Statistical Physics: Modern Trends and Applications Steady ...
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
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
Szilárd´s daemon
L. SZILÁRD
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
McDonald´s daemon
J. McDONALD
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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
Tμ
j
j
jj
,0
simulation
classical theory
new correlation
—
- - -
PROF. DR.-ING. HABIL. JADRAN VRABEC
THERMODYNAMICS AND ENERGY TECHNOLOGY
DEPARTMENT
OF MECHANICAL
ENGINEERING
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.