Slippy and sticky microtextured solids

The aim of this paper is to describe the possibility of achieving super-hydrophobic materials by tailoring their surface topography. Water droplets easily slip or roll down on such surfaces. However, it is found that microtextures on a solid can generate sticky surfaces as well, and the conditions for avoiding such an effect are discussed. 1. Adhesive properties of drops When a drop is deposited onto a solid surface, it develops a contact with the solid. The expansion of this contact can be deduced from the volume of the droplet and the so-called contact angle: the drop meets the surface with an angle fixed by the nature of the three phases coexisting at that place. Since a surface tension can be associated with each interface (we denote the surface tension between phases I and J by γIJ), the balance of these forces can be written as (as proposed by Young in 1805) cos θ = γSV − γSL γLV (1) where the letters S, L, V designate the solid, liquid and vapour. Since a small drop is a spherical cap of radius R, the solid/liquid interface is just a disc of radius R sin θ . For common values of θ , a drop thus develops with its substrate a contact of about its own size. The concept of equilibrium contact angle does not allow us to understand the ability of drops to stick to their substrates (a very common behaviour). On vertical window panes, for example, we can all see millimetric rain droplets stuck in spite of gravity. This phenomenon arises from fluctuations of the static contact angle: the observed value of the contact angle is not unique as expected from equation (1) but spans a range of a typical amplitude of 10◦–50◦. The maximum and minimum static values of the angle are respectively called the advancing and receding contact angles, and the difference θa−θr between them the contact angle hysteresis (CAH). For a drop at rest on a tilted plane, the contact angle is larger at the front and than at the rear; this generates a force opposing the weight of the drop (and able, if the drop is small enough, to balance it): the liquid is pinned [1]. CAH is due to heterogeneities (in topography and chemical composition) which are always present at a solid surface, and induce fluctuations of the quantities γSV and γSL [2]. In this short review, we first discuss how the CAH generates an adhesion force; then we describe how the contact angle and the CAH can be controlled by tailoring the surface topography of the solid substrate. The derivation of the adhesion force acting on a drop is quite subtle, and must generally be done numerically, as shown in a very comprehensive way by Dussan and Chow [1]. However, there is one special case where the adhesion force can be calculated exactly, which is the case for a drop inside a capillary tube. Then the (maximum) angle at the front is θa, while the (minimum) angle at the rear is θr, which generates a maximum sticking force of 2πbγLV(cos θr − cos θa) (where b is the tube radius). We easily deduce that the maximum length L for a stuck drop is 2(cos θr − cos θa)κ−2/b, introducing the capillary length κ−1 (κ−1 = (γLV/ρg), with ρ the liquid density). L can be larger than 10 cm for a tube radius of the order of 100 μm. For a drop on a (tilted) plate, calculation of the maximum drop size is complicated by the variation of the angle all along the contact line. A convenient approximation consists of dividing the drop into two halves, and assuming that each half joins the solid with an angle of (at least) θr = θ̄ − θ/2 at the rear, and (at most) θa = θ̄ + θ/2 at the front (where θ̄ is the mean contact angle) [3]. To a first approximation, the sticking capillary force can be written as πbγLV(cos θr −cos θa), where b is the mean radius of the solid/liquid contact (quasi-circular for θ θ̄ ). This force is bounded, which means that for a given tilt angle there is a threshold in size above which a drop starts moving. Quite commonly, this size is of the order of κ−1, i.e. a few millimetres for water. Yet this threshold can be considerably lowered by decreasing either the CAH (responsible for the sticking) or the solid/liquid contact b— which of course implies an increase of the contact angle. We discuss here how a microtexture at a surface reinforces the 0957-4484/03/101109+04$30.00 © 2003 IOP Publishing Ltd Printed in the UK 1109