## 1 Introduction

[2] Partial melting is an essential process involved in both material differentiation and heat transportation by the production and migration of melts, and causes present day mid-ocean ridge, hotspot, and subduction zone magmatism. Partial melting in the early Earth may have been even more important than in the present day [e.g., *Labrosse et al*., 2007; *Lee et al*., 2010], meaning that detailed knowledge of the processes involved in melting is an important step in improving our understanding of the material differentiation and the thermal evolution of the Earth.

[3] In general, melting is associated with dynamic processes, such as relative motion and chemical reactions between melts and solids, both of which redistribute mass and energy, with these processes being coupled and evolving with time. This means that modeling of these processes needs an approach that simultaneously solves both mass and energy balance. One consistent approach to modeling of these systems is thermodynamic calculations that minimize the free energy of the system.

[4] Several thermodynamic models have been presented to calculate melting phase relations, including MELTS and pMELTS [*Ghiorso*, 1994; *Ghiorso and Sack*, 1995; *Ghiorso et al*., 2002], and Perplex [*Connolly*, 2005]. MELTS and pMELTS have been widely used to calculate equilibrium phase relations for melt-present systems. Interaction parameters for silicate melt end-member components are calibrated assuming a regular solution model, with reference state properties derived by extrapolation from pure end-member component systems at 1 bar pressure [*Ghiorso and Sack*, 1995; *Ghiorso et al*., 2002; *Hirschmann et al*., 1998b]. Phase relations and compositions at fixed bulk compositions are then calculated to satisfy equilibrium melt and mineral compositions by evaluating “affinity”, which represents the degree of saturation of each phase in a melt [*Ghiorso*, 1994]. Put simply, the energy minimization algorithm used in MELTS/pMELTS modeling calculates the saturation state and equilibrium composition of solid phases in a melt. In this sense, MELTS and pMELTS could be referred to as “composition-based” models constructed specifically for melt-present systems.

[5] In comparison, Perplex calculates the phase equilibria of an arbitrary chosen system based on direct minimization of total Gibbs free energy [*Connolly*, 2005]. The nonlinear relationship between chemical potential and composition of the phase is described at finite discrete compositions for each phase, instead of as a continuous function. The thermodynamic models and properties can be chosen from a previously compiled database, all of which are readily incorporated into Perplex. As defaults, the THERMOCALC data set [e.g., *Holland and Powell*, 1998] for rock-forming minerals, and pMELTS or MELTS for mafic silicate melts is recommended [*Connolly*, 2005]. Although the flexibility of the system is a useful advantage, internally consistent thermodynamic models and properties need to be used to ensure accurate modeling, especially when modeling silicate melts.

[6] These thermodynamic models use different energy minimization algorithms, thermodynamic configurations, and parameters. These models should ideally reproduce experimental results within experimental uncertainties. However, as will be shown in this paper, these models exhibit significant variability in reproducibility. In order to improve these modeling approaches, we propose a new algorithm for system energy minimization and a relatively simple thermodynamic model that is appropriate for modeling of melt-present systems and that incorporates newly calibrated parameter values. For bulk-composition-constrained minimization of system energy, we have developed a straightforward formulation in continuous pressure, temperature, and composition space, and as an example, we model a peridotite system at a fixed pressure using the formulation and algorithm presented here. Our model accurately reproduces the relationship between temperature and phase fractions, including solidus temperature and degree of melting, variables that if constrained can be useful for dynamical modeling of melt generation, including melting and melt migration. The widespread nature of polybaric melting, for example at mid-ocean ridges and hotspots, means that we are currently expanding the calibration database and the range of pressures that our modeling encompasses, using the energy minimization algorithm and thermodynamic configuration outlined in this study; this expansion will be presented separately.