Contact Mechanics Modeling of Pull-Off ... - ACS Publications

Anal. Chem. 2002, 74, 3096-3104

Contact Mechanics Modeling of Pull-Off Measurements: Effect of Solvent, Probe Radius, and Chemical Binding Probability on the Detection of Single-Bond Rupture Forces by Atomic Force Microscopy Hjalti Skulason and C. Daniel Frisbie*

Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455

Pull-off forces for chemically modified atomic force microscopy tips in contact with flat substrates coated with receptor molecules are calculated using a Johnson, Kendall, and Roberts contact mechanics model. The expression for the work of adhesion is modified to account for the formation of discrete numbers of chemical bonds (nBonds) between the tip and substrate. The model predicts that the pull-off force scales as nBonds1/2, which differs from a common assumption that the pull-off force scales linearly with nBonds. Periodic peak progressions are observed in histograms generated from hundreds of computed pull-off forces. The histogram periodicity is the signature of discrete chemical interactions between the tip and substrate and allows estimation of single-bond rupture forces. The effects of solvent, probe tip radius, and chemical binding probability on the detection of single-bond forces are examined systematically. A dimensionless parameter, the effective force resolution, is introduced that serves as a quantitative predictor for determining when periodicity in force histograms can occur. The output of model is compared to recent experimental results involving tips and substrates modified with self-assembled monolayers. An advantage of this contact mechanics approach is that it allows straightforward estimation of solvent effects on pull-off forces. This paper presents a contact mechanics model, based on the Johnson, Kendall, and Roberts (JKR) formalism,1 for predicting when forces due to discrete chemical bonds can be detected in microcontact pull-off experiments. Our focus is on pull-off measurements in which a chemically modified atomic force microscopy (AFM) probe tip contacts a flat substrate decorated with complementary molecules (Scheme 1). Upon contact, molecules on the tip and substrate can bind to each other. Because of the sharpness of the tip, the contact area is on the order of a few square nanometers, so that the junction involves a small integer number of bonds. Fluctuations in the number of discrete bonds * Address correspondence to this author. E-mail: [email protected]. (1) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A 1971, 324, 301.

3096 Analytical Chemistry, Vol. 74, No. 13, July 1, 2002

Scheme 1. Microcontact Pull-Off Experiment in Which an AFM Probe and Substrate Are Coated with Complementary Molecules Capable of Forming Chemical Bonds Across the Probe-Substrate Interface

formed in consecutive pull-off measurements (in which the adhesive contact between the tip and substrate is made and broken multiple times) give rise to a distribution of pull-off forces. In principle, this distribution can be analyzed to determine the “force quantum” required to rupture single chemical bonds. In practice, the extraction of single-bond rupture forces from microcontact pull-off measurements is challenging. The goal of this work is to present a model that identifies the key experimental requirements necessary to achieve single-bond sensitivity in AFM pull-off measurements. AFM studies of molecular adhesion have been extensively reported in the literature,2 and it is well known that pull-off forces depend on the chemical functionality present on the tip and substrate.3 Path-breaking studies in the 1990s established that measurements involving large biological macromolecules grafted to AFM tips or substrates can detect single-bond rupture forces.2a,k,m,p,q For example, in 1994, Gaub and co-workers reported studies in which a biotin-modified tip was brought into contact with avidin-coated polystyrene spheres.2q Histograms showing the frequency of specific rupture forces for hundreds of consecutive contacts displayed a periodic peak progression ascribed to rupture of discrete biotin-avidin complexes. Based on these data, they reported a mechanical rupture strength of 115 pN for a single complex. Subsequently, the binding forces associated with other biological adducts, such as single DNA duplexes2p and antibodyantigen complexes,2a,k,m have been measured. AFM pull-off mea10.1021/ac020075g CCC: $22.00

© 2002 American Chemical Society Published on Web 05/18/2002

surements are a productive strategy for examining biological binding forces. In contrast, there have been very few reports of single-bond detection in pull-off experiments involving AFM tips and substrates modified with small nonbiological molecules capable of simple intermolecular-bonding interactions (e.g., hydrogen bonding, charge-transfer complexing). For example, direct detection of single bonds in rupture force histograms has not been reported for experiments involving tips and substrates modified with functionalized self-assembled monolayers (SAMs).4 Yet SAMmodified tips and substrates appear to be ideal systems for examining binding forces associated with simple bonding interactions because the exposed functional groups can be changed easily and systematically. It is also already known that the pull-off forces for SAM-modified tips are sensitive to the surface energies of the SAMs,3 meaning that the functional groups exposed on the tips and substrates do affect the measured rupture force. Why then is it difficult to measure single-bond forces between SAM-modified tips and substrates? There are many possible reasons for this, but perhaps the foremost is the role of solvent in the measurements. Most AFM pull-off measurements are performed under solvent to eliminate the effects of capillary forces due to water condensation on the substrate. However, the presence of solvent can also dramatically affect the measured pulloff forces. Solvents having large interfacial tensions prefer to minimize their contact area with the substrate and tip. Exclusion of solvent from the tip-substrate contact is therefore favorable and increases the work of adhesion.3e Solvent exclusion is a thermodynamic driving force for adhesion and is often essentially entropic in nature, analogous to depletion forces in polymer and colloidal science.5 The difficulty with solvent exclusion, in terms (2) (a) Harada, Y.; Kuroda, M.; Ishida, A. Langmuir 2000, 16, 708. (b) Kidoaki, S.; Matsuda, T. Langmuir 1999, 15, 7639. (c) Ong, Y.-L.; Razatos, A.; Georgiou, G.; Sharma, M. M. Langmuir 1999, 15, 2719. (d) Ito, T.; Citterio, D.; Bu ¨ hlmann, P.; Umezawa, Y. Langmuir 1999, 15, 2788. (e) McKendry, R. A.; Theoclitou, M.-E.; Rayment, T.; Abell, C. Nature 1998, 391, 566. (f) Marszalek, P. E.; Oberhauser, A. F.; Pang, Y.-P.; Fernandez, J. M. Nature 1998, 396, 661. (g) Oberhauser, A. F.; Marszalek, P. E.; Erickson, H. P.; Fernandez J. M. Nature 1998, 393, 181. (h) Rief, M.; Gautel, M.; Oesterhelt, F.; Fernandez, J. M.; Gaub, H. E. Science 1997, 276, 1109. (i) Ikai, A.; Mitsui, K.; Furutani, Y.; Hara, M.; McMurty, J.; Wong, K. P. Jpn. J. Appl. Phys. 1997, 36, 3887. (j) Green, J.-B. D.; McDermott, M. T.; Porter, M. D. J. Phys. Chem. 1996, 100, 13342. (k) Dammer, U.; Hegner, M.; Anselmetti, D.; Wagner, P.; Dreier, M.; Huber, W.; Gu ¨ ntherodt, H.-J. Biophys. J. 1996, 70, 2437. (l) Mitsui, K.; Hara, M.; Ikai, A. FEBS Lett. 1996, 385, 29. (m) Hinterdorfer, P.; Baumgartner, W.; Gruber, H. J.; Schilcher, K.; Schindler, H. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 3477. (n) Lee, G. U.; Kidwell, D. A.; Colton, R. J. Langmuir 1994, 10, 354. (o) Radmacher, M.; Fritz, M.; Cleveland, J. P.; Walters, D. A. Hansma, P. K. Langmuir 1994, 10, 3809. (p) Lee, G. U.; Chrisey, L. A.; Colton, R. J. Science 1994, 266, 771. (q) Florin, E.-L.; Moy, V. T.; Gaub, H. E. Science 1994, 264, 415. (3) (a) Smith, D. A.; Wallwork, M. L.; Zhang, J.; Kirkham, J.; Robinson, C.; Marsh, A.; Wong, M. J. Phys. Chem. B 2000, 104, 8862. (b) Noy, A.; Vezenov, D. V.; Lieber, C. M. Annu. Rev. Sci. 1997, 27, 381. (c) van der Vegte, E. W.; Hadziioannou, G. Langmuir 1997, 13, 4357. (d) Vezenov, D. V.; Noy, A.; Rozsnyai, L. F.; Lieber, C. M. J. Am. Chem. Soc. 1997, 119, 2006. (e) Sinniah, S. K.; Steel, A. B.; Miller, C. J.; Reutt-Robey, J. E. J. Am. Chem. Soc. 1996, 118, 8925. (f) Noy, A.; Frisbie, C. D.; Rozsnyai, L. F.; Wrighton, M. S.; Lieber, C. M. J. Am. Chem. Soc. 1995, 117, 7943. (g) Thomas, R. C.; Houston, J. E.; Crooks, R. M.; Kim, T.; Michalske, R. A. J. Am. Chem. Soc. 1995, 117, 3830. (4) By direct detection of single bonds we mean observation of an evenly spaced peak progression in a histogram of rupture forces. Beebe,8a for example, has reported a statistical approach to extracting single-bond forces from rupture force histograms that do not show periodic peak progressions. (5) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1992.

of single-bond detection in pull-off measurements, is that it masks the contribution of the chemical bonding between the tip and substrate to the total work of adhesion. To detect fluctuations in the number of discrete bonds formed, the pull-off forces must be sensitive to the tip-substrate chemical bonding, characterized by the interfacial energy. If the magnitude of the solvent interfacial energy overwhelms the tip-substrate interfacial energy, detecting single-bond forces is not possible. Pull-off measurements that involve large macromolecules circumvent this problem because as the molecules are stretched, the tip and substrate separate, the contact area effectively drops to zero (there are only a few molecular strands connecting the two surfaces), and the contribution of the solvent to the adhesion therefore becomes negligible.6 Pull-off measurements that involve a finite tip-substrate contact area at the point of rupture, however, are naturally susceptible to solvent exclusion effects. Pull-off experiments between SAM-modified tips and substrates, as shown in Scheme 1, fall into this latter category. We previously emphasized the need for large, negative interfacial energy between the tip and substrate and low solvent surface tensions to detect single bonds in pull-off measurements.7 However, there are other factors, such as tip radius and bond formation probability, that clearly should affect the ability to detect single-bond forces. Here we present a simple contact mechanics model that allows us to calculate pull-off forces and thus to examine systematically the role of solvent surface tension, tip radius, and bonding probability in our ability to detect single-bond force quanta in pull-off force histograms. Computer simulations of AFM pull-off measurements have been reported previously,8 but to our knowledge, none of these previous studies has been based on a contact mechanics formalism that allows the effect of solvent to be introduced straightforwardly. The Model. The interaction of an AFM tip with a surface can be approximated as the interaction of a sphere with a flat. We are interested in the adhesion of SAM-coated spheres under solvent. The spheres have radii of roughly 40 nm, comparable to typical AFM tip radii. The contact areas are on the order of a few square nanometers at pull-off, depending on the details of the various interfacial energies and the elastic moduli of the sphere and substrate. For a molecular surface coverage of 4 × 10-10 mol/ cm2, a contact area of 4 nm2 corresponds to ∼10 interacting molecules, so it is clear that discrete fluctuations in the pull-off force are observable in principle. A key feature of our model is that we take the probe substrate interfacial energy, γProbe-Substrate, to be a function of the number of discrete bonds, as discussed below. (6) (a) Garnier, L.; Gauthier-Manuel, B.; van der Vegte, E. W.; Snijders, J.; Hadziioannou, G. J. Chem. Phys. 2000, 113, 2497. (b) Grandbois, M.; Beyer, M.; Rief, M.; Clausen-Schaumann, H.; Gaub, H. E. Science 1999, 283, 1727. (c) Ikai, A.; Mitsui, K.; Furutani, Y.; Hara, M.; McMurty, J.; Wong, K. P. Jpn. J. Appl. Phys. 1997, 36, 3887. (7) (a) Skulason, H.; Frisbie, C. D. J. Am. Chem. Soc. 2000, 122, 9750. (b) Skulason, H.; Frisbie, C. D. Langmuir 2000, 16, 6294. (8) (a) Stevens, F.; Lo, Y.-S.; Harris, J. M.; Beebe, T. P., Jr. Langmuir 1999, 15, 207. (b) Vijayendran, R.; Hammer, D.; Leckband, D. J. Chem. Phys. 1998, 108, 7783. (c) Attard, P.; Schulz, J. C.; Rutland, M. W. Rev. Sci. Instrum. 1998, 69, 3852. (d) Evans, E.; Ritchie, K. Biophys. J. 1997, 72, 1541. (e) Izrailev, S.; Stepaniants, S.; Balsera, M.; Oono, Y.; Schulten, K. Biophys. J. 1997, 72, 1568. (f) Unger, M. A.; O’Connor, S. D.; Baldeschwieler, J. D. J. Vac. Sci. Technol. B 1996, 14, 1302. (g) Girard, C.; Van Labeke, D.; Vigoureux, J. M. Phys. Rev. B 1989, 40, 12133.

Analytical Chemistry, Vol. 74, No. 13, July 1, 2002

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Figure 1. Total interfacial energies (γA) and total work of adhesion (WAdA) as a function of contact area (A) for two objects capable of forming discrete chemical bonds between each other while in contact. Since γSolventA is not influenced by the formation of chemical bonds, it has a simple linear dependence on A with slope γSolvent (dashed line). In comparison, γProbe-SubstrateA increases in discrete steps with increasing A (dotted line), representing the formation of integer number of chemical bonds. The average slope of γProbe-SubstrateA (gray solid line), represented by 〈γProbe-Substrate〉, can be used for comparison with γSolvent. The total interfacial energy, WAdA, is the sum of the two interfacial energy terms. Due to the staircase shape of γProbe-SubstrateA vs A, WAdA can only take on values corresponding to its sloped parts, extrapolated in gray toward the y-axis.

combined effect on the total work, WAdA, to separate the tip and substrate is also shown in Figure 1. The WAdA term incorporates both the stepwise increases originating from γProbe-Substrate and the constant slope from γSolvent. As A can take on any value, possible values for WAdA are represented by the intersection of the gray areas (originating from the sloped segments) with the ordinate of the Figure 1 plot. Importantly, there are values for WAdA that cannot be obtained.9 Those values are associated with the discrete increases arising from the addition of single chemical bonds and are represented by WAdA values spanning the white regions. Our model for the pull-off force is based on the JKR formalism for mechanical contact between a sphere and a flat surface.1 The JKR model balances the interfacial adhesive energy and the energy stored in the elastic deformation of the objects. For a sphere of radius R (m) in adhesive contact with a flat substrate, the JKR model gives the pull-off force required to rupture the contact as

FRupture ) 3πRWAd/2

(2)

and the radius of the contact area at rupture, a (m), as

a3 ) 3πR2WAd/2K

(3)

Work of adhesion is defined as the free energy change per unit area associated with separating two objects from contact to infinity. For an AFM probe and a substrate submerged in a solvent, WAd is defined as

where K (Pa) is the mean elastic modulus of the contact.10 Equation 2 can be rewritten to show the explicit contact area dependence by using eq 3 to eliminate R, giving

WAd ) γSubstrate-Solvent + γProbe-Solvent - γProbe-Substrate

FRupture ) xWAdA × x3Ka/2

) γSolvent - γProbe-Substrate

(1)

where γSubstrate-Solvent (J/m2) and γProbe-Solvent (J/m2) are the interfacial free energies of the solvent with the substrate and the probe and γProbe -Substrate is the interfacial free energy between the probe and the substrate. For convenience, we define a new term, γSolvent, which combines the two terms involving the solvent interface. The total free energy change associated with exposing a contact area of size A (m2) to the medium is therefore WAdA (J). If discrete chemical bonds are formed between the probe and the substrate, then small changes in A cause variation in the integer number of bonds formed across the interface. Assuming that the number of chemical bonds formed increases with regular intervals as A increases, then a plot of the absolute value of the interfacial energy for a given area (|γProbe-Substrate|A vs A) yields the step function shown in Figure 1. The horizontal distance between each step is the area per chemical bond at the interface, and the step height is the single-bond contribution to the interfacial energy. The average slope of the trace, 〈γProbe-Substrate〉, is then the conventional interfacial energy per unit area, which can be compared directly to the surface-solvent interfacial energies (per unit area). We have assumed here that all bonds have identical energies, i.e., all steps in the Figure 1 plot have the same height. Our model also assumes the surface-solvent interfacial energy is a smooth function of contact area so that a plot of γSolventA versus A is simply a straight line with slope γSolvent (Figure 1). The 3098


(4)

where A ) πa2 is the area of the contact at rupture. Equation 4 shows that the rupture force does not scale linearly with the free energy change, WAdA. Because of the elastic energy of deformation that is recovered upon pull-off, FRupture scales with the square root of the free energy change.11 This has some important implications for the detection of single-bond forces, as discussed below. Substituting eq 1 for WAd into eq 4 and rearranging gives

FRupture ) xγSolventA - γProbe-SubstrateAx3Ka/2

(5)

The maximum number of bonds that can be formed within contact area A is then given by

nMax-bonds ) A/σ

(6)

(9) This is strictly true only when the probability for bond formation is 100%. We discuss the role of bond probability later. (10) K is given as K ) 4((1 - ν12)/E1 + (1 - ν22)/E2)-1/3, where E and ν are the Young’s modulus and Poisson ratio of the two objects, respectively. (11) The scaling of FRupture with (WAd)1/2 is a general result that also appears in the theory of brittle fracture (see, for example: Dieter, G. E. Mechanical Metallurgy, 3rd ed.; McGraw-Hill, New York, 1986; pp 246-249). The scaling is a consequence of two factors. First, the elastic strain energy stored in the tip and substrate is proportional to the square of the load stress (σ2). Second, at pull-off the elastic energy loss (upon recovery) is just balanced by the surface energy gain (γ). Thus, σ2 ∝ γ at pull-off, so σ ∝ γ1/2, and by analogy FRupture ∝ (WAd)1/2. Alternatively, eq 4 can be manipulated by dividing both sides by the contact area πa2 and defining σRupture ) FRupture/πa2, to obtain σRupture ) (3WAdK/2πa)1/2, which has nearly the same form as Griffith criterion for brittle fracture.

where σ is the area per bond (the inverse of the molecular surface coverage).12 However, every bond within the contact area may not necessarily form; the actual number of bonds formed, nBonds, can be less than nMax-bonds. To account for this, the bonds are assigned a formation probability, p (0 e p e 1). To calculate nBonds for each value of the contact area A (and thus nMax-bonds), we define a Bernoulli random variable, Xi, for each possible bond within the contact area as

separation (the force quantum, fquantum) and the width of the peaks (fwidth) in the rupture force histogram. To be able to detect force quanta in a histogram, ξ must be larger than 1 (the force quantum must be larger than the width of each peak). The force quantization is the increase in force associated with increasing the contact area by σ and thus forming one more bond. From eqs 5 and 9 we see that the force quantum, fquantum, scales as14

P{Xi ) 0} ) 1 - p

fquantum ∝ x〈WAd〉σ

P{Xi ) 1} ) p

(7)

where Xi ) 0 signifies that a bond number i is not formed and Xi ) 1 signifies that a bond i is formed. The actual number of bonds formed is therefore

The width of the force quantum stems from variations in the contact area. For p ) 1, the maximum the contact area can change without adding another bond is σ. The width of each force quantum, fwidth, scales therefore as

fwidth ∝ xγSolventσ

nMax-bonds

nBonds )

∑

Xi

which by definition makes nBonds a binomial random variable. The area covered by nBonds number of bonds is therefore nBondsσ. Substituting these variables into eq 5 above yields the working equation of the model

However, when p < 1, there is an additional spread in force values due to bonds that are not formed and contribute therefore only to the γSolvent term in eq 9. The number of bonds that are not formed (nMax-bonds - nBonds) can vary considerably between pulloffs having the same nMax-bonds. We can approximate this number using the standard deviation of nBonds (a binomial random variable) around its maximum (nMax-bonds) which is given by

(9)

This equation allows us to assess the effect of variations in contact area on the pull-off force. Furthermore, eq 9 emphasizes the nonlinearity associated with the contact mechanics of elastic objects. It shows that FRupture does not depend linearly on the number of bonds formed across the probe-substrate interface but instead scales as nBonds1/2. This scaling prediction is a direct consequence of the relationship between FRupture and the interfacial free energy change predicted by JKR (see eq 4) and is an important distinction between the contact mechanics viewpoint presented here and the assumption of other workers.8a For each simulation, specific but realistic values are assigned to the variables R, K, γSolvent, 〈γProbe-Substrate〉, p, and σ. These assignments determine both the stepwise increasing variable γProbe-SubstrateA (vs A) and the effective work of adhesion, 〈WAd〉 (via eq 1). A Gaussian distribution of 250 contact area values is generated with an average value of 〈A〉, given by eq 3, and an arbitrary but realistic standard deviation, sA. After the maximum number of bonds is determined for each value of A using eq 6, the actual number of bonds formed is calculated using eq 8 and the rupture force is calculated using eq 9. All calculations were performed in Igor Pro (Wavemetrics, Lake Oswego, OR). Histograms of FRupture values were generated using a bin width of 15 pN, a value close to the average resolution of common AFM measurements. Resolving Force Quanta. It is advantageous to define a single parameter that indicates whether observation of discrete interactions in rupture force histograms is expected. We define the effective force resolution,13 ξ, as the ratio between the peak (12) The division by σ denotes an integer division.

(11)

(8)

i)1

FRupture ) xγSolventA - 〈γProbe-Substrate〉nBondsσx3Ka/2

(10)

sn ) xnMax-bonds(1 - p)

(12)

Then eq 11 becomes

fwidth ∝

xγ

Solventσ(1

+ xnMax-bonds(1 - p))

(13)

The effective force resolution is defined as

ξ)

fquantum ) fwidth

x〈WAd〉σ

xγ

Solventσ(1

+ xnMax-bonds(1 - p))

(14)

which after rearrangement becomes

ξ)

x

〈WAd〉 γSolvent (1 +

1

xnMax-bonds(1 - p))1/2

(15)

It is worth emphasizing the main properties of eqs 10, 13, and 15. The bar graph in Figure 2A shows fquantum and fwidth as a function of nMax-bonds.15 As the figure demonstrates, the force quantum for each added bond becomes smaller as the total number of bonds (13) The effective force resolution is distinguished from the instrument resolution. Instrument resolution refers to the inherent force sensitivity of the AFM, typically ∼15 pN for a commercial instrument. (14) Where 〈WAd〉 is given as 〈WAd〉 ) γSolvent - 〈γProbe-Substrate〉. (15) The exact values for fquantum and fwidth can be calculated from the differential increase in FRupture (eq 9) as A and nBonds are increased alternately in steps of σ and 1, respectively. Other variables used were as follows: γSolvent ) 5 mJ/m2, γProbe-Substrate ) -5 mJ/m2, σ ) 0.5 nm2, and K ) 25 GPa.


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Figure 2. (A) Bar graph showing fquantum and fwidth (with p ) 1.0, 0.9, and 0.8) for each bond as a function of the number of bonds ruptured calculated using eq 9 and variables given in ref 14. The smaller p is, the lower nMax-bonds becomes for which fwidth > fquantum. (B) Effective force resolution, ξ (in units of (WAd/γSolvent)1/2 ), as a function of p for various values of nMax-bonds.

ruptured gets larger. This is a direct consequence of the fact that FRupture scales as n1/2 bonds. For p ) 1, the width of the force quantum (fwidth) decreases at the same rate, so that increasing nMax-bonds has no effect on the ratio of fquantum to fwidth (i.e., ξ is constant and equal to (WAd/γSolvent)1/2). For p < 1, fwidth is larger by the scaling factor in parentheses given in eq 13. A detailed comparison of fquantum and fwidth in Figure 2A shows that, for p ) 1, fquantum > fwidth for all values of nMax-bonds, while for p ) 0.9, fwidth > fquantum at nMax-bonds ) 17 (not shown), and for p ) 0.8, this crossover happens at nMax-bonds ) 10. Thus, detection of singlebond rupture forces in pull-off measurements places limits on the size of nMax-bonds when p < 1. To get a better understanding of how the effective force resolution, ξ, is affected by p and nMax-bonds, Figure 2B shows ξ (in units of (WAd/γSolvent)1/2) plotted as a function of p for various values of nMax-bonds. As the figure shows, ξ decreases sharply as p decreases from 1 to 0.98, but decreases almost linearly with p below 0.95. Furthermore, as nMax-bonds increases, ξ decreases, lowering the likelihood of observing quantized rupture forces. Although the size of nMax-bonds depends on multiple variables, an obvious manifestation of the effect of nMax-bonds on resolution is a restriction on the probe radius. When p < 1, the AFM probe must be small in order to detect the difference in rupture forces between integer numbers of chemical bonds. Equation 15 also shows that for ξ > 1, 〈WAd〉 > γSolvent. Referring to eq 1, this means that 〈γProbe-Substrate〉 must be negative, a point we stressed previously.7 RESULTS AND DISCUSSION Effect of γSolvent. Our first objective was to evaluate the effect of γSolvent on the ability to differentiate between ruptures of integer numbers of bonds formed across the AFM probe-substrate 3100 Analytical Chemistry, Vol. 74, No. 13, July 1, 2002

Figure 3. Effect increasing γSolvent (dashed line) has on WAdA (bold line) while keeping γProbe-SubstrateA (thin line) the same. As γSolvent increases from 0 (A) to 5 mJ/m2 (B) and further to 10 mJ/m2 (C), the relative contribution of γProbe-SubstrateA to WAdA decreases. Here γProbe-SubstrateA is defined by 〈γProbe-Substrate〉 ) -5 mJ/m2 and bond area of 0.5 nm2/bond.

interface. Thus, in the first set of calculations, γSolvent was varied while keeping other variables constant. In our calculations, we used R ) 33 nm, K ) 25 GPa, and 〈γProbe-Substrate〉 ) -5 mJ/m2 with a bond surface area of 0.5 nm2/bond, values that are close to those commonly encountered in AFM force measurements using SAM-modified substrates. We examined three scenarios with γSolvent values of 0, 5, and 10 mJ/m2. The first scenario with γSolvent ) 0 mJ/m2 represents an ideal case in which the measured rupture force is due purely to discrete probe-substrate interactions. In the two other scenarios, the γSolvent term is equal to, and twice as large as, the |〈γProbe-Substrate〉| term, respectively. The effect on WAdA is shown in Figure 3 which shows the contact area dependence of the total surface free energies (γA) and WAdA for the three different values of γSolvent. With increasing γSolvent, the effect of the stepwise increases in WAdA originating from γProbe-Substrate becomes less significant. For each γSolvent value, the average contact area, 〈A〉, and average rupture force, 〈FRupture〉, can be calculated using eqs 2 and 3, respectively. The results are shown in Table 1 along with other variables used in the simulation. Figure 4 shows three histograms of 250 random contact area values, each generated following a normal distribution consistent with variables 〈A〉 and sA shown in Table 1. Figure 5 shows the histograms of rupture forces calculated using eq 9 and the corresponding contact area values shown in Figure 4. The histogram in Figure 5A represents the ideal case where the adhesion stems completely from the discrete interaction between the probe and the substrate. The histogram consists of individual peaks, each representing a single FRupture value corresponding to an integer number of bonds formed. The autocorrelation of the histogram, shown in the inset to Figure 5A, highlights the periodicity of the peaks. As the γSolvent term increases, shown in Figure 5B and C, the mean rupture force increases and the individual peaks broaden. The broadening reflects a distribution

Table 1. Overview of the Variables Used in the Three Sets of Calculations (Results Shown in Figure 5) 〈γProbe-Substrate〉 (mJ/m2)

γSolvent (mJ/m2)

p

〈WAd〉 (mJ/m2)

R (nm)

〈A〉 (nm2)

sA (nm2)

〈F〉 (nN)

ξ

-5 -5 -5

0 5 10

1 1 1

5 10 15

33 33 33

3.2 5.1 6.6

0.6 0.85 1.0

0.8 1.6 2.3

Inf 1.41 1.22

Figure 4. Histograms of randomly generated contact areas using variables given in Table 1. Increasing γSolvent and thus increasing 〈WAd〉 leads to a larger contact area. The average contact areas are 3.2 (A), 5.1 (B), and 6.6 nm2 (C).

of contact area values that yield the same number of discrete bonds but slightly varying rupture force due to the finite size of γSolvent. Peak broadening is reflected in ξ (shown in Table 1), which decreases from infinity (ideal case) to 1.41 and 1.22 in Figure 5B and C, respectively. There are two other, more subtle but important observations associated with Figure 5. First, the peaks in Figure 5 are not evenly spaced; the peak separation decreases as FRupture increases. This is a direct consequence of the fact that FRupture scales as nBonds1/2 (or equivalently as (WAdA)1/2; compare eqs 4 and 9). Thus, the change in FRupture between breaking 4 and 5 bonds is not the same as the change for breaking 9 and 10 bonds, for example. However, the changes in the peak separation in the histograms in Figure 5 are hard to detect; the autocorrelation returns an “average” periodicity (i.e., an average force quantum) for each histogram. The second observation is that the “mean” peak separation increases with increasing γSolvent. This is a consequence of the solvent contribution to the step height changes in WAdA (Figure 3). Detailed comparison of the three histograms and autocorrelations in Figure 5 shows that the mean peak separation increases by 10% between Figure 5A and C. We are thus left with an interesting conclusion about the detection of discrete bonds in microcontact pull-off measurements, namely, that the size of force quanta is not fixed, but depends on the contact area and the solvent. The contact area dependence might suggest that the force quanta would be washed out in pulloff force histograms. However, we find that, for the parameters

Figure 5. Histograms of rupture forces calculated using eq 9 and the contact area values shown in Figure 4 and the variables in Table 1. The rupture forces in the three histograms were calculated using γSolvent values of 0 (A), 5 (B), and 10 mJ/m2 (C). As γSolvent increases, the average rupture force increases and the evenly spaced peaks in the histograms become broader. The insets show the sinusoidal autocorrelation function of the respective histogram that emphasizes the periodicity of the rupture force.

used to generate the Figure 5 data, the autocorrelation procedure easily detects a periodicity due to the rupture of discrete bonds. Furthermore, the calculations show that γSolvent values in excess of 30 mJ/m2 (a very high value) are needed to completely wipe out periodic structure in the autocorrelations. Effect of Binding Probability. The histograms in Figure 5 demonstrate that although the solvent interfacial energy affects the ability to distinguish between the rupture of n and n + 1 bonds, some additional factors must be involved that would cause a complete disappearance of peaks from the histogram. One important consideration is the binding probability of each bond within the contact area. It is likely that every possible bond is not formed due to incorrect orientation of the participating functional groups, for example. This can be accounted for by assigning a fixed probability, p, for the formation of each bond within the contact area.16 We compared three scenarios with binding probabilities of 1.0, 0.96, and 0.92 for each bond inside the contact area. The system is otherwise the same as the one shown in Figures 3B and 5B and is described in Table 2.17 Panels A-C of Figure 6 show three histograms of rupture forces corresponding to binding probabilities of 1.0, 0.96, and 0.92, (16) This probability is independent of the contact area, which may not be strictly accurate, as one could imagine that certain conditions that lead to large contact areas, such as surface roughness, could somehow affect p.


3101


γSolvent (mJ/m2)

p

〈WAd〉 (mJ/m2)

R (nm)

〈A〉 (nm2)

sA (nm2)

〈F〉 (nN)

ξ

-5 -5 -5

5 5 5

1 0.96 0.92

10 9.8 9.6

33 33 33

5.1 5.0 4.9

0.85 0.85 0.85

1.6 1.5 1.5

1.41 1.10 1.02

Figure 6. Histograms of rupture forces calculated using eq 9 and the variables in Table 2. The rupture forces in three histograms were calculated using p values of 1.0 (A), 0.96 (B), and 0.92 (C). The evenly spaced peaks in the histograms start to overlap as p decreases and have completely disappeared for p ) 0.92 (C). This is emphasized in the autocorrelation function that does not have the characteristic sinusoidal shape.

respectively, with |〈γProbe-Substrate〉| ) γSolvent ) 5 mJ/m2. A dramatic difference is observed between the three histograms with respect to resolving individual peaks corresponding to discrete interactions. While the peaks are completely resolved in the histogram in Figure 6A, they have started to overlap in Figure 6B and have completely disappeared in Figure 6C. This may at first seem somewhat surprising given that this occurs at binding probabilities larger than 0.9. However, a binding probability of 0.9 means that if a contact area covers 10 bonds, on average all the bonds will form only 35% of the time, 39% of the time 9 of them will form, and 19% of the time 8 of them will form, causing a considerable spread in rupture forces originating from the same contact area. The resolution of individual peaks in the histograms in Figure 6 coincides with the corresponding ξ values, shown in Table 2. With ξ ) 1.0, no peaks are observed in the histogram in Figure 6c. As the histograms in Figure 6 demonstrate, binding probabilities slightly lower than 1 can cause peaks in rupture force histograms to vanish completely. However, it is not the binding probability alone that causes this, but rather the combination of p and γSolvent. This can easily be seen by comparing calculations with p < 1 and varying γSolvent. Figure 7 shows histograms of (17) A side effect of the lowered binding probability is that the effective 〈γProbe-Substrate〉 scales with p, which results in slightly smaller 〈WAd〉 and smaller 〈FRupture〉.


Figure 7. Histograms of rupture forces calculated using eq 9 and the variables in Table 3. The rupture forces in three histograms were calculated using p ) 0.94 and γSolvent as 0 (A), 5 (B), and 10 mJ/m2 (C). The effect of γSolvent on the resolution of the evenly spaced peaks is amplified when p < 1, emphasizing constructing effect of the two variables.

rupture forces from three calculations summarized in Table 3. In these calculations p and 〈γProbe-Substrate〉 were kept constant at 0.94 and -5 mJ/m2, respectively, while γSolvent was varied from 0 to 10 mJ/m2. With γSolvent equal in magnitude to 〈γProbe-Substrate〉, the evenly spaced peaks in Figure 7B are clearly visible although they have started to overlap. But with γSolvent twice as large, shown in Figure 7C, the evenly spaced peaks have completely disappeared from the histogram, as is emphasized in the complicated autocorrelation function shown in the inset. These results confirm that the interplay between p and γSolvent leads to the disappearance of peaks from discrete interactions. The larger γSolvent is compared to |〈γProbe-Substrate〉|, the higher p needs to be in order for singlebond rupture forces to be determined successfully. Similarly, the lower p is, the smaller γSolvent must be in comparison to |〈γProbe-Substrate〉|. Effect of Probe Radius. Figure 8 shows histograms of rupture forces calculated using the values shown in Table 4. With a 10nm probe radius, the histogram in Figure 8A shows three welldefined peaks, representing rupture of two, three, and four chemical bonds. With a probe radius of 33 nm (Figure 8B) the average rupture force increases from 0.5 to 1.5 nN and the peaks, although still visible, have started to overlap. Figure 8C shows the histogram obtained with a probe radius of 100 nm. The peak progression that is observed in Figure 8A and B is no longer apparent, as emphasized in the irregular autocorrelation function


γSolvent (mJ/m2)

p

〈WAd〉 (mJ/m2)

R (nm)

〈A〉 (nm2)

sA (nm2)

〈F〉 (nN)

ξ

-5 -5 -5

0 5 10

0.94 0.94 0.94

4.7 9.7 14.7

33 33 33

3.1 5.0 6.6

0.6 0.85 1.0

0.7 1.5 2.3

Inf 1.06 0.88


γSolvent (mJ/m2)

p

〈WAd〉 (mJ/m2)

R (nm)

〈A〉 (nm2)

sA (nm2)

〈F〉 (nN)

ξ

-5 -5 -5

5 5 5

0.98 0.98 0.98

9.9 9.9 9.9

10 33 100

1.0 5.0 22

0.3 0.85 3

0.5 1.5 4.7

1.28 1.17 1.01

Table 5. Overview of the Variables Used in the Two Sets of Calculations (Results Shown in Figure 9)a 〈γProbe-Substrate〉 (mJ/m2)

γSolvent (mJ/m2)

p

〈WAd〉 (mJ/m2)

R (nm)

〈A〉 (nm2)

sA (nm2)

〈F〉 (nN)

ξ

-6 -6.45

9.8 9.8

1 0.93

15.8 15.8

33 33

3.7 3.7

0.45 0.45

2.43 2.43

1.27 0.90

a

Other variables were K ) 64 GPa and molecular area of 0.25 nm2/bond, which along with the above variables were obtained from ref 3c.

Figure 8. Histograms of rupture forces calculated using eq 9 and the variables in Table 4. The rupture forces in three histograms were calculated using R values of 10 (A), 33 (B), and 100 nm (C). As R increases, the number of chemical bonds within the contact area increases, leading to smaller force quantization due to nonlinear relationship between the rupture force and the total energy change upon rupture.

shown in the inset. The absence of a detectable peak progression is consistent with the calculated value of the effective force resolution (ξ ) 1.01), which because it is ∼1 indicates that force quanta would be very difficult to detect in Figure 8C. It is clear from these results that the radius of the AFM probe plays a major role in the ability to distinguish between the rupture of integer numbers of chemical bonds. The reason is best illustrated in Figure 2B, which shows that, at a given binding probability p, the effective force resolution ξ decreases as nMax-bonds increases.

Because nMax-bonds scales with R (contact area A ∝ R2/3; see eq 3), larger tip radii result in a lower effective force resolution. Comparison of Model to Experimental Data. For many common chemically tailored surfaces that have been utilized in AFM rupture force measurements, γSolvent is known, which allows 〈γProbe-Substrate〉 to be determined from the pull-off measurement.3,5 Unfortunately, assigning an a priori value to the binding probability, p, of two functional groups across the probe-substrate interface is difficult. Because of this, the model cannot be used to predict in advance whether a single-bond rupture force of a particular system can be measured by AFM. It can, however, be used to give an idea of the required minimum binding probability. To test this, we chose to compare the results of the model to experimental measurements by van der Vegte and Hadziioannou.3c They measured pull-off forces between Au-coated AFM probes and substrates modified with alkanethiol SAMs having different terminal groups, and they calculated SAM-SAM interfacial energies, and the interfacial energies of the SAMs with ethanol, using the JKR model. Not surprisingly, the system that gave the largest rupture forces (〈FRupture〉 ) 2.2 nN) and the largest negative probe-substrate interfacial energy (〈γProbe-Substrate〉 ) -6.0 mJ/ m2) consisted of acid- and amide-terminated SAMs on probe and substrate. Upon contact of the probe with the substrate, strong hydrogen bonds are expected to form across the interface between these SAMs. The authors estimated 130 pN as the rupture force for a single acid-amide hydrogen bond by dividing the average rupture force by the maximum number of possible bonds within the average contact area (determined by JKR). This 130 pN value is clearly well within the instrument resolution of the AFM, but still no quantization was observed in the measured rupture force histogram. Table 5 shows the parameters used in two calculations based on the experimental results of van der Vegte and Hadziioannou. In the first calculation, we assume that every possible hydrogen bond is formed within the probe-substrate contact area Analytical Chemistry, Vol. 74, No. 13, July 1, 2002

3103

Figure 9. Histograms of rupture forces calculated using eq 9 and variables based on the measurements made by van der Vegte and Hadiioannou,3c shown in Table 5. The rupture forces in the two histograms were calculated using p ) 1.0 (A) and 0.93 (B). This demonstrates that the probability for each acid-amide hydrogen bond to form within the probe-substrate contact area must be larger than 0.93 in order to be able to resolve the difference in rupture force due to the rupture of n and n + 1 bonds.

(p ) 1). The resulting histogram of rupture forces, shown in Figure 9A, reveals a clear quantization of the rupture force as predicted by the value of ξ, which is 1.27. Both the periodicity of the peaks (125 pN) and the average rupture force (2.4 nN) are in excellent agreement with the experimental values. To determine the minimum value of p for peaks to be visible, several calculations were performed with gradual lowering of p. At the same time, 〈γProbe-Substrate〉 was increased to accommodate for the lowered binding probability and thus keep the effective interfacial energy constant. Calculations with p e 0.93 consistently gave rupture force histograms with no peaks; an example with p ) 0.93 is shown in Figure 9B (ξ ) 0.90 in this case). Absence of periodic structure in the histogram is emphasized in the nonperiodic autocorrelation function shown in the inset. The histogram in Figure 9B is similar to the one obtained by van der Vegte and Hadziioannou. This comparison of the model and experiment underscores the importance of binding probability on the ability to detect singlebond rupture forces in cases where γSolvent > |〈γProbe-Substrate〉|.


CONCLUSION We have shown by contact mechanics modeling that the ability to successfully detect single-bond rupture forces by AFM strongly depends on probe size, the binding probability of an individual bond, and the relative magnitudes of the probe-substrate and solvent interfacial energies. For example, when γSolvent is approximately twice the size of |〈γProbe-Substrate〉|, each bond within the probe-substrate contact area must have a probability of formation larger than 0.9 and the radius of curvature of the probe must be less than 35 nm in order for a peak progression to appear in the rupture force histogram. The model illustrates the importance of these basic factors on the ability to detect single-bond rupture forces. However, our model does not take into account all factors that affect detection of single bonds in AFM pull-off measurements. Variability in the single-bond binding energy due to steric factors, variability in substrate topography, the uneven stress distribution within the probe-substrate contact area, and the thermal energy of the AFM cantilever can add to variability in the measured rupture forces. The limits on p and R for detecting single-bond forces are likely to be even more restrictive when these additional factors are taken into account. ACKNOWLEDGMENT C.D.F. thanks the Center for Interfacial Engineering (an NSFfunded Engineering Research Center) at the University of Minnesota for financial support of this work. SUPPORTING INFORMATION AVAILABLE The Igor Pro simulation code used for this work is available as Supporting Information. To run it, a free demo version of Igor Pro for Macintosh and Windows operating systems can be downloaded from http://www.wavemetrics.com/Products/IGORPro/Demo.html.

Received for review February 4, 2002. Accepted April 2, 2002. AC020075G

Contact Mechanics Modeling of Pull-Off ... - ACS Publications

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