Onsager sphere

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

Various treatments of these effects have been developed over a period of years. The conductance equations of Fuoss and Onsager l, based on a model of a sphere moving through a continuum, are widely used to interpret conductance data. Similar treatments n 3, as well as more rigorous statistical mechanical approaches 38>, will not be discussed here. For a comparison of these treatments see Ref. 11-38) and 39>. The Fuoss-Onsager equations are derived in Ref.36), and subsequently modified slightly by Fuoss, Onsager and Skinner in Ref. °). The forms in which these equations are commonly expressed are... 

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 value of the parameter L entering equation (6.4.1) defines whether the Coulomb attraction or recombination is predominant as L effective recombination sphere equals the Onsager radius). 

Figure 6.39 shows the time development of particle concentrations. At long times the kinetics for a symmetric (Da = Db) and asymmetric (Da = 0) cases differ significantly in the latter case reaction proceeds more quickly. Note that the choice of the parameter L — 1 corresponds to the weak electrostatic field the Onsager radius R is small and coincides with the recombination sphere radius r0. The initial dimensionless concentration n(0) = 0.1 is not also too large it is only 10 percent of the maximum concentration which could be achieved under irradiation [12], The magnitudes of these two parameters were chosen to make our computations more time-saving. 

The physical significance of these variables is apparent when they are evaluated in the Onsager cavity description of solvation, which treats the solute as a sphere (which we will assume here is unpolarizable) of radius a. The solvent is modeled as a uniform dielectric medium with a static dielectric constant s and an optical dielectric constant op. The following relationships apply in the Onsager cavity description... 

For comparison, the results obtained using the Maier-Meier theory [4] are also shown this is a generalization of the Onsager model [13] to uniaxial media. The same dipole moment used for the calculations with the molecular shaped cavity was assumed, and the radius a was taken to be 3.9 A, a value derived from the density of the system. Improvement of the predictions, when the sphere is replaced by a molecular shaped... 

For CS and CR processes, an alternative is provided by the Onsager model [46] of a point dipole in a sphere (of mean radius rD/A). In the limiting case of CS (CR), the initial (final) state dipole moment (/z) is zero, and the shift in dipole (Ap,) is given by the final (initial) p value. This leads to the Lippert-Mataga (LM) dipolar analog of Equation (3.93) [47]... 

The definition of the cavity (shape and size) is an intricate and delicate question that may have a considerable influence on the results (even qualitatively). In the original Onsager s theory, the molecular cavity was defined as a sphere and the volume was taken equal to the partial molecular volume of the solute in the solution. In practice, this volume can be assumed to be equal to the average volume in the pure liquid. Experimental values are then easily deduced from the experimental density of the liquid at 20°C when this quantity is available. Obviously, in SCRF applications, it became rapidly necessary to achieve a theoretical definition of the cavity applicable to any molecular structure. In former works carried out by our group [28,62], it was shown that a simple linear relationship exists between the experimental volume derived from the liquid density (Onsager s recipe) and the van der Waals volume, i.e., the volume enclosed by a set of overlapping atomic spheres with Bondi radii [63], Roughly, this relationship is... 

Onsager 0 equations (Section 5.10), one can extract the dipole moments of polar molecules and the polarizability of any solute molecule. One needs a capacitance cell whose electrodes are as close to each other as practical (for higher capacitances) and reasonable solubilities. If the shape of the solute is very different from the sphere used in the Debye model, then the ellipsoidal cavity has been treated theoretically [13] and applied to hypsochromism [14]. 

In dilute electrolyte solutions ion-ion interaction as function of electrolyte concentration is fully explained by the Debye-Hiickel-Onsager theory and its further development. The contribution of ion solvation is noticed, if, for instance, the mobilities at infinite dilution of an ion in different solvent media or as function of ionic radii as considered. Up till now the calculation of that dependence has been only rather approximateAn improvement is quite probable, though, theoretically very involved if the solvent is not regarded as a continuum, but the number and arrangement of solvent molecules in the primary solvation shell of an ion is taken into consideration. Also the lifetime of molecules in the solvation shell must be known. Beyond this region a continuum model of ion-solvent interaction may be sufficient to account for the contributions of solvent molecules in the second or third sphere. 

Onsager derived an improved formula by adopting a better model for the calculation of the local field at a molecule. His model consists of a spherical cavity which is excised in the dielectric material and which is just large enough to accommodate one molecule. The molecular dipole is supposed to be a point dipole fj, at the centre of the sphere, radius a. Onsager then said that the local field operating on the dipole at the centre of the cavity could be resolved into two components, a cavity field G and a reaction field R ... 

A similar treatment for a point dipole of magnitude p, in a sphere of radius a yields the Kirkwood—Onsager result > ... 

The Onsager theory was first successfully applied to the static permittivities of electrolyte solutions by Ritson and Hasted, who showed that almost the entire depression of the pemuttiidty arises from the region lying between two spheres, of radius 2 and 4 A, centred on the ion. Dielectric deorements were calculated, and in a subsequent paper plausible hydration numbers were obtained. In their calculation a discontinuous model was used, with the assumption that the first sheath of water molecules was fiilly saturated, i.e. oriented to a (positive) ion. 

Onsager and Fuoss viscosity equation, 125 order in liquids, 1 oriented molecules, 152, 155 0rsted s piezometer, 58 orthobaric density, 48, 327 Oseen correction for falling sphere equation, 87... 

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. 

Other ions in the solution. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager s theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechaiucs from which the adsorption isotherm is derived. 

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