Nuclei, critical number

Both terms in Eq. (7.1) are functions of the size of the cluster N. The first term increases linearly with N, and the second increases as Dependence of the energy of formation of a cluster AG(A0 on the number of adions A/ in a two-dimensional (2D) cluster is shown in Figure 7.1. It is seen from the figure that AG initially increases, reaches a maximum, and then decreases with increasing N. At the maximum, the cluster size is N. The size of the critical nucleus (the number of atoms in the cluster) in 2D nucleation is given by... 

Figure 19.10 A critical number of interactions are required to form a folding nucleus. The figure is oversimplified in that the enthalpy is shown to decrease throughout folding. In practicej there will be some gain in enthalpy as the transition state is reached because of the loss of energy of solvation and the incomplete formation of the interactions present in the native state.

From a critical value of S = S., given by the a -intercept, the nucleation rate increases with a veiy steep slope and then asjmiptotically approaches its maximum value. The critical value of S depends on A and n, the critical number of atoms, molecules, or ions in the critical nucleus. This critical number is obtained finm the following equation,... 

The process of particle formation from dissolved ions can be represented in the following order ions—monomers— nuclei—particles. After a stable nucleus is formed, it can grow by the following processes (a) incorporation of ions and aggregation of primary particles or nuclei to form bigger particles. In order to form a stable nucleus, a cluster containing a critical number of monomer (N J must form. An important parameter in this connection is the ion occupancy number, i.e., the number of reactant species in an inverse micelle. A nucleus is formed if the ion occupancy number is greater than [3]. 

It is well established that in order to obtain a stable nucleus, a cluster containing a critical number of monomers ( c) must form [126-128]. It follows, therefore, that for microemulsion-mediated synthesis, if the water pools are viewed as isolated microreactors, then a relevant question is, How many of the available water pools contain the minimum number of monomers needed for nucleation Here the concept of occupancy number ( oc) is helpful [13]. For a reactant solubilized in the reverse micellar pseudophase, the quantity Hoc represents the number of solubilisate molecules present in a given water pool. 

The fact that not N - 50 (the magic number for isolated nuclei) but iV = 48 is the critical number has to do with the fact that the forming fragment is not an isolated nucleus but is part of a scissioning system. In this context, it is worth remembering that in the transition from asymmetric fission to symmetric fission the limiting neutron number is 158 for Fm (Z = 100) and 154 for Rf (Z = 104) (O Fig. 4.16), i.e., a few neutrons less than 164, which would represent twice the magic number of 82. 

In this relationship, the excess free energy for the formation of clusters AG first increases then decreases with the radius of the particles, r. The critical radius, r, is associated with a maximum excess free energy, AG. Accordingly, there exists a critical number of atoms, , in cluster A at the radius of r. When r < r, the system can lower its free energy by dissolution of clusters. Thus, these clusters are not thermodynamically stable and dissolve quickly, whereas some new clusters form due to spontaneous collisions. These unstable particles (A < A ) are known as clusters or embryos, and their numbers follow the Boltzmann distribution and decrease exponentially with increases of AG as described in Equation 10.4. When the radius of a cluster is larger than the critical value (r> r ), it becomes stable and is referred to as a nucleus. Thus, the expressions of critical radius r and maximum excess free energy AG can be obtained mathematically when dAG,/dr is equal to zero [22, 23, 28] ... 

The observed minimum in the curve was adequately explained, provided a critical number of Ni(ll) ions is assumed for the formation of one nucleus. This number was determined to be equal to 2 for the formation of Ni2B and Co2B (refs. 1,9,10,15,16). 

The dynamic picture of a vapor at a pressure near is then somewhat as follows. If P is less than P , then AG for a cluster increases steadily with size, and although in principle all sizes would exist, all but the smallest would be very rare, and their numbers would be subject to random fluctuations. Similarly, there will be fluctuations in the number of embryonic nuclei of size less than rc, in the case of P greater than P . Once a nucleus reaches the critical dimension, however, a favorable fluctuation will cause it to grow indefinitely. The experimental maximum supersaturation pressure is such that a large traffic of nuclei moving past the critical size develops with the result that a fog of liquid droplets is produced. 

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].

Pure titanium is cooled from a temperature at which the b.c.c. phase is stable to a temperature at which the c.p.h. phase is stable. As a result, lens-shaped nuclei of the c.p.h. phase form at the grain boundaries. Estimate the number of atoms needed to make a critical-sized nucleus given the following data AH = 3.48 kJ moT atomic weight = 47.90 - T = 30 K = 882°C y= 0.1 ]ra density of the c.p.h. 

Neutrons produced in a chain reaction are moving very fast, and most escape into the surroundings without colliding with another fissionable nucleus. However, if a large enough number of uranium nuclei are present in the sample, enough neutrons can be captured to sustain the chain reaction. In that case, there is a critical mass, a mass of fissionable material above which so few neutrons escape from the sample that the fission chain reaction is sustained. If a sample is supercritical,... 

It should be noted that the critical nucleation process does not depend on M. This can be explained by our model of surface diffusion (Fig. 27). In the model a nucleus will be formed from the absorbed chains. We can estimate the number of repeating units within a critical nucleus (N ) using parameters a, ae, and Ah given in [14]. N is the order of 102-103 for the range of AT in our experiment, which is much smaller than the number of repeating units within a molecule (103-104). This indicates that a critical nucleus should be formed by a part of a molecular chain. Therefore, the nucleation process of the critical nucleus will not depend on M. Thus, it is a natural result that B does not depend on M in this study. This is consistent with the discussion by Hoffman et al. [28] on FCC. They showed that the nucleation process of an FCC does not depend on Mn in the case of Mn > 104. On the contrary they showed that it depends on Mn for Mn < 104, because ae depends on Mn due to the effect of chain ends on the end surface of the critical nucleus. 

---