Deconvolution of dissipative pathways for the interpretation of tapping-mode atomic force microscopy from phase-contrast

Phase-contrast in tapping-mode atomic force microscopy (TM-AFM) results from dynamic tip-surface interaction losses which allow soft and hard nanoscale features to be distinguished. So far, phase-contrast in TM-AFM has been interpreted using homogeneous Boltzmann-like loss distributions that ignore fluctuations. Here, we revisit the origin of phase-contrast in TM-AFM by considering the role of fluctuation-driven transitions and heterogeneous loss. At ultra-light tapping amplitudes <3 nm, a unique amplitude dependent two-stage distribution response is revealed, alluding to metastable viscous relaxations that originate from tapping-induced surface perturbations. The elastic and viscous coefficients are also quantitatively estimated from the resulting strain rate at the fixed tapping frequency. The transitional heterogeneous losses emerge as the dominant loss mechanism outweighing homogeneous losses at smaller amplitudes for a soft-material. Analogous fluctuation mediated phase-contrast is also apparent in contact resonance enhanced AFM-IR (infrared), showing promise in decoupling competing thermal loss mechanisms via radiative and non-radiative pathways. Understanding the loss pathways can provide insights on the bio-physical origins of heterogeneities in soft-bio-matter e.g., single cancer cell, tumors, and soft-tissues. Phase-contrast in tapping-mode atomic force microscopy has been interpreted using homogeneous distributions that ignore fluctuations. Here, the validity of this assumption is challenged, revealing tip-induced heterogeneous contributions.

S ince inception 1 atomic force microscopy (AFM) has emerged as a key nanoscale surface metrology tool for scientists and engineers alike 2,3 . Tapping mode AFM 4 (TM-AFM) deserves a special mention in this regard for its ability to nondestructively characterize soft bio-interfaces 5,6 at the nanoscale. In recent years it has found renewed popularity in the study of nonequilibrium (NE) energetics of protein and DNA/RNA 7,8 folding landscapes. NE energy routes underpin fundamental biophysical mechanisms like cell migration, cell signaling, tissueself-repair, damage evolution, and toughening, among others [9][10][11][12] . An apparently homogeneous surface locally deviates off equilibrium to metastable states and eventually restores back via NE dissipative routes. Fluctuations mediate the thermodynamic cost of such excursions. These are pivotal to the evolution of mechano-biological functions 3 that closely imitate soft glassy dynamics 13 yielding to viscous relaxations at nanoscale heterogeneities. New knowledge in macroscopic and biological phenomena from nanoscale and molecular heterogeneity perspectives has emerged [14][15][16] . Yet, our predictability of NE pathways in biology is still in its infancy. This is because of our general lack of understanding of the role of heterogeneities in NE dissipative pathways. Methods that can connect local fluctuations at heterogeneities to energy losses are thus becoming more general and broader to elucidate insights on such mechanisms 17,18 . TM-AFM with microcantilevers, which have tapping frequencies in the order of $ 10 2 kHz, is a unique platform to study such intricate energetics in soft-bio-matter at the nanoscale 19 . Typically, nondestructive bio-TM-AFM has been executed at the large tip-surface separations with high amplitude tapping ensuring that the tip-surface interaction potentials remain predominantly conservative 20,21 . Yet, losses originate from inelastic interactions and are reflected as a phase-contrast 22 . These constitute loss contributions that are intrinsic to the cantilever and from tip-surface interactions including the hysteretic losses from indentations that cannot be decoupled. This is because the tip is invariably exposed to nonconservative interaction potentials every cycle on tip-approach 20,21 , even if for a fraction of its oscillation-period 22 . The conservative interactions typically overshadow the nonconservative effects at large oscillation amplitudes since a greater percentage of cantilever motion occur at large separations. However, at small separations nonconservative or inelastic effects dominate which are of particular interest from a bio-physical perspective, since they carry the signatures of the NE loss pathways. Deconvoluting the nonconservative loss mechanisms is thus essential to gain insights on the role of nanoscale heterogeneities. Surfaces begin to deform even without indentation at small separations 20 . The very nature of such induced deformation introduces two possibilities depending on the material's relaxation characteristics at the tapping perturbation rate. If the perturbation rate is high enough, the surface deformations would exhibit inelastic viscous relaxations manifesting over multiple-length and timescales 13 adopting NE dissipative routes. Biophysical mechanisms adopt such NE energy routes in bio-evolutionary processes, relentlessly competing over local fluctuations. For example, to maintain equilibrium at room temperature, single biomolecule dissipates energy at the rate of 10 À20 À 10 À19 J=s, competing over thermal fluctuations that are in the order of $ 10 À20 J=s. Interrogating such nonconservative losses without indentation may prove invaluable in deciphering signatures of such multiscale NE relaxation mechanisms 13,23 .
TM-AFM's operation typically relies on 1 pN À 100 nN forces that manifest at a tip-surface junction in the limit where constitutive continuum laws breakdown and nanoscale effects takeover. Complex multiscale energy interplay takes predominance linking molecular (or atomic scale) forces at the tip to macroscale cantilever dynamics 20,24 . The dynamics capture surface deformation modalities as topography, amplitude, and phase-contrast images through raster scans that form the backbone of any TM-AFM study. At each oscillation cycle, the tip experiences a gradient of forces with loss of energy. The oscillation amplitude decreases linearly starting with negligible interaction (mostly fluid media loss) when the tip is furthest from the surface ! longrange attractive force (non-contact regime) ! repulsive forces (contact regime) (Fig. 1). The reverse happens in the retraction cycle. The energy losses in the repulsive and attractive regimes have been explained to appear as a phase-change Δϕ in the cantilever dynamics with respect to the external drive. Δϕ essentially displays compositional contrast of the heterogeneous boundaries. By convention, the repulsive and attractive regimes are characterized by positive (þΔϕ) and negative (ÀΔϕ) phase shifts, respectively, arising from the energy dissipated, E dis , during tip-surface interactions 25,26 as E dis presents an accurate approximation of the net dissipated energy when the free amplitude quality factor Q ¼ ω 0 2Γ is relatively high $ 100 À 1000 (oscillation decay rate Γ widely separated from the resonance frequency of the AFM cantilever ω 0 ), and the damped tapping amplitude A ¼ A 0 À A sp is sufficiently large >15 nm ensuring that the tip-motion accesses the contact regime every cycle 22,27,28 (Fig. 1). Here, k is the spring constant with A 0 the free amplitude set by tuning and A sp the operational amplitude setpoint in nanometers, both calibrated from volt units (V) from the AFM's optical deflection measurements. True atomic 29 and bond-level 30,31 resolutions have been demonstrated exploiting large-amplitude techniques relying on dynamic tip-surface interactions. However, the steady-state dissipation approximation of Eq. (1) necessarily rests on the assumption of a constant Q even on intermittent contact at large A. In addition, for large A, the effective tip-surface interaction time is a fraction of the cantilever oscillation time-period 22 (Fig. 1). Thus, in principle, a softsurface gets enough time to relax to its equilibrium state before the tip interacts again in the next cycle. This essentially results in homogeneous Boltzmann-like distributions 32 in Δϕ reflecting equilibrium tip-surface interactions (black curve in phase histogram plot of Fig. 1). The captured surface modalities end up being analyzed from a conservative interactions perspective. E.g., relatively hard materials with interatomic spacing $ 2 À 3 Å are typically modeled as simple-solids exhibiting pure elasticity (Young's-moduli in the order of ≥10 GPa), with no dissipation.
The assumption is that deformations at the molecular length scales ($Å) relax so quickly (picoseconds) in comparison to the cantilever's timescale (micro-milliseconds) that the microscopic components of the material behave as if they are locally at equilibrium 33 . This is intuitively valid for atoms in crystal lattices ($Å) and small molecules like H 2 O (mean diameter ' 2:7 Å) that invariably adsorb at nanometric imperfections of biointerfaces. However, in bio-materials with larger constituent components (typically ≥ 10 Å) relaxation rates become comparable to nanometer order strain rates. The assumption of local equilibrium on deformation no longer holds in such a case and is bound to have metastable states 13 . A more intricate experimental description thus becomes necessary to correctly account for the metastable relaxation phenomenon from the resulting phasecontrast. Dissipative features have so far been qualitatively explained in terms of an apparent lighter contrast in the noncontact regime (ÀΔϕ phase shift) and a darker contrast in the contact regime (þΔϕ phase shift). Such interpretations directly follow from the sudden changes in phase observed in large amplitude tapping experiments 34-37 agreeing to Eq. (1). Yet, the origin of phase shift remains ambiguous for apparent erroneous invoking of elastic and/or inelastic interactions 34 . Inherent tip-surface hysteretic interactions from capillary effects 35,36 , chemical affinity 37 , solid-mechanics assumptions of tip-surface indentation models 24 , artifacts from control system feedback 38 , presence of intrinsic stochasticity at the nanoscale 39 , the combination of many several possible mechanisms like the hydrodynamic effects 40,41 and the controversial nature of friction at the nanoscale [42][43][44] , all of these could contribute simultaneously to the ambiguities. It must be noted here that previous efforts in generating decent phase-contrast images have necessarily adopted surface indentation methodology with high amplitude oscillations $ A>15 nm but without the means to decouple the hysteretic effects. This is to specifically conform to steady-state loss descriptions that satisfy Eq. (1) despite compromising on the tip-surface interaction time. In principle, to accurately quantify viscous relaxations the tip-surface interaction time per oscillation cycle should sufficiently be close to the viscous relaxation timescale. Previously, contact time has been argued to be independent of topographic features 22 , while others contradicted with models decomposing phase-contrast into moments of topography 45 . Both, however, agreed on the criticality of tip-surface equilibrium separation z c , typically >20 nm, to correctly reflect on the qualitative interpretation of soft and hard features from the generated Δϕ. Nevertheless, viscoelasticity has an inherent time dependence and phase data from high-amplitude tapping experiments makes such interpretations erroneous and inconclusive for the insufficient tip-surface interaction time. The cantilever dynamics basically fails to follow the surface relaxation dynamics that are crucial for decoupling the viscous loss pathways originating from the strain rate or strain history of the interface deformations.
On the other hand, matching tip-surface interaction time to the oscillation timescale invokes the promise of reconceptualizing phase-contrast where the microcantilever tip has access to the surface relaxations during its entire tapping cycle, without necessarily indenting the surface.
Here, with our ultra-light tapping amplitudes A < 3 nm, we demonstrate the emergence of a complex double-stage phasecontrast (red curve in phase histogram plot in Fig. 1) as evidence of energy losses from metastable relaxation in a soft-material. We decouple a heterogeneous loss component from the appearing Δϕ distributions. Previously ultra-low amplitude tapping has been implemented at subharmonics, successfully decoupling hysteretic interactions in metal lattices 21 . Our experimental outcome reveals an intricate phase distribution response (Fig. 1) where the single Boltzmann peak (black curve A 0 > 15 nm) bifurcates to two peaks (red curve A 0 < 10 nm). We explain this from fluctuational excursions of microcantilever dynamics in the limit of Δϕ ! 0 ( Fig. 1). Such distribution in context has never been observed or explained. We must note here that Δϕ appearing from dynamic losses of an oscillating microcantilever is expected to be maximized at the condition of resonance since loss pathways maximize variationally at resonance 46 . Uniquely, resonance provides access to two timescales: the oscillator timescale (faster-resonance frequency) referred to here as τ osc and a dissipation timescale (slower-the resonance width), that essentially reflect the cumulative result of phase-trajectory excursions in the underlying dynamics. We exploit these two timescales with controlled tip-surface interactions and demonstrate the emergence of the intricate distribution of phase-change Δϕ that surprisingly deviates from a standard Boltzmann distribution. We analyze it in detail to reflect on the dissipative pathways at the tip-surface junction. We go on to explain that the key to obtaining such a distribution lies in the close matching of the tip-surface interaction timescale τ to the cantilever resonance timescale, whereby, the multi-time scale relaxation processes are encoded in the resonance width as an integral of dissipation functions; the secondary peak signifying a unique dissipation timescale corresponding to metastable relaxations of the soft surface. We further comment on the origins of Δϕ from the aspect of energy exchange through time-dependent adiabatic and anti-adiabatic routes and connect them to fluctuational dynamics of the AFM microcantilever tip in response to heterogeneous interactions at the nanoscale. Earlier implemented nonresonant frequency scanning based AFM techniques do not provide the means to access this dissipation or non-conservative timescale since they follow steady-state deformation dynamics. Such steady-state deformations are expected to be predominantly elastic since the probe cantilever behaves as an elastic element at the dynamic frequencies employed in such a study. The smallamplitude resonant tapping mode method we expound here will particularly be applicable for a broad class of soft-bio matter such as soft tissues, cells, DNA, proteins, and drug molecule clusters with intercalated H 2 O, that exhibit means to simultaneously store (solid-like) and dissipate (liquid-like) energy from the tip-surface interactions.

Results and discussion
Experimental technique. At the nanoscale, viscosity manifests from the diffusion of atom centers when perturbed. Heterogeneities accentuate such manifestations. The perturbation rate is important since it continuously works against the material's stress relaxation or dissipation mechanisms. On tapping, a phasechange, Δϕ, thus appears from the mismatch of the applied and induced surface strain rates owing to the soft-material dynamics. Considering resonance, Δϕ signify a delay-bandwidth product Δϕ ¼ δω Á τ vide A ¼ A 0 e Ài δωÁτ ð Þ , where δω ¼ ω À ω 0 is the shift in fundamental resonance frequency ω 0 on tip approach. The steady state damped amplitude A evolves as a function of the delay bandwidth product with ÀiΔϕ necessarily capturing the time dependance. Therefore, in principle, Δϕ have both a steady state solution (contact and noncontact regime) and a transitional component that depends on how Δϕ gets influenced as a function of tip-surface interactions. We focus on the transitions in the limit Δϕ ! 0 with small tapping amplitudes. We are specifically considering the effect of small fluctuations on an otherwise deterministic dynamical system that have been described by phenomenological laws of equilibrium energetics 32 . We determine the dynamic crossover Δz ffi 4 À 7 nm from snap-in lengths having equilibrium tip-surface separation z c % 2:5 À 3:5 nm for different samples under study as shown in Fig. 1. We carefully restrict the dynamics in this near-contact crossover regime ensuring that the damped amplitude A stayed within the fluctuational Δz as shown in Fig. 1. This we enforce with ultralight tapping with a gradual controlled tip-surface approach as a function of A sp and an integral time optimization. On optimization, constant amplitude TM images were captured as a function of varying A sp . The phase at free amplitude A 0 is zeroed at each run corresponding to a particular A sp allowing the precise monitoring of the phase changes Δϕ at the ultra-light tapping condition. Fluctuations would be dominant in this operational regime with z c as the boundary of crossover in our consideration. In the short run, we cannot expect large deviations from a deterministic equilibrium behaviour [33][34][35][36][37] . Yet, if enough equilibrium time is allowed per scanning point (pixel) for the system to settle down, the cumulative effect of the fluctuations may become pronounced, albeit the rare events become more probable on account of synchronization of tip oscillation with the surface relaxations 47 . This we ensure by an optimized scan rate and controller integral time that caters to the necessary timescale conditions. In view of the fluctuational phenomenon in context, the recorded amplitude, phase, and topography data channels at each A sp have important statistical significance in the appearance of a surprising two-stage phase contrast as shown in Fig. 2a-f. We have selected Doxorubicin (DxO) and Minoxidil (MxD) drug clusters on multilayered graphene oxide (mGO) as the model DxO@mGO; c, f MxD@mGO. Experimental free amplitude A 0 for mGO is 6.2 nm while that for DxO@mGO is 6.32 and 6.34 nm in the case of MxD@mGO. Curiously enough, the soft DxO@mGO and MxD@mGO surfaces show the multi-peak distribution in comparison to simpler single peak distribution in mGO. The appearance of the secondary peak is direct evidence of the interplay of the metastable relaxation timescale of a soft-surface as interrogated by ultra-light tapping conditions enforced here. The relative position and amplitude of the distribution peaks change as a function of setpoint amplitude A sp . At higher A sp , i.e., lower tapping A at tip-surface approach, the fluctuational probability transition from surface heterogeneity is expected to be high since damping is low at small A. This is evident in a relative increase in phase delay Δϕ À peak with respect to Δϕ þ at higher A sp in both DxO@mGO and MxD@mGO samples. DxO-Doxorubicin, MxD-minoxidil, mGO-multilayered graphene oxide.
soft-matter systems (DxO@mGO and MxD@mGO, respectively). Invariably, the samples will have intercalated H 2 O at the nanometric imperfections. DxO ($ 15 Å) 48 and MxD ($ 10 Å) [ChemDraw] molecules are expected to exhibit finite relaxation rates making it an ideal soft-matter candidate to investigate. Such drug@mGO systems are gaining interest in drug delivery systems 49,50 , where the correct interpretation of the loading condition is crucial for drug delivery efficiency and their therapeutic success. Accurate estimations of surface energy play a vital role in controlling drug release. From the distribution results of DxO@mGO and MxD@mGO samples, we quantify the surface energy in terms of the elastic and viscous components and compare them to bare mGO samples, where the appearance of a secondary peak is not prominent. On the fundamental side, exploiting the fluctuational regime serves us two purposes: (a) determine the basis of the intermediate energetics in the origin of Δϕ and (b) link them to the dissipative pathways.
Timescales and significance of fluctuations. Soft surfaces typically relax over multiple timescales with the system hopping between metastable and equilibrium states attesting to their unique NE loss mechanisms 13 . The energy loss corresponding to the metastable hops typically signifies a heterogeneous loss path that rides over the equilibrium homogenous loss path and may appear as small fluctuations of the steady-state energy landscape. Necessarily, the timescale separation of the states needs to be small for both the loss pathways to reflect simultaneously in the dynamics. In its limiting case, when the states are widely separated in timescales, the excursions amongst the states become statistically independent. In that case, the energy loss follows a typical homogenous Gaussian distribution with vanishingly small relative fluctuations σ dis E dis h i $ 0:1, E dis and σ 2 E dis being the mean and variance of the dissipated energy as described by Eq. (1). This is apparently evident in the generated Δϕ distributions in our experimental outcomes at high amplitude ( Fig. 2b, d), similar to the previous experiments 35,40,51,52 with surface indentation and where the interaction time τ is negligible compared to the perturbation timescale. However, subject to the condition that the timescale separation is short, the energy landscape of a softmaterial deviates from a single-stage Boltzmann potential. A kinetic variant becomes essential to account for and explain a succession of NE states. This was postulated in Onsager's principle of least dissipation of energy. This manifests at near contact conditions where the tip interaction time τ (Fig. 1) renders fluctuations relevant to the underlying dynamics making the states and their probability of excursions in between the states statistically correlated. Onsager attributed this statistical significance to the time integral of dissipation functions 47 . Variationally this has the means to be effectively encoded in resonance width broadening timescale 46 . This is because when enough interaction time is allowed the microcantilever dynamics essentially start to follow the multi-time scale dynamical relaxations of the soft viscous material 13,23 typically adopting two or more timescales. In principle, the shortest timescale (resonance timescale) relates to dynamical trapping of oscillator around one of the equilibrium states (contact/non-contact in our case) while the longest (resonance broadening timescale) signifies relaxation time towards a (unique) dynamic steady state. For soft-material, this typically signifies a structurally disordered metastable state 13,23 . Phenomenologically, the energy states of soft material can be modeled to be confined in a double-well free energy landscape with the system spontaneously hopping between the potential wells when externally perturbed, i.e., by the periodic tapping mechanism in our case. Specific to a material, the doublewell free-energy landscape will exhibit a characteristic relaxation time that is inversely proportional to Kramer's rate of hop in between the wells. When the interaction time τ relevant to the perturbations, closely matches this inherent rate of hop, a state of dynamic stochastic resonance emerges; the excursion between the minima states starts to synchronize with the external perturbation dynamics. I.e., the microcantilever dynamics sync to fluctuational dynamics about ϕ ¼ 0 (in the near-contact regime) deviating from equilibrium (Fig. 1). Physically, the microcantilever follows the surface's excursions to the metastable state in time. These transitional excursions amongst the fluctuational and the steady states thus become relevant to the dynamics reflecting phase distributions that deviate from standard Gaussian as two-stage distribution in Δϕ (Fig. 2). Basically, two markedly distinct regimes of energy interactions are possible (i) the anti-adiabatic case: when the cantilever dynamics does a fast transition between equilibrium states remaining oblivious to the intermediate metastable state 53 and (ii) the opposite adiabatic case: where the oscillator dynamics is very slow compared to the faster relaxation processes at the interface, thus susceptible to nonlinear damping effects 54 . A third crossover regime at a comparable timescale τ osc ffi τ sur has the fluctuational relevance as described above. I.e., when the enforced dynamics is slow enough to reversibly follow the transitional excursions at small-separation deformations where τ sur is the inherent metastable relaxation timescale of the surface in question. Our experimental description institutes this premise to faithfully follow the transitional adiabatic " antiadiabatic crossovers in between the equilibrium and metastable states. In effect, enforcing oscillations within the crossover regime renders the tip-surface interaction correlation time τ c finite, allowing the interrogation of a heterogeneous interaction scale at the tip-surface junction (see "Methods") in addition to inherent homogeneous loss timescale from the cantilever's oscillatory dynamics.
The appearance of two-stage phase contrast. The enforcement of a finite correlation time τ c ! τ osc % τ sur is optimized in our experiments in terms of A sp and the integration time of the PI controller. The choice of P and I gains is critical at our operational low scanning rates $ 0:05 Hz, giving the tip enough time to equilibrate at every pixel of the image (pixel data is sampled over an average of 40,000 cycles). The P and I gain are set such that the integral timescale is of the order τ c ¼ 6:430 ms; sufficiently close to τ osc ¼ 2 ' 0:55 À 0:64 ms, within 1 order of magnitude, corresponding to the fundamental resonance frequencies $ 302 and 309 kHz and Q $ 355 and 254 of the tapping mode cantilevers employed in cases of DxO@mGO and MxD@mGO, respectively. Such sampling satisfies the condition of ergodicity making the collected data every channel and their probability distributions statistically relevant to the fluctuational transitions about ϕ ¼ 0. In addition, when τ osc % τ sur within same order of magnitude, the underlying dynamics start following the viscous relaxation scale of the soft-surface. Thus, for τ c >τ osc in the limit τ c ! τ osc at our ultra-light tapping implementation, the mean surface deformation and the fluctuations about the mean also become a function of the cantilever's mean amplitude evolution at every pixel. The frequency fluctuations and the time delay associated with such evolution thus should reflect a unique distribution relevant to the enforced conditions. Contrarily, large amplitude oscillations approximate steady-state dynamics when τ c <τ osc . The phase crossover between the equilibrium states is sudden, necessarily satisfying the Gaussian distribution approximation of the central limit theorem in the limit that the fluctuations become statistically insignificant and dynamically irrelevant.
Under our enforced fluctuational conditions (τ c >τ osc ), however, we may infer that the dynamic amplitude measured at each pixel will be proportional to a phase lag Δϕ¼τ c Á ω À Δ ω ð Þ(see "Methods") as Δa being the mean deformation at each amplitude setpoint.
Herein lies the significance of fluctuations or the adiabatic " anti-adiabatic crossover as the fundamental basis of origin of Δϕ in our experimental description. The mean deformation Δa for each operational A sp were determined from the Lorentzian width (heterogeneous broadening) of the normalized amplitude histograms as shown in Fig. S1. Following the premise of Eq. (2), phase lag Δϕ is expected to follow a complex probability distribution in the zeroth order of τ c Δ ω , signifying energy losses via both homogeneous (inherent steady state losses) and heterogeneous (dynamic interaction losses) mechanisms. Indeed, complex distributions in Δϕ appear in the form of convoluted peaks centered at Δϕ þ and Δϕ À changing in form as a function of A sp for DxO@mGO and MxD@mGO samples as evident from the phase histograms (Fig. 2e, f) of the phase images ( Fig. 2b and 2c).
For the control sample mGO, the phase responses resemble homogeneous distributions ( Fig. 2d and Fig. S2), since deformations are expected to be predominantly elastic at this perturbation frequency. The two-stage distribution for the drug@mGO samples is significant from the fact, that it appears from dynamic tip interactions at the nanoscale and not from typical resonance bi-stability on account of amplitude nonlinearity 55 . The span of the two fluctuational peaks ϕ * ¼ Δϕ þ À Δϕ À becomes a measure of the apparent phase-contrast that highlights distinctive phase features obtained in our experiments (Fig. 2b, c). The individual peaks encode the dynamic loss mechanisms which essentially materialize from two disparate timescale events relative to the operational correlation timescale τ c : It must be noted here that τ c is not strictly independent of A sp at constant P and I, since cantilever steady-state time has a strong frequency and Q dependence when dissipation dominates and deviates from linearity. Operationally in our experiments P and I were optimized to an order of few % at each A sp to ensure conformed trace-retrace scans. The separation of the obtained two-stage phase peaks ϕ * signifies the separation of interaction timescales as captured by the underlying dynamics via fluctuational crossovers as explained in detail in the "Timescales and significance of fluctuations" section before. A longer timescale ensues from slower surface relaxation events. The tip follows the strain history via interactions that are finite time-correlated every successive oscillation cycle at a pixel. These reflect as a phase-lag Δϕ À probability transition. The very nature of time-correlated interactions is suggestive of a heterogeneous (Cauchy) distribution. A second inherently faster oscillatory timescale results from surface interactions that are uncorrelated (spatial and temporal) down to nearest neighbor pixels. A phase-lead Δϕ þ probability transition results with an apparent homogeneous (Gaussian) distribution. This heterogeneous phase-lag transition peak centered at Δϕ À appears in the case of viscoelastic DxO@mGO and MxD@mGO but is absent in mGO ( Fig. 2d and Fig. S2). The loss mechanisms at either peak demand elaboration. We consider a minimal scalar model here with the least number of parameters to explain the transitions phenomenologically. A more detailed exact solution model can be formulated in the likes of Sollich's exposition 23 incorporating multiscale dynamics, essentially giving the same conclusions. Normalized cumulative density graphs (Fig. 3a, c) for each A sp as a function of Δϕ highlight a clear two-stage transition in dissipated energy ΔE dis . In comparison, the same for mGO depicts a more homogeneous Boltzmann-like transition as discussed in Fig. S2. For our case where the distributions deviate from a single homogeneous distribution, it is important to consider how a change in a parameter, A ! A þ Δa, would affect a change in the average value of its dependent variable Δϕ h i. Under a generalized assumption of coupled homogeneous and heterogeneous loss mechanisms 56 from disparate interaction timescales as elucidated before, a measure of the net energy dissipation as a function of appearing Δϕ can be formulated as where σ AG and σ AL are the respective homogeneous and heterogeneous standard deviations of Δϕ þ & Δϕ À for a surface deformation Δa at the oscillation amplitude of A. In principle, Δϕ h i x is the ensemble average that is proportional to its variance at the parameter value x; x taking the values A and A þ Δa. For example, Δϕ h i A explicitly represents a probability density Δϕ Þthe normalized distribution function. The integral range Ω denotes the Δϕ span obtained in an experiment for a particular A sp , thus accounting for an estimate of the net dissipated energy ΔE dis as in Eq. (3). We further re-cast Eq. (3) to a fitting function of coupled cumulative energy densities (see Supplementary Section C and Fig. S3 for the goodness of fit) from which the respective homogeneous and heterogeneous probability densities centered at Δϕ þ and Δϕ À are determined. P Δϕ þ À Á and P Δϕ À À Á thus essentially represent the relative energy losses via homogeneous and heterogeneous routes, respectively.
The phase image and histograms for DxO@mGO and MxD@mGO obtained at high amplitude oscillations are also shown in comparison (Fig. 3b, d). The absence of any secondary transition at Δϕ À is apparent. Imaging at lower amplitude A increases the probability of tracking the Δϕ À transitions at the heterogeneities since the correlation time τ c theoretically tends to 1 when Δϕ ¼ 0. This promotes a relative skew in the Δϕ À transition density at very high A sp 's (Fig. 2e, f). In principle, the secondary Δϕ À transition suggests a negative shift in the eigen frequency with acute losses. This is rather very interesting from the aspect of fluctuational trajectories and Lyapunov exponents in phase space. An in-depth theoretical treatment on this will be taken up elsewhere. This secondary transition is clearly absent in mGO, which is expected to exhibit predominant equilibrium behavior at our perturbation rates $ 10 2 s À1 . The probability distribution in Δϕ appearing for bare mGO is centered around Δϕ ¼ 0 with a longer positive tail (Fig. S2). The phase-contrast and peak height decrease at lower A, signifying lower energy losses as expected for materials with dominant elasticity. The positive skew in Δϕ is suggestive of a positive shift in eigenfrequency. For pure elastic deformations, this is reminiscent of an effective increase in surface stiffness as has been considered earlier 22 at high amplitude tapping. Further, to decouple noise correlation effects, the correlation length results of the height and phase images were taken into consideration and compared to previous results 32 (see Supplementary section D and Fig. S4 for details).
Importance of captured strain rate. The energy losses as captured in Fig. 3a, c provides access to both the elastic and timedependent viscous interactions as a function of Δϕ distributions since the cumulative energy losses have nonconstant zerocrossovers. Specifically, ΔE dis at Δϕ ¼ 0 o provides a measure of conserved energy in the underlying dynamics. In our notation, the conserved energy is the same as energy stored ΔE stored . Elastic deformation caters to this conserved energy scale providing a means to determine variations in elastic surface energy γ el on tip interactions. The viscoelastic deformation on the other hand is inelastic, and causes variations in surface stress as a function of the strain rate. This can be estimated from the net steady state ΔE dis as a function of apparent phase-contrast ϕ * obtained in experiments using Eq. (1). The effective surface deformation in each case changes with the effective interaction area over which tip-surface energetics take root. The deformation-dependent changes in the interaction area were approximated as A interac ¼ 2πR Á Δa with R % 12 nm (Fig. S5) alongside the deformation rates Δ _ a ffi ω Á Δa in units of m Á s À1 . The deformation rate represents a measure of the strain rate scale in the underlying dynamics as highlighted in Fig. 4 color bars. ΔE stored and ΔE dis are plotted against the estimated deformation dependencies, to determine the elastic γ el and viscosity coefficient η as ΔE stored ¼ Àγ el ΔA interac and ΔE dis ¼ exp ÀηA interac Δ _ a À Á vide 58 as shown in Fig. 4. The relative variations in energy stored and dissipated vs the strain rate amplitude (s −1 ) are also suggestive as clear from the color bars in Fig. 4. For an elastic material, higher strain rate amplitude leads to more energy storage and loss, while in case of a viscoelastic material the energy storage reverses at higher strain rate amplitudes with the dissipated energy remaining higher for higher strain rate amplitudes as expected. It must be noted here that since TM-AFM is a fixed frequency method, the variation in strain rate in our experiments is obtained via a statistically relevant deformation scale Δa that corresponds to a fixed oscillating amplitude A as set by the amplitude setpoint A sp . In essence, Δa is the mean deformation the surface encounters at a fixed tapping amplitude and frequency, depicting the viscoelastic dependencies in our experiments. The variations in Δa are thus indeed a function of different strain amplitude at a fixed frequency. The energy losses or the equivalent viscoelastic behavior show an exponential dependence as a function of the strain rate scale Δ _ a in units of m Á s À1 for a fixed perturbation frequency.
Theoretical consideration. For a detailed theoretical understanding, a comprehensive stochastic-dynamics model of motion is necessary to explain the probability of transitions or hopping between the states Δϕ þ and Δϕ À as captured with ultra-light tapping. One approach can be the probability of achieving states U Δϕ þ À Á and U Δϕ À À Á as a function of τ c , U Δϕ À Á being the potential function defining the energy of the states as depicted in (Fig. 5). Establishing such dependence will be crucial for identifying the stochastic parameters that are relevant to the underlying fluctuational dynamics appearing from tip-surface interactions in the limit Δϕ ! 0. However, such a detailed theoretical exposition is avoided here for the sake of brevity. Instead, we postulate and outline an energy partitioning approach that faithfully explains our results and institutes a general framework for explaining similar stochastic-resonance conditions 21 that may arise in an experimental realization. The energy ΔE osc required to attain the minima at Δϕ þ and Δϕ À appear from the losses in the driven oscillatory dynamics that are above the thermal noise potential E noise . It goes without saying that the operational parameters in the experiments are set to ensure that the deformations Δa that result in the Δϕ distributions in the first place are greater than the noise equivalent amplitude A noise ¼ ffiffiffiffiffi ffi where ΔE fluc represents the energy losses from the transitional probabilities corresponding to P Δϕ À À Á appearing from interactions at the interface heterogeneities. This probability density P Δϕ À À Á signifies the relative strength of transitional loss pathways. ΔE osc on the other hand, corresponds to energy losses from the oscillatory dynamics with P Δϕ þ À Á signifying the probability density of achieving the equilibrium states while E 0 accounts for the average energy conserved. The rationale behind casting Eq.
(3) in the generalized Gibbs measure form (Supplementary Section G) is the assumption that fluctuations from interactions at the interface heterogeneities become more probable in the limit Δϕ ! 0 at small amplitudes A and at smaller equilibrium separations z c . The fluctuational transitions augment the homogeneous energy losses centered at Δϕ þ with an additional phase lag probability centered at Δϕ À reflecting the overall phase contrast ϕ * ¼ Δϕ þ À Δϕ À in our phase images (Fig. 3b, d). The appearing Δϕ þ and Δϕ À distribution peaks, respectively, correspond to the faster resonance and the slower resonance broadening timescale affected by the tip-surface interactions. Equation (4) remarkably agrees with our experimental results for DxO@mGO and MxD@mGO normalized on the same scale, showing that the ratio of the fluctuational probability transitions to the equilibrium losses in the limit Δϕ ! 0 is exponentially related to the apparent phase-contrast ϕ * as shown in Fig. 6a justifying our Gibbsian approach. As evident from this result, the cantilever tip is more susceptible to experience fluctuational losses at low A near-contact imaging, though yielding a smaller net phase contrast ϕ * (Fig. 6b). The molecular clusters of DxO and MxD on mGO indeed are expected to exhibit surface heterogeneities, as the results indicate, owing to its viscoelastic properties which pronounce transitional losses from the viscous tip-surface interactions. The probability transition ratio P Δϕ À ð Þ P Δϕ þ ð Þ (Fig. 6a) provides the Gibbs measure partitioning of heterogeneous losses over homogeneous losses. mGO, on the other hand, undergoes predominant elastic deformation and thus the heterogeneous effects are nonsignificant at the probing frequency. The transition ratio curve thus flattens out to an extent of small oscillations around a linear dependence in ϕ * (Fig. 6a inset).
Conclusions on ultra-light tapping mode AFM. Exploiting the fluctuational regime in TM-AFM with ultra-light tapping allows us to make the following important conclusions: (i) Phase imaging with ultra-light tapping (A<3 nm) can discern both heterogeneous and homogeneous loss mechanisms as unique probability distributions centered at Δϕ À and Δϕ þ , but at the cost of low apparent phasecontrast ϕ * ¼ Δϕ þ À Δϕ À . ϕ * marks out the separation of the characteristic relaxation timescales that describe the underlying transitional dynamics. (ii) Homogeneous losses stem from hopping between equilibrium states while heterogeneous losses originate from fluctuational transitions to metastable states of equilibrium. The latter dominates at small A but are averaged out in high amplitude hard tapping. Fast transition between equilibrium states at high A approximates monostable dynamics 55 . (iii) Despite fluctuations at small A, a unique steady state irrespective of its evolving tip-surface potential U Δϕ ð Þ in Δϕ is exhibited, and is approached throughout the transitional dynamics following Gibbs measure of energy partitioning. The unique steady-state denotes the metastable relaxation state in the case of soft-matter. (iv) The exponential fitting coefficient signifies a dimensionless damping parameter characterizing the underlying fluctuational dynamics off equilibrium. Our obtained experimental fit value = 0.11 on imaging DxO@mGO and MxD@mGO (Fig. 6a) matches well to a generalized theoretical estimate of damping coefficient of order 10 À1 that considers similar dynamics 53 ; attesting merit to our analysis.
Application: transitional phase-contrast at contact resonanceenhanced AFM-IR can decouple thermal loss pathways in AFM-IR. The relevance of the crossover regime and the underlying transitional dynamics as argued above becomes even more crucial in studying and understanding thermal dissipative pathways in contact resonance-enhanced AFM-IR. Resonanceenhanced AFM-IR is emerging as a critical tool in characterizing compositional heterogeneities in addition to mechanical heterogeneities in bio-interfaces. This has been attempted with extracellular vesicles to aid with early diagnosis of diseases 59 , miscibility of pharmaceutical blends for therapeutic drug delivery applications, and with the composition of extra-cellular-matrix at the nanoscale boundaries of cells 60 to study cell kinetics and cell signaling. The correct interpretation of the heterogeneities is crucial, and this is where our fluctuational understanding as argued above can create inroads.
Pseudo-ultra-light AFM-IR tapping mode. In contact resonanceenhanced AFM-IR, while a tip is scanned in the contact mode, a Potential function U Δϕ À Á for the generalized theoretical consideration. The AFM-cantilever translates U Δϕ À Á in the ultra-light tapping mode operation. U Δϕ À Á in the case of soft-matter is expected to typically exhibit two minima 13 that correspond to steady-state and second metastable energy. This is dynamically captured by the cantilever as minima at Δϕ þ and Δϕ À while translating between the states. The minima are separated with a potential barrier ΔE fluc . It must be noted here, the potential barrier can consist of multiple transitional levels each having its own characteristics of relaxation timescale that would account for a phase delay Δϕ proportional to the bandwidth product Δ ω Á τ c ; where Δ ω is the frequency dispersion and τ c is the correlation time. Energy losses from fluctuations in the limit Δϕ ! 0 compensates the thermodynamic cost of transition to the metastable state Δϕ À from the steady state Δϕ þ . focused IR beam is simultaneously pulsed at the tip-surface junction. The IR pulsing rate is tuned to the contact resonance harmonic mode of the tip; the tip-surface interface essentially behaving like a viscoelastic contact junction. At the right IR wavelength, the absorption of IR energy resonantly excites vibrational states of molecules on the surface at the tip-surface junction. Nonradiative decay of these excited states causes thermal changes at the tip junction through multi-photon relaxation processes. The locally generated heat needs to dissipate at the nanoscale junction and there arise two possible routes: (i) a longtimescale conduction pathway and (ii) a faster dynamic dissipation route via a radiative coupling. Typically, in response to the long-timescale conduction pathway, the cantilever bends on heating generating a steady-state deflection giving IR-contrast as contact resonance (CR) amplitude (CR-amplitude in Fig. 7a, b). The dynamic dissipation route (fast timescale) on the other hand, manifests in the underlying CR dynamics. Details on optimization steps for the AFM-IR experiments is provided in the methods section. Essentially, thermal fluctuations at the tip-surface junction from IR absorption, instigate the probe cantilever to oscillate in a pseudo tapping mode with the active drive being at the tip of the cantilever. This establishes the pseudo-ultra-light AFM-IR tapping mode giving us the AFM-IR phase images. Radiative phonons instigated from the thermal fluctuations at the tip-junction have a finite probability to couple to acoustic phonon states of the probe tip (Fig. 8a), the tip-surface being a very low thermal mass junction. Such phonon couplings may promote transitional fluctuations in the underlying CR dynamics (Fig. 8b). The mechanism surrounding the energy dissipation probability from fluctuational transitions is illustrated in Fig. 8. Typically, radiative phonons relax over timescales that are in the order of ns, much faster compared to the contact resonance timescale in the order of a few microseconds. Thus, vide discussions of interaction timescales before, phonon coupling would result in an inherent fast interaction timescale in the underlying dynamics with respect to IR drive frequency. An apparent þΔϕ phase shift would result in giving a phase-contrast image (CR-phase in Fig. 7a, b). The fluctuational phase characteristics as a function of contact resonance amplitude are shown in Fig. 8d, e. Surface phonon excitations by IR and their subsequent coupling to acoustic phonon states of the cantilever, perturb the contact potential gradient. This enhances the probability transition into the transitional states from minute changes in heat. The modification of tip-surface potential function as shown in Fig. 8b in principle leads to multiple relaxation timescales. This is reflected as an apparent broadening of the fluctuational regime (Fig. 8b). The interplay of these multiple timescales is reflected in the apparent skewness of the phase lag distributions (Fig. 8d, e). In principle, following Eq. (3), the net dissipated energy is proportional to the standard deviation of the phase lag fluctuations. The slowest timescale being the rate-determining step of the underlying fluctuational dynamics, the effective phase contrast ϕ * becomes the harmonic mean of the asymmetric Gaussian distribution fits (Fig. 8d, e) and is plotted in Fig. 8c inset. An apparent skewness toward þΔϕ is evident in both DxO and MxD molecules due to inherent viscous and compositional heterogeneities. Such behavior is absent in the case of bare mGO as detailed in Fig. S7 along with IR contact resonance-enhanced dynamics. The longtimescale conduction dissipation path is measured from the cantilever deflection as IR amplitude contrast while the transitional dissipation path is measured from resonance fluctuations from the phase signal locked in at the contact resonance IR pulsing frequency. As clear from the results DxO has IR absorbance peak at 1416 cm −1 , while MxD has an absorption peak at 1455 cm −1 . The IR peak observed at 1416 cm −1 for DxO@mGO is a representative signature of the vibration modes of skeletal rings of a DxO molecule (four rings per molecule) 64,65 . The 1455 cm −1 peak of for MxD@mGO is attributed to the vibration mode corresponding to aromatic C=C stretch (three bonds per molecule) 66,67 . In contrast, the generated IR-contrast and phase lag contrast in the case of mGO shows no variations as a function of contact setpoint or its equivalent A cr /A 0 , where A 0 , in this case, is the contact mode mean deflection and A cr is the contact resonance amplitude at 100% IR power. Contact resonance characteristics relevant to the experiments are discussed in the Supplementary section H. DxO and MxD clusters on the other hand have appreciable IR absorbance and are reflected in the IR and phase-contrast images. Essentially IR-contrast (CR-Amplitude) and phase-contrast (CR-Phase) are complementarylocations having higher long-timescale dissipation reflect lesser dissipation by the transitional path and vice versa. These are more evident in the phase-contrast probability distributions showing an increase in phase contrast as a function of the contact mode setpoint in the case of DxO@mGO and MxD@mGO, respectively. mGO on the other hand does not show any variation since the contact resonance is not initiated.
Conclusions and broad applicability. The exploitation of the transitions can thus be crucial in determining surface mechanical and chemical heterogeneities of pharmaceutical blends with relevance to drug delivery and therapeutic applications. A further detailed study on the transitional dynamics from the aspect of probability distributions is imperative which can link the coupling of IR-induced surface phonon and cantilever tip phonon states. These will be taken up elsewhere. A higher resolvability of the noise and contact resonance amplitudes and exploitation of higher contact harmonic modes may prove beneficial in nanoscale imaging of phonon density of states from surface heterogeneities. High resolution Δϕ distributions at heterogeneous biointerfaces of DNA, proteins, cells, and tissues augmented via CR enhanced AFM-IR contrast signatures, can provide insights on mechanical cues that bio-systems adopt to bring about irreversible evolutionary changes.

Methods
Materials. Silicon wafer (Si-w) was immersed in piranha solution for 1h to introduce an oxide layer on the surface of the wafer. The oxidized Si-w was immersed in 3% of (3-aminopropyl) triethoxysilane (APTES) ethanol solution to produce a positively charged amine group on the surface. After washing with fresh ethanol, the negatively charged GO solution at concertation of 1mg/mL was poured on the APTES functionalized Si-w for 10 min. Then, the prepared sample was spincoated at 100 rpm for 2 s and 2000 rpm for 1 min to obtain mGO coated Si-w. Doxorubicin hydrochloride solution at a concentration of 0.1 mg/mL was prepared in deionized (DI) water and the pH of the solution was adjusted to pH 9 using NaOH. Then, the DxO solution was loaded on the mGO coated Si-w (mGO-Si-w) surface for 24 h. The DxO loaded mGO-Si-w (DxO@mGO) was washed with fresh DI water thrice to remove the unbounded DxO from the surface. Minoxidil loaded mGO (MxD@mGO) was prepared by the same method used for DxO@mGO, where minoxidil solution was initially prepared in ethanol at a concentration of 1 mg/mL and then diluted in DI water with a final concentration of 0.1 mg/mL before loading on mGO. Before each run samples were dried, and AFM was conducted in a continuous dry nitrogen environment. A comparative result of DxO@Si is provided in Fig. S8-small and high amplitude tapping highlighting the multi-stage phase distribution response. In our experiments, the tip-radius of the AFM probes are in the order of $ 12 nm. This limits imaging DxO and MxD clusters of roughly 8-15 molecules with intercalated water in our case.
Equipment. Bruker's ANASYS Nano IR2 system was used to run experiments. The tapping mode AFM images were obtained with soft tapping mode-PR-EX-TnIR-A probe cantilevers supplied by Bruker. Low amplitude runs within the fluctuational regime were optimized in steps starting with high amplitude tapping A 0 in the order of 10-12 nm till cantilever dynamics stabilize to give decent overlap of trace and retrace plots every scan. Subsequently, A 0 is brought down to the operational free amplitudes as annotated in Fig. 1 to achieve desired ultra-light tapping at amplitudes A in the order of 1-2 nm. Low damping at the operational small A gives us control over the dynamics to navigate the transitional excursions about the steady-state as highlighted in the Fig. 1. Setpoint amplitude was similarly optimized from low to high to obtain clean scans via overlap of trace and retrace scans. Images were captured with alternate low and high amplitude setpoint A sp following bisection rule to negate the charge saturation effects at near-contact closest separations of tip and sample. The representative frequency response characteristics of the tapping mode cantilevers used in our analysis of mGO, DxO@mGO, and MxD@mGO data are shown in Fig. S9. Multiple scans were conducted at each optimized setpoint to ensure minimization of drift and artifacts, though at the cost of low throughput.
For the contact resonance-enhanced AFM-IR experiments, the same Nano IR2 system was used in conjunction with a tunable quantum cascade IR laser from Pranalytica as the high-power IR source. Bruker's PR-EX-TnIR contact mode probes were used for getting the AFM-IR images. IR laser focus at the tip-surface junction was optimized with tip-engage to ascertain maximum contact resonance amplitude and establish the contact resonance-enhanced operation for AFM-IR runs (Fig. S7). The contact force was varied as a function of the contact mode setpoints at each run and the corresponding contact resonance-enhanced amplitudes A cr were determined from the amplitude spectra collected as an independent data channel. The contact resonance amplitude A cr varies as a function of the contact mode setpoint set in an experiment and a representative response spectrum (first contact resonance-enhanced mode) is shown in Fig. S7. The mean contact mode deflection for a given amplitude setpoint is obtained as a function of the spring constant of the probe cantilever. This along with the contact resonance noise spectra (Fig. S7) is used to determine the pseudo-ultra-light tapping AFM-IR free amplitudes A 0 . These corresponding AFM-IR response characteristics as a function of A cr Exploiting fluctuations by the correlation timescale τ c . To explain how fluctuations affect dynamics, it is convenient to start with the oscillator's susceptibility X ω ð Þ; which relate the average value of damped amplitude A as A h i ¼ X ω ð ÞA 0 e Àiωτ ; with A h i ¼ 0 in the absence of external drive and ω the frequency on tip-surface approach from amplitude damping. To capture the transients, correct ensemble averaging over the timescale τ is important such that the correlation timescale τ c of two successive data cycles at every pixel is finite. A linear amplitude damping on tip approach is a direct outcome of the oscillator's frequency dispersion Δ ω ¼ Δ ω ð Þ À Δ ω ð Þ Â Ã 2 D E 1 2 accentuated by frequency fluctuations Δ ω ð Þ near resonance. The regime of interest is the condition where the reciprocal of correlation timescale τ À1 c becomes comparable to the standard deviation of the fluctuations Δ ω . With the free amplitude frequency ω 0 , the largest frequency in the system, the rest of frequency scales Γ; Δ ω % τ À1 c and δω j j satisfy the condition to be ( ω near resonance. In effect, the frequency fluctuations Δ ω ð Þ becomes the parameter of significance dictating the linewidth of X ω ð Þ near resonance ω À ω 0 ( ω when simultaneously Γ; τ À1 c % Δ ω is satisfied. In the limit when τ À1 c is finite and nonzero, and τ c ! τ osc (τ c >τ osc as in our case), the energy interactions at the tip-surface junction becomes heterogeneous; leading to a Lorentzian linewidth spread in ImX ω ð Þ centered at ω À Δ ω with no averaging of the eigenfrequency 57 . Corresponding variations in amplitude of oscillation would then resolve as A þ Δa with Δa being the standard deviation (Lorentzian width) of the Fig. 8 Contact resonance-enhanced infrared atomic force microscopy schematic. a Illustration of IR-induced fluctuational enhancement. b describes the change in interaction potential function as a function of external IR excitation that brings about higher fluctuational probability transitions. c Energy dissipation probability as a function of phase contrast, the gray areas represent 98% confidence bands; Insets depict the apparent phase-contrast as function of the contact resonance amplitude (CR-Amplitude) ratios. d, e shows the IR phase-contrast characteristics in DxO@mGO and MxD@mGO, respectively. Error bars are computed from the standard deviations of the measured phase histograms and the dynamic amplitude A.
normalized amplitude histogram obtained in an experiment for a particular A sp . Δa essentially becomes the surface deformation at the interaction timescale of τ c ! τ osc τ c >τ osc À Á that is in the same order as τ sur % 10 2 À 10 4 ns 61 . In the opposite limit τ c <τ osc however, the oscillator cannot resolve the frequency variations and are thus averaged out giving Δ ω ffi 0. The linewidth shape of ImX ω ð Þ remains a Lorentzian Γ centered at ω À Δ ω with Δ ω ffi 0. The limit τ c <τ osc 35,40,51,52 has been the premise of hard tapping operation that satisfies Eq. (1) for both hard elastic and soft viscoelastic surfaces at higher tapping amplitudes. However, the data captured under such conditions, convey energetics that asymptotically tend to the steady state dynamics overlooking the fluctuations. The fluctuations near equilibrium are thus averaged out as explained, missing out on the critical NE energetics of tip-surface interactions.