Two methods are used: for all forms of input (moles or masses of ions or components, or dry particle radii and volume
fractions of solids) the calculations can be done with the *E-AIM* models. Additionally, for the case where the
user enters the dry particle radius and volume fractions of the constituent solids, the κ-Köhler equations of Peters and Kreidenweis (1) can be used.

The fundamental relationship in these calculations, in addition to the ability to calculate water activities of the aqueous particle, is the Kelvin equation which describes the increase in equilibrium vapour pressure of the volatile components of an aqueous particle, in this case the water, due to surface curvature. The saturation vapour pressure of water above the aqueous particle (S) is given by:

S = *a*W · exp[2σV_{w} / (RT·r)]

where *a*W is the water activity of the liquid particle, σ (N m^{−1}) is its surface
tension, V_{w} (m^{3} mol^{−1}) is the partial molar volume
of liquid water in the particle (for the current particle composition), r (m) is the particle radius, R (8.3144 J mol K^{−1})
is the gas constant, and T (K) is temperature.

All calculations are carried out for a range of particle liquid water contents, from that corresponding to a low (fixed) water activity limit up to a water content corresponding to an extremely dilute solution and water activity of almost unity.

Here, the water activity of the particle is calculated by one of Models I – IV, as appropriate, and the surface tensions and densities from the work of Dutcher et al. (2) and Clegg and Wexler (3), respectively. Both their models can be run on this site (see the home page), and calculations using them are incorporated into the output of Models I – IV. The partial molar volume of water in the particle is also calculated using the work of Clegg and Wexler (3), by partial differentiation of the total volume of the particle in the usual way.

Densities of the dry solids are
needed to determine the amounts (moles) of chemical components present in the particle for the case where dry particle
radius is input by the user. This is true for both for both *E-AIM* and κ-Köhler calculations. Values
are taken from Table 16 of Clegg and Wexler (3). For convenience they are listed on the
inputs page.

By default, solids are allowed to form within the aqueous particles. They are most likely to occur at low values of S (corresponding to high solute concentrations within the aqueous phase of the particle). The option to prevent the formation of solids is provided, which will yield a fully liquid particle at all values of S. Such particles may be be supersaturated with respect to one or more solids. The assumed geometry of the particle is "core-shell": a spherical core containing any solid salts that have formed, surrounded by a shell of aqueous solution.

The densities referred to above are also used to estimate the dry particle radius, which is needed for *E-AIM*
calculations of the hygroscopic growth factor (named HGrFac in the results output) for all types of input
except particle size plus volume fractions of solids. This dry particle radius is determined
by first assuming that all possible 'simple' solids (i.e., those containing a single cation, a single anion, and no water)
will exist, in amounts predicted using eq (4) of Clegg and Simonson (4). For example, in a chemical system containing
NH_{4}^{+}, Na^{+}, SO_{4}^{2−}, and
Cl^{−} the following solids would occur:
(NH_{4})_{2}SO_{4},
Na_{2}SO_{4}, NH_{4}Cl, and NaCl. In acidic particles,
which also contain SO_{4}^{2−},
H_{2}SO_{4} will be predicted to occur.
In such cases, if
NH_{4}^{+} and/or Na^{+} are also present, then it is
assumed that the maximum possible amounts of the bisulphates NaHSO_{4} and
NH_{4}HSO_{4} (in that order) will be formed, in preference to the acids. If any
H_{2}SO_{4} remains, after assignment to these acid salts, then
its density and molar volume are assumed to
have the values listed in Table 1 on the inputs page. (If HNO_{3} and/or
HCl are predicted in the dry particle, its assumed volume properties are also as listed in Table 1.) Further details are
given in section 6 of the chemical systems page (the link is on the
*E-AIM* home page).

Here the saturation vapour pressure of water is calculated using the simplified expression of Petters and Kreidenweis (1):

S = (D^{3} – D_{d}^{3}) /
[D^{3} – D_{d}^{3}(1 − κ)] **·**
exp[4σ_{w} V_{w}^{o} / (RT·D)]

where D and D_{d} are the wet and dry particle diameters, respectively, and κ is the overall "kappa"
for the particle composition (see eq (7) of Petters and Kreidenweis (1)). The quantity σ_{w} is the surface tension of pure water at temperature T, and V_{w}^{o}
the molar volume of pure water. The kappa parameter incorporates, in a highly simplified way, non-ideal effects on solution
properties as well as the relationship between solute amounts (moles) and volumes.

The equation above is eq (6) of Petters and Kreidenweis (1) and, as well as being used in this equation to calculate S, the
parameter κ is
also output in the *E-AIM* results. This is done for each value of the particle water content. The variation of this calculated κ with S
demonstrates the approximate nature of the parameter and the fact that any value should only be applied over a
limited range of water activities (close to those for which the κ value was determined).

Calculations with the κ-Köhler equations assume a fully liquid particle at all relative humidities.

(1) M. D. Petters and S. M. Kreidenweis (2007) A single parameter representation of hygroscopic growth and cloud
condensation nucleus activity. *Atmos. Chem. Phys.* **7**, 1961-1971.

(2) C. S. Dutcher, A. S. Wexler and S. L. Clegg (2010) Surface tensions of
inorganic multicomponent aqueous electrolyte solutions and melts. *J. Phys. Chem. A*, **114**, 12216-12230.

(3) S. L. Clegg and A. S. Wexler (2011) Densities and apparent molar volumes of atmospherically
important electrolyte solutions. I. The solutes
H_{2}SO_{4}, HNO_{3}, HCl,
Na_{2}SO_{4}, NaNO_{3}, NaCl,
(NH_{4})_{2}SO_{4},
NH_{4}NO_{3}, and NH_{4}Cl
from 0 to 50 °C, including extrapolations to very low temperature and to the pure liquid state,
and NaHSO_{4}, NaOH and NH_{3} at 25 °C.
*J. Phys. Chem. A* **115**, 3393-3460.

(4) S. L. Clegg and J. M. Simonson (2001) A BET model of the thermodynamics of aqueous
multicomponent solutions at extreme concentration. *J. Chem. Thermodyn.* **33**, 1457-1472.