Theory of Homogeneous Nucleation
At the very first stage of phase transition at a particular super-saturation a few no. of small-sized nuclei (cluster) of new phase is generated in the sea of old meta-stable phase. To undergo phase transition the system needs to cross a barrier at a certain cluster size which is called critical cluster size. This barrier crossing is activated process and is accompanied by thermal fluctuation of cluster size. Once system crosses this barrier the size of the critical cluster starts growing spontaneously and leads to the phase transition. The first activated process is called “nucleation” whereas the second spontaneous step is known as “growth”. Although nucleation is initiated by local fluctuation of order parameter (such as density in case of gas-liquid condensation), to cross the barrier it needs a large thermal fluctuation which is rare at small super-saturation or super cooling. However, as super-cooling/super-saturation is increased, the height of barrier decreases, and at some well defined value of the super cooling/super saturation the barrier height becomes comparable to the magnitude of the thermal fluctuation, and the formation of the new phase becomes rapid.
Let us consider a liquid drop formed in a super cooled vapor of the same substance starting from homogeneous vapor phase. We consider the situation where the drop is in equilibrium with the vapor. Now, we want to calculate the free energy difference due to formation of the liquiddrop.
Let, be the bulk free energy difference to obtain a liquid drop of unit volume.
Then the free energy to formation a liquid drop of radius ‘r’ is given by,
where ‘’ is the surface tension of liquid-gas interface
.
figure1: free energy of nucleation as a function of radius of nucleus (green line is surfacefree energy, red is bulk and black line is total free energy of formation)
This function is sketched in fig. It has a maximum at , where
and the value of corresponding to this maximum is,
Note that as super cooling increases, decreases, as increases and does not change with increasing super cooling.
figure2: free energy at different super-cooling (ΔT)
Since the nucleation is an thermally activated process this is the slowest and rate determining step of phase transition. In other words, the formation of a droplet of critical size constitutes the bottleneck of the whole phase transition process.
Assuming that an equilibrium embryo distribution is maintained at any stage of super cooling one can correlate the population of nuclei of critical size. The no. of embryo of size ‘k’ is given by,
where ‘’ is the total no. of particles.
Rate of Nucleation
In order to talk about the dynamics of the formation of clusters, it is convenient to talk in terms of the no. of particles in the cluster because then we can talk in terms of addition of particles to cluster or removal of particles from cluster. Nucleation theory aims at quantifying the net rate at which embryos of the stable phase(liquid) are constantly being formed and destroyed. The usual dynamical treatment is based on the following assumptions:
1. The clusters are developed by addition or removal of single particle.
2. The process is completely isothermal i.e. there is no rate problem involved in dissipation of heat of condensation.
3. The equilibrium concentration of embryos of ‘n’ particles can be expressed as
where is the equilibrium concentration of clusters having ‘i’ no. of particles,
is the free energy of formation of cluster of size ‘n’ relative to the cluster of size ‘1’.
Now, is given by
where is the change in bulk free energy per particle to form new phase.
Let us plot as a function of ‘n’
figure3: equilibrium free energy as a function of cluster size
figure4: Number density profile of clusters
The minimum is due to the maximum in at the critical size. The super critical nuclei are in unstable equilibrium and can not exists in number given by equilibrium theory.
For any ‘n’,, the difference between the rate at which droplets containing ‘n+1’ molecules are formed by single-molecule condensation onto n-molecule droplets and the rate at which they are destroyed by single-molecule evaporation, can be written as
where is the flux of per unit time and area of single molecules onto the n-molecule droplet, s(n) and s(n+1) are surface area of ‘n’ and ‘n+1’ sized clusters, respectively. f(n,t) and f(n+1,t) are concentrations of clusters of size ‘n’ and ‘n+1’ at time ‘t’, respectively. w(n+1) is the flux of single molecules leaving the ‘n+1’ molecule droplet. Of the two rate constants, can be calculated from kinetic theory, but ‘w’ is not known in general. To overcome this difficulty, equilibrium considerations are invoked. Assuming that an equilibrium distribution of droplets can be established in the bulk metastable phase, we can write, because of microscopic reversibility,
where we have assumed that the single molecule fluxes are independent of embryo size, and c(n+1) and c(n) are the equilibrium concentrations of droplets composed of ‘n+1’ and ‘n’ molecules, respectively. It is also assumed that the values of ‘w’ and ‘’ do not change in going from an equilibrium to a non-equilibrium situation. We will use this expression of ‘w’ to get
=
= -
= -
Now,
So,
This is a generalized diffusion flux for in a unidimensional space under a potential gradient. Remember that the diffusion coefficient in a discrete process (such as in random walk) is given by
is jump frequency and is mean square displacement. Here
[in this case =1 and ]
Then,
[ leads to diffusion equation]
Next we want to evaluate the rate of nucleation.
For steady state is independent of n.
or,
Boundary Conditions:
, as (equilibrium prevails for small clusters)
as ( true concentration is very small)
So,
J =
c(n) goes through a pronounced minimum at the critical size . Accordingly, it is a good approximation to remove the surface area from the integral
J =
Now, if we do Taylor series expansion of around then we get
= +
goes through a maximum at , so
< 0
Let,
Then
so,
with < 0
for all practical purposes
=
so,
Now, J =
=
Zeldovich factor Z is given by
Z =
so, Z =
The final rate of nucleation is,
J = Z
Limitations of Classical Nucleation Theory:
1. It is unable to predict rate of nucleation for a range of temperature. At low temperature it predicts low rate and at high temperature high rate compared to experimental rate of nucleation.
2. One of the assumption of nucleation theory is that nucleation happens through one particle addition or removal to the nucleating nucleus. At high super-saturation or high super-cool there exists other kinds of mechanisms like coalescence of two clusters of breaking up of one large cluster to several small sized clusters. Nucleation theory can not capture all these kind of pictures. As a result it fails badly at high super-saturation/super-cooling.
References:
1. Zettlemoyer, A. C. Nucleation (Dekker, New York, 1969).
2. Debenedetti, P. G. Metastable Liquids: Concepts and Principles (PrincetonUniversity Press, 1996).
3. Wolde, P. R. and Frenkel, D. Computer simulation study of gas-liquid
nucleation in a Lennard-Jones system. J. Chem. Phys. 109, 9901-9918 (1998).
4. Bhimalapuram, P., Chakrabarty, S. and Bagchi, B. Elucidating the mechanism of nucleation near the gas-liquid spinodal, Phys. Rev. Lett., 98, 206104, (2007) .