successfully explain the shape of nanodroplets 共Checco et al., 2003兲.
To manufacture surfaces that are better characterized, model surfaces can be produced using photolithographic techniques. For example, the substrates used byMouli- net, Guthmann, and Rolley 共2002, 2004兲, are made of random patches of chromium on glass; cf. Fig9. These surfaces are, however, not perfectly flat since the chro- mium defects deposited on the glass have a thickness of the order of a few hundred angstroms. However, their thickness is much smaller than their lateral size 共a few micrometers兲, so that the effects of the chemical hetero- geneity should largely dominate those due to the rough- ness. The use of such model substrates have permitted a precise characterization of the contact line roughness, as discussed below.
Independent of the origin and the length scale of the chemical heterogeneities, one anticipates that in simple cases the properties of the substrate can be described by a few parameters, namely, the mean value of the spread- ing coefficient S¯, the amplitude of the disorder ⌬S
⬅
冑
共S−S¯兲2, and a correlation length
giving the charac- teristic length scale of the disorder. If the substrate is made of a surface with spreading coefficientS0which is covered by well-defined defects with spreading coeffi- cientSd, the natural parameters are the defect sizedand the defect force 共Sd−S0兲d, as discussed in the next sec- tion. The important point here is that, once the local spreading coefficientS共x,y兲is known for all positions on the surface, the local force balance on the contact line can easily be written down as a local version of Young’s law, provided of course the surface is flat. From this point of view, the second case of rough substrates is much more difficult to handle. A phenomenological at- tempt to understand how roughness affects wetting is contained in Wenzel’s equation 共44兲, discussed in Sec.II.D.2. This relation for rough surfaces is similar to the Cassie-Baxter relation 共13兲 for chemically heteroge- neous substrates: it provides a prediction for the equilib- rium state, which is usually out of reach, but does not tell us anything about the hysteresis.
In order to understand how the roughness can lead to hysteresis, we consider the simplest geometry: a surface z共y兲 with grooves parallel to the contact line 共assumed along Oy兲. Application of Young’s law leads to a local contact angle
eqbetween the liquid-vapor interface and the substrate. However, on average, the contact angle
between the liquid-vapor interface and the average sur- face is
=
eq−dz/dx, wheredz/dx is the local slope of the interface. This equation shows that the substrate is more or less wettable depending on the sign ofdz/dx.It was shown byCox共1983兲that, to first order in the local slope, the roughness can be described in terms of a local spreading coefficient. However, for a large ampli- tude of the roughness, the situation is more complicated and can lead to nonintuitive behavior of the contact angle. For instance, it has been observed that the contact angle hysteresis
a−
r can be a nonmonotonic function of the roughness共Ramoset al., 2003兲; this was explainedby the existence of air bubbles trapped between the liq- uid and the substrate. The encapsulation of air therefore provides us with a completely different way of manipu- lating the contact angle. It was shown that, by a clever tuning of the surfaces, superhydrophobic surfaces with contact angles close to 180° for water can be created by trapping a significant amount of air between the liquid and the substrateBicoet al.共2001兲.
In what follows, we discuss chemical and physical het- erogeneities in the same manner. Since it is the simplest case to understand theoretically, we restrict the discus- sion to the situation in which the inhomogeneous wet- ting properties of the substrate can be described by S共x,y兲 or equivalently by a local equilibrium contact angle
locdefined byS共x,y兲=␥
共cos
loc− 1兲.We focus on macroscopic or mesoscopic defects 共d
⬎10 nm兲. This is necessary if we want to describe the disorder of the substrate in terms of a local spreading coefficient. This restriction also enables us to neglect the thermal noise which is much smaller than the
“quenched” noise due to the disorder. This is easily checked in experiments on patterned substrates, where the motion of the contact line is deterministic: when re- peating experiments on the same substrate, the line al- ways remains pinned on the same defects and therefore in the same configurationPrevostet al.共2002兲.
B. The single defect case
We first discuss the simplest situation of a single de- fect which permits to understand how hysteresis arises from the interplay of the defect pinning force and the contact line elasticity. A natural extension is the case of sufficiently dilute defects.
1. Pinning on a single defect
The pinning on a single defect has been discussed by Joanny and de Gennes 共1984a,1984b兲. Consider a con- tact line whose average direction is along thex axis; cf.
Fig.45. Its distortion
共x兲with respect to its asymptotic position y⬁ is due to a defect of characteristic size d located atx= 0, y= 0. Because of the defect, the contact line is submitted to a total extra capillary force:Fd=
冕
关S„x,
共x兲+y⬁…−S0兴dx, 共101兲where S0 is the spreading coefficient of the bare sub- strate, and
eqdenotes the corresponding contact angle.x y
η(x) y∞
yM
FIG. 45. A contact line pinned on a single defect.
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Rev. Mod. Phys., Vol. 81, No. 2, April–June 2009
The integral is dominated by the central regionx⬃d:Fd depends mainly on the position yM of the line in this region, henceyMhas to be determined as a function of y⬁. Note that the maximum value ofFdis on the order of Fmax=d共Sd−S0兲, whereSdis the spreading coefficient on the defect.
The deformation of the meniscus induces an elastic restoring forcefel共x兲per unit length, where to first order in deformation,
fel共x兲=
␥
sin2
eq 冕
共x
−共xx⬘ ⬘
兲兲2dx⬘
共102兲 共we use a lower casef for a force per unit length andF for an actual force兲. This expression underlines the non- local character of the contact line elasticity, which results in the distortion of the whole meniscus. Of course, a small cutoff length scale共⬃d兲is necessary to prevent the singularity that occurs forx⬘
→x. The deformation of the contact line extends up to a macroscopic cutoff lengthL, which is usually taken to be the capillary length ᐉc for large enough drops.It then follows that at equilibrium the response
共x兲 to the localized forceFdis
共x兲= Fd␥
sin2
eqln共L/兩x兩兲. 共103兲 The logarithmic shape of the meniscus was confirmed experimentally; see Fig.46共Nadkarni and Garoff, 1992兲. The maximum deformation
M=yM−y⬁ is reached for x⬃d so that
M=
共d兲. Consequently, the contact line behaves as a spring: Fd=K
M, with a spring constant 共the stiffness of the contact line兲K⬅␥
sin2
eq/ ln共L/d兲.We are now in a position to determineyMas a func- tion of the position y⬁ far from the defect, given the pinning force Fd共yM兲 of the defect, which is determined by its shape. The force balance
K共yM−y⬁兲=Fd共yM兲 共104兲 is best understood by considering the graph shown in Fig. 47. When the defect is very smooth, the equation has only a single root and the shape of the contact line is the same, not depending on whether the contact line is advancing or receding. Such defects are called “weak” in the language of random fields. For patterned substrates, S共x,y兲 is discontinuous and the defects are strong
共“mesa” defects according to de Gennes兲. In this case, Eq. 共104兲 has three roots, and the system displays con- tact angle hysteresis. In an advancing experiment, the line is initially at y⬁⬍0 and y⬁ increases. The position yMof the center of the line is close toy⬁untily⬁reaches a positionyawhere the center of the line jumps forward over the defect. In a receding experiment, a backwards jump occurs for a position yr⬍ya. Note that the maxi- mum distortionyrmeasured in a receding experiment on a wettable defect共Sd= 0兲is
yr/d=1 − cos
eq
sin2
eqln共L/d兲. 共105兲
It follows that the maximum distortion is always at most of the order of the defect size.
The energyWadissipated in an advancing jump of the contact line is shown as the hatched area in Fig. 47; a similar construction can be made for the receding jump which dissipatesWr. For a strong defect, one finds
Wa⬃Wr⬃ 关共Sd−S0兲d兴2/
␥
sin2
eq. 共106兲2. Dilute defects
Consider now a surface with a small but finite density n of defects. Strictly speaking, defects can only be con- sidered as independent if the distance 1 /ndbetween two defects along the line is larger than the capillary length ᐉcover which the contact line is distorted by an isolated defect. This is not a stringent condition, however, the result that we derive below for the amplitude of the hys- teresis agrees empirically with experiment up to cover- age Xof the order of 0.03共Ramoset al., 2003兲.
Far from the contact line, the liquid-vapor interface is flat and shows an apparent contact angle, which is
afor a line advancing at a vanishingly small velocityU, so the dissipated power per unit length is given by Eq.共15兲. For Uⲏ0, this energy is dissipated in the pinning and depin- FIG. 46. Receding meniscus pinned on a defect. The experi-mental shape of the contact line 共dots兲 is well fitted by Eq.
共103兲withL= 2.4 mm. FromNadkarni and Garoff, 1992.
FIG. 47. Distortion of the contact line at equilibrium for a strong defect. Top: For a giveny⬁there may be three equilib- rium positions of the contact line, among which one 共y2兲 is unstable. Middle: For an advancing line far enough from the defect,yM⯝y⬁. Bottom: Wheny⬁=ya,yMjumps forward; the energy dissipated in the jump is the hatched area.
786 Bonnet al.: Wetting and spreading
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