Correction for Heterogeneous Nucleation

heterogeneous nucleation

Frequently studies which use classical nucleation theory account for heterogeneous nucleation using a heterogeneous nucleation correction factor. This is normally denoted f(θ) or something similar, and is inserted into the classical (homogeneous) nucleation rate equation to account for the effect of nucleation on a surface. In this blog entry I’m going to cover this factor. The “flat surface” correction is common because in most of the cases commonly studied the foreign nuclei is much larger than the critical cluster that is forming on its surface, so the contact surface between them can be considered flat. A more complicated version of f(θ) can be derived for a curved foreign nucleus, derivation is covered in Vehkamaki (2006) ch. 7. This subject is something of a departure from the normal subjects discussed on this blog, as this correction to my knowledge is not available in default settings of any particle formation laws etc. in Fluent (Fluent is the software I currently use exclusively for CFD). But It’s still of interest to the particle formation stuff I’m working on currently. 

Heterogeneous nucleation has an effect that is similar to catalysis in reactions. It lowers the energy barrier. Heterogeneous nucleation happens faster at lower degrees of supersaturation due to this change in the critical Gibbs free energy.

Figure 1 – A graph showing the effect of heterogeneous nucleation on critical energy (Zang, 2015).

The correction factor was originally derived by Fletcher in 1958, and is sometimes referred to as the “Fletcher factor” (Kalikmanov, VI. 2013). But Equation 1 reduces to the flat-surface correction when the curved surface is very large.

 

Equation 1 – The long-winded version of Fletcher factor, for curved surfaces. Dimensionless versions are common. Here a is the ratio between the foreign nucleus and the cluster; m is the cos of the (cluster) contact angle; w = √1+a2-2ma.

The flat surface correction is:

Equation 2 – The flat-surface correction. υ is the contact angle (between the critical droplet on the surface).

For the heterogeneously nucleating critical droplet, the radius is exactly the same as for the homogeneous case, but the volume of the nucleating particle changes because the particle becomes a spherical cap instead of a sphere. Equation 2 corrects for this reduced volume (which reduces the critical Gibbs free energy, the critical Gibbs free energy is the point where the (-ve) volume free energy change balances the (+) free energy change needed to create a surface). The angle can be found by young’s equation if the surface tensions between the gas, cluster and substrate are known.

Figure 2 - Effect of heterogeneous nucleation on the nucleation rate. shamelessly copied-and-pasted from Coulson 2.

How it is applied in the nucleation rate equation (shown in Equation 3). The biggest difference comes in the form of the effect on ΔG*het. Typically the pre-exponential factor may be known experimentally (typical value is around 1017 cm-2s-1 for heterogeneous vapour condensation). The monomer flux (W*) is the sum of both the molecular-bombardment style growth present in homogeneous nucleation (R* = monomer flux, s* = surface area of critical droplet) but also adatom impingement (W*s) – surface diffusion of adsorbed atoms to the nucleus periphery.

Equation 3 - Difference between heterogeneous and homogeneous nucleation, (Cooper, 2015).

Most sources simply state that the dependency of Jhet on the corrected free energy, and do not discuss the pre-exponential factor in detail. However, Fletcher (1959) himself states that heterogeneous nucleation requires changes to this pre-exponential factor (though he does not cover what these changes are in the source). Talanquer and Oxtoby (1996) give an example of an analytical expression for this pre-exponential factor in their paper. However, overall this is all there is to the classical treatment of heterogeneous nucleation.

Cooper, S. (2015). Crystallisation Kinetics. Durham University. Available: http://community.dur.ac.uk/sharon.cooper/lectures/cryskinetics/handoutsalla.html accessed: 17 Aug 2015.

Fletcher, NH. (1959). On ice-crystal production by aerosol particles. Journal of Meteorology. 16(2). pp. 173-180.

Kalikmanov, VI. (2013). Nucleation theory. Berlin: Springer.

Vehkamaki, H. (2006). Classical nucleation theory in multicomponent systems. Berlin: Springer.

Zang, L. (2015). Lecture 12 Heterogeneous Nucleation: a surface catalysed process. College of engineering University of Utah. Available: http://www.eng.utah.edu/~lzang/ accessed: 12 Aug 2015.