an event-driven hybrid molecular dynamics and direct

An Event-Driven Hybrid Molecular Dynamics and DirectSimulation Monte Carlo Algorithm

Aleksandar Donev1 , Berni J. AlderLawrence Livermore National Laboratory1

Alejandro L. GarciaSan Jose State University

1Lawrence Postdoctoral Fellow

CMLS Postdoc Symposium4th September 2007

1This work was performed under the auspices of the U.S. Department of Energy bythe University of California Lawrence Livermore National Laboratory under Contract No.W-7405-Eng-48 (UCRL-PRES-234157).

A. Donev (LLNL) SEDMD 2007 1 / 16

Abstract

A novel Stochastic Event-Driven Molecular Dynamics (SEDMD) algorithm is developed for thesimulation of polymer chains suspended in a solvent. The polymers are represented as chains ofhard spheres tethered by square wells and interact with the solvent particles with hard corepotentials. The algorithm uses Event-Driven Molecular Dynamics (EDMD) for the simulation ofthe polymer chain and the interactions between the chain beads and the surrounding solventparticles. The interactions between the solvent particles themselves are not treateddeterministically as in event-driven algorithms, rather, the momentum and energy exchange inthe solvent is determined stochastically using the Direct Simulation Monte Carlo (DSMC)method. The coupling between the solvent and the solute is consistently represented at theparticle level, however, unlike full MD simulations of both the solvent and the solute, the spatialstructure of the solvent is ignored. The algorithm is described in detail and applied to the studyof the dynamics of a polymer chain tethered to a hard wall subjected to uniform shear. Thealgorithm closely reproduces full MD simulations with two orders of magnitude greater efficiency.Results do not confirm the existence of periodic (cycling) motion of the polymer chain.


Introduction

Hydrodynamics of Polymer Solutions

We consider atomistic modeling of a polymer chain in a flowingsolution, for example, DNA in a micro-array.

Most methods use molecular dynamics to model the solute (polymerchains):

Bead-Spring Kuhn segments of the chain are point particles (beads)connected by non-linear elastic springs (FENE,worm-like, etc.)

Bead-Link The beads are free joints between inextensible links

Classical MD, even for simple polymer chains, requires a very smalltime-step ∆t.

The solvent (fluid, liquid) can be modeled implicitly via Browniandynamics as a linear frictional damping and uncorrelated stochasticforcing on the beads. But we want reverse coupling of the polymermotion on the flow!


Introduction

Explicit Solvent Models

The solvent can be modeled explicitly and coupled bidirectionally tothe flow:

(Fluctuating) Navier-Stokes CFD solver coupled to the MD (ex.,Trebotich et al.)

Lattice-Boltzmann solver coupled to the MDMolecular Dynamics fully resolving the motion of all solvent and

solute particles (very slow!)DSMC Variants Mesoscopic model for the solvent not fully resolving

the structure and motion of the solvent particles (ex.,MPCD of Kapral, Yeomans, and others)


Introduction

Coupling of Polymer to Solvent

The coupling between the solute and solvent is phenomenologicaland approximate for most methods in use:

Point beads with artificial friction coefficients based on Stokes lawUncorrelated fluctuating forces on the beadsPoint beads exerting δ-function forces on the solvent

Methods that rely on a constitutive equation for the solvent (CFDor LB) may not work well at micro and nano-scales.

There is a need for fast atomistic model of both the solvent andsolute with direct coupling (Kapral and Lee).


Event-Driven Molecular Dynamics (EDMD)

Time-Driven (TD) Molecular Dynamics

Time-Driven Molecular Dynamics (TDMD) for soft particles:

1 All of the particles are displaced synchronously in small time steps ∆t,calculating positions and forces on each particle at every time step.

2 The linked-list cell method or near-neighbour list (or combinations)are used for efficient neighbor searches.

3 TDMD is not rigorous (there is an error ∼ ∆t), but it is verywell-understood and widely implemented.

4 For discontinuous interactions (ex., hard-spheres), discontinuouschanges of the state, aka events, occur a posteriori, in the middle oftime steps.



Asynchronous Event-Driven (AED) Algorithms

Event-Driven Molecular Dynamics (EDMD) for hard particles:

Time is advanced from one event to the next event.Asynchronous: Each particle is at the point in time when the lastevent involving it happened.Given infinite numerical precision, this kind of approach can rigorouslyfollows the dynamics of the system.

There also exist synchronous event-driven algorithms, for example,dynamic Markov chain Monte Carlo algorithms.

Asynchronous event-driven algorithms naturally handle variabletime-scales.



Basic Algorithm

Each particle has its own current time t predicts its impending event(te , pe).

Types of events: binary collision, boundary events, internal events,geometrical events, etc.

Each particle i predics events with particles and objects in itsneighbourhood N (i).

Collision predictions must be kept symmetric, that is, if i predicts anevent with j , it changes j ’s prediction as well.

Event schedule consists of a priority queue of time-ordered impendingevents, one for each particle.

Neighbor List Collision-Driven Molecular Dynamics Simulation for Nonspherical ParticlesA. Donev, F. H. Stillinger, and S. Torquato, J. Comp. Phys, 2005



EDMD for Polymer Solutions

(MNG)

Tethered (square-well) hard-spherechain polymers are the simplest butuseful model.

Most of the computation is “wasted”on the unimportant solvent particles!

Over longer times it ishydrodynamics (local momentum andenergy conservation) and Browniandynamics (fluctuations) that matter.

We can use a stochasticrepresentation of the momentumtransfer in the fluid ignoring thedetails of the solvent structure.


Stochastic Event-Driven MD (SEDMD)

Direct Simulation Monte Carlo (DSMC)

(MNG)

No fluid structure: Solvent particlespass through eachother freely (nocollision predictions)!

Stochastic conservativecollisions with randomly chosennearby solvent particles (in thesame cell): Direct SimulationMonte Carlo (DSMC).

Stochastic DSMC collisions area new type of event scheduled asa Poisson process (a random celland a pair of particles is selectedusing rejection techniques).

Classical time-driven DSMC isfaster because of the overheadof event queue management.



Mixed Event-Time-Driven Approach

Partition the cells into interior(event-driven I-ED, or time-drivenI-TD), boundary (B), and exterior(E) cells.

A novel Stochastic Event-DrivenMolecular Dynamics (SEDMD)method.

Interior cells within theinteraction range of non-DSMCparticle are event-driven cells.Only particles in such cells arein the event queue.

All other particles aretime-driven at times n∆t.Stochastic collisions are onlyprocessed at a time-step event.



SEDMD Algorithm

(MNG) (MNG)

We implement open (stochastic) boundary conditions: Reservoirparticles are inserted every timestep in the boundary cells withappropriately biased velocities (local Maxwellian or Chapman-Enskogdistributions).



Tethered Polymer in Shear Flow

The speedup over full MD is impressive: The number of particles isgreatly reduced and the simulation is 50-200 times faster! Thisallows us to reach much longer time scales than classical MD.

As a test problem we study a polymer tethered to a hard wall andsubject to simple shear flow. The polymer exibits cyclic dynamics:Moving closer to the wall, contracting, moving away from the wall,extending, ad infinitum.

Experiments and previous simulations have claimed the existence of acharacteristic period for this cycling that is about 10τ

We find no evidence of a characteristic cycling time for the tetheredpolymer larger than the internal relaxation time.



Tethered Polymer Contd...

-10 -5 0 5 10 15 20 25t (τ=6.2)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Cxy

Wi=0Wi=0.6Wi=1.2Wi=1.8Wi=3.1Wi=5.6

-20 -10 0 10 20t

-1

-0.75

-0.5

-0.25

0

Cxφ

Figure: Cross correlation function Cxy (t) for chains of Nb = 30 small beads.A. Donev (LLNL) SEDMD 2007 14 / 16


Future Directions

The DSMC fluid is too compressible: Can we make it partiallyincompressible but maintain thermodynamic consistency(fluctuation-dissipation theorems)?

In the future we will focus on the interaction between non-steady(near-turbulent) flow and polymer chains:

To reach high Reynolds number we need to have a less compressibleDSMC fluid.But three dimensional flows are still out of reach because of the serialalgorithm. Parallelization is challenging.Using the open BC implementation we can couple the (atomistic)SEDMD domain to a (mesoscopic) fluctuating hydrodynamics solver(Garcia and Bell), possibly also using adaptive mesh refinement (AMR).



Conclusions

Event-driven algorithms are a very efficient alternative to traditionaltime-driven simulations in situations where the evolution of a systemis dominated by discontinuous state changes (events).

DSMC can easily be incorporated into an asynchronous event-drivenalgorithm.

Time-driven and event-driven handling can be combined together forgreater efficiency.

A DSMC fluid is just as good as any other fluid for meso- andmicro-scopic hydrodynamics.

Reference: An Event-Driven Hybrid Molecular Dynamics and DirectSimulation Monte Carlo Algorithm, A. Donev, A. L. Garcia and B. J.Alder, 2007, submitted to Journal of Computational Physics


an event-driven hybrid molecular dynamics and direct

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