Debye-Htickel interaction

There is ample experimental evidence that identically charged polymers in the presence of ordinary univalent counterions have a tendency to form loose clusters in solution [65-70], and we have asked whether the attractive polyion-polyion potential discussed in Sec. Ill can stabilize a finite-sized cluster of parallel rodlike polyions without leading to precipitation [71,72]. The theoretical problem is complicated by a failure of pairwise additivity the work of assembling N polyions is not equal to the work of assembling the N(N — l)/2 polyion pairs, each in isolation from the other N 2 polyions. To be sure, the Debye-Htickel interaction term for a cluster (the generalization of Eq. 5 above) takes the form of a pairwise sum over polyions,... 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

The above equation assumes that the ions are point charges and interact in a continuous dielectric. It is essentially correct in the limit, but problems arise when considering finite concentrations of solute where an extended Debye-Htickel expression may be more appropriate... 

Before discussing mathematical formalism we should stress here that the Kirkwood approximation cannot be used for the modification of the drift terms in the kinetics equations, like it was done in Section 6.3 for elastic interaction of particles, since it is too rough for the Coulomb systems to allow us the correct treatment of the charge screening [75], Therefore, the cut-off of the hierarchy of equations in these terms requires the use of some principally new approach, keeping also in mind that it should be consistent with the level at which the fluctuation spectrum is treated. In the case of joint correlation functions we use here it means that the only acceptable for us is the Debye-Htickel approximation [75], equations (5.1.54), (5.1.55), (5.1.57). 

However, experimentally observed y =f c) functions usually first decrease, pass through a minimum, and then increase at high concentrations. In order to explain the increase of y with concentration, Stokes and Robinson modified the Debye-Htickel theory by introducing the effect of ion-solvent interaction. Thus, the modified theory is based on ion-ion and ion-solvent interactions. The modified theory is in good agreement with experimental results, up to an ionic strength of about 4, as shown in Figure 5.14. 

Using Monte Carlo simulations and a Debye-Htickel pah-repulsive potential, effective interaction potentials between identical charged particles were calculated and the phase structures were obtained for various charges per particle and various particle volume fractions. The simulation re-... 

In solvents of high dielectric constant such as water, the deviations from ideality caused by ion-ion interactions are reasonably small below concentrations of 0.1 ilf for 1 1 electrolytes and can be treated adequately by means of the Debye-Htickel theory. For polyvalent electrolytes or for higher concentrations of 1 1 electrolytes, or for either in solvents of lower dielectric constant, the situation is less fortunate. The deviations from ideality can become rather large, and there is no adequate theory for either correlating them or predicting them. 

The Debye-Htickel ionic-cloud model for the distribution of ions in an electrolytic solution has permitted the theoretical calculation of the chemical-potential change arising from ion-ion interactions. How is this theoretical expression to be checked, i.e., connected with a measured quantity It is to this testing of the Debye-Huckel theory that attention will now be turned. 

The correction factor,/, relates the actual mobility of a fully charged particle at the ionic strength under the experimental conditions to the absolute mobility. It takes ionic interactions into account and is derived for not-too-concentrated solutions by the theory of Debye-Htickel-Onsager using the model of an ionic cloud around a given central ion. It depends, in a com-... 

The Debye-Htickel model considered the solvent to be a structureless medium whose only property is to reduce the interactions between ions in a vacuum by a factor given by the macroscopic relative permittivity, e. No cognisance was taken of the possibility of ion-solvent interactions, and the size of the ion was assumed to be that of the bare ion. Gurney in the 1930s introduced the concept of the co-sphere and this has proved to be a useful concept in the theory of electrolyte solutions. Many recent theories of conductance are based on the Gurney co-sphere concept (see Section 12.17). 

The model and theory, like that of the Debye-Htickel treatment of non-ideality, were based on consideration of long range electrostatic coulombic interactions only. The model was most likely to be inadequate because it did not take into account specific short range interactions corresponding to ion-ion, ion-solvent, and solvent-solvent interactions. 

The specific interaction parameter b is also different from the corresponding Ae value because of the different Debye-Htickel terms. Up to the review of Baes and Mesmer [1976BAE/MES], there were no data available to model the ionic strength dependence of the equilibrium constants for polynuclear species in one of the ionic media. The common method to estimate equilibrium constants at zero ionic strength was to use an experimental value at finite ionic strength and then to use only the Debye-Htickel term to estimate the value at zero ionic strength. 

All pairs of charged monomers within the polyelectrolyte interact via a screened Debye-Htickel long-range potential ... 

Soon after the appearance of the Debye-Htickel theory, it was found that the theory did not work well for many electrolytes. In 1926, Bjerrum suggested that electrostatic attraction between pairs of oppositely charged ions resulted in ion-pairs, which would account for the lower measured activity coefficients in these solutions. The problem then, as now, was how best to define and measure the extent of ion-pairing. How close must two ions be to become an ion-pair What is the difference between an ion-pair and a complex Is it really necessary to know this to use thermodynamics Helgeson (1981) notes that The distinction between ion association and short-range ionic interaction is nebulous at best. ... 

The first virial coefficient /(/) is some function of the ionic strength and is not 0 as it would be for an ideal solution, but is in fact a version of the Debye-Htickel equation, which represents departure from ideality in very dilute solutions. The following term is a function of the interactions of all pairs of ions, and the third term a function of the interactions of ions taken three at a time. The second coefficient. Ay, is a function of ionic strength, but the third coefficient ju-y - is considered to be independent of ionic strength and equals zero if /, j, and k are all anions or cations. Later extensions to the model published by Pitzer and co-workers allow for an ionic strength dependence to the third coefficient. Pitzer (1987) and Harvie and Weare (1980) note that higher virial coefficients are required only for extremely concentrated solutions, so the series is stopped at the third coefficient. 

In 1973, Pitzer el al. began presenting a series of papers (PI - P12) reporting their development of a system of equations for electrolyte thermodynamic properties. In expanding the Debye-Htickel method, terms were added to account for the ionic strength dependence of the short-range forces effect in binary interactions. 

Chemical equilibrium in a closed system at constant temperature and pressure is achieved at the minimum of the total Gibbs energy, min(G) constrained by material-balance and electro-neutrality conditions. For aqueous electrolyte solutions, we require activity coefficients for all species in the mixture. Well-established models, e.g. Debye-Htickel, extended Debye-Hiickel, Pitzer, and the Harvie-Weare modification of Pitzer s activity coefficient model, are used to take into account ionic interactions in natural systems [15-20]. 

Lastly, the uniformity of ionic strength provided to a solution by the presence of ample supporting electrolyte limits effects due to the non-ideality of the solution. According to the Debye-Htickel theory, the presence of electrostatic interactions between ions causes solution non-ideality because these forces are on average stabihsing. Therefore, activity, the quantity appearing in the Nernst equation, differs from concentration by a factor known as the activity coefficient, y, in a maimer which for a dilute (<0.01 M) solution is given by a simplified formula ... 

The theory of strong electrolytes due to Debye and Htickel derives the exact limiting laws for low valence electrolytes and introduces the idea that the Coulomb interactions between ions are screened at finite ion concentrations. 

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