Double logarithmic coordinates

In figure 1 the dependence pA(t) in double logarithmic coordinates, corresponding to the relationship (2), for solid state imidization reaction without filler at the four mentioned above imidization temperatures 7) are shown. As can be seen, the received dependences are linear and according to their slope the value ds can be obtained. The 7) increase in the range 423-523... 

These results could be complemented well with the curve slopes in the double logarithmic coordinates as plotted in Fig. 6.33(a) using idea of the intermediate critical exponent a(t), equation (4.1.68). In the traditional chemical kinetics its asymptotic limit ao = a(oo) = 1 is achieved already during the presented dimensionless time interval, t 104. For non-interacting particles and if one of two kinds is immobile, Da = 0, it was earlier calculated analytically [11] that the critical exponent is additionally reduced down to ao = 0-5. However, for a weak interaction (curve 1) it is observed that in the time interval t 104 amax 0.8 is achieved only for a given n(0) = 0.1, i.e., the... 

The group on the left side of this equation is a form of dimensionless film thickness and has been termed the Nusselt film thickness parameter Nt (D12). Equation (97) indicates that a plot of Nt against Nne on double-logarithmic coordinates should give a straight line of slope for the... 

The polymers were fractioned into 8-14 components by coacervate extraction from the benzene - methanol system. For fractions and nonfractioned polymers, characteristic viscosities [t ], were me-asured. Because that was the first example of studying conformations of macromolecules of this ty-pe in diluted solutions, authors of the work [56] paid much attention to selection of an equation, which would adequately describe hydrodynamic behavior of polymeric chains. Figure 10 shows de-pendencies of [q] on molecular mass (MM), represented in double logarithmic coordinates. Parame-ters of the Mark-Kuhn-Hauvink equation for toluene medium at 25°C were determined from the slo-pe and disposition of the straight lines. 

It has been experimentally demonstrated that the principles of temperature and concentration superposition of flow curves are applicable to melted keroplasts. In other words, in the fixed matrix FC compositions with different filler content, tp and T measured at different temperatures, can be interpolated through a flat and parallel displacement along the coordinate axes (in double logarithmic coordinates). 

Shear stress-shear rate plots of many fluids become linear when plotted on double logarithmic coordinates and the power law model describes the data of shear-thinning and shear thickening fluids ... 

Table 17.24 shows data for a series of regrinds all derived from the same composition, that is, HDPE (MFI 0.5) filled with rice hulls and Biodac , 29% of each filler. The storage and loss moduli were plotted against the frequency in double logarithmic coordinates, and the respective slopes are shown in Table 17.24. In all cases the storage modulus and the loss modulus increased with frequency. The same data were analyzed earlier, in Table 17.12, in terms of their zero-shear viscosity.

Particle sizes were established on the basis of atomic-power microscopy data (see Figure 6.2). For each from the three-studied nanocomposites no less than 200 particles were measured, the sizes of which were united into 10 groups and mean values N and p were obtained. The dependences TV(p) in double logarithmic coordinates were plotted, which proved to be linear and the values were calculated according to their slope (see Figure 6.5). 

In Figure 6.6 the dependences of TV on S. in double logarithmic coordinates for the three-studied nanocomposites, corresponding to the relationship Eq. (6.25), is adduced. As one can see, these dependences are linear, which allows to determine the value from their slope. The determined values D, according to Eq. (6.25), are also adduced in Table 6.2, from which a good correspondence of dimensions obtained by the two... 

Figure 10.14 Dependence of a degree of conversion a, on time of reaction t, in double logarithmic coordinates corresponding to Equation (10.1) for system 2DPP + HCE/DDM at 353 (1), 373 (2) and 513 K (3).

FIGURE 63 The dependence of reaction rate constant on microgels gyration radius in double logarithmic coordinates for system EPS-l/DDM. 

FIGURE 77 The scaling relationship between PDMDAAC concentration Cj and its average viscous molecular weight ATM in double logarithmic coordinates. 

The authors [226] constructed the dependence x MM) in double logarithmic coordinates for PS-Cgg solutions in two solvents— tetrahydro-fliran and chloroform. These dependences proved to be linear, that allows to estimate the exponent for the studied polymers, which proved to be equal to -0.06 0.07 for tetrahydrofuran and 0.2 0.2—for chloroform. However, this construction causes a questions number. Firstly, in the well-known Mark-Kuhn-Houwink equation (the formula (1)) coefficientdepends on both exponent and macromolecular repeated link molecular weight m. One from the relationship between K, and variants is the Eq. (2). The plots v MM) in double logarithmic coordinates construction and subsequent the exponent determination from their slope assumes the condition K =const, that is not far from obvious. 

Then by the dependences (t.) in double logarithmic coordinates construction the value h can be determined from their slope and the experimental values of effective spectral dimension d], can be calculated according to the Eq. (6). In Fig. 4 the comparison of theoretical values d calculated according to the Eq. (11), and evaluated according to the described method magnitudes is adduced. As one can see, a theory and e5q)eriment good correspondence is obtained (Ihe mean discrepancy of (dly and d] values makes up 4.5% and maximiun one 9.3%). 

In Fig. 21 the kinetic curves conversion degree—reaction duration Q-t for two polyols on the basis of ethyleneglycole (PO-1) and propylene-glycole (PO-2) are adduced. As it was to be expected, these curves had autodecelerated character, that is, reaction rate was decreased with time. Such type of kinetic curves is typical for fractal reactions, to which either fractal objects reactions or reactions in fractal spaces are attributed [85], In case of Euclidean reactions the linear kinetics (i> =const) is observed. The general Eq. (2.107) was used for the description of fractal reactions kinetics. From this relationship it follows, that the plot Q t) construction in double logarithmic coordinates allows to determine the exponent value in this relationship and, hence, the fractal dimension value. In Fig. 3.22 such dependence for PO-1 is adduced, from which it follows, that it consists of two linear sections, allowing to perform the indicated above estimation. For small t t 50 min) the linear section slope is higher and A =2.648 and for i>50 min A =2.693. Such A increase or macromolecular coil density enhancement in reaction course is predicted by the irreversible... 

FIGURE 22 The dependence of conversion degree Q on reaction duration t in double logarithmic coordinates for polyol on the basis of ethyleneglycole. 

In Fig. 24, the dependenee in double logarithmic coordinates is adduced. As it was to be expected, this dependenee proves to be linear, that allows one to determine from its slope the value h=0.56. Thus, this result confirms polyesterification reaction proceeding in heterogeneous (fractal) medium. For such mediums (particularly at elevated temperatures) the connectivity is characterized by not spectral dimension but by its effective value, which takes into account temporal (eneigetic) disorder availability in the system [22]. value is linked with heterogeneity exponent h by the Eq. (6), which for h=0.56 gives the value <, =0.88 [83],... 

The relationship between shear stress and shear rate (plotted on double logarithmic coordinates) for a shear-thinning fluid can often be approximated by a straightline over a limited range of shear rate (or stress). For this part of the flow curve, an expression of the following form is applicable ... 

The limited information reported so far suggests that the appment viscosity-shear rate data often result in linear plots on double logarithmic coordinates over a limited shear rate range and the flow behaviom may be represented by the power-law model, equation (1.13), with the flow behaviom index, n, greater than one, i.e. 

Within the frameworks of fractal analysis the polycondensation kinetics is described by the general Eq. (27). If the indicated relationship describes polycondensation kinetics correctly, then the dependence Q(t) in double logarithmic coordinates should give a straight line, from the slope of which the value Dp changing within the limits of 1 < Dj,< 3 [4], is determined. The determined by the indicated mode Df values are also adduced in Table 2. 

In Fig. 4, the dependences of (1 - Q) on t in double logarithmic coordinates for DMDAACh polymerization at two values of monomers initial concentration cO 4.0 and 4.97 moles/1 (0.646 and 0.728 by the mass) are adduced. As one can see, in both cases the linear dependences with negative slope, having absolute values 0.26 and 0.38, are obtained. Hence, these dependences correspond to the Eq. (6) and allow to calculate the values ds of reactive medium, which for the used monomer concentrations are varied within the limits of 0.52-2.95. 

In Fig. 46, the dependences S n (SF) in double logarithmic coordinates are shown for DMDAACh, synthesized at different c . As it follows from the data of this figure, all four MWD curves, shown in Fig. 45, are described by a sole generalized curve. This is the most important result, confirming an irreversible aggregation models using correctness for polymerization process description. 

Further the dependences MM, in double logarithmic coordinates, corresponding to the Eq. (21), can be plotted for determination of the exponent y (t) experimental values in the indicated relationship. The plotted by the indicated mode dependences MM for four T are shown in Fig. 14 and the values f, are adduced in Table 3. 

FIGURE 19 The dependence of branching center number per one macromolecule m on molecular weight MM in double logarithmic coordinates for PHE synthesized at T = 333 K. 

Let us consider further the Eq. (27) of Chapter 1 application for the system 2DPP+HCE/DDM curing kinetics description. In Fig. 18 the dependences 0, in double logarithmic coordinates corresponding to the Eq. (27) of Chapter 1, were adduced. As one can see, they ate linear, that allows to determine from their slope the value of fractal dimension of microgels, forming in curing process. As the... 

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