Rigid-rod model

The first molecular theory of liquid-crystalline ordering was proposed by Onsager on a solution of cylindrical, long, rigid rods. The theory may be described as follows. Let N2 be the number of rods in a volume V. Each rod has length L and cross-sectional area d. The volume faction of rods, V2, in the solution can be calculated as 

The first term is the free energy of the translational motion of the rods the second term is the loss of orientational entropy, and the last term the fi ee energy of interaction of the rods in the second virial approximation. The term B y) is the 

When only steric interactions of the rods are present, the term B y) is expressed as 

The next step is to find the equilibrium distribution function of/(n). Onsager used a standard approximation method, namely, the variation principle (well known in mathematical physics) to attack the problem. Taking h as a variational parameter, a new angle 9 is defined between n and the direction of the anisotropy axis. He then took as a trial function 

At Vj V2 v the solution separates into isotropic and anisotropic phases, the values of v 2 and v being 

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A) Onsager s rigid-rod model (1949) was the first correct model of an athermal Isotropic —> Nematic phase transition. It is a relatively simple model that predicts phase transitions. It is based on excluded volume between two rigid rods, which reads... 

In this case if the molecule is large enough the sum will be time-dependent. The rigid rod model, an example of this case, is discussed in detail below. 

The calculation may be completed by substituting the explicit forms of the spherical Bessel functions into Eqs. (8.B.26) and (8.B.27) and then performing the summation. Since for the rigid rod model there is a constant anisotropy per unit length, the summation may, as in the isotropic case in Section (8.2), be replaced by an integration, that is,... 

Table II shows the values of these constants for five different rigid rod models. The predicted reduced moduli from the theory of Marvin and...

The question now arises, what model(s) make these data understandable in molecular terms In Figure 2 is shown the non-Newtonian intrinsic viscos- ity for the rigid rod model, with length 1.5 m and midpoint diameter 1.9 nm, as calculated by Whitcomb and Macosko (2)- The model gives a satisfactory fit to the experimental data, including the new data for the first Newtonian plateau. 

Figure 3. Diagram of the effect of shear gradient on shape and orientation of a wormlike xanthan chain and a rigid-rod model.

Summary. The non-Newtonian intrinsic viscosity of xanthan can be explained either by a bead-spring model or by a rigid rod model with appropriate parameters. A Kuhn-equivalent chain with about 200 repeating units per link and about 50 links per molecule is in my view more consistent with all the data than is a rigid rod model. 

A rigid rod model has been proposed for the meningococcal polysaccharides ( ) as well as the bacterial polysaccharide produces by Xanthomonas campestris (12) (Xanthan gum). Such a model assumes the form of the correlation function to be (10)... 

We have taken two approaches in con iaring our results to the rigid rod model. First, we have used the literature value for the axial ratio (6), 50 to 60, and our measured value of the... 

In the present paper, we first review briefly the rigid rod models for liquid crystalline phase transitions. In these models, emphasis is placed on the anisotropic form and on the orientation dependent intermolecular interactions between rigid particles. Conformational studies on isolated chains have shown that liquid crystalline polymers are rather semi-rigid in character although only a narrow range of deformations is possible due to intrachain interactions. The effect of chain flexibility on the formation of liquid crystalline phases has been pointed out both experimentally and the-oretically J. 

Secondly, the flexibility gives a relaxation in the high-firequency region. For example, according to the rigid rod model, the complex intrinsic viscosity [ (a>)] approaches the finite value... 

The parameter X reflects the linear charge density of the polyion in a rigid rod model, X is given by... 

Figure 48. Origin of the spontaneous polarization in a smectic C phase. The illustration shows the directional bias for a lateral dipole in the rigid rod model of a molecule, with the directionality diagram for the short axis when the molecule is rotating about its long axis (from Blinov and Beresnev [45]).

These mesophases are defined and characterized by maity physical parameters such as long- and short-range order orientational distribution functions, and so oa They are explained in greater detail in the following chapters. Here we continue to use the rigid-rod model and pictorially describe these phases in terms of their molecular arrangement... 

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