Surface region

Secondly, a short pulse duration is required in order to achieve a good axial resolution, i.e. two signals close together should be detected without interference. The task can be, for example, to detect a small reflector close to the surface or back wall of the test object, as the inspection has to cover the total volume as complete as possible, including the near-surface regions. 

One may consider a molecule in the surface region as being in a state intermediate between that in the vapor phase and that in the liquid. Skapski [11] has made the following simplified analysis. Considering only nearest-neighbor interactions, if n, and denote the number of nearest neighbors in the interior of the liquid and the surface region, respectively, then, per molecule... 

The next point of interest has to do with the question of how deep the surface region or region of appreciably unbalanced forces is. This depends primarily on the range of intermolecular forces and, except where ions are involved, the principal force between molecules is of the so-called van der Waals type (see Section VI-1). This type of force decreases with about the seventh power of the intermolecular distance and, consequently, it is only the first shell or two of nearest neighbors whose interaction with a given molecule is of importance. In other words, a molecule experiences essentially symmetrical forces once it is a few molecular diameters away from the surface, and the thickness of the surface region is of this order of magnitude (see Ref. 23, for example). (Certain aspects of this conclusion need modification and are discussed in Sections X-6C and XVII-5.)... 

It must also be realized that this thin surface region is in a very turbulent state. Since the liquid is in equilibrium with its vapor, then, clearly, there is a two-way and balanced traffic of molecules hitting and condensing on the surface from the vapor phase and of molecules evaporating from the surface into the vapor phase. From the gas kinetic theory, the number of moles striking 1 cm of surface per second is... 

There is also a traffic between the surface region and the adjacent layers of liquid. For most liquids, diffusion coefficients at room temperature are on the order of 10 cm /sec, and the diffusion coefficient is related to the time r for a net displacement jc by an equation due to Einstein ... 

In the case of solids, there is no doubt that a lateral tension (which may be anisotropic) can exist between molecules on the surface and can be related to actual stretching or compression of the surface region. This is possible because of the immobility of solid surfaces. Similarly, with thin soap films, whose thickness can be as little as 100 A, stretching or extension of the film may involve a corresponding variation in intermolecular distances and an actual tension between molecules. 

A. Further Development of the Thermodynamic Treatment of the Surface Region... 

For the case where the curvature is small compared to the thickness of the surface region, d(c - C2) = 0 (this will be exactly true for a plane or for a spherical surface), and Eq. III-28 reduces to... 

The foregoing serves as an introduction to the detailed thermodynamics of the surface region the method is essentially that of Gibbs [35], as reviewed by Tolman [36, 37]. An additional relationship is... 

As in Section III-2A, it is convenient to suppose the two bulk phases, a and /3, to be uniform up to an arbitrary dividing plane S, as illustrated in Fig. Ill-10. We restrict ourselves to plane surfaces so that C and C2 are zero, and the condition of equilibrium does not impose any particular location for S. As before, one computes the various extensive quantities on this basis and compares them with the values for the system as a whole. Any excess or deficiency is then attributed to the surface region. 

Here, the distances x and a are relative to planes A and B located far enough from the surface region so that bulk phase properties prevail. The actual amount of component i present in the region between A and B will be... 

An approach developed by Guggenheim [106] avoids the somewhat artificial concept of the Gibbs dividing surface by treating the surface region as a bulk phase whose upper and lower limits lie somewhere in the bulk phases not far from the interface. 

The most widely used experimental method for determining surface excess quantities at the liquid-vapor interface makes use of radioactive tracers. The solute to be studied is labeled with a radioisotope that emits weak beta radiation, such as H, C, or One places a detector close to the surface of the solution and measures the intensity of beta radiation. Since the penetration range of such beta emitters is small (a ut 30 mg/cm for C, with most of the adsorption occurring in the first two-tenths of the range), the measured radioactivity corresponds to the surface region plus only a thin layer of solution (about 0.06 mm for C and even less for H). 

It was pointed out at the beginning of this section that ir could be viewed as arising from an osmotic pressure difference between a surface region comprising an adsorbed film and that of the pure solvent. It is instructive to develop... 

Let us now suppose that the surface region can be regarded as having a depth T and an area and hence volume V. A volume V of surface region, if made up of pure solvent, will be... 

Now, nfV /T is just the surface area, and, moreover, V /t and Vi/t have the dimensions of molar area, (/"the surface region is considered to be just one molecule thick, V /t and V /t becomes A and A2, the actual molar areas, so that Eq. III-115 takes on the form... 

Molecular dynamics calculations have been made on atomic crystals using a Lennard-Jones potential. These have to be done near the melting point in order for the iterations not to be too lengthy and have yielded density functioi). as one passes through the solid-vapor interface (see Ref. 45). The calculations showed considerable mobility in the surface region, amounting to the presence of a... 

Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

An equation algebraically equivalent to Eq. XI-4 results if instead of site adsorption the surface region is regarded as an interfacial solution phase, much as in the treatment in Section III-7C. The condition is now that the (constant) volume of the interfacial solution is i = V + JV2V2, where V and Vi are the molar volumes of the solvent and solute, respectively. If the activities of the two components in the interfacial phase are replaced by the volume fractions, the result is... 

Many of the adsorbents used have rough surfaces they may consist of clusters of very small particles, for example. It appears that the concept of self-similarity or fractal geometry (see Section VII-4C) may be applicable [210,211]. In the case of quenching of emission by a coadsorbed species, Q, some fraction of Q may be hidden from the emitter if Q is a small molecule that can fit into surface regions not accessible to the emitter [211]. 

TWo limiting conditions exist where lubrication is used. In the first case, the oil film is thick enough so that the surface regions are essentially independent of each other, and the coefficient of friction depends on the hydrodynamic properties, especially the viscosity, of the oil. Amontons law is not involved in this situation, nor is the specific nature of the solid surfaces. 

As load is increased and relative speed is decreased, the film between the two surfaces becomes thinner, and increasing contact occurs between the surface regions. The coefficient of friction rises from the very low values possible for fluid friction to some value that usually is less than that for unlubricated surfaces. This type of lubrication, that is, where the nature of the surface region is... 

An important aspect of the stabilization of emulsions by adsorbed films is that of the role played by the film in resisting the coalescence of two droplets of inner phase. Such coalescence involves a local mechanical compression at the point of encounter that would be resisted (much as in the approach of two boundary lubricated surfaces discussed in Section XII-7B) and then, if coalescence is to occur, the discharge from the surface region of some of the surfactant material. 

The Donnan effect acts to exclude like-charged substrate ions from a charged surface region, and this exclusion, as well as the concentration of oppositely charged ions, can be expressed in terms of a Donnan potential pD. Thus for a film of positively charged surfactant ions S one can write... 

The total moles in the surface region is given by C t/1000, where C is the sum of the concentrations of the ionic species present ... 

A still different approach to multilayer adsorption considers that there is a potential field at the surface of a solid into which adsorbate molecules fall. The adsorbed layer thus resembles the atmosphere of a planet—it is most compressed at the surface of the solid and decreases in density outward. The general idea is quite old, but was first formalized by Polanyi in about 1914—see Brunauer [34]. As illustrated in Fig. XVII-12, one can draw surfaces of equipo-tential that appear as lines in a cross-sectional view of the surface region. The space between each set of equipotential surfaces corresponds to a definite volume, and there will thus be a relationship between potential U and volume 0. 

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