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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