Expanded chains

Expanded chains are found in dilute solution in good solvents. The effective interaction energy between two monomers is always repulsive here, and, as a consequence, chains become expanded. Expansion will come to an end at some finite value since it is associated with a decreasing conformational entropy. The reason for this decrease is easily seen by noting that the number of accessable rotational isomeric states decreases with increasing chain extension. The decrease produces a retracting force which balances, at equilibrium, the repulsive excluded volume forces. 

The distribution function p R) has a general shape as indicated in Fig. 2.16. When compared to the properties of ideal chains there is first a change in the asymptotic behavior at large R. It is now given by 

The described distribution function refers to the asymptotic limit of large degrees of polymerization. It is important to note that, as for ideal chains, p R) includes one parameter only, now the quantity called in the literature Flory-radius . Rp is a measure for the diameter of the volume occupied by the expanded polymer chain, with the identical definition as for ideal chains 

Of central importance for the discussion of the properties of expanded chains is the relation between Rf and the degree of polymerization N. It is given by the scaling law 

Although the rigorous solution was not presented until 1972, Flory offered arguments in support of the scaling law Eq. (2.83) much earlier. The point of concern is the equilibrium conformation of a chain, being the result of a balance between repulsive excluded volume forces and retracting forces ar-ing from the decreasing conformational entropy. Above, with Eq. (2.78), we have already introduced Flory s expression for the potential produced by the excluded volume forces 

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Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

In summary, we see now how tire change from tire expanded chains in dilute solutions to tire ideal chains in a melt is accomplished. Witli increasing polymer concentration, tire chain overlap increases and tire lengtli scale over... 

There is no doubt that the asymptotic large Q-behavior at average contrast conditions agrees better with the expanded chain conformation than with the ideal one. However, since under these contrast conditions the scattering is weak, the data at large Q are affected by large errors, thereby limiting the precision of this statement. 

To obtain a closed expression for A2, suitable for all values of z, two types of theories have been developed by several authors in recent years. The first type of theory is based on the uniformly expanded chain model and on a spherically symmetrical distribution of segments about the molecular center of mass. The segment distribution is taken to be a spherical cloud of constant density in Flory s first theory 101), a Gaussian function about the center of mass in Fi.ory and Kkigbaum s (103 ) and in Orofino and Flory s (204) theories, and a sum of N different Gaussian functions in Isihara and Koyama s theory (132 ). All of these theories may be summarized in the following type of equation given by Orofino and Flory,... 

The second type of theory is based on the uniformly expanded chain model and on a spherical distribution of segments about the locus of an initial interchain context, rather than about the molecular center of mass. Casassa and Markovitz (50) developed this type of theory and obtained... 

Cross-linked polystyrene beads (Section 25-1) are most commonly used in conjunction with organic solvents for the separation and characterization of polymers. Like the rigid gel Styragel, it is available with pore sizes that have exclusion limits for molecules ranging from 4 to 10 nm (expanded chain length), corresponding to molecular weights from 200 to 5 x 10 . 

The attention paid to the polymer solid state is minimized in favour of the melt and in this chapter the static properties of the polymer are considered, i.e. properties in the absence of an external stress as is required for a consideration of the rheological properties. This is addressed in detail in Chapter 3. The treatment of the melt as the basic system for processing introduces a simplification both in the physics and in the chemistry of the system. In the treatment of melts, the polymer chain experiences a mean field of other nearby chains. This is not the situation in dilute or semi-dilute solutions, where density fluctuations in expanded chains must be addressed. In a similar way the chemical reactions which occur on processing in the melt may be treated through a set of homogeneous reactions, unlike the highly heterogeneous and diffusion-controlled chemical reactions in the solid state. 

A number of studies have considered the nature of the PAn conformation in solution as an effect of solvent/dopant/oxidation state. As described in Chapter 5, the PAn can form either tight coils or expanded chains depending on the nature of the solvent used. This behavior is typical of polymers in either poor or good solvents, but has a significant impact on the electrical properties of conjugated polymers, as described in Chapter 5 for PAn. 

P. Anilkumar and M. Jayakannan, Single-molecular-system-based selective micellar templates for polyaniline nanomaterials control of shape, size, solid state ordering, and expanded chain to coillike conformation. Macromolecules, 40, 7311-7319 (2007). 

All associations are intended to initiate or expand chains of thought, sequences of ideas, or lines of evolution of ideas (using the terminology of the theory of evolution of engineering systems [Zlotin and Zusman 2006]). They are like catalysts in the process of chain thinking. 

FIGURE 14.7. If an aqueous polymer-surfactant interaction occurs via the surfactant tail, the resulting complex will generally exhibit modified solution characteristics, usually reflecting a greater interaction with water and an expanded chain conformation. 

Similar expansion factors exist for e.g. the ratio of intrinsic viscosities, a. At temperatures just above the 0-temperature, we have the regime of the perturbation treatment. Its extension, the two-parameter theory, relates the expanded chain dimensions to the parameter with P the binary... 

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