Wetting Transition on Textured Surfaces: A Thermodynamic Approach

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Wetting Transition on Textured Surfaces: A Thermodynamic Approach Junfei Ou, Guoping Fang, Wen Li, and Alidad Amirfazli J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b05477 • Publication Date (Web): 29 Aug 2019 Downloaded from pubs.acs.org on August 30, 2019

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The Journal of Physical Chemistry

Wetting Transition on Textured Surfaces: A Thermodynamic Approach

Junfei Ou,† Guoping Fang,# Wen Li,*† and A. Amirfazli*†‡

†School

of Materials Engineering, Jiangsu University of Technology, Changzhou, P. R. China

‡Department

of Mechanical Engineering, York University, Toronto, ON, M3J 1P3, Canada

#CEMATRIX

Inc. 5440 - 53rd Street SE, Calgary, AB, T2C 4B6, Canada

*Corresponding Author: Tel: 416 736 5901; e-mail: [email protected] *Corresponding Author: e-mail: [email protected]

1

ACS Paragon Plus Environment

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ABSTRACT: Transition between various state of wetting can have a profound effect on adhesion of a droplet onto a surface. In this study, a 3-D free energy (FE) model was developed to investigate the wetting transition for microstructured surfaces with various microgeometries, i.e., arrays of pillars (from upright frustum to inverted frustum geometries). The comparison of FE levels for a drop progressively penetrating into troughs of a textured surface allows one to understand the effects of parameters such as contact angle, feature size, and shape, on the wetting transition. In particular, transition free energy barrier (FEB) between composite and non-composite wetting states is determining the transition behavior. Preferred wetting states can be determined for a given microstructured surface by comparing FEBcom-non and FEBnon-com. A method to generate wetting maps for designing surface microstructures to keep drops in superhydrophobic (or low adhesion) state is provided. Stable and metastable superhydrophobicity can be seen only if the edge angle for surface features is set at a smaller value than the intrinsic contact angle. Moreover, FE analysis for arrays of two typical re-entrant structures, i.e., ones with convex and concave side wall, shows FE curves with minimum and maximum FE states, describing stability for drop penetration process. As such, we provide a framework for designing textured surfaces with superhydrophobic or low adhesion properties.

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1. INTRODUCTION Wettability of coatings can be designed by either surface chemistry, suitable roughness (or microor nano-texture), or a combination of both. Chemistry has been the traditional way to manipulate wetting, but more and more textured surfaces have been used in a variety of applications when either enhanced wettability is needed,1,2 or reduced wettability is desired.3-5 As such, study of the fundamentals of wetting from a fundamental perspective using thermodynamic principles is important to develop a deeper understanding of wetting for textured surfaces. For textured surfaces mainly two wetting states has been studied extensively:6 (i) noncomposite state where liquid completely penetrates into the troughs of the surface texture with only solid-liquid interface under the drop; and (ii) composite state where drop is suspended on tips of a surface texture with the air pockets trapped under the drop. Depending on what state of wetting the drop assumes, its adhesion can be low (composite state) or high (non-composite sate). The composite wetting state also known as superhydrophobicity (water-repellent) or superoleophobicity (oil-repellent), is of particular interest; due to applications in fields of electronics for transparent and self-cleaning surfaces,3 anti-fouling,4 printing,5 anti-icing,7 etc. Regarding superhydrophobic surfaces (SHS) the issue of contact angle hysteresis and robustness of superhydrophobicity has been studied extensively.8-14 This is so, as contact angle hysteresis (the difference in contact angle when drop advances and recedes on a surface) is related to drop mobility or adhesion.15 The robustness of superhydrophobicity (not in mechanical terms) concerns with drop penetration into the troughs of a textured surface; this is important since wetting transition from initial composite to noncomposite states or vice versa can determine usefulness of a textured surface in above applications. The wetting transition is especially important when one is interested in dynamical aspects of drop interactions with such surfaces, e.g. drop impact,16 drag 3



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reduction for marine vessels,17 or in general when adhesion of a surface needs to be tuned for a particular application. A wetting diagram,18 is usually used to demonstrate the wetting states on textured surfaces, considering the intrinsic contact angle (CA) of the material (i.e. Young’s CA, θY) and the apparent CA, θ’, as shown in Figure 1. In the hydrophobic/oleophobic region of Figure 1 (i.e., when θY > 90º) with a decreasing value of θY, Equation 1 (Cassie equation) and then Equation 2 (Wenzel’s equation) will be applicable for the composite (solid line) and the noncomposite (dashed line) states, respectively (see Figure 1). (1)

cos 𝜃′ = 𝑓cos𝜃Y ― (1 ― 𝑓) (2)

cos 𝜃′ = 𝑟cos𝜃Y

where f is the area fraction of the liquid in contact with the solid; and r is the roughness factor defined as the ratio between the actual wetted area of a textured surface, and the geometrically projected area of the wetted surface. Equation 1 is a simplified form of Cassie equation, since it is valid only for when a drop is sitting atop a textured surface with flat-top surface features (e.g. cylindrical pillars).19 The Young’s contact angle that divides the two wetting states is named as: the critical CA (θC); for a surface with flat-top texture it can be found from:21 cosθC = (f-1)/(r-f). In the hydrophilic/oleophilic region (when θY < 90º) with a decreasing value of θY, the wetting states changes from the noncomposite state (dashed line) to a hemi-wicking state (dotted line). In this case the Young’s contact angle that divides the two wetting states, θC’ , can be calculated as:20 cosθC’ = (1-f)/(r-f). The apparent contact angle, θ’, of the hemi-wicking state for surface s with flattop features can be found using a modified Cassie equation as: cos 𝜃′ = 𝑓cos𝜃Y + (1 ― 𝑓) 4


(3)

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If θY is larger than θC, then the composite state can exist; this means that for water which has a maximum θY value of approximately 120°,21 only the narrow-hatched area in Figure 1 is expected to show a composite state for water (i.e. superhydrophobicity). Experimental works in the literature,22-25 show that it is possible to have composite wetting state with materials that are moderately hydrophobic ((f-1)/(r-f) < cosθY < 0º) or even hydrophilic (cosθY > 0º); this can be achieved using e.g., re-entrant microstructures.26-28 From a wetting diagram it is not possible to explain such observations, since a wetting diagram does not provide any information regarding the stability of the wetting states. A thermodynamic analysis of the wetting stability is required to understand how to design superhydrophobic (or superoleophobic) surfaces from hydrophilic (or oleophilic) materials.

Figure 1. Schematic of a wetting diagram for textured surfaces. The narrow-hatched area (-0.5 < cosθY < (f-1)/(r-f)) denotes the stable composite state for water. Note that the general form of the wetting diagram is not limited to water. 5



By comparing the interfacial energy between noncomposite and composite states, in20 it was shown that the thermodynamically stable composite state is possible only if θY > θC. As such, the observed composite state for the moderately hydrophobic and hydrophilic regions in Figure 1 (the dash-dotted line) should represent a system in local energy minima. If this is so, then there should be a transition free energy barrier (FEB) between one local energy minima and the other, or the absolute energy minima, which should fall within the hatched area of Figure 1. Study of such FEB is important to understand the stability of the composite states for a drop on textured surfaces, e.g. as was shown for the role that the edges play in a textured surface.29 To date as it will be explained below, there has not been a general thermodynamic analysis based on surface free energies to study the details of the FEB, with respect to the surface texture, and the role it plays in the transition FEB from composite to non-composite state of wetting. Patankar11,30 and Marmur10,31 were amongst the first researchers to theoretically discuss the importance of transition FEB for designing robust SHS. They showed that a drop may remain in the composite state due to the transition FEB even when the energy state for the noncomposite state was lower. Despite this important insight, their models could not provide detailed information regarding the effect of geometry/shape of the surface texture on FEB since they used r and f in their models. This is so, as it is known that surfaces with different textures can have the same r and f values.32 The issue of the shape of surface texture was nicely dealt with in the work by David and Neumann33 (even for randomly rough surface texture). But unfortunately, an analysis on how shape and size of geometrical features of a surface texture affects FEB and hence robustness of the superhydrophobic state was not provided. Their work was focused on determining the equilibrium contact angles. Barbieri et al.12 and Bormashenko et al.14,34 studied the transition FE barriers for cylindrical 6


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pillars and parallel grooves (i.e., a 2D model), respectively. The former however, did not consider the effect of re-entrant structure in drop wetting transition; whereas the latter studied a system that may not be so consistent with the practical case of 3D pillars; also, many simplifications were required for the anisotropic wetting of parallel grooves.35,36 The 3D model for wetting transition on pillars with re-entrant geometry (Φ, i.e. the edge angle, was set to be smaller than 90°) was considered in,34 but a number of simplifications made limited the generality of the model. For example, the transition FE barrier equation obtained depended on the opening angle of the cone, which does not exist for upright frustum geometries and cylindrical pillars. Moreover, the works in12,14,34 did not consider the change of liquid volume in the main drop as a result of liquid penetration into troughs of the microstructures during transition (depending on the size of surface texture this can violate the mass conservation). One work which considered in detail the FEB and its effect on wetting transition is by Kim et al.;37 however, they only considered liquid penetration into a single cavity; the thermodynamic analysis therefore cannot be directly extended to textured surfaces which usually have open structures (see Figure 2a). As discussed above re-entrant surface features with concave and convex side walls are basic elements for having robust SHS, as seen for animal feathers,38 oleophobic fabrics,39 and plants.40 Using a 2D concave and convex protrusions as basic elements Marmur10 analyzed the wetting of such textured surfaces, but due to using r and f in building the model (with its drawbacks discussed earlier) conclusions remain at a general level. As such, still a detailed 3D model to analyze transition FEB analysis taking into account the specific micro-geometrical parameters, are needed. Another point that needs attention is that the studies on transition FEB in literatures are all concerned with the transition from composite to noncomposite states (with the exception for a single cavity37). This is partly due to the fact that experimentally observing a transition from 7



noncomposite to composite state is rare, so study of its FEB has been overlooked. Nevertheless, a transition from noncomposite to composite state exist,41-45 and it is important to understand when designing robust SHS, or when maintaining a low adhesion surface is critical. An example is condensation where tiny drops initially form in a noncomposite states and later as they grew they will experience a transition to a composite state.43 This means that there should be a relatively small transition FEB for from noncomposite to composite state which can be exploited in designing robust SHS. This issue will be addressed in this paper. We will however leave the somewhat controversial line tension effect46 out of this study (line tension is the one-dimensional analogue of the surface tension, which is a force that operates along the three-phase contact line). This is so as there is still a lively debate about the magnitude and sign of line tension, which we feel will take away from core message of this work; it is also of note that line tension can be primarily important for the nano-scale features47 which is not the focus of this work. But interested readers can learn about how line tension may affect the transition between wetting states should refer to the work of Bormashenko and Whyman47 who used a simplified 2D model to study the topic. They found that the tension may stabilize or destabilize the initial composite state depending on the geometry of surface features at the nano-scale level. Another interact factor that has been discussed in the literature, is the role of disjoining pressure for partial filling of troughs in a textured surface. In48 it was shown that the long-range surface forces can promote partial wetting depending on the specific form of the Derjaguin isotherm. A cylindrical capillary was used as a surrogate to model troughs of a textured surface.48 It was shown that the interplay of capillary and disjoining pressures governs the penetration of the liquid, and the level of liquid penetration depends on the radius of the cylindrical capillary representing the trough of a textured surface. Thus, small capillaries will show the non-composite wetting behavior, while large 8


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capillaries can promote composite wetting in the specific case of a textured surface made of cylindrical pores. Taking a first-principle 3D thermodynamic analysis approach, we will investigate the wetting transition by study of a number of typical surface textures. Various factors, such as intrinsic contact angle, edge angle of surface pillars, and length scale factor for features within a textured surface, are considered to systematically study their effect on wetting transition. The role of transition FEB from composite to noncomposite state (low to high adhesion for a droplet), and from noncomposite to composite state are also studied.

2. THEORETICAL METHODS 2.1 System Definition. The textured surfaces considered in this work are made of micropillars, see Figure 2a. The micropillars were either straight walled, i.e., upright or inverted frustum, or cylindrical, see Figure 2b; or had one of the two typical re-entrant geometries (arrays of pillars with convex or concave side walls, see Figure 2d). A three-dimensional model was developed from a first-principle thermodynamics basis to account for surface parameters, such as, intrinsic contact angle, θY; edge angle, Φ; pillar width, a; and pillar spacing, b (see Figure 2) to study the FE states for wetting transition.

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Figure 2. (a) A droplet placed on a typical surface texture considered in this study. (b) Side view of the microstructures of the surface shown in (a); by changing Φ from an acute to an obtuse angle, respectively, an inverted or upright frustum geometry were considered. (c) Top view of the pillars shown in (a) or (b). Since all pillars have a symmetrical form in 3D, their top view will be the same as shown in (c). (d) Side view of the two typical re-entrant pillars studied (for convex walled pillar, the bottom half of the pillar is considered a re-entrant geometry).

The following assumptions were made in developing the 3D thermodynamic model: (1) we ignored the line tension and the gravity to simplify the model and bring to focus the effect of basic factors such as, θY, Φ, and pillar size, affecting the transition FEB; (2) the drop size is considered much larger than the surface pillars and their spacing; (3) as a consequence of the first assumption we can consider that the drop profile is a spherical cap; also we assume that the liquidair interface under the drop is flat; (4) the solid surface was considered to be homogeneous, isotropic, and rigid.

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2.2 Thermodynamic Analysis. Our thermodynamic approach here is an extension of our previous 2D model.49 First, the model for wetting of textured surfaces made of pillars with straight side wall is discussed. Consider a drop with the base radius of LA that touches the texture of the surface at an arbitrary position A, as shown in Figure 3a; we consider this as the reference state for free energy analysis (the drop is in composite state). The profiles of the drop near the three-phase-line shown in Figure 3 are not necessarily in equilibrium state (i.e., the contact angle seen is an apparent contact angle); the profiles shown are for the purpose of comparing the free energy states to find the minimum energy state for a drop of constant volume. For straight wall pillar surfaces, three possible cases shown in Figure 3a to 3c, will be discussed in turn. When the liquid penetrates into troughs from positions A to B (see Figure 3a), an intermediate composite state is formed; in this case, the base radius LB, remains equal to LA, but the contact angle changes to θB (note that none of θA or θB, are necessarily an equilibrium contact angle). In such virtual movement, the system free energy (F) will change from FA to FB as interfacial areas (liquid-air, solid-air, and liquid-solid) change. Equations 4 and 5 can be used to find the FE of the drop at positions A and B, respectively. sa sa ls ls 𝐹A = 𝛾la𝐴la A + 𝛾 𝐴A + 𝛾 𝐴A +𝐾

(4)

sa sa ls ls 𝐹B = 𝛾la𝐴la B + 𝛾 𝐴B + 𝛾 𝐴B +𝐾

(5)

where subscripts A and B refer to state A and B; 𝐴𝑙𝑎, 𝐴𝑠𝑎and 𝐴𝑙𝑠 are the liquid-air, solid-air and liquid-solid interfacial area, respectively. FA and FB are the free energy of drops at the positions A and B, respectively; 𝛾𝑙𝑎, 𝛾𝑠𝑎and 𝛾𝑙𝑠 are the liquid-air, solid-air and liquid-solid interfacial tension (energy), respectively; K denotes the FE of the portion of the system that remains unchanged, e.g., from bulk phases. Equation 6 gives the energy cost of transiting form state A to B; in Equation 6, 11



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cos 𝜃Y represents the magnitude of the term (𝛾sa ― 𝛾ls)/𝛾la, through simplifications considering Young equation.

(

𝛥𝐹A→B 𝛾la = (𝐹B ― 𝐹A)/𝛾la = 2𝜋𝐿20 𝐻𝐵𝜋2𝐿20

1 ― cos𝜃B sin2𝜃B

―

1 ― cos𝜃A sin2𝜃A

𝜋2𝐿20

)

( (

+ (𝑎 + 𝑏)2

𝑎2 4

―

𝑎 2

2

𝐻𝐵

― sinΦ cosΦ

) )―

𝐻𝐵

(6)

(𝑎 ― sinΦ cosΦ)cos𝜃Y sinΦ(𝑎 + 𝑏) 2

where L0 is the drop base radius at A or B (note that LA = LB); HB is the penetration height (see Figure 3a); note that in subsequent plots shown using Equation 6, the free energy is normalized with respect to 𝛾la (J/m2), so it is unitless. For a drop of constant volume (V), 𝑉A = 𝑉B. Since drop has a spherical cap shape, the contact line of the drop, for both zero penetrated (wetting state A) and partially penetrated (wetting state B) cases is a circle with radius L0 (see Figure 3a). The number of the unit cell (see Figure 2c) under the drop, then can be expressed as: (𝜋𝐿20)/(𝑎 + 𝑏)2, which was used to drive the Equation 6. The free energy cost for transiting form an initial composite state (solid lines in Figure 3b) to a noncomposite state – dashed line in Figure 3b, i.e., 𝛥𝐹com→non, when the drop contact line remains at the same arbitrary position, can be found as: 𝛥𝐹com→non 𝛾la =

(

𝜋ℎ(𝑎 ― ℎcot Φ) sinΦ

𝐹non ― 𝐹com 𝛾la

(

= 2𝜋𝐿20

1 ― cos𝜃non sin2𝜃non 2

―

)

1 ― cos𝜃com sin2𝜃com

𝜋𝐿20

(𝑎2 ― ℎcot Φ) )cos𝜃Y

+ (𝑎 + 𝑏)2 ― 𝜋

(

― (𝑎 + 𝑏)2 (𝑎 + 𝑏)2 ―

𝜋𝑎2 4

)

𝜋𝐿20

― (𝑎 + 𝑏)2 (7)

where 𝐹com and 𝐹non are the free energy at the composite and noncomposite states, respectively; the corresponding apparent CAs for drops in composite and noncomposite states, respectively, are: θcom and θnon.

12


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The free energy cost for drop receding form an initial arbitrary position, A to another composite state C (with drop base radius LC; see Figure 3c), i.e., 𝛥𝐹A→C , can be shown to be as:

(

𝛥𝐹A→C/𝛾la = (𝐹C ― 𝐹A)/𝛾la = 2𝜋

𝐿2C(1 ― cos𝜃C) sin2𝜃C

―

𝐿2A(1 ― cos𝜃A) sin2𝜃A

)

+

𝜋(𝐿2C ― 𝐿2A) (𝑎 + 𝑏)2

((𝑎 + 𝑏)2 ―

𝜋2𝑎2(𝐿2C ― 𝐿2A) 4(𝑎 + 𝑏)2

𝜋𝑎2 4 )

― (8)

cos𝜃Y

where FA and FC are the free energy of drops at the positions A and C, respectively; the corresponding apparent CAs for positions A and C are: θA and θC, respectively. Arbitrarily choosing the value of the reference energy state to be zero, when LA = 0, (i.e. when apparent CA, θA, is 180°), we will calculate the energy states of all other possible wetting cases on the textured surfaces (see next section). Considering Figures 3d and 3e, one can formulate Equations 9 and 10, similar to above, to find the energy cost of drop partially penetrating from an arbitrary position A to B. Note that Equations 9 and 10, are for surface features with convex and concave side walls, respectively. Equations 9 and 10 are equivalent to Equation 6 for straight wall micropillars.

(

𝛥𝐹A→B 𝛾la = (𝐹B ― 𝐹A)/𝛾la = 2𝜋𝐿20

1 ― cos𝜃B 2

sin 𝜃B

―

1 ― cos𝜃A 2

sin 𝜃A

)+

𝜋2𝐿20

((𝑅 ― 𝐻𝐵)2 ― (𝑅 ― 𝐻𝐴)2) ―

(𝑎 + 𝑏)2

2(𝐻𝐵 ― 𝐻𝐴)𝜋2𝑅𝐿20 (𝑎 + 𝑏)2

(9)

cos𝜃Y

(

𝛥𝐹A→B 𝛾la = (𝐹B ― 𝐹A)/𝛾la = 2𝜋𝐿20

[

2

𝑎

𝑎

1 ― cos𝜃B sin2𝜃B

―

(2 ― 𝑅2 ― (𝑅 ― 𝐻𝐴)2) ― (2 ― 𝑅2 ― (𝑅 ― (𝑅 ― 𝑥)2

1 + 𝑅2 ― (𝑅 ― 𝑥)2𝑑𝑥

1 ― cos𝜃A sin2𝜃A 2 𝐻𝐵)2)

)

]

𝜋2𝐿20

+ (𝑎 + 𝑏)2

―

2𝜋2𝐿20cos𝜃Y 𝐻𝐵 𝑎 ∫ (𝑎 + 𝑏)2 𝐻𝐴 2

(

)

― 𝑅2 ― (𝑅 ― 𝑥)2

(10)

In Equations 9 and 10, R is the radius of curvature of the side wall. To find the equivalent FE equations to Equation 7 for pillars with concave and convex side walls, Equations 11 and 12 can be derived, respectively: 13



𝛥𝐹com→non 𝛾la =

𝐹non ― 𝐹com 𝛾

la

(

= 2𝜋𝐿20

1 ― cos𝜃non 2

sin 𝜃non

―

) ―𝜋𝐿 ― (𝜋𝐿

1 ― cos𝜃com 2

sin 𝜃com

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2 0

2 0

4𝜋2𝐿20𝑅2

)

+ (𝑎 + 𝑏)2 cos𝜃Y (11)

𝛥𝐹com→non 𝛾la =

(

ℎ 𝑎 2

∫0

𝐹non ― 𝐹com 𝛾la

)

― 𝑅2 ― (𝑟 ― 𝑥)2

(

= 2𝜋𝐿20

1 ― cos𝜃non sin2𝜃non

(𝑅 ― 𝑥)

―

)

1 ― cos𝜃com sin2𝜃com

𝜋𝐿20

[

― (𝑎 + 𝑏)2 (𝑎 + 𝑏)2 ―

2

(12)

1 + 𝑅2 ― (𝑅 ― 𝑥)2𝑑𝑥

where h is equal to 2R.

14


𝜋𝑎2 4

]―

2𝜋2𝐿20cos𝜃Y (𝑎 + 𝑏)2

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(a)

(b) LA = LB = L0

(c)

(d) LA = LB = L0

R

(e)

LA = LB = L0

R

Figure 3. (a) Partial liquid penetration of liquid into troughs of a textured surface, and the corresponding parameters used to do the FE analysis for a drop at arbitrary positions A when in composite state (solid line) and when the liquid is partially penetrated (dotted line). (b) Droplet with a fixed contact line at an initial composite state (solid line), and noncomposite state (dashed line). (c) Illustration of a drop receding from an arbitrary composite state (A) to another composite state (C). (d) and (e) Illustrations of FE analysis for drops penetrating from an arbitrary position A (solid line) to B (dotted line) on arrays of micropillars with convex or concave side walls; σ𝐴 and σ𝐵 are the microscopic contact angle seen on the side wall of the pillars at positions A (solid line) and B (dotted line), respectively.

15



2 RESULTS AND DISCUSSION 3.1 Free Energy Analysis for Straight-Wall Surface Pillars. Using Equations 6-8 we can generate plots of normalized FE as a function of apparent CA for straight-wall pillars of a textured surface, e.g. Figure 4. All FE curves in Figure 4 show an absolute minimum corresponding to the equilibrium CAs, for the initial composite and noncomposite states (an “initial composite state” can be thought of when a droplet is gently placed on a surface in this discussion). In Figures 4a (Φ=110°) and 4b (Φ=60°), the FE curve for the initial composite state is always above that of the noncomposite state at any apparent CA. This means any external disturbances (vibrations, or if the droplet is not gently placed on the surface, e.g. it impacts the surface) leads to a transition to Wenzel wetting state (not a very robust SHS especially for Φ = 110° where a spontaneous transition will occur since the curve for the intermediate composite state lies between the two sates of Cassie and Wenzel; see Figure 4a). Interestingly, however, in Figure 4b, the curve for the intermediate composite state (when liquid has partially penetrated into the troughs of the surface texture) lies higher than the FE curve for the noncomposite state. This implies an energy barrier that prevents the wetting transition from initial composite to noncomposite state to be spontaneous. In other words, a metastable initial composite state can exist when Φ = 60°, even if the surface material is hydrophilic (θY = 80°). This explains from a thermodynamics perspective, the experimental observations for the super-hydrophobic or oleophobic wetting on rough surfaces constructed with high energy materials.22-25 Analysis of the transition FEB will be further discussed below. But before that note that the intersection of all FE curves at CA = 180° in Figure 4 is due to selection of reference point for FE at CA = 180°.

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(a)

(b)

Figure 4. Normalized FE as a function of the apparent CA for textured surfaces shown in the inset. For both (a) and (b) the drop volume was 9.7μl; and θY = 80°; the pillar size and spacing -6

-6

were: a=b=10×10 m; h = 8×10 m; Φ is 110° for (a) and 60° for (b). FE curves are for initial composite, noncomposite, and partially penetrated troughs (intermediate composite) states. The Wenzel and Cassie equilibrium contact angles for (a) are: 74.2°, and 140.3°, respectively; and for (b) are: 74.0°, and 140.3°, respectively.

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3.2 Wetting Transition for Straight-Wall Surface Pillars. Considering the three representative cases, in Figure 5 insets, Equations. 6 and 7, can be used to calculate the normalized FE as a function of drop penetration depth (from the initial composite to noncomposite states). Three wetting transition behaviors is seen in Figure 5: (1) In the example of a hydrophobic material, Figure 5a where θY = 100°, for a surface texture with re-entrant features, the FE increases progressively as drop penetrates into troughs of the texture. The FE reaches the maximum just before the drop touches the bottom of the troughs (h=8 μm) - creating a noncomposite state. The sudden drop of the FE in Figure 5 is due to vanishing of the liquid-air and solid-air interfaces and the simultaneous formation of liquid-solid interface. The magnitude of increase in FB from base value of noncomposite to just before composite wetting, is the FE barrier to overcome for transition from composite to noncomposite states, FEBcom-non, see Figure 5a. For transition from noncomposite to composite state, a larger free energy barrier FEBnon-com exists; as seen in Figure 5a: FEBnon-com>FEBcom-non >0. So, one can conclude that in this example, the initial composite state is a metastable wetting state. (2) In the example of a hydrophilic material, Figure 5b where θY=80° and when pillars are frustums, e.g. Φ=100°, the FE decreases progressively as the liquid penetrates into the troughs from an initial composite state. As the drop forms a noncomposite state, a sudden drop in FE is seen and the minimum value of the FE is reached since FEBnon-com > 0 > FEBcom-non. In this case the energy barrier for transition from the initial composite to noncomposite states does not exist (note: 0 > FEBcom-non), and thus the transition will be spontaneous. (3) If we do not confine ourselves to organic liquids, e.g. consider molten metals or mercury, then the intrinsic contact angle can be very high, e.g. θY = 130°. For such a system consider a textured surface with a re-entrant pillar, see Figure 5c where Φ = 60°; one can see that FEBcom-non 18


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> FEBnon-com > 0, and the composite state will become the most stable energy state for the drop. As seen from Figure 5c, the transition from noncomposite to composite state can be spontaneous with the input of an external mechanical vibration and/or thermal energy (e.g., liquid condensation in the troughs of a textured surface).

(a)

(b)

(c)

Figure 5. Examples of the three possible wetting scenarios using the FE and FEB analysis: (a) metastable composite wetting: θY = 100°, Φ = 60°; (b) stable noncomposite wetting: θY = 80°, Φ = 100°; and (c) stable composite wetting: θY = 130°, Φ = 60°. For examples, the drop volume

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was 9.7μl; the pillar size and spacing were: a=b=10×10 m; h = 8×10 m; and L0 = 4.5×10-4m. Solid lines are to guide the eyes. One can think of FEBcom-non as a measure of robustness of superhydrophobicity as it denotes the thermodynamic stability of the composite state. In the same manner, FEBnon-com can be thought of the ability to recover superhydrophobicity. As such, to have a robust SHS, the texture of the surface should be designed such that a large FEBcom-non and small FEBnon-com are obtained. The magnitudes of the FE and FEB for the wetting system will change, if the drop volume is changed, as the interfacial area in Equations 4 and 5 (e.g. liquid-air or liquid-solid interfaces) depends on the drop volume. However, the relative trend of FE and FEB as demonstrated will be independent on the drop volume. As such, the above observations still hold for different drop volumes.

3.3 Contact Angle Effect. In this section we will probe the effect of intrinsic contact angle (θY) on the FEB and the resulting effect on robustness of the superhydrophobic surfaces. For a surface decorated with inverted frustums (see Figure 2a), Figure 6a shows the normalized FE as a function of drop penetration depth for surfaces with different θY. The FE increases regardless of the wetting states (i.e., composite, noncomposite, or partially penetrated, i.e. intermediate composite), as θY increases. The FE also increases as drop penetration depth increases regardless of the value of θY. However, with an increase in θY, Figure 6b shows that the normalized FEBcomnon increases,

whereas FEBnon-com decreases. This means that to have a robust SHS, a large value of

θY is preferred; this is so, since a large value of θY results in a large FEBcom-non and a small FEBnoncom.

Having a small FEBnon-com has the advantage of even if the system transits to a non-composite

state, a small external perturbation can bring it back to the composite state. It is of note that below the dashed line in Figure 6b, there is no FEB. Thus, in this example, FEB for preventing the drop 20


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transition from the initial composite to noncomposite states will exist only for when θY > 60° (noted by the dotted line). In Figure 6b where curves for FEBcom-non and FEBnon-com intersect, once can see that for θY > 125° always FEBcom-non > FEBnon-com > 0 meaning a stable composite state will result. As stated earlier for water the maximum θY is ~120° (denoted by the dash-dotted line in Figure 6b), also considering the curve for FEBcom-non intersecting with the FEB =0 line at 60°, then the superhydrophobicity can be realized only in the region 60° < θY < 120° in this example. However, if material science can provide us with a substance that shows a water CA larger than 125°, Figure 6b points to the existence of a super robust superhydrophobic state.

(a)

(b)

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Figure 6. (a) Changes in normalized FE with respect to penetration of liquid into surface troughs depth for various intrinsic contact angle. (b) Changes in normalized FEB with respect to intrinsic contact angle. Note FEB below the dashed line (FEB = 0) does not exist. The system dimensions are: a = b = 10×10-6m; h = 8×10-6m; drop volume = 9.7μl; θY = 10° ‒ 160°; Φ = 60°; and L0 = 4.5×10-4m. Solid lines are to guide the eyes.

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3.4 The Pillar Geometry Effect. The normalized FE is shown in Figure 7a as a function of drop penetration depth for various pillar geometries, from an inverted frustum, i.e. edge angle is less than 90° to a straight wall (edge angle = 90°), to an upright frustum (edge angle > 90°). The FE curves for when liquid partially penetrates into troughs, shifts up, with an increase in Φ. This is different for composite (i.e. zero penetration) and noncomposite states that FE curves remain unchanged; this means that the transition FEB is a constant, in this example, FEBnon-com FEBcom-non = constant > 0 (see Figure 7a). In terms of adhesion, this means that there will be no change in adhesion characteristics of the surface. For the system shown in Figure 7b, the FEBcom-non curve is always below that of FEBnon-com, which means a stable composite wetting state does not exist. But, if Φ is decreased, FEBcom-non can change to a positive value (see thick dotted line in Figure 7b). This means that even for a hydrophilic material a metastable composite state can exist. However, there is a price to be paid when Φ is decreased, since FEBnon-com will increase and consequently, the transition from a noncomposite to a composite state will be difficult. Considering the above, then a small Φ is helpful to have a metastable composite state; but, if drop transits into noncomposite state, its recovery back to a composite state will be difficult. This point is important for designing SHS.

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(a)

(b)

Re-entrant pillar geometry Frustum pillar geometry

Straight wall

Figure 7. (a) Changes in normalized FE with respect to penetration of liquid into surface troughs depth for various Φ (edge angle). (b) Changes in normalized FEB with respect to Φ (edge angle). Note FEB below the dashed line (FEB = 0) does not exist. The system dimensions are: a = b = 10×10-6m; h = 8×10-6m; drop volume = 9.7μl; θY = 80°; L0 = 4.5×10-4m. Solid lines are to guide the eyes.

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3.5 The Effect of Length Scale. Here we define the “length scale factor” as the multiplier for a and b, as shown in Figure 2. This way we will be able to study what happens to the transition FEB, if the size of surface features changes. As seen in Figure 8, the FE analysis indicates that the wetting transition FEB is influenced by the length scale factor. Changes in normalized FE with respect to drop penetration depth is shown in Figure 8a for different length scale factors. Three observations can be made: 1) initial composite FE is insensitive to the length scale factor; this can be well understood from Cassie factor, f, in Equation 1 for the surface texture example analyze here (a = b and the value of f (= πa2/(4×(a+b)2) will be insensitive to the length scale factor as defined). 2) the FE for intermediate composite states (liquid penetration into surface roughs) decreases progressively as the length scale factor increases; 3) the FE of noncomposite state increases as the length scale factor increases. These observations taken together means that both FEBcom-non and FEBnon-com decrease as the length scale factor increases. From Figures 8b and 8c one can see that the smaller the length scale factor, the more stable the drop will be in a composite state. But the FEBcom-non and FEBnon-com plots also show that once the drop transits into the noncomposite state, it will be difficult for the drop to recover to a composite state. The edge angle (Φ) however plays an important role in changing the trend of the curves for FEBcom-non and FEBnon-com. For example, if the Φ is higher than 80°, FEBcom-non will be negative so the energy barrier for preventing the transition from a composite to a noncomposite state vanishes; see the shaded area in Figure 8b. For length scale factor is less than 3, a steeper slope is observed for the FEBcom-non and FEBnon-com plots. This means that at length scale factor values below, one can control the energy needed for wetting transition more effectively 3 (in the example here when pillar width and spacing are less than 30μm).

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(b)

(a)

(c)

Figure 8. (a) Changes in normalized FE with respect to penetration depth for different length scale factor. The insets show the enlarged views of segments of the FE curves. (b) Changes in normalized FEBcom-non with respect to the length scale factor. The FEB do not exist in the shaded area as FEB < 0. (c) Changes in normalized FEBnon-com with respect to the length scale factor. For all plots: a = b = Length scale factor×10-5m; h=8×10-6m; drop volume = 9.7μl; θY = 80°; L0 = 4.5×10-4m; and Φ = 70°‒110° (except in (a) that Φ is 70°). Solid/dotted lines are to guide the eyes. 26


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3.6 Wetting Maps. Using the thermodynamic approach explained in this paper, one can generate wetting maps to decide on the design of the texture for a surface; such maps can help to understand how to achieve the desired robustness for the SHS, e.g. depending on the application, or how to tune the adhesion of the surface to droplets, e.g. for self-cleaning surfaces. An example of such map is provided in Figure 9 for a straight-wall micro-textured surface. The dash-dotted lines in Figure 9a delimit the valid region of analysis due to the geometrical constraints (Φmax=141° and Φmin=39°). Three wetting regions is seen in Figure 9a; generally, for edge angle larger than the value of the intrinsic contact angle, one can observe only a noncomposite state. But if the edge angle is less than the value of the intrinsic contact angle, a metastable composite state is seen within the θY range of 39°-130°; the wetting state changes, however, to a stable composite state when θY > 141°. When θY is in the range of 130°-141°, lowering Φ changes the metastable wetting state to a stable composite one. As noted by the dotted line in Figure 9, θYmax is 120° for water in this example, so attaining a stable composite state for the microstructured surface with pillar dimensions given is difficult. A wetting map can be constructed to probe the role of length scale factor, e.g. Figure 9b for the same surface as in Figure 9a (in the example here regardless of length scale factor value, the Cassie’s fraction f remains unchanged at 0.2). For the region where edge angle values (Φ) are larger than the intrinsic contact angle (θY) a noncomposite state is energetically favorable. However, when Φ FEBcom-non > 0). The microscopic contact angle, σ, (see Figure 3) at this metastable state is calculated as 90º, which is equals the θY, in this example. Likewise, if θY is 60°, σ for the metastable state again is calculated to be 60º, i.e. the θY for he system (see Supporting Information for details). In general, our calculations show that the smaller the θY, the larger will be the penetration, i.e. a shifting of the local metastable minimum energy to the right. For the concave walled pillars the situation is different, see Figure 10b; the FE curve is convex meaning a maximum FE state is seen for partial penetration of liquid into troughs. This means that compared to the noncomposite state, the initial composite state will be a metastable one since FEBnon-com > FEBcom-non > 0. Again, looking at the calculated σ, at the extremum, its value is equal to θY for the system here where θY=90° (see Supporting Information for another example).

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(a)

(b)

Figure 10. Changes in normalized FE with respect to the drop penetration depth. (a) For pillars with a convex side wall; the pillar dimensions are: a = b = 10×10-6m; 2R = 10×10-6m; the drop volume and size are: 9.7μl, and L0 = 4.5×10-4m, respectively; the material has an intrinsic contact angle, θY , of 90°. (b) For pillar with concave side wall. System parameters for (b) are the same as (a).

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3.8 Practical Considerations. Not all surfaces are ideal, so in practice defects may be present on a surface. A surface defect may be chemical, geometrical (e.g. a surface feature that is out of norm compared to other surface features), or both. Any such anomalies will affect surface wettability locally which means within a limited area, the global droplet wetting behavior discussed, will not hold. Let's consider the case of a chemical defect as an example. Consider a local chemical anomaly that is hydrophilic in an otherwise hydrophobic surface. This means that intrinsic contact angle for a number of surface features will be less than 90 deg.; if we consider the Figure 6b, this leads to a local transition to noncomposite state that will be easier near the local chemical defect, compared to the rest of the surface having a higher θY (i.e. FEBcom-non locally will be smaller than its global value). If the size of chemical defect is not large (e.g. limited to a few micro-pillars), our (practical) experience has shown that overall the droplet wetting behavior will not be affected. The same local influence can be said for when there is a geometrical defect or a combination of geometrical and chemical defect.

4. CONCLUSIONS Using the free energy approach, we analyzed the wetting states and its stability for a number of typical textured surfaces. In our analysis we considered the 3D nature of surface features (pillars), and its shape. The focus of the study was to use the developed model, to understand the relationship between the energy barrier for transiting from composite to noncomposite state, or vice versa, with the intrinsic contact angle of the material, shape of surface pillars/texture, and its size. We demonstrated with various examples, how the values and trends for transition FEBcom-non and FEBnon-com changes, depending on the intrinsic contact angle of the material, shape of surface 31



pillars/texture, and its size. We found that increasing intrinsic contact angle leads to an increase of FEBcom-non (i.e. a more stable composite state) and the decrease of FEBnon-com (i.e. an easier recovery from noncomposite to composite state). The decrease of edge angle led to an increase of both FEBcom-non and FEBnon-com (i.e. a more stable composite state and a more difficult recovery from noncomposite to composite state). The effect of length scale factor (size of texture) is best understood through a wetting map highlighting the relationship between edge angle, intrinsic contact angle, and stability of a particular wetting state. From the wetting map, one can identify that when the edge angle is smaller than the value of the intrinsic contact angle, a stable, or metastable composite states can be found. We also examined wetting of two examples of textured surface having typical re-entrant pillars, i.e. ones with convex and concave side wall. We showed that for surface features with convex walls a minimum FE energy exists, which means a drop prefers to stay stably/metastably on the side wall of the pillars. For concave side walls on the other hand, a maximum FE states exists, which means a drop prefers to stably/metastably stay on top of the pillars. The results of this work can be very useful for thinking about designing surface pillars (microstructures) to keep drops in a robust superhydrophobic state (or when one needs to manipulate the adhesion of a drop onto a surface).

ACKNOWLEDGEMENTS The authors acknowledge the financial support of QingLan Project of Jiangsu province, as well as NSERC.

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SUPPORTING INFORMATION DESCRIPTION Normalized Free Energy (FE) plots as a function of the drop penetration depth for the microstructures with concave and convex side walls. This material is available free of charge via the Internet at http://pubs.acs.org.

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Wetting Transition on Textured Surfaces: A Thermodynamic Approach

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