Nucleation number

Nucleation Number N = 0.5A//mn0 l D(fpIT) Compare nucleation time to impact time Dykhuizen [390]... 

Shape factor, defined by Eq, (27.16), dimensionless Nucleation rate, number/cm -s or number/ft -h Frequency factor in nucleation, number/cm -s also, mass production rate of crystals, kg/h or Ib/h... 

A metal particle forms inside two fused droplets (dimer) when both the micelles contain free metal atoms if the total number of free metal atoms is equal to or greater than a critical nucleation number n. ... 

The importance of the water core size in determining the final particle size, discussed in some detail in Chapter S, has been emphasized in the model simulation [160]. Another factor that affects the particle size and polydispersity is the critical nucleation number (n, the number of metal atoms in the reverse micelles required for nucleation to occur). The model simulation shows that both average and maximum particle sizes, as also polydispersity, increase with increase in n. The issue of polydispersity has been taken up again in Chapter 5 (especially Section 5.5). 

The model of Natarajan et al, [ 160] introduced a parameter , defined as the critical nucleation number for metal particle formation. It has been shown that with an increase in n, particles of a particular size should grow in number, and the polydispersity in size should also increase. The reason behind this is that with higher values of n, more free metal atoms will be available for growth. 

Figure 1-29 Nucleation number (A/) and nucleation rate (/) as functions of time (f), according to Gutzow 1980. (a) homogeneous steady state, (b) homogeneous non-steady state, (c) heterogeneous steady state, (d) heterogeneous non-steady state.

Table 63 The nucleation number density of Nd nuclei from j-t curves at each applied polmitial in [P2225][TFSA] including 0.5 mol dm [Nd(TFSA)5] ...

The number density of nuclei was also investigated as follows the nucleation number density, Nq, can be calculated from the maximum current, /max. and time, max. of the j-t curve based on the following equation [45] ... 

However as the amounts of matter concerned in nucleation are often very low, we prefer to use, in the place of the rate, / the frequency of nucleation, that is, the number of nuclei created per unit of time and, in the place of reactivity g, y the specific frequency of nucleation (number of nuclei created per unit of time and per unit of area) that can replace the reactivity. If is the number of Avogadro, wq the amount of matter of reference, and n the size of the stable nucleus (in number of structure elements), we have at aity time... 

Once nuclei form in a supersaturated solution, they begin to grow by accretion and, as a result, the concentration of the remaining material drops. There is thus a competition for material between the processes of nucleation and of crystal growth. The more rapid the nucleation, the larger the number of nuclei formed before relief of the supersaturation occurs and the smaller the final crystal size. This, qualitatively, is the basis of what is known as von Weimam s law [86] ... 

For a general dimension d, the cluster size distribution fiinction n(R, x) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between andi + dR. Assuming no nucleation of new clusters and no coalescence, n(R, x) satisfies a continuity equation... 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

For these sequences the value of Gj, is less than a certain small value g. For such sequences the folding occurs directly from the ensemble of unfolded states to the NBA. The free energy surface is dominated by the NBA (or a funnel) and the volume associated with NBA is very large. The partition factor <6 is near unify so that these sequences reach the native state by two-state kinetics. The amplitudes in (C2.5.7) are nearly zero. There are no intennediates in the pathways from the denatured state to the native state. Fast folders reach the native state by a nucleation-collapse mechanism which means that once a certain number of contacts (folding nuclei) are fonned then the native state is reached very rapidly [25, 26]. The time scale for reaching the native state for fast folders (which are nonnally associated with those sequences for which topological fmstration is minimal) is found to be... 

An increase in the time required to form a visible precipitate under conditions of low RSS is a consequence of both a slow rate of nucleation and a steady decrease in RSS as the precipitate forms. One solution to the latter problem is to chemically generate the precipitant in solution as the product of a slow chemical reaction. This maintains the RSS at an effectively constant level. The precipitate initially forms under conditions of low RSS, leading to the nucleation of a limited number of particles. As additional precipitant is created, nucleation is eventually superseded by particle growth. This process is called homogeneous precipitation. ... 

Those exponents which we have discussed expUcitly are identified by equation number in Table 4.3. Other tabulated results are readily rationalized from these. For example, according to Eq. (4.24) for disk (two-dimensional) growth on contact from simultaneous nucleations, the Avrami exponent is 2. If the dimensionality of the growth is increased to spherical (three dimensional), the exponent becomes 3. If, on top of this, the mechanism is controlled by diffusion, the... 

A larger number of smaller spherulites are produced at larger undercoolings, a situation suggesting nucleation control. Various details of the Maltese cross pattern, such as the presence or absence of banding, may also depend on the temperature of crystallization. 

Increases in the appHed static pressure increase the acoustic intensity necessary for cavitation, but if equal number of cavitation events occur, the coUapse should be more intense. In contrast, as the ambient pressure is reduced, eventuaUy the gas-fiUed crevices of particulate matter which serve as nucleation sites for the formation of cavitation in even "pure" Hquids, wiU be deactivated, and therefore the observed sonochemistry wiU be diminished. 

Fig. 3. Curve ihustrating the activation energy (barrier) to nucleate a crystalline phase. The critical number of atoms needed to surmount the activation barrier of energy AG is n and takes time equal to the iacubation time. One atom beyond n and the crystahite is ia the growth regime.

Polypropylene molecules repeatedly fold upon themselves to form lamellae, the sizes of which ate a function of the crystallisa tion conditions. Higher degrees of order are obtained upon formation of crystalline aggregates, or spheruHtes. The presence of a central crystallisation nucleus from which the lamellae radiate is clearly evident in these stmctures. Observations using cross-polarized light illustrates the characteristic Maltese cross model (Fig. 2b). The optical and mechanical properties ate a function of the size and number of spheruHtes and can be modified by nucleating agents. Crystallinity can also be inferred from thermal analysis (28) and density measurements (29). 

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