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Time evolution equations for the condensation and evaporation of SOA¶

Mass transfer coefficient¶

As mentioned earlier, the SOAG in MAM4 refers to a single lumped semi-volatile organic gas-phase species. It is assumed that

SOAG (or SOA) has a fixed molecular weight of \(M_{w,SOAG} = 150~\rm kg~kmol^{-1}\);
The function \(f(K_{n,L}, \alpha_{L})\) in the expression of the mass transfer coefficient \(C_{L,i}\) is the same for SOA and sulfuric acid.

It then follows that the mass transfer coefficients of SOA are proportional to those of the sulfuric acid, and their relationship is

\[ \begin{equation} C_{SOAG,i} = \frac{\mathbb{D}_{g,SOAG}} {\mathbb{D}_{g,H_2SO_4}} C_{H_2SO_4,i} \end{equation} \]

It is further assumed that SOAG's molecular diffusion volume is proportional to that of the sulfuric acid gas, i.e.,

\[ \begin{equation} \mathbb{V}_{D,SOAG} = \frac{M_{w,SOAG}}{M_{w,H_2SO_4}} \mathbb{V}_{D,H_2SO_4} \end{equation} \]

Using this assumption as well as typical values of temperature and air pressure, we further simplify \(\mathbb{D}_{g,SOAG}/ \mathbb{D}_{g,H_2SO_4}\) into a constant of 0.81, i.e.,

\[ \begin{equation} C_{SOAG,i} = 0.81 C_{H_2SO_4,i} \end{equation} \]

Equilibrium mixing ratio¶

For the condensable organic vapor SOAG, the equilibrium vapor concentration must be considered. MAM4 makes the simplifying assumption that the equilibrium mixing ratio is independent of the composition of SOAG and the surface curvature of the particles (i.e., there is no curvature effect). The ambient equilibrium molar mixing ratio of the SOAG, \(q_{v,SOAG,equ}\), is assumed to be 1.0E-10 mol/mol at 298 K and 1 atm pressure, so it follows that

\[ \begin{equation} q_{v,SOAG,equ} = \frac{101325}{p} \exp \left[ \frac{-\Delta H_{vap}}{R} \left( \frac{1}{T} - \frac{1}{298} \right) \right] \end{equation} \]

where the unit of \(p\) is Pa, \(\Delta H_{vap} = 156~\rm kJ~mol^{-1}\) is the approximated mean enthalpy of vaporization of SOAG, and \(R\) is the universal gas constant.

The solute effect is considered. Following Raoult’s law, the ambient equilibrium molar mixing ratio of SOAG over the surface of aerosol particles in mode \(i\), denoted by \(q_{v,SOAG,sat,i}\), is reduced by the presence of other organic molecules in the particles that form a liquid solution with the SOA molecules. MAM assumes that 10% of the primary organic aerosol (POA) is oxygenated (OPOA) and participates in the solution, while the remainder does not. Raoult’s law then gives:

\[ \begin{equation} \label{soagsfc} q_{v,SOAG,sfc-equ,i} = \frac{q_{m,SOA,i}} {q_{m,SOA,i}+q_{m,OPOA,i}} q_{v,SOAG,equ} = \frac{q_{m,SOA}} {q_{m,SOA,i}+0.1q_{m,POA,i}} q_{v,SOAG,equ} . \end{equation} \]

Note that here we use an assumed molecular weight of 150 kg~kmol\(^{-1}\) for POA.

Time evolution equations¶

Considering only the condensation and evaporation of SOA, the time evolution equations of relavant prognostic variables in MAM read

\[ \begin{align} % \frac{dq_{m,SOA,i}}{dt} & = 0.81C_{H_2SO_4,i} \left(q_{v,SOAG} - q_{SOAG,sfc-equ,i} \right) ,\quad i = 1,...,I \label{eqn:ddt_soa_mass} \\ % \frac{dq_{v,SOAG}}{dt} & = - \sum_{i=1}^{I} \left[0.81C_{H_2SO_4,i} \left( q_{v,SOAG} - q_{v,SOAG,sfc-equ,i} \right)\right] \,, \label{eqn:ddt_soag_mass} \\ \frac{dq_{m,SP,i}}{dt} &= 0, \quad \text{where}\,\, SP \neq SOA, \,\, i = 1,...,I, \label{eqn:ddt_mass_not_SOA}\\ % \frac{dq_{n,i}}{dt} & = 0, \quad i = 1,...,I \label{eqn:ddt_number_2} \end{align} \]