that density being selected either arbitrarily or as a fluid density at the location of the equimolar dividing surface for one of the one-dimensional FDDs employed in procedure P1. In the third procedure, P3, which is suggested for the first time in this paper, the one-dimensional FDDs are taken along the straight lines passing through a selected point inside the drop (radial line). Then, the drop profile is calculated like in procedure P1. It is shown, that procedure P3 provides a drop profile which is more reasonable than the other ones. Relationship of the discussed procedures to those used in image analysis is briefly discussed.

A nanodrop on the surface of a lubricating liquid covering a rough solid surface

A two-component fluid consisting of a lubricating fluid (LF) that covers a rough solid surface (surface decorated by periodic array of identical pillars) and a test fluid (TF) as a nanodrop over LF is considered. A horizontal external perturbative force is applied to TF and the density functional theory is used for the treatment of the system. The concepts of advancing and receding contact angles as well as of leading edges of the drop are revisited. Three different definitions of the contact angles are analyzed and the most plausible selected. The contact angles are calculated as functions of drop size and magnitude of the perturbative force. For small drops, both angles change nonmonotonously with increasing perturbative force. For larger drops, the advancing contact angle has the tendency to increase and the receding contact angle to decrease with increasing force. The sticking force which maintains the drop equilibrium in the presence of an external perturbative force is determined as function of the contact angles. It is shown that this dependence is similar to that for a drop on a rough solid surface in the absence of LF. A critical sticking force, defined as the largest value of the perturbative force for which the drop remains at equilibrium, is determined.

Nanodrop on a smooth solid surface with hidden roughness. Density functional theory considerations

A nanodrop of a test fluid placed on a smooth surface of a solid material of nonuniform density which covers a rough solid surface (hidden roughness) is examined, on the basis of the density functional theory (DFT), in the presence of an external perturbative force parallel to the surface. The contact angles which the drop profile makes with the surface at the leading edges of the drop are determined as functions of drop size and perturbative external force. A critical sticking force, defined as the largest value of the perturbative force for which the drop remains at equilibrium, is determined and its dependence on the size of the drop is explained on the basis of the shape of the interaction potential generated by the solid in vicinity of the leading edges of the drop. For even larger values of the perturbative force no drop-like solution of the Euler-Lagrange equation of the DFT was found. The upper bound of the inclination angle of a surface containing a macroscopic drop is estimated on the basis of results obtained for nanodrops and some experimental results are interpreted. The main conclusion is that the hidden roughness has a similar effect on the drop features as the traditionally considered physical and chemical roughnesses.

Evaluation of macroscale wetting equations on a microrough surface

The wettability of critical droplets on microscale geometric rough surfaces has been investigated using a density functional theory approach. In order to analyze the effect of roughness on nucleation free-energy barriers, the local density fluctuations at liquid-solid interfaces induced by the multi-interactions of a corner substrate are presented to interpret the interfacial free-energy variations, and the vapor-liquid-solid contact line tensions are derived from the contact angles of nuclei to account for the three-phase contact energies. The corresponding wetting diagrams have been constructed in Cassie, Wenzel, and impregnation regions. It is shown that, under the same condition, modest deviations between the microscale and the macroscale models can be observed within the Cassie region, whereas these deviations have been enlarged in the Wenzel and impregnation regions as well as the Cassie-Wenzel transition region. These deviations are also correlated to the roughness of the surface. The reason can be attributed to the cooperative effect of the liquid-solid interfacial free energy and line tension. This study offers a fundamental understanding of wettability of ultrasmall droplets on a microscale geometric rough surface.

Contact angle of a nanodrop on a nanorough solid surface

The contact angle of a cylindrical nanodrop on a nanorough solid surface is calculated, for both hydrophobic and hydrophilic surfaces, using the density functional theory. The emphasis of the paper is on the dependence of the contact angle on roughness. The roughness is modeled by rectangular pillars of infinite length located on the smooth surface of a substrate, with fluid-pillar interactions different in strength from the fluid-substrate ones. It is shown that for hydrophobic substrates the trend of the contact angle to increase with increasing roughness, which was noted in all previous studies, is not universally valid, but depends on the fluid-pillar interactions, pillar height, interpillar distance, as well as on the size of the drop. For hydrophilic substrate, an unusual kink-like dependence of the contact angle on the nanodrop size is found which is caused by the change in the location of the leading edges of the nanodrop on the surface. It is also shown that the Wenzel and Cassie-Baxter equations can not explain all the peculiarities of the contact angle of a nanodrop on a nanorough surface.

A novel approach to the theory of homogeneous and heterogeneous nucleation

A new approach to the theory of nucleation, formulated relatively recently by Ruckenstein, Narsimhan, and Nowakowski (see Refs. [7-16]) and developed further by Ruckenstein and other colleagues, is presented. In contrast to the classical nucleation theory, which is based on calculating the free energy of formation of a cluster of the new phase as a function of its size on the basis of macroscopic thermodynamics, the proposed theory uses the kinetic theory of fluids to calculate the condensation (W(+)) and dissociation (W(-)) rates on and from the surface of the cluster, respectively. The dissociation rate of a monomer from a cluster is evaluated from the average time spent by a surface monomer in the potential well as obtained from the solution of the Fokker-Planck equation in the phase space of position and momentum for liquid-to-solid transition and the phase space of energy for vapor-to-liquid transition. The condensation rates are calculated using traditional expressions. The knowledge of those two rates allows one to calculate the size of the critical cluster from the equality W(+)=W(-) as well as the rate of nucleation. The developed microscopic approach allows one to avoid the controversial application of classical thermodynamics to the description of nuclei which contain a few molecules. The new theory was applied to a number of cases, such as the liquid-to-solid and vapor-to-liquid phase transitions, binary nucleation, heterogeneous nucleation, nucleation on soluble particles and protein folding. The theory predicts higher nucleation rates at high saturation ratios (small critical clusters) than the classical nucleation theory for both solid-to-liquid as well as vapor-to-liquid transitions. As expected, at low saturation ratios for which the size of the critical cluster is large, the results of the new theory are consistent with those of the classical one. The present approach was combined with the density functional theory to account for the density profile in the cluster. This approach was also applied to protein folding, viewed as the evolution of a cluster of native residues of spherical shape within a protein molecule, which could explain protein folding/unfolding and their dependence on temperature.

Contact angles on surfaces using mean field theory: nanodroplets vs. nanoroughness

With emerging systems and applications accessing features within the nanoregime, whether due to droplet size or feature size, understanding the wetting behaviours of these materials is an area of ongoing interest. Theoretical studies, providing a fundamental understanding of how contact angle behaviour changes at these length scales, are important to further such work. This study provides a comprehensive examination of the application of lattice density functional theory (LDFT) to a pillared surface to confirm the suitability of LDFT for studying more complex surfaces. Incorporation of the correct level of details for the fluid-wall interaction was found to produce all of the qualitative changes that have been observed in off-lattice theories. Though previous reports have provided apparently conflicting results, the more comprehensive examination of feature sizes provided here demonstrates that those behaviours are consistent with one another. The well-studied failure of macroscopic models that results from non-negligible line tension contributions and small droplets to feature size ratios was demonstrated using LDFT. Furthermore, the failure of macroscopic models resulting upon reduction in feature size, which has been considered less often, is clearly demonstrated. A key assumption of the macroscopic models is the consistent interaction between surface and fluid regardless of the flatness or roughness of the surface. The density functional results presented here show that, for the smallest features, this is not the case and demonstrate that macroscopic models do not predict the correct contact angle for droplets of any size on nanorough surfaces.

Adsorption isoterms and capillary condensation in a nanoslit with rough walls: a density functional theory

Adsorption isoterms and capillary condensation in an open slit with walls decorated with arrays of pillars are examined using the density functional theory. Compared with the main substrate, the pillars can have the same or different parameters in the Lennard-Jones interaction potential between them and the fluid in the slit. The roughness of the solid surface, defined as the ratio between the area of the actual surface and the area of the surface free of pillars, is controlled by the height of the pillars. It is shown that the capillary condensation pressure first increases with increasing roughness, passes through a maximum, and then decreases. The amount of adsorbed fluid at constant volume of the slit has, in general, a nonmonotonic dependence on roughness. These features of adsorption and capillary condensation are results of increased surface area and changes in the fluid-solid potential energy due to changes in roughness.

Nanodrop of an Ising magnetic fluid on a solid surface

The density functional theory of inhomogeneous simple fluids is extended to an Ising magnetic fluid in contact with a solid surface, which is subjected to an external uniform or nonuniform magnetic field. The system is described by two coupled integral equations regarding the magnetic moment and fluid density distributions. The dependence of the contact angle that a nanodrop makes with the solid surface on the parameters involved in the magnetic interactions between the molecules of fluid and between the molecules of fluid and an external magnetic field is calculated. For the uniform magnetic field, the contact angle increases with increasing magnetic field, approaching an asymptotic value that depends on the strength of the fluid-fluid magnetic interactions. In the nonuniform field generated by a permanent magnet, the contact angle first increases with increasing magnetic field B(M) and then decreases, with the decrease being almost linear for large values of B(M). The obtained results are in qualitative agreement with the experimental data on the contact angle of magnetic drops on a solid surface available in the literature.

Size dependence of the contact angle of a nanodrop in a nanocavity: density functional theory considerations

The dependence of the contact angles of nanodrops of Lennard-Jones type fluids in nanocavities on their sizes are calculated using a nonlocal density functional theory in a canonical ensemble. Cavities of various radii and depths, various temperatures, as well as various values of the energy parameter of the fluid-solid interactions were considered. It is argued that this dependence might affect strongly, for instance, the rate of heterogeneous nucleation on rough surfaces, which is usually calculated under the assumption of constant contact angle.

The influence of molecular-scale roughness on the surface spreading of an aqueous nanodrop

We examine the effect of nanoscale roughness on spreading and surface mobility of water nanodroplets. Using molecular dynamics, we consider model surfaces with sub-nanoscale asperities at varied surface coverage and with different distribution patterns. We test materials that are hydrophobic, and those that are hydrophilic in the absence of surface corrugations. Interestingly, on both types of surfaces, the introduction of surface asperities gives rise to a sharp increase in the apparent contact angle. The Cassie-Baxter equation is obeyed approximately on hydrophobic substrates, however, the increase in the contact angle on a hydrophilic surface differs qualitatively from the behavior on macroscopically rough surfaces described by the Wenzel equation. On the hydrophobic substrate, the superhydrophobic state with the maximal contact angle of 180 degrees is reached when the asperity coverage falls below 25%, suggesting that superhydrophobicity can also be achieved by the nanoscale roughness of a macroscopically smooth material. We further examine the effect of surface roughness on droplet mobility on the substrate. The apparent diffusion constant shows a dramatic slow down of the nanodroplet translation even for asperity coverage in the range of 1% for a hydrophilic surface, while droplets on corrugated hydrophobic surfaces retain the ability to flow around the asperities. In contrast, for smooth surfaces we find that the drop mobility on the hydrophilic surface exceeds that on the hydrophobic one.

Multiscale morphology of organic semiconductor thin films controls the adhesion and viability of human neural cells

We investigate how multiscale morphology of functional thin films affects the in vitro behavior of human neural astrocytoma 1321N1 cells. Pentacene thin film morphology is precisely controlled by means of the film thickness, Theta (here expressed in monolayers (ML)). Fluorescence and atomic force microscopy allow us to correlate the shape, adhesion, and proliferation of cells to the morphological properties of pentacene films controlled by saturated roughness, sigma, correlation length, xi, and fractal dimension, d(f). At early incubation times, cell adhesion exhibits a transition from higher to lower values at Theta approximately 10 ML. This is explained using a model of conformal adhesion of the cell membrane onto the growing pentacene islands. From the model fitting of the data, we show that the cell explores the surface with a deformation of the membrane whose minimum curvature radius is 90 (+/- 45) nm. The transition in the adhesion at approximately 10 ML arises from the saturation of xi accompanied by the monotonic increase of sigma, which leads to a progressive decrease of the pentacene local radius of curvature and hence to the surface area accessible to the cell. Cell proliferation is also enhanced for Theta < 10 ML, and the optimum morphology parameter ranges for cell deployment and growth are sigma <or= 6 nm, xi > 500 nm, and d(f) > 2.45. The characteristic time of cell proliferation is tau approximately 10 +/- 2 h.

Microscopic description of a drop on a solid surface

Two approaches recently suggested for the treatment of macro- or nanodrops on smooth or rough, planar or curved, solid surfaces, based on fluid-fluid and fluid-solid interaction potentials are reviewed. The first one employs the minimization of the total potential energy of a drop by assuming that the drop has a well defined profile and a constant liquid density in its entire volume with the exception of the monolayer nearest to the surface where the density has a different value. As a result, a differential equation for the drop profile as well as the necessary boundary conditions are derived which involve the parameters of the interaction potentials and do not contain such macroscopic characteristics as the surface tensions. As a consequence, the macroscopic and microscopic contact angles which the drop profile makes with the surface can be calculated. The macroscopic angle is obtained via the extrapolation of the circular part of the drop profile valid at some distance from the surface up to the solid surface. The microscopic angle is formed at the intersection of the real profile (which is not circular near the surface) with the surface. The theory provides a relation between these two angles. The ranges of the microscopic parameters of the interaction potentials for which (i) the drop can have any height (volume), (ii) the drop can have a restricted height but unrestricted volume, and (iii) a drop cannot be formed on the surface were identified. The theory was also extended to the description of a drop on a rough surface. The second approach is based on a nonlocal density functional theory (DFT), which accounts for the inhomogeneity of the liquid density and temperature effects, features which are missing in the first approach. Although the computational difficulties restrict its application to drops of only several nanometers, the theory can be applied indirectly to macrodrops by calculating the surface tensions and using the Young equation to determine the contact angle. Employing the canonical ensemble version of the DFT, nanodrops on smooth and rough solid surfaces could be investigated and their characteristics, such as the drop profile, contact angle, as well as the fluid density distribution inside the drop can be determined as functions of the parameters of the interaction potentials and temperature. It was found that the contact angle of the drop has a simple (quasi)universal dependence on the energy parameter epsilon(fs) of the fluid-solid interaction potential and temperature. The main feature of this dependence is the existence of a fixed value theta(0) of the contact angle theta which separates the solid substrates (characterized by the energy parameter epsilon(fs) of the fluid-solid interaction potential) into two classes with respect to their temperature dependence. For theta>theta(0) the contact angle monotonously increases and for theta<theta(0) monotonously decreases with increasing temperature. For theta=theta(0) the contact angle is independent of temperature. The results obtained via DFT were also applied to check the validity of the macroscopic phenomenological equations (Cassie-Baxter and Wenzel equations) for drops on rough surfaces, and of the equation for the sticking force of a drop on an inclined surface.

Contact angles of nanodrops on chemically rough surfaces

The experimental observations of Gao and McCarthy [Gao, L.; McCarthy, T. Langmuir, 2007, 23, 3762] that only the interfacial area near the leading edges of the drop on physically smooth but chemically rough solid surfaces affects the contact angle and that most of the contact area has no effect is checked for nanodrops on the basis of a density functional theory. The contact angle was calculated for three cases: (i) the leading edges of the drops are located on much higher or (ii) much lower hydrophobic surfaces than the remaining surface beneath the drop; (iii) the surface is composed of a periodic array of two kinds of stripelike solid plates. In the first two cases, if the distance between the leading edges and the region which has higher or lower hydrophobicity is sufficiently large, there is agreement with the experiments mentioned. However, when those distances are sufficiently small, the internal part affects the value of the angle. In the third case, we found that the internal part always affects the wetting angle. All these peculiarities, as well as the contact angle hysteresis, can be explained by accounting for the local conditions in the vicinity of the leading edges of the drop.