Derivation of Size Distributions

Since Nx is synonymous with the mole or number fraction of molecules in the polymer mixture that are x-mers (i.e., that contain x structural units), then 

Neglecting the weights of the end groups, the weight fraction Wx of x-mers (i.e., the weight fraction of the molecules that contains x structural units) is given by Wx xNx/Nq and Eq. 2-88 becomes 

Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a 

The limit of 4 at high values of x is 1 and is reached at progressively higher values of x for higher conversions. 

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Nj. is synonymous with the mole- or number-fraction distribution. When x is an odd number, there are x pathways and the (x — 1) term in Eq. 3-190 should be replaced by x. For polymerizations yielding high polymer, the difference between x and (x + 1) is negligible and can be ignored. [Some derivations of size distributions show exponents y and z, respectively, in Eqs. 3-186 and 3-187 instead of (y — 1) and (z — 1), which results in an exponent of x instead of (x — 1) in Eqs. 3-189 and 3-190). The differences are, again, unimportant for systems that yield high polymer.]... 

Bergstroem, M. 1996. Derivation of size distributions of surfactant micelles taking into account shape, composition, and chain packing density uctuatidng.oll. Interface Scil81 208-219. 

Modern N2 sorption porosimeters are very sophisticated and generally reliable. Typically they come supplied with customized user-friendly software which enables the experimental data to be readily computed using the above models and mathematical expressions. Usually the raw isotherm data is displayed graphically along with various forms of the derived pore size distribution curve and tabulated data for surface area, pore volume and average pore diameter. 

It is evident from the above considerations that the use of the physisorption method for the determination of mesopore size distribution is subject to a number of uncertainties arising from the assumptions made and the complexities of most real pore structures. It should be recognized that derived pore size distribution curves may often give a misleading picture of the pore structure. On the other hand, there are certain features of physisorption isotherms (and hence of the derived pore distribution curves) which are highly characteristic of particular types of pore structures and are therefore especially useful in the study of industrial adsorbents and catalysts. Physisorption is one of the few nondestructive methods available for investigating meso-porosity, and it is to be hoped that future work will lead to refinements in the application of the method -especially through the study of model pore systems and the application of modem computer techniques. 

Validity of the derived pore size distribution should not be expected unless certain conditions are met and the following practice is recommended for mesopore size analysis if the isotherm is of Type IVa, the desorption branch should be adopted if the isotherm is of Type IVb, the adsorption branch is likely to provide a more reliable overall estimate of die pore size distribution. However, in the latter case the pore shape may be a critical factor. The procedure adopted for mesopore size analysis together with the branch of the hysteresis loop should always be clearly stated. 

The applied pressure is related to the desired pore size via the Washburn Equation [1] which implies a cylindrical pore shape assumption. Mercury porosimetry is widely applied for catalyst characterization in both QC and research applications for several reasons including rapid reproducible analysis, a wide pore size range ( 2 nm to >100 / m, depending on the pressure range of the instrument), and the ability to obtain specific surface area and pore size distribution information from the same measurement. Accuracy of the method suffers from several factors including contact angle and surface tension uncertainty, pore shape effects, and sample compression. However, the largest discrepancy between a mercury porosimetry-derived pore size distribution (PSD) and the actual PSD usually... 

Other workers [10,11] matched the NMR and mercury porosimetry derived pore size distributions to estimate p. More recently, Davis and co-workers [12] have shown that p can be found via a series of Ti experiments, varying the quantity of fluid sorbed on the solid surface. In that work it was shown that a plot of inverse average T, versus the surface area (as determined via conventional methods) times solid concentration (SA C) will give a line with slope (p/2) and intercept a. This value of p can then be applied to find unknown surface areas and pore size distributions using experimentally determined T, s for similar material at the same fluid, frequency and temperature. 

Measurements indicate that in natural waters values of /3 range from 2 to 5. Lerman (1979) reports measurements of size distributions at four locations in the North Atlantic. Fifty-three size distributions derived from samples taken at depths ranging from 30 to 5100 m yielded a mean value of /3 = 4.01 0.28. Fillela and Buffle (1993) report on size distributions based on particle number for different natural aquatic systems. 

We have found systematic quantitative discrepancies between BJH-derived pore size distributions and geometrically defined ones, in a scries of model porous materials. The isotherm-based PSDs are shaiper than the geometric ones, and are shifted by approximately 1 nm to smaller... 

Although DFT is now rapidly replacing the HK method, there remain a number of fundamental problems to overcome. For example, energetic heterogeneity and hysteresis phenomena are generally not taken into account in the application of DFT for pore size analysis. On the other hand, in principle DFT should be applicable to both microporous and mesoporous solids. The derived pore size distributions are shown in Figure 3. It is of interest that the results of the DFT analysis and the Os-plots are at least consistent, but further progress will depend on the application of DFT to a number of well-defined pore structures. 

For polymerization where termination occurs by all three modes, namely, coupling, disproportionation, and chain transfer, one can derive the size distribution as a weighted combination of the above two sets of distribution functions. For example, the weight distribution can be obtained as (Smith et al., 1966) ... 

An improved method of deriving pore-size distributions from adsorption isotherms is described which is also believed to provide information on pore shapes. The theory is similar in principle to that of Barrett, Joyner, and Halenda (JS), but the method of calculation is more precise. 

Thus, to determine the frequency of collision of particles or drops, it is necessary to determine the forces of particle interaction first, and then to find the trajectories of their motion and the collision cross-section or the diffusion flux. In the latter case, it is necessary to And the turbulent diffusion factor. As a result, the kernel of the kinetic equation is determined. If the kernel thus derived appears to be asymmetric, it should be symmetrized. After that, one can proceed to study the kinetics of coalescence for the considered process, including the time rate of change of size distribution of particles and the parameters of this distribution. 

FIG. 4 Plots of the derived micropore size distribution for Ajax activated carbon and Nuxit activated carbon. (From Ref. 126.)... 

Coke sampling is marginally less problematic because the product from a single source derives from coal or blend of coals that have been prepared to a specification for ash, moisture, particle size distribution, etc. The final coke produced will be relatively homogeneous in all properties, with the exception of size distribution. Standard methods are available for coke sampling that reflects the somewhat less rigorous requirements for this material. 

Both the RBC distribution (8) and the geometric distribution (11) are defined only for specific integer bubble sizes, and derivatives of their distribution functions do not exist. For subsequent developments we need an equivalent continuous distribution. Fortunately, for N and k large with respect to m, both discrete distributions can be closely approximated by the exponential distribution if its mean is set to the RBC mean volume given by (10). The exponential probability density is... 

This method gives a volume distribution and measures a diameter known as the laser diameter. Particle size analysis by laser diffraction is very common in industry today. The associated software permits display of a variety of size distributions and means derived from the original measured distribution. 

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