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Wetting Phenomena on Structured Surfaces**

By Peter Lenz*

Wetting phenomena on structured surfaces are reviewed. The interplay between the two-dimensional pattern of surface do-mains and the three-dimensional morphology of the wetting fluid is theoretically discussed. The results presented illustratethe novel morphological aspects in the wetting behavior. It is shown that these systems have potential both as models for fun-damental research and as a templates for technologically relevant microscale structures with unique geometric and topologi-cal properties.

1. Introduction

With modern experimental methods it has become possi-ble to control the process of shape formation in physical,chemical, or biological systems. This can be achieved bystructuring surfaces that act as adsorption sites for liquidssuch as water or complex fluids such as colloidal solutionsor biological cells. Then, the emerging three-dimensionalshape of these systems is crucially influenced by the two-di-mensional structure of the substrate. In this way, novelªsoftº materials of complex geometry and with prescribedphysical properties can be produced in a controlled fashion.The subject of this article is the interplay between thestructure of the surface and the morphology and physicalproperties of the corresponding systems. A special empha-sis lies on the wetting behavior of these surfaces, since theassociated phenomena are fundamental both for the studyof more complex systems and for the design of novel tech-nologies.

Several experimental methods have been developed bywhich a substrate can be laterally structured, that is, en-dowed with a pattern of spatial regions (domains) withmodified chemical or physical properties. Some importantexamples are 1) elastomer stamps,[1±3] 2) vapor depositionthrough grids,[4] 3) photolithography of amphiphilic mono-layers that contain photosensitive molecular groups,[5,6]

4) lithography with colloid monolayers,[7] 5) atomicbeams,[8] and 6) microphase separation in diblock copoly-mer films.[9] Surface patterns with micrometer-size domainscan be produced by methods 1±3, while patterns with nano-meter-size domains can be produced by methods 4±6.

If such a structured or imprinted surface is in contactwith a liquid, the corresponding interface has a position-de-

pendent free energy that reflects the underlying surfacepattern. If, for example, a thin wetting layer of water isplaced on a hydrophobic substrate with hydrophilic do-mains, the water tends to wet the hydrophilic domains butto dewet the hydrophobic matrix. This modulation of theshape of the wetting layer by the surface pattern leads towetting properties differing significantly from those ofhomogeneous substrates. First of all, for a special volumeregime droplets on surface domains exhibit contact anglesthat do not have to fulfil the classical Young equation. Thisgeneral property gives rise to a variety of effects: i) unex-pected droplet shapes; ii) novel instabilities of the droplets;and iii) new morphological transitions between differentwetting states, which might be accompanied by sponta-neous symmetry breaking. None of these aspects are pres-ent on homogeneous substrates, where the equilibriummorphology of the wetting layer is always one single drop-let with the shape of a spherical cap. Other droplet shapesare unstable and patterns of several distinct droplets areonly metastable.

2. Theoretical Description

To proceed, the vapor and liquid phases are denoted by(a) and (b), respectively, and the hydrophilic and hydro-phobic surface regions by (g) and (d), respectively. The in-terfacial region between phase (i) and phase (j) has surfacearea Aij and interfacial tension Sij. The equilibrium state ofthe wetting layer with prescribed volume V corresponds tothe global minimum of the total interfacial free energy asgiven by[10,11]

F(A,V) = SabAab + Abs(Sbs ± Sas) ± (Pb ± Pa)V (1)

where s = g or d. The pressure term (Pb ± Pa) is included tofulfil the constraint on the volume. Such an ensemble, wherethe volume of liquid is fixed, describes various experimentalsituations: non-volatile liquids, binary fluids, or liquid±vaporsystems at, or slightly off, two-phase coexistence.

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[*] Dr. P. LenzMax-Planck-Institut für Kolloid- und GrenzflächenforschungAm Mühlenberg, Haus 2D-14476 Golm (Germany)

[**] Valuable discussions with C. Bechinger, W. Fenzl, G. Gompper, S. Her-minghaus, U. Seifert, P. Swain, and especially with R. Lipowsky aregratefully acknowledged.

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The macroscopic description used here is valid for inter-mediate length scales, that is, domains and correspondingdroplets in the micrometer-size range if the wetting liquidis water. At larger scales gravity plays a role, while forsmaller scales corrections arising from line tensions and in-termolecular interactions have to be taken into account.

The free energy (Eq. 1) has a functional dependence onthe droplet morphology. On homogeneous substrates, itsstationary states correspond to droplet shapes fulfillingYoung's and Laplace's equations, that is, droplets having afixed contact angle and an (ab) interface of constant meancurvature. On patterned surfaces the stationary dropletsstill fulfil the Laplace equation. However, the free energy(Eq. 1) has an additional functional dependence on thestructure of the substrate. This can be taken into accountby considering position-dependent surface tensions

Sas ± Sbs º Sas(x) ± Sbs(x) (2)

that is, surface tensions that depend on the position x onthe surface, which then lead to a position-dependent Youngequation.[10] Here, only the limiting case shall be consid-ered where the width of the transition region between (g)and (d) is much smaller than the lateral dimension of thedomain. Then, droplets may exhibit contact angles that donot have to obey the Young equation.

The physical origin of this fundamental property can bemade clear by considering a hydrophobic substrate (d) witha single circular hydrophilic domain (g) with area Ag. Thetwo surface regions are then characterized by two contactangles yg and yd, with yg < p/2 < yd and which satisfy theusual Young relations (Eq. 3), where s = g or d.

Sabcosys = Sas ± Sbs (3)

A small amount of water placed on this substrate willtend to wet only the (g) domain by forming a spherical capwith contact angle y = yg. For sufficiently low volume onlyparts of (g) are covered, i.e., Abg < Ag. With increasing vol-ume the droplet spreads, that is, Abg grows until its contactline reaches the domain border. On a homogeneous sub-strate the droplet would keep on spreading as its volume isfurther increased. Here, it does not have enough volume tofulfil the Young equation y = yd on (d). Hence, the dropletcannot spread further but has to stay on (g). However,since its volume is increased, its contact angle has to grow,implying yg < y < yd. As the volume is further increased, ycontinues to grow until it reaches the limiting value y = yd.Then, the contact line detaches from the domain borderand parts of (d) are also wetted. Thus, depending on its vol-ume a droplet on a structured surface belongs to one ofthree droplet regimes depicted in Figure 1.

Since in regime 2 the domain is completely covered, thedroplet morphology can be influenced in a controlled wayby prescribing the contact area by choosing an appropriatedomain geometry. The structure of the substrate then stabi-

lizes even droplet morphologies that are unstable on homo-geneous substrates.

Such new droplet morphologies have been observed ex-perimentally on striped surfaces.[12] Here, in regime 2 thecontact area is a rectangle. Due to the cylindrical symmetrythe liquid forms a channel with a surface shaped as a cy-lindrical segment. On homogeneous substrates such chan-nels are unstable against displacements of their contact lineand decay into a chain of spherical caps. Generally, onstructured surfaces droplets that cover the domain com-pletely and particularly these channels are stabilizedagainst displacements of the contact line. Nevertheless,they can exhibit a new kind of instability characterized by arearrangement of the liquid within the droplet at fixed con-tact line.[12] In the case of liquid channels the homogeneousstate with a spatially constant cross-section undergoes ashape instability to a state with a single bulge when theliquid volume reaches a critical value, see Figure 2.

On a hydrophobic substrate with many hydrophilic do-mains the wetting layer can exhibit several distinctmorphologies: a) a homogeneous droplet pattern where alldroplets have the same size; b) a heterogeneous dropletpattern consisting of one large and many small droplets;and c) a film state where the wetting layer covers both thehydrophilic and the hydrophobic surface regions. Transi-tions between these states can be induced by changingeither the amount of adsorbed volume or the geometry ofthe substrate, see Figure 3.

The instability of the droplets on a single domain and thetransitions between the various morphologies of the wet-ting layer on substrates with many domains are intimatelyrelated. Both represent a new type of wetting transition be-tween different morphologies of constant mean curvature.These volume-induced transitions are characterized by arearrangement of the liquid within the system. They are ageneral phenomenon for a liquid confined to a hydrophilicregion in a hydrophobic surrounding and only occur be-cause the contact angles of the droplets do not have to fulfilthe Young equation.

The droplet patterns shown in Figure 3 exist in a wideclass of domain geometries, provided the hydrophobic re-gions are (topologically) connected. In particular they

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Fig. 1. Depending on its volume, V, a droplet belongs to regime 1, 2, or 3. Indroplet regime 1 only parts of (g) are wetted. In regime 2 the (g) domain iscompletely covered, and in regime 3 also parts of (d). Whereas in regimes 1and 3 the Young equation is fulfilled, the contact angle y is determined in re-gime 2 by the volume only and can adopt any value within the range yg < y< yd. In the limit of a very hydrophilic domain (yg = 0) on a very hydropho-bic substrate (yd = p) all droplets belong to regime 2.

should be experimentally observable on substrates with hy-drophilic circular domains or stripes. The phase diagram ofa topologically different system consisting of a hydrophilicsubstrate with many circular hydrophobic domains is domi-nated by the film state. Only for an intermediate volumerange does the wetting layer dewet the (disconnected) hy-

drophobic regions. The corresponding liquid±vapor inter-face is here given by a two-dimensional, periodic, and con-nected surface of high topological genus, see Figure 4.

Fig. 4. Morphology of the wetting layer dewetting the hydrophobic domainson a hydrophilic substrate.

3. Outlook

The understanding of the interplay between the chemicalstructure of a substrate and the morphology of its wettinglayer allows controlled manipulation of the behavior of thesystem. Thus, this opens new perspectives in the study offundamental problems of interfacial science and in the de-sign of physical systems with prescribed properties.

As an example the controversially discussed problem ofthe magnitude and the sign of the line tension can be inves-tigated here in a completely new context. On appropriatelychosen structured substrates the precise determination ofthe line tension becomes possible via the accurate measure-ments of droplet morphologies.[13]

Furthermore, due to the various control possibilities asystematic investigation of the dynamics of (de)wetting ispossible. For example, the stages of the dynamics can beexperimentally ªfreeze-framedº by, for example, solidifyingthe system. This allows a detailed comparison with theoret-ical investigations and so makes a systematic classificationof the influence of the boundary conditions on the hydro-dynamical behavior achievable. Such studies might alsoopen new approaches to related classical problems such asthe Rayleigh-Plateau instability.

On the other hand, the systems described here can beseen as a first step towards the design of soft materials withunique geometric and topological properties. The three-di-mensional morphologies of the liquid layer can be tem-porarily or permanently stabilized by, for example, poly-merization, freezing, or sol±gel reactions.[14] Themacroscopic properties of the artificial microstructuresproduced in this way will depend on their morphology. Thisallows the physical behavior of the system to be influencedby its geometry. As a next step, systems with more complexinteractions should be looked at. Indeed, on-going experi-mental investigations are correspondingly concentratingon, for example, the wetting behavior of liquid crystals,[15]

colloidal systems,[16] or the adhesion of biological cells[17] inthe presence of such special boundary conditions or of sim-ilar confining geometries.

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Fig. 2. Instability of liquid channels on striped surfaces as observed experi-mentally (a,b) and calculated theoretically (c,d). a) Low volume regime.The liquid forms channels of constant cross-section. b,c) High volume re-gime. The channels develop a single bulge. d) Theoretically calculated sideview of a channel with a bulge.

Fig. 3. Morphological phase diagram for the wetting layer on a substratewith N hydrophilic domains. It is determined by the water volume per hy-drophilic domain v and the area fraction of the hydrophilic domains X. Thenumerical values correspond to the case of four circular domains with diam-eter ag.

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Some of these systems might also become technologi-cally important. For example, the bulge instability could beused to build microreactors. Our results show how the posi-tion of the contact line of liquid channels can be controlledby the substrate structure and by the amount of adsorbedvolume. By volume-induced coalescence of the bulges ofneighboring microchannels contact via microbridges can beestablished. If the channels are filled with different reac-tants such bridges lead to a well-mixed state without anystirring but simply by volume control. More generally, simi-lar techniques of non-mechanical fluid control could beuseful in the design of so-called microfluidic devices,[18,19]

where the static and dynamic control of small amounts ofliquid is fundamental. Finally, the interaction of complexfluids with appropriately structured surfaces is believed tobe important for the fabrication of microsensors by whichthe mechanisms of molecular recognition can be studied.[20]

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[4] K. Jacobs, H. Gau, S. Schlagowski, W. Mönch, T. Pompe, A. Fery,S. Herminghaus, in Proc. 2nd European Coating Symp., Strasbourg,July 22±25, 1997.

[5] G. Möller, M. Harke, H. Motschmann, D. Prescher, Langmuir 1998,14, 4955.

[6] R. Wang, K. Hashimoto, A. Fujishima, Nature 1997, 388, 431.[7] F. Burmeister, C. Schäfle, T. Matthes, M. Böhmisch, J. Boneberg,

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