Volume relaxation function

The relaxation function has been calculated and is compared with experimental data in Figure 5.16. The agreement between the model and the data is reasonable. The storage and loss moduli for a polystyrene latex have also been measured and compared to the model for the relaxation spectra. The data was gathered for a dispersion in 10 2M sodium chloride at a volume fraction of 0.35 is shown in Figure 5.20. 

An experimental and theoretical study of the degassing of an LDPE high-density foam is presented. Measurements of the mass, dimensions, and density as a function of storage time are reported. A geometrical model is described to represent the basic mass transport and volume relaxation processes in a cellular system. Model predictions were compared with experimental results. 12 refs. 

Figure 1 The shear stress relaxation function, C(t), obtained from a molecular dynamics simulation of500 SRP spheres at a reduced temperature of 1.0 and effective volume fraction of 0.45. Note that n = 144 and 1152 (from Equation (1)) cases are superimposable with the analytic function of Equation (4) ( Algebraic on the figure) for short times, t (or nt here)...

The existence of molecules often creates permanent intramolecular optical anisotropy. The optical anisotropy of the liquid is then due to fluctuations in the orientations of the molecules or molecular subunits. If we assign a symmetric traceless anisotropy tensor a to each molecule or molecular subunit in the scattering volume, then the relaxation function for collective optical anisotropy fluctuations can be expressed as... 

The best explanation of the good results for peptide syntheses in ice-water mixtures are based on the freeze-concentration-model, which just provides for a volume-reducing function for the ice while the liquid aqueous part is still the only relevant phase for the reaction. All observed enhancements of reaction rate would then have to be attributed to an increase in effective concentration. H-NMR relaxation time measurements have been used to determine the amount of unfrozen water in partially frozen systems, thus quantifying the extent of the freeze-concentration effect (Ullmann, 1997). Comparative studies in ice and at room temperature verify the importance of freeze-concentration which, however, is not sufficient for a complete understanding of the observed effects. 

The presence of a temporary network structure manifests itself through the first factor in the right hand side of Eq. 17. It comes from the h (t) component of the relaxation function. The second factor results from fast non-isotropic motions of monomeric imits which give rise to the (t r(t) function defined in equations (6) and (9). It must be noticed that the reference temperature Ti = Tg(,xi,2) +50 which appears in NMR properties, is close to the reference temperature Tj introduced from viscoelastic measiurements [11]. The second factor reflects a free volume effect it depends necessarily on an expansion coefficient which is about equal to 10-3K. 

However, many experiments observe that the amount of stress relaxed at the time scale of the reptation time ts of shorter chains is much larger than the volume fraction of short chains. This is shown in Fig. 9.27(a), where the loss moduli of binary blends are compared with the predictions of Eq. (9.92) using the Doi-Edwards reptation model predictions for G(t) [Eq. (9.21)] for the Gl(0 Gs(t) relaxation functions. Recall from Section 7.6.5 that the magnitude of G" lj) directly reflects the amount of relaxation occurring at each frequency u. Hence, Eq. (9.92) strongly underestimates the amount of relaxation occurring when the short chains relax [the high-frequency peak in G"((u)j. 

Pioneering work of Tool [1946, 1948] on inorganic glasses using dilatometry indicated that volume relaxation after a temperature jump from an initial equilibrium state could not be described simply by a kinetic model in which the relaxation time T was solely dependent on the temperature. Tool therefore proposed that r was also a function of the structure of the glass, and this led to the definition of the Active temperature Tf. 

This study shows that none of the various forms of relaxation function used to describe ageing are completely satisfactoiy and TRS is inappropriate. Correlation between results, (Figure 9) indicates the inherent connectivity between the processes. Curro et al (24,25) have studied the change in density fluctuation with temperature and annealing time for PMMA (25) and compared it with specific volume data. Positron annihilation data on PMMA (27,28) has been interpreted in terms of free volume. For a distribution of hole sizes there will exist many decaying exponentials each with a different characteristic lifetime. The composite of these many exponentials can itself be approximated to an exponential, and it is this decay constant that is used to represent the mean lifetime, and therefore mean hole size. 

Fig. 27 Relaxation function R t) (filled circles) and/exch(0 (open squares) of C24H49PEO5 in a log-ln plot at room temperature. The line represents the single exponential fit for R(t). Inset. Concentration dependence of R(t) at 0.25% (diamonds), 0.5% (squares), and 1% (circ/es) polymer volume fraction. [103]. Reproduced by permission of The Royal Society of Chemistry...

Thus, the distribution of free volume, or the LL environments, and the distributed material property affected, such as the local fluidity or relaxation behavior, reflects the variations in the local atomic packing discussed in Section 1.3. Such property variations have long been of interest (Scherer 1990). For the case presented above, in which the viscosity at T2 needs a certain relaxation time from that of Ti, the change in the time-dependent property, p (e.g., viscosity), is given by a relaxation function Mp(t),... 

To assess how the presence of specific interactions affects the physical aging behavior, Robertson and Wilkes carried out volume relaxation measurements on PMMA/SAN blends as a function of blend composition (Robertson and Wilkes 2001). In this system, there are no attractive interactions between the components, and miscibility is a result of the so-called copolymer repulsion effect. The dependence of the volume relaxation rate on blend composition was found to be consistent with the enthalpic measurements of Mijovic et al. (1989) on the same system both values of volume and enthalpy relaxation rates were intermediate for the blends compared to the pure components and composition dependent except for... 

The shape of the curves is highly nonexponential. One obvious reason is that the rate of relaxation depends on the free volume that is changing during the volume relaxation. However, attempts to explain the observed relaxation functions in terms of a single volume-dependent relaxation time have not been successful. Because equilibrium liquids can be obtained at... 

Khater et al. calculated the time dependence of TRM for an assembly of noninteracting uniaxial particles with a Poisson volume distribution function [P(V) dV= (4V/VQ)exp(-2VIVg) dV]. Transforming the volume distribution in a distribution over relaxation times t, described... 

The model has some limits, mainly as far as the calculation of in the blocked state ( t j = M, I2>K for uniaxial symmetry) is concerned. The calculation does not account for vibrations of the magnetic moment m in the potential well, that is, for transverse relaxation. This leads to an abrupt variation of x close to Tg) (for a single particle the variation should be steplike). Such a very rapid variation is never observed experimentally, neither for samples with a narrow volume distribution. In our opinion it is important to take into account the effect of transverse relaxation, which should smooth the x variation below Tg. Moreover, the volume distribution function assumed in the Gittleman calculation is not realistic. Finally, interactions between particles, almost always present in real systems, are neglected. 

Tensile experiments are often easy to perform but have the disadvantage that simultaneous changes in both shape and volume make the behavior more difficult to interpret on a molecular basis moreover, perceptible volume changes may significantly modify the relaxation functions as mentioned above. However, in polymeric systems, in certain broad ranges of time scale, K t) is often greater than G t) by two orders of magnitude or more. This condition corresponds to a Poisson s ratio ju very close to (e.g.,.0.499), and in this case equations 51 and 55 become... 

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