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Sandia is a multi-program laboratory operated by Sandia corporation, a fully owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy’s National Nuclear Security Administration under

contract DE-AC04-94AL85000.

Molten salt eutectics from atomistic alchemical simulations

Sai Jayaraman1,*, Aidan Thompson1 and Anatole von Lilienfeld2

1Advanced Device Technologies, Sandia National Laboratories2Leadership Computing Facility, Argonne National Laboratory

Lammps workshopAugust 11, 2011

*sjayara@sandia.gov

Common tangent at eutectic point between liquid and solid phases

Chemical thermodynamics of Materials - C. H. P. LupisNorth-Holland, 1983

2

Computing the free energy of a liquid mixture

Free energy of binary mixture at

T, P, xA, xB

Free energy of Pure

species A at T,P

Free energy of Pure

species B at T,P

Free energy contribution arising from entropy of mixing

Residual contribution from energetic

interaction between molecules of A and B.

Gmix = xAGA + xBGB + RT�

i=A,B

xi lnxi + Gexcess

3

B

NA, NB

λ = 0

NA+1, NB-1A

λ = 1

Alchemical Transformation

∂Gexcess

∂x≈ ∆G

∆x

∆µAB =1�

0

dλ

�∂U

∂λ

�

λ

=1�

0

dλ

��

i

(uiA − uiB)

�

λ

4

uij(λ) = λuiA + (1− λ)uiB

B

NA, NB

λ = 0

NA+1, NB-1A

λ = 1

Alchemical Transformation

∂Gexcess

∂x≈ ∆G

∆x

∆µAB =1�

0

dλ

�∂U

∂λ

�

λ

=1�

0

dλ

��

i

(uiA − uiB)

�

λ

4

uij(λ) = λuiA + (1− λ)uiB

0 0.2 0.4 0.6 0.8 1!

-80

-60

-40

-20

0"#U/#!$

!

LiNO3-KNO3

Reversibility of transformation: Red (Li K) and Black (K Li)

coincide.

xLiNO3 = 0.5

B

NA, NB

λ = 0

NA+1, NB-1A

λ = 1

Alchemical Transformation

∂Gexcess

∂x≈ ∆G

∆x

∆µAB =1�

0

dλ

�∂U

∂λ

�

λ

=1�

0

dλ

��

i

(uiA − uiB)

�

λ

4

uij(λ) = λuiA + (1− λ)uiB

fix 1 all adapt 1 pair lj/cut epsilon 5 * v_lj scale yescompute 1 all ti lj/cut 5 v_lj v_dlj tail v_lj v_dlj

gex� (xA, xB) =

xA�

0

dx�A ∆µAB(x�

A)− xA

1�

0

dx�A∆µAB(x�

A)

gex� (xA, xB, xC) =

xA�

0

dx�A∆µAB(x�

A, xC)− xA

xA + xB

xA+xB�

0

dx�A∆µAB(x�

A, xC)

+xA

xA + xBgexAC�(xC) +

xB

xA + xBgexBC�(xC)

Excess free energies from ∆μAB

Binary:

Ternary: Area under ∆μAB

C

A B

x A=

1−x C

xB

=1−

xC

At each xC, alchemical changes conducted at xA mesh points

5

gex� (xA, xB) =

xA�

0

dx�A ∆µAB(x�

A)− xA

1�

0

dx�A∆µAB(x�

A)

gex� (xA, xB, xC) =

xA�

0

dx�A∆µAB(x�

A, xC)− xA

xA + xB

xA+xB�

0

dx�A∆µAB(x�

A, xC)

+xA

xA + xBgexAC�(xC) +

xB

xA + xBgexBC�(xC)

Excess free energies from ∆μAB

Binary:

Ternary: Area under ∆μAB

C

A B

x A=

1−x C

xB

=1−

xC

At each xC, alchemical changes conducted at xA mesh points

0 0.2 0.4 0.6 0.8 1xLiNO3

-27

-26.5

-26

-25.5

-25

-24.5

-24

!µ

AB (k

cal/m

ol) LiNO3-KNO3

5

LiNO3-KNO3 mixture - Free energy vs composition

0 0.2 0.4 0.6 0.8 1xLiNO

3

-1.5

-1

-0.5

0

Fre

e en

ergy (

kca

l/m

ol)

400 K423 K450 K

KNO3 LiNO3

�

i

xi∆s�g◦i

gmix� (xA) = gid(xA) + gex

� (xA)

• Approximation: “Simple Eutectic” - Solid phases approximated by pure solids.

• Location of eutectic : Point of tangency between liquid free energy of mixing and solid-liquid connecting line

410 K

M.J. Maeso and J. Largo, Thermochimica Acta, 223, 145-156 (1993)

6

Jayaraman et al. IECR (2010)

LiNO3-NaNO3-KNO3 ternary

S. Jayaraman, A. P. Thompson and O. A. von Lilienfeld, PRE rapid communication, accepted (2011)

7

1M.J. Maeso and J. Largo, Thermochimica Acta, 223, 145-156 (1993) - Binaries

2A.G. Bergman and K. Nogoev, Zh. Neorg. Khim., 9 1423 (1964) - Ternary 3H. R. Carveth, J. Phys. Chem., 2, 209 (1898) -

Conclusions

• Thermodynamic integration based method developed to compute free energies of mixing from MD simulations

• Tangent method extended to compute approximations of eutectic compositions assuming a “simple eutectic”

• Need per-atom contributions from pppm!

8

fix 1 all adapt 1 pair lj/cut epsilon * 5 v_lj scale yescompute 1 all ti lj/cut 5 v_lj v_dlj tail v_lj v_dlj