Hierarchical Pattern Discovery in Stochastic Lattice Systems Markos - - PowerPoint PPT Presentation

Hierarchical Pattern Discovery in Stochastic Lattice Systems Markos Katsoulakis University of Crete & FORTH, Heraklion, Greece, and University of Massachusetts, Amherst, USA

Joint work with:

• Petr Plech´

aˇ c, Mathematics, University of Tennessee, Knoxville, & Oak Ridge National Laboratory

• Dionisios G. Vlachos, Chemical Engineering, University of Delaware

Supported in part by:

• National Science Foundation: DMS and CMMI CDI-Type II
• U. S. Department of Energy
• Marie-Curie IRG Programme

Towards guiding engineering tasks for nanopattern formation in heteroepitaxial processes

• Patterns in such systems have rich morphologies at mesoscales

that change dramatically as control parameters vary

• Typically they form as a result of microscopic particle dy-

namics in a complex energy landscape, in the presence of stochastic fluctuations Applications:

• Formation of nanopatterns in heteroepitaxy: templating,
• ptical magnetic, and electronic devices.
• Self-assembly resulting from an interplay of short-range

attractive and long-range repulsive

MODEL SYSTEM: Plass et al Nature ’01 Self-assembly of Pb on Cu(111): LEEM of the Cu(111) surface at 673 K with increasing area fractions: i: Droplet pattern after cooling down to room temperature and air exposure. Scale bar: 0.3µm

Why study this model system?

• It is two-dimensional:

easier modeling, simulation, com- parison to data.

• Stable upon exposure to air for long times, rendering it an

ideal template for hierarchical self-assembly.

• Low-energy electron microscopy (LEEM) delivers:
• spatial resolution of ≃7 nm for patterns in experiments.
• provides dynamic information for modeling and control.
• Finer resolution, atomic force microscopy (AFM) can be

done at selected times.

• Accumulated knowledge about the system, e.g., the diffu-

sion mechanism of Pb on Cu has been studied in depth.

First task: comparisons of experimental data to simulations , , , , Top: Experimental snapshots of pattern formation. Bottom: Results of CGMC simulations on the same system. Intractable with conventional KMC due to µm scales Need a statistical comparison!

Goals of the presented research are

• to obtain an approximate phase diagram by using a deter-

ministic mesoscopic PDE

• to use Adaptive Coarse-Grained Monte Carlo (AdCGMC)

simulations in order to refine the phase diagram by includ- ing properly coarse-grained interactions and fluctuations.

• Address systems tasks:

– What is the phase diagram for nanopatterns ? – What controls the particle spacing and size ? – What causes the non-uniformity in shape and size ?

• A. Coarse-graining microscopic systems: continuous limits

Thermodynamic/Hydrodynamic limits (equilibrium/nonequilibrium) Coarse quantities: density, average velocity, one-point pdf, etc. Examples

• Deterministic ODE/PDE: Mean-field approximations, Ginzburg-

Landau models, kinetic equations, field theories in poly- mers, etc.

• Stochastic Corrections as SPDEs: Stochastic Allen-Cahn,

Cahn-Hilliard-Cook, diblock co-polymer models, etc.

• B. Hierarchical Coarse-graining
• 1. Coarse-graining of polymers; proteins; biomembranes

CG procedure, ”super-atoms”:

• 2. Stochastic lattice dynamics/ KMC

Examples: Catalysis, epitaxial growth, micromagnetics, etc. Patterning through self-assembly

Microscopic lattice Coarse lattice Time (s)

Model reduction → ”systems tasks”: control and optimization for materials engineering

Theory and General Mathematical Framework

• N, M – the size of the microscopic and CG systems, re-

spectively.

• q – the parameter that defines the level of coarse-graining,

e.g., number of spins in the coarse variable (a block spin)

• r number of atoms in the “meta-particle”.
• X ∈ SN = (Σ)ΛN – the microscopic configuration space.
• Q = TX ∈ Sc

M,q – the coarse-graining operator defining

configurations on the coarse configuration space.

Basic Steps

• A. Microscopic equilibrium Gibbs measure

µN,β(dX) = 1 ZN,β e−βHN(X)PN(dX) , PN(dX) =

• i∈ΛN

ρ(dXi) .

• B. Coarse-grained measure is given by the projection operator:

µc(dQ) := µ{X ∈ SN | T(X) = Q} New CG Hamiltonian: e−βHc(Q) = E[e−βHN | Q] ≡

• SN

e−βHN(X) PN(dX | Q) , Computing Hc(Q) directly = ⇒ integration on a high-dimensional space. Solution: Approximate Hc or better µc by a convergent expan- sion in a small parameter.

• C. Approximating CG Hamiltonian Hc

Step 1: identify a base state and a “small” parameter ǫ Step 2: choose a suitable expansion and expand the approxi- mation error in ǫ Expand the coarse-grained measure µc

q,β around the approximate

equilibrium measure defined by the approximate coarse Hamilto- nian ¯ H(0) rather than around the product measure PN(dX | Q): Hc(Q) = ¯ H(0) − log E[e−(HN− ¯

H(0)) | Q] .

Suitable choice for H(0)

m ? What is the error?

Step 3: The zero-th order approximation ¯ H(0) should satisfy (1)

E

HN − ¯ H(0)

m | Q

= 0 .

• Heuristics: Expansion of e∆H and log:

= E [∆H | Q] + E (∆H)2 | Q − E [∆H | Q]2 + O((∆H)3) formal calculations inadequate since: ∆H ≡ HN − ¯ H(0)

m

= N · O(ǫ)

• Rigorous analysis – Cluster expansion: around ¯

H(0)

m .

Cluster expansions developed in statistical physics for controlling measures on high-dimensional spaces. K., Plechac, Rey-Bellet, Tsagkarogiannis, [M2AN, ’07]

• D. Numerical Analysis of CG
• I. Equilibrium: Bounds on the “distance” between µc

q,β(dQ) and

approximate ¯ µq,β(dQ)) Error control in terms of relative entropy estimates:

• 1. Information loss in terms of relative entropy of π1 and

π2: R (π1 | π2) =

• S

log dπ1 dπ2 π1(dσ) .

• II. Observables: Bounds on the weak error:

ETX0[f(TXT)] − EQ0[f(QT)]

• E. Microscopic Reconstruction – Reverse CG map

Stochastic lattice dynamics and nanopattern formation

• Configuration space on the lattice ΛN:

SN = {0, 1}ΛN, σ = {σ(x) ∈ Σ | x ∈ ΛN}

• Microscopic Hamiltonian

H(σ) = 1 2

• x,x′∈ΛN ,x=x′

U(x − x′, σ(x), σ(x′)) +

• x∈ΛN

V (x, σ(x)) , pair-wise interactions: U = Jlong(x − x′)σ(x)σ(x′) + Kshort(x − x′)σ(x)σ(x′) single site interactions: V = Ψ(σ(x)) + h(x)σ(x)

• CG transformation:

η(k) = (Tσ)(k) :=

• x∈Ck

σ(x)

• CG Hamiltonian:

e−βHc(η) =

• e−βHN(σ) PN(dσ | η) ≡ E[e−βHN | η]

The canonical CG Gibbs measure on Sc

M,q

µΛc

M,q,β(dη) =

1 ZΛc

M,q,β

e−βHc(η)PM,q(dη) with coarse-grained prior PM,q(dη) = Πk∈Λc

Mρq(dη(k)).

• Base Hamiltonian ¯

H(0): pair-interactions between block-spins η(k) and η(l) with the in- teraction potential ¯ J(k − l) given explicitly (compressed interac- tion kernel J using Haar basis) ¯ J(k, l) = 1 q2

• x∈Ck
• y∈Cl,y=x

J(x − y) .

Corrections to the Hamiltonian ¯ H(0)→Multi-body terms ¯ Hm(η) = ¯ H(0)

m (η) + ¯

H(1)

m (η) + ...

¯ H(1)(η) = β

• k1
• k2>k1
• k3>k2

[j2

k1k2k3(−E1(k1)E2(k2)E1(k3) + ...

• Er(k) ≡ Er(η(k)) = (2η(k)/q − 1)r + oq(1)
• “Moments” of interaction potential J:

j2

k1k2k3 =

• x∈Ck1
• y∈Ck2
• z∈Ck3

(J(x − y) − ¯ J(k1, k2))(J(y − z) − ¯ J(k2, k3))

• Computational complexity of the corrections?
• Compression of ¯

H(1)?

Computational complexity-Compression of ¯ H(1)

• Evaluation of the Hamiltonian:

Count Speed-up Microscopic: HN(σ) O(NLd) 1 CG0: ¯ H(0) O(MLd/qd) O(q2d) CG1: ¯ H(0) + ¯ H(1) O(ML2d/q2d) O(q3d/Ld)

• Decay of J (e.g. Coulomb) → J− ¯

J decays faster; thus, further truncations/splittings within the correction ¯ H(1) are possible: N = 1000, βJ0 = 6.0, Process CPU (secs) q=1 (no coarse-graining) 322192.06 q=8 5232.62 q=8c (no splitting) 69473.09 q=8c (splitting) 6900.72 [Are, K. Plechac, Rey-Bellet, SIAM J. Sci. Comp. (2009)]

Arrhenius dynamics: (a) deposition – spin-flip, (b) diffusion – spin exchange CG Markovian Dynamics approximation:

η η η η η block spin η(k) =

• x∈Ck σ(x)

q q diffusion adsorption desorption

Birth-Death type process, with interactions. Lcg(η) =

• k∈Λc

ca(k, η) g(η + δk) − g(η) + cd(k, η) g(η − δk) − g(η) .

• Coarse-grained rates and Detailed Balance: ¯

c(α)

a

and ¯ c(α)

d

com- putable using ¯ H(α).

Coarse observable at resolution q: ηt(k) = Tσt(k) :=

• y∈Dk

σt(y) In general, it is non-markovian. Mori-Zwanzig formulation. Stochastic closures: when can we write a new approximating Markov process for ηt? Ergodicity: Are the long-time dynamics reproduced? Rare events?

• Errors can contaminate the simulation at long times; wrong

switching times in bistable systems: Hanggi et al PRA ’84 (well- mixed systems), K. Szepessy, CMS ’06.

Error Quantification in CG Schemes Theorem 1: (A priori error analysis) Define the “small” parameter ǫ ≡ β q

L∇J1

• 1. Approximation of the CG free-energy landscapes

¯ Hm(η) = ¯ H(0)

m (η)−1

β log E[e−β(HN− ¯

H(0)

m ) | η] = ¯

H(0)

m (η)+ ¯

H(1)

m (η)+NO(ǫ3) .

• 2. Loss of information during coarse-graining
• Specific relative entropy:

R (µ | ν) := 1 N

• σ

log

µ(σ)

ν(σ)

• µ(σ)

.⋄ R ¯ µ(α)

M,q,β | µN,βoT−1

= O ǫα+2 .

• Tσ = Projection on coarse variables=

y∈Dk σ(y).

K., Plechac, Rey-Bellet, Tsagkarogiannis, [M2AN, ’07, J. Non.

• Newt. Fluid Mech. ’08]

• II. Error Estimates for observables – Dynamics

[K., P. Plechac, A. Sopasakis, SIAM Num. Anal. ’06] Theorem 2: coarse grained observables/quantity of interest: ψ, microscopic dynamics: σt, coarse-grained dynamics: ηt Then for any fixed time 0 < T < ∞ |Eψ(TσT) − Eψ(ηT)| ≤ CTǫ2 ,

• Tσt = Projection on coarse variables=

y∈Dk σt(y).

• Error accumulation as T → ∞? 2nd order error estimates at

equilibrium

Hierarchical Pattern discovery

• Applications of nanopatterns in heteroepitaxy
• Self-assembly results from an interplay of short-ranged attrac-

tive and long-ranged repulsive (elastic and/or electronic) inter- actions Key questions:

• Phase diagram for nano-patterns
• What controls the particle spacing and size
• What causes the non-uniformity in shape and size
• Performing ”systems tasks”

Prototype potentials: Competing attractive and repulsive in- teractions Gaussian potentials: J(r) = J0

• e−r2/R2

r − χe−r2/R2 a

Morse potentials: J(r) = J0

• e−r/Rr − χe−r/Ra

Arrhenius dynamics: (a) deposition – spin-flip, (b) diffusion – spin exchange

Hierarchical coarse-grained models derived from kinetic Monte Carlo

Step 1: Obtain a rather “crude” phase diagram by employing deterministic mesoscopic PDE (local mean-field approximation). Example: For the Arrhenius diffusion dynamics the coverage c(x, t) solves ∂tc = −∇·(e−βV (∇c−c(1−c)∇V ) , where V (x) =

• J(x−y)c(y) dy .

Variational formulation+Dynamics: ct − ∇ ·

• µ[c]∇
• δE[c]

δc

+ h

• ,

E[c] = −1 2 J(r−r′)c(r)c(r′)drdr′+

• 1

β[c ln c+(1−c) ln(1−c)]dr ,

µ[c] =

Dβc(1 − c) ,

D = Ψ(0) , Metropolis-type , βc(1 − c) exp(−βJ ∗ c) , Arrhenius

• Metropolis-type:

Lebowitz, Orlandi, Presutti JSP ’91; Gia- comin, Lebowitz, etc

• Connections to Cahn-Hilliard fo attractive interactions (J > 0)

Stability analysis: c(x, t) = c0 + ǫ exp(ωt + iξ · x) where c0 is a constant state and ǫ < < 1. Dispersion relation: λ = D|ξ|2 exp(−βJ0c0)βc0(1 − c0) ˆ J(ξ) − 1 βc0(1 − c0)

• ,

J0 = J(r)dr and ˆ J is the Fourier transform of J. λ > 0 ∼ formation of clusters/patterns