Shapes of Polymer Molecules in Solution

In solution the molecules of a polymer undergo various segmental motions, changing rapidly from one conformation to another, so that the molecule itself effectively takes up more space than the volume of its segments alone. As we have seen, the size of the individual molecules depends on the thermodynamic quality of the solvent in good solvents chains are relatively extended, whereas in poor solvents they are contracted. 

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. 

Because of the random nature of the typical conformation, the size of the molecule has to be expressed in terms of statistical parameters. Two important indications of size are  

Root-mean-square end-to-end distance, which effectively takes account of the average distance between the first and the last segment in the macromolecule, and is always less that the so-called contour length of the polymer. This latter is the actual distance from the beginning to the end of the macromolecule travelling along the covalent bonds of the molecule s backbone. Radius of gyration, which is the root-mean-square distance of the ele- 

If we consider the size of a polymer molecule, assuming that it consists of a freely rotating chain, with no constraints on either angle or rotation or of which regions of space may be occupied, we arrive at the so-called unperturbed dimension, written (r)o Such an approach fails to take account of 

Rool-mean-square end-to-end distance, (r ) -, which effectively takes 

If we consider the size of a polymer molecule, assuming that it consists of a freely rotating chain, with no constiaints on either angle or rotation or of which regions of space may be occupied, we arrive at the so-called unperturbed dimension, written (r), /. Such an approach fails to take account of the fact that real molecules are not completely flexible, or that the volume element occupied by one segment is excluded to another segment, i.e. in terms of the lattice model of a polymer solution, no lattice site may be occupied twice. Real molecides are thus bigger than the unperturbed dimension, which may be expressed mathematically 

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As may be expected, polymers behave differently toward solvents than do low-molecular-weight compounds. Studies of the solution properties of polymers provide useful information about the size and shape of polymer molecules. In this section we discuss how some of the molecular parameters discussed in the previous sections are related to and can be calculated from thermodynamic quantities. We start with a discussion of the simplest case of an ideal solution. This is followed by a treatment of deviations from ideal behavior. 

Even in the absence of flow, a polymer molecule in solution is in a state of continual motion set forth by the thermal energy of the system. Rotation around any single bond of the backbone in a flexible polymer chain will induce a change in conformation. For a polyethylene molecule having (n + 1) methylene groups connected by n C — C links, the total number of available conformations increases as 3°. With the number n encompassing the range of 105 and beyond, the number of accessible conformations becomes enormous and the shape of the polymers can only be usefully described statistically. 

Ultracentrifuge data can also be interpreted to give information about the effective shape of the dissolved polymer molecules. The actual frictional coefficient (f) can be calculated from the sedimentation constant, and this compared with the theoretical figure (f0) calculated from Stokes law (assuming a spherical molecule). The frictional ratio (f/fo) has been called the asymmetry coefficient. It expresses the deviation of the molecule in solution from a spherical shape, and can be interpreted in terms of the axial ratio of idealized particles. 

A polymer molecule in solution has a certain shape that strongly depends on the type of polymer, type of solvent, temperature, and other conditions. Usually a polymer forms some kind of globular species whose size is dependent on the degree of solvation by solvent molecules. 

Before discussing the detailed chemistry, kinetics, and mechanisms of the various pathways of polymer synthesis, it is necessary to introduce some of the fundamental concepts of polymer science in order to provide essential background to such a development. We need to know what a polymer is and how it is named and classified. It is also necessary to obtain an appreciation of the molecular size and shape of polymer molecules, the molar mass characteristics, the important transition temperatures of polymers, and their distinctive behavior both in solid state and in solution. These concerns are addressed in the first four chapters of the book while the remaining six chapters deal with the important categories of polymerization processes and their mechanisms and kinetic aspects. Throughout this journey the narrative in the text is illuminated with thoughtfully worked out examples which not only complement but also supplement, where necessary, the theoretical development in the text. 

The soluble blend system is a single phase material in which two components (such as two polymeric species or a polymer and a solvent) are dissolved molecularly as a homogeneous solution in the thermodynamic sense. A miscible polymer blend, a block copolymer in a disordered state, and a polymer solution are examples. Whether a homogeneous solution of this kind is regarded as a soluble blend system or as a dilute particulate system discussed above is often simply a matter of viewpoint. When there is a dilute solution of polymer molecules in a solvent and the focus of interest is the size and shape of the polymer molecules, the theoretical tools developed for the dilute particulate systems are more useful. If, on the other hand, the investigator is interested in the thermodynamic properties of the solution, the equations developed for the blend system are more appropriate. 

Polymer molecules in solution also display Brownian motion. Because the polymer molecule is not a simple sphere, each polymer conformation has its own diffusion characteristics. For rigid molecules, the shape of the molecule, spherical or rodlike, for instance, makes a difference. For a Unear flexible molecule, connectivity... 

The success of the rotational isomeric model lies in predicting the time averaged shape and size of polymer molecules in dilute solution. However, we have to look further to find the way in which the molecule moves between the various states that are averaged in the model. 

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