APPLIED PHYSICS LETTERS 98, 013101 共2011兲
Energy dissipation in a dynamic nanoscale contact Sergio Santos1,a兲 and Neil H. Thomson1,2 1
School of Physics and Astronomy and Faculty of Biological Sciences, Structural Molecular Biology, University of Leeds, Leeds 9JT, United Kingdom 2 Department of Oral Biology, Leeds Dental Institute, University of Leeds, Leeds LS2 9JT, United Kingdom
共Received 15 November 2010; accepted 2 December 2010; published online 3 January 2011兲 The size of an atomic force microscope tip can vary rapidly during interaction with a surface but this is typically overlooked. Here, we treat the tip radius as a dynamic variable. Comparison with nanoscission of DNA molecules shows that the pressure in the nanoscale contact cannot increase without bound, rather the tip gradually blunts as the energy in the cantilever is increased. We develop a method to stabilize the tip and then reliably calculate the effective area of interaction and dissipation of eV/atom in situ. Tip radius typically stabilizes around 20–30 nm and keeps the maximum pressure below ⬃1 GPa. © 2011 American Institute of Physics. 关doi:10.1063/1.3532097兴 The atomic force microscope 共AFM兲 is a well established tool for nanoscale characterization.1–4 While topography can be routinely characterized and sensible numerical data can be easily obtained, routinely acquiring numerical data about the mechanical and chemical properties of the samples during the scanning process remains challenging.5–8 AFM phase imaging can provide other than just topographical constrast,4,8–10 however, its interpretation is not straightforward.7,8 Another major field of research in AFM consists of simultaneously analyzing the data involved in the higher harmonics11–14 sometimes by directly exciting higher modes of the lever.12 In general, for a technique involving chemical and/or material properties mapping the stability of the tip is crucial since the data strongly depends on the state of the tip at a given time. The difficulties of accounting for a variable size of the tip while scanning are clearly stated everywhere in the literature.2,10,15–17 Nevertheless, with simplified approaches only the magnitude of the tip-sample force can typically18,19 be used to predict or interpret tip and sample elastic and plastic deformation.19–26 Here we show how to characterize and stabilize the tip in situ and use this finding to interpret and predict elastic and plastic deformation of soft matter from a more fundamental mechanism, namely, the energy dissipated per atom and a limit in pressure during dynamic nanoscale interactions. It can be shown9,27 that from energy considerations and with some reasonable assumptions, a simple algebraic expression giving the average energy dissipated per cycle 具Edis典 in the tip-surface interaction Eq. 共1兲 can be found. 具Edis典CYCLE =
冋
Asp kA0Asp sin共兲 − A0 Q
册
= 0 .
共1兲
The experimental parameters are readily obtained; the cantilever free amplitude A0, the amplitude set-point Asp, the spring constant k, Q 共or Q factor兲, the sine of the phase shift 共⌽兲 the angular driving frequency 共兲 and the natural angular frequency 0. Here = 0 has been used throughout where Eq. 共1兲 is valid.7–10,28 Starting from Eq. 共1兲, note that no dependency on tip radius R or the effective area of interaction 具Area典 can be a兲
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deduced. Nevertheless, the importance of R and 具Area典 when interpreting energy dissipation can be demonstrated with a highly reproducible experiment as described next 共Figs. 1 and 2兲. The topography of two DNA molecules on a mica surface is shown in Fig. 1共a兲. The images have been obtained in the noncontact 共nc兲 mode using a standard but relatively sharp tip 共R ⬍ 10 nm, Olympus AC160TS兲. Then m1 has been centered 关Fig. 1共b兲兴 and cut using a nanoscission technique by temporarily driving the tip into the repulsive regime. The nanomanipulation scan took place for ten lines in the slow scan axis 共scan rate 2 Hz兲 with A0 = 12 nm and Asp / A0 = 0.4. The dissection of m1 shows that imaging in the repulsive regime can result in molecular damage 关Figs. 1共c兲 and 1共d兲兴. From Eq. 共1兲, we find 具Edis典 = 18 eV has been responsible for plastic deformation and scission of the dsDNA molecule. Moreover, we have observed in similar experiments that molecules can be completely wiped from the surface in this way.29
FIG. 1. 共Color online兲 共a, b兲 Topography images 共Z-piezo兲 of two dsDNA molecules imaged in the nc mode. Molecule m1 has then been cut and folded up by scanning in the repulsive regime. 共c, d兲 This outcome is verified with subsequent scans of both molecules in the nc mode. Experimental parameters: f = f0 = 302 kHz 共driving and natural frequencies, respectively兲, k = 40 N / m, R ⬍ 10 nm, Q = 500. All the images shown have been taken with Asp / A0 = 0.9 and A0 = 3 nm.
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FIG. 2. 共Color online兲 Topographic sequence of dsDNA molecules on a mica surface with the same cantilever-sample system as that shown in Fig. 1. The value of A0 has been systematically increased from 共a兲 4.5 to 共l兲 165 nm while keeping Asp / A0 = 0.80. The molecules submitted to the sequence have been labeled m3 to m7. For easy comparison only a zoomed view of m4 is shown whereas the full scans are shown in the supplementary material 共See Ref. 29 for a full description兲. The values of A0 are shown at the bottom of each panel.
The relationship between plastic deformation and tip sharpness, however, cannot be reduced to the regime of operation. This is due to the fact that 具Area典, 具Edis典, the pressure in the contact Pm and R might greatly differ from system to system and, in particular, Fts might be large but Pm might be small due to a large value of 具Area典.29 For example Round and Miles24 conducted experiments with dsDNA similar to those conducted by San Paulo and Garcia with antibodies and concluded that DNA could not be permanently deformed in the repulsive regime. Nevertheless, the simple experiment shown in Fig. 1 argues otherwise. Also, Thomson imaged antibodies with considerably high resolution in the repulsive regime with no sign of plastic deformation.26,30 An experimental example of the relevance of 具Area典 and R is shown in Fig. 2 where nearby molecules to those in Fig. 1 have been imaged systematically by discretely increasing the value of A0 and keeping Asp / A0 constant with the same tip as in Fig. 1. The minimum A0 inducing severe observable plastic deformation has been reached at approximately A0 ⬃ 110– 130 nm 关Figs. 2共h兲–2共j兲兴 but critical deformation could only be observed above A0 ⬎ 120– 130 nm 关c.f. Figs. 2共i兲 and 2共j兲兴. The average energy dissipated per cycle 具Edis典 has been calculated for every value of A0 in the sequence 关Fig. 3共a兲兴. For comparison, the average energies per cycle dissipated to the medium 具Emed典 and the mean energy stored 2 and in the cantilever 具Ec典 are also shown; 具Ec典 = 21 kAsp 2 具Emed典 = 共k / Q兲Asp assuming viscous damping. The three energy variables rapidly grow with A0 reaching hundreds and thousands of eV. Significantly, tips submitted to these imaging sequences can later be used to scan biomolecules such as DNA and antibodies with relatively high values of A0, i.e., A0 ⬍ 60– 70 nm, and any given set-point without inducing molecular plastic deformation.29
FIG. 3. 共Color online兲 共a兲 Experimental values of 具Edis典 共squares兲, 具Emed典 共circles兲 and 具Ec典 共triangles兲 for the sequence in Fig. 2. The values of A0 are also shown and errors of ⫾12%, ⫾10%, and ⫾5% have been allowed for k, Q, and ⌽, respectively. The error bars for 具Emed典 and 具Ec典, almost coincided with the average values in this logarithmic scale, thus they have not been shown for clarity. 共b兲 Predictions 共simulations兲 of the increase in Pm with increasing A0 共filled squares兲 when the tip radius is set to a constant value 共R = 7 nm兲 and predictions of the increase in R 共outlined rhombuses兲 when a limit to the pressure is set 共Pm = 0.9 GPa兲. 共c兲 Prediction of the eV dissipated per nm2 and per atom in both cases when inputting the experimental data from 共a兲 into the model. The parameters are: f = f0 = 300 kHz, k = 40 N / m, Q = 500, ␥ = 40 mJ 共surface energy兲, E = 10 GPa 共elastic modulus of the surface兲, Et = 120 GPa 共elastic modulus of the tip兲.
The above shows that R and 具Area典 have to increase with A0. Note however, that failure to increase A0 smoothly and systematically can result in the tip mechanically fracturing in an uncontrolled fashion. Nevertheless, scanning electron microscopy 共SEM兲31,32 can be used 共data not shown兲 to show that tips with curvatures in the range of 5 nm become stable when reaching values of R = 20– 30 nm provided A0 ⬍ 60– 70 nm. We term this critical value A0L or limiting free amplitude where for mechanical stability A0 ⬍ A0L is required. Thus, provided A0 is smoothly increased as in Fig. 2 the tip mechanically stabilizes at R = 20– 30 nm and, in principle, other values of A0L could be used; SEM can be used to verify that for A0L = 20– 30 nm stability occurs for R = 10– 20 nm whereas for A0L = 150– 200 nm, the range is R = 30– 40 nm 共data not shown兲. Furthermore, these data agree with our predictions in the simulations below. We proceed to model the interaction by including long range van der Waals forces33 共vdW兲 and contact forces from the Deraguin-Muller-Toporov 共DMT兲34 model. While more sophisticated models are sometimes used to account for the higher modes of vibration and the study of harmonics,4,13,14 the point mass is a good approximation to the phenomena in
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ambient conditions with sufficiently stiff cantilevers.35 Modeling 具Area典 in the dynamic mode can prove challenging but we use a method that produces consistent results.36 The effective area for the vdW and the DMT forces are given by 共2兲 36 and 共3兲34,37 respectively. Then, 共4兲 and 共5兲 can be used to finally obtain 具Area典 in the dynamic mode 共6兲29 AreavdW,0.95共d兲 = 共1.5R1/3d3/5 + 0.25R兲2 ,
共2兲
AreaDMT共d兲 = ␦R,
共3兲
具AreavdW典 =
1
i=n
兺 AreavdW,i ⫻ 兩FvdW,i兩,
具FvdW典 i=1
共4兲
i=n
共5兲
i=1
and similarly for the DMT force. Finally 具Area典 =
We would like to thank Sergio’s sponsors BBSRC and Asylum Research and Tony Fischer 共Fischer-Cripps laboratories兲, Hugo K. Christenson, Daniel Billingsley, William A. Bonass 共University of Leeds兲, Antonio Vazquez 共Universidad de la Coruña兲, Victor Barcons and Josep Font 共Universitat Politècnica de Catalunya兲. C. F. Quate, Surf. Sci. 299–300, 980 共1994兲. Y. Gan, Surf. Sci. Rep. 64, 99 共2009兲. 3 A. Alessandrini and P. Facci, Meas. Sci. Technol. 16, R65 共2005兲. 4 R. García and R. Perez, Surf. Sci. Rep. 47, 197 共2002兲. 5 O. Sahin, S. Magonov, C. Su, C. F. Quate, and O. Solgaard, Nat. Nanotechnol. 2, 507 共2007兲. 6 R. Proksch, Appl. Phys. Lett. 89, 113121 共2006兲. 7 N. Martínez and R. Garcia, Nanotechnology 17, S167 共2006兲. 8 R. Garcia, C. J. Gómez, N. F. Martinez, S. Patil, C. Dietz, and R. Magerle, Phys. Rev. Lett. 97, 016103 共2006兲. 9 J. P. Cleveland, B. Anczykowski, A. E. Schmid, and V. B. Elings, Appl. Phys. Lett. 72, 2613 共1998兲. 10 X. Chen, M. C. Davies, C. J. Roberts, S. J. B. Tendler, P. M. Williams, and N. A. Burnham, Surf. Sci. 460, 292 共2000兲. 11 R. W. Stark, Nanotechnology 15, 347 共2004兲. 12 T. Rodríguez and R. Garcia, Appl. Phys. Lett. 84, 449 共2004兲. 13 R. Stark and W. Heckl, Rev. Sci. Instrum. 74, 5111 共2003兲. 14 O. Sahin, C. Quate, O. Solgaard, and A. Atalar, Phys. Rev. B 69, 165416 共2004兲. 15 L. Zitzler, S. Herminghaus, and F. Mugele, Phys. Rev. B 66, 155436 共2002兲. 16 M. L. Bloo, H. Haitjema, and W. O. Pril, Measurement 25, 203 共1999兲. 17 M. Kopycinska-Mueller, R. H. Geiss, and D. C. Hurley, Size-related plasticity effect in AFM silicon cantilever tips 共Materials Research Society 2006 Spring Meeting, San Francisco, CA, 2006兲, Vol. 924, p. Z3.2.1. 18 S. Patil, N. F. Martinez, J. R. Lozano, and R. Garcia, J. Mol. Recognit. 20, 516 共2007兲. 19 C.-W. Yang and I.-S. Hwang, Nanotechnology 21, 065710 共2010兲. 20 J. Tamayo and R. Garcia, Langmuir 12, 4430 共1996兲. 21 B. Drake, C. B. Prater, A. L. Weisenhorn, S. A. Gould, T. R. Albrecht, C. F. Quate, D. S. Cannell, H. G. Hansma, and P. K. Hansma, Science 243, 1586 共1989兲. 22 A. l. Weisenhorn, P. K. Hansma, T. R. Albrecht, and C. F. Quate, Appl. Phys. Lett. 54, 2651 共1989兲. 23 C.-W. Yang, I.-S. Hwang, Y. Fu Chen, C. Seng Chang, and D. Ping Tsai, Nanotechnology 18, 084009 共2007兲. 24 A. Round and M. Miles, Nanotechnology 15, S176 共2004兲. 25 A. San Paulo and R. Garcia, Biophys. J. 78, 1599 共2000兲. 26 N. H. Thomson, Ultramicroscopy 105, 103 共2005兲. 27 J. Tamayo and R. Garcia, Appl. Phys. Lett. 73, 2926 共1998兲. 28 M. Stark, C. Möller, D. J. Müller, and R. Guckenberger, Biophys. J. 80, 3009 共2001兲. 29 See supplementary material at http://dx.doi.org/10.1063/1.3532097 for details on molecular scission, Pm and R dependencies, the model and other experimental data. 30 N. H. Thomson, J. Microsc. 217, 193 共2005兲. 31 S. Santos, V. Barcons, J. Font, and N. H. Thomson, J. Phys. D: Appl. Phys. 43, 275401 共2010兲. 32 S. Santos, V. Barcons, J. Font, and N. H. Thomson, Nanotechnology 21, 225710 共2010兲. 33 H. C. Hamaker, Physica 4, 1058 共1937兲. 34 B. V. Derjaguin, V. Muller, and Y. Toporov, J. Colloid Interface Sci. 53, 314 共1975兲. 35 T. R. Rodríguez and R. García, Appl. Phys. Lett. 80, 1646 共2002兲. 36 S. Santos, V. BarconsD. J. Billingsley, H. K. ChristensonW. A. Bonass, and N. H. Thomson 共unpublished兲. 37 A. C. Fischer-Cripps, Nanoindentation 共Springer, New York, 2004兲. 38 M. Grandbois, M. Beyer, M. Rief, H. Clausen-Schaumann, and H. E. Gaub, Science 283, 1727 共1999兲. 1 2
where 具FvdW典 = 兺 兩FvdW,i兩
ing A0. Quantification of the interaction area between the tip and sample will lead to better mapping of local chemistry and mechanics and shed light on the mechanism of elastic and plastic deformation and manipulation at the nanoscale.
具AreavdW典 ⫻ 兩具FvdW典兩 + 具AreaDMT典 ⫻ 兩具FDMT典兩 . 兩具FvdW典兩 + 兩具FDMT典兩 共6兲
In Fig. 3共b兲, R⫽7 nm and A0 has been gradually increased while keeping Asp / A0 and R constant; Pm increases with A0 共filled squares兲, with no upper bound which is physically unrealistic. Alternatively an upper bound on Pm can be set 共outlined rhombuses兲. In Fig. 3共b兲, the initial radius is R = 7 nm but the maximum pressure is 共Pm兲max = 0.9 GPa while R has been allowed to increase with A0. The value 共Pm兲max = 0.9 GPa has been obtained from simulations by setting A0L = 60– 70 nm and R = 20– 30 nm32 in the model where Pm ⬃ 0.8– 1 GPa. The predicted value is also consistent with other values of A0L 共data not shown兲. In Fig. 3共c兲, the experimental values of 具Edis典 in Fig. 3共a兲 have been introduced in the model. The value of 具Area典 has been calculated according to Eq. 共6兲. This allows one to estimate eV/ nm2 and/or eV/atom if 共1兲 the tip radius remains constant 共R = 7 nm兲 and the pressure is allowed to increase without upper bound 共filled squares兲 and 共2兲 if the pressure has an upper bound 共i.e., Pm = 0.9 GPa兲 and R is allowed to increase 共outlined rhombuses兲 关Fig. 3共c兲兴. If R is not allowed to increase several eV/atom are predicted. This situation is physically unrealistic since a covalent bond has a typical value of 2 eV.38 If R is allowed to increase the number of eV per atom remains always below or close to 1 eV. Figures 3共b兲 and 3共c兲 can be readily used to interpret elastic and plastic deformation and their relationship with 具Area典, R, and A0 and eV/ atom. More thoroughly, from the above and in order to keep the tip radius stable and in the range of R ⬍ 20– 30 nm, values of A0 ⬍ 60– 70 nm= A0L should be used. Experimentally, the tip should be initially submitted to a sequence similar to that in Fig. 2 with A0L = 60– 70 nm. Then reliable experimentation with the tip can be performed provided A0 ⬍ 60– 70 nm throughout, guaranteeing mechanical stability. The energy quantification presented here is fundamental for reliable characterization of nanoscale processes in dynamic AFM. Importantly, we have shown experimentally how the tip can be made reliably stable by smoothly increas-
