Supersaturation intervals

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). 

Being an integer, ATcrit changes with changing supersaturation taking discrete values only. Hence, to every discrete value of iVcrit a supersaturation interval of stability can be ascribed. TTiis is clearly seen in Fig. 4.6. In the supersaturation interval between A// = 2 and 2.5 U mol ], for instance, the critical cluster consists of 8 atoms. At higher supersaturations, e.g., between Aju = 2.5 and 3 iV/vv i, a 6-atomic cluster forms the nucleus, followed by a 2-atomic in the interval A/j = 3 and 4 N ln For A// > 4 the nucleus is one atomic, showing that each adsorbed atom can be considered as the nucleus of the new phase. 

Increasing the supersaturation above the value 3 y/ changes the size itc of the critical nucleus from 6 to 3 atoms and the 3-atomic cluster remains the critical nucleus within the supersaturation interval 3 < A fi 3.75ip. Lines b and c in Figure 1.28 correspond to supersaturations Aju = 3.25 i... 

Having revealed the discrete character of the nJiAp) relationship it is important to derive a general expression for the length L of the supersaturation interval within which a given cluster plays the role of a critical nucleus [1.115,1.116]. For the purpose we consider again Figure 1.28. 

Figure J.30 Supersaturation intervals corresponding to critical nuclei consisting of q, and Pc atoms.

Concluding, we should emphasize that equations (1.147) and (1.148) allow us to obtain useful information on the free energy excess (n) of small clusters. This can be achieved without the implication of any assumptions concerning the cluster stmctuie and the interatomic interactions if the limiting supersaturations Aju and Ap" are determined from experimental studies of the nucleation kinetics performed within sufficiently wide supersaturation intervals [1.67, 1.115-1.117]. Making use of equations (1.149-1.151) and assuming some model ofthe cluster stmcture, information can also be obtained on the separation works and on the bond energies ofthe atoms in the critical nuclei. 

These simple considerations show that the temperatures T and T2 mark the upper and the lower limit of a supersaturation interval within which the 2-atomic cluster plays the role of a critical nucleus. Of course, we should note that in a real physical system the choice of a stable configuration cannot be made by means of such elementary geometric arguments, particularly if the clusters consist of a larger number of atoms. Therefore, a real nucleation experiment may not necessarily show exactly the sequence of critical nuclei presented in Figure 2.5. 

The difference between the two models, as far as such difference does exist, should be sought for in the supersaturation dependence of the critical nucleus size. In the classical model this dependence is determined by the Gibbs-Thomson equation which juxtaposes a different critical cluster to each supersaturation. An analytical expression is proposed also for the supersaturation dependence of the stationary nucleation rate. The atomistic model takes into account the discrete character of the clusters size alteration at small dimensions (see Chapter 1.4) and does not propose a simple analytical relation between c and Ap. Beside that, it accounts for the fact that a supersaturation interval and not a fixed supersaturation corresponds to each critical cluster. This changes also the shape of the supersaturation dependence of the stationary nucleation rate which, in coordinates In/ / vs. Ap, is a broken straight line (Figure 2.6) representing the intervals of... 

Since for small clusters the size tie of the critical nucleus is constant within a given supersaturation interval and the free energy excess term ) of the... 

As we have already seen in Chapter 1.4, if clusters consisting of c> c and pc atoms are critical nuclei in three neighboring supersaturation intervals, the quantity Z( c) equals where 2l and " are the Hmiting... 

Suppose that clusters consisting of JCo. Xi, Xi,..., x atoms are critical nuclei in neighboring supersaturation intervals and that the limiting supersaturations are correspondingly... 

Update the crystallization set W(t +i) Activate the supersaturated salts SM tn+i) that are expected to crystallize in the time interval (tn,tn+l)... 

Increase in temperature causes a slight decrease in refractive index. The heat of dissolution 4 of the octahedral form is - 7530 calories at 18° C. The velocity of crystallisation from supersaturated solutions corresponds with 5 - dcjdt = fre4, where c is the concentration the temperature coefficient for the interval 0 to 25° C. is zero. 

These interacting process parameters are illustrated in Fig. 5-3, where supersaturation is plotted against an average solution concentration that would be experienced during addition over the indicated time interval (each point represents an average—one point per run, not a sequence of points in one mn). The amount of seed is shown as a parameter in allowing an increased addition rate. This concept is also valid for antisoivent addition time and reactive reagent addition time. 

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