Nucleation critical radius

In heterogeneous systems Looking for an industrially accessible route to develop polymer nanoporous materials, other approaches were developed based on heterogeneous precursors (ie, systems in which heterogeneous nucleation takes place). On the one hand, addition of particles to the polymer (such as talc, titanium oxide, kaolin, nanosilica, and other nanoparticles) can increase the nucleation ratio [77-81]. The size of the individual particles is amain issue They should present a size of the same order of magnitude, or higher, than the critical nucleation radius of the polymer-C02 system [82]. In addition, the particles should be weU dispersed to increase the potential nucleation sites individual particle volumetric density should be of the same order, or higher, than the desired nucleation density. However, the low... 

However, qualitative estimations based on the CNT equation were performed sustaining the hypothesis of a predominant pore nucleation inside the PBA core-shell nanodomains. It was established that, depending on the exact critical nucleation radius for each foaming conditions (further studies are required to determine accurately these values), this nucleation should happen inside the PBA phase or around the PMMA inner core (Fig. 9.31). This hypothesis was partially confirmed by some TEM micrographs of the nanostructuration after the foaming process, showing blown micelles in the foamed samples (Fig. 9.32) [101],... 

FIGURE 7.6 Critical nudeus radius of silicon calculated by homogeneous nucleation theory for 0.2 atm and 0.7 atm. Atomic radius of silicon is 0.146 run. From Sawano [18]. 

Obviously, A F p) increases for small p (where the surface term SdPd X fmi dominates), reaches a maximum AF at a critical droplet radius / , and then decreases again due to the negative volume term. In this classical nucleation theory, it is straightforward to obtain the critical droplet radius R from... 

Fig. 1. Calculated homogeneous nucleation rate and critical nuclei radius as a function of supersaturation...

FIGURE 13.6. In nucleation processes there wiU exist a critical particle radius, below which free energy considerations will drive the incipient aerosol particle to disappear. Above rc particle growth should occur. 

The radius of a nucleation exclusion zone can be calculated on the basis of the following discussion, taking into account the charge transfer overpotential also. If there is a half-spherical nucleus on a flat electrode, the extent of the deviation in the shape of the equipotential surfaces which occurs around it depends on the crystallization overpotential, current density, a resistivity of the solution and on the radius of the nucleus, r. If the distance from the flat part of the substrate surface to the equipotential surface which corresponds to the critical nucleation overpotential, rj, is /n, then this changes around defect to the extent where A is a number, as is presented in Fig. 2.18. 

The activation energy barrier against heterogeneous nucleation (AG het) is smaller than AG hom by the shape factor fid). In addition, the critical nucleus radius (r ) is unaffected by the mould wall and only depends on the undercooling. This result was to be expected since equilibrium across the curved interface is unaffected by the presence of the mould wall. 

Using y—= lO " J/m, a, the critical crack radius, computes to 6 nm from Eq. (19). Lin [10] observed fatigue cracks as small as 200 nm along slip bands in copper, and Kwon [19], using replicas, observed fatigue cracks that were only 100 nm deep along slip bands in copper. The just-nucleated cracks in copper could have been smaller because of the limitations of the observation techniques. If y—ya = 10 J/m, a is predicted to be 60 nm. The model thus predicts a ball park value for the critical crack radius from the observed decrease in (Tmax per cycle. 

Figure 1.9 The balance of endothermic surface energy and the exothermic formation of the stable condensed phase during nucleation from the vapour phase. The critical radius, above which the nuclei become stable, is where the resultant Gibbs energy change has zero slope...

When the nucleus is a liquid, the angle 6 is called tire wetting angle. It can be seen that the critical radius in heterogeneous nucleation is given by the same equation as tlrat for homogeneous nucleation, but the radius now refers... 

Figure 10.2 The time-temperature-nucleation curve showing the balance between the rate of nucleation and the critical radius which produces a maximum rate...

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 nucleation rate is, in fact, critically dependent on temperature, as Fig. 8.3 shows. To see why, let us look at the heterogeneous nucleation of b.c.c. crystals at grain boundaries. We have already looked at grain boundary nucleation in Problems 7.2 and 7.3. Problem 7.2 showed that the critical radius for grain boundary nucleation is given by... 

Grain boundary nucleation will not occur in iron unless it is cooled below perhaps 910°C. At 910°C the critical radius is... 

Nucleation of solids from liquids critical radius for homogeneous and heterogeneous nucleation... 

Here R is a critical radius for nucleation [20,56] on a faceted surface... 

At higher temperatures, other degrees of freedom than the radius R must also be considered in the fluctuation. However, this becomes critical only near the critical point where the system goes through a phase transition of second order. The nucleation arrangement described here is for heterogeneous or two-dimensional nucleation on a flat surface. In the bulk, there is also the formation of a three-dimensional nucleation, but its rate is smaller ... 

Figure 17. Energy for the nucleation of a surface film on metal electrode. M, metal OX, oxide film EL, electrolyte solution. Aj is the activation barrier for the formation of an oxide-film nucleus and rj is its critical radius. 7 a is the interfacial tension of the metal-electrolyte interface, a is the interfacial tension of the film-electrolyte interface. (From N. Sato, J. Electro-chem. Soc. 129, 255, 1982, Fig. 5. Reproduced by permission of The Electrochemical Society, Inc.)...

Using the properties of water Li and Cheng (2004) computed from the classical kinetics of nucleation the homogeneous nucleation temperature and the critical nu-cleation radius ra. The values are 7s,b = 303.7 °C and r nt = 3.5 nm. However, the nucleation temperatures of water in heat transfer experiments in micro-channels carried out by Qu and Mudawar (2002), and Hetsroni et al. (2002b, 2003, 2005) were considerably less that the homogeneous nucleation temperature of 7s,b = 303.7 °C. The nucleation temperature of a liquid may be considerably decreased because of the following effects dissolved gas in liquid, existence of corners in a micro-channel, surface roughness. 

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