superhydrophobic & superhydrophilic surfaces sanjay ... · derived in terms of a dimensionless...

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/301677106

Superhydrophobic & Superhydrophilic Surfaces

Technical Report · June 2013

DOI: 10.13140/RG.2.1.3772.9685

CITATIONS

0

READS

172

2 authors, including:

Some of the authors of this publication are also working on these related projects:

upstream flow, oil spill, degradation of organic molecules, bio polymeric membrane View project

Dhurjati Prasad Chakrabarti

University of the West Indies, St. Augustine

34 PUBLICATIONS 379 CITATIONS

SEE PROFILE

All content following this page was uploaded by Dhurjati Prasad Chakrabarti on 27 April 2016.

The user has requested enhancement of the downloaded file.

https://www.researchgate.net/publication/301677106_Superhydrophobic_Superhydrophilic_Surfaces?enrichId=rgreq-ba2c061c5d0291f6f1afd8f417068a00-XXX&enrichSource=Y292ZXJQYWdlOzMwMTY3NzEwNjtBUzozNTU2MDgxNjQzNTYwOThAMTQ2MTc5NTAwMTY3Ng%3D%3D&el=1_x_2&_esc=publicationCoverPdf

https://www.researchgate.net/publication/301677106_Superhydrophobic_Superhydrophilic_Surfaces?enrichId=rgreq-ba2c061c5d0291f6f1afd8f417068a00-XXX&enrichSource=Y292ZXJQYWdlOzMwMTY3NzEwNjtBUzozNTU2MDgxNjQzNTYwOThAMTQ2MTc5NTAwMTY3Ng%3D%3D&el=1_x_3&_esc=publicationCoverPdf

https://www.researchgate.net/project/upstream-flow-oil-spill-degradation-of-organic-molecules-bio-polymeric-membrane?enrichId=rgreq-ba2c061c5d0291f6f1afd8f417068a00-XXX&enrichSource=Y292ZXJQYWdlOzMwMTY3NzEwNjtBUzozNTU2MDgxNjQzNTYwOThAMTQ2MTc5NTAwMTY3Ng%3D%3D&el=1_x_9&_esc=publicationCoverPdf

https://www.researchgate.net/?enrichId=rgreq-ba2c061c5d0291f6f1afd8f417068a00-XXX&enrichSource=Y292ZXJQYWdlOzMwMTY3NzEwNjtBUzozNTU2MDgxNjQzNTYwOThAMTQ2MTc5NTAwMTY3Ng%3D%3D&el=1_x_1&_esc=publicationCoverPdf

https://www.researchgate.net/profile/Dhurjati_Chakrabarti?enrichId=rgreq-ba2c061c5d0291f6f1afd8f417068a00-XXX&enrichSource=Y292ZXJQYWdlOzMwMTY3NzEwNjtBUzozNTU2MDgxNjQzNTYwOThAMTQ2MTc5NTAwMTY3Ng%3D%3D&el=1_x_4&_esc=publicationCoverPdf


https://www.researchgate.net/institution/University_of_the_West_Indies_St_Augustine?enrichId=rgreq-ba2c061c5d0291f6f1afd8f417068a00-XXX&enrichSource=Y292ZXJQYWdlOzMwMTY3NzEwNjtBUzozNTU2MDgxNjQzNTYwOThAMTQ2MTc5NTAwMTY3Ng%3D%3D&el=1_x_6&_esc=publicationCoverPdf



i Superhydrophobic & Superhydrophilic Surfaces Sanjay Babooram*, Dhurjati Prasad Chakrabarti*

*The Department of Chemical Engineering, The University of The West Indies, St. Augustine

Trinidad and Tobago

Abstract Superhydrophobicity is a very interesting phenomenon that has recently been extensively

researched. While superhydrophobicity describes the non-wetting characteristic of a surface,

superhydrophilicity describes the converse i.e. the attraction or spreading out of water on a

surface. The key aspect in obtaining superhydrophobic surfaces is the roughness of the surface.

Generally, roughening a surface enhances the hydrophilicity or hydrophobicity of the surface.

Since roughness is such an important factor in superhydrophobicity and superhydrophilicity, then

detailed knowledge of the structures that influence the roughness ratio of these surfaces should

be thoroughly understood. Therefore, the scope of this paper is to determine the optimal surface

topography or structure that would enhance superhydrophobic or superhydrophilic properties.

In order to successfully determine this, the Wenzel and Cassie-Baxter equations were

derived in terms of a dimensionless base radius, R, and a dimensionless height, h, for the six (6)

different surface topographies that were considered in this study. These geometric parameters

were normalised with respect to the side of a unit square cell in which the protrusion was

considered to be within. It is important to remember that the Cassie-Baxter equation only holds

for the hydrophobic case and does not hold for the hydrophilic case. The six (6) topographies

considered were: cylindrical, truncated cone, paraboloidal, hemispherical, cuboid and cube-type

protrusions. The optimal surface topography was determined based on comparisons made

between the transition contact angles obtained for each structure as well as mechanical criteria.

To effectively determine the optimal surface topography, graphical representations of the results

were developed for both the hydrophobic and hydrophilic study. These graphs included the

transition contact angle θW=CB versus the dimensionless base radius R, the dimensionless height,

h, versus the dimensionless base radius, R, as well as the transition contact angle θW=CB versus

the equilibrium contact angle θY.

From the results, it was determined that for the superhydrophobic study, the optimal surface

topography was that of the paraboloidal protrusions. This finding was particularly interesting

since this somewhat resembles the shapes of the protrusions found on the lotus leaf (the most

ii well known superhydrophobic surface). The results also stress that at least two degrees of

freedom (height and base radius of the protrusion) is required for effective superhydrophobic

properties since the cube and hemisphere, which contains only one degree of freedom due to the

radius being equal to the height (hemisphere) and the base length being equal to the height

(cube), displayed very ineffective superhydrophobic properties.

Also, for the superhydrophilic study, the optimal surface topography was determined to be the

cuboid since it always gives the smallest required height for a given base radius of all the

topographies tested to bring about superhydrophilicity. Therefore, it would be the least prone to

mechanical breakage of the structures than the other shapes and so, would be the most durable.

Again, the superhydrophilic study reiterates the need for at least two degrees of freedom since

the cube and hemisphere were deemed ineffective as superhydrophilic surface topographies.

In addition, the study also showed a discrepancy in that the developed Wenzel equation for the

cuboid does not hold when R = 0.5. Therefore, the cuboid would be the optimal topography at all

dimensionless base radii except at R = 0.5.

iii Table of Contents

Introduction .................................................................................................................................................. 1

Objective ....................................................................................................................................................... 3

Literature Review/Theory ............................................................................................................................. 4

Methodology ............................................................................................................................................... 18

Cylinders .................................................................................................................................................. 20

Truncated Cones ..................................................................................................................................... 21

Paraboloids ............................................................................................................................................. 23

Hemisphere ............................................................................................................................................. 25

Cuboid ..................................................................................................................................................... 26

Cube ........................................................................................................................................................ 27

Results ......................................................................................................................................................... 29

Tabulated Results For Superhydrophobic Optimisation ......................................................................... 29

Graphs for Superhydrophobic Optimisation ........................................................................................... 33

Tabulated Results For Superhydrophilic Optimisation ........................................................................... 35

Graphs for Superhydrophilic Optimisation ............................................................................................. 37

Sample Calculations .................................................................................................................................... 38

Superhydrophobic Optimisation ............................................................................................................. 38

Superhydrophilic Optimisation ............................................................................................................... 39

Discussion.................................................................................................................................................... 41

Conclusion ................................................................................................................................................... 47

Recommendations ......................................................................................... Error! Bookmark not defined.

References .................................................................................................................................................. 48

v List of Figures

Figure 1: Water Droplet on a Superhydrophobic Surface (Left) and a Superhydrophilic Surface (Right) .... 4

Figure 2: The Interfacial Surface Tensions for a Water Droplet with Equilibrium Contact Angle θ ............. 5

Figure 3: Distinction Between Hydrophobic & Hydrophilic Surfaces based on Equilibrium Contact Angle . 6

Figure 4: Hierarchical Roughness Structures on the Lotus Leaf displaying (a) micro-sized pillars and (b)

nano-sized structures on each pillar. Adapted from "Hierarchical roughness optimisation for biomimetic

superhydrophobic surfaces," by M. Nosonovsky and B. Bhushan, 2007, Ultramicroscopy, 107, p. 974. .... 7

Figure 5: Difference in Water Droplet Behaviour on a smooth surface (left), a Wenzel surface (middle)

and a Cassie-Baxter surface (right). .............................................................................................................. 8

Figure 6: Advancing angle and Receding angle on a Tilted Surface. Adapted from "Fabrication of

Superhydrophobic Surface,” by S.H. Kim, Journal of Adhesion Science and Technology, 22, p. 239. ........ 11

Figure 7: Water Droplet rolling off a Superhydrophobic Surface Carrying Impurities along with it (Self-

Cleaning Property). ..................................................................................................................................... 12

Figure 8: Spherical Water Droplets on a Lotus Leaf .................................................................................... 12

Figure 9: Various Controlled Surface Topographies Manufactured by Lithographic Techniques. Adapted

from "Progress in superhydrophobic surface development," by P. Roach, N.J. Shirtcliffe and M.I. Newton,

2008, Soft Matter, 4, p. 234. ....................................................................................................................... 14

Figure 10: Paraboloidal Papillae on Lotus Leaf with an Ogive-shaped apex. Adapted from

"Superhydrophobicity in perfection: the outstanding properties of the lotus leaf," by Ensikat et al., 2011,

Beilstein Journal of Nanotechnology, 2, p. 154. .......................................................................................... 15

Figure 11: Topology Optimisation Design. Adapted from "Topology Optimisation of robust

superhydrophobic surfaces," by A. Cavalli, P. Boggild and F. Okkels, 2013. .............................................. 16

vi Figure 12: The Six (6) Different Surface Topographies Considered in this Study. (a) Cylinder, (b) Truncated

Cone, (c) Paraboloid, (d) Hemisphere, (e) Cuboid and (f) Cube. ................................................................ 20

Figure 13: Graph showing Variation of θW=CB with the Dimensionless Base Radius, R, for the Different

Surface Topographies Under Study for Superhydrophobic Optimisation. ................................................. 33

Figure 14: Graph showing Dimensionless Height, h, Variation with respect to the Dimensionless Base

Radius, R, at the Corresponding Values of θW=CB for the Surface Topographies Under Study for

Superhydrophobic Optimisation ................................................................................................................. 34

Figure 15: Graph showing Transition Contact Angle θW=CB at Different Equilibrium Contact Angles θY for

the Cube for Superhydrophobic Optimisation Study.................................................................................. 35

Figure 16: Graph showing Dimensionless Height, h, vs Dimensionless Base Radius, R, for

Superhydrophilic Surface Optimisation Comparison .................................................................................. 37

Figure 17: Graph showing Dimensionless Base Radius, R, Required at Different Equilibrium Contact

Angles θY to Bring About Superhydrophilicity ............................................................................................ 38

vii List of Tables

Table 1: Calculated Transition Contact Angle θW=CB and Dimensionless Height, h, at Different

Dimensionless Base Radii, R, for the Cylindrical Protrusions for Superhydrophobic Optimisation Study. 29


Dimensionless Base Radii, R, for the Truncated Cone at an 85° Slant angle for Superhydrophobic

Optimisation Study. .................................................................................................................................... 29


Dimensionless Base Radii, R, for the Truncated Cone at an 80° Slant angle for Superhydrophobic



Dimensionless Base Radii, R, for the Truncated Cone at a 75° Slant angle for Superhydrophobic



Dimensionless Base Radii, R, for the Paraboloidal Protrusions for Superhydrophobic Optimisation Study.

.................................................................................................................................................................... 30


Dimensionless Base Radii, R, for the Hemispherical Protrusions for Superhydrophobic Optimisation

Study. .......................................................................................................................................................... 31


Dimensionless Base Radii, R, for the Cuboid Protrusions for Superhydrophobic Optimisation Study. ...... 31


Dimensionless Base Radii, R, for the Cube Protrusions for Superhydrophobic Optimisation Study. ......... 32

Table 9: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Cylindrical

Protrusions to Achieve Superhydrophilicity ............................................................................................... 35

Table 10: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Paraboloidal


viii Table 11: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Cuboid


Table 12: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the

Hemispherical Protrusions to Achieve Superhydrophilicity ....................................................................... 36

1 Introduction

Superhydrophobic surfaces have caught the attention of many researchers within the past

recent years due to its various practical applications. Superhydrophilicity, however, have been

studied to a lesser extent [28]

. Much work on superhydrophobic surfaces has come from

biomimetics since many plants and animals use this phenomenon for survival. The most famous

of these are the lotus leaves and so, superhydrophobicity is sometimes referred to as the lotus

effect.

The distinction between these two surfaces is an equilibrium contact angle of 90°.

Generally, surfaces that displayed equilibrium contact angles less than 90° are considered to be

hydrophilic. A hydrophilic surface exhibits good wettability characteristics and so, water spreads

out and is attracted to the surface. A surface in which the equilibrium contact angle is greater

than 90° is described as being hydrophobic. Hydrophobic surfaces exhibit poor wettability

characteristics and so, a water droplet appears more spherical in shape on a hydrophobic surface.

Since water is a polar molecule, then it would be attracted to charged surfaces such as

metals. Therefore, metals cause water droplets to spread out over the surface and are generally

characterised as hydrophilic surfaces. Non-polar surfaces such as many polymers, which are not

charged, exhibit hydrophobic properties. However, surface energies contribute to hydrophilic or

hydrophobic characteristics as well. Surfaces with high surfaces energies are usually hydrophilic

while surfaces with low surface energies are hydrophobic.

It is also possible to achieve, so called superhydrophilic and superhydrophobic surfaces.

A superhydrophilic surface is one in which the static contact angle is less than 5° while a

superhydrophobic surface is one in which the static contact angle is greater than 150°. US patent

2013/0059123 A1 [36]

on superhydrophobic surfaces defines this surface as having a static

contact angle of greater than 150°, droplet hysteresis less than 10°, and roll-off tilt angles

typically less than 2°, resist wetting and exhibit self-cleaning properties [36]

.

Two factors contribute to superhydrophilicity or superhydrophobicity. These are the

surface chemistry as well as the surface roughness. For currently known materials, the highest

2 possible static contact angle is 120°. Roughening the surface enhances its repellent or wetting

properties, resulting in superhydrophobic or superhydrophilic textures [28]

. This roughening

process can be either stochastic, in the case of scraping of the surface, or it may be controlled, in

the case of lithographic techniques. The degree of roughness determines whether the surface

would be in the Wenzel state or the Cassie-Baxter state. Therefore, the shape and height of the

rough structures play an important role in determining the wettability characteristics and should

be studied in more detail.

Potential applications of superhydrophobic surfaces vary widely. If superhydrophobic properties

are imparted to fabrics, one can make weather resistant fabrics and garments. If glass or display

surfaces are made superhydrophobic, they will be resistant to water condensation and exhibit an

anti-fogging capability. These materials could be useful for vehicle windshields, display panels,

etc [23]

. Due to water droplets being able to easily slide off superhydrophobic surfaces, dust

particles are removed along the water droplet sliding path. These surfaces are thus referred to as

self-cleaning surfaces and as such, are used in solar cell panels or satellite dishes. The water

repellence can also be used to move liquid in contact with the surface with no or reduced drag.

This property could be used for drag reduction in microfluidics, piping, or boat hulls. With these

potential applications, various ways of producing superhydrophobic surfaces as well as scientific

understanding of superhydrophobic behaviour have recently been developed [23]

.

3 Objective

The importance of roughness is well known for its ability to enhance both hydrophilicity

and hydrophobicity. As already mentioned, roughness structures can be made on a surface in a

stochastic manner or in a controlled manner. The features of the structures (shape and height) are

very important in determining the roughness ratio for the surface and by extension, the degree of

hydrophilicity or hydrophobicity that is achieved by the surface.

The objective of the research is thus to:

1. Develop the Wenzel and Cassie-Baxter equation for the six (6) different surface

topographies under study,

2. To compare the six (6) different topographies in terms of the contact angles that are

achieved by each as well as the mechanical properties, which determines the durability

and strength of the surface and ultimately,

3. To determine theoretically, the optimal surface topography for a superhydrophobic and

superhydrophilic surface by examination of these six (6) different surface topographies.

The shapes tested in this paper can all be produced by lithographic techniques and so, the height

and base radius of the shapes can be controlled. Therefore, it is important to know which

topographic features give the best superhydrophobic and superhydrophilic properties.

4 Literature Review/Theory

Figure 1: Water Droplet on a Superhydrophobic Surface (Left) and a Superhydrophilic Surface (Right)

The pictures shown above, both display a water droplet. However, the wettability

characteristic is different for each droplet in that, in the first picture, the water droplet takes on an

almost spherical shape while in the second picture, the water droplet spreads out. What causes

this difference in wettability behaviour?

It all comes down to surface tension. Surface tension is caused by the cohesive

interactions between water molecules. Water molecules in the bulk of the fluid experience more

cohesive interactions between neighbouring molecules than molecules at the surface of the

liquid. This is because, in the bulk of the liquid, a molecule is completely surrounded by other

molecules while a molecule at the surface of the liquid, would only experience interactions with

water molecules beneath it since there would be no water molecules above the liquid surface.

These cohesive interactions lower the energy state of the molecules so that, the molecules in the

bulk of the fluid, which experience more cohesive interactions, are at a lower energy state than

those molecules at the surface of the liquid. Water molecules, however, prefer to be at a lower

energy state, so to minimise the total number of higher energy state molecules, water droplets

would usually take on the shape of a sphere (assuming there are no other forces acting on the

water droplet eg. gravity). Therefore, a liquid droplet suspended in a gas phase would take on a

spherical shape if it is smaller than the capillary length. This is the reason why blobs of water in

space appear spherical. The sphere is the shape that exposes the smallest surface area for a given

volume therefore, the least amount of higher energy state molecules would be achieved from a

spherical water droplet.

5 Gravitational forces also affect the shape of the water droplet. Gravity acts downwards

and tries to flatten the droplet. However, the body force (weight) can be neglected if the liquid

drop size is smaller than the so-called capillary length, kc [23]

:

�� =�� (1) where γLG is the surface tension at the liquid-gas interface and ρ is the liquid density.

The capillary length for clean water at ambient conditions is ≈ 2.7mm [23]

. Therefore, if

the water droplet is smaller than this capillary length, gravitational forces are ignored and the

shape of the drop depends on the surface tensions of the surfaces shown below:

Every single surface has surface tension/surface energy. A surface can be thought of as the

interface between two phases. Figure 2

illustrates the interfacial surface tensions

between liquid-gas ��, liquid-solid ��

and solid-gas �� interfaces. Balancing

these interfacial tensions on the tangential

direction of the non-deformable solid

surface yields an equilibrium relation

(Young’s equation) [3, 23]

:

�� + �� = �� (2) where θ represents the equilibrium contact angle, which for a small liquid drop on a solid

surface, should be able to tell the shape and wettability of the liquid drop [23]

. Young’s relation

shows that the equilibrium contact angle is dependent on the interfacial surface energies.

The equilibrium contact angle θ is used to classify a surface as hydrophobic or hydrophilic. The

contact angle distinction between these two surfaces is usually 90°, such that, if the equilibrium

contact angle is less than 90°, the surface is described as hydrophilic. However, if the equilibrium

contact angle is greater than 90°, the surface is hydrophobic [30]

. Superhydrophilicity and

superhydrophobicity is described by contact angles less than 5° and greater than 150°

Figure 2: The Interfacial Surface Tensions for a Water Droplet with

Equilibrium Contact Angle θ

6 respectively

[14]. However, although 90° is the distinction

between hydrophilic and hydrophobic surfaces,

experiments conducted by Guo et al. 2008 [17]

, suggests

that this critical value should be revised. Their findings

are discussed later in this section.

Methods also exist to transform

superhydrophobic surfaces to superhydrophilic surfaces.

One means of doing this is by chemical wettability

switching with hot chromosulphuric acid which also

forms a nanostructure on the surface [22]

. Moveover, the

surface transition between superhydrophobicity and superhydrophilicity can be easily achieved

by the alternation of UV irradiation and closed thermal heating [10]

.

Superhydrophobicity is a very interesting phenomenon that has grasped the attention of

researchers within the past recent years because of its various practical applications. As the name

suggest, a hydrophobic surface is one that has “fear off” water so that water does not wet the

surface.

Two factors determine the wettability characteristic of a surface. These are:

1. Surface Chemistry – in general, surfaces with low surface energies are hydrophobic

while surfaces with high surface energies are hydrophilic. If the liquid surface energy is

significantly above that of the substrate, the substrate does not wet as well [1]

. The surface

energy of water is about 72 mJ/m2 while that of Teflon is about 18.5 mJ/m

2

[21]. Thus,

Teflon will display hydrophobic properties. Other surfaces with surface energies lower

than that of water and hence, would display hydrophobic properties include other

polymers such as polystyrene, polyethylene, polypropylene and PVC. However, surfaces

such as aluminium and glass whose surface energies are ≈500 mJ/m2 and ≈1000 mJ/m

2

respectively, would show hydrophilic properties.

2. Surface Roughness – One of the ways to increase the hydrophobic or hydrophilic

properties of the surface is to increase surface roughness; so roughness-induced

hydrophobicity has become a subject of intensive investigation [29]

. Both hydrophobicity

Figure 3: Distinction Between Hydrophobic &

Hydrophilic Surfaces based on Equilibrium

Contact Angle

7 and hydrophilicity are reinforced by roughness

[32], that is, the intrinsic hydrophobicity or

hydrophilicity of a surface can be increased by roughening the surface. Since currently

known materials only display intrinsic hydrophobicity with contact angles less than 120°,

roughness serves as a mechanism to achieve higher contact angles [25]

. In other words,

superhydrophobicity can only be achieved through the roughening of a surface [6]

.

Hydrophobic materials include many well-known, commercially available polymers and

the largest contact angles are for those of perfluorinated hydrocarbons, which display

contact angles to about 120° on a smooth surface [11]

. Superhydrophobic surfaces are

brought about by micro- and nano- hierarchical structures as demonstrated by natural

surfaces, including leaves of water-repellent plants such as the lotus leaf, legs of insects

such as the water strider, and butterfly wings etc. This micro- and nano- hierarchical

structure is shown in the figure (4) below. Figure 4(a) shows micro-sized pillars while

figure 4(b) shows nano-sized structures are present on each micro-sized pillar. However,

it is not clear why the lotus leaf and other natural hydrophobic surfaces have a multiscale

(or hierarchical) roughness structure, that is, nanoscale bumps superimposed on

microscale asperities [29]

.

Figure 4: Hierarchical Roughness Structures on the Lotus Leaf displaying (a) micro-sized pillars and (b) nano-sized structures

on each pillar. Adapted from "Hierarchical roughness optimisation for biomimetic superhydrophobic surfaces," by M.

Nosonovsky and B. Bhushan, 2007, Ultramicroscopy, 107, p. 974.

8 A water droplet can exists in either of two different states when it contacts a rough surface.

These are as follows:

1. The Wenzel state (homogeneous state) – In this state, the liquid is in complete contact

with the surface so that there are no air bubbles underneath the surface. Wenzel derived

the following equation: cos �� = ��(3) Where � is a dimensionless roughness factor, which is the ratio of the actual surface area to

its nominal (apparent) surface area, �� is the apparent contact angle in the Wenzel state and �� is the equilibrium or Young’s contact angle [23]

. Since no surface is ideal and would have

some degree of roughness, this implies that � > 1 so that �� > θY for hydrophobic surfaces

and �� < θY for hydrophilic surfaces. The following example demonstrates this:

Assuming � = 1.2 and θY = 120°, then:�� =��(1.2��120°) = 127°

This showed that for a hydrophobic surface (contact angle > 90°), a rough surface increases the

hydrophobicity (increases the contact angle). However,

Assuming � = 1.2 and θY = 45°, then:�� =��(1.2��45°) = 32°

This calculation shows that for a hydrophilic surface (contact angle < 90°), a rough surface

increases the hydrophilicity (decreases the contact angle). Therefore, this model predicts that

both hydrophobicity and hydrophilicity are reinforced by roughness [28]

.

Figure 5: Difference in Water Droplet Behaviour on a smooth surface (left), a Wenzel surface (middle) and a Cassie-Baxter

surface (right).

9 2. The Cassie-Baxter state (heterogeneous state) – In this state, as the surface roughness

(height-to-area aspect ratio of the surface topographic features) increases, the liquid is no

longer able to fill the surface texture. As such, air pockets are created as shown in Figure

(5) (right). The water droplet, therefore sits on top of the solid surface as well as air

pockets. The Cassie-Baxter state is expressed as [2,16,23]

:

cos �$% = & cos �� + & − 1(4) Where & is the fraction of solid area wetted by the liquid and �$% is the apparent contact angle in

the Cassie-Baxter state. From this equation, we can see that as & approaches zero, so that the

water droplet is resting on mostly air pockets, then ��$% approaches negative one and

therefore �$% would approach 180° (resulting in a perfect sphere). The Cassie-Baxter (CB) state

is sometimes also seen expressed as [6,14,27,28]

:

cos �$% =�(& cos �� + & − 1(5) Where �( is the roughness ratio of the wetted area. Since the water droplet generally sits flat on

top of the pillars in the CB state, the value of �( is almost always close to 1 so that it is usually

omitted from the CB equation so the CB equation can be seen written either way. However, the

latter equation is the more corrected form and shall be considered in this study.

Since in hydrophilic surfaces, the water is attracted to the surface so that the water droplet

spreads out over the surface, then the Cassie-Baxter equation does not hold for the hydrophilic

case since this model considers air pockets underneath the liquid. The Cassie-Baxter equation is

only applicable to the hydrophobic case.

As mentioned above, the distinction between a hydrophobic and hydrophilic material is usually

90°. However, experiments conducted by Guo et al. [17]

suggest that this critical value needs to be

revised. They carried out a simple and practical method of scraping to construct large-scale

rough structures on polymer surfaces so as to control their surface wettability effectively. They

concluded that for the polymers with a contact angle on the smooth surface larger than 65°, the

polymer tends to show increased contact angle and even can convert into a superhydrophobic

surface when rough structures are fabricated on its surface. For the polymers with a contact angle

on the smooth surface less than 65°, the polymer tends to show decreased contact angle and

10 increased hydrophilicity. As a result, for polymers with an initial contact angle greater or less

than 65°, both hydrophobicity and hydrophilicity can be enhanced by the scraping method. They

showed that their results are consistent with Vogler [35]

. In his paper “Structure and Reactivity of

Water at Biomaterial Surfaces”, Vogler stated that “water structure is a manifestation of surface

hydrophobicity that can be directly measured using the surface force apparatus. The collective

evidence gleaned from more than a decade of intermolecular force research using the surface

force apparatus suggest that hydrophobic surfaces exhibit water contact angles greater than 65°

whereas hydrophilic surfaces exhibit water contact angles less than 65°” [35]

. Therefore, the

critical distinction angle should be 65° and Guo et. al suggested that the Wenzel equation be

revised into cos �� = � cos(�� + 25°) [17]. Active recent research on superhydrophobic

materials might eventually lead to industrial applications. The potential for industrial application

requires surfaces to be processed rapidly and to be reproducible and cheap [33]

. An example in

which superhydrophobicity can be useful in industries can be seen if an ammonia plant is

considered. If vessels are made of a hydrophilic material, then when rain falls and if ammonia is

in the air, it will dissolve in the water (since ammonia has a high affinity for water) and begin to

corrode the vessel. After many years, this corrosive effect can be a problem. However, with a

vessel made from a superhydrophobic material, the vessel shall remain dry and corrosion shall

not be an issue. This scraping method has advantages such as construction of rough

microstructures over a large area, which can be easily applied on an industrial scale [17]

.

Besides a high contact angle, the easy sliding-off behaviour of liquid droplet is another criterion

related to superhydrophobicity. The sliding behaviour of the droplet is again governed by the

balance between surface tension and gravity. On a tilted surface, the liquid drop becomes

asymmetric and the contact angle of the lower side becomes larger and the upper side gets

reduced. The difference between these two contact angles (hysteresis) reaches the maximum

when the drop begins to slide down the tilted surface. The contact angles of the forefront and

trailing edges of the liquid drop just prior to movement of its contact line are called the

advancing (θa) and receding (θr) contact angles respectively. For a given mass of water droplet, a

smaller contact angle hysteresis will result in a smaller sliding angle and easier roll-off. The

smaller the roll-off angle, the more hydrophobic is the material [23]

.

11

Figure 6: Advancing angle and Receding angle on a Tilted Surface. Adapted from "Fabrication of Superhydrophobic Surface,”

by S.H. Kim, Journal of Adhesion Science and Technology, 22, p. 239.

Recently, there has been a significant amount of research in the use of superhydrophobic

surfaces to reduce drag. Experimental research with superhydrophobic surfaces has uncovered

the possibility of dramatic drag reduction in both laminar and turbulent flow regimes.

One of its widespread uses is in microfluidic devices. The problem arises such that high pressure

drops are associated with flows through micro-channels. Particularly in small scales, which

means that channels have an increased surface area to volume ratio, flow is severely resisted

between the moving flow and channel walls. As such, micro pumps are usually used in order to

overcome this resistance if drag reduction methods are not used. Decreasing the drag due to

surface resistance in these channels would have serious implications for reducing operating

power for microfluidic devices, that is, the use of superhydrophobic surfaces would imply a

reduction in operating power for microfluidic devices. On a larger scale, the use of

superhydrophobic surfaces in high speed marine vessels would drastically reduce drag forces by

water resulting in substantial energy savings as well as increased speed and performance

improvements [4]

.

Superhydrophobic properties can also be imparted in fabrics to make weather resistant garments.

Superhydrophobic glass will be resistant to water condensation and exhibit an anti-fogging

capability. Thus, they can be use in car windshields and display panels. No residues would be

left behind by water. In addition to this, the glass would always be kept clean and free of dust

and other minute particles as superhydrophobic surfaces have the property of being self-cleaning

as seen in Figure 7. Water droplets simply roll off easily, trapping any dust and small particles

that may be on the surface [23]

.

Figure 7: Water Droplet rolling off a Superhydrophobic Surface Carrying Impurities along with it (Self

Some impressive and elegant examples of superhydrophobic

most widely known example is the lotus leaf for its self

bounce off after impact, carrying along any dirt particles from the leaf thus allowing the leaf to

photosynthesise most efficiently.

surface of butterfly wings. This roll

stick together. Certain animals also

the morning dew for subsequent use

Figure

Many methods have been developed to produce rough structures on polymer surfaces, such as

solidification, plasma polymerisation/etching, chemical vapour deposition, solvent

phase separation, molding and template

: Water Droplet rolling off a Superhydrophobic Surface Carrying Impurities along with it (Self-

Some impressive and elegant examples of superhydrophobic surfaces come from nature. The

most widely known example is the lotus leaf for its self-cleaning action. Rain drops simply


photosynthesise most efficiently. Another example will be the roll-off of water droplets from the

surface of butterfly wings. This roll-off effect is essential so that the butterfly’s wings do not

stick together. Certain animals also use superhydrophobicity as a means of collecting water

morning dew for subsequent use [7]

.

Figure 8: Spherical Water Droplets on a Lotus Leaf


solidification, plasma polymerisation/etching, chemical vapour deposition, solvent

phase separation, molding and template-based extrusion [17]

. Other preparation methods for

12

-Cleaning Property).

surfaces come from nature. The

cleaning action. Rain drops simply


off of water droplets from the

off effect is essential so that the butterfly’s wings do not

uperhydrophobicity as a means of collecting water from


solidification, plasma polymerisation/etching, chemical vapour deposition, solvent-mediated

tion methods for

13 superhydrophobic surfaces include electrochemical deposition, electrospinning method, wet

chemical reaction, hydrothermal synthesis and sol-gel method [26]

. Etching is an easy way to

make rough surfaces. Micrometer scale topographic structures such as grooves can be created

with controlled width and depth with a high power pulsed laser beam [23]

.

According to US patent 2013/0059123 A1 [36]

, hierarchical structures can be fabricated using

self-assembled biological structures such as viruses. Viruses can be genetically engineered to

impart desired properties, such as affinity for a surface. A genetically-modified virus can serve as

a nanoscale template for the synthesis of a hierarchically structure surface. The surface can be

superhydrophobic with static contact angles greater than 170°, contact angle hysteresis of less

than 2°, and roll-off angles of less than 0.25°. The surface can also exhibit advantageous

condensation mass and heat-transfer properties [36]

.

Photo-lithography and electron beam lithography techniques developed in semi-conductor

processing have extensively been utilized to create periodic topographic patterns. The use of

lithographic patterning allows precise control of dimensions (width, height, and separation

distances) and shapes (pillars vs. Holes, facetted vs. rounded, continuous vs. discontinuous). The

production of these controlled topographic features has played significant role in advancing

fundamental understanding of the geometric effect on superhydrophobicity [23]

. However, the use

of lithography for scaling up of superhydrophobic surfaces for industrial applications is very

problematic perhaps due to the cost of productivity of such nano-scaled technology [24]

.

Therefore, although lithography can be used to control the shape and size of rough structures, it

is expensive in comparison to the method of scraping which is inexpensive but the rough

structures imparted on the surface cannot be controlled.

One of the first reports using photolithography to produce 3D surface features for the

investigation of wetting was by Kawai and Nagata in 1994, although these features were of low

aspect ratio (height/width) they did show a change in wettability with respect to feature height.

Oner and McCarthy produced a larger range of feature sizes with patterns etched in silicon,

including square posts from 20–140 µm height and side lengths 2–128 µm, Fig. 9(a), as well as

staggered rhombus- and star-shaped structures, Fig. 9(b). A similar silicon-processing technique

was reported by Zhu et al., giving square pillars in the range 10–85 µm, Fig. 9(d), and more

recently by Dorrer and Ruhe to generate smaller posts. Fig. 9(e) illustrates paraboloidal pillars.

14 Cylindrical pillars were produced with diameters from 2 to 40 µm and up to 80 µm in height Fig.

9(g) [33]

.

Figure 9: Various Controlled Surface Topographies Manufactured by Lithographic Techniques. Adapted from "Progress in

superhydrophobic surface development," by P. Roach, N.J. Shirtcliffe and M.I. Newton, 2008, Soft Matter, 4, p. 234.

With all these possible shapes and sizes of protrusions that can be created in a controlled manner

by lithographic techniques, it raises the questions as to which of these structures would display

the best hydrophobicity. From Figure (4) above, we see that the lotus leaf has somewhat

paraboloidal shaped pillars. However, Ensikat et al.[13]

in their paper “Superhydrophobicity in

perfection: the outstanding properties of the lotus leaf”, stated that the epidermis cells of the

upper leaf side form papillae of varying height and with a unique shape. The diameter of the

papillae is much smaller than that of the epidermis cells and each papillae apex is not spherical

but forms an ogive [13]

as seen in the Figure (10) below. An ogive is a roundly tapered end.

Again, from Figure (4), we notice that the lotus leaf is made up of nano-scale rough structures.

15 There was no experimentally-determined optimal surface topography that could have been found

so that surface hydrophobicity in this area of study is relatively limited.

Figure 10: Paraboloidal Papillae on Lotus Leaf with an Ogive-shaped apex. Adapted from "Superhydrophobicity in perfection:

the outstanding properties of the lotus leaf," by Ensikat et al., 2011, Beilstein Journal of Nanotechnology, 2, p. 154.

Topology optimisation was investigated by Cavalli, Boggild and Okkels [8]

. To keep the fluid

from penetrating the space between the posts (pillars), they searched for an optimal post cross

section, which minimises the vertical displacement of the liquid-air interface at the base of the

drop when a pressure is applied. Their investigation was aimed towards optimisation of the

Cassie-Baxter (CB) model and was carried out for a cylindrical pillar. When pressure is applied

to the droplet, the water droplet can be pressed downwards so that it transitions from the CB state

to the Wenzel state. This would not be desired since contact angles in the CB state is larger than

contact angles in the Wenzel state so that water droplets in the CB state is generally more

hydrophobic. For their optimisation investigation, they used a Matlab code that relies on the

commercial software COMSOL to solve partial differential equations at every step [8]

. They

obtained the following results shown in figure (11):

16

Figure 11: Topology Optimisation Design. Adapted from "Topology Optimisation of robust superhydrophobic surfaces," by A.

Cavalli, P. Boggild and F. Okkels, 2013.

Figure 11(a) shows the top view of a cylindrical pillar in which the fraction of solid/liquid

contact, &, in the CB equation is 0.25. However, in Fig.11(c), the top view of the optimised pillar

is shown. The support structure is still cylindrical but the top face is now a branched feature in

which the fraction of solid/liquid contact is still 0.25 but now; the top surface is more widely

distributed. This branched structure minimises the vertical displacement of the liquid-air

interface since the pressure acting downwards is now better distributed.

Due to the intensive computation that would have been required for a three-dimensional study of

this optimisation, they limited their study to the two-dimensional case. They concluded that the

fractal-like structures resemble several biological surfaces (such as the lotus leaf), which use

analogous (although three-dimensional) multi-scale structures to achieve their superhydrophobic

properties [8]

.

E. Bittoun and A. Marmur also investigated the optimisation of superhydrophobic surfaces. They

considered 4 different surface topographies in their study; a cylinder, a truncated cone, a

paraboloid and a hemisphere. They focused on the wetting criteria, namely the apparent contact

angle and the wetted area, and use them to compare the mentioned surface topographies. The use

of the wetted area is suggested as a replacement for the criterion of the low “roll-off” angle, since

the wetted area is much simpler to calculate than the roll-off angle. The mechanical criteria that

are related to the strength and durability of the surface topography, and the criteria related to

productivity and cost are also considered in the optimal design [6]

. From their study, they

17 concluded that the paraboloidal protrusions seem to be most advantageous. Quite interestingly

again, is the relation to the structures of the lotus leaf which is also somewhat paraboloidal.

Abraham Marmur also investigated the effect of steepness and protrusion distance on the contact

angle in the Wenzel and CB states for a paraboloidal protrusion. He also investigated the wet

area in the Wenzel and the CB regimes against the Wenzel and CB contact angles, respectively.

His results lead to the conclusion that the heterogeneous wetting regime is practically preferred

by nature as the superhydrophobic state on Lotus leaves for several related reasons: (a) for most

combinations of steepness and protrusion distance, the CB contact angle is higher than the

Wenzel one; (b) the CB contact angle is insensitive to the protrusion distance and mildly

sensitive to the steepness; and (c) the heterogeneous wetting regime yields a much lower wet

area even when the Wenzel and CB contact angles are equal [27]

.

In this present study, the criterion that was used to determine superhydrophobicity was a very

high contact angle. However, another criterion can be used to determine superhydrophobicity.

This criterion is a very low roll-off angle. However, since it is usually very difficult to calculate

the roll-off angle for rough surfaces, the approach is to use a very low wet (solid-liquid) contact

area as a simple, appropriate substitute for the roll-off area criterion. This approach was

investigated by both Abraham Marmur, 2004 [27]

and again by Marmur and Bittoun, 2009 [6]

in

order to determine the optimal surface topography for superhydrophobicity. Marmur developed a

graph illustrating the dimensionless wetted area in the homogeneous and heterogeneous regimes

versus the Wenzel and CB contact angles, respectively. From his findings, it stated that the

heterogeneous wetting regime yields a much lower wetted area even when the Wenzel and CB

contact angles are equal [27]

. This suggests that the heterogeneous wetting regime displays better

properties of superhydrophobicity.

18 Methodology

In this study, six (6) different topographies are being considered to determine the optimal

surface topography for a superhydrophobic surface. These include a cylinder, a truncated cone, a

paraboloid, a hemisphere, a cuboid and a cube. Also, the optimal surface topography for a

superhydrophilic surface was determined considering the surface topographies of a cylinder,

paraboloid, cuboid, cube and hemisphere. The choice of these shapes were made since the

cylinder, cuboid and cube are surfaces commonly made from lithography (as mentioned above);

the paraboloid represent the shape similar to that of the protrusions on the Lotus leaf and the

truncated cone is an intermediate between the cylinder and the paraboloid for comparison. The

hemisphere is similar to the paraboloidal shape however; it has only one degree of freedom. For

these reasons, the choice of shapes for the analysis was considered to be appropriate.

Superhydrophobicity is defined by two criteria: a very high water contact angle and a

very low roll-off angle [27]

. For this analysis, the former will be used as the criteria for comparing

the different topographies. The latter was investigated by Bittoun and Marmur [6]

and shall be

mentioned later on. Although the comparison is being made with respect to a high contact angle,

factors such as mechanical design will be considered so that the surface would be durable and

long-lasting. From the literature presented, it is known that as the roughness is increased, the

contact angle increases so that there would exist a transition from the homogeneous (Wenzel)

wetting regime to the heterogeneous (Cassie-Baxter) wetting regime. When the roughness ratio

is not significant, the droplet is usually in the Wenzel state so that the liquid fills the grooves in

the surfaces. However, as the height to base radius ratio of the protrusion increases, the water

droplet eventually starts obeying the CB state so that it sits on the solid protrusions as well as air

pockets. Therefore, there is a transition point from the Wenzel state to the CB state. From

previous studies, the heterogeneous regime is more advantageous than the homogeneous regime

in terms of superhydrophobicity characteristics such that the CB regime demonstrates higher

contact angles and lower roll-off angles than the Wenzel regime [27]

. However, when the

roughness is increased beyond the transition point between the two regimes, the surface structure

may become more susceptible to mechanical breakage. Moveover, it seems that increasing the

roughness beyond the transition point does not increase the contact angle by much [6]

. For these

19 reasons, the contact angle at the transition point between the homogeneous and heterogeneous

regime will be used for the comparison of the different surface topographies under consideration.

This contact angle shall be denoted as θW=CB. The surfaces with higher values of θW=CB means

that they display better superhydrophobic properties.

Firstly, all the topographies are assumed to be in a square unit cell in which the center of

the single protrusion lies in the center of the unit cell. Each protrusion will have a base radius R,

and a height, h as shown in figure (12) below. The side of the square is defined to be of unit

length and therefore, all the geometric parameters are normalized with respect to the side of this

square [6]

. As a result of the square being of unit length, this means that the maximum

dimensionless base radius R can be is 0.5 and as such, the range of R considered in this analysis

would be R ≤ 0.5. θW=CB would be found as follows: a dimensionless base radius R would be

selected and then the dimensionless height, h would be varied until θW and θCB become equal.

This contact angle would represent the transition contact angle θW=CB. However, in order to do

this, the equations for the Wenzel and CB models would have to be developed in terms of R and

h for each protrusion under study. The methods in which these equations are developed are

discussed below.

20

Figure 12: The Six (6) Different Surface Topographies Considered in this Study. (a) Cylinder, (b) Truncated Cone, (c)

Paraboloid, (d) Hemisphere, (e) Cuboid and (f) Cube.

Cylinders

The equations for the cylindrical pillar, as seen in Figure 12(a) above shall be developed

first. Recall, from equations (3) and (5) above, that the Wenzel and CB equations are given as:

cos �� = ��(3) Where: r is the roughness ratio which is the area of the surface that is exposed to the liquid

divided by the projected area.

cos �$% =�(& cos �� + & − 1(5) Where: & is the fraction of the projected solid area that is wetted by the liquid

�( is the roughness ratio of the wetted area.

For the cylinder, the area of the surface that is exposed to the liquid would consist of the upper

face and the lateral surface given by *+, + 2*+ℎ, as well as the remainder of the unit cell that is

not covered by the cylinder i.e. 1 − *+,. The roughness ratio is therefore, the sum of these areas

divided by the projected area, which is the area of the unit cell (always equal to 1) [6]

. Therefore:

21 �$ = *+, + 2*+ℎ + (1 − *+,)

�$ = 1 + 2*+ℎ(6) The Wenzel equation now becomes:

cos ��$ = (1 + 2*+ℎ)�� (7) The superscript C stands for cylindrical pillars.

For the CB equation, the water droplet sits on the solid surface as well as air pockets. It will be

assumed that the water droplet is sufficiently large so that when the water droplet sits on the

pillar, the liquid-air interface would be almost flat so that the liquid does not wet the lateral sides

of the protrusion. Hence if this assumption is made, then the fraction of solid that is wetted

would only be the upper face of the protrusion divided by the projected area (which is 1).

Therefore:

& = *+,

The roughness ratio of the wetted surface �( would be 1 since the water droplet is considered to

be resting perfectly flat on the upper face of the protrusion.

Therefore, the resulting CB equation is given as:

cos �$%$ = *+,(1 + cos ��) − 1(8)

Truncated Cones

The truncated cones are shown in Figure 12(b). For the surface made from truncated cones,

similarly, the area of the surface that is exposed to the liquid would be the upper face of the cone,

its lateral sides and the remainder of the unit cell that is not covered by the cone. The area of the

upper face is given by *+�,. The lateral area of a cone is given as [5]

:

*(+ + +�)�(9) Where the slant height s can be obtained from Pythagoras’ Theorem as:

� = 2(+ − +�), + ℎ,(10)

22 Finally, the area of the unit cell that is not covered by the cone is given by 1 − *+,. Therefore,

the roughness ratio, �3$ is given by:

�3$ = 1 − *+, + *+�, + *(+ + +�)�

Rearranging and substituting the equation (10) into �3$ gives �3$ in terms of R and h as:

�3$ = 1 + *(+�, − +,) + *(+ + +�)�

�3$ = 1 + *(+� + +)(+� − +) + *(+ + +�)�

�3$ = 1 + *(+ + +�)4� + +� − +5 �3$ = 1 + *(+ + +�) 62(+ − +�), + ℎ, + +� − +7(11)

And as a result, the Wenzel equation for the truncated cone would be:

cos ��3$ = 81 + *(+ + +�) 62(+ − +�), + ℎ, + +� − +79 cos �� (12) The superscript TC represents truncated cone.

Using the same assumptions as for the cylinder for the CB equation, the fraction of solid that is

wetted would only be the upper face of the protrusion divided by the projected area (which is 1).

Therefore:

& = *+�,,

While the roughness ratio �( would again be 1 since the water droplet rests perfectly flat on top

of the upper face of the truncated cone. As a result, the CB equation for the truncated cone is:

��$%3$ = *+�,(1 + cos ��) − 1(13) Using simple trigonometric functions +�can be shown to be:

+� = + − ℎtan =(14) Where = is the slant angle [See figure 12(b)].

23

Paraboloids

The shape of the paraboloids is shown in Figure 12(c). To develop the Wenzel equation, the area

exposed to the liquid would consist of the surface area of the paraboloid plus the area of the unit

cell that is not covered by the protrusion. The latter if given as 1 − *+,. The surface area of a

paraboloid (excluding the bottom face) as given as [34]

:

>*6? @ +ℎ,A B(+, + 4ℎ,)C, − +CD(15) The height ℎ of a paraboloid is related to its radius + by

[27]:

ℎ = E+,,(16) where E represents the steepness of the paraboloid. Substituting equation (16) into the equation

(15), then equation (15) can be manipulated as follows in order to make it easier to work with to

determine the roughness ratio. Equation (15) becomes:

*+ B(+, + 4E,+G)C, − +CD6E,+G

=*+ H4+,(1 + 4E,+,)5C, − +CI6E,+G

= *+ J+C(1 + 4E,+,)C, − +C6E,+G K = *+ L+C B(1 + 4E,+,)C, − 1D6E,+G M

= *+, J(1 + 4E,+,)C, − 16E,+, K(17) Therefore, using equation (17) above as the surface area for a paraboloid (excluding the base),

the roughness ratio can be determined to be:

24 �N = 1 − *+, + *+, J(1 + 4E,+,)C, − 16E,+, K This equation can be written as:

�N = 1 + *+, J(1 + 4E,+,)C, − 16E,+, − 1K(18) To obtain the roughness ratio �N in terms of ℎ and +, rearrange equation (16) to get equation

(19) below and substitute equation (19) into equation (18) to get equation (20):

E, = ℎ,+G (19) Therefore:

E,+, = Oℎ,+GP �+, =@ℎ+A,

�N = 1 + *+,QRRRS@1 + 4>ℎ +T ?,AC, − 1

6>ℎ +T ?, − 1UVVVW (20)

The Wenzel equation for the paraboloid is thus:

cos ��N =XYZY[1 + *+,

QRRRS@1 + 4>ℎ +T ?,AC, − 1

6>ℎ +T ?, − 1UVVVW\Y]Y cos �� (21)

Equation (21) was also developed by Bittoun & Marmur, 2009 [6]

. Also, using equations (22) and

(23) below from Marmur, 2004 [27]

for the CB model:

& = *4E, @ 1��,�� − 1A(22) ∅ represents the product of �(& and is given as:

25 ∅ = *6E, J@1 + 4* E,&AC, − 1K(23)

Bittoun and Marmur, 2009 [6]

gave the CB equation for a paraboloid as:

cos �$%N = *2 O+,ℎ P, B 56��,�� − cos ��3 − 12D − 1(24) The superscript P represents the paraboloid.

Hemisphere

For the case of the hemisphere, shown in Figure 12(d), the height is equal to the base radius so

that only one degree of freedom exist. Again, the area exposed to the liquid would be the surface

area of the hemisphere i.e. 2*+, plus the rest of the cell that is not covered by the hemisphere

i.e. 1 − *+,. Therefore, the roughness ratio for the hemisphere, �`is given by:

�` = 1 − *+, + 2*+,

�` = 1 + *+,(25) The Wenzel equation is thus:

cos �� = (1 + *+,) cos �� (26) For the CB state, the degree to which the water droplet sits on the hemisphere is given as a

function of the angle α. The water droplet cannot exist resting on a point at the top of the

hemisphere. Due to the weight of the droplet, it must press downwards. For the flat top surfaces

such as the cylinder and truncated cone, the water droplet could have rested on the top and not be

pressed downwards but this is not so for the hemisphere. Thus, &` and �( was given as [6]

:

&` = *+,�ab,c(27) �( = 2(1 − cos c)�ab,c (28)

The resulting CB equation was [28]

:

cos �$% = *+,(1 + cos ��), − 1(29)

26 Therefore, if equations (26) and (29) were equated, then the dimensionless base radius R at the

transition contact angle ��d$% can be expressed as:

+ = cos �� + 1*(��,�� + cos �� + 1)(30)

Cuboid

For the cuboid, Figure 12(e), if the distance from the centre to the edge is considered to be R,

then the total length of one side of the cuboid shall be 2R. Therefore, the area exposed to the

liquid shall be the area of the top face, the lateral sides and the rest of the cell that is not covered

by the cuboid.

e�fE�&g�hiE�f = 2+�2+ = 4+,

e�fE�&jEkf�Ej�alf� = 4�(2+ℎ) = 8+ℎ

e�fEb�k��mf�flno�pn�al = 1 − 4+,

Therefore, the roughness ratio for the cuboid in the unit cell would be:

�$qrstu = (1 − 4+,) + 4+, + 8+ℎ

�$qrstu = 1 + 8+ℎ(31) The resulting Wenzel equation is:

cos ��$qrstu = (1 + 8+ℎ) cos �� (32) Using the same assumptions as for the cylinder for the CB equation, the fraction of solid that is


Therefore:

& = 4+,

Also, since the droplet rests on the top of the cuboid, which is flat, �( is equal to 1. As a result,

the CB equation is:

27 cos �$%$qrstu = 4+,(cos �� + 1) − 1(33)

Cube

Similar to the cuboid, the distance from the centre to the edge is defined as R, as seen in Figure

12(f). Therefore, the length of one side of the cube shall be 2R. Also, similar to the hemisphere,

this geometrical shape has only one degree of freedom since the length of one side is equal to the

height. Again, the area exposed to the liquid would be the upper face, the lateral sides and the

area of the unit cell that is not covered by the cube. These values are as follows:

e�fE�&g�hiE�f = 2+�2+ = 4+,

e�fE�&jEkf�Ej�alf� = 4�(4+,) = 16+,

e�fEb�k��mf�flno�pn�al = 1 − 4+,

Therefore, the roughness ratio � for the cube in the unit cell would be:

�$qrv = (1 − 4+,) + 4+, + 16+,

�$qrv = 1 + 16+,(34) The resulting Wenzel equation would be:

cos ��$qrv = (1 + 16+,) cos �� (35) Using the same assumptions as for the cylinder for the CB equation, the fraction of solid that is


Therefore:

& = 4+,

Also, since the droplet rests on the top of the cube, which is flat, �( is equal to 1. As a result, the

CB equation is:

cos �$%$qrv = 4+,(cos �� + 1) − 1(36)

28 Having developed all the Wenzel and CB equations for the six (6) protrusions under

investigation, in which the Wenzel contact angle or CB contact angle is given in terms of ℎ and +, the transition contact angle ��d$% can be found at different values of +. As mentioned above,

a dimensionless base radius R would be selected (where + ≤ 0.5 and R cannot be negative) and

then the dimensionless height, h would be varied until θW and θCB become equal. This would be

a very tedious trial and error process and so, the use of Excels’ Solver function is advised.

Therefore, using Excels’ Solver, the dimensionless height h required to equate both the Wenzel

contact angle and CB contact angle at a particular base radius R can be found.

In order to effectively compare the surface topographies, graphs of ��d$% versus + (Figure 13)

and ℎ�d$% versus + (Figure 14) were plotted.

The optimal surface topography for a superhydrophilic surface was also determined using the

following surface topographies: the cylindrical protrusion, paraboloid, cuboid, cube and

hemisphere. Since the CB equation does not apply to hydrophilic surfaces, only the Wenzel

equation for these protrusions was used in this study. An initial intrinsic hydrophilic contact

angle of 60° was used. The height required at different base radii to give a sufficiently rough

surface to cause superhydrophilicity (contact angle less than 5°) was investigated. Therefore,

substituting θY and θW as 60° and 5° respectively, and setting various base radii, the

corresponding height can be determined. Again, the use of Excel’s solver function was utilised.

A graphical representation of the results for comparison was generated.

29 Results

Tabulated Results For Superhydrophobic Optimisation

All calculations for the determination of the optimal surface topography for the

superhydrophobic surface were performed using a value of ��= 110°.

For the Cylindrical Protrusions:

Table 1: Calculated Transition Contact Angle θW=CB and Dimensionless Height, h, at Different Dimensionless Base Radii, R, for

the Cylindrical Protrusions for Superhydrophobic Optimisation Study.

R hW=CB Cos θW Cos θCB θW=CB (°)

0.05 6.07 -0.9948 -0.9948 174.17

0.1 2.97 -0.9793 -0.9793 168.33

0.2 1.34 -0.9173 -0.9173 156.54

0.3 0.73 -0.8140 -0.8140 144.49

0.4 0.38 -0.6693 -0.6693 132.01

0.5 0.13 -0.4833 -0.4833 118.90

For Truncated Cones:

For Truncated Cone protrusions at a slant angle of 85°. Note that the values for R start at 0.25

because the slant angle limits the height of the cone such that for values less than R = 0.25, the

required height necessary to equate the Wenzel and CB equations could not be reached.


the Truncated Cone at an 85° Slant angle for Superhydrophobic Optimisation Study.

R hW=CB R1 Cos θW Cos θCB θW=CB ( ° )

0.25 2.05 0.0704 -0.9898 -0.9898 171.79

0.3 1.18 0.1966 -0.9201 -0.9201 156.95

0.35 0.80 0.2801 -0.8378 -0.8378 146.91

0.4 0.54 0.3527 -0.7429 -0.7429 137.98

0.45 0.34 0.4201 -0.6353 -0.6353 129.44

0.5 0.18 0.4844 -0.5150 -0.5150 121.00

30 For the Truncated Cone protrusions at a slant angle of 80°. Values of R start at 0.36 because of

reason stated above.


the Truncated Cone at an 80° Slant angle for Superhydrophobic Optimisation Study.

R hW=CB R1 Cos θW Cos θCB θW=CB (°)

0.36 1.82 0.0397 -0.9967 -0.9967 175.43

0.4 0.97 0.2293 -0.8913 -0.8913 153.03

0.425 0.74 0.2953 -0.8197 -0.8197 145.06

0.45 0.56 0.3520 -0.7438 -0.7438 138.06

0.475 0.41 0.4034 -0.6636 -0.6636 131.58

0.5 0.28 0.4513 -0.5790 -0.5790 125.38

For the Truncated Cone protrusions at a slant angle of 75°. Values of R start at 0.47 because of

reason stated above.


the Truncated Cone at a 75° Slant angle for Superhydrophobic Optimisation Study.

R hW=CB R1 Cos θW Cos θCB θW=CB (°)

0.47 1.21 0.1470 -0.9553 -0.9553 162.85

0.48 0.95 0.2248 -0.8955 -0.8955 153.60

0.49 0.77 0.2830 -0.8344 -0.8344 146.56

0.5 0.63 0.3320 -0.7721 -0.7721 140.54

For the Paraboloidal Protrusions:


the Paraboloidal Protrusions for Superhydrophobic Optimisation Study.


0.05 9.22 -1.0000 -1.0000 180

0.1 4.67 -1.0000 -1.0000 180

0.2 2.44 -0.9998 -0.9998 178.88

31 0.3 1.73 -0.9981 -0.9981 176.47

0.4 1.39 -0.9907 -0.9907 172.19

0.5 1.18 -0.9685 -0.9685 165.57

For Hemisphere:

Since only one degree of freedom exist, then one variable cannot be varied to affect another

variable in the case of the other shapes in which R was varied and h was determined at that

particular R value. Therefore, only one value of R would be obtained at a particular θY. Hence,

the effect of different equilibrium contact angles �� on the dimensionless base radius R and thus,

the transition contact angle ��d$% was investigated. The above protrusions were studied at ��=

110°. However, �� ranging from 95° to the maximum intrinsic contact angle for any known

material i.e. 120° was tested.


the Hemispherical Protrusions for Superhydrophobic Optimisation Study.

θY Cos θY R Cos θW Cos θCB θW=CB (°)

95 -0.0872 0.562 -0.1736 -0.1736 100.00

100 -0.1736 0.554 -0.3412 -0.3412 109.95

105 -0.2588 0.540 -0.4962 -0.4962 119.75

110 -0.3420 0.520 -0.6324 -0.6324 129.23

115 -0.4226 0.493 -0.7454 -0.7454 138.19

120 -0.5 0.461 -0.8333 -0.8333 146.44

For Cuboid:


the Cuboid Protrusions for Superhydrophobic Optimisation Study.


0.05 4.76 -0.9934 -0.9934 173.42

0.1 2.31 -0.9737 -0.9737 166.83

0.2 1.01 -0.8947 -0.8947 153.47

32 0.3 0.51 -0.7631 -0.7631 139.74

0.4 0.22 -0.5789 -0.5789 125.37

0.45 0.10 -0.4671 -0.4671 117.84

0.46 0.08 -0.4431 -0.4431 116.30

0.47 0.06 -0.4186 -0.4186 114.75

0.48 0.04 -0.3936 -0.3936 113.18

0.49 0.019 -0.3681 -0.3681 111.60

0.5 0.000 -0.3421 -0.3421 110.00

For Cube:

Again, like the hemisphere, since the cube only has one degree of freedom, then only one

dimensionless base radius R would exist for each equilibrium contact angle. Hence, the effect of

different equilibrium contact angles �� on the dimensionless base radius R and thus, the

transition contact angle ��d$% was investigated. The above protrusions were studied at ��=

110°. However, �� ranging from 95° to the maximum intrinsic contact angle for any known

material i.e. 120° was tested.


the Cube Protrusions for Superhydrophobic Optimisation Study.

θY (°) Cos θY R Cos θW Cos θCB θW=CB (°)

95 -0.0872 0.425 -0.3394 -0.3394 109.84

100 -0.1736 0.369 -0.5510 -0.5510 123.44

105 -0.2588 0.323 -0.6908 -0.6908 133.69

110 -0.3420 0.285 -0.7863 -0.7863 141.84

115 -0.4226 0.252 -0.8530 -0.8530 148.54

120 -0.5 0.224 -0.9 -0.9 154.16

33 Graphs for Superhydrophobic Optimisation

110

120

130

140

150

160

170

180

190

0 0.1 0.2 0.3 0.4 0.5

θW

=C

B(

°)

Dimensionless Base Radius, R

Variation of θW=CB with the Dimensionless Base Radius, R for the Different Surface

Topographies Under Study

Cone 85

Cone 80

Cone 75

Paraboloid

Cuboid

Cylinder

85°80°

75°

Figure 13: Graph showing Variation of θW=CB with the Dimensionless Base Radius, R, for the Different Surface Topographies

Under Study for Superhydrophobic Optimisation.

34

Figure 14: Graph showing Dimensionless Height, h, Variation with respect to the Dimensionless Base Radius, R, at the

Corresponding Values of θW=CB for the Surface Topographies Under Study for Superhydrophobic Optimisation

-2

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

hW

=C

B


Dimensionless Height Variation with respect to the Dimensionless Base Radius, R, at

the Corresponding Values of θW=CB for the Surface Topographies Under Study

Cone 85

Cone 80

Cone 75

Paraboloid

Cuboid

Cylinder85°

80°

75°

35

Figure 15: Graph showing Transition Contact Angle θW=CB at Different Equilibrium Contact Angles θY for the Cube for

Superhydrophobic Optimisation Study.

Tabulated Results For Superhydrophilic Optimisation

These calculations were performed for θY = 60°.

For Cylinder:

Table 9: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Cylindrical Protrusions to Achieve

Superhydrophilicity

R hW Cos θW θW (°)

0.05 3.159 0.9962 5.00

0.1 1.579 0.9962 5.00

0.2 0.790 0.9962 5.00

0.3 0.526 0.9962 5.00

100

110

120

130

140

150

160

95 100 105 110 115 120 125

θW

=C

B(

°)

Equilibrium Contact Angle θY (°)

Transition Contact Angle θW=CB at Different Equilibrium Contact Angles for the Cube

36 0.4 0.395 0.9962 5.00

0.5 0.316 0.9962 5.00

For Paraboloid:

Table 10: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Paraboloidal Protrusions to

Achieve Superhydrophilicity

R hW Cos θW θW (°)

0.05 4.776 0.9962 5.00

0.1 2.443 0.9962 5.00

0.2 1.324 0.9962 5.00

0.3 0.984 0.9962 5.00

0.4 0.831 0.9962 5.00

0.5 0.748 0.9962 5.00

For Cuboid:

Table 11: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Cuboid Protrusions to Achieve

Superhydrophilicity

R hW=CB Cos θW θW=CB (°)

0.05 2.481 0.9962 5.00

0.1 1.240 0.9962 5.00

0.2 0.620 0.9962 5.00

0.3 0.413 0.9962 5.00

0.4 0.310 0.9962 5.00

0.45 0.276 0.9962 5.00

0.46 0.270 0.9962 5.00

0.47 0.264 0.9962 5.00

0.48 0.258 0.9962 5.00

0.49 0.253 0.9962 5.00

0.5 0.248 0.9962 5.00

For Hemisphere:

Table 12: Dimensionless Height, h, at Various Dimensionless Base Radii, R, Required for the Hemispherical Protrusions to

Achieve Superhydrophilicity

θY cos θY R Cos θW θW (°)

80 0.1736 1.228 0.9962 5.00

75 0.2588 0.952 0.9962 5.00

70 0.3420 0.780 0.9962 5.00

37 65 0.4226 0.657 0.9962 5.00

60 0.5000 0.562 0.9962 5.00

55 0.5736 0.484 0.9962 5.00

50 0.6428 0.418 0.9962 5.00

45 0.7071 0.361 0.9962 5.00

40 0.7660 0.309 0.9962 5.00

Graphs for Superhydrophilic Optimisation

Figure 16: Graph showing Dimensionless Height, h, vs Dimensionless Base Radius, R, for Superhydrophilic Surface

Optimisation Comparison

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6

Dim

en

sio

nle

ss H

eig

ht,

h


Dimensionless Height vs Dimensionless Base Radius for Superhydrophilic Surface

Comparison

Cylindrical

Paraboloid

Cuboid

38

Figure 17: Graph showing Dimensionless Base Radius, R, Required at Different Equilibrium Contact Angles θY to Bring About

Superhydrophilicity

Sample Calculations

Superhydrophobic Optimisation

Using the truncated cone at a slant angle of 85° and a dimensionless base radius, R = 0.25 to

demonstrate the sample calculations. Note, θY was used as 110°:

Using Excel’s Solver function, the dimensionless height of the protrusion at which the Wenzel

contact angle is equal to the CB contact angle was found to be ℎ�d$%= 2.05. This value can be

checked to ensure that this is true:

+� = + − ℎtan=

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

30 35 40 45 50 55 60 65 70 75 80 85

Dim

en

sio

nle

ss B

ase

Ra

diu

s, R

Equilibrium Contact Angle, θY

Dimensionless Base Radius Required at Different Equilibrium Contact Angles to

Bring About Superhydrophilicity

39 +� = 0.25 − 2.05tan 85° +� = 0.0704

Then, calculating the Wenzel contact angle:

cos ��3$ = 81 + *(+ + +�) 62(+ − +�), + ℎ, + +� − +79 cos ��

cos ��3$ = 81 + *(0.25 + 0.0704) 62(0.25 − 0.0704), + 2.05, + 0.0704 − 0.2579 cos 110° cos ��3$ = −0.9898

��3$ = 171.79° Calculating the CB contact angle:

��$%3$ = *+�,(1 + cos ��) − 1 ��$%3$ = *(0.0704),(1 + cos 110°) − 1

��$%3$ = −0.9898

�$%3$ = 171.79° Therefore, since ��3$ = �$%3$, then the dimensionless height ℎ�d$%= 2.05 calculated using Excel’s

Solver was, in fact, correct.

Superhydrophilic Optimisation

Sample calculation for cylinder at R = 0.05

Wenzel Equation for Cylinder:

cos ��$ = (1 + 2*+ℎ)��

Considering ��= 60°, we want to determine the height required to create a superhydrophilic

surface in which the dimensionless base pillar radius is 0.05. Therefore, to create a

superhydrophilic surface, the contact angle must be less than or equal to 5°. Therefore, we will

consider the height of the pillar when the surface first begins to be classified as a

40 superhydrophilic surface i.e. when the contact angle is 5°. So ��$ = 5°. The calculation would

proceed as follows:

cos(5°) = (1 + 2*40.055ℎ)��60° Therefore,

ℎ = > cos 5°cos 60° − 1?2*�0.05

ℎ = 3.159

Calculations for the paraboloid, cuboid and hemisphere were conducted in the same manner

using the relevant Wenzel equations for the paraboloid, cuboid and hemisphere.

41 Discussion

Figure (13) shows the transition contact angle between the Wenzel and the Cassie-Baxter

state, θW=CB, given as a function of the dimensionless base radius, R. This graph was developed

using the equations above, at a value of θY = 110°. The study was done for 6 different surface

topographies, that is, cylindrical protrusions, truncated cones, paraboloids, hemisphere, cuboid

and cube. Figure (13) displays results for the cylinder, the truncated cones, paraboloid and the

cuboid. The hemisphere and the cube will be discussed separately. The bold horizontal line at

θW=CB = 150° is used as a distinction above which surfaces are considered as being

superhydrophobic. Along with figure (13), figure (14), which displays the dimensionless height,

hW=CB, as a function of the dimensionless base radius, R, at the corresponding values of θW=CB

will be used in the analysis to determine the optimal surface topography for superhydrophobic

surfaces.

From figure (13), we see that the variation of θW=CB with the dimensionless base radius,

R, for the cylindrical protrusions and cuboid are linear over the entire range. It is seen that the

cuboid always gives a lower transition contact angle when both the cuboid and cylinder are

compared at the same dimensionless base radius. However, for cylindrical protrusions and the

cuboid protrusions, superhydrophobic surfaces are only achieved at small base radii (R < ~ 0.25).

When R>0.25, the contact angle becomes less than 150° and thus, the surface would no longer be

classified as a superhydrophobic surface. By comparison, the paraboloidal protrusions were

superhydrophobic over the entire range, displaying the lowest possible contact angle at the

largest base radius, measuring a contact angle of approximately 165°. The dimensionless base

radii for the truncated cones are very limited. In fact, from figure (13), θW=CB could not have

been calculated for the truncated cones for R < 0.25. This is because, at the small base radii,

usually high protrusions are required and due to the slant angle β, this limits the height of the

cone so that the necessary height hW=CB, at which the Wenzel contact angle and the CB contact

angle are equal, cannot be reached. Therefore, the slant angle affects the height of the truncated

cone and hence, the value of θW=CB. This suggests that, for the truncated cone at a slant angle of

85°, a higher protrusion height can be obtained than for the truncated cone at a slant angle of 75°.

42 For this reason, the truncated cone at 85° shows a wider range for the dimensionless base radii,

R, than the truncated cone at 75°.

From figure (13), we see that for small base radii (R < 0.25), the paraboloidal protrusions

as well as the cylindrical protrusions and cuboid protrusions shows high contact angles, above

150° and hence, within the range of being superhydrophobic. This is expected since the drop

would rest on mostly air pockets and only a small area of solid surface would support the drop.

However, the corresponding height of the protrusion to bring about this high contact angle at

these small base radii is generally large. This is illustrated in figure (14). Figure (14) indicates

that the heights of these small base radii for cylinders, cuboids and paraboloids are large,

particularly if the base radii, R, are less than 0.1. This is understandable since the roughness ratio

of the surface must be high to achieve high contact angles, and so, if the protrusion radii are

small, high pillars would be required in order to bring about a high roughness ratio (area of

wetted surface to area of projected surface). However, it should be kept in mind that high

protrusions might not be desirable, since they are more prone to erosion and mechanical

breakage. Thus, designs that do not require steep protrusions should be preferred [27]

.

For larger base radii (R > ~ 0.25), the droplet is now more supported by the solid surface

and less on air pockets. Generally, the larger the dimensionless base radius, the lower the contact

angle and the lower the dimensionless height that is required as seen from figures (13) and (14).

The lowest heights that are achieved between the dimensionless base radii range was obtained

for the cuboid protrusions and the cylindrical protrusions as seen from figure (14). However, at R

> ~ 0.25 for these types of protrusions, the contact angle is less than ~ 150°. Therefore, these

protrusions would not permit super-hydrophobicity.

It is also interesting to note however, that when R = 0.5 for the cuboid, this would mean

that the cuboid protrusion would now cover that entire unit cell and hence, this would essential

re-create a flat surface and no protrusion would actually exist. Hence, the transition contact angle

should be the original equilibrium contact angle that was initially considered i.e. 110°. From

figure (13), we see that at R = 0.5 for the cuboid, the transition contact angle is in fact 110° so

this confirms that the method and calculations are correct. Also, from figure (14), when the R =

0.5 for the cuboid, the dimensionless height equals zero so as already mentioned, the surface

43 essentially reverts back to behaving like a flat surface with no protrusions. These results are

particularly interesting since the calculated results reflect what was already thought was going to

happen.

The next lowest heights are achieved by the truncated cones. However, for the truncated

cones, their values for θW=CB are lower than the surface made by the paraboloids for most of the

range. In addition to this, the truncated cones exhibit super-hydrophobicity over a very limited

range, depending on the slant angle. It is important to notice that θW=CB of the truncated cones is

quite sensitive to the base radius, more than the other shapes studied here. It is also of interest to

notice that for β < * – θY (70° in the present calculations) only the Wenzel contact angle exists

since there is no location on the cone surface for which the contact line can make an actual

contact angle of θY [6]

.

For the case of the hemisphere and the cube, only one degree of freedom exists.

Therefore, at a particular equilibrium contact angle (say θY =110°), only one value of R would be

calculated since there is not two variables in which one can be varied and the other can be

calculated as was the case for the other topographies. Hence, the study was done at different

equilibrium contact angles and the corresponding values for R at that particular equilibrium

contact angle were determined.

The calculated results for R for the hemisphere were given in table (6). From this table,

we see that when the equilibrium contact angle is less than ~115°, the base radius required to

equate the Wenzel contact angle and the CB contact angle is greater than 0.5. As such, since a

unit cell is considered, it is clear that R cannot exceed 0.5 and hence, these values are not

acceptable. The values of R only drop below 0.5 at equilibrium contact angles greater than or

equal to ~115°. However, even at the highest possible equilibrium contact angle that is attainable

for currently known materials, that is, even when the equilibrium contact angle is 120°, the

transition contact angle only reached 146.44° which is still not even considered as a

superhydrophobic surface. Hence, the hemisphere is considered to be a very ineffective surface

topography for a superhydrophobic surface.

Similar results are obtained for the cube. Again, since the cube only has one degree of

freedom, the investigation was done varying the equilibrium contact angle and determining the

44 transition contact angle that is attainable. Figure (15) shows a graphical representation of the

equilibrium contact angle versus the transition contact angle for the cube. Unlike the hemisphere,

the dimensionless base radii are always below 0.5. However, the problem lies in that

superhydrophobic behaviour (transition contact angles > 150°) are only achieved at equilibrium

contact angle of greater than 116°. Since the highest possible equilibrium contact angle for

known materials only reach up to 120°, this means that a very limited group of materials can be

made of cube-type protrusions to obtain superhydrophobicity. Also, even at 120°, the transition

contact angle only measures 154.16° so that it just about classifies as a superhydrophobic

surface. This transition contact angle is very low when compared to those that are attained by the

paraboloids.

From figure (13), it is clear that the paraboloids exhibit excellent superhydrophobic

properties over the entire range of R. However, from figure (14), it can be seen that the highest

protrusions are required for the paraboloids. This may lead someone into thinking that the

paraboloids would be most susceptible to mechanical breakage. Although the required height of

protrusion is relatively high, the contact angle achieved is also high. Consider the paraboloidal

protrusion, the cylinder and the cuboid. If a transition contact angle of about 166° is required,

from figure (13) and (14), a dimensionless base radius of 0.5 and height 1.2 is required for the

paraboloid. However, to achieve this transition contact angle for the cylinder, R should be ~0.12

and h ≈ 2.4. Similarly, for the cuboid, to achieve a transition contact angle of 166°, R needs to be

0.105 and h ≈ 2.2. Therefore, it can be seen that to achieve a particular transition contact angle,

the height of the paraboloid is lower than that for the cylinder and cuboid suggesting that it

would be less susceptible to mechanical breakage.

The optimal surface topography for a superhydrophilic surface was also investigated.

Figure (16) shows the relationship between the dimensionless heights required to bring about

superhydrophilic at different dimensionless base radii. The figure shows results for the

cylindrical, paraboloidal and cuboid-type protrusions. The cube and hemisphere would be

discussed separately. From the figure, it is evident that at any value of R, the required height to

bring about superhydrophobicity is always less for the cuboid than either of the other two shapes.

This is understandable and can be explained. The cuboid would give the largest surface area

exposed to the liquid at the same radius for the cylinder or the paraboloid. As such, the height to

45 acquire a certain roughness ratio would be the smallest for the cuboid. Therefore, since the

cuboid gives the smallest height to achieve super-hydrophilicity, it would be least prone to

mechanical breakage. The cylindrical protrusions give heights that are smaller than the

paraboloid but larger than the cuboid.

Again, due to only one degree of freedom in the cube and hemisphere, only one value of

R is determined for each equilibrium contact angle. Therefore, the variation of the equilibrium

contact angle with the dimensionless base radius was investigated. For the cube, it was seen that

at an equilibrium contact angle of 60°, the required base radius R would have been:

cos 5° = (1 + 16+,) cos 60° Therefore:

+ =�> cos 5°cos 60° − 1?16

+ = 0.249

And the resulting height would be 2�0.249 = 0.498. This is the same height that would be

required for the cuboid if R = 0.249. Therefore, the cuboid already incorporates values that

would have been obtained by the cube.

The hemisphere’s variation between the equilibrium contact angles and the dimensionless

base radius are displayed in figure (17). Since R cannot be greater than 0.5, this implies that the

hemispherical protrusions cannot be used for materials where the equilibrium contact angle is

greater than approximately 52° because when the equilibrium contact angle is greater than 52°,

then too large a radius is required and it gives an impractical answer. Therefore, the use

hemispherical protrusions are severely restricted. Also, at an equilibrium contact angle of 60°,

the dimensionless base radius R ≈ 0.562 which is impractical so it cannot be used. Therefore, the

hemisphere is not an effective topography for super-hydrophilicity.

It is important to mention that, for the cuboid, when R = 0.5, the height required to bring

about superhydrophilicity is h = 0.248. This is incorrect since at R = 0.5, the surface would be a

flat surface with no protrusions so the expected contact angle should be the equilibrium contact

46 angle i.e. 60° in this case. A possible and likely explanation for this inconsistence is because the

Wenzel equation for the cuboid was determined by considering the lateral sides being exposed to

the liquid. If the radius is 0.5, then no lateral sides would be exposed to the liquid. Hence, we can

deduced, that the Wenzel equation for the cuboid is valid within the range 0 < + < 0.5. That is,

R cannot be equal to 0.5. When R = 0.5, the liquid is exposed only to the top face whose area

shall now be 2+�2+ = 1�1 = 1. Hence, the roughness ratio would now be 1 so that, from the

Wenzel equation, we see that cos θW = cos θY. This is what was expected at R = 0.5 for the

cuboid.

47 Conclusion

From the analysis of the six (6) different topographies, the following conclusions can be made

about the optimisation of a superhydrophobic surface:

1. The paraboloidal protrusions seem to offer the best choice for the optimal design of a

superhydrophobic surface based on the criteria of a high water contact angle. At first

glance, it is evident that at the smaller base radii, superhydrophobicity is achieved by the

cylinder, cuboid and paraboloid. However, it should be remembered that small base radii

are not desired from a mechanical point of view. At the higher base radii, the cylinder and

the cuboid do not exhibit superhydrophobicity. Also, due to the very limited range of R

for the truncated cones as well as the fact that the transition angle is generally always

lower than that of the paraboloid, it is apparent that the truncated cones do not offer the

optimal surface topography.

2. The cube and the hemisphere protrusions were shown to be very ineffective as

topographies for the manufacture of superhydrophobic surfaces. This emphasizes the

need for at least two degrees of freedom when rough structures are being produced on

superhydrophobic surfaces.

3. Therefore, all other topographies were considered obsolete in comparison with the

paraboloidal protrusions. This finding is particularly interesting because these

paraboloidal protrusions resemble the protrusions used by the lotus leaf to achieve

superhydrophobic properties.

For the optimisation of the superhydrophilic surface:

1. The cuboid was seen to display the lowest height required at the same base radius to

achieve superhydrophilicity. Therefore, it would be the least susceptible to mechanical

breakage and as such, would be more durable. Therefore, the optimal surface topography

for superhydrophilicity was the cuboid from the topographies tested.

2. The hemisphere only achieves superhydrophilicity within a small range of equilibrium

contact angles. Again, this emphasizes the need for at least two degrees of freedom.

3. There exists an inconsistency at R = 0.5 for the cuboid due to the way in which the

Wenzel equation was developed.

48 References

1. Adhesives & Sealants Overview. 2008. “Surface Energy and Wetting.” Accessed

February 6, 2013. http://www.adhesives.org/adhesives-sealants/adhesives-sealants-

overview/structural-design/surface-energy-and-wetting

2. Anastasiadis, Spiros. H., Maria I. Lygeraki, Athanassia Athanassiou, Maria Farsari, and

Dario Pisignano. 2009. “Reversibly Photo-Responsive Polymer Surfaces for Controlled

Wettability.” Journal of Adhesion Science and Technology 22 (15): 1853-1868.

doi:10.1163/156856108X320014.

3. Balu et al. 2009. “Tunability of the Adhesion of Water Droplets on a Superhydrophobic

Paper Surface via Selective Plasma Etching”. Journal of Adhesion Science and

Technology 23: 361-380. doi:10.1163/156856108X383547

4. Berglund, Sara. 2009. “Modeling Channel Flow over Superhydrophobic Surfaces.”

Master’s thesis. Brown University, Providence, Rhode Island

5. Bird, John. 2001. Newnes Engineering Mathematics Pocket Book. Burlington: Elsevier.

6. Bittoun, E., and Marmur, A. 2009. “Optimizing Super-Hydrophobic Surfaces: Criteria for

Comparison of Surface Topographies.” Journal of Adhesion Science and Technology 23

(3): 401-411

7. Carré, A., and K.L. Mittal. 2009. Superhydrophobic Surfaces. Leiden, The Netherlands:

Koninklijke Brill NV.

8. Cavalli, Andrea, Peter Boggild, and Fridolin Okkels. 2013. “Topology Optimisation of

robust superhydrophobic surfaces.” doi:10.1039/b000000x.

9. Cho, Kwun Lun. 2011. “Development of Superhydrophobic Coatings”. Ph.D Thesis.

University of Melbourne, Parkville VIC 3010, Australia.

10. Cui, H., G.Z. Yang, Y. Sun, and C.X. Wang. 2010. “Reversible ultraviolet light-

manipulated superhydrophobic-to-superhydrophilic transition on a tubular SiC

nanostructure film”. Applied Physics Letters 97 (18). Accessed May 16, 2013.

http://apl.aip.org/resource/1/applab/v97/i18/p183112_s1?isAuthorized=no

11. D’Urso, R. Brian, and Simpson, T. John. 2010. Super-hydrophobic water repellent

power. EP 2,150,650 A1, filed May 1, 2008 and issued February 10, 2010.

49

12. Duez, C., Ybert, C., Barentin, C., Cottin-Bizonne, C., and Bocquet, L. 2008. “Dynamics

of Fakir Liquids: From Slip to Splash.” Journal of Adhesion Science and Technology 22

(3-4): 335 – 351

13. Ensikat, Hans J., Petra Ditsche-Kuru, Christoph Neinhuis, and Wilhelm Barthlott. 2011.

“Superhydrophobicity in perfection: the outstanding properties of the lotus leaf”.

Beilstein Journal of Nanotechnology 2: 152-161.

14. Flemming, Marcel, Luisa Coriand, and Angela Duparre. 2009. “Ultra-hydrophobicity

Through Stochastic Surface Roughness.” Journal of Adhesion Science and Technology

23 (3): 381-512

15. Frechet, M.J. Jean, A. Pavel Levkin, and Frantisek Svec. 2011. Superhydrophobic and

superhydrophilic materials, surfaces and methods. EP 2,283,067 A2, filed April 21, 2009

and issued February 16, 2011.

16. Fresnais, J., J.P. Chapel, L. Benyahia, and F. Poncin-Epaillard. 2009. “Plasma-Treated

Superhydrophobic Polyethylene Surfaces: Fabrication, Wetting and Dewetting

Properties.” Journal of Adhesion Science and Technology 23 (3): 447-467.

doi:10.1163/156856108X370127.

17. Guo, C., Wang, S., Liu, H., Feng, L., Song, Y., and Jiang, L. 2008. “Wettability

Alteration of Polymer Surfaces Produced by Scraping.” Journal of Adhesion Science and

Technology 22 (3-4): 395 – 402

18. Harder, P.M., Shedd, T.A., and Colburn, M. 2008. “Static and Dynamic Wetting

Characteristics of Nano-patterned Surfaces.” Journal of Adhesion Science and

Technology 22 (15): 1931 – 1948

19. Hatch, Jeremy. 2006. “Switchable Surfaces; Superhydrophilic to Superhydrophobic.”

20. Hwang, Woon-Bong, Dong-Hyun Kim, Kun-Hong Lee, and Hyun-Chul Park. 2008.

Method for fabricating superhydrophobic surface and solid having superhydrophobic

surface structure by the same method. WO 2,008,004,827 A1, filed July 5, 2007 and

issued January 10, 2008.

21. Industrial Adhesives and Tapes Division. 2012. “Innovations in Bonding to Low Surface

Energy Surfaces.” Accessed February 9, 2013.

50 http://multimedia.3m.com/mws/mediawebserver?mwsId=SSSSSufSevTsZxtUo8_9nx2B

evUqevTSevTSevTSeSSSSSS--&fn=Final%20LSE%20Bonding%20Paper.pdf

22. Jennissen, H.P., S. Lüers, M. Laub. 2012. “Hyperhydrophilic Surfaces, the Inverse Lotus

Effect and Imaginary Contact Angles”. Biomed Tech 57. doi: 10.1515/bmt-2012-4464.

23. Kim,S.H. 2008. “Fabrication of Superhydrophobic Surfaces.” Journal of Adhesion

Science and Technology 22 (3-4): 235 – 250

24. Kobrin, Boris. 2012. Fabrication of Nanostructured Devices. US Patent 8,318,386 B2,

filed August 6, 2009 and issued November 27, 2012.

25. Lafuma, Aurélie, and David Quéré. 2003. “Superhydrophobic States”. Nature materials

2. doi:10.1038/nmat924.

26. Latthe, S. S., Gurav, A. B., Maruti, C. S., and Vhatkar, R. S. 2012. “Recent Progress in

Preparation of Superhydrophobic Surfaces: A Review.” Journal of Surface Engineered

Materials and Advanced Technology 2: 76 – 94

27. Marmur, Abraham. 2004. “The Lotus Effect: Superhydrophobicity and Metastability.”

Langmuir 20: 3517-3519.

28. Martines, Elena, Kris Seunarine, Hywel Morgan, Nikolaj Gadegaard, Chris D.W.

Wilkinson, and Mathis O. Riehle. 2005. “Superhydrophobicity and Superhydrophilicity

of Regular Nanopatterns”. Nano Letters 5 (10): 2097-2103.

29. Nosonovsky, Michael, and Bharat Bhushan. 2007. “Hierarchical roughness optimisation

for biomimetic superhydrophobic surfaces.” Ultramicroscopy 107: 969-979

30. Patankar, Neelesh A. 2009. “Hydrophobicity of surfaces with Cavities: Making

Hydrophobic Substrates from Hydrophilic Materials?” Journal of Adhesion Science and

Technology 23 (3): 381-512

31. Qin, Jian, Donald E. Waldroup, Constantine M. Megaridis, Thomas M. Schutzius, and

Ilker S. Bayer. 2013. Superhydrophobic Surfaces. US Patent 2013/0029551 A1, filed July

28, 2011 and issued January 31, 2013.

32. Quéré, David. 2002. “Rough Ideas on Wetting”. Physica A 313: 32-46

51 33. Roach, P., Shirtcliffe, N.J., and Newton, M.I. 2008. “Progress in superhydrophobic

surface development.” Journal of The Royal Society of Chemistry 4: 224 – 240.

doi:10.1039/b712575p

34. Tanton, James Stuart. 2005. Encyclopedia of Mathematics. New York: Facts on Files,

Inc.

35. Vogler, Ervin A. 1998. “Structure and Reactivity of Water at Biomaterial Surfaces.”

Advances in Colloid and Interface Science 74: 69-117

36. Wang, Evelyn N., Matthew McCarthy, Ryan Enright, James N. Culver, Konstantinos

Gerasopoulos, and Reza Ghodssi. 2013. Superhydrophobic Surfaces. US Patent

2013/0059123 A1, filed March 4, 2011 and issued March 7, 2013.

View publication statsView publication stats

https://www.researchgate.net/publication/301677106

superhydrophobic & superhydrophilic surfaces sanjay ... · derived in terms of a dimensionless...

Documents