Post on 12-Jul-2015
transcript
Nucleation and cavitation ofspherical, cylindrical and slab like
droplets and bubblesSlideshow for an invited seminar at the Condensed Matter Theory Group,
Johannes Gutenberg Universitat Mainz, February 2007.
by
Luis Gonzalez MacDowell
References:
√
MacDowell, Virnau, Muller, Binder, J. Chem. Phys. 120, 5293 (2004).
√
MacDowell, Shen, Errington, J. Chem. Phys. 125, 034705 (2006).Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.1/23
Nucleation and cavitation ofspherical, cylindrical and slab like
droplets and bubbles
Luis González MacDowell1, Vincent Shen2, Jeff Errington3
Peter Virnau4, Marcus Müller4, Kurt Binder4
1. Universidad Complutense de Madrid.
2. National Institute for Standards and Technology.
3. University of New York at Buffalo.
4. Johannes Gutenberg Universität, Mainz.
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.2/23
Subcritical isotherm
−0.1 0.1 0.3 0.5 0.7 0.9 ρ−1.5
−0.5
0.5
1.5
µ
Equilibrium curve
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.3/23
Subcritical isotherm
−0.1 0.1 0.3 0.5 0.7 0.9 ρ−1.5
−0.5
0.5
1.5
µ
‘Metastable’ branch
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.3/23
Subcritical isotherm
−0.1 0.1 0.3 0.5 0.7 0.9 ρ−1.5
−0.5
0.5
1.5
µ
‘Unstable’ branch
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.3/23
Grand Canonical Simulations (µVT)WµV T (N) = −kBT lnP (N)
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.4/23
Grand Canonical Simulations (µVT)WµV T (N) = −kBT lnP (N)
N
WµVT
WµVT
180 N
g l
g
l
g
l
∆ΩVT
∆ΩVT
−(pl−pg)V
b)
c) d)
a)
−(pl−pg)V Nspin
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.4/23
Grand Canonical Simulations (µVT)WµV T (N) = −kBT lnP (N)
N
WµVT
WµVT
180 N
g l
g
l
g
l
∆ΩVT
∆ΩVT
−(pl−pg)V
b)
c) d)
a)
−(pl−pg)V Nspin
Wµ′V T (N) ∝ WµV T (N)− (µ′ − µ)N
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.4/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Chemical potential v density loops
0 0.2 0.4 0.6ρ−0.5
0
0.5
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.5/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
∆A
∆A
R R
ρ<ρ*
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
∆A
∆A
R R
ρ<ρ* ρ=ρ*
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
∆A
∆A
R R
ρ<ρ* ρ=ρ*
ρ>ρ*
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Capillary drop model in a closedsystem
A(Vl) = av[V − Vl] + alVl + γS
∆A
∆A
R R
ρ<ρ* ρ=ρ*
ρ>ρ*ρ>>ρ*
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.6/23
Resulting equation of state
0 2 4 6 8 10ρ0
0.2
0.4
0.6
0.8
1
µ
√
Homogeneous branch forρ < ρt
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.7/23
Resulting equation of state
0 2 4 6 8 10ρ0
0.2
0.4
0.6
0.8
1
µ
√
Homogeneous branch forρ < ρt√
Inhomogeneous branch forρ > ρt
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.7/23
Taking into account the fluctuations
Two states model:√
system is in homogeneous state with weight 1√
system is in inhomogeneous state with weightexp(−β∆A)
〈µ(ρ)〉 =µ(ρ) + µ(ρg)e
−β∆A
1 + e−β∆A
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.8/23
Taking into account the fluctuations
Two states model:√
system is in homogeneous state with weight 1√
system is in inhomogeneous state with weightexp(−β∆A)
〈µ(ρ)〉 =µ(ρ) + µ(ρg)e
−β∆A
1 + e−β∆A
Quantitative description:√
MSA equation of state for the LJ fluid√
Simulation result for the surface tension
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.8/23
Predicting the equation of state
0 0,02 0,04 0,06 0,08 0,1 0,12
ρ-ρc
0
0,2
0,4
0,6
0,8
1
1,2
1,4
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.9/23
Predicting the equation of state
0 0,02 0,04 0,06 0,08 0,1 0,12
ρ-ρc
0
0,2
0,4
0,6
0,8
1
1,2
1,4
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.9/23
Predicting the equation of state
0 0,02 0,04 0,06 0,08 0,1 0,12
ρ-ρc
0
0,2
0,4
0,6
0,8
1
1,2
1,4
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.9/23
Some simulated subcriticalisotherms
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.10/23
Some simulated subcriticalisotherms
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.10/23
Some simulated subcriticalisotherms
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.10/23
Some simulated subcriticalisotherms
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
β∆µ
0 0.2 0.4 0.6 0.8ρ
−1.6
−0.8
0
0.8
1.6
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.10/23
Low temperature isotherm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.11/23
Low temperature isotherm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.11/23
Low temperature isotherm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.11/23
Low temperature isotherm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.11/23
Low temperature isotherm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.11/23
Low temperature isotherm
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.11/23
The Laplace Equation
(
∂Ainh
∂Vl
)
= ∆p− γ(
∂S∂Vl
)
ρV = ρv[V − Vl] + ρlVl + ΓS
Generalization: S = kgV(q−2)/(q−1)l
q Domain kg
4 spherical (36π)1/3
3 cylindrical 2(πL)1/2
2 slab 2L2
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.12/23
Simplified liquid model√
Density increments are linear in the chemicalpotential
√
The fluid is symmetric,χv = χl√
The surface tension is constant√
Adsorption at the surface of tension is negligible
Solution:
χl∆µq −∆ρ∆µq−1 +
(
nkgγ
∆ρncV1/(q−1)
)q−1
= 0
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.13/23
Scaling form of the solutions
x = χl
∆ρ∆µ Kq =(
nkgγχl
∆ρnc∆ρq/(q−1)V 1/(q−1)
)q−1
xq − xq−1 +Kq = 0
∆a = 12
χl
∆ρ2∆AV ω = 1
2(1− x) n = q−2q−1
∆a(ω) = ω2 − ω + 2n−1
n K1−nq ωn
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.14/23
Solutions for different domainshapes
transition transition density
hom → sph ρt = ρcv + 2 · 33/4∆ρc
(
ξsphV
)1/4
hom → cyl ρt = ρcv + 3 · 21/3∆ρc
(
ξcylV
)2/9
hom → slb ρt = ρcv +∆ρc(
ξslbV
)1/6
ξ ∝ γ3χ3v
∆ρ6c
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.15/23
System size features of the isotherm
volume range stable domains observed
Vξsph
> 43
π4 (43)41 hom→ sph→ cyl → slab
43
π4 (43)41 < V
ξsph< π5
27(32)22 hom→ cyl → slab
π5
27(32)22 < V
ξsph< 3427
πhom→ slab
Vξsph
< 3427
πhom
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.16/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Large system, low temperature
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ−1
−0.6
−0.2
0.2
0.6
1
µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.17/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ 0 0.1 0.2 0.3 0.4
ρ−50
−20
10
40
χ−1
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ 0 0.1 0.2 0.3 0.4
ρ−50
−20
10
40
χ−1
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (low T)
0 0.2 0.4 0.6 0.8ρ
−1.6
−1
−0.4
0.2
0.8
1.4
2
β∆µ 0 0.1 0.2 0.3 0.4
ρ−50
−20
10
40
χ−1
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.18/23
Increasing system size (high T)
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.19/23
Increasing system size (high T)
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.19/23
Increasing system size (high T)
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.19/23
Increasing system size (high T)
0.05 0.25 0.45 0.65ρ
−0.2
−0.1
0
0.1
0.2
β∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.19/23
A look at the volume scale
Properties are governed by scaled volumeV/ξ, with
ξ ∝ χ2γ3∆ρ−6c
For the temperature approachingTc:
ξ ∝ |T − Tc|−3ν
ξ1/3 is a meassure of the correlation length
The scaled volume decreases asT approachesTc
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.20/23
Increasing Temperature =Decreasing volume
−0.5 −0.25 0 0.25 0.5(ρ−ρ1/2)/∆ρc
−1
−0.5
0
0.5
1
∆µ/∆
µ s
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.21/23
Increasing Temperature =Decreasing volume
−0.5 −0.25 0 0.25 0.5(ρ−ρ1/2)/∆ρc
−1
−0.5
0
0.5
1
∆µ/∆
µ s
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.21/23
Increasing Temperature =Decreasing volume
−0.5 −0.25 0 0.25 0.5(ρ−ρ1/2)/∆ρc
−1
−0.5
0
0.5
1
∆µ/∆
µ s
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.21/23
Increasing Temperature =Decreasing volume
−0.5 −0.25 0 0.25 0.5(ρ−ρ1/2)/∆ρc
−1
−0.5
0
0.5
1
∆µ/∆
µ s
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.21/23
Approaching infinite system size ...
0 0.2 0.4 0.6 0.8ρ
−2
0
2
∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.22/23
Approaching infinite system size ...
0 0.2 0.4 0.6 0.8ρ
−2
0
2
∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.22/23
Approaching infinite system size ...
0 0.2 0.4 0.6 0.8ρ
−2
0
2
∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.22/23
Approaching infinite system size ...
0 0.2 0.4 0.6 0.8ρ
−2
0
2
∆µ
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.22/23
Approaching infinite system size ...
0 0.2 0.4 0.6 0.8ρ
−2
0
2
∆µ
∆A∗ =∆ρ2cχl
ξsph
(
V
ξsph
)1/2
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.22/23
Conclusions√
Droplet states obey a universal scaling law√
Different sequences of domain transitions occurdepending onV/ξ
√
Small ‘scaled’ systems follow a continuous loopisotherm
√
Stable states are possible inside coexistence loop(for small systems)
√
Apparent spinodal points are small system dewand bubble points
√
Young-Laplace equation (capillary model)provides accurate description
Nucleation and cavitation ofspherical, cylindrical and slab likedroplets and bubbles – p.23/23