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INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 16 (2005) 901–907
doi:10.1088/0957-4484/16/6/046
Experimental determination of conservative and dissipative parts in the tapping mode on a grafted layer: comparison with frequency modulation data P Martin1,2 , S Marsaudon1 , J P Aim´e1 and B Bennetau2 1
Centre de Physique Mol´eculaire Optique et Hertzienne, UMR 5798 CNRS, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405 Talence Cedex, France 2 Laboratoire de Chimie Organique et Organom´etallique, UMR 5802 CNRS, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405 Talence Cedex, France
Received 19 November 2004, in final form 15 March 2005 Published 19 April 2005 Online at stacks.iop.org/Nano/16/901 Abstract The aim of the present work is to extract the conservative and dissipative parts of the tip–sample interaction from the atomic force microscope amplitude modulation mode (AM, often called tapping). To do so, analytical expressions are used to transform the experimental amplitudes and phase variations with tip–sample distance to frequency shift and damping coefficient. The experimental procedure for the separation is detailed. The separated conservative and dissipative parts from the AM mode are compared for two driving frequencies, and then they are compared to the frequency modulation mode (FM, often called resonant non-contact) measurements for two different amplitudes. The conservative parts from the AM measurements are similar to the frequency shifts measured in the FM mode and also the dissipative parts from the AM measurements are very close to the dissipation measured in the FM mode. Experimentally, this good agreement is related to the small amplitude variation in the AM data on a chemically controlled grafted layer. Those results show experimentally that the AM and FM modes are just two different ways to probe the same tip–sample interaction. They also validate the AM data treatment to separate the conservative and dissipative parts as the only hypothesis needed is that only the fundamental harmonic contributes to the signal. (Some figures in this article are in colour only in the electronic version)
1. Introduction The possibility of local force probe methods to explore any sample surface at the nanometre scale is nowadays well known [1]. In the development of those methods, the dynamical modes [2] are commonly used with their ability to probe soft samples and minimize the sample shear. The latter was the main historical reason for their development. 0957-4484/05/060901+07$30.00 © 2005 IOP Publishing Ltd
Two dynamical modes of operating are possible, with the same physics of an oscillator made of a cantilever and a tip interacting with a surface, but with different ways to probe it. In the amplitude modulation (AM) mode [3], often called the ‘tapping’ mode as its commercial name, the tip–cantilever system is excited by a sine generator at a fixed frequency and excitation. AM is the most common mode and is often used to probe soft samples such as biological samples or polymers [4].
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Figure 1. Illustration of the intrinsic difficulty of the AM mode with fixed frequency and excitation: theoretical resonance curves (Lorentzian shape of a harmonic oscillator with values of the resonance frequency and quality factor equal to the experimental values of the tip–cantilever system used (see figure 2)) for three cases: without interaction (dark solid curve corresponding to Q = 350, resonance amplitude of 10 nm) with a driving frequency smaller than the resonance one (working amplitude of 7 nm), when the oscillator interacts with the surface, a change of amplitude A (giving 4.2 nm in the case represented) can be due to two complete different interactions: a change of dissipation (light solid curve, corresponding to Q = 175: doubled dissipation as compared to the non-interacting case) or an attractive force gradient that, as a first level of approximation, shifts the resonance curve toward the lower frequencies (dark dashed line). Because working at fixed excitation and frequency is an indirect way to probe the oscillator behaviour, it is difficult to analyse the amplitude variations in AM mode.
This mode offers a relatively easy way to obtain images at the nanometre scale with a main drawback due to the physical variables that are recorded: the tip amplitude and phase delay with the excitation are quite complicated to interpret due to bistable and hysteretic behaviour of the tip–cantilever system [5]. An example of AM mode intrinsic difficulty linked to fixed driving frequency and excitation is given with the resonance curves of figure 1. When the tip is far from the sample, the driving frequency is fixed at a frequency smaller than the resonance one, giving the oscillator amplitude. When the tip interacts with the sample, two complete different types of interaction can give the same amplitude variation A: a dissipation process (corresponding to a quality factor divided by two) or a displacement of the resonance curve towards smaller frequencies due to attractive interaction. Because any variation of the oscillation amplitude at a fixed driven frequency is always a mixing of at least two processes, it is then difficult to interpret the amplitude variations. On the other hand the frequency modulation (FM) mode, often called the non-contact resonant mode [6], is mostly used by physicists under high vacuum and at low temperature, as it enables one to obtain atomic resolution [7]. Originally, the FM mode was conceived to reduce the influence of the oscillator transient time τ that can be larger than 20 ms in ultrahigh vacuum [6]. In this FM dynamical mode, the tip– cantilever system is inserted in an electronic loop with constant oscillation amplitude. The measured data are the resonance 902
frequency variation with interaction and the damping signal which is the excitation necessary to maintain the amplitude constant. The high sensitivity of this mode requires controlled samples (nearly flat at the image scale surfaces with small pollution) as compared to the AM mode, and probing samples in repulsive mode is usually more difficult than for AM [8]. The measured data are a more direct way to probe the tip– cantilever interaction. Provided that the phase lag on the oscillator is fixed at the resonance one and that the amplitude control gain time constant is smaller than the characteristic time of variation of the interaction [9], the frequency is linked to conservative forces only, whereas the damping is linked to dissipative forces only. The interpretation of the data can then be a priori simplified as illustrated by some works [10, 11] where the tip–sample force is extracted from the FM approach curves frequencies and damping signals. Though the physics for both modes has been known to be the same for many years, few attempts have been made to compare experimental data from AM and FM modes [8, 12, 13]. Images of copolymers in AM and FM modes have shown the mechanical origin of the height contrast between soft and hard areas in AM, whereas the FM height image is flat corresponding to the sample topography, and the damping image reveals the viscous differences between the two areas [12]. Another approach to show the similarity between the two modes through the resonance curves has been done recently [13]. In this paper we focus on a way to separate the conservative and dissipative parts of the interaction of the tip with a surface from AM measurements of amplitude and phase variation with tip–sample distance. The separation of the two contributions is based on analytical expressions derived from the least action principle described in the experimental methodology. Results of this separation into conservative and dissipative parts are first compared for two AM experiments done at the same amplitude but at different driving frequencies recorded with the same tip. Then the AM data are compared with FM measurements for two amplitudes. The selected surface is a fluid-like monolayer of silanes grafted onto a silicon surface with controlled surface chemistry as it prevents the tip modification after many cycles of approaches and retracts. Also, the amplitude variation during the crossing of the tip in the grafted layer is small, which enables an easy comparison between AM and FM data.
2. Experimental details The experiments were performed with a modified atomic force microscope made with a Digital Instruments contact NII head and nanoscope E controller with a Perkin Elmer 7280 DSP lock-in amplifier performing the excitation of the cantilever piezo-ceramic and the amplitude and phase measurements for the AM mode. The FM data with constant oscillation amplitude are recorded with an Omicron electronics that has been modelled with a virtual NC-AFM [14]. The cantilever–tip used is a supersharp tip ‘SSS-NCL’ from NanoAndMore [15]: the resonance curve is very close to the harmonic oscillator curve. The resonance curve has been recorded at about 100 nm from the surface, giving a quality factor Q of 340 and a resonance frequency of 159 186 Hz (see figure 2). The tip has a
Extraction of conservative and dissipative parts in tapping: comparison with FM data Normalized amplitude a=A/Ao
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Frequency (Hz) Figure 2. Experimental normalized resonance curve recorded at 100 nm from the surface. For the sake of clarity only one point out of four has been represented. The quality factor given by the Lorentzian fit is Q = 340. The two AM driving frequencies used are marked, giving the coupled values: (a = 0.707, ϕ = −45◦ ) and (a = 0.867, ϕ = −60◦ ).
A mixed monolayer of n-octadecyltrichlorosilane (Cl3 Si(CH2 )17 CH3 ; OTS) and 21-aminohenicosyltrichlorosilane (Cl3 Si(CH2 )21 NH2 ; AHTS) was grafted on a silicon wafer by immersing the Si substrate for two hours, by coadsorption of OTS and AHTS (90/10 respectively). The experimental procedure was performed as previously reported [14]. The synthesis of long chain terminally aminated silicon compounds and full characterizations of the modified surfaces will be presented in a forthcoming paper [15]. For this study, the most important feature of the sample is its flatness to ease the AFM analysis (see figure 3), its chemical homogeneity to prevent the tip pollution with organic components and its mechanical liquid-like behaviour to prevent tip morphological change due to wear. The thickness of the layer has been measured on a hole in the grafted layer (data not shown) and is of about 3.5 nm. All the data have been recorded at the same location on the surface with the same tip. No noticeable change was seen between the beginning and ending of the measurements.
3. Analytical transformation of the experimental data: determination of the frequency shift and dissipation from recorded AM amplitude and phase variations
very small attractive interaction with the sample corresponding to a small radius of the apex. For the AM experiments, to reach contact situations between the tip and the grafted layer, the driving frequency of the oscillation was slightly smaller than the resonance one. To check our analytical transformation, the AM data are recorded at two frequencies as displayed in figure 2:
To understand the following, it is useful to start with the expressions of the cosine and sine of the tip–cantilever oscillator without any interaction as a function of the tip normalized amplitude a = A/ A0 where A0 is the resonance amplitude: cos(φ) = Qa(1 − u 2 ) (1) sin(φ) = −ua
• a frequency√ (158 961 Hz) giving the resonance amplitude divided by 2 corresponding to a phase delay of −45◦ because at this point the slope of the variation of the amplitude with the frequency is the largest one; • a frequency (159 058 Hz) giving the amplitude divided by 1.15 corresponding to a phase of −60◦ to compare two different settings for the AM mode.
where Q is the quality factor of the oscillator and u the frequency normalized by the resonance frequency: u = ν/ν0 . To extract the conservative part from the AM amplitude and phase signals, we use analytical expressions derived from the least action principle [18]. The main hypothesis is that we do not consider frequencies other than the fundamental one [19], nor any chaos appearing [20].
Those two driving frequencies have another benefit for our AM experiments: when driven at a frequency lower than the resonance one, the oscillator resonance curve shape deformation, due to attractive interaction (see [5]), leads to an amplitude jump that helps to locate the sample position as it appears within 0.5–1 nm away from the sample for organic materials. It is thus possible to transform the piezo-ceramic vertical displacement toward the tip–sample distance with a precision better than 0.5 nm for the AM data. To reach the same amplitude value for the two driving frequencies, two different excitations were used. For the FM data, the origin of the sample–tip distance is found on the frequency curves, at a frequency smaller than the resonance one: at the inflexion point where the repulsive contribution starts to be noticeable. The damping signals were normalized with the value of the damping far from the sample for comparison with the AM data. Typical piezo vertical displacements were of 10 nm with a frequency sweep of 0.5 Hz, and thus a piezo-ceramic speed of about 10 nm s−1 .
cos(φ) − Qa(1 − u 2 ) = Qa f (A, D, C) where C is a constant depending of the type of interaction. As an example, we can consider an attractive interaction between a spherical tip and a flat surface. With a Hamaker constant H and R the tip radius, the expression for the cosine of the phase is cos(φ) = Qa(1 − u 2 ) −
HR Qa 3 (d 2 − a 2 )3/2 3kc A0
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where d is the reduced tip–sample distance d = A/ A0 , and kc the cantilever stiffness. The first term is due to the harmonic oscillator as seen previously, and the second term is due to the attractive sphere–plane interaction. To compare the AM data with the FM data, this attractive term can be linked to the frequency shift through the expression [21] f 2 − f 02 ∼ f − f 0 ∼ HR 1 . =2 =− f0 f 02 3kc A30 (d 2 − a 2 )3/2
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Figure 3. AM height (left) and phase (right) image of 5 µm per 5 µm of the grafted layer on silica. The scale bar corresponds to 5 nm for the height image and 20◦ for the phase image. Both the height and phase signals are flat and homogeneous over the whole surface, showing the homogeneity of the grafted surface.
Thus we can get the frequency shift from the experimental amplitude and phase variations with f ≈ 2 f0
cos(φexp ) − Q exp aexp (1 − u 2exp ) Q exp aexp
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where φexp and aexp are recorded experimental data, u exp is specified with the driving excitation reduced with the resonant frequency as measured with the resonance curve, and Q exp is adjusted to zero the term without interaction cos(φ) − Qa(1 − u 2 ). The value of Q obtained is 360 (for data recorded at 158 961) or 355 (for data recorded at 159 058 Hz), values close to the one obtained from the resonance curve fits: 340. Note that the expression (4) is independent of the type of interaction, and thus independent of the model used to describe the interaction. An example is given for an attractive interaction between a spherical tip and a flat surface, but it can be used for any other interactions, such as an interaction between a spherical tip and a melted sample, for example [17, 22]. When assuming the description of an added dissipation due to the tip interaction with the sample, to first order in velocity and whatever the physical origin of the damping [23], the energy dissipated per period is given by γint int E dis (5) = πω Al2 (γ0 + γint ) = πω Al2 γ0 1 + γ0 where γ0 is the tip–cantilever intrinsic damping and γint is the additional damping due to the interaction. The sine of the phase is then given by [18] γint . sin(φ) = −ua 1 + γ0 Thus the normalized dissipation from the experimental amplitude and phase variation is sin(φexp ) γtot γint =1+ =− . γ0 γ0 u exp aexp
(6)
The results of equation (6) can be directly compared to the normalized damping signal. For the sake of clarity, the theoretical expressions are presented with reduced coordinates, but they obviously can be rewritten with raw data. 904
Then the experimental AM data at two driving frequencies are transformed with expressions (4) and (6) to extract the conservative and dissipative parts of the interaction. To compare the AM and FM data, the plots are displayed as a function of the tip indentation into the sample given by the difference δ = A − D.
4. Determination of the conservative and dissipative parts from AM amplitude and phase signals To validate the separation of conservative and repulsive parts from AM measurements, we first compare the results of expression (4) and (6) on the experimental AM curves displayed in figure 4(a) for the amplitude and 4(b) for the phase variation with tip–sample distance. As explained in the methodology part, the transformation of the piezo-ceramic displacement to the tip–sample distance is possible thanks to the driving frequencies being smaller to the resonance one. The curves are recorded at the same amplitude (12.5 nm), but at two different driving frequencies corresponding to a free phase (when the cantilever is far from the surface) of −45◦ and −60◦ . Both amplitude curves present an amplitude transition to a larger value due to attractive interaction [5]. After the transition, the amplitude changes slowly for about 3–4 nm (corresponding to a small interaction with the liquid-like grafted layer thickness of about 3.5 nm) and then drops down more rapidly as the piezo moves towards the surface, corresponding to the silica surface. The two phase variation shows, after the jump due to attractive interaction, the contribution of the fluid layer, which then starts to increase sharply due to the strong repulsive contribution of the silica surface. The result of the separation using relations (4) and (6) is displayed in figure 5. To compare the same situations (same tip–sample distance and same amplitude), the horizontal axis has been changed for the indentation δ into the sample given by δ = A − D. When moving the surface towards the tip, the frequency resonance first decreases due to attractive forces. Then the repulsive forces arise, leading to a change of frequency variation, and finally a frequency shift increasing rapidly as the tip touches the silica surface. The smallest frequency shift is about 70 Hz. Again, we note that the smooth
Extraction of conservative and dissipative parts in tapping: comparison with FM data
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change, between −20 and −70 Hz, due to the grafted layer, occurs over a distance of 3.5 nm, corresponding to the grafted layer thickness. Then, a large repulsive contribution due to the silica surface is observed. For the dissipative part, the additional dissipation in the grafted layer is a small percentage higher than the dissipation of the non-interacting tip. For the two data sets obtained at two frequencies, the conservative and dissipative contributions to the AM measurements are very similar, as expected. This result is encouraging for us to go further and compare the AM data to FM measurements.
5. AM/FM comparison Figure 6 shows the comparison of the AM extracted conservative part with FM frequency shift (a) and of the AM extracted dissipative part with FM normalized damping signal (b) by using equations (4) and (6). The calculated AM data display very similar frequency shifts and normalized damping to the FM data with the characteristic length of the grafted layer of about 3.5 nm. This
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Figure 5. Extraction of conservative (‘ν’, (a)) and dissipative (‘normalized dissipation’, (b)) from the AM measurements at two different driving frequencies displayed in figure 4, by using expressions (3) and (4). The shift frequency minimum is −67 Hz for the driving frequency 158 961 and −72 Hz for the driving frequency of 159 058 Hz, quite usual values for an amplitude of 12.5 nm. The conservative and dissipative parts for the two data sets are very similar, as expected: the physics of interaction revealed by the separation of conservative and dissipative parts depends on the oscillator, on the amplitude and the tip–sample distance, but not on the driving frequency.
excellent agreement shows that the two modes are just different ways to probe the same tip–sample interaction. Figure 7 shows one more comparison between AM extracted conservative parts and FM frequencies and AM extracted dissipative parts and FM normalized dissipation for a larger amplitude (26 nm) and for the two preceding driving frequencies. Again, the AM extracted parts for the two driving frequencies and FM measurements are very similar.
6. Discussion The experimental comparisons all agree with each other, showing experimentally that the two dynamical AFM modes AM and FM are different ways to measure the same tip to sample interaction (provided that the amplitude and the distance are the same), as expected since late 1990s. This good agreement validates the methodology used, which is not model dependent, so long as the other harmonic 905
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Figure 6. Comparison of conservative parts extracted from AM measurements of figure 4 with FM frequency shift (a) and dissipative parts extracted from AM measurements of figure 4 with normalized damping signals (b) recorded at an amplitude of 12.5 nm at the same location on the grafted surface. There is about 10 Hz discrepancy between the AM extracted frequency shift minimum and the FM value of about −81 Hz. For the dissipation, the FM data level of discrepancy with respect to the AM data is similar to the discrepancy between the two AM data and indeed very small: the largest normalized dissipation difference measured at about 4 nm indentation is γ = 1.046 − 1.025 = 0.0021 = 0.21%, with the γ0 mean energy dissipated per oscillation evaluation E = kcQA = 40 × 10−16 /400 = 10−17 J; this discrepancy gives a energy difference of 2.1 × 10−20 J.
contributions remain negligible. In addition, one has to emphasize that there is no fit; only experimental data are used and compared via analytical expressions (4) and (6). However, there is an obvious difference between the AM and FM curves: the amplitude is not constant in the AM mode whereas it is kept constant for the FM mode. Here the comparison was made easy because the reduction of the amplitude is small in the grafted layer: less than 10%. This is the reason why the liquid-like behaviour of the grafted layer, which leads to a small change of the oscillator amplitude, is a good system to compare the two modes. This kind of separation of the conservative and dissipative parts is helpful in understanding more complicated data sets 906
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Figure 7. Comparison of AM and FM data for a larger amplitude than figure 6. (a) Comparison of the extracted conservative part from AM measurements at two frequencies with FM frequency shift: the discrepancy between the two AM calculated frequency shifts and the FM measured frequency shift is very small; see for example the minimum frequency of −29 Hz for the AM and −31.5 Hz for the FM data. (b) Comparison of the extracted dissipative part of AM data with FM normalized damping. The dissipation variation is nearly identical for AM extracted and FM data with a step on the grafted layer and a sharp increase for the silica.
with, for example, the jump of a soft sample towards the surface [17, 24]. As an example here, the grafted layer added dissipation in figure 5 is about 2%–3% of the free cantilever and tip dissipation. We can estimate the energy level corresponding to this percentage. Taking a stiffness of 40 N m−1 , an amplitude of 10 nm and a Q factor of 400 gives E=
kc A = 40 × 10−16 /400 = 10−17 J Q
and an added energy due to the tip dissipation into the layer of 2 × 10−19 J, thus nearly 1 eV or 40 kb T . Consider a dissipation of kb T per silane molecule, then 40 silane molecules are underneath the tip. With an area per molecule of 0.3 nm2 (a value a little higher than the close packing of the alcane chains of polyethylene), the contact area between the tip and the grafted layer is about 12 nm2 , and thus a contact radius of 2 nm as can be expected for a supersharp tip.
Extraction of conservative and dissipative parts in tapping: comparison with FM data
7. Conclusion The aim of this paper is to separate experimentally the conservative and dissipative parts in the dynamical amplitude modulation mode of an atomic force microscope. The mixing of those two parts in the measured amplitude and phase signals is a major drawback in the analysis of AM data despite the large popularity of this mode for soft sample investigations. This separation is possible by using analytical expressions derived from the least action principle that only use experimentally available data. The main hypothesis for the use of this separation procedure is the absence of other harmonic contributions. The procedure for this separation and the transformation of the piezo-ceramic displacement to the tip– sample distance necessary for the comparison of two data sets is described in detail. To validate the procedure, two sets of experiments were done: we first compared tapping experiments recorded at two driving frequencies, and then compared the results with those obtained with the frequency modulation mode for the same amplitude. The FM data measure directly the conservative part in the frequency shift and the dissipative part in the damping signal. A grafted layer was used for the recording of AM and FM data at the same amplitude and same location for two amplitude values: 12 and 26 nm and two driving frequencies for the AM measurements. Because of the liquid-like behaviour of the grafted layer, it was possible to directly compare AM and FM curves as the AM amplitude variation on the grafted layer is relatively small. The agreement obtained without any fitting procedure between transformed AM measurements recorded at two driving frequencies and between the AM and FM measurements is very good. The good agreement between the FM and AM data at two driving frequencies measured at the same location and with the same amplitude is an experimental proof that the two AM and FM modes are two different ways to probe the same interaction between a tip and a sample. This agreement also validates the extraction of conservative and dissipative parts in AM measurements, which can help in understanding peculiar AM results such as those recorded on soft samples.
Acknowledgments This work was supported by grants from the Conseil Regional d’Aquitaine and University of Bordeaux 1 (BQR).
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