Statistical Physics: Modern Trends and Applications Steady ...

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PROF. DR.-ING. HABIL. JADRAN VRABEC THERMODYNAMICS AND ENERGY TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Statistical Physics: Modern Trends and Applications Steady-state molecular dynamics simulation of vapour to liquid nucleation with McDonald‘s daemon Lviv, June 25, 2009 M. Horsch, S. Miroshnichenko, and J. Vrabec SFB 716

Transcript of Statistical Physics: Modern Trends and Applications Steady ...

PROF. DR.-ING. HABIL. JADRAN VRABEC

THERMODYNAMICS AND ENERGY TECHNOLOGY

DEPARTMENT

OF MECHANICAL

ENGINEERING

Statistical Physics: Modern Trends and Applications

Steady-state molecular dynamics simulation

of vapour to liquid nucleation with McDonald‘s daemon

Lviv, June 25, 2009

M. Horsch, S. Miroshnichenko, and J. Vrabec

SFB 716

PROF. DR.-ING. HABIL. JADRAN VRABEC

THERMODYNAMICS AND ENERGY TECHNOLOGY

DEPARTMENT

OF MECHANICAL

ENGINEERING

250 500 750

fre

e e

ne

rgy

in u

nits o

f kT

-10

0

10

nucleus size in number of molecules

2.55 mol/l

2.8 mol/l

ethane at T = 280 K ( s = 1.88 mol/l)

-

-*

j*

*

j*

Free energy of formation

Positive surface contribution:

Negative volume contribution:

The critical nucleus

It is essential to know the supersaturation in terms of Δµ.

… is defined by a stable or unstable equilibrium with the vapour.

μμdjd jV

l iqΩ

s

liqlim μμ jj

jA γdAdΩ

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free e

nerg

y in

units o

f kT

-300

-200

-100

0

100

NpT: N = 6000, p = 1170 kPa

NVT: N = 5000, = 1.40 mol/l

NVT: N = 2000, = 2.16 mol/l

droplet size in number of molecules10 100 1000

p / k

Pa

0

1000

2000

methane at 149 K

Equilibrium vapour pressure

Equilibrium condition for

a droplet containing j

molecules:

jTpp ,

ΔG at constant p and T:

1 unstable equilibrium

ΔF at constant V and T:

1 unstable equilibrium

1 stable equilibrium

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pressure in units of -30.04 0.05

critica

l siz

e in

nu

mb

er

of

mo

lecu

les

1000

10000

100000

0.004 0.008 0.016 0.020

100

1000

10000

0.7 /k

0.65 /k

0.8 /k

0.9 /k 0.95 /k

Single droplet in equilibrium

present simulation data

classical theory—

Classical theory:

Good prediction of the critical

size for low temperatures

Above 0.8 Tc, significant

deviations are present.

3

1

Δ3

2

μ

Aγj

PROF. DR.-ING. HABIL. JADRAN VRABEC

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droplet size in number of molecules0 5000 10000

surf

ace tensio

n in u

nits o

f -2

0.0

0.2

0.4

0.6T = 0.65 /k

T = 0.8 /k

T = 0.95 /k

Surface tension

Integral over normal pressure:

Size dependence (Tolman):

Correlation from simulation data

for T = 0.65, 0.70, … 0.95 ε/k:

● simulation

— planar interface

- - - new correlation

drdr

rdpr

ppγ

)()( N

0

32

l3 8

2T21 γ

γ

ROR

δ

γ

γ

31

c

T 9.01

7.0j

TTR

δ

γ

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excess pressure in units of -3

0.004 0.008 0.012

su

pers

atu

ration

1

2

3

0.7 /k

0.8 /k

S

Sp

S

density in units of -3

0.02 0.04 0.06

ch

em

ica

l p

ote

ntia

l su

pers

atu

ratio

n

1.0

1.5

2.0

2.5

3.0

0.85 / k

0.7 / k

Supersaturation from NVT and µVT simulation

µVT

NVT

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Szilárd´s daemon

L. SZILÁRD

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DEPARTMENT

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ENGINEERING

McDonald´s daemon

J. McDONALD

PROF. DR.-ING. HABIL. JADRAN VRABEC

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DEPARTMENT

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nucleus size in number of molecules0 10 20 30 40 50 60

in u

nits o

f -3

10-8

10-7

10-6

10-5

10-4

10-3

CNT

µVTi=50

tNVT = 400

tNVT = 1050

Nucleus size distribution

t-s-LJ fluid at T = 0.7 ε/k:

µVT (Sµ = 2.866) and NVT (ρ = 0.004044 σ-3) simulation

Good agreement with CNT for the number of small nuclei.

nucle

i per

volu

me

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pressure in units of -3

0.008 0.012 0.016 0.03 0.04 0.05

nucle

ation r

ate

in u

nits o

f -4

-0.5m

0.5

10-6

10-9

10-12

10-15

0.7 /k

0.85 /k

0.95 /k

0.65 /k

GCMD simulation of nucleation: Results

McDonald´s daemon

classical theory

critical scaling (Hale)

Reiss-Reguera

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pressure in units of -3

0.048 0.050critical siz

e in n

um

ber

of

mole

cule

s

0

5000

10000 T = 0.95 /k

chemical potential supersaturation2.0 2.5 3.0 3.5 4.0 4.5

nucle

atio

n r

ate

lo

ga

rith

m

-30

-27

-24

-21

-18

-15

0.7 / k 0.65 / k

Effective surface area

New correlation, using surface

tension data from simulations with

and an effective surface with

124.0

1185.01

31

1

c

Ο j

TT

A

A

j

j

djj

Aγζ

j

j

jj

,0

simulation

classical theory

new correlation

- - -

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Conclusion

• Stable droplets in the canonical ensemble in equilibrium with

supersaturated vapour have the properties of critical nuclei.

• A supersaturated vapour near the spinodal line can be

stabilized by grand canonical MD simulation with McDonald´s

daemon.

• The classical theory leads to acceptable results for the t-s-LJ

fluid. However, it does not take into account curvature effects

on the surface tension.

• A consistent description is given by postulating an increased

effective surface area.