H.2 Euler-Maclaur in Sum Formula
14.4 Faceting of a Large Planar Face
Herring [44] has also considered the possibility that a large planar surface of a crystal could break up into a hill-and-valley structure composed of facets. Such a consideration is important for kinetic reasons because a large amount of transport would be required to convert a large crystal to its equilibrium shape. It therefore makes sense to consider a state that could occur on a time scale that is very short compared to the time needed to transform an entire crystal to its equilibrium shape.
To analyze this problem, consider a small areaa0on the planar face of a large crystal having unit normalnˆ0 and free energyγ (nˆ0)≡ γ0per unit area. We then investigate the stability of this planar area with respect to being replaced by a pyramid10 having three noncoplanar orientationsnˆ1,nˆ2, andnˆ3, corresponding to facets having respective areas a1,a2, anda3, as illustrated inFigure 14–7. From Gauss’s theorem in the form
V∇ ãkd3x=
Akã ˆnd2x, wherekis an arbitrary but constant vector, we can deduce that
Anˆd2x =0.
By applying this result to the pyramid just described, we obtain ˆ
n0=f1nˆ1+f2nˆ2+f3nˆ3, (14.76) wherefi =ai/a0are area fractions.11By using reciprocal vectorsτidefined such thatτiã
ˆ
nj =δijfori,j=1, 2, 3, we deduce thatfi=τiã ˆn0. Thus theτi, but not necessarily thenˆi
as required by Herring [45], must have positive projections onnˆ0in order to obtain a real pyramid with positivefi. The free energy associated with the three faces of the pyramid, measured per unit area of the large planar face, is
γh=f1γ1+f2γ2+f3γ3. (14.77) Thus
γh=cã ˆn0, (14.78)
ˆ n1 ˆ
n2 ˆ
n1 ˆ n2 ˆ
n3
ˆ n0 a3 a2
a1 a0
(a) (b)
FIGURE 14–7 (a) Typical pyramid for faceting of a surface. Thenˆiare unit normals andaiare respective areas of the faces. (b) Faceted surface in two dimensions, showing facets of different sizes but the same orientation.
10For the entire planar face, this is to be done by using many pyramids but without a change in volume.
11Note thatnˆ0is the outward normal to the large planar face so it is an inner normal to the pyramid.
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where c:=τiγ1 + τiγ2 +τiγ3 can be interpreted geometrically as the vector from the origin of theγ-plot to a point defined by the intersections of three Wulff planes drawn perpendicular to nˆ1, nˆ2, and nˆ3 at the points where they intersect the γ-plot. This interpretation follows because c ã ˆni = γi for i = 1, 2, 3. We observe that c lies along the diameter of a sphere that passes through four points, nˆ1γ1, nˆ2γ2, nˆ3γ3, and the origin.
From the above considerations, it follows that the large planar face will be stable against faceting ifγh> γ0, which means that the point(cã ˆn0)nˆ0=γhnˆ0will lie outside theγ-plot.
On the other hand, if the pointγhnˆ0 lies inside theγ-plot, the large planar face will be unstable with respect to this type of faceting. However, if the orientationnˆ0 occurs on the equilibrium shape, it is impossible for pointγhnˆ0to lie inside theγ-plot because at least one of the Wulff planes corresponding tonˆ1,nˆ2, ornˆ3would cut it off. This results in Herring’s theorem[44]:
If a given macroscopic surface of a crystal does not coincide in orientation with some portion of the boundary of the equilibrium shape, there will always exist a hill-and- valley structure which has a lower free energy than a flat surface, while if the given surface does occur in the equilibrium shape, no hill-and-valley structure can be more stable.
With keen geometrical insight, Frank [42] observed that Herring’s faceting criterion has a very simple interpretation in terms of the invertedγ-plot. In particular, the tip of the inverted vectornˆ/γhlies on the plane that passes through the points nˆ1/γ1,nˆ2/γ2, and
ˆ
n3/γ3. To see this, letpˆbe a unit vector perpendicular to that plane and pointing away from the origin. Then the distance from the origin to that plane is given byd = ˆpã ˆn1/γ1 = ˆpã
ˆ
n2/γ2= ˆpã ˆn3/γ3, from which we deduce thatpˆ =d(τiγ1+τiγ2+τiγ3)=cd. Thuspˆã ˆn/γh= d, confirming Frank’s observation. Therefore, we can comparenˆ/γhwithnˆ/γ and deduce that the free energy will be lowered by faceting only ifn/γˆ lies inside the plane (nearer to the origin) that passes throughnˆ1/γ1,nˆ2/γ2, andnˆ3/γ3. This analysis also clarifies that the orientations that are unstable with respect to faceting are those that lie on the ears of the ξ-plot, which result from nonconvex portions of the 1/γ-plot. Indeed, the very notion of a value ofγ for unstable orientations requires the concept of a constrained equilibrium state for which faceting is prevented.
Herring’s analysis was extended by Mullins and Sekerka [45] by using linear program- ming theory to analyze faceting into shapes having an arbitrary number of orientations. It was shown that a minimum value ofγhcan always be obtained by usingno more than three orientations; however, degeneracies can occur such that more than three orientations can lead to the same minimum value of γh. Moreover, the minimum value of γh that can be achieved by faceting corresponds to the distance (nˆ)from the origin to a so-called contact planeof the Gibbs-Wulff shape, the latter being a plane that is perpendicular to
ˆ
nand touches but does not cut that shape. In fact,nˆ(nˆ)is theminimum gamma-plot (contained in all others) that gives the same Gibbs-Wulff shape as γ (n).ˆ Figure 14–8
−1.5 −1.0 −0.5 0.5 1.0 1.5
−1.5
−1.0
−0.5 0.5 1.0
1.5 γ
Γ ξ
FIGURE 14–8 Aγ-plot (outer curve) and aξ-plot (inner curve) in two dimensions forγ =1+
2n2xn2y+0.08 in arbitrary units. The equilibrium Gibbs-Wulff shape is the convex shape found by truncating the ears of theξ-plot.
The middle curve is the-plot, which is the smallest that will lead to the same Gibbs-Wulff shape. The distance along anynˆbetweenγ (ˆn)and(ˆn)represents the maximum possible energy reduction by faceting. Orientations for which this difference is zero appear on the Gibbs-Wulff shape.
illustratesγ (nˆ),(nˆ), andξ(nˆ)in two dimensions. For orientations such that a contact plane is actually tangent to the Gibbs-Wulff shape, that orientation appears on the shape and the corresponding plane is not unstable with respect to faceting. The inverted plot
ˆ
n(n)ˆ is just the convex plot obtained by enveloping the plotn/γ (ˆ n)ˆ by portions of planes, as illustrated inFigure 14–3b. The portions of planes invert to portions of spheres on(nˆ) that correspond to orientations for which the contact plane is not tangent to the Gibbs- Wulff shape.
It is important to recognize that this analysis of faceting provides no size scale for the facets; it deals only with their orientation. In other words, surfaces with large facets have the same free energy as those with small facets. However, one would expect there to be a mixture of facet sizes on a given surface (e.g., colonies of large facets and small facets, as suggested byFigure 14–7b in two dimensions) and the resulting configurational entropy would further lower the free energy of a faceted surface. Modification of the theory to allow for excess energies at edges and corners would change the invariance to size scale. Of course it would also require modification of our concept of an equilibrium shape, which would only be valid for crystals sufficiently large that excess energies at edges and corners are negligible.
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