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Unformatted text preview: Comment on Wulff construction Relevant for Equilibrium shape of crystals, all sizes. (You need to consider the next terms in the Taylor series at very small sizes, not often.) Equilibrium shape of precipitates Equilibrium shape of grains Equilibrium shape of grain boundaries One comment, faceting. In the case below only <111> and <001> facets exist. As T>Tm the surface free energy becomes isotropic so goes to a sphere. As it cools a direction in between these will break up into regions of the two, what is called microfacetting. This can occur for all the different cases, e.g. dislocation loops are ellipsoidal at high T but can be faceted at low T. Grain Boundaries Relevance: Can act in essence as another phase of the material with it’s own thermodynamic properties In some materials (e.g. ceramics) the weak point Can dominate properties, for instance dominant source for resistance in highTc superconductors; for some electrical properties in conducting ceramics. Go over the basic description in terms of two single crystals with a orientation change. Detail the degrees of macroscopic freedon: 3 for rotation that describes the difference in orientation between the grains, and two for the grain boundary plane. There are also what are called microscopic degrees of freedom; two for a translation in the plane of the boundary between the two grains and one for a translation normal (i.e. an expansion/contraction) Often there is an expansion and the local density at the grain boundary is lower than that of the bulk. However, this assumes that the grain boundary is a flat plane and it does not have to be; it can be energetically favorable for it to facet (describe, more later). Low Angle Grain Boundaries The simplest type of grain boundary (tilt) is one where the angle between the two is small. A simple case is when there is a rotation about a vector that lies in the plane of the boundary. By geometry (shown in class), if θ is the rotation angle and we assume that the interface is symmetric, if b is the Burgers vector of an edge dislocation (assumed to lie normal to the boundary) and the separation is D, tan( θ /2) = (b/2)/D Similarly if the rotation is in a vector normal to the boundary plane (twist boundary) the misfit can be described in terms of an array of screw dislocations with the same formula. In general one can have arrays along more than one direction depending upon the symmetry, e.g. Burgers vectors of ½(110) and ½(110) for a tilt boundary with a common (001) boundary, and three for a (111) common plane. The dislocations don’t have to be full but could be partial.full but could be partial....
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 Winter '10
 MatSci
 Thermodynamics, Entropy, Surface tension, energy cost, grain boundary

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