Mass exponent

In a number of experimental studies of polymer diffusion, molar mass exponents close to 2 have been found, though always with some deviations. For example, using radio-labelled molecules, the diffusion coefficient of polystyrene in dibutyl phthalate was found to follow the relationship... 

Experimental values of the molar mass exponent close to 2 have been obtained. For example, for poly(methyl methacrylate), a value of 2.45 has found (see P. Prentice, Polymer, 1983, 24, 344—350). As with values of selfdiffusion coefficient, this has been regarded as close enough to 2 for reptation to be considered a good model of the molecular motion occurring at the crack tip. 

For small molecules (metabolites, monomers, and small oligomers) the mobility equation may be empirically approached with the Offord model (linear relation to charge-to-size ratio, the charge being obtained directly from the ionization constants and the size being approached with the molecular mass exponent a factor a) [see Offord (1966)]. 

Nelson, D.E., A. Angerbjorn, K. Liden, and I. Turk (1998). Stable isotopes and the metabolism of the European cave bear. Oecologia 116 177-181. Nevill, A.M. (1994). The need to scale for differences in body size and mass an explanation of Kleiber s 0.75 mass exponent. J. Appl. Physiol. 77 2870-2873. 

The differences in metabolic rate between ectotherms and endotherms thus are due to the preexponential term a, which is approximately four- to fivefold higher in birds and mammals relative to reptiles. On a mass-specific basis (for instance, oxygen consumption per gram wet mass) the mass exponent is near... 

The mass exponent is related to the generalized dimension D(q) by the relation... 

The local Holder exponent h varies with the -dependent mass exponent through the equality... 

Figure 9. (a) The mass exponent as a function of the -moment obtained from a numerical fit to... 

As mentioned above, a time series is mono-fractal when the mass exponent is linear in q, otherwise the underlying process is multifractal. We apply the partition function measure to numerically evaluate... 

A second method for determining the singularity spectrum, the one we use here, is to numerically determine both the mass exponent and its derivative. In this way we calculate the multifractal spectrum directly from the data using Eq. (86). It is clear from Fig. 9b that we obtain the canonical form of the spectrum that is, f(h) is a convex function of the scaling parameter h. The peak of the spectrum is determined to be the fractal dimension, as it should. Here again we have an indication that the interstride interval time series describes a multifractal process. We stress that we are only using the qualitative properties of the spectrum for q < 0, due to the sensitivity of the numerical method to weak singularities. 

The scaling relation in Eq. (149) determines the q h order structure function exponent p(g). Note that when Mq) is linear in q the underlying process is monofractal, whereas when it is nonlinear in q the process is multifractal, because we can relate the structure function to the mass exponent [88] ... 

The nonlinear form of the mass exponent in Fig. 9a, the convex form of the singularity spectmm/(/i) in Fig. 9b, and the fit to f(q) in Fig. 13, are all evidence that the interstride interval time series are multifractal. This analysis is further supported by the fact that the maxima of the singularity spectra coincide with the fractal dimensions determined using the scaling properties of the time series using the allometric aggregation technique. 

Heusner, A. (1982a). Energy metabolism and body size. is the 0.75 mass exponent of Kleiber s equation a statistical artifact Respiratory Physiology and Neurobiology, 48, 1-12. 

Where the slope b yields x(q). Then the mass exponent can be found from the linear slope, that is t(q) = b. If x(q) is nonlinearly related to q, the research object has multifractal character. 

The f(a) curve is not the only way to characterize the properties of multi-fi actal measures. Another, widely used, approaeh to describe their features involves sequences of the mass exponents, x(q). To define these exponents, it is convenient to consider a set i covered with -mesh cubes , as in Figure 2. A iii). Within each one of these 5-mesh cubes, one may define a measure /x that is similar to that of Equation (2.22) and represents the probability of finding an element (a point) of set within the 5-mesh cube. With this probability, one may construct, for any real number q, the measure... 

The mass exponent value x decreases as the temperature increases to reach a value of 3-45 at 55 C when cyclohexane can be considered as a good solvent for polystyrene. One has to note that the exponent x is larger than the predicted va e x = 3. Self-diffusion coefficient measurements lead to Dg -- Af. Assuming that is the time required for the polymer chain to diffuse on a distance of order of its radius of gyration R - (R = Tffig) we should have Present results for relaxation time therefore disagree with self-diffusion coefficient measurements. From viscosity and relaxation time measurements, we can deduce the shear elastic modulus at short times G We find that G is mass-independent and increases with concentration ... 

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