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

Basic molecular dynamics

Lecture 2

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

Goals of part I (particle methods)

You will be able to …

 Carry out atomistic simulations of various processes (diffusion, deformation/stre tching, materials failure)

Carbon nanotubes, nanowires, bu lk metals, proteins, silicon crystals, etc.

 Analyze atomistic simulations (make sense of all the numbers)

 Visualize atomistic/molecular data (bring data to life)

 Understand how to link atomistic simulation results with continuum models within a multi-scale scheme

Lecture 2: Basic molecular dynamics

Outline:

1. Introduction

2. Case study: Diffusion

2.1 Continuum model

2.2 Atomistic model

3. Additional remarks – h istorical perspective

Goals of today’s lecture:

 Through case study of diffusion, illustrate the concepts of a continuum model and an atomistic model

 Develop appreciation for distinction of continuum and atomistic approach

 Develop equations/models for diffusion problem from both perspectives

 Develop atomistic simulation approach (e.g. algorithm, pseudocode, etc.) and apply to describe diffusion (calculate diffusivity)

 Historical perspective on computer simulation with MD, examples

from literature 4

1. Introduction

5

Relevant scales in materials

 (atom) <<

 (crystal)

<< d (grain)

<< dx , dy , dz <<

y A y

yy

H , W , D

Image from Wi ki medi a Common s , h t t p ://c o m mons

Courtesy of Elsevier, Inc.,

Fi g. 8. 7 i n : Bue h l e r, M.

n y

 yz

z Image by MIT

 yx

 xz

 xy

n x

A x

 xx

x

.wikime d ia .o r g .

http://www.scien c edirect. com . Us ed w i t h p e r m iss i on .

A t o m is t ic Mod e lin g o f M a t e ri a ls Failu re . Sp ring er, 2 008 . © Sp ring er. All rig h ts reserv e d . Thi s co nte n t is ex cl u d e d from

our Creative Common s licen se.

Op en Cou r seW a r e .

Bottom-up

Atomis tic viewpoint :

For more i n fo rmati on, see h t t p :// oc w . m i t . e d u/ f a ir use .

•Explic i tly consider discrete atomistic structure

•Solve for atomic trajectories and infe r from these about material properties & behavior

•Features internal length scales (atomic distance)

“Many-particle system with statistical properties” 6

Relevant scales in materials

<<

y

dx , dy , dz

 yy

<<

A y

H , W , D

n y

 yz

 yx

 xy

n x

 xz

 xx

x

z

Image by MIT Op en Cou r seW a r e .

A x

RVE

 (atom) <<

 (crystal) << d (grain)

Image from Wi ki medi a Common s , h t t p ://c o m mons

.wikime d ia .o r g .

Courtesy of Elsevier, Inc., http://www.scien c edirect. com . Us ed w i t h p e r m iss i on .

Fi g. 8. 7 i n : Bue h l e r, M. A t o m is t ic Mod e lin g o f M a t e ri a ls Failu re. Sp ring er, 2 008 . © Sp ring er. All rig h ts reserv e d . Thi s co nte n t is ex cl u d e d from our Creative Common s licen se. For more i n fo rmati on, see h t t p :// oc w . m i t . e d u/ f a ir use .

Top-down

Continuum viewpoint :

•Treat material as matter with no internal structure

•Develop mathematical model (governi ng equation) based on representative volume element (RVE, contains “enough” ma terial such that internal structure can be neglected

•Features no characteristic length scales, provided RVE is large enough

“PDE with parameters” 7

2. Case study: Diffusion

Continuum and atomistic modeling

Goal of this section

 Diffusion as example

 Continuum description ( top-down approach ), partial differential equation

 Atomistic description ( bottom-up approach ), based on dynamics of molecules, obtained via numerical simulation of the molecular dynamics

Introduction: Diffusion

 Particles move from a domain with high concentration to an area of

low concentration

 Macroscopically , diffusion measured by change in concentration

 Microscopically , diffusion is process of spontaneous net movement of particles

Result of random motion of particles (“Brownian motion”)

High concentration

Low concentration

c = m/V = c ( x, t ) 10

Ink droplet in water

hot cold

© s o ur c e u n k n ow n. Al l rights res e rv ed. Thi s co ntent i s ex cl u d ed from our Creati ve

Co mmo ns lice n se . F o r mo r e in fo r m a t io n , s e e http:/ /o c w .mi t .edu /fai ru se . 11

Microscopic observation of diffusion

12

Microscopic mechanism: “Random walk” – or Brownian motion

 Brownian motion was first observ ed (1827) by the British botanist Robert Brown (1773-1858) when studying pollen grains in water

 Initially thought to be sign of life, but later confirmed to be also present in inorganic particles

 The effect was finally explained in 1905 by Albert Einstein , who realiz ed it was caused by water molecules randomly smacking into the particles.

Publ i c domai n i m age.

13

Brownian motion leads to net particle movement

time

Particle “slowly” moves a w ay from its initial position

Robert Brown’s microscope

Robert Brown's Mic roscope

1827

Instrument with whic h Robert Brown studied Brownian motion and which he used in his work on identifying the nucleus of the living cell

Instrument is preserved at the Linnean Society in London

It is made of brass and is mount ed onto the lid of the box in which it can be stored

http://www.brianjford.com/pbrownmica.jpg

Simulation of Brownian motion

 http://www.scienceisart.com/A_Diffus/Jav1_2.html

Macroscopic observation of diffusion

Macroscopic observation: concentration change

S e e al so:

Ink dot

© source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .

18

ht t p :// www. ui c. edu/ classes/phy s / phys45 0 /MA R K O /N0 04. html

Brownian motion leads to net particle movement

Time t

c 1 =m 1 /V (low)

c 2 =m 2 /V (high)

© s o ur c e u n k n o w n. Al l rights res e rv ed. Thi s co ntent i s ex cl u d ed from our Creative Common s licen s e. For more in fo r m a t io n , se e h t t p ://ocw.m i t .e d u / f a i r u se .

© T h e Mc Gr a w H ill Co mp a n ies . A l l r i g h ts reserved . T h i s content is e x c l ud 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 .

19

Diffusion in biology

Image removed due to copyright restrictions. See the image now: http://www.biog1105-1106.org/demos/105/unit9/media/diffusion-after-act- pot.jpg .

20

http://www.biog1105-1106.org/demos/105/unit9/r oleofdiffusion.html

2.1 Continuum model

How to build a continuum model to describe the physical phenomena of diffusion?

Approach 1: Continuum model

 Develop differential equation based on differential element

m L m R

x

J

W= 1 H= 1

 x  x

J: Mass flux (mass per unit time per unit area)

Concept: Balance mass [here], force etc. in a differential volume element; much greater in dimension than inhomogeneities (“ sufficiently large RVE ”)

Approach 1: Continuum model

 Develop differential equation based on differential element

m L m R

x

J

W= 1 H= 1

 x  x

Probability of mass m

crossing the central boundary during a period  t is equal to p

Approach 1: Continuum model

 Develop differential equation based on differential element

m L m R

x

J

W= 1 H= 1

 x  x

Probability of mass m

crossing the central boundary during a period  t is equal to p

J  1 p m

Mass flux from left to right

L 1  1  t L

J  1 p m

Mass flux from right to left

[ J ] = mass per unit time per unit area

R 1  1  t R

Approach 1: Continuum model

 Develop differential equation based on differential element

m L m R

x

J

W= 1 H= 1

 x  x

Probability of mass m

crossing the central boundary during a period  t is equal to p

J  1 p m

Mass flux from left to right

Effective mass flux

L 1  1  t L

J  1 p  m  m 

J  1 p m

Mass flux from right to left

1  1  t L R

R 1  1  t R

More mass, more flux ( m L is ~ to number of particles) 25

Continuum model of diffusion

Express in terms of mass concentrations

c  m

V

m  c V

J  p  m

 t L

 m R 

c L c R

x

J

W= 1 H= 1

 x  x

Continuum model of diffusion

Express in terms of mass concentrations

c  m

V

m  c V

J  p  m

 t L

 m R 

J  1 p  c  c    x  1  1

c L c R

x

J

1  1  t L R

   c  V

W= 1 H= 1

 x  x

Continuum model of diffusion

Express in terms of mass concentrations

c  m

V

J 

1 p  c

 c R    x  1  1

x

J

c L c R

   c

 V

Concentration gradient

W= 1 H= 1

J   p

 t

 c  x

  p

 t

 x 2  c

 x

 x  x

expand  x

Continuum model of diffusion

Express in terms of mass concentrations

c  m

V

J 

1 p  c

 c R    x  1  1

x

J

c L c R

   c

 V

Concentration gradient

W= 1 H= 1

J   p

 t

 c  x

  p

 t

 x 2  c

 x

 x  x

D  p

 x 2

 t

Parameter that measures how “fast” mass moves (in square of distance per unit time)

Diffusion constant & 1 st Fick law

Reiterate: Diffusion constant D describes th e how much mass moves per unit time

Movement of mass characterized by square of displac ement from initial pos ition

Flux

J

  p

 t

 x 2 dc

dx

  D dc

dx

1 st Fic k law

(Adolph Fick, 1829-1901)

D ~ p

Diffusion constant relates to the ab ility of mass to m o ve a distance

 x 2 over a time  t (strongly temperature dependent, e.g. Arrhenius) 30

2 nd Fick law (time dependence)

c

dc

J   D

dx

1 st Fic k law

J 1 x

 c 

 t

 x

x 0  x x 1

J: Mass flux (mass per unit time per unit area)

2 nd Fick law (time dependence)

c

dc

J   D

dx

1 st Fic k law

J 1 x

x 0  x x 1

 c J 1  J 2 1  dc  dc  

 t   x

   D

 x dx

|    D dx

|  

 x  x 0  x  x 1  

J 1 

J ( x 

x )   D dc

0 dx

|

x  x 0

2 nd Fick law (time dependence)

c

dc

J   D

dx

1 st Fic k law

 J

J 1

x 0  x

x

x 1

 c  1   D dc

|  D dc | 

 t  x 

dx x  x 0 dx x  x 1 

 c   d  J    d   D dc 

 t dx dx dx

Change of concentration in time equals change of flux with x (mass balance)

2 nd Fick law (time dependence)

c

dc

J   D

dx

1 st Fic k law

 J

J 1

x 0  x

x

x 1

 c  1   D dc

|  D dc | 

 t  x 

dx x  x 0 dx x  x 1 

 c   d  J    d   D dc 

 t dx dx dx

Change of concentration in time equals change of flux with x (mass balance)

Solve by apply ing ICs and BCs…

PDE

34

Example solution – 2 nd Fick’s law

 c 

 t

d 2 c D dx 2

Concentra t ion

c 0

c 0 x

time t

c ( x , t )

BC: c ( x = 0) = c 0 x

IC: c ( x > 0, t = 0) = 0

Need diffusion coefficient to solve for distribution!

How to obtain diffusion coefficient?

 Laboratory experiment

 Study distribution of concentrations (previous slide)

 Then “fit” the appropriate diffusion coefficient so that the solution matches

 Approach can then be used to solve for more complex geometries, situations etc. for which no lab experiment exists

 “Top down approach”

Matching with experiment (parameter identification)

Concentra t ion

c 0

time t 0

experiment

continuum model solution

c ( x , t 0 )

x

Summary

 Continuum model requires paramet er that describes microscopic processes inside the material

 Typically need experimental measurements to calibrate

Time scale

“Empirical”

or experimental parameter feeding

2.2 Atomistic model

How to build an atomistic bottom-up model to describe the physical phenomena of diffusion?

Approach 2: Atomistic model

 Atomis tic model provides an al ternative approach to describe diffusion

 Enables us to directly calculat e the diffusion constant from the trajectory of atoms (“microscopic definition”)

 Approach: Cons ider set of atoms/molecules

 Follow their trajectory and calculate how fast atoms leave their initial

position

Follow this quantity over time

Recall : Diffusion constant relates to th e “abil i t y ” of partic le to move a distance  x 2 over a time  t

Molecular dynamics – s imulate trajectory of atoms

Goal: Need an algorithm to predict positions , veloc ities, accelerations as

function of time 41

Solving the equations: What we want

Solving the equations

Recall : Taylor expansion of function f around point a

Solving the equations

Recall : Taylor expansion of function f around point a

Taylor series expansion r i ( t ) around

a  t 0

x  t 0   t

x  a  t 0   t  t 0   t

Solving the equations

Recall : Taylor expansion of function f around point a

Taylor series expansion r i ( t ) around

a  t 0

x  t 0   t

x  a  t 0   t  t 0   t

45

Taylor expansion of

r i ( t )

r i ( t 0

  t ) 

r i ( t 0

)  v i

( t 0

)  t 

1

2 a i

( t 0

)  t 2

 ...

r ( t

 r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

i 0

i 0 i 0

2 i 0

a  t 0 a  t 0

x  t 0 x  t 0

  t

  t

x  a x  a

 t 0

 t 0

  t

  t

 t 0

 t 0

  t

   t

Taylor expansion of r i ( t )

r ( t   t )  r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

i 0 i 0 i 0

2 i 0

r ( t   t )  r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

+ i 0

i 0 i 0

2 i 0

r i ( t 0

  t ) 

r i ( t 0

  t )

 2 r i ( t 0

)  v i

( t 0

)  t

 v i

( t 0

)  t

 a i

( t 0

)  t 2

 ...

Taylor expansion of r i ( t )

r ( t   t )  r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

i 0 i 0 i 0

2 i 0

r ( t   t )  r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

+ i 0

i 0 i 0

2 i 0

r i ( t 0

  t ) 

r i ( t 0

  t )

 2 r i ( t 0

)  v i

( t 0

)  t

 v i

( t 0

)  t

 a i

( t 0

)  t 2

 ...

r ( t   t )  2 r ( t )  r ( t   t )  a ( t

)  t 2

 ...

i 0 i 0 i 0 i 0

Taylor expansion of r i ( t )

r ( t   t )  r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

i 0 i 0 i 0

2 i 0

r ( t   t )  r ( t )  v ( t

)  t  1 a ( t

)  t 2

 ...

+ i 0

i 0 i 0

2 i 0

r i ( t 0

  t ) 

r i ( t 0

  t )

 2 r i ( t 0

)  v i

( t 0

)  t

 v i

( t 0

)  t

 a i

( t 0

)  t 2

 ...

49

Physics of particle interactions

Laws of Motion of Isaac Newton (1642 – 1727):

1. Every body continues in its stat e of rest, or of uniform motion in a right line, unles s it is compelled to change that state by forces impressed upon it.

2. The change of motion is proportional to the motive force impresses, and is made in the direction of the right line in which that force is impressed.