Fall and rise of small droplets on rough hydrophobic substrates
aa r X i v : . [ c ond - m a t . s o f t ] N ov epl draft Fall and rise of small droplets on rough hydrophobic substrates
M. Gross , , F. Varnik , and D. Raabe Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-Universit¨at Bochum, Stiepeler Strasse129, 44801 Bochum, Germany Max-Planck Institut f¨ur Eisenforschung, Max-Planck Str. 1, 40237 D¨usseldorf, Germany
PACS – Wetting
PACS – Interface structure and roughness
PACS – Droplets and Bubbles
Abstract. - Liquid droplets on patterned hydrophobic substrates are typically observed either inthe Wenzel or the Cassie state. Here we show that for droplets of comparable size to the roughnessscale an additional local equilibrium state exists, where the droplet is immersed in the texture,but not yet contacts the bottom grooves. Upon evaporation, a droplet in this state enters theCassie state, opening the possibility of a qualitatively new self-cleaning mechanism. The effect isof generic character and is expected to occur in any hydrophobic capillary wetting situation wherea spherical liquid reservoir is involved.
Introduction. –
The fact that roughness at the mi-crometer level can drastically increase the water-repellantproperties of a hydrophobic substance has been known fora long time [1, 2] and is a frequent phenomenon in na-ture – the most prominent example being the Lotus leaf.Technical advancement in surface fabrication and novel in-dustrial applications, such as self-cleaning materials, hasbrought the phenomenon of superhydrophobicity again inthe focus of research during the last years (see [3, 4] forrecent reviews).Until now, mostly droplets that are much larger thanthe typical roughness scale of the surface have been inves-tigated. In that case, the droplet either appears in theWenzel state, where it completely wets the substrate [1],or in the Cassie state, i.e. on top of the roughness struc-tures [2]. The characteristic low-adhesion, water-repellantproperties of superhydrophobic surfaces are associatedwith droplets in the Cassie state. Contrarily, the Wen-zel state leads to sticky, highly pinned droplets [5].The fact that usually both the Cassie and the Wenzelstate can be observed on the same substrate implies thatboth are separated by a free energy barrier, which canbe overcome by external forces or kinetic energy [5–7].It is known that the transition from the Cassie to theWenzel state proceeds through the nucleation of contactbetween the liquid and the substrate at the grooves [8],initiated e.g. via the increase of internal droplet pressureduring evaporation [9–11]. It is unclear, however, how deep a droplet can really sink into the texture and howthis process is modified for droplets of similar size as theroughness scale.The validity and applicability of the classical two-statepicture of Wenzel and Cassie has been questioned sev-eral times [12–14]. In particular, for the case of dropletsthat are of comparable size to the surface roughness –and which therefore directly feel the influence of the sur-face geometry – it can not be expected to hold a priori.Due to the fact that such sub-micron-sized droplets havea very short lifetime, only few experimental hints on theirbehaviour have been provided so far [11, 15, 16]. Recentstudies, focusing explicitly on this situation, confirmed atleast the existence of the two equilibrium states [16, 17].Nevertheless, droplets of this length scale are not onlyimportant for a better understanding of the wetting prop-erties of microscale systems [18], but also in many indus-trial applications, as, for example, in the production ofefficient self-cleaning surfaces [19], robust metal coatings,or in plasma spraying techniques. Moreover, since thesedroplets naturally occur in any condensation or evapora-tion process [20, 21], their phenomenology is also funda-mental for a better understanding of the water-repellentproperties of many plant-leaves or insect eyes and legs.In this letter, we investigate the behaviour of dropletsin three dimensions both analytically and via numericallattice Boltzmann (LB) computer simulations. We showthat, for a droplet of comparable size to the surface rough-p-1. Gross, F. Varnik, D. Raabe(a) (b)
Fig. 1: A small droplet on a hydrophobic pillar array. (a)shows the impaled state as a typical situation in the presentsimulations. In the analytical model (b), the droplet is assumedto be pinned at the edges of the pillars, i.e. it has a fixed baseradius a . For a given droplet size, the only way to minimize theoverall free energy is thus through a change in the penetrationdepth p . This also fixes the apparent contact angle θ . ness, besides the possible Wenzel and Cassie states, thereexists a further generic state (hereafter called impaledstate ), characterized by almost complete immersion of thedroplet into the texture, yet not touching the bottom ofthe grooves. To the best of our knowledge, such a statehas not been reported before. It is important to realizethat this state is different from the “partially impaled”conformations of the Cassie state, where the liquid-vaporinterface below the macroscopic contact line is curved,but the droplet essentially remains on top of the asper-ities [10, 22]. This statement is supported by the fact thatwe indeed find a coexistence regime for the new impaledstate and the (partially impaled) Cassie state. By virtueof this new state, a droplet can in fact be saved from pen-etrating into the texture completely and going over to theWenzel state. Instead, it can reenter a Cassie state upon,e.g., evaporation. We demonstrate the generic characterof this finding, which has important implications for thewetting behaviour of droplets and for the effectiveness ofself-cleaning surfaces. Model. –
The roughness of a surface is modeled bya regular array of cuboidal pillars with width b , height h and spacing d (Fig. 1). The intrinsic hydrophobicity of theflat parts of the surface is described by the Young contactangle θ Y . In this work we only focus on the case θ Y > ◦ ,which also is a necessary condition for the existence of aCassie state. Gravity will be neglected throughout, as weonly consider droplets that are smaller than the capillarylength ( ∼ a = b/ d = R sin θ (with R being the ra-dius of the cap and θ the apparent contact angle). Theimpaled part is approximated as a cylindrical liquid col-umn with radius a and height p (penetration depth), sur-rounding the central pillar. The macroscopic contact lineof the droplet is assumed to remain pinned at the edgesof the outer pillars. Note that in this model the Wenzel state would correspond to p = h and the Cassie state to p = 0. Therefore, the model also neglects a possible finitepenetration depth of a droplet in a “partially impaled” Cassie state. As further simulations [23] have shown, thesymmetric droplet configuration considered here is stableagainst moderate perturbations.Since the mechanism of the Wenzel transition has beendiscussed in detail in previous publications, we will ingorethe Wenzel state completely, and, for the rest of this work,assume the pillars to be so tall that no contact betweenthe liquid and the bottom of the grooves is possible.The total volume of the droplet shall be fixed, V tot =const = V sph ( θ ) + V cyl ( p ), with V sph = πa (2 − θ + cos θ ) / sin θ the volume of the cap and V cyl = ( πa − b ) p the volume of the penetrating cylin-der. In the following, instead of the droplet volume V tot ,we will usually refer to the effective droplet radius R eff that corresponds to a spherical droplet of the same vol-ume (4 π/ R = V tot ). We consider p as the free variableand determine the dependence of θ on p via the fixed vol-ume condition.Since the volume of the drop (and the temperature)is constant, only surface energy contributions play a rolefor a change in the total free energy. The free energy f ( p ) of the model droplet, neglecting gravity and termsassociated with the Wenzel transition, and normalizingsuch that f (0) = 0, is then given by f ( p ) = σ LV h S sph ( p ) − S sph (0) + S cyl,LV ( p ) − bp cos θ Y i . (1)Here, S sph = 2 πa (1 − cos θ ) / sin θ is the surface area ofthe spherical cap S cyl,LV = (2 πa − b ) p is the lateral liquid-vapour surface area of the cylinder and − σ LV bp cos θ Y isthe (positive, since θ Y > ◦ ) free energy associated withthe wetting of the (eight) side walls of the pillars. Analytical results. –
Figures 2a,b show the depen-dence of the free-energy on the penetration depth p forvarying Young contact angles and droplet sizes. Several in-teresting observations can be made: Firstly, as also foundin the case of droplets large compared to the roughnessscale [5, 10, 11, 22, 24], the stability of the Cassie state, de-termined by the slope of f at p = 0, depends not only onthe contact angle but also on the size of the droplet.The novel feature is the appearance of a local mini-mum of the free energy at large penetration depths, ex-isting in addition to the possible minimum associatedwith the Cassie (and Wenzel) state. From the conditiond f / d p = 0, which is easily evaluated with the help of thefixed volume constraint, a necessary condition for the ex-istence of a minimum of the free energy arises, namely, θ Y < arccos( − + b a ), with a = b/ d being the baseradius of the spherical cap. Interestingly, this conditiondoes not depend on the droplet size.The origin of this new state can be understood by imag-ining the pillars to represent a (partly open) hydrophobiccapillary tube, wetted by a small droplet that is placedat its entry. In this situation, the equilibrium state ofthe droplet is a consequence of the balance between theLaplace pressure within the spherical cap (pushing thep-2all and rise of small droplets on rough hydrophobic substrates(a) - - - p (cid:144) a F r eeene r g y Θ Y = Θ Y = Θ Y = Θ Y = Θ Y = (b) - - - p (cid:144) a F r eeene r g y R eff = R eff = R eff = R eff = (c) - - - p (cid:144) a F r eeene r g y Apparent contact angle Θ H ° L ff sph f cap Fig. 2: Predictions of the analytical model. (a) Dependence of the free energy f/ πa σ LV on the penetration depth p and theYoung contact angle θ Y for a droplet of fixed size R eff = a . (The inset shows a magnification of the curve for θ Y = 108 ◦ .)(b) Dependence of the free energy f/ πa σ LV on the penetration depth p and the droplet size R eff for a fixed contact angleof θ Y = 100 ◦ . (c) Contributions to the free energy f/ πa σ LV for R eff = a and θ Y = 100 ◦ . f sph is the surface free energy ofthe liquid-vapour interface of the spherical cap, f cap is the free energy due to the wetting of the “capillary” constituted by thepillars and f = f sph + f cap is the total free energy. The insets sketch the droplet configuration for different p according to theanalytical model. All curves in (a-c) are given for b/d = 1 and plotted up to a value of p where no further volume is left in thespherical cap. droplet into the capillary) and an opposing capillary forcedue to the hydrophobicity of the substrate.To illustrate this idea, we split the free energy [Eq. (1)]into the contributions of the spherical cap and the remain-ing “capillary” part. As shown in (Fig. 2c), an increase ofthe droplet penetration p leads to a linear increase of cap-illary free energy, while the free energy associated with thespherical cap decreases in a non-linear fashion. As a re-sult, the total free energy f may exhibit a local minimum.This simple reasoning suggests that the intermediate min-imum constitutes a generic equilibrium state of a droplet,occurring in any situation of filling hydrophobic capillariesby a spherical liquid reservoir. Indeed, further simulationsusing various surface geometries (e.g. omitting the centralpillar) clearly underline this assertion [23].These results also show under which conditions we areallowed not to consider the Wenzel state in the first place:A transition to this state can be inhibited, if the pillarheight h is larger than the penetration depth p of a dropletat the local minimum, plus a small correction of the orderof d /R eff [3] that accounts for the curvature of the lowerdroplet interface.Figure 3 presents a morphological phase diagram dis-playing the theoretically expected regions of existence forthe Cassie and the impaled state. The Cassie state is(meta-)stable for the set of points ( θ Y , R eff ) that fulfilld f / d p | p =0 >
0, while the phase boundary for the im-paled state is determined from d f / d p = 0 combined withd f / d p >
0. Note that below a certain droplet size, thelocal minimum of the free energy shifts continuously to p = 0 (Fig. 2b), hence the impaled state now effectivelyappears as a Cassie state and the phase boundaries for theimpaled and Cassie state are identical. In that case, theCassie state becomes the only possible state (disregardingthe Wenzel state).Interestingly, the stability region of the Cassie stateshows a characteristic shape, which is also largely inde-
95 100 105 110 1150.60.81.01.21.4 Contact angle Θ Y H ° L D r op l e t s i z e R e ff (cid:144) a çæ un s t ab l e i m pa l ed m e t a s t ab l e i m pa l edun s t ab l e C a ss i e stable Cassie m e t a s t ab l e C a ss i e s t a b l e i m p a l e d (cid:144) b = eva po r a t i on Fig. 3:
Phase diagram:
Regions of stability for the Cassie andthe impaled state (intermediate minimum) as predicted by theanalytical model. Focusing on the case of d/b = 1 (thick lines),the impaled state is expected to exist in the complete shadedregion, while the Cassie state is predicted to be (meta-)stableright to the dashed curve. An unstable impaled state (whiteregion) automatically implies an absolutely stable Cassie state.Similarly, a unstable Cassie state goes along with an absolutelystable impaled state. In the lighter shaded area, a coexistenceof metastable Cassie and impaled states is predicted. Thindotted and dot-dashed curves give the theoretical predictionsfor other values of d/b . The arrow illustrates the path of aquasi-statically evaporating droplet as it enters the region ofan unstable Cassie state ( ◦ ), and finally goes over to a stableCassie state again ( • ). pendent of the surface geometry: There exists a certaindroplet size where the Cassie state is unstable for a max-imal range in contact angle, and both towards larger andsmaller radii the stability region increases.Noting that in the phase diagram a quasi-statically Here, quasi-static refers to a sufficiently slow evaporation, suchthat the droplet assumes its optimum shape at any instant in time. p-3. Gross, F. Varnik, D. Raabeevaporating droplet would move on a vertical line fromlarge towards small R eff , we infer the existence of a reentrant transition : A droplet, initially existing in a(meta-)stable Cassie state, can (following the arrow inFig. 3) become unstable upon a reduction of its size andthereupon adopt an impaled state ( ◦ ). However, due tothe fact that the position of the local free energy minimumshifts towards smaller p with decreasing volume (Fig. 2b),further evaporation will always result in the droplet tore-appear in a stable Cassie state ( • ). Simulation results. –
We now compare the predic-tions of the analytical model to computer simulations. Incontrast to standard single phase LB models [25–27], weemploy here a free energy based two-phase (liquid-vapor)LB approach [28, 29]. Details of the algorithm can befound in [30, 31]. The relaxation time is fixed to τ = 0 . T = 0 . ρ L ≈ . ρ V ≈ .
9, respectively. Note that this rather smalldensity ratio (a well-known limitation of the present LBmodel) is not expected to adversely affect our results, sincewe are only concerned with the quasi-static (thermal equi-librium) behavior of a droplet. For the interface parameter κ we use a value of 0 . L x × L y × L z = 64 × ×
64 lattice nodes, but is enlargedappropriately for the smaller droplets. The substrate atthe top ( z = L z ) of the simulation box is flat. At the bot-tom ( z = 0), it is decorated with an array of equidistantcuboidal pillars. Typically, we will use b = 12, d = 12 and h around 30 lattice units. Periodic boundary conditionsare applied along the x and y directions.The relation between LB and physical units [30] (as-suming the simulated liquid is some viscous silicon oil)shows that the capillary time in our simulations is t c =8 · LB time steps ≈ × − s and our droplets wouldbe of micron scale. However, since it can be argued thatthe actual value of the physical viscosity is not importantin the present case, it can be used to tune the unit oflength, allowing the simulation results to be applied to abroad range of length scales, as long as thermal fluctua-tions and gravity can be disregarded.We first of all demonstrate the metastability of the dif-ferent wetting states, thereby establishing also the exis-tence of the new impaled equilibrium state. Indeed, as pre-dicted by the corresponding theoretical free energy curve(Fig. 4a), the equilibrium position of a droplet dependson where it is placed at the beginning: A droplet de-posited close to the top of the pillar array moves further tothe top (Fig. 4b), while a droplet that initially penetratesdeeper into the grooves becomes trapped in the interme-diate minimum of the free energy (Figs. 4c,d) . This fig-ure also nicely shows that the impaled state reported here Note that the free energy curves are just approximate descrip-tions of the real droplet behavior and, for example, do not predictthe residual penetration depth of a droplet in the Cassie state. (Figs. 4c,d) is indeed different from a partially impaledCassie state (Fig. 4b).Figure 5 shows a simulation of the reentrant transition,achieved through a quasi-static evaporation process. Inthe beginning (Fig. 5a), the droplet resides in a (“partiallyimpaled”) Cassie state and only slightly increases its pen-etration depth (thus confirming [10, 22]). However, afterreaching a critical size, the droplet suddenly penetratesinto the pillar grooves and goes over to the intermediateminimum of the free energy (Fig. 5b). During the impaledphase (Figs. 5b,c) the droplet gradually climbs up the pil-lars again, still residing in the local minimum. Note thatits penetration depth is in nice agreement with the ana-lytical predictions (Fig. 5e). The final transition from theimpaled to the Cassie state (Fig. 5c,d) is, in contrast tothe predictions of the analytical model (Fig. 2), not con-tinuous, but happens with the droplet depinning of fromthe outer pillars.According to the common understanding of self-cleaning, impalement is considered unfavorable and thedroplet cleans the surface by rolling over the top of the tex-ture. Interestingly, the existence of a reentrant transitionsuggests the possibility of a qualitatively new self-cleaningmechanism, since the droplet not only touches the top ofthe substrate, but also its inner parts.The existence of a reentrant transition can also explainsome recent experimental observations, that found thatsmall evaporating droplets indeed tend to remain close tothe top of the substrate [9, 11] and not get trapped insideof the texture.In Figure 6, the stability regions of the different dropletstates predicted by the analytical model are investigatedmore closely. Due to limited computational power, only apart of the full phase diagram (Fig. 3) is covered in thepresent case. The Wenzel state is also again not consideredexplicitly. Hence, for small contact angles and dropletsizes, the impaled state is expected to be the only stablestate (region I). Conversely, for large contact angles, theCassie state should be the only stable state (region III).Between these two extremes, the model predicts a regionwhere both a (meta-)stable Cassie and impaled state exist(region II).As shown in Figure 6, despite its approximate nature,the overall droplet behaviour is described correctly by theanalytical model and the three different regions are indeedrecovered in the LB simulations. The deviations betweenthe simulated results and the theoretical phase diagramare not surprising, once the simplifications of the ana-lytical model are taken into account. In particular theassumption of pinning at the inner pillar edges will be vi-olated for larger droplets that are close to the top of thetexture. They are observed to spread laterally into thegrooves (increase their base radius) in order to fulfill theYoung condition (cf. Figs. 4b and 5a).The boundary for the existence of the intermediate min-imum of the free energy is found to be independent of thedroplet size, as predicted by the analytical model. How-p-4all and rise of small droplets on rough hydrophobic substrates(a) - p H LB units L F r eeene r g y b c d (b) (c) (d) Fig. 4: Droplet behaviour for different initial configurations. We take a fixed droplet size of R eff = 1 . a , a contact angle of θ Y = 103 ◦ and b = d = 12 LB units. (a) shows the theoretical free energy f/ πa σ LV in dependence of the penetration depth p . In (b-d), the results of the LB simulations are displayed. The labeled dots in (a) mark the initial penetration depth of thedroplet in the cases (b-d). The arrows indicate the predicted time evolution. The (partially impaled) Cassie state is observedin (b), while new impaled state is found in (c,d). (a) (b) (c) (d) (e) t H LB units L p H L B un i t s L a b c dCassie stateimpaled state theory Fig. 5: Reentrant transition through evaporation of a droplet. In the simulation, the droplet is initially (t=0) prepared in a(meta-)stable (partially impaled) Cassie state (a). Evaporation is then switched on and proceeds by reducing, at a sufficientlylow rate satisfying quasi-static equilibrium, the mass of the vapor phase across a xy − plane close to the top of the simulationbox. In the course of the evaporation process, the Cassie state becomes unstable and adopts an impaled state (b), from where itgradually climbs up the pillars (c) until it reaches again a Cassie state (d). (e) shows the penetration depth of the lower dropletinterface as it is observed in the simulation ( • ) and predicted by the analytical model (—). Simulation parameters: contactangle θ Y = 103 ◦ , initial droplet size R eff = 1 . a , b = d = 12 LB units, evaporation rate 5 × − /LB time steps. All lengths aregiven in units of the LB grid spacing. ever, this line lies at θ Y ≈ . ◦ , instead of the theoreti-cally expected value of θ Y ≃ . ◦ . This deviation arisesfrom the approximation of the impaled part of the dropletas a cylinder, which actually overestimates the contribu-tion of this part to the total droplet volume. In the caseof much larger droplets, contacting many pillars, the ar-guments given in our work suggest that an impaled statecan always be realized provided that a contact betweenthe droplet and the bottom substrate is inhibited.Based on the Wenzel-Cassie model it can be shown [32]that the free energy of a droplet is smaller in the Cassiestate if r > ( φ cos θ Y − (1 − φ )) / cos θ Y . Here, φ is thepillar density (ratio of area covered by pillars and totalprojected area), r is the surface roughness (ratio betweenentire surface area that can be wetted and its horizontalprojection). For the present setup we have r = 3 . φ = 0 .
25, leading to a value of θ Y ≃ ◦ below which theCassie state would be unstable, regardless of the dropletsize. As can be seem from Fig. 6, this estimate is clearlytoo crude for the small droplets considered here. Besidesthat, it also neglects possible free energy barriers betweenthe Cassie and Wenzel state, which, as our work demon-strates, are crucial to the droplet phenomenology. Summary. –
In conclusion, via analytical calcula-tions and numerical simulations, we have uncovered theexistence of an additional local equilibrium state, wherethe droplet partially wets the inside of the grooves, yet isnot touching the base of the substrate. This finding qual-itatively modifies the two-state paradigm of Wenzel andCassie states in the case of droplets of comparable size tothe surface roughness. Interestingly, droplets in this newstate appear to have Wenzel-like properties (e.g. small ap-parent contact angles) – but on the other hand possess theinherent capability to reenter the Cassie state again. Wehave demonstrated that our results are largely indepen-dent of the particular surface geometry and are expectedto hold whenever a hydrophobic capillary is wetted by asmall droplet.The insights presented here can serve as a valuable guid-ance for the fabrication of surfaces with specific wettingproperties. Since we have explicitely shown that there isa maximal depth that a droplet can penetrate into thesubstrate, the optimal geometry of a surface can be easilyassessed, depending whether complete (Wenzel) or incom-plete wetting (Cassie) is desired. Furthermore, our resultsare expected to contribute to a better understanding ofhow many surfaces occurring in nature can remain so per-p-5. Gross, F. Varnik, D. Raabe èèèèèèè ôôôôôôô
98 100 102 104 106 108 110 1120.91.01.11.21.3 Contact angle Θ Y H ° L D r op l e t s i z e R e ff (cid:144) a I II III unstable Cassie H meta - L stable Cassie H m e t a - L s t ab l e i m pa l ed un s t ab l e i m ap l ed Fig. 6: Stable droplet configurations in dependence of thedroplet size and contact angle. In the analytical model, theCassie state is expected to be unstable in region I (left to thedashed line), and (meta-)stable in regions II and III (right tothe dashed line). The intermediate minimum of the free energyis predicted to exist in regions I and II (i.e. left to the solid line).Symbols ( • , H ) depict the phase boundaries as they result fromour numerical simulations. There, the Cassie state is found tobe unstable left to the • . Similarly, the impaled state is foundto exist left to the H . Although deviating by a few degrees,both the numerical simulations and the approximate analyti-cal model predict that the condition for the existence of theimpaled state is independent of the droplet size. fectly clean and dry.It is important to stress the scale invariance of our re-sults (see Eq. (1)). This may, firstly, significantly widenthe range of possible applications, and secondly, consider-ably simplify experimental verification of our predictions.One could e.g. study sub-millimetric drops on pillars ofcomparable size, thereby avoiding problems such as prepa-ration and fast evaporation of micron-sized droplets. ∗ ∗ ∗ We thank David Qu´er´e who raised our interest in thistopic and Alexandre Dupuis for providing us a versionof his LB code. Financial supports by the DeutscheForschungsgemeinschaft (DFG) under the grant numbersVa205/3-2 and Va205/3-3 (within the Priority ProgramNano- & Microfluidics SPP1164) as well as from the in-dustrial sponsors of the ICAMS, the state of North Rhine-Westphalia and European Union are gratefully acknowl-edged.
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