Droplets and bubbles are thought to be two sides of the same coin; this work determines how true this is at the molecular scale. Stable cylindrical nanodroplets and nanobubbles are obtained in Molecular Dynamics (MD) simulations with three-phase contact lines pinned by alternate hydrophobic and hydrophilic patterns. The surface tension and Tolman length for both types of curved interfaces are obtained with the Kirkwood–Buff method, based on the difference between normal and tangential pressure components. Both bubble and droplet cases are compared to the flat interface case for reference. Results show that the surface tension decreases linearly while the Tolman length increases linearly with the gas/liquid density ratio. By running a careful parameter study of the flat interface over a range of densities, the effect of the density ratio can be corrected isolating the effects of curvature on the surface tension and Tolman length. It is found that such effects start to be seen when the equimolar curvature radius goes down to 20 reduced Lennard–Jones (LJ) units. They have the same magnitude but act with opposite signs for nanodroplet and nanobubble interfaces. Considering effects of the density ratio and curvature, a fitted Tolman equation was obtained, which predicts the surface tension of a curved interface. Results obtained by the fitted Tolman equation agree well with those obtained by the MD simulations except at very small curvature radius (<10 reduced LJ units) due to the accumulation of the curvature dependence of the Tolman length.
References
1.
T.
Ito
, H.
Lhuissier
, S.
Wildeman
, and D.
Lohse
, “Vapor bubble nucleation by rubbing surfaces: Molecular dynamics simulations
,” Phys. Fluids
26
, 0032003
(2014
).2.
S.
Prestipino
, A.
Laio
, and E.
Tosatti
, “Shape and area fluctuation effects on nucleation theory
,” J. Chem. Phys.
140
, 094501
(2014
).3.
M.
Foroutan
, F.
Esmaeilian
, and T.
Rad Morteza
, “The change in the wetting regime of a nanodroplet on a substrate with varying wettability: A molecular dynamics investigation
,” Phys. Fluids
33
, 032017
(2021
).4.
X.
Wu
, N.
Phan-Thien
, X.-J.
Fan
, and T. Y.
Ng
, “A molecular dynamics study of drop spreading on a solid surface
,” Phys. Fluids
15
, 1357
(2003
).5.
E. R.
Smith
, P. E.
Theodorakis
, R. V.
Craster
, and O. K.
Matar
, “Moving contact lines: Linking molecular dynamics and continuum-scale modeling
,” Langmuir
34
, 12501
(2018
).6.
W.
Ren
and E.
Weinan
, “Boundary conditions for the moving contact line problem
,” Phys. Fluids
19
, 022101
(2007
).7.
J. S.
Rowlinson
, “Statistical thermodynamics of small systems and interfaces
,” Pure Appl. Chem.
59
, 15
–24
(1987
).8.
Y. A.
Lei
, T.
Bykov
, S.
Yoo
, and X. C.
Zeng
, “The Tolman length: Is it positive or negative?
,” J. Am. Chem. Soc.
127
, 15346
(2005
).9.
H. M.
Lu
and Q.
Jiang
, “Size-dependent surface tension and Tolman's length of droplets
,” Langmuir
21
, 779
(2005
).10.
A. E.
van Giessen
and E. M.
Blokhuis
, “Direct determination of the Tolman length from the bulk pressures of liquid drops via molecular dynamics simulations
,” J. Chem. Phys.
131
, 164705
(2009
).11.
Ø.
Wilhelmsen
, D.
Bedeaux
, and D.
Reguera
, “Tolman length and rigidity constants of the Lennard-Jones fluid
,” J. Chem. Phys.
142
, 064706
(2015
).12.
T.
Young
, “An essay on the cohesion of fluids
,” Philos. Trans. R. Soc. London
1
, 171
(1832
).13.
R. C.
Tolman
, “The effect of droplet size on surface tension
,” J. Chem. Phys.
17
, 333
(1949
).14.
R. C.
Tolman
, “The superficial density of matter at a liquid‐vapor boundary
,” J. Chem. Phys.
17
, 118
(1949
).15.
T. V.
Bykov
and X. C.
Zeng
, “A patching model for surface tension and the Tolman length
,” J. Chem. Phys.
111
, 3705
(1999
).16.
A. A.
Onischuk
, P. A.
Purtov
, A. M.
Baklanov
, V. V.
Karasev
, and S. V.
Vosel
, “Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: Metal vapor homogeneous nucleation
,” J. Chem. Phys.
124
, 014506
(2006
).17.
Y.-Q.
Xue
, X.-C.
Yang
, Z.-X.
Cui
, and W.-P.
Lai
, “The effect of microdroplet size on the surface tension and Tolman length
,” J. Phys. Chem. B
115
, 109
(2011
).18.
N.
Bruot
and F.
Caupin
, “Curvature dependence of the liquid-vapor surface tension beyond the Tolman approximation
,” Phys. Rev. Lett.
116
, 056102
(2016
).19.
P.
Montero de Hijes
, J. R.
Espinosa
, V.
Bianco
, E.
Sanz
, and C.
Vega
, “Interfacial free energy and Tolman length of curved liquid–solid interfaces from equilibrium studies
,” J. Phys. Chem. C
124
, 8795
(2020
).20.
S. S.
Rekhviashvili
, E. V.
Kishtikova
, R. Y.
Karmokova
, and A. M.
Karmokov
, “Calculating the Tolman constant
,” Tech. Phys. Lett.
33
, 48
(2007
).21.
P.
Rehner
and J.
Gross
, “Surface tension of droplets and Tolman lengths of real substances and mixtures from density functional theory
,” J. Chem. Phys.
148
, 164703
(2018
).22.
P.
Rehner
, A.
Aasen
, and I.
Wilhelmsen
, “Tolman lengths and rigidity constants from free-energy functionals: General expressions and comparison of theories
,” J. Chem. Phys.
151
, 244710
(2019
).23.
S.
Burian
, M.
Isaiev
, K.
Termentzidis
, V.
Sysoev
, and L.
Bulavin
, “Size dependence of the surface tension of a free surface of an isotropic fluid
,” Phys. Rev. E
95
, 062801
(2017
).24.
F.
Calvo
, “Molecular dynamics determination of the surface tension of silver-gold liquid alloys and the Tolman length of nanoalloys
,” J. Chem. Phys.
136
, 154701
(2012
).25.
A.
Malijevský
and G.
Jackson
, “A perspective on the interfacial properties of nanoscopic liquid drops
,” J. Phys.
24
, 464121
(2012
).26.
A.
Ghoufi
, P.
Malfreyt
, and D. J.
Tildesley
, “Computer modelling of the surface tension of the gas–liquid and liquid–liquid interface
,” Chem. Soc. Rev.
45
, 1387
(2016
).27.
G.
Jiménez-Serratos
, C.
Vega
, and A.
Gil-Villegas
, “Evaluation of the pressure tensor and surface tension for molecular fluids with discontinuous potentials using the volume perturbation method
,” J. Chem. Phys.
137
, 204104
(2012
).28.
S.
Paliwal
, V.
Prymidis
, L.
Filion
, and M.
Dijkstra
, “Non-equilibrium surface tension of the vapour-liquid interface of active Lennard-Jones particles
,” J. Chem. Phys.
147
, 084902
(2017
).29.
S.
Masuda
and S.
Sawada
, “Molecular dynamics study of size effect on surface tension of metal droplets
,” Eur. Phys. J. D
61
, 637
(2011
).30.
S. H.
Park
, J. G.
Weng
, and C. L.
Tien
, “A molecular dynamics study on surface tension of microbubbles
,” Int. J. Heat Mass Transfer
44
, 1849
(2001
).31.
R.
Lovett
and M.
Baus
, “A molecular theory of the Laplace relation and of the local forces in a curved interface
,” J. Chem. Phys.
106
, 635
(1997
).32.
J. W.
Gibbs
, The Collected Works of J. Willard Gibbs
(Longmans, Green, and Co.
, New York
, 1928
), Vol. 1
.33.
F. O.
Koenig
, “On the thermodynamic relation between surface tension and curvature
,” J. Chem. Phys.
18
, 449
(1950
).34.
S.
Plimpton
, “Fast parallel algorithms for short-range molecular-dynamics
,” J. Comput. Phys.
117
, 30940
(1995
).35.
K. P.
Travis
, B. D.
Todd
, and D. J.
Evans
, “Departure from Navier-Stokes hydrodynamics in confined liquids
,” Phys. Rev. E
55
, 4288
(1997
).36.
S.
Maheshwari
, M.
van der Hoef
, X.
Zhang
, and D.
Lohse
, “Stability of surface nanobubbles: A molecular dynamics study
,” Langmuir
32
, 11116
(2016
).37.
F.
Varnik
, J.
Baschnagel
, and K.
Binder
, “Molecular dynamics results on the pressure tensor of polymer films
,” J. Chem. Phys.
113
, 4444
(2000
).38.
D. M.
Heyes
, E. R.
Smith
, D.
Dini
, and T. A.
Zaki
, “The equivalence between volume averaging and method of planes definitions of the pressure tensor at a plane
,” J. Chem. Phys.
135
, 024512
(2011
).39.
B. D.
Todd
, D. J.
Evans
, and P. J.
Daivis
, “Pressure tensor for inhomogeneous fluids
,” Phys. Rev. E
52
, 1627
(1995
).40.
41.
M. P.
Moody
and P.
Attard
, “Curvature-dependent surface tension of a growing droplet
,” Phys. Rev. Lett.
91
, 056104
(2003
).42.
J. H.
Weijs
and D.
Lohse
, “Why surface nanobubbles live for hours
,” Phys. Rev. Lett.
110
, 054501
(2013
).43.
D.
Lohse
and X.
Zhang
, “Surface nanobubbles and nanodroplets
,” Rev. Mod. Phys.
87
, 981
(2015
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.