Nucleation Gibbs energy

Figure 1.9 The balance of endothermic surface energy and the exothermic formation of the stable condensed phase during nucleation from the vapour phase. The critical radius, above which the nuclei become stable, is where the resultant Gibbs energy change has zero slope...

According to homogeneous nucleation dreory, dre critical Gibbs energy to form a nucleus is given by... 

The nucleation rate kjv is the inverse of the average time between two pulses. By varying the overpotential r/, the dependence of the nucleation rate on r can be obtained. In Section 10.3 we showed that for three-dimensional nucleation the Gibbs energy of formation is proportional to r) 2. A similar analysis for the two-dimensional case gives a proportionality to if1 instead (see Problem 10.1). Hence a plot of In k/y versus ry"1 should result in a straight line, which is indeed observed (see Fig. 10.8). 

The decomposition of a solution with composition outside the spinodal region but within the metastable region can be analyzed in a similar way. Let us assume that a sample with composition in this region is cooled to low temperatures. Small fluctuations in composition now initially lead to an increase in the Gibbs energy and the separation of the original homogeneous solution must occur by nucleation of a new phase. The formation of this phase is thermally activated. Two solutions with different composition appear, but in this case the composition of the nucleated phase is well defined at all times and only the relative amount of the two phases varies with time. 

In classical nucleation theory the Gibbs energy of a nucleus is considered as the sum of contributions from the bulk and the surface. Let us consider nucleation of a spherical crystal from its liquid below its melting temperature at 1 bar. The difference in Gibbs energy between a nucleus with radius r and its liquid is... 

Fig. 7.135. Gibbs energy of 2D cluster formation as a function of size N, with O(N) = 2e(bQ/y)1/2, e = 2 x 10-13 J cm-1, Jb = 4 for a square 2D cluster form, and 2 = 4 x 10-16 cm-2. The data are taken as most probable values from nucleation rate experiments on an Ag (100) quasi-perfect face. (Reprinted from E. Budevski, G. Staikov. and W. J. Lorenz, Electrochemical Phase Formation and Growth, p. 202, copyright 1996, John Wiley Sons. Reproduced by permission of John Wiley Sons, Ltd.)...

Figure 10-12. The Gibbs energy vs. nonstoichiometry 6 for binary compounds (e.g., Ag2+(5S). Nucleation of fi at d needs activation, in contrast to nucleation of /J at

Figure 5.1 Gibbs energy of nucleation of water drops at different water vapor oversaturations of at 25°C. Curve 1 is plotted for oversaturation p/poo = 8.1 = 0.5...

So-called underpotential deposited species arise when an electrochemical reaction produces first, on a suitable substrate adsorbent metal, a two-dimensional array or in some cases two-dimensional domain structures (cf. Ref 100) at potentials lower than that for the thermodynamically reversible process of bulk crystal or gas formation of the same element. The latter often requires an overpotential for initial nucleation of the bulk phase. The thermodynamic condition for underpotential deposition is that the Gibbs energy for two-dimensional adatom chemisorption on an appropriate substrate must be more negative than that for the corresponding three-dimensional bulk-phase formation. Underpotential electrochemisorption processes commonly involve deposition of adatoms of metals, adatoms of H, and adspecies of OH and O. 

According to the classical nucleation theory, the Gibbs energy for the formation of a critical 2D nucleus of a 2D Meads phase on an atomically flat terrace (T), AGcrit,T, is given by [3.249] (cf. eq. (4.25a)) ... 

The rate of nucleation / is a probability process connected with the energy of formation of the critical cluster, AGo-ib which owing to the creation of the new crystal/solution and crystal/substrate interfaces, is always positive. The probability of a fluctuation connected with an increase of the Gibbs energy AG of a system is given in the case of nucleation, with AG = AGcdt, by... 

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