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Molecular dynamics simulation of nanowetting
TCU-USM Joint Postgraduate Symposium
on Nanotechnology & Nanoscience
December 22, 2016
Tiem Leong Yoon*School of Physics, Universiti Sains Malaysia
*E-mail: tlyoon@usm.my
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Abstract
Results of a series of molecular dynamics (MD) simulation of nano droplets interacting via Lenard-
Jones interaction is reported. The size-dependence of the phases of the nano droplets can be determined
from their radial distribution functions at various temperature. Nanodroplets of various sizes in liquid
form are then equilibrated at a constant temperature on surfaces with various geometries so that the contact
angles, surface tension and line tensions can be numerically measured in these MD experiments.
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Wetting phenomena
• A liquid droplet wets a substrate
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Classical wettability
• Resulted from intermolecular interactions when the liquid and solid surface are brought together
• Addressed by looking at the angle of contact at the edge of the interface between a liquid and a solid
• Macroscopic drop is treated classically
• Drop curve is approximated as a circle
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Young’s equation
• The contact angle or the over-all shape of the simulated liquid drop is roughly explained by the Young’s equation
cos 𝜃𝑌 =𝛾SG − 𝛾SL
𝛾LG
𝛾SG, 𝛾SL and 𝛾LG surface energies between solid-gas, solid-liquid, liquid-gas
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Line tension contribution to contact angle
• Small droplets may exhibit deviation from classical Young equation which ignores the contribution of line tension, 𝜏
cos 𝜃 = cos 𝜃𝑌 −𝜏/𝛾
𝑅• Contact angle of small droplet with line
tension contribution can be derived by minimizing the free energy
𝑑𝐹 = 𝑃𝑑𝑉 +
𝑖
𝛾𝑖𝑑𝑆𝑖 + 𝜏𝑑𝐿6
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Magnitude of line tension
• Estimated order ~ 10−12 − 10−10J/m
• Inferred by measuring the contact angle of droplet
• Due to the small scale, it is difficult to measure experimentally
• Experimental values vary from 10−5 − 10−12J/m
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Significance of line tension
In the nanoscale limit:
• Line tension contributes significantly to the determination of the morphology of a droplet
• Line tension varies with curvature (and hence size of the nanodroplet)
• Contact angle becomes a strong function of the system size
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Issues in theoretical modelling
• Theoretical modelling of nanodroplets with statistical mechanics approach of minimizing free energy may suffer complication due to
1. Granulality of the solid surface
2. Geometry of realistic droplets-substrate deviates from simple symmetric shape
3. Non-trivial structure on the surface
Nano-effect9
𝑑𝐹 = 𝑃𝑑𝑉 +
𝑖
𝛾𝑖𝑑𝑆𝑖 + 𝜏𝑑𝐿
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Molecular dynamics simulation of droplets
• MD – an alternative approach to predict the interaction of a generic droplet’s with a generic surface
• ‘Computational experiment’
• Do not need simplified assumptions made in theoretical modelling e.g., the assumed contributions of various terms in construction of the free energy and symmetrical geometries
• Compliment theoretical and experimental investigation of nanowetting phenomena
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Literature review: Early simulations
• Maruyama, S., Matsumoto, S. and Ogita, A., "Surface Phenomena of Molecular Clusters by Molecular Dynamics Method," Therm. Sci. Eng., 2-1 (1994), 77-84.
• Matsumoto, S., Maruyama, S. and Saruwatari, H., "A Molecular Dynamics Simulation of a Liquid Droplet on a Solid Surface, " Proc. ASME/JSME Therm. Eng. Conf.,Maui, (1995), 557-562.
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Simulation of realistic systems
• Water on platinum
• Spohr & Heinzinger; Zhu & Philpott potential
• Kandlikar, S. G., Maruyama, S., Steinke, M. E.
and Kimura, T., "Measurement and Molecular
Dynamics Simulation of Contact Angle of
Water Droplet on a Platinum Surface," HTD
(Proc. ASME Heat Transfer Division 2001),
369-1 (2001), 343-348.
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Simulation of realistic systems
• Water on graphite surface with specific surface orientation
• SPC/Fw potential
• Danilo Sergi, Giulio Scocchi, Alberto Ortona,
Molecular dynamics simulations of the contact
angle between water droplets and graphite
surfaces, Fluid Phase Equilibria 332 (2012)
173– 177.
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Forcefield for water
• TIP4 forcefield
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LJ forcefield
• Lennard-Jones liquid with solid surface: 6-12 potential
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Typical density profile
• A common 2D density profile of a MD droplet at equilibrium
16
• ‘Layering’ in the density profile closed close to
the substrate (oscillates as a function of height)
Weijs et al., Origin of line tension for a Lennard-Jones nanodroplet, PHYSICS OF FLUIDS 23, 022001 (2011)
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Sample MD profile from the literature
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Variation of the contact angle with 𝜖
18B. Shi and V. K. Dhir J. Chem. Phys. 130, 034705 2009
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More realistic scenario
19Erik E. Santiso 1,2, Carmelo Herdes 1 and Erich A. MüllerEntropy 2013, 15, 3734-3745
(a) 63.77° (b) 60.52° (c) 64.56° (d) 54.93°.
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Contact angles in graphite-water system
• Both experimental results and numerical simulations
20
Erik E. Santiso, Carmelo Herdes and Erich A. Müller, On the Calculation of Solid-Fluid Contact Angles from Molecular Dynamics, Entropy 2013, 15, 3734-3745
Large uncertainty in determining the contact angle in MD simulations
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Difficulties in MD simulations
• Boundary of droplet is too fuzzy, asymmetrical, and fluctuating spatially and temporally
• Contact angles may vary over ~10°
• Surface roughness and energetic heterogeneities
• “layering effect” inside the liquid
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Collapsing contours for precise definition of shape boundary
• Ignore the ‘layering’ part at the bottom
• Different isodensity contour gives different contact angle
• Isodensity contours are collapsed into a unique shape by fitting the
contour against 𝜌 =1
2[1 + tanh(
𝑅0−𝑟
𝑤)], allowing precise definition of the
curve boundary.
22
Weijs et al., Origin of line tension for a Lennard-Jones nanodroplet, PHYSICS OF FLUIDS 23, 022001 (2011)
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Constructing number density 𝜌(𝑟) at fixed 𝑧
23
Dzr
z
z
𝑑𝑉 = Δ𝑧(𝜋𝑟2)𝜌 𝑟 = 𝑑𝑁𝑑𝑉
count 𝑑𝑁 from MD
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Reconstructing the crossectional surface profile
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Determine boundary 𝑅drop at a fixed z
A sharp surface profile is reconstructed
Contact angle is easily measured at 𝑧 = 0
𝜌0 = 0.2
Fit it to a parabola
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Measuring line tension from MD
• Weijs et al., Origin of line tension for a Lennard-Jones nanodroplet, PHYSICS OF FLUIDS 23, 022001 (2011)
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Procedure
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cos 𝜃 = cos 𝜃𝑌 −𝜏/𝛾
𝑅cos 𝜃𝑌 =
𝛾SG − 𝛾SL𝛾LG
1. Fix a value of 𝜖 in the LJ potential. That fixes the surface energies, hence 𝜃𝑌.
2. Measure 𝜃 for various radius 𝑅 in the 2D constructions (and expect 𝜃 to be independent of R)
3. Measure 𝜃 for various radius 𝑅 in the 3D constructions (and expect 𝜃 to be dependent of R)
4. Plot cos 𝜃 vs.1
𝑅for both 1 and 2.
5. Differences in the slop of both plots gives −𝜏/𝛾
𝑅
W e l e a dcos 𝜃 vs. 𝑅−1 for 𝜃𝑌 = 127°
• ℓ = 0.36𝜎
27
Dotted points - MD Solid lines - DFT
The difference between the slopes of the 2D and 3D fits quantifies the tension length (ℓ = 0.36𝜎 In this case)
Diamond: Young’s law measured independently with MD from surface tensions
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Droplet on infinitely flat surface
• From existing literature, there are many MD simulations of nanodroplets on flat infinite surfaces
• Contact angle can be measured in these simulations by appropriately sampling and statistically post-processing the MD.
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Droplet on structured surfaces
• MD experiments can be designed to measure contact angle and line tension for:
• droplet size at any 𝑅
• at any temperature 𝑇
• at any 𝜖
• on surfaces with any geometry
• on surfaces with designed structures / roughness/corrugations
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MD examples
• Switching behavior in the wettability of a droplet from complete wetting to complete drying states induced by nanostructures e.g., gold (111) surface carved out with spherical cavities
(Zhang et. al., Langmuir 2016 32 (37), 9658-9663)
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MD examples
31
Cassie-Baxter
(CB) stateWenzel (WZ)
state
droplet: 13.2 nm
Cavities: 6 nm
droplet: 13.2 nm; cavities: 6 nm
81,170 water molecules; 149,000 gold atoms
30 fs time step; 12 ns duration simulated300 K temperaturewater as LJ particles
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MD results from Zhang et. al., Langmuir 2016
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Nanodroplet on a curve surface, with line tension
33
Masao Iwamatsu, Size-dependent contact angle and the wetting and
drying transition of a droplet adsorbed onto a spherical substrate: Line-
tension effect, PHYSICAL REVIEW E 94, 042803 (2016)
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Non-trivial predictions from thermodynamics theory by Iwamatsu
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Measurable via MD simulations
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MD simulation of droplet on curve surface
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Maheshwari, S., van derHoef, M., Lohse, D., Line Tension and Wettability of Nanodrops on Curved Surfaces Langmuir 32, 316 (2016)
• Nanodrops on curved surfaces • Lennard-Jones particles• Change in curvature of the
drop in response to the change in curvature of the substrate
• Measure line tension for different size drops on convex and concave surfaces
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LJ Droplet on a solid column
36
MD experiment to vary
1.𝑟𝑏
𝑟𝑐
2. 𝜖𝑏𝑐3. T
𝑟𝑏
2𝑟𝑐
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Constructing initial MD input configuration
• Cuprum column
• cut out from a fcc lattice with lattice
parameter 3.61 Å
• 𝜎Cu−Cu = 2.27Å
• 𝜖Cu−Cu = 0.583 eV
• Equilibrate at 83 K
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Constructing initial MD input configuration
• LJ ball (Ar)
• Cut out from a fcc lattice with lattice parameter 5.260 Å
• mass = 39.948 a.u
• 𝜎Ar−Ar = 3.345 Å
• 𝜖Ar−Ar/kB = 125.7 K
• 𝜎Ar−Cu = 3.345 Å
• 𝜖Ar−Cu = 0.077 eV
• Equilibrate at various temperatures
3850 K, ~30 nm 83 K, ~30 nm 150 K, ~30 nm
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Determination of phase transition temperatures from radial distribution function
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Solid-liquid transition between 50 – 60 K; Liquid-gaseous transition between 87 – 100 K
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Phase transition temperatures
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Sample MD out
• LJ ball size: ~30 nm
• Copper column size ~19 nm
• timestep = 1 fs
• Equilibrated at 83 K for 5 × 106 steps (= 5 ns)
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Sample MD out 𝜖Cu−Ar = 10 × (0.077 eV)
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Sample MD out 𝜖Cu−Ar = 4 × (0.077 eV)
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Sample MD out 𝜖Cu−Ar = 2 × (0.077 eV)
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Cross section of averaged density profilefor 𝜖Cu−Ar = 10 × (0.077 eV)
45
Work still in progress
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Conclusion
• MD simulation can be quite conveniently used to verify wetting phenomenology of nonodropletson surfaces
• Line tension can be measured by measuring contact angles for different droplet sizes (both 3D and 2D )
• Can simulate nanoscale effects without simplified assumptions as required by thermodynamical models
• Surface with roughness, orientations, structure
• Surfaces with curved or arbitrary geometry 46
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