Random-flight analysis

The long-range interactions become apparent in good solvents. Flory showed early that the experimental molar mass dependence of the second moment of the end-to-end distance, oc can be derived according to the following scheme. 

The repulsive energy Hr ) is proportional to the volume of the molecular sphere and the number of pairs of bonds  

In section 2.4 expressions were derived for the average end-to-end distance. Random-flight analysis yields an expression for the distribution of the end-to-end distance. 

It is assumed here that the polymer chain takes discrete steps in three dimensions. For simplicity it is assumed that only two types of step exist in each direction forward and backward. The first task is to determine the average step length. The origin of the chain segment is located in the centre of the sphere shown in Fig. 2.17. The other end of the segment is located on the surface of the sphere and it is assumed that the chain segment may have any direction. All parts of the sphere are equally probable. If we cut the 

The average forward (and backward) step length in any direction (x, y or 2) is l/ fi. The next task is to derive the statistics of positive and negative steps. The end-to-end distance is proportional to the net balance of forward (+) and backward (—) steps. In one dimension (x)  

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The statistical variation of the end-to-end distance is considered in so-called random flight analysis. This... 

The concentration of crosslink junctions in the network is also important if too low, flow will be possible if too high, the maximum attainable elongation will be decreased. From the point of view of theoretical analysis, the length of chain between crosslink points must be long enough to be described by random flight statistics. 

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

As a final step in this analysis it may be assumed that these exciton migrations occur by a sequence of random flights for which the mean square displacement in unit time is equal to 6D, The mean square displacement is the product of the frequency of migratory hops (f) and the square of the length of each hop (v). Thus, fi = 6D and, using the mean separation distance of solute molecules a and abitrarily setting q = 1, one finds f values between 500 sec and 600 sec for the three systems of Table II. 

The conformation of polymer chains (in effect, their size) in solution is an important characteristic of a system in solution and at an interface. The detailed analysis of polymer conformations is a very complex process requiring powerful computer facilities. A simpUfied treatment based on random-flight (or random-walk) statistics allows for the estimation of chain dimensions to a degree adequate for most practical situations. 

Jump reorientation models may involve activation over barriers to rotation or the migration of lattice defects or holes. Reorientation is in both cases discontinuous and changes in orientation occur-ing in one step are assumed to be large. Both types of jump reorientation models have been discussed by O Reilly [68], In his quasilattice random flight model, for example, O Reilly [69 70] assumes that the liquid structure up to the first coordination shell may be approximated by a lattice. Some of the properties of the solid state such as vacancies and translational diffusion by vacancy migration are considered present. In general difficulties arise when these jump reorientation models are compared with experimental data because several parameters are needed in the analysis. Furthermore, it appears that O Reilly [71] employs results obtained by Huntress [55] which apply only in the limit of small-step reorientation to treat the case of... 

Clearly, in this case, use of the square-root analysis based on Equation 8.46 for a monodisperse random-flight chain would not be appropriate. In the limit of large Rl-X, the expression for Fj s pC. O) is given by ... 

The first step in the analysis involved checking the recombination yield for different number of P particles randomly distributed on a spherical surface using a dissociation rate of 0.1 ps" From the simulation results presented in Fig. 9.19, both the IRT and random flights results show the same recombination yield for the set of parameters chosen. In order to thoroughly test the algorithm, a wide parameter range was sampled to ensure that no bias was introduced within the IRT framework. The three parameters which most influence the kinetics are (i) the encounter distance on the spherical surface (ii) rate of association (via the surface reactivity parameter) and (ill) the rate of dissociation. A sample of the results for the recombination probability for each case are shown in the Appendix (Sects. D.4-D.6). In all cases, no significant deviations are noticed in the IRT simulation. 

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