3.320 Lecture 22 (5/3/05)Problems of Time and SpaceBrute Force ApproachesCoarse-grainingCan Inhomogeneously Coarse-GrainExample: Crack Impinging on Grain BoundaryThe frontier of coarse-graining: DynamicsSome suggestionsThermodynamic Integration over Degree of Freedom of Removed Atoms2D-3D Migdal-KadanoffEvaluation Criteria and Problems with Dynamics in Coarse-Grained ModelsEvaluation Criteria and Problems with Dynamics in Coarse-Grained ModelsHow the model works: static propertiesHow the model works: thermal propertiesHeat conduction: finite size effectHeat conduction: large system“constant pressure” propertiesAccelerating time (without Einstein’s help)Do I need to explain why you want faster MD ?The ProblemDifferent Approaches within Molecular Dynamics to Study Infrequent Event SystemsA quick review of Transition State Theory (simplified)Parallel Replica MethodExample of Parallel Replica Simulation: Island on Island on Ag(111) at T=400KHyperdynamics (briefly): Elevate the potential wells to make system transition out fasterTemperature-Accelerated Dynamics (TAD)How to extrapolate to low-T ?Approximations of the methodApplication of TAD: Cu on Cu depositionSome “events” during the simulationKinetic Monte CarloExample 1: Diffusion in LixCoO2Dilute Diffusion TheoryStrategyInteractions obtained from Energy Calculations of about 60 configurationsStrategy (continued)Kinetic Monte CarloStrategy (continued)Getting activation Barriers: The elastic band methodComplicationsComplicationsUse as input in continuum modelsReferences5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari3.320 Lecture 23 (5/3/05)Faster, faster ,faster …Bigger, Bigger, Bigger Accelerated Molecular DynamicsKinetic Monte CarloInhomogeneous Spatial Coarse Graining5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariProblems of Time and SpaceYour simulation will always be “too small” and “too short”TimeSpaceIntegrated overAtomistic/ElectronicThermodynamicsContinuum matter5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariBrute Force ApproachesConquer more space with more CPU’s: parallelization over spaceHow to parallelize time ?5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCoarse-graininge.g. relaxation around a defectDo you really need all the atoms far away ?Spaceassume homogeneous deformation of groups of atomsIf displacement field for the corner atoms is known, one can interpolate the displacements for the “internal” atoms Homogeneous deformation of cell -> can calculate energy without explicitly keeping track of positions of internal atoms5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCan Inhomogeneously Coarse-GrainThis is the idea of the quasi-continuum approach (*)•Atomic extensions of Finite Elements: quasicontinuum(*) V. B. Shenoy, et al, Journal of the Mechanics and Physics of Solids 47, (1991) 611-42.5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariExample: Crack Impinging on Grain BoundaryPhoto removed for copyright reasons.from R. Miller et al. Modeling and Simulation in Materials Science and Engineering 6, (1998) 607.5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariThe frontier of coarse-graining: Dynamics MicroscopicdynamicsHamiltonian description {p ,q } of particles, timeiiThermodynamics:{p,q} of boundariesT,V,S,E,p ....p,V,T,F,S,E ...coarse grained modelswith thermodynamicsand dynamicsthermothermoslow timeThermodynamics, ElasticityMultiscaledynamics:new frontier5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariSome suggestions lump mass of removed atoms into node atoms and do MDUse free energy rather than energy to determine deformation laws inside the elements5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariThermodynamic Integration over Degree of Freedom of Removed Atoms Renormalization of the potentialS. Curtarolo, G. Ceder. Dynamics of an Inhomogeneously Coarse Grained Multiscale System. Physical Review Letters 88(25). pp. 255504 - (2002).5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariRenormalized Potentials5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari2D-3D Migdal-Kadanoff5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariEvaluation Criteria and Problems with Dynamics in Coarse-Grained ModelsPhonon reflection into fine regionsCoarse-grained regions can not sustain phonons with short wavelength5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariEvaluation Criteria and Problems with Dynamics in Coarse-Grained ModelsRemoving degrees of freedom = removing entropyDerivatives of entropy may be wronge.g. heat capacity, thermal expansion5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariHow the model works: static properties2D systemoriginal lattice with 6975 atoms & renormalized lattice with 1510 nodestensile strain/stress------- original--- renormalized4% accuracyshear strain/stress------- original--- renormalized5% accuracy5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariHow the model works: thermal properties5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariHeat conduction: finite size effectRun 2D systems with two regions and one interfaceRatio vs length.Regions with nodes>500give “accurate” results.)(/)(hhnhnhNNκκ5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariHeat conduction: large systemCompare 2D systems• homogeneous with 3.5.105atoms• coarse grained with 6.3.104nodes • Works better at high temperature• 15% underestimation• interface scatteringhomogeneous (o), coarse grained (+)5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari“constant pressure” properties• Thermal expansion and specific heat depend on the free energy.• Models with ”approximated” free energy have “approximated” thermal expansion ⇒ build up large internal strains upon changing temperature !pαpC5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariAccelerating time (without Einstein’s help)5/3/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariDo I need to explain why you want faster MD ?•MD is nano seconds•For systems where only the equilibrium behavior is of interest, use coarse methods or sampling methods