An estimate of the penetration distance for the error-function solution (Eq. 5.23) is the distance where , or equivalently, = , which corresponds to
Since the climb distance that results from the destruction of vacancies is now , using Eqs. 3.64 and 3.66,
Comparing the last result with Eq. 11.10, we see that the curvature given by is constant everywhere and that the principal normal vector at any point on the helix is pointed toward the helix axis and is perpendicular to it. Also, the vectors and are constant vectors independent of . If we now take the axis of the helix to be along so that and put the results above into Eq. 11.46, we find that it is satisfied if
Experimentally determined plot of vs.tex2html_image_mark>#tex2html_wrap_inline8542# for a (111) tilt boundary with a tilt angle in Al containing Fe solute atoms. boundary mobility. From Molodov et al. .
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Standard relationships for volumes and areas show that the volume of the nucleus is given by while the area of the two facets is and the area of the spherical portion of the interface is . The free energy to form the nucleus is therefore
Let , , and be the energies (per unit area) of the liquid/particle, liquid/solid, and solid/particle interfaces, respectively. From Section 19.2.1 the volume of the solid nucleus is , the spherical liquid/solid cap area is , and the solid/particle area is . The free energy of nucleus formation on the particles is then
On the other hand, for the homogeneous nucleation of a spherical solid nucleus in the bulk liquid, .
Using the linear approximation in Fig. 20.14, . Also, the conservation of atoms requires that the two shaded areas in the figure be equal. Therefore, . Putting these relationships into Eq. 20.103 gives