lecture 5 - university of michigan

Computational Nanoscience of Soft [email protected]

ChE/MSE 557 Lecture 5 October 24, 2006 1

Event-based or collision-basedmethods and

Brownian dynamics methods

Lecture 5



Tim

e Sc

ale

(sec

)

Length Scale

pico

Angstroms nanometers microns mm

AbInitio

metersfemto

nano

micro

millise

c MacroscaleSimulation

Electronic Structure - MO & DFTAb initio MDQuantum MC

MesoscaleSimulation Finite

element

ClassicalMolecularSimulation

Brownian dynamicsLattice BoltzmannCellular automataDPDDDFTMolecular dynamics

Monte Carlo

CFDMech

Simulating Across the Scales

The methods of computational materials science.



Simulations of colloidal nanoparticles

❏ Colloidal particles suspended in solution (“colloids”)❖ Diameters can be nanometers to microns❖ E.g. PMMA, PS, silica, gold, ….❖ Effective interaction potential

• Can be charged, so “bare potential” is Coulomb 1/r– Screened by adding salt to solution, grafting short polymers

to surface, etc. (resulting interaction: Yukawa, etc.)• Also van der Waals: -1/r when summed

– Can be screened by matching indices of refraction betweenparticles and solvent

• If both interactions screened, only repulsive hard coreinteractions left.



“MD” Simulations of Hard Spheres

❏ Discontinuity in potential❖ Potential not differentiable to get forces❖ Discontinuous change in velocities

❏ Cannot use traditional molecular dynamics

U(r)U(r) = 0 for r > σU(r) = ∞ for r ≤ σ

rσ

σ




❏ Since no force acts on particles between collisions, thenbetween collisions a particle’s position changes by viδt, whereδt is the time between collisions.

❏ Event-based collision dynamics:

(a) Identify location and time of next collision(b) Calculate new positions of all particles at collision time(c) Implement collision dynamics for colliding pair

• Calculate new positions and velocities for colliding pairbased on conservation of linear momentum for elasticcollisions

(d) Return to (a)

cf. Allen and Tildesley, pp 105-106.EtomicaDemo




❏ First “MD” simulation: Alder & Wainright 1957❖ Predicted the hard-sphere phase diagram prior to expts.

❏ Used for studying❖ Dense fluids and glasses❖ Nucleation and crystallization❖ Colloidal suspensions❖ Granular materials

0 0.494 0.545

0.58 0.64

0.74φ1 φ2

φg φrcp

φhcp

Fluid Fluid+Crystal

Crystal

Glass

Why do crystal phases form?

Dynamicsdifferent!



“MD” Simulations of Attractive Hard Spheres

❏ Square well potential (sticky hard spheres)

❏ Used to model colloidal nanoparticles with short-rangedelectrostatic interactions, which can be caused by depletionforces or surface treatment.

❏ “Stickiness” causes gelation of particles.

U(r)U(r) = 0 for r > σ2U(r) = ε for σ1 < r < σ2U(r) = ∞ for r ≤ σ1

r

σ1

σ2σ1ε

Also not differentiable.



“MD” Simulations of Attractive Hard Spheres

❏ Algorithm similar to hard spheres, but now there are two“collision times” to calculate, one at σ1 and one at σ2.❖ Collisions at the inner sphere obey normal hard-sphere

dynamics.❖ Collisions at outer sphere follow conservation of energy as

well as momentum:• For particles approaching each other, U drops when r

< σ2, so KE increases.• For particles moving away from each other, there are

two possibilities:– If KE is sufficient, the particles cross the boundary with a

loss in KE to compensate the increase in U.– If KE is not sufficient, reflection occurs at σ2 and the

particles remain “bound”.

EtomicaDemo 1



Hard Sphere Model

❏ The hard sphere and attractive hard spheremodel may also be solved using Monte Carlomethods.

❏ Both discontinuous MD and MC methodsgive the same phase diagram (e.g. samethermodynamics), but kinetics could bedifferent.



Brownian dynamics simulationsof nanoparticles

❏ In many instances of solute-solvent systems, the behavior ofthe solute is desired, and the behavior of the solvent isuninteresting (e.g. proteins, DNA, dendrimers, nanoparticles insolution.)

❏ Instead of modeling solvent explicitly (by individualmolecules), include its effects implicitly, and use appropriate,simplified force fields to describe interactions between solute.

❏ For nano-objects suspended in solvent, diffusive motion(Brownian dynamics) typically observed in quiescent state:stochastic dynamics.

Reference: Allen and Tildesley is good place to start.Also: Grest, et al JCP 105, 10583 (1996)



Brownian Dynamics Simulations of Nanoparticles

❏ To model nanoparticle-nanoparticle interaction, can usepotential of mean force appropriate to system.

❏ The solvent influences the dynamics of the nanoparticles viarandom collisions, and by imposing a frictional drag force onthe motion of the nanoparticle in the solvent.

❏ Stochastic dynamics models incorporate these two effects viathe Langevin equation of motion.



Brownian dynamics simulations

❏ In Brownian dynamics, the force on a particle is assumed tocome from three sources:❖ Conservative potential of mean force Fc between particles❖ Random collisions❖ Drag force

❏ Conservative force: Fc depends only on distance betweenparticles (e.g. soft sphere 1/r12, vdW 1/r, etc.)

❏ Random force: R(t) due to random fluctuations caused byinteractions with solvent molecules, which bombard theparticle constantly on all sides, giving rise to Brownian motionof the particle.



❏ Drag force: Frictional drag force on a particlearises from motion of particle in solvent.❖ Ffrictional = - ξ v❖ Friction coefficient ξ is related to collision frequency γ of

molecules in the solvent by γ = ξ / m.❖ 1/ γ is the velocity autocorrelation time associated with the

solvent.❖ For a spherical particle of radius a, ξ is related to viscosity

of solvent by Stokes’ law:

❖ The frictional force is then

!

" = 6#$a

!

F frictional = 6"a#v

Brownian dynamics simulations



Brownian dynamics method

❏ “Particle”-based MD-like simulation method❖ One type of Langevin dynamics method❖ Reproduces canonical [NVT] ensemble.❖ Equation of motion for particle positions

contains Newtonian and stochastic terms.

Usual NVE MD Addt’l stochastic termsLangevin equation

Total Force Frictional Force

Conservative Force Random Force

!

mi

d2xi(t)

dt2

= Fi{x

i(t)}" #

i

dxi(t)

dtm

i+ R

i(t)




❏ Stochastic terms❖ If solvent present, included implicitly through stochastic

terms:• Drag force• Random force

NB: Momentum not conserved, so hydrodynamics not fullyincluded.

❖ For a neat system (no solvent), stochastic terms represent a(non-momentum-conserving) thermostat with heat sourceand sink. Kremer and Grest 1996

Total Force Frictional Force

Conservative Force Random Force

!

mi

d2xi(t)

dt2

= Fi{x

i(t)}" #

i

dxi(t)

dtm

i+ R

i(t)

Usual NVE MD Addt’l stochastic terms




❏ Stochastic terms❖ Assumes no spatial or temporal correlations (i.e. Gaussian

white noise process).❖ Allows access to longer time scales than MD.

• No explicit solvent• Friction term dampens equation of motion, allows larger time

step

❏ Implementation❖ Particle positions and velocities updated each

time step by calculating forces acting on eachparticle due to other particles, and due to stochastic terms.



❏ One method of implementation is called the DirectForce method.❖ Calculate conservative force on i due to all particles j.❖ Calculate frictional force on i.❖ Calculate random force on i such that it satisfies the

fluctuation dissipation theorem:

❖ Use any MD integration scheme (e.g. leapfrog or velocityVerlet) to calculate new position and velocity of particle i.

!

Fi

Rt( )F j

R " t ( ) = 6kBTmi# i$ij$ t % " t ( )

Fi

Rt( ) = 0 spatially

uncorrelatedtemporally uncorrelated


Random number (zeromean, unit variance)



!

vi( t+"t) = c

ovi(t)+ c

1ai( t)"t+"vR

!

"xR = A "xR( )2

!

"vR = "vR( )2

cxvA+ B 1# c

xv

2( )( )

!

xi(t + "t) = x(t) + c

1vi(t)"t + c

2ai(t)"t 2 + "xR

!

cxv"x"v

= #xR#vR =kT

$m1% e%$#t( )

2


❏ Or use direct integration method via leapfrog (or velocity verlet):

❏ Random position and random velocity are calculated via thefollowing equations with “A” and “B” random numbers of zeromean and unit variance generated from a bivariate Gaussiandistribution.

Use Box-Mueller toget two Gaussiannumbers at once



MD vs. BD

❏ For the right problems, BD wins in three ways:❖ Eliminating many atoms and including them implicitly

means far fewer computations per time step.• If solvent not important, then BD may be a good choice.

❖ Can choose timestep δt roughly 2-3 times larger than inMD due to dissipative term, because damping termstabilizes the equations of motion.

❖ Because in BD the fastest frequency motions in the realsystem are replaced by stochastic terms, δt is now chosento resolve the slower degrees of freedom, and thus δt isseveral hundred times larger than in MD.• Thus BD starts at picoseconds and can access into

microseconds, but for effectively larger systems than couldbe done with MD.



BD Issues

❏ Simple BD contains no hydrodynamic interactions.❏ Excluded volume effects of solvent not included.❏ Not trivial to implement drag force for non-spherical

particles.❏ Solvent molecules must be small compared to

smallest molecules explicitly considered.



Examples

❏ BD for a monoatomic mixture❖Motion diffusive, except at high density



Example: Binary Mixture Phase Diagram

T

φφ1 φs1 φs2 φ2

one phase(miscible)

For each T < Tc, plot φ1, φ2,

φs1, and φs2.

This defines boundaries between (i) a miscible and immiscible regionand (ii) a metastable and unstable immiscible region.

two-phase(immiscible)

binodal or coexistence curve

spinodal

unstable metastable

UCST

Demo Glotzilla



BD simulations for nanoscience

❏ Example of a Brownian dynamics simulation❖Tethered nanospheres❖Tethered nanorods❖Tethered nanotriangles



Model tethered nano building blocks

❏ Use a coarse-grainedrepresentation whereby agroup of atoms in thenanoparticle or tether isreplaced by a single “atom”❖ Reduces force calculations❖ Retains nanoscale roughness

❏ Empirical pair potentialsbetween “atoms”❖ van der Waals interactions❖ Excluded volume interactions

❏ Minimal model❖ Thermodynamic immiscibility❖ Geometrical constraints

Our goal: discover trends and construct general design strategies.



Minimal model of a tethered nanoparticle

❏ Spherical nanoparticles modeled as spheres.Non-spherical nanoparticles modeled as smallerspheres rigidly bound to one another.

❏ Polymer tethers modeled as bead-spring chains of Nmonomers connected by FENE springs.

❏ Tethers connected to specific “atoms” on thenanoparticles via a FENE spring.



Minimal model of a tethered nanoparticle

❏ In neutral solvent, “atoms” or monomers of the sametype interact via 12-6 LJ potential.

❏ “Atoms” or monomers of different type: WCA.❏ Solvent selectivity modeled by describing interactions

between favored species via WCA and those betweenunfavored species via LJ.

❏ Degree of immiscibility and solvent quality determinedby reciprocal temperature ε/kBT.



Increasing level of detail

❏ Ab initio computations of portions of a nanocrystal ornanostructured molecule near solvent to getinteraction energies.

❏ Classical, explicit-atom MD simulations of severalnanostructured molecules or portions ofnanocrystals with explicit solvent to obtain preferreddistances, orientations, etc.

❏ Use this info to parameterize mesoscopic interactionpotentials.

lecture 5 - university of michigan

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