Characteristic relaxation

Figure IV-10 illustrates how F may vary with film pressure in a very complicated way although the v-a plots are relatively unstructured. The results correlated more with variations in film elasticity than with its viscosity and were explained qualitatively in terms of successive film structures with varying degrees of hydrogen bonding to the water substrate and varying degrees of structural regularity. Note the sensitivity of k to frequency a detailed study of the dispersion of k should give information about the characteristic relaxation times of various film structures.

Dielectric Behavior of Adsorbed Water. Determination of the dielectric absorption of adsorbed water can yield conclusions similar to those from proton NMR studies and there is a considerable, although older literature on the subject. Figure XVI-7 illustrates how the dielectric constant for adsorbed water varies with the frequency used as well as with the degree of surface coverage. A characteristic relaxation time r can be estimated... 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

The simplest method that keeps the temperature of a system constant during an MD simulation is to rescale the velocities at each time step by a factor of (To/T) -, where T is the current instantaneous temperature [defined in Eq. (24)] and Tq is the desired temperamre. This method is commonly used in the equilibration phase of many MD simulations and has also been suggested as a means of performing constant temperature molecular dynamics [22]. A further refinement of the velocity-rescaling approach was proposed by Berendsen et al. [24], who used velocity rescaling to couple the system to a heat bath at a temperature Tq. Since heat coupling has a characteristic relaxation time, each velocity V is scaled by a factor X, defined as... 

In this expression. Ait is the size of the integration time step, Xj is a characteristic relaxation time, and T is the instantaneous temperature. In the simulation of water, they found a relaxation time of Xj = 0.4 ps to be appropriate. However, this method does not correspond exactly to the canonical ensemble. 

Here, At is the size of the time step, Tp is a characteristic relaxation time, and Pg is the pressure of the external constant-pressure bath. The instantaneous pressure can be calculated as follows ... 

Example consider polystyrene with M = 245,000 and welded to itself at 118°C, the characteristic relaxation times are te 10 s, tro 21 min and Ty 1860 min [15], At these relaxation times, the respective average monomer interdiffusion distance is as follows [1 j ... 

A representative measure of rubbery elasticity of a material may be two quantities dimensionless ratio (ct/t) and characteristic relaxation time 9 = ct/2ty. According to the data of works [37, 38] when fibers are introduced into a melt, ct/t increases (i.e. normal stresses grow faster than stresses) and 0 also increases on a large scale, by 102-103 times. However, discussing in this relation the papers published earlier, we noted in the paper cited that the data were published according to which if fibers were used as a filler (as in work [37]), 9 indeed increased [39], but if a filler represented disperse particles of the type Ti02 or CaC03, the value of 0 decreased [40],... 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

The longest mode (p=l) should be identical to the motion of the chain. The fundamental correctness of the model for dilute solutions has been shown by Ferry [74], Ferry and co-workers [39,75] have shown that,in concentrated solutions, the formation of a polymeric network leads to a shift of the characteristic relaxation time A,0 (X0=l/ ycrit i.e. the critical shear rate where r becomes a function of y). It has been proposed that this time constant is related to the motion of the polymeric chain between two coupling points. 

We can perform spatially resolved Carr-Purcell-Meiboom-Gill (CPMG) experiments, and then, for each voxel, use magnetization intensities at the echo times to estimate the corresponding number density function, P(t), which represents the amount of fluid associated with the characteristic relaxation time t. The corresponding intrinsic magnetization for the voxel, M0, is calculated by... 

We represent the NMR relaxation distribution by the continuous number density function, P( t), of characteristic relaxation time t. Our measurements correspond to a series of CPMG echoes, represented by... 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

The shape of the curves for the dilational modulus (Figures 7 and 8) suggests a single relaxation mechanism, probably the unfolding of the demulsifier molecules at the interface. The frequency peak in the e"(f) plot is a measure of the characteristic relaxation time. A shorter relaxation time, by inducing faster film drainage, increases demulsification efficiency. 

The dynamic response data for PI and P2 (Figure 7) are similar. They are, however, quite different from that of 0P1 (Figure 8). The characteristic relaxation times for PI and P2 are 50 and 69 seconds respectively, whereas with 0P1 it is 158 seconds. This indicates that with PI and P2, the oil-water interface will have much shorter response time leading to an improved demulsification effectiveness. 

Unlike for diffusion, where the characteristic relaxation rate is proportional to Q2, here the fourth power of Q is found. 

Fig. 7. Characteristic relaxation rate for the Rouse relaxation in polyisoprene as a function of momentum transfer. The insert shows the scaling behavior of the dynamic structure factor as a function of the Rouse variable. The different symbols correspond to different Q-values. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

A small square wave modulation of the current is applied in order to disturbe the non-equilibrium steady state of the discharge. The exponential decay of the concentrations leads to characteristic relaxation times, which allow the calculation of rate constants used in the modeling of the deposition mechanism. 

The correlation of deposition rate with disilane concentration and the zero-barrier of the reaction of silane with silylene to disilane lead to the conclusion that the latter reaction is the dominant subsequent pathway following the silane fragmentation. Disilane shows two characteristic relaxation times, the slower being identical with the relaxation time of silane. In conclusion, the formation of... 

Let s consider an SMM, which is magnetized to a magnetization, M0, with a polarizing magnetic field, H. After (fast) removal of the field, the magnetization will decay exponentially with a single characteristic relaxation time (t) ... 

Using the time-dependent aspect of state diagrams, Roos (2003) illustrated the effects of temperature, water activity, or water content on relaxation times and relative rates of mechanical changes in amorphous systems (Figure 36). This diagram can be considered as a type of mobility map, where mobility increases (relaxation time decreases) as temperature and/or water content/activity increases. Le Meste et al. (2002) suggested the establishment of mobility maps for food materials showing characteristic relaxation times for different types of molecular motions as a function of temperature and water content. 

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 ... 

The data has been superimposed by dividing the relaxation function G(t) by G(t = 0), the limiting short time value, and the time has been divided by the characteristic relaxation time Tr. The first feature to notice is that the stress relaxation function overshoots and shows a peak. This is an example of non-linear behaviour. It is related to both the material and the instrumental response (Section 4.5.1). The general shape of the curves (excluding the stress overshoot) can be described using two approaches. 

The other important physical assumption is that the friction is local (hydro-dynamic interactions are screened in the melt [2]) so that D -(N /N)D with the diffusion constant in the melt of an unentangled chain of segments. Now the characteristic relaxation (Rouse) time of an entanglement segment % is just a /D. so that... 

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