Quasi-lattice

Figure A2.5.14. Quasi-lattice representation of an equimolar binary mixture of A and B (a) randomly mixed at high temperature, and (b) phase separated at low temperature.

It was supposed at the derivation of Eq. (13) that the number of places in quasi-lattice decreases twice. Such supposition describes perfectly the relatively large chains, where (r/2 + /) — rl2. Here t is the difference in the number of solvent molecules surrounding the chain. In comparison with rl2, t is stipulated by contacts of the solvent with the ends, forming at chain breaks. Then ... 

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

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... 

Icosahedral Al-Mg-Zn type with a quasi-lattice constant of about 520 pm and a concentration of free electrons of about 2.1 electrons per atom. Examples are represented by Li3CuAl6 (ar = 504 pm, super-space-group Pm35), Li3AuA16, Pd13Mg44Al43, CuMg4Al6, Mg32Zn52Ga16, Y-Mg-Zn andY-Zr-Mg-Zn alloys, etc. 

In an ionic melt, coulombic forces between charges of opposite sign lead to relative short-distance ordering of ions, with anions surrounded by cations and vice versa. The probability of finding a cation replacing an anion in such ordering is effectively zero and, from a statistical point of view, the melt may be considered as a quasi-lattice, with two distinct reticular sites that we will define as anion matrix and cation matrix. 

This quasi-lattice formulation of fused salts is known as Temkin s equation. Its application to silicate melts was provided by Richardson (1956), but it is inadequate for the compositional complexity of natural melts, mainly because, in a compositionally complex melt, the types of anions and consequently the entity of the anion matrix vary in a complicated way with composition. 

First, consider a binary liquid A-B in which A and B atoms mix substitutionally on a quasi-lattice with coordination number Z. There is the possibility that A-B pairs will be formed from A-A and B-B pairs by the following relation... 

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. 

The effects of coupling of the DTO and RB units in not only one- but also three-dimensional arrays are discussed below and molecular weight trends illustrated. A fundamental connection between relaxation times and normal mode frequencies, shown to hold in all dimensions, allows the rapid derivation of the common viscoelastic and dielectric response functions from a knowledge of the appropriate lattice vibration spectra. It is found that the time and frequency dispersion behavior is much sharper when the oscillator elements are established in three-dimensional quasi-lattices as in the case of organic glasses. 

As for all the systems relegated to Section 2 the attenuation function for structural H2O in the microwave and far-infrared region, as well as that for free H2O, can be understood in terms of collision-broadened, equilibrium systems. While the average values of the relaxation times, distribution parameters, and the features of the far-infrared spectra for these systems clearly differ, the physical mechanisms descriptive of these interactions are consonant. The distribution of free and structural H2O molecules over molecular environments is different, and differs for the latter case with specific systems, as are the rotational dynamics which govern the relaxation responses and the quasi-lattice vibrational dynamics which determine the far-infrared spectrum. Evidence for resonant features in the attenuation function for structural H2O, which have sometimes been invoked (24-26,59) to play a role in the microwave and millimeter-wave region, is tenuous and unconvincing. 

Biopolymers Rotational Diffusion and Quasi-Lattice Vibra-... 

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). 

Alan Mackay made the connection with crystallography [139], He designed a pattern of circles based on a quasi-lattice to model a possible atomic structure. An optical transformation then created a simulated diffraction pattern exhibiting local tenfold symmetry (see, in the Introduction). In this way, Mackay virtually predicted the existence of what was later to be known as quasicrystals, and issued a warning that such structures may be encountered but may stay unrecognized if unexpected ... 

In an ideal quasi-lattice, no ion-pairs can be distinguished over statistically significant time intervals, since all sites are equivalent. This is also the condition that leads to the maximum ionic conductivity for a given fluidity (at least in the absence of decoupling, see below). This is because the presence of ion-pairs lowers the conductivity by permitting diffusion, hence the fluid flow without ionic current flow. 

Structure is not the same after an ion has entered it near the ion. Some of the water molecules are wrenched out of the quasi-lattice and appropriated by the ion as part of its primary solvation sheath. Further off, in the secondary solvation sheaths, the ions produce the telltale effects of structure breaking. 

The influence of the interaction in binary solvents on AG r ions was analyzed by Y. Marcus [261], who assumed a quasi-lattice model for the electrolyte in such solutions. Free energies of transfer of various ions were collected and discussed [75, 76]. Ion solvation including mixed solvent media has been reviewed by several authors [45, 76, 262-265]. 

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