melting point and solid-liquid free energy computation using thermodynamic integration ·...
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Melting point and solid-liquid free energy computation using thermodynamic integration
Sai Jayaraman1435 Multiscale Dynamic Materials Modeling
Sandia National Laboratories
LAMMPS Users’ Workshop @ CSRI February 24-26, 2010
Monday, March 1, 2010
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Frenkel & Ladd’s method: H. Zhao, and N. R. Aluru, ``Molecular Dynamics Simulation of Bulk
Silicon Under Strain'', Journal of Interaction and Multiscale Mechanics,1(2), 303-315, 2008.
Free energy calculations in LAMMPS
Umbrella sampling
Atomistic computation of liquid diffusivity, solid-liquid interfacial free energy and kinetic coefficient in Au and Ag, J. J. Hoyt, M. Asta, Phys
Rev B, 65, 214106 (2002).
Method for Computing the Anisotropy of the Solid-Liquid Interfacial Free Energy, J. J. Hoyt, M. Asta, A. Karma, Phys Rev Lett, 86, 5530
(2001).
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Free energy calculation in MD:Free energy perturbation (FEP)Potential of mean force (PMF)
Thermodynamic Integration (TI)
Computing melting points:Direct methods: defect induced melting, simulation of solid-
liquid interface
Free energy-based methods: Frenkel’s Einstein crystal, Hoover & Ree’s single occupancy cell
Pseudosupercritical path sampling method1-3
1D. M. Eike, J. F. Brennecke, and E. J. Maginn, J. Chem. Phys. 122, 014115 (2005)2D. M. Eike and E. J. Maginn, J. Chem. Phys. 124, 164503 (2006)3S. Jayaraman and E. J. Maginn, J. Chem. Phys. 127, 214504 (2007)
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Pseudosupercritical path method
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
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Pseudosupercritical path method
U1!2(!) = [1 + !(" ! 1)]mUVDW + [1 + !(" ! 1)]nUELEC + UNS
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
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Pseudosupercritical path method
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
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Pseudosupercritical path method
U3!4(!) = "mUVDW + "nUELEC + UNS ! !!
i
!
j
aij exp(!bijr2ij)
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
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Pseudosupercritical path method
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
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Pseudosupercritical path method
U4!5(!) = [" + !(1! ")]mUVDW + [" + !(1! ")]nUELEC + UNS ! (1! !)!
i
!
j
aij exp(!bijr2ij)
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
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Pseudosupercritical path method
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
Monday, March 1, 2010
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Pseudosupercritical path method
!A2!3 =! V S
V !
!"P # dV!Ai!j =! 1
0
"!U
!"
#
!
d" !Gs!! =!
i,j
!Ai"j
Coupling parameter from state!i!j i! j
fix/alchemy, lj/cut/alchemy, coul/long/alchemy, pppm/alchemy
Monday, March 1, 2010
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LiNO3 NaNO3
KNO3
Contributions to ∆Gs-l : nitrate salts
400 450 500 550 600 650 700Temperature (K)
-20
-10
0
10
20
30
40
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
400 450 500 550 600 650Temperature (K)
0
20
40
60
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
400 500 600 700Temperature (K)
0
10
20
30
40
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
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LiNO3 NaNO3
KNO3
Contributions to ∆Gs-l : nitrate salts
400 450 500 550 600 650 700Temperature (K)
-20
-10
0
10
20
30
40
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
400 450 500 550 600 650Temperature (K)
0
20
40
60
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
400 500 600 700Temperature (K)
0
10
20
30
40
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
1: Experimental values were obtained from: Takahashi Y, Sakamoto R, Kamimoto M, Int. J. Thermophys., 9, No. 6, 1081-1090 (1988)
Melting point (in K) LiNO3 NaNO3 KNO3
Experimental1 526 578 607Simulation 459±3 592 ±7 511±5
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LiNO3 NaNO3
KNO3
Contributions to ∆Gs-l : nitrate salts
400 450 500 550 600 650 700Temperature (K)
-20
-10
0
10
20
30
40
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
400 450 500 550 600 650Temperature (K)
0
20
40
60
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
400 500 600 700Temperature (K)
0
10
20
30
40
Ener
gy u
nits
(kJ/m
ol)
ΔGs-lΔHs-lTΔSs-l
1: Experimental values were obtained from: Takahashi Y, Sakamoto R, Kamimoto M, Int. J. Thermophys., 9, No. 6, 1081-1090 (1988)
Melting point (in K) LiNO3 NaNO3 KNO3
Experimental1 526 578 607Simulation 459±3 592 ±7 511±5
S. Jayaraman, A. Thompson, O. A. von Lilienfeld, and E. J Maginn, I&EC Res., 49, 559-571 (2010)
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Melting point of [C4mim][Cl]
270 300 330 360 390T (K)
-2
-1
0
1
2
ΔG
/RT
CMJJ: Ortho
300 320 340 360 380 400T (K)
-1.5
-0.75
0
0.75
1.5
ΔG
/RT
CMJJ: Mono
Tmexpt
339 KTmsim
308 K
Tmexpt
340 KTmsim
333 K
270 300 330 360 390T (K)
-2
-1
0
1
2
3
ΔG
/RT
CLP: Ortho
Tmexpt
339 KTmsim
302 K
S. Jayaraman and E. J. Maginn, J. Chem. Phys. 127, 214504 (2007)
N NN1
C2
N3
C4C5
C6C7
C8
C9
C10
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Crystal polymorph stabilities [C4mim][Cl]
• Monoclinic thermodynamically stable relative to orthorhombic form at all temperatures below melting points of either polymorph.
• Free energy differences within accuracy of simulations
Computed Gibbs free energy difference between monoclinic and orthorhombic polymorphs of [C4mim]
[Cl] using the CMJJ potential
S. Jayaraman and E. J. Maginn, J. Chem. Phys. 127, 214504 (2007)
200 400 600 800T (K)
-4
-3
-2
-1
0
1
2
ΔG
/RT
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Free energy of binary mixtures
NA, NB NA+1, NB-1
P,T P,T
ZidNA,NB ,T,P =
1
NA!NB !qNAA qNB
B
ZNA,NB ,T,P = ZidNA,NB ,T,P
! !
0dV exp(!!PV )
!
V. . .
!
VdrN exp(!!U(NA, NB))
ZNA+1,NB!1,T,P
ZNA,NB ,T,P=
ZidNA+1,NB!1,T,P
ZidNA,NB ,T,P
!exp("!!U)A+B!#NA,NB ,P,T
!g
!xA= !kBT ln
!qAqB
"+ kBT ln
!xA
1! xA
"! kBT ln"exp(!"!U)#NA,NB
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Free energy of binary mixtures
NA, NB NA+1, NB-1
P,T P,T
ZidNA,NB ,T,P =
1
NA!NB !qNAA qNB
B
ZNA,NB ,T,P = ZidNA,NB ,T,P
! !
0dV exp(!!PV )
!
V. . .
!
VdrN exp(!!U(NA, NB))
ZNA+1,NB!1,T,P
ZNA,NB ,T,P=
ZidNA+1,NB!1,T,P
ZidNA,NB ,T,P
!exp("!!U)A+B!#NA,NB ,P,T
!g
!xA= !kBT ln
!qAqB
"+ kBT ln
!xA
1! xA
"! kBT ln"exp(!"!U)#NA,NB
0 0.2 0.4 0.6 0.8 1xLiNO3
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0G
mix
(kca
l/mol
)
Gmix
Gexcess
Gmix = x1G1 + x2G2 +RT!
i
xi lnxi +Gexcess
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Summary
• Implemented a rigorous method for computing the melting point and crystal phase stability in LAMMPS.
• Verified implementation and computed melting points of alkali nitrate salts using LAMMPS.
• Thermodynamic Integration can be used in alchemical transformations - free energies of mixtures.
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Acknowledgements
• University of Notre Dame: Prof. Ed Maginn
• Maginn group members
• Center for Research Computing
• Air Force Office of Scientific Research
• Sandia National Labs: Steve Plimpton, Aidan Thompson, Anatole Lilienfeld, Nate Siegel, Bob Bradshaw, John Aidun
•S. Jayaraman and E. J. Maginn, J. Chem. Phys. 127, 214504 (2007)
•S. Jayaraman, A. Thompson, O. A. von Lilienfeld, and E. J Maginn, I&EC Res., 49, 559-571 (2010)
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NaCl melting point calculations
0 0.001 0.002 0.003 0.004 0.005-1
-0.5
0
0.5
1
ΔG
s-l (k
cal/m
ol)
1070 K1100 K1150 K
0 0.001 0.002 0.003 0.004 0.0051/N
1065
1080
1095
1110
Mel
ting
poin
t (K
)
1070 K1100 K1150 K
Monday, March 1, 2010