Gaussian chain contour length

In addition to the above result for the size exponent, several quantities such as the probability distribution function for finding a particular end-to-end distance can be derived for the Kuhn model. In particular, the results become simple if the end-to-end distance is smaller than the chain contour length Nl.ln this limit, for example, the probability distribution function for the end-to-end distance is a Gaussian function. In view of this, a Kuhn model chain with large enough N is called a Gaussian chain. The major properties of a Gaussian chain are now summarized. 

A schematic representation of the Gaussian function of Equation (4b) is given by the dashed curve in Figure 1. The abscissa is normalized by dividing the end-to-end distance by the contour length of the chain, and the ordinate is made nondimensional (unitless) by multiplying xv(r) by the contour length. 

Chain for which the contour length is greater than the persistence length but for which their ratio is still below the Gaussian limit. 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

The conformation of the primitive chain becomes Gaussian on a large length scale. This means that if the position of two points on the primitive chain are r(, t) and r(s, t), where 5 and s are the contour lengths measured from the chain end, then... 

Close to the maximum height we obtain a stretching of the chains up to about 2/3 of their contour length. This is certainly out of the range where Gaussian elasticity can be applied. For ZB=0.1 a, Fig. 5 shows snapshots from the equilibrium trajectories. 

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given by... 

We consider a spring-bead chain consisting of N Gaussian springs. Its contour length L is expressed by... 

Equation 2.9 indicates that the orientation correlation of the tangents at contour points 5 and s diminishes exponentially with the contour length As = s — s l between the two points, and the decay is more rapid for smaller q, i.e., for more flexible chains. Thus, in a Gaussian chain (q = 0), there exists no correlation between the orientations of any paired segments. On the other hand, in a rod chain, the reverse is the case as can be intuitively understood. 

An equally necessary reason is that the Gaussian distribution is frequently an exact solution to a stochastic problem. For example, m the tossing of a fair coin, as the number of tosses increases, the Gaussian distribution becomes an increasingly precise description. Or, for the polymer chain, when it is not highly extended and r is considerably less than the contour length, the Gaussian distribution is a useful solution. 

The measured dependence of a on A is shown in Figure 3.7 for natural rubber. The fit is good up to A = 1.2. The fit for compression (A < 1.0) is excellent. The lack of fit above A = 1.2 is due to several factors which include (i) rather simple assumptions in the model (ii) that the chains cannot be Gaussian at high extensions since they cannot extend further than their own contour length (see (3.N.6)) and (iii) at high extensions vulcanized natural rubber crystallizes. 

In order to define the effective gaussian link chain corresponding to (2.36), the maximum contour length is required. The greatest elongation of the chain occurs when all of the bonds lie in a plane, and... 

Although the Wiener integral formulation for the distribution functions of flexible polymer chains rests upon general considerations of random walks and Brownian motion, it is easily introduced, heuristically, through the concept of an equivalent chain. In this section, only those flexible polymer chains are considered which are composed of equivalent gaussian links. Here L is the maximum contour length of the real chain at full extension, and (R ) for the equivalent chain is taken to be that for the real chain. Thus we have... 

The Gaussian statistics leading to equation (9.34) are valid only for relatively small strains—that is, under conditions where the contour length of the chain is much more than its end-to-end distance. In the region of high strains, where the ratio of the two parameters approaches j to j, this limit is exceeded. 

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