Surface curvature effect

The important difference between single-phase and two-phase flows is the interface latent heat effect involved with the latter. Also, in cases with strong curvature effects, surface tension needs to be taken into account. Because of latent heat, the heat transfer rates in two-phase problems are an order of magnitude larger than those in single phase. 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

Kofman R ef a/1989 Solid-liquid transition of metallic clusters occurrence of surface melting Physica A 157 631 Kofman R ef al 1994 Surface melting enhanced by curvature effects Surf. Sc/. 303 231... 

Fig. 3.21 The effect of meniscus curvature on surface tension. Plot of /) against r . y is the surface tension of the meniscus having the mean radius of curvature and y that of a plane surface of liquid, according to Melrose. The value of y/y was calculated by the equation V = /x(l - with a . = 3 a.

Stuart SJ, Berne BJ (1999) Surface Curvature Effects in the Aqueous Ionic Solvation of the Chloride Ion. J Phys Chem A 103(49) 10300-10307... 

SPT provides a conceptual basis relating the nonpolar free energy contribution to the solvent-exposed surface area. An attractive approximation is to ignore curvature effects and write... 

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. 

For small particles with large curvature the surface has, as previously stated, a significant effect on the thermodynamics, and the concepts developed apply to all types of interfaces between solid, liquid and gas. 

Given the thinness of a diffusion flamelet, it is possible to neglect as a first approximation curvature effects, and to establish a local coordinate system centered at the reaction interface. By definition, X is chosen to be normal to the reaction surface. Furthermore, because the reaction zone is thin compared with the Kolmogorov scale, gradients with respect to X2 and X3 will be much smaller than gradients in the x direction (i.e., the curvature is small).112 Thus, as shown in Fig. 5.18, the scalar fields will be locally onedimensional. 

For moderately doped substrates, when the surface is free of oxide the change of potential is mostly dropped in the space charge layer and in the Helmholtz double layer. The reactions are very sensitive to geometric factors. The reaction that is kinetically limited by the processes in the space charge layer is sensitive to radius of curvature, while that limited by the processes in the Helmholtz layer is sensitive to the orientation of the surface. Depending on the relative effect of each layer the curvature effect versus anisotropic effect can vary. 

Up to now we discussed planar grafting layers. It is of much interest to consider the case where polymers are grafted to curved surfaces. The first study taking into account curvature effects of stretched and tethered poly-... 

The effect of curvature or surface energy on the osmotic pressure and thus on the solubility is thus removed when... 

For large Peelet numbers and small Hamaker s constants, appreciable concentration variation occurs only very near to the collector. Then Stokes s expressions for the fluid velocity may be expanded in a Taylor series about the collector surface and higher order terms together with curvature effects may be neglected, yielding... 

The form eqn (3.1) is to hold strictly for planar surfaces. We have already alluded to curvature corrections to the effective surface tension contribution. If the micellar surface is spherical, the effective surface tension y for both a drop and a hole in the liquid is less than that for a planar interface by a factor (1 —d/R), where d is of the order of one or two molecular radii and R is the position of the Gibbs dividing surface. Corresponding corrections to the electrostatic contributions are expected to be of much more importance, and can be handled within the framework of a capacitance description. Thus for a spherical capacitance the energy per unit area is from electrostatics... 

An outer surface of a vertical cylinder can be treated as a vertical plate when the diameter of the cylinder is sufficiently large so that the curvature effects are negligible. This condition is satisfied if... 

Nusselt number relations for various surfaces are given in Table 9-1. All fluid properties are evaluated at the film temperature of Tf = 1(T, -T r ). The outer surface of a vertical cylinder can be treated as a vertical plate when the curvature effects are negligible. The characteristic length for a horizontal surface is Lg = AJp, where A, is the surface area and p is the perimeter. 

The Saam-Cole approach has several approximations, among which are the neglect of the sohd-adsorbate interaction and curvature effects on the adsorbate chemical potential, and curvature effects on surface tension in symmetrical and asymmetrical states, while modeling the multilayer region. Here, a more accurate version of the above approach has been introduced and tested for explaining the reversibihty of adsorption in MCM-41. For fluid molecules inside a cyhndrical pore of radius R, the incremental potential function has been expressed as [4,6,7]... 

As long as the range of the interaction and the separation h is much less than the radius of curvature of the system, it is a valid approximation for interaction between surfaces quadratically curved in the vicinity of the point of closest approach. The condition makes curvature effects, higher than second order (via truncation of a Taylor series expansion), on the approximated energy significantly small. However, the... 

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... 

Blade curvature (tubular surface reduced to a quarter or to a half) had no effect upon stirrer power, but the rotation direction (concave or convex) did [256]. In the turbulent flow range concave blades exhibited an Neo = 2.9 whereas convex blades exhibited an Neo = 3.3, which compared with flat perpendicular blades in non-gassed liquids only represented 52 to 60% of the power number (Neo = 5.5). 

Derouane, E.G., Andre, J.M., and Lucas, A.A. 1988. Surface curvature effects in physisoiption and catalysis by microporous solids and molecular sieves. Journal of Catalysis 110, 58-73. 

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