Contact model examples

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FIGURE 21.20 Crystal structures of the metallic elements at 2S C and 1 atm pressure. Atomic radii (A) are calculated as one half the closest atom-atom distance in each structure in most cases this is the same radius as calculated using the hard sphere contact model of Example 21.4. There are no known crystal structures for those elements for which atomic radii are not listed. 

FIGURE 3.2 Example of analyses with (a) the Hertzian contact model and (b) the JKR contact model. A conical probe with a tip radius of less than 20 nm was used for (a), while a spherical probe with a tip radius of 150nm was used for (b). The specimens were NR for (a) and PDMS for (b), respectively. Both specimens were moderately cross-hnked. 

A manifestation of the action of surface forces is the disjoining pressure. To explain the nature of the disjoining pressure, let us consider the interaction of two thick, plain, and parallel surfaces divided by a thin liquid layer of thickness h (aqueous electrolyte solution, for example). The surfaces are not necessarily of the same nature as two important examples show (1) one is air, one is a liquid film, and one is solid support, and (2) both surfaces are air, and one is a liquid film. Example 1 is referred to as a liquid film on a solid support and models the liquid layer in the vicinity of the three-phase contact line. Example 2 is referred to as a free liquid film. There is a range of experimental methods to measure the interaction forces between these two surfaces as a function of the thickness, h (gravity action is already taken into account) (Figure 1.7) [1,3,4]. 

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

The simple model given above does not take account of the facts that industrial refractories are poly crystalline, usually non-uniform in composition, and operate in temperature gradients, both horizontal and vertical. Changes in the coiTosion of multicomponent refractories will also occur when there is a change in the nature of tire phase in contact with the conoding liquid for example in Ca0-Mg0-Al203-Cf203 refractories which contain several phases. 

Langmuir-Blodgett films (LB) and self assembled monolayers (SAM) deposited on metal surfaces have been studied by SERS spectroscopy in several investigations. For example, mono- and bilayers of phospholipids and cholesterol deposited on a rutile prism with a silver coating have been analyzed in contact with water. The study showed that in these models of biological membranes the second layer modified the fluidity of the first monolayer, and revealed the conformation of the polar head close to the silver [4.300]. 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

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