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The size dependence of the vapour -liquid interfacial tension

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 f ür Thermodynamik Imperial College London, Molecular Systems Engineering

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The size dependence of the vapour -liquid interfacial tension

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  1. The sizedependenceofthevapour-liquid interfacialtension M. T. Horsch, G. Jackson, S. V. Lishchuk, E. A. Müller, S. Werth, H. Hasse TU Kaiserslautern, LehrstuhlfürThermodynamikImperial College London, Molecular Systems Engineering NTZ Workshop on Computational PhysicsLeipzig, 30th November 2012

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

  3. Equilibrium vapour pressure of a droplet Canonical MD simulation of LJTS droplets Down to 100 mole-cules: Agreement with CNT (γ = γ0). Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  4. Equilibrium vapour pressure of a droplet Canonical MD simulation of LJTS droplets Down to 100 mole-cules: Agreement with CNT (γ = γ0). At the spinodal, the results suggest that Rγ = 2γ / Δp→ 0. This implies as conjectured by Tolman (1949) … Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  5. Surface tension from molecular simulation Integral over the pressure tensor Test area method: Small deformations of the volume (Source: Sampayo et al., 2010) virial route testarea surface tension / εσ-2 LJSTS fluid (T = 0.8 ε) equimolar radius / σ Mutuallycontradicting simulationresults! Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  6. Analysis of radial density profiles • Thethermodynamicapproach of Tolman (1949) reliesoneffectiveradii: • EquimolarradiusRρ (obtainedfromthedensityprofile) with • Laplace radiusRγ= 2γ/Δp (defined in terms of thesurfacetensionγ) • Sinceγ and Rγ are under dispute, this set of variables isinconvenienthere. T = 0.75 ε Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  7. Analysis of radial density profiles • Various formal dropletradii can be consideredwithinTolman’sapproach: • EquimolarradiusRρ (obtainedfromthedensityprofile) • CapillarityradiusRκ= 2γ∞/Δp (definedbytheplanarsurfacetensionγ∞) • Laplaceradius Rγ= 2γ/Δp (definedbythecurvedsurfacetensionγ) • Thecapillarityradius can be obtainedreliablyfrom molecular simulation. T = 0.75 ε Approach: Use γ/Rγ = Δp/2 instead of1/Rγ, use Rκ = 2γ0/Δpinstead of Rγ. Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  8. The Tolman equation Tolman theory in Rρ, Rγ, and 1/Rγ Tolman length: Tolman equation: First-order expansion: Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  9. The Tolman equation in terms of Rκ Tolman theory in Rρ, Rκ, and γ/Rγ Tolman theory in Rρ, Rγ, and 1/Rγ Tolman length: Tolman equation: First-order expansion: Excess equimolar radius: Tolman equation: First-order expansion: How do these notations relate to each other? Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  10. Extrapolation to the planar limit 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. Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  11. Interpolation to the planar limit Radial parity plot Nijmeijer diagram Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  12. Simulation of planar vapour-liquid interfaces Lennard-Jones fluid rij = rcut surface tension / εσ -2 rik = yk - yi yk yj yi temperature / ε Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  13. Size dependence of liquid slab properties By simulating small liquid slabs, curvature-independent size effects can be considered. 0.8 0.6 local density / σ -3 0.4 T = 0.7 ε 0.2 0 y / σ -6 0 6 As expected, the density in the centre of nanoscopic liquid slabs deviates significantly from that of the bulk liquid at saturation. liquid slab diameter d / σ Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  14. Size dependence of liquid slab properties By simulating small liquid slabs, curvature-independent size effects can be considered. 0.8 0.6 local density / σ -3 0.4 T = 0.7 ε 0.2 0 y / σ -6 0 6 = 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. liquid slab diameter d / σ Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  15. Curvature-independent size effect on γ Surface tension for thin slabs: Relation with γ(R) for droplets? δ0 is small and probably negative: Ghoufi, Malfreyt (2011): δ0 = -0.3 or -0.008 Tröster et al. (2012): -0.27 < δ0 < +0.19 reduced tension γ(d)/γ0 Malijevský & Jackson (2012): δ0 = -0.07 Nonetheless, the surfacetension approaches zero forextremely small droplets. reduced tension γ(R)/γ0 liquid slab diameter d / σ R / σ Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  16. Curvature-independent size effect on γ Surface tension for thin slabs: Relation with γ(R) for droplets? δ0 is small and probably negative: Ghoufi, Malfreyt (2011): δ0 = -0.3 or -0.008 Tröster et al. (2012): -0.27 < δ0 < +0.19 reduced tension γ(d)/γ0 Malijevský & Jackson (2012): δ0 = -0.07 “an additional curvaturedependence of the 1/R3form is required …” reduced tension γ(R)/γ0 liquid slab diameter d / σ Correlation: R / σ Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

  17. Conclusion • Mechanical (virial) and thermodynamic (test areaand grand canonical)routes lead tocontradicting results for thecurvaturedependence of γ. • Without knowledge of the surface tension, it is impossible to determine the Laplace radiusRγ. In terms of the capillarity radiusRκ andthepressure difference Δp(orμ), Tolman’sapproach 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, forextremelysmallliquidphases, thesurfacetensiondecreasesdueto a curvature-independenteffect. Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse

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