Fuoss distribution

This distribution has some inconsistencies - for example it diverges when R is large - and was modified by Fuoss (1934) see Figure 4.6. 

Fuoss, R. M. (1934). Distribution of ions in electrolyte solutions. Transactions of the Faraday Society, 30, 967-80. 

In solution theory the specialized distribution functions of this kind should appear in the theory of ion pairs in ionic solutions, and a form of the Bjerrum-Fuoss ionic association theory adapted to a discrete lattice is generally used for the treatment of the complexes in ionic crystals mentioned above. In fact, the above equation is not used in this treatment. Comparison of the two procedures is made in Section VI-D. 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

In order to obtain better agreement with experimental results, the concept of a distribution of correlation times was introduced in nuclear magnetic relaxation. Different distribution functions, G(i c), such as Gaussian, and functions proposed by Yager, Kirkwood and Fuoss, Cole and Cole, and Davidson and Cole (asymmetric distribution) are introduced into the Eq. (13), giving a general expression for... 

Density functions used earlier to interpret the relaxation data of polymers were the Cole-Cole function,70 the Fuoss-Kirkwood function,71 and the log 2) function.72 These functions, particularly the skewed log ( 2) distribution, were accounted for by 13C T, and n.O.e. data of some polymers, but the physical significance of the adjustable parameters has been questioned by some authors.68... 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

The inversion of transform (5.3.2) and the determination of L (t ), when the analytical dependence Y j is known, have been considered previouslyThe results were based on the well-known inversion equations of Kirkwood and Fuoss (see also Ref. ) establishing a relationship between L(t ) and the corresponding dynamic compliance function Z(ico). Indeed, the reduced complex dynamic compliance corresponding to a distribution L j) is given by... 

Fuoss, R.M., Katchalsky, A., and Lifson, S. The potential of an infinite rod-like molecule and the distribution of the counter ions. Proceedings of the National Academy of Sciences of the United States of America, 1951, 37, p. 579-589. 

Fuoss [26] revised the distribution function taking into account the fact that the probability should go to zero for large values of r. However, the modified function describing P(r) is essentially the same as that given by Bjerrum for small values of r. Therefore, the simpler theory is that usually used. 

Treating the problem as one of rotary Brownian movement, Kirkwood and Fuoss were able to calculate the distribution functions and F (r) of Eqs. 22 and 29. In their case F(r) was a symmetrical function and they identified the average relaxation time with the value corresponding to the maximum in the loss curve. Unfortunately their theory is incompatible with existing experimental data on dilute solutions, since it specifies that should be proportional to the degree of polymerization. 

The expression for AX/X is obtained by a series of successive approximations yielding the terms of different order which contribute to the relaxation field. The first order term arises from eqn. 5.2.13 putting all the T terms equal to zero. Fuoss and Onsager obtain, in this way, an equation equivalent to 5.2.15. The first order expression for is then replaced in the T terms and a further approximation to the perturbed distributions and ionic potentials is calculated. 

Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New York Freed KF, Dudowicz J, Stukalin EB, Douglas JF (2010) General approach to polymer chains confined by interacting boundaries. J Chem Phys 133 094901 Fuoss RM, Katchalsky A, Lifson S (1951) The potential of an infinite rod-like molecule and the distribution of the counter ions. Proc Natl Acad Sci USA 37 579-589 Golestanian R, Kardar M, Liverpool TB (1999) Collapse of stiff polyelectrolytes due to counterion fluctuations. Phys Rev Lett 82 4456-4459 Guggenheim FA (1952) Mixtures. The Clarendon Press, Oxford... 

Extended laws are available for the variation with concentration of the transport coefficients of strong and associated electrolyte solutions at moderate to high concentrations. Like the CM calculations, this work is based on the Fuoss-Onsager transport theory. The use of MSA pair distribution functions leads to analytical expressions. Ion association can be introduced with the help of the chemical method. A simplified version of the equations, by taking average ionic diameters, reduces the complexity of the original formulas without really reducing the accuracy of the description and is therefore recommendable for practical use for up to 1-M solutions. 

Two other distribution functions are due to Kirkwood and Fuoss [1941] and Davidson and Cole [1951] (see also Davidson [1961]). 

In the last two decades, new extended laws have been obtained for the concentration dependence of transport properties. It was possible [ 15,16] to use the Fuoss-Onsager theory together with new, more accurate equilibrium pair distribution functions as obtained with the help of the hypemetted chain (HNC) or mean spherical approximation (MSA). 

Fuoss inferred that the long-range of Coulomb forces tends to establish an ordered distribution . However, the technical difficulty of performing the light scattering experiments for salt-free polyelectrolyte solutions hindered quantitative consideration of the striKture formation of macroions. 

---