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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
Florian Theil Energy-driven pattern formation
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 →∞.
Florian Theil Energy-driven pattern formation
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).
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
Diblock copolymers
Diblock copolymer melts are well-known pattern formationsystems.
Florian Theil Energy-driven pattern formation
Schematic picture
Material parameters:χ: Flory-Huggins interactionN: Index of polymerizationf : Relative abundance
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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).
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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
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
6Ground state energy Emin = −8.642719e+01 for N =121.
Recent results: Alicandro-De Luca-Garroni-Ponsiligone,Ortner-Hudson
Florian Theil Energy-driven pattern formation
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
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
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
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
6Ground state energy Emin = −8.642719e+01 for N =121.
Recent results: Alicandro-De Luca-Garroni-Ponsiligone,Ortner-Hudson
Florian Theil Energy-driven pattern formation
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)
Florian Theil Energy-driven pattern formation
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
Florian Theil Energy-driven pattern formation
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
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
Contact graphs
Fcc Hcp
Figure : Contact graphs of Qco and Qtco respectively
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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
Florian Theil Energy-driven pattern formation
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)) =?
Florian Theil Energy-driven pattern formation
Finite temperature: β 1
Gas phase, dimension irrelevant
Expected: The dimension d matters if β 1.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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)
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation
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.
Florian Theil Energy-driven pattern formation