Static lattice model

Given the character of the water-water interaction, particularly its strength, directionality and saturability, it is tempting to formulate a lattice model, or a cell model, of the liquid. In such models, local structure is the most important of the factors determining equilibrium properties. This structure appears when the molecular motion is defined relative to the vertices of a virtual lattice that spans the volume occupied by the liquid. In general, the translational motion of a molecule is either suppressed completely (static lattice model), or confined to the interior of a small region defined by repulsive interactions with surrounding molecules (cell model). Clearly, the nature of these models is such that they describe best those properties which are structure determined, and describe poorly those properties which, in some sense, depend on the breakdown of positional and orientational correlations between molecules. 

P. W. M. Jacobs and D. A. Mac Ddnaill, Solid State Ionics, 23, 307 (1987). Computational Simulations of 8-61203. 111. A Comparative Study of Static Lattice Models. 

The Mott-Littleton method is now a routine tool in computational solid-state chemistry and physics, and is implemented, together with other static lattice modelling tools, in the GULP code [24], written by Gale. Two recent applications serve to illustrate the range and diversity of current applications. 

High-frequency Hookean shear moduli of ordered latices have been compared with predictions based on pairwise-additivity of DLVO-type potentials. A simple static lattice model gives a simple expression for the modulus,... 

Many phenomena in solid-state physics can be understood on the basis of a static lattice model. In this model, the atoms of the solid are taken to constitute a fixed, rigid, immobile periodic array. Within this framework it is, for example, possible to account for a wealth of equilibrium properties of metals dominated by the behaviour of the conduction electrons. To some extent it is also possible to account for the equilibrium properties of ionic and molecular insulators. 

The static lattice model is, of course, an approximation to the actual ionic configuration, because the atoms or ions are not fixed to their equilibrium positions, but rather oscillate about them with an energy which is governed by the temperature of the solid. The reason for these oscillations is due to the fact that the ions are not infinitely massive, nor are they held in place by infinitely strong forces. In classical theory the static lattice model can therefore be valid only at zero temperature. At nonzero temperatures, each ion must have some thermal energy and therefore a certain amount of motion in the vicinity of its equilibrium position. In quantum theory, even at zero temperature, the static lattice model is incorrect, because according to the uncertainty principle AXAp h, localized ions possess some nonvanishing mean square momentum. 

The dynamics of atoms in solids is responsible for many phenomena which cannot be explained within the static lattice model. Examples are the specific heat of crystals, thermal expansion, thermal conductivity, displacive ferroelectric phase transitions, piezoelectricity, melting, transmission of sound, certain optical and dielectric properties and certain aspects of the interaction of radiation such as X-rays and neutrons with crystals. The theory of lattice vibrations, often called lattice dynamiosy and its implications for many of the above mentioned phenomena is the subject of this two-volume book. 

Taken together, the static lattice and cell model theories strongly suggest that ... 

In the previous sections, we have considered that the optical center is embedded in a static lattice. In our reference model center ABe (see Figure 5.1), this means that the A and B ions are fixed at equilibrium positions. However, in a real crystal, our center is part of a vibrating lattice and so the environment of A is not static but dynamic. Moreover, the A ion can participate in the possible collective modes of lattice vibrations. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Figure 4 Results from classical trajectory calculations for in-plane scattering of Ar from Ag(l 11) with an incidence angle of 40° measured with respect to the surface normal. In the panels a and c results for the relative final energy Ef/Ei are shown, where E is the initial energy. Lines indicate the energy transfer computed with the cube model (parallel momentum conservation) and a binary collision model. In panels b and d angular distributions are shown. Calculations for 0.1, 1,10 and lOOeV are shown. The panels a and b are calculated for a zero temperature, static lattice panels c and d for Ts = 600 K. From Lahaye et al. [43].

A description of the percolation phenomenon in ionic microemulsions in terms of the macroscopic DCF will be carried out based on the static lattice site percolation (SLSP) model [152]. In this model the statistical ensemble of various... 

We shall consider, in detail, studies of MgO lattice properties using the potential-induced breathing model and three complementary band-theoretical studies of electron distribution. Early studies using the modified electron-gas method (Cohen and Gordon, 1976) gave reasonably good agreement with experiment for equilibrium static lattice properties and... 

Using atomistic (static lattice) methods, Sayle et first modeled the (110), (310) and (111) surfaces of <7e02- The (110) and (310) surfaces are known as type I surfaces i.e. they are charge neutral with stoichiometric proportions of anions and cations in each plane (parallel to the surface). The potential for each plane is exactly zero due to the cancellation of the effects of the positive and negative charges and therefore there is no dipole moment perpendicular to the surface. The (111) surface of ceria is a type II surface, i.e. the surface terminates with a single anion plane and consists of a neutral Chree-plane repeat unit. 

Gallego R. and Ortiz M., A Harmonic/Anhamionic Energy Partition Method for Lattice Statics Computations, Modelling Simul. Mater. Sci. Eng. 1, 417 (1993). 

First, atomistic (static lattice) methods determine the lowest energy configuration of the crystal structure by employing efficient energy minimization procedures. The simulations rest upon the specification of an interatomic potential model which ex-... 

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