Statistical chain-elements

According to Equation (4-27), the chain end-to-end distance depends on parameters independent of the constitution and configuration, such as the number of bonds, as well as on parameters which are dependent, like bond length, valence angle, and steric hindrance parameter. A certain stiffness of the chain can be caused by increased bond length and valence angle, as well as 

In this model (see Section 4.4.1) the contour length is, of necessity, given by the product of the length Ls and the number of statistical chain elements Ns  

The length of the statistical chain element can thus be calculated from the contour length L ont and the experimentally determined chain end-to-end distance. 

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The Gaussian distribution (Eq. III-l) arises if the number of statistical chain elements per chain is large and if the extension of the chain is not too extreme. Treloar 171) has shown that the exact distribution for a chain of N links of length A for any value of N is given by... 

Carr and ZlMM (207), for example, correspond to this choice. A more common procedure is that of Kuhn (153), whose statistical chain element is obtained by applying the condition1... 

It remains for us to discuss the dimensions of polypeptides and other helix-forming chain molecules. Most of the theoretical works on models of such chains have been primarily concerned with the equilibrium of the helix-random coil transition and have not specifically treated the chain dimensions. An exception is found in the work of Nagai (195 ), who combined his theory of the transition with a very simple model of the chain dimensions. It amounts to the assumption that each helical sequence behaves like a rigid statistical chain element without correlation in direction with the randomly coiled sections which are adjacent to it. Then, if a fraction f of the monomer units are members of helical sequences, we may at once write... 

This result may be compared with Eq. (9.7), from which it follows that the persistence length is equal to half the length of a statistical chain element or Kuhn length ap = i A. This representation of the wormlike chain is of particular importance for the description of stiff polymers. 

Kuhn was the first to point out that the dimensions of a chain with given persistence p may always be described as if it were completely flexible (see (5.1.1)) by grouping a number of monomer units together into statistical chain elements (s.c.e.) or Kuhn segments. The number a of bonds in such an s.c.e. is the larger the stlffer the chain. The basic idea is that such s.c.e. s may be considered as orlentatlonally independent they are then independent subsystems as defined in sec. 1.3.6. The real chain of N bonds is now modelled as an equivalent ideal chain of = N/a s.c.e. s and the Kuhn length becomes bt (where a > 1, b > 1). Then (r ) = vdilch equals = 6pN(, provided that a... 

It is also seen in Table 6.1 that the number of statistical chain elements nLjb may be fairly small. If n is smaller than about 25, Eq. (6.3) is not valid any more because the average distribution of the segments is not gaussian any more. Instead, the molecule assumes an elongated form, and rm will be larger than predicted by Eq. (6.3). The extreme is a stiff rodlike molecule of length nL. 

FIGURE 6.4 The effect of the stiffness of a polymer chain on its conformational freedom. This is illustrated for a two-dimensional case, with a fixed, obtuse bond angle, implying two possible conformations at each bond. Although over a distance of two or three segments, the position cannot vary at random, this is possible over a distance of, say, 5 segments, as illustrated. The broken lines would then indicate the statistical chain elements. 

Moreover, the equation can only be accurate for small strains, since considerable change in the end-to-end distance of the cords would distort the Gaussian distribution of statistical chain elements. This happens more readily for a smaller value of It also implies that at increasing strain, the chemical bonds in the primary chain become increasingly distorted. Consequently, the increase in elastic free energy is due not merely to a decrease in conformational entropy but also to an increase in bond enthalpy. If the value of is quite small, even a small strain will cause an increase in enthalpy. (In a crystalline solid, only the increase in bond enthalpy contributes to the elastic modulus.)... 

FIGURE 17.11 The effect of ri (the number of statistical chain elements in a cord between cross-links) on the relation between stress and strain of a polymer gel in elongation. a0 is the force divided by the original cross-sectional area of a cylindrical test piece, v is twice the cross-link density, L is the length, and L0 the original length of the test piece. (After calculations by L. R. G. Treloar. The Physics of Rubber Elasticity. Clarendon, Oxford, 1975.)... 

Let us now try to describe in more detail the shape and flexibility of the molecules. Usually the nature and magnitude of the factors restricting free rotation are unknown or known incompletely. In order to arrive at a general theory, Kuhn has therefore introduced the so-called statistical chain-element, which will be called chain element for short. 

Now let us apply this result to the N statistical chain-elements composing the macromolecule. If they were all moving independently of each other, they would acquire the velocity... 

SEDIMENTATION CONSTANT O OF CELLULOSE NITRATE IN ACETONE M = molecular we ht, N = number of (statistical) chain-elements. 

It has been assumed so far that the behaviour of the molecules could be described with sufficient accuracy by the distance between the two ends. The fact that these ends are connected by a series of chain-elements was expressed by the introduction of the fictive force K [equation (11)]. The further development of the theory required some additional assumptions. We may mention, for instance, that the friction experienced in the motion of the two ends towards each other was assumed to be proportional to the total number N of statistical chain-elements, since on the average all these chain-elements will be involved. Further, a quantitative treatment of the birefringence of flow was only made possible by assuming that the molecules on the average retain their symmetry of rotation in the streaming liquid. 

As a result of the sedimentation, a concentration gradient is set up, and this gives rise to a diffusion which is superimposed on the sedimentation. The formula (11) assumes that the diffusion is not (or not yet) effective. The frictional constant w depends on M. With compact spherical particles w is proportional to the radius (Stokes law) and therefore to MV . We know from the chapter on random coiling, p. 103, that w in dilute solutions of kinky long-chain molecules is approximately proportional to M(1 -f ]/iv) , where N is the number of statistical chain-elements in the molecule. The quantity w can be eliminated, however, if we measure, in addition, the diffusion constant... 

The crystallites act as the junction points of the network, the amorphous portion behaves more or less rubberlike, but the flexibility of the chains, the degree of convolution and further, the average number of statistical chain elements between the junction points, seem to be different, all these quantities being smaller than in rubber. Furthermore, energetic factors, bound up with the much greater intermolecular cohesive forces in cellulose, change the picture and render it far more complicated. 

The value of K will be the higher, the larger G and hence, the greater the number of junction points per unit volume. From equations (60) and (61) follows that. the retractive force and the birefringence should be proportional. This is actually borne out by experiments in the initial phased of extension up to not too large extensions These theories were invariably based on the assumption that the number of statistical chain elements between the junction points is at%e. Calculations for a... 

As will be set forth later in more detail, there is, however, an intrinsic difference between the cellulose gels and rubber, in as far that cellulose chains are considerably stiffer than those of rubber. The length of the statistical chain element comprises about 20 glucose residues (cf. Ch. IV, p. 109). A cellulose chain of given length is... 

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