Ionic fluctuation forces

Fuzzy spheres. Radially varying dielectric response, 79 "Point-particle" interactions, 79 Point-particle substrate interactions, 85 Particles in a dilute gas, 86 Screening of "zero-frequency" fluctuations in ionic solutions, 89 Forces created by fluctuations in local concentrations of ions, 90 Small-sphere ionic-fluctuation forces, 91... 

Just as a charged sphere in saltwater surrounds itself with a number of mobile ions different from what would occupy the same region in its absence, so does a charged cylinder. As with spheres, there are low-frequency ionic fluctuations that create attractive forces between like cylinders. In the special case of thin cylinders whose material dielectric response is the same as that of the medium and the distance between cylinders is small compared with the Debye screening length, this ionic-fluctuation force has appealing limiting forms. 

MONOPOLAR INTERACTIONS, IONIC-FLUCTUATION FORCES, BETWEEN SMALL CHARGED PARTICLES... 

Following the strategy for extracting small-particle van der Waals interactions from the interaction between semi-infinite media, we can specialize the general expression for ionic-fluctuation forces to derive these forces between particles in salt solutions. Because of the low frequencies at which ions respond, only the n = 0 or zero-frequency terms contribute. In addition to ionic screening of dipolar fluctuations, there are ionic fluctuations that are due to the excess number of ions associated with each particle. 

MONOPOLAR IONIC-FLUCTUATION FORCES BETWEEN THIN CYLINDERS... 

Following the Pitaevskii strategy for extracting small-particle van der Waals interactions for the interaction between suspensions, we specialize the general expression for ionic-fluctuation forces to derive forces between cylinders (Level 3). As with the extraction of dipolar forces between rods, consider two regions A and B, dilute suspensions of parallel rods immersed in salt solution interacting across a region of salt solution m (see Fig. L2.19). 

Again, as with ionic-fluctuation forces in planar and spherical geometries, the exponential shows the effect of double screening, the only difference being the distance dependence in the denominator. 

This is valid only in the case of an effectively infinite medium in which no walls limit the flow of charges. Conductors must be considered case-by-case under the limitations imposed by boundary surfaces. See, for example, the treatment of ionic solutions (Level 1, Ionic fluctuation forces Tables P.l.d, P.9.C, S.9, S.10, and C.5 Level 2, Sections L2.3.E L2.3.G and Level 3, Sections L3.6 and L3.7). 

For ionic-fluctuation forces, the e s are now the dielectric constants in the limit of zero frequency (f = 0). The integration over wave vectors u, v can be converted to a p, ir integration ... 

Not only are there fluctuations in the electric fields that create the dipolar fluctuations of most van der Waals forces but there are also fluctuations in electric potential with concomitant fluctuations in the number density of ions and the net charge on and around these small spheres. Monopolar charge-fluctuation forces occur when ion fluctuations in the spheres differ from ion fluctuations in the medium. Perhaps it is better to say that these forces occur when ion fluctuations around the suspended particles differ from what they would have been in the solution in the absence of particles. To formulate these interactions, we allow the ionic population of the spheres to equilibrate with the surrounding salt solution and to exchange ions with that surrounding solution. Then we compare the ionic fluctuations that occur from the presence of the small spheres with those in their absence. To do this we must have a way to count the number of extra ions associated with each sphere compared with the number of ions in their absence. 

Because they have so many unexpected features and because they are kind of a hybrid with electrostatic double-layer forces, ionic-charge-fluctuation forces deserve separate consideration. 

The wave equation is built from V E cx pext/ - Because electrostatic double-layer equations are easier to think about in terms of potentials rather than electric fields E = -V0, we set up the problem of ionic-charge-fluctuation forces in terms of potentials. Charges pext come from the potential 0 through the Boltzmann relation... 

There also exists a range of possibilities that fall between the continuum and molecular models. These are approaches that combine some features of both. For example, in the description of Brownian motion of macromolecules or latex beads in the 10-100 pm size range, one uses classical hydrodynamic results such as the Stokes law for drag on a sphere while at the same time modeling fluctuating forces from molecular collisions that arise at the level of the molecular description. In the theory of ionic... 

Dispersive forces are more difficult to describe. Although electric in nature, they result from charge fluctuations rather than permanent electrical charges on the molecule. Examples of purely dispersive interactions are the molecular forces that exist between saturated aliphatic hydrocarbon molecules. Saturated aliphatic hydrocarbons are not ionic, have no permanent dipoles and are not polarizable. Yet molecular forces between hydrocarbons are strong and consequently, n-heptane is not a gas, but a liquid that boils at 100°C. This is a result of the collective effect of all the dispersive interactions that hold the molecules together as a liquid. 

The I term is of particular relevance since, in anisotropic media such as liposomes and artiflcial membranes in chromatographic processes, ionic charges are located on the polar head of phospholipids (see Section 12.1.2) and thus able to form ionic bonds with ionized solutes, which are therefore forced to remain in the nonaqueous phase in certain preferred orientations. Conversely, in isotropic systems, the charges fluctuate in the organic phase and, in general, there are no preferred orientations for the solute. Given this difference in the I term (but also the variation in polar contributions, less evident but nevertheless present), it becomes clear that log P in anisotropic systems could be very different from the value obtained in isotropic systems. 

The mobility of ions in melts (ionic liquids) has not been clearly elucidated. A very strong, constant electric field results in the ionic motion being affected primarily by short-range forces between ions. It would seem that the ionic motion is affected most strongly either by fluctuations in the liquid density (on a molecular level) as a result of the thermal motion of ions or directly by the formation of cavities in the liquid. Both of these possibilities would allow ion transport in a melt. 

The forces involved in the interaction al a good release interface must be as weak as possible. They cannot be the strong primary bonds associated with ionic, covalent, and metallic bonding neither arc they the stronger of the electrostatic and polarization forces that contribute to secondary van der Waals interactions. Rather, they are the weakest of these types of forces, the so-called London or dispersion forces that arise from interactions of temporary dipoles caused by fluctuations in electron density. They are common to all matter. The surfaces that are solid at room temperature and have the lowest dispersion-force interactions are those comprised of aliphatic hydrocarbons and fluorocarbons. 

A major ingredient for an RG treatment is a simple and transparent characterization of the molecular forces driving phase separation. This situation calls for mean-field theories of the ionic phase transition. The past decade has indeed seen the development of several approximate mean-field theories that seem to provide a reasonable, albeit not quantitative, picture of the properties of the RPM. Thus, the major forces driving phase separation seem now to be identified. Moreover, the development of a proper description of fluctuations by GDH theory has gone some way to establish a suitable starting point for RG analysis. Needless to say, these developments are also of prime importance in the more general context of electrolyte theory. 

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