We  are  interested  in  predic-ng  the  outcomes  of  long  -me-­‐scale  thermal  &  diffusion  processes  with  atomic  detail,  e.g.          -­‐  [Wagner  et  al,  CMAME,2008]          -­‐  [Templeton  et  al,MSMSE,2010]  We  have  developed  methods  for  -me  integra-on:            -­‐  frac-onal-­‐step  integra-on            -­‐  mul-ple  -me-­‐scale  integra-on            -­‐  projec-ve  integra-on  as  well  as  parallel  implementa-ons  of  parallel  replica  dynamics,  temperature  accelerated  dynamics,  etc.      

Time-­‐scale  issues  in  molecular  dynamics-­‐finite  element  coupling  applica5ons  R.  Jones,  J.  Templeton,  G.  Wagner  

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND No. 2011-XXXXP

Mul-ple  Time-­‐scale  Integra-on  

200

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600

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1200

1400

0 10 20 30 40 50 60 70 80 90 100

TE

MP

ER

AT

UR

E

TIME [ps]

LEFT RESERVOIRTUBE

RIGHT RESERVOIR

Two  temperature  model:  

Assume  short  &  long  -mescales:  

Expand  and  subs-tute:  

Result  :  electrons  interact  with  filtered/long  -me-­‐scale  phonon  evolu-on  

Accelerated  MD  Methods  

Accelerated MD methods focus on infrequent events that drive dynamics (e.g. atom-hopping) •  Ignore long periods of thermal

vibration within energy basins

Saddle  point  

Local  Minimum  

Projec-ve  Integra-on  To  model  surface  diffusion  with  the  Equa-on-­‐Free  Method  [I.  Kevrekidis  et  al,  2003],  fine/coarse  scale  descrip-ons  are  used  to  extend  to  longer  -mescales  

LIFT   RESTRICT  

MICRO-­‐  SIMULATOR  

tΔ

h(x,t+Δt)  h(x,t)  

Approximate:   ( ) ( )( )1h h t t h tt t∂

≈ + Δ −∂ Δ

Project:   ( ) ( ) hh t T h t Tt∂

+ Δ = + Δ∂

An  Improved  Li^  Operator:    The  Maximum  Entropy  Method  

The lift operator must preserve higher-order system statistics (like spatial correlations) to replicate dynamics but this is problem specific & difficult in higher dimensions  The  goal  is  to  constrain  only  the  coarse  scale  values  of  interest,  but  allow  other  variables  to  come  into  correct  equilibrium  

Given  “goal”  average  profile  shape  (top),  generate  many  individual  realiza-ons  (bocom)  that,  when  ensemble  averaged,  reproduce  the  goal  profile  while  also  preserving  the  correct  dynamics  (right).      

Frac-onal  Step  Integra-on  

6

�

J

∂

∂t(MIJθJ) =

�

J

KIJθJ + 2�

α

NIαvα ·�fMDα +

1

2fλα

�

mαv̇α = fMDα − mα

2

�

I

NIαλIvα

Atoms  contribute  to  nodal  heat  equa-on  

Heat  at  nodes  affects  MD  energy  through  a  spa-al  thermostat    

Gear  update  for  FE  dynamics,  predictor/corrector  requires:  

Velocity-­‐Verlet  update  for  MD,  two-­‐step  update  requires:  

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

Update  Lagrange  mul-pliers  a^er  the  predic-on  phase  by  solving  the  non-­‐linear  equa-on:  

8

∆t�

α

NαI Kα

�

JNα

J λJ − ∆t2

4

�

α

NαI Kα

��

J

NαJ λJ

���

K

NαKλK

�= Rc

I

∆tλ < 1

d

dtNIα =

d

dtNI (xα) = ∇NI · vα

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

Apply  resul-ng  force  at  three  -me  increments:      ,                          ,  and    

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

8

∆t�

α

NαI Kα

�

JNα

J λJ − ∆t2

4

�

α

NαI Kα

��

J

NαJ λJ

���

K

NαKλK

�= Rc

I

∆tλ < 1

d

dtNIα =

d

dtNI (xα) = ∇NI · vα

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

7

θ̇FE = M−1IJ KIKθK

v̇α = m−1α fMD

α

∆θI = M−1IJ

�

α

NIαeα∆Vα

��tn+1 −M−1

IJ

�

α

NIαeα∆Vα

��tn

NI∆EI =�

α

NIα

�∆tvα · fα +

∆t2

2m−1

α fα · fα + φ�α

��tn+1 − φ�

α

��tn

�

tn

tn+1/2

tn+1

tn+1,∗

∆θFE

∆θMD

∆vα

∆λ

Sintered Pd

M.  Ong,  Sandia  

Coarsening after heating to 200 C for 5 hr

GaN nanowire with top contacts A. Talin, Sandia

Photoluminescence and Joule heating under power

Methods have been implemented in Sandia’s LAMMPS [S.Plimpton,, A. Thompson, P. Crozier] code: •  Nudged Elastic Band (NEB), Parallel Replica Dynamics (PRD),

Temperature Accelerated Dynamics (TAD) •  Innovations in parallelization of methods •  New features can be used with any LAMMPS interatomic potential •  Open source release to broad research community See http:/lammps.sandia.gov (and http://www.sandia.gov/~sjplimp/spparks.html)

MD/FE  thermal  coupling  requires  a  stable  interleaving  of  integrators  for  the  molecular  and  coarse  scale  dynamics  

Given  the  rela-ve  thermal  proper-es  of  electrons  and  phonons,  an  efficient  coupling  scheme  needs  to  operate  on  two  disparate  -mescales  

Laser  hea-ng  of  a  CNT  connect  to  thermal  reservoirs  

[R.Jones  et  al,  IJNME,  2010]  [J.  Templeton  et  al,  in  prepara5on]  

(1)  Use  fine-­‐scale  simula-ons  as  “computa-onal  experiments”  to  get  approximate  course-­‐scale  -me  deriva-ves  (2)  Project  coarse  scales  in  -me  over  longer  -mestep,  and  reini-alize  fine  scales  

 

[G.  Wagner  et  al,  Int.  J.    Mul*scale  Comp.  Eng.,  2010]  [J.  Deng  et  al,  in  prepara5on]