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.

Kantrowitz Factor

Kantowitz Factor

Kantowitz factor, (often simply called the non-isothermal correction factor), is applied as a correction to the nucleation rate from classical nucleation theory. In most cases (the example here is taken from the wet steam model chapter of the fluent theory guide) it appears as follows:

Equation 1 –typical nucleation rate given by Classical Nucleation Theory, with Kantowitz factor 1/1+θ up front.

The non-isothermal factor is found from the enthalpy and the ratio of specific heat capacities:

Equation 2 – non isothermal correction factor.

According to Bakhtar et al (2005) this is a simplification that can be made at low pressure, when the critical radius is much smaller than the mean free path of the gas. Really it should look like:

Equation 3 – nonitsothermal correction factor is a simplification of this.

In the presence of a large volumes of inert gas, isothermal assumptions are justified because collisions with the inert gas, (which transfer heat) are much more common than those with condensing species.

The derviation of the Kantrowitz factor (and Feder’s correction factor) assume that every cluster is at the same elevated temperature. Wyslouzil and Sienfield (1992) criticise this, on the basis that the temperature of small (subcritical) clusters is more sensitive to the addition of more molecules, and the growth of these small clusters affects the final result.

Feder also came up with their own correction factor of the form shown below, that is used in a similar way to the Kantrowtiz factor.

Equation 3 – feder's factor - similar to 1/1+θ .

It results is around 0.01 or a hundredth of the isothermal (Wu, 1973), though it’s worth mentioning that most of these flows are very fast – discussions in the literature centre exclusively on nozzles and steam turbines. b and q are defined in terms of thermodynamic state of the supersaturated vapour. q is basically mostly latent heat. The b is calculated from β, gas kinetic collision frequency (Dawson et al. 1969).

Pandey 

Equation 3 – feder's factor - nuts and bolts.

(2014, p. 15) discusses the steady-state assumption in connection to the Kantrowitz factor. τ = 10-7 to 10-6 s is the timescale for ‘steady state nucleation rate’. Cooling rate ranges from 0.2-0.8 K/µs. The characteristic time in nozzle flows is typically τ = 10-3 s. Kantrowitz factor uses a steady-state assumption, so it is justified on this basis, for use even for unsteady state assumptions.

So to conclude on Kantrowitz factor, corrects for non-isothermal effects which make the nucleation rate about 50 to 100 times slower in rapidly expanding steam flows, such as steam turbines and nozzles. It assumes a steady-state conditions but this is valid as the time to reach steady state is so small. However, it’s the problem of non-isothermal particles itself is less of a big deal when the bulk of the gas is an inert component (such as air) – in these cases the inert gas collisions conduct heat much faster than the particle can grow. Also, my brief lit review hasn’t shown a Kantrowitz factor being applied to something that isn’t a turbine or a nozzle. Feder’s factor is used an alternative, but is basically for similar purposes.

Bakhtar, F. et al. 2004. Classical nucleation theory and its application to condensing steam flow calculations. Proceedings of the IMechE, 219 C. p. 1315-1333.

Pander, A (2014). Numerical modelling of non-equilibrium condensing steam flows. Delft University.

Wu, B. 1973. Possible condensation in rocket exhaust plumes 2. Yale University.

Wyslouzil, BE. Sienfield, JH. 1992. Nonisotermal homogeneous nucleation, 97 (4). pp. 2661-2670.