Non-interactive sphere

Collision Cross-Section The model of gaseous molecules as hard, non-interacting spheres of diameter o can satisfactorily account for various gaseous properties such as the transport properties (viscosity, diffusion and thermal conductivity), the mean free path and the number of collisions the molecules undergo. It can be easily visualised that when two molecules collide, the effective area of the target is no1. The quantity no1 is called the collision cross-section of the molecule because it is the cross-sectional area of an imaginary sphere surrounding the molecule into which the centre of another molecule cannot penetrate. 

The correlation length in semidilute solution can be experimentally determined by measuring the diffusion coefficient of very dilute colloidal spheres of various sizes, provided that the spheres do not interact with the polymers. Consider diffusion of a non-interacting sphere in a semidilute unentangled solution. 

It is frequently desirable to be able to describe and/or predict dispersion viscosity in terms of the viscosity of the continuous phase (i/q) and the amount of dispersed material. A very large number of equations have been advanced for estimating emulsion, foam, suspension or aerosol viscosities. Most ofthese are empirical extensions of Einsteins equation for a dilute suspension of non-interacting spheres ... 

The SEC partition coefficient ( sec) was measured on a Superose 6 column for three sets of well-characterized symmetrical solutes the compact, densely branched nonionic polysaccharide, Ficoll the flexible chain non-ionic polysaccharide, pullulan and compact, anionic synthetic polymers, carboxylated starburst dendrimers. All three solutes display a congruent dependence of Ksbc on solute radius, R. In accord with a simple geometric model for SEC, all of these data conform to the same linear plot of Ksec vs. / . This plot reveals the behavior of non-interacting spheres on this column. The mobile phase for the first two solutes was 0.2 M NaH2P04-Na2HP04, pH 7.0. In order to ensure the suppression of electrostatic repulsive interactions between the dendrimer and the packing, the ionic strength was increased to 0.30 M for that solute. 

Major results In non-interacting spheres, the effective permittivity, eff where e is polymer permittivity and ( ) is filler firaction. Figure... 

This energy-based method of calculation of viscosity rj is due to Einstein [87], who considered hydrod)mamic dissipation in a very dilute suspension of non-interacting spheres. Tanaka and White [86] base their calculations on the Frankel and Acrivos [88] cell model of a concentrated suspension, but use a non-Newtonian (power law) matrix. The interaction energy is considered to consist of both van der Waals-London attractive forces and Coulombic interaction, i.e. 

For non-interacting spheres D pp = D independent of q. The product of the two experimentally accessible quantities I(q) Dapp(q) exhibits the q-dependence of the particle form factor... 

The viscosities of suspensions of large non-interactive spheres have been made by various investigators [56 to 60]. The agreement of experimental data with the hydrodynamic theory seems not clear in either dilute or concentrated systems. The best correlation of data with a model over wide range concentrations is with Mooney s [36] Eq. 2.13. 

Results from fits to a Gaussian distribution of non-interacting spheres... 

The viscosity of a suspension of rigid non-interacting spheres is given by the Einstein equation... 

In 1873 van der Waais pointed out that real gases do not obey the ideal gas equation PV = RT and suggested that two correction terms should be included to give a more accurate representation, of the form (P + ali/) V - b) = RT. The term a/v corrects for the fact that there will be an attractive force between all gas molecules (both polar and nonpolar) and hence the observed pressure must be increased to that of an ideal, non-interacting gas. The second term (b) corrects for the fact that the molecules are finite in size and act like hard spheres on collision the actual free volume must then be less than the total measured volume of the gas. These correction terms are clearly to do with the interaction energy between molecules in the gas phase. 

Generally, these behave as Newtonian fluids and, for the case of an extremely dilute suspension of spherical non-interacting particles having a density equal to that of the continuous medium, we can apply the Einstein formula for a suspension of spheres ... 

At high enough electrolyte concentrations the electric potentials are quickly dissipated and this effect vanishes. Since droplets and bubbles are not rigid spheres, they may deform in shear flow. Also, with the presence of emulsifying agents at the interface, the drops will not be non-interacting, as is assumed in the theory. 

It follows from the preceding section that the limiting case xa 1 (double layer thin as compared with radius of curvature) is simple then we can simply apply the flat layer theory, discussed extensively in secs 3.5a-d. Beyond this limit, the appropriate Poisson-Boltzmann equation (with p in (3.5.631 depending on the geometry) has to be solved with the appropriate boundary condition, l.e. dy/dr for r = 0, so in the centre of a sphere or infinitely long cylinder, the field strength is zero because of symmetry. However, at that location y is not necessarily zero, because double layers from the opposite sides may overlap. This Is a new feature as compared with convex double layers around non-interacting particles. 

Wertheim [19] showed that c(r) within a non-interacting hard sphere on the basis of the MSA is given by... 

Two types of reference fluids have been studied extensively by computer experiments. One system is composed of the non-interacting hard spheres discussed... 

The above analysis demonstrates the importance of the pair correlation function in estimation of the thermodynamic properties of simple liquids. In the following section, the properties of the simplest fluid, namely, one based on non-interacting hard spheres, are developed on the basis of the relationships presented in this section. 

It is useful to examine the properties of a fluid made up of non-interacting hard spheres. Such a system may be regarded as an important reference liquid, albeit fictitious, with respect to which the properties of real systems can be compared. Its properties are most easily obtained on the basis of the Percus-Yevick (PY) approximation. Since the spheres do not interact, the interaction energy u(r) is zero outside any sphere ... 

The discussion in this chapter has largely concerned very simple liquids such as a hypothetical fluid composed of non-interacting hard spheres, or spheres interacting via the Lennard-Jones potential function. The most common liquid, namely, water, is much more complex. First, it is a molecule with three atoms, and has a... 

The techniques used to describe the properties of pure liquids can be extended in a fairly straightforward fashion to liquid solutions [26, 27]. This treatment is normally restricted to liquids in which the molecules behave as non-interacting hard spheres or as dipolar species interacting via a Lennard-Jones potential. The discussion here is limited to two-component mixtures but it is easily extended to more complex systems. 

In order to complete the MSA estimate of Iny,- one must add the hard-sphere contribution, which accounts for the fact that work must be done to introduce the ions as hard spheres into the solution. It is obtained from the Percus-Yevick model for non-interacting hard spheres. For the case that all ions (spheres) have the same radius, the result is (see equation (3.9.22))... 

L. G. Leal, The slow motion of slender rod-like particles in a second-order fluid, J. Fluid Mech. 69, 305-37 (1975) B. P. Ho and L. G. Leal, Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid, J. Fluid Mech. 76, 783-99 (1976) P. C. H. Chan and L. G. Leal, The motion of a deformable drop in a second-order fluid, J. Fluid Mech. 92, 131-70 (1979) L. G. Leal, The motion of small particles in non-Newtonian fluids, J. Non-Newtonian Fluid Mech. 5, 33-78 (1979) R. J. Phillips, Dynamic simulation of hydro-dynamically interacting spheres in a quiescent second-order fluid, J. Fluid Mech. 315, 345-65 (1996). 

Many theoretical and empirical models have been proposed to predict the effective thermal conductivity of two phase mixtures. Comprehensive review articles have discussed the applicability of many of these models that appear to be more promising [34-36]. First, using potential theory. Maxwell [20] obtained a simple relationship for the conductivity of randomly distributed and non-interacting homogeneous spheres in a homogeneous medium. Maxwell model is good for low solid concentrations. Relative thermal conductivity enhancement (ratio of the effective thermal conductivity keffO nanofluid to base fluid kj) is. 

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