Lattice diagram method

The simplest unit cell. When the indices of all reflections on the X-ray photographs of a crystal have been obtained by any of the methods described—indices based, it will be remembered, on morphologically chosen axes - -the whole set of indices can be surveyed to see whether any simpler cell would account for all the reflections. One way of doing this is to look at reciprocal lattice diagrams or models. For instance, in Fig. 107 it is obvious that the larger, heavily outlined... 

Du et al studied the equilibrium phase diagram in the CeOa-ZrOg system using the CALPHAD (CALculation of PHAse Diagram) method. First the lattice stability of Ce02 was evaluated to reproduce the measured data for the mean heat capacity (Fig. 1.16). For example, the lattice stability of cubic CeO, (298 [Pg.17]

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

Before considering the principles of this method, it is useful to distinguish between anodic protection and cathodic protection (when the latter is produced by an external e.m.f.). Both these techniques, which may be used to reduce the corrosion of metals in contact with electrolytes, depend upon the electrochemical mechanisms that result from changing the potential of a metal. The appropriate potential-pH diagram for the Fe-H20 system (Section 1.4) indicates the magnitude and direction of the changes in the potential of iron immersed in water (pH about 7) necessary to make it either passive or immune in the former case the stability of the metal depends on the formation of a protective film of metal oxide (passivation), whereas in the latter the metal itself is thermodynamically stable and egress of metal ions from the lattice into the solution is thus prevented. 

Chapter S examines various models used to describe solution and compmmd phases, including those based on random substitution, the sub-lattice model, stoichiometric and non-stoichiometric compounds and models applicable to ionic liquids and aqueous solutions. Tbermodynamic models are a central issue to CALPHAD, but it should be emphasised that their success depends on the input of suitable coefficients which are usually derived empirically. An important question is, therefore, how far it is possible to eliminate the empirical element of phase diagram calculations by substituting a treatment based on first principles, using only wave-mecbanics and atomic properties. This becomes especially important when there is an absence of experimental data, which is frequently the case for the metastable phases that have also to be considered within the framework of CALPHAD methods. 

Thermochemical methods generate lattice stabilities based on high-temperature equilibria that yield self-consistent multi-component phase-diagram calculations. However, as they are largely obtained by extrapolation, this means that in some cases they should only be treated as effective lattice stabilities. Particular difficulties may occur in relation to the liquid — glass transition and instances of mechanical instability. 

There are, obviously, no compounds to illustrate lattice-induced strains with GII 3> 0.2 vu. Such structures are unstable and cannot exist, but if it is possible to model structures of any arbitrary composition using the methods described in Chapter 11, it is possible to determine which compositions give rise to stable structures and which ones do not. A systematic exploration of different compositions occurring between a group of elements would then lead to an understanding of the phase diagram. For example, on the basis of a few simple rules, Skowron and Brown (1994) were able to predict most of the structures in the Pb-Sb-S phase diagram and their relative stabilities (Section 11.2.2.2). 

Indexing rotation photographs by reciprocal lattice methods. Orthorhombic crystals. First of all, the coordinates and for each reflection on the photograph (Fig. 86) are found in one of the ways just described these coordinates may be plotted as in Fig. 87 a to form the reciprocal lattice rotation diagram. The problem now is to decide which point of the reciprocal lattice itself corresponds to each spot on the rotation diagram. 

Many polymer blends or block polymer melts separate microscopically into complex meso-scale structures. It is a challenge to predict the multiscale structure of polymer systems including phase diagram, morphology evolution of micro-phase separation, density and composition profiles, and molecular conformations in the interfacial region between different phases. The formation mechanism of micro-phase structures for polymer blends or block copolymers essentially roots in a delicate balance between entropic and enthalpic contributions to the Helmholtz energy. Therefore, it is the key to establish a molecular thermodynamic model of the Helmholtz energy considered for those complex meso-scale structures. In this paper, we introduced a theoretical method based on a lattice model developed in this laboratory to study the multi-scale structure of polymer systems. First, a molecular thermodynamic model for uniform polymer system is presented. This model can... 

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