Relaxation spectrum Rouse theory

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeavuc [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time Xj, proportional to M2, which can be considered as the Rouse time of an entangled polymer movii in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced Xj by an empirical combination of... 

In a good solvent, where there are excluded-volume effects, G and G" can be fit to the Zimm theory simply by adjusting h downward, for finite Ns (Ferry 1980). Thus, as the solvent quality improves, the relaxation spectrum becomes more Rouse-like (since... 

The Doi-Edwards theory provides expressions for G%, r[, and D that contain two adjustable parameters the friction coefficient and the primitive path step length L. The friction coefficient can be obtained from the relationship between the viscosity and molecular weight in the Rouse theory [Eq. (11.36)] or from the relaxation spectrum discussed below. Moreover, the step length a can be determined from the plateau modulus G%. Actually, according to the Doi-Edwards theory... 

Fig. 2.3. Relaxation spectrum of Ogasa-lmai theory (47) compared with those erf Rouse theory (27) and Zimm theory (29). Strength Gp of p-th relaxation mode is plotted against logarithm erf reduced relaxation time t,/tp. Strength of mode of the longest relaxation time at left side is nkTon each panel...

A relaxation spectrum similar to that of Fig. 4.2 is obtained for the diffusional motion of a local-jump stochastic model of IV+ 1 beads joined by N links each of length b, if a weak correlation in the direction of nearest neighbor links is taken into account for the probability of jumps (US). On the other hand, relaxation spectra similar to that of the Rouse theory (27) are obtained for the above mentioned model or for stochastic models of lattice chain type (i 14-116) without the correlation. Iwata examined the Brownian motion of more realistic models for vinyl polymers and obtained detailed spectra of relaxation times of the diffusional motion 117-119). However, this type of theory has not gone so far as to predict stationary values of the dynamic viscosity at high frequencies. 

Fig. 11.9 Comparison of the measured storage-modulus spectrum (o and ) and that calculated from the Rouse theory (solid line) for sample A. The dashed line indicates the separation of the contributions from the G1 and GIO components. The arrow at 1/ti(1) indicates the frequency that is the reciprocal of the relaxation time of the first Rouse mode of the G1 component calculated from Eq. (7.57) with K = x 10, wheresis the arrow at l/ri(2) indicates the same for the GIO component.

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, G,(t), in equation [7.2.4]. According to the Rouse theory, a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic interaction of a molecule with a solvent while the springs model elastic linkage between the beads. The polymer macromolecule is subdivided into a number of equal segments (submolecules or subchains) within which the equilibrium is supposed to be achieved thus the model does not permit to describe small-scale motions that are smaller in size than the statistical segment. Maximal relaxation time in a spectrum is expressed in terms of macroscopic parameters of the system, which can be easily measured ... 

The retardation spectrum corresponding to the Rouse theory is also discrete, equivalent to a generalized Voight model, but not with equal magnitudes of compliance contributions the pth contribution Jp to the compliance falls ofif rapidly with increasing p, being approximately proportional to the retardation time itself as well as to l/nkT. They have been calculated by Betry the retardation times are spaced between the relaxation times, as shown in the lower panel of Fig. 9-7. From the discrete retardation times and Jp values, the creep compliance can be calculated in principle by equation 18 of Qrapto 3. 

Analysis of the normal modes and transformation of the coordinates leads to another discrete spectrum which differs from that of the Rouse theory only in the values of the relaxation times. Thus, equations 15,16,21, and 26 are unchanged, but the relaxation times are obtained from the eigenvalues of a complicated matrix which includes the pairwise hydrodynamic interactions. - The strength of the latter is measured by either of two parameters ... 

Equation 30 gives the experimentally observed proportionality of [i ] to A/ /2 in a 0-solvent. Again the reduced intrinsic moduli as functions of (otj are independent of M. They are plotted in Fig. 9-8-II. As with the Rouse theory, the results are insensitive to N so long as the frequency is not too high. The slope of on the logarithmic plot at higher frequencies corresponds to a relaxation spectrum in which all Tp except the first few are proportional to instead of to as in the free-draining case. 

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... 

Although equation 3 was obtained from the Rouse theory, the application of reduced variables in Fig. 11-3 is based on a much more general hypothesis, namely, that the modulus contributions are proportional. opT and the relaxation times to whether or not the specific spectrum of times predicted by the Rouse theory is applicable. (In fact, the shape of the curve in Fig. 11-3 deviates considerably from the Rouse theory predictions.) This hypothesis is widely fulfilled but must be carefully examined each time it is used. An important criterion is that the shapes of the original curves at different temperatures must match over a substantial range of frequencies other criteria will be discussed in Section B. 

On this model the submolecule is the shortest length of chain that can undergo relaxation and the motion of segments within the submolecules are ignored. But such motions contribute to the relaxation spectrum for values of m s 5. Thus, we would only expect the Rouse theory to be applicable for m 1 where the equation for Xp reduces to... 

Rouse s theory is the simplest molecular theory of polymer relaxation. A later theory of Zimm [28] does not assume that the velocity of the liquid solvent is unaffected by the movement of the polymer molecules (the free draining approximation ). The hydrodynamic interaction between the moving submolecules is taken into account and this gives a modified relaxation spectrum. 

Figure 6 Inversion of the dielectric loss data for the normal mode spectrum of the type-A polymer polyisoprene. In the inset, the dielectric loss data show the spectrum of normal modes and at higher frepuencies the segmental mode. The distribution of relaxation times shows peaks at times that are characteristic of the different normal modes. However, the obtained peak positions differ from the Rouse theory predictions (shown by vertical lines).

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

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