Boltzmann formalism

In this section we will consider only equilibria in which the number of moles of solute does not change. In such cases the population of a given contributor to the equilibrium may be calculated by using a standard Boltzmann formalism, i.e., the fraction of species A is... 

There are also other effects, which cannot be incorporated easily in the Poisson-Boltzmann formalism, such as the non-planarity of the interface, its thermal fluctuations, the charge fluctuations and so on. 

A more complete treatment should include in the Poisson-Boltzmann formalism all the interactions between the electrolyte ions and the interface, summarized in Section 2. Here it will be shown that, by accounting only for the hydration interaction of the electrolyte ions, one can capture the experimental dependence of the surface potential on pH and electrolyte concentration. 

In general, Bt = B2 and a double layer is generated by the asymmetric distribution of the electrolyte ions even if there is no charge on the plate surface. A treatment based on the modified Poisson-Boltzmann formalism will be presented in Section... 

The polarization model is extended to account for the ion-ion and ion-surface interactions, not included in the mean field electrical potential. The role of the disorder on the dipole correlation length A, is modeled through an empirical relation, and it is shown that the polarization model reduces to the traditional Poisson Boltzmann formalism (modified to account for additional interactions) when X, becomes sufficiently small. 

Use Boltzmann formalism to calculate entropy (sometimes difficult to do). 

In the following sections the basic concepts of the transport theories within the scope of the Boltzmann formalism are described. In the succeeding discussion the validity and the applicability of the Matthiessen and the Nordheim-Gorter rule are considered. A description of transport phenomena (electrical and thermal resistivity, thermopower) without an external magnetic field is given in sect. 2.3. There, the influence of the various scattering mechanisms on the temperature dependence of the electrical and the thermal resistivity as well as on the thermopower will be briefly discussed. 

The calculation of the coefficients /Sy can be carried out in the scope of the Boltzmann formalism. The Boltzmann equation can be written in its general form as ... 

The Boltzmann formalism is based on the assumption that a distribution function, f y,r), can be defined. However, we are dealing with a strongly interacting, quantum-mechanical system, and it is by no means clear, because of the Heisenberg uncertainty principle, that a distribution function in six-dimensional k,r) phase space can be correctly defined. However, in many cases one considers the distribution function as a function of k only the Heisenberg uncertainty principle does not preclude the use of this function. When a positional dependence is considered at all, it usually occurs through the dependence of the distribution function on external parameters, such as the dependence of the temperature on r in the case when a temperature gradient exists = (8/o/8r)(0T/5x). This expression makes sense if the electron can be... 

The previous section concerned the application of the Boltzmann formalism. An exact solution of the linearized Boltzmann equation (eq. (11)) is very difficult. However, there are many examples of calculations which are in good agreement with experiment. 

The Poisson-Boltzmann formalism is used to compute the electric potentials and charge distributions. In general, this requires numerical work. The Debye-Htickel theory assumes a weak electrical energy compared to the thermal energy allowing closed form analytical formulas for various quantities of interest in electrolyte solutions. 

An analogous situation arises for what concerns the ability of CG models to correctly reproduce system entropy. According to the Boltzmann formalism, the entropy of a system depends on the number of available microscopic phase space microstates. Since several degrees of freedom are discarded in the CG representation, the entropy of a system may thus be (sometimes dramatically) wrong. Even though the overall free energy can be set as to reproduce that of atomistic systems or of... 

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