Hard-rod theory

These contributions should be added to the elastic viscosities [the af s in Eq. (11-19b)] calculated by Kuzuu and Doi, who neglected the viscous stress. Table 11-1 shows values of ratios of viscosities computed from the hard-rod theory both without the viscous term, and with a viscous term with fiy = 0.03, computed from Eqs. (ll-19a) and (11-20), for PBG of molecular weights M = 70,(XX) and 130, 0. The molecular lengths are giverr by L = M/Ml = 48 nm and 89 nm, where Ml = 1460 nm for PBG molecules. The... 

Generally speaking, compounds exhibiting the Sc phase have transverse components of permanent electric dipole moments. A number of molecular statistical models (including hard rod theories for systems composed of oblique cylinders) have been developed. " Goossens " has proposed a model composed of ellipsoidal molecules with attractive interactions arising from anisotropic dispersion forces as well as permanent quadrupole moments. His calculations show that the interaction between the permanent quadrupole moments can produce a tilting of the molecules, but a detailed comparison of the predictions with experimental data has yet to be made. 

A very simple model that predicts lyotropic phase transitions is the hard-rod model proposed by Onsager (Friberg, 1976). This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder where the... 

Comparison with Other Theories for Hard Rods.100... 

This conclusion was reached, tentatively, by Frenkel, Shaltyko and Elyashevich A phenomenological analysis presented by Pincus and de Gennes predicted a first-order phase transition even in the absence of cooperativity in the conformational transition. These authors relied on the Maier-Saupe theory for representation of the interactions between rodlike particles. Orientation-dependent interactions of this type are attenuated by dilution in lyotropic systems generally. In the case of a-helical polypeptides they should be negligible owing to the small anisotropy of the polarizability of the peptide unit (cf. seq.). Moreover, the universally important steric interactions between the helices, regarded as hard rods, are not included in the Maier-... 

A. Lattice Theory for Hard Rods with Exact Treatment of the Orientation Distribution... 

Crystalline or orientational orderings are mostly controlled by repulsive forces, such as excluded-volume forces. The crystalline transitions in ideal hard-sphere fluids and the nematic liquid crystalline transitions in hard-rod suspensions are convenient simple models of corresponding transitions in fluids composed of uncharged spherical or elongated molecules or particles. The transition from the isotropic to the nematic state can be described theoretically using the Onsager, Maier-Saupe, or Rory theories. 

Kimura, H. Hosino, M. Nakano, H. Statistical theory of cholesteric ordering in hard-rod fluids and liquid crystalline properties of polypeptide solutions. J. Phys. Soc. Jpn. 1982, 51 (5), 1584-1590. 

A realistic theory of nematics should, of course, incorporate the attractive potential between the molecules as well as their hard rod features. There have been several attempts to develop such hybrid models. Equations of state have been derived based on the Percus-Yevick and BBGKY approximations for spherical molecules subject to an attractive Maier-Saupe potential.However, a drawback with these models is that they lead to y = 1 (see (2.3.18)). 

The first consistent statistical theory of the dipolar flexoeffect was developed by Straley for an athermal nematic composed of polar hard rods. Helfrich and Petrov and Derzhanski also proposed explicit expressions... 

In this chapter we have presented the free volume theory for hard spheres plus depletants and focused on the simplest possible case of hard spheres + penetrable hard spheres. In the next chapters we will extend the free volume theory to more realistic situations (Chap. 4 hard spheres + polymers. Chap. 5 hard spheres -I- small colloidal particles. Chap. 6 hard rods -I- polymers) and compare the results with experiments and simulations. 

Later, Burning and Lekkerkerker [37] observed isotropic—nematic phase separation in a dispersion of sterically stabilized boehmite rods, which approximate hard rods, in cyclohexane. Buitenhuis et al. [43] studied the effect of added 35 kDa polystyrene (/ g = 5.9nm) on the hquid crystal phase behaviour of sterically stabilized boehmite rods with average length L = 1.1 nm and average diameter D = ll.lnm in ortho-dichlorobenzene. Different phase equihbria were observed. Two biphasic equilibria dilute isotropic phase Ij + nematic N, concentrated isotropic phase I2 + nematic N and a triphasic equilibrium 1 -F I2 + N (see photo. Fig. 6.20). In this system the boehmite rods are quite polydisperse. Therefore comparison with theory should be done with an approach including polydisperse rods. We further note no li +12 coexistence was observed experimentally but... 

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