Carbon geometries as optimal configurations. Ulisse Stefanelli

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1 Carbon geometries as optimal configurations Ulisse Stefanelli In collaboration with Edoardo Mainini, Hideki Murakawa, Paolo Piovano Ulisse Stefanelli (Vienna) ESI Vienna, / 18

2 Carbon structures Graphene Nanotubes Fullerenes local minimizers Ulisse Stefanelli (Vienna) ESI Vienna, / 18

3 Quantum? Schrödinger 60 carbon atoms solve Schrödinger in R 1080 (R {x 1,..., x 10 } grid points) Ulisse Stefanelli (Vienna) ESI Vienna, / 18

4 Configurational energy Nuclei: {x 1,..., x n } with charges {z 1,..., z n } Electrons: {y 1,..., y m } Time-independent, Born-Oppenheimer temperature 0 Admissible waves: A = } {ψ : (R 3 Z 2 ) m H 1 C, ψ L 2 = 1, antisymm. Electronic hamiltonian H = 1 y 2 2 α + 1 y α α y β + α β i j z i z j x i x j i,α z i y α x i Energy E(x 1,..., x n ) = min ψ (y, s)h(x, y)ψ(y, s) dy ds ψ A (R 3 Z 2) m Ulisse Stefanelli (Vienna) ESI Vienna, / 18

5 Carbon nanostructures Ulisse Stefanelli (Vienna) ESI Vienna, / 18

6 Carbon nanostructures [Tersoff 89] E = 1 v 2 ( x i x j ) i j v 3 (θ ijk ) ijk NN x i θ ijk x k Two-body interactions x j Three-body interactions v 2 v 3 1 [E & Li 09] 1 2π/3 4π/3 Ulisse Stefanelli (Vienna) ESI Vienna, / 18

7 2D Crystallization Three-body-interactions hexagonal lattice [E & Li 09] thermodynamic limit [Mainini & S. 14] finite crystallization [Davoli, Piovano, & S. 1?] Wulff shape, isoperimetric Two-body-interactions triangular lattice [Heitman & Radin 80] sticky potentials [Radin 81] [Wagner 83] soft potentials [Theil 06] long-range interactions [Au Yeung, Friesecke, & Schmidt 12], [Schmidt 13] Wulff shape Three-body-interactions square lattice [Mainini, Piovano, & S. 14] finite crystallization, Wulff shape Ulisse Stefanelli (Vienna) ESI Vienna, / 18

8 Graphene E = 1 v 2 ( x i x j ) i j v 3 (θ ijk ) ijk NN Ground states are honeycomb in 2D E n = 3n/2 3n/2 (exact surface energy geometry) Grounds states are perimeter-minimizers Differ from Wulff by O(n 3/4 ) atoms Ulisse Stefanelli (Vienna) ESI Vienna, / 18

9 Rolling up λ w newly activated bonds = λ θ nanotube = 2π/3 + cw 1 v 3 (θ nanotube ) = cw 2 number of angles = cwλ E nanotube = E graphene #(new bonds) + #(angles)v 3 (θ nanotube ) = E graphene λ + cλw 1 w 2 < E graphene Aspect ratio: n = wλ, minimize λ + c λ2 n λ n, w c Ulisse Stefanelli (Vienna) ESI Vienna, / 18

10 Rolling up Rolling up a nanotube sufficiently large diameter needed armchair b a pa+qb zigzag Growing a nanotube constant diameter by adding atoms Ulisse Stefanelli (Vienna) ESI Vienna, / 18

11 Fullerenes Local minimality v 3 strictly convex and decreasing around 3π/5 C 20 and C 60 strict local minimizers [video] Ulisse Stefanelli (Vienna) ESI Vienna, / 18

12 Fullerenes Ẽ #(bonds) #(bonds) pent i=1 pent 5 v 3 (π i ) v 3 (h i ) 2 hex i=1 ( ) 1 5 v 3 π i i=1 hex v 3 ( 1 6 ) 6 h i #(bonds) v 3(3π/5) v 3(2π/3) = E(C 60 ) i=1 Ulisse Stefanelli (Vienna) ESI Vienna, / 18

13 Fullerenes! Planarity of faces needed Planarity stability? [Kamatgalimov et al. 10] corannulene pentaindenocorannulene graphene Ulisse Stefanelli (Vienna) ESI Vienna, / 18

14 Nanotubes β α α Two competing periodic models: Rolled-up α = 2π/3 > β [Dressehaus et al. 95] Polyhedral α = β < 2π/3 [Cox-Hill 07] Ulisse Stefanelli (Vienna) ESI Vienna, / 18

15 Nanotubes Minimize α E 3 (α) = 2v 3 (α) + v 3 (β(α, γ)) γ β α v 3 ( )+v 3 ( ( )) Rolled-up Cox-Hill β(α) αp 0 α αru Ulisse Stefanelli (Vienna) ESI Vienna, / 18

16 Nanotubes C(α ) is a strict local minimizer Ulisse Stefanelli (Vienna) ESI Vienna, / 18

17 Nanotubes Local minimality v 3 strictly convex at 2π/3 C(α ) is a strict local minimizer Ẽ 3,density = ) (2v 3 (α i ) + v 3 (β i ) /#(atoms) 2v 3 (α mean ) + v 3 (β mean ) (convexity of v 3 ) 2v 3 (α mean ) + v 3 (β(α mean, γ mean )) (concavity of β) 2v 3 (α mean ) + v 3 (β(α mean, γ )) (if γ mean γ ) 2v 3 ( α) + v 3 (β( α, γ )) 2v 3 (α ) + v 3 (β(α, γ )) = E 3 (α ) (numerically checked) (minimality) Ulisse Stefanelli (Vienna) ESI Vienna, / 18

18 Nanotubes F * Random Modified E stefanelli Ulisse Stefanelli (Vienna) ESI Vienna, / 18