Subscriber access provided by LONDON METROPOLITAN UNIV
Article
Multiscale Modeling of the Three-Dimensional Meniscus Shape of a Wetting Liquid Film on Micro-/Nano-Structured Surfaces Han Hu, Monojit Chakraborty, Taylor Allred, Justin A. Weibel, and Suresh V. Garimella Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02837 • Publication Date (Web): 27 Sep 2017 Downloaded from http://pubs.acs.org on September 30, 2017
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Multiscale Modeling of the Three-Dimensional Meniscus Shape of a Wetting Liquid Film on Micro-/Nano-Structured Surfaces Han Hu, Monojit Chakraborty, Taylor P. Allred, Justin A. Weibel1, and Suresh V. Garimella1 School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA
ABSTRACT The design of structured surfaces for increasing the heat flux dissipated during boiling and evaporation processes via enhanced liquid rewetting requires prediction of the liquid meniscus shape on these surfaces. In this study, a general continuum model is developed to predict the three-dimensional meniscus shape of liquid films on micro-/nano-structured surfaces based on a minimization of the system free energy that includes solid–liquid van der Waals interaction energy, surface energy, and gravitational potential. The continuum model is validated at the nanoscale against molecular dynamics simulations of water films on gold surfaces with pyramidal indentations, and against experimental measurements of water films on silicon Vgroove channels at the microscale. The validated model is used to investigate the effect of film thickness and surface structure depth on the meniscus shape. The meniscus is shown to become more conformal with the surface structure as the film thickness decreases and the structure depth increases. Assuming small interface slope and small variation in film thickness, the continuum model can be linearized to obtain an explicit expression for the meniscus shape. The error of this linearized model is quantitatively assessed and shown to increase with increasing structure depth and decreasing structure pitch. The model developed can be used for accurate prediction of three-dimensional meniscus shape on structured surfaces with micro-/nano-scale features, which is necessary for determining the liquid delivery rate and heat flux dissipated during thin-film evaporation. The linearized model is useful for rapid prediction of meniscus shape when the structure depth is smaller than or comparable to the liquid film thickness.
KEYWORDS: multiscale modeling, meniscus shape, thin liquid films, micro-/nano-structured surfaces, disjoining pressure
1
Corresponding author. E-mail:
[email protected] (J.A. Weibel),
[email protected] (S.V.
Garimella)
ACS Paragon Plus Environment
1
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 2 of 25
INTRODUCTION Thin liquid menisci are ubiquitous in nature and play a critical role in various physical processes, including DNA linearization,1,
2
metal casting,3,
4
dip-pen nanolithography,5
convective self-assembly,6 thin-film evaporation7-9, and nucleate boiling10, among others. In particular, the critical heat flux (CHF) during nucleate boiling is limited by the liquid delivery to thin evaporating films on structured surfaces. Although nucleate boiling is a highly nonequilibrium process, the equilibrium meniscus shape plays a critical role in the heat transfer performance as it determines the disjoining pressure and capillary pressure that are the driving forces for this liquid delivery. In recent years, micro- and nano-structured surfaces have been used to improve boiling CHF by maintaining the wetting of the surface at high heat fluxes via enhanced rates of liquid delivery to dry patches.10-17 For example, Rahman et al.
15
fabricated structured surfaces offering a range
of liquid delivery rates and showed that the CHF in pool boiling increases proportionally with the liquid delivery rate. Dhillon et al.
16
studied pool boiling on micro-/nano-structured surfaces
and developed a thermal-hydraulic model to relate CHF enhancement to liquid delivery. Shim et al. 17 investigated nucleate boiling on aligned and random nanowire arrays and observed a linear correlation between the boiling CHF and a volumetric wicking rate. However, these empirical studies have not explained the relationship between the surface properties and the liquid delivery rate, preventing the rational design of surfaces for enhanced boiling heat transfer performance. In order to design optimized micro-/nano-structures for enhancement of the liquid delivery rate and boiling CHF, the three-dimensional meniscus shape on structured surfaces must be accurately predicted. Various experimental techniques have been developed to measure the meniscus shape of thin liquid films. Interference reflection microscopy (IRM), a mature characterization technique, is capable of measuring the meniscus shape with a resolution down to several nanometers in the thickness direction and tens of nanometers in the lateral dimensions. It has been widely used to study the liquid meniscus shape of an evaporating corner meniscus,18 during thin-film wicking in a square pillar array,19 of the extended meniscus under electrowetting,20 and of thin water films between an air bubble and hydrophobic surfaces.21 Recent studies have integrated IRM with other techniques such as optical microcopy and total internal reflection microscopy (TIRM). Zhang et al.
22
developed the Interferometry Digital Imaging Optical Microscopy (IDIOM)
ACS Paragon Plus Environment
2
Page 3 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
method that integrated IRM with optical microscopy to visualize and characterize the size, shape, and evolution kinetics of nanoscale interfacial features in free-standing films. Pack et al. 23 used a combination of TIRM and IRM to further enhance the resolution of the measurement in the thickness direction. Atomic force microscopy (AFM) has also been used to measure the meniscus shape. Checco et al. 24, 25 measured the meniscus shape of thin organic liquid films (~5 nm thick ethanol and octane films) on a nonwettable surface with wettable nanostripes using amplitude modulation atomic force microscopy (AM-AFM) in the noncontact regime. Fukuzawa et al. 26 measured the meniscus shape of a perfluoropolyether lubricant film with a thickness of a few nanometers on a silicon surface with oxide nanostructures using AFM. Seemann et al.
27
used AFM to measure the wetting morphologies of polystyrene on silicon microstructures. Despite their ability to accurately measure the meniscus shape, these experimental approaches are costly and time-consuming, and therefore impractical for optimizing the design of surface structures for enhanced liquid delivery. Molecular dynamics (MD) simulations have been used to investigate the meniscus shape on two-dimensional structures including V-grooves28 and square channels,28, 30
29
as well as three-
31
dimensional structures including spheres and pillars. In particular, Seyf and Zhang30 observed conformal non-evaporating argon films on heated spherical copper nanostructures. Hu et al.
28
investigated the effect of structure geometry and film thickness on the water meniscus shape on gold nanostructures. However, MD simulations are limited to specific materials systems for which the intermolecular potentials are well-known. Also, the length scales of MD-simulation systems are usually smaller than ~100 nm due to the high computational cost involved. As a result, while MD simulations can be a useful tool to investigate the meniscus shape for specific materials systems at the nanoscale, it is difficult to generalize MD results to other combinations of materials and to larger scales, without complementary continuum models. As a result, MD simulations have been integrated with continuum modeling to investigate thin film-related phenomena in recent years. 28, 32 Theoretical and numerical approaches model the meniscus shape by balancing the dominant forces in the liquid film, including surface tension, gravity, and the solid–liquid intermolecular forces. Surface tension flattens the meniscus to minimize the surface energy, while gravity acts to minimize the height of the center of mass of the liquid to reduce the gravitational potential. The solid–liquid intermolecular forces, including van der Waals and electrostatic interactions,
ACS Paragon Plus Environment
3
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 25
make the meniscus more conformal with the surface structure to reduce the disjoining pressure gradient in the film. The equilibrium meniscus shape is a result of competition between these forces. Meniscus shapes of liquid films may be modeled using a variety of open-source and commercial software packages that analyze capillary flows in micro-/nano-structures. In particular, surface Evolver,33 an open-source software that uses an energy-gradient-based evolution of the surface shape to minimize the energy of a surface, has been widely used to predict the meniscus shape. Ranjan et al. 34 modeled the free-surface shapes of the liquid menisci in several different canonical microstructure geometries to evaluate their thin film evaporation characteristics. Xiao et al.
19
modeled the meniscus shape on micropillars to calculate the
capillary pressure in the meniscus. The solid–liquid interactions are represented by the contact angle in Surface Evolver and other software codes; while this works well for systems with a three-phase contact line, it cannot be applied for continuous liquid films. The roles of van der Waals interactions and surface tension in wetting of liquid films are discussed in the seminal review by de Gennes. 35 To predict the meniscus shape of a continuous liquid film, Andelman et al.
36
developed a model by minimizing system free energy and introduced a healing length to
characterize the competition between the van der Waals interactions and surface tension. Robbins et al. 37 developed a theoretical model for rough and chemically heterogeneous surfaces based on the balance of disjoining pressure and capillary pressure, and applied this model for two-dimensional menisci on square channels. More recently, Hu et al.
28, 38
modeled the
meniscus shape on two-dimensional structured surfaces by minimizing system free energy, and using Fourier series to represent the meniscus shape; however, these models have not been directly validated against experiments. Moreover, they primarily provide implicit expressions for the meniscus shape, which makes them difficult to employ for design of surface structures or to couple with other models. In addition, three-dimensional structures are more commonly used in practice for nucleate boiling and thin-film evaporation application. In this work, a general continuum model is developed to predict the meniscus shape of liquid films on three-dimensional micro-/nano-structured surfaces based on minimization of system free energy, accounting for the solid–liquid van der Waals interactions, surface tension, and gravity. The continuum model is validated against molecular dynamics simulations of a water film on gold pyramidal indentations at the nanoscale, and against experimental measurements of a water film on silicon V-grooves at the microscale. The atomically smooth
ACS Paragon Plus Environment
4
Page 5 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
crystalline surfaces of the V-grooves fabricated by wet etching are ideal for model validation; in addition, the V-groove geometry stresses all of the physics contained in the model, as the local film thickness varies from nanoscale near the edge to microscale near the center of the Vgrooves. The effects of the liquid film thickness and solid geometry on the meniscus shape are discussed. A linearized model is derived based on the assumptions of small meniscus slope and small thickness variation, and the error of the linearized model is analyzed for varying film thickness and structure depth.
METHODS Continuum Model for Three-Dimensional Meniscus Shape. A theoretical model is developed to predict the three-dimensional meniscus shape of a liquid film on micro-/nanostructured surfaces. All symbols used in the continuum model and molecular dynamics simulations are summarized in Table 1. Fig. 1 shows a schematic drawing of a liquid film on a three-dimensional structured surface of known profile, Ζ ( x, y ) . The surface structure is characterized by depth D and pitches Lx and Ly in the x and y directions, respectively. The meniscus shape, ζ ( x, y ) , is determined by minimizing the system free energy. The local film thickness is defined as δ ( x, y ) = ζ ( x, y ) − Ζ ( x, y ) , and the mean film thickness
δ0 =
1 Lx Ly
∫
Lx 2
∫
Ly 2
− Lx 2 − Ly 2
δ ( x, y )dxdy . The following basic assumptions are used in this model:
i) The liquid film completely wets the structured surface (i.e., there is not a three-phase contact line). ii) The Derjaguin approximation39 is adopted in the model; namely, the local solid-liquid interaction of a curved surface can be approximated by that of a flat substrate with the same local film thickness. iii) The van der Waals forces dominate interactions between the liquid film and the solid surface. The total system free energy consists of three energy contributions according to W total = W vdW + W γ + Wg , where WvdW is the van der Waals interaction energy between the solid
surface and the liquid film, W γ the surface energy, and Wg the gravitational potential of the
ACS Paragon Plus Environment
5
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 25
liquid film. Following the Derjaguin approximation,39 the solid–liquid van der Waals interaction energy WvdW =
can A 12π
be
∫
Lx 2
∫
calculated
based
1
Ly 2
− Lx 2 − L y 2
(ζ
−Z)
3
on
the
theory
for
a
flat
surface
following
40 dxdy , where A is the Hamaker constant. If the substrate is
chemically heterogeneous, A varies spatially following, A ( x, y ) = A0 Z% A ( x, y ) , where Z% A ( x, y ) is a known distribution function that scales the reference magnitude of Hamaker constant, A0 . The surface energy is represented as W γ = γ AL-V , where AL-V is the liquid–vapor interfacial area and γ the surface tension of the liquid. For a three-dimensional meniscus ζ ( x, y ) , the surface energy can be expressed as Wγ = γ ∫
Ly 2
∫
Lx 2
1 + ζ x2 + ζ y2 dxdy , where ζ x and ζ y are the spatial
− Ly 2 − Lx 2
derivatives of the meniscus in the x and y directions, respectively. The gravitational potential energy is Wg = ρ l g ∫
Lx 2
∫
Ly 2
− Lx 2 − L y 2
(
)
1 2 ζ − Ζ 2 dxdy , where ρ l is the liquid density, and g the 2
gravitational acceleration. The three-dimensional meniscus can be represented with double Fourier series 41 ∞ ∞ 2π m 2π n y + ∑ ∑ bmn cos x sin n =0 m =0 L n =0 m=0 x Ly ∞ ∞ 2π n ∞ ∞ 2π m 2π m 2π n + ∑ ∑ cmn sin x cos y + ∑ ∑ d mn sin x sin y Ly n = 0 m = 0 n =0 m =0 Lx Lx Ly ∞
2π n 2π m x cos L Lx y
∞
ζ ( x, y ) = ∑ ∑ amn cos
y
(1)
By definition, the Fourier coefficients are related to the meniscus shape ζ ( x, y ) as
β mn =
∫
Lx 2
∫
Ly 2
− Lx 2 − Ly 2
ζ
∂ζ dxdy ∂β mn
(2)
2
∫
Lx 2
∫
∂ζ dxdy 2 ∂β mn
Ly 2
− Lx 2 − Ly
where β mn is a general symbol for the Fourier coefficients amn , bmn , cmn or d mn . Based on the expressions for the individual energy terms, the system free energy, Wtotal , is a function of the meniscus shape ζ and its first-order spatial derivatives, ζ x and ζ y , for a specific system. Therefore,
the
variation
of
the
ACS Paragon Plus Environment
free
energy
is
6
Page 7 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
δ Wtotal =
∂Wtotal ∂ζ ∂Wtotal ∂ζ x ∂Wtotal ∂ζ y + + δβ mn ζ β ζ β ζ β ∂ ∂ ∂ ∂ ∂ ∂ = a ,b ,c , d n = 0 m = 0 mn x mn y mn ∞
∞
∑ ∑∑ β
.
Based
on
the
independence of β mn for different m and n,
∂W ∂Wtotal ∂ζ x ∂Wtotal ∂ζ y ∂ζ + + total ζ β ζ β ∂ ∂ ∂ ∂ ∂ζ y ∂β mn mn mn x
=0 β ∈[a ,b ,c,d ]
(3)
Substituting the expression of Wtotal into eq 3 yields,
∫
Lx 2
∫
Ly 2
− Lx 2 − Ly 2
ζx
∂ζ y ∂ζ x +ζ y ∂β mn ∂β mn
1 + ζ x2 + ζ y2
dydx −
A0 6πγ
∫
Lx 2
∫
Ly 2
− Lx 2 − Ly 2
∂ζ % ΖA ∂β mn
(ζ − Ζ )
3
dydx +
ρl g L 2 L 2 ∂ζ ζ dydx = 0 ∫ ∫ γ − L 2 − L 2 ∂β mn x
y
x
y
(4) The Fourier coefficient for m = 0 and n = 0, a00 , can be calculated based on the mean film thickness as a00 = δ 0 +
1 Lx Ly
∫
Lx 2
∫
Ly 2
− Lx 2 − L y 2
Ζdxdy
(5)
Equations 4 and 5 form a complete set of equations to calculate the Fourier coefficients that describe the three-dimensional meniscus shape of a thin liquid film on a structured surface.
Fig. 1 (a) Schematic drawing of a thin liquid film on a three-dimensional structured surface and a (b) zoomed-in schematic drawing of one unit cell in this system.
Most practical structured surfaces can be represented by symmetric geometries (e.g., pillars, rods, and channels) and have uniform wetting properties. The generalized model represented by
ACS Paragon Plus Environment
7
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 8 of 25
eq 4 can be further simplified by assuming symmetric structures and uniform wetting properties. Symmetric structures, viz., ζ ( x, y ) = ζ ( − x, y ) and ζ ( x, y ) = ζ ( x, − y ) , yield bmn ≡ 0 , cmn ≡ 0 , ∞ ∞ 2π n 2π m and d mn ≡ 0 ; thus, the Fourier series reduces to ζ ( x, y ) = ∑ ∑ amn cos x cos L n =0 m = 0 Lx y
y .
For chemically homogeneous surfaces, the Hamaker constant A does not vary spatially (viz., Ζ% A ≡ 1 ); therefore, eq 4 becomes
∫
Lx 2
∫
Ly 2
− Lx 2 − Ly 2
ζx
∂ζ y ∂ζ x +ζ y ∂amn ∂amn 1+ ζ + ζ 2 x
2 y
dydx −
A
6πγ ∫
Lx 2
∫
Ly 2
− Lx 2 − Ly 2
∂ζ ∂amn
(ζ − Ζ )
3
dydx +
ρ l g L 2 L 2 ∂ζ ζ dydx = 0 γ ∫− L 2 ∫− L 2 ∂amn x
y
x
y
(6)
Table 1. Nomenclature for symbols in the continuum model and molecular dynamics simulations Definition Symbol Unit Definition Symbol Unit Hamaker constant A J Surface tension γ J/m2 L-V, S-L surface area AL-S, AL-V m2 Local film thickness δ m 2 Footprint area AF m Mean film thickness δ0 m Fourier coefficient a, b, c, d, β m L-J energy constant ε J Conformity C 1 Liquid profile ζ m Structure depth D m Solid profile Ζ m Gravitational Spatial derivatives of g m/s2 ζx, ζy 1 acceleration liquid profile Structure pitch L, Lx, Ly m Capillary length λc m Wenzel roughness r 1 Healing length ξ m ratio System free energy Wtotal J Disjoining pressure Π Pa van der Waals free WvdW J Liquid, vapor density ρl, ρv kg/m3 energy Surface energy, Solid, liquid number gravitational free Wγ, Wg J ρN,s, ρN,l m-3 density energy x, y, z Coordinates x, y, z m L-J distance constant σ m While the meniscus shape can be determined by solving eq 6 implicitly, an explicit expression of the Fourier coefficients (and thus the meniscus shape) can be derived by linearizing eq 6 with the following two additional assumptions:
ACS Paragon Plus Environment
8
Page 9 of 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
2 2 i) The slope of the meniscus is small: ζ x + ζ y
