Stability of Evaporating Droplets on Chemically Patterned Surfaces

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Stability of Evaporating Droplets on Chemically Patterned Surfaces Maximilian Hartmann, and Steffen Hardt Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00172 • Publication Date (Web): 15 Mar 2019 Downloaded from http://pubs.acs.org on March 19, 2019

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Stability of Evaporating Droplets on Chemically Patterned Surfaces Maximilian Hartmann and Steen Hardt∗ Institute for Nano- and Microuidics, Technische Universität (TU) Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany E-mail: [email protected]

Abstract

Introduction

The stability of water droplets on striped surfaces exposing regions of dierent wettability is studied experimentally, numerically, and based on a scaling model. Dierent values of the stripe widths, and dierent contact angle contrasts between the hydrophilic and hydrophobic stripes are considered. The boundary between the contact angle contrasts leaving the droplets intact and those leading to droplet breakup is computed numerically. The minimum contrast for which breakup occurs increases with increasing hydrophobic contact angle. The existence of an unstable and a stable regime is conrmed experimentally. In the unstable regime, when approching droplet breakup, a conguration with two liquid ngers on the hydrophilic stripes connected by a capillary bridge on the hydrophobic stripe is found. For decreasing volumes, the width of this capillary bridge decreases until a critical value is reached at which the droplet breaks up. The critical width depends on the ratio of the hydrophilic and the hydrophobic stripe width. A simple scaling model is presented with which the critical width can be predicted. According to the model, the droplet becomes unstable when the increasing Laplace pressure inside the bridge can no longer be balanced by the pressure inside the liquid ngers on the hydrophilic stripes. All related data can be found under the PID http://hdl.handle.net/11304/ 82d159a0-7b8f-478f-9c54-62f8efbc9752.

In recent years, wetting on chemically patterned surfaces gained more and more attention in the scientic community. In printing technology, it can be distinguished between technologies where surfaces with dierent wettabilities are produced prior to the actual printing process, and during printing itself. The latter case occurs in multilayer inkjet printing where dierent layers of ink are printed above each other. This technology is used for the fabrication of fuel cells 1 , solar panels 2 or electric circuits 35 , among others. While in this case dierent wettabilites lead to a more complex and less predictable fabrication process 6,7 , dewetting of the hydrophobic areas of a pre-patterned exible web can be exploited to direct ink to the hydrophilic areas of a pre-patterned substrate. 8 Especially in inkjet printing, chemically patterned surfaces can be used to repel droplets from pre-dened hydrophobic regions. This can lead to droplet splitting due to a non-negligible impact velocity of the droplets. 911 Kuang et al. published a review where they focused on inkjet-printed dots and lines. 12 One strategy in that context is to pre-pattern corresponding substrates. In another review by Xia et al. 13 , the focus was on one-dimensional and directionally patterned surfaces (e.g. surfaces with a stripe pattern). The authors also summarized the research on striped chemically patterned surfaces up to the year 2012 and mentioned potential applications in micro- and nanouidics,

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self-cleaning and biomedical devices. As an example of the use of striped surfaces in microuidics, their ability to move droplets in a certain direction was analyzed by Bliznyuk et w al. 14 They introduced the parameter α = wphob phil which is the ratio of the hydrophobic and the hydrophilic stripe widths wphob and wphil , respectively. The authors found that a sessile droplet's contact angle measured parallel to the stripes can be calculated by the Cassie-Baxter model as a function of α and the contact angles of the hydrophilic and hydrophobic stripes, as long as the droplet size is at least 1-2 orders of magnitude larger than the stripe width. The contact angle perpendicular to the stripes is slightly less than the equilibrium contact angle on a pristine hydrophobic substrate. This knowledge can be utilized to guide droplets along stripes by decreasing α in the desired direction of motion. 15,16 This can be exploited in printing technology to assemble gold nanorods on such types of surfaces. 17 It was then Jansen et al. 18 who studied the shapes and evaporation behavior of water droplets on chemically patterned striped surfaces. The stripe widths were in the range between 2 µm and 30 µm and the α-values between 0.1 and 6. The authors found that a 1 µl-droplet assumes an elongated shape in the direction parallel to the stripe pattern. It then evaporates at a rst stage in the constant contact radius (CCR) mode in both directions, parallel and normal to the stripe pattern, until in the parallel direction (after about 33% of total evaporation time) an evaporation in the constant contact angle (CCA) mode occurs, while in the normal direction the radius is still constant. This leads to a droplet shape which more and more approaches a spherical cap, since the contact line in the parallel direction recedes, while it is pinned in the normal direction. This continues until the wetted widths in both directions are the same. In a last step, the droplet evaporates with a shape that can be approximated by a spherical cap. The authors report that the contact line in normal direction to the stripe pattern recedes in a stick-slip-like manner. Stick-slip-jump behavior of evaporating droplets on chemically patterned striped sur-

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faces was also investigated numerically using the Lattice-Boltzmann method 1921 or molecular dynamics simulations. 22 Also, numerical solutions of the Young-Laplace equation were used to gain more insight into this behavior. 23 However, in none of these studies the breakup of droplets on striped surfaces was observed or studied. For droplet sizes of the order of the stripe width, Leopoldes et al. 24 investigated the impact of micron scale droplets on chemically patterned striped surfaces experimentally and numerically using Lattice-Boltzmann simulations. They found that a droplet's nal shape depends on the deposition position and on the droplet size relative to the stripe width. The authors also observed that small portions of liquid remain on neighbouring hydrophilic stripes when the drop diameter is comparable to the stripe width. This is due to a disruption that has its origin in inertial forces. In a separate publication of Jansen et al. 25 , the ndings of Leopoldes et al. concerning the droplet shape dependence on the impact location could be conrmed. In this paper the shape of picoliter droplets on stripes with a width of the order of 10 µm was studied, covering between 3 and 15 stripes. Although it was not the main focus of this publication, the authors observed experimentally and numerically that a satellite droplet remains on the outer hydrophilic stripe. The underlying mechanism is the inertia-driven spreading of the droplet, followed by evaporation, but these phenomena were not studied in more detail. We performed evaporation experiments similar to those of Jansen et al. 18 in the same region of stripe widths and aspect ratios, i.e. when the droplet size is much bigger than the stripe width. In that context we observed that when the droplet recedes from an outer hydrophilic stripe, a satellite droplet remains on it (see gure 1). Basically, this is the same bahavior as observed by Jansen et al. in their 2014 paper. 25 These results provide motivation to study the stability of evaporating droplets on chemically patterned striped surfaces in more detail. Key questions arising in that context are those connecting the stability of a droplet to the problem parameters such as the contact angle contrast


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a

b

C1:

Phil Phob Phil

C2 C2: Trig

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PC

Figure 2: Sketch of the experimental setup. Two triggered Photron highspeed cameras are used to record the droplet shape from the bottom and from the side. For illumination, two cold light sources are used. The images at the right-hand side show an evaporating droplet on a pattern with 500 µm stripe width and α = 1 about 4 s before breakup. In C1-view, hydrophilic and hydrophobic stripes are marked. The C2-view is normal to the stripe pattern and shows the wetting behavior parallel to the stripes.

Figure 1: Evaporating water droplet on a chemically patterned striped surface with a hydrophilic stripe width of 25 µm and α=0.5. In (a), the droplet volume is large enough to wet N stripes. When evaporation proceeds, the droplet breaks up into a large droplet wetting N − 2 stripes, leaving a small droplet on the outer hydrophilic stripe, as shown in (b). or the stripe width ratio. Furthermore, we ask for the physical mechanisms driving the droplet breakup. For that purpose, we chose congurations where the droplet diameter is of the order of the stripe width, since this allows studying the droplet breakup quite systematically. We performed experiments and numerical studies based on the minimization of the surface energy of droplets.

vapor. This is illustrated in C1-view in gure 2. Further details on substrate preparation can be found in the supplementary information. We denote the resulting hydrophilic stripe width wphil and the hydrophobic/hydrophilic width ratio α = wphob /wphil . 14 Two dierent silanes, 1H,1H,2H,2H-Peruorodecyltrichlorosilane (PFDTS, CAS: 78560-44-8, abcr GmbH, Germany) and 3,3,3-Triuoropropyltrichlorosilane (TFPTS, CAS: 592-09-6, abcr GmbH, Germany), are used. To characterize the surfaces, all contact angles are measured on unpatterned wafers. Advancing contact angles are measured by inating a water droplet on the corresponding substrate using a needle attached to a syringe. After removing the needle, the change in contact angle during evaporation can be recorded and the receding contact angle can be measured (see gure 1 of the supplementary information). Experiments on stripe patterns are carried out by placing water droplets on substrates with a pipette. We use a Milli-Q device delivering de-ionized water with a resistivity of 18.2 MΩcm at 25 ◦ C. The evolution of the shapes and contact lines of evaporating water droplets is recorded with two triggered high-

Experimental and Numerical Details

Experiments Glass substrates with alternating covered and uncovered stripes of dierent widths and width ratios are produced by spin coating a positive photoresist (AZ 9260, MicroChemicals GmbH, Germany) onto a borooat 33 glass wafer (SIEGERT WAFER GmbH, Germany) and common photolithography steps. In a low pressure chemical vapor deposition process, these wafers are exposed to a silane atmosphere. The vacuum chamber in which the silanization takes place is designed to match the conditions reported by Mayer et al. 26 After removing the photoresist with acetone and isopropanol, the bare glass regions are more hydrophilic than those regions exposed to silane


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speed cameras (see gure 2). For the bottom view, a Photron FASTCAM SA-1.1 camera is attached to a microscope body, and illumination is performed with a separate cold light source through the condenser of the microscope. This assures sucient light to record the breakup process with high frame rates. For side view we use a Photron FASTCAM SAX1 camera attached to a Navitar-12X macro objective. Illumination from the back is also achieved with a cold light source. In some experiments for which less light is required, also images from the top are taken. In this case, the Photron FASTCAM SA-1.1 is mounted above the substrate and an additional Navitar-12X macro objective which allows co-axial illumination is used. Triggered side view imaging is still being performed as described above. Since the focus of the present experiments is on the stability behavior of the droplets rather than on the speed of evaporation, the humidity of the atmosphere around the droplets is not controlled. The lab temperature is about 25 ◦ C. The evaluation of image data is performed using a selfwritten software. Experiments are performed at least seven times for each set of parameters, which allows computing average values and error bars representing the standard deviations.

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bphob

bphil

Figure 3: Droplet shape calculated by Surface Evolver. Since the evaporating droplet has a symmetric shape with respect to two mirror planes - one in the middle of the hydrophobic stripe and one normal to the stripe pattern only one quarter of the droplet is considered in the simulations (dark blue). Hydrophilic and hydrophobic stripes are indicated in green and red, respectively. bphil and bphob refer to the wetted width of the hydrophilic stripe and wetted length on the hydrophobic stripe. be written as

dE = (σsl − σs ) · dAsl + σl · dAl

(1)

where A denotes specic interfacial areas, σ denotes interfacial tension, and the subscripts sl, s and l indicate solid-liquid, solid and liquid, respectively. The latter two refer to interfaces with the gas phase. Surface Evolver triangulates the liquid surface with an adaptive mesh and minimizes functional (1) until convergence is reached. For the present simulations, the surface is evolved for a user-dened volume until the maximum cell size reaches 1/100(wphil + wphob ), and convergence is achieved if the coecient of variation of the change in energy ∆E of the last 7 iterations is smaller than 10−7 . Since the shape of the droplet is symmetrical with respect to the plane cutting through the centerline of the hydrophobic stripe and the plane orthogonal to that dividing the capillary bridge in two halves, the computational domain can be chosen to be one quarter of the droplet, as it is shown in gure 3, where the computed quarter droplet is visualized by a dark blue surface. At the sym-

Numerics The equilibrium shape of a sessile droplet corresponds to a conguration with minimal interfacial energy, where the gas-liquid, solidliquid and solid-gas interfaces need to be considered. Surface Evolver 27 is a software tool that can be used to minimize the corresponding Gibbs free energy. It relies on gradient descent implemented by the conjugate-gradient method under given boundary conditions, e.g. a given liquid volume and given contact angles. In the present case, no other energies than the interfacial energies are taken into account. Specically, the gravitational energy is neglected, which is justied for droplets signicantly smaller than the capillary length. Therefore, the innitesimal change in energy dE can


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metry planes, the contact angle is 90◦ . Surface Evolver is employed to nd the critical contact angle contrast ∆Θ = Θphob − Θphil , i.e. the maximum dierence between the two contact angles for which no breakup is observed. To judge whether a droplet breaks up or evaporates as a whole, two criteria are used:

the number of iterations required signicantly increases when the threshold value is reduced. The implemented algorithm is as follows: First the initial model geometry with given hydrophilic and hydrophobic stripe widths and contact angles is loaded with a certain liquid volume. Then the surface is evolved until convergence is achieved or one of the two above-dened criteria is fullled. If the latter is the case the calculation is stopped and the results are saved. If convergence is achieved but none of the two criteria is fullled, then the droplet volume is reduced. These steps are repeated until either the breakup or the nobreakup criterion is fullled. It is not necessary to further evolve a droplet that fullls the no-breakup criterion to convergence, since our preliminary studies showed that when convergence is reached, still no breakup has occurred.

1. Breakup: An evolved liquid surface is considered as broken up if the minimum width of the liquid bridge, i.e. twice the width of the wetted length in the middle of the hydrophobic stripe directly at the symmetry plane of the quarter droplet (bphob in gure 3), is less than 0.1wphil . 2. No breakup: No breakup is expected when the wetted width of the hydrophilic stripe (bphil in gure 3) is less than 0.25wphil . The breakup criterion (1.) was chosen in such a way that the corresponding threshold is signicantly smaller than the critical width we found in our experiments and in the calculations with the model we present below. The no-breakup criterion (2.) is an empirical criterion we developed after we performed preliminary simulations. In these simulations we found that for a given parameter set (wPhil , α, Θphil and Θphob ) and if the dierence between the hydrophilic and the hydrophobic contact angle is small enough, bphil decreases with volume. This nding is based on fully converged Surface Evolver simulations. Consequently, in this situation it is energetically more favourable for a droplet to recede from the hydrophilic stripe than to break up and wet only the hydrophilic stripes. The value of 0.25wphil for the no-breakup criterion results from a parameter study for wphil = 500 µm and α = 1 we performed prior to our nal calculations. In this study we subsequently decreased the threshold value for bphil and observed that the calculated hydrophilic contact angle marking the transition to no breakup changes by maximum 1◦ compared to the one obtained with the no-breakup criterion. Having a not too small threshold value for bphil also helps to limit the CPU-time requirements of the simulations, since for a given set of parameters

Results and Discussion In the experiments, the contact angles are determined by the silanes used and the corresponding carrier substrate. The resulting receding and advancing contact angles on the silanized glass surfaces are Θphob,rec = 100◦ and Θphob,adv = 120◦ in case of PFDTS and Θphob,rec = 72◦ and Θphob,adv = 94◦ in the case of TFPTS, respectively. On the bare glass regions, contact angles of Θphil,rec = 12◦ and Θphil,adv = 49◦ are achieved. The corresponding contact angle contrasts for evaporating droplets on striped surfaces can be inferred from these values. Results on the droplet stability as obtained using Surface Evolver are shown in gure 4. The gure contains colored rectangles showing the contact angle range given by the advancing and receding contact angles of the combination of hydrophilic/hydrophobic materials used. The advancing and receding contact angles of water on the hydrophobic surface can be read o from the abscissa, those on the hydrophilic surface from the ordinate. Numerical calculations with Surface Evolver were performed for dierent values of α to determine the critical contact angle contrasts


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100 80

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not entirely correct for large contact angle contrasts (glass-PFDTS). Nevertheless, the deviations from the measured receding contact angle on the bare glass surface are relatively small, so that the contact angles of the Surface Evolver calculations can be interpreted as the receding contact angles. Figure 4 can be used to predict if an evaporating water droplet on a striped surface breaks up or stays intact. To validate the numerical predictions, experiments in the stable and unstable regime were performed. Figure 5 shows the behavior of evaporating water droplets on TFPTS-PFDTS (a) and glass-PFDTS (b) surfaces for 4 dierent stages (I-IV) of evaporation. These experiments correspond to the blue and purple rectangles in gure 4. The rst row shows the bottom view, the second the side view. As mentioned in the experimental section, the side view shows the wetting behavior parallel to the stripes. While in case (a) the droplet evaporates as a whole, it breaks up in case (b). This behavior agrees well with the results of the Surface Evolver simulations. Also for the third material combination (glass-TFPTS, yellow rectangle in gure 4) droplet breakup is observed. In the following, the breakup regime will be further discussed based on the experimental data obtained for the combination glass-PFDTS. The time evolution of a droplet that stays intact is as follows. In stage I, both hydrophilic stripes and the hydrophobic area between them are wetted. Then the liquid recedes from the bottom hydrophilic stripe (stage II), until only parts of the hydrophobic stripe are wetted in stage III. The contact angle on the hydrophilic stripe does not change during stages I - III since it is in the receding state. Evaporation continues and the liquid completely retreats to the upper hydrophilic stripe (stage IV). After this point has been reached, the contact angle becomes smaller than the receding contact angle on the hydrophilic stripe. This is a known phenomenon for evaporating droplets on small scales that we also observed in our contactangle measurements. Corresponding details can be found elsewhere. 28,29 In the case of gure 5b (glass-PFDTS), the droplet breaks up. In that case it is energet-

TFPTS PFDTS

60 stable

40 glass TFPTS

20

glass PFDTS

unstable

0 20

40

60

80

100

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140

◦

Θphob ( )

Figure 4: Stability of droplets wetting three stripes, one hydrophobic one between two hydrophilic ones. The symbols show the boundary between the stable (no breakup) and unstable (breakup) regions as obtained with Surface Evolver. A droplet evaporates without breaking up if its contact angles are located above the line connecting the corresponding data points. Otherwise it breaks up. The corrsponding regions are marked with arrows. The rectangles denote the range of advancing and receding contact angles of water droplets on dierent combinations of substrates. where breakup occurs. The data points in gure 4 mark the last stable congurations obtained from these calculations, i.e. the contact angle contrasts ∆Θcrit where no breakup is expected for decreasing volumes. It can be seen that ∆Θcrit is smaller for α = 0.5 in comparison to α ≥ 1. Since this dierence is relatively small and almost not measurable in potential experiments, the inuence of α can be neglected for most practical purposes. Then, the only relevant parameter to predict if an evaporating droplet breaks up is the contact angle contrast as a function of the hydrophobic contact angle. ∆Θcrit increases with increasing hydrophobic contact angle. Note that in the Surface Evolver calculations constant contact angles are used on the hydrophobic and the hydrophilic stripe, respectively. No eects of contact angle hysteresis are implemented since contact angles are expected to be in the receding state. Later on it will be shown that this is


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breakup, the capillary bridge on the hydrophobic stripe is considered. As an example for a contact angle contrast in the unstable regime, experiments with hydrophobic stripes treated with PFDTS were performed. Figure 6a shows the inuence of the hydrophilic stripe width and α on the minimal width of the capillary bridge, i.e. its width at the neck. To compare the results, both the minimal width d of the capillary bridge and the time t before breakup are made dimensionless (see in gure 6b). Since the breakup process is expected to be dominated by inertia and surface tension, the time 3 )1/2 is chosen, with the hyscale τ = (ρ/σwphob drophobic stripe width wphob as the characteristic length and ρ and σ being the density and the surface tension, respectively. The dimensionless time is denoted T = t · τ −1 . The minimal width of the capillary bridge is multiplied by the inverse of the hydrophilic stripe width −1 (Dphil = d · wphil ) to compare the corresponding inuence. In gure 6b, the slope of the curves for large dimensionless times T > 1 is relatively small. When approaching breakup, at approximately T ≈ 1, the slope of the curves becomes steeper. In subsequent paragraphs we will show that this is the point where the critical width of the capillary bridge Dcrit is reached. The critical width marks the point beyond which no conguration of the droplet surface with constant mean curvature can be found. From gure 6 it is obvious that, to a good approximation, for dierent wphil but constant α the breakup behavior is the same. The slightly dierent slopes before approaching the critical width, which can exemplarily be seen in the experimental data for α = 1.5, can be explained by the fact that the humidity is not controlled during the experiments and therefore the volume of dierent droplets decreases at dierent rates. Owing to the high frame rate of 75,000 fps used in these experiments, no pictures from the side were taken since this was not possible with the light source and the objective used. For values of the minimal width smaller than Dcrit , the dynamics is the same for all stripe patterns with the same α, and the data collapse onto a single curve. After Dcrit has been reached, the breakup process happens very fast, so that

a

b

I

II

III

IV

Figure 5: Water droplets on a striped surface at dierent stages of evaporation (indicated by I - IV). Hydrophilic and hydrophobic stripes are formed by TFPTS and PFDTS in (a) and glass and PFDTS in (b). While in (a) the droplet evaporates as a whole, it breaks up in (b). The width of the hydrophilic stripe is 500 µm and α = 1 in both cases. The rst row in each case shows the bottom view, the second the side view. ically more favorable to wet two hydrophilic stripes instead of receding onto one. Dierent from case (a), Θphil increases as the minimum width d of the capillary bridge on the hydrophobic stripe decreases. While Θphil is slightly bigger than the receding contact angle on bare glass in stage I, it increases as the capillary bridge approches its critical width (stage III). This happens due to an increased pressure in the capillary bridge for decreasing d, which propages into the rest of the droplet, since the pressure needs to be constant over the entire liquid volume. Since the liquid on the hydrophobic stripe has retracted in stage IV - this is immediately after droplet breakup - Θphil increases again in comparison to stage III. A satellite droplet in the middle of the hydrophobic stripe is found in all of our experiments directly after the breakup of the liquid bridge. The wetted length on the hydrophilic stripe stays constant over the total evaporation time before breakup. For further characterization of the droplet


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200

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t0 − t

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1

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Figure 6: Inuence of the hydrophilic stripe width on the breakup behavior for dierent values of α. The data are plotted in non-dimensionless form in part (a) and in dimensionless form in part (b). The hydrophobic stripes were obtained by functionalization with PFDTS. t0 − t and T0 − T measure the time before breakup of the liquid bridge. t0 and T0 mark the points in time when the width of the liquid bridge becomes zero. In (b), the dimensionless minimal width of the liquid −1 bridge is denoted by Dphil = dwphil . The assignment of symbols is as follows: α = 0.5 (squares), α = 1 (circles), α = 1.5 (triangles). wetting the hydrophilic stripe, pphil , based on the Young-Laplace equation

evaporation plays no role. The fact that the data points for the same value of α collapse onto a single master curve indicates that the breakup process of the capillary bridge is dominated by inertial and capillary forces. Note q that this is ρ 3 w as only achieved when choosing τ = σ phob the time scale for non-dimensionalization. While the dimensionless critical width is virtually independent of wphil , it depends on α. This can be explained by geometric similarity. For xed wphil , the evaporating droplets are geometrically dissimilar when α is varied. Dcrit increases with increasing α. However, when α is xed, a droplet shape for a specic value of wphil can be obtained from the droplet shape corresponding to a dierent value simply by rescaling. For this reason, the curves representing the time evolution of Dphil do not depend on wphil . It is worth noting that from gure 6 it can be inferred that for constant hydrophobic stripe width the critical width of the liquid bridge decreases with increasing α. A simple geometrical model has been developed to predict the critical width of the capillary bridge. It relies on comparing the pressure within the bridge, pphob , and within the liquid

∆pi = σ(

1 1 + ). ri,k ri,⊥

(2)

The subscripts i denote phil and phob, while k and ⊥ label the directions parallel and normal to the stripe orientation. In the following we formulate scaling relationships to compute the pressure inside the bridge and inside the liquid on the hydrophilic stripe. For the latter, the relevant length scales are wphil and lphil,k . lphil,k is the the wetted length on the hydrophilic stripe which, corresponding to experimental observations, is assumed to be xed. The radii of curvature are obtained from these length scales when taking into account the contact angles:

rphil,⊥ =

wphil 2 · sin(Θphil,⊥ )

(3)

rphil,k =

lphil,k . 2 · sin(Θphil )

(4)

and

The relevant geometrical quantities are indicated in gure 7. In equation (3), Θphil,⊥ is the contact angle of the liquid normal to the


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function of Θphil and lphil,k . We made a sensitivity analysis, and it turned out that lphil,k does not have a signicant inuence on the breakup diameter. The magnitude of lphil,k used in the model was determined from the experiments performed for α = 1. To proceed, we write the volume V of the droplet as the sum of the liquid volume above the hydrophilic (Vphil ) and above the hydrophobic (Vphob ) stripe. It is assumed that before breakup almost the entire volume of the droplet is located above the hydrophilic stripes, i.e. Vphil >> Vphob . This assumption can be justied if stage III (right before bridge breakup) of gure 5b is taken into account, where the capillary bridge wets less than 10 % of the entire area wetted by the droplet. Then the droplet volume can be approximated by V ≈ Vphil , which is a monotonically increasing function of Θphil if lphil,k is constant (as observed in our experiments): V ∼ Θphil . It is well known that a constrained capillary surface becomes unstable dp changes its sign from positive to negaif dV dp dV 30 it foltive. Therefore, with dΘdpphil = dV dΘphil lows that the liquid bridge becomes unstable if dp changes its sign. dΘphil In gure 8a, the inverse of the width of the capillary bridge, 1/d, is plotted as a function of Θphil for dierent values of α. Since 1/rphob,⊥ is almost constant over the total time of evaporation, 1/d can be used as a measure for the pressure inside the capillary bridge. As already shown in gure 5b, Θphil increases when approaching the point of instability. This corresponds to decreasing Vphob while increasing Vphil to balance the pressure inside the bridge with that on the hydrophilic stripes. The model does account for that fact. Before breakup, volume is added to Vphil , when the system is still in the dp > 0 and therefore dΘdpphil > 0. stable region dV

hphil

Θphil d

rphob,⊥

wphob wphil

Θphil,⊥ lphil,k

Figure 7: Shape of a droplet as it is assumed in the model, together with relevant model parameters. Green and red areas denote hydrophilic and hydrophobic stripes, respectively. stripe orientation at the boundary between a hydrophilic and a hydrophobic stripe

Θphil,⊥ = 2 · arctan(2 ·

hphil ) wphil

(5)

with

lphil,k Θphil · tan( ) (6) 2 2 being the height of the liquid wetting the hydrophilic stripe. For an estimation of ∆pphob , rphob,⊥ is calculated using the t function 1/rphob,⊥ (α) = 2.75/(wphil α), which is valid for contact angles Θphob ≥ 90◦ as it is the case in the present work. If the receding contact angle on the hydrophobic stripe is smaller than 90◦ (Θphob < 90◦ ), the same t function can be used, but the lefthand side must be supplemented by the factor sin(ΘPhob ), so that the curvature is calculated normal to the plane of the liquid surface. The t function is based on experimental data obtained for dierent values of α. For the two different hydrophobic surfaces used in our study (TFPTS and PFDTS), no dierences in curvature could be observed. Details can be found in the supplementary information. The radius of curvature parallel to the hydrophobic stripe hphil =

rphob,k =

d 2 sin(Θphob )

(7)

However, Θphil only increases until dΘdpphil < 0. This is where the liquid bridge becomes unstable. Correspondingly, the critical width dcrit dd−1 can be found by setting dΘ = 0. phil From gure 8b it can be seen that the critical width calculated with the model agrees well with the experimental results. This is indicated by the fact that the critical widths ac-

is modeled as a function of the minimum width of the capillary bridge d, with Θphob being the receding contact angle on the hydrophobic stripe. In a droplet in equilibrium, the pressure is uniform. Therefore we demand ∆pphil = ∆pphob , from which d can be calculated as a


9

Langmuir 0.008

500

a

0.006

350

0.004 0.003

α = 0.5, dcrit α = 1, dcrit α = 1.5, dcrit α = 2, dcrit α = 2.5, dcrit

0.002 0.001 0

b α = 0.5

400

d (µm)

0.005

450

Θphil,adv

Θphil,rec

0.007

d−1 (µm−1 )

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

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= 127µm = 199µm = 247µm = 281µm = 308µm

300

α=1 α = 1.5 α=2 α = 2.5

250 200 150 100 50

-0.001

0 0

5

10

15

20

25

Θphil

30

35

40

45

50

0.01

◦

0.1

1

t0 − t

( )

10

100

(ms)

Figure 8: (a) Inverse width of the liquid bridge as obtained from the model. A change in slope from dd−1 dd−1 > 0 to dΘ < 0 marks the critical width. The experimental values of the advancing and dΘphil phil receding hydrophilic contact angles are indicated. (b) Width of the capillary bridge as a function of time before breakup (t0 −t) as obtained from the experiments. Exemplarily, data for wphil = 500 µm and dierent values of α are shown. The black lines show the corresponding results for the critical width as calculated with the model. cording to the model approximately mark the points where the experimental curves change their slope. Therefore it can be concluded that the droplet becomes unstable if the pressure in the capillary bridge exceeds the maximum pressure that can be sustained in the liquid above the hydrophilic stripes.

numerical ndings are conrmed by experimental studies performed with three dierent combinations of surfaces, glass-TFPTS, glassPFDTS and TFPTS-PFDTS. Furthermore, experiments in the unstable regime of contact angle contrasts were carried out to disclose the details of the droplet breakup. In this regime, two liquid ngers of identical length form on the hydrophilic stripe, connected by a capillary bridge on the hydrophobic stripe. With ongoing evaporation, the width of the capillary bridge d decreases, until a critical width dcrit is reached. It was found that the dimension−1 less critical width Dcrit = dcrit wphil only depends on α and increases with increasing α. To better understand the criteria responsible for droplet breakup, a model was set up that balances the Laplace pressures inside the capillary bridge and inside the liquid ngers on the hydrophilic stripes. According to the model, there is a maximum pressure that can build up in the liquid ngers. When the width of the capillary bridge decreases due to evaporation, a point is reached where the Laplace pressure inside the bridge becomes so high that it can no longer be balanced by the pressure inside the liquid ngers. This marks the point where the droplet becomes unstable. The corresponding

Conclusion Detailed studies of the breakup of evaporating water droplets on striped surfaces with a wettability contrast were carried out. As a key conguration, droplets wetting three stripes in total (one hydrophobic one between two hydrophilic ones) were chosen. The experiments were complemented by Surface Evolver simulations allowing to determine the critical contact angle contrast ∆Θcrit (the dierence between the contact angles on the hydrophilic and the hydrophobic stripes) forming the boundary between the regimes where the droplet breaks up and where it stays intact. This critical contrast increases with increasing hydrophobic contact angles. No signicant dependence on α - the ratio between hydrophobic and hydrophilic stripe width - can be observed. These


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model predictions agree well with the experimental data for droplet breakup.

(4) McKerricher, G.; Vaseem, M.; Shamim, A. Fully inkjet-printed microwave passive electronics. Microsystems & Nanoengineering 2017, 3, 16075.

Acknowledgement Financial support by

the German Research Foundation (DFG) within the Collaborative Research Centre 1194 "Interaction of Transport and Wetting Processes", project A02b, is kindly acknowledged. Furthermore, the authors are grateful to KlausDieter Voss, Bioinspired Communication Systems, Technische Universität Darmstadt, for carrying out the photolithography steps, to Jörg Bültemann and Frank Plückebaum for helping with the construction of the setup, and to Michael Eigenbrod for fruitful discussions (all with the Institute for Nano- and Microuidics, Technische Universität Darmstadt).

(5) Correia, V.; Mitra, K.; Castro, H.; Rocha, J.; Sowade, E.; Baumann, R.; Lanceros-Mendez, S. Design and fabrication of multilayer inkjet-printed passive components for printed electronics circuit development. Journal of Manufacturing Processes 2018, 31, 364371. (6) Calvert, P. Inkjet Printing for Materials and Devices. Chemistry of Materials 2001, 13, 32993305. (7) Cho, C.-L.; Kao, H.-l.; Chang, L.-C.; Wu, Y.-H.; Chiu, H.-C. Fully inkjetprinting of metal-polymer-metal multilayer on a exible liquid crystal polymer substrate. Surface and Coatings Technology 2017, 320, 568573.

Supporting Information Available Document including experimental protocol for substrate preparation, contact angle measurement, curvature measurement and parameters used in the model.

(8) Bower, C. L.; Simister, E. A.; Bonnist, E.; Paul, K.; Pightling, N.; Blake, T. D. Continuous coating of discrete areas of a exible web. AIChE Journal 2007, 53, 1644 1657.

References

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(11) Zou, L.; Wang, H.; Zhu, X.; Ding, Y.; Chen, R.; Liao, Q. Droplet splitting on chemically striped surface. Colloids and

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(13) Xia, D.; Johnson, L. M.; López, G. P. Anisotropic Wetting Surfaces with OneDimensional and Directional Structures: Fabrication Approaches, Wetting Properties and Potential Applications. Advanced Materials 2012, 24, 12871302.

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(17) Ahmad, I.; Jansen, H. P.; van Swigchem, J.; Ganser, C.; Teichert, C.; Zandvliet, H. J.; Kooij, E. S. Evaporative gold nanorod assembly on chemically stripe-patterned gradient surfaces. Jour-

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Graphical TOC Entry 300 stable

250 200

d (µm)

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

150

unstable

100 d(t)

50 0 0.001

0.01

0.1

1

10

100

1000

t0 − t (ms)

For Table of Contents Only


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Stability of Evaporating Droplets on Chemically Patterned Surfaces

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