Particle radius distribution function

Site occupation probability in percolation theory (dimensionless) Percolation threshold of site occupation probability (dimensionless) Pdclet number. Equation 1.30 Particle radius distribution function Capillary pressure (Pa)... 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

Figure 14. Change in the theoretical radius distribution function of the growing particles L(S,t) with time t the numbers in this figure Indicate t in seconds.

One can distinguish between the differential and integral (or cumulative) particle size distribution functions. These two types of functions are related to each other by the differentiation and integration operations, respectively. The adequate description of distribution function must include two parameters the object of the distribution (i.e. what is distributed), and the parameter with respect to which the distribution is done. The first parameter may be represented by the number of particles, their net weight or volume10, their net surface area or contour lengthen some rear cases). The second parameter typically characterizes particle size. It can be represented as a particle radius, volume, weight, or, rarely, surface area. Consequently, the differential function of the particle number distribution with respect to their... 

With respect to the fact that coc s co/jo, ( >s(R) = oor (see Eq. 3.95) and substituting (3.97) into (3.96), one can find the integral intensity dependence on particle radius. The observed lineshape can be obtained by averaging of this intensity with particles size distribution function determined by (3.83), i.e. 

Twenty years ago, Bogolubov3 developed a method of generalizing the Boltzmann equation for moderately dense gases. His idea was that if one starts with a gas in a given initial state, its evolution is at first determined by the initial conditions. After a lapse of time—of the order of several collision times—the system reaches a state of quasi-equilibrium which does not depend on the initial conditions and in which the w-particle distribution functions (n > 2) depend on the time only through the one-particle distribution function. With these simple statements Bogolubov derived a Boltzmann equation taking into account delocalization effects due to the finite radius of the particles, and he also established the formal relations that the n-particle distribution function has to obey. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

Criticize or defend the following proposition The data give the time required for particles to fall 20 cm, making it easy to convert time to sedimentation velocity for each point. Equation (11) may then be used to convert the velocity into the radius of an equivalent sphere. The resulting graph of W versus radius is a cumulative distribution function similar to that shown in Figure 1.18b. 

Although this form accounts for the distribution of particles of arbitrary shape, the theory is well developed for spheres. In this case, one can also define the distribution function in terms of the particle radius (or diameter),... 

The shape of the size distribution function for aerosol particles is often broad enough that distinct parts of the function make dominant contributions to various moments. This concept is useful for certain kinds of practical approximations. In the case of atomospheric aerosols the number distribution is heavily influenced by the radius range of 0.005-0.1 /xm, but the surface area and volume fraction, respectively, are dominated by the range 0.1-1.0 fxm and larger. The shape of the size distribution is often fit to a logarithmic-normal form. Other common forms are exponential or power law decrease with increasing size. 

It corresponds to the number of particles less than or equal to the radius R. Since n = dN(R)/dR the distribution function can be calculated in principle by differentiating... 

Let us now consider the relative motion of two particles of the same radius Rp and mass mp, and denote by W(r, Ci r2, c2)dr dcidr2dc2 the probability of finding the first particle between r and n + drt, with the velocity between c and Ci + dc, and the second particle between r2 and r2 + t/r2, with the velocity between c2 and c2 + r/c2. The distribution function W satisfies the steady-state Fokker-Plank equation... 

One of the most attractive features of colloidal semiconductor systems is the ability to control the mean particle size and size distribution by judicious choice of experimental conditions (such as reactant concentration, mixing regimen, reaction temperature, type of stabilizer, solvent composition, pH) during particle synthesis. Over the last decade and a half, innovative chemical [69], colloid chemical [69-72] and electrochemical [73-75] methods have been developed for the preparation of relatively monodispersed ultrasmall semiconductor particles. Such particles (typically <10 nm across [50, 59, 60]) are found to exhibit quantum effects when the particle radius becomes smaller than the Bohr radius of the first exciton state. Under this condition, the wave functions associated with photogenerated charge carriers within the particle (vide infra) are subject to extreme... 

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