Rouse spectrum

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

The form of the spectrum is the same for Rouse chains fixed at one or both ends, although t, in these cases is no longer given by Eq. (4.32). This is also the spectrum at long times for various models with explicit forms for the local motions (91,92). The Rouse spectrum can also be represented by a continuous function for times which are small compared to xx. For N- 00 ... 

The exponent b is a matter of some current dispute, being 2 in nearly all studies but appearing to range as high as 3 in a few instances (Section 5.4.4). In any case, the explicit molecular weight dependence is lost, and concomitantly the concentration dependence is increased (173). If all relaxation times had been scaled up by the same factor (the relative spacings and intensities of the Rouse spectrum being retained), the Rouse forms would have been preserved for M > (Mc)soln. 

If the terminal spectrum were simply a shifted Rouse spectrum, t would be the same as that given in Eq. (6.1) but tw and tw/t would be different ... 

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. 

The reptation model (225) also appears to produce a Rouse spectrum at long times. In order to renew its configurations a chain must diffuse out of the tunnel defined by the fixed obstacles along its length. De Gennes calculates the autocorrelation function for the end-separation vector, obtaining... 

The longest unshifted Rouse relaxation time is only proportional to M2. The extra power of M in the relaxation time is consistent with the circulation contribution to t]0 which was described in the preceeding section. This modification together with Eq.(6.57) suggests a simple Rouse spectrum shifted to longer times. 

Thirion (239a) has suggested that the plateau and terminal regions are the result of diffuse interchain interactions in a viscoelastic medium. He obtains a modified Rouse spectrum by replacing the subchain frictional coefficient by a time dependent micro-memory function. The theory is partly phenomenological since the memory function is not specified. However, reasonable choices lead to forms for G (co) and G"(o>) which are similar to those observed experimentally. 

The beads represent entanglement sites which are distributed uniformly along the chain contour the frictional coefficients increase rapidly with distance from the chain ends. The spring constant also depends on contour position, being governed by the mean equilibrium distance of that position from the center of gravity. The resulting spectrum is narrower than the Rouse spectrum, and for E > 1 ... 

Parameter of order unity in the continuous form of the Rouse spectrum. Concentration exponent in Jt° and xm correlations. 

Very recently Monnerie31 has described Monte Carlo calculations for a quite realistic lattice model nonintersecting chains confined to tetrahedral lattices performing local stochastic processes involving the simultaneous motion of three or four bonds. Without volume exclusion and with no correlations in the orientation probabilities of neighboring bonds, the model has also been treated analytically,32 with application to the fluorescence depolarization experiment. It is easy to show that this model also leads to the long-time Rouse spectrum. 

By now it may have dawned on the reader that the long-time Rouse spectrum (i.e., proportionality of xp to p 2) is to be expected for any chain model in which the correlation lengths for both equilibrium conformations and frictional processes are small compared to the chain dimensions (and thus to the wavelength of the slow normal modes). A possible exception is that of the continuous wormlike chain of invariant contour length, which has been studied by Saito, Takahashi, and Yunoki.33 In this latter case, the low-frequency spectrum makes xp proportional to p A, which resembles our special one-dimensional model in the limit 1 — p 1. 

Both experimentally and theoretically the relaxation spectrum for entangled stars differs quite substantially in form from the Rouse spectrum. Thus, at first sight it seems remarkable that the values of should resemble so closely the Rouse model predictions. However, the reason is quite simple. The Rouse form (J c M ) is expected... 

Its fonr parameters Ge, S, Xmax, and n all depend on the bond probability p. In the liquid below GP and at GP, the equilibrium modulus is equal to zero, Ge = 0. For n = 0.5 and Ge = 0, this spectrum reduces to the well-known Rouse spectrum. It is remarkable that depending on the value of A,max, the Rouse spectrum describes a viscoelastic liquid that includes the Newtonian liquid (A,max 0) and the critical gel (A.max oo) as limiting cases. 

The shortest relaxation time in the Rouse-spectrum depends on the choice of the Rouse-sequence and follows, for m = Vr —1, from Eqs. (6.28) and (6.38)... 

In the range of time or frequency where the mecheinical response of an amorphous polymer varies from that of a soft rubber to that of a hard glass, the relaxation spectrum (which determines time or frequency dependence) has been obtained experimentally for about two dozen polymers Its form depends somewhat on chemical structure, but these relations are still not well understood. At the low frequency or long time end, it can be approximated by the Rouse spectrum and a match here can furnish values of the monomeric friction coefficient, a measure of the location of the transition zone on the time or frequency scale. 

The division of spatial scales separating the very high-frequency, local Rouse processes, with time scales less than t, from the longitudinal Rouse modes, with time scales greater than T, has been represented by Likhtman and McLeish [18] using an approximate jragmented Rouse spectrum to calculate the Rouse relaxation modulus Gj j of chain i ... 

Thimm, W., Friedrich, Ch., Marth, M. On the Rouse spectrum and the determination of the molecular weight distribution from rheological data. /. Rheol. (2000) 44, pp. 429-438... 

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