Nucleation critical nucleus

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. 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

If we compare eqns (7.11) and (7.3) we see that the expressions for the critical radius are identical for both homogeneous and heterogeneous nucleation. But the expressions for the volume of the critical nucleus are not volume is... 

The primary nucleation process is divided into two periods in CNT one is the so called induction period and the other is the steady (or stationary) nucleation period (Fig. 2) [16,17]. It has been proposed by CNT that small (nanometer scale) nuclei will be formed spontaneously by thermal fluctuation after quenching into the supercooled melt, some of the nuclei could grow into a critical nucleus , and some of the critical nuclei will finally survive into macroscopic crystals. The induction period is defined as the period where the nucleation rate (I) increases with time f, whereas the steady period is that where I nearly saturates to a constant rate (fst). It should be noted that I is a function of N and t,I = I(N, t). In Fig. 2, N and N mean the size of a nucleus and that of the critical nucleus, respectively. The size N is defined... 

Classical nucleation theory (CNT) shows that I is a product of the probability of diffusion and that of formation of a critical nucleus [1,4],... 

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. 

The free energy necessary for the formation of a critical nucleus AG in both primary and secondary nucleation processes does not depend on Mn, i.e., AG ps const, while only the diffusion coefficient D depends on Mn, i.e., I ex D(Mn). Therefore, the Mn dependences of I and V are not controlled by the formation process of a critical nucleus but are mainly controlled by the chain sliding diffusion process. 

There is a point at which these aggregates reach a critical size of minimum stability r and the free energy of formation AG is a maximum. Further addition of material to the critical nucleus decreases the free energy and produces a stable growing nucleus. The nucleation rate is the product of the concentration of critical nuclei N given by... 

The models incorporate two microscopic parameters, the site density and the critical nucleus size. A fit of experimental current transients to the models allows conclusions, for example, concerning the effect of additives on nucleation rate. Fabricus et al. found by analysis of current transients that thiourea increases the nucleation density of copper deposited on glassy carbon at low concentration, but decreases it at higher concentration [112], Schmidt et al. found that Gold nucleation on pyrolytic graphite is limited by the availability of nucleation sites [113], Nucleation density and rate were found to depend on applied potential as was the critical nucleus size. Depending on concentration, critical nuclei as small as one atom have been estimated from current transient measurements. Michailova et al. found a critical nucleus of 11 atoms for copper nucleation on platinum [114], These numbers are typical, and they are comparable to the thermodynamic critical radii [86],... 

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... 

The size of the spherical critical nucleus in 3D nucleation as a function of the... 

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 rate, Rn, of random nucleation is therefore obtained from Eqns. (6.4) and (6.5) by recognizing that the addition of one more particle i to the critical nucleus makes it supercritical, which means that it will grow further. A simple way to represent the transfer frequency s of i across the surface of a critical nucleus is as follows... 

Up to this point we have dealt with random nucleation processes in a homogeneous phase. However, in solids with many structural imperfections, it is very likely that nonrandom, heterogeneous nucleation takes place. The basic idea of this mode of nucleation is that the energy of the imperfection is brought into the energy balance of the critical nucleus. Let us demonstrate the basic idea with a dislocation line as the preferred nucleation site. We assume that a cylindrical precipitate (p) forms along the dislocation line and, in the spirit of Eqn. (6.2), we obtain per unit length of the nucleus... 

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