energy-driven pattern formation: optimal crystals · energy-driven pattern formation: optimal...

Energy-driven pattern formation: Optimalcrystals

Florian Theil

1Mathematics InstituteWarwick University

WienOctober 2014

Florian Theil Energy-driven pattern formation

Polymorphism

It is amazing that even complex molecules like proteinscrystallize without significant interference.

The anti-AIDS drug ritonavir wasdiscovered in 1992From 1996 onwards the drug wassold in the form soft gelatin capsulesIn 1998 the drug failed thedissolution test because a newcrystal form of ritonavir precipitatedThe observation triggered acomplete withdrawal, redevelopmentand reapproval of the drug


Energy minimization

Standard modeling approach:

EΩ(u) ∈ R

is the energy of a subsystem in a box Ω ⊂ Rd , d ∈ 2,3. Freeenergy:

f (β,Ω) = − log(∫

ΩNexp (−β EΩ(u)) du

)The probability of a state u at temperature T is given by

PΩ,T (u) = exp (−β EΩ(u) + f (β,Ω)) ,

where β = 1kB T is the inverse temperature.

The study of the thermodynamic limit N →∞ with |ΩN | = ρN isa hard problem.

Easier: Exchange the limits β →∞, N →∞ and study theasymptotic behavior of the minimizers of EΩ as N →∞.


Energetic crystallization

Assumption: ρ = 1, N := Ω ∩ L, limN→∞1N |ΩN | = 1 and E is

permutation-invariant:

E(u1, . . . ,uN) = E(uπ(1), . . . ,uπ(N)) for each permutation π.

A lattice L ⊂ Rd is a crystalline ground state of E if

e0 := limN→∞

1N

minu

EΩ(u) = limN→∞

1N

EΩ(Ω ∩ L).


Models dominated by local interactions

Strategy: A groundstate is periodic if it is almost everywherelocally periodic.Define the Cauchy-Born energy density function

W CB(F ) = limN→∞

det FN

EΩ(Ω ∩ F L)

and the Cauchy-Born approximation

ECBΩ =

∫Ω

W (∇v(x)) dx ,

if v is piecewise affine and v(L) = u1,u2, . . ..EΩ is dominated by local interactions if

EΩ(u) ≥∫

ΩW (∇v) dx

for suitable v (depends on triangulation).Florian Theil Energy-driven pattern formation

Wasserstein distance

Let Ω ⊂ R2 bounded, µ, ν ∈ P(Ω) probability measures. Theassociated Wasserstein-distance is defined as

W2(ν, µ) = inf∫

Ω|x − y |2 dΨ(x , y) : Ψ ∈ P(Ω× Ω)∫

ΩΨ(x , y) dx = µ,

∫Ω

Ψ(x , y) dy = ν

.

Energy (in 2 dim) if |Ω| = 1:

Fλ(µ) = λ∑

z∈supp(µ)

µ(z)12 +W2(LebΩ, µ).

Key property: W2 is local.

W2(LebΩ, µ) =

∫W (x ,∇v)

if v is induced by Vornoi tesselation.Florian Theil Energy-driven pattern formation

Numerical Minimization

0 10

15 points

0 10

17 points

0 10

110 points

0 10

164 points

0 10

1128 points

0 10

1256 points

Figure : An illustration of the appearance of a hexagonal pattern asλ→ 0.


Diblock copolymers

Diblock copolymer melts are well-known pattern formationsystems.


Schematic picture

Material parameters:χ: Flory-Huggins interactionN: Index of polymerizationf : Relative abundance


Results: Minimum energy

Theorem (Bourne/Peletier/T.’14)A If Ω is a polygon with at most 6 sides, then

λ−43 Fλ ≥ c6,

where c6 is the energy of a hexagon.B If ∂Ω is Lipschitz, then

limλ→0

infλ−43 Fλ = c6.

The limit is achieved by a scaled triangular lattice.


Configurational crystallization

Theorem

(a) If Fλ(µ) = λ43 c6, then µ is an atomic measure with all

weights equal to 1, and suppµ a translated, rotated anddilated triangular lattice.

(b) Define the dimensionless defect of a measure µ on Ω as

d(µ) := λ−43 Fλ(µ)− c6.

There exists C > 0 such that for λ < C−1 and for all µ withd := d(µ) ≤ C−1, suppµ is O(d1/6) close to a triangularlattice.


Mathematical tools

Theorem (Kantorovich (1942), Optimal mass transportation)If ν is absolutely continuous with respect to the Lebesguemeasure, then for each µ there exists an optimal transport planT : Ω→ Ω such that µ is the push-forward of ν under T and

W2(ν, µ) =

∫Ω|x − T (x)|2 dν(x).

Lemma (Fejes Tóth (1972))If ∂χ is a polygon with n sides, then

infz

∫χ|x − z|2 dx ≥ cn |χ|

12 ,

where cn = 12n

(13 tan π

n + cot πn).

Equality is attained if and only if ∂χ is a regular n-gon.Florian Theil Energy-driven pattern formation

Bounds on holes and masses

LemmaIf µ is a minimizer of Eλ, then µ(z) > 0 impliesµ(z) > 2.4 · 10−4.

The result is certainly not optimal and relies on the followingupper bound bound on the size of holes:

µ(B(z,R) \ z) > 0

if µ(z) > 0 and R > 3.3.


3d Crystallization230 (17) different space groups in 3d (2d).

Most common crystalline lattices are fcc (face-centeredcubic), hcp (hexagonally close packed) and bcc (bodycentered cubic).Only fcc and hcp are close packings.


Multiplicity of close packings: Stacking sequence

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


Two- and three-body energies

E(Y ) =∑

y ,y ′⊂Y

V2(y , y ′)+∑

y ,y ′,y ′′⊂Y

V3(y , y ′, y ′′).

Y ⊂ R3 particle positions.

E invariant under rigid body motion:V2(y , y ′) = V (|y − y ′|),

The Lennard-Jones potential

VLJ(r) = r−12 − r−6

accounts for van-der-Waals interac-tions at long distances.

ϕ(r)

r


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 ⊂ C2ρ([0,∞)) such that for all all

V2 ∈ Ω the identity

limn→∞

min#Y =n

1n E(Y ) = E∗ = min

r

12

∑k∈L\0

V (r |k |)

holds.


Visualisation of Ω

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

>1/αv

v’’>1

v>−1/2

|v’’(r)|<r(−7)

v=−1/2

r=1− r=1+ αα

Example: Morse potential V (r) = e−2a(r−r0) − 2e−a(r−r0).


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 ⊂ C2ρ([0,∞))× C2([0,∞)× [0,∞)) such that the

projection of Ω to C2ρ([0,∞)) is open and for all all (V2,V3) ∈ Ω

the identity

limn→∞

min#Y =n

1n E(Y ) = E∗

= minr

12

∑k∈L\0

V (r |k |) +16

∑k ,k ′∈L\0

V3(0, r k , r k ′)

.

holds.


Ground states

Theorem C.L Bravais lattice, Y ⊂ Rd countable and L-periodic,

1#Y

Eper(Y ) ≤ E∗.

Then 1#Y Eper(Y ) = E∗ and there exists dilation r > 0, trans-

lation t ∈ R3 and rotation R ∈ SO(d) such that

R Y + t = r Lfcc or r Ltri.


Dislocations in the case of pair interaction potentials

Equilibria can have many defects.

−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4

6Ground state energy Emin = −8.642719e+01 for N =121.

−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


Recent results: Alicandro-De Luca-Garroni-Ponsiligone,Ortner-Hudson


Dislocations in the case of pair interaction potentials

Equilibria can have many defects.

−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


−8 −6 −4 −2 0 2 4 6 8−6

−4

−2

0

2

4


Recent results: Alicandro-De Luca-Garroni-Ponsiligone,Ortner-Hudson


Previous results

Radin ’79 Periodicity of groundstates for Lennard-Jonestype potentials (1d)Radin ’81 Periodicity of triangular lattice for compactlysupported potentials (2d)Friesecke-T’02 Periodicity of ground states for mass-springmodels (2d)Suto ’05 Stealthy potentials (3d)T’06 Proof of minimality of the triangular lattice forLennard-Jones type models (2d)E-Li ’08 Minimality of hexagonal lattice for three-bodyinteractions (2d)


Sketch of the proof

Strategy: A groundstate is periodic if it is almost everywherelocally periodic. → Local problems

Step 1: Local analysis: Neighborhood graph, defects.Step 2: Treat non-local terms using cancelations and

rigidity estimates


Step 1: Local analysis

Minimum particle spacing:|yi − yi ′ | > 1− α for all i 6= i ′

since particles can be movedto infinity.Construction of a graph:i , i ′ ∈ B ⇔ |yi − yi ′ | ∈(1− α,1 + α)

Musin’s (2005) proof of the3d-kissing problem:

#b ∈ B | i ∈ b ≤ 12


Local Analysis

Theorem. (Harris/Tarasov/Taylor/T.’13)If S is a unit sphere in R3 and S1 . . .Sn pairwise exclusive unitspheres touching S. Then the number of tangency points issmaller or equal 36.Equality is obtained if and only if n = 12 and S1 . . .Sn form thecorners of a either a cuboctahedron or a twistedcuboctahedron.

Conjecture: Let S1,S2 be two touching unit spheres in R3.Then there can be at most 18 pairwise exclusive unit spherestouching S1 or S2.


Contact graphs

Fcc Hcp

Figure : Contact graphs of Qco and Qtco respectively


Sketch of the proof

There are 1,430,651 2-connected graphs with 12 vertices, atleast 24 edges, at most 5 edges adjacent to any given vertex.Contact graphs are those graphs where edges are induces byvertex positions.For each contact graph we define angles uj ∈ [0,2π] subtendedby edges.The angles satisfy linear inequalities such as

minj

uj ≥ arccos(13),∑

j adjacent to vertex i

uj = 2π,

...

Only for two graphs the corresponding linear programmingproblems admit a solution.


Step 2: Nonlocal terms

Two different ideas are needed.

A) Discrete rigidity estimates

B) Cancelation of the ghost forces

Use geometric rigidity of thesystem in order to control er-rors:

If the small triangles areundistorted, then the big tri-angle is undistorted.


Continuum rigidity results

minR∈S(d)

∫Ω|∇u − R|2 dx ≤ C(Ω)

∫|∇u − SO(d)|2 dx .

(Friesecke, James & Müller, CPAM 2002). The rigidity constantC(Ω) is invariant under rotations, dilations and translations:

C(RΩ + t) = C(Ω) for all R = λQ, λ ∈ R,Q ∈ O(d), t ∈ Rd


Summary

Convergence of the minimum energy for block co-polymermodelMethods from optimal transportation theoryOpen set of interaction potentials which admits periodic(fcc) minimizers in R3

Result based on mixture of results from Geometry andPDE theory.Open problem: Finite temperature.

lim|Ω|→∞

pβ,Λ(Y ) = lim|Λ|→∞

exp (−β E(Y ) + f (β,Ω)) =?


Finite temperature: β 1

Gas phase, dimension irrelevant

Expected: The dimension d matters if β 1.


Finite temperature: β 1

d = 3: Long-range order, one large cluster iff β > βcrit(ρ)d = 2: Local order, one large cluster iff β > βcrit(ρ). Nolong-range order due to the Mermin-Wagner Theoremd = 1: Several finite clusters, no phase transition. Cluster sizesare bounded.


Local and global order

Local (orientational) orderLet ϕ be testfunction with compact support such that∫

SO(d) ϕ(R x) = 0 and define for R ∈ SO(d)

f (R) =1|Y |

∑y 6=y ′∈Y

ϕ(R(y ′ − y)).

Y exhibits local order if f (R) = O(1), |Y | 1.Long range orderLet ϕ be a L-periodic testfunction such that

∫Rd/L ϕ = 0 and

define for R ∈ SO(d)

g(R) =1|Y |2

∑y 6=y ′∈Y

ϕ(R(y ′ − y)).

Y exhibits long-rande order if g(R) = O(1), |Y | 1.Florian Theil Energy-driven pattern formation

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 = 2and global order for d = 3.


Surface energy for d = 1

Binding energy per particle:

e0 = limN→∞

1N

min#Y =N

E(Y ) < 0.

Surface energy:

esurf = limN→∞

(min

#Y =NE(Y )− N e0

)> 0.

The surface energy accounts for missing bonds and surfacerelaxation and is different from −e0.


Cluster size distribution

A configuration Y = y1 < y2 < . . . < yN ⊂ Λ is decomposedinto clusters Ci which have the properties

yj , yj+1 ∈ Ci then yj+1 − yj < RY = ∪iCi

dist(Ci1 ,Ci2) ≥ R if i1 6= i2.

νβ(k) = limN→∞

#clusters C : #C = k#clusters

(cluster length histogram)


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 < 1a

p(β, ρ) chosen so that p−1 νβ( · /p) is tight andnon-singular as β →∞.

Thenlimβ→∞

1β log(p(β, ρ)) = −1

2esurf,

limβ→∞

p−1 νβ( · /p) = exp1.


Appendix References

References I

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 Systemwith Wasserstein Interaction.Com. Math. Phys, doi: 10.1007/s00220-014-1965-5.

S. Jansen, W. König, B. Schmidt & F. T.In preparation.


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