A parametric study of the hydrophobicity of rough surfaces based on finite element computations

Accepted Manuscript Title: A parametric study of the hydrophobicity of rough surfaces based on finite element computations Author: Muhammad Osman Roger A. Sauer PII: DOI: Reference:

S0927-7757(14)00635-9 http://dx.doi.org/doi:10.1016/j.colsurfa.2014.07.029 COLSUA 19358

To appear in:

Colloids and Surfaces A: Physicochem. Eng. Aspects

Received date: Revised date: Accepted date:

25-4-2014 17-7-2014 21-7-2014

Please cite this article as: Muhammad Osman, Roger A. Sauer, A parametric study of the hydrophobicity of rough surfaces based on finite element computations, Colloids and Surfaces A: Physicochemical and Engineering Aspects (2014), http://dx.doi.org/10.1016/j.colsurfa.2014.07.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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*Graphical Abstract (for review)

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A Parametric study of the hydrophobicity of rough surfaces based on finite element computations Muhammad Osman and Roger A. Sauer

• We model and simulate wetting of multi-scale rough surfaces, using FEM. • The numerical models are validated against experimental data.

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• Hydrophobicity is evaluated through the wetted area and the contact angle.

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Research highlights

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• The effect of surface and droplet parameters on wetting of surfaces is studied.

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• The multi-scale nature of hydrophobic surfaces significantly influences self-cleaning.

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A parametric study of the hydrophobicity of rough surfaces based on finite element computations Muhammad Osman and Roger A. Sauer ∗

Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University,

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Templergraben 55, 52056 Aachen, Germany

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Abstract

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Several biological and artificial hydrophobic surfaces exhibit self-cleaning mechanisms, which ensure smooth sliding/rolling of liquid droplets, allowing them to sweep pollutant particles away from the surface. While enhancing of the

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self-cleaning property is a major topic in numerous industrial applications, the underlying physics is not yet fully understood. The first step towards analyzing this mechanism is studying the surface hydrophobicity, which is characterized by the contact angle and surface topography. In this article, we investigate the wetting of hydrophobic surfaces at three

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different length-scales, over a range of surface and droplet parameters. A mathematical model describing multi-scale surface topographies is presented, using exponential functions. The contact between liquid droplets and these surfaces is numerically studied using a finite element (FE) droplet model. The introduced models are verified numerically and ex-

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perimentally. Numerical examples are shown for axisymmetric droplets of different size on surfaces with various contact angles and various levels of surface roughness, representing different length-scales. The corresponding wetted area is

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computed in order to evaluate the hydrophobicity of the surface. Significant differences are observed in contact angles and wetted areas captured at different length-scales. This highlights the importance of the multi-scale structure of hydrophobic

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surfaces on wetting. Furthermore, the presented work provides guidelines for the design of artificial hydrophobic surfaces.

Keywords: Self-cleaning mechanism, contact angle, static wetting, nonlinear finite element analysis, droplet membranes, rough surface contact

ticles. The hydrophobicity is highly influenced by the sur-

1. Introduction

face topography ([3], [4]). Therefore, manipulating sur-

Some surfaces such as the lotus leaf exhibit self-cleaning

face morphology over different length-scales, has signif-

effects (also called lotus effects [1]), when water droplets

icantly increased the industrial ([5], [6], [7], [8]) and re-

pass over it. The superhydrophobic nature of these sur-

search interest to this problem.

faces [2] imposes a large contact angle, and therefore minimum contact area, which forces water droplets to form

While studying hydrophobicity of surfaces, it is quite

small semi-spherical shapes. This allows water droplets to

common to consider a system of a liquid droplet in contact

roll-off the inclined surface and sweep away pollutant par-

with a substrate surface. The first fundamental equation

∗ Corresponding

that quantifies the static contact angle of a liquid droplet

author: e-mail: [email protected]

1

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plex surface geometries. One of the first numerical for-

[9]. However, he did not account for surface roughness

mulations for static droplets in contact with flat surfaces

captured at the microscopic scale. Wenzel [10] extended

was introduced in 1980 by Brown et al. [18], [19]. They

the Young equation to account for surface roughness, and

computed the membrane shape of a droplet resting on flat

modeled a droplet in intimate contact with a rough surface

surfaces, by solving the Young-Laplace equation, based on

(non-composite state). Cassie and Baxter [3] introduced

the finite element method (FEM). For more general rep-

the composite contact state, where air is trapped between

resentation of membrane surfaces, differential geometry

the droplet and the rough surface. They observed that the

based formulations were introduced by Steigmann et al.

hydrophobicity of the surface is enhanced by increasing

[20], Agrawal and Steigmann [21], [22], and Sauer et. al

the air-surface fraction. Surface properties which influ-

[23].

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on a solid flat surface was proposed by Young in 1805

ence the wetting state were investigated by Johnson and

Hydrophobicity is influenced by several factors such as

ness are more likely to be in composite state. Kavou-

surface energy, electro-magneticity [24], surface chem-

sanakisa et al. [11] quantified the effect of geometric

istry [25], surfactants [26], surface topography, contact

characteristics of micro-structured solid surfaces on the

angle, and droplet size. In this work, we focus on the

wetting state, considering single-level roughness. Simi-

last three, which are structural aspects, and show how sur-

lar study was performed by Raeesi et al. [12] for tubes

face and droplet parameters affect the hydrophobicity of

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Dettre [4]. They argued that surfaces of higher rough-

surfaces, considering its multi-scale structure. This study

[13] observed that nano-protrusions decrease the contact

serves as the first step to understand self-cleaning mech-

area between a water droplet and the surface, causing the

anisms. We model a static system of a liquid droplet

contact angle to increase considerably, and thus enhance

in contact with a rigid substrate surface of three differ-

the hydrophobicity. They observed that contact angles

ent topographies; flat, single-roughness and two-level-

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of uniform cross-section. On the nano-level, Lee et al.

roughness (double-roughness) surfaces. These can be also

multi-scale structured surface. Static and dynamic wetting

seen as different resolutions of the same physical surface.

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change considerably among different length-scales of a

The hydrophobicity is evaluated through two parameters:

by Ramiasa [14].

the contact angle and the contact area (wetted area). Due

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on nano-topographical surfaces were recently investigated

to the local deformation of the droplet membrane at the

An experimental study of drops on inclined surfaces

region of contact with rough surfaces, local and global

was performed by El-Sherbini et al. [15]. They investi-

contact angles are distinguished, depending on the scale

gated the geometry of drops on various surfaces, for dif-

at which they are measured. We use the term apparent for

ferent inclinations, contact angles, contact lines, volumes,

global contact angles captured at larger length-scales (e.g.

and surface conditions. Callies et al. [16] discuss the re-

macro-scale) where the rigid surface appears flat, while

cent advances in water repellency, and examine wetting of

the term true is used for local contact angles captured lo-

surfaces, experimentally. While some experimental mea-

cally at the individual asperities of the rigid surface, at high

surements of wetting areas on small scales are too diffi-

resolutions (similar definition used in [11]). Similarly,

cult, or even impossible [17], numerical treatment of the

we distinguish apparent and true contact areas to describe

problem provides suitable solutions, especially for com-

global and local surface wetting. Based on a developed FE

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pressed as

model, we compute the evaluation variables: the apparent contact angle and the true wetting area, for given droplet

4p = pf − pc ,

and surface parameters, which are discussed in section 4.

pf = p0 + ρgy,

(2)

We investigate superhydrophobic surfaces with true conwhere pf is the fluid bulk pressure comprising the capil-

tact angles in the range 140◦ ≤ θtr ≤ 180◦ . Droplets are

lary pressure p0 and the hydrostatic pressure in terms of

considered thermodynamically stable with zero hysteresis,

the liquid density ρ, gravity g and the surface height y.

so only the Young-equilibrium contact angle appears.

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A contact pressure pc appears where contact between the membrane and other surfaces take place. Since it is more

2. Numerical model

convenient in computations to use normalized quantities, we multiply Eq.(1) by L/γLG to obtain the dimensionless

scribing the two models; liquid droplet model and the rigid

quantities marked with tilde,

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In this section, we present the governing equations de-

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surface model. The implementation of the first model us-

˜ = p˜f − p˜c 2H

ing the FEM is discussed in detail by Sauer et al. [23]

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and Sauer [27]. We use the stabilization scheme for liquid

p˜f = λ + B0 y˜,

membranes, introduced in [23], with a stabilization parameter µ = 0.01.

[−],

B0 =

ρgL2 , γLG

(3)

(4)

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where λ is the Lagrange multiplier accounting for the capillary pressure, B0 is the so called Bond number, and L is

2.1 Droplet model

a characteristic length, which can be related to the droplet diameter, if it is spherical, or the droplet width otherwise.

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The liquid droplet can be modeled as a structural membrane whose deformation is governed by the Young-

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Laplace equation, and an internal liquid flow governed by

2.2 Rigid surface model

the Stokes equation. Different approaches can be used to Multi-scale surfaces involve multi-level roughness. While

solve the two problems. A simple approach is solving the

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a hydrophobic surface is observed to be flat on the macro-

two problems in a decoupled manner, where each problem

scale level, roughness is captured at the meso-scale level,

is solved separately ([28], [29] and [30]). In this article,

and multi-level roughness at finer length-scales.

we consider quasi-static droplets, where no internal flow takes place.

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mathematical modeling of the substrate surface at three different topographies is presented in this section. These

The difference between the internal and the external

are: flat, single-roughness, and double-roughness topogra-

pressure 4p on the membrane interface Sd is balanced

phies (Fig.1), which can be interpreted as three different

by the mean surface curvature 2H, through the Young-

length-scales. While most authors use simple sinusoidal

Laplace equation,

functions for representation of roughness ([31] and [32]), 2HγLG = 4p

[N/m2 ],

we use super-imposed exponential functions, which pro-

(1)

vide more flexibility in surface description, and better apwhere γ is the surface tension at the liquid-gas interface.

proximation of the real topography. This is because more

The pressure difference across the interface can be ex-

parameters are used to describe exponential functions. A

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point xm on the surface is defined through the relation,

spond to Cr1 = 0, while a single-level rough surface correspond to Cr2 = 0. Furthermore, in this work we fix the

xm = y1 n0 + y2 n1 ,

distance between each two consequent asperities to three

(5)

times their width, 4xi = 3hi for i = 1, 2. We emphasize where n0 and n1 are the normals w.r.t surfaces S0 (the

here that both 4xi and hi are fixed parameters defined

flat horizontal surface) and S1 , respectively (see Fig.1).

in terms of the characteristic length L. In spite of being

The following set of equations describe surfaces of up to

captured at different length-scales, the two surfaces might

two-level roughness,

np 2 X

 (s − j4x )2  2 A2 exp − , 2 h 2 j=1 Z r  ∂y 2 1 s= 1+ dx1 . ∂x1 S1

y2 (s) =

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 (x − j4x )2  1 1 A1 exp − , 2 h 1 j=1

is distinguished through the variable hi and not Cri . (6)

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np 1 X

(7)

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y1 (x1 ) =

have the same roughness factor. Therefore the length-scale

(8)

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The exponential function y1 describes the first level of

Figure 1: Parameterization of the multi-scale surface

roughness on surface S1 , while y2 describes the rough-

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ness on the second level (surface S2 ). The latter is defined

After defining the droplet and substrate surface models,

face S1 . Ai and hi are the amplitude and asperity width

the numerical treatment of contact is introduced. In or-

of the corresponding surface Si , respectively, for i = 1, 2.

der to compute the contact pressure pc between the droplet

A summation over the number of asperities on the corre-

membrane surface Sd and the substrate surface Si (Eq.(2)),

sponding surfaces np1 and np1 is taken, considering 4x1

the normal gap gn between both surfaces must be com-

and 4x2 , which define the spacing between asperities on

puted. For that, each point on the droplet surface is pro-

S1 and S2 , respectively. Setting A2 = 0 yields a single-

jected normally onto the substrate surface. The impenetra-

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in Eq.(7) in terms of the arc length s, measured on sur-

roughness surface described only by y1 , while setting both

bility constraint then reads,

A1 = A2 = 0 yields a flat surface. The width of the asgn = (xs − xp ) · np ≥ 0,

perities on each level is related to a characteristic length

∀ xs ∈ Sd ,

(10)

L through the relations h1 = f1 L and h2 = f2 L, where f1 &f2 < 1 are factors which determine the two length-

where xp is the closest projection of the membrane point

scales of surface S1 and S2 , respectively.

xs onto the substrate surface Si in the direction np , normal to Si . Several numerical approaches are introduced by

In order to describe the roughness, we define the dimen-

Wriggers [33] to enforce the contact constraint in Eq.(10).

sionless surface roughness factor,

While the computation of gn and its spatial derivatives on Cri = Ai /hi ,

a flat surface is straight forward, it is challenging in the

(9)

case of curved surfaces, since the surface normal varies which is the ratio of the asperity amplitude to its width on

w.r.t xs . An iterative solution is therefore necessary to

the corresponding surface i. A flat surface would corre-

compute all possible projection points xm , satisfying the 4

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vided.

orthogonality condition,

(11)

am · (xs − xm ) = 0,

3.1 Validation against experiments The numerical solution is validated against the experimen-

where am is the surface tangent on Si at xm . Generally,

tal solution obtained by Callies et al. [16], and plotted several projection solutions for xs could exist, and therein Fig.2, which depicts an image of an axisymmetric wafore a minimum distance problem has to be solved to obter droplet of given volume V , resting on a flat (observed tain the closest projection point xp ,

xp (xs ) = min (xs − xm ),

(θc ≈ 180◦ ). Droplet parameters are: surface tension

(12)

γ = 0.0728 N/m, temperature 20◦ C, and density of water

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∀xm ∈Si

∀ xs ∈ Sd .

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at the macro-scale) rigid superhydrophobic surface with

ρ = 998.2 kg/m3 . The corresponding numerical quanti-

S2 is computed in terms of the arc length s corresponding

ties are Bond number B0 = 0.1363, and the characteristic

to the point x1 on S1 , which is the projection of xp on S1 .

length L = 1mm.

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In the case of a double-roughness surface, the point xp on

Numerical solution

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The three-phase boundary (contact line) changes as the contact area varies, depending on the surface roughness. While most of the literature treat the contact line as a pre-

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defined boundary given by Dirichlet boundary conditions, we treat it as an internal interface defined through a force

balance based on Young-Dupr´e equation, where the posi-

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tion is a-priori unknown. The force balance in the tangen-

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tial and normal directions, respectively, can be written as

(14)

Figure 2: Numerical solution against the experimental solution of Callies et al. [16], for an axisymmetric droplet, V ≈ 0.52 µl, θc ≈ 180◦ . [Adopted with permission from Royal Society of Chemistry, Soft matter, RF14]

where θc is the contact angle measured inside the liquid

Fig.2 shows good agreement of the numerical solution

between the solid and the liquid interfaces, while γSG and

with experiments for the case of contact on flat surfaces.

γSL denote the surface tension at the solid-gas and the

Furthermore, the experimental solution of Zhang et al.

solid-liquid interfaces, respectively. qn is the force which

[34] has been considered, and also gave an equally good

counterbalances the normal component of the surface ten-

agreement. Numerical solutions for contact on rough sur-

sion γLG . Details on the numerical implementation of the

faces are verified only numerically in the following sub-

contact line can be found in [30] and [27].

section.

3. Model validation

3.2 Convergence of the numerical solution

The droplet model is validated experimentally. A conver-

Since no analytical solution for a deformed droplet under

gence study to verify the numerical solution is also pro-

gravity exists, we study the convergence of the numerical

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γSG − γLG cos θc − γSL = 0,

qn − γLG sin θc = 0,

(13)

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solution based on the relative error computed at each mesh

by minimizing the true contact area, therefore allowing

density. This error is defined as the relative difference in

smoother sliding/rolling motion of a droplet on the sur-

the true contact area computed at two consequent mesh

face. This minimization can be achieved by: 1) increasing

densities Ahi+1 and Ahi ,

the surface roughness, which results in more air gaps between the droplet and the substrate surface, 2) maximizing the contact angle, and 3) minimizing the droplet size, through chemical or electrical treatments. In order to find

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local and global minima, an optimization study is necessary, based on predefined constraints. Such a study is not

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addressed in this article. In this section, we study the effect of these three parameters on: 1) the difference between the

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true and apparent contact angles (θtr − θap ), and 2) the ratio of the true to apparent contact area (Atr /Aap ). The Figure 3: Convergence of the true contact area of a droplet resting on a rigid surface (flat and rough), for θc = 180◦

latter can be seen as a quantification of the composite wet-

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ting state introduced by Cassie and Baxter [3]. Alternative to Atr /Aap , the surface roughness effect can be studied − Ahi

Ahi

through the ratio of the true contact area (Atr ) to the total

(15)

,

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4Ahtrue =

Ahi+1

surface area of the droplet (AT ) defined as the sum of the

which converges to zero as the number of elements nel

true contact area and the free surface area Af , shown in

ed

goes to infinity (element size goes to zero), as shown in

Fig.4.

Fig.3. For a very coarse mesh, the error in the true contact area of both flat and rough surfaces is almost iden-

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tical. This is because the mesh is too course to cap-

While the true contact area Atr and true contact

dimensional finite elements are used in these computa-

angle θtr are usually captured at scales smaller than

tions.

or equal to the micro-scale, where surface roughness

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ture the curved rough surface. Quadratic Lagrange two-

appears, the apparent contact area Aap and apparent

4. Results and discussion

contact angle θap are observed at the macro-scale, where

The introduced numerical model can predict the static wet-

the surface appears flat (see Fig.4 and Fig.5).

The

ting state on hydrophobic surfaces, based on predefined

dotted line in Fig.5, representing the apparent surface,

surface roughness and droplet parameters. Whether it is

is chosen to be tangent to the peak of the asperity. θtr

a Wenzel or Cassie-Baxter wetting state, is therefore a re-

is defined locally at each asperity as the angle between

sult and not an assumption. As often adopted in litera-

the surface tangents of both the asperity and the droplet

ture for quasi-static modeling, axisymmetric droplets are

surface, at the contact line. This definition is also valid

considered in this section, so that the 3-D case can be

for surfaces with second-level roughness. We note that

reduced to a simple 2-D model. In fabrication of artifi-

considering a perfect flat surface, we have θtr = θap = θc .

cial hydrophobic surfaces, the hydrophobicity is enhanced

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use 200 quadratic Lagrange finite elements, while for the case of contact with rough surfaces, 2000 quadratic finite elements are used at the contact region. Fig.7 illustrates the single-level roughness effect on the wetting area. As seen, the ratio Atr /Aap decreases as the surface roughness Cr1 increases. Furthermore, curves in Figure 4: Schematic of true contact area Atr , apparent contact area Aap , and free surface area of the droplet Af

Fig.7 have higher slopes at regions with small roughness

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factors (Cr1 < 0.4), and lower slopes at higher roughness factors. This is because the increase of Cr1 in the first re-

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gion correspond to the transition from the non-composite (Wenzel) to the composite contact state (Cassie-Baxter),

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where the variation in the wetting area (Atr /Aap ) is considerably higher than that in the second region where con-

Figure 5: True contact angle θtr and apparent contact angle θap

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tact is already in a composite state. In the latter state, the droplet is pinned on the asperities of the rough surface.

4.1 Surface parameters

Therefore, with 4x1 being fixed, further increase in Cr1 would correspond to elevating the droplet, with negligible

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We investigate the effect of the surface roughness factor

change in the wetting area. The transition from the Wenzel

Cri and the true contact angle θtr on the hydrophobicity.

wetting state (W) to Cassie-Baxter (C-B) state is marked in

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Computations are done for a defined range of contact angles 140◦ ≤ θtr ≤ 180◦ , for a fixed droplet volume V , and fixed spacing between asperities 4xi = 3hi , for i = 1, 2.

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Adaptive finite element meshes are used in the computa-

Fig.7 by the blue lines. This transition occurs at different Cr1 as the contact angle changes. For a fixed contact angle θtr = 180◦ , considering an additional level of roughness, parameterized by the factor

tions, in order to provide finer mesh size in the region of

Cr2 , results in a further decrease in the true contact area,

face roughness factor is chosen in the range 0 ≤ Cri ≤ 2,

as shown in Fig.8. The roughness factor Cr2 here is ex-

which covers a reasonable range of surface profiles rep-

pressed in terms of Cr1 only for simplification, however

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contact and coarser in the region of free surface. The sur-

arbitrary values can also be assigned to the model.

resenting roughness at the length-scales under consideration. Fig.6 shows a droplet with a fixed volume V = πL3

The second criteria for evaluating the hydrophobicity is

in contact with rigid flat and rough surfaces at three dif-

the contact angle. For the same droplet volume, larger con-

ferent contact angles. On a flat surface, a decrease in the

tact angles result in smaller contact areas (less wetting),

contact angle from 180◦ to 140◦ results in a considerable

as in Fig.7, and smaller differences between the appar-

increase in the true contact area (almost four times). In

ent and the true contact angle, as in Fig.9. In analogy to

other words, at θtr = 140◦ , Atr is 25% of AT ( Fig.6.a).

the effect of roughness on the wetting area, the difference

This percentage drops to 5.5% in the case of contact with

(θtr − θap ) increases with a higher slope in the transition

a rough surface of roughness factor Cr1 = 0.9 (Fig.6.b).

region (Cr1 <0.4), compared to the region where contact

For computations of contact with flat surfaces in Fig.6, we

is already in composite state (Cr1 >0.4). 7

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a)

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Figure 7: Ratio of true to apparent contact area over surface roughness factor Cr1 = A1 /h1 , V = 2πL3 , for a single-roughness surface (Cr2 = 0)

b)

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θtr = 180◦

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a)

b)

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θtr = 160◦

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Figure 8: Effect of double-level of roughness on the true to apparent contact area ratio, for V = 2πL3 , at a fixed contact angle θtr = 180◦

and apparent contact angles are observed when a second level of roughness is considered, at a fixed θtr = 180◦ .

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a)

Fig.10 shows that larger differences between the true

b) θtr = 140◦ Figure 9: Effect of single-roughness on the true contact angle, V = 2πL3 , for various θtr

Figure 6: FE solutions for a droplet of volume V = πL3 , in contact with: a) flat and b) rough surface with Cr1 = 0.9. True contact angles for both surfaces ={180◦ , 160◦ , 140◦ }. The ratio of true to total contact area (up to down): a) Atr /AT ={6.1, 15.3, 25}% , b) Atr /AT ={0.9, 2.5, 5.5}%

4.2 Droplet size Small droplets of small size have larger surface curvature, and therefore higher capillary pressure, compared to larger

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Page 10 of 13

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Figure 10: Effect of double-level of roughness on the true to apparent contact angle ratio, V = 2πL3 , at θtr = 180◦

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Figure 12: Effect of droplet size on the true to total contact area at different surface roughness factors Cr1 , for θtr = 180◦ and Cr2 = 0

droplets. This limits the deformation of a small droplet, as it preserves a semi-spherical shape with a minimum area

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cussed in section 2.2. At smaller scales, where finer as-

of contact (see Fig.11). We use the expression size for

perities are captured on the rough surface, smaller contact

the volume of an axisymmetric droplet, which is initially

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area and more air gaps are observed, compared to larger

a sphere. In 2D, this volume correspond to an area of a

scales. Furthermore, local wetting regions are observed on

circle.

the single asperities, which differ from the global wetting

Now, we study the effect of variation of size (i.e vol-

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region observed at larger scales. Therefore considering the

ume of an axisymmetric droplet) on the ratio of true con-

multi-scale nature of hydrophobic surfaces provides a re-

tact area to the total droplet area Atr /AT , at contact angle ◦

alistic modeling of wetting states. The wetting behaviour

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θtr = 180 , for single-roughness and flat surfaces. As observed in Fig.12, a linear relationship between the volume

is highly influenced by the surface and droplet parameters described in previous sections.

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and the contact area is obtained for all surfaces, in the de-

fined range of surface roughness and droplet volume. The

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wetted area increases with the volume at lower slopes for higher surface roughness.

a)

a)

b)

c)

Figure 13: FE results for a droplet in contact with a superhydrophobic surface at different length-scales [30], with contact angle θtr = 180◦

5. Summary and Conclusion

b)

Figure 11: Effect of droplet size on the area of contact with a) flat surface, Cr1 = 0 and b) rough surface, Cr1 = 1

Based on finite element computations, we perform a study on a range of surface and droplet parameters for which the hydrophobicity of rough surfaces is enhanced. This

4.3 Multi-scale representation of hydrophobic surfaces

enhancement is mainly achieved by minimization of the

A multi-scale view of wetting on hydrophobic surfaces is

wetting area, through manipulating the surface topogra-

depicted in Fig.13, based on the general surface model dis-

phy, contact angle and the droplet size. The effect of these

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Page 11 of 13

three parameters on the wetted area is studied.

[2] A. Carre and K.L. Mittal, Superhydrophobic Surfaces. VSP/Brill, Leiden, (2009).

Three types of surfaces are considered here; flat,

[3] A.B.D. Cassie and S. Baxter, Wettability of porous surfaces, Trans.

single- and double-roughness surface, representing three

Faraday Soc., 40, 546–551, (1944).

length-scales: macro-, meso- and micro-scale, respec-

[4] R.E. Johnson and R.H. Dettre, Contact angle, wettability and adhesion, Adv. Chem. Ser., 43, 112–135, (1964).

tively. These surfaces are mathematically modeled using

[5] M. Ma, R.M. Hill, and G.C. Rutledge, A review of recent results

super-positioned exponential functions, and parameter-

on superhydrophobic materials based on micro- and nanofibers, J.

ized by a surface roughness factor Cr , and a length-scale L.

Adhes. Sci. Technol., 22, 1799–1817, (2008). [6] A.D. Sommers and A.M. Jacobi, Creating micro-scale surface

times the length-scale. Further study of the effect of this

topology to achieve anisotropic wettability on an aluminum sur-

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The spacing between asperities is fixed to three

face, J. Micromech. Microeng., 16, 1571–1578, (2006).

parameter on wetting can be considered in the future.

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[7] T. Onda, S. Shibuichi, N. Satoh, and K. Tsujii, Super-waterrepellent fractal surfaces, Langmuir, 12, 2125–2127, (1996).

surface topography changes across different length-scales.

[8] R. F¨urstner, W. Barthlott, C. Neinhuis, and P. Walzel, Wetting

Furthermore, true contact angles captured locally at small

and self-cleaning properties of artificial superhydrophobic surfaces,

us

Results show significant changes in the wetted area, as the

Langmuir, 21(3), 956–961, (2005).

scales are more precise than the global contact angle

[9] T. Young, An essay on the cohesion of fluids, Phil. Trans. R. Soc.

an

measured at a large scale, where the surface appears flat.

London 95, 65, (1805).

[10] R.N. Wenzel, Resistance of solid surfaces to wetting by water, Ind.

These conclusions agree with the physical observations of

Eng. Chem., 28, 988–994, (1936).

Cassie and Baxter [3], and Lee et al. [13] on the behavior

M

[11] Michail E. Kavousanakisa, Carlos E. Colosquib, and Athanasios G.

of wetting on rough surfaces. Based on these conclusions,

Papathanasiou, Engineering the geometry of stripe-patterned surfaces toward efficient wettability switching, Colloids Surf. A:

roughness observed at small length scales while modeling

Physicochem. Eng. Aspects, 436, 309–317, (2013).

ed

we highlight the importance of considering the surface

wetting of hydrophobic surfaces. The provided study is

pt

specially useful for manufacturing and design of artificial hydrophobic surfaces. Only static wetting is considered

fect of surface roughness on wettability and displacement curvature in tubes of uniform cross-section, Colloids Surf. A: Physicochem. Eng. Aspects, 436, 392–401, (2013). [13] S.M. Lee, Im.D. Jung, and J.S. Ko, The effect of the surface wetta-

in this study. Including inertial effects will result in a

Ac ce

bility of nanoprostrusions formed on network type microstructures,

different wetting behavior, and therefore different surface design.

[12] Behrooz Raeesi, Norman R. Morrow, and Geoffrey Mason, Ef-

J. Micromech. Microeng, 18, 125007 (7pp), (2008). [14] M. Ramiasa, J. Raiston, R. Fetzer, and R. Sedev, The influence of

Acknowledgment

topography on dynamic wetting, Adv Colloid Interface Sci., 206, 275–293, (2014). [15] A.I. ElSherbini and A.M. Jacobi, Liquid drops on vertical and in-

The authors are grateful to the German Research Foun-

clined surfaces: I. an experimental study of drop geometry, J. Col-

dation (DFG) for the financial support, under grant num-

loid Interface Sci., 273, 556–565, (2004). [16] M. Callies and D. Qu´er´e, On water repellency, Soft matter, 1, 55–

bers SA 1822/ 3-2.

61, (2012). [17] H.J. Ensikat, A. Schutle, K. Koch, and W. Barthlott, Droplets on superhydrophobic surfaces: visualization of the contact area by

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