Double logarithmic plot

The general function giving straight lines on double logarithmic plots is ... 

Both sets of experiments seem to support the proportionality of crack opening displacement 5C = 2w and molecular mass Mc between crosslinks as indicated by the slope 1 in the double logarithmic plot (Fig. 7.5). Even if Mc had to be adjusted due to doubts about the front factor in Eq. (4.3), the proportionality would stay unaffected. Consequently, the size of the deformation zone ahead of the crack is determined by the length of the molecular strands in the chemical network. 

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

RIA PREC.dat Two hundred thirty-eight calibration data sets were collected and analyzed for repeatability (within group CV) and plotted against the mean concentration. In a double-logarithmic plot the pattern seen in Fig. 4.6 appears. 

Fig. 141.—Double logarithmic plot of [77]0 against M for several polymer series polyisobutylene in benzene at 24°C, O polystyrene in cyclohexane at 34°C, cellulose tricaprylate in T-phenylpropyl alcohol at 48 C, Q and...

Fig. 5. Double logarithmic plot of zero-shear rate viscosity against concentration and molar mass... 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Fig. 60. Crossover from single-chain to many-chain relaxation at T = 343 K. Lineshape analysis for PDMS/d-benzene at c = 5 and 18% double logarithmic plot of — ln/S(Q,t)/S(Q,0) vs. t/s. (Reprinted with permission from [116]. Copyright 1982 J. Wiley and Sons, Inc., New York)...

Indeed, in electrolytes containing no aggressive anions (as are the halides), over a wide pH range between 0 and 12 (from sulfuric acid through acetate to phthalate and borate), a double logarithmic plot of jQ2- versus 7A13+, shown in Fig. 5, yielded straight lines, with slopes of 1.38 0.14. 

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

Figure 8.14. Sketch of diffuse scattering in the double-logarithmic plot that is used for the determination of the fractal dimension. Superior and lower cut limit the fractal region that should be followed by an interval in which POROD s law is valid... 

It is often convenient to plot the Langmuir isotherm in a double logarithmic plot (Fig. 4.3). 

Plot of adsorption data in a double logarithmic plot. In a Langmuir isotherm the initial slope is unity. A Freundlich isotherm shows in a double log plot a slope of n < 1. Such a Freundlich isotherm is obtained if the adsorbent is heterogeneous (decreasing tendency for adsorption with increasing 6). (Modified from Morel, 1983)... 

Data on the adsorption of caprylic acid on a hydrophobic (mercury) surface in terms of a double logarithmic plot of Eq. (4.13) Panel a) compares the experimental values with a theoretical Langmuir isotherm, using the same values for the adsorption constant B for both curves. Panel b) shows that the adsorption process can be described by introducing the parameter a, which accounts for lateral interaction in the adsorption layer. Eq. (4.13) postulates a linear relation between the ordinate [= log [0/ 1 - 0)] - 2a 0 / (In 10)] and the abscissa (log c). If the correct value for a is inserted, a straight line results. For caprylic acid at pH 4, a value of a = 1.5 gives the best fit. 

Fig. 19. Weight fraction molar mass distributions w(x) of the Schulz-Zimm type for various numbers of coupled chains in a double logarithmic plot. Note fory=l the Schulz-Zimm distribution becomes the most probable distribution in the limit of/ l the Poisson distribution is eventually obtained. In all cases the weight average degree of polymerization was 100. The narrowing of the distribution with the number of coupled chains is particularly well seen in the double logarithmic presentation...

A careful examination of the Stockmayer and the percolation distributions reveals that both theories gives the same type of distribution [110]. In terms of the two exponents in Eqs. 52 and 53, the percolation calculation yields t=2.2 and 0 0.44, and the Stockmayer distribution yields r 2.5 and o 0.50. These differences in the exponents appear to be small in a double logarithmic plot, but they cause significant differences in the absolute values for w(x) when 3-4 decades in the degree of polymerization are covered. Another point is that the cut-off function could be calculated analytically in the FS-theory to be a single exponential function [110], while the percolation theory could only make a guess about its shape [7]. 

The vs Mj curve often gives a straight line in the double logarithmic plot and can be represented by... 

The intrinsic viscosity vs molecular weight dependence extracted from the SEC experiments curve of the intrinsic viscosity also often gives a straight line in the double logarithmic plot and can be described by the (KMHS) relationship ... 

This calculation is repeated over a range of box sizes, and a double - logarithmic plot of the lacunarity versns the size of the shding box is then produced. FracLac then outputs a text file containing the valnes of r and A for each image. 

Fig. 46. Double-logarithmic plot of the lineshapes of the specular He intensity from a Ni( 113) surface along the (3,3,2) direction with the temperature as parameter. (After Ref. 117.)...

Fig. 47. Double-logarithmic plot of the lineshape of the purely elastic specular He intensity from a Pt(lll) surface along the F M azimuth. The primary beam energy was 18.3 meV.

Figure 1. Sketch of double logarithmic plot of (r) — r]o)/ o o)/ o versus c in (a)...

Fig. 6 Electron time-of-flight photocurrent transients of solution-cast film of EHO-OPPE (L=8 pm), measured at 295 K and an electric field of 2.5-10 V cm in (a) linear and (b) double logarithmic plots. Reproduced with permission from [61]...

Typical photocurrent transients are shown in Fig. 6 for electrons and in Fig. 7 for holes. The shape of these curves is representative for all transients observed in the study and is characteristic of dispersive transport [64-68]. The carrier mobility p was determined from the inflection point in the double logarithmic plots (cf. Fig. 6b and Fig. 7b) [74]. TOF measurements were performed as a function of carrier type, applied field, and film thickness (Fig. 8). As can be seen from Fig. 8, the drift mobility is independent of L, demonstrating that the photocurrents are not range-limited but indeed reflect the drift of the carrier sheet across the entire sample. Both the independence of the mobility from L, and the fact that the slopes of the tangents used to determine the mobility (Fig. 6 and Fig. 7) do not add to -2 as predicted by the Scher-Montroll theory, indicate that the Scher-Montroll picture of dispersive transients does not adequately describe the transport in amorphous EHO-OPPE [69]. The dispersive nature of the transient is due to the high degree of disorder in the sample and its impact on car-... 

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