Isolated ion pairs

Mozumder (1971) calculated F(t) by the prescribed diffusion method. For the isolated ion-pair case, the solution appears in (7.28) for the multiple ion-pair case, further approximation was introduced in the nature of mean force acting on an electron, by which the problem was reduced to that of a collection of isolated... 

The limitation of the prescribed diffusion approach was removed, for an isolated ion-pair, by Abell et al. (1972). They noted the equivalence of the Laplace transform of the diffusion equation in the absence of scavenger (Eq. 7.30) and the steady-state equation in the presence of a scavenger with the initial e-ion distribution appearing as the source term (Eq. 7.29 with dP/dt = 0). Here, the Laplace transform of a function/(t) is defined by... 

The methods discussed so far are essentially limited to isolated ion-pairs or, in the admittedly crude approximation, to cases when a multiple ion-pair spur can be considered to be a collection of single ion-pairs. Additionally, it is difficult to include an external field, as that will destroy the spherical symmetry of the problem. Stochastic treatments can incorporate both multiple ion-pairs and the effects of an external field. 

Pimblott (1993) has used MC and ME methods for the external field (E) dependence of the escape probability (Pesc) for multiple ion-pair spurs. At low fields, Pesc increases linearly with E with a slope-to-intercept ratio (S/I) very similar to the isolated ion-pair case as given by Onsager (1938). Therefore, from the agreement of the experimental S/I with the Onsager value, one cannot conclude that only isolated ion-pairs are involved. However, the near equality of S/I is contingent on small Pesc, which is not expected at high fields. 

Figure 4 of Mozumder (1971) compares the kinetics of neutralization in an isolated ion-pair, a spherical blob, and a short track. For the isolated ion-pair there is vey little recombination until 5 x 10 u s, but most recombination is over by -Ins. The main difference in the recombination kinetics is between the isolated and multiple ion-pair cases. There is not such a great difference among the different multiple ion-pair blobs or short tracks. In the multiple-ion-pair cases, the neutralization is gradual and much faster than that in the isolated ion-pair case at short times. However, this could be an artifact of the model predicated by close proximity of the positive ions having essentially zero mobility (vide infra). 

These results imply that the use of the representative single ion-pair distribution in the ionization produced by low-LET irradiation in liquid hydrocarbons can be approximately justified even though the track itself has considerable contribution from multiple-ion-pair spurs and short tracks. It also means that even in the case of an isolated ion-pair, the long-time limit of the existence probability is perturbed by the long-range coulombic field. 

Let us suppose that an isolated ion-pair is formed at time t0 with a separation r0. The rate of creation of the ion-pair is 5(r—r0)5(f — f0) (Chap. 6, Sect. 2.2) and the appropriate diffusion equation must include... 

The observed survival probability depends not only on the survival probability of an ion-pair formed with an initial separation, r0, but also on the distribution of initial separations, w(r0). As will be discussed in Sects. 3 and 4, the rate of loss of energy of ions after heterolytic bond fission (or, alternatively, the range of electrons formed by ionisation of a molecule) depends sensitively on the energetics of the solvent molecules. These separation distances can range up to 10 nm or more and are specifically discussed in Sect. 3. The survival probability of a collection of isolated ion-pairs, P(t), formed at a time t0 — 0, and with a distribution of initial distances w(r0) is... 

An isolated ion-pair, of initial separation r0 at time t0 = 0, in a nonpolar solvent may recombine or separate and ultimately escape. At a time t, the probability that the ion-pair will have recombined is < (t r0, t0 = 0) and that it is still extant p(t r0) f0 = 0). A short while later, the probability that the ion-pair has not recombined is p t + df r0, t0 =0). The change in survival probability is the probability that the ion-pair recombined during the time interval t to t + df, that it had a lifetime between t and t -f df. Defining the lifetime distribution function as f(f), then... 

Mozumder for the neutralization of an isolated ion pair in polar media [74]. The predictions are considerably more complex for solvents with a distribution of relaxation times and barrier energies that are comparable or larger than the available thermal energy. Intramolecular vibrational effects can also be important. 

Today, an ionic bond is recognized as the nondirectional electrostatic attraction between oppositely charged ions. For an isolated ion pair, the Coulomb potential energy, U, is simply... 

CaC03.6 H2O. In addition to the well-known anhydrous forms CaCOs also crystallizes with 1 and 6 H2O. In the remarkable structure of the hexahydrate there are isolated ion-pairs surrounded by an envelope of 18 water molecules. Of these,, six complete the 8-coordination group around Ca " and the remainder are bonded to other cations. (Each water molecule is adjacent to (only) one Ca ion.) Tliis type of hydrate, in which ion pairs (resembling the classical picture of a CaC03 molecule ) are embedded in a mass of H2O molecules, may be contrasted with... 

Cryptand, 37, and its reduced form, 38, come from the extensive work of Nelson and McKee. Other cryptands incorporating aromatic spacers are also described in their papers and can be made in a similar fashion. The compounds bind cations, specifically transition metals, and anions. As such they perhaps represent the ultimate three-dimensional supramolecular systems as the compounds encrypt isolated ion pairs. 

Figure 10 shows the 23Na MAS NMR spectra of some sodium salt ethylene ionomers at room temperature. A broad peak at about —10 to 12 ppm is clearly observed for all samples. The 23Na NMR spectra of the styrene ionomers have been reported44 (details are given in the next section, see Fig. 14). Peaks are observed at about 7, 0, and -12 to -23 ppm, which are assigned to isolated ion pairs, hydrated ions and aggregated ions respectively. The sodium cations in ethylene ionomers, therefore, are almost in ionic aggregates. The isolated ion pairs...

Polystyrene-based ionomers have no crystalline phase in a host polymer matrix. Figure 13 shows a schematic model of the styrene ionomer (compare with Fig. 1). One of the characteristics of styrene ionomers is that, in addition to the ionic aggregates, there are also some isolated ion pairs distributed throughout the polystyrene matrix, which do not contribute to the crosslinking of the polystyrene chains. 

Cooper et al. reported that solid-state 23Na NMR is useful for observing the sodium cation in sulfonated polystyrene ionomers (NaSPS).44 Three NMR peaks were detected, corresponding to isolated ion pairs, aggregated ions and hydrated ions (Fig. 14). The distributions of these three types of sodium cations are systematically influenced by hydration treatment, sulfonation level and neutralization level. Fully dried NaSPS at low ion content shows that the isolated and aggregated sodium ions in NaSPS are available for hydration. As the sulfonation level increases, the fraction of sodium ions held in isolated ion pairs decreases while the fraction of ionic species in the ionic aggregates increases. This coincides with a shift in the peak position of the aggregated sodium ions to low frequency, indicative of increased quadrupolar interactions. 

Before we consider an ionic lattice, let us review the appropriate equation for the change in internal energy when two oppositely charged ions M and X are brought together from infinite separation to form the isolated ion-pair, MX (equation 5.8). 

The change in internal energy can be estimated from equation 5.9 by considering the Coulombic attraction between the ions. For an isolated ion-pair ... 

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