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, optical 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 . 3 µm

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 Time (s) Microscopic lattice Coarse lattice 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) or number of atoms in the “meta-particle”. • X ∈ S N = (Σ) Λ N – the microscopic configuration space. • Q = T X ∈ S c M,q – the coarse-graining operator defining configurations on the coarse configuration space.

Basic Steps A. Microscopic equilibrium Gibbs measure 1 e − βH N ( X ) P N ( dX ) , µ N,β ( dX ) = Z N,β � P N ( dX ) = ρ ( dX i ) . i ∈ Λ N B. Coarse-grained measure is given by the projection operator: µ c ( dQ ) := µ { X ∈ S N | T ( X ) = Q } New CG Hamiltonian: � e − βH c ( Q ) = E [ e − βH N | Q ] ≡ e − βH N ( X ) P N ( dX | Q ) , S N Computing H c ( Q ) directly = ⇒ integration on a high-dimensional space. Solution: Approximate H c or better µ c by a convergent expan- sion in a small parameter.

C. Approximating CG Hamiltonian H c 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 µ c q,β around the approximate equilibrium measure defined by the approximate coarse Hamilto- H (0) rather than around the product measure P N ( dX | Q ): nian ¯ H (0) − log E [ e − ( H N − ¯ H (0) ) | Q ] . H c ( Q ) = ¯

Suitable choice for H (0) m ? What is the error? H (0) should satisfy Step 3: The zero-th order approximation ¯ E � H (0) m | Q � H N − ¯ (1) = 0 . • Heuristics: Expansion of e ∆ H and log: (∆ H ) 2 | Q � − E [∆ H | Q ] 2 + O((∆ H ) 3 ) = E [∆ H | Q ] + E � formal calculations inadequate since: H (0) ∆ H ≡ H N − ¯ = N · O( ǫ ) m H (0) • Rigorous analysis – Cluster expansion : around ¯ m . Cluster expansions developed in statistical physics for controlling measures on high-dimensional spaces. K., Plechac, Rey-Bellet, Tsagkarogiannis, [ M 2 AN, ’07 ]

D. Numerical Analysis of CG I. Equilibrium: Bounds on the “distance” between µ c q,β ( dQ ) and approximate ¯ µ q,β ( dQ )) Error control in terms of relative entropy estimates: 1. Information loss in terms of relative entropy of π 1 and π 2 : � log dπ 1 R ( π 1 | π 2 ) = π 1 ( dσ ) . dπ 2 S II. Observables: Bounds on the weak error: E T X 0 [ f ( T X T )] − E Q 0 [ f ( Q T )] E. Microscopic Reconstruction – Reverse CG map

Stochastic lattice dynamics and nanopattern formation • Configuration space on the lattice Λ N : S N = { 0 , 1 } Λ N , σ = { σ ( x ) ∈ Σ | x ∈ Λ N } • Microscopic Hamiltonian H ( σ ) = 1 � � U ( x − x ′ , σ ( x ) , σ ( x ′ )) + V ( x, σ ( x )) , 2 x,x ′ ∈ Λ N ,x � = x ′ x ∈ Λ N pair-wise interactions: U = J long ( x − x ′ ) σ ( x ) σ ( x ′ ) + K short ( x − x ′ ) σ ( x ) σ ( x ′ ) single site interactions: V = Ψ( σ ( x )) + h ( x ) σ ( x ) • CG transformation: � η ( k ) = ( T σ )( k ) := σ ( x ) x ∈ C k • CG Hamiltonian: � e − βH c ( η ) = e − βH N ( σ ) P N ( dσ | η ) ≡ E [ e − βH N | η ]

The canonical CG Gibbs measure on S c M,q 1 e − βH c ( η ) P M,q ( dη ) µ Λ c M ,q,β ( dη ) = Z Λ c M ,q,β with coarse-grained prior P M,q ( dη ) = Π k ∈ Λ c M ρ q ( 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 � � ¯ J ( x − y ) . q 2 x ∈ C k y ∈ C l ,y � = x

Corrections to the Hamiltonian ¯ H (0) �→ Multi-body terms H (0) H (1) H m ( η ) = ¯ ¯ m ( η ) + ¯ m ( η ) + ... � � � H (1) ( η ) = β ¯ [ j 2 k 1 k 2 k 3 ( − E 1 ( k 1 ) E 2 ( k 2 ) E 1 ( k 3 ) + ... k 1 k 2 >k 1 k 3 >k 2 • E r ( k ) ≡ E r ( η ( k )) = (2 η ( k ) /q − 1) r + o q (1) • “Moments” of interaction potential J : � � � j 2 ( J ( x − y ) − ¯ J ( k 1 , k 2 ))( J ( y − z ) − ¯ k 1 k 2 k 3 = J ( k 2 , k 3 )) x ∈ C k 1 y ∈ C k 2 z ∈ C k 3 -Computational complexity of the corrections? -Compression of ¯ H (1) ?

Computational complexity-Compression of ¯ H (1) • Evaluation of the Hamiltonian: Count Speed-up O( NL d ) Microscopic: H N ( σ ) 1 CG0: ¯ H (0) O( ML d /q d ) O( q 2 d ) H (0) + ¯ CG1: ¯ H (1) O( ML 2 d /q 2 d ) O( q 3 d /L d ) • Decay of J (e.g. Coulomb) �→ J − ¯ J decays faster ; thus, further H (1) are possible: truncations/splittings within the correction ¯ N = 1000 , βJ 0 = 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 adsorption desorption η η q diffusion block spin η ( k ) = x ∈ C k σ ( x ) � q

Birth-Death type process, with interactions. � c a ( k, η ) � g ( η + δ k ) − g ( η ) � L c g ( η ) = + k ∈ Λ c c d ( k, η ) � g ( η − δ k ) − g ( η ) � . c ( α ) c ( α ) • Coarse-grained rates and Detailed Balance: ¯ and ¯ com- a d putable using ¯ H ( α ) .

Coarse observable at resolution q : � η t ( k ) = T σ t ( k ) := σ t ( y ) y ∈ D k In general, it is non-markovian . Mori-Zwanzig formulation. Stochastic closures : when can we write a new approximating Markov process for η t ? 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.