thermodynamics 2009 steady state molecular dynamics simulation of … · 2017. 2. 6. ·...
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PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
ENGINEERING
ThEt
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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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ThEt
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)
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
ENGINEERING
ThEt
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
'
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
ENGINEERING
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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
ENGINEERING
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 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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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ThEt
Interactive presentation: MCDONALD‘s dæmon
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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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
DEPARTMENTOF MECHANICAL
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ThEt
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
PROF. DR.-ING. HABIL. JADRAN VRABECTHERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENTOF MECHANICAL
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ThEt
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