Plateau modulus concentration dependence

Roughly speaking, the plateau modulus G depends on the number density of the entanglement points, and so the large deviation in the exponent for the viscosity from the prediction may originate from the deviation in the exponent for the stress relaxation time. It can be seen from Figure 7.7a that the slope of the double-logarithmic plot of the relaxation times Tr and vs. concentration at 35 °C is close... 

The problem of concentration dependence of yield stress will be discussed in detail below. Here we only note that (as is shown in Figs 9 and 10) yield stress may change by a few decimal orders while elastic modulus changes only by several in the field of rubbery plateau and, moreover, mainly in the range of high concentrations of a filler. 

Fig. 10. Concentration dependence of a modulus in the region of low-frequency plateau (i.e. yield stress , measured by a dynamic modulus). Dispersion medium poly (butadiene) with M = 1.35 x 105 (7), silicone oil (2) polybutadiene with M = 1 x I04 (3). The points are taken from Ref. [6], The straight line through these points is drawn by the author of the present paper. In the original work the points are connected by a curve in another manner...

Figure 8.26 Double logarithmic plots showing the concentration dependence of both the creep compliance and the relaxation modulus at the plateau. (From Ref. 8.)...

It is expected that the same picture that gives a good account of the linear viscoelastic behavior of polymer melts should also hold for semidilute and concentrated solutions. In the case of semidilute solutions some conclusions can be drawn from sealing arguments (19,3, p. 235). In this way, concentration dependence of the maximum relaxation time tmax the zero shear rate viscosity r Q, and the plateau modulus G% can be obtained, where t is the viscosity of the solvent. The relevant parameters needed to obtain Xmax as a function of concentration are b, c, N, kgT, and Dimensional analysis shows that... 

Finally, by taking into account that G%, the plateau modulus dependence of the concentration is expressed by... 

For high concentrations the dynamics of the chains are the same as in melts. Excluded volume effects and the hydrodynamic interaction are not important. However, both the radius of the tube and the friction coefficient strongly depend on the concentration. Because the depedence of on G is nontrivial, there are no scaling laws relating x ax nnd r o lo the concentration. For the plateau modulus, it was found experimentally that (8)... 

The plateau modulus in entangled polymer solutions has practically the same concentration dependence in all solvents, ... 

While ionomers of many types have been made and characterized [1,2,3], there is little work on the overall relaxation mechanisms. For polymers with low ionic concentrations, there is general agreement on the fundamental relaxation step. The stress relaxes by detachment of an ion pair from one cluster and reattachment to another. For the styrene/methacrylic acid Na salt (ST/-MAA-Na) system, there is a secondary plateau in the relaxation modulus which depends on the ionic content and can be described as a rubbery modulus [4], While a rubbery modulus with stress relaxation due to ionic interchange has been invoked earlier, it does not adequately describe the relaxation curves. A different approach is taken here. 

It is illuminative to point out two characteristics in the transformation. One is the decrease in the plateau modulus (denoted by for the blend solution) with dilution. Prom Eqs. (11.4) and (11.5), one obtains the concentration dependence of G),... 

Figure 7.7b shows the concentration dependence of the plateau modulus obtained from the fitting of the viscoelastic spectra. We have also calculated GL from Tls and 770 by using the relation... 

Fig. 7 Left. Frequency dependence of G squares) and G" (circles) of a 50 g L aqueous solution of associative polymers from [50] lines are fits to the Maxwell model [Eqs. (6) and (7)]. Right Plateau modulus (Go) and relaxation time (tq) versus the associative-polymer concentration lines are power law fits to the data. Reprinted figure with permission from [50]. Copyright 2008 by the American Physical Society...

The exponent n in the concentration dependence of the storage modulus, E = Ac , is much larger than 2. However, the values of E measured at 2.5 Hz are in general not the equilibrium rubber plateau values. Hence, it is difficult to interpret these high values of n. 

In order to illustrate the typical nonlinear mechanical response of wormlike micelles under steady shear flow, we chose to focus on the cetylpyridinium (CPCl)/sodium saUcylate (NaSal) system. It is often considered as a model system since it follows the right scaling laws for the concentration dependence of the static viscosity and plateau modulus [32]. Moreover, for concentrations ranging from 1 to 30wt. %, the samples behave, in the linear regime, as almost perfect Maxwellian elements with a single relaxation time Tr and a plateau modulus Go- This system has been... 

The plateau compliance J% or its reciprocal, the modulus G%, can be obtained from stress relaxation, creep, or dynamic data by integrations such as equations 3 to 5 of Chapter 13 in order to determine its concentration dependence without complication of time scale shifts. Examples are shown in Fig. 17-14 for concentrated solutions of poly(methyl methacrylate) and m-polyisoprene. The... 

C < Cn and the presence of the dynamic coupling between concentration fluctuation and elastic stress of the transient network for C > Cn- From these results, we drew the picture on a growing process of associating PVA chains to a temporally cross-hnked network with increasing C and proposed that Cn corresponded to the gel point of the chemically cross-linked gel in the short-time domain. The concept is found to be useful for interpretation of the concentration dependence of the plateau modulus. 

Some other experimental data on the concentration dependence of the plateau modulus i and of diffusion fall closer to the prediction D oc C. Thus the correct theoretical interpretation for this case remains an open question. 

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