Wulff diagrams

Figure 1.7 Molecular orbital diagram for molecular oxygen, O2. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

Figure 4.5. Wulff s polar diagram based on ref. [8]. The equilibrium form is obtained by drawing inscribed lines at the cusps.

Fig. 7. Detailed models of surface free energies based on quasi-chemical metal-metal interactions allow detailed Wulff plots, and hence particle shapes, to be predicted as a function of temperature, (a) Interfacial phase diagram for simple cubic lattice model with nearest-neighbor and next-nearest-neighbor attraction, (b) Representative Wulff plots and equilibrium crystal shape of (a) (103).

This function can be plotted on a polar diagram and used to predict the shape of the surface energy plot cusps in the Wulff constmction. The results are semi-quantitative but useful for finding the relative anisotropic surface energy, in that for cubic crystals, minima are found at low-index (111), (110), and (100) orientations. The interested reader is referred to Venables (2000) and Howe (1997) for details. 

Figure 6. Dendrite formation. Top Rapid two dimensional crystal growth in halocarbons. Different degrees of supersatnration (or undercooling) lead to fluctuations and limitations in nutrient supply to growing faces. Center Wulff plots with orientation appropriate for each type of deiidrite. Bottom left three dimensional Wulff plot for this system. Bottom right diagram of dendrite head structure showing eqtrihbrium face and fluctrrations in growth rate. From www.gps.jussieu.fr/engl/cell.htm.

The factors that determine the crystal structure of particles formed in aerosol reactors have not been studied systematically. In this section, we identify key theoretical concepts and review relevant experimental observations. Consideration is limited to single-component systems. Panicle crystal structure depends on a combination of thermodynamic (equilibrium) factors and rate processes. The equilibrium shape of a particle is detennined by the surface energies of its crystal face.s according to the Wulff construction (Chapter 8). Another factor that inay enter into the process is the excess pressure inside small particles according to the Laplace formula (Chapter 9). Thus the equilibrium form may vary with panicle size depending on the phase diagram,... 

To determine the equilibrium shape from a y-plot, connect all of the y points in the polar diagram to the centre of the diagram, and construct perpendicular planes (Wulff planes) on the y points. Then, the minimum volume enveloped by the Wulff planes exhibits the equilibrium shape of the crystal. This is because the surface energy of a plane is proportional to the distance from the centre to the y point, and, therefore, the total surface energy is proportional to the volume enveloped by Wulff planes. When a minimum cusp is present in a y-plot, a facet plane appears. For a point and a line of energy maximum in a y-plot, a corner and an edge appear in the equilibrium shape. 

Figure 4.19 A schematic diagram showing the process offaceting, (a) the initial surface, with surface energy yo at an angle to the low-index planes of the crystal (b) the crystal after it has rearranged to two sets of planes with surface energies y 1 and yz (c) a schematic Wulff plot for this system and (d) polar plot of 1/y for this...

Figure 10.14 Shows the constraction of a Wulff plot for Si based on data in Eaglesham, Reference 8. Note that because the four planes indicated do not lie perpendicular to a single plane, the perpendiculars shown in the figure are projected such that they are in the same plane. Therefore, the distances are the projected lengths on the diagram. The shape of the Wulff plot should be approximately correct based on the observed void pubished in Eaglesham et al. [8]...

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