Dependence of the surface tension on curvature

Transcription

1 Dependence of the surface tension on curvature rigorously determined from the density profiles of nanodroplets Athens, st September M. T. Horsch,,, S. Eckelsbach, H. Hasse, G. Jackson, E. A. Müller, G. eina, 4 and J. Vrabec TU Kaiserslautern Universität Paderborn Imperial College London 4 Universität Stuttgart

2 VLE at a curved interface Droplet + metastable vapour Δp Spinodal limit: For the external phase, metastability breaks down. st September Martin Thomas Horsch

3 VLE at a curved interface Droplet + metastable vapour Bubble + metastable liquid Δp Spinodal limit: For the external phase, metastability breaks down. Planar limit: The curvature changes its sign and the radius diverges. st September Martin Thomas Horsch

4 Stability in the canonical ensemble free energy / kt p vap / MPa Ar, T = 7 K NpT NVT (large) NVT (small) Unstable and stable phase equilibria CNT, ideal vapour droplet size (atoms) st September Martin Thomas Horsch 4

5 Equilibrium vapour pressure over a droplet Canonical MD simulation of LJTS droplets Down to molecules: Agreement with CNT ( = ). st September Martin Thomas Horsch 5

6 Equilibrium vapour pressure over a droplet Canonical MD simulation of LJTS droplets Down to molecules: Agreement with CNT ( = ). At the spinodal, the results suggest that = / Δp. This implies lim, as conjectured by Tolman (949) st September Martin Thomas Horsch 6

7 Surface tension from molecular simulation surface tension / Integral over the pressure tensor T =.8 /k Virial route (LJTS fluid) T =.7 /k T =.95 /k 4 4 droplet size (molecules) surface tension / εσ - Test area method: Small deformations of the volume virial route test area LJSTS fluid (T =.8 ε/k) equimolar radius / σ Mutually contradicting simulation results! st September Martin Thomas Horsch 7 (Source: Sampayo et al., )

8 The Tolman length characterizes the curvature dependence of the surface tension according to the Tolman equation in terms of and /. It is defined by the deviation = ρ -, between the equimolar radius ρ with ρ ln ln T ρ ρ d ρ ρ d, and the Laplace radius. ρ surface tension / Laboratory of LJTS fluid T =.7 /k capillarity Tolman's scenario =.5 spinodal limit pressure difference / - st September Martin Thomas Horsch 8

9 Analysis of spherical density profiles The Tolman approach is based on the quantities: density / - Equimolar radius ρ (from the density profile).. LJTS fluid T =.75 /k equimolar radius 5 5 distance from the centre of mass / Laboratory of Laplace radius = /Δp of the surface of tension (determined via ) Surface tension as a function of / (which requires ) Without previous knowledge of ( ), this set of variables is inconvenient. st September Martin Thomas Horsch 9

10 Analysis of spherical density profiles The Tolman approach is based on the quantities: density / - Equimolar radius ρ (from the density profile).. LJTS fluid T =.75 /k equimolar radius capillarity radius = /p 5 5 distance from the centre of mass / Laboratory of Laplace radius = /Δp of the surface of tension (determined via ) Surface tension as a function of / (which requires ) Without previous knowledge of ( ), this set of variables is inconvenient. Novel approach: Use Δp instead of /, use κ = /Δp instead of. st September Martin Thomas Horsch

11 The excess equimolar radius st September Martin Thomas Horsch How do these notations relate to each other? lim lim η ρ Δp ρ Δp Tolman theory in ρ,, and / Tolman theory in ρ, κ, and / Tolman length: Tolman equation: First-order expansion: Excess equimolar radius: Tolman equation: First-order expansion: ρ κ ρ η ln ln T ln ln η T O O η

12 Extrapolation to the planar limit adial parity plot Laboratory of Nijmeijer diagram st September Martin Thomas Horsch

13 Conclusion The virial (Irving-Kirkwood) and test area approaches lead to contradicting results for the curvature dependence of. Without knowledge of the surface tension, it is impossible to determine the Laplace radius. In terms of the capillarity radius κ (instead of ) and the pressure difference Δp (instead of / ), Tolman s approach can still be applied. For the LJTS fluid, the planar limit of the Tolman length is smaller in magnitude than σ. This result is consistent with Tolman s scenario ( and ) for the spinodal limit, if is assumed to be curvature independent. st September Martin Thomas Horsch