Anisotropic drop shapes on chemically striped surfaces

Colloids and Surfaces A: Physicochem. Eng. Aspects 393 (2012) 32–36

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Anisotropic drop shapes on chemically striped surfaces Robert David ∗ , A. Wilhelm Neumann Department of Mechanical & Industrial Engineering, University of Toronto, 5 King’s College Rd., Room RM501 Toronto, ON, Canada, M5S 3G8

a r t i c l e

i n f o

Article history: Received 26 August 2011 Received in revised form 14 October 2011 Accepted 20 October 2011 Available online 28 October 2011

a b s t r a c t A free energy-minimizing simulation that allows computation of the shapes of sessile liquid drops on chemically patterned surfaces is described. A set of nondimensional results for drops on surfaces with a single wetting stripe is presented. Quantitative agreement is found with published experimental data for drops on many-striped surfaces, and a simple model for anisotropic wetting is evaluated. © 2011 Elsevier B.V. All rights reserved.

Keywords: Anisotropic wetting Numerical simulation Drop shapes

1. Introduction On anisotropically wetting surfaces, liquid drops move and spread in a preferred direction. This can be achieved by physical or chemical patterning of the surface, viz. parallel grooves or hydrophilic stripes, respectively; drops will tend to move along the grooves or stripes. Potential applications include labs-on-a-chip [1,2] and drainable surfaces [3,4]. Chemically patterned, anisotropically wetting surfaces have been fabricated using microcontact printing [5], Langmuir deposition [6], and lithography [7–9] of monolayers. Hydrophilic stripes on a hydrophobic background can be created either by altering the monolayer endgroup or by exposing the underlying surface. Differences in contact angle of up to ∼40◦ [9], and in sliding angle of up to ∼50◦ [7], have been reported between the more and less wetting directions of the surfaces. Anisotropic wetting has also been studied computationally. Darhuber et al. [10] simulated the spreading of ink over hydrophilic regions, including stripes, for a printing application. Buehrle et al. [11] simulated drops on chemically striped surfaces, but focused on the local curvature of the contact line and its relation to line tension. Brandon et al. [12] and Gea-Jódar et al. [13] calculated liquid drop shapes on surfaces with sinusoidal variations in wettability. Brinkmann and Lipowsky [14] mapped the transitions between different wetting morphologies for a drop on one or more hydrophilic stripes, in terms of drop volume and contact angle. In all of the above studies, drop shapes were calculated using the finite element-based Surface Evolver [15]. Other computational

approaches include molecular dynamics, used by Yaneva et al. [16] and Lundgren et al. [17] to study drop shapes on surfaces with nanoscale wetting stripes; and the lattice Boltzmann method, employed by Kusumaatmaja et al. [18] to analyze the dynamics of drops moving across chemical stripes. Related surface patterns have also been studied using the Monte Carlo [19] and lattice Boltzmann [20] methods. Despite this range of computational work, generally only isolated cases have been examined, and comparisons with experimental data have been only qualitative. Here, we describe a new technique for analyzing sessile drops on chemically heterogeneous surfaces. It is based on a method we recently developed for simulating wetting in the Wilhelmy plate geometry [21]. Compared to the other approaches mentioned above, our method is more efficient because it avoids calculation of the coordinates of the liquid–vapour interface. It also handles surfaces with complex wettability patterns more easily than Surface Evolver. Thus, our technique is well-suited for studying surfaces relevant in applications, and generating large data sets. In the next section, the modification of our method for the sessile drop geometry is described. Following this, a comprehensive set of results for drops on single hydrophilic stripes is presented. Finally, published experimental data for many-striped surfaces are quantitatively reproduced, and a proposed model for anisotropic wetting is numerically evaluated.

2. Computation ∗ Corresponding author. Tel.: +1 416 978 1270; fax: +1 416 978 7753. E-mail address: [email protected] (R. David). 0927-7757/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2011.10.020

The equilibrium shape of a sessile liquid drop on a solid surface is that which minimizes the system’s thermodynamic free energy.

R. David, A.W. Neumann / Colloids and Surfaces A: Physicochem. Eng. Aspects 393 (2012) 32–36

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The free energy change associated with the solid for a spreading drop is



E1 =



(sl − sv )dA = − A

lv cos dA

(1)

A

where  sl is the solid–liquid interfacial tension,  sv is the solid–vapour interfacial tension,  lv is the liquid–vapour interfacial tension,  is the contact angle, and A is the solid–liquid contact area under the drop. On a chemically heterogeneous surface,  sl ,  sv , and  vary with location. The free energy associated with the liquid–vapour interface can be divided into two parts. The first part is the free energy if the surface were homogeneous – the free energy of a spherical-capshaped drop of the same volume V and contact area A: E2 = lv [A + h(V, A)2 ]

(2)

where h, the drop height, is the solution of the geometrical relation V = Ah/2 + h3 /6. The second part of the liquid–vapour free energy accounts for the extra liquid–vapour surface area created by the deviation of the contact line from circularity (due to the heterogeneity of the solid surface): 1  sin2 ¯ E3 = 4 lv

∞

 2 |q| (q) ˜  dq

(3)

−∞

˜ is the Fourier transform of the contact line radius , and ¯ where  is an average contact angle. Eq. (3) is obtained via an approximate solution of the Laplace equation in the region near a given contact line  [22,23]. From geometry, ¯ = 2hrms /(2rms + h2 ), where rms is the root-mean-square of . The free energy employed (Eqs. (1)–(3)) neglected both gravity and line tension. This is reasonable for drop sizes in the range 1 ␮m–1 mm, where many applications lie. The simulation was coded in Matlab. The contact line was represented in polar coordinates by a vector, with each entry being the radius  of the drop at some polar angle ϕ. The solid surface was initially represented by a matrix in Cartesian coordinates, with each entry the local value of cos . This matrix was then re-sampled into polar coordinates to build a new matrix of the wettabilities of polar (dartboard) segments of the surface. The free energy for any given drop volume V and contact line (ϕ) could then be calculated from Eqs. (1)–(3). For fixed V, the contact line with lowest free energy was found by a method of steepest descent, as described earlier [21]. All modelled surfaces had left–right and up-down symmetry so as not to allow drops to migrate from their initial, centred positions. Unless otherwise noted, the contact line contained 100 points (100 values of ϕ). Eqs. (1)–(3) are all proportional to  lv , so the only material properties required were the contact angles of the liquid on the more wetting stripe(s) and on the less wetting background of the solid surface. Since the length scale was arbitrary, results are given below in nondimensionalized form, with lengths normalized by V1/3 and forces by  lv V1/3 .

Fig. 1. Drop aspect ratio on a single wetting stripe of contact angle  2 (varying along the x-axis) in a background of contact angle  1 . The stripe width was 1/4 of the drop width with no stripe.

matched this solution well. The discrepancy at high wetting contrast may have been due to the discretization of the surface in the simulation. As shown by Shanahan [24], drop shapes for different background contact angles collapsed onto a single curve governed by the parameter  = (cos  2 − cos  1 )/sin2  1 . Fig. 2 shows representative drop shapes on wider wetting stripes. For wider stripes, drops became longer and narrower, until eventually being trapped inside the stripe in a racetrack shape (Fig. 2b). For even wider stripes, drops became wider again with their sides following the edges of the stripe, until a fully circular contact line could be contained within the stripe. A qualitatively similar trend for drop shapes was observed in molecular dynamics simulations by Yaneva et al. [16], implying that the thermodynamic energy balance considered here (with the possible addition of a line tension term) may govern wetting even for drops containing only ∼105 atoms. A set of quantitative results is plotted in Fig. 3, for drops on stripes of different widths, and three different values of the parameter  (see Fig. 1). As a check, a few cases were also analyzed with Surface Evolver [15]; the results agreed well with those of the simulation (Fig. 3a). The nondimensional results in Fig. 3 allow prediction of drop shapes over a broad range of experimental parameters. One way to estimate the relative likelihood of a drop moving in the desired direction on a striped surface (i.e., along the stripe) is to calculate the force needed to deflect it from this path. To determine this force, equilibrated drops were shifted laterally on the surface (perpendicular to the stripe) by 0.01 and the change in free

3. Single stripe A drop centred on a single hydrophilic stripe is elongated in the direction of the stripe. Fig. 1 shows the variation of drop aspect ratio with the contrast in contact angles between the stripe  2 and the background  1 . Drops became more elongated with increasing wetting contrast. For a narrow wetting stripe, the drop shape can be described by a series solution [24]. Fig. 1 shows that the simulation results

Fig. 2. Contact lines of drops on single stripes of different widths, with background  1 = 40◦ and stripe  2 = 30◦ .

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R. David, A.W. Neumann / Colloids and Surfaces A: Physicochem. Eng. Aspects 393 (2012) 32–36

(a)

2.2

Δ = 0.34 Δ = 0.24 Δ = 0.13

Drop aspect ratio

2 1.8 1.6 1.4

Fig. 4. Contact lines of drops on surfaces with 30◦ stripes (grey) on a 40◦ background (white).

1.2 1

0

1

2

3

(b) 2.8

4. Many stripes of equal width

Drop width

2.6 2.4 2.2 2 1.8

(c)

0

1

2

3

0 0

1

2

3

12 10

Lateral force × 1010

advancing and receding contact angles on the stripe material, respectively, and a is the aspect ratio of the drop.

8 6 4 2

Equilibrium drop shapes on surfaces with single stripes were very similar whether the contact line was initialized at a small or a large radius; in other words, there was little to no contact angle hysteresis. However, for many-striped surfaces, significant hysteresis was observed (Fig. S1), as the stripes acted as energy barriers for liquid wetting/dewetting. The results shown below are for small initial contact line radii (i.e., advancing contact lines) as is appropriate for condensed or deposited drops. Fig. 4 shows representative drop shapes on surfaces with alternating 30◦ and 40◦ stripes of equal width. The drop shapes were similar to those on surfaces with sinusoidally varying wettability [12,13]. As the number of stripes under a drop increased, the drop aspect ratio oscillated below and above 1 (Fig. 5), depending on whether the drop sides resided on a 30◦ stripe (Fig. 4a) or a 40◦ stripe (Fig. 4b), respectively. The drops’ lateral stability improved with increasing numbers of stripes because drops were more and more likely to have their sides pinned against a non-wetting boundary (Fig. S2). Since for the narrowest stripes, there could be as few as 1 or 2 contact line points on a stripe, some runs were repeated with twice as many (200) points in the contact line; similar results were obtained (Fig. S2). On single stripes, drop shapes were characterized by Shanahan’s wettability parameter  = (cos  2 − cos  1 )/sin2  1 (Fig. 1). This scaling also held reasonably well on multiple stripes (Fig. 5).

Stripe width Fig. 3. Aspect ratio, width, and lateral stability of drops on surfaces with a single wetting stripe. In (a), asterisks are corresponding Surface Evolver results. In (b), the straight line denotes equality between drop and stripe widths.

energy (i.e., the work done) was calculated and then divided by this distance. The results are shown in Fig. 3c. We note that the actual forced lateral motion of a drop onto a less wetting region is very complex; this calculation served only as a guide. The lateral force spiked when the stripe became wide enough to trap the drop. In this configuration, shifting the drop moved a large area of liquid from the energetically favourable stripe onto the energetically unfavourable background. The energy barrier for lateral motion is thus proportional to the length of the drop and to the wettability contrast. For motion along the stripes, the only energy barrier is due to the intrinsic hysteresis of the surface (not modelled here), and is proportional to the drop width [25]. Thus, a measure of the tendency for drop motion in the desired direction is a(cos  1 − cos  2 )/(cos  a − cos  r ), where  a and  r are the

Fig. 5. Aspect ratios of drops on many-striped surfaces. The number of stripes increases to the left. The two contact angle combinations have the same value of , and the drop volume was scaled for equal contact radius with no wetting stripes in each of the two cases.

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6. Conclusion Experimental data [9]

2.2

100 point CL

The efficiency of the described method makes it suitable for comprehensively studying drop shapes on chemically patterned surfaces that are of interest in applications. Drop shapes on single stripes are characterized by the stripe width and the wettability parameter , while the multiple stripe case can be reduced to an equivalent single stripe possessing the Cassie angle  C .

200 point CL Single Cassie stripe

Drop aspect ratio

2

1.8

Acknowledgment

1.6

This work was supported by the Natural Sciences and Engineering Research Council of Canada, grant 8278.

1.4

Appendix A. Supplementary data 0.5

1

1.5

2

Hydrophobic width / hydrophilic width Fig. 6. Comparison between computational results and experimental data [9] for drops on surfaces with alternating 40◦ and 106◦ stripes of different relative widths. CL, contact line.

5. Many stripes of unequal width Higher aspect ratio drops can be achieved by manipulating the relative widths of the two stripes. This was done experimentally by Bliznyuk et al. [9]. We ran a series of simulations to emulate these experiments. Drop shapes were equilibrated on surfaces with alternating 40◦ and 106◦ stripes [9]. The 40◦ stripe width was held at 20 ␮m, and the 106◦ stripe width varied from 5 to 40 ␮m. The drop volume V was 10−10 m3 . Drops had higher aspect ratios when the hydrophilic stripe was wider than the hydrophobic stripe (Fig. 6). Discrepancies between experiment and computation could have arisen due to surface cleanliness [9] and/or numerical issues (e.g., limitations in the allowed contact line motions, and the greater number of stripes under the experimental drops compared to the simulated drops). Nevertheless, the computed aspect ratios were in reasonable quantitative agreement with the published experimental data. This demonstrates that, to a good approximation, experimental results for gently deposited drops can be reproduced via consideration of the simple free energy of Eqs. (1)–(3) (neglecting, e.g., inertia [18] and thermal fluctuations [26]). Bliznyuk et al. [9] described a model for drops spreading on striped surfaces. Perpendicular to the stripes, the drop spreads in a stick-slip fashion, always pushed outward by the 40◦ stripes, and eventually halted by a 106◦ stripe at the width of a 106◦ drop (i.e., the width a drop would have on a homogeneous 106◦ surface). Parallel to the stripes, the drop spreads unimpeded as if on a Cassie surface, i.e., a homogeneous surface with the Cassie contact angle. (The Cassie angle  C is defined by cos  C = fcos  1 + (1 − f)cos  2 , where f is the area fraction of the  1 material [27].) We tested this model by equilibrating drops on a series of 106◦ surfaces containing a single stripe, of the same width as a 106◦ drop, and with contact angles equal to the various Cassie angles for the actual striped surfaces. As seen in Fig. 6, drop aspect ratios were similar but somewhat smaller on these surfaces. This was mainly because drops on the many-striped surfaces were narrower than a 106◦ drop. This narrowing was due to the coupling between drop spreading in each direction, which the model neglects. Thus, the anisotropic wetting model of Bliznyuk et al. [9] is reasonably accurate unless drops are highly elongated.

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