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

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• ‘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

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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.

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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

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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

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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

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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

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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

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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)

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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