GGchem Phase-Equilibrium Model

• GGchem Phase-Equilibrium Model is a thermo-chemical equilibrium code that computes gas-phase and condensate abundances from elemental data, temperature, and pressure using iterative methods.
• It employs the law of mass action and Gibbs free energy minimization to simulate complex astrophysical environments such as stellar atmospheres, brown dwarfs, and exoplanets.
• The model integrates robust numerical techniques with updated thermochemical databases to handle low-temperature regimes and guide atmospheric retrievals and dust formation analyses.

Searching arXiv for GGchem and closely related phase-equilibrium papers to ground the article in current and original sources. arxiv_search.query{"search_query":"all:GGchem OR ti:GGchem","max_results":10,"sort_by":"relevance","sort_order":"descending"} GGchem is a thermo-chemical equilibrium code for astrophysical gases that computes both pure gas-phase chemical equilibrium and gas–condensed-phase equilibrium, described in the source literature as “equilibrium condensation” and “phase equilibrium.” In Woitke et al., the code is presented as a fast public implementation designed to remain robust down to $100\,\mathrm{K}$, with applications to stars, brown dwarfs, exoplanets, and dust-forming outflows; the same codebase later appears in exoplanet-retrieval and self-consistent atmosphere frameworks, where it functions either as a condensate-capable equilibrium solver or, when embedded differently, as a gas-phase equilibrium kernel (Woitke et al., 2017).

1. Concept, scope, and scientific role

GGchem—expanded in the paper as “Gleich-Gewichts-Chemie”—solves for the equilibrium composition of a chemically reacting mixture from elemental abundances, gas temperature $T$, and either total pressure $p$ or gas density $\rho$. Its intended output is the abundance pattern of atoms, ions, molecules, liquids, and solids under thermo-chemical equilibrium. The original low-temperature paper emphasizes two operating modes: pure gas-phase equilibrium and gas-plus-condensate phase equilibrium. The latter is the defining feature of the “GGchem phase-equilibrium model” in the strict sense (Woitke et al., 2017).

The code’s scientific use is broad within astrophysical equilibrium chemistry. The source material explicitly associates it with stellar atmospheres, substellar atmospheres, exoplanets, oxygen-rich AGB-star winds, and the search for the first condensate in astrophysical dust formation. In later exoplanet work, GGchem is treated not as an isolated chemistry code but as an interchangeable component inside TauREx 3.1, allowing direct comparison with ACE and FastChem under identical radiative-transfer and Bayesian-sampling conditions. In the 2024 MSG framework, by contrast, GGchem remains part of a self-consistent atmosphere model but is used only for gas-phase particle densities, while condensates are handled by a kinetic cloud model outside GGchem (Al-Refaie et al., 2021, Jørgensen et al., 2024).

This division of labor is important because the label “GGchem phase-equilibrium model” can denote two distinct practical usages. In the original 2017 formulation, it refers to a full gas–condensed-phase equilibrium treatment with equilibrium condensation. In later frameworks, it may instead refer to the GGchem solver as one module inside a broader architecture, sometimes with its condensate capability switched on and sometimes with cloud or condensed matter treated elsewhere. A common misconception is that every GGchem-based atmospheric model uses GGchem itself as the final condensate solver. The MSG paper explicitly states otherwise (Jørgensen et al., 2024).

2. Thermodynamic formulation

The general equilibrium problem is described in later exoplanet-retrieval work as minimizing Gibbs free energy subject to elemental and charge conservation, with a practical alternative based on the law of mass action and equilibrium constants $K$ (Al-Refaie et al., 2021). GGchem’s original formulation adopts that practical route. For a molecule composed of elements A, B, and C with stoichiometric coefficients $a,b,c$, the law of mass action is written as

$$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$

with partial pressures $p_i=n_i kT$, standard pressure $p^\ominus$, and Gibbs free energy of formation $\Delta G^\ominus$ from neutral gas-phase atoms at the same $T$0. GGchem defines the equilibrium constant through

$T$1

and

$T$2

In logarithmic form,

$T$3

where $T$4 (Woitke et al., 2017).

Element conservation is imposed through

$T$5

where $T$6 is the abundance of element $T$7 relative to hydrogen, $T$8 is the total hydrogen nuclei density, $T$9 is the number density of species $p$0, and $p$1 is the stoichiometric coefficient of element $p$2 in species $p$3. Charge neutrality is implemented by treating charge as an additional conserved “element” with $p$4. Gas density and pressure are related by

$p$5

If pressure rather than density is prescribed, GGchem iterates on the mean molecular weight $p$6 (Woitke et al., 2017).

The phase-equilibrium extension introduces a supersaturation criterion for condensates. For a condensate $p$7,

$p$8

with vapor pressure

$p$9

In equilibrium,

$\rho$0

The equilibrium state therefore forbids $\rho$1. For most minerals, where no corresponding free gas molecule exists, GGchem uses a generalized supersaturation ratio such as

$\rho$2

with $\rho$3 defined from the condensed phase and neutral gas atoms (Woitke et al., 2017).

A later TauREx comparison paper restates the same equilibrium-constant logic in more generic notation,

$\rho$4

and lists fitting families including the Tsuji form,

$\rho$5

and a NASA-polynomial form. It also notes that GGchem is unusually flexible because it supports several fitting-function families, including these forms (Al-Refaie et al., 2021).

3. Numerical realization and thermochemical databases

GGchem does not solve equilibrium by direct multidimensional Gibbs minimization. The original paper explicitly contrasts its approach with Gibbs-minimization methods and instead eliminates molecular abundances via $\rho$6, reducing the gas-only problem to a nonlinear root system in atomic partial pressures and, when needed, the electron partial pressure. That system is solved by Newton–Raphson iteration. If $\rho$7 rather than $\rho$8 is supplied, the code iterates on $\rho$9, typically requiring $K$0–$K$1 chemistry calls (Woitke et al., 2017).

The low-temperature regime is numerically stiff. The paper states that some equilibrium constants exceed $K$2 and some partial pressures and electron abundances fall below $K$3. GGchem addresses this with quadruple precision for $K$4 and with a pre-iteration strategy based on a hierarchy of elemental abundances. Elements are sorted by abundance, introduced progressively, and repeatedly refined; the authors state that best performance was found with $K$5 jointly refined elements. They also store correction factors from nearby converged solutions. According to the paper, this combination is what makes equilibrium calculations robust down to $K$6. The practical lower limit appears near $K$7, where the equilibrium constant of $K$8 reaches about $K$9, exhausting quadruple-precision range (Woitke et al., 2017).

The phase-equilibrium extension uses a nested Newton–Raphson scheme. The outer iteration adjusts gas-phase elemental abundances $a,b,c$0; the inner chemistry solve computes the gas equilibrium for those depleted abundances; the condensate set is then updated by enforcing $a,b,c$1 for all condensates and $a,b,c$2 for the selected stable set. The condensate concentrations are not the primary iterated variables. Instead, GGchem iterates on elemental abundance corrections, from which condensate concentrations follow. The paper states that the phase-equilibrium solver generally converges in fewer than $a,b,c$3 iteration steps when a good initial guess is available (Woitke et al., 2017).

Thermochemically, the code relies on a compiled and refitted database. For molecules, the final dataset is stated in the summary to contain $a,b,c$4 molecules and ions made from $a,b,c$5 elements and to be selected to remain safe from $a,b,c$6 to $a,b,c$7; elsewhere, a selected benchmark subset of $a,b,c$8 molecules is mentioned. The molecular data are taken mainly from Stock (2008) fits to JANAF and from Barklem & Collet (2016) for species absent in JANAF. The preferred fitting form is the Stock representation,

$a,b,c$9

which the paper argues extrapolates more smoothly to low temperature than older forms (Woitke et al., 2017). For condensed phases, the database is built chiefly from NIST-JANAF and SUPCRTBL, with the summary stating $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$0 condensates including $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$1 liquids (Woitke et al., 2017). The 2024 MSG paper independently describes GGchem as based on minimization of the total Gibbs free energy, including condensates, and states that it uses thermochemical data from JANAF and Barklem & Collet (2016), with comparisons discussed by Worters et al. (2017) (Jørgensen et al., 2024).

Performance is one of the code’s distinctive attributes. In the 24-element gas-only benchmark described in the 2017 paper, GGchem requires about $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$2 CPU s per $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$3 point on a 2.8 GHz laptop and about $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$4 CPU s per point above $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$5 in double precision; in the cited low-temperature comparison regime, the authors state that GGchem is more than $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$6 times faster than TEA (Woitke et al., 2017).

4. Condensation, depletion, and astrophysical consequences

The condensed-phase formulation changes the chemistry by depleting gas-phase elemental abundances. Appendix D writes the balance as

$$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$7

where $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$8 is the total elemental abundance before condensation, $$\frac{p_{\rm ABC}}{p^\ominus} = \left(\frac{p_{\rm A}}{p^\ominus}\right)^a \left(\frac{p_{\rm B}}{p^\ominus}\right)^b \left(\frac{p_{\rm C}}{p^\ominus}\right)^c \exp\left(-\frac{\Delta G^\ominus}{RT}\right),$$9 is the remaining gas-phase abundance, $p_i=n_i kT$0 is the condensate concentration per H nucleus, and $p_i=n_i kT$1 is the stoichiometric coefficient of element $p_i=n_i kT$2 in condensate $p_i=n_i kT$3. The mechanism is therefore local elemental sequestration into solids or liquids, with immediate feedback on all gas-phase species through the law of mass action (Woitke et al., 2017).

For solar abundances at $p_i=n_i kT$4, the paper gives a detailed condensation sequence. The first stable condensate is metallic tungsten, W[s], at $p_i=n_i kT$5; it is followed by ZrO$p_i=n_i kT$6[s] at $p_i=n_i kT$7, Al$p_i=n_i kT$8O$p_i=n_i kT$9[s] at $p^\ominus$0, CaTiO$p^\ominus$1[s] at $p^\ominus$2, gehlenite at $p^\ominus$3, and Fe[l] at $p^\ominus$4. Major Mg silicates then appear, notably Mg$p^\ominus$5SiO$p^\ominus$6[s] at $p^\ominus$7 and MgSiO$p^\ominus$8[s] at $p^\ominus$9. At lower temperature, FeS[s] appears at $\Delta G^\ominus$0, NaCl[s] at $\Delta G^\ominus$1, and LiCl[s] at $\Delta G^\ominus$2. Phyllosilicates follow, including phlogopite at $\Delta G^\ominus$3, sodaphlogopite at $\Delta G^\ominus$4, and lizardite, Mg$\Delta G^\ominus$5Si$\Delta G^\ominus$6O$\Delta G^\ominus$7H$\Delta G^\ominus$8[s], which replaces forsterite at $\Delta G^\ominus$9. Finally H$T$00O[s] forms at $T$01 and NH$T$02[s] at $T$03 (Woitke et al., 2017).

The dust/gas ratio responds sharply to this sequence. Once Ca, Fe, Si, and Mg compounds—especially the major Mg silicates—have formed, the ratio rises to about $T$04 around $T$05. With sulfur condensation around $T$06, it rises to about $T$07. After phyllosilicates form below about $T$08, it reaches about $T$09. Only after water and ammonia ice condensation does it approach $T$10. The paper therefore argues that the often assumed dust/gas value of $T$11 is too high for a solar-composition gas before volatile ices condense (Woitke et al., 2017).

A central scientific result is the modification of the gas-phase carbon-to-oxygen ratio. The initial solar value is about $T$12. After major silicate condensation, the gaseous C/O rises to about $T$13. After the additional intake of water and hydroxyl into the solid matrix through phyllosilicate formation below about $T$14, the gaseous C/O rises further to about $T$15. The paper attributes this directly to oxygen sequestration: silicates remove oxygen into condensed solids, and phyllosilicates remove still more oxygen by incorporating hydroxyl or water into the crystal structure, while carbon remains in the gas. For solar composition at $T$16, carbon does not condense down to $T$17; it remains mainly in gaseous CH$T$18 at low temperature (Woitke et al., 2017).

The tungsten result has separate significance. The paper identifies W[s] as the first thermodynamically stable condensate, appearing several hundred Kelvin before ZrO$T$19 or corundum. The authors briefly discuss whether tungsten, despite its abundance of $T$20 times the silicon abundance, could act as the first seed particle for astrophysical dust formation. They also emphasize that actual nucleation is kinetic rather than equilibrium-limited. This suggests that GGchem’s phase-equilibrium solution supplies a thermodynamic upper structure for condensation, not a cloud-microphysical prediction by itself (Woitke et al., 2017).

5. GGchem in retrieval frameworks and self-consistent atmosphere models

TauREx 3.1 made GGchem part of a generalized plugin architecture in which chemistry models are interchangeable components. In that framework, GGchem is wrapped as a true Python interface and exposed through a dedicated chemistry interface; it can be selected in the input file with chemistry_type = ggchem, and condensation is activated by equilibrium_condensation = True. Hydrogen, helium, and oxygen are mandatory selected elements, and elemental abundances can be initialized from solar, earth, ocean, or meteorite defaults. Metallicities are then defined relative to the initial abundance profile, while each metal element can be varied through an oxygen-relative ratio such as $T$21 (Al-Refaie et al., 2021).

The TauREx study used this architecture to compare ACE, FastChem, and GGchem under identical forward-model and retrieval conditions. Seven synthetic JWST spectra were generated and then cross-retrieved, giving $T$22 retrievals in total. Within matched assumptions, the result was that “like-for-like, all chemical codes retrieve the correct parameters to $T$23 of the truth.” For a C/H/O/N truth spectrum, GGchem retrieved

$T$24

and GGchem+condensation gave effectively identical results because condensation is inactive in that restricted chemistry at $T$25 K. For heavy chemistry without condensation, FastChem and GGchem again agreed closely. The dominant biases therefore arose not from the equilibrium-solver implementation itself, but from assumptions about the atmospheric process set, especially missing elements and condensation (Al-Refaie et al., 2021).

The same paper provides explicit examples of assumption-driven failure. When a heavy GGchem model without condensation is fitted to a C/H/O/N truth, metallicity is pushed to the upper prior boundary and C/O to the lower boundary: $T$26 Against a heavy non-condensed truth, GGchem(heavy)+cond yields

$T$27

showing a large temperature bias and a strongly subsolar metallicity. The paper summarizes the generic issue as follows: in mismatched cases, temperature uncertainties can be less than $T$28 even when the deviation from truth exceeds $T$29, and chemistry parameters can deviate by over $T$30 while having extremely small formal uncertainties (Al-Refaie et al., 2021).

The GGchem-specific phase-equilibrium behavior in TauREx is most visible in Ti- and V-bearing chemistry. In the heavy-condensation run, TiO is depleted in the lower atmosphere because Ti is incorporated into MgTiO$T$31 in the troposphere and Ti$T$32O$T$33 in the stratosphere; for vanadium, solid VO forms and removes gas-phase VO from the lower atmosphere. The same analysis also shows that the selected element inventory changes condensate stability. When Si and Ca are absent, MgTiO$T$34 becomes overabundant because silicon would otherwise tie up magnesium in forsterite, Mg$T$35SiO$T$36. When Si and Ca are added, CaTiO$T$37 becomes important below $T$38 bar and changes the vertical Ti partitioning. This is a direct demonstration that condensation outputs in GGchem depend materially on the chosen elemental subset (Al-Refaie et al., 2021).

The practical cost is also substantial. In TauREx, average retrieval times are $T$39 min for GGchem C/H/O/N, $T$40 min for GGchem heavy, and $T$41 min for GGchem heavy+cond. The paper attributes the last figure—just over $T$42 hours per retrieval—to CPU-dominated equilibrium-condensation calculations, which limit the benefit of GPU acceleration in the rest of the forward model (Al-Refaie et al., 2021).

A different integration pattern appears in the MSG atmosphere grid. There, GGchem is coupled self-consistently to MARCS and StaticWeather, but the authors state that “in this paper we use GGchem solely for the purpose to calculate the gas phase particle densities based on temperature, pressure and the gas phase elemental abundances.” Cloud condensates in the final MSG workflow are not taken from GGchem equilibrium condensation. Instead, StaticWeather computes nucleation, growth, sublimation, settling, and mixing, while GGchem is also used internally by StaticWeather to obtain the molecular densities needed for effective growth rates and supersaturation ratios. In that framework, GGchem remains fundamental to gas chemistry, electron density, and ionic abundances, but the cloud model is explicitly kinetic rather than equilibrium-condensational (Jørgensen et al., 2024).

6. Limitations, interpretation, and related methodologies

GGchem is an equilibrium model. The exoplanet-retrieval paper defines chemical equilibrium in the usual thermochemical sense: chemical reaction timescales are assumed to be faster than dynamical timescales, and photochemistry is ignored (Al-Refaie et al., 2021). The original low-temperature paper is equally explicit about omissions. The equilibrium-condensation solver does not model condensation kinetics, nucleation, grain growth, metastability, rainout, transport-induced quenching, photochemistry, UV/X-ray/cosmic-ray chemistry, grain-size distributions, cloud microphysics, or aqueous solution chemistry. It also treats condensed phases as pure stoichiometric phases and neglects some pressure corrections in SUPCRTBL data for pressures below a few bar (Woitke et al., 2017).

This limitation matters in two different ways. First, the thermodynamic prediction of condensate stability is not a kinetic prediction of cloud formation. The tungsten example is the clearest case: W[s] is the first thermodynamically stable condensate, but the paper separately notes that actual nucleation would require supercooling and sufficiently high pressure (Woitke et al., 2017). Second, equilibrium condensation is not identical to every condensed-phase treatment used in later atmosphere models. In MSG, cloud formation is kinetic, non-local, and transport-coupled: gas-phase elements are consumed by nucleation and grain growth, replenished by sublimation and convective mixing, and the total elemental abundances present in solid and gaseous states in a cloudy layer do not in general sum to the input abundances. That is fundamentally different from local equilibrium condensation (Jørgensen et al., 2024).

A second common misconception is that discrepancies among atmospheric analyses chiefly reflect different equilibrium solvers. The TauREx comparison argues the opposite. Under matched assumptions, GGchem, FastChem, and ACE agree at the $T$43 level on retrieved parameters; the dominant biases instead come from missing species, fixed elemental ratios, and condensation assumptions. The authors therefore recommend that free chemistry or simpler parametric approaches be used first to explore the information content of the data before imposing self-consistent equilibrium models. This suggests that GGchem is most reliable when the chosen elemental set and condensation treatment are themselves physically appropriate (Al-Refaie et al., 2021).

Methodologically, GGchem belongs to the family of equilibrium models rooted in Gibbs free-energy minimization and mass-action constraints, but it is distinct from several adjacent approaches. FastChem, for example, is explicitly a gas-phase equilibrium solver for neutral and ionized species and “is not a full phase-equilibrium model with condensates” (Stock et al., 2018). Conversely, convex-envelope methods for $T$44 flash calculations construct lower convex envelopes of Gibbs-energy graphs over composition space and determine phase splits for nonreactive mixtures with arbitrary component count and aggregate states, but the cited papers state that such methods are not drop-in replacements for a GGchem-style reactive gas-condensate equilibrium solver (Göttl et al., 13 Feb 2025, Göttl et al., 2023). A plausible implication is that GGchem’s distinctive niche is neither generic flash calculation nor gas-only speciation, but reactive thermo-chemical equilibrium with explicit element depletion into stoichiometric condensates over astrophysically relevant temperature ranges.

In that sense, the GGchem phase-equilibrium model is best understood as a fast equilibrium-condensation framework whose central variables are gas-phase atomic abundances, equilibrium constants, and condensate stability conditions. Its scientific influence derives both from its original low-temperature condensation analysis—including the silicate and phyllosilicate control of gaseous C/O—and from its later use as a benchmarkable chemistry engine inside retrieval and atmosphere architectures (Woitke et al., 2017).