Concentrated isotropic regime

Thus, (f) = 0.01 is in the semidilute concentration regime. (It is actually at the lower end of the concentrated isotropic regime, which begins at a concentration = nd/4L = 0.008. However, even in the concentrated isotropic regime, the viscosity versus concentration behavior does not change much from that of the semidilute until near the transition to the nematic regime at 0 = 3.3d/L = 0.033.)... 

Fig. 1 Illustrations of the different concentrations regimes encountered in wormUke micellar solutions with increasing concentration. is the mesh size of the entangled network in the semidilute regime and d denotes the average distance between colinear micelles in the concentrated isotropic, nematic and hexagonal phases. An estimate of d can be gained from the position of the structure peak in the scattering function...

Larson s results [154] are divided into the three shear rate regimes - tumbling, wagging, and steady-state - as explained below. He chose the strength of mean-field potential 2L2dc in Eq. (41) to be 10.67, which corresponds to the concentration cA of the nematic phase coexisting with the isotropic phase (in the second virial approximation), and expressed the shear rate in terms of T defined by... 

Figure 6.18 Concentration regimes for rodlike molecules, (a) Dilute (b) semidilute (c) isotropic concentrated and (d) nematic. (From Doi and Edwards, copyright 1986 by Oxford Univcrsiiy Press, Inc. Used by permission of Oxford University Press, Inc.)...

In the semidilute regime, at very low salt concentration, the chains are entangled and can form an isotropic transient network. In the scaling picture [56,57,58] the solution is described by an assembly of blobs of size with... 

FIGURE 1.14 Viscosity as a function of polymer concentration of a solution of a 50 50 random copolymer, poly(w-hexylisocyanate-co-n-propylisocyanate) (M , =41,000 g/mol) in toluene, encompassing the isotropic, biphasic, and anisotropic (liquid crystal) regimes. (Adapted from Aharoni [1980a].)... 

Figure 8.3 Concentration regimes of rod-like polymers (a) dilute solution, (b) semi-dilute solution, (c) isotropic concentrated solution, and (d) liquid crystalline solution (from Doi and Edwards [2], The Theory of Polymer Dynamics, 1986, reprinted with permission from Oxford University Press).

As the concentration increases further, the excluded volume effect becomes important. The excluded volume of the rod is approximately bL, where b is the diameter of the rod. When the concentration is greater than / bL ), the excluded volume interaction increases, and, consequently, both the static and dynamic properties change substantially. Although the solution is well above the concentration regime where the excluded volume effect is significant, it remains isotropic until it reaches a critical concentration v [see Figure 8.3(c), and note that the rods are thicker to signify the excluded volume interaction] ... 

We consider now the case when the surface is saturated with isotropic blobs from long chains. As discussed previously, this may be realized even when the bulk concentration in probe chains is very sm l. In the plateau regime, the surface is covered... 

When we now go from the regime of dilute solutions and basically isolated chains to the r me of semidilute and dense solutions of semi-flexible chains we can expect two main effects. Due to the stiffness of the chains and at sufficiently large aspect ratio the chains will prefer a nematic alignment and we expect to see an isotropic to nematic transition. In the limit of almost rigid rods this transition happens at very dilute concentrations. When the chains become more flexible the transition density increases. This behavior was studied for a lyotropic system in [22]. The Hamiltonian employed in this simulation of the bond-fluctuation model was given by... 

The review is organized as follows Sect. 2 deals with the shear-thickening behavior found in dilute and very dilute surfactant solutions. Section 3 examines the shear-banding instability and the isotropic-to-nematic transition revealed in the semidilute and concentrated regimes, respectively. The last part focuses on the wormlike micellar nematics under shear, and emphasizes the analogy with liquid-crystalline polymers. 

In a perfectly macro-mixed batch vessel, the mean concentration is obviously constant and the variance of the fluctuations decreases exponentially with time. The latter is observed in turbulent regimes and is also predicted from isotropic, homogeneous turbulence theory (Fig. 12.6.1.A-2). 

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