Loop-bridge transition

As expected, the curves for low polymer concentrations first increase to a peak and then monotonically decrease to zero where the gel network is broken into sol pieces by the surfactant. For higher polymer concentrations, however, the curves do not show any peak because the junctions are well developed without surfactant molecules. The added surfactant only destroys the junctions. 

6 Intra- and intennolecular flowers formed by hydrophobically modified associating polymers. (Reprinted with permission from Ref. [29].) 

In general, the CMC of the flower micelles is very low it can be as low as 10 polymer wt%. At low concentrations, intramolecular flowers dominate. With an increase in the polymer concentration, loops dissociate and have more chance to form an open association with many bridges, thus eventually leading to gelation. The sharpness of the transition depends on the association constant and the aggregation number of the micelles. 

If we can assume association to be an entire equilibrium and reversible, it can be decomposed into intra- and intermolecular association. In intramolecular association, each chain has a conformation carrying several inframolecular flowers along the chain [29]. The hydrophobic cores are regarded as composite associative groups. In intermolecular association, such composite chains are connected with each other by intermolecular association. Thus, the system is modeled as a polymer solution in which polymers carry many associative groups of different sizes that may form junctions of variable multiplicity. The functionality of each chain is not fixed, but is controlled by the thermodynamic requirement. 

Let us specifically consider telechelic polymers. The intramolecular association of telechelic polymers is unique, namely, they form a single loop (petal). The probability to form such a loop is decided by the thermodynamic equilibrium condition. It is given by the cyclization parameter introduced in Section 6.2 [30] 

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Figure 9.19 Hysteresis loop for ErMn at 5.7 K [99]. (With kind permission from Springer Science + Business Media Transition Metal Chemistry, The magnetochemistry of novel cyano-bridged complexes Ln(DMF)4(H20)2Mn (CN)6 H20 (Ln = Tb, Dy, Er), 26, 2001, 287-289, B. Yan, andZ. Chen.)...

Fig. 7. Transition from a two-domain structure to a three-domain structure in an ABC triblock copolymer lamellar phase. Although state ii is thermodynamically the most stable state, transition to this state from i is hindered because of the high free energy cost in switching the orientation of the bridges and in turning the loops into bridges. Thus, a kinetically more likely process is for the A and C blocks from the bridge conformation to separate laterally, with the loops straddling the interfaces between the A and the C domains.

Other wonderful shapes can also be found. The metal phthalocyanine structure with its bridging pyrazine units, shown in Scheme 4, resembles a shiskabob. The metal and pyrazine chain forms the "skewer" which "pierces" the phthalocyanine groups. A braided struc-ture is also represented in Scheme 4. 5-Phenyltetrazolate reacts with metals from all three transition metal series to give the loops shown. The Ni(II) and Fe(II) adducts give extremely viscous aqueous solutions from which both flexible sheets and threads have been made. ... 

In summary, as we have shown in brief, and as shown in more detail in [89], a selfavoiding PS model (with unique marking, with self-avoiding bridges and (unrooted) selfavoiding loops) as defined above yields a first order phase transition in both d = 2 and d = 3. 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

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