Quantifying droplet–solid friction using an atomic force microscope

Controlling the wetting and spreading of microdroplets is key to technologies such as microfluidics, ink ‐ jet printing, and surface coating. Contact angle goniometry is commonly used to characterize surface wetting by droplets, but the technique is ill ‐ suited for high contact angles close to 180° . Here, we attach a micrometric ‐ sized droplet to an atomic force microscope cantilever to directly quantify droplet – solid friction on different surfaces (superhydrophobic and underwater superoleophobic) with sub ‐ nanonewton force resolutions. We demonstrate the versatility of our approach by performing friction measurements using different liquids (water and oil droplets) and under different ambient environments (in air and underwater). Finally, we show that underwater superoleophobic surfaces can be qualitatively different from superhydrophobic surfaces: droplet – solid friction is highly sensitive to droplet speeds for the former but not for the latter surface.


INTRODUCTION
Submillimetric or microdroplets are formed during various processes, such as condensation and human respiratory activities, with implications for heat transfer 1,2 and disease spread. 3,4At the same time, the ability to control surface wetting by microdroplets is key to optimizing the performance of various technologies, such as ink-jet printing 5,6 and enhanced oil recovery techniques. 7,8Contact angle goniometry is the most common approach to quantify the wetting properties of surfaces especially by a millimetric-sized liquid droplet. 9,10The adhesion and friction forces required to remove the droplet can also be deduced, albeit indirectly, from the advancing and receding contact angles. 11,12However, contact angle goniometry does not work well for large contact angle values close to 180°and for rough surfaces. 9,135][16] The microdroplet is attached to and spatially manipulated by a tipless AFM cantilever, and the forces acting on the droplet can be deduced from the cantilever deflection.This technique is sometimes known as droplet probe AFM and has been used to study various interfacial problems, [17][18][19][20][21][22] but its potential as a surface wetting characterization tool-especially to measure droplet-solid friction-remains untapped.
Here, we used droplet probe AFM to directly quantify the droplet-solid friction (with sub-nN force resolutions) for a submillimetric water or oil droplet moving laterally (with controlled speeds U = 2-600 μm s −1 ) on superhydrophobic and underwater superoleophobic surfaces (Figure 1).We explain how to convert the raw voltage signal from the four-quadrant sensor first into the moment M and ultimately the friction force F fric acting on the droplet.4][25] For example, the droplet probe is much more easily deformed compared to a solid tip, and special care has to be taken to disentangle droplet-solid friction from other effects, for example, viscous drag acting on the droplet. 21 this article, we chose a black silicon surface 26 and a zwitterionic poly(sulfobetaine methacrylate) (pSBMA) brush 27,28 as the archetypal superhydrophobic and underwater superoleophobic surfaces, respectively.Black silicon consists of micron-sized cones with hierarchical nanotextures to trap an air layer beneath the water droplet (Figure 1a), while the polymer brushes in pSBMA play a similar role by trapping a stable water film under the oil droplet (Figure 1b).Previously, it was thought that zwitterionic polymer brushes (and other underwater superoleophobic surfaces) are simply oil-repellent analogues of superhydrophobic surfaces. 29This view is partly based on the experimental observations that droplets on both surfaces show high contact angles close to 180°.
Here, we show how this view is inaccurate and that droplet-solid friction is not just quantitatively but qualitatively different for the two surfaces.First, we found that F fric is one to two orders of magnitude lower for the underwater superoleophobic surface compared to the superhydrophobic surface for similarly sized droplets.More interestingly, F fric increases with droplet speed U for the underwater superoleophobic surface but is independent of U for the super- hydrophobic surface, reminiscent of the distinction between lubricated and dry frictions in tribology. 30

RESULTS AND DISCUSSION
We start by addressing the deformability of the droplet probe.We can control the contact radius r by varying the normal force F N acting on the droplet.For example, increasing F N from 8 to 350 nN results in an increase of r from 10 to 18 μm for an underwater oil droplet (radius R = 50 μm) on the pSBMA surface (Figure 2a); a similar increase in F N results in an increase of r from 4 to 10 μm for a water droplet (R = 20 μm) on a superhydrophobic surface (Figure 2b).For a droplet of radius R and surface tension γ making a contact angle F I G U R E 1 Schematic of our setup.(a) Measuring friction force F fric for a water droplet moving on a black silicon superhydrophobic surface.Torsional deflection ϕ Δ of cantilever causes laser deflection, which is detected as voltage signal V lat on the four-quadrant sensor.The inset below shows the scanning electron micrograph of the surface.(b) Measuring F fric for an oil droplet on a hydrated polyzwitterionic brush surface (underwater superoleophobic).The inset below shows AFM topography of the brush layer with thickness h.AFM, atomic force microscopy.  results from a balance between the Laplace pressure γ R πr (2 ∕ ) 2 , the capillary adhesion force πrγ θ 2 sin , and the applied F N , that is, where r r R ˜= ∕ and F F πRγ ˜= ∕2 Equation (1) (Figure 2d).Note that while Equation ( 1) is initially formulated for droplet contact on flat solid surfaces, it can be applied to structured superhydrophobic surfaces if we assume that , where ϕ is the solid (area) contact fraction and θ μ is the local contact angle on solid fractions.Here, the ϕ 1∕2 term comes from line-averaging over the droplet's outer perimeter to achieve force balance, as opposed to the area-averaging approach used to derive the Cassie-Baxter relation to achieve energy balance.We also assume that the droplet is large compared to the structures, and that droplet contact can be approximated as a circle.See detailed discussions in Supporting Information: Figure S1.
We can obtain the contact angles θ from the plot of r ˜versus F ˜N since θ r F r sin = ˜− ˜∕Ñ .In the limit of small F ˜N ≈ 0, both surfaces have similar θ ≈ 170°(Figure 2e).As we increase F ˜N from 0 to 0.08, θ for the superhydrophobic surface decreases from 170°to 155°.This is likely because the water droplet penetrates deeper into the cones at higher F N , resulting in increasing solid contact fraction and decreasing θ, consistent with predictions by Cassie-Baxter. 31In contrast, θ for the underwater superoleophobic surface remains relatively constant between 165°and 170°for the same increase in F N .As explained in our previous paper, the polymer brushes become charged when submerged underwater, resulting in repulsive electric double-layer forces that can sustain strong hydration layers, 28 that is, the oil droplet cannot penetrate into the polymer brush layer even with increasing F N .
The difference between the two surface classes will become even more apparent from the droplet-solid friction measurements presented later.
Before we can quantify the friction force F fric accurately, we need to perform several calibration steps to convert the raw voltage signal V lat (due to torsional deflection of the cantilever) from the four- quadrant sensor (volts) into forces (newtons). 25To do this, we need to know (1) the sensitivity S [V ] (see the schematic in Figure 1).We can also define a lateral sensitivity The droplet height H can be obtained from side-view images such as in Figure 2a,b.To determine k ϕ and S ϕ , we follow the approach first proposed by Sader 32 and Green and Sader, 33 who realized that the thermal response of the cantilever beam (its resonant frequency and the quality factor) is determined by the equipartition theorem and damping due to the surrounding fluid, which, in our case, is either air or water.
Hence, by looking at fluctuations of V lat of an unloaded cantilever with time, it is possible to deduce k ϕ and S ϕ with about 10%-20% accuracy.
In many commercial AFM systems, there is no automatic option to implement Sader's method for torsional deflection.Here, we describe a practical implementation of Sader's method for two different cantilevers: TL-Cont and Arrow-TL used to measure friction for the superhydrophobic and underwater superoleophobic surfaces, respectively.We first measured the thermal fluctuations of V lat in air and underwater as a function of time (at a rate of 600 kHz) and performed a fast Fourier transform to obtain the power spectral density (PSD) of the frequency components f, which is then fitted with a simple harmonic oscillator model ( ) where f o is the resonant frequency, Q is the quality factor, and A DC and A o are fitting parameters. 34k ϕ and S ϕ can then be determined using the following equations: where Γ i is the hydrodynamic function described in Green and  35 for our Python programming code).We can then obtain k ϕ and S ϕ (and hence LS) by applying Equation ( 4), which we have summarized in Table 1.Note that while S ϕ and LS values in air and water can differ, the spring constant k ϕ cannot.
To confirm the accuracy of Sader's method, we used a direct contact method to obtain LS in air. 36,37We pushed the cantilever edge against a hard solid tip (i.e., at w/2 away from the cantilever central axis) to apply a known force F and hence a resultant moment M Fw = /2, while recording the signal response V lat (Figure 3e,f).
Depending on the cantilever side, we can apply either a clockwise or anticlockwise rotation/moment.The slopes of   V lat against M correspond to 1/LS.Experimentally, we found that the two methods, Sader and contact, yield similar LS values: 1.2 × 10 −11 versus 0.9 × 10 −11 N m/V for TL-Cont; 7.6 × 10 −13 versus 8.7 × 10 −13 N m/V for Arrow-TL (see Table 1).Once the calibration has been performed, the raw signal V lat can be converted into F fric using the relation lat .We first performed friction measurements for a water droplet (F = 5 N nN, R = 30 μm, and r = 6 μm) moving on the superhydrophobic surface.Figure 4a shows the force data with F = 1.9 ± 0. V lat against M corresponds to LS 1∕ .
T A B L E 1 Calibration results from Figure 3. where η is water's viscosity.A more significant effect (i.e., artifact) is likely due to the misalignment of the optical lever, which causes a signal that varies with cantilever position as can be clearly seen in Figure 4c.In any case, we can obtain F = 120 ± 20 fric pN (much smaller than F fric in the superhydrophobic surface) correctly by subtracting the reference data in Figure 4c from the raw data in Figure 4b,d).As will be explained later, F fric for the oil droplet is related to viscous drag in the thin water film beneath the droplet.
The difference between the two surfaces goes beyond the magnitude of F fric and is evident in the force fluctuations F Δ fric .Previously, we and others showed that electric double-layer forces in pSBMA can sustain a nanometric water film beneath the oil droplet, preventing contact with the polymer brushes. 18,28,38This explains why the force curves are smooth for the underwater superoleophobic surface (Figure 4d), and the observed F Δ = 80 fric pN is mostly due to thermal fluctuations (we measured similar magnitude of signal fluctuations for a stationary cantilever when performing calibration using Sader's method).In contrast, the force curve is more jagged, with larger F Δ = 0.7 fric nN for the superhydrophobic surface, which we attribute to pinning-depinning events as the moving droplet detaches from individual cones (Figure 4a and later schematic in Figure 5b).
To better illustrate the difference in friction behaviors, we repeated the friction measurements over a wide range of U = 2-600 μm s −1 and applied F = 5-580 N nN (which translates to different contact radii r).First, we observed that F fric for the underwater superoleophobic surface is one to two orders of magnitude lower as compared to the superhydrophobic surface (Figure 5a).We have chosen to present the friction force in its nondimensional form F rγ ∕2 fric to allow us to compare results between different surfaces, droplet sizes, and applied 28,39,40 For a millimetric-sized droplet on the Black Si superhydrophobic surface, we measured advancing and receding contact angles using contact angle goniometry and found that θ = 169 ± 3°a  More interestingly, we found that ∝ F U fric for the underwater superoleophobic surface (inset, Figure 5a), but is independent of U for the superhydrophobic surface.This discrepancy points to different dissipation processes for the two surfaces (Figure 5b,c).
For the superhydrophobic surface, F fric is dominated by contact-line pinning, while for the underwater superoleophobic surface where there is no contact line, F fric is dominated by viscous flow in the nanometric water film beneath the droplet.This is reminiscent of the distinction between dry and lubricated frictions in tribology. 30eviously, we experimentally verified the presence of water film thickness h ~100 water nm that is stabilized by electric double-layer forces for polyzwitterionic brushes. 28Most of the viscous dissipation occurs in the advancing and receding front of size λ Rh ~water .
Hence, we expect which is consistent with the experimental observation that ∝ F U fric .
Equation ( 5) can be rearranged to its non-dimensionalized form where Ca ηU γ = ∕ is the capillary number, where η is the water's We can also compare the relative magnitudes of the viscous drag due to the thin water film F fric beneath the droplet and the stokes drag force F stokes surrounding the oil droplet: fric stokes water . Note that F πRηU = 6 stokes will be changed near the wall and is not accounted for in our model. 41It is not trivial to determine an accurate expression for F stokes , whereby ≫ R h ∕ 1 water (the wall effect is important) and droplet deformation is significant, that is, the droplet is no longer spherical.
The friction force F fric described here (Equations 5 and 6) applies only to micro-droplets.Previously, we showed that for millimetric droplets moving on the same surface, F rγ Ca ∕2 fric 2∕3 as the water film thickens with increasing U in accordance to Laudau-Levich-Derjaguin (LLD) predictions, that is, h RCa LLD 2∕3 . 28r micro-droplets, hydrodynamic forces are insufficient to thicken the water film and h ~100 Here, we assume that the water inside the brush layer is bound and does not contribute to viscous dissipation.We also do not rule out the presence of the nonviscous component in F fric in the underwater superoleophobic surface, which can be important at lower contact line speed U ≪ 1 μm s −1 .
Finally, we would like to point out that the AFM technique described here measures surface wetting under dynamic conditions (i.e., with controlled droplet speeds), which is different from a conventional contact angle measurement, which is primarily a static/quasi-static measurement with poorly controlled contact-line speeds.This is a point often ignored in the literature, which we discussed at lengths in a recent review paper. 40

CONCLUSIONS
In this paper, we demonstrated the versatility of droplet probe AFM as a surface-wetting characterization tool by measuring droplet-solid frictions (with sub-nN resolutions) for different liquids (water and oil droplets) and under different ambient environments (in air and underwater).Our approach can quantify surface wetting properties with controlled droplet/contact-line speeds, and when combined with the sensitivity of the technique, allows us to probe the physical origins of the observed liquid repellency for superhydrophobic and underwater superoleophobic surfaces.

MATERIALS AND METHODS
Superhydrophobic surfaces: A black silicon surface was fabricated as described previously in the literature. 26,39Briefly, a maskless Zwitterionic PSBMA surface preparation: The polymer brush surfaces are prepared using surface-initiated Atom Transfer Radical Polymerization (ATRP) using a protocol adapted from Azzaroni et al. 27 and described in detail in our previous papers. 16,28obe liquid droplets: Silicone oil (polyphenyl-methylsiloxane, viscosity ~100 mPa.s, density 1.06 g/mL) was purchased from Sigma-Aldrich.The water-oil interfacial tension is 40 mN/m for silicone oil, as measured using the pendant drop method.
Deionized water with a resistivity of 18.2 MΩ cm was obtained from a Milli-Q water purification system (Millipore).We dissolved a small amount of salt (3 wt% CaCl 2 ) in water to prevent the microdroplets from drying out.

F
I G U R E 2 Increase in contact radius with increasing normal force.Side-view images of (a) an oil droplet on an underwater superoleophobic surface and (b) a water droplet on a superhydrophobic surface subjected to different normal forces F N .Scale bars are 50 μm and 25 μm for (a) and (b), respectively.(c) Contact radius r as a function of F N for droplets of different radii R. Uncertainty in radius measurement r Δ ≈ 1 μm (random error), while the corresponding (systematic) error in force measurement F F Δ ∕ ≈ 0.1 N N due to uncertainty in the cantilever spring constant.(d) When r and F N are normalized by R and πRγ 2 , the data for superhydrophobic and underwater superoleophobic surfaces collapse into their respective master curves.(e) The contact angle θ can be deduced from Equation (1) for the two surfaces.
θ r normal force, respectively.Experimentally, we observed that r increases with increasing F = 5-400 N nN for droplets of different sizes R = 11-70 μm for both surfaces (Figure 2c).More importantly, the data for superhydrophobic and underwater superoleophobic surfaces collapse into their respective master curves when r and F N are normalized by R and πRγ 2 , as predicted by

ϕ − 1
, which relates the voltage signals to the torsional deflections ϕ Δ [rad], (2) the spring constant k ϕ [N m], which converts the deflections into moment M, and (3) the droplet height H since

2 fric(F = 30 N
nN as we move the droplet back and forth at a speed U = 30 μm s −1 over three cycles, traversing a distance of 50 μm each time-several times the contact diameter r 2 = 12 μm.See Video S1 in the Supporting Information.We repeated the friction measurement for an oil droplet nN, R = 55 μm, r = 11 μm) moving on an underwater super- oleophobic surface at a speed of U = 40 μm s −1 (raw force data shown in Figure 4b; see Video S2 in the Supporting Information).To correctly measure friction for the oil droplet, it is important to disentangle F fric from other effects.To do this, we repeated the same force measurements as in Figure 4b, except with the oil droplet lifted up from the surface (by about 100 μm), and observed a clear signal, F I G U R E 3 Calibrating cantilever's spring constant.To obtain k ϕ and S ϕ for the two cantilevers TL-Cont and Arrow-TL, we fitted the power spectral density (PSD) in (a, b) air and (c, d) underwater with Equation (3) using Sader's method.Raw data are in gray, while blue lines show the best-fit curves.(e, f) Contact method: alternatively, we can obtain the lateral sensitivity LS by applying a known moment M Fw = ∕2 (where w is the cantilever width) and recording the signal response V lat .The slope of   droplet-surface interactions.The observed force is partly due to viscous drag (i.e., Stokes's law) , θ Δ cos = (4 ± 2) × 10 −2 .This is not far F I G U R E 4 Friction data for superhydrophobic and underwater superoleophobic surfaces.(a) Friction for a water droplet (F = 5 N nN, R = 30 μm) moving at a speed U = 30 μm s −1 on the superhydrophobic surface.(b) Raw force data for an oil droplet (F = 30 N nN, R = 55 μm) moving at a speed U = 40 μm s −1 on the superoleophobic surface.(c) Reference data for an oil droplet not touching the surface.(d) Friction data after subtracting c from b.

F I G U R E 5
Figure 5a).For the underwater superoleophobic surface, we measured F rγ ∕2 fric viscosity and γ is the oil-water interfacial tension.Both equations are in quantitative agreement with our experimental results.For example, Equation (5) predicts F ≈ 100 fric pN for the oil droplet in Figure 4d, consistent with the experimentally measured value of 120 pN.When plotting F rγ ∕2 fric versus U, our model also predicts a gradient of πη γ R h ∕ ∕ ~1 m water −1 s close to the slope of 1.7 m −1 s for the best-fit line in the inset of Figure 5a.
water nm is determined by electric double-layer forces: the LLD model predicts a much lower h ~10 LLD nm for R ~50 μm and U ~0.2 mm s −1 .This discrepancy in results between micro-and macro-droplets highlights the usefulness of our AFM technique.
deep reactive ion etching (Oxford Plasmalab System 100; Oxford Instruments) process was used to create the cone-shaped topography (7 min etch time, −110°C temperature, forward power 6 W, ICP power 1000 W, 18 sccm of O 2 , and 40 sccm of SF 6 ).The surface was made superhydrophobic by depositing a nominally 40 nm thick fluoropolymer coating using a plasma-enhanced chemical vapor deposition process (Oxford Plasmalab 80+; 50 W power, 5 min deposition time, 100 sccm of CHF 3 ).