Necklace Models

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

Kirkwood and Riseman (1948) did not encounter this problem, because they used the bead-rod or, in other words, pearl-necklace model of macromolecule (Kramers 1946), in which A is a number of Kuhn s stiff segments, so that N present the length of the macromolecule. 

Parameter values were chosen so that Ub models a stiff covalent bond, whereas the repulsive portion of ii j, approximates a hard sphere potential of diameter au, which was set equal to the bond length a so that the chain becomes the familiar pearl necklace model. 

The conformation parameter a (=A/Af, where Af is A of a hypothetical chain with free internal rotation) for cellulose and its derivatives lies between 2.8-7.5 2 119,120) and the characteristic ratio ( = A2Mb//2, where Ax is the asymptotic value of A at infinite molecular weight, Mb is the mean molecular weight per skeletal bond, and / the mean bond length) is in the range 19-115. These unexpectedly large values of a and Cffi suggest that the molecules of cellulose and its derivatives behave as semi-flexible or even inflexible chains. For inflexible polymers, analysis of dilute solution properties by the pearl necklace model becomes theoretically inadequate. Thus, the applicability of this model to cellulose and its derivatives in solution should be carefully examined. 

A similar relation p exists for a solution in which the hydrodynamic properties of a dissolved molecule are approximated by a rigid-diain necklace model of spherical beads. 

The bond lei h b entering in this expression ould not be confused with the c c bond length. We are using an idealized pearl necklace model, and b may be several times larger than the c — c bond length. 

A pearl necklace model in which the polypeptide chain forms the string of the necklace and the surfactant molecules form micelle-like clusters along the polypeptide chain, which passes through the micellar clusters in a a-helical conformation. In contrast to the rod-like particle model, this model assumes that the polypeptide chain is flexible. 

The pearl necklace model for the BSA-dodecylsulphate complexes is very different from the model used to interpret the neutron diffraction data for the complexes formed between the deuterated bifunctional enzyme N-5 -phosphoribosylanthranilate/indole-3-glycerol-phosphate... 

There is clearly a considerable difference between the pearl necklace and decorated micelle models, which may in part relate to the differences between the proteins, in particular the fact that BSA has a more restricted conformation because of disulphide linkages. However the most important difference is that in the pearl necklace model the polypeptide chain is believed to pass through micelles of constant size as opposed to around micelles of variable size in the decorated micelle model. However it is interesting that for the decorated micelles formed from the fragments S and L and the whole molecule the numbers of SDS molecules per amino acid residue are surprisingly uniform (0.45 (S), and 0.49 (L), and 0.48 (W)) and very close to the values in the flexible helix model stabilized by hydrogen bonding proposed by Lundahl et al. [110]. 

Application of the concept of partial shielding by Debye and Bueche, and, independently, by Brinkman,to the pearl-necklace model gave, for viscosity and sedimentation. 

Explanation of Property (viii ) of the PVME component in the 25% PVME blend with PS has been offered near the end of the section where these properties are made known. There we have mentioned that this property is shared by some neat glass-formers (Ngai, 2011) including a bead-necklace model for polymer (Bedrov and Smith, 2011) and mixtures of van der Waals liquids (Mierzwa et al., 2008 Kessairiet al., 2008), and the explanation of it readily follows from the co-invariance of r , tjg, and n to changes of pressure and temperature at either constant Tq, or constant tjg, found by experiments and computer simulations. The explanation also follows from the CM equation when combined with the relation, tjg tq, found experimentally valid for many glass-formers between the JG /3-relaxation time and the primitive relaxation time, to, of the CM. 

Two early theoretical models to rationalize this result were pursued the porous-sphere model of Debye and Bueche [1948], in which spherical beads representing the monomers are distributed uniformly in a spherical volume, and the more realistic pearl necklace model, proposed by Kuhn and Kuhn [1943], in which the beads are linked together by infinitely thin linkages. For each of these models, the principal challenge was to describe the flow of solvent around and within the volume occupied... 

Fig. 104. The Kondo-necklace model moving from the eritical coupling eonstant to the right strengthens the Kondo inter-aetion (T ) and in consequence weakens magnetism. When moving to the left, the RKKY inteiaction (Trkky) wins, leading to increasingly localized magnetism. The difieient regions, marked I, II and III, are explained in more detail in fig. 105. After Doniach (1977).

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