Topics in Soft Matter Physics Theory and Modeling

Introduction to Computer Modeling

& Lab Set-up

Topics Today

• VMD Tutorial

• Molecular Simulation• Monte Carlo (MC) Method• Molecular Dynamics (MD) Method

• Euler Method• Störmer–Verlet Method • Velocity Verlet Method• Langevin Dynamics• Brownian Dynamics

• Periodic Boundary Condition• Non-dimensionalization and Unit Conversion

• Code Explanation

VMD

VMD Introduction

• VMD: Visual Molecular Dynamics

• Viewing and analyzing the results of molecular dynamics simulations.

• VMD includes embedded Tcl and Python interpreters.

• VMD is free of charge.

VMD Download and Registration

• For downloading, go to• http://www.ks.uiuc.edu/Development/Download/download.cgi?PackageNam

e=VMD , or

• http://www.phys.tw/~b97202017/vmd-1.9.1.bin.LINUXAMD64.opengl.tar.gz

• For installing, follow the steps below• cd ~/Download/

• tar –xzf vmd-1.9.1.bin.LINUXAMD64.opengl.tar.gz

• cd vmd-1.9.1/

• ./configure

• cd src

• sudo make install

• vmd

Creating an XYZ File(Molecules’ Configuration)• cd ~/Doc[TAB]

• vi test.xyz

• i

• [ESC]

• :wq

• vmd test.xyz

• Set the drawing style: VMD main >> Graphics >> Representations >> Drawing method >> VDW (then adjust the Sphere Scale)

More Info: http://en.wikipedia.org/wiki/XYZ_file_format

VMD Movie Maker

• VMD main >> Extensions >> Visualization >> Movie Maker

• Movie Settings >> Trajectory

• Format >> MPEG-1 (ffmpeg) or MPEG-2 (ffmpeg)

• Renderer >> Tachyon (Ray Tracer)

• http://www.ks.uiuc.edu/Training/Tutorials/vmd/tutorial-html/node3.html

• It may be necessary to install ffmpeg before generating movie for the first time. Turn off vmd, type “sudo apt-get install ffmpeg” in command line.

VMD Movie Maker

• Install mplayer after successfully generating a movie• sudo apt-get install mplayer

• Type “mplayer {video name}.mpg”, or type “mplayer –loop 0 {video name}.mpg” if want to replay video.

• In the future, you may want to convert video format into .wmv so that it can be inserted into PowerPoint. There is a convenient website:

• http://video.online-convert.com/convert-to-wmv

Some Simple Measurements Using VMD

• Calculate radial distribution function g(r)• VMD main >> Extensions >> Analysis >> g(r)

• http://www.ks.uiuc.edu/Research/vmd/plugins/gofrgui/

• Calculate Root Mean Square Deviation/Displacement (RMSD)• VMD main >> Extensions >> Analysis >> RMSD Trajectory Tool

• http://www.ks.uiuc.edu/Training/Tutorials/vmd/tutorial-html/node7.html

𝑅𝑀𝑆𝐷 = 𝑖=1

𝑁 𝑟𝑖(𝑡1) − 𝑟𝑖(𝑡2)2

𝑁

g(r) of Lennard-Jones Fluid

Useful Link for VMD

• Main Page:• http://www.ks.uiuc.edu/Research/vmd/

• Document:• http://www.ks.uiuc.edu/Research/vmd/current/docs.html

• Recommended Tutorial:• http://www.ks.uiuc.edu/Training/Tutorials/vmd/tutorial-html/index.html

• Tcl/Tk Document:• http://www.tcl.tk/doc/

Molecular Simulation

Monte Carlo (MC) Method

• Named from Monte Carlo Casino

• Utilize the concept of random and probability

• Simple examples:• Estimating π

• Integration

Monte Carlo

MC Method in Molecular Simulation

• 𝑃 𝐸𝑖 ∝ 𝑒−

𝐸𝑖𝑘𝐵𝑇,

𝑃 𝐸𝑗

𝑃 𝐸𝑖= 𝑒

−𝐸𝑗−𝐸𝑖

𝑘𝐵𝑇

• Random Trail Move Calculating Energy

• Probability of Accepting Trail Move P’

• Trail Move (i j) 𝑃′ =𝑃 𝐸𝑗

𝑃 𝐸𝑖< 1

• Trail Move (i k) 𝑃′ =𝑃 𝐸𝑘

𝑃 𝐸𝑖≥ 1 → 1

E

𝐸𝑗

𝐸𝑖

𝐸𝑘

Molecular Dynamics (MD) Method

• 𝑎 = 𝐹/𝑚 = −𝛻𝑈/𝑚

• 𝑣′ = 𝑣 + 𝑑 𝑣 = 𝑣 + 𝑎𝑑𝑡

• 𝑥′ = 𝑥 + 𝑑 𝑥 = 𝑥 + 𝑣𝑑𝑡 +1

2 𝑎𝑑𝑡2

• Euler Method

• Störmer–Verlet Method

• Velocity Verlet Method

• Langevin Dynamics

• Brownian Dynamics

(N, V, E) fixed Micro-canonical Ensemble

(N, V, T) fixed Canonical Ensemble

Euler Method

• 𝑋′: 𝑋 𝑡 + ∆𝑡 , 𝑋: 𝑋(𝑡)

• 𝑎 = 𝐹/𝑚 = −𝛻𝑈/𝑚

• 𝑣′ = 𝑣 + ∆ 𝑣 = 𝑣 + 𝑎∆𝑡 + 𝑂(∆𝑡2)

• 𝑥′ = 𝑥 + ∆ 𝑥 = 𝑥 + 𝑣∆𝑡 +1

2 𝑎∆𝑡2 + 𝑂(∆𝑡3)

Störmer–Verlet Method

• 𝑋𝐵: 𝑋 𝑡 − ∆𝑡 , 𝑋𝐴: 𝑋 𝑡 + ∆𝑡 , 𝑋: 𝑋(𝑡)

• 𝑎 = 𝐹/𝑚 = −𝛻𝑈/𝑚

• 𝑥𝐴 = 𝑥 + 𝑣∆𝑡 +1

2 𝑎∆𝑡2 +

1

6𝑏∆𝑡3 + 𝑂(∆𝑡4)

• 𝑥𝐵 = 𝑥 − 𝑣∆𝑡 +1

2 𝑎∆𝑡2 −

1

6𝑏∆𝑡3 + 𝑂(∆𝑡4)

• ∴ 𝑥𝐴 = 2 𝑥 − 𝑥𝐵 + 𝑎∆𝑡2 + 𝑂(∆𝑡4)

• ∴ 𝑣 = 𝑥𝐴− 𝑥𝐵

2∆𝑡+ 𝑂(∆𝑡2)

Velocity Verlet Method

• 𝑥𝐴 = 𝑥 + 𝑣∆𝑡 +1

2 𝑎∆𝑡2

• 𝑎𝐴 = 𝐹𝐴/𝑚 = −𝛻𝑈( 𝑥𝐴)/𝑚

• 𝑣𝐴 = 𝑣 +𝑎𝐴+𝑎

2𝑑𝑡

• It can be shown that Velocity Verlet is equivalent to the original Verletalgorithm.

• 𝑥2𝐴 = 𝑥𝐴 + 𝑣𝐴∆𝑡 +1

2 𝑎𝐴∆𝑡2, 𝑥 = 𝑥𝐴 − 𝑣∆𝑡 −

1

2 𝑎∆𝑡2

• 𝑥2𝐴 + 𝑥 = 2 𝑥𝐴 + ( 𝑣𝐴 − 𝑣)∆𝑡 +1

2( 𝑎𝐴 − 𝑎)∆𝑡2

• 𝑥2𝐴 + 𝑥 = 2 𝑥𝐴 + 𝑎𝐴∆𝑡2 → 𝑥2𝐴 = 2 𝑥𝐴 − 𝑥 + 𝑎𝐴∆𝑡2 → 𝑥𝐴 = 2 𝑥 − 𝑥𝐵 + 𝑎∆𝑡2

Langevin Dynamics

• 𝑚 𝑎 = −𝛻𝑈 − 𝜁 𝑣 + 2𝜁𝑘𝐵𝑇𝑅 𝑡

• 𝜁: damping coefficient (energy dissipation)

• 𝑅 𝑡 : noise term (thermal fluctuation), a delta-correlated stationary Gaussian process with zero-mean,

• 𝑅 𝑡 = 0

• 𝑅 𝑡 𝑅 𝑡′ = 𝛿(𝑡 − 𝑡′)

Brownian Dynamics

• If the viscosity of fluid is very large or we are only interested in the long-time behavior of molecules, the inertial term can be neglected

• 𝑚 𝑎 = −𝛻𝑈 − 𝜁 𝑣 + 2𝜁𝑘𝐵𝑇𝑅 𝑡 → 𝑣 = −𝛻𝑈 𝜁 + 2𝐷𝑅 𝑡

𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛 𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛: 𝜁𝐷 = 𝑘𝐵𝑇http://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)

Periodic Boundary Condition

• right out, left in.

• Box size should be twice larger than the interparticleinteraction.

• Finite size effect: cannot obtain system property of which characteristic length is close to box size

• Shape of box can introduce artifacts, e.g. crystal formation.

Non-dimensionalization

• 𝑥 = 𝑥𝐿

• E = 𝐸𝐽

• 𝑡 = 𝑡𝑇

• 𝑥𝐴 = 𝑥 + 𝑣∆𝑡 +1

2𝑎∆𝑡2 → 𝑥𝐴 = 𝑥 + 𝑣∆ 𝑡 +

1

2 𝑎∆ 𝑡2

• After set three independent units of physical quantities, the other units can be represented by the combination of these three units, e.g. angular momentum.

MD Program Code Explanation

For a semi-flexible chain and phantom particles

Algorithm

Velocity Verlet + Langevin Dynamics

• 𝑥𝐴 = 𝑥 + 𝑣∆𝑡 +1

2 𝑎∆𝑡2

• 𝑎𝐴 = 𝐹𝐴/𝑚 = −𝛻𝑈( 𝑥𝐴) − 𝜁 𝑣 + 2𝜁𝑘𝐵𝑇𝑹𝐴 /𝑚

• 𝑣𝐴 = 𝑣 +𝑎𝐴+𝑎

2𝑑𝑡

• where• 𝑹 𝑡 = 0

• 𝑹 𝑡 𝑹 𝑡 = 1/Δ𝑡

)()( tftvUtf rand

wallevbendvib UUUUU

2

122mii

m

Bvibvib rr

TkkU

mnjimnBmnmnev rTkU ,, exp

11

111iiii

iiiiBbendbend

rrrr

rrrrTkkU

m-n: monomer-monomer, monomer-nanoparticle

DNA Model with Langevin Dynamics

• Bonded Interaction

• Uvib

• Ubend

• Non-Bonded Int.

• Uev

m= 10 nm

L = 100 - 200 m = 1 - 2 mm

Bulk Lp = 50 nm

Dt = 2.29 nanosecond

Viscosity = 1 centipoise

3,,ˆ

wallnnBnwallnwall rirTkkU

nwalln ririf |ˆ|,

n: monomer, nanoparticle

Numerical Method: Velocity Verlet

fev (t)

frand (t)

fvib (t)

fbend (t)

σm

Creating Neighbor List

• Energy/Force Calculation is necessary in Molecular Simulations. Themost time consuming part is non-bonded interaction, which is O(N2).

• For short range forces, there are many pairs who don’t interact with each other. Thus, algorithms were developed to reduce the computing cost.• Verlet List

• Cell List

• Combined List

for ( int i = 0 ; i < N ; i++ ) for ( int j = i+1 ; j < N ; j++ ) {F[i] = non_bonded_int(r[i], r[j]);F[j] = -F[i];

}

Verlet list

• First developed by Verlet in 1967 (Phys. Rev. 159, 98–103)

• Create neighbor list ~ O(N2)

• Calculation of Non-bond int. uses NBL for nu steps

2

• After optimized ~ O(N1.5)

Cell List

• Create cell list ~ O(N)

• Calculate non-bond int. with

neighbor cells (8 in 2D, 26 in 3D)

Process of Cooking Food

Preparing Tools

Cookinghaving a

tasteenjoy

your meal

Process of Molecular Simulation

Initialization Simulation Measurement End

File Dependency

util.c

initialize simulationallocate memory

measurementrelease memory

neighbor list update

init.c

main.c

initlat.c

input.dat euler.c

toterg.c

enerbend.cenervib.c ener_ev.c wall_force.c

ev_nlist.c

fileoutput.cconfig.xyz

run_time.datRg.dattotal_energy.dat

warning_message.datheader.h

Makefile

process directionlink towarddata flow directionboundary.tcl

mtwist.c/h

randistrs.c/h

Input Parameters (input.dat)

• nstep = # of steps / cycle

• lattice = initial configuration. 0=read from config.xyz, 1=stretch config, 2=random config, 3=screw config

• Ncycl = # of cycles

• Dt = integration time step

• ELL = # of monomers

• BOX_X, BOX_Y, BOX_Z = dimensions of the simulation box

Tot step = nstep*Ncycl*microstep, microstep is defined in int main(…)

Simulation & DNA Energy Calculation

• Calculating energies and forces

• Updating velocity and position of particles

• euler.c• 𝑣𝑖 𝑡 , 𝑟𝑖 𝑡 →

𝑣𝑖 𝑡 + ∆𝑡 , 𝑟𝑖 𝑡 + ∆𝑡

• toterg.c (Utot)• enervib.c (Uvib)

• enerbend.c (Ubend)

• ener_ev.c (Uev)

• wall_force.c (Uwall)fev (t)

fvib (t)

fbend (t)

σm

wallevbendvib UUUUU

Utf

Input Parameters (input.dat)

• WALL_FLAG = type of confinement. 0= bulk, 1=y walls, 2=yz walls

• TEMP = temperature

• KV = bond vibrational energy parameter

• KB = bending energy parameter

2

122mii

m

Bvibvib rr

TkkU

11

111iiii

iiiiBbendbend

rrrr

rrrrTkkU

Input Parameters (input.dat)

• EPS_M/NPexcluded volume parameter (magnitude) between monomers/nanoparticles

• ALPHA_M/NPexcluded volume parameter (range) between monomers/nanoparticles

mnjimnBmnmnev rTkU ,, exp

m-n: monomer-monomer, monomer-nanoparticle

2, NNMMMNNNMMMN

Input Parameters (input.dat)

• WALL_EPS_M/NPwall repulsion parameter (magnitude) between monomer/nanoparticles and wall

• WALL_ALPHA_M/NPwall repulsion parameter (range) between monomer/nanoparticles and wall

3,,ˆ

wallnnBnwallnwall rirTkU

nwalln ririf |ˆ|,

n: monomer, nanoparticle

Input Parameters (input.dat)

• SIGMA_M/NP = diameter of monomer/nanoparticle

• MASS_M/NP = monomer/nanoparticle mass

• SEED = random seed

• INTG = integration type, 0 = euler, 1 = implicit BD, 2=velocity verlet

File Description

• header.h: defines the global parameters and global functions.

• main.c: the main file that handles initialization, force calculation, trajectory integration, and data output

• init.c: reads in the parameters from input.dat

• initlat.c: initializes the particle positions

• fileoutput.c: analyzes particle configurations to determine polymer size, displacement, and write output to files.

• randistrs.c/h, mtwist.c/h: generating random numbers

File Description

• euler.c: integrates the particle trajectories using Euler or Verletalgorithms

• toterg.c: calls to evaluate all the energies and forces

• enerbend.c: evaluates the bonded chain bending forces

• ener_ev.c: evaluates the non-bonded interparticle forces (WCA Potential, Repulsive Morse Potential)

• enervib.c: evaluates the bonded particleforces

• ev_nlist.c: Use the neighbor list method to evaluate non-bonded interparticle forces

• wall_force.c: evaluates forces from the wall boundaries

Measurement (fileoutput.c)

• Recording position of particles, which can later be visualized by VMD.

• Measuring and outputting some properties of particles (e.g. diffusion, molecule’s structure, …)

• void fileoutput(…);• Outputting files:

config.xyz (particle configuration)

run_time.dat (program efficiency)

total_energy.dat (Etot, Ek, U)

warning_message.dat

Rg.dat (properties of polymer)

Some Useful Commands

• Makefile• make clean• make clean all• make

• vmd config.xyz –e boundary.tcl

• ctags: tracking code easily (sudo apt-get install ctags)• Ctrl + ]• Ctrl + t