Media continuous

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... 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

Nomenclature. Colloidal systems necessarily consist of at least two phases, the coUoid and the continuous medium or environment in which it resides, and their properties gready depend on the composition and stmcture of each phase. Therefore, it is useful to classify coUoids according to their states of subdivision and agglomeration, and with respect to the dispersing medium. The possible classifications of colloidal systems are given in Table 2. The variety of systems represented in this table underscores the idea that the problems associated with coUoids are usuaUy interdisciplinary in nature and that a broad scientific base is required to understand them completely. 

To prepare stable emulsions ia this way gelation of the continuous medium is necessary. The appearance of a Hquid emulsion may be retained by choosing a polymer for the continuous phase, giving a thixotropic solution with short breakdown and buildup times. The polymers used for this purpose are natural gums (qv) or synthetic polymers. Clay particles also act as viscosity enhancers. The members of the bentonite family derived from... 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

Bilayers have received even more attention. In the early studies, water has been replaced by a continuous medium as in the monolayer simulations [64-67]. Today s bilayers are usually fully hydrated , i.e., water is included exphcitly. Simulations have been done at constant volume [68-73] and at constant pressure or fixed surface tension [74-79]. In the latter case, the size of the simulation box automatically adjusts itself so as to optimize the area per molecule of the amphiphiles in the bilayer [33]. If the pressure tensor is chosen isotropic, bilayers with zero surface tension are obtained. Constant... 

Methods for evaluating the effect of a solvent may broadly be divided into two types those describing the individual solvent molecules, as discussed in Section 16.1, and those which treat the solvent as a continuous medium. Combinations are also possible, for example by explicitly considering the first solvation sphere and treating the rest by a continuum model. Each of these may be subdivided according to whether they use a classical or quantum mechanical description. 

Various novel applications in biotechnology, biomedical engineering, information industry, and microelectronics involve the use of polymeric microspheres with controlled size and surface properties [1-31. Traditionally, the polymer microspheres larger than 100 /urn with a certain size distribution have been produced by the suspension polymerization process, where the monomer droplets are broken into micron-size in the existence of a stabilizer and are subsequently polymerized within a continuous medium by using an oil-soluble initiator. Suspension polymerization is usually preferred for the production of polymeric particles in the size range of 50-1000 /Ltm. But, there is a wide size distribution in the product due to the inherent size distribution of the mechanical homogenization and due to the coalescence problem. The size distribution is measured with the standard deviation or the coefficient of variation (CV) and the suspension polymerization provides polymeric microspheres with CVs varying from 15-30%. 

The free radical initiators are more suitable for the monomers having electron-withdrawing substituents directed to the ethylene nucleus. The monomers having electron-supplying groups can be polymerized better with the ionic initiators. The water solubility of the monomer is another important consideration. Highly water-soluble (relatively polar) monomers are not suitable for the emulsion polymerization process since most of the monomer polymerizes within the continuous medium, The detailed emulsion polymerization procedures for various monomers, including styrene [59-64], butadiene [61,63,64], vinyl acetate [62,64], vinyl chloride [62,64,65], alkyl acrylates [61-63,65], alkyl methacrylates [62,64], chloroprene [63], and isoprene [61,63] are available in the literature. 

Figure 9 The schematical representation of dispersion polymerization process, (a) initially homogeneous dispersion medium (b) particle formation and stabilizer adsorption onto the nucleated macroradicals (c) capturing of radicals generated in the continuous medium by the forming particles and monomer diffusion to the forming particles (d) polymerization within the monomer swollen latex particles, (e) latex particle stabilized by steric stabilizer and graft copolymer molecules (f) list of symbols.

Where, x,- is the volume fraction of component /, S, and S/ are the solubility parameters of the homogenous solvent mixture and the component i, respectively. The solubility parameters of some solvents that are widely used as the continuous medium in the dispersion polymerization are given in Table 6. 

Almog et al. [80] studied the dispersion polymerization of styrene in different alcohols as the continuous medium by using AIBN and vinyl alcohol-vinyl acetate copolymer as the initiator and the stabilizer, respectively. Their results showed that the final particle size decreased with the alcohol type according to the following order ... 

To integrate equation (10.149) we must know how dn is related to v. Debye assumed that a crystal is a continuous medium that supports standing (stationary) waves with frequencies varying continuously from v = 0 to v = t/m. The situation is similar to that for a black-body radiator, for which it can be shown that... 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

So far in this chapter, consideration has been given to transfer taking place in a single direction of a rectangular coordinate system. In many applications of mass transfer, one of the fluids is injected as approximately spherical droplets into a second immiscible fluid, and transfer of the solute occurs as the droplet passes through the continuous medium. 

Let us imagine that the liquid cage is a spherical cavity in a continuous medium. When the molecule is in its centre, the orienting field is equal to zero. At this point the anisotropic part of the rotator-neighbourhood interaction appears only in the case of asymmetrical breathing of the... 

The Knudsen number is small enough so that the fluid is a continuous medium. 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

Since the phenomena studied in fluid dynamics are macroscopic, the fluid is considered to be a continuous medium, and the theory is not based on the behavior of individual molecules in the fluid but, rather, on their averages. Thus, fluid dynamics studies the motion of fluid volume elements which contain a large number of molecules. Such a volume element defines in the continuous medium a point which is small compared to the total system volume, but large when compared to typical intermolecular distances. 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

Much attention has been directed since olden times towards ion solvation, which is a key concept for understanding various chemical processes with electrolyte solutions. In 1920, a theoretical equation of ion solvation energy (AG ) was first proposed by Born [1], who considered the ion as a hard sphere of a given radius (r) immersed in a continuous medium of constant permittivity (e), and then defined AG as the electrostatic energy for charging the ion up to ze (z, the charge number of the ion e, the elementary charge) ... 

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