Number-average size

It should be noted that the predictions for the number average cluster size and polydispersity agree with analytical results for K(x, y) = 1, x + y, and xy. Furthermore, the short-time form of number average size in Eq. (81) matches the form of s(t) predicted by the scaling ansatz. Computational simulations (Hansen and Ottino, 1996b) also verify these predictions (Fig. 38). 

Unfortunately another factor complicates the measurement of true crystallite size. The K parameter is also a function of both crystallite size and of lattice distortion. We recently studied the effect of crystallite size and distortion on the K parameter using optical transform methods with simulated lattice images drawn by a computational method (26). The Scherrer K parameters given in Table X may be used to obtain a true number average crystallite size by any of the methods quoted, but will only be valid for crystallites with a number average size in the range 8 to 15 layer planes and a lattice distortion of 4-6%. The mixed function method (3) appears to give the best estimate of true crystallite size (K = 1.0) and the Hosemann method (4) the best estimate of lattice distortion. 

Then the number-average size ln and the weight average size 2W are given by... 

This moment is equal to the total number of floccules per unit volume, and Eq. 6.13a shows that it must always decline with increasing time if kmn > 0. The first moment M, is equal to the total number of primary particles "per unit volume, which should be constant according to the definition of the flocculation process and to Eq. 6.12 dM,/dt = 0.15 The number-average size is the ratio of M, to M0 16... 

The size and composition dispersions of droplets can be estimated by using the maximum-term method, without performing detailed distribution calculations. This is accomplished by adapting the approach used for micellar systems,17-18 as described in Appendix C. The aggregate corresponding to the maximum a Xgo or Xgy/ is considered to provide the number-average size and composition of the equilibrium droplets. For each component k (=S, A, O, or W) present in the microemulsion, the mean-square deviation o 2(k) from the number-average number of molecules g (k) can be shown to be... 

This section develops the relation between the number average size of polymers produced in a Step-growth polymerization and the fraction of functional groups which have been reacted at any point in the process. The basic equation which will be developed, Eq. (5-19), is called the Carothers equation. It illustrates the fundamental principles which underlie the operation of such polymerizations to produce good yields of polymers with the desired molecular weights. 

Since the relations between the number average sizes and the number geometric mean are known [equations (2.77) to (2.80)], these can now be expressed as relationships between number average sizes and the weight geometric mean to give a similar set of equations. 

Particle characteristics particle number, average size, and size distribution. 

X-ray diffraction (XRD) is mostly used for bulk structure analysis, yet it has one feature which makes it suitable for size determination, if the concentration of the active component is large enough. Figure 7.21 shows how x-ray diffraction lines broaden as the crystallite size decreases. Quantitatively, a number average size is obtained from... 

In an aqueous dispersion, these soy products swell to different degrees and show a broad size distribution except for soy protein aggregates, which have a number average size of 2.2 pm in the wet state (Figure 2). DSF, SPC, and SSF have a similar swollen size and size distribution due to swollen soy carbohydrates. Under ultrasonic dispersion, SPI aggregates tend to breakup while DSF, SPC, and SSF retain a similar size and distribution. 

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