Supersaturation critical cluster size

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. 

Equation (13.5) shows the critical cluster size decreases with increase of the relative supersaturation S or a reduction of a by the addition of surfactants. This explains why a high supersaturation and/or addition of surfactants favours the formation of small particles. A large S pushes the critical cluster size N to smaller values and simultaneously lowers the activation barrier, as illustrated in Figure 13.2, which shows the variation of AG with radius at increasing S. 

Fig. 2. Free energy of formation of a cluster of size i from a supersaturated vapor. The free energy is a maximum at the critical cluster size i. ...

Fig. 3. Calculated time dependence of the cluster concentrations and currents after compression of water vapor to a supersaturation of p/po = 9.24 at —In this calculation, the critical cluster size for nucleation was — Note that the currents and concentrations are strongly time dependent only in the first microsecond of the calculation.

Equation (2.12) shows that the critical cluster size is inversely proportional to the degree of supersaturation. As the supersaturation increases so the critical size decreases and consequently the solutions become less and less stable. Crystal growth becomes a stable process because as the nuclei are formed the concentration of solute in solution is reduced and the system becomes thermodynamically stable. Substituting eqn (2.12) into eqn (2.7) gives... 

Note that the free energy barrier depends more strongly on surface tension than on the free energy difference. From the preceding discussion, we get the expected result that both the critical cluster size (/ ) and the free energy barrier (AG(R )) decrease with increase in supersaturation or supercooling and the rate of nucleation increases following the Arrhenius rate expression. 

Within a classical macroscopic picture the stability of a supersaturated vapor is a consequence of the surface related activation barrier associated with the growth of small clusters. As indicated in Fig. 1 the free energy change for the formation of small clusters from a vapor is positive and increases with cluster size until a "critical cluster size is attained. Beyond the critical cluster size subsequent cluster growth results in a decrease in the free energy and the process occurs spontaneously. Because the growth up to the critical cluster size is responsible for the stability of the supersaturated vapor, it is important that it be well understood. 

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 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. 

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)... 

Apart from the purely thermodynamic analysis, the description of the -> electro crystallization phenomena requires special consideration of the kinetics of nucleus formation [i-v]. Accounting for the discrete character of the clusters size alteration at small dimensions the atomistic nucleation theory shows that the super saturation dependence of the stationary nucleation rate /0 is a broken straight line (Figure 2) representing the intervals of Ap within which different clusters play the role of critical nuclei. Thus, [Ap, Apn is the supersaturation interval within which the nc -atomic cluster is the critical nucleus formed with a maximal thermodynamic work AG (nc). 

As shown in the previous chapter, molecular clusters are always present even in an unsaturaied gas. When a system becomes supersaturated, these clusters increase in concentration and pass through the critical size d by attachment of single molecule,s. The formation of stable nuclei relieves the supersaturation in the gas. Because condensation nuclei are generated by the vapor itself, the process is known as homogeneous mtdeation or self-nudeation. 

This makes it possible for clusters below some critical size to be energetically unfavorable and hence unstable with respect to disassociation. Depending on the shape of the crystal, the surface energy terms, and the supersaturation, there exists a value of j for which the free energy is at maximum and therefore the attachment of one more molecule will make the crystal stable. The size of the island corresponding to this maximum is known as the critical island size, denoted by i. The value of is determined by the relevant interaction potentials between the molecules and the substrate. The details ai e worked out explicitly in Markov [3] and Taylor et al. [16]. 

A given supersaturation thus corresponds to a crystal seed of length in labile equilibrium. Aggregation of molecules to form clusters (ca. 20-100 molecules) with the critical crystallite size is only possible in the case of a local deviation in the supersaturation of the solution. Figure 2.3.5-3 shows a schematic plot of the supersolubility curves, which lie above the saturation curve and run approximately parallel to it. They indicate to which degree of supersaturation spontaneous crystallization is not observed in a technically acceptable time (20-30 min).This experimentally determined time is an important quantity in the design of industrial crystallizers. 

Further molecular additions to the critical cluster would result in nucleation and subsequent growth of the nucleus. Similarly, ions or molecules in a solution can interact to form short-lived clusters. Short chains may be formed initially, or flat monolayers, and eventually a crystalline lattice structure is built up. The construction process, which occurs very rapidly, can only continue in local regions of very high supersaturation, and many of the embryos or sub-nuclei fail to achieve maturity they simply redissolve because they are extremely unstable. If, however, the nucleus grows beyond a certain critical size, as explained below, it becomes stable under the average conditions of supersaturation obtaining in the bulk of the fluid. 

A graph of AG, versus n displays a maximum (n ), which corresponds to the number of molecules in a critical cluster that is in equilibrium with the supersaturated solution. The addition of one more molecule to this cluster will cause it to spontaneously grow to macroscopic size. The value of n is found from the derivative of Eq. (9.21). 

Primary nudeation refers to the (homogeneons) spontaneous formation of nuclei of the crystallizing phase. The critical cluster or critical nucleus is the minimum size that a continuously growing nucleus has to surpass in order to make the transition to a stable crystalline phase. Secondary nudeation or seeding describes the process whereby nuclei are induced on or near crystals (i.e., the seeds) of the solute that are already present in a supersaturated solution. Moreover, the seeds are not required to have the same crystal... 

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