Energy-driven pattern formation. crystals

Transcription

1 : Optimal crystals 1 Mathematics Institute Warwick University Wien October 2014

2 Polymorphism It is amazing that even complex molecules like proteins crystallize without significant interference. The anti-aids drug ritonavir was discovered in 1992 From 1996 onwards the drug was sold in the form soft gelatin capsules In 1998 the drug failed the dissolution test because a new crystal form of ritonavir precipitated The observation triggered a complete withdrawal, redevelopment and reapproval of the drug

3 Energy minimization Standard modeling approach: E Ω (u) R is the energy of a subsystem in a box Ω R d, d {2, 3}. Free energy: ( ) f (β, Ω) = log exp ( β E Ω (u)) du Ω N The probability of a state u at temperature T is given by P Ω,T (u) = exp ( β E Ω (u) + f (β, Ω)), where β = 1 k B T is the inverse temperature. The study of the thermodynamic limit N with Ω N = ρ N is a hard problem. Easier: Exchange the limits β, N and study the asymptotic behavior of the minimizers of E Ω as N.

4 Energetic crystallization Assumption: ρ = 1, N := Ω L, lim N 1 N Ω N = 1 and E is permutation-invariant: E(u 1,..., u N ) = E(u π(1),..., u π(n) ) for each permutation π. A lattice L R d is a crystalline ground state of E if 1 e 0 := lim N N min u E 1 Ω(u) = lim N N E Ω(Ω L).

5 Models dominated by local interactions Strategy: A groundstate is periodic if it is almost everywhere locally periodic. Define the Cauchy-Born energy density function W CB det F (F) = lim N N E Ω(Ω F L) and the Cauchy-Born approximation = W ( v(x)) dx, E CB Ω if v is piecewise affine and v(l) = {u 1, u 2,...}. E Ω is dominated by local interactions if E Ω (u) W ( v) dx for suitable v (depends on triangulation). Ω Ω

6 Wasserstein distance Let Ω R 2 bounded, µ, ν P(Ω) probability measures. The associated Wasserstein-distance is defined as { W 2 (ν, µ) = inf x y 2 dψ(x, y) : Ψ P(Ω Ω) Ω } Ψ(x, y) dx = µ, Ψ(x, y) dy = ν. Energy (in 2 dim) if Ω = 1: F λ (µ) = λ Ω z supp(µ) Ω µ({z}) W 2 (Leb Ω, µ). Key property: W 2 is local. W 2 (Leb Ω, µ) = W (x, v) if v is induced by Vornoi tesselation.

7 Numerical Minimization 1 5 points 1 7 points 1 10 points points points points Figure : An illustration of the appearance of a hexagonal pattern as λ 0.

8 Diblock copolymers Diblock copolymer melts are well-known pattern formation systems.

9 Schematic picture Material parameters: χ: Flory-Huggins interaction N: Index of polymerization f : Relative abundance

10 Results: Minimum energy Theorem (Bourne/Peletier/T. 14) A If Ω is a polygon with at most 6 sides, then λ 4 3 Fλ c 6, where c 6 is the energy of a hexagon. B If Ω is Lipschitz, then lim λ 0 inf λ 4 3 Fλ = c 6. The limit is achieved by a scaled triangular lattice.

11 Configurational crystallization Theorem (a) If F λ (µ) = λ 4 3 c 6, then µ is an atomic measure with all weights equal to 1, and suppµ a translated, rotated and dilated triangular lattice. (b) Define the dimensionless defect of a measure µ on Ω as d(µ) := λ 4 3 Fλ (µ) c 6. There exists C > 0 such that for λ < C 1 and for all µ with d := d(µ) C 1, suppµ is O(d 1/6 ) close to a triangular lattice.

12 Mathematical tools Theorem (Kantorovich (1942), Optimal mass transportation) If ν is absolutely continuous with respect to the Lebesgue measure, then for each µ there exists an optimal transport plan T : Ω Ω such that µ is the push-forward of ν under T and W 2 (ν, µ) = x T (x) 2 dν(x). Lemma (Fejes Tóth (1972)) If χ is a polygon with n sides, then inf x z 2 dx c n χ 1 2, z χ where c n = 1 ( 1 2n 3 tan π n + cot π ) n. Equality is attained if and only if χ is a regular n-gon. Ω

13 Bounds on holes and masses Lemma If µ is a minimizer of E λ, then µ({z}) > 0 implies µ({z}) > The result is certainly not optimal and relies on the following upper bound bound on the size of holes: if µ(z) > 0 and R > 3.3. µ(b(z, R) \ {z}) > 0

14 3d Crystallization 230 (17) different space groups in 3d (2d). Most common crystalline lattices are fcc (face-centered cubic), hcp (hexagonally close packed) and bcc (body centered cubic). Only fcc and hcp are close packings.

15 Multiplicity of close packings: Stacking sequence Many non-equivalent close packings can by constructed by stacking 2d triangular lattices. An hcp-lattice is obtained if the particles in the third layer are on top of the particles in the first layer. An fcc-lattice is obtained if the particles in the third layer are on top of the holes in the first and second layer. (ABABA... = hcp, ABCABC = fcc). Third-nearest neighbors are closer in the hcp-lattice.

16 Two- and three-body energies E(Y ) = {y,y } Y V 2 (y, y )+ Y R 3 particle positions. {y,y,y } Y E invariant under rigid body motion: V 2 (y, y ) = V ( y y ), The Lennard-Jones potential V 3 (y, y, y ). V LJ (r) = r 12 r 6 ϕ(r) accounts for van-der-waals interactions at long distances. r

17 Energetic crystallization in 2 dimensions Theorem A. (T. 07) Let d = 2, L be the triangular lattice and ρ(r) = min { 1, r 7}. There exists a an open set Ω C 2 ρ([0, )) such that for all all V 2 Ω the identity holds. lim min 1 n #Y =n n E(Y ) = E 1 = min r 2 k L\{0} V (r k )

18 Visualisation of Ω r=1 α r=1+ α v> 1/2 v (r) <r ( 7) v= 1/ v >1/α v >

19 Energetic FCC crystallization Theorem B. (Harris-T. 14) Let d = 3, L be the fcc lattice and ρ(r) = min { 1, r 10}. There exists a set Ω C 2 ρ([0, )) C 2 ([0, ) [0, )) such that the projection of Ω to C 2 ρ([0, )) is open and for all all (V 2, V 3 ) Ω the identity holds. lim min n #Y =n 1 2 = min r 1 n E(Y ) = E k L\{0} V (r k ) k,k L\{0} V 3 (0, r k, r k ).

20 Ground states Theorem C. L Bravais lattice, Y R d countable and L-periodic, 1 #Y E per (Y ) E. Then 1 #Y E per (Y ) = E and there exists dilation r > 0, translation t R 3 and rotation R SO(d) such that R Y + t = r L fcc or r L tri.

21 Dislocations in the case of pair interaction potentials Equilibria can have many defects. 6 Ground state energy Emin = e+01 for N = Recent results: Alicandro-De Luca-Garroni-Ponsiligone, Ortner-Hudson

22 Dislocations in the case of pair interaction potentials Equilibria can have many defects. 6 Ground state energy Emin = e+01 for N = Recent results: Alicandro-De Luca-Garroni-Ponsiligone, Ortner-Hudson

23 Previous results Radin 79 Periodicity of groundstates for Lennard-Jones type potentials (1d) Radin 81 Periodicity of triangular lattice for compactly supported potentials (2d) Friesecke-T 02 Periodicity of ground states for mass-spring models (2d) Suto 05 Stealthy potentials (3d) T 06 Proof of minimality of the triangular lattice for Lennard-Jones type models (2d) E-Li 08 Minimality of hexagonal lattice for three-body interactions (2d)

24 Sketch of the proof Strategy: A groundstate is periodic if it is almost everywhere locally periodic. Local problems Step 1: Local analysis: Neighborhood graph, defects. Step 2: Treat non-local terms using cancelations and rigidity estimates

25 Step 1: Local analysis Minimum particle spacing: y i y i > 1 α for all i i since particles can be moved to infinity. Construction of a graph: {i, i } B y i y i (1 α, 1 + α) Musin s (2005) proof of the 3d-kissing problem: #{b B i b} 12

26 Local Analysis Theorem. (Harris/Tarasov/Taylor/T. 13) If S is a unit sphere in R 3 and S 1... S n pairwise exclusive unit spheres touching S. Then the number of tangency points is smaller or equal 36. Equality is obtained if and only if n = 12 and S 1... S n form the corners of a either a cuboctahedron or a twisted cuboctahedron. Conjecture: Let S1, S2 be two touching unit spheres in R 3. Then there can be at most 18 pairwise exclusive unit spheres touching S1 or S2.

27 Contact graphs Fcc Hcp Figure : Contact graphs of Q co and Q tco respectively

28 Sketch of the proof There are 1,430,651 2-connected graphs with 12 vertices, at least 24 edges, at most 5 edges adjacent to any given vertex. Contact graphs are those graphs where edges are induces by vertex positions. For each contact graph we define angles u j [0, 2π] subtended by edges. The angles satisfy linear inequalities such as min u j arccos( 1 j 3 ), u j = 2π, j adjacent to vertex i Only for two graphs the corresponding linear programming problems admit a solution..

29 Step 2: Nonlocal terms Two different ideas are needed. A) Discrete rigidity estimates B) Cancelation of the ghost forces Use geometric rigidity of the system in order to control errors: If the small triangles are undistorted, then the big triangle is undistorted.

30 Continuum rigidity results min u R 2 dx C(Ω) R S(d) Ω u SO(d) 2 dx. (Friesecke, James & Müller, CPAM 2002). The rigidity constant C(Ω) is invariant under rotations, dilations and translations: C(RΩ + t) = C(Ω) for all R = λq, λ R, Q O(d), t R d

31 Summary Convergence of the minimum energy for block co-polymer model Methods from optimal transportation theory Open set of interaction potentials which admits periodic (fcc) minimizers in R 3 Result based on mixture of results from Geometry and PDE theory. Open problem: Finite temperature. lim p β,λ(y ) = lim exp ( β E(Y ) + f (β, Ω)) =? Ω Λ

32 Finite temperature: β 1 Gas phase, dimension irrelevant Expected: The dimension d matters if β 1.

33 Finite temperature: β 1 d = 3: Long-range order, one large cluster iff β > β crit (ρ) d = 2: Local order, one large cluster iff β > β crit (ρ). No long-range order due to the Mermin-Wagner Theorem d = 1: Several finite clusters, no phase transition. Cluster sizes are bounded.

34 Local and global order Local (orientational) order Let ϕ be testfunction with compact support such that ϕ(r x) = 0 and define for R SO(d) SO(d) f (R) = 1 Y y y Y ϕ(r(y y)). Y exhibits local order if f (R) = O(1), Y 1. Long range order Let ϕ be a L-periodic testfunction such that R d /L ϕ = 0 and define for R SO(d) g(r) = 1 Y 2 y y Y ϕ(r(y y)). Y exhibits long-rande order if g(r) = O(1), Y 1.

35 Local and long-range order (ctd) Expected: If β 1 there is neither local, nor long range order for all d > 1. If β 1 there exists local order, but not global order for d = 2 and global order for d = 3.

36 Surface energy for d = 1 Binding energy per particle: Surface energy: 1 e 0 = lim N N min E(Y ) < 0. #Y =N e surf = lim N ( ) min E(Y ) N e 0 #Y =N > 0. The surface energy accounts for missing bonds and surface relaxation and is different from e 0.

37 Cluster size distribution A configuration Y = {y 1 < y 2 <... < y N } Λ is decomposed into clusters C i which have the properties y j, y j+1 C i then y j+1 y j < R Y = i C i dist(c i1, C i2 ) R if i 1 i 2. #{clusters C : #C = k} ν β (k) = lim (cluster length histogram) N #{clusters}

38 Convergence of the cluster length histogram Work in progress. (Jansen-König-Schmidt-T 14) Assume v(r) = r 2 p r p (Lennard-Jones potential), ρ < 1 a p(β, ρ) chosen so that p 1 ν β ( /p) is tight and non-singular as β. Then 1 lim β β log(p(β, ρ)) = 1 2 e surf, lim β p 1 ν β ( /p) = exp 1.

39 References I Appendix References L. Harris & F. T. 3d Crystallization: The FCC case. Submitted for publication. D. Bourne, M. Peletier& F.T. Optimality of the Triangular Lattice for a Particle System with Wasserstein Interaction. Com. Math. Phys, doi: /s S. Jansen, W. König, B. Schmidt & F. T. In preparation.