Ionic clouds

Note that when the concentration of added salt is very low, Debye length needs to be modified by including the charge contribution of the dissociating counterions from the polyelectrolytes. Because the equilibrium interaction is used, their theory predicts that the intrinsic viscosity is independent of ion species at constant ionic strength. At very high ionic strength, the intrachain electrostatic interaction is nearly screened out, and the chains behave as neutral polymers. Aside from the tertiary effect, the intrinsic viscosity will indeed be affected by the ionic cloud distortion and thus cannot be accurately predicted by their theory. 

The polycationic ammonium salt PEU can be chosen for increasing the ionic cloud in the electrical double layer, thus raising the electric repulsion between the gold particles in the sol. Small particles can be obtained (mean diameters between 3.1 and 4.2 nm) [24]. 

One is talking here of volumes that are large compared with the dimension K of the ionic cloud of Chapter 3. 

An analogy between the situation just described and those involved in ion-solvent and ion-ion interactions can be drawn. The solvent water, for example, normally has a particular structure, the water network. Near an ion, however, the water dipoles are under the conflicting influences of the water network and the charged central ion. They adopt compromise positions that correspond to primary and secondary solvation (Chapter 2). Similarly, in an electrolytic solution, the presence of the central ion makes the surrounding ions redistribute themselves—an ionic cloud is formed (see Chapter 3). 

Just as the central ion can perturb and cause a rearrangement of the surrounding solvent molecules and ions, the electrode itself can cause the surrounding particles to assume abnormal, compromise positions (relative to the bulk of the electrolyte). It will be seen later that an electrode also can get enveloped by a solvent sheath and an ionic cloud. There are, however, many other interesting, phenomena arising from the fact that one can connect an external potential source (e.g., a battery) to the electrode by a metallic wire and thus control the electrode charge. New possibilities emerge that do not exist in the case of the central ion. 

The Ionic Cloud The Gouy-Chapman Diffuse-Charge Model of the Double Layer... 

With the ionic cloud on the electrode, the resemblance of the Gouy—Chapman model to that of the theoiy of ion-ion interactions in solution reviewed in Chapter 3 is evident. There, it was necessary to arbitrarily choose one ion and spotlight it as the central ion, or source, of the field. Here, the discussion resolves on ion-electrode interactions with the electrode as the source of the field. The response of an ion, however, does not depend on how the electric field is produced (i.e., whether the source is a central ion or a charged electrode). It depends only on the value of the field at the location of the ion. Hence, the electrostatic arguments in the problems of ion-ion interactions and ion-electrode interactions must be similar. 

This potential decay implies that there is a field inside the semiconductor and that the excess-charge density slowly decays to zero as if there were an electronic cloud analogous to the ionic cloud adjacent to an electrode in solution. It can be seen that the potential due to the atmosphere of holes and electrons is characterized by the same parameter... 

The Stern surface is drawn through the ions that are assumed to be adsorbed on the charged wall. (This surface is also known as the inner Helmholtz plane [IHP], and the surface running parallel to the IHP, through the surface of shear (see Chapter 12) shown in Figure 11.9, is called the outer Helmholtz plane [OHP]. Notice that the diffuse part of the ionic cloud beyond the OHP is the diffuse double layer, which is also known as the Gouy-Chapman... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

The Structured Ionic Cloud. We base our treatment on the general solution of the linearized Poisson-Boltzmann equation (LPBE, Equation 1). 

A well-known result of the DH theory is that the charge dq in the volume element 4ttr2dr has a maximum value at a distance k 1 from the central ion (Figure la)2. The ionic cloud can be reduced to a charge on an infinitesimally thin shell placed at a distance (a + k 1) from the center of the central ion such that the potential caused by the reduced ionic cloud is... 

A structure may be imposed on the ionic cloud by supposing that dq in the volume element dv = r2 sin OdOdipdr has a finite number, n, of maxima similarly situated at k 1 from the surface of the central ion (Figure lb). By analogy, this non-radial atmosphere is reducible to a corresponding array of point charges, and this device later enables us to formulate the necessary boundary conditions. 

All symbols have their usual meaning in the c.g.s. system of units, as given in Ref. 3. The common interpretation that the central ion sees its ionic cloud at a distance k 1 away is valid for the point-charge model only. For the DH second approximation the ionic cloud can be reduced to a charge located on a spherical surface at k 1 so as to maintain a constant potential at the surface of the central ion. Therefore, it cannot be replaced by a point charge. 

Figure la. Left a segment of the spherical ionic atmosphere. Right the reduced ionic cloud of Debye and Hiickel. 

Thus the contribution of the structured ionic cloud to the total potential at the surface of the central ion will not be as it is in the DH theory, and because the electrostatic model requires an equipotential surface to be maintained there, a new model is needed. We therefore approximate an ion to a dielectric sphere of radius a, characterized by the dielectric constant of the solvent D, and having a charge Q, residing on an infinitesimally thin conducting surface. This type of model has been exploited by previous workers (17,18) and may be reconciled with a quantum-mechanical description (18). 

The mutual polarization of ions is equivalent to the redistribution of surface charge on the central ion in response to the nonhomogeneous field of the ionic cloud. We need not speculate here on the physical nature of the equipotential surface, except to emphasize that it refers to a solvated species, and one of our... 

The conventional viewpoint, which assumes that the ionic atmosphere is spherically symmetric, does not take account of the inevitable effects of ionic polarization. From an analysis of the general solution (19), however, it is evident that the ionic atmosphere must be spherically symmetric for nonpolarizable ions, and the DH model is therefore adequate. (Moreover, in very dilute solution polarization effects are negligibly small, and it does not matter whether we choose a polarizable or unpolarizable sphere for our model.) But once we have made the realistic step of conferring a real size on an ion, the ion becomes to some extent polarizable, and the ionic cloud is expected to be nonspherical in any solution of appreciable concentration. Accordingly, we base our treatment on this central hypothesis, that the time-average picture of the ionic solution is best represented with a polarizable ion surrounded by a nonspherical atmosphere. In order to obtain a value for the potential from the general solution of the LPBE we must first consider the boundary conditions at the surface of the central ion. 

Boundary Conditions Solution of the Problem of the Reduced Ionic Cloud. For equipotential conditions the total potential at the surface of the... 

The first and second terms on the right are self-energies of the central ion and the ionic atmosphere, and the third contains the interaction energies of the central ion with its ionic cloud and vice versa. According to Green s reciprocal theorem, these energies are equal and are given by... 

Since the ideal solution is defined by the absence of interactions between ions, the total energy WIS required to charge the central ion and its ionic cloud in an ideal solution is obtained from Equation 14 by setting Wint = 0 ... 

For charging up a central polarizable ion and its ionic cloud we thus derive... 

For 1 1 electrolytes the simplest choice for n is unity (as in Figure lb) and is shown to be appropriate by comparison with experiment. Thus we have n = 1, X = 1 (cos 0i = 1, 0i = 0), and can take any value, since m = 0 and does not depend on (p. Variants of Equation 39 are easily obtained for other than uni-univalent salts by choosing a structure for the reduced ionic atmosphere in the light of symmetry and chemical intuition. This is illustrated with reference to the divalent ion of a 1 2 electrolyte, where it is reasonable as a first approximation to suppose that the ionic cloud will have two diametrically opposed maxima, each at a distance 1 /k from the reference ion. It is easy to see that dipoles induced on the central ion by these two charge centers will cancel, as well all higher terms of odd Z, but that quadrupolar effects (Z = 2) and other terms of even Z will not. For the structure factor the coordinates of the two maxima in dq are 0i = 0 and 02 = 7r, while the atmosphere is still symmetrical with respect to the angular coordinate [Pg.211]

Multivalent Electrolytes. For 2 1 electrolytes we assume n = 2 for the divalent ion and n = 1 for the monovalent ion. The latter is a necessary choice which supposes that a monovalent ion sees a singly charged ionic cloud represented by a reduced ionic atmosphere with one maximum. This model gives log 7 vs. a/7 plots which are remarkably like the experimental pictures. The... 

From a consideration of the electrostatic free energy alone it is not immediately obvious how the arrangement of ions in the solution as a whole is related to the moving polarized central ion and its structured ionic cloud. It is reasonable to think that the induced multipoles impose restrictions on the mixing of the ions, so that the energy and entropy of configurations described by the structured ionic cloud are lower than in the DH model, as envisaged earlier by Frank and Thompson (16). Such considerations do not lead directly to predictions of the... 

Ion Association. One feature of the model is that it requires no vaguely defined breakdown of the DH ionic cloud at some critical concentration, but predicts an even gradation of properties over the whole concentration range, or at least up to concentrations where the co-spheres start to overlap. 

Relation of Electrostatic and Statistical-Mechanical Approaches to Interionic Theory. We believe that the ionic cloud concept is appropriate for the... 

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