Approximate Mobility Formula

For the limiting case of a 0, the particle core vanishes, so a spherical soft particle becomes a spherical polyelectrolyte. In this case, Eq. (25.27) tends to 

In this section, we treat the practically important case where potential is not very high so that dynamic relaxation effect is negligible. In this case, we have 5pei(T) = 0 or 5rij(r) = 0. Consider first the case where potential is low and a — 0 and where Pflx = constant, which corresponds a uniformly charged spherical polyelectrolyte of radius b. 

We thus obtain the following expression for the dynamic mobility of a spherical polyelectrolyte  

In this case, the potential inside the polyelectrolyte layer can be approximated by Eq. (21.47). 

Equation (25.45) is the required expression for the dynamic mobility of a soft particle, applicable for most practical cases. When co 0 fi 2, y 0, and F 0), Eq. (25.45) tends to Eq. (21.51) for the static case. When the polyelectrolyte layer 

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We derive approximate mobility formulas for the simple but important case where the potential is arbitrary but the double-layer potential still remains spherically symmetrical in the presence of the applied electric field (the relaxation effect is neglected). Further we treat the case where the following conditions hold... 

Although the theory of polyelectrolyte dynamics reviewed here provides approximate crossover formulas for the experimentally measured diffusion coefficients, electrophoretic mobility, and viscosity, the validity of the formulas remains to be established. In spite of the success of one unifying conceptual framework to provide valid asymptotic results, in qualitative agreement with experimental facts, it is desirable to establish quantitative validity. This requires (a) gathering of experimental data on well-characterized polyelectrolyte solutions and (b) obtaining the relationships between the various transport coefficients. Such data are not currently available, and experiments of this type are out of fashion. In addition to these experimental challenges, there are many theoretical issues that need further elaboration. A few of these are the following ... 

Equation (2.16) is applicable for all for 10 < ka < oo. To obtain an approximate mobility expression applicable for ka < 10, it is convenient to express the mobility in powers of and make corrections to higher powers of in Henry s mobility equation (2.6), which is correct to the first power of Ohshima [25] derived a mobility formula for a spherical particle of radius a in a symmetrical electrolyte solution of valence z and bulk (number) concentration n under an applied electric field. The drag coefficient of cations, A+, and that of anions, A, may be different. The result is... 

The first term on the right-hand side of Equation (2.26) corresponds to an approximate Equation (2.10) for Henry s function (Equation (2.6)). Equation (2.26) excellently agreed with exact numerical results [15] especially for small Ka, in which region no simple analytic mobility formula is available other than Henry s equation (2.6). Thus Equation (2.26) is a considerable improvement of Henry s equation (2.8). Eor example, the relative error is less than 1% for < 7 at ka = 0.1 and for < 3 at ka = 1. 

Bromine is a dense, mobile, dark red liquid at room temperature. It is moderately soluble in water (33.6 g L 1 at 25°C) and miscible with nonpolar solvents such as CS2 and CCI4. Like Q2 it gives a crystalline hydrate, which appears to have a unique, noncubic structure, with a formula approximating to Br2-7.9H20. 

Equations 1-4 can easily be set up on a PC and used to simulate or fit temperature-dependent mobility data. In an n-type sample, the only undetermined parameter is the acceptor concentration Na, so usually Na is varied to give the best fit to the data. (For this fit, the approximate carrier concentration, nn = 1/Re, can be used when n is required in the various scattering formulas.) Then, the Hall factor r = can be calculated at... 

Hence, in the first approximation, the extra entropy of the mobile adsorbate compared with that of the localized adsorbate exponentially decreases with the higher barrier of diffusion. The formula is probably not very accurate for the barriers... 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

This condition is guaranteed for the correct mobility matrix. However, the mobility matrix given by eqn (4.40) is an approximate one, and does not satisfy the inequality (4.1 ) in a certain configuration in which the beads are too close to each other. An improved formula which guarantees the inequality is proposed by Rotne and Prager. However, this correction is irrelevant for the asymptotic behaviour of N 1, which is determined by the hydrodynamic interaction between beads far apart from each other. Thus we shdl use eqn (4.40) for H, . 

HOMO and LUMO levels of the NCs are calculated based on reported bulk values, effective electron and hole mobilities of the bulk material, and the relative permittivities following the formula for the effective mass approximation from Brus [34]. 

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