Mean distance: between particles

In its turn Fig. 6.2 illustrates the effect of the initial concentration on the static tunnelling recombination kinetics. The latter is defined by a competition of three distinctive scales - the tunnelling recombination radius ro, mean distance between particles Iq = n(0) 1/d and lastly, the time-dependent correlation radius . At long time curves corresponding to different initial concentrations could be coincided by their displacements along ordinate axis, which confirms existence of the universal asymptotic decay law. 

The value of R = 1 [by the definition of R and the dimensionless concentration n(R), such that n(0) = 1] corresponds to the case when the reaction has destroyed pairs AB separated by the relative distances r less or equal to the mean distance between particles. In other words, according to equation (6.1.73), a new asymptotic law with a = 1/2 occurs already at very small reaction depths, r 0.5 ... 

Here r is a mean distance between particles in the cluster (an AB pair in-staneously recombines if the distance r between A and B is r < r0). The values were obtained in the model being discussed of r0/f = 3.43, N se 143. We can record the results of this study by using the parameter a0. Thus, we easily note that... 

There is currently considerable interest in processing polymeric composite materials filled with nanosized rigid particles. This class of material called "nanocomposites" describes two-phase materials where one of the phases has at least one dimension lower than 100 nm [13]. Because the building blocks of nanocomposites are of nanoscale, they have an enormous interface area. Due to this there are a lot of interfaces between two intermixed phases compared to usual microcomposites. In addition to this, the mean distance between the particles is also smaller due to their small size which favors filler-filler interactions [14]. Nanomaterials not only include metallic, bimetallic and metal oxide but also polymeric nanoparticles as well as advanced materials like carbon nanotubes and dendrimers. However considering environmetal hazards, research has been focused on various means which form the basis of green nanotechnology. 

As measured, the Hs term means that a smaller particle size reduces the distance between particles, and thus the spreading due to diffusion of sample molecules is minimized. From Equations 5.18 and 5.19, a decrease in H value is achieved by increasing the diffusion speed (elevating the column temperature,... 

The relaxation rates calculated from Eq. (15) are smaller than the measured ones at low field, while they are larger at high field. OST is thus obviously unable to match the experimental results. However, water protons actually diffuse around ferrihydrite and akaganeite particles and there is no reason to believe that the contribution to the rate from this diffusion would not be quadratic with the external field. This contribution is not observed, probably because the coefficient of the quadratic dependence with the field is smaller than predicted. This could be explained by an erroneous definition of the correlation length in OST, this length is the particle radius, whilst the right definition should be the mean distance between random defects of the crystal. This correlation time would then be significantly reduced, hence the contribution to the relaxation rate. 

To explain this behaviour, physicists appeal to the very foundations of quantum theory. Because of their much reduced freedom to move in space, the particles can be considered to be more and more localised. Then, by Heisenberg s uncertainty principle, the spread in their velocities has to grow. In other words, some particles may have much higher velocities than those allowed by the temperature. A quantum pressure arises at high densities, when the mean distance between electrons becomes comparable with their associated wavelength... 

When two similarly charged colloid particles, under the influence of the EDL, come close to each other, they will begin to interact. The potentials will detect one another, and this will lead to various consequences. The charged molecules or particles will be under both van der Waals and electrostatic interaction forces. The van der Waals forces, which operate at a short distance between particles, will give rise to strong attraction forces. The potential of the mean force between colloid particle in an electrolyte solution plays a central role in the phase behavior and the kinetics of agglomeration in colloidal dispersions. This kind of investigation is important in these various industries ... 

We recall here that estimates on the basis of a simple model of the mean number of defects in a cluster [25, 26, 28] gave Uq = ro/r = 3.43, where r is a mean distance between defects. The mean number of particles in a cluster is N = 120. These values correlate with the values of Uq from the computer experiment, which obtained Uq 5 and a mean number of defects in a cluster, respectively, of about 100. (As follows from the pattern of accumulation for L = 2000 and l — 5 with a total number of creation events of 5 x 105 [107].)... 

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

The Fundamentals of Acoustic Agglomeration of Small Particulates. Let us consider a polydisperse aerosol consisting of submicrometer and micron sized particles. The mean separation distance between particles would typically be about 100 micrometers. Brownian movement of the particles is caused by the collision of the thermally agitated air molecules with the particles. Also any convection currents or turbulence in the carrier gas will of course cause the particles to be partially entrained and moved in the air. If we next impose an acoustic field of acoustic pressure p, the acoustic velocity u will be given by... 

Collision frequencies, mean free paths, and cross sections. Collision processes are central to the description of chemical reactions in the plasma. The mean free path /Ia/b of particle A is the mean distance this particle travels before encountering particle B. AA/b is related to the mean velocity of particle A and to the mean collision frequency between A and B. 

Volumetric coefficient of expansion in general. With subscripts C, M. PB, PS, P composite, matrix, polybutadiene, polystyrene, particle, respectively Mean distance between crazes in the volume With subscripts. A, B solubility parameter With subscripts. A, B, solubihty parameter... 

The first term (kinetic energy) is a summation over all the particles in the molecule. The second term (potential enctgy) uses Coulomb s law to calculate the interaction between every pair of particles in the molecule, where e, and cj ate the charges on particles i and j. For electrons, the charge -e, while the charge for a nucleus is Ze, where Z Is the atomic number. The summation nutation ipairwise interaction terms in the summation (e.g., eg / = e e, and should only appear in the potential energy term once). The denominator r in the. second term is the distance between particles i anil j. J. i is understood to be the electronic wave function for a many-atom system. 

In the case of carbon black, the aggregates are distributed in the matrix rather than individual particles, it is therefore important in some applications (e.g., conductive plastics) to evaluate the distance between these aggregates. It is now possible to measure these distances by atomic force microscopy coupled with straining device. There is a linear relationship between the parallel distance between aggregates dispersed in SBR and strain value. For 10 phr of N 234, the mean distance between aggregates varied in a range from 1.85 to 3.42 jm. For practical purposes, a modified equation [5.4] is used to determine the interaggregate distance ... 

DLVO theory explained major principles of coagulation of hydrosols by electrolytes and brought to common grounds all previous observations (primarily of qualitative nature) that related to individual cases and often seemed to be contradictory. In years that followed further extensions of DLVO theory that took into account the possibility of reversible particle aggregation were developed. At very small distances between particles in addition to the usual long-range interaction, molecular attraction and electrostatic repulsion, one must account for other factors that play role at a direct particle contact. The formation of peculiarly structured hydration layers in the vicinity of solid surface, the appearance of elastic forces that are responsible for the Born repulsion between surface atoms at the point of contact, the repulsion between the adsorbed surfactant molecules in contact zone between two particles, all represent the so-called non-DLVO stability factors . This means that more or less deep primary minimum remains finite. 

Fig. 10.6 Model sketches depicting the influence of surface roughness on the approach of particles to each other (left) and wetting (liquid bridge formation, right). The cross-hatched area is the actual bridge, a represents the mean distance between the particles, the outlines of the ideal particles (averaging out roughness), and the theoretical bridge contours assuming perfect wetting...

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