Quasi-regular solution

Figure 9.3 lnyA of a quasi-regular solution A-B for xA = = 05 as a function of tempera-... 

T entropy term in the quasi-regular solution model... 

The entropy of mixing of many real solutions will deviate considerably from the ideal entropy of mixing. However, accurate data are available only in a few cases. The simplest model to account for a non-ideal entropy of mixing is the quasi-regular model, where the excess Gibbs energy of mixing is expressed as... 

While a random distribution of atoms is assumed in the regular solution case, a random distribution of pairs of atoms is assumed in the quasi-chemical approximation. It is not possible to obtain analytical equations for the Gibbs energy from the partition function without making approximations. We will not go into detail, but only give and analyze the resulting analytical expressions. 

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

Statement 3. If the concentrations are fixed, iVa = const, Nb = const, the set of kinetic equations (8.2.17), (8.2.22) and (8.2.23) as functions of the control parameter demonstrates two kinds of motions for k k the stationary (quasi-steady-state) solution holds, whereas for k < k a regular (quasi-regular) oscillations in the correlation functions like standing waves... 

For this purpose it was assumed that the molecules are located on a quasi-crystalline lattice and can be moved about on this lattice without affecting the spacing of the energy levels of the system. This will be discussed again in Chapter 14 in connexion with regular solutions and the equations above will also be applied to the problem of adsorption. 

Essentially, we are looking at the solubility of metals in other metals - i.e. monophase metal alloys. The most commonly used solution models are the models with similar atomic volumes, which give us the perfect solution, the infinitely-dilute solution and the strictly-regular solution. Thus, we will then look at Guggenheim s quasi-chemical model, which includes the notion of short-distance order. 

In order to evaluate the functions g(5) and E s), we need to know the distribution of the atoms on the lattice for the given value of s. Two models have been developed the Gorsky, Bragg and Williams model and the quasi-chemical model. The hypotheses upon which these models are based are similar, respectively, to those used for the model of a strictly-regular solution (see section 2.3.3) and those used for Fowler and Guggenheim s quasi-chemical solution model (see section 2.3.5). 

Figure 3.2. Comparison of the excess Gibbs energies of a strictly-regular solution and the quasi-chemical model (reproducedfrom [DES10], p.62 - see Bibliography)...

The thermodynamic data of polymer solutions were first derived from the approach of regular solutions of small molecules by Flory [29] and by Huggins [30]. In the original quasi-lattice model of Flory, the interaction parameter, x, was representative of the enthalpic difference of polymer/polymer, sol-vent/solvent and polymer/solvent contacts. Progressively % lost this simple signification and represented the shift with an ideal behavior (including entropic effects) but nevertheless the model was established for... 

Introduction 53. 2 Strictly Regular Solutions 55 3. Quasi-Chemical Approximation 59. 4 General Remarks concerning the Order-Disorder Problem 64 5 Mole-... 

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... 

Statement 2. A set (8.3.22) to (8.3.24) with fixed concentrations = P/K and N = VIP has two kinds of motions dependent on the value of parameter n. As k kq, the stationary (quasi-steady-state) solution occurs, whereas for /c < kq the correlation functions demonstrate the regular (quasiregular) oscillations of the standing wave type. The marginal magnitude is Ko = o(p,/3)-... 

According to the basic principles of the regularization method, we have to find a quasi-solution of the inverse problem as the model ma that minimizes the parametric functional... 

For example, let be a solution, obtained for the given ao. Then wo can use this solution as the initial apjuoximation mo for the regularized MRM in the next cycle, when we use th< next value of rcgularizal ion parameter Oi, etc. The calculations arc repeated until we find the quasi-optimal value of the regularization parameter, based on the rnislil condition (2.81) ... 

Therefore, the linear inverse problem (10.4) can have an infinite number of equivalent solutions. All these nonradiating currents form a so-called null-space of the linear inverse problem (10.4). In principle, we can overcome this difficulty using a regularization method, which restricts the class of the inverse excess currents j in equation (10.4) to physically meaningful solutions only. We will discuss an approach to the solution of this problem below, considering a quasi-linear method. 

Unlike the above mentioned methods, another Floquet-theorem-based approach, the many-electron many-photon theory (MEMPT) of Mercouris and Nicolaides (71,72) does not involve complex rotated Hamiltonians. The complex coordinate rotation is used only to regularize that part of the wave functions which describes unbound electrons (see the CESE method). This allows efficient description of bound or quasi-bound states, involved in a problem under consideration, by MCHF solutions and therefore enables ab initio application to many-electron systems (71,72,83-87). 

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