Calculation of phase diagrams method

In order to examine the possible relationship between the bulk thermodynamics of binary transition metal-aluminum alloys and their tendency to form at underpotentials, the room-temperature free energies of several such alloys were calculated as a function of composition using the CALPHAD (CALculation of PHAse Diagrams) method [85]. The Gibbs energy of a particular phase, G, was calculated by using Eq. (14),... 

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]

The state of the art has been summarized by Colinet (2003) who reported a description of the ab initio calculation methods of energies of formation for intermetallic compounds and a review of the aluminium-based compounds studied. In its conclusions, this paper underlined that the complete ab initio calculation of complex phase diagrams is not close at hand. However, calculation of phase diagrams in systems, where experimental data are missing, could, in the future, be performed by combination of CALPHAD routines and ab initio calculations of formation energies or mixing energies. 

From a theoretical standpoint, the traditional approach for the determination of an alloy structure implies, in principle, a search through any possible configuration until the most energetically favorable is found. While current first-principles methods, coupled with a substantial increase in computational power, have made this approach a standard practice for the calculation of phase diagrams of (mostly binary) bulk alloys, the complexity of surfaces makes quantum approximate methods a necessary tool to supplement the existing techniques and the growing body of experimental data. 

M. Hilled, Methods Of Calculating Phase Diagrams , in Calculation of Phase Diagrams and Thermochemistry of Alloys, Y. A. Chang and J. F. Smith, Eds., Proc. Conf. AIME Fall Meeting, Sept. 17-18, 1979, Milwaukee, 1979, The Metallurgical Society, pp. 1-13 (1979). 

Saunders, N. 1998. CALPHAD (Calculation of Phase Diagrams) A Comprehensive Guide. Oxford, U.K./New York Pergamon. This is an in-depth treatment of the method of computer coupling of phase diagrams and thermochemistry, which makes it possible to calculate the phase behavior of multicomponent materials. See also the journal CALPHAD that publishes new developments quarterly. Available online through various services. 

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. 

Calculation, thermodynamic optimization of phase diagrams. The knowledge of phase equilibria, phase stability, phase transformations is an important reference point in the description and understanding of the fundamental properties of the alloys and of their possible technological applications. This interest has promoted a multi-disciplinary and multi-national effort dedicated not only to experimental methods, but also to techniques of optimization, calculation and prediction of... 

Figure 2.43. Optimization, calculation, prediction of phase diagrams an indication of the working scheme of the CALPHAD method is shown.

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. 

A phase diagram is often considered as something which can only be measured directly. For example, if the solubility limit of a phase needs to be known, some physical method such as microscopy would be used to observe the formation of the second phase. However, it can also be argued that if the thermodynamic properties of a system could be properly measured this would also define the solubility limit of the phase. The previous sections have discussed in detail unary, single-phase systems and the quantities which are inherent in that sjrstem, such as enthalpy, activity, entropy, etc. This section will deal with what happens when there are various equilibria between different phases and includes a preliminary description of phase-diagram calculations. 

Lacher and Anderson discussed this phase diagram from the viewpoint of statistical thermodynamics based on the Fowler-Guggenheim treatment. On the other hand, Libowitz and Simon and Flanagan also calculated this phase diagram based on Libowitz s method mentioned in Section 1.3.7. The result obtained by the latter is the same as eqn (1.124). 

The values of the chemical potentials in each phase, together with their pressure derivatives (available through the measured number densities), can be exploited to home in on the coexistence pressure. The method has been successfully applied to calculate the phase diagrams of a number of simple fluids and fluid mixtures [79, 80]. 

Estimates of phase diagrams can be made on the assumption of ideal behavior or with activity coefficient data based on binary measurements that are more easily obtained. In such cases, clearly, it should be known that intermolecular compounds do not form. The freezing behaviors of ideal mixtures over the entire range of temperatures can be calculated readily. The method is explained for example by Walas (1985, Example 8.9). 

The diffusion path method has been used to interpret nonequilibrium phenomena in metallurgical and ceramic systems (10-11) and to explain diffusion-related spontaneous emulsification in simple ternary fluid systems having no surfactants (12). It has recently been applied to surfactant systems such as those studied here including the necessary extension to incorporate initial mixtures which are stable dispersions instead of single thermodynamic phases (13). The details of these calculations will be reported elsewhere. Here we simply present a series of phase diagrams to show that the observed number and type of intermediate phases formed and the occurrence of spontaneous emulsification in these systems can be predicted by the use of diffusion paths. 

First-Principles Approach to Guinier-Preston Zones. We have already seen that the combination of first-principles calculations with Monte Carlo methods is a powerful synthesis which allows for the accurate analysis of structural questions. In chap. 6 we noted that with effective Hamiltonians deduced from a lower-level microscopic analysis it is possible to explore the systematics of phase diagrams with an accuracy that mimics that of the host microscopic model. An even more challenging set of related questions concern the emergence of microstructure in two-phase systems. An age-old question of this type hinted at in the previous chapter is the development of precipitates in alloys, with the canonical example being that of the Al-Cu system. 

Our purpose in these last two subsections has been to show how the simplest fundamental description of SEE for van der Waals solids can emerge from the hard-sphere model and mean field theory. Much of the remainder of the chapter deals with how we extend this kind of approach using simple molecular models to describe more complex solid-fiuid and solid-solid phase diagrams. In the next two sections, we discuss the numerical techniques that allow us to calculate SEE phase diagrams for molecular models via computer simulation and theoretical methods. In Section IV we then survey the results of these calculations for a range of molecular models. We offer some concluding remarks in Section V. 

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