1.021 , 3.021, 10.333, 22.00 I ntroduc tion to Modeling and Simulation
Spring 2011
Introduction
Lecture 1
Markus J. Buehler
1
Laboratory for Atomistic and Molecular Mechanics Department of Civil and Environmental Engineering Massachusetts Institute of Technology
Subject structure and grading scheme
Part I: Continuum and particle methods (Markus Buehler)
Lectures 2-13
Part II: Quantum mechanics (Jeff Grossman)
Lectures 14-26
The two parts are based on one another and will be taught in an integrated way
The final grade will be based on:
Homework (50%) and exams (50%)
A few things we’d like you to remember…
The goal is to provide you with an excellent foundation for modeling and simulation, beyond the applications discussed in IM/S.
Our goal: Discover the world of Modeling and Simulation with you
– using a bottom-up approach.
We will cover multiple scales -- t he atomic scale, using Newton’s laws, statistical mechanic s and quantum mec hanics (involving electrons), as well as continuum methods.
You will be able to apply the knowle dge gained in IM/S to many other complex engineering and scienc e problems
Subject content: Big picture
Subjec t provides an introduction to modeling and simulation .
Sc ientists and engineers have long used models to better understand the system they study, for analysis and quantification, performance prediction and design . Howev er, in recent years – due to the advanc e of computational power, new theor ies (Density Func tional Theory, reactive force fields e.g. R eax FF), and new experimental methods (atomic force microscope, optical tw eezers, etc.) – m ajor advances have been possible that provide a fundament ally new approach to modeling materials and structures.
This subject will provide you with the relevant theoretical and numerical tools that are neces sary to build models o f co mplex physical phenomena and to simulate thei r behavior using computers.
The physical system can be a collection of electrons and nuclei/core shells, atoms, molecules, structural elements, grains, or a continuum medium: As such, the methods discussed here are VERY FLEXIBLE!
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The lectures will provide an exposure to several areas of application, based on the scientific exploitati on of the power of computation,
Engineering science paradigm: Multi-scale view of materials
Buehler and Ackbarow, Materials Today , 2007
Courtesy of Elsevier, Inc., 5
http://www.scien c edirect. com . Us ed w i t h p e r m iss i on .
Characteristic scale o f technology frontier (materials)
m
Bridging the scales
cm Ax es Weapons
mm Equipment
tools weapons
Machines Mass production
m Agriculture
Building
materials
IT revolution
nm Industrialization
Å
Transistors Integrated circuits
AFM, SEM
CNTs as electronic devices
…
Biology & nanotech
B i o-X revolutio n
Stone age bronze age se mic onductor age nanotechnology
Fi g. 1. 1 i n Buehl e r, Mark u s J. Atomisti c Mo del in g of Mater ial s Fail ur e . Spri n g er , 2008. © S pri n ger. Al l ri ghts r e ser v ed. This content is ex cl u d ed from ou r Creat i ve Common s licen se. For more inform ation, see htt p :/ / o cw.mi t .edu /fai ru se .
Content overview
I. Particle and continuum me thods
1. Atoms, molecul e s, chemistry
2. Continuum modeling approac hes and solution approaches
3. Statistical mechanics
4. Molecular dynamics, Monte Carlo
5. Visualization and data analysis
6. Mechanical proper ties – applic ation: how things fail (and how to prevent it)
7. Multi-scale modeling par adigm
8. Biological systems (simulation in biophysics) – h ow proteins work and how to model them
II. Quantum mechanical methods
1. It’s A Q uantum World: T he Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
3. The Many-Body Problem: Fr om Many-Body to Single- Particle
4. Quantum modeling of materials
5. From Atoms to Solids
6. Basic pr operties of mater i als
7. Advanced proper ties of materials
8. What else can we do?
Lectures 1-13
Lectures 14-26
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Engineering science paradigm: Multi-scale view of materials
“ molec u lar ” ( explicitly resolve molecules/atoms) Molecular Dynamics
Part I
Part II
“ continuum ” ( m a t t e r infinitely divis i ble, no internal structure)
e.g. finite element methods
“ quantum ” (explic itly resolve electrons);
e.g. Dens ity Functional Theory
© s o ur c e u n k n o w n. Al l rights res e rv ed. Thi s
content is ex cl uded from our Creative 8
Co mmo ns lice n se . F o r mo r e in fo r m a t io n , s e e
A few important concepts in modeling and simulation
What is the difference between modeling and simulation?
Modeling and simulation
The term modeling refers to the development of a mathematical representation of a physical situation.
On the other hand, simulation refers to the procedure of solving the equations that resulted from model development.
What is a model?
Mike Ashby (Cambridge University):
A model is an idealization. Its relationship to the real problem i s like that of the map of the London tube trains to the real tube systems: a gross simplification, but one that captures certain essentials.
“Physical situation” “Model” 11
© G oo gl e, Inc. Al l ri ghts r e se r v e d . Thi s co nte n t i s e x cl u d ed from our Creative Common s licen se . F o r m o re in fo rm a t io n , s ee h t t p :// oc w . m i t . e d u/ f a ir use .
© M as sa c h u s etts Bay Trans p ortati on Authori t y. Al l rights res e rv ed. Thi s c o ntent i s ex cl u d ed from our Creative Common s lic ense . F o r mo re in fo r m a t io n , s ee htt p :/ / o cw.mi t .edu /fai ru se .
What is a model?
Mike Ashby (Cambridge University):
The map misrepresents distances and directions , but it elegantly dis p lays the connectivity .
The quality or usefulness in a model is measured by its ability to capture the governing physical f eatures of the problem. All successful models unashamedly distort the inessentials in order to capture the features that really matter.
At worst, a model is a concise description of a body of data . At best, it captures the essential physics of the problem , it illuminates the princ iples that underline the key observations, and it predicts behavior under conditions wh ich have not yet been studied .
What is a simulation?
S imulation refers to the procedure of solving the equations that resulted from model development.
For example, numerically solve a set of differential equations with different initial/boundary conditions.
+ B C s , I C s
1.021 , 3.021, 10.333, 22.00 I ntroduc tion to Modeling and Simulation
Spring 2011
Part I – C ontinuum and partic le me thods
Introduction part I
Markus J. Buehler
Laboratory for Atomistic and Molecular Mechanics Department of Civil and Environmental Engineering Massachusetts Institute of Technology
Content overview
I. Particle and continuum me thods
1. Atoms, molecul e s, chemistry
2. Continuum modeling approac hes and solution approaches
3. Statistical mechanics
4. Molecular dynamics, Monte Carlo
5. Visualization and data analysis
6. Mechanical proper ties – applic ation: how things fail (and how to prevent it)
7. Multi-scale modeling par adigm
8. Biological systems (simulation in biophysics) – h ow proteins work and how to model them
II. Quantum mechanical methods
1. It’s A Q uantum World: T he Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
3. The Many-Body Problem: Fr om Many-Body to Single- Particle
4. Quantum modeling of materials
5. From Atoms to Solids
6. Basic pr operties of mater i als
7. Advanced proper ties of materials
8. What else can we do?
Lectures 2-13
Lectures 14-26
Multi-scale view of materials
Buehler and Ackbarow, Materials Today , 2007
Co u r t e sy El se vier , I n c ., 16
http://www.scien c edirect. com . Us ed w i t h p e r m iss i on .
Example application: Stiffness of materials (Young’s modulus)
Objective : Illustrate the significance of multiple scales for material behavior and introduce multi-scale modeling paradigm
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Beam deformation problem – c ontinuum model
A
Question: Dis p lacement field
Governing equation (PDE)
Integration & BCs
Geometry
BC - load:
E = unknown parameter
E is parameter called “Young’s modu lus” that relates how force and 18
deformation are related (captur es properties of material)
How to determine Young’s modulus E ?
Measurement (laboratory):
Rod/beam (e.g. plastic, metal, nanowire)
A =cross-section
Young’s modulus E (~stiffness=proportionality between
force and dis placement) 19
How to determine E ? - alternative approach
Atomistic simulation – new engineering paradigm
Idea: Consider the behavior of a collec tion of atoms inside the beam as
deformation proceeds 20
Molecular dynamics simulation
Newton’s laws: F =m a
Chemistry: Atomic interactions – calculate interatomic forces from atomic interactions, that is, calculate F from energy landscape of atomic configuration (note that force and energy are related…)
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Linking atomistic and continuum perspective
Atomis tic viewpoint enables us to calculate how force and deformation is related, that is, we can predict E once we know the atomic structure and the type of chemical bonds
Example, in metals we have meta llic bonding and crystal structures – thus straightforward calculation of E
Atomis tic models provide fundam ental perspective, and thereby a means to determine (solely from the atomistic / chemical structure of the material) important parameters to be used in continuum models
Image from Wikimedia C o mmo n s , h t t p : //co mmons .wikime d ia .o rg .
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Ener gy U
r
1/r 12 (or Exponential)
Repulsion
Radius r (Distance between atoms)
e
Attraction
1/r 6
Image by MIT OpenCou r seWare.
Quantum mechanics
Deals with fundamental view of chemical bonding, based on electrons in atoms
diene + dienophile
conjugated (substituted) diene + (substituted) olefin (substituted) cyclohexene
“Schroedinger equation”
Developing a potential energy from quantum mechanics
O
O r
r
Image remo v e d du e to co py ri ght restri cti o n s . See: http://www. kressw ork s .com /kressw o rk sorg/Quantu m_Chemi stry /Pote nti al _ E n er g y _S u r face s /water _ di m er /R e s ou r c e s / c harts /D FT_v s _V QZ_HF_and _ MP 4SD T Q _ r e si ze d. gi f .
copper
Example: Stretching nanowire
force
deformation 26
© s o ur c e u n k n o w n. Al l rights res e rv ed. Thi s co nt ent is ex clu d ed from our C r eative Common s
Multi-scale simulation paradigm
y
yy
n y
yz
yx
xz
xy
n x
A y
xx
x
z A x
div + f = 0
Image by MIT OpenCou r seWare.
“continuum scale” Matter is indefinitely Divisible
Youn g ’s modulus as p arameter
Molecular m odel (fundamental) P a rameters (Young’s modulus) Use in model with PDE that involves
Courtesy of Elsevier, Inc . 2 , 7 http://www.scien c edirect. com . Us ed w i t h p e r m iss i on .
Beam deformation problem – c ontinuum model
A
Question: Dis p lacement field
Governing equation (PDE)
Integration & BCs
Geometry
BC - load:
E = parameter (obtained from atomistic simulation)
E is parameter called “Young’s modu lus” that relates how force and 28
deformation are related (captur es properties of material)
Applications of continuum methods
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Cloth modeling for animated movies
Image of fl ag remov e d du e to copy ri gh t restri cti on s . See http : / / w ww.moma .org/c ollec t ion/objec t .p hp ?ob j ec t_ id =78805 .
Aivaz i s, Lombeyda and RR, 2003
Airbag deployment dynamics
Image co u r te s y of Hig h Co ntrast . L i cen s e: CC-B Y .
Image co u r te s y of High Co ntrast . L i cen s e: CC-B Y .
Publ i c domai n i m age.
Benefits of atomistic models
Other material properties
Atomistic models are not limited to calculation of E (or generally, elastic properties)
Atomistic models also enable us to predict failure, fracture, adhesion, diffusion constants, wave speeds, phase diagram (melting), protein folding (structure), …
Glass – b rittle (breaks easily) Metal – ductile (deformable)
Failure of materials and structures
Failure = uncontrolled response of a struct ure, often leading to malfunction of entire device, system
Earthquake
Publ i c domai n i m age .
Image by qui n n. anya o n F lic kr . L i c e ns e: C C - B Y .
Collapse of buildings
Engineering materi als fracture (ceramics, tiles)
Bone fracture
Image by di g i tal s ad hu on Flickr. L i cen s e: CC-NC .
Cost of failure of materials : > > $100 billion (1982)
Image by Wh a’ppe n on F l i c kr .
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Failure proceeds by rupture and tear of molecular and atomistic structures
Breaking of chemical bonds
Courtesy of Elsevier, Inc., http://www.scien c edirect. com . Used with permissi on.
Failure of materials observed at macroscale is due to repeated breaking, shearing, tearing of bonds at atomistic scale
Nanoscopic response of material’s building block is key for materials failure 35
Pl ease s e e: Buehl e r, Mark u s J., F ari d F . Abraham, et al . "Hyper el asti ci ty Go ve rn s D y nami c Fractu re at a Criti c al L e n g th Scal e.” Nature 4 2 6 (200 3): 14 1-6 .
Supersonic fracture: Disc overed in atomistic simulation on supercomput e 3 r 8 s
Theory/M D experimen t
Image remo v e d du e to co py ri ght restri cti o n s .
Pl ease s e e Fig. 9 i n Buehl e r, Mark u s, an d Huaji an Gao. "Mo d el i n g D ynami c Fractu re Us ing L a r g e-Sc a l e At o m is t i c Simu la t i o n s . " Ch a p t e r 1 in Sh u k la , Ar u n .
Dynamic Fractu r e Mecha n ics .Ha c ke ns ack, N J : W o r l d Sc ie n t ific , 2 006 .
Image remo v e d du e to co py ri ght restri cti o n s .
Pl ease s e e Fig. 2 i n Peters an, Paul J. , R o b e rt D. D eegan,
M. Marder, a n d Harry L. Swi n n e y. " C ra c k s i n Rubber u n d e r Ten si o n Ex c e ed the S h ear Wave S p ee d. " Phy s Re v Lett 9 3 ( 2004) : 01550 4 .
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Failure of biological structures in diseases
Failure of materials is critical for understanding function and malfunction of biology
Example: Rapid aging disease progeria - Single point mutations (changes) in protein structure causes severe diseases
Cell nucleus loses mechanic al stability under loading (heart, muscles)
Patient
Image re mo ved due to cop y r i gh t res t r i c t ions .
Fracture in cell’s nucleus Created under mechani cal deform ati o n
Failure of prote in mo le cu les Building blocks of life
R ep r i n t e d by p er m i s s i on fr o m M a c m illa n Publ i s he r s Ltd : Nature Ma terial s.
Sour c e : Buehl e r, M., and Y. Yung. " D e f ormati on a n d F a ilu r e o f P r o t e i n M a t e r i a l s in P h ysio lo g i c a lly Extr eme Con d i ti o n s and Di sea s e. " Na ture Mater ial s 8 , no. 3 ( 2 009): 17 5-88 . © 200 9 .
Courtesy of Elsevier, Inc., http://www.scien c edirect. com . Used with pe rmi s si o n .
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Cells
Vimentin intermediate filament
Filaments
Protein molecule
Chemical bonding
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Sour c e : Qi n, Z . , L . Krepl a k, and M. Bu ehl e r. "H i e rar c hi cal Struct ur e Co n t r o l s N a no mech an ic a l P r op er t i es o f Vime n t in I n t e r m e d ia t e Fi laments." PL o S O N E 4 , no. 1 0 ( 2009) . d o i : 10 .1 371/journal.p o ne .000 7294 . L i c e nse CC B Y .
How structural building blocks of cells break
• Genetic diseases
Courtes y of Nati onal Academ y of Sci e n c e s , U. S. A. Use d w i th permi s si o n .
Sour c e : Ackbarow, Theo dor, et al. " H i e rar c hi es, Mul t i p l e Ener gy Barri er s, and Robu stnes s Go v e r n the F ract u r e Mec h ani c s of Al ph a-Hel i c al and Beta-Sheet Protei n Dom ai n s." PNAS 104 (2 007): 1 6410 -15 . Cop y rig h t 2 007 N a tional Ac ad emy of Sc iences , U. S . A .
• M olecular m e chanisms of biology
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Unfolding of titin molecule
X : breaking
Force (pN)
X
X
Titin I27 domain: Very resistant to unfolding due to parallel H- bonded strands
Displacement (A)
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Keten and Buehler, 2007
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Folding of beta-sheet protein structure
Movie
45
S. Keten and M.J. Buehler , in submission
A New Approach to Molecular Simulation
Vijay Pande, Associate Professor of Chemistry, Struct ur al Biology, and Computer Science, Stanford University
Folding@home distributed computing
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Opportunity: Experimental techniques
Single cells Single molecules
Courtesy of Elsevier, Inc., http://www.scien c edirect. com . Us ed w i t h p e r m iss i on .
Integration with experimental techniques
R ep r i n t e d by p er m i s s i on fr o m M a c m illa n Publ i s he r s Ltd : Nature Ma terial s.
Sour c e : Buehl e r, M., and Y. Yung. " D e f ormati on a n d F a ilu r e o f P r o t e i n M a t e r i a l s in P h ysio lo g i c a lly Extr eme Con d i ti o n s and Di sea s e. " Na ture Mater ial s 8 , no. 3 ( 2 009): 17 5-88 . © 200 9 .
For most applications, we will use a website-driven simulation framework developed in collaboration with MIT’s Office for Undergraduate Education
nanoHUB: https://nanohub.org
More than 160 tools: https://nanohub.org/resources/tools
Technical assistance: Justin Riley