Nucleation work

We have already stated in Chapter 1.1.2 that to supersaturate the parent phase may not be enough to initiate the phase transition. Why is the nucleus formation not an instantaneous process In order to answer this question we shall consider the formation of an n-atomic nucleus of the new phase from n single metal ions of the electrolyte solution. The supersaturation is given by equation (1.15) and the nucleus is formed on the surface of an inert foreign substrate. 

To immerse the working electrode in the electrolyte solution means to create a new dividing surface in the bulk of the liquid phase. The work F done in this process accounts for the energy excess due to the creation of the new interface and takes into consideration the energy contribution of all physical processes that may take place at the solution-substrate phase 

For sufficiently large, finite working electrodes, G can be split into a 

In the final state when the n-atomic metal nucleus is already formed (Fig. 1.1b) the Gibbs free energy of the whole system changes to  

The term G (n) in equation (1.30) accounts for the energy contribution of the foreign substrate, as changed by the presence of the n-atomic nucleus thereon. Thus it is the difference G (n) - G that gives the Gibbs free energy G(n) of the n-atomic cluster on the electrode surface. It depends on the cluster structure, which may not necessarily coincide with the stmcture of the bulk new phase. 

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Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

Instead of supramolecular synthons, in nucleation work it is more common to talk about a growth unit defined as the es-sential building block that transfers structural information from the solution to the crystal surface. The relationship between supramolecular synthons, which come from observation of complete crystal structures, and growth units, is at present poorly understood. 

Nucleation — Atomistic theory of nucleation — Figure 1. Dependence of the nucleation work AG (ft) on the cluster size n (a) and dependence of the critical nucleus size nc on the supersaturation Ap (b) according to the atomistic nucleation theory (a schematic representation)... 

Note that the quantity AG (n) is called also a nucleation work. 

Eqs. (3.110) and (3.111) show that both the nucleus size and nucleation work become infinitely large if A// = 0, i.e. if C = Ce Physically, this means that formation of nucleus holes in the bilayer is then impossible. Accordingly, the bilayer is then truly stable (and not metastable) in respect to rupture by hole nucleation. It must be emphasised that the bilayer also retains this true (or infinite) stability for A/r < 0, i.e. for C > Ce, since both terms in Eqs. (3.108) and (3.109) are then positive and W, can only increase with increasing i (Fig. 3.84). Thus, as it requires more and more work to be done, the overgrowth of the randomly formed holes is then suppressed and the bilayer cannot rupture despite the presence of a certain population of holes in it. 

With W, from Eq. (3.109) it follows from Eq. (3.127) that the nucleation work Wei in the presence of an applied electric field is of the form... 

Parameters of hole nucleation and molecular characteristics of bilayers. Theoretically, the fitting constants A, B and Ce in Table 3.12 contain important information. Using their values in Eqs. (3.107), (3.110), (3.111), (3.126) and (3.129), the following characteristic parameters of the process of hole nucleation in the foam bilayers can be evaluated nucleation work W, number C of surfactant vacancies in the nucleus hole, specific... 

Kane DB, Johnston MV (2000) Size and composition biases on the detection of individual ultrafine particles by aerosol mass spectrometry. Environ Sci Technol 34 4887-4893 Ka rcher B, Turco RP, Yu F, Danilin MY, Weisensdn DK, Miake-Lye RC, Busen R (2000) A unified model for ultrafine aircraft particle emissions. J Geophys Res 105 29379-29386 Kashchiev D (1982) On the relation between nucleation work, nucleus size, and nucleation rate. J Chem Phys 76 5098-5102... 

Oxtoby DW, Kashchiev D (1994) A general relation between the nucleation work and the size of the nucleus in multicomponent nucleation. J ChemPhys 100 7665-7671 Park PW, Ledford JS (1997) Characterization and CH4 oxidation activity of Cr/Al203 catalysts. Langmirir 13 2726-2730... 

In nudeation theory, Wf, i is the heterogeneous nucleation work as compared with the homogeneous nucleation work W. The heterogeneous nucleation refers to the fact that the gas bubble forms at the interface of a liquid and a solid phase. Note that 0 ,, is simply which varies in the range of 0... 

According to the classical thermodynamics, the difference AG n) = G2 -G gives the energy barrier for the formation of the nucleus consisting ofn atoms and is called nucleation work. Thus from equations (1.28)-(1.30) for AG(n) one obtains ... 

Equation (1.31) tells us that what we call nucleation work is the difference between the Gibbs free energy G(n) of n atoms, when they form an individual n-atomic cluster ofthe new phase on the foreign substrate, and the... 

The general expressions for the nucleation work AG n), equations (1.31) and (1.32), are valid for condensed, liquid and solid, phases when the total volume of the supersaturated system does not change essentially if an n-atomic nucleus forms. In this form they apply without any restrictions even to clusters consisting of a very small nunAer of particles, including the limiting case n= Similar formulae for AG in) were derived by Kaischew [1.14,1.15] who considered the homogeneous and heterogeneous formation of crystalline nuclei from vapours and melts. 

The above thermodynamic derivation of the nucleation work raises the important question for the meaning of the concept electrochemical potential of the atoms of a small cluster. In section 1.1.1 we defined the... 

The above considerations have been performed expressing the supersaturation Ap by the difference p ° (a )-p if) (equation 1.15). The same general formula for the nucleation work AG in) (equation 1.32)... 

Jhe general expression for the nucleation work, equation (1.32), says that AG(n) has a minimal value for clusters formed with a minimal excess energy ( ). In terms of the classical nucleation theory developed in the pioneering works of Gibbs [1.1], Volmer [1.11], Volmer and Weber [1.16], Kossel[1.17], Stranski [1.12, 1.13, 1.18, 1.19], Farkas [1.20], Stranski and... 

Before deriving explicit expressions for the nucleation work it is necessary to define the concept specific free surface energy [1.25]. For that purpose let us consider two finite solid bulk phases with equal size, Phasei and Phase2 (Figure 1.3a) immersed in a solution of metal ions which represents a third, liquid phase. The pressure P and the temperature T are kept constant and the two solid phases are polarised to the same potential E. 

The general formula for the nucleation work AG(n) = -nAji +0 n) (equation (1.32)) provides the possibility to obtain explicit expressions for this quantity if the surface free energy 0(n) is evaluated accounting for the supersaturation dependence of the nucleus size MAji). To illustrate the thermodynamic method developed by Gibbs [1.2] and Volmer [1.11,1.16] we shall calculate the woik of formation of two-dimensional (2D) and three-dimensional (3D) liquid and crystalline nuclei on flat foreign substrates. For the sake of simplicity in these calculations we shall consider the idealized case of a stmctureless substrate thus neglecting any lattice mismatch between the electrode surface and the nuclei of the new phase. 

Figure 1.15- Dependence of the nucleation work A G(n) on the size of the nucleus according to the classical nucleation theory (equation (1.55)). The dependencies of the terms nA p and K(ro) in equation (1.55) on the size n are given by dashed lines.

It must be pointed out that equation (1.56) can also be derived without calculating the nucleation work [1.65-1.67], In order to demonstrate this we consider an electrolyte solution containing metal ions with... 

Thus far we derived analytical expressions for the nucleation work and for the size of the critical nucleus neglecting the line tension effect. The latter can be taken into consideration if the general equation (1.32) is combined... 

Equation (1.70) tells us that the line tension may increase or decrease the nucleation work depending on the sign of. What is more, for negative values ofthe line tension the condition = 0 applied to equation... 

For high supersaturations and/or strong adhesion between the substrate and the new phase the classical nucleation theory predicts the possibility to form two-dimensional nuclei on a foreign substrate. Brandes [1.74] was the first who in 1927 considered this case of phase formation and found that the nucleation work equals one half of the total edge energy of the critical nucleus. Here we shall derive explicit expressions for the nucleation work... 

If two-dimensional nuclei with a crystalline stmcture are formed on the foreign substrate the nucleation work and the size of the critical nucleus are... 

Under these conditions the nucleation work AGiNio. = -N acA/I + 0 (.N3d,c) becomes ... 

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