Phase-Equilibrium Calculations

Models for Thermodynamic and Phase Equilibria Calculations, edited by Stanley I. Sandler... 

In the sections below, we describe several studies in which flat-histogram methods were used to examine phase equilibria in model systems. The discussion assumes the reader is familiar with this general family of techniques and the theory behind them, so it may be useful to consult the material in Chap. 3 for background reference. Although the examples provided here entail specific studies, their general form and the principles behind them serve as useful templates for using flat-histogram methods in novel phase equilibria calculations. 

Clearly CALPHAD has now come of age and is at a watershed where complex phase equilibria calculations can now be performed as a routine operation and yet have also been placed on a sound physical basis. Computer programmes exist which can perform complex calculations on a PC and which can therefore be operated at any location without extensive prior expertise. Furthermore, it is possible in many cases to predict phase equilibria in multi-component alloys to a degree which is close to that expected from experiment (see Chapter 10). It is therefore a branch of science that is mature and, indeed, has already entered the next stage of development, which emphasises the need to concentrate on extending its range of applicability. 

The purpose of phase equilibria calculations is to predict the thermodynamic properties of mixtures, avoiding direct experimental determinations, or to extrapolate the existing data to different temperatures and pressures. The basic requirements for performing any thermodynamic calculation are the choice of the appropriate thermodynamic model and knowledge of the parameters required by the model. In the case of high pressure phase equilibria, the thermodynamic model used is generally an equation of state which is able to describe the properties of both phases. 

In phase-equilibria calculations different combinations of input variables are possible but, in general, the following choices are the most commonly used in order to solve typical... 

I. C. Sanchez and C. G. Panayiotou, Equation of State Thermodynamics of Polymer and Related Solutions, chapt. 3 in "Models for Thermodynamic and Phase equilibria Calculations, S. L. Sandler ed., M. Dekker, New York, 1994. 

Fussell, D.D. and Yanosik, J.L. An Iterative Sequence for Phase Equilibria Calculations Incorporating the Redlich-Kwong Equation of State, Soc. Pet. Eng. J. (June 1978) 18, 173-182. 

Our aim is to derive the chemical potential of water to enable phase equilibria calculations. Note that, while Equation 5.7 is the grand canonical partition function (2guest) an(j with respect to the solute (guest) molecule, it is the canonical partition function (<2host) with respect to the host (water) because k = 1, so that we have... 

Rafal et al. (Chapter 7, "Models for Electrolyte Solutions, in Models for Thermodynamic and Phase Equilibria Calculations, S. I. Sandler, ed., Marcel Dekker, New York, 1994, p. 686) have provided a comprehensive discussion of speciation and thermodynamic models. 

Bogdanic, G., Fredenslund, A. Revision of the Group-Contribution Flory Equation of State for Phase Equilibria Calculations in Mixtures with Polymers. 1. Prediction of Vapor-Liquid Equilibria for Polymer Solutions. Ind. Eng. Chem. Res. 1994,33 1331-1340. 

Refs. [i] van tHoffJH (1887) Zphys Chem 1 481 [ii] Aseyev GG (1998) Electrolytes, equilibria in solutions and phase equilibria calculation of multicomponent systems and experimental data on the activities of wa-... 

Chapter 4 describes the polymer data bases. This chapter is organized into sections discussing the experimental methods available for measuring the thermodynamic data of polymer solutions with an overview of the advantages and disadvantages of each method. The next section, Data Reduction Methods, describes how the experimental measurements from these experiments can be used to calculate the activity coefficients that are necessary for phase equilibria calculations. Finally, a summary of all the systems that are available on the data diskettes is provided. A user can scan this section or use the computer program POLYDATA to find if data are available for a particular system. 

Equation 11.32 is used to model a single-phase liquid in a ternary system, as well as a ternary substitutional-solid solution formed by the addition of a soluble third component to a binary solid solution. The solubility of a third component might be predicted, for example, if there is mutual solid solubility in all three binary subsets (AB, BC, AC). Note that Eq. 11.32 does not contain ternary interaction terms, which ate usually small in comparison to binary terms. When this assumption cannot, or should not, be made, ternary interaction terms of the form xaXbXcLabc where Labc is an excess ternary interaction parameter, can be included. There has been httle evidence for the need of terms of any higher-order. Phase equilibria calculations are normally based on the assessment of only binary and ternary terms. 

Figure 11.15. Neglecting the existence of a high-melting line compound (AS) in the phase equilibria calculations (left) can result in a largely erroneous liquidus or solidus in an extrapolated phase diagram (right).

Supersaturation and deposition temperature (diffusion) are the two critical factors that can be effectively used to control the morphology of the coating as can be seen in Figure 16a. The effect of changes in the supersaturation of the reactive gases combined with thermodynamic phase equilibria calculations can be effectively used to understand and therefore control the morphology of the deposit (Figure 16b). It was established that the formation of whiskers or polycrystalline TiC is dependent on the level of free carbon present in the TiCl -CH -H gas mixture. A typical cross-... 

The problem arises also if the mole fractions are not all equal to each other. Let a, be defined as NX/. One obtains C °/RT = Xa/lno/ + ln(l/A0, where the first term is always finite and in fact small, because la, = N. The offending term ln(l/A0 cancels in all phase equilibria calculations. 

Luks, K. D., Turek, E. A., and Krogas, T. K., Asymptotic effects using semicontinuous vis-a-vis discrete descriptions in phase equilibria calculations. Ind. Eng. Chem. Res. 29,2101 (1990). 

Smith, W. R., Chemical and phase equilibria calculations for complex mixtures. In Chemical Reactions in Complex Systems (A. V. Sapte and F. J. Ktambeck, eds.). Van Nostrand Reinhold, New York, 1991, p. 298. 

Eredenslund, Aa., Sprensen, J.M., 1994. Group contribution estimation methods. In Sandler, S.I. (Ed.), Models for Thermodynamic and Phase Equilibria Calculations. Dekker, New York (Chapter 4). 

In selecting a particular form of the equation to be used for phase equilibria calculations, you must select a method of solving the equation that has desirable convergence characteristics. Convergence to the solution should... 

---