Line tension and the shape of nanodroplets

Effect of molecular branching and surface wettability on solid-liquid surface tension and line-tension of liquid alkane surface nanodroplets

HYPOTHESIS

Surface nanodroplets have important technological applications. Previous experiments and simulations have shown that their contact angle deviates from Young's equation. A modified version of Young's equation considering the three-phase line tension (τ) has been widely used in literature, and a wide range of values for τ are reported. We have recently shown that molecular branching affects the liquid-vapour surface tension γlv of liquid alkanes. Therefore, the wetting behaviour of surface nanodroplets should be affected by molecular branching. This study conducted molecular dynamics (MD) simulations to gain insight into the wetting behaviour of linear and branched alkane nanodroplets on oleophilic and oleophobic surfaces. We aim to examine the Young equation's validity and branching's effect on fundamental properties, including solid-liquid surface tension γsl and line tension τ.

SIMULATIONS

The simulations were performed on a linear alkane, triacontane (C30H62), as well as four of its branched isomers: 2,6,13,17-tetrapropyloctadecane,2,6,9,10,13,17-hexaethyloctadecane, 2,5,7,8,11,12,15-heptaethylhexadecane and 2,3,6,7,10,11-hexapropyldodecane. Nanodroplets with a diameter of approximately 15 nm were released onto the surfaces, and their contact angles were measured. Additionally, using a novel approach, the solid-liquid surface tension (γsl), the validity of Young's equation and line tension for all alkane and surface combinations are determined.

FINDINGS

It was discovered that the calculated γsl, deviated from the theoretical γsl,Young predicted from Young's equation for all alkanes on oleophilic surfaces. However, this deviation was minimal for branched alkanes on the oleophobic surfaces but more significant for the linear alkane. The findings indicated that γsl < 0 for oleophilic surfaces and γsl > 0 for oleophobic surfaces. Moreover, it was observed that |γsl| was lower for branched molecules and decreased as branching increased. Line tension values were then determined through a novel method, showing τ was positive for oleophilic surfaces ranging from 1.30 × 10-10 to 6.27 × 10-11N. On an oleophobic surface, linear alkane shows a negative line tension of -1.15 × 10-10N and branched alkanes up to two orders of magnitude lower values ranging from -2.09 × 10-12 to 2.43 × 10-11N. Line tension values between -1.15 × 10-10 and + 1.1 × 10-10N are calculated for various linear alkane and surface combinations. These findings show the dependence of line tension on the contact angle and branching, demonstrating that for linear alkanes, τ is significant, whereas, for branched alkanes, line tension is smaller or negligible for large contact angles.

Line Tension of Nanodroplets on a Concave Surface

The contact angle of a nanodroplet on a surface may deviate from that of a macrodroplet on a surface. Even though there are many studies regarding line tension, it is not well understood. In this paper, molecular dynamics simulation is performed for nanodroplets on a concave solid wall. The Lennard-Jones (L-J) potential is directly used or modified to simulate the force interaction between argon atoms and between solid-liquid particles. The initial droplet radius is 4, 5, and 6 nm, respectively. The k coefficient is defined as the ratio of the initial droplet radius with respect to the curvature radius of concave walls, which is in the range of 0-0.9, in which k = 0 refers to a flat surface. We found that indeed the contact angle θ of a nanodroplet on a concave wall deviates from that of a macrodroplet on a flat surface. Contact angles display a two region distribution, in which θ increases with increasing k for k < 0.5 and decreases with increasing k for k > 0.5. The k coefficient influences the droplet morphology. With k in the range of 0-0.9, the vapor-liquid interface is switched from a convex shape to a flat shape and finally reaches a concave shape. The line tension generally behaves in an increasing trend with the increase of k but becomes constant when k is beyond 0.7. The liquid densities, radial distribution functions, and coordination numbers show that the liquid particles are more closely packed with each other with the increase of k. The line tension achieves a positive sign and on the magnitude of 10^-11 N, which has a linear increase with respect to the peak density of the first liquid layer.

Do the contact angle and line tension of surface-attached droplets depend on the radius of curvature?

Results from Monte Carlo simulations of wall-attached droplets in the three-dimensional Ising lattice gas model and in a symmetric binary Lennard-Jones fluid, confined by antisymmetric walls, are analyzed, with the aim to estimate the dependence of the contact angle [Formula: see text] on the droplet radius [Formula: see text] of curvature. Sphere-cap shape of the wall-attached droplets is assumed throughout. An approach, based purely on 'thermodynamic' observables, e.g. chemical potential, excess density due to the droplet, etc, is used, to avoid ambiguities in the decision which particles belong (or do not belong, respectively) to the droplet. It is found that the results are compatible with a variation [Formula: see text], [Formula: see text] being the contact angle in the thermodynamic limit ([Formula: see text]). The possibility to use such results to estimate the excess free energy related to the contact line of the droplet, namely the line tension, at the wall, is discussed. Various problems that hamper this approach and were not fully recognized in previous attempts to extract the line tension are identified. It is also found that the dependence of wall tensions on the difference of chemical potential of the droplet from that at the bulk coexistence provides effectively a change of the contact angle of similar magnitude. The simulation approach yields precise estimates for the excess density due to wall-attached droplets and the corresponding free energy excess, relative to a system without a droplet at the same chemical potential. It is shown that this information suffices to estimate nucleation barriers, not affected by ambiguities on droplet shape, contact angle and line tension.

Water graphene contact surface investigated by pairwise potentials from force-matching PAW-PBE with dispersion correction

A pairwise additive atomistic potential was developed for modeling liquid water on graphene. The graphene-water interaction terms were fit to map the PAW-PBE-D3 potential energy surface using the adaptive force matching method. Through condensed phase force matching, the potential developed implicitly considers the many-body effects of water. With this potential, the graphene-water contact angle was determined to be 86° in good agreement with a recent experimental measurement of 85° ± 5° on fully suspended graphene. Furthermore, the PAW-PBE-D3 based model was used to study contact line hysteresis. It was found that the advancing and receding contact angles of water do agree on pristine graphene, however a long simulation time was required to reach the equilibrium contact angle. For water on suspended graphene, sharp peaks in the water density profile disappear when the flexibility of graphene was explicitly considered. The water droplet induces graphene to wrap around it leading to a slightly concave contact interface.

Multiscale Simulation Method for Quantitative Prediction of Surface Wettability at the Atomistic Level

The solid-liquid interface is of great interest because of its highly heterogeneous character and its ubiquity in various applications. The most fundamental physical variable determining the strength of the solid-liquid interface is the solid-liquid interfacial tension, which is usually measured according to the contact angle. However, an accurate experimental measurement and a reliable theoretical prediction of the contact angle remain lacking because of many practical issues. Here, we propose a first-principles-based simulation approach to quantitatively predict the contact angle of an ideally clean surface using our recently developed multiscale simulation method of density functional theory in classical explicit solvents (DFT-CES). Using this approach, we simulate the surface wettability of a graphene and graphite surface, resulting in a reliable contact angle value that is comparable to the experimental data. From our simulation results, we find that the surface wettability is dominantly affected by the strength of the solid-liquid van der Waal's interaction. However, we further elucidate that there exists a secondary contribution from the change of water-water interaction, which is manifested by the change of liquid structure and dynamics of interfacial water layer. We expect that our proposed method can be used to quantitatively predict and understand the intriguing wetting phenomena at an atomistic level and can eventually be utilized to design a surface with a controlled hydrophobic(philic)ity.

Size-Dependent Submerging of Nanoparticles in Polymer Melts: Effect of Line Tension

Adhesion of nanoparticles to polymer films plays a key role in various polymer technologies. Here we report experiments that reveal how silica nanoparticles adhere to a viscoelastic PMMA film above the glass transition temperature. The polymer was swollen with CO₂, closely matching the conditions of nanoparticle-nucleated polymer foaming. It is found that the degree by which the particles sink into the viscoelastic substrate is strongly size dependent and can even lead to complete engulfment for particles of diameter below 12 nm. These findings are explained quantitatively by a thermodynamic analysis, combining elasticity, capillary adhesion, and line tension. We argue that line tension, here proposed for the first time in elastic media, is responsible for the nanoparticle engulfment.

Intrusion and extrusion of water in hydrophobic nanopores

Molecular springs, constituted by nanoporous materials immersed in a nonwetting liquid, are compact, economical, and efficient means of storing energy, owing to their enormous surface area. Surface energy is accumulated during liquid intrusion inside the pores and released by decreasing liquid pressure and thus triggering confined cavitation. State-of-the-art atomistic simulations shed light on the intrusion and extrusion of water in hydrophobic nanopores, revealing conspicuous deviations from macroscopic theories, which include accelerated cavitation, increased intrusion pressure, and reversible intrusion and extrusion processes. Understanding these nanoscale phenomena is the key to a better design of molecular springs as it allows relating the characteristics of the materials to the overall properties of the devices, e.g., their operational pressure and efficiency.

Heterogeneous systems composed of hydrophobic nanoporous materials and water are capable, depending on their characteristics, of efficiently dissipating (dampers) or storing (“molecular springs”) energy. However, it is difficult to predict their properties based on macroscopic theories—classical capillarity for intrusion and classical nucleation theory (CNT) for extrusion—because of the peculiar behavior of water in extreme confinement. Here we use advanced molecular dynamics techniques to shed light on these nonclassical effects, which are often difficult to investigate directly via experiments, owing to the reduced dimensions of the pores. The string method in collective variables is used to simulate, without artifacts, the microscopic mechanism of water intrusion and extrusion in the pores, which are thermally activated, rare events. Simulations reveal three important nonclassical effects: the nucleation free-energy barriers are reduced eightfold compared with CNT, the intrusion pressure is increased due to nanoscale confinement, and the intrusion/extrusion hysteresis is practically suppressed for pores with diameters below 1.2 nm. The frequency and size dependence of hysteresis exposed by the present simulations explains several experimental results on nanoporous materials. Understanding physical phenomena peculiar to nanoconfined water paves the way for a better design of nanoporous materials for energy applications; for instance, by decreasing the size of the nanopores alone, it is possible to change their behavior from dampers to molecular springs.

Macroscopic Domains within an Oriented TQ1 Film Visualized Using 2D Polarization Imaging

Large-area self-assembly of functional conjugated polymers holds a great potential for practical applications of organic electronic devices. We obtained well-aligned films of poly[2,3-bis(3-octyloxyphenyl)quinoxaline-5,8-diyl-alt-thiophene-2,5-diyl] (TQ1) using the floating film transfer method. Thereby, a droplet of the TQ1 solution was injected on top of the surface of an immiscible liquid substrate, at the meniscus formed at the edge of a Petri dish, from where the polymer solution and the film spread in one direction. Characterization of the TQ1 film using the recently developed two-dimensional polarization imaging (2D POLIM) revealed large, millimeter-sized domains of oriented polymer chains. The irregular shape of the contact line at the droplet source induced the appearance of disordered stripes perpendicular to the spreading direction. A correlation of polarization parameters measured using 2D POLIM revealed the microstructure of such stripes, providing valuable information for further improvement and possible upscaling of this promising method.

Silicon-wall interfacial free energy via thermodynamics integration

We compute the interfacial free energy of a silicon system in contact with flat and structured walls by molecular dynamics simulation. The thermodynamics integration method, previously applied to Lennard-Jones potentials [R. Benjamin and J. Horbach, J. Chem. Phys. 137, 044707 (2012)], has been extended and implemented in Tersoff potentials with two-body and three-body interactions taken into consideration. The thermodynamic integration scheme includes two steps. In the first step, the bulk Tersoff system is reversibly transformed to a state where it interacts with a structureless flat wall, and in a second step, the flat structureless wall is reversibly transformed into an atomistic SiO₂ wall. Interfacial energies for liquid silicon-wall interfaces and crystal silicon-wall interfaces have been calculated. The calculated interfacial energies have been employed to predict the nucleation mechanisms in a slab of liquid silicon confined by two walls and compared with MD simulation results.

How a Surface Nanodroplet Sits on the Rim of a Microcap

The location and morphology of femtoliter nanodroplets that nucleate and grow on a microcap-decorated substrate in contact with a liquid phase were investigated. We experimentally examined four different wetting combinations of the flat area and the microcaps. The results show that depending on the relative wettability, the droplets sit either on the plain surface or on the top of the microcap or on the rim of the microcap. The contact angle and, for the last case, the radial positions of the nanodroplets relative to the microcap center were characterized, in reasonable agreement with our theoretical analysis, which is based on an interfacial energy minimization argument. However, the experimental data show considerable scatter around the theoretical equilibrium curves, reflecting pinning and thus nonequilibrium effects. We also provide the theoretical phase diagram in parameter space of the contact angles, revealing under which conditions the nanodroplet will nucleate on the rim of the microcap.

Line Tension and Wettability of Nanodrops on Curved Surfaces

In this work we study the formation of nanodrops on curved surfaces (both convex and concave) by means of molecular dynamics simulations, where the particles interact via a Lennard-Jones potential. We find that the contact angle is not affected by the curvature of the substrate, in agreement with previous experimental findings. This means that the change in curvature of the drop in response to the change in curvature of the substrate can be predicted from simple geometrical considerations, under the assumption that the drop's shape is a spherical cap, and that the volume remains unchanged through the curvature. The resulting prediction is in perfect agreement with the simulation results, for both convex and concave substrates. In addition, we calculate the line tension, namely, by fitting the contact angle for different size drops to the modified Young equation. We find that the line tension for concave surfaces is larger than for convex surfaces, while for zero curvature it has a clear maximum. This feature is found to be correlated with the number of particles in the first layer of the liquid on the surface.

Dry-Surface Simulation Method for the Determination of the Work of Adhesion of Solid-Liquid Interfaces

We introduce a methodology, referred to as the dry-surface method, to calculate the work of adhesion of heterogeneous solid-liquid interfaces by molecular simulation. This method employs a straightforward thermodynamic integration approach to calculate the work of adhesion as the reversible work to turn off the attractive part of the actual solid-liquid interaction potential. It is formulated in such a way that it may be used either to evaluate the ability of force fields to reproduce reference values of the work of adhesion or to optimize force-field parameters with reference values of the work of adhesion as target quantities. The methodology is tested in the case of water on a generic model of nonpolar substrates with the structure of gold. It is validated through a quantitative comparison to phantom-wall calculations and against a previous characterization of the thermodynamics of the gold-water interface. It is found that the work of adhesion of water on nonpolar substrates is a nonlinear function of the microscopic solid-liquid interaction energy parameter. We also comment on the ability of mean-field approaches to predict the work of adhesion of water on nonpolar substrates. In addition, we discuss in detail the information on the solid-liquid interfacial thermodynamics delivered by the phantom-wall approach. We show that phantom-wall calculations yield the solid-liquid interfacial tension relative to the solid surface tension rather than the absolute solid-liquid interfacial tension as previously believed.

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.

Influence of contact-line curvature on the evaporation of nanodroplets from solid substrates

The effect of the three-phase contact-line curvature on the evaporation mechanism of nanoscopic droplets from smooth and chemically homogenous substrates is studied by molecular dynamics simulations. Spherical droplets, whose three-phase contact line is curved, and cylindrical droplets, whose contact radius is infinite, are compared. It is found that the evaporation of cylindrical droplets takes place at constant contact angle, while spherical droplets evaporate by simultaneous reduction of their contact area and their contact angle. This is independent of the substrate-liquid interaction strength. The dependence of the evaporation mechanism on the contact-line curvature can be rationalized with the help of the concept of a contact-line tension, and the evaporation simulations of the spherical droplets are used to extract the line tension on each surface. The corresponding values for the Lennard-Jones systems studied here are of the order of 10(-11)N, which is in a good agreement with previous theoretical and experimental estimates. With this order of magnitude, the line tension is expected to have an effect on the contact angle of spherical droplets only, when their diameter is less than about 100 nm. The observed difference in evaporation mechanism is interpreted as a manifestation of the line tension whose existence has been controversial.

Theoretical model of droplet wettability on a low-surface-energy solid under the influence of gravity

The wettability of droplets on a low surface energy solid is evaluated experimentally and theoretically. Water-ethanol binary mixture drops of several volumes are used. In the experiment, the droplet radius, height, and contact angle are measured. Analytical equations are derived that incorporate the effect of gravity for the relationships between the droplet radius and height, radius and contact angle, and radius and liquid surface energy. All the analytical equations display good agreement with the experimental data. It is found that the fundamental wetting behavior of the droplet on the low surface energy solid can be predicted by our model which gives geometrical information of the droplet such as the contact angle, droplet radius, and height from physical values of liquid and solid.

Measurement of line tension on droplets in the submicrometer range

Wetting is a universal phenomenon in nature and of interest in fundamental research as well as in engineering sciences. Usually, wetting of solid substrates by liquid drops is described by Young's equation, which relates the contact angle between the liquid and the substrate to the three interfacial tensions. This concept has been widely used and confirmed for macroscopic droplets. On the contrary, it is still matter of debate to what extent this concept is able to explain relations on the micrometer scale and below. The so-called extended Young's equation, which takes account of the specific arrangement of the molecules in the three-phase contact line by implementing a term called "line tension", is frequently used to characterize deviations from the "ideal" Young's case. In this work we tried to look into the dependence of measured contact angles of droplets on their size for a close to ideal system. We measured contact angles of ionic liquid droplets with radii between some tens and some hundreds of nanometers by atomic force microscopy on an ideally flat silicon wafer. We found that the contact angles decreased with decreasing droplet size: smaller droplets showed stronger wetting. This dependence of the contact angle on the droplet radius could not be described with the concept of line tension or the modified Young's equation. We propose simple arguments for a possible alternative concept.

Elastic modulus of a polymer nanodroplet: theory and experiment

We redevelop a theoretical model that, in conjunction with atomic force microscopy (AFM), can be used as a noninvasive method for determination of the elastic modulus of a polymer nanodroplet residing on a flat, rigid substrate. The model is a continuum theory that combines surface and elasticity theories for prediction of the droplet's elastic modulus, given experimental measurement of its adsorbed height. Utilization of AFM-measured heights for relevant droplets reported in the literature and from our own experiments illustrated the following: the significance of both surface and elasticity effects in determining a polymer droplet's spreading behavior; the extent of a continuum theory's validity as one approaches the nanoscale; and a droplet size effect on the elastic modulus.

Wetting and contact-line effects for spherical and cylindrical droplets on graphene layers: a comparative molecular-dynamics investigation

In molecular dynamics (MD) simulations, interactions between water molecules and graphitic surfaces are often modeled as a simple Lennard-Jones potential between oxygen and carbon atoms. A possible method for tuning this parameter consists of simulating a water nanodroplet on a flat graphitic surface, measuring the equilibrium contact angle, extrapolating it to the limit of a macroscopic droplet, and finally matching this quantity to experimental results. Considering recent evidence demonstrating that the contact angle of water on a graphitic plane is much higher than what was previously reported, we estimate the oxygen-carbon interaction for the recent SPC/Fw water model. Results indicate a value of about 0.2 kJ/mol, much lower than previous estimations. We then perform simulations of cylindrical water filaments on graphitic surfaces, in order to compare and correlate contact angles resulting from these two different systems. Results suggest that a modified Young's equation does not describe the relation between contact angle and drop size in the case of extremely small systems and that contributions different from the one deriving from contact line tension should be taken into account.