LTD Lehrstuhl fr Thermodynamik Prof Dr Ing H

LTD Lehrstuhl für Thermodynamik Prof. 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, 30 th November 2012

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse 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. 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 2

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Equilibrium vapour pressure of a droplet Canonical MD simulation of LJTS droplets Down to 100 molecules: Agreement with CNT (γ = γ 0). 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 3

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Equilibrium vapour pressure of a droplet Canonical MD simulation of LJTS droplets Down to 100 molecules: Agreement with CNT (γ = γ 0). At the spinodal, the results suggest that Rγ = 2γ / Δp → 0. This implies as conjectured by Tolman (1949) … 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 4

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Surface tension from molecular simulation Test area method: Small deformations of the volume virial route test area LJSTS fluid (T = 0. 8 ε) (Source: Sampayo et al. , 2010) surface tension / εσ -2 Integral over the pressure tensor equimolar radius / σ Mutually contradicting simulation results! 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 5

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Analysis of radial density profiles 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. T = 0. 75 ε 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 6

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Analysis of radial density profiles 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. T = 0. 75 ε Approach: Use γ/Rγ = Δp/2 instead of 1/Rγ, use Rκ = 2γ 0/Δp instead of Rγ. 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 7

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse The Tolman equation Tolman theory in Rρ, Rγ, and 1/Rγ Tolman length: Tolman equation: First-order expansion: 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 8

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse The Tolman equation in terms of Rκ Tolman theory in Rρ, Rγ, and 1/Rγ Tolman theory in Rρ, Rκ, and γ/Rγ Tolman length: Excess equimolar radius: Tolman equation: First-order expansion: How do these notations relate to each other? 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 9

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse 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. 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 10

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Interpolation to the planar limit Radial parity plot 30 th November 12 Nijmeijer diagram Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 11

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Simulation of planar vapour-liquid interfaces surface tension / εσ -2 Lennard-Jones fluid rij = rcut rik = yk - yi yj 30 th November 12 yi yk temperature / ε Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 12

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Size dependence of liquid slab properties By simulating small liquid slabs, curvature-independent size effects can be considered. local density / σ -3 0. 8 0. 6 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. 30 th November 12 liquid slab diameter d / σ Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 13

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Size dependence of liquid slab properties By simulating small liquid slabs, curvature-independent size effects can be considered. local density / σ -3 0. 8 0. 6 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. 30 th November 12 = 1 – a(T)d -3 liquid slab diameter d / σ Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 14

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Curvature-independent size effect on γ Surface tension for thin slabs: Relation with γ(R) for droplets? reduced tension γ(d)/γ 0 δ 0 is small and probably negative: liquid slab diameter d / σ reduced tension γ(R)/γ 0 Ghoufi, Malfreyt (2011): δ 0 = -0. 3 or -0. 008 Tröster et al. (2012): -0. 27 < δ 0 < +0. 19 Malijevský & Jackson (2012): δ 0 = -0. 07 Nonetheless, the surface tension approaches zero for extremely small droplets. R/σ 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 15

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Curvature-independent size effect on γ Surface tension for thin slabs: Relation with γ(R) for droplets? reduced tension γ(d)/γ 0 δ 0 is small and probably negative: liquid slab diameter d / σ Correlation: reduced tension γ(R)/γ 0 Ghoufi, Malfreyt (2011): δ 0 = -0. 3 or -0. 008 Tröster et al. (2012): -0. 27 < δ 0 < +0. 19 Malijevský & Jackson (2012): δ 0 = -0. 07 “an additional curvature dependence of the 1/R 3 form is required …” R/σ 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 16

LTD Lehrstuhl für Thermodynamik Prof. Dr. -Ing. H. Hasse Conclusion • 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 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. 30 th November 12 Martin Horsch, George Jackson, Sergey Lishchuk, Erich Müller, Stephan Werth, Hans Hasse 17