Visualizing thermodynamic stability and phase equilibrium through computer graphics

Abstract

<p>In a landmark paper, J. W. Gibbs introduced a fundamental thermodynamic surface for pure materials and, later, expanded that analysis to mixtures. While Gibbs certainly had images of these surfaces in his mind, he described them in words rather than drawings. Since Gibbs time, a number of researchers have produced representations of these surfaces, but their studies have been limited because of the difficulty of creating and visualizing the data sets required. In this work, computer graphics has been used to visualize a variety of these models for pure, binary, and ternary systems;Fundamental-equation surfaces produced from Legendre transforms of the energy function are well suited to illustrate the criteria for phase equilibrium through contact structures: Y[superscript](n-1) transforms display tangent planes, Y[superscript](n) transforms display tangent lines, and Y[superscript](n+1) transforms display points of self-intersection. Extensive variables (those homogeneous to order one) are scaled by either the total or component mole-numbers, the volume, or the entropy to produce molar, mole-ratio, volume-density, or entropy-scaled quantities. The details of entropy scaling are presented, and the generality of the procedure is confirmed;Equation-of-state surfaces derived from first derivatives of Y[superscript](n) transforms are well suited for displaying stability limits as local extrema along isotherms, isobars, or other intensive level curves. Binary-mixture functions illustrate the pseudo-pure stability limit, where the surface becomes vertical and folds over onto itself. Surfaces for a ternary illustrate the controlling, pseudo-binary, and pseudo-pure stability limits. Such limits lie, respectively, outside of one another and establish a stability hierarchy;The data sets were generated by a program developed specifically for this work. The program reads an input file that specifies the thermodynamic variable mappings, the ranges, and the quatitities to be held fixed. One extensive variable is held constant to scale the data, and for mixtures, (n-1) intensive variables are fixed so that the surfaces can be plotted in three dimensions;Recommendations are given for extending this work to systems containing both solid and fluid phases, to mixtures displaying complex critical behavior, and to reacting systems.</p>