A Gibbs energy minimization method for constrained and partial equilibria

Abstract

The conventional Gibbs energy minimization methods apply elemental amounts of system components as conservation constraints in the form of a stoichiometric conservation matrix. The linear constraints designate the limitations set on the components described by the system constituents. The equilibrium chemical potentials of the constituents are obtained as a linear combination of the component-specific contributions, which are solved with the Lagrange method of undetermined multipliers. When the Gibbs energy of a multiphase system is also affected by conditions due to immaterial properties, the constraints must be adjusted by the respective entities. The constrained free energy (CFE) minimization method includes such conditions and incorporates every immaterial constraint accompanied with its conjugate potential. The respective work or affinity-related condition is introduced to the Gibbs energy calculation as an additional Lagrange multiplier. Thus, the minimization procedure can include systemic or external potential variables with their conjugate coefficients as well as non-equilibrium affinities. Their implementation extends the scope of Gibbs energy calculations to a number of new fields, including surface and interface systems, multi-phase fiber suspensions with Donnan partitioning, kinetically controlled partial equilibria, and pathway analysis of reaction networks.

Combined thermodynamic–geochemical modeling in metamorphic geology: Boron as tracer of fluid–rock interaction

“…The mathematical treatment of this approach involves (1) finding the phase assemblage with the lowest Gibbs energy among a large number of solid solution phases (so called global Gibbs energy minimum) and (2) finding the tangent plane that touches the G-X curves of all stable solid solution phases, which is prerequisite for finding the thermodynamically stable compositions of solution phases. For both aspects, different mathematical approaches are published (see Koukarri and Pajarre (2011) for a detailed review) and are implemented into a number of commercial and open source software packages, including MELTS (Ghiorso and Sack, 1995;Asimow and Ghiorso, 1998), pMELTS (Ghiorso et al, 2002) PERPLE_X (Connolly, 2005), THERIAK/DOMINO (de Capitani and Brown, 1987) and GEM-Selektor (Kulik et al, 2004;.…”

Thermodynamic equilibrium at heterogeneous pressure

“…Typically, the constraints in the minimization are the amounts of chemical components of the system. Koukkari and Pajarre (2011) demonstrate that the minimization procedure can include system or external potential variables with their conjugate coefficients as well as non-equilibrium affinities.…”

EChem++ – An object-oriented problem solving environment for electrochemistry. Part 71Part 6, see Ref. [1].1: Simulation of equilibrium electron transfer processes with implicit Dirichlet boundary conditions

“…Modeling of polyelectrolyte swelling and deformation by a polarizing potential was supported by enforcing an incompressibility constraint through Lagrange multipliers [47]. To the authors' knowledge applications of FEM calculations with the use of Lagrange multipliers in electrochemistry have been reported previously only in contexts different from the modeling of equilibrium electron transfers emphasized in the present work: for example, tasks related to optimization with respect to constraints and for systems in electrochemical equilibrium [48], or accurate flux calculations [49]. The latter reference is also an example for the use of the COMSOL Multiphysics commercial software that is not restricted to electrochemical simulations.…”