BCNucleation-Aggregation Workshop Grand canonical molecular dynamics - - PowerPoint PPT Presentation

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BCNucleation-Aggregation Workshop Grand canonical molecular dynamics simulation Grand canonical molecular dynamics simulation

• f homogeneous nucleation

Universitat de Barcelona, Departament de Física Fonamental, June 18, 2009

• M. Horsch, H. Hasse, and J. Vrabec

SFB 716

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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Molecular simulation of nucleation

Indirect simulation: T iti th li Transition path sampling Determination of the critical size … by observing single droplets in non-equilibrium by observing single droplets in equilibrium … by observing single droplets in equilibrium Direct simulation: … of a metastable state far from the spinodal line … of nucleation at a high supersaturation, decreasing over time … of a metastable state near the spinodal line … of nucleation at a constantly high supersaturation

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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The critical nucleus

ethane at T = 280 K (ρs = 1.88 mol/l)

Free energy of formation … is defined by a stable or unstable equilibrium with the vapor.

ts of kT 10 2.55 mol/l ethane at T 280 K (ρs 1.88 mol/l)

• Δμ

ΔΩ*

Free energy of formation Positive surface contribution:

ergy in uni 2.55 mol/l

• Δμ

ΔΩ*

N ti l t ib ti

j A

γ dA d = Ω

free en

• 10

2.8 mol/l j* j*

Negative volume contribution:

μ μ dj d

j V

− =

liq

Ω

li

250 500 750 nucleus size in number of molecules j j

It i ti l t k th t ti i t f Δ

s liq

lim μ μ j

j

=

∞ →

• PROF. DR.-ING. HABIL. JADRAN VRABEC

THERMODYNAMICS AND ENERGY TECHNOLOGY

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

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Equilibrium vapor pressure

• f kT

100 methane at 149 K

Equilibrium condition for a droplet containing j molecules:

gy in units

• 100

molecules:

( )

j T p p , =

free energ

• 200

NpT: N = 6000, p = 1170 kPa NVT: N = 5000, ρ = 1.40 mol/l NVT: N = 2000 ρ = 2 16 mol/l

ΔG at constant p and T: 1 unstable equilibrium

f

• 300

NVT: N = 2000, ρ = 2.16 mol/l kPa 1000 2000

ΔF at constant V and T: 1 unstable equilibrium

10 100 1000 p / 1000

1 unstable equilibrium 1 stable equilibrium

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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droplet size in number of molecules

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Systems containing a single droplet

Vapor and liquid are

equilibrated separately. q p y A small (j < 10000) droplet is inserted into droplet is inserted into the vapor.

If the droplet cannot If the droplet cannot

evaporate completely, an equilibrium is established ithin a fe established within a few nanoseconds. t LJ fl id ( 2 5 )

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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t-s-LJ fluid (rc = 2.5 σ)

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2

Surface tension

nits of εσ-2 0.6 T = 0.65 ε/k

Integration of the pN(r) profile:

dr r dp r p p γ ) ( ) (

N 3 2 l 3

∫

∞ −

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − =

sion in un 0.4 T = 0.8 ε/k

Size dependence (Tolman):

dr γ 8

∫

⎥ ⎦ ⎢ ⎣

urface ten 0.2 T = 0 95 ε/k

p ( )

( )

2 T

2 1

− ∞

+ + =

γ γ

R O R δ γ γ

droplet size in number of molecules 5000 10000 su 0.0 T = 0.95 ε/k

Correlation from simulation data for T = 0.65, 0.70, … 0.95 ε/k:

p

, ,

• simulation

— planar interface

3 1 c T

9 . 1 7 .

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = j T T R δ

γ

• PROF. DR.-ING. HABIL. JADRAN VRABEC

THERMODYNAMICS AND ENERGY TECHNOLOGY

• - - new correlation

c

⎠ ⎝

γ

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Tolman equation

gth 10 0.8 ε/k

Vapor-liquid interface of the t-s-LJ fluid

man leng 0.9 ε/k

simulation

us / Tolm 5

correlation for γ, using

δ γ

T

2 1+ =

∞

5000 10000 radi

using

γ

R γ 1+ =

droplet size in number of particles

The higher order terms of the Tolman equation should not be neglected.

• PROF. DR.-ING. HABIL. JADRAN VRABEC

THERMODYNAMICS AND ENERGY TECHNOLOGY

e g e o de te s o t e o a equat o s ou d

• t be

eg ected

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Direct MD simulation of nucleation

Integration time step typically between 1 and 5 fs; Feasible simulation time: on the order of nanoseconds. A d i h V 10 20

3

i A saturated vapor with V = 10-20 m3 contains: 800,000 molecules (methane at 114 K = 0.6 Tc) 7,000,000 molecules (CO2 at 253 K = 0.83 Tc) 7,000,000 molecules (CO2 at 253 K 0.83 Tc) Minimal nucleation rate accessible by direct simulation: #nuclei / (volume V x time Δt) = nucleation rate J 10 1030 / m3s ( 10-20 m3 10-9 s ) = x /

Experiment Direct MD simulation up to 1023 / m3s above 1030 / m3s

( ) /

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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up to 10 / m s above 10 / m s

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Grand canonical molecular dynamics

Algorithm according to Cielinski: fi d l f V d T

• fixed values of μ, V und T
• test insertion of a molecule at a random position

( )⎥

⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 Λ Δ exp , 1 max

3 ins ins

N V kT U μ P

• test deletion of a random molecule

⎥ ⎤ ⎢ ⎡ ⎟ ⎞ ⎜ ⎛ − − = V U μ P

ins d l

Δ exp 1 max

• equal number of test insertions and deletions (10-5 – 10-3 / step)

⎥ ⎦ ⎢ ⎣ ⎟ ⎠ ⎜ ⎝ = N kT P

3 del

Λ exp , 1 max

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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Supersaturation from NVT and µVT simulation

3 ration 3.0

µVT NVT

ation 0.7 ε/k Sρ supersatur 2.5

NVT

upersatura 2 Sp Sμ potential s 2.0 0.7 ε / k s 0.8 ε/k chemical p 1.5 0.85 ε / k excess pressure in units of εσ-3 0.004 0.008 0.012 1 density in units of σ-3 0.02 0.04 0.06 c 1.0

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excess pressure in units of εσ density in units of σ

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Szilárd‘s demon

SZILÁRD

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SZILÁRD

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McDonald‘s demon

McDONALD

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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McDONALD

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Interactive presentation: McDonald‘s demon

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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Comparison: NVT and µVT simulation

80 iNVT = 25

t-s-LJ fluid at T = 0.7 ε/k

x 10

6σ3

40 60 iNVT = 50 iμVT = 50 iNVT = 150

NVT

ρj

20 40 iNVT 150 jμVT > 25 10

3σ3ε-1

15 20 μVT

µVT: S = 2.866 NVT: ρ = 0 004044 σ-3

i l ti ti i it f ( / )1/2 250 500 750 1000 1250 p x 1 10 NVT

NVT: ρ = 0.004044 σ-3

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simulation time in units of σ(m/ε)1/2

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Nucleus size distribution

t-s-LJ fluid at T = 0.7 ε/k: µVT (S = 2.866) and NVT (ρ = 0.004044 σ-3) simulation

• f σ-3

10-4 10-3 y in units 10-6 10-5 CNT tNVT = 400 density 10-8 10-7 CNT µVTi=50 tNVT = 1050 nucleus size in number of molecules 10 20 30 40 50 60

G d t ith CNT f j* d th b f ll l i

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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Good agreement with CNT for j* and the number of small nuclei.

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Threshold dependence of the intervention rate

Statistical probability for a nucleus of growing from size j to infinite size:

m

• 15

j*CNT

t-s-LJ fluid, T = 0.7, S = 2.496 to infinite size:

e logarithm

• 20

( )

2 exp 1 dj j P

∫

∞ ∞

⎟ ⎞ ⎜ ⎛ Ω − = ω

h h

vention rat

• 25

( )

, exp 1

Z

dj kT j P

j

∫

⎟ ⎠ ⎜ ⎝ = ω

such that

20 40 60 80 interv

• 30

CNT (ln J = -26.4)

( )

2 1

=

∗ ∞ j

P

threshold size in number of particles 20 40 60 80

CNT predicts an acceptable value for j* and underestimates J significantly

( )

2

j P

• PROF. DR.-ING. HABIL. JADRAN VRABEC

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CNT predicts an acceptable value for j* and underestimates J significantly.

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GCMD simulation of nucleation: Results

ε-0.5m

0.5

10-6 0.85 ε/k 0.65 ε/k s of σ-4ε- 10-9 e in units 10-12 McDonald‘s demon l i l th NVT (YM method) ation rate 10 0.7 ε/k 0.95 ε/k classical theory critical scaling EMLD-DNT pressure in units of εσ-3 0.008 0.012 0.016 0.03 0.04 0.05 nuclea 10-15 EMLD DNT

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pressure in units of εσ

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Conclusion

• MD simulation of equilibria allows sampling over an arbitrary

time interval eventually leading to the desired level of accuracy time interval, eventually leading to the desired level of accuracy.

• Single droplets can be stable in the canonical ensemble.
• A supersaturated vapor near the spinodal line can be

stabilized by grand canonical simulation with McDonald‘s demon.

• The classical theory leads to acceptable results for the t-s-LJ
• fluid. However, it does not take into account curvature effects
• n the surface tension.
• PROF. DR.-ING. HABIL. JADRAN VRABEC

THERMODYNAMICS AND ENERGY TECHNOLOGY