Theoretical Consideration of Wetting on a Cylindrical Pillar Defect

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Theoretical Consideration of Wetting on a Cylindrical Pillar Defect: Pinning Energy and Penetrating Phenomena Hiroyuki Mayama*,† and Yoshimune Nonomura‡,§ † ‡

Research Institute for Electronic Science, Hokkaido University, CRIS Building, N21W10, Sapporo 001-0021, Japan Department of Chemistry and Chemical Engineering, Graduate School of Engineering and Science, Yamagata University, 4-3-16 Jonan, Yonezawa 992-8510, Japan ABSTRACT: Wetting on a cylindrical pillar defect is discussed in terms of the free-energy difference ΔG. Wetting is divided into wetting on a flat surface, a pinning effect at the apex of the defect, and wetting on a pillar wall. First, we confirmed that ΔG between before and after ideal wetting on a flat surface can be derived as a function of the contact angle θ in which the free-energy minimum is obtained as the equilibrium contact angle θeq described by Young’s and Wenzel’s laws. Second, the pinning effect at the apex in the cross section of the pillar defect is discussed in ΔG, where the pinning effect is shown to originate from the energy barrier by an increase in the air-liquid interfacial area of a pinned droplet induced by deformation. Next, the ΔG profiles of wetting on the pillar wall are drawn based on the theory of Carroll (Carroll, B. J. J. Colloid Interface Sci. 1976, 57, 488-495) to better understand the ΔG profile during penetration. Differences in the manner of wetting between the wetting state on a flat surface and the pillar wall are reflected in ΔG. Finally, penetration of a droplet into a pillar defect is comprehensively discussed on the basis of wetting on a flat surface and a pillar wall. If we consider a simple manner of penetration, another type of energy barrier resulting from an anomalous deformation of the air-liquid interface of the penetrating droplet can be theoretically suggested. Consequently, two types of energy barrier are found. These energy barriers should play a significant role in the hysteresis of wetting, the liquid-repellent Cassie-Baxter state (CB), and the CB-Wenzel wetting transition on a microtextured surface.

1. INTRODUCTION Wetting phenomena are always seen in nature. For example, wetting and dewetting on leaves1-6 and insects7-10 are familiar for us. On the other hand, various wetting phenomena have been found in waxes and fats,11-17 surfaces covered by needles,18-21 agar gels,22 polymer films,23-26 glass,27 and multipillar surfaces.28-33 In this Article, we focus on wetting on a cylindrical pillar defect where the pinning effect and the penetration of a droplet into a cylindrical pillar defect, as shown in Figure 1, are included. Similar penetration is a familiar phenomenon, where a water droplet is initially repelled by fine feathers of down and fine hairs but eventually penetrates into them. Penetrating phenomena are related to the Cassie-Baxter (CB) state and the CB-Wenzel transition,2,4,5,28-38 but the underlying causes of such wetting phenomena are still unclear. To capture the essence of this phenomenon, we simplify it as the penetration of a droplet into a cylindrical pillar defect. This encompasses different manners of wetting, including wetting on the flat cross section of pillar defects, a pinning effect arising from the apex of the pillar defects, and wetting on a pillar wall. The first can be treated as ordinary wetting on a flat surface with an equilibrium contact angle θeq expressed by Young’s law, which occurs under a suitable droplet volume where the contact line (three-phase line) of the droplet does not reach the apex. The second is usually interpreted as a situation in which the contact angle at the apex changes with a finite change in the contact angle between θeq and θeq þ R,2 where R is the tilting angle r 2011 American Chemical Society

Figure 1. Schematic representation of wetting on a cylindrical pillar defect, including the pinning effect and penetration.

of the pillar wall (usually, R = 90°). The third is closely related to wetting on a fiber, and the geometry of the droplet on a fiber has been considered theoretically and experimentally.39-41 While each manner of wetting has been independently studied by various approaches, in this study we comprehensively reconsidered them to determine the essence of penetration in terms of the free-energy difference ΔG, where ΔG profiles generally lead us to a direct understanding of thermodynamics and kinetics. We would like to emphasize the importance of ΔG before the discussion. Wetting phenomena have usually been discussed in terms of the surface tension, surface energy with a minimum Received: November 18, 2010 Revised: December 24, 2010 Published: February 22, 2011 3550

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Figure 2. Schematic representations of before (a) and after (b) ideal wetting and the geometry of the system after ideal wetting (c). Hemispherical geometry is assumed after wetting.

surface area, and the free energy G.42-52 However, we should recognize that ΔG determines the absolutely stable state between before and after wetting in thermodynamics while G only determines the relatively stable state in wetting, and a discussion in terms of the surface tension cannot argue a thermodynamically stable state. In this Article, based on ΔG, we systematically discuss wetting on a flat surface in section 2, the pinning effect at the apex of a cylindrical pillar defect in section 3, wetting on a fiber (pillar wall) in section 4, the penetration of a droplet into a pillar defect in section 5, and the ΔG profile in the overall process from wetting on the flat cross section of the defect to penetration in section 6. We offer a conclusion in section 7. In the theoretical treatments, we neglect the effect of gravity.

2. FREE-ENERGY DIFFERENCE OF WETTING ON A FLAT SURFACE We consider the situations shown in Figure 2. Before ideal wetting, a droplet is a sphere with radius r0, while the substrate has surface area AS, as shown in Figure 2a. After ideal wetting, the drop becomes a hemisphere to maintain a minimum surface area with a unique contact angle θ, as shown in Figure 2b and c, where we neglect the wicking front. The areas of the air-liquid, air-solid, and solid-liquid interfaces are denoted as A0L, A0S, and A0SL, respectively. The interfacial energy densities of the air-liquid, air-solid, and solid-liquid interfaces are denoted as γL, γS and γSL, respectively. To consider the effect of surface roughness at the solid-liquid interface, we introduce a roughness factor rr in the free-energy description. Let us first consider the free energy of ideal wetting. Since the free energies before and after wetting, G and G0 , respectively, are described as ð1Þ G ¼ γS AS þγL AL G

0

¼ γS AS0 þγL AL0 þγSL ASL0

the free-energy difference, ΔG = G0 - G, is ΔG ¼ γSL -γS ASL0 þγL AL0 -AL

ð2Þ ð3Þ

Since A0S and A0SL are functions of θ, ΔG can be formulated by θ. Let us discuss the relation of θ with AL0 and A0SL. First, r1 (θ, r0) can be obtained from the volume of the droplet VL = (4/3)πr02 (r0: the radius of the droplet before wetting): 1=3 4 ð4Þ r0 r1 ðθ,r0 Þ ¼ 2 - 3cos θþcos3 θ From the relations r2(θ, r0) = r1(θ, r0) sin θ and r3(θ, r0) = r1(θ, r0) cos θ, A0SL and A0L can be formulated as functions of θ. A0SL(θ, r0) is 2=3 4 0 ð5Þ r0 2 sin2 θ ASL ðθ,r0 Þ ¼ rr π 2 - 3cos θþcos3 θ

Figure 3. Free-energy profiles of wetting states from complete wetting to complete dewetting on flat surfaces. The profiles from bottom to top have an equilibrium contact angle θeq = 0, 30, 60, 90, 120, 150, and 180°, respectively, while the dashed line represents the trace of the free-energy minimum described by eq 9. γL and rr are assumed to be 72 mJ/m2 and 1, respectively, and γS - γSL = γL cos θeq.

where rr = 1 and rr > 1 for smooth and rough surfaces, respectively. On the other hand, A0SL(θ, r0) is 2=3 4 0 r0 2 ð1-cos θÞ ð6Þ AL ðθ,r0 Þ ¼ 2π 2 - 3cos θþcos3 θ From eqs 3, 5, and 6, ΔG is obtained as ΔGðθ,r0 Þ=4πr0 2 ¼ ΔgðθÞ¼ rr γSL -γS sin2 θ 4-1=3 þ2γL ð1-cos θÞ -γL ð2 - 3cos θþcos3 θÞ2=3

ð7Þ

This is the first description of the dependence of the free-energy difference on the contact angle, where ΔG can be represented as a product of a θ-dependent function, Δg(θ), and 4πr02 (= AL). Here, Δg(θ) is convenient for discussing the θ-dependence of wetting phenomena. From ∂[Δg(θ)]/∂θ = 0, the thermodynamically stable contact angle θeq can be derived as rr ðγS -γSL Þ cos θeq ¼ ð8Þ γL Young’s law (rr = 1) and Wenzel’s law (rr > 1) are thus obtained as the condition of the free-energy minimum. Here, we realize again that θ at the contact line is a unique factor that determines the thermodynamically equilibrium state through the balance between A0SL and A0L in ideal wetting. The depth of the free-energy profile at θeq is formulated as35,36 ΔGðθeq ,r0 Þ=4πr0 2 ¼ Δgðθeq Þ 1=3 ¼ 4-1 = 3 2 - 3cos θeq þcos3 θeq γL -γL ð9Þ Figure 3 shows free-energy profiles from complete wetting to complete dewetting, where we assume that γL = 72 mJ/m2 (water), γS - γSL is set to be θeq = cos-1[(γS - γSL)/γL] and rr = 1, and we neglect the Cassie-Baxter state. In Figure 3, the θ-range of Δg e 0 at arbitrary θeq is always distributed between smaller θ and 180°, which means that metastable wetting states appear under suitable conditions. The trace of the free-energy minimum is also shown. It is obvious that the depth of ΔG/4πr02 becomes shallower with an increase in θeq. In particular, the 3551

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Figure 4. Distribution of the depth of ΔG/4πr02 in the θeq-θ plane. The contour lines are drawn for ΔG/4πr02 = -70, -60, -50, -40, -30, -20, -10, -5, and 0 mJ/m2. The dashed line represents the trace of the free-energy minimum, i.e., θeq.

profile for θeq = 150°, the lowest limit of the empirical condition of super water-repellency, is very close to that for θeq = 180°. Figure 4 shows the distribution of depth in the θeq-θ plane. Reflecting the asymmetry of the free-energy profiles for θeq, the distribution of depth is asymmetric for the trace of θeq on the diagonal line. This discussion based on the free-energy difference allows us to determine the absolute depth of ΔG, while we can determine the relative stability of wetting by a discussion in terms of free energy after wetting, G0 .

3. PINNING ENERGY IN A PINNING EFFECT AT THE APEX OF A CYLINDRICAL PILLAR DEFECT Based on ΔG in wetting on a flat surface, let us discuss the pinning effect. Here, we focus pinning energy. To capture its essence, we consider wetting situations shown in Figure 5a-c, where rr = 1. When the contact line is far from the defect (Figure 5a), the wetting manner behaves as wetting on a flat surface. However, when the contact line reaches the apex, the droplet is then pinned along the apex and the volume of the droplet increases until the contact angle reaches θeq þ R. After that, the contact line is depinned and advancement along the slope occurs spontaneously. Figure 5d shows the geometry of the pinned droplet. If we consider that ΔG(θeq, r0) is the product of Δg(θeq) and 4πr02(θeq) as shown in eq 9, it is possible to independently consider their dependences on the distance between the contact line and the starting point with an increase in the drop volume, Δg(θeq d)) and 4πr02(θeq(d), d), respectively, where d is the distance between the contact line and the starting point along the reference axis and r0 can be evaluated from the volume. As shown in Figure 5e, the ideal contact angle at an arbitrary position in d < R, θeq(d) is constant at θeq. However, at R, it varies between θeq and θeq þ R, where R is the angle of the slope and now R = 90°. Therefore, as shown in Figure 5f, the free-energy density at an arbitrary position, Δg(θeq(d)), except at R, is constant at Δg(θeq), while at R it changes between Δg(θeq) and Δg(θeq þ R). On the other hand, Figure 5g and h qualitatively shows that the volume and surface area of the pinned droplet increase discretely with a change in θeq(R), as shown in Figure 5d, when the contact line is pinned. From the volume, we can determine 4πr02(θeq(R), R). In the case of 90° e θeq þ R e180°, the

Figure 5. Schematic representations of the wetting on the cross section of the pillar defect (a-c) and the geometries in the pinning (solid shape) and depinning states (dashed shape) (d), where d ∼ R is assumed at the depinning state. (e-i) Schematic dependences of θ, Δg, V, 4πr02, and ΔG on d, respectively.

volume of the pinned drop VL(θ(R), R) at d = R is VL ðθeq ðRÞ,RÞ ¼

2π R 3 R 3 π R 3 3 -π cos θþ cos θ 3 sin θ sin θ 3 sin θ

ð10Þ where the geometrical relation between r0, r1, R (=r2), and θ follows that in Figure 2c, r1 = R/sin θ, θ = θeq(R) = θeq ∼ θeq þ R. The AL of the pinned droplet is 2=3 ð11Þ 4πr0 2 ðθeq ðRÞ,RÞ ¼ ð4πÞ1=3 3VL ðθeq ðRÞ,RÞ Note that Δg and 4πr02 are connected each other through VL. When VL is smaller (larger), both Δg and 4πr02 are smaller (larger). Therefore, the stepwise ΔG(θeq, r0) profile is obtained as shown in Figure 5i. However, we can understand that the pinned droplet can be depinned when we give some perturbation such as pushing, shaking, and vibration to the droplet. Along the usual explanation of the pinning effect,2 the droplet would be depinned if the contact angle of the droplet becomes more than the threshold contact angle (θeq þ R) by deformation, where it is easy to understand that the air-liquid interfacial area increases and play as an energy penalty in ΔG. To capture the essence of the pinning energy, we consider a simple situation as shown in Figure 6. Figure 6a and b represents the situations before and after perturbation, respectively, where we assume that the droplet has a pancake-like geometry as shown in Figure 6c when we push the droplet downward until the contact angle reaches the threshold contact angle. We calculate A0L and ΔG based on the geometry. 3552


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ΔGpc ∼ -100R2 (mJ) from eq 3, where γL = 72 mJ/m2 and γS γSL = γL cos θeq. On the other hand, the A0L of the hemispherical cap with θeq = 30° is 3.48R2 from eq 6 and ΔG is calculated to be ca. -150R2 (mJ). The pinning energy Epin is estimated as Epin ¼ ΔGpc -ΔG

Figure 6. Schematic representations of the droplet before (a) and after deformation (a pancake) (b), the geometry (c), and pinning energy in ΔG profile (Figure 5i) (d). Here, we assume θeq = 30°.

First, let us calculate the radius of curvature of the side of the pancake, r4, from constrain of the constant volume. We assume that r4 is related to the geometry as ð12Þ ðxþAr4 -RÞ2 þz2 ¼ r4 2 = 30°, which means A In the following discussion, we assume θ eq √ = cos θeq = 3/2 in eq 12. The volume is calculated from Z þr4 sin θeq pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 VL ¼ π r4 2 -z2 -Ar4 þR dz -r4 sin θeq

2 2ð1þA2 Þsin θeq - sin3 θeq 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi -2A 1-sin2 θeq -2Aθeq r4 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2R 1-sin2 θeq -4AR sin θeq 2 2 þ2Rθeq r4 þ2R r4

¼π

ð13Þ

From constant volume, the following relation is obtained.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 2 2 π 2ð1þA Þsin θeq - sin θeq -2A 1-sin θeq -2Aθeq r4 3 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2R 1-sin2 θeq -4AR sin θeq 4π þ2Rθeq r4 2 þ2R 2 r4 - r0 3 ¼ 0 ð14Þ 3 For example, since the volume of the pinned droplet in the case of θeq = 30° is 0.43R3 from eq 10, it is numerically obtained from eq 14 that r4 = 0.135R. Next, let us calculate A0L of the pancake, where we assume that the upper surface of the pancake faces ambient environment (air) and A0SL = πR2 in the base. The surface area of the side A0Lside is # Z þr4 sin θeq " Ar4 2 Rr4 0 ALside ¼ 2π r4 -pffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffi dz r4 2 -z2 r4 2 -z2 -r4 sin θeq ¼ 4π r4 2 sin θeq þ Rr4 -Ar4 2 θeq ð15Þ

ð16Þ

As a result, Epin is ∼50R2 (mJ). If R = 1 mm, Epin = 5 10-5 J, which corresponds to the difference in kinetic energy when we stop wetting systems moving with velocity ∼14.4 m/s. This order is close to our feeling when we quickly shake water drops off a hydrophilic plate. Figure 6d illustrates the energy barrier on the ΔG profile. As well-known in first-order phase transition, the energy barrier in ΔG affects the time evolution of the system. The existence of the energy barrier in the pinning effect and the correlation with the time evolution would be clarified in future studies. Thus, it is roughly revealed that the deformation of the droplet on the cross section of the pillar defect to reach the threshold contact angle is the origin of the pinning effect. The above discussion allows us to estimate Epin of a pinned droplet with arbitrary θeq (e90°), θ(R), R, and VL including a smaller droplet which does not reach the apex. This is the first estimation of the pinning energy, and it readily explains why water droplets pinned to a down feather penetrate when we flick the feather, since the flick corresponds to the perturbation energy required to deform the droplet to go over the threshold contact angle. The freeenergy profile after depinning (d > R) is discussed in section 5. On the other hand, the above scenario suggests experimental strategies to enhance the pinning energy and to design liquidrepellency. An effective approach would be to design higher R for a highly threshold contact angle. In fact, a superoleophobic surface has been achieved through the use of thin disks (R ∼ 180°) on the tops of micropillars.53

4. ΔG PROFILES OF THE WETTING ON A CYLINDRICAL PILLAR WALL Next, we consider the wetting state on a pillar wall, as shown in Figure 7, before we discuss the penetration of a droplet into a cylindrical pillar defect. This problem is well-known as “wetting on a fiber”. A theory of wetting on a fiber (a pillar wall) is already established,39 which formulates the shape and the air-liquid and solid-liquid interfacial area of the droplet on the fiber. Since the geometry of the system is quite different from that in wetting on a flat surface, the thermodynamically stable state is not consistent with Young’s law, as shown below. The essence of the theory of wetting on a fiber is summarized in the Appendix. Based on the theory, we describe the ΔG profiles. The normalized L, A0L, A0SL, and VL of a droplet on a fiber are defined as L, A0L, A0SL, and VL, as follows: _ L L ¼ ¼ 2½aFðj, kÞþnEðj, kÞ ð17Þ x1 _ AL0 A 0 0 ¼ SL ¼ 2πL ¼ 4πðaþnÞnEðj, kÞ, A SL x1 2 x1 2

VL 2πn 2 VL ¼ 3 ¼ 2a þ3anþ2n2 -3 Eðj, kÞ 3 x1 2 1=2 2 3a 2 1=2 - aþ 1-a Fðj, kÞþ n -1 n

AL0 ¼

From r4 = 0.135R, it is obtained that A0Lside = 0.89R2. Since the upper interfacial area A0Lup = πR2, it is calculated that A0L = 4.03R2 and the free-energy difference in the pancake shape 3553

ð18Þ

ð19Þ


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Figure 7. Schematic representations of before (a) and after wetting on a pillar wall (b) and the geometry of the system after wetting (c).

where x1 (= R in Figure 5) and x2 are the radius of a pillar defect (a fiber) and the thickness of the droplet from the center axis, respectively, a = (x2 cos θ - x1)/(x2 - x1 cos θ), n = x2/x1 g 1, x2 = x22(1 - k2 sin2 j) with k2 = (x22 - a2x12)/x22, sin2 j = (x22 - x2)/(x22 - a2x12). The free-energy difference in the wetting on the pillar wall normalized by x12, ΔGpw, can be obtained as ΔGpw ¼ ΔGpw =x1 2 h i ¼ ðγSL -γS ÞASL0 þγL AL0 -ð4πÞ1=3 ð3VL Þ2=3 ð20Þ

Figure 8. Dependences of n (a), normalized L (b), A0L (c), and A0SL (d) in the wetting on a pillar wall on the contact angle θ at VL = 1, 10, and 100 from bottom to top.

where (4/3)πr30 = VLx31. On the other hand, ΔGpw/4πr02 is helpful for comparison with wetting on a flat surface: h i ΔGpw =4πr0 2 ¼ ð4πÞ-1=3 ð3VL Þ-2=3 ðγSL -γS ÞASL0 þγL AL0 -γL ð21Þ Figure 8 shows the numerical results of the dependences of n, L, A0L, and A0SL on θ at VL = 1, 10, and 100. The minimum in the A0L profiles clearly suggests the existence of a stable wetting state on the pillar wall in ΔG. To demonstrate its existence, we attempt to draw the profiles of ΔGpw/4πr02 at VL = 1, and the fitting equations of A0L and A0SL are useful for this purpose. They are described by eqs 22 and 23, respectively: C0 C2 AL0 ¼ þC4 þC5 θþC6 θ2 þC7 sinðC8 θþC9 Þ 2þ θþC ðθþC1 Þ 3 ð22Þ ASL0 ¼

Figure 9. Free-energy profiles of wetting states on a pillar wall at VL = 1. The profiles from bottom to top are drawn under γL = 72 mJ/m2, γS γSL = γL cos θeq, and θeq = 30, 60, 90, 120, 150, and 180°, respectively.

h i2 D0 D2 þD4 þD5 θþD6 θ2 þD7 e-D8 ðθ-D9 Þ 2þ ðθþD1 Þ θþD3 ð23Þ

where C0 = 8974.07, C1 = 145.22, C2 = 108.403, C3 = 3.0961, C4 = 6.401, C5 = -5.3383 10-2, C6 = 3.678 10-4, C7 = -0.2347, C8 = 2.0190, and C9 = 44.748°, while D0 = 11079.4, D1 = 38.224, D2 = 72.731, D3 = 2.269, D4 = 2.823, D5 = -3.911 10-3, D6 = -6.569 10-5, D7 = 0.5555, D8 = 1.325, and D9 = 0.492. Figure 9 illustrates the ΔGpw/4πr02 profiles at VL = 1, where γL = 72 mJ/m2 and γS - γSL = 62.35, 36, 0, -36, -62.35, and -72 mJ/m2 from bottom to top, respectively, and γS - γSL corresponds to the conditions where θeq = 30, 60, 90, 120, 150, and 180° on the flat surface, respectively. The ΔGpw/4πr02 profiles seem to be almost the same as the ΔG/4πr02 profiles on the flat surface, but the difference between ΔGpw/4πr02 and ΔG/ 4πr02 is quite distinct in the distribution of the depth of ΔGpw/ 4πr02 profiles, as shown in Figure 10, which shows the numerically obtained results of the contour lines of ΔGpw/4πr02 = -50,

Figure 10. Distribution of the depth of ΔGpw/4πr02 at VL = 1 in the θeq-θ plane. The contour lines are drawn for ΔGpw/4πr02 = -50, -40, -30, -20, -10, and 0 mJ/m2, from left to right. The dashed line represents the trace of the free-energy minimum at the equilibrium contact angle on the pillar wall θpw eq .

-40, -30, -20, -10, and 0 mJ/m2 at VL = 1 in the θeq-θ plane. There are three significant differences in comparison with 3554


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Figure 11. Free-energy profiles of wetting states on a pillar wall at VL = 10. The profiles from bottom to top are drawn under γL = 72 mJ/m2, γS - γSL = γL cos θeq, and θeq = 30, 60, 90, 120, 150, and 180°, respectively. Figure 13. Scheme of the penetration of a droplet into a cylindrical pillar defect. (a) Initial state. (b, c) Early stage, θ g 90°. (c, d) Intermediate stage, θ < 90° and Lpen e (1/2)Lfinal. (d-f) Late stage, θ < 90° and (1/2)Lfinal < Lpen e Lfinal.

slightly with an increase in VL. Thus, it becomes obvious that wetting on a pillar wall is different from that on a flat surface in ΔG.

Figure 12. Distribution of the depth of ΔGpw/4πr02 at VL = 10 in the θeq-θ plane. The contour lines are drawn for ΔGpw/4πr02 = -40, -30, -20, -10, and 0 mJ/m2, from left to right. The dashed line represents the trace of the freeenergy minimum at the equilibrium contact angle on the pillar wall θpw eq .

Figure 4. First, the dependence of the equilibrium contact angle on the pillar wall θpw eq on θ is quite different. θeq is consistent with a diagonal line as shown in Figure 4, while θpw eq weaves along the diagonal line in Figure 10, where θpw eq is larger than θeq in the range θ = ca. 23-90° and < ca. 10°, and smaller than θeq at θ = ca. 90-180° and ca. 10-23°. The second difference is that the area of ΔGpw/4πr02 e 0 in the θeq-θ plane depends on VL, which is indirectly described by eq 21. The area increases with an increase in VL, as shown later. Third, the depth of the ΔGpw/4πr02 profiles is always shallower than that of ΔG/4πr02. This shows that the wetting on a pillar wall is relatively unstable in comparison with the wetting on a flat surface. To investigate the volume effect on the wetting on a pillar wall, we draw the ΔGpw/4πr02 profiles at VL = 10, as shown in Figure 11, which is obtained by eqs 22 and 23 with C0 = C1 = 0, C2 = 1308.820, C3 = 26.918, C4 = 7.1701, C5 = 3.0634 10-3, C6 = 5.576510-4, C7 = C8 = C9 = 0, D0 = D1 = 0, D2 = 1515.950, D3 = 28.956, D4 = -5.735, D5 = 6.999 10-2, D6 = 3.448 10-4, and D7 = D8 = D9 = 0. In comparison with Figure 9, the depth becomes slightly shallower than that at VL = 1, as shown in Figure 12, which shows the contour lines of ΔGpw/4πr02 = -40, -30, -20, -10, and 0 mJ/m2. As speculated in the above discussion, there is a volume effect in that the area of ΔGpw/4πr02 e 0 is enlarged. The dependence of θpw eq for θ > 40° is almost the same in the θeq-θ plane, but θpw eq decreases

5. THEORETICAL CONSIDERATION OF THE PENETRATION OF A DROPLET INTO A CYLINDRICAL PILLAR DEFECT In the above sections, we independently discussed two types of wetting, that is, wetting on a flat surface and a pillar wall, in terms of ΔG. In this section, we consider the penetration of a droplet into a cylindrical pillar defect from the point of view of both types of wetting. Here, we try to draw the ΔG profile in the penetration to obtain a semiquantitative understanding. As suggested in section 3, we also expect that the deformation of the droplet induced during penetration plays a significant role because such deformation increases the air-liquid interfacial energy as an energy penalty in ΔG. While no experimental results on the manner of penetration have ever been reported, we roughly assume that penetration occurs with quasi-statically continuous deformation, as shown in Figure 13. We speculate that deformation plays a significant role in ΔG not only in our model but also for other manners of penetration. We assumed the following manner of penetration in Figure 13: When a droplet at the apex with θ g θeq þ R (Figure 13a) is depinned, its contact line with θpw eq advances on the pillar wall spontaneously and quasi-statically (Figure 13b and c). Here, the volume is constant and the upper and lower parts can be approximated as a hemisphere and a hemispindle, respectively, where the boundary between the hemisphere and hemispindle is continuously connected and the penetrating state (the hemispindle) is defined by the penetration length Lpen and width x2. In the penetration, the hemispindle part gradually grows and Lpen and x2 reach (1/2)Lfinal and (x2)max (Figure 13d), respectively, where Lfinal and (x2) max are the length and width, respectively, of the finally formed spindle. They can be evaluated in terms of the final volume of the spindle, as shown later. For Lpen > (1/2)Lfinal, the hemispindle part becomes a bell which tapers off with penetration (Figure 13e). The width of the mouth of the bell is x3 and the 3555


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length of the missing part is Lmiss. Finally, a hemispherical cap appears on the flat cross section and a spindle appears on the pillar wall after penetration (Figure 13f). We attempt to describe this process in terms of ΔG. For convenience in the calculation, we divide the penetration into three stages: early, θ g θeq þ R f 90° (Figure 13a f c), intermediate, Lpen e (1/2)Lfinal (Figure 13c f d), and late (1/2)Lfinal < Lpen e Lfinal (Figure 13d f e f f). In the calculation, we assume that the pillar material is hydrophilic, VL = 5.44R3 and AL = 14.93R2 from eqs 10 and 11, respectively, which are the critical volume and surface area of a pinned droplet with θeq = 30°, θeq þ R = 120° and R = 1 mm, and θpw eq ∼ 45° in Figures 9 and 11. With regard to the volume, 0.43R3 is the volume of the spherical cap remaining on the flat cross section and ∼5R3 is the final volume of the spindle part on the pillar wall. We also assume that γL = 72 mJ/m2 and γS γSL = γL cos θeq = 62.35 mJ/m2. Early Stage in Penetration. Let us consider ΔG in the early stage. First, we assume that the drop has a geometry as shown in Figure 13b. A0L and A0SL in ΔG are divided into the hemispherical and hemispindle parts, respectively: AL0 ¼ AL10 þAL20 ,

ASL0 ¼ ASL10 þASL20

Figure 14. Schematic representations of the dependences of L, A0SL, A0L and VL of a droplet on n in the theory of wetting on a fiber by Carroll (ref 39). The dotted line represents the dependences of L and A0SL, while the solid line shows the dependences of A0L and VL.

because they rapidly increase at n = 1, as shown in Figure 14. As a result, the penetrating part has 1_ 1 Vpen ¼ L lðn-1Þ1=δ ð31Þ 2 2 _ 1 ASL20 ¼ 2πðL Þπlðn-1Þ1=δ ð32Þ 2

ð24Þ

where A0L1 and A0SL1 are due to the hemisphere and A0L2 and A0SL2 are due to the hemispindle. Note that A0SL and A0L are determined under constant volume because penetration occurs under the condition. Let us consider A0L1, A0SL1 and the volume of the hemispherical part, VL1. They are derived as functions of x2 in the range r g x2 and θ g 90°: ð25Þ AL10 ¼ 2πr 2 ð1-cos θÞ ASL10 ¼ πx1 2 VL1

ð26Þ

2 x2 3 x2 3 π x2 3 3 ¼ π þπ cos θcos θ 3 3 sin θ sin θ sin θ ð27Þ

where r = (R/sinθ) and R = x1. For convenience in the later discussion, the normalized A0L1, A0SL1 and VL1, A0L1, A0SL1, and VL1, are defined as follows: AL10 ð28Þ AL10 ¼ 2 ¼ 2πm2 ð1-cos θÞ R ASL10 ¼

ASL10 ¼π R2

ð29Þ

VL1 R3 3 3 3=2 2 n 3 n 2 1=2 π n ¼ π þπ 1-x 1-x2 3 x x 3 x

VL1 ¼

ð30Þ where x1 = R, m = r/R = 1/x, and x = sin θ. Next, let us consider A0L2, A0SL2, and VL2. Here, we assume that the penetrating part is half of a spindle with length 2Lpen. However, the dependences of A0L2, A0SL2, and VL2 on x2 are quite complicated for the treatment along eqs A10-A12 in the Appendix. Therefore, we simplify the relations to capture the essence of penetration. In ref 39, we find that the dependences of L, ASL, AL, and V on n are roughly approximated by power laws

1 AL20 aL ðn-1Þβ 2

ð33Þ

1 VL2 vL ðn-1Þν 2

ð34Þ

where Lpen, A0SL2, A0L2, and VL2 are the length, the air-liquid and solid-liquid interfacial areas, and the volume of the hemispindle (the penetrating part) normalized by x1(R), x12(R2), and x13(R3), respectively, l, aL and ν are coefficients, and the exponents β, δ, ν > 1. For wetting on a fiber with θeq = 30°, we assume that θpw eq is ∼45° in Figures 9 and 11 and that νL ∼ 23.684, ν ∼ 2.292, aL ∼ 48.0, β ∼ 1.414, l ∼ 4.317, and δ ∼ 1.254. For convenience in the following discussion, n is formulated as a function of L: _ !δ L n∼ þ1 ð35Þ l In the early stage, 2Lpen n∼ l

!δ þ1

Since the hemispindle part is still growing, νL2 is _ !δν 1 L VL2 vL 2 l

ð36Þ

ð37Þ

where L = 2Lpen. As a result, νL normalized by R3, VL, in the early stage is _ !δν 1 L VL ¼ VL1 þ vL ð38Þ 2 l From the constant VL, we obtain 2 3 1 πn þπn3 ð1-x2 Þ1=2 - πn3 ð1-x2 Þ3=2 3 3 2 3 _ !δν 1 L þ 4 vL -5:445x3 ¼ 0 2 l 3556

ð39Þ


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With a change in Lpen, we can first calculate n0 and x by eqs 36 and 39, respectively, and then θ, r, A0L1, A0SL2, and A0L2 from x = sin θ, r = R/x, and eqs 28-30. Based on the dependences of A0L1, A0SL2, and A0L2 on Lpen, normalized ΔG, ΔG, can be obtained as ΔG ¼ ΔG=R 2 ¼ γSL -γS ASL10 þASL20 þγL AL10 þAL20 -γL AL

where n0 = x3/x1 and x3 is the width of the mouth of the bell and is obtained from x3=R[(2Lmiss/l)δþ1] along eq 36. The volume balance is 2 0 3 1 πðn Þ -πðn0 Þ3 ð1-x2 Þ1=2 þ πðn0 Þ3 ð1-x2 Þ3=2 3 3 2 3 !δν !δν L 1 2L final miss þ4vL - vL -5:445x3 ¼ 0 2 l l

ð40Þ Equation 40 is valid for calculating ΔG in all three stages. Intermediate Stage in Penetration. In the intermediate stage with θ < 90°, as shown in Figure 13 (d), VL1 and VL1 in which the signs of the second and third terms change are as follows: 3 2 x2 3 x2 1 x2 3 -π ð1-x2 Þ1=2 þ π ð1-x2 Þ3=2 VL1 ¼ π 3 3 x x x

With a change in Lmiss, we can first calculate n0 and x by the equation of x3 and eq 47, respectively, and then obtain θ, m, r, A0L1, A0L2, and A0SL2, where A0L2 and A0SL2 are !βδ !βδ L a 2L final L miss AL20 ¼ aL ð48Þ l 2 l

ð41Þ VL1

3 2 n 3 n 1 n 3 ¼ π -π ð1-x2 Þ1=2 þ π ð1-x2 Þ3=2 3 x x 3 x ð42Þ

On the other hand, we assume that the hemispindle part is still growing, as shown later, and VL2 is expressed by eq 37. Therefore, VL and the volume balance are 3 2 n 3 n VL ¼ π -π ð1-x2 Þ1=2 3 x x _ !δν π n 3 L 2 3=2 1 þ ð1-x Þ þ vL ð43Þ 3 x 2 l 2 3 πn3 πn -πn3 ð1-x2 Þ1=2 þ ð1-x2 Þ3=2 3 3 2 3 _ !δν 1 L þ4 vL -5:445x3 ¼ 0 2 l

ð44Þ

With a change in Lpen, we can first calculate n0 and x by eqs 36 and 44, respectively, then obtain θ, r, A0L1, A0L2 and A0SL2 as shown in the early stage. The ΔG profile in the intermediate stage is obtained by eq 40. Late Stage in Penetration. In the late stage, we assume that the penetrating part with (x2)max changes from a hemisphere to a bell, which can be approximated as a spindle that misses a tip of length Lmiss (a hemispindle). Since θ < 90°, VL1 and VL1 are described by eqs 41 and 42 with the exchange x2 f x3, respectively. On the other hand, due to tapering of the bell, VL2 changes to the following relation (Figure 13c): !δν !δν Lfinal 1 2Lmiss VL2 vL - vL ð45Þ 2 l l Therefore, VL is described as 0 3 2π n0 3 n π n0 3 VL ¼ -π ð1-x2 Þ1=2 þ ð1-x2 Þ3=2 3 x 3 x x !δν !δν Lfinal 1 2Lmiss þvL - vL ð46Þ 2 l l

ð47Þ

ASL20 ¼ 2πðLfinal -Lmiss Þ

ð49Þ

The ΔG profile in the late stage is obtained by eq 40. ΔG Profile in Penetration. Figure 15 summarizes the changes in A0L1, A0L2, A0SL1, and A0SL2 (Figure 15a), θ (Figure 15b), m (= r/R), n (= x2/R), n0 (= x3/R) (Figure 15c), and the ΔG profile (Figure 15d) in the early, intermediate, and late stages of penetration. Consequently, these results suggest the existence of another type of energy barrier, different from that in the pinning effect. In the early stage, ΔG gradually decreases with penetration because the droplet shrivels (γL A0L1 decreases). θ reaches 90° from 120°, r is equal to x2, and the system reaches the local free-energy minimum, which suggests the possibility of “small penetration”.29 In the intermediate stage, the hemispherical part is anomalously deformed and the hemispindle part grows rapidly as shown in the changes in θ, m, and A0L2. These are necessary for further penetration, where volume has to be supplied from the hemispherical part to the growing hemispindle part. The inset in Figure 15a shows the deformation as the dependences of A0SL (upper) and A0L (lower) on Lpen, which helps us to understand the origin of the energy barrier in Figure 15d, where A0SL = A0SL1 þ A0SL2 and A0L = A0L1 þ A0L2. This obviously indicates that the increase of the air-liquid interfacial area is steeper than that of the solid-liquid interfacial one in the intermediate stage. In the late stage, the growing A0L2 and A0SL2 dominate ΔG, which gradually decreases with penetration. Consequently, the existence of an energy barrier in penetration, which appears as a cusp in ΔG, is theoretically suggested as shown in Figure 15d. On the other hand, the scenario also suggests that no energy barrier exists when the defect is shorter than R. While we described a scenario according to the scheme shown in Figure 13, an energy barrier resulting from deformation of the droplet should also be seen for other manners of penetration. Future experimental studies on the penetration should provide helpful findings. In other experiments, a finite Laplace pressure ΔP is observed to penetrate a droplet into a multitextured surface,32 that is, the CB-Wenzel transition, where deformation of the air-liquid interface occurs. The energy barrier resulting from the deformation must be considered to explain the origin of ΔP and the CB-Wenzel transition.35-37 While we discussed the penetration into a hydrophilic pillar, the energy barriers also appear in the penetration into a hydrophobic pillar.

6. ΔG PROFILE OF THE OVERALL PROCESS Figure 16 summarizes the numerical results of ΔG (Figure 16b) and Δg profiles (Figure 16c) of the overall process from the wetting on a flat surface to penetration including the process after full 3557


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Figure 16. Changes in VL (a), ΔG (b), and Δg (c) of the overall process. In the calculation, γL = 72 mJ/m2, γS - γSL = γL cos θeq, θeq = 30°, R = 90°, θpw eq ∼ 45°, and R = 1 mm.

Figure 15. Changes in A0L1 (closed and opened squares), A0L2 (closed and opened circles), A0SL1 (closed and opened rhombuses) and A0SL2 (closed and opened triangles) (a), θ (b), m (closed and opened squares), n (closed triangles) and n0 (opened triangles) (c), and the ΔG profile during penetration (d). The shaded areas show the range of the intermediate stage. Except n, the closed symbols denote the early and intermediate stages, while the opened symbols show the late stage. In the calculation, γL = 72 mJ/m2, γS - γSL = γL cos θeq, θeq = 30°, θpw eq ∼ 45°, and R = 1 mm. The inset in (a) presents dependences of A0L (upper) and A0SL (lower) on Lpen, where A0L = A0L1 þ A0L2 and A0SL = A0SL1 þ A0SL2.

penetration together with a change in volume (Figure 16a), where R = 1 mm, θeq = 30°, R = 90°, γL = 72 mJ/m2, γS - γSL = γL cos θeq, and Δg = ΔG/AL = Δg. Complex behavior is suggested in the penetration. Let us note the ΔG profile in Figure 16b. First, ΔG gradually decreases as the droplet grows on the cross section of the cylindrical pillar defect until the contact line reaches the apex of the pillar defect. However, a discrete jump in ΔG can be described due to the deformation of the droplet in which θeq(R) reaches the threshold contact angle, θeq þ R, when the contact line reaches the apex, while the jump is hidden in Δg. After the depinning, ΔG again gradually decreases with the penetration into the pillar defect. However, a cusp in ΔG, as another type of energy barrier, is

suggested in penetration. This is caused by the rapid increase of the air-liquid interface in the intermediate stage. While this scenario roughly follows the scheme shown in Figure 13, such deformation would be observed to a greater or lesser degree with other manners of penetration. On the other hand, the scenario also suggests that no energy barrier exists if the length of the defect is shorter than R. After that, the state goes over the energy barrier, and ΔG gradually decreases again. Let us mention ΔG after full penetration. After full penetration, the spindle part grows when we add the liquid. Roughly, ΔG in L . R can be described as a function of L: _ ΔGðγSL -γS ÞASL20 þγL ðAL20 -AL ÞµðγSL -γS ÞL h _ i _ þγL ðL Þβδ -ðL Þ2δν=3 ð50Þ where the relations A0SL2 µ L, A0L2 µ (n - 1)β, VL µ (n - 1)ν, and AL ≈ (VL)2/3 are used and the contribution from the spherical cap is neglected. Since βδ = 1.77 and 2δv/3 = 1.92, ΔG obviously decreases with an increase in volume as shown in Figure 16b. However, Δg in Figure 16c shows a different tendency after full penetration in comparison with ΔG . This reflects the change in the geometry of wetting. Throughout the entire process, two kinds of energy barrier are suggested in ΔG. Since the overall ΔG profile is a nucleation-and-growth type because of the energy barriers, this profile suggests that hysteresis of wetting occurs at the level of a single pillar defect. We should emphasize the importance of the scenario based on the ΔG profile to understand the hysteresis of wetting, the liquidrepellent CB state, and the CB-Wenzel transition on a multitextured surface. As addressed above, the existence of an energy 3558


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barrier leads to hysteresis of wetting. An example can be observed through hysteresis of contact angle, that is, the advancing and receding angles in wetting on a multitextured surface.54-56 This may be originated from the energy barrier in the pinning effect of the contact line. Furthermore, this scenario points out that the liquid-repellent CB state on a multitextured surface is a thermodynamically metastable state because it is separated from the Wenzel state by an energy barrier that arises from the deformation of the droplet in the “air-pocket” state defined in refs 35-37. This explains that the CB-Wenzel wetting transition is a first-order phase transition. This consistency between the scenario, the hysteresis of wetting, the CB state, and the CB-Wenzel transition should be examined in further studies.

7. CONCLUSION We have considered penetrating phenomena in terms of a free-energy difference. The free-energy difference between before and after ideal wetting on a flat surface can be described as a function of the contact angle, where Young’s and Wenzel’s laws are obtained as the free-energy minimum condition. Next, the pinning energy was discussed, where the origin of the pinning energy was caused by the deformation of the pinned droplet to go over the threshold contact angle. Furthermore, we described the ΔG profiles for wetting on a cylindrical pillar wall based on the theory of Carroll. We found that the ΔG profile and the equilibrium contact angle are different from those in wetting on a flat surface. Finally, we attempted to understand the penetration of a droplet into a pillar defect based on a rough scheme. The result suggests that another type of energy barrier may exist due to the increase in the air-liquid interface of the droplet during penetration. The existence of these energy barriers should play a significant role in the hysteresis of wetting on a multitextured surface, the Cassie-Baxter (CB) state, and the CB-Wenzel wetting transition. Further experimental and theoretical studies should help to clarify these issues. ’ APPENDIX The essence of the theory of Wetting on a fiber39 is summarized as follows. From the Laplace pressure, ΔP, the curvature K1 of the droplet on the fiber is defined as 1 1 ΔP=γL ¼ þ ðA1Þ ¼ K1 R1 R2

respectively. The shape of the droplet can be obtained from dz - ¼ tan θ dx x2 ðx2 -x1 cos θÞþx1 x2 ðx2 cos θ-x1 Þ ¼h i1=2 x2 ðx2 -x1 cos θÞ2 ðx22 -x2 Þ-x21 ðx2 cos θ-x1 Þ2 ðx22 -x2 Þ ðA5Þ By substituting a = (x2 cos θ - x1)/(x2 - x1 cos θ), this can be modified as dz x2 þax1 x2 ðA6Þ - ¼ 1=2 dx ðx2 2 -x2 Þ-ðx2 -ax1 2 Þ Here, the variable transformation of x2 = x22(1 - k2 sin2 j) with k2 = (x22 - a2x12)/x22 provides the drop profile as z ¼ ( ax1 Fðj, kÞþx2 Eðj, kÞ ðA7Þ where F(j, k) and E(j, k) are the incomplete elliptic integrals of the first and second kind, respectively. Z j 1 Fðj, kÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dj, 1-k2 sin2 j 0 Z j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eðj, kÞ ¼ ðA8Þ 1-k2 sin2 j dj 0

The width of the droplet L is L ¼ 2 ax1 Fðj, kÞþx2 Eðj, kÞ

where sin2 j = (x22 - x2)/(x22 - a2x12), which rules out the range of j. A0L, A0SL, and VL are derived as ðA10Þ AL0 ¼ 4πðax1 þx2 Þx2 Eðj, kÞ ASL0 ¼ 2πx1 L

ðA11Þ

2πx2 2 2 VL ¼ 2a x1 þ3ax1 x2 þ2x2 2 Eðj, kÞ-a2 xFðj, kÞ 3 1=2 2 1=2 x þ x2 2 -x2 x -x1 2 ðA12Þ -πx1 2 L x2 For convenience, the normalized L, A0L, A0SL, and VL are defined as L, A0L, A0SL, and VL as follows: _ L ðA13Þ L ¼ ¼ 2½aFðj, kÞþnEðj, kÞ x1

where R1 and R2 are the radii of the liquid surface from the center of the spherical approximation and the center axis of the pillar, respectively, as shown in Figure 7. The angle φ and the radii are correlated to each other as ðA2Þ R1 dφ ¼ dx sec φ, R2 ¼ x cosec φ Through the use of eq 18, eq 17 becomes 1d ðxsin φÞ ¼ K1 x dx From the boundary conditions, x2 -x1 cos θ 2 x2 cos θ-x1 x sin φ ¼ xþ x1 x2 x2 2 -x1 2 x2 2 -x1 2

ðA9Þ

_ AL0 A 0 0 ¼ SL ¼ 2πL ¼ 4πðaþnÞnEðj, kÞ, A ðA14Þ SL 2 2 x1 x1

V 2πx2 VL ¼ 3 ¼ ð2a2 þ3anþ2n2 -3ÞEðj, kÞ x1 3 3a - a2 þ Fðj, kÞþðn2 -1Þ1=2 ð1-a2 Þ1=2 ðA15Þ n

AL0 ¼

ðA3Þ

where n = x2/x1 g 1. ðA4Þ

where x1 (= R in Figure 5) and x2 are the radius of a pillar defect (a fiber) and the thickness of the droplet from the center axis,

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Telephone & Fax: þ81-11706-9346. 3559


Langmuir Present Addresses §

Department of Biochemical Engineering, Graduate School of Engineering and Science, Yamagata University, Yonezawa 992-8510, Japan.

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Theoretical Consideration of Wetting on a Cylindrical Pillar Defect

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