Random-flight model

The relation (1 10) leads to a number of interesting consequences. In a theta solvent, in which the shape of the chain is described by the random flight model, is proportional to M2, so that the intrinsic viscosity should be proportional to M /2. And this prediction has been applied and verified. In solvent media better than 0-solvents, the theory of Flory [11,46] predicts that the linear expansion factor a increases for any polymer - homologous series with chain length. Thus the exponent v in the empirical equation should be larger than 0.50. 

The factor Cw> the Flory characteristic ratio of the actual end-to end distance to that predicted on the basis of a random flight model, obviously depends on chain stiffness or bond rotational freedom (Equation 8-11). 

The random-flight model used in the previous section underestimates the true dimensions of polymer molecules, because it ignores restrictions to completely... 

Barzykin A. V. and Taehiya M. (2002), Meehanism of eharge reeombination in dye-sensitized nanocrystalline semieonduetors random flight model , J. Phys. Chem. B 106,4356 363. 

The limitations of the random-flight model when applied to real polymer chains arise from... 

Nevertheless, the monomer moment calculated from the random flight model for polybutyl methacrylate in CC14 at 40°, is 1.90 D, a value which is very close to that of most aliphatic esters. This would seem to prove that, at temperatures higher than 40°, the moment has reached its highest possible value. 

In defence of the artifice under criticism, it can be argued that the uncertainty introduced by this artificial separation is significantly less than that stemming from the simulation of a real polymer chain by the primitive random flight model, which doesn t even recognize the necessity for such a separation. It should also be recalled that this separation was exploited by... 

The reason why the random flight model has proved so popular theoretically stems from its simplicity, which offers hope for the development of analytic solutions. The problem can usually be cast in the form of a diffusionlike or a Schrodinger-wave-equation-like differential equation, the solutions of which are reasonably well explored. A tendency has developed in recent times to apply extremely sophisticated mathematical procedures to what are really very primitive models for polymer chains (see, e.g. Levine et al., 1978). Whether the ends merit the means in such instances cannot yet be assessed objectively. A strategy that might be more productive in terms of the development of a practical theory for steric stabilization is to aim for a simpler mathematical description of more complex models of polymer chains. It should also be borne in mind in developing ab initio theories that a simple model that may well suffice in polymer solution thermodynamics may be quite inadequate for the simulation of the conformational properties of polymers. Polymer solution thermodynamics seem to be relatively insensitive to molecular architecture per se whereas the conformation of a polymer chain is extremely sensitive to it. 

The result is characteristic for all processes based on a random flight model. Such processes also include diffusion. The random flight model is distinguished by the fact that each step (in the above case, bond direction) is independent of the previous one. 

The summary of dimensions at the bottom of Fig. 1.33 demonstrates the effects of the various restrictions on the model compound. The random flight model gives for polyethylene already an answer within a factor of about 2.5. Comparison with the experiment is possible by analyzing the dimensions of a macromolecule in solution, as will be discussed in Chap. 7. One can visualize a solution by filling the vacuum of a random flight of the present discussion with the solvent molecules. The 0-temperature listed as condition for the experiment is the temperatme at which the expansion of the molecule due to the excluded volume is compensated by compression due to rejection of the solvent out of the random coil. This compensation of an excluded volume is similar to the Boyle-temperature of a real gas as illustrated in... 

Next we consider the statistical distribution of the end-to-end vector of the random flight model. Let (H, N) be the probability distribution function that the end-to-end vector of the chain consisting of N links is R. Given the conformational distribution for W( r ), [Pg.11]

Chain End-to-End Distance and Radii of Gyration 4.4.2.1. The Random-Flight Model... 

The freely jointed chain model (known also as random flight model) was proposed for polymers by Kuhn in 1936. The chain is assumed to consist of n bonds of equal length I, jointed in linear succession, where the directions 6, cf>) of bond vectors may assume all values (0 tt ... 

Fig. 1.2 Typical coarse-grained models of a polymer chain (a) random flight model, (b) bead-spring model, (c) lattice model.

Fig. 1.3 Random coil formed with the random flight model with 200 bonds produced in three dimensions...

The single chain partition function <2i (/, T) under the action of the tension / for the random flight model chain with a bond length a takes the asymptotic form... 

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