Macromolecule freely-jointed segment model

Formulae (1.2) and (1.3) determine the model of a freely-jointed segment chain, which is frequently used in polymer physics as a microscopic heuristic model (Mazars 1996, 1998, 1999). A Kuhn segment in the flexible polymers (polyethylene, polystyrene, for example) usually includes a few monomer units, so that a typical length of the Kuhn segment is about 10 A or 10-7 cm and, at the number of segments 2 = 104, the end-to-end distance (R2)1 2 of a macromolecule is about 10-5 cm. 

In such a way, there are two universal, (that is, irrespective of the chemical nature) methods of description of a macromolecule either as a flexible thread or as freely-jointed segments. Either model reflects the properties of each macromolecule long enough to be flexible. A relation... 

It is noteworthy that Eq. (3.1), as well as the other results of this section, can be applied not only to the model of freely jointed segments but also to any other model of semiflexible macromolecules it is necessary only to replace p in all equations by the ratio of the effective segment length to its width. In fact, the translational entropy... 

To consider the relative permittivity tensor of polymeric system, one makes use of the heuristic model of a macromolecule as freely-jointed segments each macromolecule consists of z segments and is surrounded by solvent molecules (Sect. 2.1). Then, the simple old-fashion [111, 112] speculations allow us to determine the relative permittivity tensor of polymeric system in terms of the mean orientation of anisotropic segments of the macromolecules (6,6 ). The relative permittivity tensor is formulated below to within first-order terms in the orientation tensor... 

One can use another way to describe the long macromolecule. One can see that at high temperatures there is no correlation between the orientations of the different parts of the macromolecule, which are not close to each other along the chain. This means that the chain of freely-jointed rigid segments reflects the behaviour of a real macromolecule. This model carries the name of Werner Kuhn who introduced it in his pioneering works (Kuhn 1934). 

For the model of semiflexibie macromolecule accepted in Sect. 3.1, the attractive part of the second virial coefficient of the interactions of segments is given directly by Eqs. (2.16) and (2.21). Thus, the procedure for the determination of the most stable homogeneous phase of the solution of freely jointed semiflexibie macromolecules is absolutely identical with the corresponding procedure for the solution of disconnected rods (compare Eqs. (2.3) and (3.1)) consequently, this procedure leads to Eqs. (2.25) (see also Fig. 5). 

In these formulae, n is the density of the number of macromolecules in the solution, while the coefficient of the anisotropy of the macromolecular coil T is given by the following expression in the case where the macromolecule is modelled by a freely-jointed chain of Kuhn segments... 

The diffo ence in the character of the nematic ordering in solutions of semiflexible macromolecules with diffaent mechanisms of flexibility is not only manifested in the thermodynamic characteristics of the phase transition itself, but also in the conformations of the polymer chains in the liquid-crystalline phase. For example, the dependences of the root-mean-square distance between chain ends (/ 2) on the concentration of polymer in the solution for semiflexible freely jointed and persistent chains calculated in [43,44] are shown in Fig. 1.4. Note that for the freely jointed model, the value of (jf) is almost independent of the concentration of the solution in the anisotropic phase (i.e., orientation of the segments but not uncoiling of the macromolecules takes place), while for a solution of persistent chains, the increase in (/ ) in the anisotropic phase with an increase in the concentration is very signiflcant (exponential). A solution of chains with the rotational-isomeric mechanism of flexibility (cf. Fig. 1.2c) behaves analogously in this case, as demonstrated in [35], in the... 

---