Relaxation curve

Fig. 38. Stress—relaxation curve for a lightly vulcanized rubber (242). To convert Pa to dyn/cm, multiply by 10.

Fig. 8.90 Load relaxation curves for a maraging steel stress corroded in 0.6 m NaCl at pH 2...

All relaxation curves exhibited more than one phase at various degrees of conversion and at different temperatures. This clearly rules out the all-or-none mechanism (AON) although the AON model is able to fit easily to the measured equilibrium transition curve. However, a mechanism has been proposed which allows the existence of side... 

R.L. Bohon, AnalChem 35 (12), 1845-52 (1963) CA 60,1527 (1964) Approx heats of expin, Qv were detd on mg amounts of propints and expls by differential thermal analysis (DTA). Small-screw-cap metal cupsi sealed with a Cu washer served as constant vol sample containers the initial cup pressure could be controlled from 0 to approximately lOOOpsia. The calibration constant was calcd for each run from the total heat capacity of the cup and the relaxation curve, thereby compensating for equipment variations. 

Figure 46. Separation of the overall oxidation curve into its two components a relaxation curve, responsible for the initial slope and the position of the chronoamperometric maximum, and a diffusion curve that controls the overall shape of the chronoampero-gram. (Reprinted from T. F. Otero, H.-J. Grande, and J. Rodriguez, J. Phys. Chem. 101, 3688, 1997, Figs. 1, 3, 6, 7, 13. Copyright 1997. Reprinted with permission from the American Chemical Society.)...

Figure 7.11 Stress relaxation curve for vulcanized natural rubber showing characteristic upswing at higher stresses...

A linear regression of A (ul, x) for < 0.8 was performed for each value of x, the slope giving A °(x) which is found to be linear in x. From the linear dependence of A °(x) on x, the original relaxation curves can be retraced. Inserting Eq. (86) into the modified Arrhenius equation ... 

Fig. 17. High-spin low-spin relaxation of [Fe(ptz)j](BF4)2 a normalized relaxation curves at temperatures between 57.5 and 66.5 K b activation energy AE as a function of n. The solid line represents a simultaneous fit of the five experimental curves displayed under a. The broken line is the result of a linear regression determined by AE° = — 164 cm" = — 0.47 kcal mol" as slope and AE = 797 cm" = 2.28 kcal mol" as intercept. Notation in the figure uses yi, A instead of Hl,...

A more complex but faster and more sensitive approach is polarization modulation (PM) IRLD. For such experiments, a photoelastic modulator is used to modulate the polarization state of the incident radiation at about 100 kHz. The detected signal is the sum of the low-frequency intensity modulation with a high-frequency modulation that depends on the orientation of the sample. After appropriate signal filtering, demodulation, and calibration [41], a dichroic difference spectrum can be directly obtained in a single scan. This improves the time resolution to 400 ms, prevents artifacts due to relaxation between measurements, and improves sensitivity for weakly oriented samples. However, structural information can be lost since individual polarized spectra are not recorded. Pezolet and coworkers have used this approach to study the deformation and relaxation in various homopolymers, copolymers, and polymer blends [15,42,43]. For instance, Figure 7 shows the relaxation curves determined in situ for miscible blends of PS and PVME [42]. The (P2) values were determined... 

It is clear from the above equations that numerous parameters (proton exchange rate, kcx = l/rm rotational correlation time, tr electronic relaxation times, 1 /rlj2e Gd proton distance, rGdH hydration number, q) all influence the inner-sphere proton relaxivity. Simulated proton relaxivity curves, like that in Figure 3, are often used to visualize better the effect of the... 

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) ... 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

Thus (he time scale / at /, divided by an is equivalent to the scale at On a log scale, log a, is thus the horizontal shift factor required for superposition. An important consequence of equation (22) is that a, or log (ii is the same for a given polymer (or solution) no matter what experiment is being employed. Thai is. creep and stress-relaxation curves are shifted by the same amount. 

The temperature-time superposition principle is illustrated in Figure 8 by a hypothetical polymer with a TK value of 0°C for the case of stress relaxation. First, experimental stress relaxation curves are obtained at a series of temperatures over as great a time period as is convenient, say from 1 min to 10 min (1 week) in (he example in Figure 8. In making the master curve from the experimental data, the stress relaxation modulus ,(0 must first be multiplied by a small temperature correction factor/(r). Above Tg this correction factor is where Ttrt is the chosen reference... 

Beyond Tfl, whole molecules are moving and contributing to viscous flow [i.e., equation (44) describes the long-time tail of the stress relaxation curve or the onset of the flow regime]. 

An example of experimental stress-relaxation data is shown in Figure 14 (160). Master stress-relaxation curves made from the experimental data on different molecular weight materials are shown in Figure 15. The temperature-shift factors used in making the master curves are shown in Figure 16. Note that the shift factors a, are the same for all molecular weights... 

Figure 23. Stress-relaxation curves of amorphous bisphenol A polycarbonate at the different temperatures shown by the curves. The numbers in brackets are the maximum deformations used in the tests. (From Ref. 217.)...

Another quite delicate step is the determination of the relaxation time constant r from the relaxation curve T(t). In this method, instead of the electrical heater, other heating sources, such as an optical source, can be used [13],... 

It should also be noted that the LIESST phenomenon has been recently observed on these materials [53-55]. This discovery may lead to a new wave of photomagnetic investigations of these bistable materials in view of potential applications. The shape of the relaxation curves after LIESST could be modelled within the framework of a revised ID Ising like model [56]. 

NMR imaging Images (one-, two-, or three-dimensional array of voxel intensities) pixel-specific relaxation curves maps, and movies (i.e., density, 7), T2, or D weighted) Hills (1998) McCarthy (1994) Price (1998b)... 

FIG. 23 A schematic illustration of the molecular motions and associated T2 relaxation curve behavior for the three major domains in foods—liquid, viscous liquid, and solid (crystalline and glassy). Typical H T2 NMR relaxation time values observed in these domains, and values specific for water in liquid and crystalline domains, are listed. 

We can see that the different positions along the chain show distinct temperature-dependent relaxation curves. To further analyze these relaxation functions, we must Fourier transform them to determine their spectral density, which is best done employing an analytic representation of the data that... 

Figure 4.9 A stress relaxation curve for a viscoelastic solid...

Figure 1. Typical relaxation curve in the aqueous y-A1203 -Cu(N03)2 suspension observed by using the pressure-jump method. [P] = 30 g/dm3, and I = 7.5 x10-3 at 25 °C sweep 20 ms/div.

Fig. 6.16 NSE relaxation curves obtained from a 16% volume fraction poly(fluorosilicone) gel in acetone using tbe bigb resolution NSE spectrometer INI5 at tbe ILL, Grenoble. Tbe existence of plateaus that represent tbe level of excess scattering from static inhomogenieties at low wave vector Q is clearly visible. Tbe decay rates times of tbe dynamic parts yield tbe collective diffusion coefficient of tbe gel. (Reprinted with permission from [291]. Copyright 2002 American Chemical Society)...

Fig. 6.29 S(Q,t)/S(Q) measured at the IN15 on a 156 mg/mL apoferritin solution with 0.01 mM salt. Despite the paracrystalline order no permanent restriction of motion is present as indicated by the virtually full decay of the relaxation curve (Q=0.09 A ). (Reprinted with permission from [333]. Copyright 2003 Elsevier)...

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