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Morphological Transitions of Droplets Wetting a Series of Triangular Grooves Marcin Dokowicz* and Waldemar Nowicki Faculty of Chemistry, Adam Mickiewicz University in Poznań, Umultowska 89b, 61-614 Poznań, Poland ABSTRACT: Morphology and thermodynamics of a microdroplet deposited on a grooved inhomogeneous surface with triangular cross section of the grooves were studied by computer simulations with the use of Surface Evolver program. With increasing volume of the droplet, it initially spreads along the series of grooves assuming the filament-like morphology. After reaching a certain volume, the surface wetted by the droplet is reduced and the droplet assumes the bulge morphology or spreads over the surface bordering on the groove initially occupied (it can be either a neighboring groove or a flat surface). The character of the process is determined by the geometry of the edge of the inhomogeneity studied. The effect described also depends on the number of grooves G and the Young contact angle θY. The change in the shape of the droplet becomes more pronounced with decreasing θY and G. Above a certain number of grooves, in the range of contact angles studied (e.g., G > 6 if θY = 70° and G > 4 if θY = 75°), no morphological transition of the droplet was observed. or rectangular grooves.9,16,21,22 They have been mainly concerned with an attempt at description of behavior of a droplet settled on a single linear inhomogeneity. The droplet behavior on surfaces with 1D geometric patterns, made of a few linear inhomogeneities, has not been studied in detail yet. Brinkmann and Lipowsky4 have observed a morphological transition on substrates with striped surface domains. They have shown that with increasing distance separating the striped surface domains, the filament-like morphologies wetting simultaneously more than one strip disappear together with morphological transition between the filaments and the bulges. Therefore, particularly interesting seem to be the substrates with one-dimensional geometric patterns made of inhomogeneities not separated with a stripe of flat separatory surface. A good example of such a substrate is that covered with a series of grooves of triangular cross section. The morphological transition of a droplet on such a substrate has not been described in literature yet. As shown in ref 16, the shape of the droplet settled on a single triangular groove of the dihedral angle α = 90° and at the contact angle θY = 75° depends in a complex way on the droplet volume: (1) The droplet of the volume at which the linear size of the droplet is smaller than the width of the groove assumes the shape of a part of a spherical cap. (2) With increasing volume of the droplet, its shape changes up to a certain critical volume (VC). The droplet elongates along the groove, and its free surface (liquid/gas interface) assumes the filament shape. (3) At VC, because of the Rayleigh−Plateau

1. INTRODUCTION In recent years, much effort has been devoted to investigation of wetting of inhomogeneous surfaces which has brought many applications of the phenomena (e.g., micro- and nanofluidics, biomedical devices, and self-cleaning),1,2 and this phenomena still remains a scientific challenge. The hitherto studies have been concerned with the surfaces of chemical and topological inhomogeneities: 1D (i.e., strips, posts, and grooves)3−9 and 2D (i.e., pillars and chemical patterns).3,6,10−12 The studies have been performed by experimental and theoretical methods and by computer simulations. Particularly interesting results have been obtained for the surfaces with single linear inhomogeneities and patterns made of them. A droplet deposited on such a surface shows a natural tendency to pinning to physical or chemical heterogenities.13 Because of the pinning effect, the droplet shows anisotropy of shape.14 Along the pinning line, the angle made by the surface of the solid and the plane tangent to the droplet surface takes a different value than the Young contact angle θY. This angle is called the pinning angle ϕ, and its value varies in the range determined by the canthotaxis condition13 θY < ϕ < π − β + θY

(1)

(where βis the opening angle at the extreme edges of the grooves). A recent review covers the latest advances in anisotropic wetting surfaces field.1 Gau et al.15 were the first to observe that a droplet settled on linear inhomogeneities can show the morphological wetting transitions. Many authors have been interested in spontaneous and reversible change in shape of a droplet deposited, for example, on substrates with striped surface domain,4,5,15−17 between two posts or parallel fibers,18−20 and also in triangular © 2016 American Chemical Society

Received: April 3, 2016 Revised: May 24, 2016 Published: June 27, 2016 7259

DOI: 10.1021/acs.langmuir.6b01275 Langmuir 2016, 32, 7259−7264

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dihedral angle α = 90° and different Young contact angles θY. The simulations were performed for the following set of θY values: 70°, 72.5°, 75°, 77.5°, and 80°. For smaller θY, the number of mesh nodes needed to map the liquid surface abruptly increased since the droplet underwent strong elongation. Accordingly, the computational time increased enormously as a result of poor convergence of energy, and the obtained results were inaccurate. The applied scale of geometrical elements (the droplet volume, the height of grooves) involves the capillary range, that is, both the gravitational effects (the Bond number → 0) and the line tension effects (line tension → 0) were neglected. Since, at assumed values of α and θY, the Concus−Finn condition24 was not met, no unlimited spreading of the droplet along the groove was observed. Minimization steps included a sequence of conjugated descents and saddle and Hessian_seek commands. In the vicinity of the instability point, the number of the optimizing steps was increased to ensure good convergence of the free energy of the system. At the end of each simulation, the Hessian command (switching on the Newton−Raphson method of the free-energy minimization) was executed. Convergence of computations was regarded as satisfactory when the energy change was less than 10−7 units. The final free droplet surface consisted, dependently on G, of roughly 2500− 13 000 individual nodes. At the end of each simulation, when the droplet reached its final equilibrium morphology, several selected quantities were calculated by means of internal variables and attributes of geometric elements of SE. These quantities include geometrical dimensions of the droplet, Laplace pressure, free energy, pinning angle, and so forth. To characterize the geometrical droplet dimensions, the reduced height of the droplet over the bottom of grooves H̃ = H/h and the reduced length of the triple line L̃ = L/h, where H and L are the droplet height over the elevation of surface edges and the length of the droplet base on the grooved surface, respectively, were calculated. The reduced Laplace pressure was obtained as Δp̃ = Δph/γL where Δp is defined by

instability,15 the droplet assumes the bulged shape and the further increase in its volume involves the increase in the size of the bulge. The abrupt change in the droplet morphology at the VC is reversible and, because of a jumpwise change in the thermodynamical parameters of the system, it resembles a phase transition.16 Exploration of new areas of morphological transitions is particularly interesting because of potential use in the microfluidal systems. This study was aimed at examination of the possibility of similar morphological transitions of the droplet deposited on several grooves without flat separators. We also undertook to answer the question on the influence of canthotaxis condition on the observation of the morphological transition. The aim of this study is to discuss the parameters governing these transitions and their thermodynamics. The study was performed by computer simulation based on minimization of energy of free surface of the droplet settled on a solid surface of a defined geometry with the aid of the public domain software Surface Evolver (SE).23

2. SIMULATION METHOD The surfaces modeled by SE are represented by a mesh of triangles. The program minimizes the free energy of the modeled system in defined steps including procedures of mesh refinement, vertex averaging, and polishing up the triangulation and the energy minimization by means of the conjugated gradient descent method and the Newton−Raphson method. Mechanically stable surface/interface configurations are obtained by minimizing the sum of all surface/interface energies, which is a function of the coordinates of the nodes. In the simulations, the effect of increasing volume of the droplet deposited onto grooved surface on its morphology and thermodynamics was modeled. The exemplary morphology of the droplet deposited onto such a surface is shown in Figure 1

⎛1 1 ⎞ Δp = pint − pext = 2γL ϑ = γL⎜ + ⎟ R= ⎠ ⎝ R⊥

(2)

and pint is the internal pressure of the droplet, whereas pex is the ambient pressure (assumed to be equal to zero), ϑ is the mean curvature at all mesh nodes mapping the liquid surface, R⊥ and R= are the radii of liquid surface curvature measured on the planes perpendicular and parallel to the y axis, respectively, and γL is the liquid surface tension. Both Δp and ϑ are the internal variables of SE. The pinning angle ϕ (see Figure 1) and the maximum pinning angle ϕmax found along the whole pinning edge were assumed to be equal to the slope of each mesh triangle adjoining the pinning edge and were calculated from the equation

Figure 1. Droplet equilibrated at pinning angle ϕ and apparent contact angle ξ deposited on triangular grooves of dihedral angle α and height h (Ṽ = 400, θY = 75°, G = 3). The opening angle β at the extreme edge is also depicted.

together with symbols of geometric quantities further referred to in the text. The increase in the droplet volume was modeled by a series of independent simulations performed at a constant reduced volume from the range Ṽ = V/h3 = 10−3000. The grooved surface was represented by adjacent parallel grooves of the height h and dihedral angle α (see Figure 1) identical in all simulations. The length of the grooves was unlimited. In each simulation, the droplet was initially settled on the arbitrarily chosen number of adjacent grooves G and was pinned to the assumed boundary edges along the y direction. The value of the pinning angle ϕ was unlimited. The study was performed for G ranging from 1 to 5. The solid surface was characterized by

⎛z⎞ ϕ = arcsin⎜ ⎟ ⎝S⎠

(3)

where z and S denote the vertical coordinate and the length of normal vector of a triangle (the length of normal vector of the surface in the SE representation is equivalent the surface area). The reduced free energy of the droplet Ẽ = E/(γLh2) was found from the formula 7260

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Langmuir E = Esurf + E int = γL(SL − SSL cos(θY ))

abrupt transition from the filament-like to the bulge morphology (type II) is observed. This transition manifests itself in simultaneous sharp changes of H̃ and L̃ (i.e., increase and decrease, respectively). Further increase in Ṽ C, causes a slow monotonic increase in H̃ and L̃ values. The extent of changes accompanying the transition from filament to bulk morphologies weakens with increasing G. For G = 4, the sharp transition is still observed, but the differences between parameters characterizing morphologies of type I and II are small. The droplet at G > 4 evolves without steeper shape changes; morphologies I and II smoothly transform from one to another. The occurrence of morphological transition of the droplet deposited on grooved surface can be found also at other contact angles, as exemplary shown in Figure 3 for G = 4 and α = 90°.

(4)

where Esurf is the liquid surface energy, Eint is the solid−liquid interfacial energy, and SL and SSL are the areas of the liquid surface and the interface. Since all simulations were made for unlimited value of the pinning angle ϕ, the obtained values have physical meaning only for geometrical models fulfilling some assumed limitations related to the groove geometries discussed further in section 3.1.

3. RESULTS WITH DISCUSSION 3.1. Morphology of the Droplet. The morphology of the droplet deposited onto a grooved surface is strongly affected by the pinning of the triple line on the edges. As a result of geometrical constraints, the increase in droplet volume can cause its elongation along the grooves. However, even for a very simple geometry, when the droplet is deposited into a deep single wedge so its contact lines are not pinned, the necking instability can be observed if the droplet surface is long enough. Thus, in the system studied, which satisfies the canthotaxis condition, the occurrence of the Rayleigh−Plateau instability involving transformation of the droplet shape from filament-like to bulge, as reported for a droplet sitting on a single groove,16 can be expected. The droplet morphologies on different numbers of wetted grooves are shown in Figure 2 presenting the dependencies of

Figure 3. Dependence of H̃ on Ṽ at different θY (G = 4, α = 90°).

However, for θY > 75°, the dependence of H̃ and L̃ versus Ṽ becomes continuous and no sudden morphological transition is observed. On the other hand, the decrease in θY to 70° causes an increase in abruptness of transition between morphologies I and II at Ṽ C. Further decrease in θY did not produce plausible results as mentioned in section 2. The dependence of Ṽ C values on the number of wetted grooves is shown in Figure 4. Irrespectively of the Young

Figure 4. Dependence of Ṽ C on G at two different values of θY (α = 90°).

Figure 2. Effect of Ṽ on (a) H̃ and (b) L̃ at different G (θY = 75°, α = 90°).

contact angle, the relationship between Ṽ C and G is linear in the range of G in which the sharp morphological transition occurs. However, the slope of the line obtained for θY = 70° is higher than for θY = 75°, which indicates that at smaller contact angle the filament morphology is stable in the wider range of Ṽ as a result of a smaller value of the free interfacial energy. The changes in linear dimensions of the droplet induced by an increase in Ṽ are accompanied by the appropriate adjustment of the pinning angle. For morphology of type I, the maximum pinning angle takes values 75°, 83°, and 105° for 1, 2, and 3 grooves, respectively. The value of ϕ = 75° obtained

reduced height of the droplet H̃ and the reduced length of the triple line L̃ on the droplet reduced volume Ṽ at the assumed values of α = 90° and θY = 75°. As seen, the evolution of droplet morphology upon increasing volume strongly depends on the number of grooves. In the range of G from 1 to 3, at small Ṽ values, the droplet takes the shape of a filament (morphology of type I). In this region, the increase in Ṽ practically does not cause any change in H̃ , whereas it involves a linear increase in L̃ . At Ṽ equal to the critical volume Ṽ C, an 7261

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Langmuir for G = 1 is the smallest pinning angle allowed by the canthotaxis condition (inequality 1). This low value of ϕ results from the fact that the initial pinning angle of the growing droplet, reached when the triple line touches the groove edge at one point, must be equal to θY. Further increase in the droplet volume expands this point to a section of wetted edge without any change in the pinning angle. The explanation of such a behavior of the pinning angle at G = 1 can be related to the fact that when the droplet touches the convex bottom edge of the groove the surface wettability increases. Namely, when the Concus−Finn inequality24 2θY < π − α

(5)

is fulfilled, the liquid spreads unlimitedly over the groove. Increase in θY or α deteriorates the wettability, but for θY < 90° and α < 180°, the effective contact angle (the slope angle25)

⎛ ⎞ ⎜ cos(θY ) ⎟ ξ = arccos⎜ α ⎟ ⎝ sin 2 ⎠

()

(6)

is still smaller than θY. For example, at θY = 75° and α = 90°, the slope angles obtained in simulations and calculated from eq 5 take the value 68°. The effect is equivalent to the force exerted along the direction parallel to the groove. For that reason, the increase in the droplet volume causes only an increase in the length of the cylindrical part of the liquid surface at the same pinning angle. For higher G values, when the droplet touches both convex and concave edges, the elongating force is smaller and pinning angles can assume higher values. Figure 5 illustrates the influence of the droplet volume on the pinning angle observed at G = 2. As shown, for type I

Figure 6. (a) The dependence of ϕmax on Ṽ at different G values (θY = 75°, α = 90°). The maximum pinning angles resulting from the inequailty (1) are marked as dashed lines. (b) β denotes the angles of extreme edges at different geometries of the solid surface.

equivalent to the fanlike folded plane of zero thickness. In other cases (models A−C), pinning angles are limited by open angles β at the extreme edges. Thus, no morphological transition can be observed on the flat surface with an isolated inhomogeneity in the form of a series of grooves (model A) because for Ṽ > Ṽ C, the canthotaxis condition is exceeded and the droplet spreads over the neighboring flat surface. In model B, the morphological transition is observed only for G = 4. In model C, the sharp morphological transition occurs at G = 1−4, but the further increase in the bulge diameter can be disrupted by spreading of the droplet over the side walls (G = 1−3). 3.2. Thermodynamics of the Droplet. Figure 7a shows the reduced free surface/interface energy Ẽ of the droplet versus its reduced volume Ṽ at different G values in the logarithmic scale. As expected, the free energy increases with droplet volume. For G = 1−4, the Ẽ = f(Ṽ ) function seems to be slightly discontinuous, but the energy difference between the filament-like and bulge morphologies decreases or even diminishes with increasing G. The slope of the curve determined for G = 1 takes values of 1 and 2/3 for morphology I and II, respectively, as one can predict for a cylinder of growing height and a sphere of growing radius. As also follows from Figure 7a, with increasing G the slope of curves decreases. In the wide range of the droplet volume, its spreading over the surface adjacent to the studied grooves is more energetically favorable then remaining in them. However, it does not happen because of the pinning effect; the droplet remains in the metastable state as long as the canthotaxis condition is preserved. Changes in the droplet morphology after reaching Ṽ C can lead to the state at which the canthotaxis condition is no longer satisfied, and then the droplet undergoes depinning from the external edges of the series of grooves, crosses them, and wets adjacent surfaces. The process cannot be withdrawn by the decrease in the droplet volume because the reverse crossing of

Figure 5. Values of the pinning angle ϕ along the pinning line of the droplet sitting on two adjacent grooves (G = 2) obtained for different droplet volumes (α = 90°).

morphology, the values of ϕ are practically identical along the whole length of the triple line. Just after Ṽ C is exceeded, the pinning line shrinks and the ϕ versus Ṽ dependence gets a curve shape with a symmetrically located maximum ϕmax. The value of ϕmax increases with increasing Ṽ . The values of maximum pinning angles ϕmax are collected in Figure 6. Their dependencies on Ṽ are monotonic and for G from 1 to 4 are also discontinuous. Since, as mentioned earlier in section 2, all simulations were performed without limitation imposed on the pinning angle ϕ, the obtained results should be addressed to the physical systems with some strong geometry limitations caused by the canthotaxis condition (inequality 1). Thus, all results can be applied without any limitation to the surface presented as model D (see Figure 6b), that is, 7262

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f(Ṽ ) dependencies are nonincreasing functions (according to the Gromov theorem27) and that the mean curvature ϑ calculated at each node mapping the free liquid surface is uniform. Nevertheless, in the vicinity of Ṽ C, some metastable states can be expected, similar to those described in ref 5. The studies of such systems require an algorithm4 different from the one applied here. The energy of the droplet deposited onto a grooved surface was also characterized by the effective contact angle θ*. This is a thermodynamic parameter describing the apparent wettability of the studied surface. Namely, it is the contact angle of the equivalent droplet, that is, the droplet of the same volume as the droplet under study, placed on the ideally smooth surface and being in thermodynamic equilibrium with the droplet studied. By thermodynamic equilibrium, we understand the same vapor pressure over the droplet surfaces. Both droplets hold the same surface tension and the mean curvature of free liquid surface. The angle θ* was calculated on the basis of the Laplace pressure in the studied droplet (eq 2) from the equation ⎛ ⎛ ⎛ 3V Δp arccos⎜ 16π γ ⎜ ⎜ ⎝ L θ* = arccos⎜2 cos⎜4π + ⎜ ⎜ 3 ⎜ ⎜ ⎝ ⎝

⎞⎞

( ) − 1⎞⎠ ⎟⎟⎟⎟

Figure 7. Dependencies of (a) ln(Ẽ ) and (b) ln(Δp̃) on ln(Ṽ ) for a droplet settled onto G triangular grooves (θY = 75°, α = 90°).

⎟

⎟⎟ ⎟⎟ ⎠⎠

(7)

The dependence of the effective contact angle θ* on Ṽ is presented in Figure 8. As shown, at small Ṽ , the equivalent

edges requires the opposite cantotaxis conditions to be overcome. The dependence of the reduced Laplace pressure inside the droplet Δp̃ versus Ṽ is more revealing (see Figure 7b). Three different parts of the curve obtained at G = 1−4 can be identified: (1) the common descending part of the slope of 1/6 which is a result of evolution of the droplet of a very small volume when its length along the grooves is smaller than the distance between pinned parts of the triple line, (2) the almost plateau part of the ln(Δp̃) = f(ln(Ṽ )) dependence related to the filament morphology, and (3) the common, descending part of the slope of 1/3 characteristic of the sphere (the bulge morphology). At G = 5, the ln(Δp̃) = f(ln(Ṽ )) dependence is smooth. The ln(Ẽ ) = f(ln(Ṽ )) and ln(Δp̃) = f(ln(Ṽ )) dependencies obtained for G = 1−4 contain some characteristic irregularities at Ṽ C, especially evident as the breaking point on the Δp̃ = f(Ṽ ) curve, which is indicative of the phase transition (in the Ehrenfest classification26). The free energies of the droplet Ẽ gathered in Figure 7a and other geometric and thermodynamic parameters of the morphology of the droplet deposited onto the grooved surface were obtained by the minimization of energy Ẽ by means of SE without any preferences to a particular morphology. The search of the minimum was performed carefully using several methods with the SE internal commands saddle and hessian_seek and with some jiggle commands pushing out the droplet from a local minimum and helping it to find a global one. Therefore, we believe that the obtained morphologies are absolute energy minimizing surfaces at the assumed number of grooves. It should be stressed here that the obtained morphologies are not equivalent to the equilibrium state of the whole system; in some cases, this equilibrium can be achieved by depinning of the droplet edge and wetting the surfaces adjacent to the initial grooves. The achievement of a global minimum for the assumed groove set is confirmed by two facts: that the Δp̃ =

Figure 8. Values of θ* of the equivalent droplet (θY = 75°, α = 90°).

droplet wets the surface even better than the flat surface of the contact angle equal to θY. However, the spreading of the filament along grooves causes an increase in energy and, as a consequence, the deterioration of wettability. The effective contact angle increases to the value of about 180° at Ṽ C. At this point, the instable droplet is characterized by the large excess of free energy. The droplet is in the labile state, and even a very weak stimulation can switch the droplet to another type of morphology. At higher Ṽ , the droplet reaches the bulge morphology; it wets the surface, however, the contact angle is higher than that of the filament. As shown in Figure 8, the higher the number of grooves, the better the wettability of the surface by the droplet of the bulge morphology.

4. CONCLUSIONS Specific properties of the droplet settled on a triangular groove or strip are a consequence of the pinning to the external edges of the inhomogeneity. This effect causes an anisotropy of the droplet shape. However, with increasing volume of the droplet, its morphology changes. This study maps the morphologies of the droplet deposited onto several grooves on the solid surface 7263

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(10) Tóth, T.; Ferraro, D.; Chiarello, E.; Pierno, M.; Mistura, G.; Bissacco, G.; Semprebon, C. Suspension of Water Droplets on Individual Pillars. Langmuir 2011, 27, 4742−4748. (11) Semprebon, C.; Forsberg, P.; Priest, C.; Brinkmann, M. Pinning and Wicking in Regular Pillar Arrays. Soft Matter 2014, 10, 5739− 5748. (12) Varagnolo, S.; Schiocchet, V.; Ferraro, D.; Pierno, M.; Mistura, G.; Sbragaglia, M.; Gupta, A.; Amati, G. Tuning Drop Motion by Chemical Patterning of Surfaces. Langmuir 2014, 30, 2401−2409. (13) Langbein, D. Canthotaxis/Wetting Barriers/Pinning Lines. In Capillary Surfaces; Höhler, G., Ed.; Springer Tracts in Modern Physics, Vol. 178; Springer: Berlin, 2002; pp 149−177. (14) Chen, Y.; He, B.; Lee, J.; Patankar, N. A. Anisotropy in the Wetting of Rough Surfaces. J. Colloid Interface Sci. 2005, 281, 458− 464. (15) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Liquid Morphologies on Structured Surfaces: From Microchannels to Microchips. Science (Washington, DC, U. S.) 1999, 283, 46−49. (16) Dokowicz, M.; Nowicki, W. Morphology of a Droplet Deposited on a Strip and Triangular Groove. Surf. Innovations 2014, 2, 228−234. (17) Klingner, A. Electrowetting-Induced Morphological Transitions of Fluid Microstructures. J. Appl. Phys. 2004, 95, 2918. (18) Lipowsky, R. Morphological Wetting Transitions at Chemically Structured Surfaces. Curr. Opin. Colloid Interface Sci. 2001, 6, 40−48. (19) Broesch, D. J.; Frechette, J. From Concave to Convex: Capillary Bridges in Slit Pore Geometry. Langmuir 2012, 28, 15548−15554. (20) Protiere, S.; Duprat, C.; Stone, H. A. Wetting on Two Parallel Fibers: Drop to Column Transitions. Soft Matter 2013, 9, 271. (21) Seemann, R.; Brinkmann, M.; Herminghaus, S.; Khare, K.; Law, B. M.; McBride, S.; Kostourou, K.; Gurevich, E.; Bommer, S.; Herrmann, C.; et al. Wetting Morphologies and Their Transitions in Grooved Substrates. J. Phys.: Condens. Matter 2011, 23, 184108. (22) Khare, K.; Herminghaus, S.; Baret, J.-C.; Law, B. M.; Brinkmann, M.; Seemann, R. Switching Liquid Morphologies on Linear Grooves. Langmuir 2007, 23, 12997−13006. (23) Brakke, K. A. Surface Evolver. Exp. Math. 1992, 1, 141−165. (24) Concus, P.; Finn, R. On the Behavior of a Capillary Surface in a Wedge. Proc. Natl. Acad. Sci. U. S. A. 1969, 63, 292−299. (25) Shuttleworth, R.; Bailey, G. L. J. The Spreading of a Liquid over a Rough Solid. Discuss. Faraday Soc. 1948, 3, 16. (26) Jaeger, G. The Ehrenfest Classification of Phase Transitions: Introduction and Evolution. Arch. Hist. Exact Sci. 1998, 53, 51−81. (27) Berthier, J.; Brakke, K. A. The Physics of Microdroplets; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2012. (28) Herminghaus, S.; Brinkmann, M.; Seemann, R. Wetting and Dewetting of Complex Surface Geometries. Annu. Rev. Mater. Res. 2008, 38, 101−121.

in the equilibrium state at different droplet volumes. However, some metastable states can be expected near the morphological transition point. Verification of this hypothesis needs further studies. The studied morphologies are of type I (filament-like) or II (bulge), if the number of grooves is not too high (G = 1−4 for θY = 75°, G = 1−6 for θY = 70°). The abrupt transition from the filament-like to the bulge morphology similar to a phase transition is observed at a certain Ṽ C whose value depends on the number of grooves. In the vicinity of the transition point Ṽ C, the morphology of the droplet is labile, and even a small disturbance can switch the droplet from morphology I to II. This process seems to be reversible. It has been found that the bulgelike droplet can undergo depinning from the external edge of the groove and wet the neighboring surface. The range of the existence of bulge morphology depends then on the geometry of the external edges and the number of grooves G. For β → 0, no depinning from the external edge is expected. If 0 < β < α + π/2, the range of bulge morphologies is narrower and depends on the number of grooves. If the grooves are surrounded by a flat surface, no bulge morphology should be observed; the droplet spreads over the flat surface, as found for a single groove.28 We believe that the phenomenon described can be potentially used for construction of microfluidal systems, in particular, microfluidal switches.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ABBREVIATIONS SE, Surface Evolver REFERENCES

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DOI: 10.1021/acs.langmuir.6b01275 Langmuir 2016, 32, 7259−7264