Heterogeneous Dissipation and Size Dependencies of Dissipative

Article pubs.acs.org/Langmuir

Heterogeneous Dissipation and Size Dependencies of Dissipative Processes in Nanoscale Interactions Karim R. Gadelrab,† Sergio Santos,*,† and Matteo Chiesa Laboratory for Energy and Nanosciences, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates S Supporting Information *

ABSTRACT: Here, processes through which the energy stored in an atomic force microscope cantilever dissipates in the tip−sample interaction are first decoupled qualitatively. A formalism is then presented and shown to allow quantification of fundamental aspects of nanoscale dissipation such as deformation, viscosity, and surface energy hysteresis. Accurate quantification of energy dissipation requires precise calibration of the conversion of the oscillation amplitude from volts to nanometers. In this respect, an experimental methodology is presented that allows such calibration with errors of 3% or less. It is shown how simultaneous decoupling and quantification of dissipative processes and in situ tip radius quantification provide the required information to analyze dependencies of dissipative mechanisms on the relative size of the interacting bodies, that is, tip and surface. When there is chemical affinity, atom− atom dissipative interactions approach the energies of chemical bonds. Such atom−atom interactions are found to be independent of cantilever properties and tip geometry thus implying that they are intensive properties of the system; these interactions prevail in the form of surface energy hysteresis. Viscoelastic dissipation on the other hand is shown to depend on the size of the probe and operational parameters.

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calibration of the natural frequency of the cantilever ω0, the quality factor Q, the spring constant k, the phase lag φ of the oscillation amplitude A relative to the drive, and the conversion of A from volt units to meters. The expression reads

INTRODUCTION Nanoscale heterogeneity may influence phase transformations, provide materials with nucleation sites for the evolution of processes,1,2 and even drive the self-repairing mechanisms involved in biomolecular processes such as ductility enhancement, damage evolution, and toughening.3 The interest in these studies is becoming more general and broader, and it is leading to a true understanding of macroscopic and biological phenomena from nanoscale and molecular points of view.2−7 Due to its versatility,8−11 atomic force microscopy (AFM) has recently been employed in several studies to determine the heterogeneity of samples in terms of variations in the dissipative channels1,6,9,12,13 sometimes utilizing higher modes of vibration in liquid.13,14 The reconstruction and understanding of the dissipative terms however is particularly challenging partly due to the presence of hysteretic interactions15 that might relate to capillary effects or chemical affinity,10,15−19 artifacts related to the feedback and/or control system,20 the overall presence of processes that are intrinsically challenging,18 stochastic,1 or might include several possible mechanisms,9,17,21 and the fact that the nature of friction in the nanoscale is still relatively unknown and controversial.22 Thus, it could be argued that these difficulties legitimate the continuous efforts in both theoretical12,17,21 and experimental studies of nanoscale dissipation.7,22,23 The field of energy dissipation in amplitude modulation atomic force microscopy (AM AFM) was rigorously introduced in a robust fashion that allowed for energy quantification in 1998.24,25 In principle, the method simply requires the © 2013 American Chemical Society

=

πkA 0A ⎡ A⎤ ⎢sin(φ) − ⎥ Q ⎣ A0 ⎦

(1)

where is the mean energy dissipation per cycle and A0 is the free amplitude or unperturbed oscillation of the cantilever. From here onward, is simply written as Edis. Practically, A0 is measured when the cantilever−sample separation zc is large enough that the tip−sample forces do not influence its decay; that is, 3/2A0 < zc < 2A0. Note that zc refers to the unperturbed or rest cantilever−sample separation,26 and it is a constant in the steady state. Typically, this separation is written as the sum of the instantaneous cantilever deflection z and the tip−sample distance d; that is, d = zc + z.17,26 In order to use eq 1, the drive frequency ω should be set to ω0. While energy might dissipate in the tip−sample junction via several dissipative modes,9,10 eq 1 provides the net energy dissipated per cycle with great accuracy. That is, while better approximations might be used,14,27 eq 1 provides sufficient accuracy in ambient environments where the Q factor is relatively high. Methods were later proposed to identify the physical origin of the different dissipative channels involved in Received: November 7, 2012 Revised: January 21, 2013 Published: January 22, 2013 2200

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the interaction,9,11,21 with some aiming at quantifying the relevant parameters.10,23 Two main mechanisms controlling the net dissipation during mechanical contact were initially identified8,9,28 as (1) viscoelastic interactions where the viscosity η is the characteristic parameter of the sample under study and (2) variations in surface energy γ where during tip approach the surface energy is γa and during tip retraction it is γr (γr > γa for hysteresis). More recently,29 a methodology has been proposed to quantify these parameters, that is, η (viscoelasticity), Δγ = γr − γa (surface energy hysteresis), and maximum sample deformation δM, directly from experimental dissipation curves. While one would be too optimistic to expect that a relatively simple theory would satisfactorily and accurately account for energy dissipation in the nanoscale for the whole variety of samples, here, we show that the accuracy of the proposed method29 is enough to yield quantitative predictions. It could also be argued that size dependency is the landmark of nanotechnology. In this respect, this work also shows how simultaneous in situ calibration of the tip radius R30 further allows discerning the dependencies of variations in the dissipative processes on R.

given prescribed tip−sample deformation−magnitude relationship, namely, that it is proportional to the viscous coefficient η and the contact radius r = (Rδ)1/2 as given by the Derjaguin Muller Toporov (DMT) model.36 Viscoelasticity is also assumed to be linear in and an odd function of velocity ż. In short, the viscoelastic force is here written as Fη = −ηrż. This is the common expression in the dynamic AM AFM literature.28,33,37 The odd character of viscoelasticity and its linearity in velocity could be theoretically established as a requirement rather robustly.16,33 On the other hand, its proportionality to r is an assumption that is typically made in the literature and shown to be significantly accurate, at least qualitatively, via numerical integration of the equation of motion and by comparison with experimental data.9,10 Finally, the main assumption in terms of the hysteretic force is that both γr and γa remain constant during tip−sample deformation. These parameters, namely, γr and γa,, quantify the chemical affinity between the tip and the sample in a given medium and could therefore be thought of as intensive properties of the system. Thus, as they provide information about atom−atom interactions,38 it is reasonable to assume that they remain constant. In short, and despite the relatively limiting restrictions placed by the above assumptions, it will be next shown that, provided they hold, these allow obtaining quantitative information about the sample from the energy dissipation curves alone and for a variety of samples. Let us start by recalling that, since A can be arbitrarily set in AM AFM (0 ≤ A ≤ A0), there are many points for which eq 4 must apply. Furthermore, a given value of δM, say δMi, corresponds to a given value of A, say Ai. Thus, a relationship for two points δMi and δMj (Ai and Aj) can be found directly from experimental parameters by writing

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RESULTS AND DISCUSSION Let us first define the tip−sample interaction in terms of the energy dissipated via viscoelasticity Eη and surface energy hysteresis Eγ. By assuming that higher harmonics can be neglected, it follows that (limitations are detailed in the literature and later below29) Eη = BηA1/2 δM2

(2)

and

Eα = C ΔγδM

(3) 1

δj = δi − Δdmij

where B = (√2/4)πR ω, C = 4πR, and δM is the maximum tip−sample deformation within the interaction for a given amplitude A. Then,29 Edis =

BηA1/2 δM2

(5)

/2

+ C ΔγδM

and Δdmij = Δzcij + Δz 0ij − ΔAij

(4)

(6)

where Δz0ij is the increment in cantilever mean deflection z0 as the cantilever approaches the surface and ΔAij and Δdmij are the respective increments in oscillation amplitude A and in distance of minimum approach dm. The distance of minimum approach dm is the distance at which the tip turns from tip approach to tip retraction in an oscillation cycle, hence the name distance of minimum approach. Details on its use can be found in the literature.33 Here, the terms dm and δM are used interchangeably because tip−sample deformation is always assumed to occur. Nevertheless, care should be taken if the tip oscillated in the attractive or noncontact regimes. The increment in cantilever separation Δzcij can be obtained with the precision of the zpiezo of AFM, that is, subangstrom.37 While eqs 5 and 6 alone can be used to reconstruct the shape of δM as a function of the normalized amplitude A* = A/A0, the reference values are arbitrary. On the other hand, if the parameters in eq 1 can be calibrated39,40 and provided R is known,30 then the only unknowns in eq 4 are the sample’s parameters η and Δγ and the tip−sample deformation δM. It will be shown below that the geometrical relationships in eqs 5 and 6 alone can be used to calibrate the conversion of the oscillation amplitude A from volts, which is the experimental observable in AM AFM, to nanometers with a precision of 1 nm or less, that is, errors of 3% or less. Since this is arguably one of the main challenges in terms of the use of eq 1,41 such accurate calibration should

Equation 4 assumes that (1) no higher harmonics are excited, (2) viscoelasticity and/or surface energy hysteresis are the two dissipative mechanisms present in the interaction, and (3) that these two mechanisms can be properly modeled in dynamic AFM with the use of eqs 2 and 3, respectively. The first assumption is legitimate in ambient AM AFM where the Q factor is relatively high (Q ∼ 102−103).31,32 The second assumption requires that viscoelasticity and/or surface energy hysteresis are at least the dominant dissipative processes in the interaction during mechanical contact.29 It is worth noting that these two processes have been employed in dynamic AFM studies by a number of groups for several years now10,28,33−35 and that they have also been experimentally tested.9,10,35 Furthermore, their possible presence and/or dominance in the tip−sample interaction can in fact be qualitatively established in AM AFM by monitoring the behavior of energy curves.6,9,11,21,23 These qualitative methodologies will be explained in detail below, and experimental examples will be shown and discussed. Assuming that long-range dissipative processes are also present in the interaction, their relative importance can be inhibited by, for example, driving with sufficiently large amplitudes and inducing large enough tip− sample deformation δ.10,29 The third assumption is the most restrictive, since it imposes that the viscoelastic force has a 2201

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otherwise lead to robust energy measurements. By taking the above into account, it turns out that, by using two values for A in eq 4 and the relationships in eqs 5 and 6, a constant K can be defined that fully determines δM, viscoelasticity, and surface energy hysteresis.29 In fact, K is independent of R, and reads29 K = K ij =

EdisiδM−i1 − EdisjδM−j1 Ai1/2 δMi − A1/2 j δ Mj

(7)

From eq 7, the tip−sample deformation δM for a given amplitude A can be recovered. That is, from the relationship Kij − Kik = 0, the absolute value δM can be found.29 In terms of the sample’s parameters, the following follows η = DK

where

D=

2 2 πR1/2ω

(8)

Figure 1. Experimental results for a silicon tip interacting with a silicon sample with a tip radius of R = 12 ± 2 nm as a function of normalized amplitude A*. (a) Conservative (dashed blue) and dissipative (continuous black) phase branches for the system where ΔΦ is the phase difference. (b) Qualitative dE*dis/dA* (red circles) and E*dis × ΔΦ* (blue squares) methods have been used to hint at the nature of the dissipative processes. In this case, surface energy hysteresis controls the interaction according to both methods.9,11,19 (c) Reconstructed distance of minimum approach δM with the use of eqs 5−7. Where δM ≥ 0, tip−sample deformation occurs. Note that δM becomes positive only in the repulsive regime, i.e., after the force transition, implying that, in the attractive regime, there is no sample deformation. (d) Decoupling of surface energy hysteresis Eγ* (red squares) from viscoelasticity Eη* (blue triangles) from total energy Edis* (black circles); the raw data is presented in light gray. The maximum energy is Edis ≈ 110 eV, and the free amplitude A0 ≈ 24 nm.

and Δγ =

1 (EdisδM−1 − A1/2 δMK ) C

, C = 4πR

for any pair A and δM (9)

where it should be observed that, while η and Δγ are dependent on R, the expressions η/D and ΔγC can be found experimentally from energy dissipation measurements and the oscillation amplitude alone, that is, K and A. Here, this formalism has been implemented in Matlab42 and applied to a variety of samples to quantify η, Δγ, and δM experimentally. The experimental data has been obtained with a commercial Cypher AFM from Asylum Research. In Figure 1, some results are shown for a silicon tip interacting with a silicon sample. Figure 1a shows the conservative (discontinuous blue lines) and dissipative (continuous black line) phase branches as a function of A*.11,25 The concept of phase difference11,23 ΔΦ (shown in Figure 1a) has recently been shown to provide information about the presence or absence of viscoelasticity and surface energy hysteresis11,23 (asterisks imply normalized values throughout). More thoroughly, if the product E*dis × ΔΦ* presents maxima11 (or convexity11,19,23) at intermediate values of A* (i.e., 0.2 < A* < 0.8), then viscoelasticity is present. If E*dis × ΔΦ* monotonically increases with decreasing A* at intermediate values of A* (or presents concavity), then surface energy hysteresis dominates. The predictions for the silicon tip−silicon sample system are shown in Figure 1b (blue squares) in terms of E*dis × ΔΦ*. Note that concavity and a monotonous increase in E*dis × ΔΦ* with decreasing A* are observed indicating that Eγ dominates over Eη. The dominance of Eγ can be further confirmed (red circles in Figure 2b) with the method proposed by Garcia et al.,9 that is, dE*dis/dA*, where if dE*dis/dA* ≈ 0 at intermediate values of A*, then Eγ controls the interaction. The reconstruction of δM as a function of A*, according to eqs 5−7, is shown in Figure 1c. Maxima in δM (≈1 nm) occurs at intermediate values of A* where there is a relatively flat plateau. This plateau tilts with positive slope and/or negative slope if the conversion from amplitude in volts into nanometers is not correct (see the Supporting Information for details). In particular, such tilting can be detected with the use of the geometrical relationships (eqs 5 and 6) alone implying that the conversion can be calibrated by minimizing the tilting (Figure S3, Supporting Information). Where δM is positive in Figure 1c, deformation occurs. In fact, the definition of tip−sample deformation given here is rather peculiar. It

defines as deformation the range of tip−sample distances for which mechanical dissipation occurs. This provides an unambiguous definition of spatial boundaries in terms of dissipation channels. Where δM is negative, the formalism presented in this work (eqs 7−9) should not be used to recover material properties. In practice, one should initially employ the data from the repulsive branch only to get an indication (with the help of eq 7) of whether δM is negative in the attractive branch. In the case of Figure 1c, note that, before the attractive to repulsive transition occurs26 (gray highlighted area in Figure 1c), δM is negative. Hence, the data in the repulsive regime only, that is, A* < 0.85 (Figure 1d), has been employed to find, for the silicon tip−silicon sample system, δM = 1.1 ± 0.2 nm of peak deformation (eq 7), η=35 ± 30 Pa·s (eq 8), and Δγ = 90 ± 30 mJ (eq 9). These are the median, maximum, and minimum values, respectively, found for the particular tip− sample system discussed in Figure 1. The range is a consequence of amplitude and phase experimental errors and possibly deviations in the behavior of the actual dissipative phenomena relative to the model proposed in eq 4 and as discussed above regarding its assumptions. In particular, the data, including errors, have been obtained by assuming, for the system in Figure 1, that R = 12 ± 2 nm and also by comparing the results of multiple data sets or curves. The raw energy (normalized) dissipated, according to eq 1, is shown in Figure 1d (light gray). From the recovered values for η, Δγ, and δM and with the use of eqs 2 and 3, the energy dissipated in each contribution, that is, viscoelasticity Eη (eq 2) in blue triangles and surface energy hysteresis Eγ (eq 3) in red squares, can then be recovered as a function of normalized amplitude A*. It can be observed in Figure 1d that Eη accounts for less than 10% of the total energy Edis and that Eγ overwhelms it. It is clear that 2202

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surface energy hysteresis arises from a rearrangement of atoms, electronic or physical, due to the free energy or chemical affinity of the surfaces, then these relatively large values imply that bonds with interaction energies close to those of chemical bonds might be forming and rupturing intermittently. In Figure 2, the results obtained for a silicon tip (AC160TS) interacting with a variety of samples and where R = 8 ± 2 nm are shown. The samples are silicon (black squares), polypropylene isotactic (blue triangles), ferrite (red circles), and graphite (brown crosses). Again, the dE*dis/dA* and the E*dis × ΔΦ* methods are shown in Figure 2a and b, respectively. In particular, the convexity of the E*dis × ΔΦ* signal and the presence of maxima at intermediate values of A* indicate the presence of strong viscoelastic interactions Eη relative to surface energy hysteresis Eγ for the polypropylene and ferrite samples. This effect is particularly pronounced for the polypropylene sample and agrees with previous studies on polymers using the dE*dis/dA* method.9 Note however that the E*dis × ΔΦ* signal for the ferrite sample tends to increase and displays some concavity near A* ≈ 0.2. The implication is that the contribution due to Eγ for the ferrite sample should also be significant.11 The concavity of the E*dis × ΔΦ* signals for the silicon and graphite samples and the monotonous increase for the same range of A* indicate that Eγ controls dissipation for these samples. The corresponding deformations are shown in Figure 2c. Note the relatively large values of δM (peak) ≈ 4−5 nm for the graphite sample. While these values are relatively large, DC force curves indeed show that this relatively sharp tip could induce such large tip−sample deformations (see the Supporting Information for details). For the silicon sample here (black squares), the deformation is predicted to be positive, even if very small, even in regions of the attractive regime (note the vertical step in δM for A* ≈ 0.95). Physically, it is plausible that, for this sharper tip (compare with Figure 1c), the adhesion forces are so small that the tip can induce larger deformations and go straight into mechanical contact even in the attractive regime (see DC force curves for this tip−sample system in the Supporting Information). The corresponding energy contributions for the two dissipative channels, namely, Eγ and Eη, are shown in Figure 3 for the graphite, ferrite, and polypropylene samples. The results for the silicon sample are not shown, since they are similar to those discussed in Figure 1. Note that the predictions are in agreement with the above qualitative methods (Figure 2a,b). In short, for the graphite sample (Figure 3a), the contribution due to Eγ is more than twice that due to Eη. Note that the energies are normalized with respect to maxima in Edis. For the ferrite sample (Figure 3b), the contributions are similar, that is, Eγ ≈ 80 eV ∼ Eη ≈ 105 eV of maxima. Finally, for the polymer (Figure 3c), the maxima in Eη (≈45 eV) approximately triple those in Eγ (≈15 eV). The predictions for the range shown in Figure 3 are as follows: silicon: η = 25 ± 20 Pa·s, Δγ = 80 ± 50 mJ, ργ = 3 ± 0.7 eV/nm2, ρη = 0.27 ± 0.03 eV/nm2; graphite: η = 7 ± 2 Pa·s, Δγ = 30 ± 10 mJ, ργ = 1.2 ± 0.3 eV/ nm2, ρη = 0.5 ± 0.1 eV/nm2; ferrite: η = 61 ± 15 Pa·s, Δγ = 45 ± 10 mJ, ργ = 1.6 ± 0.5 eV/nm2, ρη = 2.4 ± 0.3 eV/nm2; polypropylene: η = 61 ± 15 Pa·s, Δγ = 6 ± 4 mJ, ργ = 0.24 ± 0.06 eV/nm2, ρη = 0.8 ± 0.1 eV/nm2. Note that, for the graphite sample, deformations for a larger range of A* have been calculated since δM is positive even in the attractive regime (Figure 2c).

Figure 2. Experimental results for a silicon tip with a tip radius of R = 8 ± 2 nm interacting with silicon (black squares), ferrite (red circles), polypropylene (blue triangles), and graphite (brown crosses) samples. The x axis is the normalized amplitude A*. (a) The dE*dis/dA* and (b) E*dis × ΔΦ* methods have been used to hint at the nature of the dissipative processes for each sample. (c) Reconstructed distance of minimum approach δM with the use of eqs 5−7 for each sample. Note that δM is positive throughout only for the graphite sample (brown crosses). The free amplitudes are ≈30, 45, 14, and 50 nm, respectively.

this result agrees with the qualitative E*dis × ΔΦ* and dE*dis/ dA* methods (Figure 1b). Note however that eqs 8 and 9 depend on R. The value of R has been estimated with the use of the critical amplitude or Ac method (cantilever model AC160TS where k ≈ 40 N/m and f 0 ≈ 300 kHz). Here, transitions to the repulsive regime were observed at Ac ≈ 20− 22 nm implying that R = 12 ± 2 nm. This follows from the relationship R = 4.75Ac1.12 and by assuming that transition values for silicon and mica samples are similar.30 Thus, once δM has been found (Figure 1c) and the energy dissipated quantitatively decoupled into contributions (Figure 1d), the density due to each dissipative channel in electronvolts per square nanometer can be estimated.19,43 For the system in Figure 1 and by taking the mean values of Δγ and η, the results are ργ = 3.7 ± 0.7 eV/nm2 and ρη = 0.3 ± 0.03 eV/nm2 for surface energy hysteresis and viscoelasticity, respectively. The errors are due to the uncertainties in tip radius, R = 12 ± 2 nm, and the values are estimated at peak deformations. By assuming that the atomic layers of the tip and the sample at the interface consist of 12 atoms, approximately 0.3 ± 0.05 eV are dissipated per atom due to surface energy hysteresis. These energies are about 1 order of magnitude smaller than those due to covalent bonds and 1 order of magnitude larger than those corresponding to van der Waals interactions. In fact, this order of magnitude more closely corresponds to that of hydrogen bonds.38,44 Thus, if the energy dissipated due to 2203

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(with the relatively small peak forces involved in AM AFM, that is, ∼1−102 nN) indicate deformations in the nonlinear elastic regime. In this respect, we have found close agreement between the deformations predicted with the use of DC force curves (see the Supporting Information for details) and the deformations predicted via the present formalism (Figure 2c). In particular, relatively sharp tips, that is, R < 10−20 nm, seem to induce relatively large deformations according to DC curves, whereas blunter tips, that is, R > 20−30, induce smaller deformations. This relationship between tip radius R and tip− sample deformation δM is reproduced in the AC curves here. For example, for graphite, and when R ≈ 25 nm, maxima in deformation lie in the 3 nm range rather than in the 4−5 nm range as in Figure 2c. As a side note, it is worth mentioning that the presence of a dissipative channel such as surface energy hysteresis might contribute to an increase in tip−sample deformation according to standard AM AFM modeling10 even with the use of larger tips. Furthermore, with the larger tips (see the Supporting Information for details), the contributions due to surface energy hysteresis might overwhelm any viscoelastic processes in terms of the magnitude of the energy dissipated and the energy per square nanometer or atom. This phenomenon should be related19,29 to (1) the increase in Eη 1 with R /2 (eq 2) and the linear increase in Eγ with R (eq 3) and (2) the fact that larger tips induce smaller deformations thus also decreasing the contributions from Eη as predicted by eq 2 where Eη increases with the square of δM . In terms of densities (these are in joules/m2),

Figure 3. Experimental decoupling of surface energy hysteresis Eγ* (red squares) and viscoelasticity Eη* (blue triangles) from total energy (smoothened) Edis* (black circles); the raw data is presented in light gray. The maximum energies are ≈60, 125, 180, and 55 for the silicon, (a) graphite, (b) ferrite, and (c) polypropylene samples, respectively; the results for the silicon sample are not shown but are similar to those discussed in Figure 1.

ρη =

5 2 ω 1/2 ηA δ M 12R1/2

(10)

and

ργ = 6Δγ

It is interesting to note that viscosities 2 or 3 orders of magnitude larger than the ones reported in this work have been recently predicted.10 Nevertheless, such work required (1) that the magnitude of the energy dissipated in the simulations matched the energy dissipated experimentally and (2) an appropriate choice of the conservative forces. In particular, the choice of forces, that is, large bulk values of elastic modulus for the tip and the samples that prevented deformation, led to very small deformations, that is, δM < 1 nm. Such small deformations implied that very large values of η were required in the simulations in order to match the experimental results. On the other hand, the present results (Figures 1−3) do not depend on assumptions for the conservative part of the force and have led to the prediction of much larger deformations (see Figures 1c and 2c). In turn, these larger deformations directly lead to smaller values of η that are in closer agreement with standard values.9,27,33 Moreover, despite the disparate predictions between the values of η predicted here and those in other10,19,23 studies, the magnitude of the energy dissipated via viscoelasticity is similar. As above, this is a result of the compensation in the increase in deformation with decreasing η. Note however that ργ and ρη should decrease with decreasing Δγ and η, respectively, as discussed below. From the above arguments, and since the tip−sample deformation is directly related to the elasticity of the tip−sample system, it is clear that the dramatic discrepancies between the results reported here (η ∼ 1−102) and in ref 10 (η ∼ 103−104), where bulk values of elastic moduli where employed, must relate to deviations in elasticity on the surface. Moreover, it could be argued that deformations as large as 1−2 nm in stiff samples such as silicon

(11)

That is, ρη is predicted to decrease with increasing R for a given amplitude A and increase with increasing deformation δM (eq 10). This implies that both conditions reduce the contributions of ρη with increasing R. On the other hand, ργ should not depend on R (eq 11). The physical implication is that ργ is independent of the number of atoms interacting and velocity; that is, ργ should be related to the specific atom−atom interactions (intensive property) or chemistry. We have experimentally observed that the relationships involving ργ, R, and δM in eqs 10 and 11 agree for the tip−graphite interaction (see the Supporting Information for details). Also note that, for the graphite sample, ργ is about a factor of 2−3 smaller than for the silicon sample. For the ferrite sample, the value of ργ is halved relative to that of the silicon sample. These values however are still slightly larger than purely van der Waals interactions and might be due to the chemical affinity of the oxide layers on the tip and the samples. On the other hand, for the polypropylene sample, ργ is about an order of magnitude inferior and thus more closely agreeing with purely van der Waals interactions, that is, London forces.

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CONCLUSION It has been shown that accurate calibration of the atomic force microscope, together with advances made in electronics and low noise data acquisition, allow obtaining quantitative information about dissipative processes in the nanoscale even with the use of standard equipment (Cypher Asylum Research). In particular, a relatively simple model has been 2204

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presented that leads to experimental quantification of relevant dissipative parameters in the nanoscale such as surface energy hysteresis, viscoelasticity, and tip−sample deformation. It is also particularly interesting that simultaneous accurate energy (magnitude) dissipation measurements and in situ tip radius characterization have now been made readily available in the field. The main contribution made here in terms of the accuracy of energy dissipation measurements relates to the method of converting the oscillation amplitude from units of volts to meters with the use of geometrical relationships alone. This method leads to conversions with errors of 3% or less. Simultaneous in situ tip radius characterization has further allowed deducing relationships between dissipative phenomena and the size of the tip. The prediction is that the energy dissipated per atom due to surface energy hysteresis should be independent of operational parameters, cantilever properties, and tip radius while that dissipated via viscoelasticity should not.19,29 If the energy dissipated per atom due to surface energy hysteresis is related to the formation of chemical or physical bonds between the tip and the sample,1,23 and thus chemical affinity and intensive system properties, the prediction would seem physically reasonable. These relationships have been verified here experimentally for a silicon tip interacting with a graphite sample with the present methodology. Future measurements for a range of tip radii should either validate or contradict these results and possibly hint at whether the present models have intrinsic limitations that invalidate them, at least in some aspects, in the nanoscale. It is also expected that experiments with challenging samples and more controlled environmental conditions will lead to the appearance of several other dissipative channels, for example, capillary dissipation, that mandate the adaptation of the present formalism. Nevertheless, the fundamental procedure presented in this work should still be valid in such cases, since it is based on energy conservation principles and experimental data only.

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ASSOCIATED CONTENT

S Supporting Information *

Details and a discussion of DC force curves obtained with the cantilevers employed in this work; the effects of the volts to nanometers conversion are shown for the system discussed in Figure 1; an example of the effects of increasing tip radii for a silicon tip−graphite sample system are shown where R ≈ 25 nm, and the results are compared to those discussed in Figures 2 and 3 for the graphite sample where R ≈ 8 nm; a beta version of the code used in this work to convert the raw experimental data into quantitative parameters can be downloaded; examples of raw data are also given. This material is available free of charge via the Internet at http://pubs.acs.org.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions †

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Maritsa Kissamitaki for designing the visual abstract. 2205

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