The activated fraction AF can be expressed as followed:
AF = 1-exp(-J.A.tnuc)
(1)
Where tnuc is the residence time, A the surface area and J the nucleation rate.
For a similar AF and residence time (AF1=AF2 and tnuc1=tnuc2):
A1J1=A2J2
(2)
The nucleation rate can be expressed as followed:
(3)
where K is the kinetic factor, fhet the geometric compatibility factor. K is assumed to be constant for
the range of RHi where ice nucleation happened. The Gibbs free energy ΔG can be expressed as
follow:
(4)
With k=1.38.10-23 J/K, ϑice the volume of water molecules in an ice embryo, σi/v the surface tension at
the ice/vapour interface, Si the saturation ratio with respect to ice, k the Boltzmann constant.
By combining equations (1), (2), (3) and (4), we can obtain the following expression:
(5)
Any combination of A1=πD12 and A2= πD12, Si,1 and Si,2 for a given temperature can lead to a similar
AF with adjustment of fhet.
In our study, D1=130nm and D2=180nm, Si,1=1.58 and Si,2=1.47 and T=238.15K
We derive fhet:
(6)
fhet= 0.00057 for our study.
We then derive the kinetic factor K from equation (3).
With the kinetic factor we can calculate the contact angles α1 and α2 with A1 and A2 their respective
surface area:
(7)
Which gives us: α1=30.1° and α2=27.6°