Nucleus, stable clusters

It is not obvious how AF varies with the size of a cluster, because vv depends on the size, but an indirect scheme is available for determining the desired information. For one particular size, R0, there is assumed to be a value of AF for a condition of stability. This means that for a superheated liquid at a stated temperature and pressure, one and only one cluster size is capable of existence for long. This cluster is called a nucleus. A stable cluster is really in a metastable state, as discussed later. However, for any degree of equilibrium, AF must be unaffected by infinitesimal changes in the cluster size. So d(AF)/dR = 0. If vv is defined as the volume occupied by one vapor molecule, then ny = 47r.fi o8/(3ty)- These two manipulations produce a solution for the quantity r v — vl in Eq. (40)... 

Generally, nucleation is a kinetic process where a small number of atoms form a stable cluster of atoms arranged closely to the structure of the new phase or phases, named nucleus, within the old phase. This nucleus subsequently operates as the primary construction block for a growing grain (see Figure 3.1). 

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. 

Nucleation is the process by which atoms (or ions) that are free in solution come together to produce a thermodynamically stable cluster. The cluster must exceed the critical nucleus size, n, and then it becomes a supercritical nucleus capable of further growth. If the nucleus is smaller than the critical size, spontaneous dissolution can occur. In the simulation, if the number of products inside the same droplet is smaller than the n parameter, they are considered free inside the droplet. When the number of products is equal or greater than n, they come together forming a stable nucleus. This nucleus has to be exchanged as a whole. 

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. 

In the same time, the condition co. loXri< means that a single atomjoins the 3-atomic cluster before its disintegration and the configuration in Figure 3c, in which each atom is connected with three bonds is relatively stable at the temperature T. In terms of the atomistic model the 2-atomic cluster is defined as a critical nucleus, the 3-atomic one is defined as the smallest stable cluster and the stationary nucleation rate is ... 

The above considerations raise a very important question What do we measure in a progressive nucleation experiment Is dN t)/dt a nucleation rate, a rate of appearance of active sites or it is a combination of the tree rate constants As we have seen, depending on the values ofK , Ka and Ka ihs. experimentally accessible physical quantities ht and to may have completely different physical significance although what we only can and do measure in a direct nucleation experiment is the number of nuclei vs. time, N(t) vs. t, relationship. Note that for sufficiently high values of Kf, or Kf (cases 2.4.2.2.2, equation (2.141) or 2.42.2.3, equation (2.144), respectively) the induction time to may become too short to be experimentally detected. Therefore even in the case of experimental N t) relationships without induction periods [2.72, 2.176] one would not be able to assert that the constant rate / of appearance of stable clusters on the electrode surface would relate to the actual process of nucleus formation. Clearly, we are in exactly the same situation when data on the nucleation kinetics are obtained indirectly, by measuring the current of progressive nucleation [2.169]. 

In order to obtain a homogenous and stable latex compound, it is necessary that insoluble additives be reduced in particle size to an optimum of ca 5 )Tm and dispersed or emulsified in water. Larger-size chemical particles form a nucleus for agglomeration of smaller particles and cause localized dispersion instabiHty particles <3 fim tend to cluster with similar effect, and over-milled zinc oxide dispersions are particularly prone to this. Water-soluble ingredients, including some accelerators, can be added directly to the latex but should be made at dilute strength and at similar pH value to that of the latex concentrate. 

Nucleation is the growth of clusters of molecules that become a thermodynamically stable nucleus. This process is dependent on the vapor pressure of the condensable species. The molecular clusters undergo growth when the saturation ratio, S, is greater than 1, where saturation ratio is defined as the actual pressure of the gas divided by its equilibrium vapor pressure. S > 1 is referred to as a supersaturated condition (14). 

Knowledge concerning the mechanism of hydrates formation is important in designing inhibitor systems for hydrates. The process of formation is believed to occur in two steps. The first step is a nucleation step and the second step is a growth reaction of the nucleus. Experimental results of nucleation are difficult to reproduce. Therefore, it is assumed that stochastic models would be useful in the mechanism of formation. Hydrate nucleation is an intrinsically stochastic process that involves the formation and growth of gas-water clusters to critical-sized, stable hydrate nuclei. The hydrate growth process involves the growth of stable hydrate nuclei as solid hydrates [129]. 

Much work will be required before the details of these electron reactions and the structure of (HCl) are fully understood. However, Raff and Pohl111 have estimated that the binding energy of an electron in an HC1 dimer complex (i.e. ClH-e-HCl) would be 22 kcal.mole-1. Since the electron affinity of HC1 cannot be more112 than a few kcal.mole-1, a complex of this type would be expected to form a more stable nucleus for (HC1)(-) than localisation of the excess electron on a single HC1 molecule in the centre of the cluster. Nevertheless, there may be a stage in the formation of (HC1)(-) at which the electron is associated with only one molecule. 

This may continue until eventually the cluster is large enough to be thermodynamically stable (i.e., will not redissolve). However, if the cluster is smaller than the critical nucleus size, then there is the possibility that the nucleus will redissolve. The lifetime of the nucleus will then depend on its size and also on the temperature lower temperatures will slow the redissolution step. Thus lower temperature increases the chance that a subcritical nucleus will eventually grow to a stable size rather than redissolve. This kinetic stabilization of small nuclei results in a greater total density of nuclei and therefore smaller crystal size, since the total quantities of reactants are fixed. 

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