The Persistent Chain

Making use of Eq. (2.117), we reahze what z actually means. We obtain 

It is always useful to check for the number of independent parameters. For the Brownian chain there is only one parameter, namely Rq. For the expanded chain in general, we find two parameters, Rp and t, but in the Kuhnian limit 

The basic structure of the persistent chain has, indeed, already been introduced in Fig. 2.5 at the beginning of Sect. 2.3. It shows a chain with varying curvature being represented by a curve of length l t, which possesses at each point a well-defined tangent vector, e(/). In order to describe the chain structure, statistics was employed and the orientational correlation function iFor(A/) introduced by Eq. (2.5) 

The persistent chain model is obtained by an obvious choice for Kot, namely the exponential function 

The persistence length is included as the characteristic parameter and is equal to the integral width of Kot (Eq. (2.7)). A basic property of Kot is 

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In Sect. 3, we will consider the orientational ordering in the solution of semiflexible macromolecules. In general, semiflexible macromolecules can have different flexibility distributions along the chain contour compare, for example, the freely-jointed chain of the long thin rods (Fig. 1 b) and the persistent chain, which is homogeneous along the contour (Fig. lc). We will see what properties of the liquid-crystalline transition do depend on the flexibility distribution along the drain contour and what properties are universal from this point of view. 

It should be emphasized that these features of the transition between coil and liquid-crystalline globule, as well as the expression (3.7) for the temperature of this transition, are not sensitive to the specific polymer chain model in the limits p S> 1 and N > 1. In particular, the above results remain valid for each of the models shown in Fig. 7b-d, i.e. the chain composed of rods connected by flexible spacers (Fig. 7 b), the flexible chain with the rodlike side groups (Fig. 7 c) and the persistant chain (Fig. 7d). Such universality can be proved by means of the following simple arguments. 

Expression (5.14) gives the operator g for the persistent model. A similar operator has already been used for the analysis of the persistent coils in Ref.35-37 1Z. In particular, in these references it was shown that the effective segment of the persistent chain described by the operator (5.14) is equal to... 

It is clear that the macroscopic characteristics of the persistent chain can depend only on the combination (5.16) of the microscopic parameters i and 6. 

In the persistent chain there are no points of easy bending the persistent macromole-cule prefers the conformations exhibiting a constant small curvature. Thus, as soon as the chain elements in the liquid-crystalline globule are oriented mainly in the tangential direction (see Sect. 4), the small globule formed by the persistent chain must display the tendency to form the cavity in the middle of the globule. In this case, the globule assumes the shape of either a torus or a spherical layer. 

Upon application of the smoothing procedure (see Sect. 3.3) to Eq. (5.3), it is possible to find the following expression for the conformational entropy of the globule formed by the persistent chain ... 

Different from the ideal chain description, the persistent chain model is not only able to deal with short chains, but also addresses the effect of temperature. Another model, which will introduced as the Ising chain in the next section, accomplishes these tasks in an even more detailed, rather perfect manner. 

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