Relaxation longest

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] ... 

All these methods have in common that the receptor is not detected so that no size limit applies. Relaxation-based exchange-transferred experiments actually show best performance for longest correlation times, i.e. very large receptors. 

The longest mode (p=l) should be identical to the motion of the chain. The fundamental correctness of the model for dilute solutions has been shown by Ferry [74], Ferry and co-workers [39,75] have shown that,in concentrated solutions, the formation of a polymeric network leads to a shift of the characteristic relaxation time A,0 (X0=l/ ycrit i.e. the critical shear rate where r becomes a function of y). It has been proposed that this time constant is related to the motion of the polymeric chain between two coupling points. 

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 ... 

CC- critical exponent for longest relaxation time before LST... 

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...

With increasing distance from the gel point, the simplicity of the critical state will be lost gradually. However, there is a region near the gel point in which the spectrum still is very closely related to the spectrum at the gel point itself, H(A,pc). The most important difference is the finite longest relaxation time which cuts off the spectrum. Specific cut-off functions have been proposed by Martin et al. [13] for the spectrum and by Martin et al. [13], Friedrich et al. [14], and Adolf and Martin [15] for the relaxation function G(t,pc). Sufficiently close to the gel point, p — pc <4 1, the specific cut-off function of the spectrum is of minor importance. The problem becomes interesting further away from the gel point. More experimental data are needed for testing these relations. 

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...

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

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 diverging longest relaxation time, Eq. 1-6, sets the upper limit of the integral. The solid (gel) contribution is represented by Ge. The crossover to any specific short-time behavior for A < A0 is neglected here, since we are mostly concerned with the long-time behavior. 

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. 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

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... 

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