Contact Angle Hysteresis at the Nanometer Scale

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Contact Angle Hysteresis at the Nanometer Scale

Mathieu Delmas, Marc Monthioux, and Thierry Ondarc¸uhu* CEMES-CNRS, 29 rue Jeanne Marvig, 31055 Toulouse cedex 4, France

(Received 10 November 2010; published 29 March 2011)

Using atomic force microscopy with nonconventional carbon tips, the pinning of a liquid contact line on individual nanometric defects was studied. This mechanism is responsible for the occurrence of the contact angle hysteresis. The presence of weak defects which do not contribute to the hysteresis is evidenced for the first time. The dissipated energy associated with strong defects is also measured down to values in the range ofkT, which correspond to defect sizes in the order of 1 nm.

DOI:10.1103/PhysRevLett.106.136102 PACS numbers: 68.08.Bc, 68.37.Ps, 61.48.De

Whereas thermodynamics predicts a single contact angle value for a liquid droplet at rest on a perfect solid surface, experiments on real surfaces show that this contact angle is not univocal but depends on the droplet history [1].

This phenomenon, known as contact angle hysteresis (CAH) is due to topographical or chemical surface defects which pin the contact line (CL) [2]. A quantitative correlation between the CAH and the shape, dimensions, and repartition of the surface defects is still lacking. Model experiments have been performed focusing on artificial defects a few tens of microns in size [3,4], with random or regular patterns [4,5] and more recently with sizes down to 20–200 nm [6,7]. Nevertheless, the measurements are perturbed by the fact that the surface between defects presents some intrinsic hysteresis. Indeed the topography at molecular level [8] and very small densities of defects [4] are sufficient to create appreciable CAH. A precise description of the pinning on individual defects, down to nanometer scale, is therefore essential to fully describe the mechanisms responsible for the occurrence of the CAH.

In order to address this challenging issue, we considered an original approach by studying the wetting of a carbon nanocone dipped in a liquid (Fig. 1). This surface with nanometric dimensions can be homogeneous and accurately defined over length scales which are larger than the length probed by the CL (several tens of nanometers).

Hence, portions of the surface are exempt of hysteresis, a situation extremely difficult to obtain with a macroscopic surface. The 1D geometry is also optimal for probing the defects individually since the short CL can interact with only one defect at once (or few of them only). This situation also leads to a favorable defect over CL force ratio.

The wetting properties of nanoneedles [9] or nanotubes [10,11] have been investigated recently by attaching these objects to an atomic force microscope (AFM) tip and measuring the force when the tip is dipped into, then withdrawn from a liquid bath. This nanoscale replica of the Wilhelmy balance technique allowed the contact angles, meniscus height, and surface energies to be accurately characterized. In this work, the fluctuations of the force which originate from the pinning of the CL on

nanosized surface defects are investigated and the results are discussed in the framework of an existing theory.

Mounting single carbon nanotubes onto AFM tips has recently attracted a lot of attention [12]. Here, we used nanotube-supported nanosized carbon cones instead [13].

The structure of these nanocones suits well the needs for our study because they present a well defined surface with diluted defects with sizes ranging from a nanometer to tens of nanometers [13]. Details of the experimental procedure are given in [14]. The notations used in the following are reported in Fig.1.

A scanning electron microscopy (SEM) image of a typical carbon tip is shown in Fig. 2(a). A 300 nm-long nanotube with a diameter of ca. 20 nm, protruding at the end of a sharp cone with an apex angle of about 4 is observed. The force curves measured by AFM when dipping in and then withdrawing the same carbon tip from two different liquids with low volatility are reported in Figs. 2(b) and 2(c). The two curves within each plot correspond to the advancing (in grey) and receding (in black) CL, respectively. The main features of the force curve can be interpreted using the formula of the capillary force

F¼2rcos0; (1) where ris the fiber radius,is the liquid surface tension and0is the contact angle of the liquid on the fiber surface.

FIG. 1. Sketch of the experiment. (a) Contact line pinned on a defect on the nanoneedle: the contact line and meniscus shape are modified compared to the equilibrium shape (dotted line);y represents the position of the CL on the tip,dthe position of the tip extremity with respect to the free liquid interface; (b) AFM measurement method:zis the AFM piezoelectric extension and the cantilever deflection.

PRL106,136102 (2011) P H Y S I C A L R E V I E W L E T T E R S week ending 1 APRIL 2011

0031-9007=11=106(13)=136102(4) 136102-1 Ó2011 American Physical Society

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At z¼0, the tip extremity reaches the liquid surface.

The force remains roughly constant as the CL proceeds on the nanotube surface. Atz¼280 nm, the force starts to increase mainly due to the increasing cone radius. When the motion is reversed, the CL starts to recede along the cone. The capillary force follows rather closely the same path as the advancing one but is slightly shifted towards larger values, thereby creating a CAH. Regardless of this shift, there is a very good correlation between the jumps observed at both advancing and receding steps. When the CL is back on the nanotube, no appreciable hysteresis is observed except on the two defects observed at z¼ 50–100 nm. For negativezvalues, the CL is receding on the nanotube until it reaches the tip end. The ultimate increase of the force is attributed to the meniscus being stretched before it snaps off [9,10]. In Fig.2(c), we report the force curve measured with the same tip on a glycerol droplet. We observe similar main features as for heptadecane except that the change from advancing to receding motion is associated with a very strong hysteresis, and that the force fluctuations are much more important. Note that the curves were reproducible over continuous cycling with no influence of the velocity (in the range 10 nm=s to 1m=s).

Measuring the tip profile by transmission electron microscopy (TEM) allowed the liquid contact angles to be estimated for heptadecane and glycerol, using Eq. (1) with heptadecane¼27:5 mN=m and glycerol¼64 mN=m, respectively: for heptadecane,av¼rec ¼385 on the nanotube,av ¼265andrec¼235on the cone, and for glycerol, av ¼805 and rec ¼755, on the nanotube, av¼705 and rec¼405 on the cone. Smaller contact angle values on the cone than on the

nanotube is a general feature in our experiments. It may come from the fact that, unlike the nanotube surface which mainly exposes hydrophobic graphene planes, the cone surface mostly exhibits graphene edges [13] which have higher surface energy.

Figure 2evidences that heptadecane is more favorable for studying individual events. In most of the experiments in heptadecane, the advancing and receding curves could barely be discriminated [see Fig. 3(a)]: both curves are perfectly superposed on the major part of the displacement range. Nevertheless, hysteresis cycles appear locally. This is clearly evidenced by plotting the hysteresis force H¼ FrecFadvas a function ofz[see Fig.3(b)]. We attributed these cycles to the pinning of the line on individual (or a small number of ) defects of the tip surface (see the discussion on this issue in [14]).

The situation shown in Fig. 4(a) [enlarged detail of Fig.3(a)] evidences two qualitatively different behaviors.

On the right part of the curve (z >50 nm), a modulation of the force amplitude gives rise to a hysteresis cycle between the advancing and receding curves. On the contrary, on the left part, a smaller modulation of the force without any noticeable difference between both curves is observed.

This latter case demonstrates that not every surface defect induces a CAH. Some defects lead to a local change in the contact angle whose related force is fully reversible.

Since it was established early that the CAH originates in the pinning of the CL on surface defects, the single defect situation attracted much attention. A pioneer paper by Joanny and de Gennes [15] described the case as a balance between the elastic response of the CL [15,16] and the external force due to a local heterogeneity of the surface.

In our experimental geometry, the pinning of the CL may also involve a deformation of the whole liquid meniscus.

Interestingly, the spring constantkof the CL anchored on a defect of sizea[15] and that of a meniscus around a fiber of radiusa[11] have similar expressions :

FIG. 3. (a) Force curve measured on the conical part of a carbon tip; (b) Hysteresis force defined as the difference between the advancing and receding curves in (a).

FIG. 2. (a) SEM image of the extremity of a carbon cone tip;

(b) Force curve measured when dipping in (grey line) and withdrawing (black line) the carbon tip from heptadecane;

(c) same with glycerol.

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k¼sin²0=lnL

a (2)

whereLis a large scale cutoff. Indeed, the meniscus spring constant can be measured experimentally from the force curves at the very end of the separation process corresponding to the stretching of the liquid meniscus (see [14]). We found k¼83 mN=m for heptadecane and k¼184 mN=mfor glycerol. Comparing with formula (2) leads toLnðL=aÞvalues comprised between 5 and 10, for heptadecane and glycerol, respectively, corresponding to values of L ranging from 2 to 200m. Despite no precise experimental estimation, the L value is usually taken as the capillary length or the size of the droplet, but may reduce, in the particular case of the pinning on many dilute defects, to the average distance between defects [15]. This latter situation may be relevant in the case of macroscopic measurements where the contact line is pinned on many defects in parallel but is not in our 1D geometry where the short contact line cannot pin on many (dilute) defects at the same time. Our results suggest a rather large cutoff length, an issue which requires more precise study.

Two cases can be distinguished, as represented sche- matically in Fig.4(b)by plotting the defect force (presum- ably Gaussian) and the (linear) elastic energy of the CL as a function of the positionyof the CL on the fiber (see Fig.1):

(i) In the weak heterogeneity regime [left defect in Fig. 4(b)], for any tip position d, only one equilibrium position of the CL exists. The contact angle is locally modified but in a fully reversible way. (ii) On the contrary,

for strong heterogeneities [right defect in Fig.4(a)], three equilibrium points may exist, the intermediate one being unstable. The equilibrium point follows two different branches depending on the motion direction, leading to a hysteresis cycle shaded grey in Fig.4(b). Using a Matlab routine, we replotted in Fig.4(c)the expected force curves as a function of d(see [14] for the detailed construction of the curves in the different variablesy,dandZ) which explains qualitatively the experimental observations.

Hence, depending on the defect strength, either a reversible modulation [as in the left part of the curve in Fig. 4(a)]

or a hysteresis cycle [as in the right part of the curve in Fig. 4(a)] is observed, corresponding toweak and strong defects respectively, according to Joanny and de Gennes’s definition [15]. To our knowledge, this is the first direct experimental evidence of the weak regime on an individual defect. Such an observation was impossible at macroscopic scale due to the difficulty of having hysteresis-free surfaces. In the particular case of superhydrophobic substrates which are claimed to exhibit no CAH [17], the methodol- ogy used could not probe individual defects.

In order to quantitatively investigate the pinning on single defects, a simplified model of the strong defect regime is proposed here. Considering the case of a defect with small dimension and sharp edges whereF0andaare the force and lateral extension of the defect, respectively, a FðdÞcurve with a triangular shape is expected (Fig. 5, inset) (see [14] ). The dissipated energy is given by the cycle area shaded grey which writes W ¼^F_2k²⁰^ka₂² or W ^F_2k²⁰ in the case of a localized defect (aD¼ F0=k). Both the pinning forceF0and the dissipated energy W were measured experimentally (with accuracies of 10%

and 15%, respectively) as the jump height and the area of the local hysteresis cycle, respectively, for a whole set of different events attributed to individual defects, for various tips dipped in heptadecane. The results evidence a qua- dratic relationship betweenWandF0, which is compatible with the expression given above. A value of the meniscus spring constantk¼92 mN=mis obtained, in very good agreement with the value deduced above from the meniscus stretching at the end of the separation process. The FIG. 4. (a) Detail of a portion of the force curve reported in

Fig.3; (b) model construction of the force curve for two different defects with different strength as a function ofy; (c) same curve as (b) plotted as a function ofd.z,d, andyrefer to Fig.1.

FIG. 5. Plot of the dissipated energy W as a function of the defect forceF0; inset: schematic representation of the pinning on a strong defect, in the simplified model.

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single defect case is therefore quantitatively described by this simple model. Interestingly, this description assuming a Hookean elasticity of the CL—or meniscus—accounts for pinning forces as small as 10 pN and dissipated energies in the range of10²⁰J. For such values close to the thermal energy kT¼410²¹ J, thermal fluctuations may play an important role.

In order to interpret these values in terms of defect dimensions, we considered a topographical defect with a maximal slope’(similar estimation can be performed with chemical defects). The maximum forceF0can be written as [6] F0¼aðcosð0’Þ cos0Þ asin0sin’. The experimental F0 values (10 pN–1 nN) would thus correspond to defect sizesaranging from 2 nm to tens of nm.

Despite the fact that’is unknown, this rough estimate is consistent with the range of sizes observed by TEM [13]

and SEM (Fig.2) and validates the interpretation of individual defects as the origin of localized hysteresis cycles (see more detailed discussion in [14] ). The formula given above can also interpret the large difference of force amplitude observed between heptadecane and glygerol. For a given defect, the force is proportional to sin0 which leads to a ratio Hglycerol=Hheptadecane¼6 as observed in the experimental data. In this simple model, we also estimated the conditions necessary to obtain a hysteresis on a single defect. The strong defect regime requires a slope of theFðyÞcurve larger thank. This leads to sin’ >_LnðL=aÞ^sin⁰ . This value is small (’¼8) and explains why weak defects are actually scarce. The other condition to obtain CAH is that the dissipated energy W should be larger thankT. Using theW¼^F_2k²⁰relation, a minimal defect size a_Ccan be defined asa_C¼ ð_LnðL=aÞsin² 2’^kTÞ¹⁼². In the case of a heptadecane, with cohesive Van der Waals interactions, the surface tension can be written¼_2b^kT2 where bis the dimension of the molecules in the liquid [18]. This leads to aC¼ ð_LnðL=aÞsin⁴ ²_’Þ¹⁼²b1;5b. This simple estimate shows that the critical defect dimensions necessary to obtain a hysteresis is of molecular size, in agreement with previous results [8,19]. This explains why it is so challenging to obtain hysteresis-free surfaces.

As a conclusion, using an AFM-based method, the contact angle hysteresis was investigated at nanometer scale for the first time. Two different types of defects were revealed, and the minimal defect size to introduce CAH was demonstrated to be of molecular dimension. The dissipated energy per defect was measured over four decades down tokTand could be interpreted using a model devel- oped by Joanny and de Gennes [15], demonstrating that the description of the CL as a continuous elastic system is valid down to the nanometer scale. By its ability to probe the CL response at nanometer scale, this experimental procedure may open the way to systematic studies on important yet unanswered questions in wetting science. According to [20], pinning of the CL on nanometric defects may be the unique elementary mechanism controlling both CAH

and activated spreading dynamics [21]. It is therefore important to extend our results to dynamic processes [22]

whose investigation should be possible by using dynamic AFM modes.

This work was partly made possible thanks to the fund- ing by the ANR program HD-Strain, and by the CNRS specific support for technology transfer.

*To whom all correspondence should be addressed:

[email protected]

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