Other Time-Dependent Phenomena

In addition to the normal stress differences and manifestations of non-Newtonian flow mentioned above, all the viscoelastic experiments described in Section C above (stress relaxation after sudden strain, stress relaxation after cessation of steady flow, creep, creep recovery, behavior in oscillating deformations, etc.) become much more complicated when deformations are large. A few examples will be shown in Chapter 2 and others in Chapters 13 and 17. 

The writer is particularly indebted to Professors R. B. Bird, N. W. Tschoegl, and M. W. Johnson for advice in formulating this chapter. 

The various moduli and compliances which have been introduced for infinitesimal deformations are summarized in Table l-I, including the glasslike modulus and compliance which will be explained in Chapter 2. All moduli have the dimensions of stress (units usually dynes/cm or alternatively Pa) all compliances have the dimensions of reciprocal stress (usually cm /dyne or Pa )- Certain quantities in any row of the table can be interrelated by equations of the form of 55 and 58. The symbols follow rather closely the recommendations of a Committee of the Society of Rheology.  

Deformation Simple Shear Bulk Compression Simple Extension Bulk Longitudinal 

SUPPLEMENT 2 COMPLEX NOTATION FOR DYNAMIC (SINUSOIDAL) STRESS-STRAIN RELATIONS 

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The results of two catalyst life tests, one in a laboratory autoclave and the other in the pilot plant (Run E-3), are shown in Rg. 1. Both tests were conducted with the same catalyst at a total pressure of 5.27 MPa. (750 psig.) and a temperature of 523X The catalyst concentration was 15 wt.% In the laboratory autoclave and 25 wt. % in the pilot plant the gas hourly space velocities were 5000 and 10,000 liter/kg.(cat.),hr., respectively. Rg. 1 is a plot of a normalized rate constant, k /k (0, versus the time onstream. The normalized rate constant was formed by dividing the rate constant at any time, k, by the rate constant at the beginning of the experiment. k (0). The correspondence between the two sets of data is reasonably good, suggesting that catalyst deactivation is a time-dependent phenomenon that behaves simileu ly in the two reactor systems. 

The concept of intramolecular vibrational energy redistribution (IVR) can be formulated from both time-dependent and time-independent viewpoints (Li et al., 1992 Sibert et al., 1984a). IVR is often viewed as an explicitly time-dependent phenomenon, in which a nonstationary superposition state, as described above, is initially prepared and evolves in time. Energy flows out of the initially excited zero-order mode, which may be localized in one part of the molecule, to other zero-order modes and, consequently, other parts of the molecule. However, delocalized zero-order modes are also possible. The nonstationary state initially prepared is often referred to as the bright state, as it carries oscillator strength for the spectroscopic transition of interest, and IVR results in the flow of amplitude into the manifold of so-called dark states that are not excited directly. It is of interest to understand what physical interactions couple different zero-order modes, allowing energy to flow between them. A particular type of superposition state that has received considerable study are A/-H local modes (overtones), where M is a heavy atom (Child and Halonen, 1984 Hayward and Henry, 1975 Watson et al., 1981). 

The influence which relaxation processes and other time-dependent effects have on the Mossbauer spectra is a function of the relative timescales associated with the effects themselves and the timescales of the nuclear transitions and hyperfine interactions. In order to interpret the Mossbauer spectra in terms of time-dependent effects each type of relaxation phenomenon must be considered in the context of the appropriate timescale. Thus, as with the hyperfine interactions themselves, the time dependence results from the interplay of both nuclear and extra-nuclear factors. 

Many explosives, which have solid carbon as a detonation product, exhibit behavior that is not described adequately without including some time-dependent phenomenon, such as diffusion-controlled carbon deposition or some other kinetic behavior of the detonation products. A time-dependent carbon deposition is the only process known that could account for the large energy deficits required by the build-up model. The observed velocity constancy and large C-J pressure variations can be reproduced by the time-dependent carbon deposition mechanism. 

Thixotropy and Other Time Effects. In addition to the nonideal behavior described, many fluids exhibit time-dependent effects. Some fluids increase in viscosity (rheopexy) or decrease in viscosity (thixotropy) with time when sheared at a constant shear rate. These effects can occur in fluids with or without yield values. Rheopexy is a rare phenomenon, but thixotropic fluids are common. Examples of thixotropic materials are starch pastes, gelatin, mayoimaise, drilling muds, and latex paints. The thixotropic effect is shown in Figure 5, where the curves are for a specimen exposed first to increasing and then to decreasing shear rates. Because of the decrease in viscosity with time as weU as shear rate, the up-and-down flow curves do not superimpose. Instead, they form a hysteresis loop, often called a thixotropic loop. Because flow curves for thixotropic or rheopectic Hquids depend on the shear history of the sample, different curves for the same material can be obtained, depending on the experimental procedure. 

The time dependence of the various species concentrations will depend on the relative magnitudes of the four rate constants. In some cases the curves will involve a simple exponential rise to an asymptote, as is the case for irreversible reactions. In other cases the possibility of overshoot (exists, as indicated in Figure 5.2. Whether or not this phenomenon will occur depends on the relative magnitudes of the rate constants and the initial conditions. However, the fact that both roots of equation 5.2.34 must be real requires that there be only one maximum in the curve for R(t) or S(t). 

The excited state of a molecule can last for some time or there can be an immediate return to the ground state. One useful way to think of this phenomenon is as a time-dependent statistical one. Most people are familiar with the Gaussian distribution used in describing errors in measurement. There is no time dependence implied in that distribution. A time-dependent statistical argument is more related to If I wait long enough it will happen view of a process. Fluorescence decay is not the only chemically important, time-dependent process, of course. Other examples are chemical reactions and radioactive decay. 

All types of time evolution are present in dynamical solvation effects. It is difficult, and perhaps not convenient, to define a general formulation of the Hamiltonian which can be used to treat all the possible cases. It is better to treat separately more homogeneous families of phenomena. The usual classification into three main types adiabatic, impulsive and oscillatory, may be of some help. The time dependence of the phenomenon may remain in the solute, and this will be the main case in our survey, but also in the solvent in both cases the coupling will oblige us to consider the dynamic behaviour of the whole system. We shall limit ourselves here to a selection of phenomena which will be considered in the following contributions for which extensions of the basic equilibrium QM approach are used, mainly phenomena related to spectroscopy. Other phenomena will be considered in the next section. 

Stretched exponential relaxation is a fascinating phenomenon, because it describes the equilibration of a very wide class of disordered materials. The form was first observed by Kohlrausch in 1847, in the time-dependent decay of the electric charge stored on a glass surface, which is caused by the dielectric relaxation of the glass. The same decay is observed below the glass transition temperature of many oxide and polymeric glasses, as well as spin glasses and other disordered systems. 

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