Facet plot

Figure 3 Measured time dependences of the facet widths for Cases A and B are shown in a log-log plot. All data in each case fall into a line indicating that the reconstructed terrace width increases as w. The a values of the fitting lines are 1/2 for Case A and 1/4 for... 

The arguments advanced here apply equally well to 110 facets. This conclusion is illustrated schematically in Fig. 4. Figure 4a shows the y-plot obtained by Heyraud and Metois for pure Pb . Figure 4b is a modification of Fig. 4a, in which the cusps at 111 and 100 have been deepened, and new cusps have been created at 227) and 110) orientations. The depth of the cusps has been adjusted to produce an equilibrium form by means of the Wulff constmction (Fig. 4c) that matches the experimental equilibrium form of Fig. 2d. The deepening of the cusps needed to produce the desired shape is of the order of a few percent, and falls within the general range of anisotropy measured previously in the case of pure Pb. 

Transmission electron micrographs (TEM) of submicrometer-size particles show faceted particles, and selected area electron diffraction (SAED) patterns of isolated particles show that they are formed by a small number of crystallites (Fig. 9.2.14a), This result is consistent with the mean size of the crystallites, which can be inferred from the x-ray diffraction lines broadening analysis using a William-son-Hall plot (35) in order to take into account the contribution of microstrains to the line broadening. Over the whole composition range, the mean crystallite size is in the range 40-60 nm for particles with a mean diameter in the range 200-300 nm (Table 9.2.5) (33). 

For surfaces with the two-dimensional 7-plot shown in Fig. 14.6, certain inclinations will be unstable and will be replaced by other inclinations (facets), even though this increases the total surface area. Whether a certain inclination is unstable and prone to facet into other inclinations can be determined by a simple geometrical construction using the 7-plot [8]. The surface will consist of two different types of facets, as in Fig. 14.7a. The energy of such a structure per unit area projected on the macroscopically flat surface, 7, is... 

It may also be seen that the energies of all other surfaces with inclinations varying between those at B and C will fall on the dashed circle. All of these surfaces will therefore be faceted. On the other hand, a similar construction shows that all surfaces with inclinations between those at C and D will be stable against faceting into the inclinations at C and D. Points such as those at B and C where the dashed circle is tangent to the 7-plot therefore delineate the ranges of inclination between which the surface is either faceted or nonfaceted. The construction indicated in Fig. 14.6 is readily generalized to three dimensions three facet planes could then be present, as in Fig. 14.76, and c then terminates at the point of intersection of three planes rather than two lines. 

Figure 14.8 shows a three-dimensional 7-plot comprised of eight equivalent spherical surface regions. The shape of this 7-plot is consistent with all surfaces represented by the plot being composed of various mixtures of the three types of facets,... 

Other constructions employing the 7-plot are reviewed in Section C.3.1. These include the reciprocal 7-plot, which is also useful in treating the faceting problem above, and the Wulff construction, which is used to find the shape (Wulff shape) of a body of fixed volume that possesses minimum total surface energy. 

Note that when the interfacial energy is isotropic and the 7-plot is a sphere, the Wulff shape will also be a sphere. However, if the 7-plot possesses deep depressions or cusps at certain inclinations such as in Fig. C.4a, the planes normal to the radii of the plot at these inclinations will tend to dominate the inner envelope, and the Wulff shape will be faceted. In such cases, the system is able to minimize its total interfacial energy by selecting patches of interface of particularly low energy even though the total interfacial area increases. 

The means to determine the minimum-energy shape for a crystal of fixed volume was developed by Wulff (38), who showed that the equilibrium shape can be determined if the surface tension, y, at all crystallographic orientations is known. As illustrated in Fig. 2, on a polar y plot of the surface tension as a function of orientation, the inner envelope of the planes drawn perpendicular to and at the ends of the radius vectors gives the equilibrium shape of a crystal of constant volume. Faceting in the equilibrium crystal shape is due to cusps in the polar y plot. 

Fig. 2. A schematic Wulff construction for an equilibrium crystal shape using the polar y plot of the surface tension, (a) The equilibrium shape is that found from the inner envelope of tangents to the y plot, (b) An ECS with (001) facets produced by cusps in the y plot (39). (Reprinted from Prog. Surf. Sci., Volume 39, E. H. Conrad, Page 65, Copyright (1992), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 IGB, UK.)...

As might be expected, this is easiest to visualize in two dimensions, but the principles apply directly to three dimensions as well. For sodium chloride, at low temperatures, a plot qualitatively similar to Figure 2.5a is obtained. In this two-dimensional section [taken through the (100) plane], the center square corresponds to the Wulff construction described above and it is the absolute minimum energy. The 100 facets are lower in energy than the 110 or 111 facets, the latter not shown in this 2D section. Accordingly, in three dimensions, a cube represents the lowest-energy crystal form for sodium chloride at low temperatures. 

The example of this reaction demonstrates another important facet of kinetics. Figure 5.4 shows side by side the experimental data plotted as a first order-first order reversible reaction and as an irreversible reaction of order 1.5. Over a limited conversion range (here about two thirds of the way to equilibrium) the second plot is linear within the scatter of the data points. Although evaluation of the full conversion range leaves no doubt that the reaction is indeed reversible and first order-first order, its rate up to a rather high conversion is approximated surprisingly well by the equation for an irreversible reaction of higher order, in this instance of order 1.5 ... 

If a certain value of Z is regarded as acceptable, the bounds on c for a given number, d, of a factors can be read off. The facetting is not an artifact of the plotting the curves for c < 0 really are piecewise linear, and so exact bounds for given values of d and Zoo are accessible. 

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