Quantitative Analysis of the Interaction between an Atomic Force

Nov 23, 2010 - Hubert Gojzewski , Michael Kappl , Hans-Jürgen Butt , and Arkadiusz ... Kircher , Wojciech Koczorowski , Hans-Jürgen Butt , Arkadiusz P...
1 downloads 0 Views 457KB Size
Download PDF
21572

J. Phys. Chem. C 2010, 114, 21572–21578

Quantitative Analysis of the Interaction between an Atomic Force Microscopy Tip and a Hydrophobic Monolayer Arkadiusz Ptak,*,† Hubert Gojzewski,† Michael Kappl,‡ and Hans-Ju¨rgen Butt‡ Institute of Physics, Poznan UniVersity of Technology, Nieszawska 13A, PL-60965 Poznan, Poland, and Max-Planck-Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, Germany ReceiVed: August 22, 2010; ReVised Manuscript ReceiVed: NoVember 5, 2010

Atomic force microscopy has become an important technique for studying the adhesion of nanometer-sized contacts. However, there is still no good method to quantitatively characterize adhesive interactions. With a modified atomic force microscope, adhesion forces between a silicon nitride tip and a self-assembled monolayer of 1-dodecanethiol on gold(111) have been measured at different loading rates. Adhesion force-versus-loading rate curves revealed two regimes. Various interpretations of the two-regime character, like the existence of an inner transition state in the interaction potential or rebinding processes, are discussed. An explanation related to the cooperativity of individual van der Waals bonds responsible for effective adhesion is proposed. A way to extract the kinetic and interaction potential parameters for an individual van der Waals bond is demonstrated using a microscopic model to the analysis of adhesion force-versus-loading rate dependence. Introduction 1

Force experiments using an atomic force microscope (AFM) are widely used to probe adhesive and mechanical properties of materials at the nanoscale.2 Physicists and chemists study molecular mechanisms of surface interactions, biologists investigate protein unfolding and ligand dissociation, and engineers are interested in practical aspects of nanocontacts for construction of micro- and nanodevices. In the force spectroscopy mode, the tip is approached to the sample surface and subsequently retracted while the force is measured. From the retracting part of the force-displacement curve, it is possible to detect adhesion and mechanical effects like multistep surface separation or molecular conformational changes.3,4 Models based on Kramers’ theory5 for bond dissociation rates, like the Bell-Evans model6,7 and models by Hummer and Szabo8 and Dudko et al.,9 show that association and dissociation rates of a bond depend on the shape of the tip-sample interaction potential, from which both rates can be extracted. Thus, the interaction potential determines not only the strength of interactions but also their kinetics. The conventional way to reconstruct a tip-sample interaction potential from force-distance curves would be to integrate the force times the distance covered by the tip approaching (retracting) the sample surface. Unfortunately, this approach will usually not allow full reconstruction of the tip-sample interaction potential. When the force gradient of the tip-sample attraction is higher than the spring constant of the cantilever used, an instability occurs and the tip snaps into contact with the sample. Upon retraction the adhesion force keeps the tip in contact with the surface until it is overcome by the spring force of the cantilever. Then the tip snaps off the contact with the sample. As a consequence, no information is provided over the distance of snapping and hence it is impossible to use the integration method to reconstruct the interaction potential. An alternative approach for measuring the interaction potential was proposed by Cleveland et al.10 They used the Boltzmann * Corresponding author, [email protected]. † Institute of Physics, Poznan University of Technology. ‡ Max-Planck-Institute for Polymer Research.

distribution function to calculate the free energy of the tip and sample surface from the probability distribution of the tip position, while the cantilever was maintained at a fixed position. In this approach the total potential of the tip, which is the sum of the interaction potential and the harmonic potential of the cantilever, is measured. Unfortunately, it is not possible to extract the contribution of both potentials because the relative position of the cantilever base and sample surface is not known. A modification of the method was presented by Willemsen et al.11 They generated force-distance curves at low approach rates to enable the tip to explore a large part of the continuously varying potential energy curve. Since the tip was brought into contact with the sample during each approach, the researchers knew the absolute position of the cantilever potential. However, there are some disadvantages to their method. Because of the low approach rates, drifts in the measurement setup can deform the shape of the measured interaction potential and transition states located close to the sample surface may not be detected. Moreover, the accuracy of the methods based on the thermal noise detection is limited by the spring constant of applied cantilevers. To avoid a tip snap-off, contact with the cantilever should be stiff enough, which reduces its sensitivity for thermal noise detection. The classic work by Bell6 established a framework for understanding the kinetics of forced dissociation by recognizing that an external loading force reduces the activation barrier. Evans and Ritchie7 showedsusing Bell’s formulasthat the force required to rupture an adhesive bond is not unique but depends on the loading rate and the shape of the interaction potential. Therefore, measuring the bond rupture force (or adhesion force) over a range of loading rates, i.e., performing the so-called dynamic force spectroscopy (DFS) measurements, provides a way to determine such parameters as the kinetic off-rate and the distance to the transition state. Recently Dudko et al.,12 however, showed that the Bell-Evans model is a particular case of a more general theory and its application to the analysis of DFS data can produce unreliable results. We start our paper by summarizing theories which have been used to quantitatively describe bond rupture. Then, we critically

10.1021/jp107948q  2010 American Chemical Society Published on Web 11/23/2010

Quantitative Analysis of Adhesion Forces

J. Phys. Chem. C, Vol. 114, No. 49, 2010 21573

discuss various interpretations of adhesion force-versus-loading rate dependence (DFS measurement data), which assume the existence of an inner transition state or rebinding processes for instance. We propose a novel explanation related to the cooperativity of individual van der Waals bonds responsible for effective adhesion. We also demonstrate a way to extract the kinetic and interaction potential parameters for an individual van der Waals bond using the microscopic model by Dudko et al.12 We hope that our work contributes to better understanding of the issue of surface interactions and interaction potential reconstruction and will prevent misinterpretations of dynamic force spectroscopy data. Theory A common approach to analyze adhesion force-versus-loading rate curves is to apply the Bell-Evans model.6,7 The model treats the unbinding process as an escape from a potential well under the influence of the external loading force. This external force tilts the interaction potential and facilitates a thermally activated escape from the bound state. The rate constant of bond rupture koff(F) scales with the exponent of the applied force F according to6

( )

0 koff(F) ) koff exp

Fxβ kBT

(1)

0 where koff is the intrinsic (force-free) off-rate constant, xβ is the distance between the bound state and the transition state along the direction of the external pulling force, kB is the Boltzmann constant, and T the absolute temperature. The loading force can be ramped in time

F(t) ) rFt

( ) rF

0 Fβkoff

p(F) )

{ ∫[ ] }

koff(F) exp F˙

(3)

Here, Fβ ) kBT/xβ is the so-called thermal fluctuation force.13 Thus, Fad is predicted to grow linearly with ln(rF) and eq 3 0 from provides a simple way to extract the value of xβ and koff the slope and intercept of the Fad versus ln(rF) curve fitted to experimental data. The model has been widely applied by several research groups to study intermolecular interactions between or within biomolecules14-17 and by our group to study nanoadhesion between an AFM tip and self-assembled monolayers.18-20 The attractive feature of this phenomenological model seems to be its generality, which is, unfortunately, only apparent. The kinetics 0 . The underlying free-energy surface is is subsumed into koff characterized by a single parameter (xβ), which is, however, according to Bell’s formula (eq 1), assumed to be constant during external force application. This assumption is not fulfilled by potentials describing real interatomic or intermolecular interactions like the Morse or van der Waals potentials. Moreover, it was shown by analyzing simulated data obtained

F

0

k(F′) dF′ F˙

(4)

where F˙ ≡ dF/dt. In AFM experiments dF/dt ) rF, where rF is the force loading rate. The adhesion force is then the mean rupture force, 〈F〉 ) ∫Fp(F) dF. The rate constant of bond rupture can be written on the base of the Kramers theory as

(

0 koff(F) ) koff 1-

νFxβ ∆Gβ

)

1/(ν-1)

×

{ [ (

exp ∆Gβ 1 - 1 -

(2)

where rF is the force loading rate being a product of the effective spring constant and the pulling speed. For AFM experiments the retraction of the base of the cantilever is constant. The solution of eq 1 for such a constant loading rate leads to a mean rupture force:

Fad ) Fβ ln

from a simple microscopic model of pulling experiments that the phenomenological description is inadequate for the experimentally relevant loading rate.8 Dudko et al.12 showed that the phenomenological approach by Evans and Ritchie7 as well as the microscopic models by Hummer and Szabo,8 and Dudko et al.9 are particular cases of the more general approach, which we will call the DudkoHummer-Szabo (DHS) model. This unified model allows 0 , xβ, andsaddionally to the Bell-Evans extraction of koff modelsthe free energy of activation in the absence of external forces, ∆Gβ from DFS experiments. We have applied the DHS model to analyze the adhesion force-versus-loading rate curve for nanoadhesion between a silicon nitride tip and a methylterminated self-assembled monolayer (SAM). Although this model is much more sophisticated than the Bell-Evans model, it is still analytically tractable. Since the escape process is stochastic, the rupture forces differ from one experiment to another, resulting in the following distribution

νFxβ ∆Gβ

) ]} 1/ν

(5)

where the parameter ν describes the shape of free-energy surface, e.g., ν ) 2/3 corresponds to the linear-cubic free-energy surface9 and ν ) 1/2 to the cusp free-energy surface.8 For ν ) 1 and ∆Gβ . Fxβ, which is not the case for DFS experiments, the expression reduces to Bell’s formula. The distribution of rupture forces can be obtained from eqs 4 and 5 analytically, and the expression for the mean rupture force is

{ [

0 ∆Gβ koff exp(∆Gβ + γ) 1 〈F〉 = 1ln νxβ ∆Gβ xβrF

]} ν

(6)

Here, γ ) 0.577 is the so-called Euler-Mascheroni constant.21 The DHS model provides also an expression for the rupture force variance. Since our experimental data of force distribution did not exhibit a clear dependence on the loading rate, we omit this line of analysis. Dynamic force spectroscopy measurements usually do not provide a sufficient number of pull-off forces, because the experimental series should be performed in reasonable time to avoid tip wear as well as deformation and contamination of the sample. Experimental Section Self-Assembled Monolayers and Their Characterization. As a model adhesion system, we studied a well-known and characterized self-assembled monolayer (SAM) of methyl-

21574

J. Phys. Chem. C, Vol. 114, No. 49, 2010

terminated thiols interacting with a silicon nitride tip.20 In the case of hydrophobic SAMs and low humidity, the interactions responsible for the effective adhesion are limited mostly to van der Waals forces. Results for this model system should be easier to interpret than the commonly studied interprotein adhesion, which is a result of various interactions covering hydrogen bonds and electrostatic forces in addition to van der Waals forces. SAMs of 1-dodecanethiol (Sigma-Aldrich) were formed on Au(111) substrates by immersion in 1 mM thiol in ethanol solution for 18-24 h. After the incubation, the samples were rinsed with ethanol, dried under a nitrogen flow, and immediately used for experiments. We applied scanning tunneling microscopy (STM) and contact angle measurements to characterize the SAMs. STM images showed that a hexagonal (3 × 3)R30° lattice structure dominated our samples. The high and stable (no change for days) value of the advancing contact angle, 96 ( 2°, confirmed that a stable hydrophobic monolayer was formed. (Precise description of the characterization by STM imaging and contact angle measurements has been placed in the Supporting Information.) Modified Atomic Force Microscopy. The modified AFM setup for DFS measurements was described by us previously.20,22 This system has two modifications: first, to extend the range of accessible separation rates up to 107 nm s-1; second, to control the humidity of the experimental environment. To achieve high loading rates a high-frequency piezo-actuator together with a fast acquisition and data-processing system was added to a commercial AFM. A specially designed Teflon sample chamber allows control of relative humidity by mixing flows of pure N2 and N2 saturated with water vapor. The relative humidity was kept constant below 20%, which limited the presence of capillary forces. The cantilevers (model MLCT-F, Veeco, USA) of high nominal resonance frequency (120 kHz) and relatively low nominal spring constant (0.5 N m1-) were used. Prior to each measurement the cantilevers were cleaned using Ar plasma for 25 s. The spring constant of the cantilevers was measured based on the thermal noise method.23 (Details and a schematic of the modified AFM setup as well as the methodology of force spectroscopy measurements can be found in the Supporting Information.) Results Adhesion force increased almost monotonically with increasing loading rate in the measured range of loading rates, i.e., from 30 to 7 × 105 nN s-1 (Figure 1). The experiments were performed in air; therefore the influence of the fluid viscosity can be neglected. Fitting with the DHS model (eq 6) has been performed for two loading rate regimes: (1) from 30 to 8 × 104 nN s-1 and (2) from 105 to 7 × 105 nN s-1 (Figure 1, the red solid curves). The fit to the whole measured range, i.e., from 30 to 7 × 105 nN s-1, did not give a good agreement (Figure 1, the blue dash curve). The fits for the two separated regimes were performed for different values of parameter ν (1/2, 2/3, and 1) and with the Bell-Evans model to compare results (Table 1). The coefficient of determination, R2, which measures the quality of a fit, is close to 1.00 (perfect fit) for all the fits. This indicates that the DHS model, which was developed for an individual bond, can be applied to nanoadhesion. In such a case the adhesive bonds within the nanocontact are not considered individually but lumped into one effective tip-sample interaction potential. According to Dudko et al.12 the most reliable fitting to DFS data is with the model assuming the linear-cubic free-energy surface, i.e., for ν ) 2/3. The fitting parameters

Ptak et al.

Figure 1. Adhesion force-versus-loading rate for nanoadhesion between a silicon nitride tip and a methyl-terminated SAM (1dodecanethiol on gold (111)). The red solid lines are a result of the fitting with the DHS model (ν ) 2/3) separately for two loading rate regimes: (1) from 30 to 5 × 104 nN s-1 and (2) from 7 × 104 to 7 × 105 nN s-1. The blue dash curve is a result of the fitting with the DHS model for the whole measured regime, i.e., from 30 to 7 × 105 nN s-1. The error bars were calculated as a standard deviation of mean.

obtained with the ν ) 2/3 DHS model and the model assuming the cusp free-energy surface (ν ) 1/2) are similar to each other for both loading rate regimes (Table 1). The small differences are similar to those observed by Dudko et al. for the simulated data describing the unfolding of the titin and RNA.12 The xβ 0 and ∆Gβ values are slightly higher and koff values are lower for ν ) 1/2 than for ν ) 2/3. All fitting parameters obtained for ν ) 1 are significantly different from those for ν ) 1/2 and ν ) 2/3. The xβ value for ν ) 1 is the same as that calculated directly with the Bell-Evans model, which is expected since the slope of the adhesion force-versus-loading rate in the ν ) 1 DHS model reduces to that in the Bell-Evans model (eq 3). Analyzing the fitting parameters for regime 1 of loading rates we have found that they are not unique and the fitting is ambiguous. There are two sets of parameters for which convergence occurs. We present such data for the ν ) 2/3 DHS model. One fitting converges for ∆Gβ ≈ 70 kBT, whereas the other for ∆Gβ ≈ 664 kBT, which is the limit of convergence for this parameter in the DHS model. If one fixes ∆Gβ (normally being fitted) to values higher than 665 kBT, there will be no convergence. Additionally, the fitting parameters for regime 1 are very sensitive to a chosen range of loading rates. If fitting was performed for a range with one or two points more, the result differed significantly. Therefore parameters obtained from fitting adhesion forces which depend only weakly or not at all on the loading rate are of limited significance. The situation is different for regime 2. Here, the DHS model results in stable fits, and the data obtained with the ν ) 2/3 DHS model are reliable. The calculated parameters of the effective tip-sample interaction potential were xβ ) 1.9 × 10-3 0 nm, koff ) 22 s-1, ∆Gβ ) 9 kBT. It is important to emphasize that these parameters, which were calculated for a macrobond, have not the same meaning as for a single bond. For example, xβ is extremely small and would be unrealistic for a single bond. Before we try to extract further information from the results, especially regarding the van der Waals interactions, we need to explain the two-regime character of the adhesion force-versusloading rate curve. Discussion The common interpretation combines the two regimes of the adhesion force-versus-loading rate curve with two barriers on

Quantitative Analysis of Adhesion Forces

J. Phys. Chem. C, Vol. 114, No. 49, 2010 21575

TABLE 1: The Fitting Parameters for the Data of Figure 1 Obtained with the DHS and Bell-Evans Models Dudko-Hummer-Szabo model regime 1

2

parameters xβ (nm) k0off (s-1) ∆Gβ (kBT) R2(COD) xβ (nm) k0off (s-1) ∆Gβ (kBT) R2(COD)

ν ) 1/2 -2

1.5 × 10 10-18 80 0.90 2.8 × 10-3 6 10.8 0.99

ν ) 2/3 -2

1.4 × 10 5 × 10-18 70 0.90

the interaction potential describing effective adhesion. Accordingly, the second regime is due to an inner barrier.14,16,17,24 Inner barriers are often observed for ligand-protein specific adhesion. Such adhesion is a sum of various interactions, including electrostatic repulsion, which can cause the formation of a transition state between the bound state and the dissociated state in the interaction potential. Such a transition state does not need to have a barrier character to be demonstrated as a second regime. It can be a certain curvature on the interaction potential, which becomes a barrier when bending the potential curve by applying an external load (Figure 3). What could be the repulsive interaction, which is necessary to form such a curvature in the potential curve for the AFM tip-alkanethiol monolayer interaction? One candidate is the elastic repulsion of the alkanethiols compressed by the apex of the AFM tip.19 Some of the molecules in the contact center remain compressed during separation of the tip from the sample surface (Figure 2). Such an elastic repulsion would be governed by a semiharmonic potential contributing to the effective interaction potential (Figure 3). The ∆Gβ of the inner transition state would equal about 10 kBT (Table 1).

Figure 2. Schematic of the interactions between the AFM tip and a self-assembled monolayer of alkanethiols. The attractive van der Waals forces occur in the outer ring of the contact and repulsive elastic forces due to the pressed thiol moleculessin the contact center.

Figure 3. Conceptual interaction potential of the resultant adhesion between the AFM tip and a self-assembled monolayer of methylterminated alkanethiols. The external load tilts the potential and lowers the transition states. The inner transition state dominates at high loading rates (regime 2). It can result from the superposition of the attractive van der Waalse forces (expressed by Hamaker’s potential) and the repulsive elastic forces (expressed by semiharmonic potential) due to the pressed alkanethiol molecules under the apex of the tip.

ν)1 -2

0.9 × 10 6.5 × 10-14 >664 0.90

1.9 × 10-3 22 9.0 0.97

Bell-Evans model -2

0.9 × 10-2 2 × 10-13

0.90 0.8 × 10-3 340

0.90 0.8 × 10-3 610

0.91

0.90

0.9 × 10 10-15

However, our test measurement for the NH2-terminated 11alkanethiol (the same alkyl chain as in 1-dodecanethiol but different terminal group) SAM has not exhibited a two-regime character of the adhesion force-versus-loading rate dependence (data not shown). This suggests that it is not the elasticity of the alkyl chains responsible for the complex character of the adhesion force-loading rate curve but rather the interactions between the methyl groups and the AFM tip. An alternative to the “two-barriers” interpretation was proposed by Friddle et al.25 The authors suggest that the low loading rate regime can reflect near-equilibrium unbinding with possible rebinding processes. To check the probability of rebinding, we analyzed the shape of force curves during retraction at different loading rates (Figure 4). If a clear snap off from contact occurs, we can assume that the process is far from equilibrium and rebinding is unlikely. Such a situation happens in the case of force curves recorded at loading rates higher than 104 nN/s, which is below the threshold value (105 nN/s) of regime 2 (Figure 4B,C). Therefore, the rebinding can hardly be the explanation of the weak dependence of adhesion force-versus-loading rate in regime 1. Recording the cantilever relaxing oscillations, visible in Figure 4D, was possible thanks to the high sampling rate at high loading rates. At loading rates lower than 105 nN s-1, the sampling is not efficient enough and the signal of oscillations is averaged (Figure 4B,C). At loading rates lower than 103 nN s-1, the oscillations do not occur because the cantilever is in near-equilibrium position during separation and hence the effect of snap-off is insignificant (Figure 4A). We propose a different explanation of the two-regime character of the adhesion force-loading rate dependence, which is related to the multiplicity of the naoadhesion bond. Originally, the DHS model is valid for a single bond. The energy of transition state 1 is significantly higher than that of transition state 2 and resembles a covalent bond (Table 1, the ν ) 2/3 DHS model results). However, the extremely small value of the distance between the bound state and the transition state (0.014 Å) cannot correspond to a single, covalent bond and thus one can interpret it as due to multiple, parallel bonds. The case of transition state 2 is even clearer because not only the position of this transition state but also its low energy (one order lower than covalent bond energy) and high value of the rate constant of bond rupture (several orders higher than for a covalent bond) indicate that weak interactions are related to this transition state. Capillary forces caused by condensing water should be negligible, since thiol molecules with methyl terminal groups form a hydrophobic monolayer. Therefore, we can assume that the adhesion is mainly due to van der Waals interactions, particularly the London dispersion forces. The adhesion force-loading rate curve depends on the number of individual bonds in the effective adhesion bond and also their configuration and cooperativity during separation. Evans13 and later Williams26

21576

J. Phys. Chem. C, Vol. 114, No. 49, 2010

Ptak et al.

Figure 4. Representative force-distance curves recorded at loading rates of (A) 5 × 102 nN s-1, (B) 2.1 × 104 nN s-1, (C) 9.7 × 104 nN s-1, and (D) 4.5 × 105 nN s-1. The slope of the snap-off line corresponds to the cantilever spring constant. All force curves were recorded with the same cantilever. The oscillations visible in part D have been recorded due to high sampling rate at high loading rates.

analyzed theoretically the behavior of multiple bonds. Their models can be applied in our considerations. In the case of cooperative bonds configured in parallel, the effective barrier energy is given by the sum of individual barrier energies and the bonds act as a single macrobond. Then the equation for the adhesion (total) force versus loading rate can be written as13

( )

Fad ) Fβ ln N

rF

0 Fβkoff

+

(N - 1)E0 xβ

(7)

where N is the number of individual bonds. The adhesion force is determined by a rate-independent component. Therefore the long durability of the van der Waals adhesion related to transition state 1 could be explained by van der Waals forces acting as a macrobond. In the case of N independent bonds in parallel, the equation for adhesion force versus loading rate can be given as25

[ ( ) ( )]

Fad ) NFβ ln

rF

0 Fβkoff

- ln

Fad Fβ

( )

) NFs - NFβ ln

Fad Fβ (8)

where Fs is the force for a single bond. If the van der Waals bonds behave in this way at higher loading rates, it could explain the strong adhesion force-loading rate dependence in regime 2. Transformation of eq 8 into force distributed per individual bond gives

Fs )

( )

Fad Fad + Fβ ln N Fβ

(9)

Applying the DHS model to the adhesion force per individual molecule versus loading rate would make possible the extraction of kinetic and interaction potential parameters characterizing van der Waals interaction per one methyl group, i.e., an average individual van der Waals bond. The number of bonds in the adhesion contact can be estimated using the Johnson-Kendall-Roberts (JKR) model.27 In the JKR model the adhesive interaction between a sphere and the planar surface of homogeneous elastic half space is considered. The radius of the contact area at separation is given as

ac )

(

3πR2W 2Es

)

1/3

(10)

where R is the radius of tip curvature, W is the work of adhesion (per unit area), and Es is the elastic modulus of the alkanethiol monolayer on a gold substrate, assuming that the elastic modulus of the tip material is much higher. Different values of Es can be found in the literature depending on which method was used.28-30 Most values were obtained by compression of the monolayer under high load. Under such conditions the contribution of the underlying substrate is significant. The rupture of the adhesion contact, however, occurs at a much smaller negative load and thus the influence of the substrate should be negligible as well as the intermolecular interactions between neighboring alkylthiols. Therefore we assumed the value determined by DelRio et al.31 for 1-dodecanethiol SAM, i.e., 0.86 ( 0.14 GPa, to be most applicable to our situation.

Quantitative Analysis of Adhesion Forces

J. Phys. Chem. C, Vol. 114, No. 49, 2010 21577

The work of adhesion has been calculated from the adhesion force

W)

2Fad 3πR

(11)

The radii of the AFM tips were determined from SEM images as about 35 nm. The calculated work of adhesion, W ) 0.097 N m1-, is in good agreement with our previous estimations for methyl-terminated SAMs.20 The lattice constant of the hexagonal structure of thiol monolayer is 0.5 nm, and hence the area per molecule is 0.22 nm2. The contact area at snap off was calculated to be 230 nm2 for the adhesion force of 16 nN, which results in ∼1000 contacting molecules. Although it is a rough estimation, it can give information about the force applied to an individual molecule during a pulloff experiment. The adhesion force per alkylthiol molecule is roughly 15 pN, which seems to be a low but still reasonable value for van der Waals bonds. Such a calculation, however, is naı¨ve. The bond strength is dynamic and depends on the loading rate. Sharing the load between multiple bonds reduces the individual-bond loading rate and hence lowers the force at which each bond ruptures. Therefore a more complex calculation according to eq 9 should be performed. Taking the reduced, effective loading rate into account, the force per one molecule equals 35 pN at low loading rates, which is a typical value for an individual van der Waals bond.32 We have recalculated the adhesion force-versus-loading rate curve (Figure 1, regime 2) into the adhesion force per one molecule-versus-loading rate curve and performed fitting to the curve using the ν ) 2/3 DHS model (Figure 5). As a result, we obtained the following parameters of the interaction potential describing the interaction between an individual methyl group and silicon nitride tip: xβ 0 ) 1.50 ( 0.05 nm, koff ) 620 ( 20 s-1, and ∆Gβ ) (12 ( 1) kBT. Since the van der Waals potential has no transition states, xβ is an apparent quantity that depends on the potential shape.12 The value of ∆Gβ is slightly higher than typically reported for van der Waals dispersion forces. However, the strength of van der Waals forces depends on the size and shape of interacting molecules. The commonly presented, particularly in books of chemistry, values are valid rather for tiny atoms of noble gases than methyl groups, which can be much higher.32 Finally, we have verified the reliability of the calculated parameters by comparison of theoretically calculated values of

the effective adhesion force using eq 7 with experimentally obtained values of nanoadhesion force (Figure 1). Assuming that all the molecules in contact (∼1000) create van der Waals 0 bonds (characterized by the calculated parameters xβ, koff , and ∆Gβ) which behave cooperatively during rupture, the effective adhesion force should measure about 30 nN. The value is higher than the experimentally measured 15-18 nN in regime 1, but it should be treated rather as an upper limit, when all the molecules contribute to the nanoadhesion and all the van der Waals bonds behave perfectly cooperative. Taking this into consideration, we can conclude that the calculated potential parameters reliably characterize the dispersion van der Waals forces between a silicon nitride tip and a self-assembled monolayer of 1-dodecanethiol. Conclusions The common interpretations of the two-regime character of adhesion force-versus-loading rate curves, which assume the existence of an inner transition state in the interaction potential or the process of rebinding, were discussed and rejected in our analysis of the nanoadhesion between an AFM tip and a hydrophobic monolayer. The major reason of the steep adhesion force-versus-loading rate increase at higher loading rates is the decrease of cooperativity between individual van der Waals bonds responsible for the effective adhesion. We have also demonstrated a way to extract the kinetic and interaction potential parameters for an individual van der Waals bond using the microscopic model by Dudko et al.12 (the DHS model) instead of the commonly used phenomenological Bell-Evans model. Acknowledgment. The research was partially supported by the Polish Ministry of Science and Higher Education (MNiSW) within research projects No. 2859/B/H03/2008/34 and No. 2909/ B/H03/2010/38. H.G. acknowledges financial support within the project “Scholarship support for Ph.D. students specializing in majors strategic for Wielkopolska’s development”, Submeasure 8.2.2 Human Capital Operational Programme, cofinanced by European Union under the European Social Fund. Supporting Information Available: Description of the sample characterization, schematic and details of the modified AFM, and methodology of dynamic force spectroscopy experiments. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes

Figure 5. Adhesion force per single molecule-versus-loading rate in regime 2 for nanoadhesion between a silicon nitride tip and an 1-dodecanethiol SAM. The solid line is a result of the fitting with the ν ) 2/3 DHS model.

(1) Binning, G.; Quate, C. F.; Gerber, C. Phys. ReV. Lett. 1986, 56, 930. (2) Butt, H.-J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1. (3) Florin, E.-L.; Moy, V. T.; Gaub, H. E. Science 1994, 264, 415. (4) Ptak, A.; Takeda, S.; Nakamura, C.; Miyake, J.; Kageshima, M.; Jarvis, S. P.; Tokumoto, H. J. Appl. Phys. 2001, 90, 3095. (5) Kramers, H. A. Physica 1940, 7, 284. (6) Bell, G. I. Science 1978, 200, 618. (7) Evans, E.; Ritchie, K. Biohys. J. 1997, 72, 1541. (8) Hummer, G.; Szabo, A. Biophys. J. 2003, 85, 5. (9) Dudko, O. K.; Filippov, A. E.; Klafter, J.; Urbakh, M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 11378. (10) Cleveland, J. P.; Scha¨ffer, T. E.; Hansma, P. K. Phys. ReV. B. 1995, 52, R8692. (11) Willemsen, O. H.; Kuipers, L.; van der Werf, K. O.; de Grooth, B. G.; Greve, J. Langmuir 2000, 16, 4339. (12) Dudko, O. K.; Hummer, G.; Szabo, A. Phys. ReV. Lett. 2006, 96, 108101. (13) Evans, E. Annu. ReV. Biophys. Biomol. Struct. 2001, 30, 105. (14) Merkel, R.; Nassoy, P.; Leung, A.; Ritchie, K.; Evans, E. Nature 1999, 397, 50.

21578

J. Phys. Chem. C, Vol. 114, No. 49, 2010

(15) Strunz, T.; Oroszlan, K.; Shafer, R.; Guntherodt, H.-J. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 11277. (16) Neuert, G.; Albrecht, C.; Pamir, E.; Gaub, H. E. FEBS Lett. 2006, 580, 505. (17) Maki, T.; Kidoaki, S.; Usui, K.; Suzuki, H.; Ito, M.; Ito, F.; Hayashizaki, Y.; Matsuda, T. Langmuir 2007, 23, 2668. (18) Ptak, A.; Kappl, M.; Moreno-Flores, S.; Gojzewski, H.; Butt, H.J. Langmuir 2009, 25, 256. (19) Ptak, A.; Makowski, M.; Cichomski, M. Chem. Phys. Lett. 2010, 489, 54. (20) Gojzewski, H.; Kappl, M.; Ptak, A.; Butt, H.-J. Langmuir 2010, 26, 1837. (21) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1972. (22) Ptak, A.; Kappl, M.; Butt, H.-J. Appl. Phys. Lett. 2006, 88, 263109. (23) Hutter, J. L.; Bechhoefer, J. ReV. Sci. Instrum. 1993, 64, 3342. (24) Zepeda, S.; Yeh, Y.; Noy, A. Langmuir 2003, 19, 1457.

Ptak et al. (25) Friddle, R. W.; Podsiadlo, P.; Artyukhin, A. B.; Noy, A. J. Phys. Chem. C 2008, 112, 4986. (26) Williams, P. M. Anal. Chim. Acta 2003, 479, 107. (27) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301–313. (28) Leng, Y. S.; Jiang, S. Y. J. Chem. Phys. 2000, 113, 8800. (29) Burns, A. R.; Houston, J. E.; Carpick, R. W.; Michalske, T. A. Phys. ReV. Lett. 1999, 82, 1181. (30) Kiridena, W.; Jain, V.; Kuo, P. K.; Liu, G. Y. Surf. Interface Anal. 1997, 25, 383. (31) DelRio, F. W.; Jaye, C.; Fischer, D. A.; Cook, R. F. Appl. Phys. Lett. 2009, 94, 131909. (32) Clark, J. Calculations in Chemistry; Pearson Education: Upper Saddle River, NJ, 2000.

JP107948Q