### 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

**Abstract**

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.

**Contents**

**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 [F*_{i}* ^{A}*]vector . . . 43

*8.5.4 Building the [K** ^{AB}*]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 σ**ψ* . . . 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**

**Introduction**

**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 γ*SL*, γ*SV *and γ*LVfor solid-liquid, solid-vapour
and liquid-vapour, respectively. As illustrated in Figure 2.1, the force balance reads

*γ*LV*cos θ*Y*+ γ*SL*= γ*SV*,* (2.1)

from which the contact angle can be solved as

*cos θ*Y= *γ*SV*− γ*SL

*γ*_{LV} *.* (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

γLV

γSV

γSL

*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≡* *A*_{real}

*A*proj*,* (2.3)

*where A*real*and A*proj*≤ A*realare 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 γ*SV *to rγ*SV *and γ*SL *to rγ*SL. This changes the force balance from
Equation (2.2) into

*γ*LV*cos θ*W*= rγ*SV*− rγ*SL

*cos θ*W*= rγ*SV*− γ*SL

*γ*_{LV} *= rcos θ*Y*,* (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 θ*Y*< 90*^{◦}*, r will make the contact*
*angle smaller, while materials with θ*Y*> 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 90* ^{◦}*line.

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]

*γ*LV*cos θ*CB=∑

*i*

*r**i**f**i**γ*SV*,i**−*∑

*i*

*r**i**f**i**γ*SL*,i*

=∑

*i*

*r**i**f**i**(γ*SV*,i**− γ*SL*,i*)
*cos θ*CB=∑

*r**i**f**i**cos θ*Y*,i**,*

(2.5)

0 0.2 0.4 0.6 0.8 1 60

80 100 120 140 160 180

fs

θ CB

(a)

θ_{Y} = 70
θ_{Y} = 90
θ_{Y} = 100
θ_{Y} = 110
θ_{Y} = 120

1 1.5 2 2.5 3

0 20 40 60 80 100 120 140 160 180

r

θ W

(b)
θ_{Y} = 70°

θ_{Y} = 80°

θ_{Y} = 90°

θ_{Y} = 100°

θ_{Y} = 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 f**s**on 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*
*r**i* *and f**i* *are the material properties of component i of the composite surface. This general form*
may also be written as

*γ*LV*cos θ*CB*= γ*^{∗}_{SV}*− γ*SL^{∗}*,* (2.6)
*where we simply pose that the surface free energies γ*SV* ^{∗}* = ∑

*i**r**i**f**i**γ*SV*,i**and γ** ^{∗}*SL = ∑

*i**r**i**f**i**γ*SL*,i* are
*a function of the surface microstructure, which includes parameters as surface roughness r and*
*surface fractions f**i*. 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 θ*Y*and roughness r**s**; and f**s*is 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 θ*Y= 180* ^{◦}*, resulting
in a speciﬁc form of the Cassie-Baxter Equation,

*cos θ*CB *= f**s**r**s**cos θ*Y+ (1*− f**s*)cos(180* ^{◦}*)

*= f*_{s}*(r*_{s}*cos θ*Y+ 1)*− 1.* (2.7)

*Note that in the limit f**s**→ 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.

f_{s} 1-f_{s} “Air pockets”

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

a

b

c

*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 θ*A *and θ*R, respectively. The contact
angle hysteresis is then deﬁned as

*∆θ≡ θ*A*− θ*R*.* (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 θ*Yand the maximum and minimum slope obtained from the height proﬁle:

*S =*max*dz*
*dx*
*α≡ arctan S*
*θ*_{A}*= θ*_{Y}*+ α*
*θ*R*= θ*Y*− α.*

(2.9)

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 θ*R *and θ*Ato a smooth surface, they pose that the hysteresis is simply
*calculated by multiplying the difference with r and f**s*. 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 θ*A

*and θ*Rof 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.

θ_{A}

θ_{Y} θ_{R}

(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, θ**R**. (c) Maximum angle possible*
*on the surface, resulting in the advancing contact angle, θ**A**. 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**

**Microstructures**

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 =* *A*real

*A*_{proj} =*(a + b)*^{2}*+ 4ah*
*(a + b)*^{2}

= 1 + *4ah*
*(a + b)*^{2}*.*

(3.1)

Combined with the Wenzel Equation, the Wenzel contact angle would become
*cos θ*W*= rcos θ*Y=

(

1 + *4ah*
*(a + b)*^{2}

)

*cos θ*Y*,* (3.2)

L

L h

a+b

a+b

a

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 θ*Ythe 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

*f** _{s}*=

*a*

^{2}

*(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, r**s*= 1),
the contact angle can be derived from the Cassie-Baxter Equation,

*cos θ*CB*= f** _{s}*(cos θY+ 1)

*− 1 =*

*a*

^{2}

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

*cos θ*W=
(

1 + *2h*
*a + b*

)
*cos θ*Y

*cos θ*CB = *a*

*a + b*(cos θY+ 1)*− 1.*

(3.5)

**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

*E*_{int}*= γ*_{int}*A*_{int}*,* (3.6)

*with γ*int*the corresponding interfacial surface tension or surface free energy and A*intthe 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
*E*CB

*d* *= γ*_{SL}*a + γ*_{LV}*b + γ*_{SV}*(2h + b),*
*E*W

*d* *= γ*SL*(a + b + 2h),*

(3.7)

*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

*E*CB*= E*W

*γ*_{SL}*a + γ*_{LV}*b + γ*_{SV}*(2h + b) = γ*_{SL}*(a + b + 2h)*
*b(γ*LV*+ γ*SV*− γ*SL*) = 2h(γ*SL*− γ*SV)

*2h*

*b* =*γ*_{LV}*+ γ*_{SV}*− γ*SL

*γ*SL*− γ*SV

*→ 1 +2h*

*b* = *−1*
*cos θ*Y

*.*

(3.8)

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 E*CB*= E*W. 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 θ*W=*cos θ*CB

*r**c**cos θ*Y*= f**s*(cos θY+ 1)*− 1*

*= f*_{s}*− 1 + f**s**cos θ*Y*.*

(3.9)

This reduces to

*r**c* *≡ f**s*+ (1*− f**s*) *−1*
*cos θ*Y

= 1*− (1 − f**s*) + (1*− f**s*) *−1*
*cos θ*Y

= 1 + (1*− f**s*)
( *−1*

*cos θ*Y

*− 1*
)

*,*

(3.10)

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 θ*Y

*− 1*
)

*.* (3.11)

The roughness of a groove (given the Wenzel state) is given by
*r*_{groove}= *a + b + 2h*

*a + b* = 1 + *2h*

*a + b.* (3.12)

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

*r*_{groove}= 1 + *b*
*a + b*

( *−1*
*cos θ*Y

*− 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.

*f** _{s}*=

*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 f*s, i.e.

*r*groove= 1 + (1*− f**s*)
( *−1*

*cos θ*Y *− 1*
)

*= r**c**.* (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:

*E*_{CB}*= γ*_{SL}*a*^{2}*+ γ*_{LV}*((a + b)*^{2}*− a*^{2}*) + γ*_{SV}*((a + b)*^{2}*− a*^{2}*+ 4ah),*

*E*_{W}*= γ*_{SL}*((a + b)*^{2}*+ 4ah).* (3.16)

Equating the two energies yields

*E*CB*= E*W

*γ*_{SL}*a*^{2}*+ γ*_{LV}*((a + b)*^{2}*− a*^{2}*) + γ*_{SV}*((a + b)*^{2}*− a*^{2}*+ 4ah) = γ*_{SL}*((a + b)*^{2}*+ 4ah)*
*(γ*_{SL}*− γ*SV*)((a + b)*^{2}*− a*^{2}*+ 4ah) = γ*_{LV}*((a + b)*^{2}*− a*^{2})

*− cos θ*Y

(

1*−* *a*^{2}

*(a + b)*^{2}+ *4ah*
*(a + b)*^{2}

)

= 1*−* *a*^{2}
*(a + b)*^{2}*,*

(3.17)

results in

*4ah*
*(a + b)*^{2} =

(

1*−* *a*^{2}
*(a + b)*^{2}

) ( *−1*
*cos θ*Y *− 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

*f** _{s}*=

*a*

^{2}

*(a + b)*^{2}*,* (3.20)

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

*r**c**≡ 1 + (1 − f**s*)
( *−1*

*cos θ*Y*− 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 S**a* *and S**b*. The unit cell has a projected
surface area of

*S*_{proj}*= S**a**+ S**b**,* (3.22)

with real surface area

*S*real*= r**a**S**a**+ r**b**S**b**,* (3.23)

*with r**a* *and r**b**the roughness of parts a and b, respectively.*

*We pose that, if the structure is in the Cassie-Baxter state, S**a*will be in contact with the liquid and
*S**b* *with vapour. This means that our solid fraction will be f**s**= r**a**S**a**/S*real. We calculate the energy
of both states

*E*_{CB}

*S*real = *r*_{a}*S*_{a}*γ*_{SL}*+ S*_{b}*(γ*_{LV}*+ r*_{b}*γ*_{SV})

*S*real *= r*_{a}*f*_{s}*γ*_{SL}+ (1*− f**s**)(γ*_{LV}*+ r*_{b}*γ*_{SV})
*E*_{W}

*S*real = *(r*_{a}*S*_{a}*+ r*_{b}*S*_{b}*)γ*_{SL}

*S*real *= (r*_{a}*f*_{s}*+ r** _{b}*(1

*− f*

*s*

*))γ*

_{SL}

*.*

(3.24)

*Again setting E*CB *= E*_{W}results in

*r*_{a}*f*_{s}*γ*_{SL}+ (1*− f**s**)(γ*_{LV}*+ r*_{b}*γ*_{SV}*) = (r*_{a}*f*_{s}*+ r** _{b}*(1

*− f*

*s*

*))γ*

_{SL}(1

*− f*

*s*

*)(γ*LV

*+ r*

*b*

*γ*SV

*) = r*

*b*(1

*− f*

*s*

*)γ*SL

*γ*_{LV}*+ r*_{b}*γ*_{SV} *= r*_{b}*γ*_{SL}

*→ r**b*= *−1*
*cos θ*Y

*≡ r**b,c**.*

(3.25)

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

*This derivation shows that surface r**b**S**b*is the only part that determines whether the system is in
*the Cassie-Baxter or Wenzel state. Surface S**a* is still important, but it only is of use for the eventual
*contact angle. For example, in grooved structures, S**a**is the top of the groove, while r**b**S**b*represents
*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].

L

V

S θY

S

*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 γ*LVthan
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

*r**tot**= r*1*r*2*...r**m*=

∏*m*
*i=1*

*r**i**.* (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 r**i*,
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 l**i**, roughness r**i+1*

*to r**m* 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

*r**c,i*= ∏*m* *−1*

*j=i+1**r*_{j}*cos θ*Y

*.* (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 r**i**≥ r**c,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 2^{m}*combinations of states. Adapting the notation W*
*and C for the single Wenzel and Cassie-Baxter states, the hierarchical states range from W W..W ,*

(r_{1} − 1) / (r_{c,1} − 1)
(r 2 − 1) / (r c,2 − 1)

a

0 0.5 1 1.5 2

0 0.5 1 1.5 2

(r_{1} − 1) / (r_{c,1} − 1)
(r 2 − 1) / (r c,2 − 1)

b

0 0.5 1 1.5 2

0 0.5 1 1.5 2

(r_{1} − 1) / (r_{c,1} − 1)
(r 2 − 1) / (r c,2 − 1)

c

0 0.5 1 1.5 2

0 0.5 1 1.5 2

(r_{1} − 1) / (r_{c,1} − 1)
(r 2 − 1) / (r c,2 − 1)

d

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 (r*1*, r*2)*data. The r*3

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

*CW..W, ..., to CC..C. To generalize this problem, we deﬁne the total state of the system S =*
*S*1*S*2*..S**m**, with S** ^{i}*deﬁned as

*S**i*=

*Young if r**i* = 1or undeﬁned
*Wenzel if 1 < r**i**< r*_{c,i}*Cassie-Baxter if r**i**≥ r**c,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 r**i**and properly normalize to r**c*. This
way, we can predict the contact angle in both the Wenzel and Cassie-Baxter state. Utilizing the
*fact that θ** _{S}* =

*min(θ*W

*, θ*CB)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 θ*CB=∑

*i*

*r**i**f**i**(γ*SV*,i**− γ*SL*,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 = θ*Yas material property, will yield the following equations:

*• W W ↔ CW*

*r*_{1}*r*_{2}*cos θ*Y*= f*_{s,1}*((r*_{2}*cos θ*Y) + 1)*− 1*

*→ r*2= *f**s,1**− 1*

*cos θ*Y*(r*_{1}*− f**s,1*)*.* (4.5)

*• W W ↔ W C and CW ↔ CC*

*f**s,1**(r*2*cos θ*Y*+ 1) = f**s,1**f**s,2*(cos θY+ 1)*− 1*

*→ r*2= 1 + (1*− f**s,2*)
( *−1*

*cos θ*Y *− 1*
)

*= r**c,2**.* (4.6)

*• W C ↔ CC*

*r*1*(f**s,2*(cos θY+ 1)*− 1) = f**s,1**f**s,2*(cos θY+ 1)*− 1*

*→ r*1= *f**s,1**f**s,2*(cos θY+ 1)*− 1*

*f**s,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 (r*1*, r*2)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).

(a)

0 0.5 1 1.5 2

100 110 120 130 140 150 160 170 180

(r_{i}−1) / (r_{c}−1)

θ 1

1 level 2 levels 3 levels

(b)

*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 r**b,c* =*−(cos θ*Y)* ^{−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, θ*_{∥}*and θ** _{⊥}*, are given. The

*parallel angle θ*

*is the observed angle when looking parallel to the groove and the perpendicular*

_{∥}*angle θ*

*when 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
*θ*_{A}from 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 θ*

*will reach its equilibrium value which can be predicted using the Wenzel Equation.*

_{∥}*In the perpendicular view, the contact angle θ*_{⊥}*will stop at θ*A(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 θ*_{∥}*and θ** _{⊥}*.

θ_{//}

θ˔

(i)

(ii)

h

h W

L
**e**_{x}

**e**_{y}

*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.*

[60].

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 (r**i*= 1*for all i) [64]*

*cos θ*CB=∑

*i*

*r**i**f**i**cos θ*Y*,i*= *αcos θ*1+*cos θ*2

*1 + α* *.* (6.1)

*Here, θ*1*and θ*2are 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 =*

√*γ*LV

*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 θ** _{∥}*and

*θ*

*using 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.*

_{⊥}