Constrained Equilibrium Cluster Distribution

Equation (11.51) can be written in terms of the dimensionless surface tension 0 as 

TABLE 11.2 Surface Tensions and Densities of Five Organic Species at 298 K 

Recall that the constrained equilibrium cluster distribution is the hypothetical equilibrium cluster distribution at a saturation ratio 5 1. The constrained equilibrium cluster distribution obeys the usual Boltzmann distribution 

Another consistency issue arises with (11.57). The law of mass action (Appendix 11) requires that the equilibrium constant for formation of an i-mer, /A t A,- 

For small / and very large /, (10.57) departs from reality. When / = 1, N[ should be identical to N], but (10.57) does not produce this identity. The failure to approach the proper limit as / 1 is a consequence of the capillarity approximation and the specific 

FIGURE 10.2 Constrained equilibrium and steady-state cluster distributions, N- and N, respectively. 

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The constrained equilibrium cluster distribution, Nf, is based on a supposed equilibrium existing for S > 1. In actuality, when 5 > 1, a nonzero cluster current J exists, which is the nucleation rate. Let the actual steady-state cluster distribution be denoted by N,. When imax is sufficiently larger than /, the actual cluster distribution approaches zero,... 

The classical theory of homogeneous nucleation dates back to pioneering work by Volmer and Weber (1926), Farkas (1927), Becker and Doring (1935), Frenkel (1955), and Zeldovich (1942). The expression for the constrained equilibrium concentration of clusters (11.57) dates back to Frenkel. The classical theory is based on a blend of statistical and thermodynamic arguments and can be approached from a kinetic viewpoint (Section 11.1) or that of constrained equilibrium cluster distributions (Section 11.2). In either case, the defining crux of the classical thoery is reliance on the capillarity approximation wherein bulk thermodynamic properties are used for clusters of all sizes. 

It is important at this point to define three different cluster distributions that arise in nucleation theory. First is the saturated (5=1) equilibrium cluster distribution, Nf which we will always denote with a superscript s. Second is the steady-state cluster distribution at 5 > 1 and a constant net growth rate of y, N,. At saturation (5 = l)all 7,+ 1/2 =0, whereas at steady-state nucleation conditions all 7, +1 /2 = 7. There is a third distribution that we will not explicitly introduce until the next section. It is the hypothetical, equilibrium distribution of clusters corresponding to 5 > 1. Thus it corresponds to all 7,+ 1/2 = 0, but 5 > 1. Because of the constraint of zero flux, this third distribution is called the constrained equilibrium distribution, Nf. We will distinguish this distribution by a superscript e. 

Figure 19.4 Cluster size distribution, (a) System at equilibrium (Njy). (b) System in constrained equilibrium (JV q) and for nonequilibrium system during nucleation (Nm)-...

Thus the unknown steady-state cluster distribution N, disappears, allowing the nucleation rate to be expressed in terms of the constrained equilibrium distribution Nf. We see that (11.73) is equivalent to (11.11), in that (11.65), which defines the constrained equilibrium distribution Nf, is identical to (11.5) that definesf. 

The coefficients in Eq. 19.7 may be taken as constants independent of the form of the distribution of cluster sizes. The relationship between the two coefficients fa and qaa+i may then be obtained by imposing an artificial constraint on the system no clusters are allowed to grow beyond a limiting size, Af m, which is considerably larger than the critical size Afc. At the same time, all clusters of size below this limit are allowed to equilibrate with respect to one another so that detailed balance is achieved between them and all fluxes in cluster space go to zero. A new distribution of cluster sizes, N q, will be produced in this constrained system. It is assumed that the same free-energy minimization procedure used previously to find the size distribution under true equilibrium conditions, and which led to Eq. 19.6, may be used for the constrained system, resulting in... 

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