Heterogeneous Nucleation

Figure 2.1.2: Schematic of heterogeneous liquid cluster formation on an insoluble spher-ical particle (substrate). θ_contis the contact angle between the cluster and the substrate.

Table 2.1.1 lists the number of molecules in critical cluster sizes i^∗ = 4πr^∗3/3vl and the resulting nucleation rates for water and saturations from S = 2 to S = 5. Ac-cordingly, within this saturation range J strongly increases with S by more than 60 orders of magnitude. The absolute values of the nucleation rates further reveal that homogeneous nucleation of water droplets can only effectively occur for high satura-tions. For instance, at S = 3 (meaning at a relative humidity of 300%) it would still take about a week for one droplet to form in a volume of 1 cm³. As heteroge-neous nucleation of water droplets on pre-existing aerosol particles requires far smaller saturations, homogeneous nucleation and, therewith, cloud formation without aerosol particle involvement is practically not taking place in the atmosphere.

In any case, for a completely non-wettable particle θcont = 180° and f(θcont) = 1, meaning that ∆G^∗ = ∆G^∗_hom, i.e. the particle has no effect on cluster formation.

However, for a wettable particle the contact angle is smaller than 180° andf(θ_cont)<1.

As a result, ∆G^∗ < ∆G^∗_hom, which connotes that the onset of nucleation arises at smaller saturations and that the nucleation rate for a given saturation is increased compared to the homogeneous nucleation process.

Although even insoluble particles can promote droplet nucleation, atmospheric aerosol particles are usually not completely insoluble. In general, they contain both water-soluble and insoluble material. By affecting the chemical potential of the liquid phase, the water-soluble substances lead to a decrease in saturation vapor pressure over the solution compared to that over pure water. The solution saturation vapor pressure can be written as

p_s,sol =a_wp_s

with the saturation vapor pressure for pure water ps and the water activity aw. For dilute solutions a_w is given by

a_w = n_w n_w +n_sol

with n_w and n_sol representing the number of moles of water and solute in the solu-tion, respectively. The resulting equasolu-tion, describing the reduction in saturation vapor pressure over dilute solutions is known as Raoult’s law

p_s,sol = nw

n_w +n_solp_s

Inserting this modified saturation vapor pressure into Eq. (2.1.5) and solving for the logarithm of the saturation yields

lnS = 2σv_l

kT r^∗ + lna_w = 4σM_w

RT ρ_wD + lna_w (2.1.6) with the molar mass and density of water M_w and ρ_w, and the (Kelvin) equilibrium droplet diameter D = 2r^∗. Assuming that, aside from the amounts of water and solutes, the droplet contains an insoluble (spherical) core of diameter D_u, the total drop volume will satisfy

π

6D³ =nw

M_w

ρ_w +nsol

M_sol ρ_sol +π

6D_u³ which can be used to express a_w as

a⁻¹_w = 1 + n_sol

n_w = 1 + M_w ρ_w

n_sol

π

6 (D³−D³_u)−n_sol^M_ρ^sol

sol

For the approximation of dilute solutions both n_sol^M_ρ^sol

sol and n_sol^M_ρ^w

w are much smaller than ^π₆ (D³−D³_u) such that

lnS ≈ A

D − B

(D³−D³_u) (2.1.7)

with

A= 4σM_w

RT ρ_w B = 6n_solM_w

πρ_w (2.1.7a)

Equation (2.1.7) is known as the Köhler equation and combines the two components that determine equilibrium saturation over an aqueous solution droplet, i.e. the Kelvin effect (first term) and the solute effect (second term). In most cases, however, it is inconvenient to use a particle’s insoluble core diameterDu and the number of moles of solute n_sol, but it is more meaningful to use the total dry particle diameter D_dry and the mass fraction of soluble material _m giving

Figure 2.1.3: Köhler curves showing the equilibrium vapor supersaturation (S−1) as a function of water droplet diameter at T = 293 K for water-soluble ammonium sulfate particles of varying initial dry diameters. Adopted from Andreae and Rosenfeld (2008).

D³_u =D³_dry ρ_u ρ_sol

_m 1−_m + 1

!−1

n_sol = _m M_sol

ιπD³_dry

6_ρsol^m + ¹⁻_ρu^m (2.1.7b) with the density of the insoluble material ρ_u and the number of ions resulting from dissolution of one solute molecule ι. Figure 2.1.3 exemplary shows a set of so-called Köhler curves satisfying Eq. (2.1.7) for different initial dry diameters of (completely) water-soluble ammonium sulfate particles. It further shows the equilibrium supersat-uration SS = S −1 for pure water droplets. For the latter the solute term in Eq.

(2.1.7) is zero and it would require very high supersaturations for small droplet diame-ters to be in (unstable) equilibrium with the surrounding vapor. For larger pure water droplets the equilibrium supersaturation may decrease, but as discussed in Sect. 2.1.1 at the same time the Gibbs free energy barrier for homogeneous nucleation increases and, consequently, nucleation rates become vanishingly small.

For droplets containing soluble particles the solute term in Eq. (2.1.7) dominates over the Kelvin term for small droplet diameters, leading to a strong lowering of the equilibrium SS. Moreover, the gradient of the equilibrium SS(D) changes from nega-tive to posinega-tive. This implies that aqueous solution droplets in this size range are now in stable equilibrium, meaning that a small change in diameter via random loss or up-take of water molecules leads to a restoring force. If, for instance, an aqueous solution droplet that is beforehand in equilibrium with the surrounding vapor slightly shrinks due to random loss of water molecules the ambient SS is larger than the new equilib-riumSS of the shrunken droplet. This provokes a net transfer of vapor molecules back towards the droplet until equilibrium with the environment is restored. In contrast to pure water droplets, aqueous solution droplets further allow for equilibrium with the vapor phase also for unsaturated conditions (S < 1 or SS < 0). By convention, aqueous solution droplets in the ascending branch of the Köhler curves, i.e. in stable equilibrium are commonly referred to as hydrated particles rather than droplets, par-ticularly for unsaturated conditions. Unlike for droplets in unstable equilibrium, the random acquisition of water molecules by hydrated particles in stable equilibrium does not initiate (unrestricted) accelerated growth. On the other hand, for larger diameters where the Kelvin term dominates over the solute term the aqueous solution droplets behave like pure water droplets and the equilibrium state changes back to unstable.

This means that any further uptake of water molecules again leads to what so far was been termed nucleation and will hereafter be referred to as droplet “activation”.

Between the two regimes the Köhler curves exhibit a maximum that is defined by the so-called critical supersaturation SS_c, which depends on the properties of the dry aerosol particle. For positive supersaturations with SS < SS_c the Köhler curves feature two equilibrium states, the discussed stable equilibrium at smaller and the un-stable equilibrium at larger diameters. Similar to the case of homogeneous nucleation, aqueous solution droplets have to surmount an energy barrier to reach the unstable equilibrium state from the stable equilibrium state and to finally activate by random net uptake of water molecules. This energy barrier rapidly grows the farther SS falls below SS_c, so that droplet activation rates for SS SS_c become negligible. When the supersaturation approaches SSc, however, a stable hydrated particle needs only few additional water molecules to be pushed towards the unstable descending Köhler curve branch and, accordingly, to trigger its unrestricted water uptake. In summary, if for a certain aerosol particle the ambientSS fulfills SS≈SSc or evenSS > SSc, then this particle will activate into a droplet. For a given particle composition (soluble con-tent etc.), the dry diameter for which the critical supersaturation equals the ambient supersaturation (SSc = SS) is called the critical dry diameter or activation diameter D_act.

From Eq. (2.1.7), (2.1.7a) and (2.1.7b) it is apparent that a number of hardly accessible intrinsic parameters, such as Du,ρu, ρsol, m, Msol and ι, are necessary to determine SS_c for a given D_dry, and vice versa D_act for a given SS. For practical use it would, hence, be of advantage to find a simpler parametrization for the ability of a certain particle type to activate into a droplet, i.e. to act as a CCN at a given size and ambient supersaturation.