Fractional exponent estimation

Let us note, that the dependences m(MM) with fractional exponent gother polymers as well. So, in work [4] the constants of Kuhn-Mark-Houwink equation for -solvent in case of the branched polyarylate D j and its linear analog have been adduced, that allows to calculate intrinsic viscosities [tj] 0 and [Tjjj g, accordingly, for arbitrary MM. Then, the value g can be estimated according to the relationship [4] ... 

Let s consider methods of estimation of parameters in the Eq. (29). As it has been shown above (see section 1.2) for fractal objects in Euclidean spaces with d> as a fractional exponent v fractional part of D is accepted, that allows one to determine value according to the Eq. (43 of Chapter 1). In this case the value v characterizes reactive molecules fraction, not participating in evolution (reaction) process. Then the molecule fraction capable of chemical transformations in reaction course, is defined as follows [39] ... 

Fig. 26. Molar mass dependence of the g factor for three pregel and one postgel fraction of end linked PS stars. A good fit was obtained with the Zimm Stockmayer equation (Eq. 69) and an exponent in Eq. (70) of fi 0.63 [95] which agrees well with Kurata s estimation with b-0.6 [129]. Reprinted with permission from [129]. Copyright [1972] American Society...

Thus the molar mass dependence of the CP-parameter can be estimated when, besides the exponent, the fractal dimension of the clusters could be measured. This fractal dimension can be obtained from the molar mass dependence of the radius of gyration of the fractions, or from the angular dependence of the... 

Particulate size data reported for numerous particulate systems in natural waters can be modeled accurately with a two-parameter power law, given by dN/dl = AV. The exponent of the power law, p, has been shown to be a useful estimator of the relative contribution of particulate size classes to the total number, surface area, mass and volume concentration, and extinction coeflBcient of the particulate fraction. Reported values of p range from 1.8 to 4.5 in low ionic-strength solutions. 

Equation (6) defines the fractional turnover rate( 2) for the large pool (B) in terms of the exponent (n) for the slow phase of the curve, the small pool (A), the large pool (B), and the intake (/). In the case of normal vitamin B6 intake, IS nmol/g times g body wt usually yields a reasonable estimate of the total pool. Since muscle usually contains 70-80% of the pool, muscle biopsies provide a means of verifying the pool size. The input and/or excretion can usually be measured. The exponent can be obtained by fitting a curve to the data. The size of pool A must be estimated. However, assuming that A is small relative to B and that n is small, variations in A have relatively little effect on the estimates of kz- Therefore, Eq. (6) provides a means for making k2 more consistent with physiological observations. 

Harris [84] warns against placing too much faith in the accuracy of the probit over the full range. It would be most helpful to use Eq. (4) to estimate what concentrations are toxic to a smaller fraction, say 1% or less, of the population, but the values even at the 10% and 90% levels are questionable. The difference between the concentration exponent of 2 used in Eq. (4) and the value of 2.75 used by Lees illustrates the problem. 

The deviations from the mean field (classical) exponents are due to the fluctuations in the gel fraction inside one correlation volume O = We shall estimate these fluctuations, assuming that the classical exponents hold, and see if they are large or small when compared to the average gel fraction. We shall see that they are indeed small whenever the dimensionality d is larger than 6 in this case it was thus correct to assume classical behavior. 

The Avrami equationhas been extended to various crystallization models by computer simulation of the process and using a random probe to estimate the degree of overlap between adjacent crystallites. Essentially, the basic concept used was that of Evans in his use of Poisson s solution of the expansion of raindrops on the surface of a pond. Originally the model was limited to expansion of symmetrical entities, such as spheres in three dimensions, circles in two dimensions, and rods in one, for which n = 2,2, and 1, respectively. This has been verified by computer simulation of these systems. However, the method can be extended to consider other systems, more characteristic of crystallizing systems. The effect of (a) mixed nucleation, ib) volume shrinkage, (c) variable density of crystallinity without a crystallite, and (random nucleation were considered. AH these models approximated to the Avrami equation except for (c), which produced markedly fractional but different n values from 3, 2, or I. The value varied according to the time dependence chosen for the density. It was concluded that this was a powerful technique to assess viability of various models chosen to account for the observed value of the exponent, n. 

The classical method to determine and a of a given polymer is as follows. First, prepare fractions of different molecular weights either by synthesis or by fractionation. Next, make dilute solutions of different concentrations for each fraction. Measure the viscosity of each solution, plot the reduced viscosity as a function of polymer concentration, and estimate [17] for each fraction. Plot [17] as a function of the molecular weight in a double logarithmic scale. This method has been extensively used to characterize polymer samples because the exponent a provides a measure of the chain rigidity. Values of a are listed in Table 3.2 for some typical shapes and conformations of the polymer. The value of a is around 0.7-0.8 for flexible chains in the good solvent and exceeds 1 for rigid chains. In the theta solvent, the flexible chain has a = 0.5. 

Observed interatomic distances for diatomic transition-element interactions are estimates of the fraction, d = 0.783 of nearest-neighbor approaches in the metals [5] and may be considerably in error in the present context, especially for the second transition series. Apart from first-order La2 and Ce2, with = 245 30kJmol homonuclear diatomics have weak interactions with an average Dx = 70 40kJmor in agreement with our estimates. Multiple bond orders, in general, are characterized by stepwise reduction of the first-order golden exponent, such that... 

Numbers with decimal points are numerical estimates, fractions are (presumably) exact. As explained in Eq. (16) and on p. 126, classical exponents are exact for dimensionalities above 6 (left part) and above 4 (right part)... 

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