Revisiting the critical condition for the CassieâWenzel transition on

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Revisiting the critical condition for the Cassie– Wenzel transition on micropillar-structured surfaces Wei Fang, Hao-Yuan Guo, Bo Li, Qunyang Li, and Xi-Qiao Feng Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00121 • Publication Date (Web): 07 Mar 2018 Downloaded from http://pubs.acs.org on March 7, 2018

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Revisiting the critical condition for the Cassie–Wenzel transition on micropillar-structured surfaces Wei Fang†, Hao-Yuan Guo†, Bo Li†, Qunyang Li∗†‡, Xi-Qiao Feng*†‡ †

AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China ‡

State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China

Abstract Biological and engineering applications of superhydrophobic surfaces are limited by the stability of the wetting state determined by the transition from the Cassie–Baxter (CB) state to the Wenzel state (C–W transition). In this paper, we performed water droplet squeeze tests to investigate the critical conditions for the C–W transition for solid surfaces with periodic micropillar arrays. The experimental results indicate that the critical transition pressure for the samples with varying micropillar dimensions are all significantly higher than the theoretical predictions. Through independent measurements, we attributed the disparity to the incorrect assessment of the contact angle on the sidewall surfaces of the micropillars. We also showed that the theoretical models are still applicable when the correct contact angle of the sidewall surfaces is adopted. Our work directly validates and improves the theoretical models regarding the C–W transition and suggests a potential route of tuning superhydrophobicity using finer scale surface features.

Key words: Superhydrophobicity, Wetting, Micropillars, Cassie–Wenzel transition, Critical condition, Contact angle

*

Corresponding authors: [email protected] (X. -Q. Feng) and [email protected] (Q.

Li).

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1. INTRODUCTION Solid surfaces with a contact angle larger than 150o are usually called superhydrophobic surfaces.1 Superhydrophobic phenomena have been widely exploited in nature2-6 and industry7-10 to render materials with special functionalities, e.g. self-cleaning, antifogging, water transport and harvesting. Superhydrophobicity with a small sliding angle is usually achieved on microstructured surfaces with air trapped in the contact area underneath the droplet. Such a wetting state is referred to as the Cassie–Baxter (CB) state11. If the trapped air is removed from the microstructures under external stimuli, e.g., external pressure12-14, evaporation15-16, condensation17 and droplet impact18-19, the wetting behavior will change and the CB state will be converted to the Wenzel state20-21. For many biological and engineering applications, the stability of the CB state is very critical and an unexpected transition from the CB state to Wenzel state is often catastrophic.14-15, 22-23 In recent years, considerable effort has been directed toward understanding the Cassie–Wenzel transition.13-14, 23-25 For example, Lafuma and Quéré conducted water droplet squeeze tests on superhydrophobic surfaces. They found that the CB state can be irreversibly lost when the exerted pressure reaches a threshold.12 Forsberg et al. investigated

the

immersed

superhydrophobic

surfaces

experimentally

and

demonstrated that a closed film of trapped air can increase the pressure threshold and delay the occurrence of Cassie–Wenzel transition.26 To theoretically predict the critical

2


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conditions, by considering the balance between the capillary force and the exerted pressure, Zheng et al. derived an expression predicting the critical pressure at the Cassie–Wenzel transition on solid surfaces with micropillars.27 Based on energetics analysis, Afferrante and Carbone studied the impalement pressure of the CB state on surfaces with micropillars of conical, hemispherical-topped, and flat-topped cylindrical shapes.28 For wetting behavior on protruding and porous surface structures with arbitrary generatrix, Guo et al. provided a general solution to calculate the critical transition pressure while considering the effect of line tension.29 Although the theoretical models can qualitatively predict the variation trend of the critical transition pressure on surface geometries, quantitative comparison and direct experimental validation of these theories are desired and invite a more systematical study. In this work, water droplet squeeze tests were conducted on superhydrophobic surfaces consisting of micropillar arrays to experimentally identify the critical pressure at the Cassie–Wenzel transition. By comparing the experimental results with the theoretical predictions, we found that the theoretical values from traditional models significantly underestimated the critical transition pressure. Through direct geometric measurements, we attributed the disparity to the incorrect assessment of the contact angle on the sidewall surfaces of micropillars. The influence of sidewall microstructures on the Cassie–Wenzel transition also suggests a strategy to improve global stability of superhydrophobicity.

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2. EXPERIMENTS 2.1 Preparation of superhydrophobic surfaces Silicon wafers were decorated with a periodic array of micropillars on their surfaces using deep reactive ion etching (DRIE) technique. Each pillar has a height of 100 µm, a squared cross section with side length a and neighboring spacing b, as shown −2

in Fig. 1(a). The solid area fraction is calculated as φ = a 2 ( a + b ) . In the experiments, the geometric parameters (a, b) were varied to change the solid area fraction while all other parameters remained fixed. The superhydrophobicity of the silicon samples was achieved by a series of surface treatments. Briefly, the samples were first carefully rinsed in acetone for 2 minutes. After the liquid residue was completely dried under a dry nitrogen flow, the samples were treated by O2 plasma for 6 minutes. Then the silicon

surfaces

were

further

modified

with

1H,

1H,

2H,

2H-Perfluorodecyltrimethoxysilane30 through chemical vapor deposition for 2 hours under 80°C.

2.2 Measurement of contact angles We measured the water contact angles on the samples with a Drop Shape Analyzer (Krüss DSA25) by fitting the droplet profile with the Young–Laplace equation. Each measurement was repeated three times and the results were averaged. The apparent contact angles θ , the advancing contact angles θ A , and the receding contact angles 4


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θ R of the samples with different microstructural sizes (a, b) are given in Table S1 of the Supporting Information. For comparison, a clean, flat and smooth silicon wafer that had been similarly treated was measured. Its intrinsic Young’s contact angle θ 0 , the advancing contact angle θ 0A , and the receding contact angle θ 0R are 113.3±3.6°, 117.4±2.9°, and 89.1±5.8°, respectively. In addition, the smooth wafer was used as the upper plate to vertically press the droplet in our water droplet squeeze tests, which will be described in the next subsection.

2.3 Droplet squeeze tests Squeeze tests have been widely performed in experimental studies on the stability of solid–liquid–gas systems, such as superhydrophobic surfaces and liquid marbles.12, 26, 31-32

Here, we also conducted water droplet squeeze tests using a Nano Tribometer

(NTR2, Anton Paar) to determine the critical conditions for the unstable Cassie–Wenzel transition. As shown in Fig. 1(b), a drop of deionized water with volume V=4.4 µL was placed on a microstructured sample. The sample was moved vertically upwards via a step motor to compress the water droplet against a smooth, flat and hydrophobic upper plate. The upper plate was connected to a double-cantilever force sensor (HR-S 212, Anton Paar) so that the compressive force F could be measured simultaneously as a function of the vertical displacement of the lower plate, D. To minimize the effect of water evaporation, we started the squeeze test once the water droplet had been 5


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deposited on the microstructured substrate. During a squeeze test, the two plates were brought close to compress the droplet at a low speed of v=0.1 mm/s to minimize the effect of dynamics. After the CB state had been completely converted to the Wenzel state, the two plates were pulled apart with the same speed to release the compressive load. The whole squeezing–relaxing process was recorded by a camera (D750, Nikon) at 60 frames per second. The droplet squeeze tests were repeated five times for each microstructured sample.

3. RESULTS AND DISCUSSIONS 3.1 Results of droplet squeeze test Figure 1(c) shows a force F versus displacement D curve (red curve) together with a few representative snapshots obtained from a typical squeeze test on a microstructured sample with a=70 µm and b=160 µm. As indicated by the snapshots, the droplet had a sphere-like cap shape before its contact with the upper plate. The upward displacement of the lower sample, D, was measured after the droplet got an initial contact with the upper plate. During the initial squeeze stage, the droplet was gradually flattened and became drum-shaped after spreading. The force–displacement relation during this initial stage was nonlinear and F increased with a rising slope (segment OA in the curve). At the critical displacement of D=1.56 mm (point A), the Cassie wetting state became unstable and quickly evolved into a full Wenzel state at D=1.57 mm (point B) within ~50 ms. The transition caused a sudden drop in force F 6


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(33.6% reduction, from 4.02 mN to 2.67 mN). Upon further compression, the droplet was flattened more, expanded laterally, and penetrated deeper into the interspaces of the micropillars. During this stage (BC), F increased rapidly, indicating an enhanced resistance to droplet compression. Finally, water would fill all gaps at the maximum compressive load at around 7.22 mN (point C). When the upper plate started to retract from the maximum compression, the droplet would initially stay in the Wenzel state (segment CE). The apparent contact angle of the droplet in this stage decreased rapidly as the upper plate retracted, exhibiting a strong hysteresis due to the pinning effect from the micropillars. The contact angle became lower than 90° when D> h ), Eq. (1) can be approximated as V = πr 2h . As a consequence, the above model gives a scaling −2

law as F ≈ γ V ( h0 − D ) , which is consistent with the previous theoretical analysis34. As can be seen from Fig. 2(c), the force–displacement relation predicted by the above model (the blue curve) agrees well with the experimental results (the red curve) prior to the occurrence of the Cassie–Wenzel transition. A good consistency between the experimental and theoretical results was also obtained for the contact radius r and the droplet height h, as shown in Fig. 2(d) and (e), respectively (See Fig. S2 for more discussions)

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(a)

8

(c) r

h

Upper plate Droplet

Experiment Theory

6

k

θ 0A θ

F (mN)

F

A

4

F

R=0.9988

2 0

Sample

-2 0.0

0.3

0.6

0.9

1.2

1.5

1.8

D (mm)

(d) 2.0 Water

θw Air Pillar

(e) 6 Experiment Theory

1.5 1.0 0.5

h

0.0 0.0

0.5

1.0

1.5

D (mm)

2r (mm)

(b)

h (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5 4 3 2 1 0 0.0

Experiment Theory

2r 0.5

1.0

1.5

D (mm)

Figure 2. (a) The droplet squeeze model at the CB state. (b) The interface between the droplet and the micropillars at the critical Cassie–Wenzel transition. R is the correlation coefficient for the theory and the experiment, which is defined in Supporting Information. (c)–(e) Comparisons between the theoretical results and the experimental results. For the sample (70, 160), the theoretical and experimental results of force F, droplet height h, and diameter 2r with respect to the increasing displacement D in the squeezing stage OA of the droplet squeeze test are compared in (c), (d), and (e), respectively.

3.3 Stability analysis of the Cassie–Baxter wetting state As shown in Fig. 2(b), when the squeeze pressure reaches a critical value, the droplet cannot maintain its stability at the CB state and the lower liquid–air interface hanging on the micropillars will collapse, leading to Cassie–Wenzel transition. Let Fc denote the critical force at the Cassie–Wenzel transition, ∆Pc the corresponding pressure difference across the liquid–air interface, and θ w the contact angle between 12


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the liquid and the sidewall surface of the micropillars. In a previous reference,27 by assuming θ w = θ 0 , the theoretical critical pressure ∆Pctheory was proposed as

∆Pctheory = −

4γ a cos θ 0 , d 2 − a2

(5)

where d = a + b . In our experiments, the critical force Fcexp can be readily obtained from the sudden transition point in the force–displacement curve, e.g., Fcexp =4.02 mN for the example sample (70, 160). Therefore, the critical pressure ∆ Pcexp can be calculated from Eq. (3) by plugging that F = Fcexp and r = rcexp , where rcexp is the critical contact radius captured by the snapshot at the moment of the Cassie–Wenzel transition. Comparing the theoretically predicted critical pressure ∆Pctheory with the experimentally measured value ∆ Pcexp , we found that the theoretical predictions were always noticeably smaller than the experimental values for all samples we tested, as shown in Fig. 3. For example, for the sample (30, 100), our experiments give ∆ Pcexp = 466.2±28.7 Pa; while the theoretical model predicts ∆Pctheory =216.6 Pa, exhibiting a significant discrepancy. Since the Cassie–Wenzel transition is essentially caused by the depinning of the three-phase line on the micropillar sidewalls, the three-phase line starts to move only when the apparent contact angle reaches the advancing angle.35 The relatively low critical pressure predicted by the model might be caused by the difference in the static contact angle and the advancing contact angle. Therefore, we replace θ 0 with the advancing contact angle θ 0A in Eq. (5) and re-calculated the critical pressure. As 13


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shown in Fig. 3, the theoretical values are still much lower than the experimental results. It is emphasized that, though this discrepancy was rarely discussed and reported previously, it seemed to commonly exist in many previous experiments32, 36.

700

Theory, assume θw=θ0 Theory, assume θw=θ0A

600

Experiment 500

∆Pc (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

400 300 200 100 0

(a,b) (µm) (

00 ,1 0 3

)

) ) ) ) 0) 0) 0) 0) 0) 0) 0) 0) 30 60 60 30 13 16 13 10 13 13 10 16 ,1 ,1 ,1 ,1 , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 (4 (7 (8 (6 (5 (5 (7 (9 (1 (1 (1 (1

Figure 3. The theoretical (shadow red bar according to Eq. (6), shadow blue bar according to Eq. (6) by replacing θ 0 with θ 0A ) and experimental (green bar) critical pressure ∆ Pc for the samples.

3.4 Origin of discrepancy: Enlarged contact angle on the micropillar sidewalls To explore the mechanism that causes the discrepancy, we re-examined the interface configuration by considering a unit cell, as shown in Fig. 4(a). At the point of the Cassie–Wenzel transition, the force along the vertical direction comes from two contributions: One is from the external pressure ∆ Pc  (b + a ) 2 − a 2  , and the other from the surface tension 4aγ cos θ w , where θ w is the contact angle of water on the sidewalls of the micropillars.27 Since the dimensions (a, b) of the micropillars and the 14


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surface tension γ can be measured reasonably well, the inconsistent prediction of ∆Pctheory may arise from the incorrect assessment of the contact angle θ w . Although

θ w is usually assumed to be equal to the intrinsic contact angle, θ 0 ,27 it may differ from θ 0 in reality. For example, the surfaces on the microstructures (e.g., micropillars) are usually not smooth at the nanoscale due to fabrication processes. The nanoscale surface roughness may hinge the movement of the three-phase contact line,21, 37-38 and the apparent contact angles are usually dominated by three–phase contact line, instead of the contact area.39 In addition, the corners of the microstructures are usually highly curved or even with sharp edges, which may have a profound impact on its pinning capability. To examine this aspect, we back calculated the “expected” contact angle of the sidewalls θ%w using the experimental pressure at the Cassie–Wenzel transition,

( cos θ%w = −

d 2 − a 2 ) ∆Pcexp

4γ a

.

(6)

We refer to this as the expected value from ‘force-fitting’, and include them as red columns in Fig. 4(c). For the sample (30, 100), θ%w is evaluated as 148.4°±5.8°, which is significantly larger than θ 0A = 117.4 o ± 2.9 o and θ 0 = 113.3o ± 3.6o measured on a flat surface. To further confirm that the value of θ w was indeed enlarged compared to the intrinsic value of a smooth surface, we independently estimated θ w to be θˆw by examining the deformed configuration of the droplet at the critical point of transition. If we assume a circular profile for the liquid–air interface hanging between adjacent pillars, then θˆw can be directly calculated from the geometrical relation, 15


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θˆw = sin −1  (b 2 − 2δ 2 ) (b 2 + 2δ 2 )  , where δ is the maximum deflection of the liquid–air interface, as shown in Fig. 4(b). We refer this to be the “true” value from ‘deflection-fitting’ (See Fig. S3 for more discussions). Fig. 4(c) indicates that θ w obtained from the force-fitting method agrees well with the value from the deflection-fitting method and both values are larger than the intrinsic contact angle

θ0 .

γ

(a)

θw

Pillar

(c) ∆Pc

Liquid–air interface

Liquid–air interface

(b)

Force-fitting method Deflection-fitting method

180

θw (°)

120

θ0 60

θw δ ) 10

,1 30

) (1

,1 60 (1

10

13 (7 0,

16 (7 0,

0)

0)

) 30 ,1 (4 0

0,

10 0)

0

(a,b) (µm)

(3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Figure 4. Schemes of (a) the force-fitting method and (b) the deflection-fitting method. (c) θ w estimated by two methods are much larger than θ 0 .

If we denote the deviation of the contact angle on the micropillar sidewalls from the intrinsic contact angle as ∆θ = θ%w − θ 0 , we can plot the variation in ∆ θ with the solid area fraction φ in Fig. 5(a), where θ%w is estimated by the force-fitting method. It can be seen that ∆θ randomly scatters from 24° to 45° with an average about 34°. 16


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The independency of ∆ θ on the micropillar density and the distance of neighboring pillars suggest that ∆θ is unlikely to be caused by the interaction of the three-phase line from adjacent pillars40-42. To trace the origin of ∆ θ , we used atomic force microscopy (AFM) to inspect the topography of the top and sidewall surfaces of a micropillar. As shown in Fig. 5(b), the top surface is relatively smooth while the sidewall surface is rather rough and contains nanoscale periodic and semicircular–scalloping stripes. These stripes were likely formed during the microfabrication process when silicon wafers were etched.43-46 Since the apparent advancing or receding contact angle of a surface can sensitively depend on surface morphology

2, 4, 47-51

, we attribute the noticeable difference in

contact angle ∆θ to these nanosized stripes.

(a)

(b)

100

Pillar 75

∆θ = ߠ෨w－θ0 (° )

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1 µm 50

β

(c) 25

x

θw

Water

Pillar

0 0

5

10

15

20

25

Air

Solid fraction φ (%)

1 µm

Figure 5. (a) Variation of the contact angle difference ∆ θ with respect to the solid fraction φ . (b) AFM images of the typical topography of the micropillar top surface 17


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and sidewalls. The top surface is relatively smooth and has an RMS roughness of 0.73 nm (5 µm×5 µm). However, the sidewall surface is rather rough with clear periodic, semicircular–scalloping stripes of 60 nm in height and 430 nm in width along the horizontal direction. (c) Schematic of the three–phase contact line hanging on the concavoconvex sidewall surfaces with a sinusoidal profile. x represents the vertical distance between the three-phase contact line and the pillar top, β and θ w represent the local inclination angle at the three-phase contact line of the sidewall surface and the liquid–air surface, respectively.

Figure 5(c) illustrates the schematic of the liquid–air interface hanging on the pillar tops. The curvature of sidewall surface will affect the value of θ w as well as the critical pressure ∆ Pc . Based on the topography observed by AFM, the sidewall surface of pillars is approximately sinusoidal with a wavelength λ =430 nm and an amplitude

ξ =60 nm. When the three-phase contact line moves downward the sidewalls in a quasi-static process, the concavoconvex sidewall surface may introduce an unstable equilibrium of the three-phase contact line23, 29. To study the equilibrium and the stability condition, we analyze the Gibbs free energy of the system. Ignoring the effect of gravity at this scale, the Gibbs free energy of the system can be simplified as29

G = −γ cos θ 0 ASL + γ ALA − ∆PVim ,

(7)

where ASL and AL A are the surface areas of solid–liquid and liquid–air, respectively, 18


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∆ P is the pressure difference across the liquid–air interface and Vim is the volume of

liquid immersed in the interspaces of pillars. θ0 is the intrinsic contact angle of silicon wafer. Equilibrium of the three-phase contact line requires the first derivative of the Gibbs free energy with respect to ASL to be zero29, which gives θ w ( x) = θ 0 + β ( x) , with β ( x ) being the local inclination angle of the sidewall surface at the three-phase contact line (i.e., position x). Therefore, if one substitutes the contact angle θ 0 for the actual contact angle θ w ( x) in Eq. (5), the equilibrium pressure can be expressed as

∆P ( x ) = −

4γ a cos[θ 0 + β ( x )] . (a + b) 2 − a 2

(8)

Equation (8) can be solved numerically and the results for the samples (70, 160) are shown in Fig. 6. When the equilibrium pressure reaches the maximum, the pinning force of the three–contact line is insufficient to sustain a larger hydraulic pressure, which will eventually cause the loss of stability of the system. The instability point at the critical pressure can be obtained as ∆ Pc =384.48 Pa, corresponding to an apparent contact angle increase about β c = 41.2 o . Therefore, the roughness of the sidewall surface of the pillars can affect the apparent advancing angle to a large extent. It is noted that the contact angle enhancement revealed in our experiment is relatively smaller compared to the theoretical estimation, which can be partially attributed to the non-ideal sinusoidal profile of the real surface roughness. Therefore, if one uses the correct value of θ w , the theoretical predictions24, 27-29 will still be applicable. This also directly confirms that the stability of surface hydrophobicity can be enhanced by decorating the surfaces with microstructures as 19


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proposed previously36, 52-53.

400

∆Pc

300

∆P (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Stable Unstable

200 100 0

-100 -200 0.0

0.2

0.4

0.6

0.8

1.0

x/λ Figure 6. Variation in the equilibrium pressure ∆P with respect to the vertical position of the three-phase contact line at sidewalls.

4. CONCLUSIONS The stability of Cassie–Baxter wetting state on micropillar arrays was studied both experimentally and theoretically. The experimental results indicate that the theoretically predicted critical pressure, where unstable CB to Wenzel transition occurs, is significantly underestimated. Through independent measurements, we attributed the discrepancy to the incorrect assessment of the contact angle on the sidewall surfaces of the micropillars. We confirmed that the theoretical models would still hold if the correct contact angle of the sidewall surfaces is adopted. This study not only sheds light on the stability of CB state but also suggests a potential route of tuning surface superhydrophobicity using finer scale surface features. 20


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Acknowledgments Supports from the National Natural Science Foundation of China (Grant No. 11432008 and 11772169) and the National Basic Research Program of China (Grant No. 2015CB351903) are acknowledged. The authors thank J. Liu, H. Z. Li, J. Zhang, S. Zhang and S. Qiao for their help in experiments.

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12. Lafuma, A.; Quéré, D. Superhydrophobic states. Nature Materials 2003, 2 (7), 457-460. 13. Lv, P.; Xue, Y.; Shi, Y.; Lin, H.; Duan, H. Metastable states and wetting transition of submerged superhydrophobic structures. Physical Review Letters 2014, 112 (19), 196101. 14. Yao, X.; Chen, Q.; Xu, L.; Li, Q.; Song, Y.; Gao, X.; Quéré, D.; Jiang, L. Bioinspired ribbed nanoneedles with robust superhydrophobicity. Advanced Functional Materials 2010, 20 (4), 656-662.

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26. Forsberg, P.; Nikolajeff, F.; Karlsson, M. Cassie–Wenzel and Wenzel–Cassie transitions on immersed superhydrophobic surfaces under hydrostatic pressure. Soft Matter 2011, 7 (1), 104-109. 24


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27. Zheng, Q. S.; Yu, Y.; Zhao, Z. H. Effects of hydraulic pressure on the stability and transition of wetting modes of superhydrophobic surfaces. Langmuir 2005, 21 (26), 12207-12212. 28. Afferrante, L.; Carbone, G. Microstructured superhydrorepellent surfaces: effect of drop pressure on fakir-state stability and apparent contact angles. Journal of Physics: Condensed Matter 2010, 22 (32), 325107.

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two-scale surfacial structure of lotus leaves. Langmuir 2007, 23 (15), 8212-8216. 53. Su, Y.; Ji, B.; Zhang, K.; Gao, H.; Huang, Y.; Hwang, K. Nano to micro structural hierarchy is crucial for stable superhydrophobic and water-repellent surfaces. Langmuir 2010, 26 (7), 4984-4989.

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F

F (mN)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8 4

Squeezing Relaxing

Transition

0 0.0

0.6

∆P

theory c

D Pillar

1 µm

1.2

1.8

D (mm)

< ∆P

exp c

θ0

θw

1 µm

For Table of Contents Use Only.

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