Double repulsion

In the case of lithium fluoride the ratio R i /Rj>- is 0.44. In this crystal there is double repulsion the repulsive forces between anion and anion and those between anion and cation are simultaneously operative. The inter-atomic distances are determined neither by the sum of the radii for the anion and cation nor by the radius of the anion alone, but are larger than those calculated by either method. Thus the lithium-fluoride distance is 0.05 A. (2.5%) larger than the sum of the radii, and one-half the fluorine-fluorine distance is 0.06 A. larger than the fluoride radius. 

The sodium salts also show the effect of double repulsion, the increase of the observed distances over the calculated being greatest (2.7%) for the iodide, for which f Na+/- i- is 0.44. 

The ratio Rnt /Ro- is 0.46, so that in magnesium oxide the mutual repulsion of the oxide ions is beginning to have influence, and the interatomic distance is increased by 0.05 A. (2.5%) through the effect of double repulsion. 

In this discussion, two mutually canceling simplifications have been made. For the transition value of the radius ratio the phenomenon of double repulsion causes the inter-atomic distances in fluorite type crystals to be increased somewhat, so that R is equal to /3Rx-5, where i has a value of about 1.05 (found experimentally in strontium chloride). Double repulsion is not operative in rutile type crystals, for which R = i M + Rx- From these equations the transition ratio is found to be (4.80/5.04)- /3i — 1 = 0.73, for t = 1.05 that is, it is increased 12%. But Ru and Rx in these equations are not the crystal radii, which we have used above, but are the univalent crystal radii multiplied by the constant of Equation 13 with z placed equal to /2, for M++X2. Hence the univalent crystal radius ratio should be used instead of the crystal radius ratio, which is about 17% smaller (for strontium chloride). Because of its simpler nature the treatment in the text has been presented it is to be emphasized that the complete agreement with the theoretical transition ratio found in Table XVII is possibly to some extent accidental, for perturbing influences might cause the transition to occur for values a few per cent, higher or lower. 

The effect of double repulsion for the sodium chloride structure may raise this limit a few per cent. 

In lithium chloride, bromide and iodide, magnesium sulfide and selenide and strontium chloride the inter-atomic distances depend on the anion radius alone, for the anions are in mutual contact the observed anion-anion distances agree satisfactorily with the calculated radii. In lithium fluoride, sodium chloride, bromide and iodide and magnesium oxide the observed anion-cation distances are larger than those calculated because of double repulsion the anions are approaching mutual contact, and the repulsive forces between them as well as those between anion and cation are operative. 

Anion Contact and Double Repulsion.2 —The explanations of the deviations from additivity are indicated by Figure 13-6, in which the circles have radii corresponding to the crystal radii of the ions and are drawn with the observed interionic distances. It is seen that for LiCl, LiBr, and Lil the anions are in mutual contact, as suggested in 1920 by Land6.14 A simple calculation shows that if the ratio p = r+/r of the radii of cation and anion falls below /2 — 1 = 0.414 anion-anion contact will occur rather than cation-anion contact (the ions being considered as rigid spheres). A comparison of apparent anion radii in these crystals and crystal radii from Table 13-8 is given in Table 13-7. 

A Detailed Discussion of the Effect of Relative Ionic Sizes on the Properties of the Alkali Halogenides.—A simple detailed representation of interionic forces in terms of ionic radii has been formulated that leads to complete agreement with the observed values of interionic distances for alkali halogenide crystals and provides a quantitative theory of the anion-contact and double-repulsion effects. 0... 

It is to be emphasised that equilibrium interionic distances are less well defined than covalent bond lengths their values depend not only on ligancy, but also on radius ratio (anion contact, double repulsion), amount of covalent bond character, and other factors, and a simple discussion of all the corrections that have been suggested and applied cannot be given. On the other hand, we have a reliable picture of the forces operating between ions, and it is usually possible to make a reliable prediction about interionic distances for particular structures. 

In the derivation of these ionic radii, it has been assumed that the repulsion coefficient B depends only on the coordination number that is, on the number of anion-cation contacts, but if the radius ratio is close to or less than the lower limit, anion-anion contact occurs and the additional Bom repulsion will lead to equilibrium with the attractive Coulomb forces at a larger distance than that given by the sum of the ionic radii. This phenomenon of double repulsion is shown (see tabulation) by the lithium halides especially. In a more detailed treatment, Pauling 112, 114) has... 

The repulsion between two double layers is important in determining the stability of colloidal particles against coagulation and in setting the thickness of a soap film (see Section VI-5B). The situation for two planar surfaces, separated by a distance 2d, is illustrated in Fig. V-4, where two versus x curves are shown along with the actual potential. 

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Fig. V-5. The repulsive force between crossed cylinders of radius R (1 cm) covered with mica and immersed in propylene carbonate solutions of tetraethylammonium bromide at the indicated concentrations. The dotted lines are from double-layer theory (From Ref. 51).

Using the conditions of the Langmuir approximation for the double-layer repulsion, calculate for what size particles in water at 25°C the double-layer repulsion energy should equal kT if the particles are 40 A apart. 

A major advance in force measurement was the development by Tabor, Win-terton and Israelachvili of a surface force apparatus (SFA) involving crossed cylinders coated with molecularly smooth cleaved mica sheets [11, 28]. A current version of an apparatus is shown in Fig. VI-4 from Ref. 29. The separation between surfaces is measured interferometrically to a precision of 0.1 nm the surfaces are driven together with piezoelectric transducers. The combination of a stiff double-cantilever spring with one of a number of measuring leaf springs provides force resolution down to 10 dyn (10 N). Since its development, several groups have used the SFA to measure the retarded and unretarded dispersion forces, electrostatic repulsions in a variety of electrolytes, structural and solvation forces (see below), and numerous studies of polymeric and biological systems. 

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances... 

The repulsion between oil droplets will be more effective in preventing flocculation Ae greater the thickness of the diffuse layer and the greater the value of 0. the surface potential. These two quantities depend oppositely on the electrolyte concentration, however. The total surface potential should increase with electrolyte concentration, since the absolute excess of anions over cations in the oil phase should increase. On the other hand, the half-thickness of the double layer decreases with increasing electrolyte concentration. The plot of emulsion stability versus electrolyte concentration may thus go through a maximum. 

For example, van den Tempel [35] reports the results shown in Fig. XIV-9 on the effect of electrolyte concentration on flocculation rates of an O/W emulsion. Note that d ln)ldt (equal to k in the simple theory) increases rapidly with ionic strength, presumably due to the decrease in double-layer half-thickness and perhaps also due to some Stem layer adsorption of positive ions. The preexponential factor in Eq. XIV-7, ko = (8kr/3 ), should have the value of about 10 " cm, but at low electrolyte concentration, the values in the figure are smaller by tenfold or a hundredfold. This reduction may be qualitatively ascribed to charged repulsion. 

The necessity to calculate the electrostatic contribution to both the ion-electrode attraction and the ion-ion repulsion energies, bearing in mind that there are at least two dielectric ftmction discontinuities hr the simple double-layer model above. 

Here we consider the total interaction between two charged particles in suspension, surrounded by tlieir counterions and added electrolyte. This is tire celebrated DLVO tlieory, derived independently by Derjaguin and Landau and by Verwey and Overbeek [44]. By combining tlie van der Waals interaction (equation (02.6.4)) witli tlie repulsion due to the electric double layers (equation (C2.6.lOI), we obtain... 

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