Surface tension curvature dependence

Just as all the common measurements of surface tension a depend on observing the effects of the pressure difference across an interface of known curvature, via Laplace s equation, (2.2), so also the measurement... 

The interfacial tension also depends on curvature (see Section III-1C) [25-27]. This alters Eq. IX-1 by adding a radius-dependent surface tension... 

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...

On the meniscus surface the deviation of vapor pressure from the saturation pressure Psat depends on the surface tension a, liquid density p( gas constant R, temperature T, and radii of curvature r. When p( > -Psat(T) < (2[Pg.354]

Accounting for the unusual r dependence of the surface tension c(r) oc modifies the standard result from Eq. (33) by a factor of. The reason is that the peculiar surface energy dependence F uTfiR) = 4nR a = AkgoR / o / calls for the following dependence of pressure on the curvature ... 

Because surface curvature depends on radius and different atoms have different sizes, and because the atomic surface tension depends on atomic number, the atomic surface tensions also include surface curvature effects, which has recently been studied as a separate effect.7 Local surface curvature may also correlate with nearest-neighbor proximity and thus may be implicitly included to some extent when semiempirical atomic surface tensions depend on interatomic distances in the solute. 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

The equilibrium curvature of a liquid surface or meniscus depends not just on its surface tension but also on its density and the effect of gravity. The variation in curvature of a meniscus surface must be due to hydrostatic pressure differences at different vertical points on the meniscus. If the curvature at a given starting point on a surface is known, the adjacent curvature can be obtained from the Laplace equation and its change in hydrostatic pressure Ahpg. In practice the liquid... 

In neutral gels, in contrast, neither a nor T0 depend on the shape of samples at all, as shown in Fig. 11. Thus, it is certain that the unusual shape-dependent properties shown in Figs. 9 and 10 are due to ions contained in gels. Although the mechanism underlying the shape dependence is not dear at present, one possible mechanism has been proposed [31] on the basis of the surface tension of ionized gels. Denoting the surface tension as y and the radius of curvature of the gel surface as p, an additional contribution to the osmotic pressure due to... 

The force only depends on the radius of the particles and the surface tension of the liquid. It does not depend on the actual radius of curvature of the liquid surface nor on the vapor pressure This is at first sight a surprising result, and is due to the fact that, with decreasing vapor pressure the radius of curvature, and therefore also x, decreases. At the same time the Laplace pressure increases by the same amount. 

The fluid phase that fills the voids between particles can be multiphase, such as oil-and-water or water-and-air. Molecules at the interface between the two fluids experience asymmetric time-average van der Waals forces. This results in a curved interface that tends to decrease in surface area of the interface. The pressure difference between the two fluids A/j = v, — 11,2 depends on the curvature of the interface characterized by radii r and r-2, and the surface tension, If (Table 2). In fluid-air interfaces, the vapor pressure is affected by the curvature of the air-water interface as expressed in Kelvin s equation. Curvature affects solubility in liquid-liquid interfaces. Unique force equilibrium conditions also develop near the tripartite point where the interface between the two fluids approaches the solid surface of a particle. The resulting contact angle 0 captures this interaction. 

Relation between surface tension and the pressure differences across a curved liquid surface. We must now return to a most important consequence of the existence of free surface energy, which was known to Young and Laplace, and is the foundation of the classical theory of Capillarity, and of most of the methods of measuring surface tension. If a liquid surface be curved the pressure is greater on the concave side than on the convex, by an amount which depends on the surface tension and on the curvature. This is because the displacement of a curved surface, parallel to itself, results in an increase in area as the surface moves towards the convex side, and work has to be done to increase the area. This work is supplied by the pressure difference moving the surface. 

There are static and dynamic methods. The static methods measure the tension of practically stationary surfaces which have been formed for an appreciable time, and depend on one of two principles. The most accurate depend on the pressure difference set up on the two sides of a curved surface possessing surface tension (Chap. I, 10), and are often only devices for the determination of hydrostatic pressure at a prescribed curvature of the liquid these include the capillary height method, with its numerous variants, the maximum bubble pressure method, the drop-weight method, and the method of sessile drops. The second principle, less accurate, but very often convenient because of its rapidity, is the formation of a film of the liquid and its extension by means of a support caused to adhere to the liquid temporarily methods in this class include the detachment of a ring or plate from the surface of any liquid, and the measurement of the tension of soap solutions by extending a film. 

Other methods depending directly on the fundamental equation. Direct measurement of the radius of curvature of a surface, by methods similar to those used in determining the radius of curvature of mirrors, has been applied by C. T. R. Wilson1 and C. V. Boys 2 simultaneous measurement of the pressure on both sides of the surface gives the surface tension at once by (2). No convenient instrument has been designed for rapid measurement of surface tensions, on this principle, however. 

Fig. 2.4 presents a measuring cell with a porous plate made of sintered glass (similar to variant C, Fig. 2.2). Porous plates of various pore radii can be used (usually the smallest radius is about 0.5 p.m) [23]. In this case the meniscus penetrates into the pores and their radius determines the radius of curvature, i.e. the small pore size allows to increase the capillary pressure until the gas phase can enter in them. The radius of the hole in which the film is formed is usually 0.025 - 0.2 cm. To provide a horizontal position of the film the whole plate is made very thin. In the porous plate measuring cell (Fig. 2.4) the capillary pressure can be varied to more than 10s Pa, depending on the pores size and the surface tension of the solution. When the maximum pore radius is 0.5 (tm, the capillary pressure is 3- 10s Pa at a - 70 mN/m.

When attempting to relate the adhesion force obtained with the SFA to surface energies measured by cleavage, several problems occur. First, in cleavage experiments the two split layers have a precisely defined orientation with respect to each other. In the SFA the orientation is arbitrary. Second, surface deformations become important. The reason is that the surfaces attract each other, deform, and adhere in order to reduce the total surface tension. This is opposed by the stiffness of the material. The net effect is always a finite contact area. Depending on the elasticity and geometry this effect can be described by the JKR 65 or the DMT 1661 model. Theoretically, the pull-off force F between two ideally elastic cylinders is related to the surface tension of the solid and the radius of curvature by... 

Tensions of non-relaxed interfaces are sometimes known by the adjective dynamic dynamic surface tension or dynamic inteifacial tension. The term dynamic is not absolute. It depends on De (i.e. on the time scale of the measurement as compared with that of the relaxation process). Some interfacial processes have a long relaxation time (polymer adsorption-desorption), so that for certain purposes (say the measurement of y] they may be considered as being in a state of frozen equilibrium. This last notion was introduced at the end of sec. 1.2.3. Unless otherwise stated, we shall consider static tensions and interfaces which are so weakly curved that curvature energies, bending moments, etc. may be neglected. 

The polar, ionic and even non-ionic head-group interactions of lipid membranes and other surfactants, (as indeed for many polymers and polyelectrolyte intra-molecular interactions) and the associated curvature at interfaces formed by such assemblies will be dependent on dissolved gas in quite complicated ways. Fluctuating nanometric sized cavities at such surfaces will be at extremely high pressure, (P = 2y/r, y= surface tension, and r the radius) resulting in formation of H and OH radicals. The immediate formation of Cl radicals and consequent damage to phospholipids offers em explanation of exercise-induced immunosuppression (through excess lactic acid CO2 production), perhaps a clue to the aging process. 

To summarize, three conclusions transpire from the nanoscale thermodynamics results (a) The interfacial tension between protein and water is patchy and the result of both nanoscale confinement of interfacial water and local redshifts in dielectric relaxation (b) the poor hydration of polar groups (a curvature-dependent phenomenon) generates interfacial tension, a property previously attributed only to hydrophobic patches and (c) because of its higher occurrence at protein-water interfaces, the poorly hydrated dehydrons become collectively bigger contributors to the interfacial tension than the rarer nonpolar patches on the protein surface. 

Another pore filling model based upon capillary equilibrium in cylindrical pores has recently been proposed in which the condition of thermodynamic equilibrium is modified to include the effects of surface layering and adsorbate-adsorbent interactions [135-137]. Assuming that the vapor-liquid interface is represented by a cylindrical meniscus during adsorption and by a hemispherical meniscus during desorption, and invoking the Defay-Prigogene expression for a curvature-dependent surface tension [21], the equilibrium condition for capillary coexistence in a cylindrical pore is obtained as... 

According to deduction the classical Kelvin equation [44], which describes the dependence of the saturated vapor pressure on the curvature of interface in two-phase system, is rigorous one, when it is employed for the free (not confined) liquid because this equation takes into consideration only the action of the surface tension forces at the fiquid-vapor interface. 

Historically, however, considerable attention has been given to corrections to the Kelvin equation arising from the thickness of adsorbed layer and the dependence of surface tension on curvature of interface. The first problem was initially considered as monolayen by Foster [48] and more recently as a function of equHibrium pressure of the system by Cohan [49], Degaguin [50], Foster [51], and Brockhoff and de Boer [52, 53]. The initial approaches of Foster and Cohan... 

The surface tension of the Hquid film is the same as that for the macroscopic liquid and does not depend on the film thickness and the interface curvature. 

In a recent study, a new model of fluids was described by using the generalized van der Waals theory. Actually, van der Waals over 100 years ago suggested that the structure and thermodynamic properties of simple fluids could be interpreted in terms of neatly separate contributions from intermolecular repulsions and attractions. A simple cubic equation of state was described for the estimation of the surface tension. The fluid was characterized by the Lennard-Jones (12-6) potential. In a recent study the dependence of surface tension of liquids on the curvature of the liquid-vapor interface has been described. ... 

A correlation between the solubility of a solute gas and the surface tension of the solvent liquid was described based on the curvature dependence of the surface tension for CgHg, QH,2, and CCI4. This was based on the model that a solute must be placed in a hole (or cavity) in the solvent. The change in the free energy of the system, AGso, transferring a molecule from the solvent phase to a gas phase is then... 

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