Random coil molecules

Application of the lattice scheme to a system consisting of rodlike molecules, random coils and a solvent is straightforward The free energy of the nematic phase would be increased markedly by the presence of a substantial proportion of the random coil. 

Use the Simha equation and these data to criticize or defend the following proposition These polymer molecules behave like rods whose diameter is 16 A and whose length is 1.5 A per repeat unit. The molecule apparently exists in fully extended form in this solvent rather than as random coils. 

The three-dimensional radius of gyration of a random coil was discussed in Sec. 1.10 and found to equal one-sixth the mean-square end-to-end distance of the polymer [Eq. (1.59)]. What we need now is a connection between two-and three-dimensional radii of gyration. Since the molecule has spherical symmetry r, r> = V + r + r, = 3r . If only two of these contributions are present, we obtain (2/3)rg 3 = rg2o- this result and Eq. (1.59) to... 

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

One thing that is apparent at the outset is that polymer molecules in solution are very different species from the rigid spheres upon which the Einstein theory is based. On the other hand, we saw in the last chapter that the random coil contributes an excluded volume to the second virial coefficient that is at least... 

The secondary stmcture of the plasminogen molecule, as determined by circular dichroism spectra, is 80% random coil, 20% beta-stmcture, and 0% alpha-helix. Electron microscopy has demonstrated the tertiary stmcture of plasminogen to be a 22- to 24-nm long spiral filament with a diameter of 2.2 to 2.4 nm. 

In the first case, that is with dipoles integral with the main chain, in the absence of an electric field the dipoles will be randomly disposed but will be fixed by the disposition of the main chain atoms. On application of an electric field complete dipole orientation is not possible because of spatial requirements imposed by the chain structure. Furthermore in the polymeric system the different molecules are coiled in different ways and the time for orientation will be dependent on the particular disposition. Thus whereas simple polar molecules have a sharply defined power loss maxima the power loss-frequency curve of polar polymers is broad, due to the dispersion of orientation times. 

When polymer melts are deformed, polymer molecules not only slide past each other, but they also tend to uncoil—or at least they are deformed from their random coiled-up configuration. On release of the deforming stresses these molecules tend to revert to random coiled-up forms. Since molecular entanglements cause the molecules to act in a co-operative manner some recovery of shape corresponding to the re-coiling occurs. In phenomenological terms we say that the melt shows elasticity. 

A characteristic feature of thermoplastics shaped by melt processing operations is that on cooling after shaping many molecules become frozen in an oriented conformation. Such a conformation is unnatural to the polymer molecule, which continually strives to take up a randomly coiled state. If the molecules were unfrozen a stress would be required to maintain their oriented conformation. Another way of looking at this is to consider that there is a frozen-in stress corresponding to a frozen-in strain due to molecular orientation. 

Traditional rubbers are shaped in a manner akin to that of common thermoplastics. Subsequent to the shaping operations chemical reactions are brought about that lead to the formation of a polymeric network structure. Whilst the polymer molecular segments between the network junction points are mobile and can thus deform considerably, on application of a stress irreversible flow is prevented by the network structure and on release of the stress the molecules return to a random coiled configuration with no net change in the mean position of the Junction points. The polymer is thus rubbery. With all the major rubbers the... 

It is not very difficult to appreciate that if polymer molecules are aligned as in Fig. 18.10 then a much higher tensile strength will be obtained if a test is carried out in the X-X direction as opposed to the Y-Y direction. It is also not difficult to understand why such a material has a lower impact strength than a randomly coiled mass of molecules (Fig. 18.10) because of the ease of cleavage of the material parallel to the X-X direction. 

Although R2 is the easiest quantity to be obtained theoretically, there is no straigthforward experimental method for its determination. For this reason, two other quantities are widely in use to characterize the dimensions of a randomly coiled polymer molecule ... 

The inability of the strain softened molecules to recover their random coil conformation when unloaded. 

Typically in solution, a polymer molecule adopts a conformation in which segments are located away from the centre of the molecule in an approximately Gaussian distribution. It is perfectly possible for any given polymer molecule to adopt a very non-Gaussian conformation, for example an all-trans extended zig-zag. It is, however, not very likely. The Gaussian set of arrangements are known as random coil conformations. 

The typical shape of most polymer molecules in solution is the random coil. This is due to the relative ease of rotation around the bonds of the molecule and the resulting large number of possible conformations that the molecule can adopt. We should note in passing that where rotation is relatively hindered, the polymer may not adopt a random coil conformation until higher temperatures. 

In the unstressed state the molecules of an elastomer adopt a more-or-less randomly coiled configuration. When the elastomer is subjected to stress the bulk material experiences a significant deformation, as the macromolecules adopt an extended configuration. When the stress is removed, the molecules revert to their equilibrium configurations, as before, and the material returns to its undeformed dimensions. 

This thermodynamic behaviour is consistent with stress-induced crystallisation of the rubber molecules on extension. Such crystallisation would account for the decrease in entropy, as the disorder of the randomly coiled molecules gave way to well-ordered crystalline regions within the specimen. X-Ray diffraction has confirmed that crystallisation does indeed take place, and that the crystallites formed have one axis in the direction of elongation of the rubber. Stressed natural rubbers do not crystallise completely, but instead consist of these crystallites embedded in a matrix of essentially amorphous rubber. Typical dimensions of crystallites in stressed rubber are of the order of 10 to 100 nm, and since the molecules of such materials are typically some 2000 nm in length, they must pass through several alternate crystalline and amorphous regions. 

Typical materials that exhibit liquid crystalline behaviour are made up of long, thin molecules. Hence in principle polymers ought to show the basic requirement for liquid crystal behaviour. Conventional polymers, however, are too flexible and tend to adopt random coil configurations in the melt. They are thus not sufficiently anisotropic to exhibit a mesophase. 

Branching also occurs in polymers. The branches are extensions of hnked monomer units that protrude from the polymer trunk chain. Branched polymers can also form random coils, but the branches prevent a highly irregular arrangement and, therefore, less crystalhnity results because the molecules cannot line up and pack as well. 

The lowermost curve in Fig. 45 represents P(0) plotted against according to Eq. (31) for random coil molecules. The results of similar calculations for spherical and for rod-shaped particles of uniform density are shown also. The curve for the former of these is not very different from that for randomly coiled polymers at corresponding values of the abscissas the factor P(0) for rods differs appreciably, however. 

Apart from their utility in determining the correction factor 1/P( ), light-scattering dissymmetry measurements afford a measure of the dimensions of the randomly coiled polymer molecule in dilute solution. Thus the above analysis of measurements made at different angles yields the important ratio from which the root-mean-square... 

---