Planar Interfaces

In this brief review of dynamics in condensed phases, we have considered dense systems in various situations. First, we considered systems in equilibrium and gave an overview of how the space-time correlations, arising from the themial fluctuations of slowly varying physical variables like density, can be computed and experimentally probed. We also considered capillary waves in an inliomogeneous system with a planar interface for two cases an equilibrium system and a NESS system under a small temperature gradient. 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

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 interface free energy per unit area fi,u is taken to be that of a planar interface between coexisting phases. Considering a solution v /(z) that minimizes Eq. (5) subject to the boundary conditions vj/(z - oo) = - v /coex, v /(z + oo) = + vj/ oex one finds the excess free energy of a planar interface ... 

Mullins and Sekerka (17) were the first to construct continuum descriptions like the Solutal Model introduced above and to analyze the stability of a planar interface to small amplitude perturbations of the form... 

Figure 5. Schematic of nonorthogonal transformations used in finite-element/Newton algorithms for calculating cellular interfaces, (a) Monge cartesian representation for almost planar interfaces, (b) Mixed cylindrical/cartesian mapping for representing deep cells.

Figure 9. Arc length as a function of time for three time transient calculations, (a) Evolution of interface from an unstable planar interface to a shape in the (lA -family. (b) Evolution of perturbation to a shape in the (up-family for P < Continued on next page.

In recent years, high-resolution x-ray diffraction has become a powerful method for studying layered strnctnres, films, interfaces, and surfaces. X-ray reflectivity involves the measurement of the angnlar dependence of the intensity of the x-ray beam reflected by planar interfaces. If there are multiple interfaces, interference between the reflected x-rays at the interfaces prodnces a series of minima and maxima, which allow determination of the thickness of the film. More detailed information about the film can be obtained by fitting the reflectivity curve to a model of the electron density profile. Usually, x-ray reflectivity scans are performed with a synchrotron light source. As with ellipsometry, x-ray reflectivity provides good vertical resolution [14,20] but poor lateral resolution, which is limited by the size of the probing beam, usually several tens of micrometers. 

Note that the external field vanishes in the normal surface tension calculation. In the fully local approximation there is no surface tension. Thus we can obtain the surface tension y associated with a planar interface of area A by the expression... 

In the sections above, only infinite planar interfaces between air and an aqueous phase or two immiscible liquids like water and DCE were considered. Reducing the question to this class of surfaces only would be a severe limitation in the scope of the review as more reports appear in the literature debating on the SH response from small centro-symmetrical particles [107-110]. It is the purpose of this section to discuss the SHG response from interfaces having a radius of curvature of the order of the wavelength of light. 

In the theoretical section above, the nonlinear polarization induced by the fundamental wave incident on a planar interface for a system made of two centrosymmetrical materials in contact was described. However, if one considers small spheres of a centrosymmetrical material embedded in another centrosymmetrical material, like bubbles of a liquid in another liquid, the nonlinear polarization at the interface of a single sphere is a spherical sheet instead of the planar one obtained at planar surfaces. When the radius of curvature is much smaller than the wavelength of light, the electric field amplitude of the fundamental electromagnetic wave can be taken as constant over the whole sphere (see Fig. 7). Hence, one can always find for any infinitely small surface element of the surface... 

Liquid interfaces are widely found in nature as a substrate for chemical reactions. This is rather obvious in biology, but even in the diluted stratospheric conditions, many reactions occur at interfaces like the surface of ice crystallites. The number of techniques available to carry out these studies is, however, limited and this is particularly true in optics, since linear optical methods do not possess the ultimate molecular resolution. This resolution is inherent to nonlinear optical processes of even order. For liquid-liquid systems, optics turns out to be rather powerful owing to the possibility of nondestructive y investigating buried interfaces. Furthermore, it appears that planar interfaces are not the only config-... 

For the remainder of this chapter, we discuss results for various studies of interfacial solvation dynamics. We first discuss studies at liquid liquid interfaces at planar interfaces and in microheterogeneous media in Section II. In Section III, we discuss solvation dynamics at liquid solid interfaces. In Section IV, we review theoretical models and simulations of solvation dynamics at liquid interfaces. Finally, we conclude with a discussion of future studies. 

For an interface between two phases with no common charged or polarizable components, contributions of the two phases to certain properties are easily distinguished theoretically. The charge density and electrical polarization at any point in the interface are each sums of two contributions which can be assigned to the two phases. The overall electroneutrality of a planar interface may be written... 

Thus, for a planar interface one determines the one-electron wave functions according to... 

This result is to be contrasted with the standard model [53] of Fig. 13, where the fold surfaces are simply treated as planar interfaces with fold surface free energy oy per unit area. In the latter case, the free energy of the nucleus is given by... 

Consider a planar interface between two immiscible solvents with dielectric constants ei and 2- Calculate (a) the Coulomb interaction of two ions situated on different sides of the interface (b) the image energy of a single ion near the interface. 

Fig. 15.1 Schematic of the field structure when TIR occurs at a smooth planar interface...

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 ... 

One other caveat concerning the approach used here must be made. This discussion, and the studies to which it relates, are based on some version of the Stern model for the oxide-electrolyte interface. Oxide surfaces are rough and heterogeneous. Even for the mercury-electrolyte interface, or single crystal metal-electrolyte interfaces, the success of some form of the Stern model has been less than satisfactory. It is important to bear in mind the operational nature of these models and not to attach too much significance to the physical picture of the planar interface. 

We shall use the familiar Gouy-Chapman model (3 ) to describe the behaviour of the diffuse double lpyer. According to this model the application of a potential iji at a planar solid/electrolyte interface will cause an accumulation of counter-ions and a depletion of co-ions in the electrolyte near the interface. The disposition of diffuse double layer implies that if the surface potential of the planar interface at a 1 1 electrolyte is t ) then its surface charge density will be given by ( 3)... 

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