Optimal configurations for classical interacting particles

Transcription

1 Optimal configurations for classical interacting particles Edoardo Mainini DIME, Università degli Studi di Genova, & Facoltà di Matematica, Università di Vienna Joint research with Ulisse Stefanelli, Paolo Piovano (University of Vienna), Hideki Murakawa (University of Kyushu) Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 / 27

2 Geometry of optimal particle configurations Atomistic configuration: {x,..., x n } R d, d =, 2, 3 Configurational interaction energy: V : R nd R (attractive-repulsive) Describe crystal pattern formation by energy minimization: Geometry optimization and the crystallization problem Prove that, for suitable configurational energies, minimizers are periodic and arrange in particular geometric shapes From crystallization to... Global geometry of crystals Quantifications of surface tension effects, Defects and deviations from reference shapes, Stability of local minimizers Atomistic to continuum models Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 2 / 27

3 Atomistic energies Zero temperature description of atomistic interaction Ab initio: V (x,... x n ) = min eig(h x,...,x n ), where H is the electronic Hamiltonian, parametrized by the nuclei positions x,..., x n Classical pair potential V (x,..., x n ) = V l ( x i x j ) 2 i j V l r Lennard-Jones: V 2 (r) = ar 2 br 6 Lennard-Jones General multiple body potential: v 2 (x i, x j ) + v 3 (x i, x j, x k ) + v 4 (x i, x j, x k, x l ) +... i,j i,j,k i,j,k,l Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 3 / 27

4 Carbon nanostructures 3D structures which are locally 2D Orbital hybridization Carbon structures with all bonds being sp 2 covalent bonds Graphene Nanotube Fullerene Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 4 / 27

5 Carbon nanostructures For describing covalent bonding behavior of carbon atoms, we introduce a Tersoff empirical potential [Tersoff 88, Brenner 90]: V = V l ( x i x j ) 2 i,j + φ( x i x j )ψ( x k x j )V 3 (θ ijk ) 2 i,j,k x i θ ijk x j x k V l r v 3 Lennard-Jones φ, ψ: long range cut-off functions. Three-body interaction favors sp 2 bond angles of 2π/3. For instance V 3 (cos(x) + /2) 2 [Stillinger-Weber 85, E-Li 09] Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 5 / 27 2π/3 4π/3

6 Outline Finite crystallization in the plane Global geometry of planar minimizers Local minimality of nanotubes and fullerenes Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 6 / 27

7 Crystallization: some remarks Proving crystal ordering can be difficult beacuse Long range interactions: concavity of two body potentials at long range. Even in D: finite chains are not equally spaced for Lennard-Jones pair interaction [Gardner-Radin 79] Then restrict to first neighbors interaction (right picture) V l r Vl Lennard-Jones Not sufficient in higher dimension. Surface tension effect is in contrast to internal periodicity. Global geometry problem. Competition of two and three body interaction term: from triangular to hexagonal lattice. Symmetry breaking bifurcation of crystal patterns Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 7 / 27

8 Previous crystallization results in 2D [Heitman-Radin 80], [Radin 8], [Wagner 83] Sticky or soft two-body potential: minimizers are subset of the triangular lattice. V l Vl r r Sticky soft Related to discrete mathematic task Find the maximal number of tangencies of n equivalent rigid disks in the plane. Solution 3n 2n 3 [Harborth 74] The corresponding configuration is a subset of the triangular lattice Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 8 / 27

9 Previous crystallization results in 2D (only in the limit n ) V l Lennard-Jones r [Theil 06] Thermodynamic limit n : the normalized energy in the limit is the one of a (dilated) triangular lattice. This result deals with 2D long range interaction. lim n n V (x,..., x n ) = V triang > 3 [E-Li 09] Analogous result with addition of three-body Stillinger-Weber potential. Asymptotic structure is hexagonal. [Yeung-Friesecke-Schmidt 2] Overall limiting Wulff shape for any sequence of rescaled empirical measures corresponding to n-particle minimizers. Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 9 / 27

10 Crystallization In [M., Stefanelli 4] we consider a short ranged Tersoff potential V (x,..., x n ) = 2 n V l ( x i x j ) + 2 i,j= i j (i,j,k) A V 3 (θ ijk ) V l (r) = 0 for r 2, A = {(i, j, k) : x i x j < 2, x k x j < 2} Vl Vl [E-Li 09] Short range LJ Hard interaction Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 0 / 27

11 Reference assumptions on V 3 V 3 : [0, 2π] [0, ) is continuous, vanishing just in 2π/3 and 4π/3, and V 3 (2π θ) = v(θ). v3 2π/3 4π/3 Crucial features: narrowness and convexity of energy wells, right and left derivative in correspondence of minimal values. V 3 is large enough out of small intervals around 2π/3, 4π/3. This prompts local optimality of hexagonal structures (three bonds per atom). V 3 (strictly) convex on [ 2 3 π ± α] and [ 4 3 π ± α] V 3, (2π/3) < 3/π (V 3, is the left derivative). Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 / 27

12 Crystallization result A particle configuration C n := x,..., x n is honeycomb if it is a subset of the hexagonal lattice. The energy is the opposite of number of bonds. Theorem (E. M., U. Stefanelli, CMP 204) In 2D ground states are honeycomb and the corresponding energy is V (C n ) = 3n/2 3n/2. Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 2 / 27

13 Proof: 2D reference configurations We construct reference structures D k, with energy V (C n ) = for n = 6k 2, k N ( 3 2 n 3 2 n ), D, D 2, D 3 Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 3 / 27

14 Proof: 2D reference configurations Geometric interpolation For any n we obtain configurations C n with energy 3 3 V (C n ) = 2 n 2 n n/2 leading term Exact surface energy quantification Surfaces are minimized Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 4 / 27

15 Proof: boundary energy estimate and induction Decomposition of the energy of C n : C n = C blk n C bnd n, V (C n ) = V blk (C n ) + V bnd (C n ). d = d(c n ) := number of boundary vertices Crucial surface energy estimate: V bnd (C n ) 3d/2 + 3, equality implies C bnd n is honeycomb (requires convexity and linear growth around 2π/3) Conclusion with induction on bond graph layers: V (C n ) 3 2 n 3 2 n. Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 5 / 27

16 Outline Finite crystallization in the plane Global geometry of planar minimizers Local minimality of nanotubes and fullerenes Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 6 / 27

17 Square V 3 favors bond angles of π/2, π. E2 2 E3 π/2 π 3π/2 Theorem (E. M., P. Piovano, U. Stefanelli, Nonlinearity 204) In 2D ground states are square and the corresponding energy is V (C n ) = 2n 2 n, ground states are optimal in terms of edge isoperimetric inequalities in the suqare graph ground states converge to square for large n (Wulff shape) Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 7 / 27

18 Non uniqueness Ground states are unique, up to isometries, if and only if: n =, 2, n = m 2, the square, or n = m 2 + m, the m (m + ) rectangle, or n = m 2, n = m 2 + m. Non-isomorphic ground states for n = 7 Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 8 / 27

19 Isoperimetric inequality C D Monotone rearrangements in the square graph Characterization by suitable version of the edge isoperimetric inequality [Harary-Harbort 76], [Bollobas-Leader 9], [Ahlswede-Bezrukov 95]. Arguments from combinatorial mathematics and graph optimization. Area(C n ) := number of unit squares Per(C n ) := number of boundary bonds (isolated bonds are counted twice) Isoperimetric inequality: n 2 n Area(C n ) k n Per(C n ), k n := 2 2 n 2. Equality holds if and only if C n is a ground state Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 9 / 27

20 Deviation from the asymptotic Wulff shape n+n /4 + n S A ground state C n = {x,..., x n } approaches the square for large n. l 2 C n Convergence of empirical measure: µ Cn := n l n i δ x i / n L 2 [0,] 2. Fixing S n := {(i, j) : i, j = 0,, 2,..., n } (the square of n edge), for a ground state C n, and asuitable rotation and translation C n we have in total variation distance dist(µ C n, µ S n ) 3n /4 + O(n /2 ). Two n-particles ground state differ at most by O(n 3/4 ) particles: #(C n G n) O(n 3/4 ). Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27

21 Deviation from the asymptotic Wulff shape Complement the results in [Yeung, Friesecke, Schmidt 2, Schmidt 3], for the two body sticky Heitman-Radin potential. They show that suitably defined discrete surface energies of rescaled empirical measures µ = n n i= δ x i / n, are Γ-converging to a anisotropic perimeter, whose minimizer (Wulff shape) is the unit regular hexagon. This implies that the rescaled empirical measures n δxi / n, converge to the uniform measure on the Wulff shape. By the isoperimetric characterization of ground states, we directly obtain the same result Indeed, the perimeter of ground states is P(C n ) = 2 2 n 4. This gives a possible deviation of the edge length from n /2, of order at most n /4. Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/205 2 / 27

22 Outline Finite crystallization in the plane Global geometry of planar minimizers Local minimality of nanotubes and fullerenes Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27

23 Local minimality of 3D carbon structures Roll-up of a 2D ground state newly activated bonds = k new angles = 2π/3 + O(k 2 ) number of angles = O(k 2 ) V 3 (θ new ) O(k 2 ) V new = V old #(new bonds) + #(angles)v 3 (θ new ) V old k + O() < V old (for k large) Similarly, rolling up a graphene strip into a nanotube is energetically favorable, for sufficiently large diameter. Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27

24 Fullerene Stability V 3 strictly convex around 3π/5 C 60 and C 20 fullerenes are strict local minimizers Planarity of faces needed for local minimality Ẽ #(bonds) + 2 #(bonds) + 2 pent i= pent 5 v 3 (π i ) + 6 v 3 (h i ) 2 hex i= ( ) 5 v 3 π i i= hex v 3 ( 6 ) 6 h i #(bonds) v 3(3π/5) v 3(2π/3) = E(C 60 ) [M., Stefanelli 4] i= Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27

25 Zigzag Nanotube β α α Two competing models for periodic nanotubes: Rolled-up: α = 2π/3 > β [Dresselhaus et al. 95] Polyhedral α = β < 2π/3 [Cox-Hill 07] Atoms are on the surface of a cylinder of radius r. They are contained in sections (orthogonal to the axis). Each section has l atoms (say l 4), arranged on a regular l-gon. All bond lengths are unit. Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27

26 Zigzag Nanotube We introduce a distinguished family F of uniformly stretched (Cauchy-Borne) configurations, parametrized by the angle α: ( ( π sin α = 2r sin, β z = β z (α) := 2 arcsin sin α sin 2l) γ(l) ). 2 Then we are reduced to min {2v 3 (α) + v 3 (β(α)) : α (π/2, π)} The above minimization problem has a unique solution, which correspond to an intermediate configuration between rolled up and Cox-Hill configurations v 3 ( )+v 3 ( ( )) Rolled-up Cox-Hill γ β α Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27

27 Zigzag Nanotube E E a L Numerical results for local minimality: on the left: energy of randum perturbations of F, as function of the mean value of the α s. on the right: energy in the F family and corresponding random perturbation. The nanotube behaves elastically close to the optimal stress-free configuration. [M., Murakawa, Piovano, Stefanelli 5] Edoardo Mainini (Genova & Vienna) Crystallization for particle interactions 4/09/ / 27