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Thermodynamics 2009

Steady state molecular dynamics

simulation of vapour to liquid nucleation

Imperial College London, September 24, 2009

M. HORSCH, H. HASSE, and J. VRABEC

SFB 716

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MD simulation of a single droplet

Vapour and liquid areequilibrated separately

A small (Nℓ < 10000)droplet is inserted into the vapour

If the droplet cannotevaporate completely,an equilibrium isestablished within a fewnanoseconds

truncated-shifted LJ fluid (rc = 2.5)

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ThEt

( )NTpp ,=

free

ener

gy in

uni

ts o

f kT

-300

-200

-100

0

100

NpT: N = 6000, p = 1220 kPaNVT: N = 5000, ρ = 1.85 mol/lNVT: N = 2000, ρ = 2.86 mol/l

droplet size in number of atoms10 100 1000

p / k

Pa

0

1000

2000

Argon at 117 K

Equilibrium vapour pressure

Equilibrium condition for a droplet containing Nℓatoms:

ΔG at constant p and T:

1 unstable equilibrium

ΔF at constant V and T:

1 unstable equilibrium1 stable equilibrium

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pressure in units of kPa0 500 1000 2000 2500

criti

cal s

ize

in a

tom

s

100

1000

10000

90 K 97 K 110 K 124 K 131 K

Single droplet in equilibrium

“standard”classical theory

Argon (truncated-shifted LJ model)

classical theory with“pressure effect”

3

32

⎟⎟⎠

⎞⎜⎜⎝

⎛=∗

μΔaσN ( )[ ]

3

s32

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=∗

ppvμΔaσN

'

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ThEtdroplet size in atoms

100 1000 10000 100000

σ in

uni

ts o

f g/s

2

0

2

4

6

8

10

97 K

110 K

131 K

curvature 1/Rσ in units of nm-10.0 0.5 1.0 1.5 2.0

δ in

uni

ts o

f nm

0.5

1.0

1.5124 K

117 K

110 K97 K

Droplet surface tension

TOLMAN equation:

● simulation— TOLMAN equation- - - planar interface

∞

∞∞

−+

≈δRδR

σσ

e

e

For small droplets, the TOLMANlength

is significantly elevated.

σRRδ −= e

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Algorithm: MC insertion/deletion steps alternating with MD steps

• test insertion of a molecule at a random position

• test deletion of a random molecule

Grand canonical MD

( )μ⎡ ⎤− Δ⎛ ⎞

= ⎢ ⎥⎜ ⎟ Λ +⎢ ⎥⎝ ⎠⎣ ⎦

potins 3min 1,exp

1U VP

kT N

μ⎡ ⎤− − Δ⎛ ⎞= ⎢ ⎥⎜ ⎟ Λ⎝ ⎠⎣ ⎦

potdel 3min 1,exp

U VPkT N

PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY

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ThEt

Algorithm: MC insertion/deletion steps alternating with MD steps

• test insertion of a molecule at a random position

• test deletion of a random molecule

Grand canonical MD

( )μ⎡ ⎤− Δ⎛ ⎞

= ⎢ ⎥⎜ ⎟ Λ +⎢ ⎥⎝ ⎠⎣ ⎦

potins 3min 1,exp

1U VP

kT N

μ⎡ ⎤− − Δ⎛ ⎞= ⎢ ⎥⎜ ⎟ Λ⎝ ⎠⎣ ⎦

potdel 3min 1,exp

U VPkT N

Thermodynamic conditions of supersaturated state are maintained

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Algorithm: MC insertion/deletion steps alternating with MD steps

• test insertion of a molecule at a random position

• test deletion of a random molecule

Grand canonical MD with MCDONALD‘s dæmon

( )μ⎡ ⎤− Δ⎛ ⎞

= ⎢ ⎥⎜ ⎟ Λ +⎢ ⎥⎝ ⎠⎣ ⎦

potins 3min 1,exp

1U VP

kT N

μ⎡ ⎤− − Δ⎛ ⎞= ⎢ ⎥⎜ ⎟ Λ⎝ ⎠⎣ ⎦

potdel 3min 1,exp

U VPkT N

Thermodynamic conditions of supersaturated state are maintained

McDONALD

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Interactive presentation: MCDONALD‘s dæmon

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ThEtthreshold size θ in atoms

0 20 40 60 80

ln (J

θ / c

m-3

s-1)

55

60

65

standard classicaltheory (CNT)

N*CNT

CNT shiftedto ln J = 56.5

ArgonT = 97 KSμ = 2.496

Probability for a droplet ofgrowing from size θ to infinity:

and in particular:

Intervention rate Jθ and nucleation rate J

( ) 21≈∗Nq

( )( )( )

,exp

exp

∫∫

∞=

1

1

2

2

dNFβ

dNFβθq

θ

Not all of the removed dropletswould eventually reach macroscopicsize.

( )θqJJ θ=

PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY

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supersaturation in terms of the chemical potential1.3 1.4 2.0 3.0 4.0

ln(J

/ cm

-3s-1

)

50

55

60

65 117 K 97 K 90 K

GCMD simulation of nucleation: Results

MCDONALD‘s dæmon

standard CNT

CNT with pressure effect and prefactor C = 200

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Conclusion

• MD simulation of equilibria allows sampling over an arbitrarytime interval, eventually leading to the desired level of accuracy

• Single droplets are stable in the canonical ensemble and yield the critical droplet size

• A supersaturated vapour near the spinodal line can bestabilized by grand canonical MD simulation with MCDONALD‘sdæmon

• MCDONALD‘s dæmon allows to simulate the instationarynucleation process in the steady state

• The classical theory leads to excellent results for argon