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Droplet shape prediction on hierarchical and anisotropic microstructures


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Droplet shape prediction on hierarchical and anisotropic microstructures

Master’s thesis Edwin de Jong

s1705938 Supervision:

Prof.dr.ir. E. van der Giessen Prof.dr.ir. P. R. Onck

Micromechanics of Materials group Zernike Institute for Advanced Materials

University of Groningen

October 9, 2013



This master’s thesis covers two subjects in the theory on wetting. The basic theories are explained and through an introduction in contact angle calculations on microstructures, hierarchical structures are treated. An expression for the critical roughness for a droplet to be in the Cassie-Baxter state is found to be only dependent on one part of the microstructure unit cell. A generalization is made for additional hydrophobic states in hierarchical structures, resulting in state diagrams as a function of multi-level surface roughness. For two-level hierarchy, a critical four-state point is found for high nano- level roughness and relatively low micro-level roughness. In the second part, a finite-element model is developed for microstructures that exhibit anisotropy. The droplet shape is then also anisotropic, and the droplets are being modeled based only on the assumption that every path from the center to the three-phase line is circular. For this, only the three-phase line has to be modeled. The model converges as expected, but the final droplet shapes are difficult to compare due to the lack of high- resolution material for comparison. The result heavily depends on the expression for the equilibrium contact angle, which is a function of the normal vector of the three-phase line. Quantities like length- over-width ratio and dependance of the droplet shape on the line tension are also computed and discussed.



1 Introduction 5

1.1 Nature-inspired physics . . . 5

1.2 Applications and importance . . . 6

1.3 Outline of the thesis . . . 6

2 Basic theories on wetting 7 2.1 The Young Equation . . . 7

2.2 The Wenzel and Cassie-Baxter Equations . . . 8

2.3 States of hydrophobicity . . . 10

2.4 Contact angle hysteresis . . . 11

2.5 Definition of superhydrophobicity . . . 13

I Hierarchical structures 14

3 Microstructures 15 3.1 Grooves and pillars . . . 15

3.2 State transitions . . . 16

3.3 Favourable state . . . 16

3.3.1 Energy calculations for critical roughness of grooves and pillars . . . 16

3.4 Critical roughness in general . . . 19

3.5 Goals in microstructure design . . . 19

3.6 Special structures . . . 20

4 Hierarchical and self-similar geometries 21 4.1 Combinations of states . . . 21

4.2 State diagrams of multirough structures . . . 22

4.3 Self-similar structure . . . 23

4.4 Advantages of hierarchy . . . 23

4.5 Discussion . . . 23

5 Summary and conclusion of Part I 25 5.1 Summary . . . 25

5.2 Conclusion . . . 25

II Anisotropic wetting 27

6 Introduction and proposal for model 28 6.1 Anisotropic wetting . . . 28

6.1.1 The process of the shaping of a droplet . . . 28

6.2 An alternative model on anisotropic wetting . . . 29

6.2.1 Limitations . . . 30


7 Force on the three-phase line 32

7.1 General force equation . . . 32

7.2 Measurable contact angle based on circular paths . . . 32

7.3 Curvature and three-phase line tension . . . 35

7.4 Equilibrium contact angle . . . 36

8 Finite-element representation 38 8.1 Definitions . . . 38

8.2 Translation and finding an equilibrium . . . 39

8.3 Shape functions . . . 39

8.3.1 Linear shape functions . . . 39

8.3.2 Quadratic shape functions . . . 40

8.4 Nodal velocities . . . 40

8.5 Implementation . . . 42

8.5.1 Base transformations . . . 42

8.5.2 Integration using Gaussian quadrature . . . 43

8.5.3 Building the [FiA]vector . . . 43

8.5.4 Building the [KAB]matrix . . . 44

8.5.5 Boundary condition and stopping criterion . . . 44

9 Results and discussion 46 9.1 Convergence . . . 46

9.2 Model demonstration . . . 49

9.3 Droplet shape dependence on σψ . . . 51

9.4 Comparing Ξ values with literature . . . 52

9.5 Ξ dependence on line tension σ . . . 53

9.6 Full droplet shape comparison . . . 54

9.7 Discussion . . . 56

10 Summary and conclusion of Part II 57 10.1 Summary . . . 57

10.2 Conclusion . . . 57

Bibliography 59 A Analytical derivations 64 A.1 The Young Equation derived by means of energy minimization of a spherical cap droplet on an isotropic, homogeneous substrate . . . 64

B Codes 67 B.1 Main script . . . 67

B.2 Custom functions . . . 72


Chapter 1


1.1 Nature-inspired physics

Biomimicry is a very hot topic in modern science: many concepts in modern technology were inspired by properties of animals and plants, which they have used for thousands of years [1]. Optic fibers, distortion free lenses, more uses for energy from sunlight, and so on. These technologies were, until recently, not necessarily available to humans, but plants, animals and insects, evolved in time so that their properties optimized (for example, to prevent being eaten by their natural enemies) [2]. The lotus flower is among these organisms. The (much studied, e.g. [3, 4]) leaf of this flower is known for its outstanding self-cleaning properties, which is because they exhibit natural superhydrophobicity (see Figure 1.1).

Another example of natural water-repellent surfaces is the leg of a water strider. The legs contain very small hairs (∼ 20µm), which have even smaller (∼ 200nm) grooves along the length. The small creature, however, does not use its superhydrophobic legs to stay clean, but to “walk” on water (see Figure 1.2). It is reported that the superhydrophobicity of the legs enables the creature to support approximately ninety times its own body weight on the surface tension of water [5].

The lotus leaf and water strider have in common that their superhydrophobic properties do not fully emerge from the materials that are present on their respective surfaces. It is the clever combi- nation of material and microstructure that is key to water repulsion. More specifically, both exhibit

Figure 1.1: The lotus leaf. (a) SEM micrographs showing the hierarchical structure. (b) A water droplet on top of a lotus leaf. Image taken from [1].


Figure 1.2: Photograph of a water strider. The superhydrophobic legs of this creature are able to support up to ninety times the body weight.

hierarchy in the microstructure, which means that on multiple length scales (say on the microme- ter and nanometer scale), a typical structure is found [6]. It appears that this hierarchical design is required, if we want to imitate these self-cleaning surfaces [7].

1.2 Applications and importance

The foremost application of superhydrophobic materials is in self-cleaning surfaces. This is already being done commercially by several companies (e.g. Ultra-Tech Ultra-Ever Dry industrial superhy- drophobic coating [8] and Südwest Lotusan superhydrophobic wall paint [9]). Self-cleaning requires that the surface exhibits properties as high water repellency and low roll-off angles. Additional re- quirements can be robustness, transparency or resistance against heat.

However, in applications like microfluidic systems, it is also very important that the exact behaviour and shape of the droplets can be predicted, for example when the flow of small amounts of water must be controlled very precisely within the dimensions of a microchannel [10].

1.3 Outline of the thesis

This thesis consists of two parts. After a general review of the basic theories on wetting, covering the Young, Wenzel and Cassie-Baxter models, the first part will treat hierarchical structures. Several microstructures are discussed and some example calculations of the Wenzel and Cassie-Baxter predictions for these surfaces are given. The effect of adding hierarchy to the microstructure is treated. Particular attention is given to different states of hydrophobicity that hierarchical structures can have.

In the second part, a finite-element method is developed to predict the anisotropic droplet shape on surfaces with anisotropic microstructures. A short introduction to anisotropic wetting is given, the equations that govern the force at the solid-fluid-gas boundary of a droplet (the three-phase line) are derived and are represented in a finite-element model.


Chapter 2

Basic theories on wetting

In this Chapter, the basic theories on wetting and hydrophobicity will be recapitulated and it will be demonstrated how to apply them. The derivations in this chapter are partially based on [11].

2.1 The Young Equation

Robert Young was one of the first to clearly describe the contact angle of a liquid on a surface in his 1805 essay on the cohesion of fluids: (...) for each combination of a solid and a fluid, there is an appropriate angle of contact between the surfaces of the fluid, exposed to the air, and to the solid [12]. This angle is the solution to the force balance on the intersection of the solid-liquid, solid-vapour and liquid-vapour interfaces (the three-phase line). These forces are a consequence of the interfacial surface tensions, or surface free energies, denoted by γSL, γSV and γLVfor solid-liquid, solid-vapour and liquid-vapour, respectively. As illustrated in Figure 2.1, the force balance reads

γLVcos θY+ γSL= γSV, (2.1)

from which the contact angle can be solved as

cos θY= γSV− γSL

γLV . (2.2)

Here, the subscript Y emphasizes that this is the Young contact angle. The assumption was made that the interfacial surface tensions are homogeneous in the neighborhood of the three-phase line.

Vapour Liquid

Solid θ




Figure 2.1: The force balance that has to be solved for the contact angle. The γ’s denote the interfacial surface free energies, or surface tensions.


Material Contact angle Polyisobutylene (butyl rubber) 112.1

Paraffin 108.9

Polydimethlsiloxane (PDMS) 107.2

Polystyrene 87.4

Polyethylene oxide 63

Table 2.1: A few materials and their equilibrium contact angles with respect to water and air [13]. PDMS is widely used in literature.

The Young Equation (and corresponding angle) forms the basis of the theory on wetting and hy- drophobicity.

Because we have only looked at forces at the three-phase line, the “droplet” here is infinitely long and cylindrical. But the same result is found when taking into account the shape of a small droplet, as is derived in Appendix A.1. The derivation of Equation (2.2) using minimization of the surface free energy of a spherical cap droplet on a substrate is performed. The energy approach, as described in the appendix is only valid for surfaces that are homogeneous and isotropic.

2.2 The Wenzel and Cassie-Baxter Equations

There are no known smooth surfaces that have contact angles larger than 120[14], whereas several surfaces in nature are known to exhibit contact angles of up to 160 (e.g. [4]). This is due to the fact that the surface is not completely flat. Wenzel [15] introduced the dimensionless roughness parameter r, that measures the increase in the net surface area due to roughness,

r≡ Areal

Aproj, (2.3)

where Arealand Aproj≤ Arealare the real surface area and projected surface area onto the horizontal plane, respectively. This roughness parameter affects the solid-vapour and solid-liquid parts of the Young Equation only: changing γSV to rγSV and γSL to rγSL. This changes the force balance from Equation (2.2) into

γLVcos θW= rγSV− rγSL

cos θW= rγSV− γSL

γLV = rcos θY, (2.4)

where the subscript W indicates that this is the Wenzel contact angle. Note that r amplifies the contact angle with respect to the 90 line: e.g. if the material has θY< 90, r will make the contact angle smaller, while materials with θY> 90 will see an increase in contact angle if r increases. This is illustrated in Figure 2.2a, where it can be seen as diverging lines from the 90line.

However, Wenzel’s approach was not enough to describe the physics behind the self-cleaning properties of the lotus leaf. In 1944, Cassie and Baxter proposed that air pockets may exist un- derneath a droplet, increasing the contact angle significantly. They divided the surface into multiple components with their own material properties and averaged out the small, local effects to obtain the contact angle of the macroscopic surface [16]

γLVcos θCB=∑







rifiSV,i− γSL,i) cos θCB=∑

rificos θY,i,



0 0.2 0.4 0.6 0.8 1 60

80 100 120 140 160 180


θ CB


θY = 70 θY = 90 θY = 100 θY = 110 θY = 120

1 1.5 2 2.5 3

0 20 40 60 80 100 120 140 160 180


θ W

(b) θY = 70°

θY = 80°

θY = 90°

θY = 100°

θY = 110°

Figure 2.2: (a) The effect of roughness parameter r (Equation (2.3)) on the contact angle, according to the Wenzel Equation (Equation (2.4)). If the contact angle of the material is smaller than 90, roughness will only reduce the contact angle. The reverse happens if the contact angle is higher than 90. (b) The effect of surface fraction parameter fson the contact angle, according to the modified Cassie-Baxter Equation (Equation (2.7)).

If the surface fraction goes to zero, the droplet will not be in contact with the surface and will exhibit a contact angle of 180.


where the subscript CB is used to emphasize that this is the Cassie-Baxter contact angle. Here, the ri and fi are the material properties of component i of the composite surface. This general form may also be written as

γLVcos θCB= γSV− γSL, (2.6) where we simply pose that the surface free energies γSV = ∑

irifiγSV,iand γSL = ∑

irifiγSL,i are a function of the surface microstructure, which includes parameters as surface roughness r and surface fractions fi. The effects due to the microstructure are averaged out on the macro scale and may be treated as an effective material property.

A commonly used form of Equation (2.5) is when only two components are taken into account.

The first is the solid (with intrinsic contact angle θYand roughness rs; and fsis the surface fraction of the droplet that is in contact with the solid) and the second component is air. This is illustrated in Figure 2.3. It is assumed that the liquid-air interface exhibits a contact angle of θY= 180, resulting in a specific form of the Cassie-Baxter Equation,

cos θCB = fsrscos θY+ (1− fs)cos(180)

= fs(rscos θY+ 1)− 1. (2.7)

Note that in the limit fs→ 1 (full solid contact), the Wenzel Equation is retrieved.

2.3 States of hydrophobicity

The publication of Cassie and Baxter explained the lotus effect, the effect that droplets “rest” on the top of a pillar-like structure: the droplet bottom is not fully in contact with the surface. Using the modified form of the Cassie-Baxter Equation, Equation (2.7), the large contact angles of the lotus flower (and lotus-like surfaces) can be explained. For this reason, this state of hydrophobicity is denoted as the Cassie-Baxter state of hydrophobicity. See Figure 2.4c for an impression of this state.

There is one important requirement to the Cassie-Baxter state. The pillars or grooves described must be much smaller than typical droplet dimensions (∼ 1µm versus ∼ 1mm). This separation of length scales is needed because if the dimensions of the microstructure are comparable to the size of the droplet, distortions will be visible on droplet scale and the Cassie-Baxter model will not hold anymore. On the other hand, there is also the possibility that the droplet could lower along the pillars due to the gravitational force on the droplet [17].

When the droplet bottom makes full contact with the surface, the Wenzel Equation applies. That is why this state is referred to as the Wenzel state of hydrophobicity (or wetting). Because of the full contact, this state is also referred to as homogeneous wetting. See Figure 2.4b for an impression of this state and the difference with the Cassie-Baxter state.

fs 1-fs “Air pockets”

Figure 2.3: The situation for the commonly used form of the Cassie-Baxter equation, Equation (2.7).





Figure 2.4: Different states of hydrophobicity, with surface structures with exaggerated dimensions. (a) A droplet on a perfectly smooth, flat surface. This is how Young first regarded solid-liquid contact. (b) The Wenzel state of hydrophobicity, where the bottom surface of the droplet is in physical contact with the substrate material.

(c) The Cassie-Baxter state of hydrophobicity. The droplet bottom surface only makes partial contact with the substrate material, while the rest of the droplet bottom surface hovers.

2.4 Contact angle hysteresis

Surfaces usually do not have one contact angle, but can maintain a range of contact angles. The main cause of this is irregularities and heterogeneities on the surface [14]: the irregularities may have completely different wetting properties, causing the three-phase line to stick, making a certain range of contact angles possible. This range is defined by the lower and upper bound contact angles. If the contact angle is above the upper bound, the three-phase line will advance until the angle has reached this upper bound. Reversely, if the contact angle is below the lower bound, the three-phase line will recede until the lower bound contact angle is reached. This is why the upper and lower bound are called the advancing and receding contact angles θA and θR, respectively. The contact angle hysteresis is then defined as

∆θ≡ θA− θR. (2.8)

In Figure 2.5, two droplets are shown on a surface with many irregularities. This heterogeneity gives rise to a large range of equilibrium contact angles. Virtually every surface exhibits contact angle hysteresis. Only when structures are hierarchical (Chapter 4) or atomically smooth [18], the range can become so small that the hysteresis is negligable.

A large amount of theoretical studies have been conducted on contact angle hysteresis (e.g. [19, 20, 21, 22, 23, 24, 25, 26]), but unlike the Young, Wenzel and Cassie-Baxter equations, there is not a generally accepted method of predicting it in all hydrophobic states. For well-defined geometries (e.g. a sinusoidal height profile [27]), it is accepted that the advancing and receding angles are


Figure 2.5: Two water droplets on poorly prepared candle wax. Due to the heterogeneity of the surface, it shows large contact angle hysteresis. Both droplets are in static equilibrium, but do show a large difference in contact angle.

obtained from θYand the maximum and minimum slope obtained from the height profile:

S =maxdz dx α≡ arctan S θA= θY+ α θR= θY− α.


See Figure 2.6 for an illustration on this approach.

Another accepted phenomenon is pinning [14, 28]. If sharp edges on the microstructure of the surface are present, the three-phase line tends to pin to those sharp edges and is therefore hindered from moving. However, it is very difficult to quantify this effect theoretically. For example, in [29], work of adhesion is used to develop a theoretical model. Given the work per unit area needed to adhere a droplet of contact angles θR and θAto a smooth surface, they pose that the hysteresis is simply calculated by multiplying the difference with r and fs. To account for pinning, a phenomenological term is added, which has to be measured separately.

This “pinning term” seems fairly arbitrary, since it means that the surface first has to be constructed and angles have to be measured. This renders such theories unusable because they lack the ability to predict droplet behavior: the theory does not really predict, since we still need to measure both θA

and θRof the microstructured surface.

Many other theories use work of adhesion, but also line energy is sometimes used in order to find correlations (e.g. [25]). We will not go into detail on these theories here.


θY θR

(a) (b) (c)

Figure 2.6: Contact angle hysteresis on a periodic surface. (a) Droplet on a perfectly smooth surface. (b) Minimum angle possible on the surface, resulting in the receding contact angle, θR. (c) Maximum angle possible on the surface, resulting in the advancing contact angle, θA. On the macroscopic level, the difference between these angles is observed as contact angle hysteresis.


2.5 Definition of superhydrophobicity

In the above we have discussed wetting and hydrophobic materials. Some literature mentions su- perhydrophobicity, or ultrahydrophobicity for surfaces with exceptional hydrophobic properties. The definition of superhydrophobicity is fairly arbitrary (it is, in fact, nothing more than “very hydrophobic”), but usually the following definition is taken to be [14]:

Superhydrophobic surfaces are surfaces that exhibit

• θ > 150(this is very consistent in the literature); and

• ∆θ < 10(although many papers use ∆θ < 5).


Part I

Hierarchical structures


Chapter 3


Inspired by structures from nature (like the lotus leaf), experimentalists have created many different microstructures. In this chapter, we will discuss different geometries and the usage of the Wenzel and Cassie-Baxter Equations from Chapter 2 is demonstrated.

3.1 Grooves and pillars

The most basic ordered structures that are made are grooved or pillared structures. These are usually made using special etching techniques on a silicon substrate [30]. Depending on the dimensions of the structure, the contact angle for pillars or grooves can be predicted using the Wenzel or Cassie- Baxter Equations (Equations (2.4) or (2.5)). Some illustrations of cylindrical pillars, square pillars and square grooves are given in Figure 3.1.

For example, if we look at the square pillars, the possible contact angles can be calculated as follows. If the droplet is in the Wenzel state, the droplet bottom has full contact with the surface area of the pillar. The sides of the pillar therefore contribute to the real surface area, while the projected surface area is just a simple square with area (a + b)2. The roughness factor becomes

r = Areal

Aproj =(a + b)2+ 4ah (a + b)2

= 1 + 4ah (a + b)2.


Combined with the Wenzel Equation, the Wenzel contact angle would become cos θW= rcos θY=


1 + 4ah (a + b)2


cos θY, (3.2)


L h




h d

a+b a

h a

Figure 3.1: Three examples of microstructures that can be made in experiment. The square pillar and groove are used in this Chapter.


with θYthe contact angle of the used material.

If the droplet would be in the Cassie-Baxter state, the droplet would only make contact with the top of the pillar. From this, we can deduce the solid fraction

fs= a2

(a + b)2. (3.3)

If we then assume that the top of the pillars are flat (there is no roughness on top of the pillar, rs= 1), the contact angle can be derived from the Cassie-Baxter Equation,

cos θCB= fs(cos θY+ 1)− 1 = a2

(a + b)2(cos θY+ 1)− 1. (3.4) Similar analysis can be performed on grooves. The resulting contact angles then become

cos θW= (

1 + 2h a + b

) cos θY

cos θCB = a

a + b(cos θY+ 1)− 1.


3.2 State transitions

When looking at engineered structures like grooves or pillars, a new phenomenon arises that is not often seen in nature. Droplets that are in the Cassie-Baxter state can change to the Wenzel state through a state transition [31, 32, 31, 33]. This effect is considered a dynamic property of the system and it is usually triggered by external energy sources like vibration [34] or pressure [35, 36, 37] (this can be done by giving the droplet an impact velocity). Usually, the Wenzel state is not optimal, since the contact angle is lower than when the droplet is in the Cassie-Baxter state, and its contact angle hysteresis is higher.

The Wenzel and Cassie-Baxter state can both exist on the same surface because of the design of the surface microstructure [31]. As derived in Appendix A.1, the Young contact angle (also) follows from an energy minimum on a homogeneous surface. However, for engineered structures, multiple points of local minimum energy may exist. One of those energy minima corresponds to the Cassie- Baxter state and the other one corresponds to the Wenzel state. In typical microstructures (e.g.

pillars or grooves), the Cassie-Baxter state is metastable and to transfer the droplet to the other state, one has to overcome an energy barrier [38].

The reverse transition (Wenzel to Cassie-Baxter) is considered more difficult, but has been re- ported using heating by laser light as trigger [39]. It has been observed that superhydrophobic surfaces with high-roughness hierarchical structures (See also Chapter 4) do not exhibit state tran- sitions, because of the huge roughness factor [29].

3.3 Favourable state

3.3.1 Energy calculations for critical roughness of grooves and pillars

Square grooves

When looking at an interface (e.g. solid-liquid), the interfacial energy can be calculated by

Eint= γintAint, (3.6)

with γintthe corresponding interfacial surface tension or surface free energy and Aintthe interfacial surface area. If we apply this to the structure of a groove, we can obtain two equations. One for the


droplet in the Cassie-Baxter state and one for the Wenzel state. The energies can be written as ECB

d = γSLa + γLVb + γSV(2h + b), EW

d = γSL(a + b + 2h),


where the definitions of a, b, h and d can be found in the sketches of Figures 3.1 and 3.2. We want the droplet to be in the Cassie-Baxter state, since that state yields higher contact angles and lower hysteresis. To find the critical geometry of the grooves for the Cassie-Baxter state, we set


γSLa + γLVb + γSV(2h + b) = γSL(a + b + 2h) b(γLV+ γSV− γSL) = 2h(γSL− γSV)


b =γLV+ γSV− γSL

γSL− γSV

→ 1 +2h

b = −1 cos θY



In the last step, we have obtained a criterion for grooves, such that droplets on the surface have the Cassie-Baxter state as global energy minimum.

The critical point is where ECB= EW. Since both the Wenzel and Cassie-Baxter Equations predict an energy minimum (this is how they are derived), equal energies imply equal contact angle. This means that we also can rewrite to

cos θW=cos θCB

rccos θY= fs(cos θY+ 1)− 1

= fs− 1 + fscos θY.


This reduces to

rc ≡ fs+ (1− fs) −1 cos θY

= 1− (1 − fs) + (1− fs) −1 cos θY

= 1 + (1− fs) ( −1

cos θY

− 1 )



resulting in an expression for the critical roughness (which is why the subscript c has been added).

It can be shown that Equations (3.8) and (3.10) are equivalent. Say we are at the critical point, we can write from Equation (3.8)

2h = b ( −1

cos θY

− 1 )

. (3.11)

The roughness of a groove (given the Wenzel state) is given by rgroove= a + b + 2h

a + b = 1 + 2h

a + b. (3.12)

Substitution of Equation (3.11) in (3.12) results in

rgroove= 1 + b a + b

( −1 cos θY

− 1 )

. (3.13)

If the droplet was in the Cassie-Baxter state, the solid fraction is given by the ratio of the droplet that is in contact with the solid, i.e.

fs= a

a + b= 1 b

a + b, (3.14)


Figure 3.2: Cross-sectional view of a groove microstructure and the two possible hydrophobic states, Wenzel (left) and Cassie-Baxter (right).

so we can substitute for fs, i.e.

rgroove= 1 + (1− fs) ( −1

cos θY − 1 )

= rc. (3.15)

This is the same result as Equation (3.10), showing that Equations (3.8) and (3.10) are equivalent.

Square pillars

The same can be done for (square) pillars. The equations slightly change, since the roughness and solid fraction are slightly different:

ECB= γSLa2+ γLV((a + b)2− a2) + γSV((a + b)2− a2+ 4ah),

EW= γSL((a + b)2+ 4ah). (3.16)

Equating the two energies yields


γSLa2+ γLV((a + b)2− a2) + γSV((a + b)2− a2+ 4ah) = γSL((a + b)2+ 4ah) SL− γSV)((a + b)2− a2+ 4ah) = γLV((a + b)2− a2)

− cos θY


1 a2

(a + b)2+ 4ah (a + b)2


= 1 a2 (a + b)2,


results in

4ah (a + b)2 =


1 a2 (a + b)2

) ( −1 cos θY − 1


, (3.18)

which is a criterion on the pillar height.

Then, the pillar roughness can be written as (Equation (3.1))

r = (a + b)2+ 4ah

(a + b)2 = 1 + 4ah

(a + b)2. (3.19)

If the droplet is in the Cassie-Baxter state, the solid fraction can be written as

fs= a2

(a + b)2, (3.20)

which, combined with Equations (3.18) and (3.19), gives the same result as Equation (3.10),

rc≡ 1 + (1 − fs) ( −1

cos θY− 1 )

. (3.21)

A comparable analysis was done in [40].


3.4 Critical roughness in general

For the grooves and pillars of the previous Section, we assumed that part of the structure always has contact with the solid (in both Wenzel and Cassie-Baxter states), and a different part was in contact with either liquid (Wenzel) or solid (Cassie-Baxter). We can also define an arbitrary structure that has these properties, i.e., a structure with projected surfaces Sa and Sb. The unit cell has a projected surface area of

Sproj= Sa+ Sb, (3.22)

with real surface area

Sreal= raSa+ rbSb, (3.23)

with ra and rbthe roughness of parts a and b, respectively.

We pose that, if the structure is in the Cassie-Baxter state, Sawill be in contact with the liquid and Sb with vapour. This means that our solid fraction will be fs= raSa/Sreal. We calculate the energy of both states


Sreal = raSaγSL+ SbLV+ rbγSV)

Sreal = rafsγSL+ (1− fs)(γLV+ rbγSV) EW

Sreal = (raSa+ rbSbSL

Sreal = (rafs+ rb(1− fs))γSL.


Again setting ECB = EWresults in

rafsγSL+ (1− fs)(γLV+ rbγSV) = (rafs+ rb(1− fs))γSL (1− fs)(γLV+ rbγSV) = rb(1− fsSL

γLV+ rbγSV = rbγSL

→ rb= −1 cos θY

≡ rb,c.


In other words, if rb> rb,c, the system’s favourable state is Cassie-Baxter.

This derivation shows that surface rbSbis the only part that determines whether the system is in the Cassie-Baxter or Wenzel state. Surface Sa is still important, but it only is of use for the eventual contact angle. For example, in grooved structures, Sais the top of the groove, while rbSbrepresents the valley of the groove, including the additional surface area due to the wall of height h (see Figure 3.2).

3.5 Goals in microstructure design

For the design of a microstructure, hydrophobicity is usually not the only requirement. For hydropho- bic surfaces to be applicable on e.g. car windshields or solar panels, transparency is necessary and also wear and weather resistance are needed. For transparency, it is required that the typical dimensions of the microstructure are smaller than the relevant wavelength. With that in mind, pillars with height and width of 10 −20µm (a size that is very typical in laboratories, e.g. [3]) are not suitable.

To achieve this, coatings are usually applied, because the roughness details of coatings are usually much smaller and therefore suitable for transparent hydrophobic surfaces[41, 42].

A different possibility of making the surfaces robust, is by inverting the structure. That way, cavities instead of pillars are present on the surface. Cavities have as additional advantage that the Cassie- Wenzel transition is very difficult, since lowering of the droplet bottom lets the pressure increase below the droplet, adding up to the potential barrier. With pillars or grooves, the air is not trapped and the pressure will be the same. This was done by e.g. [43, 44].




S θY


Figure 3.3: An omniphobic surface with a serif-like, or mushroom-like structure. Although the material and liquid have a contact angle smaller than 90, the meniscus does find an equilibrium position, turning the substrate omniphobic. The drawing is adopted from [45].

3.6 Special structures

We have seen that intrinsically hydrophobic materials can be made more hydrophobic with the help of microstructures like pillars and grooves. Oleophobicity (repelling of oil) or omniphobicity (repelling of more than one liquid) is more difficult to achieve, since oils have a lower surface tension γLVthan water. That is why a different approach is used in order to make surfaces having these properties, since most smooth surfaces in combination with these liquids have contact angles that are much smaller than 90. The technique to achieve oleophobicity or omniphobicity is to let the liquids find an equilibrium position, without touching the bottom of the substrate. It appears that extreme roughness and porosity on the side of pillars [46, 47] and spherical cavities or serif-like (mushroom) structures [45, 48, 49] allow to achieve this. This also allows hydrophilic materials to be turned into hydrophobic substrates [50, 51]. An impression of the workings of these type of surfaces is given in Figure 3.3.

A completely different, but very noteworthy view on omniphobic surfaces is that of the liquid- infused surfaces, developed at Harvard [52]. This concept is based on immiscible fluids, rather than larger contact angles. The surface is made porous and than infused with a lubricating liquid. This lubricating liquid can be anything, as long as the material does not repel it, and it should not mix with the liquids that are meant to be repelled (e.g. the most practical applications cover oils and water).

This technique was, again, inspired by nature: it is the technique that some carnivorous plants use to let insects slip into their jaws.


Chapter 4

Hierarchical and self-similar geometries

We have seen in Chapter 3 that microstructures, like pillars or grooves, can enhance the contact angle in two different ways, giving rise to the Wenzel and Cassie-Baxter states of hydrophobicity. To summarize, the idea is to make intrinsically hydrophobic surfaces rougher to optimize the contact angle. A way to make the surface more rough, which is often used in literature (e.g. [29, 53, 54]), is to introduce roughness on multiple levels, forming a hierarchical structure.

There are multiple kinds of hierarchical structures. For example, Zhang et al [55] put nanopillars on micropillars, creating self-similar structures. However, the geometry on different length scales can be completely different: Bhushan [6] created micropillars and added some nanoroughness on top of the pillars, increasing the contact angle significantly.

4.1 Combinations of states

A consequence of defining multiple levels of roughness is the change of the hydrophobic states.

Since roughness is a dimensionless multiplication factor which describes the ratio of the real surface area and the projected area, we may multiply them to obtain the total roughness. For m levels of roughness, we obtain

rtot= r1r2...rm=

m i=1

ri. (4.1)

However, this roughness value gives us little to no information about the contact angle. For that, we have to know something about the hydrophobic state that the system is in. But, given roughness ri, what state describes the situation?

To predict the state, we first modify Equation (3.25) for critical roughness from Section 3.4 to address for different length scales. Assuming that in all length scales smaller than li, roughness ri+1

to rm is sufficiently small for the fluid to follow the contour of the surface (and is therefore in the Wenzel state), the critical roughness for the void in the structure is

rc,i= ∏m −1

j=i+1rjcos θY

. (4.2)

This equation changes slightly if some of the states are changed to the Cassie-Baxter state, which is dependent on m. If the roughness ri≥ rc,i, the system is in a stable Cassie-Baxter state on that lenght scale.

If we introduce an arbitrary number of roughness levels (on different length scales) m, which all can be Wenzel or Cassie-Baxter, we end up with 2mcombinations of states. Adapting the notation W and C for the single Wenzel and Cassie-Baxter states, the hierarchical states range from W W..W ,


(r1 − 1) / (rc,1 − 1) (r 2 1) / (r c,2 1)


0 0.5 1 1.5 2

0 0.5 1 1.5 2

(r1 − 1) / (rc,1 − 1) (r 2 1) / (r c,2 1)


0 0.5 1 1.5 2

0 0.5 1 1.5 2

(r1 − 1) / (rc,1 − 1) (r 2 1) / (r c,2 1)


0 0.5 1 1.5 2

0 0.5 1 1.5 2

(r1 − 1) / (rc,1 − 1) (r 2 1) / (r c,2 1)


0 0.5 1 1.5 2

0 0.5 1 1.5 2

Figure 4.1: State contour diagrams for three-level hierarchical structure, showing only the (r1, r2)data. The r3

data is (r3− 1)/(rc,3− 1) = a) 0, b) 1/3, c) 2/3 and d) ≥ 1. For the colors and corresponding states S = S1S2, dark blue is W W , light blue is W C, yellow is CW and red is CC. For the a, b and c diagrams,S3= Wand in d,S3 = Cbecause of r3≥ rc,3. Note that the transitions on the (r1, r2)scale happen sooner due to larger r3.

CW..W, ..., to CC..C. To generalize this problem, we define the total state of the system S = S1S2..Sm, with Sidefined as



Young if ri = 1or undefined Wenzel if 1 < ri< rc,i Cassie-Baxter if ri≥ rc,i,


. (4.3)

4.2 State diagrams of multirough structures

Using the tools from the previous section, we can produce state diagrams of multiple structures.

For this, we start by defining a pillared structure, with well-known dimensions and perfectly flat pillar tops. By varying the height of the pillars, we can linearly tune riand properly normalize to rc. This way, we can predict the contact angle in both the Wenzel and Cassie-Baxter state. Utilizing the fact that θS =min(θW, θCB)describes the (globally) energetically favourable state, the local states are predicted. In Figure 4.1a, the state diagram of a two-level hierarchical structure is shown.

The boundaries of the even surfaces in Figure 4.1 represent state transition lines. They repre- sent the lines where the Wenzel and Cassie-Baxter states will yield the same contact angle and are therefore described by equating the contact angles of the different states. Recalling

cos θCB=∑


rifiSV,i− γSL,i), (4.4)


and applying it to the specific states W W , CW , W C and CC on a single-material microstructure with intrinsic contact angle θ = θYas material property, will yield the following equations:

• W W ↔ CW

r1r2cos θY= fs,1((r2cos θY) + 1)− 1

→ r2= fs,1− 1

cos θY(r1− fs,1). (4.5)

• W W ↔ W C and CW ↔ CC

fs,1(r2cos θY+ 1) = fs,1fs,2(cos θY+ 1)− 1

→ r2= 1 + (1− fs,2) ( −1

cos θY − 1 )

= rc,2. (4.6)

• W C ↔ CC

r1(fs,2(cos θY+ 1)− 1) = fs,1fs,2(cos θY+ 1)− 1

→ r1= fs,1fs,2(cos θY+ 1)− 1

fs,2(cos θY+ 1)− 1 . (4.7) This analysis is trivially extended to more roughness scales, but is not carried out here since it will only yield more extensive equations. However, in Figure 4.1b-d, an extra roughness scale has been added and its influence on the (r1, r2)state diagram is shown.

4.3 Self-similar structure

A special type of hierarchical structures is the self-similar structure. This is a structure that repeats its microstructure on all relevant length scales (e.g. pillars on pillars or grooves on grooves), with the same relative parameters. Since the state diagram from Section 4.2 is not symmetric, we should not expect two, but three states. This is indeed the case, as was found using the same procedure as described in Section 4.2. The behavior can be seen in Figure 4.2.

4.4 Advantages of hierarchy

The self-cleaning properties of hierarchical structures are reported to be very good. Since there is almost no physical contact to the surface, the contact angle hysteresis is very low and the droplets can roll off very easily [56]. The high roughness also makes it impossible for droplets to go into the Wenzel state [29]. All of these properties are utilized by the lotus leaf, to obtain very good self- cleaning properties. Since the droplets roll (rather than slide) off the surface, dust and dirt particles are picked up and removed.

4.5 Discussion

From the theoretical analysis, it follows that hierarchical structures result in increasing contact angles and decreasing contact angle hysteresis. However, the number of hierarchical levels is limited. In nature, at most two levels are found and in the laboratory, it is very difficult to create more than two levels. On the other hand, one big question is, what is defined as “smooth”? Is it atomically flat, or is there a typical length scale to be found at which the roughness does not influence the wetting behavior of the material anymore? In other words, if we can make a material with a third hierarchical layer of roughness, what roughness is needed to significantly improve the wetting properties of the two-level structures?


Another aspect of hierarchical structures is the structural integrity. If the details are very small, the soft material can become fragile, so that one small mechanical impact can destroy the wetting behavior. This is already a problem with two-level hierarchy, let alone even more levels. In addition, dirt can fill the asperities, destroying the hydrophobic properties. The usefulness also has to compete with the difficulty of fabrication. If something is very difficult to fabricate, time and costs come into the picture, which have to be balanced (e.g. to add only the second layer of roughness on the hierarhical structures in [6] takes seven days).


0 0.5 1 1.5 2

100 110 120 130 140 150 160 170 180

(ri−1) / (rc−1)

θ 1

1 level 2 levels 3 levels


Figure 4.2: The states of different hierarchy levels of self-similar structures. a) Contact angle plots of self-similar structures. b) Sketches of the corresponding states possible in the hierarchical self-similar structures. Top: one level of roughness (no real hierarchy). S = W or C. Middle: two-level roughness. S = W W, CW or CC.

Bottom: three-level roughness. S = W W W, CW W, CCW or CCC. Note: when printed on paper, the last two states of the bottom illustrations are not distinguishable.


Chapter 5

Summary and conclusion of Part I

5.1 Summary

Several basic examples of surface topologies and their effect on the contact angle were studied.

Using the dimensions of, for example, pillars, the contact angle can be predicted theoretically by means of the Wenzel and Cassie-Baxter equations. In many structures, some of these values rep- resent local energy minima, rather than a global minimum. That is why a critical roughness can be defined, that only depends on the dimensions of the part of the droplet that should be in contact with air (in the preferable Cassie-Baxter state). This critical roughness is rb,c =−(cos θY)−1. In addition, some goals in microstructure design and some special microstructures that can be omniphobic were presented shortly.

To make the structures even more hydrophobic, i.e. superhydrophobic, the structure can be made hierarchical, which is very common in nature (e.g. the superhydrophobicity of the lotus leaf is due to the hierarchy in the structure). Hierarchical structures made from an intrinsically hydrophobic material usually have a high contact angle and very low contact angle hysteresis. In addition to the Cassie-Baxter and Wenzel states of hydrophobicity, combinations of states can exist and were presented in state contour diagrams. The state transition lines were obtained by equating the Cassie- Baxter and Wenzel equations for the critical roughness coefficient. This can be done for all levels of hierarchy. However, more than two levels of hierarchy is difficult to manufacture and it was questioned whether more levels are useful.

5.2 Conclusion

We see great improvements in contact angle by introducing hierarchical structures. Using hierarchy with a very high roughness on the smaller length scale enables us to limit the height of the pillars or grooves on the larger length scale to improve robustness, without disturbing the hydrophobic properties. The goal is to design the structures in such a way to ensure that the droplet be in the Cassie-Baxter state on all length scales, which is achieved by tuning the roughness.


End of Part I


Part II

Anisotropic wetting


Chapter 6

Introduction and proposal for model

This chapter will give an introduction to anisotropic wetting and will emphasize some articles that have been used to create the model that is treated in Chapters 7 and 8.

6.1 Anisotropic wetting

In the previous chapters, it was assumed that any microstructure resulted in an isotropic droplet shape, i.e. a droplet that is shaped as a spherical cap. However, if the microstructure is not isotropic, droplets that are put onto such a surface will not have the shape of an (isotropic) spherical cap, but it will be related to the orientation of the microstructure (e.g. [57, 58]). For example, the droplet shape of a droplet on grooves is elongated in the direction of the grooves, as shown in Figure 6.1.

We consider the most anisotropic structure possible, which is a one-dimensional grooved struc- ture. In Figure 6.1, the definitions of the parallel and perpendicular angles, θand θ, are given. The parallel angle θ is the observed angle when looking parallel to the groove and the perpendicular angle θwhen looking perpendicular to the groove.

6.1.1 The process of the shaping of a droplet

Using impressions from [27], we will first try to create some intuitive understanding of the formation of anisotropic droplet shaping. By compressively straining the surface, microscopic sinusoidal wrinkles with accurately known wavelenght and amplitude appear on the microscale (according to [59]). We assume that the wrinkles are much smaller than the droplets we are going to put on the surface. Using the interpretation of contact angle hysteresis from Equation (2.9), we can predict the advancing angle θAfrom Equation (2.9) and Figure 2.6. Since the sine function has a well-defined roughness factor, we can use the Wenzel Equation to predict the equilibrium contact angle.

We create our wrinkled (grooved) surface from an intrinsically hydrophilic material that exhibits no contact angle hysteresis when it is perfectly smooth (theoretically). After wrinkling, when the droplet is put on the surface, the forces at the three-phase line begin to shape the droplet. In the parallel view (at the points where we measure the parallel angle θ), this force will be stronger than in the perpendicular view. This happens because the three-phase line is longer there, since it follows the contour of the microstructure. Now, in the parallel view, the contact angle θwill reach its equilibrium value which can be predicted using the Wenzel Equation.

In the perpendicular view, the contact angle θ will stop at θA(there is no driving force that will push it further), finalizing the anisotropic droplet shape. We now have a droplet with not one, but a full range of contact angles, depending on the position on the three-phase line. This range is usually only characterized in literature by θand θ.







h W

L ex


Figure 6.1: The principle of anisotropic wetting, in different views. Blue arrows indicate dimensions and green arrows indicate viewing angles. (i) viewing angle to observe θ, therefore referred to as perpendicular view. An example view is given in the right of the image, along with a magnification of an example of this view on the microstructure. (ii) viewing angle to observe θ, therefore referred to as parallel view. An example view is given below the image, along with a magnification of an example of this view on the microstructure. Note that from this angle the microstructure is not really observable.

6.2 An alternative model on anisotropic wetting

In most literature, a grooved surface is characterized by only two parameters. This is enough to predict the length-over-width ratio of the small, anisotropic droplet [60]. This has been confirmed by experimental data, but also simulations using Surface Evolver [61] have matching results. Surface Evolver is a mathemetical geometry tool, which can find energy minima of a given mathematical system. By defining a bottom surface and a droplet surface, the program finds an energetic minimum by iteratively altering the surface. For this, the whole droplet and substrate surfaces are discretized.

When applied to simulation of droplets on a surface, the program solves the energy balance, like the one in Appendix A.1, numerically, albeit a bit more complex (since the bottom surface can be made heterogeneous).

In some literature (e.g. [62, 63]), this energy approach is criticized, because it is observed that only the surface structure at the three-phase line matters, hence the assumption in Appendix A.1 of an isotropic, homogeneous substrate is made as justification. That is why in this thesis the approach is made to only model the three-phase line, subject to several constraints, the most important being the assumption that from the center of the droplet, all paths to the three-phase line are circular (Section 7.2).

The goal of this model is to show that the droplet shape can be predicted by modeling solely the three-phase line, that is subject to a number of assumptions. In contrast with simulations using Surface Evolver, this ad hoc model will use much less data points and it will work from a more physical point of view, rather than mostly geometry. In addition, the model can be used to study the macro-scale behavior of the three-phase line of a droplet, when put on a surface with an anisotropic microstructure, when the dimensions of the microstructure are much smaller than the dimensions of the droplet. The latter is very difficult for the Surface Evolver, since for these simulations, the surface has to be modeled in high detail: smaller details therefore are computationally much more expensive


1 α

Hydrophilic Hydrophobic

Figure 6.2: An example of a chemically striped surface as used in [60] and [64]. Parameter α is varied to create different surface properties.


As reference, the data of papers [60] and [64] are used. They use chemically striped surfaces, which basically means that the surface is flat, but alters materials with different contact angle. Specif- ically, the papers alter hydrophobic and hydrophilic materials, as is illustrated in Figure 6.2. Because the surface can be divided into well-defined fractions, we can use the Cassie-Baxter equation to predict the contact angle. Using the ratio α from Figure 6.2, we use the Cassie-Baxter equation, assuming a flat surface (ri= 1for all i) [64]

cos θCB=∑


rificos θY,i= αcos θ1+cos θ2

1 + α . (6.1)

Here, θ1and θ2are the contact angles for the two used materials.

In Chapter 7, all parts of the governing force equation are derived to which we are going to subject the three-phase line. Chapter 8 will cover technical details on the used finite-element model.

6.2.1 Limitations

The model to be described is subject to a number of assumptions, which are the following.

• The droplet should be small enough to maintain its elliptical shape; and

• The microstructure is much smaller than the droplet.

The model is based on circular paths from the center of the droplet to the three-phase line. This is treated in more detail in Section 7.2. These circular paths are valid only when the droplet size is small enough. This is the case when the droplet diameter is smaller than the capillary length, because in that case, surface tension effects dominate gravitational effects. When gravitational effects start to dominate, the droplet will become less round. This capillary length is defined by

ℓ =


ρg. (6.2)

For water, ℓ ≈ 3 mm, which means that droplet volume V should be in the order of 1 µl. When the droplet size increases, the surface tension is not sufficient to maintain the circular shape of the droplet.

Second, when looking at the three-phase line on the micro scale, there are distortions due to heterogeneity of the surface. For example, when looking at grooves, the three-phase line is not perfectly straight, but follows the contour of the grooves. When the microstructure is too large,


these effects become significant. In the present approach it is assumed that the distortions are very small and that their effects are averaged out. Because we “zoom out”, the small distortions become invisible so we only see a different contact angle than we would expect from a smooth surface. This should be accounted for in the description of the perpendicular and parallel contact angles θand θusing the existing models of Young, Wenzel, Cassie-Baxter and on contact angle hysteresis. The angles that are calculated with these models are input parameters for the model to be developed further in the next chapters.



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