Kirkwood’s theory

Because of the formal equivalence of Fokker-Planck and Langevin methods, there is no intrinsic difficulty in translating Kirkwood s theory into Langevin terms. As far as 1 am aware, this has not yet been done. The main purpose of this article is to perform the translation. 

The inverse 1 yfc of the matrix is obviously the same as (l/p)l 8/ +v If, however, we use Kirkwood s chain space variables for a system with constraints, then the two quantities would have to be distinguished. The constraints would replace part of the internal force F and the velocities associated with these constraints would vanish. This was a serious source of confusion in earlier versions of Kirkwood s theory see Ref. 4 for further information on this point. 

It should be noted that Kirkwood s formula does not appear in his first publication [240] in the form of Eq. (5-86). Nevertheless, it is this form of Kirkwood s formula which is widely known, representing only one of the terms of a more complex equation given in reference [240] with n= Kirkwood s theory was further developed in papers by Kirkwood, Westheimer, and Tanford [241], Laidler and Landskroener [242], and Hiromi [243]. 

The other part of the molar polarization (240) is, in Kirkwood s theory, of the following form ... 

The above examples illustrate that continuum models such as the Kirkwood model are reasonably successful in describing the static permittivity, provided one has an independent means of estimating the correlation parameter Unfortunately, these estimates are available for only a few polar solvents, so that gK must be considered an independent parameter. The version of Kirkwood s theory presented here only considers orientational polarization. When distortional polarization, that is, the effect of molecular polarizability, is included, interpretation of experimental results is less clear. Since the approach taken here involves continuum concepts, it is necessarily limited. In the following section, a simple model based on a molecular description of a polar liquid is presented. 

A more general statistical treatment has been given by Kirkwood for two simplified cases. He has shown30 that for nonpolar gases at low pressures Clausius-Mosotti s equation is a first approximation. Nevertheless, by refining Kirkwood s theory,... 

More recently, Harris and Alder,24 keeping the general principles of Kirkwood s theory, have tried to calculate the polarization effects more rigorously. Unfortunately their final equation does not coincide as it should, with Onsager s equation when it is assumed that there are no short-range interactions cos y) — 0). This is because some of Kirkwood s equations are only valid when the assumptions of the author are justified, and cannot be used as was done by Harris and Alder, when a deformation polarization is superimposed on the orientation polarization. For instance, in presence of deformation effects boundary conditions cannot be introduced in the same manner as in Kirkwood s model (cf. Frdhlich 21). 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

Refs 1) Ya.B. Zel dovich, ZhEksper i TeoretFiz 10, 542(1940) (On the theory of propagation of detonation in gaseous systems) la) J.G. Kirkwood S.R. Brinkley Jr, "Theory of Propagation of Shock Waves from Explosive Sources in Air and Water , OSRD 4814(1946) 2) G.N. Abramovich ... 

An alternate approach has been attempted for describing the transport phenomena in dense gas and liquid systems by means of the methods of nonequilibrium statistical mechanics, as developed by Kirkwood (K7, K8) and by Born and Green (B18, G10). Although considerable progress has been made in the development of a formal theory, the method does not at the present time provide a means for the practical calculation of the transport coefficients. Hence in this section we discuss only the applications based on Enskog s theory. 

We will now summarize the conclusion of the Kirkwood-Bethe theory. Fig 15 shows the computed peak pressure and computed reduced tunc constant for TNT plotted VS the inverse reduced distance. The dotted lines are a power function fit thru the computed peak pressures. The x s are drawn in by the writer to compare computed and measured reduced time constants (taken from Fig 7.9, p 240 of Ref 1). Comparison of other computed and measured shock parameters on the basis of the power functions shown below (in Cole s notation and in English units) is made in Table 11 (from p 242 of Ref 1)... 

In Kirkwood s original formulation of the Fokker-Planck theory, he took into account the possibility that various constraints might apply, e.g., constant bond length between adjacent beads. This led to the introduction of a chain space of lower dimensionality than the full 3A-dimen-sional configuration space of the entire chain and it led to a complicated machinery of Riemannian geometry, with covariant and contravariant tensors, etc. 

Equations (19)—(23) provide a complete translation of Kirkwood s Fokker-Planck theory into Langevin terms, when the entire 3JV-dimen-sional configuration space is used. 

Applying Kirkwood s formula to the transition-state theory for the bimole-cular reaction A + B (AB) —> C + D and combining Eq. (5-86) with Eq. (5-75), one obtains an expression for the rate constant of a reaction between two dipolar molecules A and B with moments Ra and r to form an activated complex with dipole moment r [2] ... 

Books and Monographs.—The 1946 Faraday Society Discussion on dielectrics opened a new era of study, bringing a series of books and monographs devoted to dielectric properties. Bottcher s monograph develops in particular the theoretical electrostatics and dynamics of cfielectric materials, whereas that by Smyth deals, in addition to the simpler theory, with experimental dielectric methods and the relationship between macroscopic properties and molecular structme. The fundamental prindples of the theory of dielectrics are presented in Brown s monograph in a molecular-statistical treatment, and Frohlidi employs Kirkwood s semi-macroscopic approach. In 1969, a comprdiensive volume on Dielectric Properties and Molecular Behavioiu beceime available, in which Hill discusses the theory... 

The classical treatment of nonpolar dielectric materials is expressed by the Clausius-Mossotti equation. Polar materials in nonpolar solvents are better handled by Debye s modification, which allows for the permanent dipole of the molecule. Onsager made the next major step by taking into account the effect of the dipole on the surrounding medium, and finally Kirkwood treated the orientation of neighboring molecules in a more nearly exact manner. (See Table 2-1.) The use of these four theoretical expressions can be quickly narrowed. Because of their limitations to nonpolar liquids or solvents, the Clausius-Mossotti and Debye equations have little application to H bonded systems. Kirkwood s equation has great potential interest, but in the present state of the theory of liquids the factor g is virtually an empirical constant. The equation has been applied in only a few cases. 

H. W. Graben, G. S. Rushbrooke, and G. Stell. On the dielectric constant of non-polar Lennard-Jones fluids according to the Kirkwood-Yvon theory. Molec. Phys., 30 373-388 (1975). 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

Auer, P. L., and C. S. Gardner J. Chem. Phys. 23, 1545, 1546 (1955). Application of Kirkwood-Riseman theory to various molecular models has been reported by Ullman, R. J. Chem. Phys. 40, 2193 (1964). 

The present authors employed the Kirkwood—Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in a ternary mixture with respect to the mole fractions and applied them to ternary mixtures when the composition(s) of one (or two) component(s) was (were) small. That approach will be used here to derive new expressions that can predict the Henry s constant in a binary solvent mixture in terms of binary data. 

Shimizu, S., and C. L. Boon. 2004. The Kirkwood-Buff theory and the effect of cosolvents on biochemical reactions. J. Chem. Phys. 121 9147-9155. 

Dipole moments may also be derived by a consideration of the dielectric constant data themselves. Since amino acids and proteins are soluble only in polar solvents, the treatment which is applicable to dilute solutions of polar molecules in a non-polar medium cannot be applied here. However, the general theory of polar liquids developed by Onsager (S7) and Kirkwood (67) [see also Kirkwood in Cohn and Edsall [16), Chapter 12], is applicable here. According to Kirkwood s treatment, the dipole moment (/z) of an individual molecule in the liquid is in general different from its moment in the gaseous state because the attractions... 

Further developments in the theory of the structure of polar liquids included estimates of the correlation of a given dipole to its neighbors. Important contributions were made in this direction by Kirkwood [22] and Frohlich [23]. In Kirkwood s model, the field Ej is calculated by considering all possible orientations of surrounding dipoles in a spherical cavity for a fixed orientation of the central dipole. By averaging over these orientations, Kirkwood obtained an improved estimate of the polarization of the medium. For the case of non-polarizable dipoles the result is... 

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