Interpretation of surface-tension isotherms of n-alkanoic (fatty) acids by means of the van der Waals model

Here we apply the two-dimensional van der Waals model to interpret surface-tension isotherms of aqueous solutions of n-alkanoic (fatty) acids. We processed available experimental data for a homologous series of eight acids, from pentanoic to dodecanoic (lauric). Only three adjustable parameters have been varied to fit simultaneously all experimental curves. Excellent agreement between the theoretical model and the experiment has been obtained. The determined parameter values comply well with the molecular properties and allow one to calculate the surfactant adsorption, surface elasticity, and the surface pressure vs area isotherms. For the dodecanoic acid, the van der Waals model indicates the existence of a surface phase transition.

Latex-particle-stabilized emulsions of anti-Bancroft type

Here, we investigate water-in-oil (W/O) emulsions that are stabilized by polystyrene latex particles with sulfate surface groups. The particles, which play the role of emulsifier, are initially contained in the disperse (water) phase. The existence of such emulsions formally contradicts the empirical Bancroft rule. Theoretical considerations predict that the drop diameter has to be inversely proportional to the particle concentration, but should be independent of the volume fraction of water. In addition, there should be a second emulsification regime, in which the drop diameter is determined by the input mechanical energy during the homogenization. The existence of these two regimes has been experimentally confirmed, and the obtained data agree well with the theoretical model. Stable W/O emulsions have been produced with hexadecane and tetradecane, while, in the case of more viscous and polar oils (soybean and silicone oil), the particles enter into the oily phase, and Pickering emulsions cannot be obtained. The formation of stable emulsions demands the presence of a relatively high concentration of electrolyte that lowers the electrostatic barrier to particle adsorption at the oil-water interface. Because the attachment of particles at the drop surfaces represents a kind of coagulation, it turns out that the Schulze-Hardy rule for the critical concentration of coagulation is applicable also to emulsification, which has been confirmed with suspensions containing Na(+), Mg(2+), and Al(3+) counterions. The increase of the particle and electrolyte concentrations and the decrease of the volume fraction of water are other factors that facilitate emulsification in the investigated system. To quantify the combined action of these factors, an experimental stability-instability diagram has been obtained.

Mass transport in micellar surfactant solutions: 1. Relaxation of micelle concentration, aggregation number and polydispersity

The surfactant transfer in micellar solutions includes transport of all types of aggregates and the exchange of monomers between them. Such processes are theoretically described by a system containing tens of kinetic equations, which is practically inapplicable. For this reason, one of the basic problems of micellar kinetics is to simplify the general set of equations without loosing the adequacy and correctness of the theoretical description. Here, we propose a model, which generalizes previous models in the following aspects. First, we do not use the simplifying assumption that the width of the micellar peak is constant under dynamic conditions. Second, we avoid the use of the quasi-equilibrium approximation (local chemical equilibrium between micelles and monomers). Third, we reduce the problem to a self-consistent system of four nonlinear differential equations. Its solution gives the concentration of surfactant monomers, total micelle concentration, mean aggregation number, and halfwidth of the micellar peak as functions of the spatial coordinates and time. Further, we check the predictions of the model for the case of spatially uniform bulk perturbations (such as jumps in temperature, pressure or concentration). The theoretical analysis implies that the relaxations of the three basic parameters (micelle concentration, mean aggregation number, and polydispersity) are characterized by three different characteristic relaxation times. Two of them coincide with the slow and fast micellar relaxation times, which are known in the literature. The third time characterizes the relaxation of the width of the micellar peak (i.e. of the micelle polydispersity). It is intermediate between the slow and fast relaxation times, in the case of not-too-low micellar concentrations. For low micelle concentrations, the third characteristic time is close to the fast relaxation time. Procedure for obtaining the exact numerical solution of the problem is formulated. In addition, asymptotic analytical expressions are derived, which compare very well with the exact numerical solution. In the second part of this study, the obtained set of equations is applied for theoretical modeling of surfactant adsorption from micellar solutions under various dynamic conditions, corresponding to specific experimental methods.

Mass transport in micellar surfactant solutions: 2. Theoretical modeling of adsorption at a quiescent interface

Here, we apply the detailed theoretical model of micellar kinetics from part 1 of this study to the case of surfactant adsorption at a quiescent interface, i.e., to the relaxation of surface tension and adsorption after a small initial perturbation. Our goal is to understand why for some surfactant solutions the surface tension relaxes as inverse-square-root of time, 1/t(1/2), but two different expressions for the characteristic relaxation time are applicable to different cases. In addition, our aim is to clarify why for other surfactant solutions the surface tension relaxes exponentially. For this goal, we carried out a computer modeling of the adsorption process, based on the general system of equations derived in part 1. This analysis reveals the existence of four different consecutive relaxation regimes (stages) for a given micellar solution: two exponential regimes and two inverse-square-root regimes, following one after another in alternating order. Experimentally, depending on the specific surfactant and method, one usually registers only one of these regimes. Therefore, to interpret properly the data, one has to identify which of these four kinetic regimes is observed in the given experiment. Our numerical results for the relaxation of the surface tension, micelle concentration and aggregation number are presented in the form of kinetic diagrams, which reveal the stages of the relaxation process. At low micelle concentrations, "rudimentary" kinetic diagrams could be observed, which are characterized by merging of some stages. Thus, the theoretical modeling reveals a general and physically rich picture of the adsorption process. To facilitate the interpretation of experimental data, we have derived convenient theoretical expressions for the time dependence of surface tension and adsorption in each of the four regimes.

Particle-interface interaction across a nonpolar medium in relation to the production of particle-stabilized emulsions

Quantitative theory of the particle-interface interaction across a nonpolar medium is developed. We consider a spherical dielectric particle (phase 1), which is immersed in a nonpolar medium (phase 2), near its boundary with a third dielectric medium (phase 3). The interaction originates from electric charges at the particle surface (e.g., the surface of a silica particle immersed in oil). The theoretical problem is solved exactly, in terms of Legendre polynomials, for arbitrary values of the dielectric constants of the three phases. As a result, expressions for calculating the interaction force and energy are derived. These expressions generalize the known theory of the electrostatic image force (acing on point charges) to the case of particles that have finite size and uniform surface charge density. For typical parameter values (silica or glass particles immersed in tetradecane), the image-force interaction becomes significant for particles of radius R > 30 nm. At fixed relative particle-to-interface distance, the force increases with the cube of the particle radius. In general, this is a strong and long-range interaction. For micrometer-sized particles, the interaction energy could be on the order of 10(5) k(B)T at close contact, and, in addition, the interaction range could be about 10(5) particle radii. The sign of the interaction depends on the difference between the dielectric constants of phases 2 and 3. When phase 3 has a smaller dielectric constant (e.g., air), the interface repels the particle. In contrast, when phase 3 has a greater dielectric constant (e.g., water), the interaction is attractive. Especially, water drops attract charged hydrophobic particles dispersed in the oily phase, and thus favor the formation of reverse particle-stabilized (Pickering) emulsions. The particle-interface interaction across the oily phase is insensitive to the concentration of electrolyte in the third, aqueous phase.

On the thermodynamics of particle-stabilized emulsions: curvature effects and catastrophic phase inversion

The flexural properties of a particle adsorption monolayer are investigated theoretically. If the particles are not densely packed, the interfacial bending moment and the spontaneous curvature (due to the particles) are equal to zero. The situation changes if the particles are closely packed. Then the particle adsorption monolayer possesses a significant bending moment, and the interfacial energies of bending and dilatation become comparable. In this case, the bending energy can either stabilize or destabilize the Pickering emulsion, depending on whether the particle contact angle is smaller or greater than 90 degrees . Theoretical expressions are derived for the bending moment, for the curvature elastic modulus, and for the work of interfacial deformation and emulsification. The latter is dominated by the work for creation of a new oil-water interface and by the work for particle adsorption. The curvature effects give a contribution of second order, which is significant only for emulsification at 50:50 water/oil volume fractions. A thermodynamic criterion for the type of the formed emulsion is proposed. It predicts the existence of a catastrophic phase inversion in particle-stabilized emulsions, in agreement with the experimental observations. The derived theoretical expressions could find application for interpretation of experimental data on production and stability of Pickering emulsions.

Mixed solutions of anionic and zwitterionic surfactant (Betaine): surface-tension isotherms, adsorption, and relaxation kinetics

Here, we present experimental surface-tension isotherms of mixed solutions of two surfactants, sodium dodecyl sulfate (SDS) and cocoamidopropyl betaine (Betaine), measured by means of the Wilhelmy plate method. The kinetics of surface-tension relaxation exhibits two characteristic time scales, which have been distinguished to determine correctly the equilibrium surface tension. The transition from the zwitterionic to the cationic form of Betaine is detected by surface-tension measurements. Synergistic dependence of the critical micellization concentration on the composition of the surfactant blend is established. The experimental surface-tension isotherms are fitted by means of the two-component van der Waals model, and an excellent agreement between theory and experiment was achieved. Having determined the parameters of the model, we calculated different properties of the mixed surfactant adsorption layer at various concentrations ofSDS, Betaine, and salt. Such properties are the adsorptions ofthe two surfactants, the surface dilatational elasticity, the occupancy of the Stern layer by bound counterions, the surface electric potential, and so forth. In particular, the addition of a small amount of Betaine to SDS significantly increases the surface elasticity. The results could be further applied to predict the thickness and stability of foam films or the size of the rodlike micelles in the mixed solutions of SDS and Betaine.

On the mechanism of stomatocyte–echinocyte transformations of red blood cells: experiment and theoretical model

This study represents an attempt to achieve a better understanding of the stomatocyte-echinocyte transition in the shape of red blood cells. We determined experimentally the index of cell shape at various ionic strengths and osmolarities for native and trypsin-treated human erythrocytes. For every given composition of the outer phase, we calculated the ionic strength in the cells and the transmembrane electric potential using a known theoretical model. Next, we described theoretically the electric double layers formed on both sides of the cell membrane, and derived expressions for the tensions of the two membrane leaflets. Taking into account that the cell-shape index depends on the tension difference between the two leaflets, we fitted the experimental data with the constructed physicochemical model. The model, which agrees well with the experiment, indicates that the tension difference between the two leaflets is governed by the different adsorptions of counterions at the two membrane surfaces, rather than by the direct contribution of the electric double layers to the membrane tension. Thus, with the rise of the ionic strength, the counterion adsorption increases stronger at the outer leaflet, whose stretching surface pressure becomes greater, and whose area expands relative to that of the inner leaflet. Hence, there is no contradiction between the bilayer-couple hypothesis and the electric double layer theory, if the latter is upgraded to account for the effect of counterion-adsorption on the membrane tension. The developed quantitative model can be applied to predict the shape index of cells upon a stomatocyte-discocyte-echinocyte transformation at varying composition of the outer medium.

Synergistic sphere-to-rod micelle transition in mixed solutions of sodium dodecyl sulfate and cocoamidopropyl betaine

Static and dynamic light scattering experiments show that the mixed micelles of sodium dodecyl sulfate (SDS) and cocoamidopropyl betaine (CAPB) undergo a sphere-to-rod transition at unexpectedly low total surfactant concentrations, about 10 mM. The lowest transition concentration is observed at molar fraction 0.8 of CAPB in the surfactant mixture. The transition brings about a sharp increase in the viscosity of the respective surfactant solutions due to the growth of rodlike micelles. Parallel experiments with mixed solutions of CAPB and sodium laureth sulfate (sodium dodecyl-trioxyethylene sulfate, SDP3S) showed that the sphere-to-rod transition in SDP3S/CAPB mixtures occurs at higher surfactant concentrations, above 40 mM. The observed difference in the transition concentrations for SDS and SDP3S can be explained by the bulkier SDP3S headgroup. The latter should lead to larger mean area per molecule in the micelles containing SDP3S and, hence, to smaller spontaneous radius of curvature of the micelles (i.e., less favored transition from spherical to rodlike micelles). The static light scattering data are used to determine the mean aggregation number and the effective size of the spherical mixed SDS/CAPB micelles. From the dependence of the aggregation number on the surfactant concentration, the mean energy for transfer of a surfactant molecule from a spherical into a rodlike micelle is estimated.

Spontaneous detachment of oil drops from solid substrates: governing factors

We carried out experiments on detachment of oil drops from glass substrates in solutions of an anionic surfactant. The three-phase contact line shrinks spontaneously, and eventually the oil drop detaches from the substrate. Consecutive video frames of such drops are digitized, and the time dependencies of the contact radius and angle are determined. Three stages of detachment of a drop, situated above a horizontal substrate, can be distinguished. They correspond to three different driving factors: (1) the interfacial tension decrease because of surfactant adsorption, (2) the aqueous meniscus spontaneously advances owing to the penetration of water between the oil and solid phases, and (3) at sufficiently small contact radius the shape of the oil-water interface becomes unstable and the drop detaches under the action of buoyancy. Analyzing the experimental data, we identified two important characteristics of the drop-detachment process: the velocity of spontaneous advance of the contact line and the line drag coefficient. In the case of moving contact line, a dynamic Young equation must be used, which takes into account the line drag force. The latter is proportional to the velocity of contact-line motion. The experimental data agree with the latter dependence, from whose slope the line drag coefficient is determined.

Kinetics of Solubilization of n-Decane and Benzene by Micellar Solutions of Sodium Dodecyl Sulfate

We observed the diminishing of single microscopic oil drops to study the kinetics of solubilization of n-decane and benzene by micellar solutions of sodium dodecyl sulfate (SDS). Each drop is located in a horizontal glass capillary of inner diameter 0.06 cm filled with a thermostated surfactant solution; the small vertical dimension of the cell prevents the appearance of uncontrollable thermal convections. The experiments show that the radius of an n-decane drop decreases linearly with time, whereas for benzene this dependence is nonlinear. To interpret the data, a kinetic model of solubilization is developed. It accounts for the diffusion and capturing of dissolved oil molecules by the surfactant micelles, as well as for the finite rate of oil dissolution at the oil-water interface. By processing the data, we determined the rate constant of solubilization for a given oil and surfactant. It turns out that the elementary act of catching a dissolved oil molecule by a surfactant micelle occurs under a barrier (rather than diffusion) control. The effective rate of solubilization is greater for the oil, which exhibits a higher equilibrium solubility in pure water (benzene), despite the lower value of the solubilization rate constant for this oil.

Deposition of Oil Drops on a Glass Substrate in Relation to the Process of Washing

The attachment of emulsion drops to glass substrates is investigated in relation to the redeposition of oil drops in the process of washing. It turns out that the drops of a surfactant-stabilized oil-in-water emulsion cannot be attached to an immersed glass plate simply by the buoyancy force. However, the same drops can be deposited on the plate when the latter is pulled out of the emulsion, i.e., when the drops are pressed against the substrate by a receding meniscus. We measured the amount of the oily deposit as a function of the pH, ionic strength, and composition of an amphoteric-anionic surfactant mixture. The enhanced oil deposition at low pH correlates with the domain in which the emulsion drops and the solid substrate bear opposite electric charges. This was established by zeta-potential measurements with oil drops and glass particles. The anionic surfactant brings negative surface charge to the oil droplets and suppresses the oil deposition on the negatively charged glass. With the increase of the fraction of the amphoteric surfactant in the mixture, the zeta-potential is converted from negative to positive, and the oil deposition grows almost linearly with the potential. In general, the deposition of oil drops by a receding meniscus is governed by an interplay of electrostatic and hydrodynamic factors. Copyright 2000 Academic Press.

Capillary interactions between particles bound to interfaces, liquid films and biomembranes

This article is devoted to an overview, comparison and discussion of recent results (both theoretical and experimental) about lateral capillary forces. They appear when the contact of particles or other bodies with a fluid phase boundary causes perturbations in the interfacial shape. The capillary interaction is due to the overlap of such perturbations which can appear around floating particles, vertical cylinders, particles confined in a liquid film, inclusions in the membranes of lipid vesicles or living cells, etc. In the case of floating particles the perturbations are due to the particle weight; in this case the force decreases with the sixth power of the particle size and becomes immaterial for particles smaller than approximately 10 microm. In all other cases the interfacial deformations are due to the particle wetting properties; the resulting 'immersion' capillary forces can be operative even between very small particles, like protein globules. In many cases such forces can be responsible for the experimentally observed two-dimensional particle aggregation and ordering. An analogy between capillary and electrostatic forces enables one to introduce 'capillary charges' of the attached particles, which characterize the magnitude of the interfacial deformation and could be both positive and negative. Moreover, the capillary interaction between particle and wall resembles the image force in electrostatics. When a particle is moving bound to an interface under the action of a capillary force, one can determine the surface drag coefficient and the surface viscosity supposedly the magnitude of the capillary force is known. Alternative (but equivalent) energy and force approaches can be used for the theoretical description of the lateral capillary interactions. Both approaches require the Laplace equation of capillarity to be solved and the meniscus profile around the particles to be determined. The energy approach accounts for contributions due to the increase of the meniscus area, gravitational energy and/or energy of wetting. The second approach is based on calculating the net force exerted on the particle, which can originate from the hydrostatic pressure, interfacial tension and bending moment. In the case of small perturbations, the superposition approximation can be used to derive an asymptotic formula for the capillary forces, which has been found to agree well with the experiment. Capillary interactions between particles bound to spherical interfaces are also considered taking into account the special geometry and restricted area of such phase boundaries. A similar approach can be applied to quantify the forces between inclusions (transmembrane proteins) in lipid membranes. The deformations in a lipid membrane, due to the inclusions, can be described theoretically in the framework of a mechanical model of the lipid bilayer, which accounts for its 'hybrid' rheology (neither elastic body nor fluid). In all considered cases the lateral capillary interaction originates from the overlap of interfacial deformations and is subject to a unified theoretical treatment, despite the fact that the characteristic particle size can vary from 1 cm down to 1 nm.

Energy of adhesion of human T cells to adsorption layers of monoclonal antibodies measured by a film trapping technique

A novel method for studying the interaction of biological cells with interfaces (e.g., adsorption monolayers of antibodies) is developed. The method is called the film trapping technique because the cell is trapped within an aqueous film of equilibrium thickness smaller than the cell diameter. A liquid film of uneven thickness is formed around the trapped cell. When observed in reflected monochromatic light, this film exhibits an interference pattern of concentric bright and dark fringes. From the radii of the fringes one can restore the shape of interfaces and the cell. Furthermore, one can calculate the adhesive energy between the cell membrane and the aqueous film surface (which is covered by a layer of adsorbed proteins and/or specific ligands), as well as the disjoining pressure, representing the force of interaction per unit area of the latter film. The method is applied to two human T cell lines: Jurkat and its T cell receptor negative (TCR-) derivative. The interaction of these cells with monolayers of three different monoclonal antibodies adsorbed at a water-air interface is studied. The results show that the adhesive energy is considerable (above 0.5 mJ/m2) when the adsorption monolayer contains antibodies acting as specific ligands for the receptors expressed on the cell surface. In contrast, the adhesive energy is close to zero in the absence of such a specific ligand-receptor interaction. In principle, the method can be applied to the study of the interaction of a variety of biological cells (B cells, natural killer cells, red blood cells, etc.) with adsorption monolayers of various biologically active molecules. In particular, film trapping provides a tool for the gentle micromanipulation of cells and for monitoring of processes (say the activation of a T lymphocyte) occurring at the single-cell level.

Minimization of the Free Energy of Arbitrarily Curved Interfaces

The minimization of the free energy of a two-phase system with an interface of arbitrary curvature leads to an extremum (Laplace) condition containing the pressure difference, DeltaP, between the two sides of the interface. The expression for DeltaP is a function of the normal curvatures and of the resulting bending moments which are themselves functions of the normal curvatures, the mathematical form of which depends on the particular model for the interfacial bending energy that has been employed. On this basis, conclusions can be drawn about the equilibrium shape and curvatures of an interface, e.g., for bicontinuous microemulsions and vesicles. In addition, the pressure difference between the inside and the outside of surfactant-laden interfaces can be calculated. This pressure difference influences the work of formation of microemulsion droplets. A section devoted to the boundary conditions has also been included where in particular the case of a liquid meniscus attached to a cylindrically shaped solid surface is treated.

Lateral forces acting between particles in liquid films or lipid membranes

To investigate the mechanism of formation of 2D arrays of protein macromolecules in liquid films we carried out model experiments with micron-sized latex particles. The direct observations revealed that the process of ordering is triggered by attractive lateral capillary forces due to the overlap of the menisci formed around the particles. Two types of lateral capillary forces, flotation and immersion, can be distinguished, and a theory of these interactions is developed. Similar forces are operative between inclusions (proteins) incorporated in lipid membranes. We develop an appropriate model of a lipid bilayer, which is described as an elastic layer (the hydrocarbon chain region) sandwiched between two Gibbs dividing surfaces (the two headgroup regions). The range of the interaction between two cylindrical inclusions turns out to be of the order of several inclusion radii. The results, which are in qualitative agreement with the experimental observations, can be applied to the interpretation of membrane processes and mechanisms such as protein aggregation in lipid membranes.