Critical minimum crystal size

As seal from Eq. (7.251), the value of TV depends upon several parameters of the system, e.g., the edge surface energy, . It also depends on the overpotential T, and one can see that the size of the critical cluster decreases with an increase in hjl. For 2D nucleation on quasi-perfect silver single crystals, the number of atoms in the minimum nucleus size at which AG begins to decrease with an increase in A varies from 25 to 67 atoms as t varies from -10 to -6 mV. 

A certain minimum-sized fragment is required to induce crystallization. The size of such a critical nucleus is on the order of 100 molecules (a diameter of 100 A) (Van Hook 1961). 

In the case of structural studies with x-rays, sample size is rarely a serious problem since single crystals as small as about 0.001 mm3 (sealed x-ray tube) or 10-9 mm3 (synchrotron radiation) will do. In the case of x-ray diffuse scattering measurements with the assembly described in Section VI.D, the crystal must be 2 mm long at least but can be relatively thin. With neutrons, however, sample size is more critical. Indeed, for structure measurements, 0.1 mm3 is a minimum, with the highest neutron fluxes available nowadays, and more usual sizes are in the range of several cubic millimeters to several tens of cubic millimeters. For polarized neutrons or inelastic scattering studies, much larger samples are necessary. 

Figure 4 A commonly used conceptual model for the formation of gas hydrate from water and gas. Since gas is only poorly soluble in water, hydrate forms on surfaces as well as in the bulk. Standard nucleation theory suggests that water-gas clusters form sub-critical nuclei/clusters of the thermodynamic product which at a certain minimum size become large enough to survive. Once crystals form they will grow and/or agglomerate.

Slow addition of B to minimize the concentration of R in solution by allowing time for nucleation and/or growth (particle size is not a key issue in this case because the R is resolubiUzed after completion of the reaction for the subsequent purification by extraction. R is crystallized only for minimizing S and maximizing yield). The effect of addition rate on yield/selectivity is shown qualitatively in Fig. 11-7, where it can be seen that a critical rate can be determined at which the crystallization rate can maintain the minimum concentration of R. At higher addition rates, selectivity decreases as the R concentration increases. 

The radius of the crystal to where the free energy of formation of the spherical crystal (AGt) is at its maximum is known as the critical radius this proves to be the minimum size to begin the formation of a crystal. Determine the critical radius for water with IOC (AT = 10 [K]) of subcooling. HINT Search around 1.1 x 10 and 1.2 x 10 [m]. 

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... 

An extremely important conclusion reached from Eq. (9.113) is lhat corresponds to the minimum size for the thermodynamic stability of a mature crystallite, even if growth in the lateral dimensions is unrestricted. Thus, if the value of given by Eq. (9.113) is maintained, the mature crystallite that evolves will be thermodynamically stable at the crystallization temperature. However, such a crystal will melt at a temperature just infinitesimally above the crystallizahon temperature, irrespective of the value of p. Hence, if this type of surface, or growth, nucleus is involved in a real crystallization process, a mechanism must be provided by which the longitudinal dimension increases beyond critical size in order for a crystallite to be stable above the crystallization temperature. This mandatory requirement is an inherent property of a Gibbs type nucleus. This requirement holds irrespective of the molecular species involved since it is based on a straightforward thermodynamic requirement. It will become evident shortly that this requirement is very important in polymer crystallization. In contrast, thermodynamic stability can be achieved in a three-dimensional nucleus without any need for increasing C. ... 

---