Nucleation: Probability (basics) and rates of Nucleation
Nucleation: Probability (basics), and Rates
This blog is a continuation of three previous blogs:
- 1) Bubble Nucleation: Bubble stability: Jan 16 2017
- 2) Energetics of Nucleation: Jan 17 2017
- 3) Nucleation requirements: Jan 18 2017
I will repeat that much of this blog is taken from parts of a book I wrote (never published) a decade or so ago. Accordingly image numbers and equation numbers with adhere to the book in its current format.
Brief Simplified (for this blog) Probability discussion
I previously stated that nucleation processes generally require energy at least for bubbles and small droplets, this seems to be the case. The traditionally troublesome probability for nucleation is written in terms of work required for nucleation (Wn) and a normalization factor (B):
Where P’(n) is the probability: (I use this nomenclature so that it is not mistaken for pressure), B is a constant, Wn is the total work required for nucleation, k is Boltzmann’s constant and T is temperature
Eqn 3.3.1 is troublesome mainly because nucleation involves a cluster of molecules, hence a better approximation would use the average work per molecule (perhaps mean would be even better) i.e. the total work required (Wn) divided by the number of molecules in the nucleating cluster (X). I.e.:
Eqn 3.3.2 remains inherently weak because it fails to properly describe the path. It also wrongly assumes that all X molecules can equally extract energy, which may or may not be the case.
This author does not claim to know what probability is best for all processes, as it’s writing will actually depend upon how one envisions the process of nucleation and what is path for the energy exchange. Certainly, more possibilities for writing the probability of nucleation exist.
Perhaps nucleation is a multi-step process, with each step having its own unique probability. Consider bubble nucleation wherein the formation of the gases within is one step and the formation of tensile layer is another. It feels improbable but it still warrants consideration. Then again the probability the molecules becoming gaseous might be best based upon eqn 1.12.7 that I used for decribing vaporization processes:
P’(v)=Bexp(-Tb/T) eqn 1.12.7
which for nucleation would be written:
It must be emphasized that for bubble nucleation, the boiling temperature (Tb) will be the boiling temperature at the pressure within the bubble.
Certainly eqn 3.4.5 avoids the issue of whether or not the probability concerns the energy required by the whole bubble vs the energy required on a molecular basis. To better understand eqn 3.4.5 please see my paper “Latent heat and critical temperature: A unique perspective” Physics Essays 2013. The paper can also be seen on my website: http://www.newthermodynamics.com and click on my papers.
I cannot help but think that for those who are actually involved in research that a cascading version of eqn 3.4.5 provides the best solution. Cascading probabilities will be discussed in my book on nucleation and tensile layers (if it ever gets published). For this blog a simple analysis is all that is really needed, hence we will stop at eqn 3.4.5.
No matter what the actual probability [P'(n)] is, the nucleation rate (J) will be given by:
Where: Jn= normalization constant = NC, C = concentration, and N = normalization factor
The normalization constant (Jn) relates the probability of nucleation from an energy perspective to the system in question. The normalization constant will vary depending on what system we are dealing with. Specifically, it will depend on factors such as, the types of molecules, the concentration of various substances within a system, system dynamics etc etc.
Assume that only one size of cluster can nucleate. At any instant, the normalization constant (Jn) is a function of the number of that exact size of cluster, which exists. Rather, than thinking of an instant of time, now think over a given time frame, say a minute. Now, Jn is related to the number of that exact size of a-clusters, which exist in that system over a minute in time.
Instead of one precise size of cluster nucleating, let us say that we have 3 sizes of clusters that exist and can nucleate. The 3 sizes of clusters are represented by the subscripts “1,2,3”. Then for each size of cluster, their nucleation rates can be written as:
The total nucleation rate (Jtot) must be equal to the summation of all the nucleation rates for each size of cluster. I.e.:
Where the subscripts 1 … n represents the number of different sizes of clusters.
There is an inherent weakness to all this, as was previously discussed in Section 1.13 of my books: The normalization constant (Jn) is not some constant, which is universally applicable. Rather, it is only a constant in the graphical sense, which applies to only one system, at some specific instant in time. Moreover, if we use the wrong probability function and/or work term, often can still find some Jn that normalizes our data. This explains why the misuse of Gibbs equation (for globule nucleation), as some universal nucleation equation, went unnoticed throughout the 20th century. Basically, although researchers have been using the wrong work (Wn) function they were still able to normalize their empirical data.