kappa-Kohler Hygroscopicity Theory

kappa-Kohler theory uses a single parameter kappa to describe the hygroscopicity of aerosol particles, simplifying the complexity of traditional Kohler theory.

Basic Principles

Traditional Kohler Equation

\[\ln\left(\frac{RH}{100}\right) = \frac{4 M_w \sigma}{RT \rho_w D_{wet}} - \frac{6 n_s M_w}{\pi \rho_w D_{wet}^3}\]

The competition between the Kelvin effect term and Raoult effect term determines the equilibrium diameter of the particle.

kappa Parameterization

Simplified form proposed by Petters and Kreidenweis (2007):

\[\frac{1}{a_w} = 1 + \kappa \frac{V_s}{V_w}\]

Where: - \(a_w\) = Water activity - \(\kappa\) = Hygroscopicity parameter - \(V_s\) = Dry particle volume - \(V_w\) = Water volume

Physical Meaning of kappa Values

kappa Range	Hygroscopicity	Typical Components
0	Non-hygroscopic	Mineral dust, EC
0.01-0.1	Weakly hygroscopic	Organic matter
0.1-0.3	Moderately hygroscopic	Mixed aerosol
0.3-0.7	Highly hygroscopic	Ammonium sulfate, ammonium nitrate
>0.7	Extremely hygroscopic	Sea salt

Component kappa Values

Component	kappa Value	Source
(NH4)2SO4	0.53	Petters & Kreidenweis (2007)
NH4NO3	0.67	Petters & Kreidenweis (2007)
NaCl	1.28	Petters & Kreidenweis (2007)
H2SO4	0.90	Petters & Kreidenweis (2007)
SOA	0.1 +/- 0.05	Experimental range
POA	0.01	Estimated value
BC	0	Non-hygroscopic

Mixed Aerosol kappa Calculation

For internally mixed particles, use volume weighting:

\[\kappa = \sum_i \epsilon_i \kappa_i\]

Where \(\epsilon_i\) is the volume fraction of component \(i\).

Hygroscopic Growth Factor

Calculate growth factor GF from kappa:

\[GF = \left(\frac{D_{wet}}{D_{dry}}\right) = \left(1 + \kappa \frac{RH/100}{1 - RH/100}\right)^{1/3}\]

AeroViz Implementation

import pandas as pd
from AeroViz import reconstruct_mass, growth_factor, kappa

# 1. Reconstruct volumes from chemistry
mass_result = reconstruct_mass(df_chem)
df_volume   = mass_result['volume']

# 2. Growth factor (needs total_dry + ALWC)
df_gRH = growth_factor(df_volume, df_alwc)        # column: gRH

# 3. kappa (needs gRH + AT + RH columns)
df_kappa = kappa(
    pd.concat([df_gRH, met_data[['AT', 'RH']]], axis=1),
    diameter=0.5,
)                                                  # column: kappa_chem

# Example output
#                     gRH     kappa_chem
# 2024-01-01 00:00    1.42    0.35
# 2024-01-01 01:00    1.45    0.38

Applications

1. Dry PSD Calculation

Convert ambient PSD to dry PSD:

\[D_{dry} = D_{wet} / GF\]

2. Optical Hygroscopic Growth

Estimate f(RH) for IMPROVE equation.

3. CCN Activation

Predict the ability of aerosols to act as cloud condensation nuclei.

References

1. Petters, M. D., & Kreidenweis, S. M. (2007). A single parameter representation of hygroscopic growth and cloud condensation nucleus activity. Atmos. Chem. Phys., 7(8), 1961-1971.
2. Kohler, H. (1936). The nucleus in and the growth of hygroscopic droplets. Trans. Faraday Soc., 32, 1152-1161.