Classical nucleation theory critical cluster size

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

The mechanisms of droplet (or liquid germs) formation from a supersaturated vapour phase is still the subject of many investigations. After giving a brief account of the classical theory [64], which, as shown above, provides a simple method for estimating the energy barrier to overcome before effective nucleation is started, and permits the estimation of the critical cluster size, a complementary approach will be presented. 

Using Equations 3.3a and b, Englezos et al. (1987a) calculated the critical radius of methane hydrate to be 30-170 A. In comparison, critical cluster sizes using classical nucleation theory are estimated at around 32 A (Larson and Garside, 1986), while computer simulations predict critical sizes to be around 14.5 A (Baez and Clancy, 1994 Westacott and Rodger, 1998 Radhakrishnan and Trout, 2002). 

The classical nucleation theory of Volmer (1939), Nielsen (1964), and others assumes the addition of single molecules to a cluster until it reaches a critical size ... 

Classical nucleation theory uses macroscopic properties characteristic of bulk phases, like free energies and surface tensions, for the description of small clusters These macroscopic concepts may lack physical significance for typical nucleus sizes of often a few atoms as found from experimental studies of heterogeneous nucleation. This has prompted the development of microscopic models of the kinetics of nucleation in terms of atomic interactions, attachment and detachment frequencies to clusters composed of a few atoms and with different structural configurations, as part of a general nucleation theory based on the steady state nucleation model [6]. The size of the critical nucleus follows straightforwardly in the atomistic description from the logarithmic relation between the steady state nucleation rate and the overpotential. It has been shown that at small supersaturations, the atomistic description corresponds to that of the classical theory of nucleation [7]. 

The classical nucleation theory, based on Gibbs thermodynamics statements, uses the macroscopic properties characteristic of bulk phases, such as free energies and surface tensions, for the description of small clusters. Contradictory results arose in early studies of electrochemical nucleation [9], where the size of a critical mercury nucleus on a platinum substrate amounted to only a few atoms, with properties that could substantially differ... 

Nucleation is the initial process leading to the formation of a new phase. Classical nucleation theory [11-13] describes homogeneous nucleation as the breakdown of a metastable state that occurs at a critical activation energy, which is achieved at a critical subcooling (in melts) or supersaturation (in solution). The homogeneous nucleus is conceived of as an aggregate of critical size in unstable equilibrium with the parent phase. At concentrations below the critical level the cluster grows or dissociates reversibly. 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

The main concern of the classical homogeneous nucleation theory has been a thermodynamic description of the initial stage of nucleation from embryo to nucleus with a little larger size over the critical one (Seinfeld 1986, Pruppacher and Klett 1997, Seinfeld and Pandis 1998, Kulmala et al. 2000). The change of the free enthalpy of the cluster is at first positive because the decrease of entropy is initially larger (regular structure formation) than the decrease in enthalpy ... 

The nucleation process has been discussed above in terms of the so-called classical theories stemming from the thermodynamic approach of Gibbs and Volmer, with the modifications of Becker, Doring and later workers. The main criticism of these theories is their dependence on the interfacial tension (surface energy), 7, e.g. in the Gibbs-Thomson equation, and this term is probably meaningless when applied to clusters of near critical nucleus size. 

Nowadays it is quite clear that it does not make sense to speak about two different theories of the nucleation rate the classical and the atomistic one. In reality, what we do have is a general nucleation theory comprising two limiting cases. The classical model describes the nuclei by means of macroscopic physical quantities and can be used to predict the size and to evaluate the nucleation work of sufficiently large critical clusters. The atomistic model is valid in the case of high supersaturations and very active substrates when the critical nuclei are very small. Therefore the quantitative interpretation of experimental data on the stationary nucleation rate based on the atomistic theory provides valuable information on the specific properties of clusters consisting of a few building units. 

By using the classical theory of ion induced nucleation to describe the growth of radon daughters from the free activity mode to the nucleation mode, we loose information about the size of the subcritical clusters. These clusters are all lumped together between the size of a pure H2O ion cluster at 75% r.h. and the size of the critical H2O-H2SO4 cluster. The model only does keep track of the growth by condensation of the radon daughters once they arrived in the nucleation mode. 

Under the specific conditions of electrochemical metal deposition, the critically sized clusters of the new phase have been found to consist of only a few atoms, where classical thermodynamic bulk quantities cannot be applied. Therefore, the original kinetic theory of Becker and Doering was further developed to an atomistic theory of nucleation. 

Studying the spontaneous appearance of two-dimensional clusters on the electrode surface one obtains direct information on the average time f] needed to form a 2D nucleus at a given overpotential 7. As we have seen in Chapter 3 (equations (3.8) and (3.10)), in the case of negligible non-stationary effects the time /, equals the reciprocal stationary nucleation rate. This has been used by Budevski et al. [4.16, 4.17] to examine the overpotential dependence of the stationary rate of two-dimensional nucleation. The obtained results confirm the validity of the classical theory of nucleation on a like substrate and provide the possibility to determine the nucleation work, the size of the two-dimensional critical nucleus and the specific free edge energy at the nucleus-solution interface boundary. 

---