Logarithmic time scale

Figure 4 Chemiluminescence oxidation runs at 140°C in oxygen for films of polypropylene (PP), polyamide 66 (PA 66) and polyethylene (PE) (logarithmic time scale).

Figure 5 Chemiluminescence oxidation runs for powder of poly(2,6-dimethyl-l,4-phenylene oxide) at different temperatures, oxygen (logarithmic time scale).

Equation (3) is plotted with two different time scales in Figures 1 and 2 for values somewhat typical of an elastomer. All the initial deformation takes place in the spring at a later time the dashpot starts to relax and allows the spring to contract. Most of the relaxation takes place within one decade of time on both sides of the relaxation time, but this is shown clearly only in Figure 2. On the logarithmic time scale, the stress-relaxation curve has a maximum slope at the time / = T and the stress ratio cr/cr is 0.3679 ore. The stress relaxation may also be given in terms of a stress-relaxation modulus Er(t) ... 

Figure 2 Stress relaxation of a Maxwell model on a logarithmic time scale. Model is the same as Figure 1.

Figures 5 and h show how the shape of the creep curve is modified by changes in the constants of the model. The values of the constants are given in Table I. Curve I is the same as shown in Figure 4, curve II shows onlv a small amount of viscous creep, and in curve 111, viscous flow is a prominent part of the total creep. The same data were used in Figures 5 and 6, but notice the dramatic, change in the shapes of the curves when a linear time scale is replaced by a logarithmic time scale. In the model, most of the recoverable creep occurs "Within about one decade of the retardation time.

Figure 6 Creep of a four-element model with the same constants as in Figure 5 but with a logarithmic time scale.

F(i) is the underlying modulus spectrum for that system. As noted above, since the time scale of relaxation is so broad, results are best depicted on a logarithmic time scale. To do this, one needs the contribution to the modulus associated with or lying in the time interval between In T and In T 4- Jin T this incremental contribution to the modulus is designated as... 

The continuous function II( n T) [often simply given the symbol H(r) as in this chapter) is the continuous relaxation spectrum. Although called, by long-standing custom, a spectrum of relaxation times, it can be seen that H is in reality a distribution of modulus contributions, or a modulus spectrum, over the real time scale from 0 to < or over the logarithmic time scale from - to +. ... 

A distribution obtained by the use of equation (13) is only a first approximation to the real distribution. The corresponding distribution of retardation times is designated as L(T). It may be estimated from the slope of a compliance curve D(0 or J(t), for tensile or shear creep, respectively, plotted on a logarithmic time scale according to the equation (for shear creep)-... 

Plot the creep compliance (cm /dyn) as a function of time using a logarithmic time scale. Would the curve show the upward curvature on a linear time scale ... 

Reproducibility is important and a parameter with distinct advantages in other ways may have to be rejected if it is poor in this respect. When time scales may extend over months and the results are subject to extrapolation, poor reproducibility will result in uncertainty levels which render the conclusions meaningless. This is particularly important when logarithmic time scales are applied, in which a modest level of scatter converts to errors in time stretching from minutes to centuries. 

Figure 4.4 ((a) and (b)) displays the same relaxation data, first against a linear time scale, showing the fast drop in stress at the start of test, and second against a logarithmic time scale showing a regular decrease of stress. 

Fig. 2.9. The mass spectrometric time scale. It is important to note the logarithmic time scale for the ion source spanning over nine orders of magnitude. Reproduced from Ref. [9] with permission. John Wiley Sons Ltd., 1985.

The degrees of hydration of Ca2Si04, Ca3Si05, and alite pastes as functions of time at the three temperatures are given in Figures 2, 4, and 6, respectively. The data, obtained from pastes whose compositions are given in Tables III, IV, and V, are plotted on a logarithmic time scale only as a matter of convenience. 

FIG. 13.14 The position of different experimental techniques on logarithmic time scale (Staverman and Schwarzl, 1956). 

Figure 15 Step response of the reflection coefficient for a Cole-Cole dielectric with Bo = 80, Boo = 8 A, a = 0 B, a — 0.5. For //tq > 1 o logarithmic time scale is used (Reproduced by permission from Appl. Sci. Res. in press)...

An analysis of the ratio of 2v to s as a function of time (Fig. 23) shows that on a logarithmic time scale it decreases almost linearly from about 0.077 at short times to... 

In practice, it is more precise to evaluate the viscosity from a step strain experiment by transforming the integration of Eq. (7.117) to a logarithmic time scale using the identity td nt = dt, and the lower limit of integration changes because when = 0, In f = - 00 ... 

These two extreme models cannot be distinguished from a poor autocorrelation curve with a logarithmic time scale. Conventional analyses for cases with anomalous diffusion use alternative types of equation with fractal consideration with an exponent on D or t as... 

Fig. 5 Dynamic interfacial tension (y) measurements of a toluene-water interface during adsorption of 6-nm CdSe nanoparticles to a pendant water drop in toluene (CdSe concentration was 1.58 x 10-6 mol/L). The circles mark the time at which TEM samples shown in Fig. 6 were prepared. The inset depicts the data on a logarithmic time scale. Reprinted with permission from Physical Chemistry Chemical Physics [50], Copyright (2007) RSC Publishing...

Fig. 23. Redistribution of load on to shorter of two chains (Ar10 = 500, JVjo = 3000) for which the activation energy of saltation differ by 1.8 kT (logarithmic time scale)...

More generally, the relaxation follows generalized first-order kinetics with several relaxation times x., as depicted schematically in figure B2.5.2 for the case of three well-separated time scales. The various relaxation times determine the turning points of the product concentration on a logarithmic time scale. These relaxation times are obtained from the eigenvalues of the appropriate rate coefficient matrix (chapter A3.41. The time resolution of T-jump relaxation techniques is often limited by the rate at which the system can be heated. With typical T-jumps of several Kelvin, the time resolution lies in the microsecond range. 

Figure B2.5.2. Schematic relaxation kinetics in a T-jump experiment, c measures the progress of the reaction, for example the concentration of a reaction product as a function of time t (abscissa with a logarithmic time scale). The reaction starts at (q. (a) Simple relaxation kinetics with a single relaxation time, (b) Complex reaction mechanism with several relaxation times x-. The different relaxation times x. are given by the turning points of c as a function of ln(t). Adapted from [110].

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