Measure of high contact angles

Abstract

Water repellency is often defined as the territory of contact angles θ larger than 150° and there is some paradox between the huge number of papers devoted to this effect and the lack of specific method for determining such angles. We could think of deducing directly their values from side views of water drops but this measurement is not precise at large θ (uncertainties on the order of 10°), in particular because gravity effects then tend to modify the base of these liquid pearls. We report here that measuring the base size for a drop of known volume allows us to determine high angles with a precision on the order of 1°. We illustrate this technique by questioning the contact angle of liquid marbles (non-wetting drops encapsulated by a powder) and that of water pearls on hot solids.

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Non-wetting states have been receiving attention for about 25 years,^1,2 owing to the spectacular mobility they provide to drops. Water pearls can be typically 100 times quicker than usual drops, even when liquids are as viscous as glycerol.^3–7 A first cause for this mobility is the reduction of the solid/liquid contact (Fig. 1a): its size [small script l] at large contact angle θ is significantly smaller than the drop radius R, which minimizes viscous dissipation. If gravity is negligible, the distance [small script l] trigonometrically scales as R sin θ, that is, as Rε after introducing ε = π − θ – a distance that vanishes linearly with ε.

Fig. 1 Side views of non-wetting drops. (a) A water drop (Ω = 1.2 μL) meets a super-hydrophobic material with a very high contact angle, which reduces the apparent solid/liquid contact to submillimetric scales. Its millimetric radius R can be measured (see Fig. S1 in the ESI†), and that, [small script l], of the contact be determined by analyzing the gray level along a line drawn at the solid/liquid boundary (x-axis). The bar indicates 0.5 mm. (b) A marble with same volume obtained by covering a water drop with a hydrophobic powder (here, lycopodium). Even if repellency is of a different type, both drops look quite similar, which results from gravity. The bar is 0.5 mm. (c) Shape of drops with same volume deduced from a numerical integration of the Laplace equation. For four contact angles chosen between 150° and 180°, we superimpose the drop profiles without gravity (dotted line) and with gravity (solid line). In all cases, gravity sags the drop by a distance δ (smaller than 100 μm), which results in an increase of the contact size all the more visible since angles are high. Hence, measuring this distance can be a sensitive method for accessing high contact angles.

It is essential to accurately determine contact angles, at least to compare the degree of repellency and the drop mobility on different materials. This looks trivial in Fig. 1a: a side view of a drop should directly provide us with the value of the angle, as commonly performed with goniometers. However, large angles (say, around 160° or 170°) generate two sources of error. Firstly, optical distortions near the contact lead to uncertainties as large as 10°,⁸ which makes it hazardous to rank the non-wetting abilities of hydrophobic materials, and even more to follow how the angle varies (or not) with external parameters such as temperature or surfactant concentration.⁹ Secondly, gravity tends to increase the contact size of these liquid pearls, which generates the illusion of a contact angle, as known for Leidenfrost drops where the actual angle has to be π.¹⁰ It might be similar for liquid marbles, that is, drops coated with a hydrophobic powder. Their armors make them highly non-wetting (whatever the substrate), but their contact angle, if it exists, was not characterized.^4,11,12 In Fig. 1b, we can see a micro-liter marble sitting on glass, and it seems uncertain to assess that contact is driven by wettability or by gravity. Even more puzzling, the same could be said in Fig. 1a, for which the contact of a pearl looks similar to that a marble, despite strong differences in the nature of these objects.

As pointed out by Mahadevan and Pomeau,³ the size [small script l] of the gravitational contact of a millimetric non-wetting drop (θ = 180°) can be expressed analytically. Its Laplace pressure scales as γ/R, with γ the liquid surface tension. However, gravity deforms the base of the drop and makes it flat, so that this pressure is also that below the liquid. Since the latter quantity can be expressed as ρgR³/[small script l]², where ρ and g designate the density of the liquid and acceleration of gravity, we deduce a formula for [small script l], [small script l] ∼ R²κ, where κ⁻¹ is the so-called capillary length (about 3 mm for water).³ This flat base might be incorrectly interpreted as resulting from the angle θ = π − ε that generates such a base, with a size [small script l] ∼ Rε. Hence, a gravitational contact interpreted as a wetting one would lead (for a non-wetting drop) to a value for ε on the order of Rκ, as high as 20° for R = 1 mm.

Thinking of this artifact is productive: at high θ, we expect gravitational spreading to be the dominant mechanism for the contact. It is visible when solving numerically the Laplace equation, which expresses the balance between Laplace and hydrostatic pressures in axisymmetric coordinates (see Fig. S2 and text in the ESI†). The resulting drop profiles are shown in Fig. 1c (see also Fig. S3, ESI†) for four values of θ between 150° and 180°. The profiles are calculated without and with gravity (dotted and solid lines, respectively) for a volume Ω = 1.2 μL, which evidences that gravity lowers the height of the drop by a (small) amount δ. Hence, the contact of the drop is augmented by gravity, an effect hardly visible for θ = 150°, but significantly larger when θ tends to 180°. Geometry dictates this behavior: if we push a sphere in a plane by a quantity δ, the size of the resulting contact scales as (δR)^1/2. The distance [small script l] being critical in δ, gravity acts as a magnifying lens for the contact size at high θ. Hence, the sensitivity of [small script l] in θ at large θ should yield an accurate technique for measuring angles in this limit.

We test this idea by determining the contact size [small script l] of pearls as a function of their radius R, for Rκ > ∼0.1 (gravity plays a role) and for ε → 0 (non-wetting situations). The measurement is performed by using backlighting and by extracting the gray level along the substrate (Fig. 1a and b). This unambiguously yields the contact size, even for liquid marbles (Fig. 1b) where grains generate noise in the signal, due to clusters randomly distributed at the liquid surface. We first discuss the sensitivity of the technique in terms of contact angle, and then test it in two open questions: (1) what is the contact angle of a liquid marble (Fig. 1b)? (2) How does the angle of a pearl (Fig. 1a) vary as a function of the temperature T of its substrate, below the transition to the Leidenfrost state^13,14?

In a first series of experiments, we consider water drops sitting on a substrate made super-hydrophobic by a widely used commercial treatment called Glaco. Samples are prepared by coating a piece of polished aluminum with Glaco, a solution of hydrophobic silica nanobeads dispersed in a solvent. The surface is heated in an oven at 250 °C for 30 minutes and the process is repeated three times to ensure that the entire metallic surface is covered with the nanobeads. We measure optically the contact size 2[small script l] of liquid pearls with an accuracy of 10 to 20 μm. Drop volumes Ω vary from ∼4 μL down to ∼0.03 μL, which we achieve by heating slightly the substrate so as to enhance evaporation and decrease Ω. This large variation enables us to capture the reciprocal influences of gravity and capillarity, and results are plotted in Fig. 2 (green symbols). It appears that the transition between wetting and gravity regimes ([small script l] linear and quadratic in R) is wide: the curve is mostly convex and only data at small R follow a linear dependency ([small script l] ∼ Rε) before gradually approaching the quadratic regime ([small script l] ∼ R²κ) shown with orange dots. The numerical solution of the Laplace equation (green dots) fits the data on the whole interval of drop sizes, from which we deduce an angle of 165.5°. Fig. S4a (ESI†) shows that choosing angles larger or smaller by 1° significantly underestimate or overestimate data, suggesting that the uncertainty on the measurement of θ is around 1°. We are not aware of any technique able to reach such a precision on the values of high contact angles.

Fig. 2 Contact size of pearls and marbles. Logarithmic plot of the contact size 2[small script l] of water pearls (green data) and water marbles (orange data) placed on a Glaco-treated material, as a function of the drop radius R. Although both situations may appear similar to the naked eye (Fig. 1), their respective contacts are markedly different. The pearl data (in green) sit on the line obtained after integrating the Laplace equation (green dots). The contact size is linear in R at small R, while it tends to become quadratic at larger R (yet smaller than the capillary length). The only fitting parameter in the calculation is the contact angle, chosen here to be θ = 165.5° ± 1.0°. For marbles, all data follow the quadratic law dictated by gravity only, which suggests that these marbles non-wet their substrate, θ ≈ 180°. Marbles are made either with lycopodium (black data) or with silica grains treated by Glaco (orange data) and they are deposited on glass. Horizontal error bars show the uncertainty on the measurement of the drop radius R (see Fig. S1, ESI†), while vertical ones show that on the contact size [small script l] (Fig. 1a and b).

We further test our method by considering liquid marbles, which we obtain after gently shaking for a few seconds water drops on a bed of hydrophobic grains. The grains spontaneously stick to the water surface, yet remain dilute (no jamming that would result from a much longer shaking). This operation results in deformable drops that non-wet their substrate⁴ and keep a surface tension comparable to that of water.¹⁵ As seen in Fig. 1b, the surface of a marble is rough, due to individual grains and to clusters of grains, which generate peaks in the curve of the gray level along the solid surface. However, the contact can be unambiguously defined as the plateau zone where there are no more ‘holes’ in the gray line (Fig. 1b). Owing to the grains, uncertainties on the measure of [small script l], 10 to 40 μm, are slightly larger than that for pearls, yet without jeopardizing accurate angle measurements. We consider two kinds of grain, the spores of a moss called lycopodium, with an individual average size of 30 μm (black data) or silica beads with size between 2 and 32 μm first treated with Glaco (orange data) that make smoother the marble surface and thus improve the precision of measurements. In all cases, the substrate is glass.

If we relied only on Fig. 1 for determining angles, we would conclude that both systems are very similar. This sharply contrasts with Fig. 2, where data for marbles are not only distinct from that for pearls but also follow different trends. In this logarithmic plot, the fit is a straight line with slope 2 that reveals the gravitational contact of Mahadevan-Pomeau, [small script l] ∼ R²κ. Within the uncertainty of the measurements (∼1°), we conclude that marbles non-wet their substrate (θ ≈ 180°), a conclusion reinforced by the independence of data towards the nature of the grains (orange and black data in Fig. 2). If the solid (grain)/solid (substrate) adhesion is low, the drop shape is mainly dictated by the minimization of the liquid/air interface: it is a sphere flattened by gravity. It would be interesting to check the generality of this result by considering cases where the adhesion of grains on the solid is large enough to generate deviations from this extreme non-wetting state.

Fig. 2 also emphasizes that the contact size is highly sensitive to the contact angle when the drop radius is smaller than the capillary length. For instance, the contact of pearls and marbles can differ by a factor 2 when R is around 0.4 mm. Practically, this implies that it is not necessary to vary the drop volume by a very wide range to access contact angle – an interval of variation 0.1 μL < Ω < 0.5 μL is sufficient to deduce θ with an accuracy of ±1°, which will allow us to reduce the range of explored volumes in the rest of the paper.

We further exploit the accuracy of our technique on water pearls, either by testing different substrates or by heating a given one. In the latter case, we anticipate a transition to the Leidenfrost state (levitation on a vapor film) characterized by an increase of the contact angle from its value θ ≈ 165° at room temperature to its maximum θ = 180°. The function θ(T) is not known,¹⁶ and the amplitude of the variation (around 15°) indeed necessitates an accurate technique. Suslick et al. conducted experiments to investigate the influence of a simultaneous increase in temperature and pressure on the contact angle on various hydrophobic materials⁹: silicon wafers coated with Teflon or with zinc oxide nanoparticles, surfaces with micro-pillars, copper grids coated or not with Teflon or zinc oxide nanoparticles. For temperatures ranging from 26 to 100 °C, they found that angles remain practically unchanged but uncertainties larger than 10° make it difficult to conclude firmly on this question.

Our samples are made of polished, Glaco-coated aluminum. A fresh sample is placed on a hot plate and connected to a thermal probe that continuously monitors the surface temperature T. Once temperature stabilizes, a microliter water droplet is placed on the metallic surface and a series of photos is taken as it evaporates. For each photo, the radius R and base size [small script l] are measured, from which we deduce the value of the angle by fitting the curve [small script l](R), as presented in Fig. 3 and discussed in Fig. S4b (ESI†).

Fig. 3 Testing our method with water pearls. (a) Contact diameter 2[small script l] of water pearls with various radii R sitting on Glaco-coated samples made on two consecutive days and brought to the same temperature T = 75 °C. Data do not overlap, but each series nicely fits the numerical solution (dotted lines) where the contact angle θ is the only adjustable parameter. Hence, the two samples have different contact angles θ, whose values provided by the fits are respectively 164.0° ± 0.8° (orange data) and 167.0° ± 0.7° (green data). (b) Contact 2[small script l] for water pearls on a given Glaco-treated sample of aluminum brought to T = 75 °C (orange data) or T = 134 °C (blue data). There again, data do not overlap, and the respective angles θ deduced from the fits (dotted lines) are 164° ± 0.8° and 167.5° ± 1.0°, which suggests that θ increases with T (heat-enhancement of super-hydrophobicity).

Fig. 3 illustrates the sensitivity of our method. In Fig. 3a, we compare the contact size at fixed temperature (T = 75 °C) for samples prepared on two consecutive days (orange and green data). The random character of the coating makes us anticipate some variability in wettability, which is what we observe. Each series of data is consistent and it can be convincingly fitted by the Laplace equation, which yields θ = 167.0° ± 0.7° for the green data and θ = 164.0° ± 0.8° for the orange one (the larger the angle, the smaller the contact). This experiment thus confirms the sensitivity of the method, found here able to probe the intrinsic variability of Glaco-coated materials and to distinguish materials with a close degree of hydrophobicity.

In Fig. 3b, we study the repellency of a given sample brought to either T = 75 °C (orange data, same as in Fig. 3a) or T = 134 °C (blue data). The two series of data do not overlap, which suggests that the contact angle depends on temperature: the angles deduced from the fits (dotted lines) are 164.0° ± 0.8° and 167.5° ± 1.0°, respectively, showing a modest increase of θ with T. On the one hand, we could think of this increase as logical, on the way to the Leidenfrost regime; on the other hand, the increase deduced from Fig. 3b is weak and it needs to be established in a firmer way, which we now discuss.

In our final set of experiments, we combine and extend the results of Fig. 3 by considering four Glaco-treated samples for which we determine the contact angle of water pearls as a function of the substrate temperature, using the technique developed in this paper, where we consider the known variations of surface tension and density with temperature. In all cases, convincing fits for the curve [small script l](R) are obtained, such as in Fig. 3, allowing us to deduce contact angles for each experiment, the ensemble of which is reported in Fig. 4a, where θ is plotted as a function of T.

Fig. 4 Contact angle of heated pearls. (a) Angle θ measured by our contact-size technique as a function of the substrate temperature T, for four Glaco-treated plates of aluminum. One color corresponds to one sample, and dotted lines are guides to the eyes. In each case, θ increases with temperature, but it remains significantly below 180°, the angle above the Leidenfrost temperature, whose location is indicated by the grey zone (between 135 °C and 145 °C for these hydrophobic materials). (b) We define ϕ as the ratio between the solid/liquid contact in a super-hydrophobic situation and the apparent contact, that is, the fraction of solid actually contacting the liquid. This parameter is obtained from the so-called Cassie equation, and we plot it here as a function of the temperature, after dividing it by its value ϕ_O at 60 °C.

The results in Fig. 4a confirm the observations reported in Fig. 3. Firstly, there is a variability of about 5° in the wetting properties of similar Glaco-treated materials. Secondly, contact angles systematically increase with temperature, in a way that seems independent of the sample (dotted lines in the figure are guides to the eyes): if varying T from 60 °C to 130 °C, θ increases by about 3%. Whatever the temperature, we never observe boiling, as also reported by Bourrianne on similar surfaces.¹⁴ A grey zone in the figures indicates the region where pearl drops start to levitate (Leidenfrost state). Interestingly, contact angles close to this region are typically 10° smaller than the Leidenfrost angle of 180°. This suggests a critical or discontinuous nature of the Leidenfrost transition on hydrophobic materials, which remains to be modelled.

Water-repellent materials result from a combination of chemistry (substrates are hydrophobic) and physics (surfaces are rough), so that water mainly faces air trapped in the solid cavities. If we denote ϕ as the proportion of solid meeting the deposited liquid, 1 − ϕ is the proportion of air/liquid contacts below the liquid. Then, the contact angle is an average between θ_O, the angle on a flat solid with same chemistry, and π, that on air, with respective weights ϕ and 1 – ϕ.¹⁷ This is the Cassie relationship, cos θ = ϕ (1 + cos θ_O) – 1, from which we can deduce the value of ϕ, for a known θ_O. On a hydrophobic material, θ_O is typically 110°, which provides a ϕ of 0.05 on the Glaco-coated substrates (θ ≈ 165°). Denoting ϕ_O as the value of ϕ at ∼60 °C, and assuming that θ_O hardly varies with temperature,¹⁸Fig. 4a and Cassie formula, ϕ(T) = ϕ_O (1 + cos θ(T))/(1 + cos θ(60 °C)), allow us to access the variation of ϕ with temperature presented in Fig. 4b.

All data collapse on a single curve that decreases with T. Heating the substrate produces vapor, which either enlarges the air cavities initially present below the drop, or lifts the underlying air/liquid interfaces – in both cases producing an increase of the proportion of air/vapor below the drop, and thus higher contact angles. The variation is now found to be significant: while angles only vary by a few degrees in Fig. 4a, ϕ is found to be lowered by about 40% when approaching the Leidenfrost transition (grey zone), which inspires two remarks: (1) the Cassie equation is not linear, and in the limit of interest (θ large), we can expand it. We find that ϕ scales as ε² (where ε is π−θ), which makes it understandable that a modest decrease of ε (Fig. 4a) has a large impact on ϕ (Fig. 4b). (2) This may help us to understand the apparent quasi-discontinuity of θ at the Leidenfrost transition: a significant decrease of ϕ can trigger the percolation of the gas bubbles across the solid surface,¹⁴ which induces the transition at finite ϕ (and not at zero ϕ) – that for which the coverage of the surface by vapor bubbles is such that percolation takes place, in agreement with the ideas of Chantelot and Lohse.¹⁹

Angles were measured while drops are slowly evaporating and thus correspond to receding angles. By complementing these experiments with the classical determination of the tilt angle above which droplets depart when tilting the substrate, we can access the contact angle hysteresis (Fig. S5, ESI†) and thus the values of the advancing angles (Fig. S6, ESI†). Doing so, we find that the hysteresis decreases with temperature – a logical trend since adhesion has to vanish at the Leidenfrost temperature, where levitation eliminates the possibility of pinning. We also report that advancing angles are fairly constant, so that the reduction of hysteresis when increasing the solid temperature is primarily due to the increase of the receding angle (Fig. 4a). This confirms our scenario: temperature gradually erodes solid/liquid contacts (Fig. 4b) and thus reduces the ability of the liquid to pin (Fig. 4a).

We feel that a precise method for measuring high contact angles can be useful. It is tempting, for instance, to rank non-wetting materials according to the value of θ, which requires reliable measurements of this quantity. Here, we exploited the fact that the gravitational sagging of drops creates a significant contact in the limit θ → π. Unlike the sagging distance δ, the size [small script l] of this contact is easily accessible and it critically depends on θ, which provides a resolution on the order of 1° on the value of (high) θ. Hence, new and more refined measurements become accessible, such as how non-wetting is perturbed when heating or cooling the substrate, or when adding ethanol or surfactant to the liquid – all topics on which a lot remains to be done. As a first example, we showed how the receding angle of water on a super-hydrophobic solid increases with temperature, which allows us to understand why hotter substrates are significantly less sticky. But we could also take advantage of the technique to determine the contact angle of liquid marbles, known to be highly non-wettable drops, but actually found to reach θ = 180°, the highest degree of hydrophobicity, within the uncertainty of the experiments.

Data availability

Data used in this study will be made available upon request to the corresponding author.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

We thank Philippe Bourrianne for his help in the experiments.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5sm00103j

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