Prediction of sulphate hazes in the lower Venus atmosphere

TL;DR

This study models the composition, size distribution, and formation heights of aerosols in Venus's lower atmosphere, linking chemical processes to observed haze layers and particle charge effects.

Contribution

It introduces an integrated chemical and physical model predicting aerosol composition, size, and charge distribution in Venus's lower atmosphere, aligning with recent observational data.

Findings

Aerosol particles form at specific altitudes matching haze layers.

Particles smaller than 0.3 microns can be lifted from the ground.

Large particles (>1 micron) are confined near the surface.

Abstract

We study the amount, size distribution and material composition of (sub-)mic aerosol particles in the lower Venus atmosphere < 50 km. Our GGchem phase-equilibrium model predicts metal-chloride and metal-fluoride molecules to be present in the gas over the Venus surface in trace concentrations < 2.E-12, in particular FeCl2, NaCl, KCl and SiF4. Using an improved version of the DiffuDrift model developed by Woitke et al.2020, we find that these molecules deposit to form solid potassium sulphate K2SO4, sodium sulphate Na2SO4, and pyrite FeS2 above about 15.5 km, 9.5 km and 2.4 km, respectively. These heights coincide well with the three potential haze layers found in the Pioneer Venus Large Probe neutral mass spectrometer data by Mogul et al.2023. The particles with radius < 0.3 mic can be dredged up from the ground to reach the sulphuric acid cloud base from below by diffusion. The particle density decreases from ~ 5000/cm3 at ground level to ~100/cm3 at a height of 45 km. Particles larger than about 1 mic are found to stay confined to the ground < 10 km, indicating that the larger, so-called mode 3 particles, if they exist, cannot originate from the surface. All particles are expected to be coated by a thin layer of FeS2, Na2SO4 and K2SO4. We have included the repelling effect of particle charges on the coagulation, without which the model would predict much too steep gradients close to the surface, which is inconsistent with measured opacity data. Our models suggest that the particles must have at least 100 negative charges per micron of particle radius at ground level, and > 50/mic at a height of 45 km.

Tables

Table 1. Table 1: Trace concentrations of metal-chloride and metal-fluoride molecules over the Venus surface and first cloud condensates predicted by our GGchem model.

		main	trace	first cloud	cloud
​​element		molecule	​concentration${}^{\left(1\right)}$	condensate	​base [km]​​	hypothetical		surface reaction
Fe	$\to$	\ceFeCl2	$1.3\times {10}^{-12}$	\ceFeS2[s](2)	2.9	\ceFeCl2 + 2 \ceSO2 + \ceH2O + 5 \ceCO	​​​$\to$	\ceFeS2[s] + 2 \ceHCl + 5 \ceCO2
Na	$\to$	\ceNaCl	$1.1\times {10}^{-12}$	\ceNa2SO4[s]	9.3	2 \ceNaCl + \ceSO2 + \ceH2O + \ceCO2	​​​$\to$	\ceNa2SO4[s] + 2 HCl + CO
Si	$\to$	\ceSiF4	$7.0\times {10}^{-13}$	–	–
K	$\to$	\ceKCl	$5.2\times {10}^{-13}$	\ceK2SO4[s]	13.8	2 \ceKCl + \ceSO2 + \ceH2O + \ceCO2	​​​$\to$	\ceK2SO4[s] + 2 HCl + CO
Al	$\to$	\ceAlF2O	$6.4\times {10}^{-14}$	\ceAl2O3[s]	0.0	2 \ceAlF2O + 2 \ceH2O + \ceCO	​​​$\to$	\ceAl2O3[s] + 4 \ceHF + \ceCO2
Ti	$\to$	\ceTiF4	$1.4\times {10}^{-18}$	\ceTiO2[s]	0.0	\ceTiF4 + 2 \ceH2O	​​​$\to$	\ceTiO2[s] + 4 \ceHF
Mg	$\to$	\ceMgCl2	$8.5\times {10}^{-21}$	\ceMgF2[s]	0.0	\ceMgCl2 + 2 \ceHF	​​​$\to$	\ceMgF2[s] + 2 \ceHCl
Ca	$\to$	\ceCaCl2	$1.4\times {10}^{-21}$	\ceCaSO4[s]	0.0	\ceCaCl2 + \ceSO2 + \ceH2O + \ceCO2	​​​$\to$	\ceCaSO4[s] + 2 \ceHCl + \ceCO

Equations

$$\rho L_j={\int }_{V_{\ell }}^{\infty }f(V)V^{j/3}dV$$

$$\frac{d(\rho L_j)}{dt}$$

$$\rho L_j=\sum _{i=1}^2{n}_i{V}_i^{j/3}(j=0,1,2,3).$$

$$\frac{d(\rho L_j)}{dt}$$

$$\frac{d(\rho L_3^s)}{dt}$$

$$\frac{d(\rho {ϵ}_k)}{dt}$$

$$n_{\mathrm{mol}}=n_{\mathrm{mol}}(\rho ,T,{ϵ}_k).$$

$$q=-\text{mbox}qa300\left(\frac{a}{1\mu \mathrm{m}}\right)\left(\frac{T}{300\mathrm{K}}\right),$$

$${\kappa }_{\lambda }^{\mathrm{ext}}=\sum _{i=2}^2{n}_i\pi a_i^2{Q}_{\mathrm{ext}}(a_i,\lambda ).$$

$$D(z)=3\times 10^{4}\;{\rm\!cm^{2}s^{-1}}\;\bigg{(}\frac{p}{1\rm\,bar}\bigg{)}^{-1/2}\ ,$$

$$f(a,z\!=\!0)\,=\,\frac{n_{p}}{a\,\sigma\sqrt{2\pi}}\exp\bigg{(}-\frac{\big{(}\ln a-\mu\big{)}^{2}}{2\sigma^{2}}\bigg{)}$$

$$\frac{dn_{p}}{dt}=\frac{\partial}{\partial z}\,\left({v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\,n_{p}\right)~+~\frac{\partial}{\partial z}\,\left(D\,\rho\,\frac{\partial}{\partial z}\bigg{(}\frac{n_{p}}{\rho}\bigg{)}\,\right)\ .$$

$${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\,n_{p}~+~D\,\rho\,\frac{\partial}{\partial z}\bigg{(}\frac{n_{p}}{\rho}\bigg{)}~=~{\rm const}\ .$$

$$\frac{\partial}{\partial z}\bigg{(}\ln\frac{n_{p}}{\rho}\bigg{)}~=~-\frac{{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}}{D}\ .$$

$$\hskip-8.53581pt\langle a^{j}\rangle^{1/j}\,=\,\left(\frac{\int\limits_{V_{\ell}}^{\infty}\!a^{j}\,f(V)\,dV}{\int\limits_{V_{\ell}}^{\infty}\!f(V)\,dV}\right)^{1/j}\hskip-11.38109pt=\,\bigg{(}\frac{3}{4\pi}\bigg{)}^{1/3}\!\!\left(\frac{L_{j}}{L_{0}}\right)^{1/j}\!\!\!$$

$$⟨{v_{\mathrm{dr}}^{{}^{\circ }}}^j⟩=\frac{\underset{V_{\ell }}{\overset{\infty }{\int }}{v}_{\mathrm{dr}}^{{}^{\circ }}f(V)V^{j/3}dV}{\underset{V_{\ell }}{\overset{\infty }{\int }}f(V)V^{j/3}dV}\approx \frac{\sum _{i=2}^2{v_{\mathrm{dr}}^{{}^{\circ }}}_{,i}{n}_i{V}_i^{j/3}}{\sum _{i=2}^2{n}_i{V}_i^{j/3}}.$$

$$\displaystyle\frac{\partial}{\partial z}\Bigg{(}\sum_{s}\frac{\nu_{s,k}}{V_{0}^{s}}\sum_{i=1}^{2}n_{i}V_{i}^{s}\,{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}_{,i}\Bigg{)}$$

$$\displaystyle+\,\frac{\partial}{\partial z}\,\Bigg{(}D\,\rho\,\frac{\partial}{\partial z}\bigg{(}\sum_{s}\frac{L_{3}^{s}\,\nu_{s,k}}{V_{0}^{s}}+\epsilon_{k}^{\rm gas}\bigg{)}\Bigg{)}~=~0\ .$$

$$\rho\,\epsilon_{k}^{\rm cond}\langle{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}^{\!\!3}\rangle+\,D\,\rho\,\frac{\partial}{\partial z}\Big{(}\epsilon_{k}^{\rm cond}+\epsilon_{k}^{\rm gas}\Big{)}~=~\mbox{const}\ ,$$

$$\displaystyle\big{(}\epsilon_{k}^{\rm cond}\gg\epsilon_{k}^{\rm gas}\big{)}$$

$$\displaystyle\big{(}\epsilon_{k}^{\rm gas}\gg\epsilon_{k}^{\rm cond}\big{)}$$

$$\displaystyle\hskip-5.69054pt\big{(}D\gg\langle{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}^{\!\!3}\rangle H_{p}\big{)}$$

$$\frac{\partial(\rho L_{j})}{\partial t}\bigg{|}_{\rm settle}\!=~\frac{\partial}{\partial z}\sum_{i=1}^{2}n_{i}\,V_{i}^{\;j/3}\,{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}_{,i}$$

$$v_{\mathrm{dr}}^{{}^{\circ }}=g\frac{{\rho }_{\mathrm{m}}a}{\rho v_{\mathrm{th}}}$$

$$\frac{\partial(\rho L_{j})}{\partial t}\bigg{|}_{\rm settle}\!\!\!~=~\frac{\partial}{\partial z}\Bigg{(}\bigg{(}\frac{3}{4\pi}\bigg{)}^{1/3}g\,\frac{\rho_{\rm m}}{\rho\,v_{\rm th}}\;\sum_{i=1}^{2}n_{i}\,V_{i}^{\;\frac{j}{3}+\frac{1}{3}}\Bigg{)}~=~\xi_{\rm lKn}\,\frac{\partial}{\partial z}\bigg{(}\frac{\rho_{\rm m}}{c_{T}}\,L_{j+1}\bigg{)}\ ,$$

$$v_{\mathrm{dr}}^{{}^{\circ }}=g\frac{2{\rho }_{\mathrm{m}}{a}^2}{9{\nu }_{\mathrm{dyn}}},$$

$${\nu }_{\mathrm{dyn}}=1.48\times 10^{-4}\frac{\mathrm{g}}{\mathrm{cm}\mathrm{s}}{\left(\frac{T}{300\mathrm{K}}\right)}^{0.5}.$$

$$\frac{\partial(\rho L_{j})}{\partial t}\bigg{|}_{\rm settle}\!\!\!~=~\frac{\partial}{\partial z}\Bigg{(}\bigg{(}\frac{3}{4\pi}\bigg{)}^{2/3}g\,\frac{2\,\rho_{\rm m}}{9\,\nu_{\rm dyn}}\;\sum_{i=1}^{2}n_{i}\,V_{i}^{\;\frac{j}{3}+\frac{2}{3}}\Bigg{)}\\ ~=~\xi_{\rm sKn}\,\frac{\partial}{\partial z}\bigg{(}\frac{\rho_{\rm m}}{\nu_{\rm dyn}}\,\rho L_{j+2}\bigg{)}\ ,$$

$$\Delta (\rho L_j)$$

$$\Rightarrow$$

$$\frac{\partial V}{\partial t}=4\pi a^2\sum _r{n}_r^{\mathrm{key}}{v}_r^{\mathrm{rel}}{\alpha }_r{V}_r{\gamma }_r$$

$$\gamma_{r}=\left(1-\frac{1}{S_{r}}\right)\;\times\;\left\{\begin{array}[]{cl}1&\mbox{\!\!if $S_{r}\geq 1$}\\ \!\!b_{\rm mix}^{s}&\mbox{\!\!if $S_{r}<1$}\ ,\end{array}\right.$$

Full text

Venus (1763) — Planetary atmospheres (1244) — Surface composition (2115) — Atmospheric composition (2120) — Atmospheric clouds (2180) — Dust composition (2271)

1 Introduction

The exploration of Earth’s inner neighbour planet, Venus, with the Venera (e.g., V. I. Moroz, 1983, 2002), Mariner (e.g., C. P. Sonett, 1963; A. Kliore et al., 1967; G. Fjeldbo et al., 1971), Pioneer Venus (e.g., R. G. Knollenberg & D. M. Hunten, 1980; V. I. Oyama et al., 1980; L. Colin, 1980), Vega (e.g., R. Z. Sagdeev & V. I. Moroz, 1986; R. Z. Sagdeev et al., 1986; J.-L. Bertaux et al., 1996), Venus Express (H. Svedhem et al., 2007; D. V. Titov et al., 2009), Akatsuki (Venus Climate Orbiter) (M. Nakamura et al., 2016), and Magellan (e.g., R. S. Saunders et al., 1992) space missions included flybys, descent probes, orbiters, and balloons. These missions revealed the basic physical and chemical structure of the Venus atmosphere, clouds and aerosols, its surface, and geology (see, e.g., D. M. Hunten et al., 1983; S. W. Bougher et al., 1997; B. Fegley, 2014, for further reviews on Venus).

The atmosphere of Venus is a chemically exotic and complex system, see e.g. C. de Bergh et al. (2006), D. V. Titov et al. (2018) and L. Dai et al. (2025), involving photochemistry in the upper atmosphere, thermochemical cycles in the troposphere, and presumably a strong buffering role of the surface for various molecules like CO2, HCl, HF and some S-bearing species (e.g., B. Fegley & A. H. Treiman, 1992; M. Y. Zolotov, 2018, 2024), which tends to establish thermochemical equilibrium in the gas in direct contact with the surface (P. B. Rimmer et al., 2021). Earlier works about the buffering mechanisms include e.g. V. A. Krasnopolsky & J. B. Pollack (1994); V. A. Krasnopolsky (2007), and B. Fegley et al. (1997). A recent summary of various equilibrium models for the surface of Venus can be found in Table 16.2 of M. Y. Zolotov (2024).

Venus has a thick atmosphere that contains gaseous and particle absorbers responsible for a powerful greenhouse effect that results in a mean surface temperatures of about 735 K, with substantial variations with day/night, latitude, and elevation, see e.g. J.-L. Bertaux et al. (2007). Contrary to Earth and Mars, Venus absorbs a significant fraction of the solar energy in the optically thick clouds that surround the planet at heights of about (43-47) km to (63-75) km, depending on latitude (M. Y. Zolotov, 2024). The cloud tops include an unknown ultraviolet (UV) absorber (e.g., L. V. Zasova et al., 1981; D. V. Titov et al., 2018), for which the dimer of ferric chloride \ceFeCl3 (V. A. Krasnopolsky, 1985, 2017), two isomers of \ceS2O2 (B. N. Frandsen et al., 2016), rhomboclase and ferric sulphates (C. Z. Jiang et al., 2024), or even microbial lifeforms (e.g., D. H. Grinspoon, 1997; S. S. Limaye et al., 2018) were suggested as potential absorbing agents.

In the upper clouds, sulphuric acid is identified as a particulate component, although the chemical origin and physical processes in the cloud deck are still not fully understood (see, e.g., reviews by L. W. Esposito et al., 1983, 1997; E. Marcq et al., 2018; D. V. Titov et al., 2018). The strong depletion of SO2 within the clouds (e.g., A. C. Vandaele et al., 2017a, b) remains a conundrum, although hydroxide salts in the cloud droplets were recently shown to provide a potential sink for SO2 (P. B. Rimmer et al., 2021).

The study of aerosol particles in the Venus atmosphere provides critical insight into the planetary environment and has profound implications for the understanding of atmospheric chemistry, cloud formation, and the possibility of the existence of an aerial life cycle. The Pioneer Venus Orbiter cloud photopolarimeter (K. Kawabata et al., 1980) and descent probes (R. G. Knollenberg & D. M. Hunten, 1980) measured the density and size distribution of aerosol particles in and below the clouds, revealing a possibly tri-modal size distribution with mean radii of $\sim$ 0.2 $\mu$m, (mode 1), $\sim$ 1.0 $\mu$m (mode 2) and $\sim$ 3.6 $\mu$m (mode 3). The latter have been suggested to have a crystalline structure (R. G. Knollenberg, 1984; R. W. Carlson et al., 1993), possibly coated with sulphuric acid (D. H. Grinspoon et al., 1993), which would indicate growth within the clouds. However, the existence of mode 3 particles has been challenged and suggested to be the tail-end of mode 2 rather than an independent particle population (O. B. Toon et al., 1984). The latter also fits the data from the Soviet descent probes which did not observe a separate mode 3 particle distribution (B. E. Moshkin et al., 1986; L. V. Zasova et al., 1996). The non-existence of the mode 3 particles is further supported by P. Gao et al. (2014), who used the community aerosol and radiation model for atmospheres (CARMA) (R. Turco et al., 1983) to interpret the PICAV/SOIR data from Venus Express. In their model, they found a bimodal distribution, where the mode 3 particles are merely a larger, grown version of the mode 2 particles that evaporate below the clouds.

The particle size distribution in the Venusian clouds is compatible with the known range of cells and spores of microorganisms on Earth. Based on this, S. Seager et al. (2021) suggested an aerial life cycle around the base of the Venus liquid sulphuric acid clouds, where desiccated spores (comparable to mode 1 particles) populate the layers below the clouds, diffuse up into the clouds where they are incorporated in the cloud droplets and germinate to a metabolically active life form. The droplets further grow by coagulation until they reach droplet sizes comparable to mode 2 particles and start to settle gravitationally, evaporate and release new spores. W. Bains et al. (2021a) suggested another aerial life cycle where metabolically active microbes in cloud droplets produce NH3 which raises the droplet $p$H, traps SO2 and H2O in sulphate salts and form large semisolid mode 3 particles. These particles again settle gravitationally below the cloud deck, evaporate and release spores, SO2 and H2O, a process that could also explain the abovementioned SO2 depletion in the Venus clouds. Another study by D. Schulze-Makuch et al. (2004) links the mode 3 particles in the lower atmosphere of Venus with life by proposing that mode 3 particles are microbes coated in elemental sulphur that float through the dense gas of the atmosphere (see also, S. Seager et al., 2021). The astrobiological potential of the Venus atmosphere, the chemical anomalies, and the unexplained cloud properties have been recently reviewed by J. J. Petkowski et al. (2024).

Phosphine, potentially synthesised by microorganisms, was tentatively detected in the clouds of Venus at parts-per-billion-levels (J. S. Greaves et al., 2021). By now, no abiotic sources are known that could sufficiently explain the potential existence of PH3 in the upper Venus atmosphere (W. Bains et al., 2024). The influx of micrometeorites (P. Gao et al., 2014) is too little to, even if all phosphorous was converted into phosphine (W. Bains et al., 2021b), and volcanism (N. Truong & J. Lunine, 2021) could only produce the ppb-level signals under extreme volcanic activity (W. Bains et al., 2022). Abiotic photochemical pathways for phosphine synthesis on acidic mineral surfaces have been suggested by K. Mráziková et al. (2024), but are not yet experimentally confirmed. However, similar photochemical pathways on mineral surfaces have been experimentally verified for the synthesis of \ceCH4 (methanogenese) from \ceCO2 by S. Civiš et al. (2016) and S. Civiš et al. (2019). Such processes could mimic biosignature-like redox disequilibria through abiotic chemistry.

Understanding the amount, size distribution and material composition of aerosol particles is a crucial step toward evaluating their ability to influence the redox balance in the gas through hydride synthesis and thereby contribute to chemical anomalies. Therefore, aerosols are not merely a passive component of the atmosphere but can actively shape Venus’ chemical dynamics. Another pivotal question is whether these particles can reach the clouds from below to seed the clouds, but by now only a few studies investigated the chemical and diffusive behaviour of such aerosol particles in the lower Venusian atmosphere.

Simulations and measurements of aerosol particles in the lowest parts of Venus’ atmosphere are scarce. A. D. Anderson (1969) developed a simple model to estimate the density and maximum particle size of dust at altitudes between 0 and 25 km. Based on Mariner 5 and Venera 4 data, they found that for the same convective activity as on Earth, more dust will be in Venus’ atmosphere than on Earth with diameters as large as 130 $\mu$m. C. Sagan (1975) found that particles of sizes up to $40\,\mu$m can be lifted from the surface when the threshold frictional velocity is $>1-2\rm\,cm/s$, corresponding to a horizontal wind velocity of $>\!0.3\,$m/s above the surface boundary layer. If these particles reach a height of 10 km, they could be redistributed around the entire planet, as Venera 8 measured wind velocities of a few $10\,$m/s at these heights. Experiments in a Venus Wind Tunnel by R. Greeley et al. (1984) further suggest that dust grains below 40 $\mu$m can be lifted by the wind and float in the Venusian atmosphere (R. Greeley et al., 1984; R. Greeley & R. E. Arvidson, 1990), which is in good agreement with the calculations performed by C. Sagan (1975). See also L. M. Carter et al. (2023) for a review on aeolian transport.

Continuum absorption at 1.18 $\mu$m seen in the deep atmosphere of Venus by the VIRTIS and SPICAV-IR instruments onboard Venus Express could indeed relate to the presence of dust particles (M. Snels et al., 2014). The existence of layers of dust and/or haze in the lower atmosphere of Venus are consistent with a variety of observations. Early measurements by Mariner V and Venera 4 were interpreted as dust layer at \mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr<\cr\sim\cr}}}}\!25 km (A. D. Anderson, 1969). The detection of small backscatter signals in the Pioneer Venus night and north probes around an altitude of $\sim\!6$ km was suggested to indicate a particle-bearing layer (A. Seiff et al., 1995). Later analysis of the Pioneer Venus and Venera descent probes further support the existence of low-altitude haze layers. A reconstruction of the CO2 profile measured by the Pioneer Venus Large Probe neutral mass spectrometer indicates the presence of haze layers below an altitude of 17 km, in particular at $\sim\!15\pm 2$ km, $\sim\!10\pm 3$ km, and $\sim\!3\pm 1$ km, related to partial clogging of the instrument by particles (R. Mogul et al., 2023), an effect that was even stronger in the main sulphuric acid clouds. A reanalysis of spectrophotometer (B. Grieger et al., 2004) and discharge current measurements (R. D. Lorenz, 2018) on Venera 13 and 14 also support a dusty atmosphere below 35 km. Based on the discharge current data measured at 1-2 km altitudes, R. D. Lorenz (2018) found a near-surface charge density of the order of $\rm 1000\,pC/m^{3}$. Assuming about singly charged particles, and a particle size of $\sim\!0.3\,\mu$m, R. D. Lorenz (2018) suggested a density of charged particles of order $1000\rm\,cm^{-3}$, which would correspond to an aerosol opacity of order $\rm 1/km$, backing the earlier results of B. Grieger et al. (2004). These charge and particle densities are higher than observed for the Saharan dust at high altitudes on Earth (K. A. Nicoll et al., 2010). However, these large aerosol opacities seem to conflict with the surface pictures taken by the Venera landers (A. S. Selivanov et al., 1983; C. M. Pieters et al., 1986)111See also http://mentallandscape.com/C_CatalogVenus.htm for the Venera lander images digitally remastered by Don Mitchel., which showed a clear horizon. This could indicate that there are indeed much fewer particles that are charged much more, such that the resulting charge density remains the same while the opacity is lower.

A recent study by S. V. Kulkarni et al. (2025), which reconstructed spectrophotometric data from the Venera 13 mission by digitizing old graphic material, supports the existence of a near-surface particulate layer. Its peak density was found to be at an altitude of 3.5-5 km (in agreement with the secondary peak found by B. Grieger et al., 2004) with a log-normal particle size distribution peaking at particle radii between 0.6 and 0.85 $\mu$m. The refractory index of the particles was derived to be $\eta_{\rm\,r}\sim 1.4-1.6$, which closely matches $\eta_{\rm\,r}\sim 1.52$ of basalt particles. This indicates that the particle layer could have formed from basaltic dust lifted from the surface (S. V. Kulkarni et al., 2025).

A lower haze layer is also consistent with thermo-chemical equilibrium simulations by X. Byrne et al. (2024) using the GGchem model (P. Woitke et al., 2018) for the Venus setup as published in P. B. Rimmer et al. (2021), which indicates the formation and rainout of pyrite clouds above an altitude of about 3 km, an effect that could explain the high reflectivity of the Venus mountain tops seen in radar measurements (G. H. Pettengill et al., 1982, 1996), because pyrite is a conducting material. Pyrite was considered by K. B. Klose et al. (1992) and E. Kohler et al. (2015), together with other potential materials, to explain the radar reflectivity, see also V. L. Barsukov et al. (1982, 1986); M. Y. Zolotov & V. P. Volkov (1992). Alternatively, the formation of heavy metal frost through the condensation of volcanically degassed volatile metals, such as Cu, As, Pb, Sb, and Bi, at higher elevations was first proposed by R. A. Brackett et al. (1995) and later refined by L. Schaefer & B. Fegley (2004a). Volume scattering from a potentially porous, highly weathered soil layer was suggested as an additional alternative (e.g., G. H. Pettengill et al., 1992; K. A. Tryka & D. O. Muhleman, 1992; R. A. Brackett et al., 1995). For a review of the oxidation state of the lower atmosphere and surface of Venus, see B. Fegley et al. (1997); for a review of its surface mineralogy, see M. S. Gilmore et al. (2023).

Understanding the chemical inventory of Venus’ aerosols is essential for determining how specific chemical compounds influence the thermochemistry of the planet’s atmosphere and interact with other atmospheric species. The current study represents a crucial step toward estimating the chemical nature of the aerosol particles within Venus’ atmospheric profile.

The paper is organised as follows. In Sect. 2 we describe the results obtained with our thermo-chemical equilibrium code GGchem. We discuss (i) the concentrations found for metal-chloride and metal-fluoride molecules over the Venus surface, and (ii) the stability of condensates in the Venus atmosphere up to a height of 50 km. In Sect. 3, we introduce the DiffuDrift v2 code to model chemically active aerosol particles including gravitational settling, diffusion and coagulation. We assume that a mixture of the condensates found to be stable in Sect. 2 can deposit on these particles, and use a kinetic description of the process of deposition and sublimation to predict the chemical composition, number density and sizes of these particles. Section 4 then discusses the results of four simulations, (i) a basic model of passive aerosol particles, (ii) a model for passive particles including coagulation, (iii) a model for chemically active particles based on the second model, and (iv) a model for chemically active particles with a reduced abundance of the passive particles at $z=0$. Section 6 summarises our results.

2 The GGchem model

We first summarise our previously obtained results for the composition of the gas and the material composition of the surface when applying the phase-equilibrium model GGchem (P. Woitke et al., 2018) to the foot-point of the Venus atmosphere, see P. B. Rimmer et al. (2021) for more details. GGchem predicts these properties based on the surface pressure, surface temperature, and total (condensed $+$ gas phase) element abundances. In order to determine these total element abundances, we considered a combination of (i) the surface oxide ratios measured by the Vega 2 lander (Y. A. Surkov et al., 1986) and (ii) a set of measured molecular concentrations in the lower Venus atmosphere. The included molecular concentrations have been compiled from various missions and instruments as summarised in (P. B. Rimmer et al., 2021): \ceCO2, \ceN2, \ceSO2, \ceH2O, \ceOCS, \ceCO, \ceHF, \ceHCl, \ceH2S, \ceS3, \ceS4, and \ceNO. After carefully adjusting the total oxygen abundance, the GGchem equilibrium condensation model is capable of approximately reproducing both the element composition of the solid Venus surface and the gas phase composition over its surface. All molecules that are predicted to be abundant in our model (those with percent or ppm concentrations) have observed counterparts and their concentrations agree within a factor of 2 or better with the measurements. Other molecules, such as \ceO2, \ceCH4 and \ceNH3, have concentrations $<\!10^{-15}$ in our model and are indeed not observed, see table 3 in P. B. Rimmer et al. (2021).

Therefore, we concluded in P. B. Rimmer et al. (2021) that the near-surface atmosphere of Venus is in fact close to gas phase chemical and phase equilibrium with its hot surface, in agreement with M. Y. Zolotov (1996), V. A. Krasnopolsky (2013), and C. J. Bierson & X. Zhang (2020), and that our simple GGchem equilibrium condensation model can be confidently used to interpret the measurements.

The second part of table 3 in P. B. Rimmer et al. (2021) lists the 11 condensates found to be stable along with their mass fractions found in our model for the Venus surface: \ceMgSiO3[s] (enstatite), \ceCaAl2Si2O8[s] (anorthite), \ceNaAlSi3O8[s] (albite), \ceFe2O3[s] (hematite), \ceCaSO4[s] (anhydrite), \ceSiO2[s] (quartz), \ceAl2O3[s] (corundum), \ceKAlSi3O8[s] (microcline), \ceMn3Al2Si3O12[s] (spessartine), \ceTiO2[s] (rutile), and \ceMgF2[s] (magnesium fluoride). These condensates are all saturated ($S=1$) in the model, and present in the surface, whereas all other condensates are undersaturated $S<1$ and not present in the surface. This is a consequence of our simple phase equilibrium model that includes only pure condensates. More sophisticated models include solid solutions (e.g. V. L. Barsukov et al., 1982, 1986) and discuss the presence of \ceFe3O4 (magnetite) and \ceAl2SiO5[s] (andalusite), see e.g. B. Fegley & A. H. Treiman (1992) and M. Y. Zolotov (2024). Indeed, these two minerals obtain supersaturation ratios of 0.95 and 0.93 in our model, respectively, so they are on the verge of becoming stable in our model.

Microscopically speaking, the applicability of chemical and phase equilibrium means that there must be a local balance between sublimation and re-sublimation rates, and there must be sufficiently rapid chemical processes in the gas phase to install that equilibrium over the surface.

Table 1 shows new details from our previously published phase equilibrium model for the Venus surface. The first two columns show the main molecular carriers of Fe, Na, Si, K, Al, Ti, Mg and Ca in the gas phase, and their particle concentrations over the surface. We mainly find those element carriers to be chlorine and fluorine molecules, which is in agreement with e.g. L. Schaefer & B. Fegley (2004b). We note that M. Y. Zolotov (2021, 2025) argue that chloride molecules would decompose on the surface when in contact with solid sulphates, however in our model, the surface is too hot to host \ceNa2SO4[s] and \ceK2SO4[s] as stable condensates.

Noteworthy, the main Fe-bearing molecule in our model is iron dichloride (\ceFeCl2) but not the dimer of ferric chloride (\ceFe2Cl6) as suggested by (V. A. Krasnopolsky, 2017). Our model does include \ceFeCl3 and \ceFe2Cl6, with thermo-chemical data from the NIST/Janaf tables (M. Chase, 1998), but the concentrations of these molecules result to be 4 and 15 orders of magnitudes lower than the concentration of \ceFeCl2, respectively, which itself is more than $10^{4}\times$ lower than the estimate of 17 ppb of \ceFe2Cl6 by Krasnopolsky. According to our model, if iron-chloride molecules were indeed present in ppb-concentrations, they should immediately deposit on the surface of the aerosol particles. Additional iron molecules of potential interest are \ceFeF3, \ceFe(OH)2, \ceFe2Cl4 and \ceFeF2, which are found to have concentrations between $10^{-14}$ and $10^{-21}$.

In Fig. 1 we show the results of a new, simple GGchem model for the near-surface layers, where we seek to establish which condensates can be expected to form at which height in the Venus atmosphere. We consider a simple 1D atmospheric structure with a given pressure and temperature profile. The $(p,T)$-structure is obtained from a spline fit to the values given by S. Palen et al. (2014), using 1000 equidistant points between 0 and about 105 km. At the bottom of this structure, our previous GGchem model (P. B. Rimmer et al., 2021) applies.

The cloud structure of the Venus atmosphere is then calculated as described by O. Herbort et al. (2022). In each atmospheric layer, GGchem determines the stable condensates and the amount of elements that are consumed to form these condensates, until saturation is reached. The respective numbers of condensed units per volume $n_{\rm cond}$ are stored and the corresponding amounts of condensed elements are subtracted from the total element abundances $\epsilon_{k}$, before advancing upwards to the next atmospheric layer. Precisely speaking, $n_{\rm cond}$ is the number of condensed units per $\rm cm^{3}$ that deposit per height interval $\Delta z$, when a considered gas parcel is lifted by $\Delta z$ in the atmosphere. Since the next layer is generally cooler, additional condensation will take place and new cloud condensates may occur.

We note that we are using the assumption of chemical equilibrium in the gas phase while determining the stability of condensates in this model, which is a simplification. Chemical equilibrium is not expected to hold in the Venus atmosphere (M. Y. Zolotov, 2018, 2021), except for the near-surface layers (M. Y. Zolotov, 1996; V. A. Krasnopolsky, 2013; C. J. Bierson & X. Zhang, 2020; P. B. Rimmer et al., 2021). However, the details of the gas phase chemistry have little influence on the principle stability of condensates in the atmosphere, because of the very steep dependences of the vapour pressures as function of temperature, and our method is in line with many theoretical models used in exoplanet science to establish the cloud structures, e.g. F. Allard et al. (2001), C. V. Morley et al. (2012), and L. Schaefer et al. (2012).

We updated the GGchem code with new Gibbs free energy data from FastChem (D. Kitzmann et al., 2024), based on the NIST/Janaf data (M. Chase, 1998) of 96 additional condensed species, in particular sulphates, sulphides, carbonates, nitrides, hydroxides, hydrides, fluorides, chlorides, carbides, and cyanides. Thanks to the referee, we identified an error in interpreting the vapour pressures of di-sulphur \ceS2 and octa-sulphur \ceS8 as given by K. Zahnle et al. (2016), see their equations (4) and (5). These vapour pressures are given over solid and liquid sulphur \ceS[s/l], not over the respective pure condensed phases as required in GGchem. It seems that \ceS2 has no condensed phase at all, so we eliminated \ceS2[s] and \ceS2[l] from GGchem, and have now included only \ceS8[s] according to J. F. Kasting et al. (1989), who give the vapour pressure of gaseous \ceS8 over \ceS8[s] as required in GGchem. The alpha-phase of solid sulphur \ceS[s] (orthorhombic sulphur) is not to be confused with solid octa-sulphur \ceS8[s].

The resulting cloud structure, $n_{\rm cond}(z)$, and the element abundances remaining in the gas phase, $\epsilon_{k}(z)$, are shown in Fig. 1. The main results are:

There are four groups of elements in the Venus atmosphere with abundance hierarchy $\rm O,C,N\gg S,H\gg Cl,F\gg Fe,Na,Si,K,Al,Ti,Mg,Ca$. This hierarchy immediately suggests that the trace metals in the last group prefer to form chloride and fluoride molecules rather than oxides. 2. 2.

Below the main sulphuric acid clouds \mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr<\cr\sim\cr}}}}\!\!45 km, called the “lower thin haze” by R. G. Knollenberg & D. M. Hunten (1980), we find a sequence of three main layers of new stable condensates: pyrite \ceFeS2[s] at z\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}2.9 km, sodium sulphate \ceNa2SO4[s] at z\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}9 km and potassium sulphate \ceK2SO4[s] at z\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}14 km. As pointed out by X. Byrne et al. (2024), the prediction of the stability of pyrite, a conducting material, about 3 km above ground by our GGchem model provides a natural explanation of the strongly reflective mountain tops observed in radar measurements of the Pioneer Venus and Magellan missions (G. H. Pettengill et al., 1982, 1996), similar to the snow coverage of the mountain tops on Earth. The stability of pyrite in the highlands was assessed earlier by, e.g., V. L. Barsukov et al. (1982), K. B. Klose et al. (1992), and M. Y. Zolotov (1991); M. Y. Zolotov & V. P. Volkov (1992). The formation of sulphate clouds (\ceNa2SO4[s] and \ceK2SO4[s]) is a new finding. 3. 3.

In addition to these three condensates, the following solid materials are part of the surface and remain stable in the gas over the surface, but in less quantities: \ceFe2O3[s] (hematite), \ceCaSO4[s] (anhydrite), \ceAl2O3[s] (corundum), \ceTiO2[s] (rutile), and \ceMgF2[s] (magnesium fluoride). 4. 4.

The gas phase element abundances of all metals but Si (that is Fe, Na, K, Al, Ti, Mg, Ca) are subsequently reduced by orders of magnitude above their respective cloud bases via condensation and gravitational settling. 5. 5.

Silicon is found to stay entirely bound in Si-tetrafluoride \ceSiF4 and does not participate in cloud condensation. 6. 6.

Our GGchem model does not work for the main sulphuric acid clouds, which only form at z\!\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}\!80\,km in this model. Instead, we get crystalline sulphur, \ceS[s] at z\!\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}\!45 km. This is not unexpected, because the \ceH2SO4 clouds are known to form from the photodissociation products of \ceCO2 and \ceSO2, which is not included in our model. 7. 7.

Above the \ceH2SO4[s] cloud base, at z\!\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}\!80\,km in our GGchem model, the hydrogen abundance falls rapidly, eventually approaching the sum of the chlorine and fluorine abundances, to retain the concentrations of HCl and HF, whereas the sulphur abundance only decreases very slightly. This is because of the stoichiometry of \ceH2SO4 and the relation $\rm S>H$ present at the bottom of the atmosphere. The formation of sulphuric acid clouds is hence limited by H rather than by S. This is in contrast to observations of \ceH2O and \ceSO2 above the clouds and called the puzzle of sulphur depletion by P. B. Rimmer et al. (2021).

3 The DiffuDrift v2 model

While our GGchem model discussed in Sect. 2 can predict which cloud materials become thermally stable at which heights in the Venus atmosphere, it cannot make quantitative predictions about the amount of cloud particles nor their sizes. For example, in a completely quiescent atmosphere without any turbulent mixing, all cloud condensates would eventually settle down, and no cloud particles or droplets would remain.

In order to make predictions about the amount and sizes of the cloud particles, a kinetic description of their growth and dynamical behaviour is required. The DiffuDrift code was introduced by P. Woitke et al. (2020) to model cloud formation in diffusive atmospheres of brown dwarfs and exoplanets. The code simulates cloud particles in terms of their size moments $L_{j}\;\{j\!=\!0,1,2,3\}$, similar to P. Woitke & C. Helling (2003, 2004) and C. Helling & P. Woitke (2006),

[TABLE]

where $V\,\rm[cm^{3}]$ is the volume of a cloud particle and $f(V)\,\rm[cm^{-6}]$ is the cloud particle size distribution function. $\rho\rm\,[g/cm^{3}]$ is the gas mass density, and $V_{\ell}$ is a lower integration boundary for particles to make sure they have macroscopic properties. The unit of $L_{j}$ is $[\rm cm^{j}/g]$, for example $L_{0}$ is the number of cloud particles per gram of gas, and $L_{3}$ is the total volume of the cloud particles per gram of gas.

The modelling concept of DiffuDrift is to evolve the cloud particle moments in time, in a 1D hydrostatic atmosphere, as affected by nucleation, growth & evaporation, coagulation, gravitational settling and diffusion

[TABLE]

where $J_{\star}\rm\,[s^{-1}cm^{-3}]$ is the nucleation rate, i.e. the formation rate of seed particles of size $V_{\ell}$ that form via chemical processes in the gas phase, $\partial V/\partial t$ is the change of a particle’s volume due to the deposition and sublimation of molecules on its surface (see App. B), $\partial(\rho L_{j})/\partial t\,|_{\rm coag}$ is the change of the cloud particle moments due to particle-particle collisions, ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\rm\,[cm/s]$ is the downward equilibrium drift velocity (or terminal fall speed), and $D\rm\,[cm^{2}/s]$ is the eddy-diffusion coefficient due to turbulent mixing in the atmosphere.

The advantage of this modelling concept is that, for example in the Epstein regime, $\partial V/\partial t\propto a^{2}\rho$ and ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\!\propto\!a/\rho$ depend on particle radius $a\!=(3V/(4\pi))^{1/3}$ with certain powerlaws, so that the two integrals over particle volume in Eq. (2) can be solved analytically, leading to a term involving $L_{j-1}$ for the net growth, and a term involving $L_{j+1}$ for the settling (P. Woitke & C. Helling, 2003). The size-distribution function $f(V,z)$ is not a direct result of the DiffuDrift model, but all relevant mean particle properties, such as the mean size, total surface, etc., can be calculated from the cloud particle moments $L_{j}(z)$, which has the advantage that we do not need to solve hundreds of equations for pre-defined size bins, but can approximately solve the problem of cloud formation with only four equations.

However, in the previous version of DiffuDrift (P. Woitke et al., 2020), we explicitly used the special case of a subsonic flow of the gas around the particles and large Knudsen numbers (Epstein regime), but in dense atmospheres like the ones of Earth and Venus, or in the deeper layers of gas giants including hot Jupiters, this is incorrect. We have therefore extended our moment method to arbitrary Knudsen numbers in this paper, see App. A and B. Furthermore, coagulation caused by (i) Brownian motion, (ii) difference fall speeds, and (iii) turbulence has been included, see App. C.

The key idea for arbitrary Knudsen numbers is to use a double-$\delta$ representation of the particle size distribution function $f(V)$, inspired by T. Birnstiel et al. (2012), as

[TABLE]

$n_{1}$ and $n_{2}$ are two representative particle densities, and $V_{1}$ and $V_{2}$ their respective volumes. The mapping $\{L_{0},L_{1},L_{2},L_{3}\}\leftrightarrow\{n_{1},V_{1},n_{2},V_{2}\}$ is unique in a sense that any given set of $\{n_{1},V_{1},n_{2},V_{2}\}$ results in a particular set of moments and vice versa. In P. Woitke et al. (2020), this approximation (Eq. 3) was used anyway, as closure condition, to construct $L_{4}$ from $\{L_{0},L_{1},L_{2},L_{3}\}$. Here, we use it prior to the calculations of any of the r.h.s. terms in Eq. (2) at any height, in which case a closure condition is no more necessary.

In App. A and B we show that Eq. (3) becomes an exact representation of the size distribution when all particles are either in the Epstein or in the viscous Stokes regime. In both Knudsen number limiting cases, the respective terms for growth and settling can be computed either by (i) the integrals on the r.h.s. of Eq.(2) or by (ii) applying the double-$\delta$ representation (Eq. 3) first and then calculate the terms based on the properties of the two representative particles, as formulated on the r.h.s. of Eq. (4) – the results are identical. Only when one of the two representative particles is in one and the other particle in another hydrodynamical regime, using Eq. (3) becomes an approximation.

As explained by P. Woitke et al. (2020), DiffuDrift considers a mixture of condensed materials $V\!=\!\sum_{s}V^{s}$, where $s$ is an index for the included solid and liquid materials. We assume that all particles of any size at one point in the model are made of the same material mixture, however this mixture can change with time and space. The book-keeping of condensates works by introducing one additional equation per material, using the volumes of each kind $L_{3}\!=\!\sum_{s}L_{3}^{s}$. Furthermore, DiffuDrift keeps track of the element abundances in the gas phase, accounting for the consumption and enrichment due to nucleation, particle growth and evaporation. The complete set of modelling equations is

[TABLE]

[TABLE]

[TABLE]

where $\{j\!=\!0,1,2,3\}$ is the moment index, $\{s\!=\!1,...,S\}$ the index for the included condensates, and $\{k\!=\!1,...,K\}$ the index of the elements affected by condensation. As can easily be verified, $\sum_{s}$(Eq. 5) is identical to Eq. (4) with $j\!=\!3$. $V_{i}\!=\!\sum_{s}V_{i}^{s}$ is the total volume of the representative particle $i$ which is composed of the volumes of the different materials $V_{i}^{s}$. ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}_{,i}$ is the fall speed of the representative particle $i$. $\epsilon_{k}$ is the abundance of element $k$, i.e. the number of nuclei of element $k$ per gram of gas, $J_{\star}^{s}$ is the nucleation rate of material $s$. $V_{0}^{s}$ is the volume of one condensed unit in material $s$ and $\nu_{s,k}$ is the stoichiometric coefficient of element $k$ in material $s$. Coagulation does not change the total volume of condensates, nor does it have any influence on the gas; hence there are no terms for coagulation in Eqs. (5) and (6).

3.1 Gas phase chemistry

To determine the local net growth rates $\partial V/\partial t=\sum_{s}\partial V_{i}^{s}/\partial t$, we assume chemical equilibrium in the gas phase and call GGchem at each time step and on each atmospheric grid point as

[TABLE]

This application of GGchem is for the gas phase only, disregarding condensation, to determine all molecular particle densities $n_{\rm mol}$ and the supersaturation ratios $S$ of all considered cloud condensates. We again note that this is an approximation as we cannot account for disequilibrium effects like photodissociation this way. These supersaturation ratios are then used to determine the volume growth rates of all materials $s$ for the two representative particle sizes $i$, $\partial V_{i}^{s}/\partial t$. Equation (12) in P. Woitke et al. (2020) shows that these growth rates, in the case of large Knudsen numbers, scale with $\,n_{r}^{\rm key}(1-1/S)$, where $n_{r}^{\rm key}$ is the particle density of the least abundant reactant (the key species) in a considered surface reaction $r$. For small Knudsen numbers, the total growth rate is given by Eq. (32) in P. Woitke & C. Helling (2003), which also scales with $\,n_{r}^{\rm key}(1-1/S)$, and can be generalised to a mixture of different materials in the same way. Further details about the growth rates are in App. B. In all hypothetical surface reactions listed in Table 1, the first reactant (a metal chloride of metal fluoride molecule) is by far the least abundant reactant and hence the growth rates are determined by the abundances of these rare molecules.

3.2 Coagulation

A new coagulation module has been added to DiffuDrift v2, based on a numerical implementation of the Smoluchowski equation (M. V. Smoluchowski, 1916), taking into account particle collisions due to Brownian motions, turbulence, and different settling velocities of the two colliding particles, see App. C. We have also implemented the repelling effect of particle charges, using an electrostatic repulsion factor f^{\rm C}_{i,j}=\exp\big{(}-E_{\rm C}/E_{\rm kin}\big{)} as explained in App. C, see Eq. (C12), where $E_{\rm C}=q_{i}\,q_{j}\,{\rm e}^{2}/(a_{i}+a_{j})$ is the Coulomb energy, $q_{i}$ and $q_{j}$ are the numbers of negative charges on the colliding particles $i$ and $j$, and $a_{i}$ and $a_{j}$ are the radii of the colliding particles. Concerning the particle charges, we use the scaling behaviour found by T. Balduin et al. (2023) in UV-shielded environments, also explained in the Appendix

[TABLE]

where $a$ is the particle radius and qa300 is a numerical parameter, the number of negative charges on a 1 $\mu$m particle at $T=300\,$K, henceforth called the ’grain charge parameter’, that is varied from a few to a few tens in this paper. As shown by T. Balduin et al. (2023), even a very small but non-zero cosmic ray ionisation rate creates pairs of free electrons and molecular cations. Since the electrons move faster, they charge the particles negatively, and the effect on the particles scales with $q/a/T$. Therefore, particles of all sizes in one location obtain similar $q/a$ ratios.

3.3 Operator splitting method

Figure 2 shows the numerical method used to advance our DiffuDrift v2 model in time. Each block represents an operator, which advances a single source term on the right sides of Eqs. (4) to (6) in time. For numerical stability it is important that each operator changes the physico-chemical state only moderately at any atmospheric layer, i.e. not far from it’s linear behaviour. For example, the nucleation & growth must not entirely consume all condensible elements in one time step, in other words, the element abundances must not change significantly during $\Delta t$. Concerning the settling, the time step $\Delta t$ must remain small enough to ensure that ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}_{,i}\,\Delta t\!<\!\alpha\Delta z$, where $\Delta z$ is the vertical grid resolution. Similar criteria apply to the coagulation and diffusion operators. We choose a precision of $\alpha\!=\!0.3$ for the models presented in this paper, which results in typical $\Delta t$ of about $100-1000$ s, while we need to advance the atmosphere by a least a few 100 days to reach the stationary limit. The different operators are characterised by:

• •

The diffusion is solved by a standard $2^{\rm nd}$ order explicit numerical scheme. The lower and upper boundaries of all variables $\{L_{j},L_{3}^{s},\epsilon_{k}\}$ can be individually chosen to be either constant concentration or constant influx/outflux.

• •

The coagulation is computed in an explicit way as explained in App. C.

• •

The settling is simulated optionally by a $1^{\rm st}$ or a new $2^{\rm nd}$ order explicit upwind scheme.

• •

The nucleation and growth is advanced in time by a new fully implicit integration scheme.

• •

We have added a new module for the calculation of particle opacities based on effective mixing and Mie theory, only used for output, see Sect. 3.4.

More details about the numerical methods used in DiffuDrift can be found in the appendix of P. Woitke et al. (2020).

3.4 Particle opacities

The extinction opacities of the particles are calculated via the representative particles, which together have the correct total cross section and mass

[TABLE]

The extinction efficiencies $Q_{\rm ext}(a,\lambda)$ are calculated with effective mixing and Mie theory, based on the particle radius $a$, the wavelength $\lambda$, and the local volume composition of the particles $\{L_{3}^{s}/L_{3}|\,s=1,...,S\}$. The details are explained in P. Woitke et al. (2024), where the references for our optical data (refractory indices) are given in table A.1. Unfortunately, out of the six materials considered in this paper, only \ceTiO2 and \ceAl2O3 have valid optical data. For \ceFeS2, we use the optical data for \ceFeS (troilite) instead. Both materials are conducting and are hence very opaque in the optical. For the sulphates \ceK2SO4, \ceNa2SO4 and \ceCaSO4, we use the optical data of \ceNa2S instead, and for \ceMgF2, we use the optical data of \ceMgO. Most relevant, for the “passive” particles, we use the optical data of amorphous Mg0.7Fe0.3SiO3, a common pyroxene. At optical wavelengths, for sub-micron particles, we find extinction efficiencies $Q_{\rm ext}\approx 1.3-2.1$ depending on the choice of material, which reflects our uncertainty in the optical data.

3.5 The atmospheric structure

All models for the lower Venus atmosphere presented in this paper are based on the $(p,T)$-structure based on S. Palen et al. (2014) as introduced in Sect. 2. The eddy diffusion constant is assumed to be

[TABLE]

which is a smooth version of $D(z)$ as used by P. B. Rimmer et al. (2021).

3.6 Boundary conditions

Solving the diffusion problem requires to set boundary conditions (BCs) at the lower and upper model domain. The DiffuDrift v2 code offers various options for the choice of the BCs. For our models in this paper, we have selected fixed-concentration BCs for the gas-phase element abundances $\epsilon_{k}$ and the moments of the passive particles $L_{j}$ at the lower boundary. For the material composition of these particles $L_{3}^{s}$ we choose zero-concentration lower BCs for the thermally unstable materials and zero-flux lower BCs for the thermally stable materials (\ceAl2O3, \ceTiO2, \ceMgF2 and \ceCaSO4). Concerning the upper boundary, we use zero-flux BCs for all components of the solution vector $\{L_{j},L_{3}^{s},\epsilon_{k}\}$.

The lower BC fixed-concentration values for the element abundances $\epsilon_{k}$ are those predicted by our GGchem model at $z=0$ (see Sect. 2). In order to set the fixed values for $L_{j}$ at the lower boundary, we need to make assumptions about the total number density of the particles and their size distribution $f(a)$ at $z=0$.

From the Pioneer Venus sounder probe data, R. G. Knollenberg & D. M. Hunten (1980) estimated a mean particle diameter of 0.25 $\mu$m, i.e. $\langle a\rangle=0.125\,\mu$m (mode-1 particles) at $z=40\,$km, although the instruments were only designed to detect particles with a diameter larger than $0.6\,\mu$m. They arrived at this conclusion by fitting their size-dependent data with a log-normal distribution as visualised by S. Seager et al. (2021), see their Fig. 2. Based on the discharge current measurements by Venera 13 and 14, R. D. Lorenz (2018) estimated a total charge density of about 1500 pC/m3 at $z=0$. From that value, they estimated an optical extinction opacity of 0.5/km at $z=0$, assuming the particles to be about singly charged. Based on the spectrophotometer data on Venera 13 and 14, B. Grieger et al. (2004) found opacities of order 1/km, which drop to about 0.1/km at a height of 40 km.

Based on these observations, we use a log-normal distribution $\rm[particles/cm^{3}/\mu m]$ defined as

[TABLE]

for our lower BC with $n_{p}=5000\rm\,cm^{-3}$, $\mu=\ln(0.15)$ and $\sigma=0.5$, using 50 size-bins between $0.005\,\mu$m and $20\,\mu$m to make sure that the particles outside of this size range can be neglected. Assuming a grain charge parameter of $\rm qa300=4$, this results in a charge density of 1540 pC/m3 and an opacity of 1.8/km at $z=0$.

Our opacity model shows that the aerosol opacity is mostly scattering opacity. The optical absorption opacity is only 0.089/km at $z=0$, i.e. the albedo is about 95%. This is essential for the radiative transfer in the Venus atmosphere. For example, our model for passive aerosol particles discussed in Sect. 4.1 has a vertical extinction optical depth of $\sim 25$, but the vertical absorption optical depth is only $\sim 0.49$, still allowing for some sun light to reach the Venus surface in a highly diffusive manner. To emphasise the uncertainties in the opacities, we note that the surface pictures from the Venera probes (C. P. Florensky, 1977) show a clear horizon, which suggests that the Venus atmosphere must be relatively transparent at ground level, barely consistent with such large opacity values, as also noticed by C. Sagan (1975).

4 Results

4.1 Passive particles

Before we discuss our results for chemically active particles in the lower Venus atmosphere, we first study the case of passive particles, where we only have settling, diffusion and coagulation. This problem has an analytical solution when we neglect coagulation, and this solution can be used to test the DiffuDrift implementation. Considering one single particle size, the particle density $n_{p}\!=\!\rho L_{0}\rm\,[cm^{-3}]$ follows from Eq. (2) with $j\!=\!0$

[TABLE]

Hence, in steady state $(dn_{p}/dt\!=\!0)$, there must be a constant particle flux in time and space through the atmosphere

[TABLE]

Assuming that there is no influx nor outflux at the upper boundary (neglecting the entry of micro-meteorites), the constant is zero and we find

[TABLE]

Equation (14) means that the particle-to-gas ratio is constant throughout the atmosphere when it is well-mixed ($D\gg{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}H_{p}$), where $H_{p}$ is the atmospheric scale height. Otherwise, however, the scale height of the particles is reduced, and $n_{p}/\rho$ drops quickly with height, by orders of magnitude.

Equation (14) can be integrated numerically, from the bottom of the atmosphere upwards, to find $n_{p}(z)/n_{p}(0)$. These solutions are shown for different particle sizes in Fig. 3. Particles smaller than about $a=0.2\,\mu$m are expected to be well-mixed up to heights of about 100 km in the Venus atmosphere, but particles that are just slightly larger, a\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}1\,\mum, remain much more concentrated towards the bottom of the atmosphere, barely reaching a height of 10 km. This is because of the ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\!\propto\!a^{2}$ scaling in the Stokes regime as discussed in App. A.3, which together with Eq. (14) states that the scale height reduction is $100\times$ larger for $10\times$ larger particles. Therefore, we expect a steep cutoff of passive particles in size space around $1\,\mu$m at altitudes \mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}10\,km.

Figure 4 shows the results of a simple DiffuDrift v2 test model for chemically inert passive particles (no growth and no coagulation) in the lower Venus atmosphere up to 50 km, where we assume the main sulphuric acid clouds to begin. The numerical DiffuDrift results are in good agreement with the semi-analytic solutions obtained from Eq. (14) shown with dashed lines. The measurement values of the charge density at $z=0$ from R. D. Lorenz (2018), and the mean particle radius $\langle a\rangle$ at $z=40\,$km from R. G. Knollenberg & D. M. Hunten (1980) are shown with blue bullets. In addition, the extinction opacity data from B. Grieger et al. (2004) and R. G. Knollenberg & D. M. Hunten (1980) are shown with black and grey lines. The average particle sizes and drift velocities shown in Fig. 4 are calculated as

[TABLE]

[TABLE]

Our simple model obtains a decent fit to all these data, including the opacity slope in the lower Venus atmosphere. The wiggles on $\kappa^{\rm ext}(z)$ are because of Mie-resonances that occur around size parameters x=2\pi\,a/\lambda\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}1, as the sizes of the two representative particles change quickly near the surface, which likely is an artefact of using only two particle sizes to calculate the opacities.

Figure 4 also shows that the size distribution narrows down for increasing height, because the larger (a\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}1\,\mum) particles only make it to a height of about 10 km, whereas the $a=0.1\,\mu$m particles populate the atmosphere up to heights well above 50 km with nearly constant concentration, as already discussed in Fig. (3). The exact values for these results depend on the diffusion constant assumed. If the diffusion constant was 10$\times$ larger than assumed in Eq. (10), particles with sizes up to $\sqrt{10}\approx 3\,\mu$m would populate heights up to 10 km.

Interestingly, the Grieger et al. data shows a re-increase of the extinction coefficient above about 30 km, whereas the Knollenberg & Hunten data show a roughly constant opacity at these heights. This could be an indication for an influx of particles from above, likely from the sulphuric acid clouds, maybe seeded by the impact of micro-meteorites at the top of the atmosphere P. Gao et al. (2014). If the particles are small and do not change size nor composition, we expect an opacity slope that reflects the density slope. If there is a constant flux of particles settling down, we expect a constant opacity, because in the Stokes regime ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}$ is independent of gas density (see Fig. 9 in the Appendix).

Figure 5 shows the same model with coagulation switched on, in comparison to the same analytic model without coagulation as shown in the previous figure, for various grain charge parameter values. In this model, the particles collide to form fewer and bigger particles that settle more quickly, leading to a steep gradient of $n_{p}(z)$ close to the bottom of the atmosphere if the particles are weakly charged. However, for grain charge parameter \mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}30, the coagulation becomes increasingly inefficient, because the electrostatic repulsion between the particles prevents their collisions. Eventually, for large grain charge parameters \mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}50, the results resemble the model without coagulation as shown in Fig. 4. For medium charges, the particles form fewer and bigger ($a>0.3\,\mu$m) particles, which is in conflict with both the measured opacities and mean sizes of the particles at 40 km height. Thus, our model suggests that the particles in the lower Venus atmosphere are strongly charged ($\rm qa300\!>\!50$), which is actually in agreement with our physical expectations, see App. C.

However, the total charge density of the particles becomes too high when we increase qa300 from 4 (where it fits the Lorenz et al. data) to 50. We conclude that it is currently not possible with our model to fit both the measured opacity and the charge density data.

4.2 Active particles

Figures 6 and 7 show our results for chemically active particles including coagulation. The models have been set up with the 7 materials and surface reactions listed in Table 1, one additional chemically unspecified “passive” material component, and 15 elements (H, C, N, O, S, F, Cl, Fe, Mg, Ca, Al, Na, K, Ti, Si), out of which 10 are involved in the condensation process. The GGchem-part of DiffuDrift v2 finds 308 molecules in its database for this choice of elements. Besides the particle quantities already introduced and plotted in our models for passive particles, we now show in addition the gas element abundances $\epsilon_{k}$ as function of height, the amount of condensed solid units per gas particle $n_{\rm cond}/n$ and the supersaturation ratios $S$ of the various materials.

For the model shown in Fig. 6 we have chosen the same lower BC with $n_{p}(z\!=\!0)=5000\rm\,cm^{-3}$ as in the previous models for passive particles, and grain charge parameter $\rm qa300=50$. The particle densities and sizes found in this model are very similar to the results of the same model for passive particles. Close inspection shows, however, that the particles have grown to very slightly bigger sizes due to the deposition of pyrite \ceFeS2[s], sodium sulphate \ceNa2SO4[s] and potassium sulphate \ceK2SO4[s] on their surfaces. These coatings cause the vertical absorption optical depth to increase very slightly, from 0.49 to 0.52. However, when plotting their concentration $n_{\rm cond}/n$ as function of height, the deposits do not show a saw-tooth-like profile as predicted by our simple GGchem equilibrium condensation model (Fig. 1), but extend all the way up to the upper boundary of the model with about constant concentrations. This is because once these depositions have formed, they do not come off again easily, as the upward transport by diffusion is faster than the sublimation.

However, these depositions only amount to a layer of an average $\rm thickness\approx 0.3$Å, i.e. not even a single mono-layer. This is a straightforward consequence of the availability of condensible elements (or order $10^{-12}$) compared to the passive particles as set by our lower BC, which translates to a chemical abundance of about $10^{-8}$. Therefore, the results of this model are physically not very meaningful. However, it is still reassuring to see that (a) the condensations take place at the heights predicted by our simple GGchem equilibrium condensation model (Sect. 2), (b) all supersaturation ratios are limited by about 1, and (c) the gas element abundances of the condensing metals are reduced in the upper regions of the model as expected from our GGchem model, i.e. the condensible molecules have sufficient time to find a surface to deposit.

In order to study the behaviour of chemically active particles in more detail, we have computed another model (shown in Fig. 7) where the abundance of the passive particles at the bottom of the atmosphere is reduced by 4 orders of magnitude, using $n_{p}(z\!=\!0)=0.5\rm\,cm^{-3}$ instead of $5000\rm\,cm^{-3}$, such that the aforementioned abundances of active and passive materials become about equal. In this model, coagulation is much less important, because it scales with $n_{p}^{2}$. To bring all four principle timescales (settling, diffusion, growth and coagulation) to about the same order of magnitude, we have chosen to reduce the grain charge parameter to $\rm qa300=5$.

This model (Fig. 7) shows a rather different dynamical and chemical behaviour of the active particles. The depositions of \ceFeS2[s], \ceNa2SO4[s] and \ceK2SO4[s] cover all particles with a layer of $\rm thickness\approx 0.2\,\mu$m, i.e. about 1000 mono-layers. These larger particles settle efficiently, which reduces $n_{p}(z)$ considerably in the upper layers. The material mixture in the upper layers is characterised by about equal amounts of these three condensates, followed by the passive component and by \ceAl2O3[s], with some traces of \ceTiO2[s], \ceMgF2[s] and \ceCaSO4[s]. This behaviour is due to the incompleteness of the condensation process in this model: (i) The thermal velocities of the key molecules \ceFeCl2, \ceNaCl and \ceKCl are about equal, and therefore, about equal amounts of \ceFeS2[s], \ceNa2SO4[s] and \ceK2SO4[s] are found to condense, (ii) the abundances of the condensible elements in the gas phase does not fully follow the pattern of the GGchem equilibrium condensation model, and (iii) the gas remains highly supersaturated in the upper layers.

Close inspection of $n_{\rm cond}/n$ shows that the downward transport of the particles by diffusion and settling is comparable to the speed of sublimation, and hence the depositions on these particles, while shrinking, still exist below their nominal cloud bases, where they are undersaturated according to our simple GGchem equilibrium condensation model. We note that the particles are always mixed by diffusion in this model, so there are no sharp edges as in the equilibrium condensation GGchem model.

4.3 Timescale analysis

Figure 8 shows the relevance of the four basic processes diffusion, coagulation, settling and net growth by means of a timescale analysis. We have calculated inverse timescales as $(\Delta L_{j}/\Delta t)/L_{j}\rm\,[yr^{-1}]$ from the results of the respective operators in DiffuDrift at the final timestep $\Delta t$. At any height in the atmosphere, the sum of these inverse timescales must be zero in steady state.

In most cases, there are two leading processes that cancel each other, for example in the most basic model for passive particles without coagulation at the top of Fig. 8, diffusion replenishes the particles within about two weeks, and settling removes them at the same rate. In the model for passive particles with coagulation (second model from the top), we see that coagulation becomes increasingly more relevant in removing the particles at higher altitudes where the particles are less charged, and the gain by diffusion is balanced by a combination of settling and coagulation, in consideration of the number of the particles ($\to L_{0}$). However, since coagulation has no effect on the total condensed volume of the particles ($\to L_{3}$), the diffusion is still balanced by settling when considering $j\!=\!3$.

The timescale analysis of the third model for active particles is practically identical to the second model for passive particles, because the amount of the additional depositions by the active condensates is negligible compared to the passive material component. However, the bottom model for chemically active particles with reduced passive component reveals new insights. Growth & evaporation have no effect on the number of particles $(L_{0})$. However, for all $j\!>\!0$, the net growth becomes relevant, turning negative where one material on the particles sublimates. We mark the negative green peaks of the sublimation rates of \ceK2SO4, \ceNa2SO4 and \ceFeS2 with additional height lines at 13.5 km, 8.9 km and 1.9 km, respectively.

Between these negative sublimation peaks, the inverse growth timescale is positive as gas molecules continue to deposit on the particles’ surfaces. The wiggles in the growth timescale are counterbalanced by corresponding anti-wiggles in the diffusion timescale, i.e. the liberated molecules are transported away by diffusion, whereas the settling and coagulation timescales remain smooth.

5 Discussion

5.1 Element trapping in clouds

This paper has made some predictions about the material composition of aerosol particles in the lower Venus atmosphere based on two model approaches of increasing complexity, an equilibrium condensation model (Sect. 2) and a dynamical particle model with kinetic condensation (Sect. 3). Both models agree, that elements can be trapped in cloud layers, such that the abundance of an element above a cloud layer is significantly reduced with respect to its abundance below the clouds, provided that this element is consumed by the formation of the cloud particles. In this section, we want to shed some light on the microphysics behind this important process in planetary atmospheres.

We multiply Eq. (5) by $\nu_{s,k}/V_{0}^{s}$ and sum up these equations for all materials $s$. We then add Eq. (6). This way, the chemical source terms (nucleation and net growth) cancel out, and we find in the stationary limit

[TABLE]

Identifying $\,\epsilon_{k}^{\rm cond}=\sum_{s}\frac{\nu_{s,k}}{V_{0}^{s}}\,L_{3}^{s}=\frac{1}{\rho}\sum_{s}\frac{\nu_{s,k}}{V_{0}^{s}}\sum_{i=1}^{2}n_{i}V_{i}^{s}\,$ as the abundance of element $k$ in condensed form, i.e. the number of nuclei of element $k$ in the condensed particles per gram of gas. Using Eq. (16), we find the following remarkable equation

[TABLE]

where $\langle{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}^{\!\!3}\rangle$ is the volume-mean settling velocity, see Eq. (16). The constant in Eq. (18) is the time and height-independent flux of element $k$ through the atmosphere $\rm[nuclei/s/cm^{2}]$. Assuming this constant flux to be zero (e.g. no micro-meteoritic influx) we find the following three limiting cases

[TABLE]

The first case (Eq. 19) corresponds to the behaviour of passive particles, as already derived in Sect. 4.1 (Eq. 14), stating that the concentration of particulate matter can only decrease with height, which it does in particular when the mixing is insufficient. The second case (Eq. 20) is discussed in detail by P. Woitke et al. (2020). It states that gas element abundances decrease with height in a cloud layer, see r.h.s. of Fig. 1 and lower left plots in Figs.  6 and 7. When elements rain out, a negative $\epsilon_{k}^{\rm gas}(z)$-gradient must build up in the gas phase to cause an upward-directed diffusive element flux that counteracts the settling flux. Equation (20) states explicitly that gaseous element abundances can never increase with height in static planetary atmospheres. The third case (Eq. 21) shows, however, that this is actually possible in very well-mixed cases, for example high in the atmosphere, when a $T$-inversion causes clouds to turn back into gases. For this case to occur, we need $V_{g}\!\gg\!\langle{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}^{\!\!3}\rangle$, where $V_{g}\!=\!D/H_{p}$ is the vertical gas mixing velocity, see Eq. (C16).

The third case does not occur in this model, but the atmospheric pattern evolves from case 2 (Eq. 20) directly into case 1 (Eq. 19), as soon as the reservoir of metals like Fe, Na and K is exhausted with increasing height. Since the particles stay very small ($a\!<\!0.2\,\mu$m above 20 km), they remain coupled to the gas and the material composition of the particles does not change much upward of 20 km. This behaviour is only revealed by the kinetic condensation models, whereas the material composition of the particles keeps changing with height in the equilibrium condensation models.

6 Summary and discussion

This paper has made a number of predictions concerning the gas phase chemistry and the properties and material composition of sub-$\mu$m-sized aerosol particles in the lower part of the Venus atmosphere, below the main sulphuric acid cloud layer at a height of about 45 km.

The GGchem model discussed in Sect. 2 uses the same setup as published in P. B. Rimmer et al. (2021) for the interface between the surface and the gas at the bottom of the atmosphere. Assuming chemical equilibrium and phase equilibrium, a number of metal-chloride and metal-fluoride molecules are found to be present in the gas phase over the Venus surface, in particular \ceFeCl2, \ceNaCl, \ceKCl, \ceSiF4, \ceAlF2O, \ceTiF4, \ceMgCl2, and \ceCaCl2, with very low concentrations of order $10^{-12}$ to $10^{-21}$. The formation of these molecules is a consequence of chlorine and fluorine being available in the Venus atmosphere in form of HCl and HF molecules.

The trace concentrations of these metal molecules are sufficient to stabilise a number of solid materials, which can deposit on aerosol particles in the lower Venus atmosphere. In particular we predict potassium sulphate \ceK2SO4[s] to form above a height of about 15.5 km, and sodium sulphate \ceNa2SO4[s] above 9.5 km. We call these condensations sulphate hazes. In addition, pyrite \ceFeS2[s] can deposit above 2.4 km as already noted by X. Byrne et al. (2024). These results are in close agreement with the predictions from our GGchem model (13.8 km, 9.3 km, and 2.9 km, respectively), and coincide well with the three potential dust layers found in the Pioneer Venus Large Probe neutral mass spectrometer data by R. Mogul et al. (2023). \ceFe2O3[s] (hematite), \ceCaSO4[s] (anhydrite), \ceAl2O3[s] (corundum), \ceTiO2[s] (rutile), and \ceMgF2[s] (magnesium fluoride) are found to be stable in the gas as well, all the way down to the surface. We note that the chemical pre-conditions for condensations in the lower Venus atmosphere are distinctively different from the conditions in the Venus surface, because certain elements like Si are entirely confined in the surface and no longer available to form minerals in the atmosphere.

To model the behaviour of the aerosol particles, we have improved the DiffuDrift code originally developed by P. Woitke et al. (2020) for brown dwarf and exoplanet atmospheres. We have re-formulated the basic equations to allow for a proper treatment of settling and growth at arbitrary Knudsen numbers, including the limiting cases in the Epstein and Stokes regimes. We have included coagulation driven by Brownian motion, difference in settling velocities and turbulence, taking into account the repelling effect of particle charges. We have furthermore added an unspecified, chemical inert passive material that forms the core of the aerosol particles, and have improved the numerics.

We provide analytic solutions for the most simple case of passive particles with a given concentration at the bottom of the atmosphere, only affected by diffusive mixing and settling (Eq. 14). Particles with radius $a>1\,\mu$m cannot get to heights $>10\,$km, but sub-micron particles (a\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr<\cr\sim\cr}}}}0.3\,\mum) stay well-mixed with the gas and can reach the main sulphuric acid clouds from below. The exact values depend on the eddy diffusion coefficient assumed. For example, if the diffusion coefficient was $10\times$ larger than assumed, then $\sqrt{10}\times$ larger particles would reach the same height.

According to our models for chemically active particles, we expect the aerosol particles in the lower Venus atmosphere to be covered by a thin layer of \ceK2SO4[s], \ceNa2SO4[s] and \ceFeS2[s]. At a height of 45 km, we reach particle radii of about 0.15 to 0.25 $\mu$m and particle densities of about 10 to $\rm 100\,cm^{-3}$, depending on the assumptions about the near-surface aerosol particle concentration and size distribution, and the efficiency of coagulation connected to the particle charges. We conclude that the particles must be strongly charged negatively, or order 100 negative charges per micron of particle radius, otherwise our model would produce steep gradients above the surface, which is inconsistent with the B. Grieger et al. (2004) opacity data and the R. D. Lorenz (2018) discharge current data.

In a model like ours, where particles are dredged up from the ground, and there is an equilibrium between upward mixing and gravitational settling, the particle concentration must decrease with height at least as fast as the gas density, which is in good agreement with the slope of the B. Grieger et al. (2004) opacity data below 35 km. However, above that height, the data shows an increasing trend, which suggests that particles are inserted from above and travel down the atmosphere, meeting the dredged-up particles at around 35 km height. Since the fall velocity is independent of gas density in the Stokes regime, a constant influx of particles translates into a constant particle density, i.e. an increasing particle concentration with increasing height. Such a particle influx could be provided by the impact of micro-meteorites as discussed in P. Gao et al. (2014). Another possibility is a horizontal transport of cloud particles, which then shrink to their refractory core at the sulphuric acid cloud base and continue to settle down slowly as aerosol particles. These effects will be investigated in a forthcoming paper.

We thank the anomynous referee for his detailed list of additional references and suggestions to improve the paper, in particular with regard to questions about the surface mineral composition and the chemistry in the near-surface atmosphere of Venus. PW, MS, HL and FW thank the Austrian Science Fund (FWF) for the support of the VeReDo research project, grant I6857-N. MF, KN, PE and JK thank the Czech Science Foundation (GACR) for the support of the VeReDo research project, grant I6857-N. TC thanks the Science and Technology Facilities Council (STFC) for the PhD studentship (grant reference ST/X508299/1).

Appendix A Gravitational settling

Using the double-$\delta$ representation of $f(V)$ (Eq. 3), we can express the divergence of the vertical flux of cloud particles settling gravitationally in the atmosphere as

[TABLE]

where ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}_{,i}$ is the downward equilibrium drift velocity (or final fall speed) of the representative particle $i$, which generally depends on particle radius $a$ and gas density $\rho$ in complicated ways, see Sect. 2.3 in P. Woitke & C. Helling (2003). The main results of this paper are repeated here in the following.

A.1 Large Knudsen numbers (Epstein regime)

In the Epstein regime, the equilibrium drift velocity is given by S. A. Schaaf (1963) as

[TABLE]

where $g$ is the gravitational acceleration, $\rho_{\rm m}$ the material density of the particles, v_{\rm th}\!=\!\big{(}\frac{8\,kT}{\pi\,\mu}\big{)}^{1/2} the thermal velocity and $\mu$ the mean molecular weight. In this case, Eq. (A1) becomes

[TABLE]

where \xi_{\rm lKn}\!=\!g\,\big{(}\frac{3}{4\pi}\big{)}^{1/3}\frac{\sqrt{\pi}}{2} and c_{T}\!=\!\big{(}\frac{2\,kT}{\mu}\big{)}^{1/2}. This is exactly the result for settling obtained by P. Woitke et al. (2020), see their Eq. (25), which means that using the double-$\delta$ representation (Eq. 3) leads to the same result as working out the integral over size space.

A.2 Small Knudsen numbers (Stokes regime)

For small Knudsen numbers, the drift motion of the particles becomes viscous, see Eqs. (14), (15) and (16) in P. Woitke & C. Helling (2003). For small Reynolds numbers Re=2a\,\rho\,{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}/\nu_{\rm dyn}\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr<\cr\sim\cr}}}}1 the flow of the gas around the particle is laminar and the formula for Stokes friction applies

[TABLE]

where $\nu_{\rm dyn}$ is the dynamic gas viscosity $\rm[g/cm/s]$, which we calculate for a (97 % \ceCO2, 3 % \ceN2)-mixture according to Eq. (8) in P. Woitke & C. Helling (2003) with collisional molecular radii from Table 1 in P. Woitke et al. (2022), resulting in

[TABLE]

In the Stokes regime we find

[TABLE]

where \xi_{\rm sKn}=g\,\frac{2}{9}\Big{(}\frac{3}{4\pi}\Big{)}^{2/3}. This is exactly the result for settling in P. Woitke & C. Helling (2003), see their Eqs. (74) and (76).

A.3 General Knudsen numbers

In the general case, which is now implemented in DiffuDrift v2, we use Eqs. (12), (14), (15) and (18) in P. Woitke & C. Helling (2003) to calculate the general equilibrium drift velocities ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}_{,i}$ of both representative particles, and then use Eq. (A1) to calculate the effect of the settling on the cloud particle moments. This method only becomes approximate when the two representative particles are not in the same hydrodynamical domain, otherwise this method is exact. Figure 9 shows ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}(a,z)$ in the Venus atmosphere, where we have taken $\rho(z)$ and $T(z)$ from our underlying atmospheric structure assumed for Venus, and assume a gas consisting of 97% \ceCO2 and 3% \ceN2, see Fig. 1. These results are in excellent agreement with Fig. 3 in S. Seager et al. (2021). The figure shows the proportionality ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\!\propto\!a/\rho$ in the Epstein and ${v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}\!\propto\!a^{2}$ in the Stokes regime. In the Stokes regime, the equilibrium drift velocity is height-independent, and $10\times$ larger particles fall $100\times$ faster. The transition between the Stokes and the Epstein regime takes place at about $Kn=2a/\ell\approx 1/3$, where $\ell$ is the mean free path of the molecules in the atmosphere. For Reynolds numbers between 500 and 1500, the viscous flow of the gas around the particles changes from laminar to turbulent, which temporarily reduces the frictional force and the fall speed. This effect plays a role for particles \mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr>\cr\sim\cr}}}}\!100\,\mum, similar to rain droplets in the Earth atmosphere.

Appendix B Growth and evaporation

During a small time step $\Delta t$, the volume of a particle $i$ changes by a small increment $\Delta V_{i}\!\ll\!V_{i}$ via deposition and sublimation of molecules on/from the surface of the particle, which we call growth and evaporation. Using the double-$\delta$ representation of the cloud particle size distribution function $f(V)$ (Eq. 3), the corresponding change of the cloud particle moments is

[TABLE]

In order to use Eq. (B1), we need to know the growth rates $\frac{\partial V_{i}}{\partial t}$ of both representative particles $i\!=\!1$ and $i\!=\!2$. In P. Woitke & C. Helling (2003) we have derived expressions for $\partial V/\partial t$ in the Epstein and the Stokes regimes.

B.1 Large Knudsen numbers (Epstein regime):

For large Knudsen numbers, the impinging molecules reach the particle from the distant gas in a single flight according to a Maxwellian velocity distribution.

[TABLE]

with the saturation factor

[TABLE]

which is positive when the condensate is supersaturated (causing deposition), but becomes negative when the condensate is undersaturated (causing sublimation). $r$ is an index for the included surface reactions. $n_{r}^{\rm key}$ is the density of the growth-limiting key reactant of the surface reaction, v_{r}^{\rm rel}\!=\!\big{(}\frac{kT}{2\pi\,m_{r}}\big{)}^{1/2} its thermal relative velocity, $m_{r}$ is the mass of the key species, and $\alpha_{r}$ is a sticking probability. $V_{r}$ is the condensed volume created per sticking key molecule. $S_{\!r}$ is the reaction supersaturation ratio, which in the most simple case equals the supersaturation ratio $S$ of the material affected by reaction $r$. $b_{\rm mix}^{s}\!=\!L_{3}^{s}/L_{3}$ is the volume mixing ratio of material $s$ in the particle, which is only relevant when $S_{\!r}\!<\!1$, assuming that the sublimation only occurs from patches on the surface made of material $s$. Equation (B2) is valid in the Epstein regime, for which the moment method was originally developed by H. P. Gail & E. Sedlmayr (1988) and later adopted for brown dwarfs and exoplanet atmospheres by C. Helling & P. Woitke (2006); C. Helling et al. (2008). In this case, Eq. (B1) results in

[TABLE]

where $\chi^{\rm net}_{\rm lKn}\!=\!\sqrt[3]{36\pi}\sum_{r}n_{r}v^{\rm th}_{r}\alpha_{r}V_{r}\,\gamma_{r}$, which is exactly the result for the growth as formulated by Eqs. (11) and (12) in P. Woitke et al. (2020).

B.2 Small Knudsen numbers:

In the case of small Knudsen numbers, the molecules approaching a particle undergo multiple collisions before they reach the surface, and the transport of the molecules to the particle’s surface becomes diffusion-limited, see Eq. (32) in P. Woitke & C. Helling (2003).

[TABLE]

Here, $D_{r}$ is the bi-molecular diffusion constant $\rm[cm^{2}/s]$ of the key molecule of mass $m_{r}$ in the ambient gas with mean molecular weight $\mu$, for which use

[TABLE]

from P. M. Banks & G. Kockarts (1973), where $n\!=\!p/kT$ is the total gas particle density. We note that since $D_{r}\!\propto\!1/n$, the growth rate is independent of the gas density for small Knudsen numbers and hence roughly independent of height for small particles in the lower Venus atmosphere. The result for Eq. (B1) in this case is

[TABLE]

where $\chi^{\rm net}_{\rm sKn}\!=\!\sqrt[3]{48\pi^{2}}\sum_{r}n_{r}^{\rm key}D_{r}V_{r}\,\gamma_{r}$, which is exactly the result given by Eq. (75) in P. Woitke & C. Helling (2003) when generalised to a mixture of condensates.

B.2.1 Rapidly falling particles

For very large particles, the equilibrium drift may become so fast that the particles sweep up more molecules as they fall than they can gain by diffusion. We account for this effect by an additional term in the volume growth rate for small Knudsen numbers

[TABLE]

B.3 General Knudsen numbers

For general Knudsen numbers, as now implemented in DiffuDrift v2, we use Eqs. (B2) and (B8) to calculate the growth rates $\partial V_{i}/\partial t$ for the two representative particles in the large and in the small Knudsen number limiting cases, and then use Eq. (36) in P. Woitke & C. Helling (2003) to merge them. Next, we use Eq. (B1) to calculate the effect of these general growth rates on the cloud particle moments. As for the settling, this procedure only becomes approximate when both representative particles are not in the same hydrodynamical domain.

Figure 9 shows the growth timescale $\tau_{\rm gr}=V\,/\,(\partial V/\partial t)$ for sulphuric acid droplets in the Venus atmosphere when the sulphuric acid is highly supersaturated and the growth is limited by the concentration of water vapour (assuming a concentration of 30 ppm). We obtain a pattern as function of particle radius $a$ and height $z$ in the atmosphere that is very similar to the drift velocity.

The blue dashed line shows where the growth timescale equals the sinking timescale $\tau_{\rm sink}\!=\!H_{p}/{v^{\hskip-2.41112pt{}^{\circ}}_{\rm dr}}$. To the left of this line, the particles have plenty of time to grow until saturation is reached in the gas phase, which slows down further growth, before they fall down. Particles to the right of this line should not exist, because they fall quicker than they can grow, however coagulation might help them to cross this line temporarily.

Appendix C Coagulation and Charges

Our implementation of coagulation uses a discretised form of the Smoluchowski equation (M. V. Smoluchowski, 1916). When particles with volumes $V_{1}$ and $V_{2}$ collide and stick together, there are three additional sizes of particles after a small time interval, namely $V_{3}\!=\!V_{1}+V_{1}$, $V_{4}\!=\!V_{1}+V_{2}$, and $V_{5}\!=\!V_{2}+V_{2}$. Let

[TABLE]

denote the numbers of collisions that take place between particles of sizes $i$ and $j$ during $\Delta t$ per cm3, where $\alpha_{ij}$ is the sticking probability, $\Delta v_{ij}$ the mean relative velocity between the two particles, $a_{i}$ and $a_{j}$ their radii (a\!=\big{(}3V/(4\pi)\big{)}^{1/3}), $f^{\rm C}_{i,j}$ is the Coulomb factor for electrostatic repulsion, and $n_{i}$ and $n_{j}$ are the particles’ number densities. Then

[TABLE]

are the particle number densities after $\Delta t$. This scheme can easily be extended to $i\!=\!1,...\,,N$ particle sizes, in which case $i\!=\!1,...\,,N^{\prime}$ sizes of particles occur after a small $\Delta t$, where $N^{\prime}\!=\!N+N(N+1)/2$. The change of the cloud particle moments by coagulation is then calculated as

[TABLE]

Concerning the relative velocity $\Delta v_{ij}$ we take into account three processes: Brownian motion, difference in settling velocities, and turbulence

[TABLE]

where $m_{\rm red}\!=\!m_{i}\,m_{j}/(m_{i}+m_{j})$ is the reduced mass of the two colliding particles. If the colliding particles carry charges of equal sign, $q_{i}$ and $q_{j}$, the Coulomb factor for electrostatic repulsion is

[TABLE]

where $E_{\rm C}=q_{i}\,q_{j}\,{\rm e}^{2}/(a_{i}+a_{j})$ is the electrostatic repulsion energy at contact distance, e is the CGS unit of charge, and $E_{\rm kin}=\frac{1}{2}\,m_{\rm red}\,\Delta v_{ij}^{2}$ is the collisional kinetic energy. Equation (C12) can be derived from elemental assumptions about the collisions between charged particles when they have a Maxwellian velocity distribution (Brownian motion); it states that coagulation becomes very inefficient when $E_{\rm C}$ exceeds a few $E_{\rm kin}$.

T. Balduin et al. (2023) developed a detailed astrochemical model for dust charging in protoplanetary discs, taking into account cosmic ray ionisation, electron attachment, charge exchange with molecular ions, and photoeffect. They found that grains of different sizes in the disc midplane, which is entirely shielded from UV-photons, all obtain about the same negative charge relative to size $-q/a$, of the order of a few tens to a few hundreds of negative charges per micrometer of grain radius $a$. Since the grains in the model are mainly charged by collisions with electrons and molecular cations, this result is easy to understand. The electron attachments are faster than the charge exchange reactions with molecular cations, so the particles start to charge negatively, until their electric potential $E_{\rm C}={\rm e}^{2}(q/a)$ becomes larger than a few $kT$, and the electrons cannot reach the grain surfaces anymore. This is where the charging process stops, leading to an estimation of the particle charge as

[TABLE]

which is independent of the gas density and the cosmic ray ionisation rate. The grain charging causes the density of the free electrons to decrease by orders of magnitude. The density of negative charges on the grains is eventually balanced by the density of molecular cations in the gas, such as \ceNH4+, which have so large proton affinities that they cannot recombine when they collide with a negatively charged grain. The cosmic ionisation rate of the gas is proportional to both the electron density and the molecular ion density, but it is the ratio of the two that sets the dust charge. We have applied this model to the problem of aerosol particle charges at the bottom of the Venus atmosphere and find values compatible with Eq. (C13).

If tribo-electric charging is important, as expected in highly turbulent environments, particle charges have been observed to be even larger. Laboratory simulations of the electrification of wind-driven Martian dust particles (J. P. Merrison et al., 2012) suggest a charge of $10^{4}$ to $10^{6}$ elementary charges per grain. Similar charges per grain ($10^{5}$ to $10^{6}$) have been measured in experiments on volcano ash particles in Icelandic volcanic plumes (K. L. Aplin et al., 2014; J. Méndez Harper et al., 2021), including fragmentation charging. J. Méndez Harper et al. (2024) have found similar levels of tribo-electrification when grinding coffee grains.

Our DiffuDrift models show that if Eq. (C13) was true, the aerosol particles in the lower Venus atmosphere could not collide much, and hence the models with coagulation very much resemble the models without coagulation. However, since the charge of the aerosol particles in the lower Venus atmosphere is not known exactly, we treat $q/a$ at $T=300\,$K as a free parameter, see Eq. (8).

The turbulent relative velocity $v_{ij}^{\rm\,turb}$ is calculated according to the vertical gas mixing velocity $V_{g}$ in the three cases for small, medium and large particles according to Eqs. (26), (28) and (29) in C. W. Ormel & J. N. Cuzzi (2007), depending on the relations between the stopping times $\tau_{i}^{\rm stop}$, $\tau_{j}^{\rm stop}$ of the colliding particles and the smallest eddy turnover timescale $\tau_{\eta}$. The closed expressions in C. W. Ormel & J. N. Cuzzi (2007) are then formulated in terms of the gas Reynolds number $Re$ and the particles’ Stokes numbers ${S\hskip-0.43057ptt}_{i}$ and ${S\hskip-0.43057ptt}_{j}$.

[TABLE]

[TABLE]

We note that the turbulent relative velocity is limited by v_{ij}^{\rm\,turb}\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil\displaystyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\textstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptstyle#\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil\scriptscriptstyle#\hfil\cr<\cr\sim\cr}}}}V_{g}, and the mean vertical gas mixing velocity $V_{g}$ is assumed to be related to the local eddy diffusion coefficient $D$ via the general relation $D=V_{g}\,H_{p}$ (Eq. C16). Since $D$ is assumed to be quite small in the Venus atmosphere (Eq. 10), and hence $V_{g}$ is less than about 0.1 cm/s, turbulence is found to be rather unimportant. Instead, the relative collisional velocities $\Delta v_{ij}$ are mostly driven by the Brownian motion, unless we have millimetre-sized particles, in which case the difference in settling velocities becomes more important.

The cloud particles are expected to remain quite small, less than about 1 mm (see $\tau_{gr}\!=\!\tau_{\rm sink}$ line in Fig. 9). Since the collisional difference velocities $\Delta v_{i,j}$ are found to be $<10\,$cm/s throughout the lower Venus atmosphere in our models, the collisions are expected to be far from the bouncing and fragmentation barriers. We therefore assume perfect sticking $\alpha_{i,j}\!=\!1$.