Liquids, Onsager theory

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Onsager Theory for C(t) for Non-Debye Solvents. Generally solvents have more complex dielectric responses than described by the Debye equation (Eq. (18)). To obtain the time dependence of the reaction field R from Eqs. (12, (15), (16) and (7) an appropriate model for dielectric behavior of a specific liquid should be employed. One of the most common dielectric relaxation is given by the Debye-type form, which is applicable to normal alcohols. 

The Kamlet-Taft u polarity/polarizability scale is based on a linear solvation energy relationship between the n it transition energy of the solute and the solvent polarity ( 1). The Onsager reaction field theory (11) is applicable to this type of relationship for nonpolar solvents, and successful correlations have previously been demonstrated using conventional liquid solvents ( 7 ). The Onsager theory attempts to describe the interactions between a polar solute molecule and the polarizable solvent in the cybotatic region. The theory predicts that the stabilization of the solute should be proportional to the polarizability of the solvent, which can be estimated from the index of refraction. Since carbon dioxide is a nonpolar fluid it would be expected that a linear relationship... 

The Onsager theory for the molecular polarizability28 implies that the water molecules can rotate freely in the liquid water and that their permanent dipole moments are higher in the liquid than in the vapor phase (because... 

However, recalculating the value of y using the method described in the paper for the field factors, gives the value in brackets. The unbracketed value, for the overall microscopic nonlinearity, converts to 2859 au. In the case of associating liquids the authors argue that equation (7) can be used in modified form with the inclusion of a factor, g, which they deduce from the Kirkwood-Frohlich modification of the Onsager theory,... 

ONSAGER THEORY FOR RIGID —ROD LIQUID CRYSTALLINE POLYMERS... 

It can be easily extended to disc-like molecules (de Gennes, 1973) that form a liquid crystal phase because of their anisotropic shape. The Onsager theory has also been extended to treat the poly-disperse polymer system (Parsons, 1979) and semi-rigid chain systems (Lekkerkeker et al, 1984). 

The Flory theory discussed in the next section is another important theory on rigid liquid crystalline polymers. Because of its clear picture of the lattice model and the incorporation of the Onsager theory, it has become a basic method for the theoretical study of liquid crystalline polymers. As a result of the constant efforts of Flory and his co-workers, the theory has been applied to binary and poly-disperse systems and also includes the soft interactions. 

The Onsager theory points out that liquid crystalline polymers enter from the liquid crystal phase to the isotropic phase at the volume fraction... 

Priest (1973) and Straley (1973), in terms of the classical virial expansion, the Onsager theory (referred to in Section 2.1) and the curvature moduli theory, derived the elastic constants of rigid liquid crystalline polymers. The free energy varies according to the change of the excluded volume of the rods due to the deformation. The numerical calculation of elastic constants (Lee, 1987) are shown in Table 6.2. 

In the case of concentrated solutions or melts of liquid crystalline polymers, the Onsager theory and hence the above arguments are not applicable. It is also not applicable to the rather flexible liquid crystalline polymers. 

The Onsager theory of ionic escape is widely used for the analysis of photo- and radiation-induced ionization in nonpolar liquids. Here an electron is trying to escape from its positive parent ion. Application of the model of Brownian motion requires the mean free path of the Brownian particle to be small compared to the characteristic... 

The Onsager theory is concerned with the escape of a spatially correlated single ion pair, a situation which is only realized in photoionization experiments (see Section 4.4). Ionization of a nonpolar liquid by high LET radiation produces a string of ions... 

In this section, we shall discuss the free ion yield produced by low LET radiation and make use of the Onsager theory of escape in order to rationalize the observed values. The free ion yield depends on the temperature of the liquid, on external hydrostatic pressure, on the externally applied electric field strength, and on the molecular structure of the molecules comprising the liquid. In some liquids, where electron trapping is involved in the transport, the yield can be influenced by simultaneous illumination with infrared light (see Section 3.8.3). Free ion yields produced by low LET radiation have been determined for many liquids. The discussion of all the details would exceed the frame of this book. The interested reader is referred to the reviews by Hummel and Schmidt (1974) or Freeman (1987). 

Maier and Meier [35] worked out these ideas more quantitatively by applying Onsager s theory of static polarization for nematic liquid crystals. The Onsager theory relates the dielectric constant to molecular properties, namely the molecular polarizability a and the permanent electrical dipole moment /i. In the case of nematic liquid crystals, however, the polarizability has to be treated as a tensor with principal values an and a and an angle ]8 has to be introduced to describe the... 

Abstract A simple model is presented to describe the phase behavior in binary mixtures of a flexible polymer and a liquid crystal. By using the incompressible approximation to the Onsager theory combining both the excluded volume interactions between the hquid crystals and the attractive interaction due to the alignment of liquid crystals, we obtain the free energy of mixing of the binary... 

We consider a binary mixture of liquid crystals and Np flexible polymer molecules. On the basis of the Onsager theory, including both the excluded volume interactions and the orientation-dependent attractive interactions between liquid crystals, the free energy of mixing of the... 

The hard sphere model is based on the excluded volume of spherical particles. An excluded volume theory has been developed to account for the orientational ordering of liquid crystal molecules, assuming them to be hard rods. This is the Onsager theory and its variants, outlined in Section 5.5.2. Excluded volume interactions influence the conformation of polymer chains. The conformation of an ideal chain is described by a random walk. However,... 

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