melting point and solid-liquid free energy computation...

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

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).


http://www.matsci.northwestern.edu/faculty/ma.html




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)


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



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





!A2!3 =! V S

V !

!"P # dV!Ai!j =! 1

0

"!U

!"

#

!

d" !Gs!! =!

i,j

!Ai"j





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





!A2!3 =! V S

V !

!"P # dV!Ai!j =! 1

0

"!U

!"

#

!

d" !Gs!! =!

i,j

!Ai"j





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





!A2!3 =! V S

V !

!"P # dV!Ai!j =! 1

0

"!U

!"

#

!

d" !Gs!! =!

i,j

!Ai"j




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


LiNO3 NaNO3

KNO3


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


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


LiNO3 NaNO3

KNO3


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


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)


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


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


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


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


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.


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)


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


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