Random flight configuration

In benzene, the dipole moment of the methyl compound is smaller than that of the butyl compound. Thus, if the monomers have practically similar moments, there is a better compensation of the individual dipoles in the methyl ester where they can more easily move into the energetically favored trans positions. The bulky butyl groups, by mere steric hindrance, introduce a certain amount of disorder, and the chain tends towards its random flight configuration. 

The theory of solutions of flexible uncharged polymers with excluded volume is at present well developed, but the properties of polyelectrolytes and especially polyampholytes have been considered much less from the theoretical point of view. It is well known that polyampholytes exhibit a change in phase from the extended random flight configuration to a condensed microphase. The polyampholyte theory of Edwards et al. [6] considers the isoelectric state of polyampholytes as a microelectrolyte satisfying a Debye-Huckel-type of structure. The criterion of transition from the collapsed conformation to the extended one is described as follows ... 

The perturbation of the configuration of the polymer chain caused by its internal interactions may also be considered from the somewhat different viewpoint set forth qualitatively in Chapter X, Section 3. There it was indicated that, because of the obvious requirement that two segments shall not occupy the same space, the chain will extend over a larger volume than would be calculated on the basis of elementary random flight statistics. As a matter of fact, the overwhelming majority of the statistical configurations calculated without regard for this requirement are found to be unacceptable, on this account, to... 

Richmond and Lai (1974) considered the pressure exerted on two parallel flat plates by a random flight chain confined between them. Adsorption on the plates was not allowed. Not surprisingly, the loss of configurational entropy of the chains on close approach resulted in the generation of a repulsion. This situation is quite unphysical for colloidal systems since the polymer chains in free solution can escape from the compressional zone. Accordingly, the predictions of Richmond and Lai do not relate to depletion stabilization. 

It follows from this that Bernoulli and Markov mechanisms differ in whether the transition probabilities of the crossover, or hetero, steps are the same as or different from the homo steps (see Table 15-6). In addition, both types of mechanism can be subclassified as to whether the transition probabilities for the homo linkages are symmetric or asymmetric. In copolymerization, a symmetric Bernoulli mechanism with constitutionally different monomers is called azeotropic copolymerization with configurationally different monomers, it is called random flight polymerization and in stereocontrolled polymerization with nonchiral monomers, it is also called ideal atactic polymerization. ... 

The simplicity of (2.25) is to be contrasted with the complexity of the exact P(R n) for realistic models of flexible chains for all R (and for small n). When dealing with the complicated problems of nonideal polymer solutions, etc., it is therefore customary to replace the real polymer chain by the so-called Kuhn effective random flight chain. An effective chain is one with N (in general different from n) links of size A5 such that N lS.s = L and (R ) is as given by (2.29). This substitution of a real chain by its equivalent chain is often a necessity so that we may separate errors in principle from errors arising from a poor mathematical approximation to the exact P(R n) when dealing with problems which are not exactly soluble. This equivalent chain therefore provides us with reasonable approximations to the properties of real polymer chains, provided the physical properties of interest do not depend heavily upon those chain configurations with i > L or upon chain properties over small distances for which the real chain is stiff. 

In an actual chain, two remotely connected segments cannot occupy the same space at the same time. As a result of this long-range intramolecular interaction, the actual chain will have a more extended configuration than implied by the random flight consideration. 

The quantity P is a constant for a given series of polymer homologs. Equations (5.19a) and (5.19b) are Flory s theory to describe the polymer chain configuration, de Gennes commented that because of the assumption of Gaussian distribution with respect to R )q, Flory s theory is still intrinsically a random-flight chain in nature. 

Much of polymer theory has been propounded on the basis of the Kuhn equivalent random flight chain, with adjustment of n and Z, or of C, as required to match experimental determination of or of other configuration-dependent quantities. The validity of this model therefore invites critical examination. 

Properties of polymers are intimately related to the configurational characteristics of their molecular chains, and in principle, can be evaluated by averaging over all configurations of the chain. For a random flight chain, the mean-square end-to-end distance r of the chain is given by the familiar expression... 

This hypothetical chain is considered to consist of n bonds of length / joined in sequence without any restrictions whatever on the angles between successive bonds, and with zero volume in the sense that different bonds do not interfere with one another in space. The corresponding chain configuration problem is similar to that of random flight. Specifically, the configuration of the freely jointed chain resembles the path described by a diffusing particle such as a gas molecule. ... 

Increase in the mean-square end-to-end distance , expressed in equations (19), (24) and (26), corresponds to new restrictions imposed on the chain configuration. The mean-square end-to-end distance for real chains consisting of a large number of primary valence bonds exceeds that of their random flight analogs, as represented in equation (28), where is the characteristic ratio. This coefficient represents the degree to which a real molecule departs from the freely jointed model. It may be deduced from equations (25) and (28) that equals two for a tetrahedrally... 

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