Note

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Custom Condensates

CARMApy has a number of default condensates built in but also allows the user to define custom condensates. As a reminder, you can check what default condensates are included in CARMApy by running the following command:

import carmapy

carmapy.available_species()

['KCl', 'ZnS', 'Na2S', 'MnS', 'Cr', 'Mg2SiO4', 'Fe', 'TiO2', 'Al2O3', 'SiO']

Relevant Properties

For reasons related to how CARMA was originally designed, condensate properties are tied to the limiting gas resevoir the condensate is formed from. Thus, defining a custom condensate means creating a custom carmapy.Gas object. This unfortunately has a side effect that each gas can only form condensates of a single chemical formula (ie a gas can form Pure Fe and Fe on TiO₂ condensates but cannot also form Fe₂SiO₄ on TiO2 as the latter has a different chemical formula).

A large number of properties need to be defined for a custom condensate, we will go through them below but you can also see a reference page for the carmapy.Gas object here

For purposes of this tutorial, we will be recreating Al₂O₃

To do so we need to know the following properties:

• The density of the condensate

• The molar mass of the condensate

• The molar mass of the limiting gas species

• The collisional diameter

• The cosine of the contact angle between the gas species and any seed particles it might be condensing on

• The Hill Notation Formula for the gas species

• The formula for the saturation vapor pressure of the condensate (explained below)

• The formula for the surface tension of the condensate (expalined below)

• The stochiometry factor of the condensation reaction (explained below)

• Whether or not the reaction is considered a type III reaction (explained below)

• (Optionally) the latent heat of evaporation for the condensate

Simple Properties

• Al₂O₃ has a density of 3.99 g/cm³

• Al₂O₃ has a molar mass of 101.926 g/mol

• The limiting gas species, Al, has a molar mass of 26.98 g/mol

• Al₂O₃’s collisional diameter is approximately \(3.825\times10^{-8}\) cm (value estimated from Dobrovinskaya et al. 2009)

• The cosine of the contact angle betweel Al₂O₃ and TiO₂ is 0.724172 (value taken from P. Gao et al. 2020)

Saturation Vapor Pressure

The saturation vapor pressure of a species can be a function of both pressure and temperature. carmapy assumes that the formula for vapor pressure can be paramaterized as follows:

\[\begin{split}\begin{align*} \log_{10} \frac{p'}{10^6 \text{ barye}} &= \texttt{vp\_offset} \\ &- \frac{\texttt{vp\_tcoeff}}{T} \\ &- \texttt{vp\_metcoeff} \cdot [Fe/H] \\ &- \texttt{vp\_logpcoeff} \cdot \log_{10} \frac{P}{10^6 \text{ barye}} \end{align*}\end{split}\]

where \(p'\) is the saturation vapor pressure, \(T\) is the temperature, and \(P\) is the atmospheric pressure

Using the values from Wakeford et al. 2017 for Al₂O₃ we have:

• vp_offset = 17.7

• vp_tcoeff = 45892.6 K

• vp_metcoeff = 1.66

• vp_logpcoeff = 0

Surface Tension

CARMApy also parameterizes the surface tension of condensates as follows:

\[\texttt{surface\_tension} = \texttt{surften\_0} - \texttt{surften\_slope} * T\]

where \(T\) is the temperature in K.

From Kazasa et al. 1989 we estimate that for Al₂O₃

• surften_0 = 690 dyne/cm

• surften_slope = 0

Stoichiometry Factor

The stoichiometry factor is determined by the reaction that forms the condensate, ie for a reaction which is

\[X (\text{limiting gas}) + (\text{other stuff}) \to Y (\text{condensate}) + (\text{some other stuff})\]

the stoichiometry factor is \(X/Y\)

As we are considering Al as our limiting gas and it takes 2 monatomic Al to form 1 molecule Al₂O₃, the stoichiometry factor will be 2

Type III reaction

Condensates which form via type III reactions, as defined by Helling and Woitke 2012 have a modified super saturation ratio. Type III reactions are reactions that, besides for the condensate, have 2 or more products or 2 or more reactants.

Since the main chemical reaction which forms Al₂O₃ is

\[2 \text{AlOH} + \text{H}_2\text{O} \to \text{Al}_2\text{O}_3 [\text{s}] + 2 \text{H}_2\]

Al₂O₃ is a type III reaction

Latent heat of evaporation

You can choose to provide a latent heat of evaporation, if you don’t CARMApy will calculate one for you following the assumptions of Charnay et al. 2015 (TODO: check this citation):

\[\texttt{lat\_heat\_e} = \texttt{vp\_tcoeff} \cdot \ln(10) * \frac{R}{\texttt{wtmol\_dif}}\]

where \(R\) is the ideal gas constant

Defining the Gas Object

First we will set uo the carma simulation as we did previously, without adding the Mg₂SiO₄

carma = carmapy.Carma("custom_gases")

carma.set_stepping(dt=100, output_gap=100, n_tstep=3000)


P_levs, T_levs, kzz_levs, mu_levs = carmapy.example.example_levels()

carma.add_P(P_levs)
carma.add_T(T_levs)
carma.add_kzz(kzz_levs)

carma.set_physical_params(surface_grav=31600,
                            wt_mol=mu_levs[0])
carma.set_atmospheric_parameters_from_defaults("Pure H2")

carma.calculate_z(mu_levs)
carma.extend_atmosphere(1e10)

carma.add_hom_group("TiO2", 1e-8)

<carmapy.carmapy.Group at 0x1156d4d70>

Then to add the user defined condensate, first we add a gas to our carma object with all the relevent properties, and then we add the nucleation pathways we want to consider:

carma.add_gas("Al_oxide",
    rho_cond = 3.99,
    wtmol = 101.926,
    wtmol_dif = 26.98,
    coldia = 3.825e-8,
    hill_formula = "Al",
    vp_offset = 17.7,
    vp_tcoeff = 45892.6,
    vp_metcoeff = 1.66,
    vp_logpcoeff = 0,
    surften_0 = 690,
    surften_slope = 0,
    stofact = 2,
    is_typeIII = True
)

carma.add_het_group("Al_oxide",
                    "TiO2",
                    1e-8 * 2**(1/3),
                    mucos=0.724172)

<carmapy.carmapy.Group at 0x11545b610>

Note that we can add the nucleation pathway by using the name of the gas we defined.

Note: use a short name for the gas, long names (around 12 or more characters) have a chance of breaking CARMA

Then, as before, we can populate the gas abundances as follows:

carmapy.chemistry.populate_abundances_at_cloud_base(carma)

As with other carma objects, you can run the simulation by running the following cell:

# carma.run()