The Longest Relaxation Time

Further, we can replace the ratio f /A in the left hand side with the tangential shear stress a. The most interesting result we can draw is that the coefficient in (12.4) is nothing else but the viscosity of the polymer liquid rj. Thus, 

The time r, when the polymer s type of response to the stress changes, is called the longest relaxation time. 

Now you can see the microscopic meaning of r, which we introduced as a typical relaxation time of effective cross-links. The cross-links decay 

8 Successive stages of a polymer chain leaving its original tube. 

This relationship can help us to learn about the viscosity of a polymer solution. In particular, we can use it to figure out how the viscosity depends on the number of monomer units At in a chain (in the limit N 1). Presumably, we need to know first how E and t depend on N. So we should venture a little investigation. Let s consider E and r separately, and concentrate on the case of a pol3Tner melt, to make it easier. In principle, the same sort of logic should apply to concentrated and semi-dilute solutions. 

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This result shows that the highest modes of response have the shortest relaxation times and influence the initial response of the sample. Conversely, the longest relaxation time is ti, which we can identify with the terminal behavior of the sample. For example, in Fig. 3.9 the final collapse of the modulus at long times occurs at Ti. An example will show how we can use this idea. 

The longest relaxation time of the polymer chain in a tube, however, may be defined as the time needed by the chain to diffuse its own size, ro Rg /D = as a matter of fact this is the time... 

Combining Eqs. (24) and (25) gives the longest relaxation time for the Rouse model as ... 

In analogy with the Rouse model, the longest relaxation time (Xj) according to the Zimm model can again be put into a form which does not depend on N [44] ... 

MD simulations of melts of C44H90, based on classic techniques in continuous space, have been reported recently using united atom [146] and fully atomistic [145] representations of the chain. Time in the conventional MD simulations is expressed in seconds, whereas time in the simulation of the coarse-grained chains on the 2nnd lattice is expressed in MC steps. Nevertheless, a few comparisons are possible via the longest relaxation time, rr, deduced from the decorrelation of the end-to-end vector ... 

Fig. 3. Schematic of Chambon-Winter gel spectrum. The longest relaxation time diverges to infinity. The relaxation time X0 marks the crossover to the short-time behavior, which depends on the material. The depicted case corresponds to a low-molecular-weight precursor (crossover to glass transition region)...

In the close vicinity of the gel point, pc — p 1, the longest relaxation time diverges in a power law on both sides of the gel point (Fig. 4)... 

Fig. 4. Schematic of the divergence of the longest relaxation time as the liquid-solid transition is approached from either side...

Only the value of the relaxation exponent is needed. The critical exponent a of the longest relaxation time (compare Eqs. 1-6 and 1-7) is therefore on an equal footing with the critical exponent of the viscosity ... 

For the relaxation of the solid near the gel point, the critical gel may serve as a reference state. The long time asymptote of G(t) of the nearly critical gel, the equilibrium modulus Ge, intersects the G(t) = St n of the critical gel at a characteristic time (Fig. 6) which we will define as the longest relaxation time of the nearly critical gel [18]... 

Fig. 6. Evaluation of the longest relaxation time for a sample beyond the gel point, p > pc intersect of horizontal line for Ge with the power law of the critical gel, St n...

Fig. 11. Schematic of relaxation time spectrum of the critical gel of PBD 44 (Mw = 44 000). The entanglement and glass transition is governed by the precursor s BSW-spectrum, while the CW spectrum describes the longer modes due to the crosslinking [60]. denotes the longest relaxation time of PBD44 before crosslinking...

The liquid-solid transition for these systems seems to have the same features as for chemical gelation, namely divergence of the longest relaxation time and power law spectrum with negative exponent. 

The divergence of the longest relaxation time does not perturb the measurement. In comparison, steady state properties (the steady shear viscosity, for instance) would probe an integral over all relaxation modes and, hence, fail near the gel point. 

The start-up time does not depend on the longest relaxation time of the material even if it is orders of magnitude larger than the period In/co [115]. This is an important prerequisite for an experiment near LST. 

Spectra with a positive exponent may be explored for the ideal case of power law relaxation over all times up to the longest relaxation time, 2max ... 

The damping material does not have to be a critical gel. Many applications do not require extra low damping frequencies. The lowest vibration damping frequency comin determines the longest relaxation time, Amax. A suitable damping material would be crosslinked beyond the gel point, with a 2max of about... 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

In the steady flow regime, only the longest relaxation time is applicable, so equation (16) becomes... 

Chain branching affects the viscosity, the longest relaxation time, and the steady-state compliance and therefore influences creep and stress relaxation (19,163- 167). The effect is difficult to quantify because the length... 

The viscosity relates to the longest relaxation time in a system. If we consider Rouse diffusion along the tube with a Rouse diffusion coefficient DJ l/ NQ) then an initial tube configuration is completely forgotten when the mean-square displacement along the tube fulfils (r (t))tube=(contour length ly. Thus, for the longest relaxation time, we obtain ... 

Relaxation to steady state of the network is multiexponential, and now we are interested in estimate of the longest relaxation time r ... 

An estimate of the longest relaxation time can be obtained by applying the perturbation theory for linear operators to the degenerated case of the zero eigenvalue of the matrix K. We have. ..,Ay i)-I-fcy Q-I-o(fc,),... 

The internal modes are periodic functions of bead position along the chain the characteristic wavelength of internal mode i is N/i. Thus the lower modes govern the large scale motions and have the longest relaxation times. 

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