![]() ![]() From Formation will occur only if energetically favorable. Bottom right diagram of dendrite head structure showing eqtrihbrium face and fluctrrations in growth rate. Bottom left three dimensional Wulff plot for this system. Center Wulff plots with orientation appropriate for each type of deiidrite. Different degrees of supersatnration (or undercooling) lead to fluctuations and limitations in nutrient supply to growing faces. ![]() Top Rapid two dimensional crystal growth in halocarbons. 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).Ī two-dimensional Wulff plot for a three-dimensional crystal having both 100 and 111 faces is illustrated in Figure 12.6. Thus, the equilibrium shape generally consists almost entirely of low-index planes. Atoms in the low- index planes form the greatest number of bonds and hence have less energy than atoms in less densely packed planes. This is because the surface tension of a solid is primarily determined by the strength of the bonding of the individual surface atoms. The cusp points on the Wulff plot generally correspond to low- index planes. įigure 19.29 Cross section of Wulff plot and related nucleus form. Also, the construction is consistent with Young s equation, since from the figure, 7 = 27 cos. Since the a/f3 interface is isotropic, the top surface is spherical. ![]() Cross sections of the Wulff plot and Wulff shape consistent with the symmetry of the problem are shown in Fig. yA(= AO) represents the surface energy of a plane with the normal vector AO. Wulff-plot (110) section of a fee crystal. (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. Make a Wulff construction and determine the equilibrium shape of the crystal in the xy plane. is comparable across different crystal systems and accounts for all surfaces based on their relative importance (in terms of contribution to the Wulff shape).The surface tensions for a certain cubic crystalline substance are 7100 = 160 ergs/cm, 7110 = 140 eigs/cm, and 7210 = 7120 = 140 ergs/cm. A perfectly isotropic crystal would have. Typically, the shape factor is compared against that of an ideal sphere ( ), and a larger indicates greater anisotropy.Īn alternative definition of surface energy anisotropy used in this database given by the following equation:Ĭan effectively be viewed as a coefficient of variation of surface energies that is normalized for comparison across crystals with different average surface energies. The shape factor is a useful quantity in determining the critical nucleus size. Where and are the surface area and volume of the Wulff shape, respectively. The most commonly used general measure of anisotropy is the shape factor, which is given by the following equation: Where is the work function for a unique facet existing in the Wulff shape. Similarly the weighted work function can be defined as: Where is the surface energy for a unique facet existing in the Wulff shape, is the total area of all facets in the family in the Wulff shape, and is the area fraction of the family in the Wulff shape. We define the weighted surface energy using this fraction as given by the following equation: In this construction, the distance of a facet from the crystal center is directly proportional to the surface energy of that facet, and the inner convex hull of all facets form the Wulff shape. The Wulff construction gives the crystal shape under equilibrium conditions. Where is the electrostatic potential in the middle of the vacuum region of the slab model and is the Fermi energy of the material. Similarly, each slab model of a facet has a work function given by: Where, is the total energy of the slab model with termination, is the energy per atom of the bulk OUC, is the total number of atoms in the slab structure, is the surface area of the slab structure, and the factor of 2 in the denominator accounts for the two surfaces in the slab model. For a given slab model of a facet with Miller index, the surface energy is given by: ![]()
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