Supersaturation in clouds is rarely greater than 1%.  Because it is very unlikely for cloud drops to form by homogeneous nucleation at these supersaturations, they must be forming by heterogeneous nucleation on atmospheric aerosols.

Some aerosols are capable of serving as the nuclei for the formation of clouds.  These are called cloud condensation nuclei (CCN).

A. Nucleation of water vapor on soluble particles

          Most atmospheric aerosol particles are soluble, or at least partially so. Thus, unlike homogeneous nucleation, which is important only in a few regions
          of the atmosphere, heterogeneous nucleation on at least partially soluble aerosol particles is the main process for cloud drop formation in the
          troposphere. The first successful explanation was by Koehler in 1921.

B. Conceptual basis

          1. Start with a particle in dry air. As the relative humidity is increased, the particle deliquesces (i.e., takes up water and grows).
                With sufficient increase  in humidity, the aerosol particle becomes activated, causing nucleation of a cloud drop.
          2. The two important features of the solution drops are:
               a. They contain a solute (a trace chemical that is in the water), which reduces the equilibrium vapor pressure with respect to pure water
                (Raoult Effect);
               b. They have a finite radius of curvature, which increases the equilibrium vapor pressure relative to a flat surface (Kelvin Effect).
          3. Koehler theory is a combination of the Raoult and Kelvin Effects.
          4. The Raoult Effect says that the water vapor pressure is reduced by the presence of a solute, such that:

                                             e_eq(n_sol,r, T) / e_eq(0,r,T) =X _water = 1 - X_solute

                where e_eq(n_sol,r, T) is the equilibrium water vapor pressure with n moles of solute, a radius r_d , and at temperature T;
                            X _water = moles of water / (moles of solute + moles of water) = 1 - X_solute  (note: W&H use 'f' for 'X'.)

Recall that in equilibrium, the amount of evaporation equals the amount of condensation.  Clearly this cannot be true if the particle is growing or evaporating, but if these processes happen slowly enough, then we can assume that the cloud droplet is almost in equilibrium.

                                e_eq(0,r,T) / (e_eq(0,infinity,T) = e_s(T)) = (1 - X_solute )exp(a/r), where a = 2s/n_lRT
 

and n_l is the number of moles of water per unit volume of liquid (55.6 moles per liter).

          6. To get the combination of effects, we must multiply them together to get:

                                               e_eq(n_sol,r,T) / e_s(T) = (1 - X_solute ) exp(a/r)

          7. We can make some approximations to get this expression into a common form.
               a. First, for a dilute solution, X_solute = i n_s / n_l , where i is the degree of ionization.
               b. Second, for small drops, exp(a/r_d) =1 + a/r_d , by series expansion.
          8. The net result is an expression for Koehler Theory:

                                                    e_eq / e_s = 1 + a/r_d -i n_s / n_l

          9. By substituting in for n_s / n_l, this expression can be rewritten as:

                                                e_eq / e_s = 1 + a / r_d - b i m_solute / r_d³

          where

                                               a = 2s/n_lRT = (0.33/T) (in micormeters)

                                          and b = 3/(4pn_l) = 4.29x10^-6 m³ (moles of solute)^-1

 
     We can write the equation not in terms of the saturation ratio, e_eq/e_s but in terms of supersaturation, where

                                                        s = e_eq/e_s - 1.

     So, if we think in terms of Koehler equilibrium, where s_k is the equilibrium supersaturation defined by Koehler Theory

                                                     s_k = a/r_d - b i m_s / r_d³.

C. Koehler curves are the family of curves of s_k(r_d) for several values of i m_s. They show:

   1.increasing solute content ms causes decrease in s_k;
   2.a "Kelvin" limit, which gives the variation for pure water;
   3.counteracting effects that lead to the existence of a maximum supersaturation, called the "critical saturation", defined as the equation s_c = s_k(r_c)

D. Consider a single typical aerosol particle with a specified ms as the parcel rises and sambient increases.

   1.The solution drop grows whenever sambient > s_k(r_d).
   2."Activation" of the drop occurs when sambinet > s_c = s_k(r_c). Note that sambient is a property of the air and s_k is a property of the particle.
   3.The "growth stability" is determined by the relationship between the particle size and the critical particle size.

          haze drops: r < r_c (stable)

          cloud drops: r > r_c (unstable)

   4.The critical radius is given by

                                                      r_c = (3b i m_s / a)^1/2.

   1.The critical supersaturation is given by s_c = s_k(r_c)

                                                     s_c = (4a3 / 27 b i m_s)^1/2

D. Summary of ideas about Koehler curves

   1."Critical supersaturation" is a property of the aerosol particles, not the environment.
   2.Typically, 0.1 micron particles have a 0.1% critical supersaturation.
   3.Typically, cloud condensation nuclei (CCN) have particle diameters > 0.01 micrometer when s_ambient < 3%.
   4.Soluble aerosol particles are good cloud condensation nuclei. They take up enough water vapor that particles cannot nucleate homogeneously.