A self-consistent field study of a hydrocarbon droplet at the air-water interface

A molecularly detailed self-consistent field (SCF) approach is applied to describe a sessile hydrocarbon droplet placed at the air-water interface. Predictions of the contact angle for macroscopic droplets follow from using Neumann's equation, wherein the macroscopic interfacial tensions are computed from one-gradient calculations for flat interfaces. A two-gradient cylindrical coordinate system with mirror-like boundary conditions is used to analyse the three dimensional shape of the nano-scale oil droplet at the air-water interface. These small droplets have a finite value of the Laplace pressure and concomitant line tension. It has been calculated that the oil-water and oil-vapour interfacial tensions are curvature dependent and increase slightly with increasing interfacial curvature. In contrast, the line tension tends to decrease with curvature. In all cases there is only a weak influence of the line tension on the droplet shape. We therefore argue that the nano-scale droplets, which are described in the SCF approach, are representative for macroscopic droplets and that the method can be used to efficiently generate accurate information on the spreading of oil droplets at the air-water interface in molecularly more complex situations. As an example, non-ionic surfactants have been included in the system to illustrate how a molecularly more complex situation will change the wetting properties of the sessile drop. This short forecast is aimed to outline and to stress the potential of the method.     

Evaluating the third and fourth derivatives of spectral data

The problem of differentiating spectral data to yield the third and fourth derivatives is converted into one of solving an integral equation of the first kind. This equation is solved by Tikhonov regularization. The method of General Cross Validation is used to guide the choice of the regularization parameter that keeps noise amplification under control. The performance of this route to third and fourth derivative spectra is demonstrated by applying it to a number of published spectra. A computational problem associated with General Cross Validation has been identified.     

Measurement of emulsion droplet sizes using PFG NMR and regularization methods

The droplet size distributions of emulsions have been measured using pulsed field gradient (PFG) nuclear magnetic resonance (NMR) for many years. This technique finds particular application with emulsions that are concentrated and/or opaque, since such emulsion systems are difficult to characterize by other methods. Most studies employing PFG techniques assume a lognormal form when extracting the droplet size distribution from the experimental data. It is clearly desirable to retrieve a droplet size distribution from the experimental data without assuming such a functional form. This is achieved for the first time using regularization techniques. Regularization based on the distribution area and on its second derivative are compared and assessed along with the following techniques for selecting the optimal regularization parameter: the L-curve method, generalized cross validation (GCV), and the discrepancy principle. Regularization is applied to both simulated data sets and experimental data. It is found that when the experimental error can be estimated accurately, the discrepancy principle with area regularization is the best approach. When the error is not known the GCV method, with second derivative regularization and allowing only nonnegative values, is most effective.     

Tensile film stress of parylene deposited on liquid

We found that liquid droplets encapsulated by Parylene deposited directly on a liquid surface deformed toward spherical shapes during Parylene deposition. This deformation suggested that the film stress was tensile. We calculated the film stress of such Parylene films by studying the surface mean curvature of the droplet shape and found the film stress measured about 0.7-0.9 MPa tensile. This film stress is of opposite type to that of as-deposited Parylene films deposited on solid substrates, which was compressive. This difference might indicate a profound change of the Parylene polymer due to the use of liquid surface as deposition substrate. The tensile film stress and its effect on the droplet shape also have implications in the fabrication and operation of Parylene microdevices that have encapsulated liquid structures such as microlens or micropumps.     

Computation of constant mean curvature surfaces: Application to the gas-liquid interface of a pressurized fluid on a superhydrophobic surface

The interface shape separating a gas layer within a superhydrophobic surface consisting of a square lattice of posts from a pressurized liquid above the surface is computed numerically. The interface shape is described by a constant mean curvature surface that satisfies the Young-Laplace equation with the three-phase gas-liquid-solid contact line assumed pinned at the post outer edge. The numerical method predicts the existence of constant mean curvature solutions from the planar, zero curvature solution up to a maximum curvature that is dependent on the post shape, size and pitch. An overall force balance between surface tension and pressure forces acting on the interface yields predictions for the maximum curvature that agree with the numerical simulations to within one percent for convex shapes such as circular and square posts, but significantly over predicts the maximum curvature for non-convex shapes such as a circular post with a sinusoidal surface perturbation. Changing the post shape to increase the contact line length, while maintaining constant post area, results in increases of 2 to 12% in the maximum computable curvature for contact line length increases of 11 to 77%. Comparisons are made to several experimental studies for interface shape and pressure stability.     

Amphiphilic polysaccharides: useful tools for the preparation of nanoparticles with controlled surface characteristics

Polymeric surfactants obtained by hydrophobic modification of dextran are used as stabilizers for oil-in-water emulsions. The kinetics of interfacial tension decrease is studied as a function of polymer structural characteristics (degree of hydrophobic substitution) and at various polymer concentrations. Several hydrocarbon oils, either aliphatic (octane, decane, dodecane, and hexadecane) or aromatic (styrene), are tested. Kinetics exhibits the same general trends no matter which oil or polymer is considered. The emulsifying properties of the polymeric surfactants are illustrated by the preparation of oil-in-water emulsions. The droplet size at the preparation is correlated to the amount of oil and to the polymer concentration in the aqueous phase. For low polymer/oil ratios, it is shown that the droplet size is limited by the initial amount of polymer. On the contrary, for high polymer/oil ratios, the droplet size seems to level down, indicating that other parameters become predominant. Emulsion aging occurs by Ostwald ripening, and it is demonstrated that the theoretical equation of Lifshitz, Slyozov, and Wagner (LSW) correctly describes the experimental results. The nature of the oil has important effects on emulsion aging, as described by the LSW equation. The aging of emulsions containing oil mixtures is quantitatively described on the basis of the results with pure oils. The influence of polymer chemical structure can be conveniently correlated to interfacial tension results through the LSW equation. On the contrary, the influence of oil volume fraction seems to be overestimated by the usual correction factor, k(phi). The effect of temperature on emulsion aging is finally examined. Miniemulsions stabilized with dextran derivatives are used for the radical polymerization of styrene. Following this procedure, polysaccharide-covered polystyrene nanoparticles are prepared and characterized (size and surface coverage). The size of the particles is directly correlated to that of the initial droplets for styrene volume fractions around 10%. On the contrary, for initial styrene volume fractions around 20%, particles exhibit a larger size than the initial droplets, indicating that coalescence processes take place during polymerization. The amount of dextran at the surface of the particles is determined and compared to the adsorbed amounts resulting from emulsion preparation.     

On the curvature dependence of the interfacial tension in a symmetric three-component interface

We consider a symmetric interface between two polymers A(N) and B(N) in a common monomeric solvent S using the mean-field Scheutjens-Fleer self-consistent field theory and focus on the curvature dependence of the interfacial tension. In multi-component systems there is not one unique scenario to curve such an interface. We elaborate on this by keeping either the chemical potential of the solvent or the bulk concentration of the solvent fixed, that is we focus on the semi-grand canonical ensemble case. Following Helfrich, we expand the surface tension as a Taylor series in the curvature parameters and find that there is a non-zero linear dependence of the interfacial tension on the mean curvature in both cases. This implies a finite Tolman length. In a thermodynamic analysis we prove that the non-zero Tolman length is related to the adsorption of solvent at the interface. Similar, but not the same, correlations between the solvent adsorption and the Tolman length are found in the two scenarios. This result indicates that one should be careful with symmetry arguments in a Helfrich analysis, in particular for systems that have a finite interfacial tension: one not only should consider the structural symmetry of the interface, but also consider the constraints that are enforced upon imposing the curvature. The volume fraction of solvent, the chain length N as well as the interaction parameter chi(AB) in the system can be used to take the system in the direction of the critical point. The usual critical behavior is found. Both the width of the interface and the Tolman length diverge, whereas the density difference between the two phases, adsorbed amount of solvent at the interface, interfacial tension, spontaneous curvature, mean bending modulus as well as the Gaussian bending modulus vanish upon approach of the critical point.     

Effect of surface area on equilibrium pressure. Adsorption approach

Changing droplet radii in a liquid-vapor system is due to the phase transition on the droplet surface. As a variation of the internal energy does not depend on the way the change occurs, we can imagine that a gas condenses on a droplet surface in two stages: in the first stage, autoadsorption occurs on the liquid surface, and in the second stage, adsorbed molecules transfer into the volume by diffusion. Assuming that the energetic effects of the diffusion are independent of the surface curvature, one may conclude that if two liquid bodies differ only with respect to their geometry, the difference of enthalpies of condensation on their surfaces, DeltaH(bd), is equal to the variation of energies of autoadsorption. An estimation of DeltaH(bd) for the simple bodies is presented, and the relationship between the saturation pressure and droplet radii is derived. In the range of micrometer dimensions, the new equation and the Kelvin model lead to close results; for nanocapillaries, the Kelvin equation predicts a divergence of hysteresis loops, whereas the new equation adequately describes the observations. The classical model presumes that a surface area, A, affects the free energy, while the new approach is based on the assumption that A is the repository for the internal energy.     

The role of the surfactant head group in the emulsification process: Single surfactant systems

To clarify the effect of the surfactant head group on the emulsification process, dilute dodecane in water emulsions were prepared in a small flow-through cell with three surfactants which had the same hydrocarbon tail length but different head groups. The different surfactants types were (a) a nonionic, hexa(ethyleneglycol) mono n-dodecyl ether (C12E6), (b) an anionic, sodium dodecyl sulfate (SDS), and (c) a cationic, n-dodecyl pyridinium chloride (DPC), and the emulsions were prepared under the same conditions. From dynamic light scattering measurements, it was shown that the mean steady state droplet size of the emulsions (obtained after 20 min dispersion) could be related to the interfacial tension at concentrations in the region of the cmc. This result was in agreement with laminar and turbulent viscous flow theory. However, the particle size versus surface tension data for the different surfactant systems did not fall on a single line. This behavior suggested that the surfactant played a secondary role in defining the droplet size (in addition to reducing the interfacial tension) possibly through diffusion and relaxation, during deformation of the interface. In addition, it was found that the values of the equilibrium "surfactant packing densities" of the different surfactants at the oil/water interface were almost equal near the cmc, but the mean droplet size and the interfacial tension at the cmc decreased following the order DPC>SDS>C12E6 .     

The effect of an anionic surfactant on the rheology and stability of high volume fraction O/W emulsion stabilized by PVA

The effect of the addition of an anionic surfactant (sodium dodecyl benzene sulphonate) on the rheology and storage stability of concentrated O/W emulsions stabilized by poly (vinyl alcohol) is reported. It was found that the surfactant markedly reduced the magnitudes of the storage modulii of the emulsions. This could be attributed to a reduction in the interfacial tension resulting from the formation of polyelectrolyte type complexes between the PVA and NaDBS at the O/W interface. The results were compared to the equation (given by Princen) relating concentrated emulsion rheology to the interfacial tension and droplet size. Reasonable agreement was found, though there was a small difference in the constants in the equation given by Princen and those found here. The agreement suggested that the emulsions were deforming above a critical volume fraction and that the rheological properties were dominated by the dilation of the interface during shear. Microelectrophoresis measurements showed that the addition of the surfactant conferred a charge onto the PVA stabilized droplets as a result of the formation of the polyelectrolyte complex. The NaDBS was found to reduce the long-term stability of the emulsions compared to emulsions containing PVA alone.     

Surface tension curvature correction for fluid–vapour interface by thermodynamics of curved boundary layers

Thermodynamics of curved boundary layers is used to study surface properties of Lennard–Jones fluids and an equation is obtained for the surface tension of the spherical interfaces. The work of cavity formation inside a fluid is utilised to calculate the surface tension. The scaled particle theory (SPT) equation of Pierotti and Stillinger are properly modified. The calculations by the modified Stillinger equation predicts the surface tension in any radius of curvature and the results are in good agreement with the experimental data. But for Pierotti equation, due to low sensitivity to temperature, good agreement obtained only at low temperatures.     

The line tension and the generalized Young equation: the choice of dividing surface

Using Gibbs method of dividing surfaces, the condition of equilibrium of a sessile drop on a flat non-deformable solid substrate is investigated. The dependence of the line tension on the curvature radius of the dividing three-phase contact line is found. It has been derived a relationship between the partial derivative of the line tension with respect to the curvature radius of the three-phase contact line (which stands in the generalized Young equation) and the total derivative of the line tension with respect to the same radius along the equilibrium states. Various approximated formulas of the generalized Young equation used in the literature are analyzed.     

The evaporation/condensation transition of liquid droplets

The condensation of a supersaturated vapor enclosed in a finite system is considered. A phenomenological analysis reveals that the vapor is found to be stable at densities well above coexistence. The system size at which the supersaturated vapor condenses into a droplet is found to be governed by a typical length scale which depends on the coexistence densities, temperature and surface tension. When fluctuations are neglected, the chemical potential is seen to show a discontinuity at an effective spinodal point, where the inhomogeneous state becomes more stable than the homogeneous state. If fluctuations are taken into account, the transition is rounded, but the slope of the chemical potential versus density isotherm develops a discontinuity in the thermodynamic limit. In order to test the theoretical predictions, we perform a simulation study of droplet condensation for a Lennard-Jones fluid and obtain loops in the chemical potential versus density and pressure. By computing probability distributions for the cluster size, chemical potential, and internal energy, we confirm that the effective spinodal point may be identified with the occurrence of a first order phase transition, resulting in the condensation of a droplet. An accurate equation of state is employed in order to estimate the droplet size and the coexisting vapor density and good quantitative agreement with the simulation data is obtained. The results highlight the need of an accurate equation of state data for the Laplace equation to have predictive power.     

Adsorption of colloidal particles to curved interfaces

As a simple model for a Pickering emulsion droplet, we consider the adsorption of spherical particles to a spherical liquid-liquid interface in order to investigate the curvature effect on the particle adsorption. By taking into account both the surface and the volume energies due to the presence of a particle, we show that the equilibrium contact angle is determined by the classical Young's equation although the adsorption energy depends on the curvature. We also calculate the partitioning of the colloidal particles among the two liquids and the interface. The distribution of colloidal particles is expressed in terms of the interfacial curvature as well as the relative wettability of the particle.     

Molecular dynamics study of structural and thermodynamic characteristics of nanodroplets of simple fluid

Equilibrium configurations of Lennard-Jones nanodroplets composed of 10–15000 spherically symmetric molecules placed in the center of a spherical container are studied at constant temperature by the molecular dynamics method. The distribution of local density is found and size dependences of density in the center of droplet, first coordination number, and energy surface tension coinciding for equimolecular dividing surface with specific excess free energy of droplet are studied. Radial distribution function is also determined. It is established that the passage of structural characteristics to their macroscopic values is observed for droplets containing as little as about 300 molecules, while, for energy surface tension, analogous passage for energy surface tension occurs for droplets containing 700–6000 molecules.     

Using the Surface Evolver to model droplet formation processes in membrane emulsification

A model was developed to describe the droplet formation mechanism in membrane emulsification from the point of view of Gibbs free energy with the help of the Surface Evolver, which is an interactive finite element program for the study of interfaces shaped by surface tension. A program to test the model was written and run which allows the user to track the droplet shape as it grows, to identify the point of instability due to free energy, and thus predict droplet size. The inputs of the program are pore geometry, oil-aqueous phase interfacial tension, and contact angle. The model reasonably predicted droplet sizes for oblong-shaped pores under quiescent conditions where the force balance approach is not applicable. The model was validated against experimental conditions from the literature where the average error of the predictions compared to the mean droplet sizes was 8%.     

Model description of contact angles in electrowetting on dielectric layers

The electrowetting on dielectric (EWOD) technique has considerable potential for microfluidic and biomedical applications. The Lippmann-Young model based on the force balance concept has long been used to predict the contact angles of droplets under electrowetting. However, recent experimental evidence has indicated that this model fails to provide accurate predictions of the lower contact angles associated with saturation conditions at higher electric potentials. Hence, the study simulates the internal flow in an actuated droplet and treats it as stagnation-point flow. This kinetic energy is then taken into consideration while calculating the contact angles using an energy balance model. The energy of an actuated droplet is contributed by the combination of the side surface tension energy, the base tension energy, the dielectric energy, and the kinetic energy when deriving the energy balance model. Consequently, the new energy balance model modifies the Lippmann-Young equation, thereby providing enhanced reasonable predictions of the droplet contact angle across the higher electric potential where the contact angles are close to the saturated condition.     

The influence of asymmetry in drop shape on an interfacial tension measurement

A requirement of the drop method for interfacial tension measurement is that the drop must have an axi-symmetry. The drop shape was measured as a function of the angle of the rod from which the drop was hanging. The deviation of the interfacial tension caused by asymmetry was calculated using the selected plane method in the pendant drop technique. A sharp maximum was seen in the interfacial tension vs angle curve when the rod was at the vertical point. The maximum value was concluded as being the true value of the interfacial tension. Both decreases in the pressure difference and in the curvature of the drop associated with an increase in the rod-angle are a counterpart with each other to satisfy the Bashforth and Adams equation. The underestimation of the interfacial tension seems to be caused by the apparent inconsistency between the decreases in the curvature and pressure difference terms. The rigorous mechanical setup demonstrated here is necessary to attain the true axi-symmetric condition, and thus obtain a reliable value for the interfacial tension.     

Modeling hydrophobic solvation of nonspherical systems: Comparison of use of molecular surface area with accessible surface area

It is shown that the molecular surface and the accessible surface lead to exactly the same results when calculating solvation free energies and transfer free energies, from methods using the surface tension as a parameter if the exact geometric curvature is used with the accessible surface. However, the use of the exact curvature is not necessarily the best approach chemically. Other modifications, including an approximate curvature improves the approach. Such modifications are difficult to include in methods in which the molecular surface rather than the accessible surface is used to calculate solvent effects. A modification of a Gaussian curvature term is necessary if dissociation is to be accounted for properly. The inclusion of a Gaussian curvature term, in addition to the usual mean curvature term, reconciles the difference in magnitude of the microscopic and macroscopic surface tension in the case of the accessible surface area. © 1997 by John Wiley & Sons, Inc.