Rheology of Entangled Wormy Micellar Solutions

At this concentration of salt and of surfactant, a nearly single-relaxation-time description holds over four or more decades of frequency, even to values of (o high enough that G jG 0.01 Such nearly perfect monoexponential relaxation is rare and has only been found for a few other complex fluids, as discussed in Section 5.4.1. 

Within these three assumptions, one can identify at least two important limiting cases. In the first, the micelle breakage time tbr is long enough that the typical micelle can relax by reptation before it breaks. In this case, one obtains the linear relaxation modulus G(t) 

The linear relaxation moduli of some wormy micelle solutions can indeed be fit to a stretched exponential. The exponent a has been found to increase from 0.3 to unity as the ratio of salt to surfactant concentration increases in solutions of CPyCl/NaSal (see Fig. 12-14). An exponent of a = 1 corresponds to single-exponential relaxation, similar to that shown in Fig. 12-10. 

The narrowing of the viscoelastic spectrum as salinity increases is analogous to motional narrowing of line widths in NMR spectroscopy. The narrowing process occurs when the breakage time Tbr becomes smaller than the reptation time Trep. In the limit Tbr Trep, the Gates model predicts that G(t) is monoexponential, with a time constant t given by 

The relaxation modulus is a single exponential decay because a typical tube segment relaxes only after it has been passed off from one chain to another many times. Hence, before relaxing, each tube segment samples chains of many lengths as well as many different distances from a chain end this produces motional averaging, which results in nearly monoexponential relaxation. 

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