Biological fluid mechanics at themicro and nanoscale‐

Lecture 7:Atomistic Modelling

Classical Molecular Dynamics Simulations of Driven Systems

Anne TanguyUniversity of Lyon (France)

Atomistic Modelling:

Classical Molecular Dynamics Simulationsof Driven Systems.

I.Description II.The example of Wetting

III.The example of Shear Deformation

Classical Molecular Dynamics Simulations consists in solving the Newton’s equationsfor an assembly of particles interacting through an empirical potentiaL;

In the Microcanonical Ensemble (Isolated system): Total energy E=cst

In the Canonical Ensemble: Temperature T=cst

with if no external force

Different possible thermostats: Rescaling of velocities, Langevin-Andersen, Nosé-Hoover…more or less compatible with ensemble averages of statistical mechanics.

Equations of motion: the example of Verlet’s algorithm.

Adapt the equations of motion, to the chosen Thermostat for cst T.

• Langevin Thermostat:Random force (t)Friction force –.v(t) with <(t).(t’)>=cste.2kBT.(t-t’)

• Andersen Thermostat: prob. of collision t, Maxwell-Boltzman velocity distr.

• Nosé-Hoover Thermostat:

• Rescaling of velocities:

• Berendsen Thermostat: with

Heat transfer. Coupling to a heat bath.

after substracted the Center of Mass velocity, or the Average Velocity along Layers

0'dt

dH

( )1/2

)(. tFvdt

dvm ii

ii

Thermostats:

Examples of Empirical Interactions:

The Lennard-Jones Potential:

2-body interactions cf. van der Waals

Length scales ij ≈ 10 ÅMasses mi≈10-25 kgEnergy ij≈ 1 eV ≈ 2.10-19J ≈ kBTm

Time scale or

Time step t = 0.01≈ 10-14 s106 MD steps ≈ 10-8 s = 10 ns or 106x10-4=100% shear strain in quasi-static simulations

N=106 particles, Box size L=100 ≈ 0.1 m for a mass density =1.3.N.Nneig≈108 operations at each « time » step.

sm 12

2

10.

s

TD12

8

202

1010

10

)1(

1.0

The Stillinger-Weber Potential:For « Silicon » Si, with 3-body interactionsStillinger-Weber Potential F. Stillinger and T. A. Weber, Phys. Rev. B 31 (1985)

Melting T Vibration modes Structure Factor

The BKS Potential:For Silica SiO2, with long range effective Coulombian InteractionsB.W.H. Van Beest, G.J. Kramer and R.A. Van Santen, Phys. Rev. Lett. 64 (1990)

Ewald Summation of the long-range interactions, or Additional Screening (Kerrache 2005, Carré 2008)

OSijioùr

CeA

r

qqrE ijrB

ijji

BKSij ,),(

4)( 6

0

111 ).().(,,

)(4, ).()..(),...,2,1(

ararijkkji

arjiSW

ikijefeBrANE

2-body interactions(Cauchy Model) 3-body interactions

Example: Melting of a Stillinger-Weber glass, from T=0 to T=2.

Microscopic determination of different physical quantities:

-Density profile, pair distribution function

-Velocity profile

-Diffusion constant

-Stress tensor (Irwin-Kirkwood, Goldenberg-Goldhirsch)

-Shear viscosity (Kubo)

II. The example of Wetting

Surface Tension: coexistence beween the liquid and the gas at a given V.

(L. Joly, 2009)

The Molecular Theory of Capillarity:Intermolecular potential energy u(r).

Total force of attraction per unit area:

Work done to separate the surfaces:

(I. Israelachvili, J.S.Rowlinson and B.Widom)

Surface Tension:

h

h

h

zz

drruhrr

rfrddzhF

)()(.2

..

21

321

00

)(...2 321

hh

zS rurdrdhhFW

(Hautman and Klein, 1991)

3

.for cos. LVSLSVSLSVLV

III. The example of Shear Deformation

Boundary conditions:

Quasi-static shear at T=0.Fixed walls

Or biperiodic boundary conditions (Lees-Edwards)

Example: quasi-static deformation of a solid material at T=0°K

At each step, apply a small strain ≈ 10-4 on the boundary,And Relax the system to a local minimum of the Total Potential Energy V({ri}).Dissipation is assumed to be total during .

).10(10..

10/

418lim

12

LJusa

c

a

c

t

scat

Quasi-Static Limit

stressshear xyS

F

ux

Ly

y

xxy L

u

2strain

Rheological behaviour:

Stress-Strain curve in the quasi-static regime

stressshear xyS

F

ux

Ly

y

xxy L

u

2strain

X

y

Local Dynamics:

Global and Fluctuating Motion of Particles

stressshear xyS

F

ux

Ly

y

xxy L

u

2strain

Local Dynamics:

Global and Fluctuating Motion of Particles

Transition from Driven to Diffusive motiondue to Plasticity, at zero temperature.

cage effect (driven motion) Diffusive

y _

max

n ~ xy

p

Tanguy et al. (2006)

Driving at Finite Temperature:

The relative importance of Driving and of Temperature must be chosen carefully.

Low Temperature Simulations: Athermal Limit

Typical Relative displacement due to the external strain

larger than

Typical vibration of the atom due to thermal activation

ta ...

>>

h

B

k

Tk

Convergence to the quasi-static behaviour, in the athermal limit:At T=10-8 (rescaling of the transverse velocity vy et each step)

M. Tsamados(2010)

cste

.

4.0..

.

.

T= 0.2-0.5 Tg =0.435Rescaling of transverse velocities in parallel layers

Effect of aging

at finite T

Non-uniform Temperature Profile at Large Shear Rate

Time needed to dissipate heat created by applied shear across the whole system

Heat creation rate due to plastic deformation

Time needed to generate kBT,

LL

ctd

1

.

. xydt

dQ

.

. xy

BQ

Tkt

.

.

.

.

xy

BdQ

c

LTktt

Visco-Plastic Behaviour:

Flow due to an external force (cf. Poiseuille flow)F. Varnik (2008)

Non uniform T

End