Interfacial tension curvature effects

The effect of increasing only the radius of curvature of the oil drop on the displacement of the contact fine while keeping the interfacial tension constant at 20 dyn/cm, is illustrated in Figs. 9 and 11. Figure 11 shows that for a radius of a curvature of 100 xm, there is virtually no movement of the contact fine from the base case due to the presence of nanoparticles/micelles even at volume fraction 0.25. However, when the radius of curvature is increased to 500 xm (recall Fig. 9), thereby decreasing the capillary pressure, the presence of nanoparticles at the same concentration moves the contact fine by 1 xm. 

Surfactants form semiflexible elastic films at interfaces. In general, the Gibbs free energy of a surfactant film depends on its curvature. Here we are not talking about the indirect effect of the Laplace pressure but a real mechanical effect. In fact, the interfacial tension of most microemulsions is very small so that the Laplace pressure is low. Since the curvature plays such an important role, it is useful to introduce two parameters, the principal curvatures... 

The effect of the curvation of the micelle on solubilization capacity has been pointed out by Mukerjee (1979, 1980). The convex surface produces a considerable Laplace pressure (equation 7.1) inside the micelle. This may explain the lower solubilizing power of aqueous micellar solutions of hydrocarbon-chain surfactants for hydrocarbons, compared to that of bulk phase hydrocarbons, and the decrease in solubilization capacity with increase in molar volume of the solubilizate. On the other hand, reduction of the tension or the curvature at the micellar-aqueous solution interface should increase solubilization capacity through reduction in Laplace pressure. This may in part account for the increased solubilization of hydrocarbons by aqueous solutions of ionic surfactants upon the addition of polar solubilizates or upon the addition of electrolyte. The increase in the solubilization of hydrocarbons with decrease in interfacial tension has been pointed out by Bourrel (1983). 

Mixtures of two homopolymers (A and B) and their corresponding diblock copolymer (A-B) are polymeric counterparts of mixtures of water, oil and surfactant. The immiscible nature between water and oil is also observed in polymer blends due to the fact that most polymers are immiscible in each other. The addition of diblock copolymers into blends of homopolymers has effects similar to adding surfactants into water-oil mixtures. The resulting reduction in interfacial tension and formation of the preferred interfacial curvature yield a variety of self-assembled structures. 

This equation is valid in the entire flow field even if the material properties vary discontinuously across phase boundaries. In Eq. (1), p and p are density and viscosity, v is the velocity field, p is pressure, and /is the body force. The effects of the interfacial tension are accounted for by the last term in Eq. (1). In this term, 5 is two or three dimensional delta function, cr is surface tension coefficient, k is the curvature of two-... 

The interfacial free energy associated with the creation of the micellar core-water interface, as well as with the shielding part of that interface. This contribution is obtained from available hydrocarbon-water interfacial tension data, and the interfacial area per monomer. The effect of interfacial curvature on interfacial tension was obtained from the Tolman equation [23]. 

Qualitatively the thermodynamics of microemulsions is well understood as the interplay between a small interfacial free energy and a small entropy of mixing. However, because of these contributions being small, other small effects, such as the influence of curvature on the interfacial tension and the influence of fluctuations, become important, and quantitative understanding still leaves a lot to be desired. 

In Sec. II we discuss the mechanism by which the interfacial tension may become ultralow. After that, in Sec. Ill we mention curvature effects of the oil/water interface. Subsequently, a number of models for thermodynamic calculations are described (Sec. IV). In Secs. V-VII we discuss droplet-type microemulsions in some detail. Section Vdescribes a thermodynamic formalism that incorporates the interfacial free energy (as influenced by the curvature) and the free energy of mixing of droplets and continuous medium and ultimately leads to equations for the size distribution of microemulsion droplets. This size distribution is important because measurable properties can be calculated... 

Not only do surfactants and cosurfactants lower the interfacial tension, but also their molecular structures affect the curvature of the interface as shown schematically in Fig. 3. The hydrocarbon chains are rather closely packed (about 0.25 nm per chain) they repel one another sideways and as a result have a tendency to bend the interface around the water side. The counterions of the ionic headgroups also repel one another sideways and thus tend to curve the interface around the oil side. The bullQ polar groups of nonionic surfactants have a similar effect. So we understand qualitatively that more cosurfactant promotes W/O rather than O/W microemulsions. More electrolyte compresses the double layer, diminishes the sideways pressure of the double layer, and also promotes W/O microemulsions. The polar groups of PEO nonionics become more compact (less soluble) at higher temperatures, and so with this type of surfactants high temperature leads to W/O microemulsions. 

In Sec. Ill it was shown that if it is assumed that the interfacial tension of the droplets depends on curvature as prescribed by Eq. (4), then some essential features of microemulsion systems can already be qualitatively understood. In Ref 27 the opposite route was followed. It was assumed that the interfacial tension does not depend on curvature, and in that case severe inconsistencies between theory and experiments were observed. As also mentioned in Sec. Ill, Eq. (4) does not account for finite size effects. We take into account both curvature dependence and finite size effects by using the Ansatz for the interfacial... 

The model based on the lattice fluid SCF theory offers a means to calculate fundamental interfacial properties of microemulsions from pure component properties [25]. Because all of the relevant interfacial thermodynamic properties are calculated explicitly and the surfactant and oil molecular architectures are considered, the model is applicable to a wide range of microemulsion systems. The interfacial tension, bending moment, and interaction strength between the droplets can be calculated in a consistent manner and analyzed in terms of the detailed interfacial composition. The mechanism of the density effect on the natural curvature includes both an enthalpic and an entropic component. As density is decreased, the solvation of the surfactant tails is less favorable enthalpically, and the solvent is expelled from the interfacial region. Entropy also contributes to this oil expulsion due to the density difference between the interfacial region and the bulk. The oil expulsion and increased tail-tail interactions decrease the natural curvature. 

Nielsen AE, Bindra PS (1973) Effect of curvature on interfacial tension in liquid systems measured by homogeneous nucleation. Croat Chem Acta 45 31-52... 

Capillary phenomena arise as a result of differences in pressure across a system containing at least one liquid phase and another hquid, vapor, and/or solid phase. As illustrated below, such pressure differences may result from differences in curvature in different regions of liquid-fluid phases in a system and/or due to the presence of an effective mechanical tension in the interface, the interfacial tension. The differences in curvature giving rise to the pressure differentials may result from various sources including the application of external forces, the contacting and coalescence of two masses of the liquid phase, or from the contact of the liquid phase with an second fluid phase and a solid surface. 

In all the preceding discussion of terms having the gAy form, yhas been interpreted as a surface tension, the factor g serving to correct for the molecular-scale curvature effect. But a stuface tension is measured at the macroscopic air-liquid interface, and in the solution case we are actually interested in the tension at a molecular scale solute-solvent interface. This may be more closely related to an interfacial tension than to a surface tension. As a consequence, if we attempt to find (say) g2A2 by dividing by we may be dividing by the wrong munber. 

The last term in parentheses in Equation 6.66 represents the effect of interfacial tension and curvature on the equilibrium freezing temperature. The basic idea is that curvature produces a difference in pressure between solid and melt. Since the chemical potentials in the two phases are fimctions of pressure, and since they must be equal at equilibrium, the equilibrium freezing temperature depends on curvature (see Problem 6.7). 

The emulsifying effects of a small quantity of a block copolymer, A-B, added to immiscible blend of homopolymers A and B, were examined by Leibler (1988). The theory predicted the reduction of the interfacial tension coefficient, Vi2, caused by equilibrium adsorption of a copolymer at the interface. For well-chosen compositions and molecular weights of the copolymer, low values of Vj2 are to be expected. This suggests a possible existence of thermodynamically controlled stable droplet phase, in which the minor phase homopolymer drops are protected by an interfacial film of the copolymer, interfacing the matrix polymer. The size distribution of the droplets is expected to depend on the rigidity and spontaneous radius of curvature of the interfacial film that can be controlled by molecular structure of the copolymer. 

Figure 4.12 shows the phases, and changes occurring in interfacial tensions whilst some parameter affecting chemical composition is modified. One such parameter whose effect may be quite significant is the salinity of the aqueous phase. We have already observed that it affects the spontaneous curvature of the system by altering the molecular area uq. 

Equation (2.53), known as the Kelvin equation, reveals that the vapor pressure, P , decreases with increasing interface curvature. So far, we have assumed that the substrate is liquid-wet or that the new phase forms as a bubble. For a droplet or when gas is the wetting phase, the effect of curvature on saturation pressure is formulated shortly. When the radius of a bubble or droplet becomes very small (say r < 10 cm, for a pure substance), the interfacial tension may become a function of the radius (Defay and Prigogine, 1966). However, the derivation of Eq. (2.53) was not based on the assumption of the interfacial tension being independent of r. 

Note that the vapor pressure of a bubble is less than the vapor pressure for a flat interface. On the other hand, the vapor pressure of a droplet is more than the vapor pressure for a flat interface. In the above solution, we have assumed that the interfacial tension is independent of the curvature, even at r < 10 ° cm. For a bubble, the interfacial tension increases as the curvature increases, whereas it decreases for a droplet as the curvature increases (Defay and Prigo-gine, Chapter XV, 1966). However, this trend is only true for pure components for mixtures, the effect of curvature change on the interfacial tension may be different (see Example 3.6, Chapter 3)... 

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