Wulff plot

The surface tensions for a certain cubic crystalline substance are 7100 = 160 ergs/cm, 7110 = 140 eigs/cm, and 7210 = 7120 = 140 ergs/cm. Make a Wulff construction and determine the equilibrium shape of the crystal in the xy plane. (If the plane of the paper is the xy plane, then all the ones given are perpendicular to the paper, and the Wulff plot reduces to a two-dimensional one. Also, 7100 = 7010. etc.)... 

Figure 3-8. Wulff-plot (110) section of a fee crystal. yA(= AO) represents the surface energy of a plane with the normal vector AO.

Solution. Cross sections of the Wulff plot and Wulff shape consistent with the symmetry of the problem are shown in Fig. 19.29. Since the a/f3 interface is isotropic, the top surface is spherical. Also, the construction is consistent with Young s equation, since from the figure, 7 = 27 cos ... 

Figure 19.29 Cross section of Wulff plot and related nucleus form.

The cusp points on the Wulff plot generally correspond to low-index planes. This is because the surface tension of a solid is primarily determined by the strength of the bonding of the individual surface atoms. Atoms in the low-index planes form the greatest number of bonds and hence have less energy than atoms in less densely packed planes. Thus, the equilibrium shape generally consists almost entirely of low-index planes. 

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).

A two-dimensional Wulff plot for a three-dimensional crystal having both 100 and 111 faces is illustrated in Figure 12.6. The corresponding solid is sketched. 

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.

Formation will occur only if energetically favorable. We need to discuss the relevance of Wulff plots to understand surfaces. 

The surface energy, y, depends on the structure, which depends on the orientation. We have developed two approaches for considering this orientation dependence the Wulff plot and the inverse Wulff plot. 

FIGURE 13.1 Wulff plot looking along the [liO] direction for an fee crystal. 

The Wulff plot draws a graph of y versus 0. This construction was developed to allow the equilibrium shape of crystals to be determined when the surface energy depends on crystallography. 

The Wulff plot is the conventional plot for surfaces an example of such a plot is shown in Figure 13.1. The most important point is that you have cusps in y versus 0. In ceramics, this certainly occurs if the energy were isotropic, the Wulff plot would be a circle. What is not certain is how many cusps you have. It is instructive to compare surfaces with grain boundaries (section 14.3). In the latter case it is necessary to define the orientation of the grainboundary plane, i.e., having fixed 0, you must still fix n, the normal to both grains. 

In this chapter the special features of surfaces and internal boundaries in single-phase crystalline systems are described. The orientation dependence of the surface functions will be discussed and the use of Wulff plots and stereographic triangles to present the orientation dependence of surface energy will be demonstrated. The various types of internal boundaries (grain boundaries, twin boundaries, etc.) and their thermodynamic properties will be discussed. 

The above treatment embodies the essence of the Wulff construction but is restricted by a prior choice of which planes would contribute to the surface. A more general approach may be used if the complete Wulff plot is available. This treatment was outlined by Herring [5] as follows. 

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