Statistically coiled conformers

The disagreement between the values 3.12 and 2.27 points to the fact that Equation 45 is incorrect owing to the neglect of the concentration dependence of the interaction parameter. The chief result of Vrij (1974) is in the fact that near the spinodal maximum, macromolecules have the statistical coil conformation with sizes slightly different from those in the 0 state at c —> 0 (at Iceist, for the polystyrene-t-cyclohexane system). Generally, however, this may not be the case (Vrij and van den Esker, 1972). 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

Aside from chemical composition and chain length the properties of macromo-lecular substances are substantially determined by the conformation and configuration of the individual macromolecules. Isolated macromolecules do not take up a precisely defined three-dimensional shape they rather assume a statistically most probable form which approximates to the state of maximum possible entropy. This is neither a compact sphere nor an extended rigid chain, but rather a more or less loose statistical coil (Fig. 1.7). 

While for nematic polymers the statistical distribution of the centers of gravity of the mesogenic side chains is compatible with a more or less statistical main chain conformation, for smectic polymers a three dimensional coil conformation is no longer consistent with the layered structure of the mesogenic side chains. The backbone has to be restricted in its conformation, which will cause a more pronounced interaction between the main chain and the anisotropically ordered mesogenic side chains, compared to nematic and cholesteric polymers. 

In any case, both models have in common that owing to the positional ordering of the mesogenic side chains, the polymer backbone no longer exhibits a statistical three dimensional coil conformation. Therefore at the phase transformation isotropic to smectic or nematic (cholesteric) to smectic, in addition to the change of the anisotropic packing of the side chains, the main chain has to change its conformation, which must be consistent with the layered smectic structure. A direct interaction... 

Polypeptides and poly(a-amino acid)s have a quite unique position amongst synthetic polymers. The reason for this is that most common synthetic polymers have very little long range order in solution and their properties are the products of statistical random coil conformations. Polypeptides, in contrast, can adopt well defined, ordered structures typical of those existing in proteins, such as a-helix and P-struc-tures. Moreover, the ordered structures can undergo conformational changes to the random coil state as cooperative transitions, analogous to the denaturation of proteins. 

Initiated principally by Flory (1953), an extensive literature on the statistical description of macromolecular coil conformations has developed. An extensive survey has been written by Kurata and Stockmayer (1963). Detailed conformational calculations can be found in a monograph by Flory (1969). Here only some headlines can be mentioned. 

Besides this statistical mechanical approach to the question of helix stability, the problem has also been addressed by conformational energy calculations. First, the helix-breaking tendencies of such residues as serine and aspartic acid can be accounted for by the tendency toward formation of side chain-backbone hydrogen bonds in nonhelical conformations163 (Figures 20 and 21). Second, the free energies of the helical and statistical coil forms in water have... 

H. Meirovitch, M. Vasquez and H. A. Scheraga, Biopolymers, 27, 1189 (1988). Stability of Polypeptide Conformational States. II. Folding of a Polypeptide Chain by the Scanning Simulation Method, and Calculation of the Free Energy of the Statistical Coil. 

Most synthetic polymers in which the monomer units are connected via single bonds have rather flexible chains. The bond torsion energy is relatively small and the units can rotate around their bonds [14,30,31]. Each molecule can adopt a large number of energetically equivalent conformations and the resulting molecular geometry is that of a statistical coil, approximately described by a Gaussian density distribution. This coil conformation is the characteristic secondary structure of macromolecules in solution and in the melt. It is entropically favoured because of its... 

Figure 2 (a) Sketch of the variables determining the rotation isomeric state model of a statistical coil 9 = bond angle, = bond torsion angle [3], (b) Representation of one of the possible conformations of a two-dimensional statistical coil (33 monomers) with bond-angle restriction ( — 90° <0 <90°) [3],... 

The disordered state of a statistical coil is what is displayed by polymers in the molten and amorphous states and also in solution. To describe the conformation of a macromolecule consisting of a main chain N +l atoms, the positions of all them have to be determined. Using vectorial... 

The objective is to obtain the mean value of the end-to-end distance corresponding to the set of conformations that define the statistical coil state. For this, it is possible to think physically in two ways that might seem different at first sight but are in fact the same (ergodic hypothesis) ... 

Solid polymers can occur in the amorphous or crystalline state. Polymers in the amorphous state are characterized by a disordered arrangement of the macromolecular chains, which adopt conformations corresponding to statistical coils. The crystalline state is characterized by a long-range three-dimensional order (order extending to distances of hundreds or thousands... 

The applicability of this relationship to drain molecules depends on their length and flexibility and on the value of the scalar length r. The relationdiip applies closely to very long chains in random-coil conformations at moderate extensions, i.e., when the values of r are relatively small (46). From Eq. (4) it follows that the density of the distribution Wx(r) in the region r = 0 for long chain molecules obeying Gaussian statistics is... 

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