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Simulation of fluctuation and dissipation in dynamic force microscopy
Applied Surface Science 188 (2002) 349–354
Simulation of fluctuation and dissipation in dynamic force microscopy H. Nanjoa,*, L. Nonyb, M. Yoneyaa, N. Sanadaa, T. Iijimac, J.P. Aime´b a
AIST Tohoku, ISEM, National Institute of Advanced Industrial Science and Technology (AIST), 4-2-1, Nigatake, Miyagino-ku, Sendai 983-8551, Japan b CPMOH, Universite´ Bordeaux I, 351, Cours de la Libe´ration, F-33405 Talence, France c SSRC, National Institute of Advanced Industrial Science and Technology (AIST), 1-1, Umezono, Tsukuba 305-8556, Japan Received 2 September 2001; accepted 17 September 2001
Abstract We have simulated three possible effects of the driving force fluctuation of the cantilever, fluctuation of the surface location and contribution of an additional dissipation due to the sample, with the hope to identify effective methods of improvement of the achievable resolution during a dynamic force microscopy experiment. This study is performed through numerical simulations of the approach–retract curves. We find that in the case of soft materials, the driving force fluctuation has only a small effect on the amplitude of the cantilever oscillations, while surface fluctuation can markedly decrease the maximum amplitude. We also show that dissipation has a larger effect on the phase of oscillator than on its amplitude. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Tapping; Oscillation; Fluctuation; Dissipation; Non-contact; Atomic force microscopy
1. Introduction Good resolution can be achieved with dynamic force microscopy (DFM), but this achievement has so far been limited to flat and hard surfaces. It is more difficult to observe soft surfaces, or those with sharpedged features, at a nanometer scale such as DNA [1,2] and organic materials. In recent years, the achievable resolution has gradually increased due to the availability of sharper tips, the evolution of observation methods and detection ways [3], but it remains difficult in many cases to achieve atomic or molecular resolution. If this situation can be improved a little, it *
Corresponding author. Tel.: þ81-22-237-5211; fax: þ81-22-239-0629. E-mail address: [email protected] (H. Nanjo).
is expected that much useful information will be obtained. Disturbance factors for getting high resolution are substrate flatness, substrate pollution (coverage of hydrocarbon and water), noisy sound, air stream, thermal influence, force fluctuation of the cantilever oscillation, surface fluctuation, and dissipation by these fluctuations. Because many of these factors occur mostly in air environment, they have not been discussed in detail nor reviewed precisely. Though Morita and Sugawara [4] introduced the relation between vertical resolution and signal-to-noise ratio, they did not refer the effect of noise on force curve. Du¨rig [5] investigated hysteretic effects due to an asymmetric interaction between forward and backward tip motions and evaluated the dissipation from
0169-4332/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 9 4 9 - 7
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H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
acoustical origins. He found that phonons contribution must be negligible. Gauthier and Tsukada [6] investigated the influence of Brownian motion with the help of the Langevin equation [7]. Aime´ et al. [8] used a simple mechanical model showing that the amount of dissipate energy drastically depends on the time relaxation of the surface. The comparison between model predictions and experimental data shows a good agreement. However, they do not investigate the dissipation effect under fluctuations of drive force and surface location. The present work intends to analysis the influence of such factors which are not necessarily related to nanometer scale. We have simulated three possible effects that influence the achievable resolution; force fluctuation of the cantilever oscillation, surface fluctuation, and dissipation under force fluctuation or surface fluctuation in the hope of identifying effective methods of improvement. 1.1. Basic equations Since in this work, one wants to investigate a general DFM behavior, i.e. NC-AFM and Tapping mode rather than to focus on non-contact situations, we prefer to discuss the different oscillation behaviors corresponding to situations for which either the attractive interaction between the tip and the surface is dominant or the repulsive interaction is dominant. In that case, attractive dominant interaction simply means that the repulsive interaction is negligible even when the tip slightly touches the surface. Thus, the oscillating tip motion is described as an oscillation forced by piezoelectric transducer with viscous damping and the Van der Waals interaction force between the tip and flat sample, when the tip does not contact with sample surface. The non-contact regime or dominant attractive regime is given by [9]: o0 xðtÞ þ x_ ðtÞ þ o0 xðtÞ Q F0 HR (1) ¼ ð1 þ aÞ cos ot 6mjD xðtÞj m where o0, Q, F0, m, o, H, R and D are the resonance frequency, quality factor, external drive force, effective mass, drive frequency, Hamaker constant, tip radius, and distance between the sample surface and
the equilibrium position at the oscillating tip– cantilever system, respectively. In order to consider the fluctuation of the drive force, random factor a is introduced into F0 term. And in order to consider the surface fluctuation, random factor b is introduced into distance term and indentation term of Eq. (2). When the tip touches the sample, the Van der Waals force is not suitable and the tip is supposed to be acted by a force which is proportional to sample stiffness and indentation depth, then a dissipation occurs by tip contact and energy loss, then a dissipation coefficient is introduced into the viscous term. In this case, the equation of motion is written as follows: o0 xðtÞ þ þ x x_ ðtÞ þ o0 xðtÞ Q F0 1 ¼ ð1 þ aÞ Stiff Ind cos ot þ m m
(2)
where x, Stiff and Ind is dissipation coefficient, sample stiffness and indentation depth in the sample, respectively. When the tip intermittently contacts with the sample surface, Eqs. (1) and (2) are used for the simulation. The typical initial conditions for simulation are as follows. Resonance amplitude: 20 nm; resonance frequency: 150 kHz; quality factor: 200; working free phase: 458; effective mass: 5 ng; tip’s apex radius: 20 nm; Young’s modulus: 0.1 GPa; Poisson coefficient: 0.1; percentage of reduction of amplitude: 30% of free amplitude; vertical displacement: 25 nm; cantilever stiffness: 4.4 N/m.
2. Simulation results The Eqs. (1) and (2) are numerically computed (Cþþ numerical code). Fig. 1 shows the force fluctuation effect on the tip approach–retract curve (a–r curve) at the fluctuation of 10% of the drive force in intermittent contact situation ðH ¼ 0:01 1018 JÞ and in non-contact situation ðH ¼ 0:1 1018 JÞ. Afterward, the unit of 1018 J will be omitted. Both the situations, when the tip is far from a sample, the force fluctuation causes the large amplitude fluctuation, whereas when the tip is close to a sample, force fluctuation effect on amplitude becomes small. The phase change by
H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
Fig. 1. Effect of force fluctuation on an a–r curve: (a) intermittent contact situation; (b) non-contact situation.
Fig. 2. Effect of surface fluctuation on an a–r curve: (a) intermittent contact situation; (b) non-contact situation.
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H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
Fig. 3. Dependence of force fluctuation and surface fluctuation on the maximum amplitude and the distance at the maximum amplitude in an a–r curve.
Fig. 4. Effect of dissipation on an a–r curve in intermittent contact situation ðH ¼ 0:01Þ. x is dissipation coefficient. (a) With force fluctuation; (b) with surface fluctuation. 17.12 nm in the figure (a) shows the maximum amplitude at x ¼ 0, which is larger than 17.07 nm at x ¼ 10.
H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
force fluctuation is smaller than its amplitude even if the tip is far. Fig. 2 shows the surface fluctuation effect on a–r curve at the fluctuation of 1 nm in contact and in non-contact situation. Both the situations, when a tip is far from a sample, surface fluctuation does not cause fluctuations of amplitude and phase, whereas surface fluctuation produces large fluctuations of amplitude and phase when a tip approaches a sample. Fig. 3 shows the fluctuation dependence on the maximum amplitude in intermittent contact situation and in non-contact situation. The force fluctuation does not affect the maximum amplitude in intermittent contact situation with H ¼ 0:01, whereas in noncontact situation the maximum amplitude and its distance decrease with a force fluctuation. The surface fluctuation does not affect the maximum amplitude in non-contact situation with H ¼ 0:1, whereas in intermittent contact situation the maximum amplitude decreases with a surface fluctuation. Fig. 4 shows the dissipation effect on a–r curve in intermittent contact situation. When the dissipation and surface fluctuation are affected, the maximum amplitude is much lower than when dissipation and force fluctuation are affected. The tendency is the same with the case of no dissipation effect, as is shown in Fig. 3. The dissipation is not correlated to the fluctuations either of the driving force or of the surface location. The dissipation sensitively decreases the phase slope against distance between tip and sample as compared to the case of amplitude.
3. Discussion The force fluctuation and dissipation does not so much affect the tip amplitude, therefore the resolution does not change so much, whereas the surface fluctuation causes distance fluctuation and markedly reduced the resolution. The tip amplitude is not more sensitive than the phase of tip oscillation against the change of Hamaker constant and dissipation. This suggests that phase give a better resolution to observe a sample property than amplitude. Conversely, the amplitude is difficult to be affected by the surface property as Hamaker constant and dissipation and is suitable for observing topography.
353
Fig. 5. Transient phenomena from intermittent contact situation to non-contact situation between H ¼ 0:027 and 0.039 (1018 J). Non-contact case is at H ¼ 0:040 (1018 J). A is amplitude and P is phase.
In tapping mode, atomic force microscopy (AFM), which is one of the DFM, a bifurcation phenomenon has been sometimes reported and discussed [10]. That is, some artifacts appear on an AFM image, or a low place artificially appears as a high place on the image. However, the force fluctuation or surface fluctuation hardly generated the bifurcation. Fig. 5 shows the a–r curves around H ¼ 0:039 with no fluctuations. When H is from 0.027 to 0.039 and the tip attracts from the sample, there are two distances at one setpoint of amplitude like 18 nm. This means that the bifurcation phenomenon happens without being induced by fluctuation. We demonstrate it on the a–r curves at the boundary condition of Hamaker constant from intermittent contact to non-contact situation. On the other hand, the a–r curve of phase has just one peak in the region less than 708 at H ¼ 0:039, though that of amplitude has two peaks in the region larger than 19 nm. This is, when the phase is fixed as a setpoint, the bifurcation is hardly generated as compared with the case of amplitude.
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H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
4. Conclusion
References
We have simulated three possible effects that influence the achievable resolution of dynamic force microscopy; force fluctuation of the cantilever oscillation, surface fluctuation, and dissipation. We find that in the case of soft materials, force fluctuations have a small effect on the oscillation amplitude near sample surface, whilst surface fluctuation can markedly decrease the maximum amplitude in intermittent contact situation and results in worse resolution. We also show that dissipation effects have a large effect on the phase of tip oscillation than on its amplitude.
[1] Y. Maeda, T. Matsumoto, T. Kawai, Appl. Surf. Sci. 140 (1999) 400. [2] L. Nony, R. Boisgard, J.P. Aime´ , Biomacromolecules (ACS J.), 2 (2001) 827. [3] A.S. Paulo, R. Garcia, Biophys. J. 78 (2000) 1599. [4] S. Morita, Y. Sugawara, Appl. Surf. Sci. 140 (1999) 406. [5] U. Du¨ rig, Surf. Interf. Anal. 27 (1999) 467. [6] M. Gauthier, S. Tsukada, Phys. Rev. B 60 (1999) 11716. [7] G. Couturier, J.P. Aime´ , J. Salardenne, R. Boisgard, J. Phys. D 34 (2001) 1266. [8] J.P. Aime´ , R. Boisgard, L. Nony, G. Couturier, J. Chem. Phys. 114 (2001) 4945. [9] L. Nony, R. Boisgard, J.P. Aime´ , J. Chem. Phys. 111 (1999) 1615. [10] A. Kuhle, A.H. Sorensen, J.B. Zandbergen, J. Bohr, Appl. Phys. A 66 (1998) S329.
Simulation of fluctuation and dissipation in dynamic force microscopy H. Nanjoa,*, L. Nonyb, M. Yoneyaa, N. Sanadaa, T. Iijimac, J.P. Aime´b a
AIST Tohoku, ISEM, National Institute of Advanced Industrial Science and Technology (AIST), 4-2-1, Nigatake, Miyagino-ku, Sendai 983-8551, Japan b CPMOH, Universite´ Bordeaux I, 351, Cours de la Libe´ration, F-33405 Talence, France c SSRC, National Institute of Advanced Industrial Science and Technology (AIST), 1-1, Umezono, Tsukuba 305-8556, Japan Received 2 September 2001; accepted 17 September 2001
Abstract We have simulated three possible effects of the driving force fluctuation of the cantilever, fluctuation of the surface location and contribution of an additional dissipation due to the sample, with the hope to identify effective methods of improvement of the achievable resolution during a dynamic force microscopy experiment. This study is performed through numerical simulations of the approach–retract curves. We find that in the case of soft materials, the driving force fluctuation has only a small effect on the amplitude of the cantilever oscillations, while surface fluctuation can markedly decrease the maximum amplitude. We also show that dissipation has a larger effect on the phase of oscillator than on its amplitude. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Tapping; Oscillation; Fluctuation; Dissipation; Non-contact; Atomic force microscopy
1. Introduction Good resolution can be achieved with dynamic force microscopy (DFM), but this achievement has so far been limited to flat and hard surfaces. It is more difficult to observe soft surfaces, or those with sharpedged features, at a nanometer scale such as DNA [1,2] and organic materials. In recent years, the achievable resolution has gradually increased due to the availability of sharper tips, the evolution of observation methods and detection ways [3], but it remains difficult in many cases to achieve atomic or molecular resolution. If this situation can be improved a little, it *
Corresponding author. Tel.: þ81-22-237-5211; fax: þ81-22-239-0629. E-mail address: [email protected] (H. Nanjo).
is expected that much useful information will be obtained. Disturbance factors for getting high resolution are substrate flatness, substrate pollution (coverage of hydrocarbon and water), noisy sound, air stream, thermal influence, force fluctuation of the cantilever oscillation, surface fluctuation, and dissipation by these fluctuations. Because many of these factors occur mostly in air environment, they have not been discussed in detail nor reviewed precisely. Though Morita and Sugawara [4] introduced the relation between vertical resolution and signal-to-noise ratio, they did not refer the effect of noise on force curve. Du¨rig [5] investigated hysteretic effects due to an asymmetric interaction between forward and backward tip motions and evaluated the dissipation from
0169-4332/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 9 4 9 - 7
350
H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
acoustical origins. He found that phonons contribution must be negligible. Gauthier and Tsukada [6] investigated the influence of Brownian motion with the help of the Langevin equation [7]. Aime´ et al. [8] used a simple mechanical model showing that the amount of dissipate energy drastically depends on the time relaxation of the surface. The comparison between model predictions and experimental data shows a good agreement. However, they do not investigate the dissipation effect under fluctuations of drive force and surface location. The present work intends to analysis the influence of such factors which are not necessarily related to nanometer scale. We have simulated three possible effects that influence the achievable resolution; force fluctuation of the cantilever oscillation, surface fluctuation, and dissipation under force fluctuation or surface fluctuation in the hope of identifying effective methods of improvement. 1.1. Basic equations Since in this work, one wants to investigate a general DFM behavior, i.e. NC-AFM and Tapping mode rather than to focus on non-contact situations, we prefer to discuss the different oscillation behaviors corresponding to situations for which either the attractive interaction between the tip and the surface is dominant or the repulsive interaction is dominant. In that case, attractive dominant interaction simply means that the repulsive interaction is negligible even when the tip slightly touches the surface. Thus, the oscillating tip motion is described as an oscillation forced by piezoelectric transducer with viscous damping and the Van der Waals interaction force between the tip and flat sample, when the tip does not contact with sample surface. The non-contact regime or dominant attractive regime is given by [9]: o0 xðtÞ þ x_ ðtÞ þ o0 xðtÞ Q F0 HR (1) ¼ ð1 þ aÞ cos ot 6mjD xðtÞj m where o0, Q, F0, m, o, H, R and D are the resonance frequency, quality factor, external drive force, effective mass, drive frequency, Hamaker constant, tip radius, and distance between the sample surface and
the equilibrium position at the oscillating tip– cantilever system, respectively. In order to consider the fluctuation of the drive force, random factor a is introduced into F0 term. And in order to consider the surface fluctuation, random factor b is introduced into distance term and indentation term of Eq. (2). When the tip touches the sample, the Van der Waals force is not suitable and the tip is supposed to be acted by a force which is proportional to sample stiffness and indentation depth, then a dissipation occurs by tip contact and energy loss, then a dissipation coefficient is introduced into the viscous term. In this case, the equation of motion is written as follows: o0 xðtÞ þ þ x x_ ðtÞ þ o0 xðtÞ Q F0 1 ¼ ð1 þ aÞ Stiff Ind cos ot þ m m
(2)
where x, Stiff and Ind is dissipation coefficient, sample stiffness and indentation depth in the sample, respectively. When the tip intermittently contacts with the sample surface, Eqs. (1) and (2) are used for the simulation. The typical initial conditions for simulation are as follows. Resonance amplitude: 20 nm; resonance frequency: 150 kHz; quality factor: 200; working free phase: 458; effective mass: 5 ng; tip’s apex radius: 20 nm; Young’s modulus: 0.1 GPa; Poisson coefficient: 0.1; percentage of reduction of amplitude: 30% of free amplitude; vertical displacement: 25 nm; cantilever stiffness: 4.4 N/m.
2. Simulation results The Eqs. (1) and (2) are numerically computed (Cþþ numerical code). Fig. 1 shows the force fluctuation effect on the tip approach–retract curve (a–r curve) at the fluctuation of 10% of the drive force in intermittent contact situation ðH ¼ 0:01 1018 JÞ and in non-contact situation ðH ¼ 0:1 1018 JÞ. Afterward, the unit of 1018 J will be omitted. Both the situations, when the tip is far from a sample, the force fluctuation causes the large amplitude fluctuation, whereas when the tip is close to a sample, force fluctuation effect on amplitude becomes small. The phase change by
H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
Fig. 1. Effect of force fluctuation on an a–r curve: (a) intermittent contact situation; (b) non-contact situation.
Fig. 2. Effect of surface fluctuation on an a–r curve: (a) intermittent contact situation; (b) non-contact situation.
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Fig. 3. Dependence of force fluctuation and surface fluctuation on the maximum amplitude and the distance at the maximum amplitude in an a–r curve.
Fig. 4. Effect of dissipation on an a–r curve in intermittent contact situation ðH ¼ 0:01Þ. x is dissipation coefficient. (a) With force fluctuation; (b) with surface fluctuation. 17.12 nm in the figure (a) shows the maximum amplitude at x ¼ 0, which is larger than 17.07 nm at x ¼ 10.
H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
force fluctuation is smaller than its amplitude even if the tip is far. Fig. 2 shows the surface fluctuation effect on a–r curve at the fluctuation of 1 nm in contact and in non-contact situation. Both the situations, when a tip is far from a sample, surface fluctuation does not cause fluctuations of amplitude and phase, whereas surface fluctuation produces large fluctuations of amplitude and phase when a tip approaches a sample. Fig. 3 shows the fluctuation dependence on the maximum amplitude in intermittent contact situation and in non-contact situation. The force fluctuation does not affect the maximum amplitude in intermittent contact situation with H ¼ 0:01, whereas in noncontact situation the maximum amplitude and its distance decrease with a force fluctuation. The surface fluctuation does not affect the maximum amplitude in non-contact situation with H ¼ 0:1, whereas in intermittent contact situation the maximum amplitude decreases with a surface fluctuation. Fig. 4 shows the dissipation effect on a–r curve in intermittent contact situation. When the dissipation and surface fluctuation are affected, the maximum amplitude is much lower than when dissipation and force fluctuation are affected. The tendency is the same with the case of no dissipation effect, as is shown in Fig. 3. The dissipation is not correlated to the fluctuations either of the driving force or of the surface location. The dissipation sensitively decreases the phase slope against distance between tip and sample as compared to the case of amplitude.
3. Discussion The force fluctuation and dissipation does not so much affect the tip amplitude, therefore the resolution does not change so much, whereas the surface fluctuation causes distance fluctuation and markedly reduced the resolution. The tip amplitude is not more sensitive than the phase of tip oscillation against the change of Hamaker constant and dissipation. This suggests that phase give a better resolution to observe a sample property than amplitude. Conversely, the amplitude is difficult to be affected by the surface property as Hamaker constant and dissipation and is suitable for observing topography.
353
Fig. 5. Transient phenomena from intermittent contact situation to non-contact situation between H ¼ 0:027 and 0.039 (1018 J). Non-contact case is at H ¼ 0:040 (1018 J). A is amplitude and P is phase.
In tapping mode, atomic force microscopy (AFM), which is one of the DFM, a bifurcation phenomenon has been sometimes reported and discussed [10]. That is, some artifacts appear on an AFM image, or a low place artificially appears as a high place on the image. However, the force fluctuation or surface fluctuation hardly generated the bifurcation. Fig. 5 shows the a–r curves around H ¼ 0:039 with no fluctuations. When H is from 0.027 to 0.039 and the tip attracts from the sample, there are two distances at one setpoint of amplitude like 18 nm. This means that the bifurcation phenomenon happens without being induced by fluctuation. We demonstrate it on the a–r curves at the boundary condition of Hamaker constant from intermittent contact to non-contact situation. On the other hand, the a–r curve of phase has just one peak in the region less than 708 at H ¼ 0:039, though that of amplitude has two peaks in the region larger than 19 nm. This is, when the phase is fixed as a setpoint, the bifurcation is hardly generated as compared with the case of amplitude.
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H. Nanjo et al. / Applied Surface Science 188 (2002) 349–354
4. Conclusion
References
We have simulated three possible effects that influence the achievable resolution of dynamic force microscopy; force fluctuation of the cantilever oscillation, surface fluctuation, and dissipation. We find that in the case of soft materials, force fluctuations have a small effect on the oscillation amplitude near sample surface, whilst surface fluctuation can markedly decrease the maximum amplitude in intermittent contact situation and results in worse resolution. We also show that dissipation effects have a large effect on the phase of tip oscillation than on its amplitude.
[1] Y. Maeda, T. Matsumoto, T. Kawai, Appl. Surf. Sci. 140 (1999) 400. [2] L. Nony, R. Boisgard, J.P. Aime´ , Biomacromolecules (ACS J.), 2 (2001) 827. [3] A.S. Paulo, R. Garcia, Biophys. J. 78 (2000) 1599. [4] S. Morita, Y. Sugawara, Appl. Surf. Sci. 140 (1999) 406. [5] U. Du¨ rig, Surf. Interf. Anal. 27 (1999) 467. [6] M. Gauthier, S. Tsukada, Phys. Rev. B 60 (1999) 11716. [7] G. Couturier, J.P. Aime´ , J. Salardenne, R. Boisgard, J. Phys. D 34 (2001) 1266. [8] J.P. Aime´ , R. Boisgard, L. Nony, G. Couturier, J. Chem. Phys. 114 (2001) 4945. [9] L. Nony, R. Boisgard, J.P. Aime´ , J. Chem. Phys. 111 (1999) 1615. [10] A. Kuhle, A.H. Sorensen, J.B. Zandbergen, J. Bohr, Appl. Phys. A 66 (1998) S329.