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