Interfaces curvature

The high pressures associated with high-curvature interfaces leads directly to the use of boiling chips to help nucleate bubbles with lower curvature using the porous, angular nature of the chips (Figure 2.4). 

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

For a conserved order parameter, the interface dynamics and late-stage domain growth involve the evapomtion-diffusion-condensation mechanism whereby large droplets (small curvature) grow at tlie expense of small droplets (large curvature). This is also the basis for the Lifshitz-Slyozov analysis which is discussed in section A3.3.4. 

For model A, the interfaces decouple from the bulk dynamics and their motion is driven entirely by the local curvature, and the surface tension plays only a background, but still an important, role. From this model A... 

Equation (A3.3.73) is referred to as the Gibbs-Thomson boundary condition, equation (A3.3.74) detemiines p on the interfaces in temis of the curvature, and between the interfaces p satisfies Laplace s equation, equation (A3.3.71). Now, since ] = -Vp, an mterface moves due to the imbalance between the current flowing into and out of it. The interface velocity is therefore given by... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

Picture a small element of a curved interface between a liquid a and a vapour p, having two radii of curvature r, and (Fig. 3.6). These radii are defined by taking two planes at right angles to one another, each of them... 

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...

The surface dividing the components of the mixture formed by a layer of surfactant characterizes the structure of the mixture on a mesoscopic length scale. This interface is described by its global properties such as the surface area, the Euler characteristic or genus, distribution of normal vectors, or in more detail by its local properties such as the mean and Gaussian curvatures. 

The line = 0 can be considered as a borderline for applicability of the basic model, in which the Gaussian curvature is always negative. Recall that in the basic model the oil-water interface is saturated by the surfactant molecules by construction of the model. Hence, for equal oil and water volume fractions the Gaussian curvature must be negative, by the definition of the model. 

FIG. 3 Schematic sketch of an interface between fluid and solid, or alternatively a surface step. The interface (or surface step) is characterized locally through the surface tension, the orientation ii, and the radius of curvature R. 

The physics underlying Eqs. (74-76) is quite simple. A solidifying front releases latent heat which diffuses away as expressed by Eq. (74) the need for heat conservation at the interface gives Eq. (75) Eq. (76) is the local equilibrium condition at the interface which takes into account the Gibbs-Thomson correction (see Eq. (54)) K is the two-dimensional curvature and d Q) is the so-called anisotropic capillary length with an assumed fourfold symmetry. 

J. Holuigue, O. Bertrand, E. Arquis. Solutal convection in crystal growth effect of interface curvature on flow structuration in a three-dimensional cylindrical configuration. J Cryst Growth 180 591, 1997. 

The usual way to visualize a junction is to draw an eneigy diagram that shows the bottom of the conduction band Er and the top of the valence band Ev as a function of distance. The so-called band curvature that appears at both sides of the junction interface reveals a variation in the potential with a distance in the direction perpendicular to the junction surface. The formation of an MS barrier is depicted in Figure 14-1. 

At the end of the 1960s, Subba Rao et al. examined the influence of the interface on the CMC values [56]. They found a decrease in the CMC at the oil-water interface compared with the air-water interface. The CMC decreased by about 10% in the presence of heptane and by about 30-40% in the presence of benzene. The solubilization of the hydrocarbon in the micelle interior results in an increase in the micelle size and a slight change in the curvature of the micelle surface. The electrical potential and hence the electrical work of... 

Figure 8 shows an example of the most common behavior of AEam/0 as a function of adsorbate coverage. Linear behavior, if ever observed, is seen at the air/solution interface.93 At metal/solution interfaces, if chemical interactions with the metal can be ruled out, electrostatic interactions cannot be avoided, and these are responsible for the downward curvature.91 Upward curvatures are often observed at air/solution interfaces as a consequence of lateral interactions.95... 

More recently, the curvature at air/solution interfaces has been accounted for by Nikitas and Pappa-Louisi98 in terms of a specific molecular model that predicts a linear dependence of (lM/ ) on (1/0). The same model also reproduces the behavior at metal/solution interfaces, specifically Hg electrodes, for which most of the experimental data exist. Nikitas treatment provides a method for an unambiguous extrapolation of the adsorption potential shift to 0= 1. However, the interpretation of the results is subject to the difficulties outlined above. Nikitas approach does provide... 

Thermodynamic and mechanical equilibrium on a curved vapor-liquid interface requires a certain degree of superheat in order to maintain a given curvature. Characteristics of homogeneous and heterogeneous nucleation can be estimated in the frame of classical theory of kinetics of nucleation (Volmer and Weber 1926 Earkas 1927 Becker and Doring 1935 Zel dovich 1943). The vapor temperature in the bubble Ts.b can be computed from equations (Bankoff and Flaute 1957 Cole 1974 Blander and Katz 1975 Li and Cheng 2004) for homogeneous nucleation in superheated liquids... 

When the interface surface is expressed by a function x=cp (y, z) the general radii of curvature are found from the equation (Smirnov 1964) ... 

The calculations show that the liquid pressure monotonically decreases along the heating region. Within the evaporation region a noticeable difference between the vapor and liquid pressures takes place. The latter is connected with the effect of the Laplace force due to the curvature of the interface surface. In the superheated region the vapor pressure decreases downstream. 

The 1 (0) correlation corresponding to various Weber numbers is shown in Fig. 8.5. The shape of the interface surface in a capillary flow with phase change is presented in Fig. 8.6. As the calculations show, the curvature of the meniscus is not constant and grows toward the periphery. 

The curvature of the meniscus in a heated capillary with evaporative interface... 

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