Nuclei, critical number growth

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

Much smaller values for the minimum number of atoms to form a stable growing crystal are observed for 3D nucleation of various atoms (e.g., Hg, Cu, Pb) on PL Here the number of atoms needed for the critical nucleus to ensure that growth continues varies from 5 to 15. If the planar surface of a metal is the catalyst, it is obvious that the fraction of atoms active—the surface ones—is an exceedingly tiny portion of the total number of atoms in the metal used. If, however, one uses small spheres, the fraction of the atoms actually on the surface and hence active in catalysis greatly... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Black and Davey (1988) describe a number of the interrelationships and practical aspects of the control of nucleation, crystal growth, and polymorphic transformation of amino acids. The factors described and demonstrated for primary nucleation of L-glutamic acid include temperature, critical nucleus, relationship of interfacial tension to solubility, thermal history, induction time, agitation, and effect of additive. 

During the induction period this process continues on the surface of the nucleation site until the critical cluster has collected the next ion to be added triggers nucleation. Crystal growth then can follow. For barium sulfate, La Mer concluded that the slope of the line in Figure 8-1 is six and therefore the nucleation of barium sulfate is a seventh-order reaction overall. The critical cluster is then (Ba" " S04")3, and the addition of the seventh ion, either Ba " or 804 , constitutes the final step of the nucleation process. The question of the number of ions in the critical cluster, however, is by no means settled. Christiansen and Nielsen concluded that for barium sulfate the number is 8. Johnson and O Rourke also concluded that the nucleation rate of this salt is proportional to the fourth power of the concentration. The concept of a small critical nucleus is intuitively satisfying in that the nucleus then requires only a small number of steps for its formation. On the other hand, application of the... 

Deposition of Pb in 5mM Pb(C104)2 and lOmM HCIO4 solution at large overpotentials occurs via a 3D island growth. It follows a progressive nucleation process at low overpotentials and is mainly initiated at the surface inhomogeneities. The number of atoms in the critical nucleus is estimated to be very small, about 11. The small Pb clusters are stable within a certain anodic potential range. 

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