Flight, random

Treating each bond as a vector, an end-to-end-distance vector r can be calculated. To simplify the rather lengthy computation for which all bond- and rotation-angles 

Projection of a Random Flight of x Steps and Indication of the End-to-End Distance r and the Mean-square End-to-end Distance r  

This simple calculation gives an idea of the possible amount of coiling of the molecule. The root-mean-square end-to-end distance increases with the square root of the number of bonds, while the contour length grows linearly. To fully extend such random coil, one has to use a draw-ratio of 141 x, much more than is usually possible in drawing of polymeric fibers for textile applications which is 5- lOx. But note, that the gel-spun, high-molar-mass polyethylenes, which are discussed in Sect. 6.2.6 have a draw ratio of more than lOOx. 

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Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

We shall rely heavily on models again in this chapter this time they are of two different types. We shall consider elasticity in terms of a molecular model in which the chains are described by random flight statistics. The phenomena of... 

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. 

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

This equation shows that at small deformations individual chains obey Hooke s law with the force constant kj = 3kT/nlo. This result may be derived directly from random flight statistics without considering a network. 

In connection with Eq. (3.45) we noted that the deformation of individual chains can be studied directly from random flight statistics. Using equivalent expressions for the x, y, and z components of force and following the procedure outlined above gives a more rigorous derivation of Eq. (3.39) than that presented in the last section. 

The length of the subchain is sufficient to justify the use of random flight statistics in its description. 

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

In earlier chapters an unperturbed coil referred to molecular dimensions as predicted by random flight statistics. We saw in the last chapter that this thermodynamic criterion is met under 0 conditions. 

Next we consider the situation of a coil which is unperturbed in the hydro-dynamic sense of being effectively nondraining, yet having dimensions which are perturbed away from those under 0 conditions. As far as the hydrodynamics are concerned, a polymer coil can be expanded above its random flight dimensions and still be nondraining. In this case, what is needed is to correct the coil dimension parameters by multiplying with the coil expansion factor a, defined by Eq. (1.63). Under non-0 conditions (no subscript), = a(rg)Q therefore under these conditions we write... 

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

The same effect happens inside a random flight chain where the close proximity of the polymer segments offers mutual screening from the bulk flow field. The idea of a chain being non-drained was first considered by Debye Bueche who introduced the concept of a shielding length defined as [46] ... 

The dashed line represents values evaluated by Casassa (10) for random-flight chains with infinitely many bonds of infinitesimal length. The dot-dashed line shows results of Giddings et al. (11) for rigid rods. 

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... 

The exponent a is a function of solvent power usually a > 1/2, but for an idealized random-flight polymer, a = 1/2. 

Here x=qNb/6=qR where Ru is the radius of an undeformed Gaussian random flight chain. I deriving Eq. 10, the sum in Eq. 7 is replaced by an integral. The effect of the free chain segments, exclusive of the position of the junctions appears in the first two terms in the exponential. The deformations of the chains depend on the constraints on the junctions. Results are immediately derivable from Eq. 10. 

A great deal of work has been done by Hummel and co-workers on electron-ion recombination in hydrocarbon liquids using random flight MC simulation. This will be discussed in Sect. 7.5. 

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). 

Bartczak et al. (1991 Bartczak and Hummel, 1986, 1987,1993, 1997) have used random flight MC simulation of ion recombination kinetics for an isolated pair, groups of ion-pairs, and entire electron tracks. The methodology is similar... 

Edwards (10) has treated the concentrated region by considering a mean field approximation. The problem is to solve the random flight or diffusion equation in a uniform field provided by the segments (from all chains). This field is proportional to p, but is independent of position. It was shown that, under these conditions, 5 is again r (i.e. a = 1, for all solvents). 

Figure 3 for four chain lengths, expressed as the number of segments per chain, r. As expected, in dilute solutions A is independent of In this dilute regime, A is approximately proportional to the square root of the chain length for not too short chains A/l = 0.56 (Sr - 2), which is of the order of rg/1. Note that for a random flight chain rg/1 = /r/6 = 0.41 Sr, and a lattice chain is expected to be slightly more expanded. 

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