Freely-jointed-rod chains

Liquid crystalline polymers can be regarded as a long chain with rods connected in sequence, each rod being, in some sense, equivalent to a small molecular mass liquid crystal. This is the so-called freely-jointed-rod chain, the simplest model of polymers. It is understood that the constituent units — small molecular mass liquid crystals play an essential role in liquid crystalline polymers. Here, we introduce an important theory for small molecular mass liquid crystal — the Maier-Saupe mean field theory (Maier Saupe, 1959, 1960). 

In a very crude sense, liquid crystalline polymers can be regarded as a freely-jointed-rod chain, shown in Figure 2.16. The freely-jointed-rod chain consists of a series of repeated segments of length lo- Each segment is able to rotate freely. It is assumed that the freely-jointed-rod chain is a replica of its small molecular mass liquid crystal counterpart in the liquid crystalline properties. 

The conformation of a freely-jointed-rod chain in the nematic liquid crystal is anisotropic and its end-to-end distance component along the z axis is evaluated as follows... 

It is understood that the extension of a freely-jointed-rod chain along the director is due to the preferred orientation of each segment. The maximum (R2) is less than Nl2, which is a result of random walk. In other words, a freely-jointed rod chain cannot produce the rod-like conformation expected for liquid crystalline polymers at low temperature. 

It is possible to design a theory for a chain consisting of several freely jointed rods 73). Such a model reveals certain non-monotonous dependences of Kd on the number of rods (molecular weight) close to the critical conditions. 

The jointed-rod chain is one-dimensional system of a series of repeated segments and we naturally apply the Transfer matrix technique in the Ising model to obtain the partition function for both the freely-jointed-rods and the elastically-jointed-rods. We will not go into the details of this model but instead will show some results derived from the model. 

Rory [29] proposed a lattice theory that accommodates chain flexibility in liquid-crystalline polymers in a later version. The critical concentration for mesophase formation depends for freely jointed rods on the aspect ratio of the individual Kuhn segment, that is, instead of the contour length, the axial ratio of is used for the calculation. The critical volume fraction of the polymer V is determined by the aspect ratio x, and Eq. (8) was derived as... 

An alternative approach to phase separation for rigid and freely jointed rods (38) and for worm-like chains (39,40) has been proposed by Khokhlov and coworkers. Their work is based not on lattice models but on Onsager s theory for the phase separation of rigid rods. These theories are valid only for very low concentrations, so their applicability to the phase separation of cellulose based mesophases may be questionable. 

Fig. 1. Freely jointed bead-rod model of a chain formed by(N + 1) beads and N rigid links of length Ip...

In the case of a binary incompressible mixture of stiff homopolymers (components are named A and B), the above equations simplify. Assuming that component A is flexible (freely-jointed chains) and B is rigid (rigid rod polymers) and imposing the incompressibility condition, the following result can be obtained ... 

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. 

We will assume that the segments interact in the same way as disconnected rods, i.e. as described in Sect. 2.2. Although we will consider explicitly only the freely jointed chain, we will always indicate which of the results obtained depend on the flexibility distribution along the chain contour and which results are in this sense universal. 

At a given force, the elasticity of covalent bonds of the amino acid backbone gives rise to a length increase. But thermal fluctuations act on the backbone, which on an average pulls the cantilever closer to the membrane, a phenomenon referred to as entropic elasticity of linear polymers. The wormlike chain model [50] describes the polymer as an elastic rod with bending stiffness submitted to thermal fluctuations that decrease the end-to-end distance of the rod. Alternatively, the freely jointed chain model calculates the... 

In 1944 Kramers [1] published a phase-space kinetic theory for the steady-state potential flow of monodisperse dilute polymer systems in which the polymer molecule is modeled as a freely jointed bead-rod chain. Subsequent scholars developed kinetic theories for shearing flows of monodisperse dilute polymer solutions Kirkwood [2] for freely rotating bead-rod chains with equilibnum-averaged hydrodynamic interaction. Rouse [3] and Zimm [4] for freely jointed bead-spring chains, and others. These theories were all formulated m the configuration space of a single polymer chain. 

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