Molecular Dynamics

SK6202 Kapita Selekta Sains Komputasi

Senin, 4 Februari 2013 BSC-A Lantai 3 1

Molecular Dynamics (Part 1)

Sparisoma Viridi*

Nuclear Physics and Biophysics Research DivisionInstitut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia*[email protected]



Outline

• Short history of molecular dynamics (MD)• Introduction to MD• MD Algorithm• Example• Assignments



Molecular dynamicsContribution from some works

Short history

History of MD

• It was first introduced in studying the interact-ions of hard spheres which exhibits phase transitions (Alder et. al, 1957)

• Then, a series of paper led by Alder is then pu-blished during 1959-1980 investigating this method



B. J. Alder and T. E. Wainwright, “Phase Transition for a Hard Sphere System”, Journal of Chemical Physics 27 (5) 1208-1209 (1957)

History of MD (cont.)

• Studies of Alder and Wainwright in 1957 and 1959 induced other studies concerning beha-ior of simple liquids

• Realistic potential for liquid argon is then used (Rahman, 1964)

• Simulation of realistic system is conducted for the first time for water (Stillinger et. al, 1974)



A. Rahman, “Correlations in the Motion of Atoms in Liquid Argon”, Physical Review 136 (2A), A405-A411 (1964)F. H. Stillinger and A. Rahman, “Improved Simulation of Liquid Water by Molecular Dynamics”, Journal of Chemical Physics 60 (4), 1545-1557 (1974)



History of MD (cont.)

• The dynamics of folded of globular protein (bovine pancreatic trypsin inhibitor) is the first protein simulation (McCammon et. al, 1977)

• Many program and code are released, e.g. Chemistry HARvard Molecular Mechanics (CHARMM) (Stote et. al, 1999)

J. A. McCammon, B. R. Gelin, and M. Karplus, “Dynamics of Folded Proteins”, Nature 267 (5612) 585-590 (1977)R. Stote, A. Dejaegere, D. Kuznetsov, and L. Falquet, “Theory of Molecular Dynamcis Simulation ” in Tutori@l Molecular Dynamics Simulation CHARMM, URI http://www.ch .embnet.org/MD_tutorial /pages/MD.Part1. html [2012.02.13]

Alder’s papers

• This series of papers published during 1959-1980, a lot of time of consistency of studying something– I. General Method (1959)– IV. Behavior of a Small Number of Elastic Spheres

(1960)– III. A Mixture of Hard Spheres (1964)– IV. The Pressure, Collision Rate, and Their Number

Dependence for Hard Disks (1967)SK6202 Kapita Selekta Sains Komputasi


Alder’s papers (cont.)

– V. High Density Equation of State and Entropy for ‐Hard Disks and Spheres

– VI. Free Path Distributions and Collision Rates for ‐Hard Sphere and Square Well Molecules (1968)‐ ‐

– VII. Hard Sphere Distribution Functions and an ‐Augmented van der Waals Theory (1969)

– VIII. The Transport Coefficients for a Hard Sphere ‐Fluid (1970)

– IX. Vacancies in Hard Sphere Crystals (1971)




– X. Corrections to the Augmented van der Waals Theory for the Square Well Fluid (1972)

– XI. Correlation Functions of a Hard Sphere Test ‐Particle (1972)

– XII. Band Shape of the Depolarized Light Scattered from Atomic Fluids (1973)

– XIII. Singlet and Pair Distribution Functions for Hard Disk and Hard Sphere Solids (1974)‐ ‐




– XIV. Mass and Size Dependence of the Binary Diffusion Coefficient (1974)

– XV. High Temperature Description of the Transport Coefficients (1975)

– XVI. Fluctuation Driven Resonance (1977)– XVII. Phase diagrams for ’’step’’ potentials in two

and three dimensions (1979)– XVIII. The square well phase diagram (1980)‐





Molecular dynamicsDefinitions, use, and limitations

Introduction

Molecular dynamics

• Molecular dynamics (MD) is a computer simulation of physical movements of atoms and molecules (Wikipedia, 2011)

• MD simulation consists of the numerical, step-by-step, solution of classical equation of motion (Allen, 2004)



Wikipedia contributors, “Molecular dynamics”, Wikipedia, The Free Encyclopedia, 5 September 2011, 15:49 UTC, oldid:448597141 [2011.09.21]M. P. Allen, “Introduction to Molecular Dynamics Simulation”, in Computational Soft Matter: From Synthetic Polymers to Proteins, Lecture Notes, Norberg Attig, Kurt Binder, Helmut Grubmüller, Kurt Kremer (Eds.), John von Nuemann Institut for Computing, Jülich, NIC Series, Vol. 23, pp. 1-28, 2004

Molecular dynamics (cont.)

• It is a computer simulation technique where the time evolution of a set of interacting atoms is followed by integrating their equations of motion (Ercolessi, 1997)

• MD simulations can provide the ultimate detail concerning individual motions as a function of time (Karplus et. al, 2002)



F. Ercolessi, “A Molecular Dynamics Primer”, Spring College in Computational Physics, ICTP, Trieste, 9/10/1997 URI http://www.fisica.uniud.it/~ercolessi/md /md/node6.html [2011.09.21]M. Karplus and J. A. McCammon, “Molecular Dynamics Simulations of Biomolecules”, Nature Structural Biology 9 (9), 646-653 (2002)

Range of use

• It is used from atomic until planetoid scale• Calculation of electronic ground state as

function of time of liquid metal (Kresse et. al, 1993)

• Motion of n-Alkanes molecules (Ryckaert et. al, 1977)



G. Kresse and J. Hafner, “Ab Initio Molecular Dynamics for Liquid Metals”, Physical Review B 47 (1), 558-561 (1993)J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, “Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes”, Journal of Computational Physics 23 (3), 327-341 (1977)

Range of use (cont.)

• Nanodroplet on a surface (Sedighi et. al, 2010)• Grains in mm and cm size (Gallas et. al, 1996)• Simulation of asteroids movement (Jaffé et. al,

2002)



N, Sedighi, S. Murad, and S. K. Aggarwal, “Molecular Dynamics Simulations of Nanodroplet Spreading on Solid Surfaces, Effect of Droplet Size”, Fluid Dynamics Research 42 (3), 035501 (2010)J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and Stefan Sokolowski, “Molecular Dynamics Simulation of Size Segregation in Three Dimensions”, Journal of Statistical Physics 82 (1-2), 443-450 (1996)C. Jaffé, S. D. Ross, M. W. Lo, J. Marsden, D. Farrelly, and T. Uzer, “Statistical Theory of Asteroid Escape Rates”, Physical Review Letters 89 (1), 011101 (2002)

Use of MD

• There are three main scenarios for the use of MD (Fedman, 2006)

• In the first scenario the simulated properties are compared with experimental results, and when the two agree it is reasonable to claim that the experimental results can be explained by the simulation model.



F. Hedman, “Algorithms for Molecular Dynamics Simulations: Advancing the Computational Horizon”, Ph.D. Thesis, Avdelningen för fysikalisk kemi, Arrheniuslaboratoriet, Stockholms Universitet, Stockholm, 2006

Use of MD (cont.)

• In the second scenario, MD simulations are used to interpret experimental results. In a sense the second scenario is the inverse of the first.

• In the third scenario, simulations are used as an exploratory tool to help gain an initial understanding of a problem and give guidance among possible lines of investigation, be it theoretical or experimental.



Experiment using simulation



M. P. Allen, “Introduction to Molecular Dynamics Simulation”, in Computational Soft Matter: From Synthetic Polymers to Proteins, Lecture Notes, Norberg Attig, Kurt Binder, Helmut Grubmüller, Kurt Kremer (Eds.), John von Nuemann Institut for Computing, Jülich, NIC Series, Vol. 23, pp. 1-28, 2004

Experiment .. simulation (cont.)

• It is a bridge between microscopic and macroscopic

• It is also a bridge between theory and experiment

• Do the experiment using simulation is a smart way to reduce the financial problem

• Even all considered nature laws are input to the system, it could give the unexpected



Is MD so perfect?

• Unfortunately not• It has problem even all forces are already

considered• It can produce unreported results or

unexpected (wrong) results• It has problem in time stamp



Time stamp problem

• Nanodroplet



N, Sedighi, S. Murad, and S. K. Aggarwal, “Molecular Dynamics Simulations of Nanodroplet Spreading on Solid Surfaces, Effect of Droplet Size”, Fluid Dynamics Research 42 (3), 035501 (2010)

Time stamp problem (cont.)




• Granular oscillation



K. -C. Chen, C. -H. Lin, C. -C. Li, and J. -J. Li, “Dual Granular Temperature Oscillation of a Compartmentalized Bidisperse Granular Gas”, Journal of the Physical Society of Japan 78 (4), 044401 (2009)






Trajectory of a particlePotentials, the forms, and their physical meaning

MD Algorithm

Algorithms

• It is uses Newton’s second law of motion to get the acceleration a

• It using numerical integration to get the equation of motion, use the simple method i.e. original Euler method

• New motion parameters will cause new value of all forces

• Calculate the new forces to get new a



Algorithms (cont.)

• Newton’s second law of motion

• Left side consists of all considered forces



amF

Algorithms (cont.)

• Euler method:

• Particle position is given by at time



tavv iii

1

tvrr iii

1

ttt ii 1

ir

it

Algorithms (cont.)

• You must pay attention to influence from out- side of the system that changed with order of magnitude of chosen Δt

• Normally it is chosen that Δt must be 100 times smaller than that change



Potentials

• Lennard-Jones potential

• Coulomb potential



612

LJ 4rr

rU

r

qqrU

1

4 0

21C

Potentials (cont.)

• Gravitation potensial near large object

• Gravitation potensial



mgrrU G

r

mGmrU1

21G

Potentials (cont.)

• Morse potential

• Yukawa potential



2M 1 erre eDrU

r

egrU

kmr

2Y

Potentials (cont.)

• Harmonic oscillator potential



20HO 2

1rrkrU

Force

• Force can be obtained from potential through



UF



ExampleSequence of particle under gravitation potential

Granular memory device

System

• Granular device (D), sensor (S), particle sequence (P)

• P moves with constant initial velocity before entering D



Typical states

• Observed states are:– s10w0r0 (two configurations)– s10w1r0– s10w1r1



Typical states (cont.)

• Writing zero particle and relecting none from ten particles sequence (s10w0r0)

• gn = 0, gp = 0, b = 0, v0 = 4




• Writing zero particle and relecting none from ten particles sequence (s10w0r0)

• gn = 1, gp = -2, b = 2, v0 = 6




• Writing one particle and relecting none from ten particles sequence (s10w1r0)

• gn = 1, gp = -2, b = 2, v0 = 5




• Writing one particle and relecting another one from ten particles sequence (s10w1r1)

• gn = 1, gp = -3, b = 2, v0 = 4





Assignments

Assignments

• Make six groups of 2-3 students• Each group collects only one answer file• Answer file should be sent to

[email protected] with subject [SK6202] MD Assignment 1

• The file sould be received before 11 February 2013



Assignments (cont.)

• Question 1. Derive force formulation for following potential:(a) harmonic oscillator, (b) Coulomb,(c) gravitation,(d) Lennard-Jones,(e) Morse, (f) Yukawa



Assignments (cont.)

• Question 2. Describe the physical meaning of parameters used in each force or potential formulation

• Question 3. Tell the difference between molecular dynamics and molecular mechanics

• Question 4. Find the Euler, Verlet, Gear pre-dictor-corrector, Rattle, and Shake algorithm

• Question 5. Find a topic to be solved using molecular dynamics and explain





Thank you

Education

Molecular Dynamics