Boltzmann approximation

Other molecular thermodynamic models for protein-reverse micelle complexes have also emerged. Bratko et al. [171] presented a model for phase transfer of proteins in RMs. The shell and core model was combined with the Poisson-Boltzmann approximation for the protein-RM complex and for the protein-free RM. The increase in entropy of counterions released from RMs on solubilization of a protein was the main contribution to the decrease in free energy of com-plexation. Good agreement was found with SANS results of Sheu et al. [151] for cytochrome C solubilization and the effect of electrolytes on it. However, this model assumes that filled and empty RMs are of the same size, independent of salt strength and pH, which is not true according to experimental evidence available since then. 

Ioffe and Regel (1960) were the first to point out that small values of l such that kl < 1 /2rc are in fact impossible. This point is developed further in the next section. Moreover, when fci l/2rc, which in metals is equivalent to the condition i a where a is the distance between atoms, the Boltzmann approximation becomes a bad one because k is no longer a good quantum number. This occurs particularly... 

If eF is at least a few kT from either band edge, then we can ignore the 1 in the integrands of Eqs. (B12) and (B13) (Boltzmann approximation) and explicitly solve the respective integrals. The final results are... 

Note that this result is independent of ef in the Boltzmann approximation, used here. 

Equation (39) corresponds to the Boltzmann approximation of statistical physics (or the so-called Pauli equation). We shall discuss it in more detail in Section VII. 

After this brief survey of the general theory let us discuss in a little more detail the Boltzmann approximation (Eq. (39)) and the equation (Eq. (40)) retaining higher order effects. 

The main assumption in eq 3 is that the small particles can be treated in the framework of the Poisson—Boltzmann approximation. However, being of finite size and having a relatively large charge, the interactions among particles as well as their van der Waals interactions with the plates should also be involved in the distribution of their concentration. The latter treatment24 will allow identification of the conditions under which the present simple treatment is valid. 

An interesting consequence of the Boltzmann approximation is that the product of the electron and hole densities, viz.,... 

The osmotic pressure between two flat surfaces can be derived within the Poisson-Boltzmann approximation. The PB equation was originally developed to describe ion distributions outside a large charged surface. However, there are extended PB equations where polymers have been included [32]. The expression for the osmotic pressure given below is valid in the absence of polymers. In the PB equation the correlations between ions are neglected, which means that Pei is identically zero. Furthermore, the ions are normally treated as point particles which means that the collision term disappears. Thus for symmetric systems only two terms remain, the kinetic pressure and the bulk pressure. The net pressure can be written as... 

Wennerstrom, H., Jonsson, B., and Linse, P. The cell model for poly-electrolyte systems - exact statistical mechanical relations, Monte-Carlo simulations, and the Poisson-Boltzmann approximation. Journal of Chemical Physics, 1982, 76, No. 9, p. 4665 -670. 

Figure 4.25 shows the relevant densities-of-states functions, again for the example of an n-type electrode. On the solution side, the density-of-states functions DofE) and of the Rd, Ox couple have the usual gaussian form (and we again assume these to be independent of electrode potential). On the semiconductor side, the density DfE) of states in the conduction band increases parabohcally from the band-edge energy E. The occupancy functionpfE) has the same Fermi-Dirac form as for a metal electrode but only its tail lies in the conduction band, where the Boltzmann approximation usually suffices. 

In contrast to the sitnation for a metal electrode, the rate constant for ET or HT at a doped semicondnctor electrode is therefore independent of U. The rate (cnrrent) increases exponentially with overpotential because the concentration of charge carriers at the electrode surface does. In the absence of current-limiting slow steps such as diffusion, the current at a semiconductor electrode is therefore predicted to obey Butler-Volmer kinetics (up to the limit where the Boltzmann approximation to the Fermi probabihty function is vahd). 

The theoretical basis for such a rationale has been laid in the recent work of Pack et al [161,162]. Using the Poisson-Boltzmann approximation the pH-contour maps on and near the surface of B-DNA ( poly(dG).poly(dC)) have been constructed under simulated conditions of 45 mM tris buffer with 3mM Mg at pH 7.5. Three domains of high ET concentration (>10p.M) are predicted one is spread over the minor groove and two are localised in the major groove near N7(G) and C5(C) for a G.C base pair [114,163]. The reduction in pH by two units would translate into one hundred fold increase in TC production compared to the bulk rate. This is manifested in the accelerated rate of DNA-mediated hydrolysis. Elaborating on the two state model of Islam et al [149] in which the DE is either free or statically bound. Pack and Wong [163(a)] concluded that the catalysis by DNA is primarily an electrostatic effect of acidic domains in the surface grooves of the nucleic acid. While such computations were found satisfactory for a //-BaPDE hydrolysis, they could not adequately reproduce... 

The electrostatic charges of surfactants seriously affect the localization of host molecules in the water pool. Monte Carlo simulation in which ionic reversed micelles are treated as spherical entities showed the presence of the electrical double layer in the interface of the water pool, and the distribution of counterions followed the Poisson-Boltzmann approximation [51]. Mancini and Schiavo [52] assumed recently, by the yield of halogenation, that the specific interactions between bromide or chloride ions and an ammonium head-group in cationic reversed micelles keep the ions in a defined position on the interface. 

As it was shown in Ref [81] within the linearized Poisson-Boltzmann approximation the diffuse double-layer capacitance takes the form. 

Boltzmann approximation invalid Because the nanocrystalline film contains a high density of intraband gap states, the Boltzmann approximation for n (Eq. 18) may not be valid. Instead, n will be determined by some distribution of intraband gap states, with distribution g(E), and Eq. (17) should apply. 

The fundamental theories on EDL interactions have been substantially advanced by several researchers in the recent past in order to incorporate the effects of other pertinent physicochemical phenomena in the mathematical model and to generalize the underlying postulates. Kjellander and Mitchell [5] employed the dressed ion theory for EDL structure and interactions, which is nothing but an exact statistical mechanical formahsm for electrolyte systems. In their theory, the dressed ions took equivalent roles as the bare ions in the Poisson-Boltzmann approximation. A practical method was also derived for evaluating the effective surface charge densities of the particles. Behrens and Borkovec [6] proposed... 

The introduction of counterions and added electrolyte introduces yet another length scale to electrolyte solutions, one that manifests the enhanced decay of electrostatic interactions in the presence of screening charges. When electrostatic effects are treated within the standard Poisson-Boltzmann approximation, the Debye length k governs the screening length of electrostatic interactions by these ions ... 

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