Single-Molecule Force Spectroscopy Measurements of Interactions

Aug 16, 2007 - Introduction. Since the discovery of the first fullerene (C60) in 1985,1 the ... distribution of rupture forces p(F) from dp(F)/dF ) 0...
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12898

J. Phys. Chem. C 2007, 111, 12898-12905

Single-Molecule Force Spectroscopy Measurements of Interactions between C60 Fullerene Molecules Chao Gu, Chad Ray, Senli Guo, and Boris B. Akhremitchev* Department of Chemistry, Duke UniVersity, Durham, North Carolina 27708 ReceiVed: March 5, 2007; In Final Form: May 31, 2007

The hydrophobic effect is important for many biological and technological processes. Despite progress in theory, experimental data clarifying the structure of water and the interactions between hydrophobic solutes at the nanometer scale are scarce because of the very low solubility of hydrophobic species. This article describes single-molecule force spectroscopy measurements of interactions between single fullerene C60 molecules in water. The C60 molecules were tethered by flexible poly(ethylene glycol) linkers to AFM probes and substrates, removing the uncertainty of the aggregation state of solution-based approaches and spurious surface effects. Our analysis of the dependence of the measured most-probable rupture force on the mostprobable loading rate considered the deviations from the conventional Bell-Evans model caused by the effects of the anharmonic tether, as well as by the finite depth and shape of the potential well. The kinetic parameters of the activation barrier width, the dissociation rate of the C60-C60 dimer, and the activation energy are reported. The measured values differ significantly from predictions based on molecular dynamics simulations, indicating that further advances in computer simulations are necessary for the development of a model showing quantitative agreement with experimental results.

Introduction Since the discovery of the first fullerene (C60) in 1985,1 the unique structure and remarkable physical and chemical properties of fullerenes have made them promising new materials.2-5 Recently, interest has increased in the potential applications of C60 and its derivatives as biomedical reagents based on their hydrophobicity.6-9 However, the potential biological impact and toxicity of C60, because of its formation of aggregates in water, have not been fully characterized.10 A fundamental understanding of C60’s self-association behavior in aqueous environments is therefore of great importance. Theoretical prediction of the strength of association between fullerenes requires a quantitative description of the contributions to the potential energy landscape that governs interactions between molecules in water. Predicted changes in the nature of the hydrophobic interactions depending on the size of the hydrophobic solutes and the solute-water interactions11 make theoretical analysis without direct experimental information problematic. Small hydrophobic molecules can be incorporated within the local water structure without disrupting the hydrogen-bonding network,12 but larger hydrophobic species, such as C60 and its aggregates, will disrupt the hydrogen bonding of water molecules in the hydration shell.13,14 The predicted characteristic dimension of solute where the transition between the two regimes takes place is approximately 1 nm.11,13 The size and rigidity of fullerenes make them attractive models for testing theories of hydrophobicity. Recently, the potential of mean force has been calculated for two fullerene molecules in water.14-16 The potentials of mean force that use Lennard-Jones (LJ) potentials for water-fullerene interactions predict an activation energy of ∼20-25 kJ/mol,15,16 whereas calculations that use the repulsive Weeks-ChandlerAnderson (WCA) potential predict an activation energy of * To whom correspondence should be addressed. E-mail: boris.a@ duke.edu.

approximately 55 kJ/mol.15 The distance between the equilibrium position and the transition state along the direction of dissociation (henceforth called the barrier width for brevity) is predicted to depend strongly on the type of the fullerene-water interaction potential, as well; it changes from approximately 0.25 nm for the LJ potential to approximately 0.4 nm for the WCA potential.15,16 To clarify the applicability of these models, we experimentally investigated the hydrophobic interactions between single C60 molecules. In addition, this experimental study compared interactions between C60 molecules with previous single-molecule studies of interactions between linear hydrocarbons.17,18 In this work, atomic-force-microscopy- (AFM-) based singlemolecule force spectroscopy was employed17-22 to characterize the interactions between individual C60 molecules. The doubletether approach was used to clearly identify C60-C60 rupture events.23 Functionalized C60 molecules were tethered to a silicon nitride tip and a glass substrate via poly(ethylene glycol) (PEG) of 3400 Da average molecular weight. C60-C60 rupture forces were measured at a series of different loading rates in an aqueous buffer at pH 7. Analysis of the rupture force vs loading rate dependence was used to characterize the barrier width and activation energy,17,18 which can be compared to recent theoretical predictions. Systematic error in force spectroscopy results could affect the comparison of the theoretical models with the experimental data. The polymeric tethers used in AFM force spectroscopy can cause systematic error as a result of anharmonic tether elasticity.18,24 Moreover, the Bell-Evans model that is most often employed in the analysis of experimental data assumes that the activation barrier width is independent of applied force. It was shown recently25-29 that, for potentials with finite depth, this assumption might result in significant systematic errors in the barrier width and dissociation rate obtained from force spectroscopy measurements. Therefore, the effects of the tether

10.1021/jp0717645 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/16/2007

Interactions between Single C60 Fullerenes by AFM

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and the finite barrier depth were considered in obtaining the kinetic parameters of dissociation between C60 molecules from experimental data. Theoretical Model The statistical data analysis of rupture forces involves the most-probable force F* determined from the probability density distribution of rupture forces p(F) from dp(F)/dF ) 0. The probability distribution function of rupture forces can be calculated as30

p(F) )

[∫

k(F) exp VF(F)

F

0

k(F′) dF′ VF(F′)

]

(1)

Here, k(F) is the force-dependent dissociation rate, and VF(F) is the loading rate. The loading rate VF ) dF/dt depends on the tether dynamics, whereas the dependence of the dissociation rate k on the force can be described by different models of intermolecular potentials.28,29 When the cantilever base is moving with a constant velocity V, the overall travel of the base is the sum of the cantilever deflection and tether extension: z ) Vt ) F/κc + l, where l ) l(F) is the force-dependent length of the tether and κc is the spring constant of the force sensor. Therefore, a transcendental equation for the most-probable force can be derived from eq 1 as

κcVF*k(F*) l′′(F*) ) [κcl′(F*) + 1][k2(F*) - VF*k′(F*)] (2) Here, VF* is the most-probable apparent loading rate, l(F) is the force-dependent end-to-end length of the tether, and κc is the spring constant of the force sensor. The prime here denotes differentiation with respect to force. Equation 2 does not use any particular tether dynamics model nor does it make any assumptions about a particular dissociation rate dependence on force, and therefore, it can be used to analyze the results of force spectroscopy experiments for different types of tethers and intermolecular potentials. The data analysis below uses the standard Bell-Evans (BE) model and the model that combines the freely jointed chain (FJC) polymeric tether model31 with the approximate Morse potential model29 and models for cusp and cubic-linear potentials.28 The end-to-end length of the tether l(F) under force F was calculated according to

l(F) ) Lc{coth[Fa(kBT)-1] - kBT(Fa)-1}

(3)

where Lc is the tether contour length, a is the Kuhn length, kB is the Boltzmann constant, and T is the absolute temperature. For the BE model, the dependence of the dissociation rate on force F is32

k(F) ) k0 exp[Fxq(kBT)-1]

(4)

where k0 is the dissociation rate without the external load and xq is the barrier width. The Morse potential model that considers the finite depth of the potential has the following dependence of the dissociation rate on force29

k(F) ) k0 exp{Fxq(kBT)-1[1 - Fxq/(4∆Gq)]}

(5)

Here, ∆Gq is the activation energy, and the barrier width is xq ) 4/γ, where γ is the characteristic inverse width of the Morse potential used in the article by Hanke and Kreuzer.29 Comparison of eqs 4 and 5 shows that the finite depth of the potential causes a decrease of the force-dependent dissociation rate in

comparison to the BE model. The ratio of the rates, exp[(Fxq)2/ (4kBT∆Gq)], decreases with increasing depth of the potential and decreasing width of the potential. For sharp and deep barriers such as those found in biotin-streptavidin binding,33 the overestimation is small (below 50%). However, for rupturing molecular bonds with F ) 100 pN, xq ) 0.3 nm, and ∆Gq ) 50 kJ‚mol-1, the BE model overestimates the dissociation rate by a factor of 2. Therefore, it is expected that analysis of experimental data by the BE model will overestimate the dissociation rate. In addition, the effects of anharmonic tethers also cause the dissociation rate to be overestimated;18 thus, the cumulative systematic error might be significant and was considered in the data analysis. The systematic errors related to the depth and shape of the intermolecular potential can also be analyzed using models that describe dissociation from the cusp and cubic-linear potentials.25-28 The dissociation rate for these two potentials is given by one cumulative equation28

(

k ) k0 1 - υ

Fxq ∆Gq

)

(1/υ)-1

exp

{ [ (

) ]}

∆Gq Fxq 1- 1-υ q kBT ∆G

1/υ

(6)

where υ is a parameter that equals 1/2 for the cusp potential and 2/3 for the cubic-linear potential.28 These models predict that, in the case of force ramping with a constant loading rate, the dependence of the mean rupture force on the loading rate is of the form 〈F〉 ∼ (ln VF)υ instead of 〈F〉 ∼ (ln VF), the prediction of the BE model. It can be noted that, in the range of loading rates commonly used in AFM experiments, the difference between the cusp and cubic-linear potential models usually does not exceed the random error in the force measurements (as illustrated below). Because the exact shape of the intermolecular potential is unknown, the data analysis below includes Morse, cusp, and cubic-linear potentials in addition to the BE model. The activation energy ∆Gq (in the case of the Morse potential, it equals the depth of the potential) is usually unknown during analysis of force spectroscopy data. Therefore, when substituting eqs 5 and 6 into eq 2 for fitting of the rupture force vs loading rate dependence, the activation energy is substituted by ∆Gq ) kBT ln(A/k0), where A is the Arrhenius prefactor, which can be estimated from the potential vs distance dependence34 or taken from a separate experiment.17 The resulting transcendental equations for the most-probable force are very cumbersome though straightforward to derive; these equations can be used to fit the experimental data using the barrier width xq and dissociation rate k0 as free parameters. Figure 1 compares force vs loading rate dependencies for the BE model; the BE model with the FJC tether; and models that use the Morse, cusp, and cubic-linear potentials with the FJC tether. Calculation parameters are included in the graph. As can be seen from the figure, the tether anharmonicity and the shape of the potential barrier both affect the rupture force. The shape of the potential energy barrier affects the rupture forces noticeably less than the finite depth of the potential. The force vs log(loading rate) dependence is nonlinear; however, this nonlinearity is small, and given the typical error in the mostprobable force (∼5 pN), the experimental dependence in the range of loading rates typically found in AFM experiments does not provide strong evidence for the deviation from the BE model nor for the selection of a particular shape of the energy landscape. The results below compare applications of different models to analyze experimental data.

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Figure 1. Rupture force vs loading rate calculated for indicated force spectroscopy models by numerically solving eq 2. The graphs are shown in the range of loading rates typical for AFM experiments.

CHART 1. Functionalized C60 1 and Its Derivative C60PEG-NH2 Trifluoroacetic Acid Salt 2

Experimental Section Instrumentation and Materials. Force spectroscopy was carried out using an Asylum Research Molecular Force Probe 3D AFM (Santa Barbara, CA). Glass microscope cover slips (Fisher Scientific) and mica (Structure Probe, Inc., grade V-4) were used as substrates opposite commercial AFM cantilevers (Veeco, model NP-20). Samples prepared with the mica substrates were used for imaging. Measurements were performed in pH 7 phosphate buffer (containing disodium hydrogenphosphate and potassium dihydrogenphosphate, 0.05 M; VWR International). NMR spectra were collected on a Varian Unity Inova-400 spectrophotometer operating at 400 MHz for 1H NMR spectroscopy and 100 MHz for 13C NMR spectroscopy. Column chromatography employed 230-450 mesh silica gel (Sorbent Technologies). TLC was performed on silica gel, and visualization of spots was effected with UV light, permanganate in water stain, and/or ninhydrin in n-butyl alcohol stain. Commercial reagents were purchased from the major chemical suppliers and used as received unless indicated otherwise. C60 was purchased from SES Research (Houston, TX). N-(N-β-Bocaminoethyl)-Gly-OEt was purchased from Peninsula Laboratories Inc. (San Carlos, CA). Heterobifunctional PEG of 3400 Da average molecular weight with N-hydroxysuccinimidyl (NHS) and Boc-NH ends was purchased from Nektar Therapeutics (Huntsville, AL). Monofunctional PEG-NH2 with an average molecular weight of 2000 Da was purchased from Polymer Sources Inc. (Montreal, Canada). Reaction yields were not optimized. Synthesis. Functionalized C60 1 (see Chart 1) was synthesized as previously described.35-37 Heterobifunctional PEG of 3400 Da average molecular weight with NHS and Boc-NH ends was used as the tether. Reaction between 1 and NHS-PEG3400NH-Boc gave the C60-PEG-NH-Boc molecule. Then, the Boc

Gu et al. group was cleaved by treating C60-PEG-NH-Boc with dichloromethane/trifluoroacetic acid (1/1). The resulting C60-PEG-NH2 trifluoroacetic acid salt 2 (Chart 1) was used to attach fullerenes to the aminated substrate and AFM probe surfaces. The detailed procedure for the synthesis of 1 and 2 is included in the Supporting Information. Sample Preparation. Silicon nitride AFM cantilevers and glass substrates were cleaned in 2% Hellmanex II (Hellma GmbH & Co KG, Mu¨llheim, Germany) aqueous solution, rinsed with DI water, dried under vacuum for 12 h, and then transferred into a glove box under Ar. The mica substrates were freshly cleaved prior to amination. The substrates and the probes were aminated in a saturated ethanolamine hydrochloride solution in anhydrous DMSO for 48 h at room temperature.17,38 The aminated tips and substrates were first activated with a solution of 50 mg of 1,4-phenylenediisothiocyanate (PDITC) in 1 mL of anhydrous DMF/TEA (9/1) for 4 h and rinsed thoroughly with anhydrous DMF.39 Next, they were reacted with a solution of 15 mg of the synthesized C60-PEG3400-NH3+CF3COO- (a mixture with remaining N-hydroxysuccinimidyl-PEG-NH3+CF3COO-) in 1 mL of anhydrous DMSO/TEA (9/1) for 24 h. Then, a solution of 20 mg of PEG2000-NH2 in 0.4 mL of anhydrous DMSO/TEA (9/1) was added, and the reaction was continued for another 24 h. Finally, the tips and substrates were passivated with a solution of anhydrous DMSO/ethanol amine (9/1) for 12 h. Following the final reaction, the tips and substrates were rinsed thoroughly with DMSO and DMF, moved out of the glove box, and cleaned in hexanes/i-propanol (3/2) at room temperature for 1 h. The tips then were rinsed with preheated hexanes/i-propanol (3/2). The substrates were boiled in hexanes/ i-propanol (3/2) for 15 min and rinsed with preheated hexanes/ i-propanol (3/2) several times.40 Both the tips and substrates were immersed in deionized water of 50-60 °C for 20 min and then cleaned successively in toluene, DMF, and ethanol for 5 min each with a shaker (for tips)/ultrasonic bath (for substrates). The tips and substrates were blown dry with UHP nitrogen (National Welders Supply) and were used immediately for data collection. For the control experiments with the “empty” linker,17 samples were prepared using procedures similar to those described above and replacing C60-modified PEG with modified NHS-PEG-NH-Boc polymer that had the NHS end capped by ethanolamine instead of C60. Grafting of PEG tethers to the glass substrate was characterized by X-ray photoelectron spectroscopy (XPS). Two samples were prepared, one with covalently grafted PEG (MW 1900 Da) and one with physisorbed PEG. The results show that the sample with covalently attached PEG has a significantly higher C 1s peak, supporting the covalent attachment of the functionalized PEG tethers to the substrate. The attenuation of the Si 2p peak on the grafted sample was used to measure the average thickness of the grafted PEG layer.41 The measured average thickness of the grafted PEG layer was 0.9 Å, corresponding to the average distance between grafted PEG chains of 6 nm. These results indicate a submonolayer grafting density of PEG on the glass substrate. The XPS spectra, sample preparation procedure, and details of the data analysis are included in the Supporting Information. Data Collection and Manipulation. The cantilever spring constants were determined by the built-in thermal noise method.42 All measurements were performed in 0.05 M pH 7 phosphate buffer at 20 °C. During the force curve collection, the probe was raster scanned over a 2.5 × 2.5 µm2 square area on the substrate to obtain a good statistical average over the sample’s surface. Force curves were collected with a 10-nm

Interactions between Single C60 Fullerenes by AFM relative trigger, a 0.2-s surface dwell time, and a probe velocity ranging from 0.3 to 5 µm/s. At each velocity, 10240 and 30720 force curves were collected on the glass and mica substrate samples, respectively. Force curves exhibiting typical tether stretching events were identified. The identified individual rupture events were fit with an extended freely jointed chain (FJC) model to extract the apparent loading rates and tether Kuhn and contour lengths,43 as described previously.17-19 The force curves with tether contour lengths corresponding to approximately twice the tether length (>30 nm) were selected for subsequent analysis. The rupture forces and apparent loading rates measured at different probe velocities (the values used were 0.3, 0.7, 1.0, 2.0, and 5.0 µm/s) were binned into histograms with equal bin widths to determine the most-probable values. Histograms of rupture forces were fit with Gaussian curves multiplied by a window function to account for the limited force sensitivity.17,24 The position and width of the window function were kept the same for all histograms used in the analysis. The Supporting Information includes sample force plots and all histograms of rupture forces and the apparent loading rates used in this work. The dependencies of the most-probable forces on the apparent loading rates collected with two separately prepared samples and probes were analyzed using theoretical models as described above. The models included in the analysis considered loading with FJC tethers and several kinetic models of forced dissociation including the conventional Bell-Evans model, as well as models incorporating Morse, cusp, and cubic-linear potentials. Samples prepared by tethering C60 to the mica substrates (cleaned and dried using the same procedure as for the glass substrate samples) were imaged in air with an MFP-3D AFM in AC mode (also known as the tapping mode) using Si probes (model NSC 15, Mikromasch, Wilsonville, OR). Results and Discussion The roughness of the glass substrate samples with covalently attached fullerenes precluded imaging of individual fullerene molecules (data not shown). The fullerene-modified mica substrates exhibited a roughness of 160 pm measured over 1.8 × 1.8 µm2 area, as shown in Figure 2A. The heights of many small topographic features seen in the image are identical and close to 1 nm, as indicated by the two cross sections shown in panel B. These features are identified as single fullerene molecules.44,45 The significant width of the topographic features is attributed to the convolution of the tip shape with the shape of the fullerene molecules. Panel C shows the histogram of the height features visible in the image (two features with heights greater than 4 nm are not included in the histogram). It can be noted that many spots have heights close to 1 nm, some spots have lower heights, and a few spots have heights exceeding 1.5 nm. The latter tall spots might arise from the aggregation of C60 at the surface or might be contamination particles. The smaller features (including the apparent spots in the topographic map with heights too small to be C60) might arise from clustering upon drying of the grafted nonfunctionalized PEG tethers. This was verified by imaging of the test sample with PEG (MW 1900 Da) grafted to mica without attached C60 molecules. The test sample exhibited spots with small heights that were similar to the small-height spots observed in Figure 2A (the test sample imaging results are included in the Supporting Information). Therefore, it is likely that the small spots in Figure 2A come from the grafted PEG molecules. The topographic image does not indicate whether these fullerene molecules are attached to the sample via tethers or

J. Phys. Chem. C, Vol. 111, No. 35, 2007 12901

Figure 2. (A) Topographic map of mica surface with tethered fullerene molecules. (B) Cross sections of two molecules identified in panel A with circles showing the nearly identical heights and shapes. (C) Histogram of the heights of spots in panel A indicated with black arrows.

physisorbed on the surface. To test the attachment of the fullerenes to the mica surface, force curves were collected on the same sample in phosphate buffer. Analysis of collected force curves indicated that approximately 10 force curves with rupture tip-surface distances corresponding to the double-tether length were registered per 1 µm2 area. This number can be compared to approximately 6 topographic features with heights above 1 nm (including somewhat large aggregates visible in the image) per 1 µm2 area measured from topographic maps. Considering that the sample area used for imaging was different from the area used for force measurements, comparison of the number of detected double-tether events with the number of topographic features suggests that the number of ruptures detected during force mapping was approximately equal to the number of grafted C60 molecules. Therefore, the grafting density on the samples with glass substrate can be estimated from the frequency of detected double-tether ruptures. The histogram of rupture forces collected with the mica substrate is included in Figure 3A. Comparison of this histogram with the histogram of rupture forces collected on the glass substrate shows that the most-probable rupture force on the glass substrate closely matches the most-probable rupture force on the mica substrate. However, the distribution of rupture forces collected with the mica substrate sample shows a significant fraction of high rupture forces that exceed the most-probable value by a factor of nearly 2. These high rupture forces can be attributed to interactions between fullerenes on the probe and an oligomeric aggregate of tethered fullerene molecules on the sample when the rupture force corresponds to the dissociation of not one but several molecular bonds nearly simultaneously, as previously described for different molecular systems.46,47 This explanation is supported by a significant fraction of topographic features with heights larger than a single fullerene molecule in the topographic map shown in Figure 2A. The reason for such clustering of C60 molecules on the mica substrate in comparison to the glass substrate is currently unknown.

12902 J. Phys. Chem. C, Vol. 111, No. 35, 2007

Figure 3. (A) Histograms of C60-C60 rupture forces determined on C60-functionalized glass (at 5 µm/s probe velocity) and mica (at 6 µm/s probe velocity) substrates. (B) Histograms of tether contour lengths extracted from the force curves that were measured at 5 µm/s probe velocity. Shown histograms correspond to experiments with the “normal” sample (substrate functionalized with PEG-C60, light gray bars) and to the control experiment with the “empty” linker sample (dark bars). In both experiments, the silicon nitride probes were functionalized with PEG-C60.

The probability of detecting a rupture event at the doubletether length with the mica substrate sample is 0.2% and that with the glass substrate is 1%. This difference and the difference in the number of collected force curves explain the difference in the heights of the distributions in Figure 3A. The detection probability on the glass substrate corresponds to a surface density of ∼50 µm-2 of tethered fullerenes that were detected with force spectroscopy, indicating that the grafting density on glass substrates is approximately twice the grafting density on mica. A low grafting density facilitates the detection of interactions between single molecules. It can be noted that the employed sample preparation procedure facilitates a low grafting density: First, the substrate amination with ethanolamine was adopted because this method is reported to produce a low grafting density.38 Second, anhydrous organic solvents were used in the sample preparation to minimize the aggregation of fullerenes at the surface during sample preparation. Third, a thorough sample cleaning procedure was employed to remove the physisorbed C60 molecules. Figure 3 shows typical distributions of the (A) rupture forces and (B) tether contour lengths obtained with a probe velocity of 5 µm/s. The most-probable contour length shown in Figure 3B is 62 nm, which approximately corresponds to the sum of the lengths of two extended tethers (∼59 nm according to the mass spectrum shown in the Supporting Information).17,43 Only the force curves characterized with double-tether separation were selected to build the force and loading rate histograms; this procedure removed the events of the indiscriminate binding of C60 to the solid surfaces.17,23 An empty tether control experiment was performed to attribute the detected events to the C60-C60 interactions. In the control experiment, the number of double-tether rupture events significantly decreased (only 8 events detected out of 10240 force curves compared to 91 rupture events detected out of

Gu et al. 10240 force curves for normal sample at the same probe velocity), indicating that the majority of the detected ruptures in normal experiments can be attributed to C60-C60 interactions (in contrast to C60-tether interactions). These results are similar to the previous measurements with the “empty” tether performed with hexadecane and a 12-mer fragment of R-synuclein samples.17,20 It can be noted that the tether contour length detected during the control experiment is in the range of 3555 nm (Figure 3B). This distribution is much narrower than the distribution for a normal sample, which extends up to 100 nm. This observation hints that the ruptures detected during the control experiment might come from the residual single-tether ruptures of the tip-tethered C60 from the substrate. In addition to the arguments presented above, the singlemolecule nature of the detected rupture forces (in contrast to the detection of two molecular bonds rupturing nearly simultaneously) is supported by noting that, in the case of two bonds rupturing nearly simultaneously, the detected force will be nearly twice the single-molecule force.19,46 Therefore, if the peak of the most-probable forces detected at the low probe velocity results from multiple attachments between the tip and the substrates, the appearance of the low force peak is expected at the higher probe velocity because of rupturing of single bonds. Because no such peak was detected (as can be seen in Figure 3A and in the histograms included in the Supporting Information), it is assumed that the peak corresponding to the mostprobable forces results from the single-molecule ruptures.19,46,47 The above discussion of the nature of the rupture force can be summarized by considering several plausible molecular arrangements illustrated in Figure 4. Arrangements shown in the figure correspond to events when polymer stretching is followed by bond rupture, with the cantilever deflection force returning to the baseline because only such events are included in the data processing. Panel A shows the rupture between tethered C60 and the substrate surface. These events occur at the tip-surface separation corresponding to the length of one tether. Such events were detected in the measurements (data not shown), and the majority of such instances were excluded from the rupture force analysis by applying the 30-nm threshold to the fitted contour length of tethers. Panel B shows the events when interactions between tethers or interactions between tethered molecules and tethers are responsible for the measured rupture force. Considering that water-soluble PEG linkers were used in the experiment, this might be expected if such interactions were statistically significant and in the range of the measured rupture forces. However, the experiment with the empty tether described above indicates that such events do not contribute a statistically significant number of events. Panels C-E show the ruptures that occur between the tethered molecules with the tip-surface separation corresponding to twice the single-tether length. Panel C shows interactions between individual molecules. At sufficiently high grafting density, interactions between more than two molecules can be detected as a single rupture in the force-separation curve. Panel D shows interactions between three molecules, and panel E shows interactions between four molecules. As illustrated in the figure, detection of the rupture from aggregates comprising more than two molecules involves stretching of several tethers. Detection of such events as a single rupture requires the tether lengths of the tethers to be sufficiently close; otherwise, two ruptures would be detected. Therefore, if ruptures occur as indicated in panels D and E, the force applied by the probe will be distributed over two tethers, and the force acting on one molecular bond will be lower than the force detected by

Interactions between Single C60 Fullerenes by AFM

J. Phys. Chem. C, Vol. 111, No. 35, 2007 12903

Figure 4. Possible outcomes of the interaction between the AFM tip and substrate equipped with tethered C60 molecules. The black wavy line represents the PEG tether, and the black ball represents a C60 fullerene.

TABLE 1: Kinetic Parameters of C60-C60 Physical Bond Rupture Obtained from Two Data Sets and Different Models of Molecular Bond Rupture model BE BE-FJC Morse-FJC Cusp-FJC

kc (pN/nm)

xq (nm)

k0 (s-1)

∆Gq (kJ‚mol-1)

70 51 mean 70 51 mean 70 51 mean 70 51 mean 70 51 mean

0.29 ( 0.03 0.25 ( 0.03 0.27 ( 0.02 0.31 ( 0.03 0.29 ( 0.04 0.30 ( 0.02 0.36 ( 0.04 0.32 ( 0.04 0.34 ( 0.03 0.37 ( 0.04 0.33 ( 0.04 0.35 ( 0.03 0.35 ( 0.04 0.32 ( 0.04 0.34 ( 0.03

1.8 ( 1.5 8.6 ( 3.6 2.8 ( 1.4 0.94 ( 0.78 2.9 ( 1.6 1.3 ( 0.7 0.58 ( 0.54 2.5 ( 1.3 0.9 ( 0.5 0.55 ( 0.52 2.5 ( 1.3 0.8 ( 0.5 0.67 ( 0.60 2.6 ( 1.4 1.0 ( 0.6

55.6 ( 2.0 51.7 ( 1.0 52.5 ( 0.9 57.2 ( 2.1 54.4 ( 1.4 55.3 ( 1.2 58.4 ( 2.3 54.8 ( 1.3 55.7 ( 1.1 58.5 ( 2.3 54.8 ( 1.3 55.7 ( 1.1 58.0 ( 2.2 54.7 ( 1.3 55.6 ( 1.1

Figure 5. Measured most-probable rupture forces vs loading rates for two data sets. Shown error bars were calculated using the covariance matrix.49

Cubic-FJC

the AFM cantilever. Consequently, the rupturing of more than one bond will be detected as the high-force tail48 in the distribution of the rupture forces, as can be seen in Figure 3A at ∼130 pN. The sample prepared with the mica substrate shows the more pronounced peak of higher forces than the sample prepared with the glass substrate. Because the number of events at the high-force tail for the glass substrate sample is low, it is expected that these events do not significantly alter the mostprobable force corresponding to the main peak in the histogram. Therefore, the most-probable rupture forces determined from the histogram are considered below as the most-probable singlemolecule rupture forces. Figure 5 shows the resulting most-probable force (F*) vs loading rate (νF) dependencies collected on two separately prepared samples and with AFM probes with spring constants of 0.070 and 0.051 N/m, as indicated in the figure. The kinetic parameters xq and k0 were obtained by performing the leastsquares fits of different force spectroscopy models to the data by numerically solving eq 2. The Arrhenius prefactor A was estimated34 using the predicted curvature of the potential energy curve15 and the measured barrier width to be approximately equal to ∼1010 s-1. Possible systematic errors in parameters related to the uncertainty in the prefactor are discussed below. The figure shows fits of the experimental dependencies with the BE, BE-FJC, and Morse-FJC models. Data were also fit by models considering the cusp and cubic-linear potentials; the resulting fits are very close to the fit obtained with the Morse potential model, so the fit lines are not drawn in the figure. The fit lines corresponding to different models are close; however, the resulting kinetic parameters vary as listed in Table 1. The table reports the kinetic parameters xq, k0, and ∆Gq obtained by fitting force vs loading rate data from Figure 5 by the different force spectroscopy models (indicated in the left column). The table separately includes parameters from two different experiments (identified by the cantilever spring constant in the second column), as well as the mean value for each model. The results presented in the table show that the data analysis

with the standard BE model underestimates the width and depth of the energy barrier. The systematic errors arise from the effects of both the nonlinear tether and the finite depth of the potential. Accounting for both effects increases the barrier width by ∼25%, decreases the rate of dissociation by a factor of approximately 3, and increases the depth of the potential by 4 kJ‚mol-1. It can be noted that, among the three “nonstandard” models of potential used here, the cusp potential gives the highest xq and ∆Gq values and the cubic-linear potential gives the lowest values. However, the variation in parameters resulting from using different models of the shape of the potential energy landscape is small and within the random error. Previous studies that compare use of the cusp potential model to the BE model (without considering tether effects) also indicate that the data analysis with the cusp potential model gives higher values of the barrier width. In a study of glycoproteins, the application of the cusp potential model gives a barrier width up to 30% larger than that from the BE model.50 A study of lipid extraction from bilayers found that the most-probable force vs loading rate dependence employing the cusp potential model gives a barrier width up to 60% larger than that from the BE model.51 A single-molecule study of fibronectin adsorption found barrier widths uniformly larger upon application of cusp potential models than the BE model.52 Separate fits show that the fit parameters are influenced by the value of the prefactor (fits are not shown). Decreasing and increasing the prefactor by a factor of 10 changes the barrier width by (2% and the dissociation rate by -4%; these variations are within the random error in the parameters. The corresponding change in the activation energy is close to -kBT(ln 10) ) -5.7 kJ‚mol-1, indicating that the uncertainty in the prefactor is the largest contribution to the error in the activation energy. Taking into account possible systematic errors due to the anharmonic tether and the finite barrier depth, the measured

12904 J. Phys. Chem. C, Vol. 111, No. 35, 2007

Figure 6. Morse potential between two C60 molecules in water calculated using the experimentally derived parameters at 0 and 100 pN external force as a function of displacement from the equilibrium position x0. It can be noted that, under 100 pN force, a significant barrier (∼18kBT) to dissociation remains.

barrier width xq is 0.34 ( 0.03 nm, and the activation energy ∆Gq is 56 ( 1 kJ/mol. It can be noted that the barrier width reported here does not assume that there is a barrier toward association of two C60 molecules. The reported xq value can be interpreted as a characteristic width of the Morse potential29 multiplied by 4, as illustrated in Figure 6. The measured xq value is closer to the barrier width predicted using the WCA potential for water-fullerene interaction (∼0.4 nm)15 than to the value predicted using the LJ potential (0.2 nm).14,15 The measured width (0.34 nm) is significantly larger than that predicted using LJ potential for fullerene-water interactions. It is also larger than the size of water molecule (0.28 nm),53 so this result is consistent with the dewetting13,54,55 of fullerenes during separation. Force spectroscopy does not provide direct evidence as to whether the density of water between fullerenes at a separation of 0.34 nm is equal to the density of water vapor; however, the measured width of the potential barrier indicates a longer-range attraction than that calculated for small hydrophobic molecules (∼0.16 nm).56,57 The measured activation energy ∆Gq ) 56 ( 1 kJ/mol is approximately twice the predicted energy required to separate two fullerene molecules in vacuum15 (∼25 kJ/mol). According to simulations, the ∆Gq value exceeds the thermodynamic energy difference ∆G by less than 2 kJ/mol.15,16 Therefore, these measurements imply that approximately 25 kJ/mol of the energy required to separate two C60 molecules in water is coming from the attractive hydrophobic interactions. In addition, the measured ∆Gq value is close to the value predicted using the WCA C60water potential (∼ 55 kJ/mol)15 and deviates significantly from the prediction of the LJ C60-water potential (∼20-25 kJ/ mol).15,16 The difference in the predicted values comes mostly from the energy at the equilibrium contact position between C60 molecules; the height of the activation barrier is predicted to be small, as mentioned above. The lower equilibrium energy (corresponding to the higher activation energy barrier of dissociation) is supported by considering the solubility of fullerenes. The potential energy depths of 20 and 55 kJ/mol correspond to critical micelle concentrations (CMCs) of 4 × 10-5 and 7 × 10-16 mol/L, respectively.58 The CMC of C60 in water is very low and has not been measured experimentally; however, it is likely that the CMC is close to the solubility of C60 in water. Existing estimates indicate that the solubility of C60 in water ranges between 10-18 and 6 × 10-7 mol/L.59,60 Therefore, the activation energy of ∼20 kJ/mol is too low to explain the low solubility of fullerenes. This observation supports the higher energy difference for C60-C60 dimer

Gu et al. dissociation in water and is consistent with the results of force spectroscopy measurements reported herein. Previous force spectroscopy studies of hydrophobic interactions between hexadecane molecules indicate that the measured energy ∆Gq is close to the cavitation energy estimated using the cavity equation of state.17,53 The difference in the energy of two spherical cavities with each volume equal to the solventexcluded volume of a C60 molecule and one spherical cavity with the solvent-excluded volume of two C60 molecules is 70 kJ/mol.17,53 This energy corresponds to the spherical shape of the cavity comprising two fullerenes, and the true cavity contribution is likely to be smaller than 70 kJ/mol because of the effects of the cavity shape.61 Including the attractive interaction between water and C60 also decreases the energy difference.14,15 Therefore, the experimental results imply that the upper value for the effects of cavity shape is approximately 15 kJ/mol. The discrepancy between the measured activation energy (56 kJ/mol) and the value predicted by molecular dynamics simulations15,16 (∼20-25 kJ/mol) is significant. The random error of the measured value is small ((1 kJ/mol), and the error due to the uncertainty in the prefactor is unlikely to exceed ∼5 kJ/ mol. Therefore, the observed difference can be attributed to the inaccuracy of molecular dynamics simulations. Some part of this error might come from the inaccuracy of the parameters of the LJ potential. It has been shown that the excess chemical potential of methane in water can be accurately calculated by adjusting the LJ parameters of the water-methane interaction.62 However, because the measured activation energy is close to the activation energy calculated using the repulsive WCA water-C60 potential15 (∼55 kJ/mol), the errors in the LJ parameters necessary to account for the observed difference appear too high to be realistic. It might be suggested that the difference between the measured and predicted values comes in part from the errors of using the model describing waterwater interactions in the bulk to describe water-water interactions close to the hydrophobic surface.63 Recent theoretical calculations using quantum mechanical simulations of the hydration of aromatic molecules indicate that the electronic structure of water molecules at the interface is different from that in bulk water.64 Quantum mechanics molecular dynamics calculations of the potential of mean force for two methane molecules65 indicate that classical molecular dynamics simulations significantly underestimate the depth of the potential and (less significantly) the activation barrier. It might be suggested that similar effects are responsible for the discrepancy between the experimental results presented in this article and the interactions between C60 fullerenes in water predicted by molecular dynamics simulations. Conclusions In summary, interactions between C60 fullerene molecules have been measured using single-molecule force spectroscopy. The results support the single-molecular nature of the measured interactions. The presented model for the analysis of force vs loading rate dependence includes effects due to anharmonic tethers and the finite depth of the potential well. Experimental data analyzed using this model indicate that the standard approach to force spectroscopy data analysis underestimates the barrier width xq by ∼25% and overestimates the dissociation rate by a factor of 3. The corrected barrier width was measured to be 3.4 ( 0.3 Å, and the dissociation rate was measured to be 0.9 ( 0.5 s-1, corresponding to the activation energy ∆Gq of ∼56 kJ/mol. It is noted that the measured activation energy

Interactions between Single C60 Fullerenes by AFM is consistent with the low solubility of fullerenes in water.59,60,66 In comparison to recent theoretical calculations, these results are close to the predictions that use the repulsive WCA waterC60 potential for calculations of the fullerene-fullerene dissociation energy15,16 and differ significantly (by a factor of 2.53) from the calculations that use the LJ water-C60 potential. We suggest that the discrepancy of this magnitude requires reexamination of the computer simulation methods employed in modeling interactions between nonpolar molecules in water. Acknowledgment. The authors thank Duke University for financial support. Stimulating discussions of this work with Prof. Dor Ben-Amotz and Prof. James Skinner are greatly appreciated. Supporting Information Available: Synthetic protocols, sample preparation, data collection, MS spectrum demonstrating formation of C60-PEG, XPS/AFM control sample characterization, typical force curves with FJC fit, and force and loading rate histogram fits. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. Nature (London) 1985, 318, 162. (2) Taylor, R.; Walton, D. R. Nature (London) 1993, 363, 685. (3) Hirsch, A. The Chemistry of the Fullerenes; Thieme: Stuttgart, Germany, 1994. (4) Kelty, S. P.; Chen, C. C.; Lieber, C. M. Nature (London) 1991, 352, 223. (5) Ruoff, R. S.; Ruoff, A. L. Nature (London) 1991, 350, 663. (6) Marcorin, G. L.; Da Ros, T.; Castellano, S.; Stefancich, G.; Bonin, I.; Miertus, S.; Prato, M. Org. Lett. 2000, 2, 3955. (7) Friedman, S. H.; Ganapathi, P. S.; Rubin, Y.; Kenyon, G. L. J. Med. Chem. 1998, 41, 2424. (8) Schinazi, R. F.; Sijbesma, R.; Srdanov, G.; Hill, C. L.; Wudl, F. Antimicrob. Agents Chemother. 1993, 37, 1707. (9) Jensen, A. W.; Wilson, S. R.; Schuster, D. I. Bioorg. Med. Chem. 1996, 4, 767. (10) Fortner, J. D.; Lyon, D. Y.; Sayes, C. M.; Boyd, A. M.; Falkner, J. C.; Hotze, E. M.; Alemany, L. B.; Tao, Y. J.; Guo, W.; Ausman, K. D.; Colvin, V. L.; Hughes, J. B. EnViron. Sci. Technol. 2005, 39, 4307. (11) Chandler, D. Nature (London) 2005, 437, 640. (12) Mountain, R. D.; Thirumalai, D. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 8436. (13) Southall, N. T.; Dill, K. A. J. Phys. Chem. B 2000, 104, 1326. (14) Li, L.; Bedrov, D.; Smith, G. D. J. Chem. Phys. 2005, 123, 204504/ 1. (15) Li, L.; Bedrov, D.; Smith, G. D. Phys. ReV. E: Stat. Nonlinear, Soft Matter Phys. 2005, 71, 011502/1. (16) Hotta, T.; Kimura, A.; Sasai, M. J. Phys. Chem. B 2005, 109, 18600. (17) Ray, C.; Brown, J. R.; Akhremitchev, B. B. J. Phys. Chem. B 2006, 110, 17578. (18) Ray, C.; Brown, J. R.; Akhremitchev, B. B. J. Phys. Chem. B 2007, 11, 1963. (19) Sulchek, T. A.; Friddle, R. W.; Langry, K.; Lau, E. Y.; Albrecht, H.; Ratto, T. V.; DeNardo, S. J.; Colvin, M. E.; Noy, A. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 16638. (20) Ray, C.; Akhremitchev, B. B. J. Am. Chem. Soc. 2005, 127, 14739. (21) Schmitt, L.; Ludwig, M.; Gaub, H. E.; Tampe, R. Biophys. J. 2000, 78, 3275. (22) Auletta, T.; De Jong, M. R.; Mulder, A.; Van Veggel, F. C. J. M.; Huskens, J.; Reinhoudt, D. N.; Zou, S.; Zapotoczny, S.; Schoenherr, H.; Vancso, G. J.; Kuipers, L. J. Am. Chem. Soc. 2004, 126, 1577. (23) Ratto, T. V.; Langry, K. C.; Rudd, R. E.; Balhorn, R. L.; Allen, M. J.; McElfresh, M. W. Biophys. J. 2004, 86, 2430. (24) Friedsam, C.; Wehle, A. K.; Kuehner, F.; Gaub, H. E. J. Phys. Condens. Matter 2003, 15, S1709.

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