Particle asymmetry

The factoring of /// into two contributions in Equation (14) was introduced with this interpretation in mind. It turns out that the effect of solvation is easily handled by a physically reasonable model through/ // . Dealing with particle asymmetry through / //is more complicated, but this has also been analyzed according to a model that is plausible for many situations. Let us now consider each of these developments in turn. 

We are explicitly looking for the effect of solvation on the friction factor of a sphere particle asymmetry is handled by the second ratio ///. By Stokes s law, the ratio of the friction factors of solvated to unsolvated spheres/ //0 is equal to the ratio of their radii asolJ a0. This ratio equals the cube root of the volume ratio of the solvated to unsolvated spheres, or... 

Asymmetry as well as solvation can cause a friction factor to have a value other than /0. Next let us consider the ratio///, which, according to Equation (14), accounts for the effect of particle asymmetry on the friction factor. We saw in Section 1.5 that ellipsoids of revolution are reasonable models for many asymmetric particles. 

Figure 4.12a shows plots of the intrinsic viscosity —in volume fraction units —as a function of axial ratio according to the Simha equation. Figure 4.12b shows some experimental results obtained for tobacco mosaic virus particles. These particles —an electron micrograph of which is shown in Figure 1.12a—can be approximated as prolate ellipsoids. Intrinsic viscosities are given by the slopes of Figure 4.12b, and the parameters on the curves are axial ratios determined by the Simha equation. Thus we see that particle asymmetry can also be quantified from intrinsic viscosity measurements for unsolvated particles. 

Particle asymmetry is a factor of considerable importance in determining the overall properties (especially those of a mechanical nature) of colloidal systems. Roughly speaking, colloidal particles can be classified according to shape as corpuscular, laminar or linear (see, for example, the electron micrographs in Figure 3.2). The exact shape may be complex but, to a first approximation, the particles can often be treated theoretically in terms of models which have relatively simple shapes (Figure 1.1). 

Particle asymmetry has a marked effect on viscosity and a number of complex expressions relating intrinsic viscosity (usually extrapolated to zero velocity gradient to eliminate the effect of orientation) to axial ratio for rods, ellipsoids, flexible chains, etc., have been proposed. For randomly orientated, rigid, elongated particles, the intrinsic viscosity is approximately proportional to the square of the axial ratio. 

Departure of the limiting value of rj j4> from the theoretical value of 2.5 may result from either hydration of the particles, or from particle asymmetry, or from both as discussed in Box 8.1. 

A more general form of equation (8.6) allowing for particle asymmetry is... 

Eq. (8) has the interesting property that the shape factor / enters only as part of the product fc. It predicts, therefore, that particle asymmetry and concentration are equivalent in their effects on the modulus. This prediction turns out to be true, though very probably for other reasons. Fig. 11 shows quasi-equilibrium stress data for four carbon... 

The two main conditions of the soluble protein for which we should like to have measurements of molecular weight and particle asymmetry are a) the primary solution in urea which yields a tough clot on dialysis, and b) the materials which have become water soluble on longer extraction with urea. 

Furthermore, all of these techniques assume the particles to be spherical. This assumption is not too well satisfied for this system, as the particles are rather irregularly shaped, and often plate-like (i.e. clay minerals). Each technique thus measures a radius of an equivalent sphere, but as Figure 18.5 illustrates, the data obtained from different techniques do not differ that much and all of these methods can be used to obtain a reasonable estimate of the size distribution. More detailed measurements would require direct image analysis and consideration of particle asymmetry. While such image analysis techniques can be used in a rather straightforward fashion for monodisperse samples, they tend to become extremely difficult to use for polydis-perse ones. 

The only remaining possible cause for particle asymmetry lies in the definition of r as ri — F2, but from the expression above it is clear that this is of no consequence. In addition to the Coulomb term, the expression for the interaction between two charged particles contains two relativistic terms. These will be discussed in greater detail when we later introduce similar expressions in the relativistic Hamiltonian. 

Elution volume, exclusion chromatography Flow rate, column Gas/liquid volume ratio Inner column volume Interstitial (outer) volume Kovats retention indices Matrix volume Net retention volume Obstruction factor Packing uniformity factor Particle diameter Partition coefficient Partition ratio Peak asymmetry factor Peak resolution Plate height Plate number Porosity, column Pressure, column inlet Presure, column outlet Pressure drop... 

The most complicated case is of no asymmetry, i.e., e = 0, and it is specially this problem that we shall investigate. At e = 0 the system, described by Hq, has two energy levels E = + 2 do. If the particle is initially put into the left well, the amplitudes of the particle being in the left and right wells oscillate, respectively, as... 

One area where the concept of atomic charges is deeply rooted is force field methods (Chapter 2). A significant part of the non-bonded interaction between polar molecules is described in terms of electrostatic interactions between fragments having an internal asymmetry in the electron distribution. The fundamental interaction is between the Electrostatic Potential (ESP) generated by one molecule (or fraction of) and the charged particles of another. The electrostatic potential at position r is given as a sum of contributions from the nuclei and the electronic wave function. 

The appearance of one or the other type of characteristics was shown to be connected to the asymmetry of the system, which is controlled by the value of the locked tunneling current, exactly as in the case of CdS particles. 

Fig. 6.14 Mossbauer spectra of a-Fe nanoparticles on a carbon support. The spectra were obtained at 80 and 300 K with the indicated magnetic fields applied perpendicular to the y-ray direction. The asymmetry in the spectra is due to the presence of a small amount of iron carbide particles. (Reprinted with permission from [58] copyright 1985 by the American Chemical Society)...

The frequency of fluid oscillation at which levitation takes place is plotted in Fig. 40 against the corresponding amplitude A of oscillation, the asymmetry factor ka or ratio of the duration of the downstroke to that of the upstroke, and the resin particle diameter d. From these experimental data, the three parameters of the equation were correlated to the particle diameter ... 

Abstract The equation of state (EOS) of nuclear matter at finite temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range T < 30 MeV and the baryon density range ps < 1014 2 g/cm3. In this region, in addition to the mean-field effects the formation of few-body correlations, in particular light bound clusters up to the alpha-particle (1 < A < 4) has been taken into account. The calculation is based on the relativistic mean field theory with the parameter set TM1. We show results for different values for the asymmetry parameter, and (3 equilibrium is considered as a special case. 

We conclude that not only the a-particle but also the other fight clusters contribute significantly to the composition. Furthermore they also contribute to the baryon chemical potential and this way the modification of the phase instability region with respect to the temperature, baryon density and asymmetry can be obtained. As an example, for symmetric matter the baryon chemical potential as a function of density for T = 10 MeV is shown in Fig.3... 

It is well known that at lower densities the properties of the EoS are primarily determined by the SE [2], The latter is defined in terms of a Taylor series expansion of the energy per particle for nuclear matter in terms of the asymmetry parameter a = (N — Z)/A (or equivalently the proton fraction x = Z/A),... 

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