Quasi-lattice theory

The first term in Eq. (100) may be calculated from the free energies of formation of pure salts and the four linearly additive binary terms may be evaluated from information on the four binary systems. Only the last term cannot be directly evaluated from information on lower-order systems. A useful approximation to P in this term may be made by comparing it with analogous terms in the quasi-lattice theory.13,18 This approximation is... 

Recently, Chagnes et al. [22] treated the molar conductivity of LiCl04 in y-buty-rolactone (y-BL) on the basis of the quasi-lattice theory. They showed that the molar conductivity can be expressed in the form A = (A°°) — fe c1/3 and confirmed it experimentally for 0.2 to 2 M LiCl04 in y-BL. They also showed, using 0.2 to 2 M LiCl04 in y-BL, that the relation k = Ac = (A°°) c — k cA was valid and that /cmax appeared at cmax = [3(xf00)74fe ]3 where d/c/dc=0. 

With increase in salt concentration the approximations involved in the Debye-Hiickel theory become less acceptable. Indeed it is noteworthy that before this theory was published a quasi-lattice theory of salt solutions had been proposed and rejected (Ghosh, 1918). However, as the concentration of salt increases so log7 ,7 being the mean ionic activity coefficient, appears as a linear function of c1/3 (the requirement of a quasi-lattice theory) rather than c1/2, the DHLL prediction (Robinson and Stokes, 1959). Consequently, a quasi-lattice theory of salt solutions has attracted continuing interest (Lietzke et al., 1968 Desnoyers and Conway, 1964 Frank and Thompson, 1959 Bahe, 1972 Bennetto, 1973) and has recently received some experimental support (Neilson et al., 1975). 

Gibbs and DiMarzio (34, 33) developed a theory for a second order transition in polymers based on the quasi-lattice theory and taking into... 

There is, however, a problem in dilute solutions, when values of Xb or xx approach zero, since the activity coefficients of AY should then approach those in the binaries AX-AY or AX-BX. This problem overcomes the quasi-lattice theory. 

The quasi-lattice theory is based on a quasi-lattice, which consists of two interlocking sub-lattices of cations and anions. Only the nearest neighbor interactions are taken into account and the energy of any nearest neighbor pair is assumed to be independent of its environment (additivity of pair bond interactions). Nearest neighbor interactions are ignored, which means that all the binary systems are ideal. 

Dacre, B. Benson, G. C. Application of quasi-lattice theory to alcohol - carbon tetrachloride systems Can. J. Chem. 1963,41... 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

Figure 7.15 Two-dimensional square lattice and figure for the development of a historical quasi-chemical theory. The model assumes that each lattice site will be occupied by either a black ( O ) or gray ( W ) square. Near-neighbor pairs of the same species contribute -7 < 0 to the net interaction energy. Near-neighbor pairs of the different species contribute 7 > 0 to the net interaction energy. Our calculation will focus on this five-site figure.

III. Harmonic and Quasi-harmonic Theories of Lattice Dynamics. 149... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Lattice dynamics calculations on the plastic /3-nitrogen phase are relatively scarce because, obviously, the standard (quasi-) harmonic theory cannot be applied to this phase. Classical Monte Carlo calculations have been made by Gibbons and Klein (1974) and Mandell (1974) on a face-centered cubic (a-nitrogen) lattice of 108 N2 molecules, while Mandell has also studied a 32-molecule system and a system of 96 N2 molecules on a hexagonal close-packed (/3-nitrogen) lattice. Gibbons and Klein used 12-6... 

Let us consider a system consisting of molecules of type 1, N2 molecules of type 2,. A, molecules of type t at temperature T and external pressure P. According to the LF theory, the molecules are considered arranged on a quasi-lattice of sites, Nq of which are empty. Every molecule of type k consists of segments of volume v>l each. The total number of segments in the system is... 

It should be noted that a quasi-lattice, quasi-chemical theory of preferential solvation has been developed by Marcus (1983, 1988, 1989, 2002). In the author s opinion, this approach is not adequate to describe PS in liquid mixtures, especially when the different species have widely different sizes. 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

Marcus Y(1983) A quasi-lattice quasi-chemical theory of preferential solvation of ions. Austr J Chem 36 1719-1738... 

Marcus Y (1977) Introduction to liquid-state chemistry. Wiley, Chichester, pp 241—245,267—279 Marcus Y (1983) Ionic radii in aqueous solutions. J Sol Chem 12 271—275 Marcus Y (1983a) A quasi-lattice quasi-chemical theory of preferential solvation of ions. Austr J Chem 36 1718-1738... 

The quasi-lattice model was developed by Roe (13) and Scheutjens and Fleer (14) (SF theory) The basic analysis considered all chain conformations as step-weighted random walks on a quasi-crystalline lattice that extends in parallel layers from the surface. This is illustrated in Figure 16.2 which shows a possible conformation of a polymer molecule at a flat surface. The partition function was written in terms of the number of chain configurations that were treated as connected sequences of segments. In each layer parallel to the surface, random mixing between the segments and solvent molecules was assumed, i.e. by using... 

Some novel statistical theories of solutions of polymers use the %i parameter, too. They predict the dependence of the %i parameter on temperature and pressure. According to the Prigogine theory of deformable quasi-lattice, a mixture of a polymer with solvents of different chain length is described by the equation ... 

The entropy theory is the result of a statistical mechanical calculation based on a quasi-lattice model. The configurational entropy (S, ) of a polymeric material was calculated as a function of temperature by a direct evaluation of the partition function (Gibbs and Di Marzio (1958). The results of this calculation are that, (1) there is a thermodynamically second order liquid to glass transformation at a temperature T2, and 2), the configurational entropy in the glass is zero i.e. for T > T2, => 0 as T... 

Hence, increasing best response functions is the only major requirement for an equilibrium to exist players objectives do not have to be quasi-concave or even continuous. However, to describe an existence theorem with non-continuous payoffs requires the introduction of terms and definitions from lattice theory. As a result, we restricted ourselves to the assumption of continuous payoff functions, and in particular, to twice-differentiable payoff functions. 

The basic features of the lattice theory and structure of a quasilattice EOS and its application to fluids and fluid mixtures was reviewed by Smirnova and Victorov (2000). Victorov et al. (1991) used the hole quasi-chemical group-contribution model of Victorov and Smirnova (1985) and Smirnova and Victorov (1987) to calculate the phase equilibria in water -i- n-alkane binary mixtures. This model is essentially a generalization of the Barker lattice theory in its group-contribution formulation, the main difference being the presence of vacant lattice sites (holes). The model becomes volume-dependent, and thus the derived EOS adopted the following form (Smirnova and Victorov, 1987). 

A more general statistical thermodynamic theory can be obtained with the quasi-lattice models see Figure 3.3. If a coordination number z of the lattice is assumed, then there are z - 1 choices of where to put the next bond in the chain. (This is a little smaller with excluded volume considered.) Then the entropy varies with dimensionality of the quasi-lattice according to... 

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