research article peeling of long, straight carbon...

Research ArticlePeeling of Long, Straight Carbon Nanotubes from Surfaces

Kane M. Barker,1 Mark A. Poggi,1 Leonardo Lizarraga,1 Peter T. Lillehei,2

Aldo A. Ferri,3 and Lawrence A. Bottomley1

1 School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332-0400, USA2Advanced Materials and Processing Branch, NASA Langley Research Center, Hampton, VA 23681-2199, USA3George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,GA 30332-0405, USA

Correspondence should be addressed to Lawrence A. Bottomley; [email protected]

Received 15 July 2013; Accepted 20 December 2013; Published 23 February 2014

Academic Editor: Bobby G. Sumpter

Copyright © 2014 Kane M. Barker et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The adhesion of long, straight, single-walled carbon nanotubes to surfaces is examined using multidimensional force spectroscopy.We observed characteristic signatures in the deflection and frequency response of the cantilever indicative of nanotube bucklingand slip-stick motion as a result of compression and subsequent adhesion and peeling of the nanotube from the surface.The springconstant and the elastic modulus of the SWNT were estimated from the frequency shifts under tension. Using elastica modelingfor postbuckled columns, we have determined the static coefficient of friction for the SWNT on alkanethiol-modified gold surfacesand showed that it varies with the identity of the monolayer terminal group.

1. Introduction

The atomic force microscope (AFM) is an important toolin the characterization of the structure and mechanicalproperties of carbon nanotubes (CNTs) [1–3]. The tensilestrength and bending properties of both single- and multi-walled carbon nanotubes have been determined with thisinstrument. The high aspect ratio and small effective radiusof CNTs provided impetus for attaching or growing themonto the apex of cantilever tips. Indeed, carbon nanotube tipsare now important probes in atomic force microscopy [4–6].Theyoffer significantly improved image resolution and enableimaging of surfaces with deep crevices and trench structures[7–9].

The elastic buckling property of nanotubes is both anadvantage and a limitation when used as a scanned probe [2,10]. Imaging with an unbuckled CNT-tip provides enhancedfeature resolution compared to conventional silicon or siliconnitride tips. Imaging with a buckled CNT-tip reduces theforce applied while imaging soft samples but with loss inimage resolution. Several recent reports have documented thechallenges associated with these tips [2, 8, 10–16].

Under certain conditions, buckled CNT-tips will eitherslide freely across the surface or slip and then stick at specificlocations on the surface [17–20]. Indeed, surface roughnessand composition play a large role in determining whichphenomenon will take place. When imaging with nanotubetipped probes, it is common practice to monitor up tothree different outputs from the AFM: cantilever deflection,oscillation amplitude [12, 21–26], and/or phase versus theextent of scanner motion [5, 12, 23, 27–30]. Acquisitionof oscillation amplitude and phase data requires that amechanical perturbation be imposed on the system. Trackingeither oscillation amplitude or phase is, in fact, monitoringthe system response at a single frequency and can oftenprovide misleading information and engender incompleteinterpretation of the mechanical response of the nanotube.Thus, a more systematic method for determining whenbuckling and slip or slip-stick motion event takes place isneeded.

We report herein an improved method for determiningwhen buckling and slip-stick events occur. This methodinvolves the simultaneousmonitoring of cantilever deflectionand thermal resonance during cycled movement of the AFM

Hindawi Publishing CorporationJournal of NanotechnologyVolume 2014, Article ID 349453, 11 pageshttp://dx.doi.org/10.1155/2014/349453

2 Journal of Nanotechnology

scanner [17, 18, 30]. Changes in resonance frequency andamplitude for several flexural modes of cantilever vibrationprovide information about the nanotube’s response to com-pressive load (i.e., buckling and slip-stickmotion), tip-surfaceinteractions that may be present, and a more straightforwardmeans for quantifying the CNT’s mechanical properties[31–35]. Since comparison of the frequency response withforce curve data adds an additional dimension of informa-tion to traditional force curve analysis, we will hereafterrefer to this method as multidimensional force spectroscopy,MDFS.

We also present herein an examination of the response oflong CNT-tips to compressive loading on several chemicallymodified substrates: (i) 1-undecanethiol, (ii) 11-hydroxy-undecanethiol, and (iii) 11-amino-undecanethiol. Buchouxet al. [17, 18] have previously examined axial compressionof SWNT on graphite and mica substrates. We presentherein new evidence that adhesion between the surface andthe nanotube-modified AFM tip can significantly alter themechanical behavior of the nanotube [36]. Finally, we identifycharacteristic signatures in the deflection and frequencyresponse of the cantilever that indicate buckling and slip-stickevents of the CNT under compression and its adhesion to thesurface under tension.

2. Experimental

Cantilevers with long, single-walled nanotubes attached tothe probe tip were purchased from NanoDevices. TheseSWNT modified tips were fabricated using a chemical vapordeposition method similar to that previously described [37].Attachment of the SWNT to the tip is assured by patterneddeposition of the catalyst onto the tip and direct growthof the nanotube onto the catalyst. Prior to and followingthe AFM studies described herein, each tip was examinedwith a LEO 1550 scanning electron microscope (Carl ZeissSMT Inc.,Thornwood, NY, USA). Accelerating voltages werekept below 10 kV to reduce nanotube vibration andminimizedamage from exposure to the electron beam.

Two types of substrates were used herein: silicon ⟨100⟩chips and chemically modified template-stripped gold. Thesilicon chips were cleaned with hot aqueous Piranha solution(7 : 3 H

2SO4: H2O2), rinsed in filtered absolute ethanol,

and then stored in a desiccator until use. Freshly preparedtemplate-stripped gold substrates were immersed into 2mMsolutions of one of the following alkanethiols dissolvedin filtered absolute ethanol: 1-undecanethiol, 11-hydroxy-undecanethiol, and 11-amino-undecanethiol (the first twowere used as received from Sigma-Aldrich; the last was usedas received from Dojindo Chemicals). After 2 hours, themodified gold substrates were removed from solution, rinsedin filtered absolute ethanol, and then stored in a desiccatoruntil use.

A Veeco Instruments NanoScope IIIa scanning probemicroscope with extender electronics and signal access mod-ule was used for all force spectroscopic measurements. Thepiezo scanner was calibrated in 𝑥, 𝑦, and 𝑧 using NISTcertified calibration gratings (MikroMasch, San Jose, CA,

USA). To initiate each experiment, the cantilever-SWNTprobe was mounted in the holder and placed in the AFM.Image acquisition was commenced without allowing contactbetween the probe and the substrate surface.Themicroscopewas then placed in force spectroscopy mode and the scannerwas cycled in the 𝑧-direction for 2 hr under constant flowof nitrogen (2 Lmin−1) to allow the system to reach thermalequilibrium and a relative humidity <15%.The stepper motorwas then used to move the substrate into contact with theend of the nanotube. Finally, force curves were acquired ata fixed scanner velocity of 100 nm sec−1. For all experimentspresented herein, the scanner moved 2 𝜇m in each direction(approach and retraction); the initial scanner position wasmodified to achieve the desired compression of the SWNT.The natural resonance frequency was determined from thethermal spectrum and the spring constant of the SWNTtipped cantilever was calculated using the equipartitiontheorem [38].

The AFM software was set to record both deflection andoscillation amplitude as a function of scanner displacementin the 𝑧 dimension. Raw signals (prior to analog filtering)for the vertical deflection, horizontal deflection, and the sumsignals were taken from the microscope base into a PCI-6120data acquisition card through a BNC-2110 interface (NationalInstruments, Austin, TX, USA) with a data acquisition rate of800 kHz. To correlate these signals with scanner movementin the 𝑧 dimension, the 𝑧 scanner voltage was taken from thesignal access module and sent to the data acquisition card.The thermal resonance of the cantilever was measured usinga Dynamic Signal Analyzer program written in LabVIEWwhile simultaneously acquiring the force curve. Briefly, thetime dependent vertical deflection signal was convertedinto the frequency domain using the discrete Fourier trans-form. The resultant power spectral density (PSD) data wasensemble averaged and displayed in waterfall format. Theacquisition time per FFT was 10ms, the number of averagesin each PSDwas 16, and the number of PSDs perwaterfall plotwas 500. With respect to the scanner velocity, each averagedpower spectral density plot represents 16 nm of scannermovement. Horizontal deflection and scanner movementwere also recorded with respect to time. The reader isreferred to the dissertation of Barker for complete detailsof the experimental system and measurement apparatus[39].

3. Results and Discussion

The adhesive and mechanical response of SWNTs greaterthan 2𝜇m in length during both compression and tensionwere examined. Each SWNTwas repeatedly brought into andout of contact with the substrate by extending and retractingthe scanner in the 𝑧-direction while monitoring cantileverdeflection and thermally driven resonance. An example of theSWNTs included in this investigation is provided in Figure 1.The images displayed in this figure depict the same SWNTfrom two different angles. The results from compression andtension data can be examined in three different ways. Forclarity, each of these results is discussed individually.

Journal of Nanotechnology 3

(a) (b)

Figure 1: SEM images looking at the front (a) and the side (b) of oneof the carbon nanotube-modified AFM tips used in this study. Thescale bar in each image represents 1 𝜇m.

3.1. Nanotube Buckling. Approach of the substrate to theprobe was performed in intermittent contact mode. Thepoint of contact is established with a momentary decreasein oscillation amplitude as illustrated in Figure 2(a). Uponretraction the amplitude is even further reduced but onlyfor a brief period and then it quickly returns to the presetdrive amplitude.Thiswould suggest that the restorative forcesat contact are quickly removed. Figure 2(b) shows a typicalforce curve acquired with a SWNT tip with reversal of thedirection of the scanner movement 60 nm after contact ofthe probe with the surface and with the drive amplitude ofthe oscillation set to zero. Note that the sinusoidal patternin the curve is a resultant of the optical interference fromthe laser reflecting off both the cantilever and the substratesurface. Also note that there is no measurable deflection ofthe cantilever during approach and a small but notable down-ward deflection occurs on scanner retraction. We attributethe zero or little deflection to the high spring constant of thecantilevers (2-3N/m) relative to the spring constant of theSWNT.

Without changing or stopping scanner movement, thedrive voltage for the oscillation amplitude was turned off.Thermal resonance data was then acquired by convertingthe time dependent vertical deflection signal acquired dur-ing the force curve into the frequency domain using thediscrete Fourier transform. The resultant PSD data was col-lected, ensemble averaged, and displayed in waterfall format.Figure 2(c) is waterfall plot for the cantilever resonanceassociated with the force curve data presented in Figure 2(b).Time and frequency are the in-plane axes, with the amplitudeof thermal resonance on the 𝑧-axis. In this figure, the scannerapproaches the SWNT from 0 to 20 s and retracts from 20to 40 s. The amplitude is color encoded to help visualize

changes. To reduce complexity, the data is presented in analternative format. The plot presented in Figure 2(d) depictsthe frequency of maximum amplitude corresponding to theprimary mode of cantilever thermal resonance as a functionof time. Superimposed onto this data is the movement of thescanner over the identical time period to enable correlation ofchanges in frequency with scanner extension. For the exper-iment shown in Figure 2, the apex of the curve of scannermovement (20 sec) was themomentwhen the direction of thescanner was reversed and the smallest gap existed betweenthe cantilever tip and substrate. Note that there is no shiftin the thermal resonance frequency during approach. A shiftof 4 kHz is measured upon retraction indicating that duringretraction the nanotube is put into tension.

Contact of the nanotubewith the substratewasmonitoredboth by tracking the amplitude of the oscillation when thecantilever is externally driven (Figure 2(a)) or the primarymode resonance frequency when the cantilever was drivenonly by thermal motion (Figure 2(d)). In both cases, thesudden return to baseline following contact suggests thatthe nanotube mechanically failed; that is, it buckled. Ourinterpretation that the nanotube undergoes buckling is inagreement with the similar observations made by Jiang et al.[26]. Using a scanning electron microscope, they clearly sawthat compression of multiwalled carbon nanotubes resultedin buckling at short compression distances.

The critical buckling load for our nanotubes was cal-culated. Applying a clamped-pinned model from the Eulerequation to our system, such that the nanotube is clampedat the AFM tip and pinned by adhesive interactions at thesurface, yields the critical buckling load, 𝐹crit, defined by

𝐹crit =𝐸𝐼𝛽2

(𝐿SWNT)2, (1)

where 𝐸 is the elastic modulus of the SWNT, 𝐼 is thearea moment of inertia, 𝛽 is a constant determined by theboundary conditions, and LSWNT is the length of the SWNT.For a clamped-pinned column, 𝛽 is the first nonzero rootof the equation tan(𝛽) = 𝛽, or approximately 4.4934 [40].Using themedian values for a SWNT [41, 42], where the outerdiameter of the tube is 1.3 nm, an elastic modulus of 1.0 TPa,and a representative length from our samples of 3.0 𝜇m, thecritical buckling load is calculated to be 0.6 pN. Using thespring constants of our cantilevers, the critical buckling loadis reached in the first nanometer of compression [2, 18, 43].This strongly suggests that our nanotubes buckle immediatelyafter contact with the surface.

3.2. Tensile Loading. The shift in frequency when the nan-otube is put into tension (as shown in Figure 2(d)) can bemodeled as two springs in parallel. The nanotube along itslong axis is one spring; the AFM cantilever is the otherspring, with the effective mass of the cantilever and AFM tipseparating the two. The change in resonance frequency, Δ𝑓,for such a system is directly related to the ratio of the two


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spring constants [17, 18, 44]. For springs in parallel, thechange in resonance frequency is given by

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𝑘SWNT =𝐸𝐴

𝐿SWNT, (3)

where𝐴 is the cross-sectional area and LSWNT is the length ofthe nanotube.

Figure 3 shows the frequency response of the system asthe SWNT probe is slightly compressed (only 50 nm) ontothree different surfaces over chemically modified template-stripped gold: methyl-terminated alkanethiol (Figure 3(a)),

hydroxy-terminated alkanethiol (Figure 3(b)), and amino-terminated alkanethiol (Figure 3(c)). Under the low compres-sion regime displayed in this figure, no frequency shift ismeasured during scanner extension. During scanner retrac-tion, the nanotube-cantilever system is put into tension andshifts to a singular frequency. Note that this frequency shiftis essentially the same for all three surfaces even thoughthe strengths of adhesion of the SWNT to these surfacesare known to differ [17–19]. If the strength of adhesion islikened to an “adhesive spring,” this spring would be inseries with the SWNT probe. Since the spring constant ofthe nanotube is believed to be much smaller than that of the“adhesive spring,” the SWNTdominates the equivalent springconstant of the probe/substrate. Thus, the shifted resonantfrequency should be largely independent of the strength ofadhesion.

To compute the elastic modulus of the SWNT using (3),an accuratemeasure of tube dimensions is required.Measure-ment of the length of these long SWNTs was made with ascanning electron microscope. Measurement of tube diame-ter is best performedwith a transmission electronmicroscope


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Figure 3: Plots of primary resonance mode frequency versus timefor three different self-assembled monolayers on template-strippedgold: (a) methyl-terminated alkanethiol, (b) hydroxy-terminatedalkanethiol, and (c) amino-terminated alkanethiol. For this dataset,the length of the nanotube was 5.63𝜇m long. The cantileverresonance frequency was 82.93 kHz and had a spring constant of1.8N/m.

(TEM). Our attempts to acquire the lattermet with two unde-sirable outcomes. TEM images prior to mechanical loadingexperiments resulted in pyrolysis of the tube presumably dueto charging effects and poor grounding of the probe while

being imaged at high voltages. Pyrolysed nanotubes wereno longer suitable for mechanical testing. So we elected todelay TEM imaging until after mechanical loading studies.During this phase of the investigation, the entire inventoryof our nanotube-modified AFM tips were separated fromthe AFM tip. The repeated mechanical stresses incurredduring the experiments, including compression to 30% of thelength of the nanotube, caused the nanotubes to break at thenanotube/catalyst junction. This fact is illustrated in Figures4(a) and 4(b). In some instances, detached nanotubes werelocated on the surface while imaging with the AFM tip towhich theywere formerly attached, as depicted in Figure 4(c).In this case, tube diameters were determined from heightsmeasured through imaging using the AFM tip to which theywere formerly attached.

The cross-sectional area needed in (3) was computedin the following way. An average diameter of 1.4 ± 0.1 nmwas found from AFM images of broken nanotube tips. Themanufacturer of the SWNT probes claimed that they weresingle-walled; the measured diameter is consistent with thisclaim.

Nanotube lengths were determined from SEM imagesand ranged from 2.0 to 5.7𝜇m. Using these values and themeasured frequency shift, Δ𝑓/𝑓

0, the elastic modulus of the

single-walled carbon nanotube is found to be 1.6±1TPa.Thisresult is in good agreement with previous literature values forthe elastic modulus of SWNT [41, 45–47]. The uncertaintyin our reported result is due to the precision of frequencymeasurements and the relative magnitudes of 𝑘SWNT and𝑘CANT. In our system, and 𝑘CANT is an order of magnitudegreater than 𝑘SWNT; the uncertainty of the calculatedmodulusdecreases exponentially as the ratio of the spring constantsapproaches unity [31].

3.3. Friction Analysis. The SWNT tips were also subjected tolarger compression displacements. Thermal resonance datafor a nanotube brought into and out of contact with amino-terminated alkanethiol-modified gold surface is presented inFigure 5 as the extent of compression is increased. In eachgraph, the dashed line indicates the point of contact. It isseen that the frequency response changes in magnitude andduration with increasing compression. Figure 5(a) is similarto the previous pattern of contact with immediate bucklingduring scanner extension, followed by tension during retrac-tion. Figure 5(b) shows the frequency shift as the SWNTwas compressed for a distance of 300 nm instead of 50 nm(Figure 5(a)). After initial contact and buckling, the fre-quency rises and falls with scanner extension. Sudden shiftsto higher frequency indicate that the nanotube is pinned onthe surface and provides resistance to cantilever oscillation.Sudden shifts to lower frequency indicate that the nanotubebuckles and/or slips on the surface, removing this resistance.In Figure 5(b), the nanotube undergoes two distinct slip-stick-buckle events during approach. During retraction, sev-eral stick-tension-slip events are observed. The retractionportion of the force curve displays a sawtooth pattern thatis indicative of slip-stick motion (data not shown). As thecompression on the nanotube is further increased (710 nm),


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Figure 4: SEM images showing (a) before and (b) after mechanical failure caused by repeated compression. The AFM image (and sectionalanalysis) of one of the nanotubes that was found after fracture is shown in (c). The image was acquired in intermittent contact mode usingthe same tip from which it was detached.

additional slip-stick-buckle and stick-tension-slip events areobserved (Figure 5(c)). Note that themagnitude of frequencyshifts in Figure 5(c) is greater than that found in Figures 5(a)and 5(b).

The slip-stick phenomenon depends upon the chemistryat the SWNT-substrate interface. Figure 6 shows the thermalresonance frequency response for a single SWNT probebrought in and out of contact with three different chemically-modified gold substrates applying approximately the samecompression distance. Figure 6(a) depicts the interaction ofthe SWNT with a methyl-terminated alkanethiol. Minimalshifts in frequency are observed during retraction indicatingthat the nanotube slides freely on the surface. Figure 6(b)depicts the interaction of the SWNT with a hydroxy-terminated alkanethiol. A measurable frequency shift isobserved consistentwith an increased affinity of the nanotubefor the hydroxy-terminated surface. Figure 6(c) depicts theinteraction of the SWNT with an amino-terminated alka-nethiol.The frequency response for this surface is larger thanfor the hydroxyl-terminated surface. Taken collectively, thedata presented in Figures 5 and 6 suggests that the number

and extent of slip events is indicative of the strength ofadhesion between the nanotube and the surface; that is,–CH3 < –OH < –NH2. This trend is consistent with previ-ously published adhesion measurements [20, 30, 32–34, 40].The results in Figures 3 and 6 may, at first glance, appear tobe in conflict. At minimal compression, the SWNT bucklesbut does not slip. In the absence of slip, the surface chemistryhas no effect on the frequency response of the system. Inthe presence of slip, nanotube-substrate chemistry mediatesthe extent and duration of the observed frequency shifts.Differences in the chemistry of the interface can be distin-guished after compression-induced buckling and slip. Thus,there is no conflict between the data presented in the twofigures.

Interestingly, after the buckling event (the system is nowdescribed as postbuckled), the force required to overcomethe adhesion at the pinned end of the nanotube and enterthe slip-stick domain is, by definition, the frictional force.A prediction can be made about the shape of the bucklednanotube as it undergoes further compression by applying anelastica model of a postbuckled column [36, 39].


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Figure 5: Plots of primary resonance mode frequency versus timefor a SWNT compressed for three different distances on an amino-terminated alkanethiol monolayer: (a) 50 nm, (b) 300 nm, and (c)710 nm.The dashed line indicates the contact point of the nanotubewith the substrate. For this dataset, the length of the nanotube was2.96 𝜇m long. The cantilever resonance frequency was 72.99 kHzand had a spring constant of 2.5N/m.

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Figure 6: Plots of primary resonance mode frequency versustime at large compression for (a) methyl-terminated alkanethiol,(b) hydroxyl- terminated alkanethiol, and (c) amino-terminatedalkanethiol, on template-stripped gold. The dashed line indicatesthe contact point estimated from the oscillation amplitude whenthe system was externally driven. For this dataset, the length of thenanotube was 2.96 𝜇m long.The cantilever resonance frequency was72.99 kHz and had a spring constant of 2.5N/m.


The boundary conditions for this problem correspondto a clamped-pinned configuration: zero displacement androtation at the clamped end and zeromoment at the substrate.It is assumed that the contact point is directly beneaththe AFM tip; however, the probe tip can be moved to theleft or right by varying 𝑥. The differential equations andassociated boundary conditions were solved using theMatlabsolver for boundary-value problems, bvp4c, as previouslydescribed [36, 39]. Through the solution of the boundary-value problem, the horizontal and vertical forces at the pinnedendpoint can be calculated at different compression steps.Thedistance from the AFM tip to the substrate surface is variedthrough choice of the boundary condition.

Figure 7 shows the evolution of the nanotube shape as thescanner height is raised; that is, the tip to substrate distanceis decreased. Note in particular how the angle 𝜃 that thenanotubemakes with the normal to the substrate is predictedto increase as the nanotube becomesmore andmore buckled.This trend is further revealed in Figure 8, which shows theangle 𝜃 as a function of the normalized scanner height.Note that when the scanner displacement is near 30% of thenanotube length, it is predicted that the postbuckled shapewill meet the substrate at near 90∘ angle. The variation in theangle predicted by this model can be used to determine thefriction coefficient as described below.

At the substrate surface, the horizontal force, 𝑃𝐻, is

equivalent in magnitude to the force of static friction, 𝐹𝑓, and

the vertical load, 𝑃𝑉, is equivalent to the magnitude of the

normal force, 𝐹𝑁. Since the moment at the substrate-end of

the elastica is zero, it can be shown that

𝐹𝑓

𝐹𝑁

= tan 𝜃. (4)

It is important to note that (4) holds regardless of whetherthe normal load is due to elastic deformation of the SWNTor due to adhesion, or both. A corollary to (4) is that thecoefficient of friction, 𝜇, necessary to prevent slip can befound using the relation

𝐹𝑓≤ 𝜇𝐹𝑁. (5)

In other words, 𝜇 ≥ tan 𝜃.For each compression step, the minimum coefficient of

friction required to keep the tube pinned can be calculatedusing (5). As compression increases, the angle 𝜃 increases and,consequently, the minimum coefficient of static frictionmustincrease if the nanotube is to remain pinned to the surface.

Figure 9 shows the minimum coefficient of friction as afunction of the normalized scanner height.Themiddle (blue)curve corresponds to the nominal case where the substrateis perpendicular to the 𝑧-axis. Inclining the substrate willeither inhibit or favor slip; the neighboring lines in Figure 9correspond to ±1 degree rotation of the substrate. Figure 9can be used to estimate the friction coefficient between thenanotube and substrates having various chemical coatings.

Specifically, the scanner extension and cantilever dis-placement can be monitored in an experiment until thefirst slip event is detected. The relative change in the dis-tance between the AFM tip and the surface (normalized by

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Figure 8: Predicted angle between the elastica and the normal tothe vertical at the point of contact between the nanotube and thesubstrate.

the undeformed probe length) at the first slip event is usedas the abscissa value in Figure 9 to yield the coefficient offriction.

The procedure described above was used to estimate thecoefficient of friction for three surfaces of self-assembledmonolayers on template-stripped gold. For the methyl-terminated monolayer (11-undecanethiol), the coefficient offriction was found to be 0.11; an uncertainty of 1 degree inthe nominal surface angle yields an uncertainty of ±0.02.Similarly, the friction coefficients were estimated to be 0.34 ±0.02, and 0.47 ± 0.02 for 11-hydroxyl-, and 11-amino- unde-canethiol, respectively. These results are in the ranges seenby others who worked on friction of Si

𝑥N𝑦AFM tips and

self-assembled monolayers and different carbon materials[20, 27, 28, 35, 48].

4. Conclusions

In this report, we demonstrate the advantages MDFS offersto studying mechanical responses and adhesive interactionsat the nanoscale. By monitoring the shift in the thermalresonance frequency, MDFS enables the calculation of thespring constant and elastic modulus of carbon nanotubesunder ambient conditions. This approach delineated herein


1

0.8

0.6

0.4

0.2

00 0.02 0.04

𝜇

0.06 0.08 0.1

Nominal+1∘

−1∘

Δz

Figure 9: Minimum coefficient of friction versus normalized probeend position. Blue line is for a nominally flat substrate; greenand red lines correspond to substrates inclined +1 and −1 degrees,respectively.

is not limited to SWNTs; it is directly applicable to a widevariety of nanoobjects including nanorods, nanofibers, andbiomolecules under tension. Additionally, an elastica modelof the postbuckled state of the nanotube has been developed.By monitoring the location of the frequency shift in thermalresonance, the static coefficient of friction has been calculatedfrom experiments performed on three different chemically-modified surfaces. The results presented herein are, to ourknowledge, the first static friction results for carbon nan-otubes on self-assembled monolayers. The observed trendin the dependence of the coefficient of friction with theidentity of the terminal group on the monolayer follows thesame trend previously observed for the strength of adhesionbetween a SWNT and an alkanethiol-modified surface [40,49].

In a broader context, this work cautions that high aspectratio nanotube probes buckle under minimal compressiveloads. Nanotube buckling leads to imaging artifacts [5, 25, 30,50, 51]. Imaging in intermittent contact mode does not, nec-essarily, prevent this artifact. As demonstrated in Figure 2(a),a buckled nanotube does not necessarily dampen cantileveroscillation. Proper imaging protocol requires acquisition ofan oscillation amplitude plot and setting the set point valueto the first point of cantilever damping. The spring constantof the cantilever must also be carefully chosen. Equation (1)predicts the force at which a SWNT buckles. We suggestthe following equation for choosing a cantilever with springconstant 𝑘CANT so that it will not cause a nanotube with anaspect ratio,𝐴

𝑅, and outer diameter, 𝑑, to buckle after contact

and 10 nm of axial compression:

𝑘CANT <𝐸𝑑2

𝐴2𝑅

(9.3 × 107m−1) . (6)

This equation treats the SWNT as a column whoseinner diameter is half of its outer diameter because thatapproximates the relationship for SWNTs. During intermit-tent contact mode imaging, it is possible that the nanotubecan experience compressive loads up to 10 nN. The constant

at the end of the equation is the product of all other constantsand allows the remaining variables to be calculated using SIunits. Without careful consideration of the spring constant ofthe AFM cantilever relative to the critical buckling load, highaspect ratio carbon nanotube AFM probes will buckle duringimaging, even in intermittent contact mode.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors acknowledge fruitful discussions with ProfessorJonathan Colton and Dr. Andrew MacFarland concerningthe mechanical models for their system. They thank NickSchacher from NanoDevices for providing them with thenanotube-tipped cantilevers and Jean Jarvaise formerly fromVeeco Instruments Metrology Division for guidance inmodifying the electronics of the NanoScope IIIA. M. A.Poggi gratefully acknowledges a fellowship from the NASAGraduate Student Researchers Program (NGT-1-02002).Thisresearch was made possible by Grants from the NationalInstitutes of Health (EB000767) and the National ScienceFoundation (CMMI1130739).

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