The Onsager Theory

This theory has been used to describe nematic ordering in solutions of rodlike macromolecules, such as tobacco mosaic virus (TMV) or poly(benzyl-L-glutamate). The orientational distribution is calculated from the volume excluded to one hard cylinder by another. It assumes that the rods cannot penetrate to each other. Denoting the length of the rods by L and the diameter by D, the volume fraction of the rods is expressed by O = jCkLD. (c is the concentration of rods). 

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These are just a few of the standard examples of explicit applications of the Onsager theory to concrete cases. 

A byproduct of the preceding analysis is that the Onsager theory innnediately detennines the fonn of the fluctuations that should be added to the difhision equation. Suppose that a solute is dissolved in a solvent with concentration c. The difhision equation for this is... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Here, Vd = pE is the drift velocity. The recombination and escape probabilities are now given by PR = NR /n+° L0 and Pkc = 1 - Pr. Since Vd = i, but T /r1 these probabilities are independent of mobility. However, the initial separation r0 is expected to depend (increase) with electron mobility, thus making the escape probability indirectly dependent on the mobility. These effects are quite similar to those in the Onsager theory... 

Tachiya M (1988) Breakdown of the Onsager theory of gemmate ion recombination. J Chem Phys 89 6929... 

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. 

The most authors consider that the Onsager theory cannot be applied to the photogeneration of charge carriers in polydiacetylenes and transport processes are controlled by deep traps. 

The Onsager theory of geminate recombination was qualitatively consistent for aryl-substituted thiapyrylium salt and dialkylamino-substituted triphenyl-methane dispersed in polycarbonate film [301]. The quantum yield of Hie photogeneration was equal to 0.5 at the electric field strength of 106 V cm-1, mobility of I0 12m2 V-1 s-1. Hole and electron conductivity was established. In a triphenylamine-lexan system doped with a boron diketone acceptor, the... 

Sano (501] has discussed the probability of escape of an anion (e.g, solvated electron) from several stationary cations which are also sinks. This marks an extension of the Onsager theory of ion-pair recombination [321]. If the N cations are located at p1( p2... pw and all quite close to one another, the potential energy of the anion is... 

If we admit that the Onsager theory correctly accounts for the field-facilitated dissociation phenomenon occurring in the present system,... 

According to the Onsager theory and to computer simulations of the behavior of hard sphero-cylinders [37], in the absence of additional interactions no LC ordering is predicted for rods with L/D < 4, and therefore DNA double helices with a number of base pairs N < 24 would lack the anisotropy to display mesophase behavior at any concentration (Fig. 10). 

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... 

Huraux and co-workers, in a series of papers 67 "70), and more recently, Sheppard 7I), have attempted quantitative interpretation of er during an epoxy cure in terms of the changing concentrations of constituent polar groups. They used a theory by Onsager which improves Debye s original theory to account for local dipole fields [see, for example, Ref. 23)]. The Onsager theory, expressed below, requires some explanation ... 

Their equations for the Onsager theory put i2 in the Langevin function, instead of one A inside and one outside. However, die error is not serious since the calculations and graphs are correct... 

Figure 1 The field dependencies of the photogeneration efficiency predicted from the Onsager theory for different values of the thermalization distance. The temperature was 295 K.

The applicability of the Onsager theory can be determined from the field dependence of the photogeneration efficiency at low fields. From Eq. (20), the efficiency should have a linear field dependence. From plots of the efficiency versus field, the slope-to-intercept ratio is... 

In addition to three-dimensional models, there have been several onedimensional models based on the Onsager theory (Holroyd et al., 1972 Haberkom and Michel-Beyerle, 1973 Smetjek et al., 1973 Blossey, 1974 Singh and Bassler, 1975 Charle and Willig, 1978 Hong and Noolandi, 1978b Siddiqui, 1983, 1984). For the one-dimensional case, a field-independent slope-to-intercept ratio cannot be defined. One-dimensional models have been seldom used for organic materials. 

The effect of the TTA concentration of the transport layer on the field dependencies is shown in Fig. 4. The solid lines were calculated from the Onsager theory with r Q = 0.60 and different values of rQ. The results show that rQ increases with increasing TTA concentration while r Q remains constant. The most likely explanation of this is that the photogeneration efficiency is determined by the probability of a donor molecule being in contact with the photoexcited aggregate phase, as proposed by Umeda and Hashimoto (1992). 

The wavelength was 580 run. The solid lines were calculated from the Onsager theory. The efficiency of the dual-layer structure is in excess of an order-of-magnitude greater than the generation layer. The authors attributed this to a... 

Figure 25 The low-field slope-to-intercept ratio for a dispersion of /J-F Pc for different delay times between the exposure and the application of a collection field. The exposures were 500-550 nm strobe illumination (open circles) and 532 nm laser exposures (solid circles). The solid line represents the theoretical value derived from the Onsager theory.

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