The size dependence of the vapour -liquid interfacial tension

LTDLehrstuhl für ThermodynamikProf. Dr.-Ing. H. Hasse

The size dependence of the

vapour-liquid interfacial tension

M. T. Horsch, G. Jackson, S. V. Lishchuk, E. A. Müller, S. Werth, H.

Hasse

TU Kaiserslautern, Lehrstuhl für Thermodynamik

Imperial College London, Molecular Systems Engineering

NTZ Workshop on Computational Physics

Leipzig, 30th November 2012


230th November 12

Curvature and fluid phase equilibria• Droplet + metastable vapour• Bubble + metastable liquid

Spinodal limit: For the external phase, metastability breaks down.

Planar limit: The curvature changes its sign and the radius Rγ diverges.

γR

γpΔ

2

Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse


Equilibrium vapour pressure of a droplet

330th November 12

Canonical MD simulation of LJTS droplets

Down to 100 mole-cules: Agreement with CNT (γ = γ0).



Equilibrium vapour pressure of a droplet

430th November 12

Canonical MD simulation of LJTS droplets

Down to 100 mole-cules: Agreement with CNT (γ = γ0).

At the spinodal, the results suggest thatRγ = 2γ / Δp → 0. This implies

as conjectured by Tolman (1949) …

,0lim0

γγR



Surface tension from molecular simulation

530th November 12

102 103 104

surface tension /

-2

0.0

0.2

0.4

0.6

102 103 104

droplet size (molecules)

T = 0.7

0

Virial route(LJTS fluid)

0

0

T = 0.8

T = 0.95

Integral over the pressure tensor

(Source: S

ampayo et al.,

2010)

equimolar radius / σ

Test area method:Small deformations of the volume

Mutually contradictingsimulation results!

virial route test area

surf

ace

ten

sio

n /

εσ -

2LJSTS fluid (T = 0.8 ε)



Analysis of radial density profiles

630th November 12

The thermodynamic approach of Tolman (1949) relies on effective radii:

• Equimolar radius Rρ (obtained from the density profile) with

• Laplace radius Rγ = 2γ/Δp (defined in terms of the surface tension γ)

Since γ and Rγ are under dispute, this set of variables is inconvenient here.

0)( )( 2

0

2

ρ

ρ

R

RρRρRdRρRρRdRΓ

distance from the centre of mass / 0 5 10 15 20

dens

ity /

-3

0.01

0.1

1

T = 0.75 /k

equimolar radius R

LJTS fluidT = 0.75 ε



Analysis of radial density profiles

730th November 12

Various formal droplet radii can be considered within Tolman’s approach:

• Equimolar radius Rρ (obtained from the density profile)

• Capillarity radius Rκ = 2γ∞/Δp (defined by the planar surface tension γ∞)

• Laplaceradius Rγ = 2γ/Δp (defined by the curved surface tension γ)

The capillarity radius can be obtained reliably from molecular simulation.

Approach: Use γ/Rγ = Δp/2 instead of 1/Rγ, use Rκ = 2γ0/Δp instead of Rγ.

distance from the centre of mass / 0 5 10 15 20

dens

ity /

-3

0.01

0.1

1

T = 0.75 /k

equimolar radius Rcapillarity radius

R = 20/pLJTS fluidT = 0.75 ε



The Tolman equation

830th November 12

Tolman theory in Rρ, Rγ, and 1/Rγ

Tolman length:

Tolman equation:

First-order expansion:

γρ RRδ

1

3

3

2

2

3

2221

ln

ln

γγγT

γ

R

δ

R

δ

R

δ

γ

R

2000

112

γγ RO

Rγδγγ



The Tolman equation in terms of Rκ

930th November 12

How do these notations relate to each other?

01010

00 limlim δ

Rγ

γR

Rγ

γRη

γρ

pΔγ

ρpΔ

Tolman theory in Rρ, Rγ, and 1/Rγ Tolman theory in Rρ, Rκ, and γ/Rγ

Tolman length:

Tolman equation:


Excess equimolar radius:

Tolman equation:


γρ RRδ κρ RRη

1

3

3

2

2

3

2221

ln

ln

γγγT

γ

R

δ

R

δ

R

δ

γ

R13

011

12

3

ln

ln

γ

γRγη

γ

Rγ γ

T

γ

2000

112

γγ RO

Rγδγγ

2

2

00 2γγ R

γO

R

γηγγ



Extrapolation to the planar limit

1030th November 12

Radial parity plot

• The magnitude of the excess equimolar radius is consistently found to be smaller than σ / 2.

• This suggests that the curvature dependence of γ is weak, i.e. that the deviation from γ∞ is smaller than 10 % for radii larger than 10 σ.

• This contradicts the results from the virial route and confirms the grand canonical and test area simulations.



Interpolation to the planar limit

1130th November 12

Nijmeijer diagramRadial parity plot



Simulation of planar vapour-liquid interfaces

1230th November 12

temperature / ε

surf

ace

ten

sio

n /

εσ -

2

Lennard-Jones fluid

rij = rcut

rik = yk - yi

10 4

2 12

5LRCij ij k

ij ij

u r y yr r

yi ykyj

1.23

0 2c

( ) 2.94 1T

TT



Size dependence of liquid slab properties

1330th November 12

As expected, the density in the centre of nanoscopic liquid slabs deviates significantly from that of the bulk liquid at saturation.

By simulating small liquid slabs, curvature-independent size effects can be considered.

-6 0 6

liquid slab diameter d / σ

loca

l den

sity

/ σ

-3

0

0.2

0.4

0.6

0.8

y / σ( 0)

( )

y

T

T = 0.7 ε



Size dependence of liquid slab properties

1430th November 12

By simulating small liquid slabs, curvature-independent size effects can be considered.

-6 0 6


loca

l den

sity

/ σ

-3

0

0.2

0.4

0.6

0.8

y / σ( 0)

( )

y

T

T = 0.7 ε

= 1 – a(T)d -3


As expected, the density in the centre of nanoscopic liquid slabs deviates significantly from that of the bulk liquid at saturation.


Curvature-independent size effect on γ

1530th November 12

Relation with γ(R) for droplets?

δ0 is small and probably negative:

Ghoufi, Malfreyt (2011): δ0 = -0.3 or -0.008Tröster et al. (2012): -0.27 < δ0 < +0.19

Malijevský & Jackson (2012):

δ0 = -0.07

Nonetheless, the surfacetension approaches zero for

extremely small droplets.

R / σ

redu

ced

ten

sio

n γ

(R)/

γ 0


redu

ced

ten

sio

n γ

(d)/

γ 0

Surface tension for thin slabs:



Curvature-independent size effect on γ

1630th November 12

Relation with γ(R) for droplets?

δ0 is small and probably negative:

Ghoufi, Malfreyt (2011): δ0 = -0.3 or -0.008Tröster et al. (2012): -0.27 < δ0 < +0.19

Malijevský & Jackson (2012):

δ0 = -0.07

“an additional curvaturedependence of the 1/R3

form is required …”

R / σ

redu

ced

ten

sio

n γ

(R)/

γ 0

30

( , ) ( )1

( )

d T b T

T d


redu

ced

ten

sio

n γ

(d)/

γ 0

Correlation:

Surface tension for thin slabs:



Conclusion

1730th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

• Mechanical (virial) and thermodynamic (test area and grand canonical) routes lead to contradicting results for the curvature dependence of γ.

• Without knowledge of the surface tension, it is impossible to determine the Laplace radius Rγ. In terms of the capillarity radius Rκ and the pressure difference Δp (or μ), Tolman’s approach can still be applied.

• Results for the excess equimolar radius confirm the thermodynamic routes to the surface tension: In the planar limit, the Tolman length is small (and negative, according to the most recent literature).

• However, for extremely small liquid phases, the surface tension decreases due to a curvature-independent effect.

The size dependence of the vapour -liquid interfacial tension

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