Equilibria Theory¶

Gas-particle partitioning determines how semi-volatile organic compounds distribute between the gas phase and particulate matter. This equilibrium process is central to understanding aerosol formation, growth, and composition. This page covers the theoretical basis for equilibria calculations in particula.

Introduction¶

When organic vapors encounter aerosol particles, they can condense onto the particle phase. The extent of this partitioning depends on:

Saturation vapor pressure: How volatile the compound is
Activity in the particle phase: Non-ideal mixing effects
Available condensed mass: More organic mass means more partitioning

Understanding these equilibria is essential for predicting secondary organic aerosol (SOA) formation and properties.

Liquid-Vapor Partitioning Fundamentals¶

At thermodynamic equilibrium, the partial pressure of each species in the gas phase equals its equilibrium vapor pressure above the liquid (particle) phase:

\[p_i = a_i \cdot p_i^{\circ}\]

where:

\(p_i\) is the partial pressure of species \(i\) in the gas phase
\(a_i\) is the activity of species \(i\) in the condensed phase
\(p_i^{\circ}\) is the pure component saturation vapor pressure

This relationship links gas-phase concentrations to particle-phase composition through the activity, which accounts for non-ideal mixing.

Saturation Concentration (C*)¶

The saturation concentration (\(C^*\)) is a convenient measure of volatility that directly relates to partitioning behavior. It represents the equilibrium vapor concentration when the condensed phase is present.

Dry vs. Wet C*¶

The dry saturation concentration (\(C^*_{dry}\)) assumes an ideal, single-component system:

\[C^*_{dry} = \frac{p^{\circ} \cdot M}{R \cdot T}\]

where \(M\) is the molecular weight, \(R\) is the gas constant, and \(T\) is temperature.

In a real mixture with water and multiple organics, the effective C* depends on activity coefficients and phase composition:

\[C^*_j = C^*_{j,\mathrm{dry}} \cdot \gamma_{j,\mathrm{phase}} \cdot q_{\mathrm{phase}} \cdot \frac{C_{\mathrm{liq}}}{M_j \cdot \overline{M}_{\mathrm{phase}}}\]

where:

\(\gamma_{j,\mathrm{phase}}\) is the activity coefficient in the target phase
\(q_{\mathrm{phase}}\) is the fraction of mass in that phase
\(C_{\mathrm{liq}}\) is the total liquid concentration
\(\overline{M}_{\mathrm{phase}}\) is the mass-weighted mean molar mass

Volatility Basis Set¶

The Volatility Basis Set (VBS) framework bins organic compounds by their \(C^*\) values, typically using logarithmic bins (e.g., 0.01, 0.1, 1, 10, 100 \(\mu\)g/m\(^3\)). This simplifies the treatment of complex organic mixtures by grouping compounds with similar volatility.

\(C^*\) Range (\(\mu\)g/m\(^3\))	Classification
< 0.01	Extremely Low Volatility (ELVOC)
0.01 - 1	Low Volatility (LVOC)
1 - 100	Semi-Volatile (SVOC)
100 - 10000	Intermediate Volatility (IVOC)
> 10000	Volatile (VOC)

Partition Coefficient¶

The partition coefficient (\(\varepsilon_j\)) describes the fraction of species \(j\) in the condensed phase at equilibrium:

\[\varepsilon_j = \frac{1}{1 + C^*_j / C_{\mathrm{liq}}}\]

Key behaviors:

When \(C^* \ll C_{liq}\): \(\varepsilon \approx 1\) (mostly condensed)
When \(C^* \gg C_{liq}\): \(\varepsilon \approx 0\) (mostly gas phase)
When \(C^* = C_{liq}\): \(\varepsilon = 0.5\) (equally distributed)

This relationship shows that partitioning depends on both the compound's volatility and the total available condensed mass.

Phase Separation (LLPS)¶

Under certain conditions, organic-water mixtures can undergo Liquid-Liquid Phase Separation (LLPS), forming distinct alpha (\(\alpha\)) and beta (\(\beta\)) phases:

Alpha phase: Typically organic-rich, with lower water content
Beta phase: Typically water-rich, with higher water content

When Does Phase Separation Occur?¶

Phase separation occurs when the Gibbs free energy of a two-phase system is lower than a single homogeneous phase. This typically happens at:

Intermediate relative humidity (40-80%)
Higher O:C ratios (more oxygenated organics)
Specific organic-water composition ranges

Phase Fraction (q)¶

The phase fraction \(q_{\alpha}\) describes the distribution between phases:

\[q_{\alpha} + q_{\beta} = 1\]

Particula uses a smooth sigmoid function to model the transition between single-phase and two-phase regimes, preventing numerical discontinuities during equilibrium solving.

Equilibrium Solving Process¶

Particula solves for equilibrium using an iterative optimization approach:

flowchart TD
    A[Input: Species Properties, Water Activity] --> B[Get Activity Coefficients via BAT]
    B --> C[Compute q_alpha/q_beta Phase Fractions]
    C --> D[Initialize Partition Coefficients]
    D --> E[Calculate C* for Each Species]
    E --> F[Update Partition Coefficients]
    F --> G{Converged?}
    G -->|No| E
    G -->|Yes| H[Return Equilibrium Concentrations]
    H --> I[Alpha-phase + Beta-phase Mass Balance]

Objective Function¶

The equilibrium solver minimizes the difference between guessed and calculated partition coefficients:

\[\text{Error} = \sum_j (\varepsilon_j^{guess} - \varepsilon_j^{calc})^2 + (C_{liq}^{guess} - C_{liq}^{calc})^2\]

Convergence is achieved when this error falls below a threshold (typically \(10^{-16}\)).

Mass Balance¶

At equilibrium, mass must be conserved:

\[C_j^{total} = C_j^{gas} + C_j^{\alpha} + C_j^{\beta}\]

where the particle-phase concentrations include both organic mass and associated water.

Implementation in Particula¶

The equilibria calculations are implemented in the particula.equilibria module:

API	Purpose
`Equilibria`	High-level interface for gas–particle equilibrium calculations
`LiquidVaporPartitioningStrategy`	Strategy class implementing liquid–vapor partitioning behavior
`liquid_vapor_partitioning()`	Convenience function wrapping the main equilibrium solver
`get_properties_for_liquid_vapor_partitioning()`	Compute activity/phase inputs used by the solver

Example Usage¶

import numpy as np
from particula.equilibria.partitioning import (
    liquid_vapor_partitioning,
    get_properties_for_liquid_vapor_partitioning,
)

# Define species properties
c_star_dry = np.array([1e-1, 1e0, 1e1])  # ug/m3
c_organic = np.array([1.0, 2.0, 3.0])  # ug/m3
molar_mass = np.array([200.0, 180.0, 220.0])  # g/mol
oxygen2carbon = np.array([0.4, 0.5, 0.6])
density = np.array([1400.0, 1300.0, 1500.0])  # kg/m3

# Get activity properties at 70% RH
gamma_ab, mf_water_ab, q_ab = get_properties_for_liquid_vapor_partitioning(
    water_activity_desired=0.7,
    molar_mass=molar_mass,
    oxygen2carbon=oxygen2carbon,
    density=density,
)

# Solve equilibrium
alpha, beta, system, result = liquid_vapor_partitioning(
    c_star_j_dry=c_star_dry,
    concentration_organic_matter=c_organic,
    molar_mass=molar_mass,
    gamma_organic_ab=gamma_ab,
    mass_fraction_water_ab=mf_water_ab,
    q_ab=q_ab,
)

print(f"Total liquid concentration: {system[0]:.3f} ug/m3")
print(f"Partition coefficients: {system[2]}")

Key Equations Summary¶

Equation	Description
\(p_i = a_i \cdot p_i^{\circ}\)	Vapor-liquid equilibrium
\(a_i = \gamma_i \cdot x_i\)	Activity from coefficient and mole fraction
\(\varepsilon_j = \frac{1}{1 + C^*_j / C_{liq}}\)	Partition coefficient
\(C^_j = C^_{j,dry} \cdot \gamma_j \cdot q \cdot \frac{C_{liq}}{M_j \cdot \overline{M}}\)	Effective saturation concentration

References¶

Gorkowski, K., Preston, T. C., & Zuend, A. (2019). Relative-humidity-dependent organic aerosol thermodynamics via an efficient reduced-complexity model. Atmospheric Chemistry and Physics, 19(19), 13383-13407. https://doi.org/10.5194/acp-19-13383-2019

Previous: Activity Theory - Learn about the BAT model for activity coefficients.