Polymer Size in the Amorphous State

The subject of polymer size or chain dimensions is concerned with relating the sizes and shapes of individual polymer molecules to their chemical structure, chain length, and molecular environment. The shape of the polymer molecule is to a large extent determined by the effects of its chemical structure upon chain stiffness. Polymers with relatively flexible backbones tend to be highly coiled and can be represented as random coils. But as the backbone becomes stiffer, e.g., in polymers with more aromatic backbone chain, the molecules begin to adopt a more elongated wormlike shape and ultimately become rodlike. However, the theories which are presented below are concerned only with the chain dimensions of linear flexible polymer molecules. More advanced texts should be consulted for treatments of wormlike and rodlike chains. 

The main quantitative developments of the random coil model of flexible polymers began in 1934 with the work of E. Guth and H. E Mark [12] and W. Kuhn [13]. Using the concept of free rotation of the carbon-carbon bond, Guth and Mark developed the idea of the random walk or random flight of the polymer chain, which led to the familiar Gaussian statistics of today, and eventually to the famous relationship between the end-to-end distance of the main chain and the square root of the molecular weight, described below. 

The simplest measure of chain dimensions is the length of the chain along its backbone and is known as the contour length. For a chain of n backbone bonds each of length I, the contour length is nl. However, because of the fixed bond angle (109.5°) of carbon, the maximum end-to-end distance of the polymer chain will be somewhat less than nl (see Fig. 2.12 and Problem 2.6). For linear flexible chains that are more like random coils, the distance separating the chain ends, i.e., the end-to-end distance r (Fig. 2.13) will be even considerably less than nl. 

Problem 2.6 For a linear molecule of polyethylene of molecular weight 1.4x10 what would be the end-to-end distance of the polymer molecule in the extended (all-tra 5) state, as compared to the contour length of the molecule  

The polyethylene molecule may be represented skeletally in a planar zigzag form as shown in Fig. 2.12, where I = 0.154 nm and 9 = 109.5°. In order to perform the calculation, the number n of backbone bonds is required. It can be obtained 

The development of the random coil by H. F. Mark and many further developments by P. J. Flory led to a description of the conformation of chains in the bulk amorphous state. Neutron scattering studies revealed that the conformation in the bulk is close to that found in solution in 0-solvent (see Chapter 3), thus strengthening the random coil model. On the other hand, some workers suggested that the chains have various degrees of either local or long range order. 

The contour length is the length of the molecule along its backbone and so is given by Contour length = nl = 10,000 (0.154 nm) = 1540 nm 

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Compared with their size in the amorphous state, macromolecules are expanded in good solvents whereas they tend to shrink in poor solvents. In other words, size of macromolecules of given polymer with given molar mass in solution depends on thermodynamic quality of solvent. It is characterized by the expansion... 

The diffusion rate will depend heavily on the molecular size of the diffusion species and on the size of the gaps between polymer molecules. Crystalline structures have an ordered arrangement of molecules, and diffusion can occur in amorphous regions or through regions of imperfections. The crystalline regions in a polymer can thus be considered almost impermeable. Amorphous polymers exist in the rubbery state where there is an abundance of free volume so that diffusion can occur relatively easily. 

A radius of gyration in general is the distance from the center of mass of a body at which the whole mass could be concentrated without changing its moment of rotational inertia about an axis through the center of mass. For a polymer chain, this is also the root-mean-square distance of the segments of the molecule from its center of mass. The radius of gyration is one measure of the size of the random coil shape which many synthetic polymers adopt in solution or in the amorphous bulk state. (The radius of gyration and other measures of macromolecular size and shape are considered in more detail in Chapter 4.)... 

Unlike molecular solids, polymeric solids inherently have a very large density of defects, making the establishment of an accurate value for Tm far more difficult. For semicrystalline polymers, with the very rare exceptions of special types of polymers made by the solid state polymerization of some rather unique monomers, the key properties of the crystalline phase (Tm and AHm) must be extracted from data obtained by using samples which have at least two phases. There may be more than one type of morphology present in the crystalline phase, as well as crystallites of different sizes and levels of perfection, and possibly even interfaces between the amorphous and crystalline domains. The presence of chains of different lengths provides additional complexities, as does the presence of chain defects, deviations from the predominant stereo regularity (tacticity) of the chains, and oxidation sites along the chains. 

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