Superposition, Law

Although dielectric response data often leads to temperature-independent y/ s (so that the time-temperature or liequency-tanperature superposition law holds), this is by no means always the case (Jonscher [1983]). Further, conductive-system response, as in ionic hopping conductors, often leads to appreciable temperature dependence of yf. Surprisingly, y/s and y/z temperature responses, when apparent, are usually found to be quite different, with y/ increasing with increasing temperature and y/z decreasing. 

A creep test can be carried out with an imposed stress, then after a time have its stress suddenly changed to a new value and have the test continued. This type of change in loading allows the creep curve to be predicted. The simple law referred to earlier as the Boltzmann superposition principle, hold for most materials, so that their creep curves can thus be predicted. 

Rubbery materials beyond the gel point have been studied extensively. A long time ago, Thirion and Chasset [9] recognized that the relaxation pattern of a stress r under static conditions can be approximated by the superposition of a power law region and a constant limiting stress rq at infinite time ... 

A quantum algorithm can be seen as the controlled time evolution of a physical system obeying the laws of quantum mechanics. It is therefore of utmost importance that each qubit may be coherently manipulated, between arbitrary superpositions, via the application of external stimuli. Furthermore, all these manipulations must take place well before its quantum wave function, thus the information it encodes, is corrupted by the interaction with external perturbations. The need to properly isolate qubits but, at the same time, to rapidly... 

In order to determine the force between plates as a function of their separation, one would have to perform a series of simulations with different wall separations and with the chemical potential of the fluid fixed at the bulk value. This is technically feasible, but very computationally intensive [42]. The qualitative behavior of the force law can, however, be estimated from the density profile of a fluid at a single wall using the wall sum rule and a superposition approximation [31,43]. The basic idea is that the density profile [denoted pH(z)] of a fluid between two walls at a separation H can be obtained from the density profile [denoted pj (z) of the same fluid at a single wall using... 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

Time-temperature superposition is frequently applied to the creep of thermoplastics. As mentioned above, a simple power law equation has proved to be useful in the modelling of the creep of thermoplastics. However, for many polymers the early stages of creep are associated with a physical relaxation process in which the compliance (D t)) changes progressively from a lower limit (Du) to an upper limit (DR). The rate of change in compliance is related to a characteristic relaxation time (x) by the equation ... 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

Moreover, we note that recently in reconstructing relaxation times via the time-temperature superposition principle using double quantum nuclear magnetic resonance (DQ-NMR) the and power laws were invoked without giving the spatial information of NSE [75]. 

Equation (6.1.4) asserts that the volumetric flow rate is a superposition of two components. They are the electro-osmotic component proportional to the electric field intensity (voltage) with the proportionality factor u> and the filtrational Darcy s component proportional to —P with the hydraulic permeability factor i>. Teorell assumed both w and t> constant. Finally another equation, crucial for Teorell s model, was postulated for the dynamics of instantaneous electric resistance of the filter R(t). Teorell assumed a relaxation law of the type... 

When a mixture containing compounds with known spectra is analysed, it is possible to determine the mixture s composition. According to the additivity law (equation (11.10)), the spectrum of the mixture corresponds to the weighted superposition of each of the individual constituents. The following section recalls the classical calculation method. It is rarely used as such because it is typically incorporated into software. 

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