Critical nucleus value

Fig. 7.1 The electron density p(t) is displayed in the and

In Fig. 6.2a AGj is plotted as a function of j for a few values of the saturation ratio a/ao = S. Obviously the activation energy AG decreases with increasing saturation ratio, as does the size of the critical nucleus, r or rj. 

When a similar theory (which appears objectionable to the present reviewer also on other grounds) was applied to the formation of ice in water droplets160), the critical nucleus < was > assumed to be a hexagonal prism of height equal to the short diameter . No capillary pressure acts across plane faces of a prism. Nevertheless the author found a value (for the 7s] of water - ice) near 20 erg/cmz for drops of about 0.002 cm in diameter at —37 °C. 

Much smaller values for the minimum number of atoms to form a stable growing crystal are observed for 3D nucleation of various atoms (e.g., Hg, Cu, Pb) on PL Here the number of atoms needed for the critical nucleus to ensure that growth continues varies from 5 to 15. If the planar surface of a metal is the catalyst, it is obvious that the fraction of atoms active—the surface ones—is an exceedingly tiny portion of the total number of atoms in the metal used. If, however, one uses small spheres, the fraction of the atoms actually on the surface and hence active in catalysis greatly... 

Solution. Important assumptions include that the interfacial free energy is isotropic, that elastic strain energy is unimportant, and that the nucleation rates mentioned are for steady-state nucleation. The critical barrier to nucleation, AQe, can be calculated for the 0.3 atomic fraction B alloy using the tangent-to-curve construction on the curves in Fig. 19.18b to provide the value Aga = —9 x 107 Jm-3 for the chemical driving force for this supersaturation at 800 K. AQc is given for a spherical critical nucleus by... 

The number of guest molecules in a locally ordered arrangement exceeds that in the critical nucleus. Guest-guest and host-host cluster order parameters take on values that are very close to the clathrate hydrate phase, which results in formation of a critical nucleus. 

Figure 4.6 shows the dependence on r for both contributions with small values of r its square is predominant and AG increases with increasing r the nucleus will stop growing and (with homogeneous nucleation) it disappears. From a certain value of r, the critical nucleus size, rk, AG decreases upon growth the nucleus is then stable and continues growing. The value of rk can be easily calculated at rk ... 

The critical nucleus size is given by the values of l and that minimize AG (Equation 10-24) ... 

From a critical value of S = S., given by the a -intercept, the nucleation rate increases with a veiy steep slope and then asjmiptotically approaches its maximum value. The critical value of S depends on A and n, the critical number of atoms, molecules, or ions in the critical nucleus. This critical number is obtained finm the following equation,... 

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 ... 

In eq. (IV.3) the first term is positive and increases proportionally to r2 with increasing r. The second term may be negative (in the case of supersaturation - Ap>0), with absolute value increasing proportionally to r3. Thus, in the case of supersaturation, the W(r) curve must pass through a maximum. This maximum is characteristic to some critical particle size, rcr, corresponding to the critical nucleus of a new phase. The critical nucleus of size rzx exists in equilibrium (unstable equilibrium) with the mother medium, i.e. the p = ir condition (p,. is the chemical potential of the substance in the nucleus) stays valid. For such a nucleus, one can write in agreement with eq. (1.13) that... 

In the absence of supersaturation (pt = j.K) the W(r) dependence is parabolic, W(r)=4nr2a (Fig. IV-2) rcr °°, and also Wcr - 00. Upon penetration into the metastable region (p > pK), the maximum appears in the W(r) curve, i.e. Wcr and rcr both have finite values that decrease as the supersaturation, Ap, increases. The work of the critical nucleus formation, Wcr, can be thus viewed as the height of the energy barrier that one needs to overcome in order to make further spontaneous growth of nuclei possible. 

One may expect (a more detailed derivation will be given below) that eq.(IV. 15) remains valid for a heterogeneous nucleus, i.e. the work of a critical nucleus formation is proportional to the nucleus volume. Then the work of heterogeneous formation of critical nucleus, Wcrhe equals the work of homogeneous formation of a critical nucleus, Wcrhom, multiplied by the ratio of nuclei volumes, i.e. by the value of/(0), namely... 

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