Ionic fluctuations

The reactant R2 can also be considered to be a solvent molecule. The global kinetics become pseudo first order in Rl. For a SNl mechanism, the bond breaking in R1 can be solvent assisted in the sense that the ionic fluctuation state is stabilized by solvent polarization effects and the probability of having an interconversion via heterolytic decomposition is facilitated by the solvent. This is actually found when external and/or reaction field effects are introduced in the quantum chemical calculation of the energy of such species [2]. The kinetics, however, may depend on the process moving the system from the contact ionic-pair to a solvent-separated ionic pair, but the interconversion step takes place inside the contact ion-pair following the quantum mechanical mechanism described in section 4.1. Solvation then should ensure quantum resonance conditions. 

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... 

With an important exception, Rn relativistic screening functions go as previously, Rn = (1 + r )e rn, with the ratio r reflecting the distance / or (/ + bi). This is an approximation for which we assume that there is a negligible difference between the velocities of light in m and in Bi at those frequencies at which relativistic retardation is occurring. The exception occurs when any region is a salt solution then it is not always permissible to use this approximation for the n = 0 term. (See the subsequent section on ionic fluctuations.)... 

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. 

Table P.9.c. Low-frequency ionic-fluctuation interactions between and across anisotropic media (magnetic terms neglected)...

Table S.6. Point particles (without ionic fluctuations or ionic screening)...

Table C.4. "Thin" dielectric cylinders parallel and at all angles, interaxial separation z <

Table C.5.a. Thin dielectric cylinders in saltwater, parallel and at an angle, low-frequency (n = 0) dipolar and ionic fluctuations... 

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. 

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. 

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). 

In addition to the screening factor (1 + 2/cZ)e 2ld, ionic fluctuations create a larger ALm Ar... 

Notice that k2 differs from the Debye constant k2 by a factor e. Because the dielectric permittivity is not a simple scalar quantity, it cannot be divided out of the V (eV0) on the left-hand side of the equation. Except for this difference, we can proceed as with ionic fluctuations in isotropic media. 

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 ... 

C.4. "Thin" dielectric cylinders, parallel and at all angles, interaxial separation z << radius R Lifshitz form retardation, magnetic, and ionic-fluctuation terms not included C.4.a. Parallel, interaxial separation z C.4.b.l. At an angle 0, minimal interaxial separation z C.4.b.2. Torque r(z, 0)... 

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