Temperature programmed molecular dynamics (TPMD)method

Abhijit Chatterjee

Department of Chemical Engineering

Indian Institute of Technology Bombay

Funding: Department of Science and Technology (DST),

Indian National Science Academy (INSA) 1

[1] Divi and Chatterjee, JCP, 140, 184115 (2014)

[2] Chatterjee and Bhattacharya, JCP, 2015

[3] Venkaramana and Chatterjee, JCP 2016

Students

• Missing in group photo below: Paramita Haldar, Mohit Prateek

Collaborators

• Prof. Swati Bhattacharya, IIT Guwahati

• Prof. Arindam Sarkar, IIT Bombay

Slide 2

Arti Bhoutekar

Srikanth Divi

Venkataramana ImandiM Jaipal

Biomolecular systems

(with Prof. S. Bhattacharya)

Room temperature

Slide 3

Au

Phase transformations

during lithiation and

delithiation in Si

(lithium ion battery

anode material)

Li-Si

phase

a-Si

phase

240-330 K30-80°C

METAL

SOLUTION

Surface diffusion in solution

exp( / )eff eff B

k E k Tν= −

800-1200 K

Ion conducting oxides

(yttria stabilized zirconia)

POTENTIAL ENERGY LANDSCAPE

Selective dissolution in

metal alloys: Long-time

evolution 30-80°C

Building KMC models with TPMD…… some key features of TPMD

Steps involved in building KMC model using TPMD

• Efficiently find states, pathways/rates over a range of temperatures

• Probe whether Arrhenius assumption is reasonable and find effective Arrhenius parameters

Slide 5

MD trajectory

State (collection of potential energy basins)

STATE CONSTRAINED MD

MD set-point Temperature (K)

MD Time elapsed (ps/ns)300

400500

600700

750

4 8 12 16 20

Temperature program

S′S″

S

MD (NVT) trajectory

Slide 6

0

1x103

2x103

3x103

4x103

0.0 0.5 1.0 1.5 2.0Time (ms)

Pro

bab

ilit

y d

ensi

ty (

s-1)

STATE CONSTRAINED MD

S′

S″

S′

Stitching algorithm

S′

S′

TRAJECTORIES FROM

STATE CONSTRAINED MD

COLLECTION OF

WAITING TIMES

0 0( ) exp( )P t dt k t k dtα α= −

Probability of no

escape in time [0, t]

Probability of escape

in time [t, t+dt] x

0 nkα

ατ=Maximum likelihood

estimate for rate constant

CONS: RARE EVENTS ARE POORLY SAMPLED

S′S″

S

Rate estimation in MD @ constant temperature T0MD Time elapsed during search

Synthesized waiting times

Temperature programmed MD

• Increase temperature in steps

• Langevin thermostat is employed

• Kinetic pathways can be sought in parallel

Slide 7

200

300

400

500

600

700

800

900

0.00 0.05 0.10 0.15 0.20

Tem

per

atu

re (

K)

Time (ns)

0

1x1010

2x1010

3x1010

0.00 0.05 0.10 0.15 0.20 0.25

Pro

bab

ilit

y d

ensi

ty (

s-1)

Time (ns)

( )0

( ) exp ,

( 1)

=

= −

< ≤ +

∏n

i n

i

i

P t k k dt

n t n

α ατ

τ τ

2. Temperature programmed molecular dynamics

Dynamically relevant pathways

selected with higher probability at

low temperatures

Anharmonicity correction

With

Without

T0

T1

T2

T3

In state S

ith temperature stage

TPMD distribution

Estimating rates at different temperaturesSlide 8

S”

MD Time

d) Stitched trajectories for

Method 2

b) Independent trajectories from

state S

S”

S”

S”

S’

S”

S”

S”

S’

MD1

MD2

MD3

MD4

MD TimeTem

pe

ratu

re

T0

T1

T2

a) Temperature programc) Stitched trajectories for

Method 1 @ temperature T1

S”

Waiting Time t

S’

Pro

ba

bilit

y

Pro

ba

bilit

y

e)

f)

ΔT

τ 2τ 3τ~

Waiting Time t~

Waiting Time t~

Waiting Time t

Waiting Time t

Waiting Time t

0 1 2, , ,...k k kα α α, Eα αν0 1 2, , ,...k k kα α α

� � �

T0 T1 T2

Estimating effective Arrhenius parameters for a pathway

• Assume Arrhenius behavior for the pathway

Slide 9

0

0

exp −

= B

Ek v

k T

αα α

0

1x1010

2x1010

3x1010

0.00 0.05 0.10 0.15 0.20 0.25

Pro

bab

ilit

y d

ensi

ty (

s-1)

Time (ns)

1 2{ , ,.., }mt t t� � �

max

max

1

10

0

exp( / )

exp( / )

N

i i B i

iXN

i B i

i

T E k T

T

E k T

α

α

θ

θ

−

−=

=

−

=

−

∑

∑

1

1

/m

X nT m T αα

−

=

= ∑

max

0

exp( / )N

i a B i

i

m

E k T

ν

θ=

=

−∑

Collection of waiting times

Waiting time distribution for coarse-states

• Double benefit from TPMD

• Overcoming large activation barriers

• Access low probability states

Slide 10

max

, ,

1

( ) exp( )N

s n n s i i i

i

p t k kπ π τ=

= −∏

dT

dt

ππ=

max

max

1

,10

,

0

exp( / )

exp( / )

N

s i i i B i

i

XN

s i i B i

i

T E k T

T

E k T

α

α

π θ

π θ

−

−=

=

−

=

−

∑

∑max

,

0

exp( / )N

s i i a B i

i

m

E k T

ν

π θ=

=

−∑

State S’

State S

Coarse state

Example: Alanine dipeptideSlide 11

with Prof. S. Bhattacharya, IIT Guwahati Free energy map @ 600 K

NOTE: WE DON’T EMPLOY FREE ENERGY MAPS IN TPMD

MD Temperature (K)

Time (ps)

MARKOV STATE MODEL FOR ALANINE DIPEPTIDE

300

400500

600700

750

4 8 12 16 20

Example: Alanine dipeptideSlide 12

with Prof. S. Bhattacharya, IIT Guwahati Free energy map @ 600 K

NOTE: WE DON’T EMPLOY FREE ENERGY MAPS IN TPMD

MD Temperature (K)

Time (ps)300

400500

600700

750

4 8 12 16 20

Free energy map @ 300 K

NOTE: WE DON’T EMPLOY FREE ENERGY MAPS IN TPMD

Example: Alanine dipeptide

• x

Slide 13

with Prof. S. Bhattacharya, IIT Guwahati

A-B

A-CA-D

B-A

A-E

B-C

B-D

Example: Alanine dipeptideSlide 14

500-2000 waiting times are sufficient

Example: Surface diffusion @ metal-solvent interface

Slide 16

300 K

500 K

800 K

Slide 17

Slide 18MD Temperature (K)

MD Search Time (ps)

Hop: ν= 10.3 ps-1, Ea=0.525 eV

Exchange: ν= 19.3 ps-1, Ea=0.592 eV

Hop: ν= 1.2 ps-1, Ea=0.297 eVAg/Ag(100)

Ag/Ag(100)

0 30 60 90 120 150

300

400

500

600

700

800

Hop: Ea=0.57 eV using NEBHop: ν= 2.3 ps-1, Ea=0.448 eV

Ni/Ni(100)Ni/Ni(100)

Slide 19

In solution: 0.4 ps-1, 0.154 eV

In solution: 0.63 ps-1, 0.33 eV

Conclusions

LIST OF FEATURES OF TPMD

Application to wide range of problems – metals, semiconductor materials, ionic

materials, biomolecules

State recognition allows for off-lattice description

Rate estimation at multiple temperature for multiple pathways

Can handle rugged energy landscapes

Tackle low energy barrier problem, On-the-fly coarse-graining of states possible

No reaction coordinate/collective variables/minimum energy path

Automatic construction of local environment KMC models

Probe Arrhenius behavior (no requirement for HTST)

Parallelizable to large number of processors

Orders-of-magnitude time acceleration over MD

Error control is possible

Slide 21