Size Dependent Transitions in Nanoscale ... - ACS Publications

Article pubs.acs.org/JPCC

Size Dependent Transitions in Nanoscale Dissipation Sergio Santos,*,† Carlo A. Amadei,† Albert Verdaguer,‡ and Matteo Chiesa† †

Laboratory for Energy and Nanosciences, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates Centre d’ Investigació en Nanociència i Nanotecnologia (CIN2; CSIC-ICN), Esfera UAB, Campus de la UAB, Edifici CM-7, 08193-Bellaterra, Catalunya, Spain

‡

S Supporting Information *

ABSTRACT: The irreversible loss of energy that occurs when a nanoscale tip vibrates over a surface can be monitored and quantified in amplitude modulation atomic force microscopy (AM AFM). Furthermore, two distinct dissipative processes can be identified and related to viscous and hysteretic forces respectively. Here, experimental evidence of a transition from viscous to hysteretic prevalent dissipation during mechanical contact is provided as the size of the tip increases from a few nm to 10 nm or more. Long range dissipation, defined as distances for which mechanical contact does not occur, is also investigated and related to capillary interactions. Experiments conducted on freshly cleaved mica samples show that energy dissipation increases with tip size and relative humidity in the long-range before mechanical contact occurs. Longand short-range interactions are discussed in terms of observables both experimentally and by numerically integrating the equation of motion. visualizing water distributions at a solid−liquid interface.2b It is still widely recognized however that some of the initial challenges remain. For example, the interplay between conservative and dissipative forces is still under study,13 while establishing dependencies of forces and interactions between structures on the size of the probe remains elusive.16 Here, we report the simultaneous experimental mapping of conservative forces13c,17 and the fine discrimination between viscous and the more elusive17,18 hysteretic and long-range capillary dissipative interactions.19 Fine tuning and in situ monitoring of the tip radius16 allows establishing relationships between variations in nanoscale interactions and size.12a The conservative forces are reconstructed with the use of the SaderJarvis-Katan formalism,13c,17 while dissipative interactions are monitored and identified with the use of recently proposed methods20 and variations of these. The conservative forces are monitored as a function of tip−sample distance d, thus allowing discriminating between short-range repulsive (where mechanical contact occurs) and long-range attractive interactions. The proposed variations on the methods to probe and identify dissipative interactions allow monitoring the relevant signals as a function of the intuitive tip−sample distance d rather than the oscillation amplitude A, as commonly done in AM AFM.18b,20a In this way, both the conservative and the dissipative interactions are simultaneously probed as a function of distance d, allowing direct discrimination of range, that is, short-range repulsive and long-range attractive interactions. Mechanical contact is thus identified with short-range processes and is

I. INTRODUCTION One of the paradigms of science is to identify and determine the characteristic magnitude and the distance dependencies of the fundamental forces in nature.1 It is thus not surprising that the scientific community perceives the experimental finding and characterization and theoretical understanding of intermolecular forces as a milestone in nanotechnology development.2 Mapping nanoscale properties covers a broad range of fields with applications ranging from nanoscale materials and devices3 to molecular and cell biology.4 The nanoscale involves surfacearea-to-volume ratio, chemical reactivity and affinity, and adhesion and cohesion between bodies.2b,5 An added complexity of surfaces relates to thin films of water that strongly attach and reorient on surfaces,5 for example, as a response to local short-range electric fields, and give rise to surprising phenomena such as hydration forces.6 These forces control7 biomolecular interactions and thin film self-assembly6,8 and might have implications in the functionalization of surfaces, for example, in self-cleaning applications.8b Over 20 years ago, the atomic force microscope (AFM) came into play as the candidate to robustly measure the range and magnitude of intermolecular forces between nanostructures.9 With all, several technical challenges in the interpretation of force measurements have been identified from the beginning:10 first, estimating the absolute distance between the tip and the surface;11 second, the difficulties involved with monitoring constant tip wear;12 third, the complex interplay between conservative and dissipative forces;13 and fourth, instabilities induced by force gradients that might overwhelm the spring constant of the cantilever.10,14 Rapid developments in the technique, however, have ultimately allowed resolving the chemical structure of single molecules2a or atoms15 and © 2013 American Chemical Society

Received: April 22, 2013 Revised: May 2, 2013 Published: May 2, 2013 10615

dx.doi.org/10.1021/jp4039732 | J. Phys. Chem. C 2013, 117, 10615−10622

The Journal of Physical Chemistry C

Article

Fη = −η(Rδ)1/2 δ ̇

defined as distances past the minima in force and where repulsive forces are present. In the short-range, when mechanical contact occurs, a clear transition from viscous prevalent to hysteretic prevalent dissipation is identified when the tip of the probe increases from a few nm in diameter to 10 nm or more. The prevalence of a given interaction depending on dimensions is a manifestation of the relevance of size in the nanoscale in terms of its control over the resulting phenomena. Transitions in phenomena according to the size of the structures might provide information and allow for better control over nanoparticle−nanoparticle interactions and even the interactions between nanoparticles and biological systems.21 These transitions and dependence on size should also be considered in the study of phase transformations.22 The findings presented in this work should further hint at how chemical and mechanical contrast in AFM depend on operational parameters and tip geometry and size and might explain the diversity of reported data in the literature.12a,23 A distinct variation in the magnitude and range of long-range dissipation as a function of tip size and relative humidity is also observed by probing highly wetting samples such as mica. Such phenomena might have implications in biology and, in particular, cell dehydration and surface−cell interactions.

(1)

1/2

where (Rδ) is the contact radius as given by the Derjaguin Muller Toporov (DMT)18b model and δ̇ is the velocity of deformation. While eq 1 has been typically used to describe the response of viscoelastic materials such as polymers,18b,25 here the expression is employed phenomenologically in order to provide a qualitative interpretation about viscosity in the shortrange where mechanical contact occurs. More thoroughly, here, the term viscosity is employed to refer to dissipative forces that oppose the sense of motion. Note that the sense of the force is given by the negative sign in eq 1 and the linear dependence on velocity. Through eq 1, a geometrical parameter, that is, the contact radius, is also provided that simply assumes that viscosity linearly increases with the radius of the contact predicted by the DMT model. Viscosity is further assumed to be a linear function of a parameter, that is, the viscosity η, which is characteristic of the tip−sample pair. In summary, the interpretations of viscosity and eq 1 given here do not impose any restrictions on the bulk properties of the sample, that is, the common viscoelasticity, and only the velocity and effective geometrical and effective material properties of the tip−sample interaction are involved. The hysteretic force in the short range is interpreted here as surface energy hysteresis Fα. This force should be intimately connected to inter and intramolecular interactions of surface atoms and their geometrical equilibrium state as a system, and, more thoroughly, to possible metastability due to lack of mechanical or chemical equilibrium.1,26 It is then plausible that the velocity of the tip provides the necessary energy, through eq 1, to activate atomic processes and lead to a more stable configuration,26b,27 via a physical, chemical, or electronic reconfiguration of the atoms in the tip−sample junction.18c,20b The expression commonly employed in dynamic AFM is18b,20b

II. MODEL AND METHODS (a). Short-Range Dissipative Processes. For several years now, the two main mechanisms that control short-range dissipation between a tip and a surface have been experimentally and theoretically established to be related to viscous and hysteretic forces.18b,20b,24 In this study, short-range interactions are defined as those interactions taking place at tip−sample distances d where conservative repulsive forces are effective. Such distances are identified by direct reconstruction of conservative forces both in experiments and simulations as discussed in sections II(a) and (b). Viscosity should involve any kinetic atomic process in the effective volume of interaction acting as the tip−sample interface including the local rearrangement and displacement of atoms, atomic reorientation and relative motion between atoms.18b,24 An illustration showing possible interactions between an AFM tip and a surface is shown in Figure 1. Qualitatively, it is clear that viscosity depends on velocity, contact area and the atomic or molecular arrangement of the interacting bodies which defines material properties, that is, η in Pa·s. In dynamic AFM, viscous forces Fη that occur during mechanical contact between the tip and the sample are typically modeled as18b,20b

Fα = −4πRγα = −FADα

(2)

where γ is the surface energy on tip approach, FAD is the force of adhesion on tip approach, and α is the factor by which this force increases on tip retraction; α appears due to an increment in γ.24 The relationship is α = Δγ/γ. Note that eq 2 only acts on tip retraction. It should be noted that eq 2 is employed here qualitative in a similar fashion as eq 1. That is, eq 2 provides information about the main aspect of hysteretic forces, namely, the dependence of the magnitude of the force on the sense of motion. Also, note that in eq 2 a dependence on size is also provided. This is done by modeling the hysteretic force with a linear dependence on tip radius R similarly to the dependence of the force of adhesion on R. Integration of eqs 1 and 2 over a full cycle gives the energy dissipated by each process18c,20b Eη =

2 1/2 πR ωηA1/2 δM2 4

Eα = 4πR ΔγδM

(3) (4)

where ω is the drive frequency of the cantilever, A is the oscillation amplitude, and δM is the maximum tip−sample deformation. The ratio ER = Eη/Eα provides information about the relevance of the parameters involved Figure 1. Scheme of the possible interactions occurring between an AFM tip and a surface when (a) the tip is very sharp and (b) when it becomes blunter.

ER = 10616

2

ωA1/2 δM ⎡ η ⎤ ⎢ ⎥ R1/2 ⎣ Δγ ⎦

(5)



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Figure 2. (a−c) Simulations and (d−f) experiments conducted on a freshly cleaved mica sample where conservative forces and short-range dissipative mechanisms are present. All axes are normalized and asterisks imply normalization. A transition in the prevalent mechanism of dissipation is observed by simply increasing the tip radius from ≈5 to ≈12 nm, as verified by the E*dis × ΔΦ* method. The x-axes in (a,b) and (d,e) are normalized tip−sample distances d* and have been normalized with absolute minima in the corresponding amplitude curves. The x-axes in (c) and (f) are normalized amplitudes A* and have been normalized in terms of the corresponding free amplitude A0. Simulation: Fts (minima) ≈ −2 nN, Edis × ΔΦ (maxima) ≈ 0.4 eV°, and A0 ≈ 25 nm. In (a), the only mechanism for dissipation was viscosity, where η = 100 Pa·s and α = 0. In (b), the only mechanism for dissipation was hysteresis, where η = 0 Pa·s and α = 1. Experiment: Fts (minima) ≈ −1.5 and −3.2 nN in (d) and (e), respectively, Edis × ΔΦ (maxima) ≈ 0.4 eV° in (d) and (e), and A0 ≈ 40 nm. The minima in d was ≈−2 nm in both experiment and simulation. This corresponds to tip−sample deformations of approximately 2 nm. The force at large tip−sample deformations, where Fts* > 1, is not shown because it is not relevant in the study.

imately 40−50 nm above the surface by locking the z-piezo above the surface prior to acquiring thermal data. The resonant frequency varied by approximately 8−10 Hz relative to that calibrated at several μm above the surface. Mica samples were cleaved with tape prior to performing the experiments and the relative humidity was monitored with a home-built humidity controller. The free amplitudes employed in the experiments were A0 ≈ 40 nm throughout. These relatively large values of free amplitude implied that smooth transitions from the attractive to the repulsive regime followed throughout this work. Thus, no transients due to discontinuous transitions were present in the interaction.29 That is, bistability was avoided in this way and the relevant attractive to repulsive distances were recovered successfully without discontinuities. These distances cover both long-range attractive and short-range repulsive interactions where mechanical contact occurs. This use of large amplitudes with which smooth force transitions occur has also been employed in the numerical integration with similar outcomes. Finally, R has been characterized and monitored with the use of the critical amplitude Ac method,16 which for an AC160TS

where the term in brackets contains the sample’s properties. In eq 5 the parameters outside the brackets are experimental observables. Thus, the expression can be used experimentally to deduce useful qualitative relationships. In particular, one can probe the prevalence of viscous and hysteretic forces in the short-range in terms of the size of the tip R and the dynamic operational parameters A and ω independently of the sample’s properties. First, note that if R increases sufficiently, surface energy hysteresis Eα should overwhelm viscosity Eη. Second, increasing the drive frequency ω, oscillation amplitude A, and tip−sample deformation δM should benefit the prevalence of viscosity. In turn, it is known18b,20b,24 that for a given tip− sample pair, δM increases with A and decreases with R, thus, reinforcing the above two relationships; sharp tips are defined as R < 10 nm from now on. (b). Materials and Experimental Methods. An Asylum Research Cypher AFM operated in the AM mode has been employed in the experiments. Cantilevers (Olympus AC160TS) with resonant frequencies f 0 ≈ 300 kHz, stiffness k ≈ 40 and Q factors of ≈500 were employed throughout. The resonant frequency28 has always been calibrated at approx10617



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cantilever on a mica sample gives a relationship R = 4.75Ac1.12 (see Supporting Information for details). (c). Numerical Integration and the Equation of Motion. In all numerical simulations, k = 40 N/m, Q = 450, and ω = 2πf0 ( f 0 = 300 kHz). The energy dissipation Edis has been calculated in both simulations and experiments with the use of Cleveland et al. expression30 Edis =

πkA 0A ⎡ A⎤ ⎢sin(Φ) − ⎥ Q ⎣ A0 ⎦

(d). Probing and Identifying Conservative and Dissipative Interactions. The conservative tip−sample force Fts* has been reconstructed with the use of the SaderJarvis-Katan formalism13c,17 in the simulations (dashed blue lines in Figure 2a,b) and in experiments (continuous blue lines in Figure 2d,e) by numerically integrating eq 11 in Matlab.32 This formalism uses the variations in amplitude and phase of the cantilever but fails to provide detailed information in the presence of hysteretic interactions;18a an example is shown in the Figure 2b. The Sader-Jarvis-Katan formalism produces17,18

(6)

u =∞ ⎡⎛

where A0 is the free amplitude of oscillation, Q is the quality factor due to dissipation with the medium, and Φ is the phase lag relative to the drive force. The phase difference ΔΦ is defined as the absolute of an increment of angle. This difference is calculated by subtracting Φ from the conservative angle20a ⎛A⎞ Φcons = sin−1⎜ ⎟ ⎝ A0 ⎠

Fts(d) = 2k

−

d 2z mω dz + + kz = Fts + F0 cos ωt 2 Q dt dt

(8)

2

where k is the spring constant, m = k/ω is the effective mass, Fts is the instantaneous (net) tip−sample interaction, including conservative and dissipative forces, z is the instantaneous position of the tip relative to the unperturbed equilibrium position of cantilever, and F0 is the drive force. Long range conservative forces have been implemented in all simulations following Haymaker’s approach34 as is customary in AFM theory.33a,35 This is FvdW = −

HR 6d 2

d > a0

(9)

where H is the Hamaker constant. The term a0 is an intermolecular distance that has been taken to be 0.165 nm throughout.1 This force stands for zero frequency nonretarded London dispersion force.1,35a When contact occurs, this force is identified with the short-range adhesion force FAD1 FvdW = −

HR ≡ FAD = −4πRγ 6a02

d ≤ a0

(10)

In the region d ≤ a0, the short-range repulsive force is written from the Derjaguin Muller Toporov (DMT) model36 as FDMT(d) =

4 * E R δ 3/2 3

d ≤ a0

⎞ A ⎢⎜1 + ⎟⎟Ω(u) ⎢⎣⎜⎝ 8 π (u − d ) ⎠

⎤ d Ω(u) ⎥ du 2(u − d) du ⎥⎦ A3/2

(11)

where Ω is the normalized frequency shift that is found via observables in AM AFM.17 From here onward Fts* refers to the conservative part of the force only as recovered with eq 11 unless otherwise stated. Asterisks imply normalization throughout. Only approach curves have been employed to reconstruct the tip−sample force. This does not imply that the tip does not cover the approach and retract path during one cycle, but that only the curves for which the cantilever separation was decreased have been employed. As stated, since complex dissipative processes are investigated here, the Sadar-JarvisKatan formalism in eq 11 is employed in combination with other methods, as described below. Note also that by monitoring Fts*, long- and short-range forces can be defined at once in experiments and simulations in terms of d. Here, in the simulations, the short-range is defined as distances where d ≤ a0 (Figure 2a,b) and in experiments as distances where d ≤ 0 nm (Figures 2d,e). In experiments we set d = 0 nm when minima in Fts* occurs, thus, minima in Fts* defines the point at which mechanical contact occurs or the onset of short-range repulsive forces. Thus, mechanical contact is identified with short-range processes and is defined as distances past the minima in force and where repulsive forces are present. The prevalence of hysteretic and viscous interactions in the short-range is qualitatively established with the use of the previously reported phase difference-energy E*dis × ΔΦ* method.20 This method monitors the product of the energy dissipated per cycle, Edis, eq 6, and the absolute in the difference ΔΦ = Φ − Φcons as a function of normalized oscillation amplitude A* = A/A0. The product is further normalized in terms of maxima in Edis × ΔΦ, and because of being plotted as a function of A*, it can be termed the E*dis × ΔΦ* − A* method. A variation of the E*dis × ΔΦ* − A* method is further proposed here in order to establish the effective range of the relevant dissipative interactions. The variation consists of monitoring E*dis × ΔΦ* as a function of tip−sample distance. Hence, this could be termed the E*dis × ΔΦ* − d method, where the distance d is defined by taking as a reference the minima in Fts* as before. The E*dis × ΔΦ* − d method has the advantage of allowing monitoring the presence, absence, and onset of dissipative mechanisms as a function of the intuitive tip−sample distance d rather than oscillation amplitude A. Both the E*dis × ΔΦ* − A* and the E*dis × ΔΦ* − d methods are employed simultaneously here for the sake of robustness and to

(7)

It is important to note that eqs 6 and 7 are valid in the steady state irrespective of the dissipative mechanism.30 Our simulations showed that eq 6 is very accurate with errors of 1% or less. This outcome coincides with previous works18c,31 when high Q factors, Q ∼ 102−103, are employed. The equation of motion 8 has been solved numerically with the use of a Runge−Kutta algorithm and implemented in both C and Matlab.32 Both implementations were equivalent, while C was more than an order of magnitude faster. The equation of motion is33 m

∫u=d

(11)

where E* is the effective Young’s modulus of the contacting bodies, and the deformation δ is written as δ = a0 − d.18c,37 For all the simulations, Et = 120 GPa (Young’s modulus of the tip), Es = 1 GPa (Young’s modulus of the sample), R = 8 nm, and γ = 10 mJ/m2, where α ≠ 0 and 20 mJ/m2 otherwise. 10618



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allow comparison. The results of numerical integration of the equation of motion are first shown in Figure 2a−c to establish their validity as discussed below.

III. RESULTS In Figure 2a, a simulation where conservative forces (Fts* (Sim.) continuous gray line) are present in the interaction is shown; Sim. indicates simulation. The only mechanism of dissipation is short-range viscosity η = 100 Pa·s, as modeled via eq 1. The Sader-Katan formalism, eq 11, recovers this force, dashed blue lines, Fts* (Rec.), where Rec. implies recovered force by eq 11, rather robustly, with errors of 5% or less, in agreement with the literature.17 The E*dis × ΔΦ* − d signal is predicted to curve downward with respect to decreasing d, anticlockwise, in this case. This behavior is reproduced experimentally (continuous black lines) when a sharp tip R ≈ 5 nm is made to interact with a freshly cleaved mica surface (Figure 2d) in the repulsive regime. The fact that the E*dis × ΔΦ* − d signal acts in the short-range or repulsive region is deduced by the fact that the nonzero signal and, in particular, the rotation, occurs when d < 0 nm. The experimentally reconstructed force is shown with the use of continuous blue lines (smoothened). The raw data points are shown in the background in lighter blue. When η = 0 Pa·s in eq 1, viscosity is not active. Then, if α = 1 in eq 2, hysteresis is the only mechanism of dissipation in the short-range, as shown in Figure 2b. While two force paths, approach and retraction, are clearly distinguished in the simulation in Figure 2b as a consequence of α = 124 (continuous gray lines), the Sadar-Jarvis-Katan formalism, eq 11, provides only a single path that lies in between them. The physical implication is that the presence of hysteretic forces can not be deduced from eq 11 alone. Note, however, that the sense of rotation of the E*dis × ΔΦ* − d signal is inverted to clockwise (continuous black lines). The clockwise sense of rotation is a footprint of the presence of hysteretic forces. The inversion in the sense of rotation should be reproduced experimentally, according to eq 5, by simply increasing R. This should be the case provided short-range hysteretic forces are a characteristic dissipative processes present in the mica-silicon tip interaction. The prevalence of hysteretic forces is in fact corroborated experimentally (Figure 2e) by monitoring the E*dis × ΔΦ* − d signal with a blunter tip and noticing the inversion in the sense of rotation. Experimentally, the tip was blunted by scanning in the repulsive regime and gradually monitoring its broadening16 from R ≈ 5 nm to R ≈ 12 nm. The inversion in the prevalence of the dissipative interactions can also be observed and deduced by comparing the E*dis × ΔΦ* − A* signal in Figure 2c (simulation) and f (experiment). In summary, Figure 2 confirms the existence of a transition in the relative prevalence of nanoscale dissipative interactions that occurs as the size of tip increases. This dependence of the prevalence of dissipation processes on size could be exploited in future nanotechnology developments. To probe and identify long-range dissipative interactions, the ΔΦ* − A* and ΔΦ* − d methods can be employed.20a Again, ΔΦ* − d is a variation of the previously reported20a ΔΦ* − A* method, and the concept of ΔΦ* has already been introduced above. Only the method ΔΦ* − d is qualitatively discussed here, with the use of simulations and Figure 3, in order to later interpret and identify long-range dissipation in experiments (Figure 4). In Figure 3a,b, only conservative forces, eqs 9−11, and short-range viscosity, eq 1, and hysteresis, eq 2, are present

Figure 3. Simulations illustrating the effects of (a) short-range hysteresis α = 1, (b) short-range viscosity η = 100 Pa·s, and (c) hysteretic don/doff long-range mechanisms on the normalized force used in the simulation Fts* (Sim.), the reconstructed force according to eq 11 Fts* (Rec.), the phase shift difference ΔΦ*, and dissipation E*dis signals as a function of normalized distance d*. Fts (minima) ≈ −2 nN, ΔΦ (maxima) ≈ 6, 12, and 9° and Edis (maxima) ≈ 4, 76, and 6 eV, respectively, in (a), (b), and (c). Minima in d is ≈−2 nm, implying that maximum tip−sample deformation was approximately 2 nm. The force at large tip−sample deformations where Fts* > 1 is not shown because it is not relevant in the study.

in the simulations, respectively. The ΔΦ* − d signal is shown with the use of dashed brown lines. In the figures, the energy dissipation Edis*, eq 6, signal (black lines), the force employed in the simulations Fts* (Sim.; gray lines), and the reconstructed force Fts* (Rec.; dashed blue), according to eq 11, are shown. As expected,20a neither the Sader-Katan formalism18a 11 nor the ΔΦ* − d method20a provides clear information that can be employed to discriminate or identify short-range dissipation mechanisms. This is because, as opposed to the E*dis × ΔΦ* signal, the ΔΦ* − d signal is more sensitive to long-range than short-range hysteresis and viscosity.20a Nevertheless, in Figure 3c, a longer range hysteretic force that could be associated with the onset of a capillary bridge at d = don on tip approach and the breakoff at d = doff33b,35b,38 on tip retraction has been activated in the simulation. This force can be termed FCAP and, for simplicity and generality, in Figure 3c it is given by FCAP = −FAD on tip approach and provided d ≤ don and by FCAP = −FAD on tip retraction and provided d ≤ doff. The force is zero otherwise (see Supporting Information). Note that by necessity doff > don > 0 nm. The area under the curve (colored region) in the don−doff range corresponds to the energy dissipated by this long-range interaction. 10619



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mechanisms. The change of slope, however, is not as sharp as predicted, that is, stepwise. This could imply that both viscosity due to the presence of water and the formation and rupture of the capillary neck, that is, the don/doff mechanism, as predicted by Figure 3c, might contribute to the increase in net energy dissipation Edis*. Subharmonics might also impede observing the step-like behavior of the Edis* and ΔΦ* signals at d = don.40 Energies of approximately 20 eV have been reported to be expected for capillary interactions in some studies38 by employing tips of approximately 20 nm in radius. More recent reports31b predicted that the energy associated with the formation and rupture of the neck could be closer to 50 eV for similar values of tip radius. It could be argued that, experimentally, tip and sample asperities, the dependence of the height of the water layers on relative humidity and the wettability of the sample,41 the presence of ions42 and the possible presence of other impurities on the mica sample43 might lead to deviations from theory. From the acquired experimental data here, an upper bound to the dissipation due to the capillary don/doff mechanism can be established to be in the order of 20−30 eV for the blunter tip (R ≈ 12 nm) and in the order of 5−10 eV for the sharper tip (R ≈ 5 nm); note that only representative data curves are shown in Figure 4, and the above ranges in adhesion force and energy dissipation have been obtained by processing 10−20 data curves at three different spots of the mica sample. Similar results were obtained when comparing RH < 5% to RH ≈ 40% (see Supporting Information).

Figure 4. Experiments illustrating the effects of relative humidity and tip size on normalized force Fts* (dashed lines), eq 11, dissipation E*dis (continuous lines, top raw), eq 6, and phase shift ΔΦ* (continuous lines, bottom raw) for a silicon tip and a freshly cleaved mica sample. The y-axes are normalized throughout and the x-axes correspond to the tip−sample distance d in nm, where d = 0 nm coincides with minima in Fts*. Fts (minima) ≈ −6.2 and 2.8 nN, ΔΦ (maxima) ≈ 13 and 8°, Edis ≈ 75 and 26 eV for the blunt and sharp tips, respectively.

It can be readily observed that the Sader-Katan formalism18a (dashed blue lines in Figure 3c) fails to account for events that occur at distances larger than don on tip retraction. Detailed information about events occurring at larger distances than d = don are also absent in both the ΔΦ* and the Edis* signals. Nevertheless, a clear step-like discontinuity (dashed circle) is observed on the onset of such forces at d = don in both the ΔΦ* and the Edis* signals. These steps are a footprint of don/ doff dissipative mechanisms, which are thought to occur under certain conditions in dynamic AFM interactions.19,31b,38,39 The capillary force, with the corresponding don/doff mechanism that the capillary bridge formation and rupture introduces,31b,35b,38 should be an ideal candidate to probe the presence of these complex processes. A discussion based on experimental data is given next. In Figure 4, the interaction between an AFM silicon tip and a freshly cleaved mica sample is processed for a relatively blunt R ≈ 12 nm (left column) and a sharper tip R ≈ 5 nm (right column). The reconstructed forces Fts* (dashed lines), according to eq 11, are shown throughout, together with the corresponding Edis* (continuous lines in top raw) and ΔΦ* (continuous lines in bottom raw) signals at very low (RH < 5%) and high (RH > 80%) values of relative humidity RH. Two main outcomes are worth noting. First, for both the sharper and the blunter tip, the adhesion force or minima in Fts* decreases drastically, that is, divides by a factor of 2−4, when RH < 5% (dashed light blue). This severe decrease in adhesion force is in agreement with recent reports23 obtained with the use of spherically terminated tips. These previous experiments, however, were otherwise conducted with a much lower signal-to-noise ratio,17 where capillary dissipation could not be probed or identified38 and where, in general, information about long-range forces is typically not accurate;13c,17,18 the AFM was operated in the DC mode. Second, there is a severe change in slope in the Edis* and ΔΦ* signals at RH > 80% relative to that found at RH < 5%. This is in accordance with what was predicted in Figure 3c in the presence of don/doff

IV. CONCLUSION The standard Sader-Jarvis-Katan formalism has been implemented to reconstruct the attractive and repulsive conservative forces in terms of the intuitive tip sample distance. Amplitude curves where smooth transitions from the attractive to the repulsive regime are observed have been employed for force reconstruction thus allowing defining reference distances while also making the relevant range of distances available. In the presence of viscous and complex hysteretic dissipative interactions, this formalism has been supported with the use of variations of known methods to identify and interpret dissipative interactions as a function of distance. In the shortrange, experimental evidence of a transition from viscous prevalent to hysteretic prevalent dissipation has been provided. This transition has been shown to take place as the size of the tip increases from a few nm to 10 of nm or more for a silicon tip and a mica sample pair. In the long-range, dissipation related to capillary interactions has been discussed in terms of observables by employing a variation of a method to identify and interpret long-range dissipation as a function of tip−sample distance and the energy dissipation expression. In particular, by numerically integrating the equation of motion, it has been shown that the onset of hysteretic capillary dissipation can be related to a discrete step in both signals. Experiments conducted on freshly cleaved mica samples have provided evidence of such steps. It has been further shown experimentally that energy dissipation increases with tip size and relative humidity in the long-range. These findings should assist future developments in the field in terms of the establishment of nanoscale laws and their size dependencies. 10620



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ASSOCIATED CONTENT

S Supporting Information *

Detailed information on the methodologies employed to calibrate the cantilever parameters and the tip radius, as well as further details related to the numerical integration of the equation of motion are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank graphic designer Maritsa Kissamitaki for designing the artwork. REFERENCES

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