Random flight theory

This is shown in Figure 12.7, which demonstrates that the maximum probability eorresponds to the most probable dimension for the chain. Assuming the root-mean-square end-to-end distanee represents the most probable chain dimension, then, according to random flight theory. 

According to the random-flight theory, the size of a molecule increases in proportion to N -, where size means either the average end-to-end distance (r-) = Nh or the most probable radius Rq = 2Nb /3. We showed on pages... 

The unperturbed mean-square radius is calculable for polymers of known structure, on the assumption of random-flight chains (Section 3). It has usually been assumed that random-flight conformations are adopted at the temperature Al at which A2 is zero, according to Flory s theory (18). Light... 

N 077 "Unperturbed Dimensions of Crosslinked Histones Evaluated Using Random-Flight Statistics and Rotational Isomeric State Theory"... 

Noyes [269, 270] and, more recently, Northrup and Hynes [103] have endeavoured to incorporate some aspects of the caging process into the Smoluchowski random flight or diffusion equation approach. Both authors develop essentially phenomenological analyses, which introduce further parameters into an expression for escape probabilities for reaction, that are of imprecisely known magnitude and are probably not discrete values but distributed about some mean. Since these theories expose further aspects of diffusion-controlled processes over short distances near encounter, they will be discussed briefly (see also Chap. 8, Sect. 2.6). 

The time dependence of recombination probability in the limit of almost certain recombination on the first encounter, and for small diffusive steps (a, j3 1), arises from the time taken for the radicals to diffuse together. From the theory of random flights, the probability that an isolated pair of reactants, initially separated by r0, will encounter one another during a time from t to t + df is given by... 

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. 

Krigbaum has formulated a theory predicting the dimensions of the domains in the case of a lamellar structure. For AB copolymers53, each subchain behaves as a random flight with reflecting barriers. For BAB copolymers54, the possibility... 

Let us consider a reaction where the cage recombination occurs only through the singlet close pair but not through the triplet close pair as shown in Fig. 3-1. In such a reaction, Noyes [4] showed that the probability (f(r)) of the first re-encounter between t and r+dr for a pair, separating from an encounter at /=0, is given from the theory of random flights as... 

Levy flights are the central topic of this review. For a homogeneous environment the central relation of continuous time random walk theory is given by [14,45]... 

A simpler approach to steric and depletion stabilization is to use the predictions of the random flight chain at the same reduced distance, say, HoKr y. The chain dimensions are used here as an arbitrary reduction parameter. The validity of this simple procedure can only be assessed after an exact theory has been elaborated. Unfortunately, such an... 

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. 

Hesselink et al. based their theory on random flight statistics appropriate for a six-choice cubic lattice (Hesselink, 1969 1971). In the absence of a second interface, the segment density of the isolated chain was calculated much in the spirit of the Meier approach but using a procedure that correctly eliminates the conformations that transgress the barrier (Hoeve, 1965 Hoeve et al., 1965). This met)iod was subsequently extended to allow for the presence of the second impenetrable interface (Hesselink, 1971). 

Theories of elastic steric stabilization Dolan and Edwards (1974) have calculated the elastic free energy of repulsion for isolated polymer chains attached terminally to parallel flat plates. It was shown in Section 11.4.1.1. that probability distribution function for a random flight chain obeys the diffusion equation... 

Scheutjens and Fleer (1982) have developed a theory for depletion stabilization and depletion flocculation based upon their statistical thermodynamic approach to polymer adsorption and steric stabilization. This theory is cast in terms of the most primitive model for a polymer molecule, the random flight chain. This weakens the theory in so far as providing quaintitative predictions at the fundamental level for real systems is concerned. The theory does, however, offer qualitative results over a wide range of conditions, being especially powerful in establishing the various trends involved. 

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