Nucleation theorem

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 nucleation theorems have been used to obtain information about critical clusters from experimental data, see Ford IJ (1997) Phys Rev E 56 5615 Ford IJ (1996) J Chem Phys 105 8324. These theorems state that, from known size and internal energy of the critical cluster, the nucleation rate can be deduced as a function of the temperature and the supersaturation. 

Experimental data of the dependence of J on concentration can yield an estimate of n., especially for the conversion of vapors to condensed phases. The essential relation is provided in the nucleation theorem, as follows (Oxtoby and Kashchiev 1994) ... 

This equation states that the change in the free energy of the critical germ with the chemical potential per molecule of species / in the original phase (i.e., the mother liquor) equals the negative of the excess number An of molecules of type i in the nucleus over that present in the same volume of original space. The nucleation theorem is independent of the model and of the transition it holds true for classical nucleation theory, density functional theory, or cluster kinetic analysis and for gas-to-liquid or liquid-to-solid conversions. 

Borensen C, Kirchner U, Scheer V, Vogt R, Zellner R (2000) Mechanism and kinetics of the reactions of NO2 or HNO3 with alumina as a mineral dust model compound. J Phys Chem A 104 5036-5045 Borys RD, Lowenthal DH, Mitchell DL (2000) The relationships among cloud microphysics, chemistry, and precipitation rate in cold mountain clouds. Atmos Environ 34 2593-2602 Bowles RK, McGraw R, Schaaf P, Senger B, Voegel JC, Reiss H. (2000) A molecular based derivation of the nucleation theorem. J Chem Phys 113 4524-4532... 

Vehkamaki H, Maattanen A, Lauri A, Kulmala M, Winkler P, Vrtala A, Wagner PE (2007) Heterogeneous multicomponent nucleation theorems for the analysis of nanoclusters. J Chem Phys 126 174707... 

The nucleation theorem [48, 49] is an exact relationship between the partial derivative of the reversible work of formation of the critical nucleus with respect to the chemical potential of a component in the bulk metastable phase, and the size and composition of the critical nucleus. 

Oxtoby, D.W., and Laaksonen, A. (1995) Some consequences of the nucleation theorem for binary fluids, J.Chem.Phys. 102, 6846-6850. 

S(/) can be described by the well-known Avrami theorem [3.317], supposing multiple nucleation on a quasi-homogeneous substrate surface with a sufficient density of nuclei, statistically local distribution of nuclei, and overlapping of growing 2D islands ... 

This is the Gibbs-Wulff theorem as generalized by Kaischew for a crystal in contact with a substrate, i.e., for the case of heterogeneous nucleation and growth. Relation (4.10) allows the construction of the equilibrium form of a crystal ... 

Hartman P (1973) Crystal Growth An Introduction. North-Holland Pubhshing Co, Amsterdam Herring C (1951) Some theorems on the free energies of crystal strrfaces. Phys Rev 82 87-93 Hettema H, McFeaters JS (1996) The direct Monte Carlo method applied to the homogeneous nucleation problem. J Chem Phys 105 2816-2827... 

The same strategy was applied in the derivation of rate equations for w-step nucleation according to a power law (cf. Eq. (21)) [133, 134], the combination of nucleation laws with anisotropic growth regimes [153], as well as truncated nucleation due to time-dependent concentration gradients of monomers [136]. MC simulations verified that the Avrami theorem is valid for instantaneous [184], progressive [185], and n-step nucleation according to a power law [184-187]. 

As previously considered with respect to the three-dimensional nucleation situation, the expression for the current transient corresponding to multiple two-dimensional nucleation requires consideration of the coalescence of growth centers, which diminishes the edge length available for attachment of new atoms or molecules, and produces a decay of the current. By means of the Avrami theorem (see Sect. 5.3.4.1), the following expressions are obtained for the limiting cases of instantaneous and progressive two-dimensional nucleation ... 

In all the pseudo-steady state modes of nucleation, as the space function always has the same value for the whole of the steps, the theorem of the equahty of the speeds (see section 7.4.2) can be applied to the reactivities and thus... 

Theorem.- The rate of a monodispersed powder during a reaction with instantaneous nucleation and slow growth is equal to the rate of one grain of this powder. 

Using this theorem, we can conclude that Tables A.3.1 to A.3.5 of Appendix 3 are applicable to the powders made up of monodispersed grains and which change with a veiy fast nucleation and a very slow growth. In these tables, we find a certain number of known laws as those of Valensi [VAL 36], Ginstling and Brounshtein [GIN 50], and a certain number among those listed by Sharp et al. [SHA 66]. 

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