Droplets on Microdecorated Surfaces: Evolution of the Polygonal

Apr 27, 2017 - For example, droplet contact lines, which are typically circular on a smooth and homogeneous surface, when deposited on a microdecorate...
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Droplets on Microdecorated Surfaces: Evolution of the Polygonal Contact Line Alok Kumar and Rishi Raj* Thermal and Fluid Transport Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Patna, Bihar 801103, India S Supporting Information *

ABSTRACT: Interaction of liquids with surfaces is ubiquitous in our physical environment as well as many engineering applications. Recent advances on this topic have not only provided us with valuable insight into nature’s design, but also enabled improved fluidic manipulation for liquid-based printing applications such as biomicroarrays for protein and DNA sequencing, multicolor polymer-based LED displays, inkjet printing, and solder droplet printing, among others. For example, droplet contact lines, which are typically circular on a smooth and homogeneous surface, when deposited on a microdecorated surface may take various common polygonal shapes such as squares, rectangles, hexagons, octagons and dodecagons. These polygonal contact line shapes are highly stable due to the local energy barriers associated with the anisotropy in depinning contact angles. In addition to the knowledge of the eventual contact line shapes, liquid-based printing applications would require accurate prediction of temporal evolution of contact line on these surfaces. In this work, we model and validate the evolution of droplets on microdecorated surfaces with microgoniometry experiments reported in the literature. We show that various metastable contact line shapes are formed in-between the well-known stable polygonal contact line shapes usually observed in experiments. We elucidate that the movement of the contact line between adjacent micropillars can primarily be categorized as primary zipping and transition zipping. Primary zipping occurs when the contact line moves radially outward to the next row of pillars, often resulting in the formation of a metastable contact line shape. Conversely, metastable droplet attains stable polygonal contact line shape via transition zipping wherein the contact line advances sideways. We believe that the current simulation approach can be effectively utilized for designing optimized textured surfaces for applications where control over liquid supply via surface design is required.

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INTRODUCTION Wettability is key to nature’s interesting designs such as the selfcleaning ability of lotus leaves,1 the ability of the Namibian desert beetle to harvest water,2 and the ability of insects to walk on water,3 among others. Inspired by these, considerable efforts have focused on understanding and replicating these designs for developing various interesting fluidic-based systems. For example, engineers have designed superhydrophobic polystyrene films with novel porous microspheres and nanofibers to prepare superhydrophobic surfaces which may be useful for application such as antifouling paints for boats, self-cleaning windshields for automobiles, stain resistant textiles, and antisoiling architectural coatings.4 A synthetic mimic of the Namibian desert beetle has been developed for various potential applications including water harvesting surfaces, controlled drug release coatings, open-air microchannel devices, and lab-on-chip devices, among others.5 Similarly, biomimetic water strider robots that have microfabricated hydrophobic legs for locomotion on water surface have also been developed.6 Fundamental to all these applications is wettability, which is characterized by the angle of contact of a droplet on a surface. Contact angle of a droplet of a pure liquid deposited on a smooth © 2017 American Chemical Society

and homogeneous surface in an isothermal and vapor-saturated environment is given by Young’s equation.7 The corresponding perimeter of contact on the surface, also known as the contact line, is a perfect circle. A small value of contact angle through large solid−liquid contact area implies high surface energy and hence good wettability. Conversely, a large value of contact angle through poor solid−liquid contact implies low surface energy and hence poor wettability.8 Surfaces with contact angles below 90° are typically known as hydrophilic surfaces, whereas those with contact angles above 90° are referred to as hydrophobic surfaces. Young’s equation is rarely applicable (contact angle hysteresis, which is the difference between the advancing and the receding contact angles, is observed) to practical engineering applications where surfaces are usually rough, heterogeneous, and often contaminated. Depending on the nature of the surface energy involved, different scenarios of wetting are possible,9 namely: hydrophilic, superhydrophilic, hydrophobic and superhydrophobic. Received: February 17, 2017 Revised: April 26, 2017 Published: April 27, 2017 4854

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associated with local contact line pinning in Raj et al.22 to model the evolution of polygonal contact line of droplets in Wenzel wetting state on cylindrical micropillar arrays. We first discuss pinning and depinning operators formulated to capture the evolution of the droplets on predefined microdecorated surfaces. We then explain the phenomenon of contact line zipping, and the associated thermodynamically stable and metastable shapes of the contact line. Finally, we validate our work with the experimental results in published literature.

Wenzel modeled the equilibrium contact angle of droplets on chemically homogeneous but rough surfaces where the liquid completely wets the surface.10 The degree of wettability is known to intensify upon the incorporation of roughness, i.e., a hydrophilic surface turns more hydrophilic/superhydrophilic while a hydrophobic surface turns more hydrophobic. Subsequently, Cassie and Baxter11 extended Wenzel’s analysis to model a distinct case of droplets in equilibrium on porous (composite/heterogeneous) surfaces. The same theory was later extended to model the case of superhydrophobic droplets formed on highly nonwetting surfaces. Wenzel and Cassie−Baxter equations were originally proposed for a static droplet in equilibrium, and the validity of these equations under dynamic conditions (i.e., advancing and/or receding droplet) typically encountered in practical applications is debated to date.12−21 For example, Extrand12 measured the advancing and receding contact angles of a small droplet deposited on the center of a single chemically heterogeneous island to demonstrate that the wetting behavior is determined by the interactions at the three-phase contact line and not by the solid−liquid contact area. As liquid was sequentially added, the contact line advanced beyond the island perimeter onto the surrounding area where the advancing contact angle changed to the contact angle value exhibited on the homogeneous periphery without being affected by the heterogeneity completely contained within the contact line. Gao and McCarthy13,16 supported these observations and further proposed that the advancing and receding contact angles are a function of the local activation energies that need to be overcome in order for the contact line to move. Recently, Raj et al.20 performed experiments on various types of chemically heterogeneous surfaces to propose a modified Cassie−Baxter equation that relies on contact line fraction to predict the advancing and receding contact angles on chemically heterogeneous surfaces. It was discussed that, unlike smooth and chemically homogeneous surfaces where the contact angle is independent of the line-of-sight due to the circular contact line, the contact angles of droplets on ordered rough surfaces (square, rectangular, and hexagonal arrays of cylindrical pillars) depend upon the direction of view.22 This insight was used to develop a universal model for three-dimensional droplet shape by characterizing the droplet side and top profiles. The anisotropy in the depinning/advancing contact angle was then used to show complete control of droplet contact area shapes ranging from squares, rectangles, hexagons, and octagons to dodecagons via the design of the structure or chemical heterogeneity on the surface. This versatile and robust strategy was further used to deposit high-density droplet arrays with complex wetting shapes in a repeatable manner. The droplets in Raj et al.22 demonstrated finite contact angles in the nonpropagating regime and allowed superior control over fluidic manipulation. Similar shapes in the thin-film regime or directional spreading have been previously demonstrated on microdecorated surfaces,23 lyophilic pillararrayed surfaces,24,25 and chemically patterned micropillar surfaces26,27 as well. Similarly, droplets with polygonal contact line can also be obtained by varying the concentration of a binarymixture on a microdecorated substrate.28 While the knowledge of contact angles and the overall droplet contact line shape is crucial, a detailed understanding of the temporal evolution of contact line is also required for applications like DNA printing,29 biomicroarray,30 inkjet printing,31 liquid metal printing,32 polymer-based LED displays,33 and others. In this work, we use the understanding of energy barriers

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CONTACT LINE SHAPE MODELING The contact line of a liquid droplet deposited on microdecorated surfaces (geometric arrays of micropillars) exhibits various shapes depending on the size, shape, spacing, geometry of pillars array,16−18,22,34,35 and volume of liquid.36 We consider very small droplets in this study such that the effect of gravity can be neglected in comparison to the surface tension forces. A schematic of a well-defined rectangular (i.e., the spacing of pillars are different along the x-axis and the y-axis) array of cylindrical micropillars is shown in Figure 1a as an example.

Figure 1. (a) Isometric view of a typical rectangular array of cylindrical pillars. Pitch along 0° and along 90° lines-of-sight are L and M, respectively. The corresponding smooth surface advancing contact angle is θF,A. The local depining/advancing contact angles at the pillar edges are θF,A + 90°.22 (b) Schematic of the top view of contact line of a droplet on a rectangular array of pillars (M > L). Here ai and ai+1 represent the distance of pillars from the center of the droplet on which the contact line is lying and dωi is the angle between the two adjacent pillars. Only one quadrant is shown here due to the symmetry in contact line shape. (c) Solid element approximated as the sector of a spherical cap fitted between two adjacent pillars (side view) is used for differential volume estimation.

The interpillar spacing (pitch) along the x-axis (0°), y-axis (90°), and along ⌀° are L, M (M > L), and M2 + L2 , respectively. Accordingly, the depinning contact angles in the Wenzel wetting 4855

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Conversely, once the contact angle on any of the pillars approaches the advancing contact angle along that direction, droplet shapes become highly unstable such that contact line instantaneously advances to increase the contact area (via a decrease in height, contact angle is reduced all along the perimeter), essentially even without liquid addition. However, unlike smooth, homogeneous, and uncontaminated surface where the contact line is always circular, microdecorated surfaces offer anisotropic energy barriers22 such that the depinning contact

state along these directions can be estimated by the model presented in the work of Raj et al.22 as follows: cos θA,0 ° =

⎛ D D⎞ cos(θF,A + 90°) + ⎜1 − ⎟ cos θF,A ⎝ M M⎠

(1)

cos θA,90 ° =

⎛ D D⎞ cos(θF,A + 90°) + ⎜1 − ⎟ cos θF,A ⎝ L L⎠

(2)

cos θA, ⌀° =

D

cos(θF,A + 90°) M + L2 ⎛ ⎞ D ⎟⎟cos θF,A + ⎜⎜1 − ⎝ M2 + L2 ⎠ 2

(3)

where θF,A is the intrinsic contact angle on the smooth surface (deg), D is the diameter of cylindrical micropillars (μm), L is the pitch along the x-axis (μm), M is the pitch along y-axis (μm), ⌀° = (tan−1(1/K))°, K = M/L. Please note that ⌀° = 45° if L = M, which results in a square array. The contact line shape is mathematically presented as a function of time,S(t) = [X(t),Y(t)] where X(t) = [x1, x2,...xi...xN] and Y(t) = [y1, y2,...yi...yN] are the coordinates of the pillar where contact line is pinned (Figure 1b). Local contact angles θi(t) at time t for the points lying on S(t) are determined as follows (derivation is provided in the Supporting Information in section S1): ⎡ a 2(t ) − h2(t ) ⎤ ⎥ θi(t ) = cos−1⎢ i 2 ⎣ ai (t ) + h2(t ) ⎦ 2

(4)

Figure 2. Schematic representation of the (a) pinning (PN) (θ2 > θ1), and the (b) depinning (DPN) (θ1 > θ2) operators. (c) Demonstration of the thermodynamically unfavorable (metastable) shape of the contact line where θc,out < θc,in. (d) Thermodynamically stable contact line shape is observed when θc,out ≥ θc,in.

2

where ai(t ) = xi(t ) + yi (t ) is the distance of contact point on a pillar from the center/origin of a droplet at time t, and h(t) is the height of a droplet at time t. The shape of the contact line on these surfaces deviates from a circle and hence, we estimate the volume of the droplet at time t by summing volumes of sectors of approximate spherical caps lying between two adjacent pillars on the contact line (Figure 1c.): N−1

V (t ) =

∑ i=1

h(t ) (3aavg, i(t )2 + h(t )2 )dωi 12

where

N−1

∑ dωi = 2π

(5)

i=1 a (t ) + a

(t )

and where aavg, i(t ) = i 2 i+1 , ai(t), and ai+1(t) are the respective distances of the adjacent pillars from the origin (center of the droplet) of the computational domain at time t, dωi is the angle between adjacent pillars, and N is the number of pillars wetted by the contact line. We calculate the time increment by using eq 6. t=

Vcurrent − Vinitial V̇

(6)

where Vcurrent and Vinitial are the current and initial volume of the droplet, respectively, and V̇ is the liquid addition rate. A growing droplet on a microdecorated surface is observed to evolve either in the contact line pinning (PN) or in the contact line advancing/depinning (DPN) mode. In the PN mode, a droplet accommodates volume increase via liquid addition by increasing the contact angle all along the perimeter (via an increase in droplet height) without any change in contact area.

Figure 3. Schematic of contact line zipping over a microdecorated surface (first quadrant of a rectangular array having K = M/L = 2 and

( 12 )). (a) Occurrence of transition zipping due to a meta-

⌀° = tan−1

stable shape of contact line ((θc,out) < (θc,in)). Numeric “1” denotes the location of metastable shapes. Primary zipping along the ⌀°, 0°, and 90° lines-of-sight are demonstrated through panels b, c, and d, respectively. 4856

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Figure 4. Overview of the droplet evolution algorithm over microdecorated surfaces.

moves radially outward along one of the primary directions, i.e., along the lines-of-sight defined by 0°, ⌀°, and 90° (for a rectangular array of micropillars). Conversely, zipping along all other lines-of-sight occurs sideways along the perimeter and is termed as transition zipping. Transition zipping occurs when a metastable contact line shape wherein the outside angle (θc,out) of the contact line is less than the inside angle (θc,in) is formed (Figure 2c). Under such situations, the contact line shapes are highly unfavorable (i.e., Helmholtz and Gibbs energies increase) and hence the contact line zips to attain a stable configuration (Figure 2d, here θc,out ≥ θc,in). This is further illustrated through Figure 3, which demonstrates the zipping phenomena over a rectangular array (only the first quadrant is presented here due to symmetry). The black curve represents the initial shape of the contact line, whereas the red curve refers to the current (advancing) shape of the contact line. The localized metastable shape of contact line is represented by numeric “1”. Transition zipping is demonstrated through Figure 3a, whereas primary zipping along ⌀°, 0°, and 90° are demonstrated through Figure 3b,c,d, respectively. It can be observed from Figure 3b,c,d that the occurrence of primary zipping eventually results in metastable contact line shapes, which then proceed through further transition zipping steps until the next stable polygonal contact line shape is formed. Similar zipping phenomena was also demonstrated by Kim et al.37 during thin-liquid film propagation under imbibition conditions.

angles vary along the perimeter and contact line shape is not circular. Under such scenarios, uniform pinning or depinning of the contact line along every line-of-sight is not possible, necessitating the need for local analysis at individual energy barriers, i.e., every pillar on which the contact line falls. If the estimated local apparent contact angles on all pillars are lower than the respective prescribed advancing contact angles (DPN contact angle) for a given profile S(t) of the contact line at any time t, PN operator is called upon. In this step, the droplet accommodates the additional volume due to liquid addition by an increase in the global contact angle (θ(t + Δt) > θ(t)) via the increase in droplet height (h(t + Δt) > h(t), eq 4) without any change in the shape of the contact line (S(t + Δt) = S(t)) as shown in Figure 2a (i.e., constant contact area). Conversely, if the local apparent contact angle is observed to be equal to or incrementally greater than the corresponding prescribed advancing contact angle along the respective line-of-sight, the DPN operator is called upon wherein the corresponding contact line is advanced to the adjacent pillar (ai (t + Δt) > ai (t)) and the overall shape of contact line is changed (S(t) → S(t + Δt)) keeping the droplet volume constant (height is decreased, h(t + Δt) < h(t)). Contact area increase without liquid addition requires incremental reduction in contact angle during the DPN mode (θ(t + Δt) < θ(t)), as illustrated in Figure 2b. The DPN operator during liquid addition stage is further categorized into two parts, i.e., the primary zipping and the transition zipping. Primary zipping occurs when the contact line 4857

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(step 6). We plot all variable parameters of interest such as the contact angle, base radius, height, and the volume of the droplet (step 7) before terminating the simulation. Once the liquid addition is stopped, droplet volume decreases due to evaporation. Unlike the advancing stage, the micropillar surface does not provide any local energy barriers during the receding phase,22 and the classical Wenzel equation10 remains valid. The apparent receding contact angle of the droplet is hence estimated as follows:

We next discuss the algorithm (Figure 4) for modeling of the advancing stage (liquid addition) of the contact line.

⎛ πDH ⎞⎟ cos θR = ⎜1 + cos θF,R ⎝ LM ⎠

(7)

where cos θR is the receding contact angle on the microdecorated surfac, and θF,R is the intrinsic receding contact angle on an equivalent flat surface. However, micropillars on the surface do not allow a perfect circular contact line, and hence local analysis with −PN and −DPN operators is performed. A −PN operator implies that the droplet accommodates the volume loss via a decrease in the global contact angle (θ(t + Δt) < θ(t)) such that the droplet height decreases (h(t + Δt) < h(t), eq 4) without any change in the shape of the contact line (S(t + Δt) = S(t)). Similarly, a −DPN operator implies retraction of the contact line to the adjacent pillar (ai (t + Δt) > ai (t)), and the overall shape of contact line is changed (S(t) → S(t+ Δt)) keeping the droplet height constant (h(t + Δt) = h(t)). Unlike the advancing phase, liquid removal rate is not constant and depends on the ambient conditions. We use the diffusion controlled droplet evaporation model to predict the evolution during this phase.38,39

Figure 5. Schematic of octagonal contact line evolution on a square array of cylindrical micropillars. The time-stamped images illustrate the evolution of the contact line profile from an initial best-fitted circular shape (black color, a = 180 μm, and h = 124 μm) to an octagonal shape at t = 6.36 s). Scale bar corresponds to 100 μm.

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RESULT AND DISCUSSION The evolution of an octagonal contact line as an example (volume addition rate of V̇ = 1.3 nl/s, K = 1, D = 7 μm, L = 20 μm, H = 13 μm, and θF,A = 87° and apparent contact angle: θ0° = θ90° = 108° and θ45° = 102°) is shown in Figure 5. We start (t = 0 s) with an initial near-circular (best fit) contact line shape represented by the black curve. This initial shape of the contact line is highly unstable (Figure 5a) due to the presence of many metastable states. Hence, the contact line zips (via two transition zipping steps) quickly to a stable octagonal contact line shape at t = 0.16 s (Figure 5b). Having achieved this stable polygonal contact line shape, the droplet undergoes liquid addition via global pinning until t = 1.38 s when the next metastable shape is formed via primary zipping along the ⌀° line-of-sight. Two further transition zipping steps are required before the droplet attains the next stable octagonal contact line shape at t = 1.48 s (Figure 5d and Figure 5e). The sequence of events and the snapshots of various such stable and metastable contact line shapes until t = 6.36 s is shown in Figure 5. A plot illustrating the lifetime spent in metastable/stable shapes for a square array with a pillar density D/L of 0.35 is shown in Figure 6. We assign a value of 1 to stable (octagonal) shape and 0 to metastable (nonoctagonal) shape. It can be observed that the contact line remains globally pinned in a stable octagonal shape for around 82% of its lifetime (i.e., ∑ts = 13.53 s and ∑tm = 2.97 s, where ts and tm are the lifetimes spent in stable and metastable shapes, respectively). We perform similar calculations for a very low pillar density (D/L = 0.05) surface where stable octagonal shape is again observed for around 80% of the droplet lifetime. These results suggest that while the metastable shapes are formed intermittently during the droplet evolution, they are short-lived and are hence rarely captured after the end of liquid addition as discussed in the literature.22

A combination of PN and DPN (primary and transition zipping) operators when acted upon iteratively on a given initial shape mimics the evolution of polygonal contact line of droplets on microdecorated surfaces as illustrated in Figure 4. Modeling of the evolution process is started (step 1) by defining the intrinsic contact angles (θF,A, advancing contact angle on an equivalent flat surface) and the roughness parameters such as pillar diameter (D), height (H), and the pitch along the x-axis (L), and the pitch along the y-axis (M), and the rate of liquid addition (V̇ ). We next plot the initial shape of the contact line as a best possible circular fit on the defined array of pillars. We then compute the local contact angles along the primary directions (in step 2, using eq 4). We next move on to step 3, wherein we check for metastable shapes anywhere along the contact line. If step 3 is satisfied, depinning of the contact line will occur via transition zipping (via steps 4a and 5a1). However, the droplet may still remain in a metastable state even after a transition zipping step. In such case, we compare the current droplet height (h(t)) with the zipping height hzip of the droplet (hzip is the height at which the previous primary zipping occurred, not valid for the first iteration). If h(t) < hzip (step 4a), the PN operator is activated (in step 5b), and the droplet volume is increased via increase in droplet height. Conversely, if h(t) ≥ hzip, we execute the DPN operator in step 5a1, causing transition zipping (movement to adjacent pillar) with the formation of a new contact line shape. However, if there is no metastable shape (in step 3), the value of local contact angle is compared with the apparent depinning contact angles along the primary directions (eqs 1−3) in step 4b. If any of the local contact angle exceeds the corresponding depinning angles, primary zipping is initiated using the DPN operator in step 5a2. We next come back to step 2, and the whole cycle of operations is repeated again until the end of simulation 4858

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Figure 6. Plot depicting the respective times spent in octagonal (stable) and nonoctagonal (metastable) shapes. We assign value 1 to stable shape and 0 to metastable shape. (a) An octagonal shape is observed for around 82% of droplet lifetime for a pillar density D/L = 0.35. (b) The scenario remains unchanged even if the pillar density D/L is reduced to 0.05 wherein octagonal contact line shape is observed for about 80% of the droplet lifetime.

Figure 7. Comparison between the experimental results (Raj et al.22) and simulated results. (a) Actual octagonal contact line is masked by the top view of the droplet during the advancing phase in the case where the apparent contact angle is greater than 90°. (b) However, evaporation in global pinning mode lowers apparent contact angle below 90° along all direction, making it possible to visualize the octagonal contact line. (c) Plot of the contact angle variation along 0° and 45° during the advancing, the global pinning, and the receding phases. (d) Temporal variation of droplet height and base radii a1 and a2 along the 0° and 45° lines-of-sight. Scale bars correspond to 100 μm.

line-of-sight they are, 102° and 198 μm, respectively (Figure 7a, left). The same octagonal contact line shape was also reported in the experiments of Raj et al.22 (Figure 7a, right). The evolution of contact angle, base radius, and height (along 0° and ⌀° lines-of-sight) is further illustrated through Figure 7c,d. These plots are divided into three stages: advancing, global pinning, and receding. During the advancing (liquid addition) stage, the contact line advances through a series of pinning and depinning steps (stick−slip behavior) resulting in typical40,41 contact angle oscillations as shown in Figure 7c. The large deviations in contact angle (large jump) are the instances of primary zipping while the small fluctuations represent transition

We now compare our simulated results with experimental results published in the literature. We simulate the contact line over the microdecorated surface with the same experimental parameters as that of Raj et al.22 (i.e., K = 1, D = 7 μm, L = 20 μm, H = 13 μm, θF,A = 87°, and θF,R = 70°). The simulation starts from an initial base radius a = L = 20 μm and height h = 25 μm (please note, initial base radius, height, and shape of contact line do not affect the final shape of contact line) with liquid addition for a duration 16.5 s at a rate of 1.3 nl/s. An octagonal contact line is formed at t = 16.5 s. The local apparent contact angle and the base radius at this instant along the 0° and 90° lines-of-sight are to 108° and 180 μm, respectively, whereas along the 45° 4859

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Langmuir zipping (inset image in Figure 7c). Moreover, unlike the advancing stage, the contact line is pinned at pillar edges and contact angle decreases continuously via a decrease in height during global pinning (Figure 7c,d, from t = 16.5 s to t = 54 s). Receding mode starts at t = 54 s along the octagonal edges, and the contact line recedes along the 45° and 0° lines-of-sight only after time t = 57 s and t = 61 s, respectively. Full evolution of the droplet including the liquid addition stage until complete evaporation is further illustrated through the supplementary video SV1. Furthermore, we simulate the shape of contact line on square array of cylindrical pillars with varying pillar density in Figure 8.

Figure 9. Comparison between the experimental22 and simulated results. A rectangular (approx.) contact line is obtained for rectangular array of pillars with D = 9 μm L = 12 μm, M = 22 μm, H = 20 μm, θF,A = 87°, and θF,R = 75. The side length along the x-axis is 2a2 = 384 μm and along the y-axis 2a1 = 220 μm. Liquid addition stops at time t1= 18 s, and the actual contact line becomes visible when apparent contact angle along all directions becomes equal to or less than 90°, i.e., at time t2 = 64 s. Scale bars correspond to 100 μm.

Figure 8. Effect of pillar density on droplet contact line shapes. Current simulation results are compared with the experimental images and the model predictions of Raj et al.22 Liquid addition rate in each case is 1.3 nl/s. Scale bars correspond to 100 μm.

The x and the y axes represent pillar density (D/L) and octagon side length ratio (B/A), respectively. While an octagonal shape with A > B is formed for an intermediate pillar density of D/L = 0.35, a square shaped contact line is observed when the pillar density was increased to 0.7 (please see supplementary video SV2). Conversely, regular octagonal contact line shape (A = B) is observed at very low pillar densities. All these observations are in agreement with the experimental result in the literature.22 Considering the fact that a square contact line is formed on a high-density square array (pitch ratio K = 1; D = 9 μm, L = M = 13 μm, H = 13 μm, and θF,A = 87°), we also simulate the formation of a rectangular contact line22 on a rectangular array with high pillar density (K = 1.83, D = 9 μm, L = 12, M = 22 μm, H = 13 μm, and θF,A = 87°) in Figure 9 (please see the supplementary video SV3). The same methodology is further extended to explain the formation of hexagonal droplets on a microdecorated surface with a hexagonal array of cylindrical pillars (Supporting Information sections S2 and S3). Unlike a rectangular/square array, a hexagonal array has four principal/ primary directions, i.e., four axes of symmetry. As a result, a dodecagonal contact line shape is observed at low pillar densities (D/L < 0.33). Similar to a square array, regular hexagonal contact line shape is observed upon increasing the pillar density to D/L = 0.33 (Figure 10a and supplementary video SV4). We also form a stretched hexagon by increasing the pitch along the y-axis (Figure 10b).

Figure 10. Comparison of experimental (right) and simulated (left) droplet shapes formed on microdecorated surfaces with hexagonal array of cylindrical pillars. (a) A regular hexagonal contact line is obtained for when D = 5 μm, L = M = 15 μm, and H = 13 μm. (b) A stretched hexagonal contact line is formed when D = 9 μm, L = 15 μm, M = 24 μm, and H = 13 μm. Scale bars correspond to 100 μm.

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line shapes are formed before the droplet attains the eventual stable polygonal contact line shapes observed in experiments. Liquid addition to a stable polygonal droplet moves the contact line radially outward to the next row of pillars and is termed as primary zipping. Such contact line movements often result in the

CONCLUSION We model the temporal evolution of droplets on microdecorated surfaces to show that a series of intermediate metastable contact 4860

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Langmuir

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formation of intermediate metastable shapes, which are otherwise difficult to record experimentally. A droplet with a metastable contact line undergoes a series of transition zipping steps wherein liquid moves sideways along the contact line until the droplet attains the next stable polygonal shape. We show that droplets on microdecorated surfaces spend a majority of their lifetime in stable polygonal contact line shapes. We believe that insights provided in this work can be utilized in designing and optimizing textured surfaces, specifically for printing patterning applications across various disciplines.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00559. Derivation of contact angle in terms of droplets height and base radius is discussed in section S1, modeling of hexagonal droplet on hexagonal arrays is discussed in section S2, and the subsequent hexagonal contact line evolution is discussed in section S3 (PDF) Video SV1 illustrating the complete evolution of droplet for an octagonal contact line (AVI) Video SV2 illustrating the complete evolution of a droplet for a square contact line (AVI) Video SV3 illustrating the complete evolution of a droplet for a rectangle contact line (AVI) Video SV4 illustrating the complete evolution of a droplet for a hexagonal contact line (AVI)

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AUTHOR INFORMATION

Corresponding Author

*Mailing address: R113, Block 3, Indian Institute of Technology, Patna, Bihta, Bihar 801103, India. Ph: +91-612-302-8166; E-mail: [email protected]. ORCID

Rishi Raj: 0000-0002-3653-4288 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Alok Kumar acknowledges CSIR-UGC JRF-Fellowship support. REFERENCES

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DOI: 10.1021/acs.langmuir.7b00559 Langmuir 2017, 33, 4854−4862

Article

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DOI: 10.1021/acs.langmuir.7b00559 Langmuir 2017, 33, 4854−4862