The Expanded Chain

Expanded chains are found in dilute solution in good solvents. The effective interaction energy between two monomers is always repulsive here and, as a consequence, chains become expanded. Expansion will come to an end at some finite value since it is associated with a decreasing conformational entropy. The reason for this decrease is easily seen by noting that the number of accessible rotational isomeric states decreases with increasing chain extension. The decrease produces a retracting force, which balances the repulsive excluded volume forces at equilibrium. 

The distribution function p(i ) has a general shape as indicated in Fig. 2.16. When compared to the properties of ideal chains there is first a change in the asymptotic behavior at large R. It is now given by 

The described distribution function refers to the asymptotic limit of large degrees of polymerization. It is important to note that, as for ideal chains, p R) includes one parameter only, now the quantity Rp, called Flory radius in the literature. Rp is a measure for the diameter of the volume that encloses an expanded pol3rmer chain, with the identical definition as for ideal chains 

Of central importance for the discussion of the properties of expanded chains is the relation between Rp and the degree of polymerization N. It is given by the scaling law 

V m describes, in a mean field approximation, the potential experienced by a monomer as a result of the interaction with the other monomers in its neighborhood. Consequently, we may represent the contribution of the excluded volume interactions to the free energy density by 

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There is no doubt that the asymptotic large Q-behavior at average contrast conditions agrees better with the expanded chain conformation than with the ideal one. However, since under these contrast conditions the scattering is weak, the data at large Q are affected by large errors, thereby limiting the precision of this statement. 

In summary, we see now how the change from the expanded chains in dilute solutions to the ideal chains in a melt is accomplished. With increasing polymer concentration, the chain overlap increases and the length scale over... 

Similar expansion factors exist for e.g. the ratio of intrinsic viscosities, a. At temperatures just above the 0-temperature, we have the regime of the perturbation treatment. Its extension, the two-parameter theory, relates the expanded chain dimensions to the parameter with P the binary... 

It is always useful to check for the number of independent parameters. For the Brownian chain there is one parameter only, namely Rq. For the expanded chain in general, we find two parameters, Rp and but in the Kuhnian limit 0 we return again to the simple one-parameter case. 

Combining all the information collected so far, for the single chain pair distribution function in a semidilute solution we can predict an overall shape as indicated in Fig. 3.8. For the presentation we choose a plot of 4nr g r) versus r. The curve is a composite of different functions in four ranges, with cross-overs at the persistence length Ips, the thermic correlation length and the screening length Up to r we find the properties of the expanded chain, which... 

Simulations of monolayers have focused on internal phase transitions, e.g., between the expanded phase and the condensed phases, between different tilted phases, etc. These phenomena cannot be reproduced by models with purely repulsive interactions. Therefore, Haas et al. [148,149] represent the amphiphiles as stiff Lennard-Jones chains, with one end (the head bead) confined to move in a plane. In later versions of the model [150-152], the head bead interactions differ from those of the tail beads they are taken to be purely repulsive, and the head size is variable. 

Well-defined differences exist, for instance, between the spectra of samples in the a and (3 crystalline forms (although both contain chains of trans-planar conformation) as shown by the expanded FTIR spectra, in two different regions, of Fig. 19 [110]. 

The Gibbs equation allows the amount of surfactant adsorbed at the interface to be calculated from the interfacial tension values measured with different concentrations of surfactant, but at constant counterion concentration. The amount adsorbed can be converted to the area of a surfactant molecule. The co-areas at the air-water interface are in the range of 4.4-5.9 nm2/molecule [56,57]. A comparison of these values with those from molecular models indicates that all four surfactants are oriented normally to the interface with the carbon chain outstretched and closely packed. The co-areas at the oil-water interface are greater (heptane-water, 4.9-6.6 nm2/molecule benzene-water, 5.9-7.5 nm2/molecule). This relatively small increase of about 10% for the heptane-water and about 30% for the benzene-water interface means that the orientation at the oil-water interface is the same as at the air-water interface, but the a-sulfo fatty acid ester films are more expanded [56]. 

Micro-composites are formed when the polymer chain is unable to intercalate into the silicate layer and therefore phase separated polymer/clay composites are formed. Their properties remain the same as the conventional micro-composites as shown in Figure 2(a). Intercalated nano-composite is obtained when the polymer chain is inserted between clay layers such that the interlayer spacing is expanded, but the layers still bear a well-defined spatial relationship to each other as shown in Figure 2(b). Exfoliated nano-composites are formed when the layers of the day have been completely separated and the individual layers are distributed throughout the organic matrix as shown in Figure 2(c). 

The finding that substantial 2.5-helicity may be retained in water upon the introduction of a limited number of acyclic or y9 -amino acid residues at chosen positions in the sequence, further expand the side-chain array available for functionalization of the 2.5-helical scaffold. In water, y9-heptapeptides 107 and 108 which contain two -amino acid residues [184a] and two y9 -amino acids [184b], respectively, still display a CD spectrum and NOE coimectivities characteristic of the 2.5-helix. However, the addition of a third acycHc amino acid is detrimental to the formation of the 2.5-helix in water. 

Further detailed investigations towards new chiral ruthenium catalysts that could enhance enantioselectivity and expand the substrate scope in asymmetric RCM were reported by Grubbs and co-workers in 2006 [70] (Fig. 3.24). Catalysts 59 and 61, which are close derivatives of 56 incorporating additional substituents on the aryl ring para to the ort/to-isopropyl group, maintained similar enantioselectivity than 56b. However, incorporation of an isopropyl group on the side chain ortho to the ortho-isopropyl group 60 led to an increase in enantioselectivity for a number of substrates. 

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