Normal size distribution functions

Packer and Rees [3] extended the work of Tanner and Stejskal by the development of a theoretical model using a log-normal size distribution function. Measurements made on two water-in-oil emulsions are used to obtain the self-diffusion coefficient, D, of the water in the droplets as well as the parameters a and D0 0. Since then, NMR has been widely used for studying the conformation and dynamics of molecules in a variety of systems, but NMR studies on emulsions are sparse. In first instance pulsed field gradient NMR was used to measure sdf-diffusion coefficients of water in plant cells (e.g. ref. [10]). In 1983 Callaghan... 

We can define a normalized size distribution function tisiDp) by ri/ /(Dp) = ni (Dp)/N, such that... 

The units of Jis Dp) are /xm . The normalized size distribution function ns Dp) can also be viewed as the probability that a randomly selected particle has a diameter in the range Dp, Dp -E dDp) it is therefore equivalent to the normalized probability density of particle size. 

Fig. 1. Time-independent normalized size-distribution function given by Lifshitz and Slezov(i2). Here r is the particle radius, f is the mean particle radius.

Lifshitz and Slezov on the other hand showed that in a dispersion in which grain growth was occurring according to Eq. (4) a time-independent normalized size distribution function would be approached in which the radius of the largest particles would be only 1.5 r (See Fig. 1.) From this steady-state distribution function, however, they derived an equation that predicts growth rates of the same order as Greenwood s equation, viz.. 

The size distribution function f(D) pertaining to the event as a whole has variously been taken as a log-normal or as a power law distri-... 

In the applications of gas-solid flows, there are three typical distributions in particle size, namely, Gaussian distribution or normal distribution, log-normal distribution, and Rosin-Rammler distribution. These three size distribution functions are mostly used in the curve fitting of experimental data. 

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. 

Fio. 9. Normalized distance distribution function of the polar coordinate r for all protein atom-aromatic ring contacts as defined by the coordinate system depicted in Pig. 8 (< 10 A). Each value of the distance distribution function was normalized for sample size by dividing the observed frequency by r2. 

A better fit between simulation and experimental data of the first extension of the virgin sample is obtained if the same model as above is used, but an empirical cluster size distribution is chosen instead of the physically motivated distribution function Eq. (37). This is demonstrated in Fig. 46a,b, where the adaptation of the same experimental data as above is made with a logarithmic normal form of the cluster size distribution function (xi) with i= i Id ... 

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

FIGURE 16S0 Sintering rate constant fw the (a) initial stage and (b) intermediate stage as a function of the log-normal size distribution width param r, lattice diffusion, g-b.d. giain-boundetiy difiiision, v. viscous flow). Taken from Chappell et al. [47]. 

The normalization of the size distribution function P ( g) finally yields... 

Fig. 5 Pore size distribution function of microporous membranes, parameterization with logarithmic normal distribution, for the parameter sets specified in Table 4. (a) Differential psd,...

Examples of size distribution functions are shown ill Figs. 1.4 and 1.5. Figure 1.4 shows number distributions of commercially produced silica particles in terms of the fraction of particles in the,size range around dp, dN/N d dp) = na(,dp)fNxs where is the total particle concentration. The total particle surface area corresponding to each size distribution is shown. Commercial silica manufactured by the oxidation of SiCU is used as a filler (additive) in rubber. Both coordinate axes in Fig. 1.4 are linear, and the area under each curve should be normalized to unity. A bimodal volume distribution with a minimum near a particle size of 1 is shown in Fig. 1.5. Distributions of this type are often observed for atmospheric aerosols (Chapter 13) the volume of aerosol material per unit volume of gas above and below a micron is about the same as shown by the area under the curve. Bimodal distributions are also often observed in aerosols from industrial sources as discus.sed below. 

The growth law for a polydisperse aerosol can be determined by measuring the change in the size distribution function with lime. In experiments by Heksler and Friedlander (1977), small quantities of organic vapors that served as aerosol precursors were added to a sample of the normal atmospheric aerosol contained in an 80-m bag exposed to solar radiation. The bag was made of a polymer film almost transparent to. solar radiation in the UV range and relatively unreactive with ozone and other species. Chemical reaction led to the formation... 

In equation (21-6) for the void fraction Sp, the pore-size distribution function is given by /(r), and fir) dr represents the fraction of the total volume of an isolated catalytic peUet with pore radii between r and r +dr. This is not a normalized distribution function because... 

However, the pore-size distribution function /(r) is not normalized, so the zeroth moment must be included in the expression for the average pore radins ... 

From such measurements, surface areas (normalized cumulative and relative), pore radii (choice of three measuring units), pore volumes (raw, normalized, cumulative and relative) and pore-size distribution functions of samples can calculated. Figure 8 presents the graphs of mercury-penetrated volume versus pressure in pores of Na- and La-montmorillonite samples. Figure 9 shows pore-size distribution functions from porosimetry data. 

The log-normal distribution gives a curve skewed towards the larger sizes, and it frequently gives a good representation of particle size distributions from precipitation and comminution processes. Furthermore, the log-normal distribution is often used because it overcomes the objection to the normal (Gaussian) distribution function which implies the existence of particles of negative size. 

PDI values below 0.05 indicate a very narrow, quasi monodisperse distribution, while values above 0.2 usually imply a relatively broad, possibly multimodal distribution. Values above 0.5 mean that the experimental data are poorly reproduced by Eq. (2.34). For log-normal size distributions, one can easily relate the PDI to the geometric standard deviation of the distribution function (Babick et al. 2012). 

The function a/(x) represents the measured size distribution, and this is used for the description of the data in terms of a mathematical equation. Figure 3.3 shows the measured size distribution determined from the data plotted in Fig. 3.2. Usually, the measured size distribution function is fitted in terms of an expected size disttibution, such as the normal distribution ... 

In a subsequent analysis, Srolovitz et al. (22) showed that when the grain size is normalized to the average grain size, then the size distribution function becomes time invariant, as also predicted by the mean field theories. A comparison of the distribution function obtained from the computer simulations with the lognormal distribution, and the distribution functions derived by Hillert and that by Louat are shown in Fig. 9.13. As shown in Fig. 9.18, the distribution function can also provide an excellent fit to some experimental data. 

Another consequence of LSW theory is the prediction that the size distribution function g( ) for the normalized droplet radius u = r/rc adopts a time-independent form given by ... 

The normal distribution function is rarely used to describe aerosol particle size distributions because most aerosols exhibit a skewed (long tail at large sizes) distribution function. The normal distribution is, of course, synunetrical. It can be applied to monodisperse test aerosols, to certain pollens and spores, and to specially prepared polystyrene latex spheres. The number frequency function is given by... 

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