Curved interface

For a sphere, any small change in the surface area of the sphere dS, due to a differential change dr in the radius will lead to the following change in volume dPof the sphere  

If the volume change is due to the addition of dm,- moles of species i for aU z, then 

fy is the fugacity in a system with a spherical surhice, with the interface convex going from the droplet phase to the surroundings, and /. p corresponds to that with a planar surface. For ideal gas behavior, replace fy byPv- 

We discuss first the distribution of a solute i between a gas and a liquid at pressure P. At equilibrium, we obtain from the criterion (3.3.11) 

Further, the smaller the drop diameter, the higher the observed vapor pressure of the liquid in the vapor space, leading to a higher rate of evaporation from the drop. Equation (3.3.54) is identified cls the Kelvin equation and the phenomenon is called the Kelvin effect. This increased evaporation tendency appears because a molecule near the droplet surface is attracted to a lesser extent toward the interior by the surrounding molecules (since there are fewer of these surrounding molecules). However, this analysis breaks down as the drop size becomes too small and the number of molecules become much smaller. If the drop consists of a solution of a nonvolatile solute, the Kelvin equation applies to the solvent. Note that if the liquid surface is concave toward the vapor, the fugacity fy at the surface is less than that on a planar surface. See Section 3.3.7.5 as well as Problem 3.3.2 for the application to a solid-liquid system. 

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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... 

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

The equilibrium conditions for systems with curved interfaces [3] are in part identical to those defined earlier for heterogeneous phase equilibria where surface effects where negligible ... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Equation (6.42) introduces a new independent variable of the system the mean curvature c = (c1 +C2). This variable must be taken into account in the Gibbs phase rule, which now reads F + Ph = C + 2 + 1. The number of degrees of freedom (F) of a two-phase system (Ph = 2) with a curved interface is given by... 

Helfrich has shown [18] that the surface tension of a curved interface can be expressed as a Taylor series up to second order in the radius of curvature ... 

Equation 9.6 determines the conditions of a mechanical equilibrium of the curved interface. This can be illustrated with an example of a spherical bubble of radius r. To compete the surface tension the pressure inside the bubble should exceed the external pressure with AP, which is determined from the work, W, for virtual change of r dll APdl adl. Under equilibrium, dll = 0 and APdU=erd4, thus... 

If surface tension, analogous to that in liquids, really exists in solids, then also capillary pressure Pc must exist (Laplace 1805). The pressure at any point on the concave (convex) side of a curved interface would be by... 

It will be shown here that, due to the presence of surface tension in liquids, a pressure difference exists across the curved interfaces of liquids (such as drops or bubbles). This capillary force will be analyzed later. 

The profiles of pendant and sessile bubbles and drops are commonly used in determinations of surface and interfacial tensions and of contact angles. Such methods are possible because the interfaces of static fluid particles must be at equilibrium with respect to hydrostatic pressure gradients and increments in normal stress due to surface tension at a curved interface (see Chapter 1). It is simple to show that at any point on the surface... 

Since it is relatively easy to transfer molecules from bulk liquid to the surface (e.g. shake or break up a droplet of water), the work done in this process can be measured and hence we can obtain the value of the surface energy of the liquid. This is, however, obviously not the case for solids (see later section). The diverse methods for measuring surface and interfacial energies of liquids generally depend on measuring either the pressure difference across a curved interface or the equilibrium (reversible) force required to extend the area of a surface, as above. The former method uses a fundamental equation for the pressure generated across any curved interface, namely the Laplace equation, which is derived in the following section. 

This result is the Laplace equation for a single, spherical interface. In general, that is for any curved interface, this relationship expands to include the two principal radii of curvature, Pi and Rr-... 

Figure 2.9 The Laplace pressure generated across a curved interface as a function of contract angle.

Let us assume that a small bulge appears on a rough, curved interface, and that for some reason the interface morphology is altered. Intervals between the lines of equal temperature or concentration become narrower at the bulge hence, the... 

The presence of surface tension has an important implication for the pressures across a curved interface and, as a consequence, for phase equilibria involving curved interphase boundaries. The equation that relates the pressure difference across an interface to the radii of curvature, known as the Laplace equation, is derived in Section 6.4, and the implications for phase equilibria are considered for some specific cases in Section 6.5. 

SURFACE TENSION IMPLICATIONS FOR CURVED INTERFACES AND CAPILLARITY... 

EFFECTS OF CURVED INTERFACES ON PHASE EQUILIBRIA AND NUCLEATION THE KELVIN EQUATION... 

Adsorption hysteresis is often associated with porous solids, so we must examine porosity for an understanding of the origin of this effect. As a first approximation, we may imagine a pore to be a cylindrical capillary of radius r. As just noted, r will be very small. The surface of any liquid condensed in this capillary will be described by a radius of curvature related to r. According to the Laplace equation (Equation (6.29)), the pressure difference across a curved interface increases as the radius of curvature decreases. This means that vapor will condense... 

Up to this point we have dealt with the thermodynamics of planar boundaries. Let us add several relations for curved interfaces. First, we have to establish an equivalent to the Gibbs-Thomson equation which holds for curved external surfaces in a multi-component system. For incoherent (fluid-like) interfaces, this can be done by considering Figure 10-5. From the equilibrium condition at constant P and T, one has... 

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