Heterogeneous nucleation critical radius

For heterogeneous nucleation, critical radius of a stable solid particle nucleus... 

Figure 10.6 Schematic free-energy-versus-embryo/nucleus-radius plot on which are presented curves for both homogeneous and heterogeneous nucleation. Critical free energies and the critical radius are also shown.

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... 

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... 

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

It appears that both compatibilization and the nanostructure formation at the interface play a key role for nucleation. The supposed heterogeneous nucleation activity will therefore be discussed in more detail. Heterogeneous nucleation in general is strongly affected by the particle size and the interfacial properties [79, 80], As the particle size of the PPE phase is well above the critical radius of nucleation of several nanometers [80], the interface demands closer examination. 

When the concentration is closer to the supersaturation limit, heterogeneous nucleation occurs most often. The nucleus develops onto the substrate, with which it makes a contact angle a. Solution of the equations for the nucleus size and activation energy imply that the critical radius is the same as for homonuclear... 

The mechanism of bubble formation by nucleation requires supersaturation of the dissolved gas [11-13] and a nucleus radius greater than the critical [7], The main sources of heterogeneous nucleation are usually surface irregularities capable of containing entrapped gas, e.g. pits and scratches. The bubbles typically develop over the electrode surface, grow in size until they reach a break-off diameter and subsequently detach into the electrolyte. After detachment, some residual gas remains at the nucleation site and another bubble will form at the same place [2,13,14], In most two-phase flow simulations [15-19], it is assumed that bubbles detach with a constant diameter, although from experiments [20,21] it is know that electrochemically formed bubbles show a size distribution. 

The conditions of heterogeneous nucleation, as well as the work of formation and the curvature radius of the critical nucleus, may be significantly influenced by the linear tension of the perimeter of wetting, x if x<0, the formation of heterogeneous nuclei becomes easier... 

The discussion above applies to heterogeneous nucleation on flat surfaces. For atmospheric particles the curvature of the surface complicates the situation. Fletcher (1958) showed that the free energy of formation AG of an embryo of critical radius r on a spherical nucleus of radius R is given by... 

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... 

Nucleation by ideal nucleants should be energetically favorable in comparison with homogeneous nucleation in the polymer matrix and heterogeneous nucleation by other additives or contaminants. This favorable energetic balance could be found in surfaces with weak interactions with the polymer matrix and with radii of curvature larger than the radius of the critical nucleus in CNT. 

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. 

From Equation 10.13, it is important to note that the critical radius r for heterogeneous nucleation is the same as for homogeneous nucleation, inasmuch as -/sl is the same surface energy as y in Equation 10.3. It is also evident that the activation energy barrier for heterogeneous nucleation (Equation 10.14) is smaller than the homogeneous barrier (Equation 10.4) by an amount corresponding to the value of this S(ff) function, or... 

Critical radius for stable sohd particle (heterogeneous nucleation)... 

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 ... 

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