Thermal boundary conditions for

molecular dynamics simulations Simon P.A. Gill and Kenny Jolley

Dept. of EngineeringUniversity of Leicester, UK

A pragmatic approach to

Overview

1. Imposing a steady state temperature gradient

2. Remote boundary conditions by coarse-graining

1.0 Imposing a steady state temperature gradient

• Consider a 1D LJ chain of 100 atoms subjected to a different temperature at each end

tem

pera

tu

re

position, x

T1

TN

Tkj Heat flux

Thermalconductivity

Temperaturegradient

Classical model(Ficks law)

1.1 Thermostats• Nosé-Hoover

– a global deterministic thermostat– enforces average temperature only

• Langevin– a local stochastic thermostat– enforces temperature on each atom– no feedback from actual temperature

x

T(x)

?T1

T2

x

T(x)

T1

T2

1.2 Results for 1D LJ chain

0 20 40 60 80 1000.20

0.22

0.24

0.26

0.28

0.30

Nose-Hoover

Tem

per

atu

re, <T

>

position, i

1.2 Results for 1D LJ chain

• Kapitza effect – boundary conductivity different

from bulk conductivity

0 20 40 60 80 1000.20

0.22

0.24

0.26

0.28

0.30

Langevin

Nose-Hoover

Tem

per

atu

re, <T

>

position, i

Langevin cannot control temperature in a 1D chain away from equilibrium

1.3 Boundary effect

• Green-Kubo (assuming local equilibrium)

djjkt

iit

i

0

)0()(lim

0 20 40 60 80 1000.20

0.22

0.24

0.26

0.28

0.30

Langevin

Nose-Hoover

Te

mp

era

ture

, <T

>

position, i

• Steady state – constant heat flux at all points

Tk

Tkj

1

Reduced k at boundary for

NH

k nearly zero at

boundary for Langevin

Reduce boundary effect using Memory Kernels?

1.4 Cooling of Ubiquitin (NAMD)

100 200 300 400 500 600 7000.20

0.25

0.30

0.35

0.40

0.45

Tem

pera

ture

time

Nose-Hoover Langevin

Cooling of 1000 atom

LJ chain

0 20 40 60 80 1000.20

0.25

0.30

0.35

0.40

0.45

time

Tem

pera

ture

, T

position, x (x10)

NH : Unphysically large temp

gradient maintained at

boundary

1.5 Thermostatting far from equilibrium

• Concept – use feedback control loop to regulate temp in centre of chain by thermostating ends

T1 TN

{ {

Controlled region

Thermostat at T1c to

maintain T1

Thermostat at TNc to

maintain TN

boundaryzone

boundaryzone

1.6 Feedback Control of 1D chain

• Algorithm

{

Thermostat at T1c

1

211

1

1

21

1

1

Tk

xmTQ

Tk

xmQ

xmx

Vxm

Bcc

cB

iiiM

iii

ii

Adjust T1c to

maintain T1 here

1.7 Divergence of k

• The thermal conductivity of a body with a momentum-conserving potential scales with the system size N as

• 3D molecular chain exhibits convergent conductivity• Transverse and longitudinal waves in higher dimensions

in 1D

in 2D

in 3D.

52Nk

Nk ln

constantk

1.8 Feedback control results

0 20 40 60 80 10030

35

40

45

50

55

60

NHthermostat

target50K

target40K

NHthermostat

controlledregionte

mp

era

ture

<T

>

position, x

•3D rod (8x8x100)

•Nosé-Hoover thermostat

1.9 Feedback control results

•Langevin thermostat

0 20 40 60 80 10010

20

30

40

50

60

70

80

Langevinthermostat

target50K

target40K

Langevinthermostat

controlledregionte

mp

era

ture

<T

>

position, x

Langevin can control temperature in 3D but less effective than Nosé-Hoover

1.10 Feedback control results

0 20 40 60 80 10010

20

30

40

50

60

70

80

stadiumthermostat

target50K

target40K

stadiumthermostat

controlledregionte

mp

era

ture

<T

>

position, x

• Stadium damping thermostat

0 20 40 60 80 10010

20

30

40

50

60

70

80 NH Langevin stadium

target50K

target40K

controlledregionte

mp

era

ture

<T

>

position, x0 100 200 300 400

0

10000

20000

30000

40000

50000

60000

NH thermostats

freq

uen

cy

instantaneous temperature (K)

left atom (50K) middle atom (45K) right atom (40K)

1.11 Feedback control results

• Temperature distributions

2.0 Remote boundary conditions by coarse-

graining• Some problems are not well represented by :

– periodic boundary conditions, particularly where there are long range interactions, e.g. elastic fields in solids

– standard ensembles, especially in cases where we are doing work on the system, e.g. NVE, NVT, NPT

Work dissipated as heat.

Only N is constant

2.2 Concurrent modelling approach

• Do not model dynamics in continuum region

Dynamic atomistic region DOF : positions of atoms, q momenta of atoms, p

Punch

Elastostatic continuum region DOF : positions of nodes, q temperatures at nodes, T

2.3 Stadium damping

• Diffuse thermostatting interface• Proposed by B.L. Holian & R. Ravelo (1995)• Constant temperature simulation embedded

in elastostatic FE (Qu, Shastry, Curtin, Miller 2005)

• Shown to produce canonical ensemble• Simple solution to phonon reflection problem

MD

Damping zone

Stadium damping

)(x

2.4 Steady State Concurrent model

1T NT11 TT

{ {

unthermostatted MD region

Thermostat at T1c to maintain

continuum FD region

continuum FD region

Thermostat at TNc to maintain

NN TT ˆ

1

111

)ˆ(

Q

TTT c

1T NT

2.5 Steady state FD/MD simulation

20 40 60 80 10030

35

40

45

50

55

60

continuumregion

continuumregion

atomisticregion

tem

pe

ratu

re (

K)

position, x

actual temp thermostated temp

Blue line is MD boundary zone

Red line is FD/MD result

Blue line is full FD solution

Red line is FD at ends (1-20 and 101-120) and MD in middle

Nosé-Hoover Stadium damping

2.6 Transient Concurrent model

1T NT11 TT

{ {

unthermostatted MD region

Thermostat at T1c to maintain

continuum FD region

continuum FD region

Thermostat at TNc to maintain

NN TT ˆ

dtjjQQ

TTT

t

c )ˆ(1)ˆ(

1

0

121

111

1T NTNj

Nj

11 jj

Heat must be conserved on average

1j

1j

NN jj ˆ

2.6 Transient FD/MD simulation

Blue line is full FD solution

Red line is FD at ends (1-20 and 101-120) and MD in middle

3.0 Conclusions• Kapitza boundary effect

– conductivity near an interface (real or artificial) is less than bulk value

– can obtain desired temperature by feedback control of thermostatted boundary zone.

• Thermostats for non-isothermal boundary conditions– deterministic NH thermostat minimizes Kapitza effect– although Langevin naturally thermostats each particle

individually..– ..global NH thermostat determines average temp but not

distribution– responsiveness of global NH thermostat depends on no. of

thermostatted atoms (important for transient b.c.s)

• FD/MD coupling– Stadium damping effectively removes spurious phonon

reflection from atomistic/coarse-grained interface– simple matching conditions ensure coupling between

continuum and atomistic regions