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New information on the hydrophobic interaction revealed by frequency modulation AFM Itai Schlesinger, and Uri Sivan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03574 • Publication Date (Web): 20 Feb 2017 Downloaded from http://pubs.acs.org on February 27, 2017
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New information on the hydrophobic interaction revealed by frequency modulation AFM Itai Schlesinger and Uri Sivan* Department of Physics and the Russell Berrie Nanotechnology Institute, Technion – Israel Institute of Technology, Haifa, 3200003, Israel
ABSTRACT: Using an ultra-high resolution AFM operated in frequency modulation mode we extend existing measurements of the force acting between hydrophobic surfaces immersed in water in three essential ways. (1) The measurement range, which was previously limited to distances longer than 2-3 nanometers, is extended to cover all distances, down to contact. The measurements disclose that the long-range attraction observed also by conventional techniques, turns at distances shorter than 1-2 nanometers into pronounced repulsion. (2) Simultaneous measurements of the dissipative component of the tip-surface interaction reveal an anomalously large dissipation commencing abruptly at the point where attraction begins. The dissipation is more than two orders of magnitude larger than expected from bulk water viscosity or from similar measurements between hydrophilic surfaces. (3) The short-range repulsion is oscillatory, indicating molecular ordering of the medium as the hydrophobic surfaces approach each other. The oscillation period, ~0.5 nanometer, is larger than the ~0.3 nanometer period observed with hydrophilic surfaces. Their range, ~1.5 nanometer, is longer as well. These observations are consistent with a conspicuous change in the properties of the surrounding medium, taking place simultaneously with the onset of attraction as the two surfaces approach each other.
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INTRODUCTION The governing role of hydrophobica interactions in countless biological and technological systems has motivated extensive theoretical and experimental efforts aimed at deciphering the microscopic foundations underlying this elusive interaction1–4. More than three decades have passed since the first force measurements using the surface force apparatus (SFA)5–7 but a full, predictive theory of the hydrophobic interaction is still lacking. Some of the pioneering experimental studies of the hydrophobic force suffered from jump-to-contact (JTC) of the two surfaces from tens and even hundreds of nanometers distances8–11. These JTC were later traced to bubble formation on the surfaces12–14 or surface inhomogeneity11. Subsequent theoretical and experimental studies have focused on the "true" hydrophobic attraction originating potentially from water ordering next to the hydrophobic surfaces due to dangling OH bonds or depletion of water molecules from the surface15,16. These studies required careful surface preparation, high chemical quality of the surfaces, and molecular flatness, without which reproducible results could not be obtained. Notwithstanding these efforts, the JTC hurdle was never mitigated17,18 and the shortest distance where the force could usually be measured by SFA or static mode AFM settled at 2-5 nm. Interestingly, although the short-range interaction has generally not been measured due to JTC, it was somehow assumed to be attractive all the way to contact. This assumption withstood quite a few contradictory examples where the jump terminated at a finite distance, revealing a clear short-range repulsive component. Some examples are depicted in the inset to Fig. 3(a). Stock et al.16 disclosed a soft plateau in the force at ~0.5 nm distance, which could not be attributed to any known mechanism. Meagher et al.19 measured the force using a stiff (~50N/m) cantilever fitted with an 80 μm diameter polypropylene sphere (estimate based on their image). The measurements showed an attractive force starting at ~10 nm from the surface, followed by JTC at 4 nm and a soft repulsive force extending to ~1.5 nm from the surface. The repulsion was attributed by the authors to surface asperities compression (their surfaces were rough on the nanometer scale). Donaldson et al.18 studied the interaction between two hydrophobic surfaces using the SFA. Their data for two PDMS layers and for two dioctadecyldimethylammonium bromide (DODAB) layers showed jumps from 12.6 nm to 2.5 nm and from 10 nm to 2.0 nm, respectively. The force at the shorter distance was still attractive in both cases but as the two surfaces approached each other the attraction grew smaller rather than growing larger as expected from the postulated exponential hydrophobic attraction. These results hence a
Hydrophobicity level is defined here as the energy per unit surface-water area relative to the water-water interface energy of the same area. According to this definition, all surfaces with non-zero contact angle are
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disclosed a short-range repulsive component which eventually compensated for the longer range attraction. Finally, our own measurements using static AFM also displayed short-range repulsion following the JTC (inset to Fig. 3(a)). We see then that a short-range repulsive component is not uncommon in experiments that probe the interaction between two hydrophobic surfaces and one of the questions addressed below is whether such repulsion is inherent to the hydrophobic interaction or perhaps results from extrinsic phenomena such as surface asperities, layer inhomogeneity, hydrophilic patches, etc. The references listed above point to the main hurdles on the way to a reliable measurement of the short-range force acting between two hydrophobic surfaces by static methods. First, the JTC and then, the large interaction area with its potential roughness, bubbles, and inhomogeneity. The introduction of frequency modulation AFM (FMAFM) eliminates both hurdles. The significantly higher sensitivity of dynamic methods allows the use of two orders of magnitude stiffer cantilevers compared with static AFM, hence avoiding the mechanical instability that leads to JTC in SFA and static AFM. For example, with a 40 N/m cantilever and a 1 kHz measurement bandwidth, the force sensitivity grows from ~500 pN with a commercial Dimension V (Bruker) microscope operated in static mode to ~10 pN with the home-built FM-AFM20 used in this work. The enhanced sensitivity of FM-AFM also obviates the need for the large interaction area used in static methods. The ill-characterized micron size colloidal probes could thus be replaced by sharp AFM tips having ~5-10 nm radius of curvature. Beside their atomically smooth surface (Fig. 2) the tips' miniature footprint alleviates the requirement for surface homogeneity over micron size areas. With both hurdles removed, we were able to measure the force at all distances and found that the hydrophobic attraction, seen in all experiments, turns at short distances into pronounced repulsion, in agreement with the neglected results depicted in the inset to Fig. 3(a). Moreover, using small (Ångstroms) cantilever oscillation amplitudes, we found that the hydrophobic attraction commences more abruptly than previously envisaged. Finally, taking advantage of the extreme sensitivity of our setup we found that the short-range repulsion was oscillatory with 0.5 nm period compared with the 0.3 nm characterizing hydration repulsion between two hydrophilic surfaces. In addition to its high force resolution, the FM-AFM mode facilitates simultaneous measurement of the dissipation rate incurred by the tip-surface interaction. The measurements revealed this way that the turn-on of attraction was accompanied by an abrupt enhancement of the dissipation rate. As the tip approached the surface, and attraction was replaced by repulsion, the dissipation rate hydrophobic (E. A. Vogler, "On the origins of water wetting terminology" in “Water in Biomaterials Surface Science”, M. Morra (Editor), Wiley 2001, p. 149).
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continued to grow monotonically. The overall dissipation rate was 100-200 times higher than expected from hydrodynamic drag in water, suggesting that the hydrophobic interaction was accompanied by an acute change in the medium viscosity. Katan and Oosterkamp21 were the first to utilize FMAFM in the study of hydrophobic interactions. They resolved the complete distance dependence of the force and reported that the attraction seen at long distances smoothly turns into repulsion at distances shorter than ~1 nm. Their surface, however, comprised hydrophilic and hydrophobic patches with sizes comparable to the AFM tip. Repulsion could thus be rationalized by hydration forces originating from the underlying hydrophilic patches. Furthermore, due to technical difficulties, the force curves were averaged over several nearby locations leading to a loss of site-specific information. Suzuki et al.22 were the first to report an oscillatory force profile measured next to a hydrophobic surface (highly oriented pyrolytic graphite - HOPG), albeit with a silicon tip. They attributed the 0.5 nm oscillation period to water structure near the HOPG surface, although this period was longer than the typical 0.25-0.3 nm one measures near hydrophilic surfaces. Kaggwa et al.23 measured the force acting between a hydrophobic methylated silicon surface and two types of tips: hydrophilic and hydrophobic ones. In contrast to the case of hydrophilic surface, they did not observe hydration oscillations and concluded that those were limited to hydrophilic surfaces, regardless of the nature of the tip. As shown below, and as reported by Suzuki et al.22, such oscillations are seen also with hydrophobic surfaces. It is also worth mentioning that their force curve taken with the hydrophilic tip showed an abnormally long-ranged repulsion extending up to 16 nm from the surface while the hydrophobic tip force curve showed an unusually shortranged attraction. Both the long-range repulsion in the hydrophobic-hydrophilic case and the extremely shortrange attraction in the hydrophobic-hydrophobic case contrast previous experiments that used either static or dynamic methods. They also disagree with the present report. The mixed findings in literature – attraction versus repulsion and their range, oscillatory or monotonic force profiles, and the question of surface quality motivated the present work. We report simultaneous measurements of the conservative and dissipative components of the interaction, conducted on two types of hydrophobic surfaces – self-assembled fluoro-silanated silicon surface (Perfluorodecyltrichlorosilane, FDTS) and atomically flat HOPG. Special attention was given to the surfaces quality, their atomic smoothness and the absence of deleterious nanobubbles. Both types of surfaces displayed similar behavior regardless of the fact that one comprised selfassembled molecules while the other was crystalline. The reported findings hence seem to be characteristic of the hydrophobic force and not an artefact of the surface used. The results allow critical evaluation of existing models for the hydrophobic interaction, as discussed below. None of these models accounts for all observations and the
closest is a variant of capillary evaporation9,24,25 modified by the known accumulation of dissolved gas molecules near hydrophobic surfaces26.
EXPERIMENTAL SECTION A commercial Dimension V (Bruker) AFM was used for comparative static force measurements. The cantilever deflection was recorded and later used to infer the force by multiplying it by the measured spring constant of the cantilever27. The home-built high-resolution AFM20 was operated in FM mode28. The setup20 displayed outstanding 8.5 fm/√Hz optoelectronic noise floor and a 6 pm rms positioning noise in the full operation bandwidth. The cantilever was photothermally excited using a power modulated, 405 nm laser diode focused on its base. The cantilever phase lag relative to the excitation signal was kept constant at 90o using a phase-locked loop (OC4, Nanonis) which adjusted the excitation frequency to the cantilever resonance frequency during the entire measurement. The resonance frequency shift, caused by tip-surface interaction, was measured and converted to the conservative component of the force using the Sader-Jarvis inversion method29. During the measurement, the cantilever oscillation amplitude was kept constant using a second feedback loop. The excitation power needed to maintain a constant amplitude was recorded and, together with the resonance frequency shift, converted to the dissipative component of the force using formula A5 in the Appendix. The oscillation amplitude was set in the 0.1-1 nm (zero-topeak) range in all reported graphs. Besides smoothing effects with higher amplitudes, no amplitude dependence on the force was observed (see Fig. 6). The horizontal axes of all the figures that report tipsurface distance denote the minimal approach distance of the oscillating tip. Zero distance in these figures corresponds to a user-defined threshold in the frequencyshift channel of ~5 kHz and might miss the real tip-surface distance by 1-2 Ångstroms. This margin was necessary in order to avoid tip crashing into the surface or potential damage to the monolayer. The force\dissipation measurements were carried out using a commercial liquid cell (Bruker multimode) filled with liquid. Deionized water ( 18.2M cm , Millipore) was used in all experiments except for the hydrophilic mica ones (see text). Water was degassed by vacuum pumping of 50 ml vials using a membrane pump for at least 1 hour prior to each experiment. FDTS monolayers were deposited on silicon substrates from the vapor phase. The silicon wafers were cleaned using piranha solution (3:1 H2SO4:H2O2) for 15 minutes, rinsed with deionized (DI) water and dried with nitrogen gas. The substrates were then placed in a vapor deposition machine (MVD100E, Applied Microstructures) where they were first cleaned and hydroxylated with 100W oxygen plasma for 60 seconds at 200 sccm flow rate, followed by silanization with FDTS gas (Sigma-Aldrich) and water vapor using 2 injection-deposition cycles, 15 minutes each. At the end of each injection cycle the
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chamber pressure was measured to be ~0.2 torr. The resulting advancing\receding contact angles (with DI water) were measured to be 112o\109o, respectively (Ramé-Hart 200). These monolayers are known to be molecularly smooth30 and stable for weeks31. X-ray photoelectron spectroscopy data of such FDTS sample are shown in the Supplementary Material (Fig. S3) as a testament for its high purity. HOPG samples (SPI Supplies) were prepared by exposing a fresh layer, immediately before each experiment, using an adhesive tape. The measured contact angle with DI water immediately after peeling was 60o. High resolution images of the FDTS monolayer and the HOPG surface are shown in Fig. 1. The force measurements shown here were carried out using two types of cantilevers: SNL-10, C (Bruker) for static mode and MSS_NCHAuD_13 (Nanotools) for FM mode. The corresponding nominal spring constants were 0.3 N/m and 40 N/m, respectively. The spring constants of the individual cantilevers were measured by fitting the thermal noise spectrum of the cantilevers far from the surface but still in liquid27. The MSS_NCHAuD_13 resonance frequency in water was ~130 kHz with quality factor of 7 at ~100 nm from the surface. Another cantilever (AC-55, Olympus) with a nominal spring constant of ~80 N/m and a resonance frequency in water of 1.3 MHz was also used with similar results (see Fig. S2, Supplementary Material). The SNL-10 cantilever had a silica tip that was cleaned prior to each experiment with 100 W oxygen plasma for 60 seconds (MVD100E, Applied Microstructures), while the MSS_NCHAuD_13 had a diamond-like carbon (DLC) tip (see Fig. 2 for TEM image of the tip), cleaned with UV light for 5 minutes. The DLC was measured elsewhere32 to have contact angles in the 74o-88o range. Force curves were also taken for a reference over a hydrophilic muscovite mica (Grade V1, SPI). Fresh mica surfaces were exposed prior to each experiment by peeling layers with an adhesive tape. In order to suppress the range of double-layer forces due to charging of the mica surface, experiments incorporating mica were conducted in 100mM NaCl solution in DI water. Silica tips mounted on the same cantilever as the DLC ones were used (pppNCHAuD, Nanosensors) in this case.
RESULTS AND DISCUSSION A. Topographic imaging The history of hydrophobic force spectroscopy teaches that surface quality is of utmost importance3,11. Special care was therefore taken to prepare atomically flat surfaces, free of nanobubbles or impurities. At the beginning of each experiment the sites where force spectroscopy was later carried out were imaged at an increasing level of magnification and in many cases the same area was imaged again during or after the measurement. The potential generation of nanobubbles or nanopancakes has been established33,34, particularly on rough surfaces35,36 and those are notorious for producing
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misleading artefacts. We were therefore careful to confirm their absence on both the FDTS and HOPG surfaces. Figures 1(a-c) depict three topographic images of an FDTS surface in an increasing order of magnification. The FDTS monolayer thickness has been measured by covering part of the silicon surface with photoresist film while depositing the molecules and measuring the resulting topographic step after removal of the photoresist (Fig S1). The measured average thickness was 1.6 0.1 nm, in accord with the molecule length and previous studies37. The lowest magnification (a) discloses surface roughness of 0.11 nm rms across the entire 1 1μm 2 field, similar to other reported values30. The highest magnification (c) and the cross-section along the white line is depicted in (f), disclosing roughness of 0.06 nm rms across the 25 25 nm2 field. The latter figure is relevant for estimating the roughness across the tip footprint, a 510 nm diameter semi-sphere (Fig. 2). The quality of the self-assembled monolayer and imaging resolution can be appreciated from the discerned molecular rows of FDTS molecules spaced by 0.6 nm, which are seen in (b) and (c). The rows are responsible for most of the observed surface modulation. Figures 1(d,e) depict similar topographic images of the HOPG surface. These images were taken in different locations on the surface at two different magnifications. They uncover the graphite terraces and the atomic flatness in between two terraces, as reflected also by the crosssections along the lines in (d) and (e) plotted in (f). Since the "roughness" is so much smaller than the size of an atom we attribute it to system noise rather than to actual surface roughness. Similar images were seen across all FDTS and HOPG surfaces. Importantly, they never showed signs of nanobubbles or nanopancakes down to sub-atomic resolutions. It is important to note that while taking these images the cantilever oscillation amplitude was set to ~1-4 Å peak-topeak. This amplitude was low compared with the amplitude range used in previous works. We speculate that the higher amplitude used in these works might have created the nanobubbles since the higher forces exerted by the tip might have been high enough to overcome the nucleation barrier for nanobubble formation. This hypothesis will be tested in future experiments. The smoothness of the DLC cantilever tip can be judged from the high-resolution TEM image depicted in Fig. 2. The extracted radius, 7 1 nm, lies comfortably in the manufacturer specified curvature range, 5-10 nm. As seen, the surface roughness was limited to single atoms. Taken together, Figs. 1 and 2 prove the quality of both surfaces and the tip, ruling out surface roughness and nanobubbles as potential sources of artefacts.
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B. The conservative force
Fig. 1. (a-c) Topographic images of the FDTS surface under water, taken at an increasing level of magnification. (d,e) Topographic images of HOPG taken in different locations and magnifications on the sample, revealing the well-known HOPG atomic terraces. (f) Line profiles corresponding to the white lines shown in (c), (d) and (e).
Fig. 2. TEM image of a DLC tip. The tip was imaged aspurchased. The red overlaid circle indicates the approximate tip curvature. The inset shows a zoom on the tip apex, proving that the tip is indeed atomically smooth.
Fig. 3(a) depicts complete force vs. distance curves measured by FM-AFM with an FDTS surface. The approach speed was slow, about 1 nm / sec . The main panel shows multiple individual curves taken with a 40 N/m spring constant cantilever having a hydrophobic DLC tip. The force curves were taken at arbitrary locations, immediately after imaging the surface, to exclude the presence of nanobubbles. Three typical force-curve families (nothing special about these particular families) are marked by different colors, all showing an attraction commencing at 2-4.2 nm turning into repulsion at shorter distances. The green and yellow families were measured using the same cantilever and substrate but at different times from the beginning of the experiment (~1 hour inbetween them) which probably caused a slight shift in the tip location over the surface due to piezo drift. The red family was measured with a different tip and substrate. Similar results were obtained in half a dozen experiments carried out with different substrates and cantilevers of the same type. The force displayed marked variations between successive measurements within each family of curves and between different families. The switching from one family to the other occurred spontaneously from time to time. We found that such broad variations in the measured force characterize hydrophobic surfaces (both FDTS and HOPG). In the case of hydrophilic surfaces, such as the muscovite mica surface shown in Fig. 4(a), the variations were significantly smaller. The large variation between curves creates the impression of noisy measurements but this was not the case. As seen, despite the difference between them, the individual curves are smooth. This variation seems to be intrinsic to the hydrophobic interaction and we return to it in the Discussion section. The dashed black line in Fig. 3(a) corresponds to the non-retarded van-der Waals (vdW) attraction, calculated using Eq. 11.13 in Ref. 38, between a 5 nm radius DLC sphere ( 5 ) and an infinite PTFE plane ( 2.1, assumed close to FDTS) across water, resulting in a 2.42 1020 J Hamaker constant. The dot-dashed curve corresponds to an unrealistic upper bound on the vdW attraction corresponding to a 20 nm sphere radius and a Hamaker constant of 1 1019 J . Even this upper bound shows a much smaller attraction than observed experimentally. One is therefore led to conclude that the vdW attraction plays a minor role in these experiments. Electrostatic double-layer repulsion is also small due to the low charging of the substrate and the tip and the low ion concentration in DI water. Since both double-layer repulsion and vdW forces are negligible, the measured interaction is safely attributed to the hydrophobic force, regardless of its origin. The inset to Fig. 3(a) shows, on top of sample force curves measured by FM-AFM, several force curves measured by SFA or static AFM. Those include our unfiltered, static force
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curve measured with a soft cantilever (blue line), data retrieved from Fig. 5A of Ref. 16 (gray circles), data from Figs. 2 and 4 from Ref. 18 (purple diamonds and magenta triangles, respectively), and data retrieved from Fig. 7 in Ref. 19 (red squares). Since the latter three data sets were previously acquired with a colloidal probe or SFA techniques, and were normalized by the probe radius we multiplied them here by an arbitrary factor to fit the scale
to the other results. Fig. 3. (a) Force-distance curves over an FDTS monolayer in DI water. Main panel: individual force vs. distance curves measured by FM-AFM and a hydrophobic DLC tip. The right ordinate depicts surface energy density, calculated from the force, in the Derjaguin approximation38, assuming a 5 nm radius tip. Dashed\dotdashed curves – nominal\upper bound prediction of vander Waals attraction, respectively. The arrows mark the abrupt turn-on of attraction (see text). Inset: solid lines – representative individual force curves. Blue line - force curve measured between a soft cantilever (SNL-10 C, Bruker, measured spring constant 0.3 N/m) and an FDTS monolayer in DI water using a commercial Dimension V AFM (Bruker) in static mode. Gray circles were reproduced from Fig. 5A in Ref. 16. Purple diamonds and magenta triangles were reproduced from Figs. 2 and 4, in Ref. 18 respectively, and red squares from Fig. 7 in Ref. 19, all three scaled by a multiplicative factor (see text). These five force curves show the typical JTC, which hides b
The medium uniformity is questionable at short distances and consequently also the Derjaguin approximation. Yet,
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the true hydrophobic interaction at short distances. (b) Normalized dissipation rate vs. distance curves. Main panel: Individual normalized dissipation rate curves (defined by Eq. A5 in the Appendix), corresponding to the same data sets as (a). Blue line – normalized dissipation rate over a hydrophilic muscovite mica surface in 100mM NaCl, measured with a silica tip. Black line – theoretical normalized dissipation rate due to Stokes' drag on a 10 nm radius sphere oscillating near a solid wall with non-slip boundary condition. The curve was calculated using Eqs. 2.18 and 2.19 in Ref. 39. Inset: Magnified view of the main panel. The vertical dashed lines serve as a guide-to-the-eye for comparing the onset of attraction in (a) with the onset of dissipation increase in (b). The data from these references show that the long-range attraction leading to JTC, as well as the short-range repulsion disclosed by the FM-AFM data, are consistent with previous static colloidal AFM measurements. The same traits were reproduced by our static measurements (blue line), including the JTC at ~3.2 nm, marked by an arrow. The tip jump to the repulsive range left an unavoidable blind gap in the force curve between 0.5 and 3.2 nm. In all data sets the tip jump stops at 1-2 nm from the surface and as the two surfaces further approach each other, a clear repulsive component of the force appears. The collection of curves depicted in the inset to Fig. 3(a) thus prove that short-range repulsion is not foreign to force spectroscopy between two hydrophobic surfaces. Interestingly, this feature was overlooked in the literature, possibly due to the JTC preceding it or due to association of this repulsion with probe asperities and a mundane short-range repulsion19 upon touching the surface. Based on the similarity in range and in the distance dependence, we tend to associate the repulsion observed by static methods with the repulsion seen by FM-AFM but we cannot exclude asperities touching the surface. Definitive distinction between the two potential explanations requires then a detailed microscopic characterization of the colloidal probe used in a particular experiment. The right ordinate in Fig. 3(a) depicts the corresponding surface energy density calculated by the Derjaguin approximationb, assuming a 5 nm radius tip. For comparison, the areal energy density associated with bulk water hydrogen bonds (assuming 4 hydrogen bonds per water molecule) is about 150 mJ/m2, in the same range as the measured hydrophobic interaction and about an order of magnitude higher than typical tip-surface dispersion interaction in water. For completeness, the average force curves for three families of curves are depicted in Fig. S2 together with the force measured with a higher resonance frequency cantilever. The corresponding potential of mean force (PMF), obtained by integrating these force curves, is displayed in Fig. S3. The position of minimal PMF rests in the different samples between 0.3 nm and 1.2 nm.
we use it for approximate extraction of the interaction energy density from the force.
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The turn-on of attraction judged from the average force curves, Fig. S2, is relatively smooth due to averaging over many individual curves, each slightly shifted relative to the others. Individual curves, on the other hand, turn on steeply (see arrows in main panel of Fig. 3(a)) suggesting that attraction commences abruptly rather than continuously. We return to this important point below. C. Dissipation rate Fig. 3(b) depicts several normalized dissipation rate curves, associated with tip-surface interaction (Eq. A5), taken simultaneously with the curves in Fig. 3(a). Similarly to the conservative force, the variation between successive curves within a family as well as between families is significant. Averages over numerous such curves are depicted in Fig. S2. The curves presented in Fig. 3(b) were measured with an order-of-magnitude different amplitudes and, hence, an order-of-magnitude different tip velocities. The fact that the normalized dissipation rate was similar for all curves proved that the dissipated energy per cycle scaled quadratically with velocity, as expected. For comparison, the solid black line in Fig. 3(b) depicts the theoretical normalized dissipation rate due to Stokes' drag on a 10 nm radius sphere, oscillating with 0.5 nm amplitude perpendicularly to the surface. The calculation assumes no-slip boundary condition (partial-slip boundary condition will result in even lower dissipation rate) and bulk water viscosity, 0.894 mPa·s. Each point in the graph shows the drag force on the sphere39, multiplied by the instantaneous velocity and averaged over a complete cantilever cycle. The fact that the measured dissipation rate curves are two orders of magnitude larger than the computed one suggests that the viscosity near the surface is two orders of magnitude higher than that of bulk water. For reference, the blue curve depicts the normalized dissipation rate measured with the hydrophilic muscovite mica surface. With the exception of a dissipation peak at 0.2 nm, the measured dissipation rate agrees well with the theoretical prediction for Stokes' drag and as such, is almost two orders of magnitude smaller than for the hydrophobic surfaces, FDTS and HOPG. Note that dissipation commences simultaneously with the attractive conservative force (see dashed lines connecting Fig. 3(a) and Fig. 3(b)), indicating that the same mechanism underlies both sets of data. Any theory of the hydrophobic interaction should also account for this fact. D. Measurements with HOPG Similar force curves, obtained in DI water using a DLC tip and an HOPG surface are shown in Fig. 4(a). The corresponding dissipation rate curves are depicted in Fig. 4(b) together with a magnified view of the boxed area. Other than the somewhat smaller force magnitude, dissipation rate, and range, these curves resemble the FDTS ones shown in Figs. 3 and S2. Here, again, dissipation commences abruptly and simultaneously with
attraction. As seen in the magnified view, what seems as a sharp turn-on in the averaged curve reflects a collection of dissipation rate peaks in the individual curves, taking place at slightly different distances. The peaks disclose in an even sharper way that the turn-on of attraction is associated with an acute event taking place in the medium. Since the contact angles of HOPG and DLC are smaller than 90o while that of FDTS is larger, the similarity between the FDTS and HOPG data suggests that 90o contact angle does not play any special role. The inset to Fig. 4(a) compares the average force measured with degassed and non-degassed water. Although the two curves are qualitatively similar, the attraction in the degassed case starts at a somewhat shorter distance. This trait repeated itself in other comparative experiments with HOPG and FDTS. Interestingly, the short-range repulsion was usually similar in both cases.
Fig. 4. (a) Thin red and green lines – individual conservative force curves taken with FM-AFM over HOPG with a DLC tip in DI water. Thick lines – averages over the thin lines of the same color. Thin blue lines – individual force curves taken over muscovite mica in 100mM NaCl solution with a silica tip. Inset: comparison between averaged force curves measured with degassed and non-degassed water in two different experiments (curves of experiment #2 were shifted vertically for clarity). (b) Thin lines – individual normalized dissipation rate curves measured simultaneously with the conservative force curves depicted in (a). Thick lines – averages of the individual curves with matching colors. The inset shows a magnified view of the boxed area. The vertical dashed
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lines serve as a guide-to-the-eye for comparing the onset of attraction in (a) with the onset of increased dissipation in (b).
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nm periodicity in the muscovite mica case is attributed to water layering induced by the hydrophilic surface. F. Amplitude Dependence
E. Frequency shift oscillations The conservative force is deduced in FM-AFM from the shift in resonance frequency of the cantilever due to tipsample interaction. The conversion of frequency-shift to force involves integration of the frequency-shift data29, thereby smoothing itc. It is therefore instructive to look also at the raw frequency-shift data for additional hints to the origin of the hydrophobic force. Fig. 5 shows such raw data for both FDTS and HOPG. For reference, the green curve depicts force oscillations measured by FM-AFM over a hydrophilic muscovite mica surface in 100mM NaCl solution using a silica tip. These oscillations are commonly attributed to hydration layers40,41, with a peak spacing corresponding to the size of a water molecule, 0.28 nm. Some (~18%) of the FDTS and most (~63%) HOPG frequency-shift curves showed similar oscillations. The periodicity, however, was longer, ~0.5 nm, the oscillations persisted to longer distances, and the oscillation amplitude was significantly larger than with mica. As mentioned in the Introduction, similar oscillations were also reported for HOPG with a silica tip by Suzuki et al22. At the same time, they contradict the claim by Kaggwa et al.23 that hydrophobic surfaces do not show frequency-shift oscillations. Note the striking similarity between the FDTS and HOPG oscillations, even when using different types of cantilevers and tips, and their contrast with the hydrophilic mica case. This point is explored in the Discussion section.
Fig. 5. Raw frequency-shift vs. distance curves of the DLC tip over FDTS and HOPG and an AC-55 (silica) tip over FDTS. The corresponding curve measured with a silica tip over muscovite mica under water with 0.2 nm (peak-topeak) oscillation amplitude is shown for comparison. All curves show short-range structure near the surface. The 0.3
The observation of frequency-shift oscillations required low cantilever oscillation amplitudes, typically below 0.3 nm peak to peak, and preferably an atomically flat surface. Several measurements taken over an HOPG surface at various oscillation amplitudes are depicted in Fig. 6. All four measurements were taken at the same position over the surface within ~5 minutes. Fig. 6 (a) depicts frequencyshift curves measured with oscillation amplitudes of 150 pm, 200 pm, 400 pm and 700 pm (peak to peak). The oscillations are clearly visible in the 150 pm curve and diminish slightly at 200 pm. The 400 pm curve displays a faint trace of oscillation while the 700 pm curve no longer shows oscillations. The two extreme cases: 150 pm and 700 pm, can be compared by examining the solid red and dashed blue curves. In addition to the diminishing oscillations, the turn-on of attraction at 2.2-2.3 nm is clearly sharper with the smaller cantilever amplitude.
Fig. 6. Frequency-shift and normalized dissipation curves over an HOPG surface taken with different oscillation
c
The frequency-shift-to-force conversion formula also has a frequency shift derivative term that adds high frequency noise to the force.
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amplitudes. (a) Solid lines: frequency-shift curves translated vertically by 1.5 kHz for clarity. The dashed line is the same as the 700 pm line with no vertical translation. (b) Solid lines and dashed line: normalized dissipation curves with same description as in (a). The normalized dissipation curves of the same data sets are depicted in Fig. 6 (b). Much like the frequency-shift curves, they show sharper turn-on with decreasing oscillation amplitude. G. Discussion The results outlined in the previous sections extend past measurements in three essential ways. First, being free of JTC they provide force vs. distance data in the full distance range. Second, they supplement the measured conservative force with the dissipative force, namely, the out-of-phase component, which is proportional to the tip velocity. Finally, taking advantage of the outstanding sensitivity of the microscope, measurements done with small cantilever oscillation amplitudes reveal molecular structuring of the interface layer. As stated in the introduction, with the exception of Refs. 21-23, previous studies could not probe the full interaction range due to cantilever JTC. In the absence of dedicated data, it was implicitly assumed that the hydrophobic interaction was attractive all the way to contact. In contrast, the FM-AFM force measurements depicted in Fig. 3(a) and Fig. 4(a), as well as in the work by Katan and Oosterkamp 21 disclose a pronounced repulsion at short distances while still displaying the familiar long-range attraction. This repulsion, observed also by static methods (inset to Fig. 3(a)) has commonly been ignored. The FM-AFM data show that the attraction turns into repulsion over a range of a few nanometers and the range of repulsion is typically ~1nm. The short-range repulsion seems therefore as intrinsic to the hydrophobic interaction as the longer range attraction. Short-range repulsion - First, we rule out direct hardcore interaction between the tip and the surface as a possible explanation for the short-range repulsion. Figs. 1 and 2 teach that all surfaces were smooth on a subAngstrom level while Figs. 3(a), and 4(a) teach that repulsion was observed already at distances larger than 1nm when the two surfaces could not have touched. Second, the short-range interaction was found to decay exponentially with distance with a scale longer than the atomic one, and specifically longer than hard-core repulsion (typically, ~0.1-0.2 nm). Third, with small amplitudes the short-range interaction showed 0.5nm oscillations, suggesting a structured medium between the two surfaces. Finally, the range of repulsion was significantly larger compared with the hydrophilic case while the hard-core repulsion should have been equal in the two cases. Given these findings we can safely exclude
hard-core repulsion as an explanation for the observed repulsion. Short-range repulsion is frequently seen with hydrophilic surfaces where it is attributed to the tightly bound hydration layer. In high resolution force spectroscopy, such as the mica curves of Fig. 4(a), the hydration repulsion is accompanied by pronounced ~0.3 nm oscillations that reflect water structuring between the two approaching surfaces. Molecular dynamics calculations42 show that water stacks also against hydrophobic surfaces, forming first and second hydration layers that are similar to those next to hydrophilic surfaces, with the main difference being enhanced density fluctuations. A recent calculation43 shows that the structure of interfacial water is indeed similar for hydrophilic and hydrophobic surfaces, except for an intermediate fluctuating vapor layer separating the liquid water from the hydrophobic surface. In either case, the expected hydration oscillation period should be similar to that observed with hydrophilic surfaces. The experimental results look quantitatively different. The short-range repulsion observed with hydrophobic surfaces extends to longer distances, compared with hydrophilic mica, and display ~0.5 nm oscillations (Figs. 5, 6(a)) rather than the 0.3 nm ones seen in the hydrophilic case. The hydrophobic surface therefore seems to be solvated d, as predicted, but not by pure water. The hydration repulsion in the hydrophilic case is fitted many times with an exponential function18. As seen in Fig. S5, the same functional dependence also fits the repulsion in the hydrophobic case with characteristic lengths varying between 0.4-0.65 nm. Interestingly, these lengths are comparable to the oscillation period, 0.5 nm, while the decay length in the case of mica, 0.28 nm (Fig. S5), equals its hydrophilic hydration oscillation period. We see then that in both cases the range of repulsion is similar to the oscillation period. It is worth noting that short-range repulsion between hydrophobic surfaces emerges naturally in a lattice-gas model whenever the bulk solvation free energy of solute molecules is intermediate between their solvation energy when placed next to each of the two flanking surfaces44. The solute in our case contains dissolved gas molecules15,45–48 whose attraction to FDTS is stronger than to DLC, and then to DLC compared with HOPG. The observed short-range repulsion may thus reflect the different hydrophobicity level of the surface and tip used in a given experiment. Long range attraction - The Hydra model of Donaldson et al.18 asserts that the hydrophobic attraction grows exponentially with distance as the two surfaces approach each other. Such a dependence has been observed in many experiments starting with the first SFA measurements5. It has also been reported in the FM-AFM measurement by Katan and Oosterkamp21, albeit with averaged force curves and large cantilever oscillation
d
The term "solvation" as opposed to "hydration" is meant to reflect the fact that the medium between the two surfaces may not be pure water.
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amplitudes, 1-3 nm. At the same time, most fits were done over a small range of force and the claimed exponential dependence was by no means universal. For instance, Mastropietro and Ducker49 found good agreement with vdW attraction which is inverse power law. When measured with a relatively high cantilever oscillation amplitude, 0.7 nm peak to peak or larger, or when dealing with averaged curves, the attraction could indeed be fitted over a limited range with an exponential function also here. However, with smaller amplitudes, the turn-on of attraction becomes increasingly sharper as evident from the individual force curves displayed in Fig. 3(a) and the frequency-shift data of Fig. 6(a). The same sharpening with reduced oscillation amplitude is also seen in the dissipation curves of Fig. 6(b). We were therefore led to conclude that with small amplitudes, attraction and its accompanying dissipation commence more abruptly than previously envisaged. In the 150 pm curves of Figs. 6(a) and 6(b), for instance, the frequency-shift varies from zero to its minimal value within 0.2-0.3 nm and the dissipation grows substantially in this range. This abrupt onset, taking place when the two surfaces are still more than 2 nm away, suggest a conspicuous change in the properties of the medium, taking place abruptly below a certain distance. The existence of such transition is also consistent with the large variation between successive force curves seen in Figs. 3(a) and 4(a). Averaging (Fig. S2) clearly smoothen the abrupt turn-on of individual curves. The smooth turn-on of attraction observed in previous experiments may thus be caused by the commonly practiced averaging over multiple force curves and/or an inherent averaging caused by the large probes used by static method measurements and the large oscillation amplitudes used in dynamic techniques. Interestingly, in the case of hydrophilic surfaces, such as mica (Fig. 4(a)), successive force curves repeated each other faithfully. The large variations hence seem characteristic exclusively of the hydrophobic case, indicating sensitivity of the hydrophobic attraction to local details. Those may include adsorbed air molecules and/or local defects. Anomalously high dissipation - The two orders of magnitude enhancement in dissipation rate near hydrophobic surfaces compared to hydrophilic surfaces indicates a dramatic change in the properties of the medium between the tip and the surface. The simultaneity between this change and the onset of attraction proposes that attraction may commence as a result of this conspicuous change in the medium properties. Effect of degassing – The partial removal of dissolved gasses by pre-pumping of the water in a vacuum chamber has a clear effect on the measured force. As seen in the inset to Fig. 4(a), degassing shortens the range at which attraction commences and most of the time reduces its magnitude. This effect has been observed in previous static measurements by both colloidal AFM and SFA (e.g., Refs. 17, 19). The effect of degassing on the short-range repulsion has never been reported before. We find that the length characterizing the exponential repulsion hardly changes upon degassing. This finding concurs with a
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lattice-gas model44 where the concentration of dissolved hydrophobic molecules near a hydrophobic surface is found to decay with a characteristic length that depends essentially on their solvation energy, independent of their concentration (as long as it is small compared with water). Possible mechanisms - Equipped with these observations, we turn to examine several proposals for the mechanism underlying the hydrophobic interaction. One of the early proposals for the origin of the observed hydrophobic attraction was tip penetration into a nanobubble already sitting on the surface12. To exclude such possibility, we have imaged the surfaces before and after measuring the force (section A and Fig. 1) at increasing magnifications and height resolutions down to a fraction of atomic size. We never observed any sign of nanobubbles or nanopancakes and therefore exclude them as a potential effector on the results. Note that the cantilever oscillation amplitude in our experiments was exceptionally small and the exerted force on the solution was correspondingly minute compared with static or tapping mode techniques. It could be that those significant forces created nanobubbles in previous experiments but not in the present one. Another proposal9,24,25,50 suggested cavitation transition taking place when the tip approached the surface. Water evaporation leads in this picture to the formation of vapor capillary bridging between the tip and the surface. Hydrophobic attraction in this case is attributed to the resulting capillary force9,25. The cavitation scenario fits most aspects of the measured data such as the magnitude of the conservative force being consistent with the water surface tension coefficient given the expected capillary radius, and the abruptness at which attraction and dissipation commence, in accord with a sudden capillary nucleation. Being associated with a nano-scale phase transition, which naturally depends on local features such as defects, presence of gas molecules, and line tension, cavitation is also consistent with the significant variations between force curves measured with the same surface and tip. Finally, recent measurements of the dissipation associated with periodic modulation of air-water surface area disclosed enhanced frequency dependent dissipation, which was too large to be accounted for by bulk water properties51. The authors of Ref. 51 suggested that the excess dissipation stems from a complex surface tension coefficient which may explain the large dissipation seen here following the cavitation transition and the formation of a new water-vapor interface. Within the cavitation scenario, the tip oscillation should lead to an oscillatory variation in the capillary surface area and hence, assuming a complex surface tension coefficient, to dissipation. The complex surface tension coefficient proposal has recently been questioned52 and requires further studies. Notwithstanding the agreements listed above, the cavitation scenario is also met with several difficulties. First, it was shown9,53 that cavitation should be hindered by a significant nucleation barrier. Recent simulations54 find that the actual barrier is significantly lower than expected from a macroscopic calculation but for the
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experimental distances where attraction commences it is still higher than 50k BT . Second, at least on a simplistic level, cavitation should be limited to surfaces characterized by water contact angles larger than 90 . This condition was not met with many of the surfaces where JTC was found, and was far from being fulfilled with the HOPG surface and DLC tip of Fig 4. Finally, it is difficult to see how capillary forces alone yield short-range repulsion. Donaldson et al.18 mentioned that when two PDMS surfaces approached each other, while oscillating at 5 kHz, a cavitation transition took place at distances as large as 100 nm (the oscillation amplitude was not mentioned there). This observation suggests that substrate oscillations helped overcoming the large barrier for cavitation and as such, it might suggest that the abrupt transition found in our experiments was promoted by the driven cantilever oscillations. While we cannot rule out such a scenario, we note that the conservative force and the distance at which attraction commenced, were independent of oscillation frequency and amplitude down to ~0.1 nm peak to peak. Note that this amplitude was significantly smaller than the thermal fluctuations characterizing the soft cantilevers used with static AFM. Attard suggested9 in the context of capillary evaporation that the repulsive force observed in some experiments originated from shrinking and expansion of the vapor capillary when the tip approached and retracted from the surface. The force in this case should have been proportional to the tip velocity. Unlike previous static methods, FM-AFM distinguishes between the conservative force, being independent of velocity, and the dissipative force, scaling linearly with velocity. The effect proposed by Attard should thus have shown in the dissipation rate data rather than in the conservative force. In fact, it might account for the anomalously large dissipation rate reported here (a quantitative calculation of dissipation due to this effect is required). The conservative force curves were essentially independent of cantilever oscillation amplitude between 0.1-1 nm, except for better resolution of the short-range oscillations and the sharper onset of attraction with smaller amplitudes depicted in Fig. 6. At a fixed frequency, this amplitude range corresponded to an order of magnitude variation in tip velocity. Furthermore, a different cantilever (AC-55, Olympus) with ~1.3 MHz resonance frequency in water yielded the same conservative force as the 130kHz cantilevers (Fig. S2), indicating again that the force was independent of an order of magnitude increase in tip velocity. We were therefore led to conclude that the hydrodynamic mechanism proposed by Attard may have shown in the dissipation but not in the conservative force. The observed short-range repulsion required then a different explanation which we could not find within the simple cavitation scenario. The fact that gas molecules dissolved in water tend to accumulate at hydrophobic surfaces15,17,45–48,55–58 is well established. Such gas layering on top of HOPG was reported recently under water supersaturated with gas46,47 and is also believed to serve a precursor of nanobubble
formation and stability46,48,59–61. Since gas molecules spread to a few nanometer distances from the hydrophobic surface, they may provide the necessary nucleation center for capillary formation, lowering the free energy barrier for such a transition to realistic values. Indeed, a recent molecular dynamics simulation26 of the force acting between two graphene sheets in the absence and in the presence of dissolved gas found that in the latter case a gas capillary connecting the two surfaces forms spontaneously when the two surfaces approach each other closer than 3.8nm . Interestingly, in accordance with the experimental results, the attractive force (minus derivative of the PMF depicted in Fig. 6 of that reference with respect to distance) commences abruptly and the gas molecules in the gap between the two surfaces form molecular layers. Unlike the experimental result, the calculated force between the two graphene sheets remains attractive at all distances, even when a hard-wall repulsion between the two graphene sheets should have been observed. With gas accumulation, we attribute the experimental repulsive force at short distances to solvated gas molecules adhering to the hydrophobic surfaces, much like hydration repulsion between two hydrophilic surfaces. As the tip is pushed closer to the surface, it works against partially ordered layers of solvated gas, consistent with the frequency-shift data shown in Figs. 5, 6(a). The remarkably similar oscillations in the FDTS and HOPG curves might indicate spatial arrangement of hydrated gas molecules or gas clathrates near the hydrophobic surface. The larger size of these objects, compared with water molecules, could account for the longer periodicity compared with hydrophilic hydration. None of the last two scenarios, gas accumulation and cavitation transition, accounts by itself for all features. Cavitation explains the attraction, its abrupt onset, the marked variations between consecutive force curves, and potentially the large dissipation rate. Air accumulation is consistent with repulsion, molecular layering, and reduced barrier for the cavitation transition. To account for the full experimental data we propose a hybrid model where the cavitation transition is assisted by a relatively sudden concentration of accumulated air molecules to form a gasrich bridge between the tip and the surface as seen in the simulation of Ref. 26. The resulting capillary forces account then for the abrupt attraction, the associated enhanced viscosity and dissipation, and the graded gas concentration accounts for the short-range repulsion. Finally, the ordering of the solvated gas molecules may explain the observed 0.5 nm oscillations seen in the shortrange frequency shift data.
CONCLUSIONS Using a high resolution, low noise AFM, combined with stiff cantilevers and well-characterized hydrophobic surfaces, we were able to measure the full distance dependence of the hydrophobic force. Dynamic FM-AFM mode enabled us to measure the conservative part of the force as well as the dissipated energy per cycle of the oscillating tip. Using hydrophobic DLC tips, which charge
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only weakly in water, the electric double layer contribution to the force was safely excluded. Contribution from the vdW force was also found to be small in the measured distance range, leaving hydrophobic interaction as the dominant force. In agreement with previous measurements, the long-range interaction was found to be attractive but at shorter distances, where static methods fail, we found that the hydrophobic force turns repulsive for both FDTS and HOPG. In contrast to data averaged over multiple force curves or force measured over large areas, the individual force curves measured with our sharp tips and small cantilever amplitudes showed a fairly abrupt turn-on of attraction. The dissipation rate was also found to rise abruptly, at the same distance where attraction commenced, to values which were two orders of magnitude higher than expected from bulk water hydrodynamics or the measured dissipation rate next to hydrophilic surfaces. This anomalously high viscosity was again characteristic of both FDTS and HOPG. In many cases where low cantilever excitation amplitudes were used, the repulsion was accompanied with clear oscillations that were larger in magnitude and periodicity compared with oscillations originating from hydration forces next to hydrophilic surfaces. Extensive theoretical work elucidated the hydration of hydrophobic surfaces in the absence of dissolved gas4,42,43,54,62, but the resulting range of water density modulation was found to be considerably shorter than the experimental range of hydrophobic attraction, indicating that water density modulation alone probably cannot account for the hydrophobic force. Parallel experiments and calculations pointed to the importance of dissolved gas molecules15,17,26,45–48,56–61. Specifically, it was found that gas molecules tend to accumulate near hydrophobic surfaces and that degassing shortens the range of hydrophobic attraction. An independent line of research concentrated on the cavitation transition in the absence of gas molecules9,53,54, taking place between two hydrophobic surfaces, but those studies generally led to the conclusion that the free energy barrier associated with the nucleation of a stable vapor capillary is too high. Our experiment and the single theoretical work by Qing-Qun and Jie-Ming26 strongly suggest that accumulated gas molecules near the hydrophobic surfaces may support the instantaneous formation of a gaseous capillary spanning the gap between the two surfaces with a low nucleation barrier. As such, they may finally account for the elusive hydrophobic interaction. We hope that our findings will encourage further experimental and theoretical investigations focused on this scenario, which up till now attracted little attention. We conclude with a word of caution about the fact that the tip footprint in our experiments was orders of magnitude smaller than that used by colloidal AFM or SFA. Although the results seem qualitatively similar (where they can be compared), there could also be significant scale-dependent differences, for instance in the dehydration physics or in capillary formation. We are not aware of studies of the hydrophobic interaction as a function of probe size. It was implicitly assumed to scale with the radius (aka Derjaguin approximation) but our
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limited experience with AFM tips vis-à-vis colloidal probes indicates a weaker dependence upon probe radius. We find it interesting that the range of attraction measured with very different probe sizes (from millimeters to nanometers) vary in a remarkably small range, 3 12nm .
SUPPORTING INFORMATION The supporting information is available free of charge on the ACS publication website at DOI: XXXXXXXXXXXXX. Fig. S1. FDTS monolayer height measurement. Fig. S2. Averaged FDTS conservative force and relative dissipation curves. Fig. S3. Potential-of-mean-force (PMF) curves. Fig. S4. X-ray photoelectron spectroscopy (XPS) data for the FDTS sample. Fig. S5. Exponential short-range force fits for FDTS, HOPG and mica.
AUTHOR INFORMATION Corresponding Author *E-mail
[email protected] ACKNOWLEDGEMENTS This research was supported by the Israeli Science Foundation through grant number 10/1051 and the single molecule I-Core center of excellence, grant number 12/1902.
APPENDIX: NORMALIZED DISSIPATION RATE Assume a cantilever of mass frequency
0
m
and natural resonance
oscillating over a surface28. Its equation of
motion reads
mz z m0 2 z
F cos t Fc z z z, A1 where z, , F , , Fc , are the tip position, hydrodynamic damping of the cantilever devoid of tip-sample interaction, driving force amplitude of the cantilever, driving frequency, conservative component of the interaction force, and generalized damping coefficient due to tipsample interaction. Multiplying Eq. A1 by z and averaging over one cantilever oscillation period gives
2
2 /
z 2 dt
0
2
2 /
0
F cos t z z z 2 dt.
A2
Here, was assumed constant within the small cantilever oscillation amplitude and the inertial and conservative forces dropped out since z and z were orthogonal over a complete tip oscillation. The tip position is given by
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z t A cos t ,
A3
where A, are the oscillation amplitude and phase lag relative to the driving force. In FM-AFM, the phase lag is held constant at 2 using a phase-locked loop. Substituting Eq. A3 into Eq. A2 and evaluating the integrals one finds AF A2 2 W
2
,
2 / z z 2 dt is the dissipation rate due to 2 0 tip-sample interaction, averaged over one cantilever oscillation. Several nanometers from the surface the dissipation due to tip-sample interaction vanishes. At that point, we set the reference frequency, 0 , the reference cantilever dissipation coefficient, m 0 Q0 , where Q0 is the quality factor near the surface, without tip-surface interaction, and the reference driving force F0 A0 mA0 2 / Q0 . In terms of the normalized frequency shift, 0 / 0 , and the normalized cantilever excitation force, X F F0 / F0 , Eq. A4 reads W X X 1 2 , A5 W0 m0 2 A2 0 is the bulk dissipation rate when 2Q0 the tip is far from the surface. This quantity is plotted in Fig. 3(b), 4(b) and 6(b).
where W0
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