Persistent chain

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. 

Fig. 7. (a) Model of beads the polymer chain is represented as a long flexible immaterial filament, on which interacting beads are strung, a - mean-square spatial distance between subsequent beads (b) chain composed of rods connected by flexible spacers the rods have the length 1 and the diameter d, p = 1/d > 1, a is the mean-square distance between the ends of a flexible spacer (c) flexible chain with rod-like side groups (notations are the same as in (b)) (d) persistent chain of width d and of statistical segment length 1, p = 1/d > 1... 

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 ... 

The asymptotic case Zb—>0, 0—>0 (and Zb/ = const) corresponds to the so-called persistent chain (wormlike... 

One must bear in mind that eqn [29] can be applicable only where the Gaussian chain model itself is valid, that is, for 1/4 3> 1. The formfactor for 4I < 1 depends on the details of the chain flexibility mechanism. Consider F(4) for a long persistent chain in the regime 4Rg> 1. Here the formfaaor is defined by eqn [28] with /(4) = g. (r)e d r and for rGaussian chain regime). Hence... 

For a solution of persistent chains the above condition defines the lower boundary for the polymer fraction cp (cf eqn [13]) ... 

The same condition ensures that the requirement, eqn [75], is satisfied if 2 is identified with f In the athermal regime, when the interaction potential is mainly due to the hard core repulsion, ti is dose to p. For an athermal solution of persistent chains (diameter , persistence length /) the condition [98] can be expressed as 0 > nim- where 4> is the polymer volume... 

The continuous wormlike (persistence) chain model is more suitable for the description of large-scale conformational properties of macromolecules with large intrinsic stiffness, for example, a-helical polypeptides or DNA. In this polymer, rotation around the backbone C-C bonds is reduced to small oscillations around one (ground state) conformation because... 

Comparison of eqns [10] and [ 11 ] to eqns [2] and [3] points out that long wormlike chain acquires on the large scale the random coil conformation and its large-scale properties coincide with those of an equivalent freely jointed chain comprising Ni = L/2lp statistical segments each of length t = 2lp. Hence, both the freely jointed and the wormlike persistence chain models can be applied for desaiption of large-scale conformational properties of flexible and semiflexible chain polymers. 

Khokhlov A, Semenov A (1981) Liquid-crysudline ordering in the solution of long persistent chains. Physica A 108 546-556... 

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... 

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