Convex geometric reasoning for crystalline energies
CConvex geometric reasoning for crystalline energies
Thaicia StonaUniversity at Buffalo - State University of New Yorkthaicias@buffalo.edu
Abstract “The present work revisits the classical Wulff problem restricted tocrystalline integrands, a class of surface energies that gives rise tofinitely faceted crystals. The general proof of the Wulff theorem wasgiven by J.E. Taylor (1978) by methods of Geometric Measure The-ory. This work follows a simpler and direct way through MinkowskiTheory by taking advantage of the convex properties of the consid-ered Wulff shapes.” (Final version is published on Caspian Journalof Computational & Mathematical Engineering, 1, 2016)
Introduction
This work is a short though sufficiently self-contained incursion into theWulff construction and the Wulff theorem for faceted crystals, mathemati-cally represented by the class of crystalline integrands.The aim of the Wulff Problem is to find a surface whose total surface en-ergy is minimal for a given fixed volume. This classical problem is also ∗ This work was initiated and mostly developed during the Thematic Program on Vari-ational Problems at the Fields Institute, Toronto, Canada (Fall 2014). The author isdeeply grateful to the Fields Institute for the support received and its hospitality. Theauthor would like to thank Almut Burchard, Jean Ellen Taylor, Robert McCann and allthe supporters of the Maths, Metallurgy & Crystals project (please refer to the completelist at http://goo.gl/GTgI1Y). a r X i v : . [ m a t h . A P ] F e b ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 known as the Equilibrium Shape Problem, and the solution is also called anequilibrium shape, or simply a crystal. The problem is named after GeorgeWulff, who invented an algorithm to determine the final shape of a crystalthat grows near equilibrium, based on Josiah Willard Gibbs principle of thesurface Gibbs free energy minimization for the evolution of a crystal droplet.By convexifying γ , one induces a γ -metric on the dual space of the solu-tion space. Since such a class of problems has polyhedral solutions, wecan dismiss the geometric measure versions of Brunn-Minkowski Theoremfrom Federer and Wulff Theorem by applying the Legendre transform to thecanonical version of the Wulff construction and build our way to the ConvexGeometry version of Brunn-Minkowski Theorem through geometric inequal-ities and convexity. We show equivalences between constructions and somerelations between the crystalline integrand and the area integrand versionof the problem - isoperimetry and minimal surfaces. I. The Crystalline Variational Problem
The Wulff shape arises in surface energy minimization problems when theenergy function Φ is anisotropic. For isotropic energies and a given amountof mass, the equilibrium shape is well known: a ball, formally denoted bythe n-dimensional sphere S n − . This shape encloses the prescribed masswhereas minimizing the surface area of it. This is stated by the classicIsoperimetric Inequality.For anisotropic energies, the analogous minimizer is the Wulff shape. Suchenergies have been heuristically misrepresented by simple functions that veryoften are not well-defined, presenting many singularities and unbounded en-ergy spots. We avoid this imposture following Taylor’s Geometric MeasureTheory characterization of the energy. In this context, the surface energy,also called the energy function of the anisotropic problem, is an integrand,as defined below: Def.: [integrand] An integrand on R n +1 is a function that will represent thesurface energy functionΦ : R n +1 × G ( n + 1 , n ) −→ [0 , + ∞ ]2 ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 where the Grassmannian G ( n + 1 , n ) is the manifold that parametrizes ev-ery n-dimensional linear subspace of R n +1 , ie, all the hyperplanes of the(n+1)-dimensional Euclidean space.An integrand is defined constant coefficient iff Φ( x, π ) = Φ( p, π ) ∀ x, p ∈ R n +1 , ∀ π ∈ G ( n + 1 , n ). In this case, Φ is a function of its second variableonly. An integrand is unoriented if it independs on the orientation of π . Wewill assume all integrands are continuous, constant coefficient and positivelyoriented. Def.: [Wulff construction] Given an integrand Φ, plot it radially by takingeach direction v ∈ S n and calculating Φ on the positively oriented plane π whose normal vector is v , π = (cid:8) x ∈ R n +1 / (cid:104) x, v (cid:105) = 0 (cid:9) . We will denote π by v ⊥ and vice versa. Plot Φ( v ⊥ ) in v direction: Φ( v ⊥ ) v . Then, for each v ,define the half-space H v . = (cid:110) x ∈ R n +1 / (cid:104) x, v (cid:105) ≤ Φ( v ⊥ ) (cid:111) .Take the intersection of all half-spaces. The resulting set W Φ is the Wulffshape of Φ, also called the crystal of Φ: W Φ . = (cid:92) v ∈ S n H v For an isotropic energy, Φ ≡ constant: the Wulff problem reduces to theIsoperimetric Inequality and the crystal is an Euclidean ball; that is thecase of a soap bubble. Obs.:
We can extend homogeneously the function Φ in order to calculate iton other planes related to non unitary direction vectors by formalizing theexplained abuse of notation defining the dual function Φ (cid:63) as followsΦ (cid:63) : S n −→ [0 , + ∞ ] , Φ (cid:63) ( v ) = Φ( π )where v ⊥ π as defined above, Φ (cid:63) ( p ) . = | p | Φ (cid:63) ( p | p | ).Note that since W Φ is given by an intersection of half-spaces, then W Φ isconvex. Also we can assume 0 ∈ W Φ always. The physical meaning of theorigin is the crystal seed for growing a crystal, a tiny monocrystal that in-duces the orientation of the new crystal.3 ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 Def.: [Legendre Transform] Let ξ : S n − −→ R + be a continuous function.The (first) Legendre transform of ξ is ξ (cid:63) ( v ) . = inf (cid:104) θ,v (cid:105) > ξ ( θ ) (cid:104) θ, v (cid:105) where | θ | = 1An alternative construction of the Wulff shape is based on the Legendretransform as in [5]: Def.: [Fu’s Wulff construction] Let W be the operator over integrands W (Φ)( π ) . = inf v ∈ S n Φ (cid:63) ( v ) (cid:104) π ⊥ , v (cid:105) where (cid:68) π ⊥ , v (cid:69) >
0. Then the crystal of Φ is the set enclosed by the radialplot of W (Φ), plotted as explained before. Also, the orientation of W Φ isdefined positive. Proposition:
The two given definitions of crystal are equivalent.
Proof:
Call Z the operator defined in Fu’s construction instead of W :( W Φ ⊂ Z Φ ) : Let y ∈ W Φ , ie, (cid:104) y, v (cid:105) ≤ Φ (cid:63) ( v ) ( ∀ v ∈ S n ). Then (cid:104) y, v (cid:105) = | y || y | (cid:104) y, v (cid:105) = | y | (cid:28) y | y | , v (cid:29) ≤ Φ (cid:63) ( v )If (cid:28) y | y | , v (cid:29) > | y | ≤ Φ (cid:63) ( v ) (cid:68) y | y | , v (cid:69) . Since the inequality holds forarbitrary v , then | y | ≤ inf v ∈ S n Φ (cid:63) ( v ) (cid:104) y, v (cid:105) , ie, y ∈ Z Φ If (cid:28) y | y | , v (cid:29) (cid:3)
0, then obviously the inequality holds, with y in the samehalf-space bounded by (cid:104) x, y (cid:105) = Φ (cid:63) ( y | y | );4 ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 ( Z Φ ⊂ W Φ ) : Let y ∈ Z Φ , ie, | y | ≤ ( Z (Φ)) (cid:63) ( y | y | ), (cid:28) y | y | , v (cid:29) >
0. Then: | y | ≤ Φ (cid:63) ( v ) (cid:68) y | y | , v (cid:69) ∀ v ∈ S n ⇐⇒ | y | (cid:28) y | y | , v (cid:29) = (cid:104) y, v (cid:105) ≤ Φ (cid:63) ( v )for all v ∈ S n ; but then y ∈ H v ( ∀ v ∈ S n ) ⇒ y ∈ W Φ (cid:50) II. Pathway through convexity
It is easy to visualize what kind of Wulff shape one gets when the inter-section of half-spaces is finite: a polyhedron, except for unbounded and/orempty intersections. That is the case of anisotropic energies: we say that anintegrand Φ is crystalline if its Wulff shape, or crystal W Φ is a polyhedron.Now we take advantage of this fact: Def.: [extreme point] Given a set K ⊂ R n , x ∈ K is extreme if it cannot beexpressed as a convex combination of any two other points of K . Def.: [polytope] A polytope P ⊂ R n is the convex hull of a finite set: P = [ { p , p , ..., p k } ]. Def.: [polar body] Given a convex set K , the polar body of K is the set K (cid:63) . = { x ∈ R n / (cid:104) x, y (cid:105) ≤ ∀ y ∈ K ) } . Lemma:
A supporting hyperplane H to a bounded convex set K containsat least one extreme point of K . Proof:
Denote the set of extreme points of K by E K . Since K is convex, K = [ K ], so E k ⊂ K ⇒ [ E K ] ⊂ K . We also have that H ∩ K = H ∩ ∂K , sothe set of extreme points of H ∩ K , E H ∩ K , is the set E H ∩ E K . Now supposethe claim is true for every set with dimension ≤ m −
1. Then it is also true forall sets of dimension m , since if a given non-extreme point in m dimensioncould be written as a convex combination in dimension m −
1, then it wouldbe sufficient to write it in m dimension putting λ m = 0. But for dimension1, the claim is trivially true. Therefore it is true for any dimension. (cid:50) Theorem 1:
A bounded convex set K is the convex hull of its extremepoints. 5 ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 Proof:
Since E k ⊂ K ⇒ [ E K ] ⊂ K , we only need to prove that K ⊂ [ E K ]. Suppose some x ∈ K is not in [ E K ]. Then there exists a separatinghyperplane H that separates strictly x from E K . The parallel supportinghyperplane of K that is strictly separated from E K by H must contain apoint of E K (lemma). Contradiction. (cid:50) Corollary:
Every polytope is a finite intersection of half-spaces.
Proof: If P is finite, then so is E P . For each p ∈ E P , let A p be theset of supporting hyperplanes that contains p and also contains at leastanother extreme point of P . Then take A (cid:48) p ⊂ A p the subset that containssupp. hyperplanes intersecting the maximum number of extreme points aspossible (this number is well-defined since the very P is a majorant). Thefacets of P will be contained on those hyperplanes; for each facet definethe half-space oriented to contain the origin and take the intersection of it.Because of the theorem, P is contained in this intersection. (cid:50) Theorem 2: If K is convex, then K (cid:63)(cid:63) = K Proof: ( K ⊂ K (cid:63)(cid:63) ) Let x ∈ K . Then for any y ∈ K (cid:63) we have (cid:104) x, y (cid:105) ≤
1. But then,since x is arbitrary, it has to be in K (cid:63)(cid:63) .( K (cid:63)(cid:63) ⊂ K ) Let y ∈ K (cid:63)(cid:63) and suppose y / ∈ K . Then there is a separatinghyperplane H that separates y from K , H = { x / (cid:104) x, v (cid:105) = 1 } , (cid:104) x, v (cid:105) ≤ x ∈ K and (cid:104) y, v (cid:105) > (cid:104) x, v (cid:105) ≤ x ∈ K , then v ∈ K (cid:63) and (cid:104) y, v (cid:105) ≤ y ∈ K (cid:63)(cid:63) .Contradiction. (cid:50) Theorem 2 reveals a link between Convex Geometry and Functional Analy-sis: given a polyhedral crystal W , we apply the corollary to define a convexΦ C whose crystal coincides with W , so that Φ C is the ”smallest” enclosingfunction for W . For that, we use the theorem 2 by taking the polar of W .Since Φ is a linear operator, we know its behavior everywhere by homo-geneous extension. By Riesz representation theorem, the crystal W is thepolar of the unit ball Φ C ≡ Def.: [Steiner symmetrization] For a convex body K ⊂ R n and a θ ∈ S n − ,the Steiner symmetrization of K in the direction of θ is given by S θ ( K ) . = { x + λ.θ | x ∈ P roj θ ⊥ K, λ ∈ R } ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 where | λ | ≤ | K ∩ { x + R θ } | . Some properties are the fact that | S θ ( K ) | = | K | , S θ ( K ) is convex and the convex Minkowski sum of symmetrizationsequals to the symmetrization of the convex sum of the bodies. The sym-metrization process slices K along θ , aligning the slices by putting theirmidpoints in θ ⊥ .Figure 1: Example of Steiner symmetrization of K along the vector θ † A useful classical result is stated below without its proof, which follows di-rectly from the several interesting properties of the Steiner Symmetrizationprocess. A more curious reader might refer to [6], [20] or [21].
Theorem: [Steiner-Schwarz] Given a convex body K ⊂ R n and F ak-dimensional subspace, then there exists a sequence of symmetrizations θ j such that the limiting body K satisfies | K ∩ { x + F } | = | K ∩ { x + F } | ,where K ∩ { x + F } is a k-dimensional ball centered in x with radius r ( x ). Theorem: [Brunn’s Concavity Principle] Given K ⊂ R n a convex body and F a k-dimensional subspace of R n , the function f : F ⊥ −→ R + given by f ( x ) = | K ∩ { x + F } | n is concave on its support. Proof:
Apply the former theorem and use that supt r ( x ) = P roj F ⊥ K , f ( x ) = | K ∩ { x + F } | = V ol ( S k − ) = π k Γ( k + 1) r ( x ) k . (cid:50) † Figure adapted from [24] ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 Figure 2: Application of Steiner-Schwarz to prove Brunn’s ConcavityPrinciple, where n = 3, k = 2 ‡ The Brunn-Minkowski inequality is the crucial ingredient for proving theoptimality of the Wulff shape. We conclude this section with a proof basedon convex sum of two convex bodies and the Concavity Principle:
Theorem: [Brunn-Minkowski Inequality] Given non-empty compact subsets
A, B of R n | A + B | n ≥ | A | n + | B | n Proof:
Take the Steiner symmetrization of A and B to find two convexbodies in R n . Create their convex sum L on R n +1 by taking the convex hullof S θ ( A ) × S θ ( B ) ×
1, where 0 , L ( t ) = { x ∈ R n | ( x, t ) ∈ L } .Then L ( 12 ) = S θ ( A )2 + S θ ( B )2 = S θ ( A ) + S θ ( B )2 . By the concavity principleapplied for F = R n (cid:12)(cid:12)(cid:12)(cid:12) S θ ( A ) + S θ ( B )2 (cid:12)(cid:12)(cid:12)(cid:12) n ≥ | S θ ( A ) | n + 12 | S θ ( B ) | n (cid:50) ‡ Figure adapted from [24] ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 III. The Wulff Theorem
Wulff’s 1901 seminal article provided a method to predict crystal shapesafter Gibbs’ proposition on the minimization of surface energy; since then,many have worked on the subject. Nevertheless, it was Taylor ([1]) whoproved that the Wulff construction determines the unique minimizer W Φ forthe integral of Φ over the boundary ∂W Φ . The proof requires some conceptsfrom Geometric Measure Theory, which are now introduced: Def.: [integral current] An integral n-dimensional current S ⊂ R n +1 is arectifiable oriented hypersurface generalized through GMT so that eventualanomalous portions are still well-behaved enough to allow integration withrespect to the measure | S | on R n +1 , which is a function of the Hausdorffmeasure H n restricted to the support of S , which can be arbitrarily closelyapproximate by a n-d C manifold. An interesting property of currents isthat their boundaries also have the essential properties to allow boundaryintegration (for more see [3]). In the next theorem T will denote the cur-rent whose boundary is an integral current. The total surface energy of anintegral current S ⊂ R n +1 is given by:Φ( S ) . = (cid:90) x ∈ S Φ[ n S ( x )] dH n x We also define for h > R n +1 µ h ( x ) = hx and the inte-grand Φ the isomorphism W h Φ = µ h(cid:93) ( W Φ ) following [1]. Theorem: [Wulff] Given an integrand Φ, then for every n-dimensional cur-rent P ⊂ R n +1 Φ( ∂W Φ ) ≤ Φ( ∂P )up to translations and homotheties, such that their mass coincide, M ( P ) = M ( W Φ ) Proof:
Let P be a current with ∂P its positively oriented, piecewise C boundary. ThenΦ( ∂P ) = (cid:90) Φ( −→ ∂P ( x )) d | ∂P | x ≥ (cid:90) supt ( W Φ )( −→ ∂P ( x )) d | ∂P | x = lim h → M ( P h ) − M ( P ) h ublished on Caspian Journal of Computational & Mathematical Engineering, 1, 2016 where M ( W Φ ) = M ( P ) and M ( W h Φ ) = h n +1 M ( W Φ ) and P h is the posi-tively oriented current given by the Minkowski sum x + y where x ∈ supt P and y ∈ supt W h Φ . Then Brunn-Minkowski inequality implies:= lim h → M ( P h ) − M ( P ) h ≥ lim h → (1 + h ) n +1 M ( W Φ ) − M ( W Φ ) h = lim h → (1 + h n +1 − h M ( W Φ ) = lim h → M ( W Φ ) h n +1 (cid:88) i =1 h i . ( n + 1)! i !( n + 1 − i )!= lim h → M ( W Φ ) . ( n + 1) = ( n + 1) .M ( W Φ )In particular for P = W Φ , the above inequalities are equalities. By usingthe fact that M ( P ) = M ( W Φ ) , we conclude that Φ( ∂W Φ ) ≤ Φ( ∂P ).Such shape is unique modulo translations and homotheties, and since themass is fixed, follows the uniqueness of W Φ . (cid:50) IV. Conclusion
In this exposition, different fundamental areas of Mathematics were gath-ered to structure a simple mathematical basis for the equilibrium shapeproblem with a crystalline integrand. A natural generalization of the Wulffconstruction for non-equilibrium growth is to replace the energy function forthe correspondent potential that controls the process, the mobility function.Also, through Kinectic PDEs, a flourishing area of mathematical modellingin the Sciences, it might be of interest to study the growth and the stabilityof such shapes.
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