Critical nuclei, definition

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. 

There are two typical definitions of the induction time (ti) in CNT given by Frisch [16] and by Andres and M. Boudart [17]. x is an increasing function of N, t,(N). In previous studies, the special case N = N was usually focused on. As any critical nucleus can not be directly observed, Tj(N ) has been estimated from r (N) of macroscopic nuclei by optical microscopy by correcting the time necessary for growth from N to N. Therefore, x (N ) is named r (OM) in this work. It should be noted that there is no guarantee that the estimated Xi(N ) = r (OM) is correct, that is also an important unresolved problem. 

The definitive hydrate kinetic inhibition mechanism is not yet available. Some work suggests that the mechanism is to prevent hydrate nucleation (Kelland, 2006). However, a significant amount of evidence suggests that hydrate kinetic inhibitors inhibit the growth (Larsen et al., 1996). However, this apparent conflict is due to the definition of the size at which crystal nucleation stops and growth begins. To resolve this confusion, one may consider growth to occur after the critical nucleus size is achieved. 

The limit thexmodynamlc nonequilibrium corresponds to the attainment of liquid superheats at which intensive spontaneous boiling is observed on nucleus bubbles of fluctuation nature. The physical definiteness of this boundary is conditioned by a very shazi> dependence of the nucleation rate J(T. P ) on the Gibbs number G- = A T where is the work of formation of a critical, nucleus [4 5j. is the Boltzmaim constant. By the homogeneous nucleation theory the value of J is calculated making use of thermodynamic parameters. Thus, for water at atmospheric pressux T have J ... 

The above results allow the definition of one important feature of the assembly of discotic molecules in isotropic solution. There seem to be conditions (controlled by temperature, concentration, and solvent type) in which contact forces are weak and loose binding of the unimers produces short columns with low DP and little or no chiral amplification. Cooperative growth ensues even though a detailed mechanism is often unclear. The critical nucleus size is not readily identified from theory (cf Section n.B.3) but might be associated with the number of disks included in the pitch of the highly correlated helix forming when contact forces increase at low temperature. 

Strictly speaking, nucleation does not have a definite kinetic order, although experimental log 7 = f (log J) curves do fit approximately on a straight line over a few decades of J [22]. Within a small concentration range, it is possible to write J = k C", where n is roughly equal to n , the number of monomers in the critical nucleus. This is only an approximation because n is itself a function of S [equation (2.3)], and the probability of a simultaneous collision between n monomeric precursors is zero. More probably, the formation of the critical nucleus is the result of con.secutive bimolecular steps. However, it may depend on an j th power of the concentration, considering that, since all steps are very fast, they all contribute to the global equilibrium [30]. 

We speak about unstable equilibrium because attaching more atoms from the parent phase the critical nucleus turns into a stable cluster and grows irreversibly. On the contrary, detachment of atoms from the critical nucleus leads to its irreversible decay. In the classical nucleation theory the definitions (1) and (2) are fully identical. However, we shall show in Chapter 1.4 that the first definition is more general. 

Similarly to the classical theory the Walton s treatment is based on the idea for a critical nucleus the formation of which is the rate-determining step of the phase transition. However, the definition of the critical nucleus is conformed to the specificity of the small clusters and the high supersaturations. Thus it is assumed that the critical nucleus may consist of a very low number of atoms - two, one and even zero atoms in the case of very active substrates. The c-atomic critical nucleus is considered as a cluster having a probability of decay higher than or equal to the probability... 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

A subcritical aggregate having fewer subunit components than a nucleus. When this term is applied in the kinetics of precipitation, n refers to the number of subunits in a particle and n defines the number of subunits in a particle of critical size. This definition avoids confusion by distinguishing between subcritical (n < n subunits), critical (n = n subunits), and supercritical (n > n subunits) particle sizes. If a nucleus is defined as containing n n subunits, then an embryo contains n n subunits. Note that in this treatment, we are not using a phase-transition description to describe nucleation, and we are focusing on the smallest step in the process that leads to further aggregation. 

There are several other points of view regarding the definition of the nucleus of zeolite. For example, it was suggested that some primary structural units of the framework, such as rings and basic cages, could be defined as the nucleus of zeolites and other microporous crystals. It was also proposed that the nucleus of zeolite could be defined as particles with critical size. These particles should be stable under crystallization conditions. Compared with the classical theory of nucleation from homogeneous solution, the theory developed by Pope could well explain the significant decrease of the free-energy barrier of nucleation for zeolites and other microporous compounds.[43] This... 

Of particular importance are (3, -1) CPs. They correspond to a minimum of density along a direction and to a maximum in the two perpendicular directions. At a molecular equilibrium geometry, they are called bond critical points (BCPs) and then enable to determine which atoms are bonded in the orthodox QTAIM sense two atoms are bonded if and only if a BCP is present between them, the union of the two GPs linking the BCP to each nucleus being named the bond path . Note that such a characterization has raised some controversies in the last decades (see Ref. [93] for a recent discussion). However, the existence of a bond path is a fundamental feature that has been explicitly mentioned in the lUPAC definition [1] of halogen bonds. 

A simple inspection of equation (1.55) shows that the function AG(n) displays a maximum at a certain critical value ttc of the nucleus size (Figure 1.15). Hence the first definition of the concept cntica/nMcfeiM follows ... 

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