Molecular model relaxation spectrum

Garda-Franco, C. A., Mead, D. W. Rheological and molecular characterization of linear backbone flexible polymers with the Cole-Cole model relaxation spectrum. Rheol. Acta (1999) 38, pp. 34—47... 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

This theory was able to account for both the molecular-weight scaling of the dynamic quantities Dg, r, and x as well as for the shape of the relaxation spectrum (see Fig. 5) apart from one important feature - the constant v in the leading exponential behaviour that multiplies the dimensionless arm molecular weight needed to be adjusted. This can be understood as follows. The prediction of the tube model for the plateau modulus from the stress Eq. (7) is... 

The product r o is the characteristic relaxation time xq of the terminal region. In terms of molecular models, this time scales as the longest relaxation time. In terms of the distribution of relaxation times H(x), Xo is the "weight-average relaxation time" which is the average relaxation time related to the second order moment of the relaxation spectrum ... 

While empirically useful, the temporary network model gives no indiction of the relationship between the relaxation spectrum and the molecular relaxation processes. In the next sections, this deficiency is addressed by returning to the bead-spring models of Fig. 3-5. 

Recent numerical experiments by the method of molecular dynamics have shown that, for a chain model consisting of particles joined by ideally rigid bonds, the Van der Waals interactions of chain units cause only a little change in the dependence of relaxation times on the wave vector of normal modes of motions, i.e. in the character and shape of the relaxation spectrum. It was found that for the model chain the important relationship... 

We have seen that vibrational relaxation rates can be evaluated analytically for the simple model of a hannonic oscillator coupled linearly to a harmonic bath. Such model may represent a reasonable approximation to physical reality if the frequency of the oscillator under study, that is the mode that can be excited and monitored, is well embedded within the spectrum of bath modes. However, many processes ofinterest involve molecular vibrations whose frequencies are higherthan the solvent Debye frequency. In this case the linear coupling rate (13.35) vanishes, reflecting the fact that in a linear coupling model relaxation cannot take place in the absence of modes that can absorb the dissipated energy. The harmonic Hamiltonian... 

We assume that the above-indicated drawback of the present model can be avoided (or at least reduced) if a new paradigm [mentioned below in Section X.B.4(ii)] of the molecular model will be constructed. In our opinion, this drawback of the present model is stipulated by the following. In view of Eq. (11) the libration lifetime T0r is determined by the experimental Debye relaxation time td, so variation of Tor cannot be used for other corrections of the calculated spectra. In the proposed new paradigm it is desirable to use Tor for the latter purpose, while a correct describing of the low-frequency Debye spectrum is assumed to be reached by variation of additional parameter(s). 

Comparison of the forms of equations 58 to 61 with equations 21 to 23 of Chapter 9 and equations 23 and 24 of Chapter 3 shows that the time and frequency dependence correspond to a generalized Maxwell model as in the Rouse theory and its various modifications, but here the spring constants (or discrete contributions to the relaxation spectrum) are not necessarily all equal they are proportional to the concentrations of the various types of strands, v e. The molecular weight does not enter explicitly, but it may be expected that the higher the molecular weight the greater the concentrations of strands which find it difficult to leave the network and hence have large values of the time parameter... 

The condition of infinite differentiability can be dropped obtaining a much weaker requirements for k [ 101 ]. In order to express the reduced relaxation function k(t), a discrete relaxation spectrum, whose form was derived by various molecular models [161], is generally used. Formally,... 

The behavior predicted by Eqs. (39) for values of Ei and E2 appropriate to the glass transition in an amorphous polymer (compare Figure 14.1) is shown in Figure 14.6. The model accounts for the qualitative features of experimentally observed transitions, namely a step-like drop in modulus as to decreases below r , and a characteristic peak in tan <5. However, more complex models involving many relaxation times, that is, a discrete or continuous relaxation time spectrum, are necessary if more quantitative agreement with experiment is to be obtained (for an example of a discrete relaxation time spectrum derived from a molecular model, see Section 14.3.3) [10,11[. 

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

As mentioned above, the cross peak intensities from NOESY spectra taken at long mixing times caimot be related in a simple and direct way to distances between two protons due to spin diffusion effects that mask the actual proton distances. A possibiUty to extract such information is provided by relaxation matrix analysis that accounts for all dipolar interactions of a given proton and hence takes spin diffusion effects explicitly into consideration. Several computational procedures have been developed which iteratively back-calculate an experimental NOESY spectrum, starting from a certain molecular model that is altered in many cycles of the iteration process to fit best the experimental NOESY data. In each cycle, the calculated structures are refined by restrained molecular dynamics and free energy minimization [42,43]. 

As will be shown in Chapter 6 there are molecular theories of polymer behavior that lead to relaxation moduli of the form of Eq. 4.16 with N=oo, where the parameters are precisely specified in terms of a few measurable parameters. Small et al. [4] used these theories to derive a discrete spectrum for a linear, monodisperse, entangled polymer by considering three relaxation processes, each described by a different molecular model. 

Finally, we note that the use of a discrete spectrum involves the use of an empirical equation to fit data. The resulting constants have no physical significance, and the resulting function will have local features that are artifacts of the model and do not reflect the structure of the polymer. This can cause trouble, for example if this function is used to infer the molecular weight distribution. For such a purpose, it may thus be preferable to work with a continuous spectrum function such as H(t) or L(t). Honerkamp and Weese [42] reported a nonlinear regression with regularization technique (NLRG) that takes into account noise in the data and yields a smooth relaxation spectrum. 

Janzen foimd that he could relate the value of a to the molecular weight distribution. He also found that the characteristic time of this model was equal to an average relaxation time defined in terms of the log-Gaussian (log-normal) relaxation spectrum. 

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

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