Random flight statistics

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. 

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 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... 

If interactions between parts of the molecule separated by many links (the excluded volume effect ) is absent, so that the chains obey random-flight statistics, takes its unperturbed value, (s ). Theoretical calculations of the dimensions of branched molecules usually assume random flight chains, and values of the mean-square radius so obtained are estimates of . 

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

Fig. 5. Interrelation between critical concentrations of crosslink forming (a) and ring forming (a) functionalities at the gel point calculated on the basis of random flight statistics for trifunctional polycondensation (/ = 3) and different values of parameters A and 91 [Gordon and Scantleburry (72)]...

The Bueche model is based on random-flight statistics of freely draining polymer molecules. Accordingly, two possible relations exist between g and g". (1) For the Zimm-Kilb model, combining Equations 2, 3, 8, and 11 the relation obtained is given by ... 

It is understood that the branched and linear molecules contain the same number of bonds. If tij denotes the number of bonds in branch /, the application of random flight statistics leads to a very simple expression for gJ28 ... 

In the absence of both types of interaction, except for the covalent binding forces which fix the lengths of the chain links, a long chain molecule obeys Gaussian or random-flight statistics. Under these conditions, the mean square value of the (spherical) radius of gyration, for example, is given by... 

Gas-polymer-matrix model, 687 Gaussian or random-flight statistics, 246 Gel layer, 697... 

Gupta SK, Forsman WC. A general treatment of random-flight statistics with and without excluded volume. Macromolecules 1972 5 779. 

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). 

The characteristic dimensions of the blocks were calculated assuming random flight statistics. Some of the interesting conclusions reached by Inoue et al. (1970a,b) are as follows. 

A somewhat more sophisticated approach to domain formation and fine structure was taken by Meier (1969,1970). As with Inoue et al. (1970a,f ) and Krause (1969,1970,1971), the A-B junction was restricted to a location somewhere in the interfacial region. Meier s model (Figure 4.11) assumes that random flight statistics and regular solution theory hold, that statistical chain segments (not block lengths) are of equal size, and that chain perturbation is characterized by the usual parameter a ... 

The theory of cyciization in condensation polymerization was first investigated by Kuhn in the 1930s with the introduction of the concept of effective concentration (Ceff), which is the local concentration of two chain ends of the same molecule for a Gaussian chain. This measurement provides a relationship between the end-to-end length of a polymer chain and that same chain s propensity to cyclize. Therefore, C s provides a method of quantification for the propensity of intramolecular interactions and cyciization, and Kuhn predicted that the cyciization probability decreases as where N is the number of bonds in the chain. Several other treatments have addressed the calculation of C s as a function of chain length using either random-flight statistics or a particle-in-a-sphere approximation. ... 

When the chains are extended, their conformations may be considered as being determined by equilibrium between the forces of expansion due to excluded volume and the forces of contraction due to chain segments expanding into less probable conformations. Based on random flight statistics, the chains are extended linearly by a factor a over their dimensions. The acmal root-mean-square end-to-end distance is equal to The change in the elastic part of free energy is... 

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