Nuclei phase boundary

The existence of the phase boundary between the solid and liquid phase complicates matters, since a phase boundary is associated with an increase in free energy of the system which must be offset by the overall loss of free energy. For this reason the magnitudes of the activated barriers are dependent on the size (i.e. the surface to volume ratio of the new phase) of the supramolecular assembly (crystal nucleus). This was recognized in 1939 by Volmer in his development of the kinetic theory of nucleation from homogeneous solutions and remains our best model today (Volmer 1939). 

If it is assumed that nucleation of phase B is equally probable on all inner and outer surfaces of the crystal of A, and that the rate of the phase boundary reaction for the growth of phase B is constant, then the rate law for the dissociation reaction can be calculated, provided that the nucleation probability is known as a function of time. In the simplest case, there will be An possible nucleation sites, each of which has an equal a priori probability of becoming an actual nucleus. If n is the number of nuclei already present, then the rate of nucleation is ... 

For homogeneous nucleation in condensed systems diffusion to the phase boundary becomes an important factor and A/, the energy of activation for this diffusion, must be added to AF, the free energy of formation of a critical size nucleus, AF = 6na l3 AGy. A detailed treatment of this problem has been given by Turnbull and Fisher ( ). 

In this relation. No is the number of crystallizable elements that can give rise to nuclei (entities of size higher than the critical one), Eo is the energy of activation of diffusion of the crystallizable matter through phase boundary, and, finally, AG is the free energy of crystallization of a nucleus that has reached the critical size. These two terms vary in opposite directions depending on the temperature considered. 

Particle growth occurs by long-range atomic diffusion, which normally involves several steps—for example, diffusion through the parent phase, across a phase boundary, and then into the nucleus. Consequently, the growth rate G is determined by the rate of diffusion, and its temperature dependence is the same as for the diffusion coefficient (Equation 5.8) —namely,... 

Similar MC calculations were used by Trout s group to study the carbon dioxide-liquid water interface at 220 K and 4 MPa near the phase boundary of a carbon dioxide hydrate (273 K and 4MPa). Nucleation was achieved by seeding the system with a cluster of carbon dioxide hydrate. It was found that a small cluster with diameter <9.6 A dissolved into the solution readily. A hydrate crystal started to grow, however, when a hydrate cluster twice that size (19.3 A) was implanted into the system. The crystal eventually spanned the whole system (Figure 22). Thus the critical nucleus size for hydrate nucleation is estimated to be about 19 A consisting of approximately 200 water molecules. This is a considerably smaller number than that estimated from the local harmonic model of around 600 molecules. The theoretical results refuted the labile cluster hypothesis.This hypothesis speculates the agglomeration... 

Pure titanium is cooled from a temperature at which the b.c.c. phase is stable to a temperature at which the c.p.h. phase is stable. As a result, lens-shaped nuclei of the c.p.h. phase form at the grain boundaries. Estimate the number of atoms needed to make a critical-sized nucleus given the following data AH = 3.48 kJ moT atomic weight = 47.90 - T = 30 K = 882°C y= 0.1 ]ra density of the c.p.h. 

The appearance of a twin nucleus associated with phase transitions occurs preferentially, in many cases, on free surfaces, internal surfaces of inclusions, boundary surfaces of solid inclusions, or in dislocations where strain is concentrated. 

Fig. 1. The H — T phase diagrams for different S/F systems, (a) Structure with narrow domains. The solid (dashed) line corresponds to an isolated superconducting nucleus at the domain boundary (far from the domain boundary), (b) Isolated superconducting nucleus in a structure with wide domains BoD2/To = 25). (c) Periodic domain structure for ttBow2/To = 5 (solid line) and -nBow2 /To = 1 (dashed line), (d) Ferromagnetic dot over the superconducting film (N = 10).

Clem and Fisher (1958) use a similar treatment as above to derive the solid state nucleation kinetics for new phases at grain boundaries. They neglect orientation of the critical nucleus with respect to the host, strain energy, and coherency effects. Nucleation at the grain boundary interface removes boundary energy. Their treatment yields the following critical values ... 

The existence of metastable states is caused by the activation character of the initial stage of a first-order phase transition. Homogeneous nucleation determines the upper boundary of the liquid superheat and supercooling. The appearance of a viable new-phase nucleus in a metastable liquid is connected with the performance of the work IT. determined by the height of the thermod5mamic potential barrier, which is to be overcome for the subsequent irreversible growth of a new phase. The dimensionless complex W, k T, where kg is the Boltzmann constant and T is the temperature, is the stability measure of the metastable phase. ... 

The formation of a new phase inside a solid phase is very difficult, because the transition generally implies a change in density, hence in volume. This leads to a change in pressure, and thereby to an additional, generally large and positive, term in free energy for nucleus formation. Except for some solid —> solid transitions where the change in density is small, this tends to prevent nucleation any formation of a new phase will occur at the boundary of the system. Sublimation then does not need nucleation the new phase (gas) is already present. The same holds for a solid phase (crystals) in a solution. When crystals in air tend to melt, a liquid... 

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