IOP Publishing doi:10.1088/1742-6596/61/1/011
Journal of Physics: Conference Series 61 (2007) 51–55 International Conference on Nanoscience and Technology (ICN&T 2006)
Measurement of friction coefficients with the atomic force microscope Phil Attard1 , Johanna Stiernstedt2 and Mark W. Rutland2 1
School of Chemistry F11, University of Sydney, NSW 2006 Australia. Department of Surface Chemistry, Royal Institute of Technology, SE-100 44 Stockholm, Sweden and Institute for Surface Chemistry, Box 5607, SE-114 86 Stockholm, Sweden
2
E-mail:
[email protected] Abstract. A new axial method for measuring the friction coefficient with the atomic force microscope is given. This axial method requires no calibration steps and is performed simultaneously with a normal force measurement by measuring the difference between the constant compliance slopes of the extend and retract force curves. The algorithm can be applied retrospectively to extract the friction coefficient from preexisting force measurements. Results are in quantitative agreement with the more established lateral method. The method can be used for both tipped cantilevers and for attached spherical probes.
1. Introduction The atomic force microscope, (AFM), has been extensively used in lateral scanning mode to measure or to image friction qualitatively. Comparatively fewer quantitative measurements have been made due to the difficulty in calibrating the torsional spring constant of the cantilever and the voltage response of the photodiode to cantilever twist. (See Ref. [1] for a critical review of calibration techniques.) The main methods that have proved quantitatively reliable [2, 3, 4, 5, 6] all involve an ex situ calibration that is separate to the friction measurement, and they all require special modifications to the substrate or to the cantilever, and very particular care in reassembling the cell for the actual measurement to preserve the calibration parameters. Here a simple method for quantitatively measuring the friction coefficient with the AFM is presented and shown to be reliable by comparison with lateral measurements. The friction coefficient is measured when the colloid probe slides in the axial direction of the AFM cantilever, similar to earlier suggestions [7, 8, 9, 10], but here only a normal AFM force measurement need be performed. The friction coefficient is extracted from the change in slope of the constant compliance regions. This new axial friction method requires no extra calibration beyond that of a normal force measurement. The method exploits the fact that the AFM cantilever is pitched at an angle of about 10◦ to the substrate, which causes the probe to slide in the axial direction as the substrate is driven upwards. The amount of axial sliding can be substantial, of the order of 50–100% of the vertical motion, depending upon the size of the probe and length of the cantilever, (see Eq. (4) below). This sliding causes a friction force in the axial direction that in turn exerts a torque on the tip of the probe, causing a bend in the cantilever in addition to that due to the vertical surface force.[9] The friction force reverses direction when the piezo-drive changes from extend to retract, and © 2007 IOP Publishing Ltd
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Photo-diode Voltage (V)
2.5 2 1.5
L1
L0
a
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b
c
L2
1 0.5 0 -0.5 -1 -900
-800
-700
-600
-500
Piezo-drive Displacement (nm)
Figure 1. Cantilever deflection for a silica colloid probe of diameter (a) 36 µm, (b) 16 µm, and (c) no attached probe. The difference in the constant compliance slopes on the extend and the retract branches when the surfaces are in contact is due to sliding friction.
the difference in torque changes the amount by which the cantilever is bent at a given point on the two branches. Hence the slopes of the extend and retract constant compliance lines are different. This effect is shown in Fig. 1, where it can be seen that a measurable difference in the slopes occur. The effect is larger for larger probes due to the greater torque that the friction force exerts. Usually this discrepancy in slopes is ignored, and typically one chooses one or other as the constant compliance factor to convert the vertical photo-diode voltage into cantilever deflection and thence into force. Our method exploits the difference in slopes to obtain the friction coefficient, as well as the correct calibration factor for the force conversion. The results below are based upon the analysis and geometry given previously [9, 10, 11]. The model, (see Fig. 1), consists of a rigid tip, (or colloid probe), of length L2 attached at a point L1 + L0 from the fixed base of the cantilever, where L0 is the length of the flexible part of the cantilever, and L1 is the length of the rigid base of the tip, (or the extent of the glue attaching the probe). Usually, L0 L2 ≈> L1 . The cantilever makes an angle θ0 < 0 to the horizontal. Although not required here, for force measurement in general one should take into account the length used to measure the spring constant of the cantilever, Lcal . If the resonance method is used [12], this is the length of the cantilever at which the masses are attached. If the thermal method is used [13, 14, 15, 16], it is the full length of the cantilever. The effective spring constant that should be used to obtain the normal force in an AFM measurement is[9, 10] keff =
kcal cos2 θ0
Lcal L0
3
,
(1)
where kcal is the calibrated spring constant. Note the very strong dependence on the lengths; in a typical situation the effective spring constant can be a factor of two greater than the calibrated one. Existing force measurements reported in the literature should be corrected by this factor. 2. Experimental Method A Nanoscope III AFM (Veeco) was used, together with a pico-force unit to measure the position of the piezo-drive independently. The cantilevers were uncoated, tipless, rectangular silicon cantilevers (MicroMasch, Tallinn, Estonia), with silica beads (Duke Scientific Corporation, USA), attached in-house with Casco Araldite Rapid epoxy adhesive. The dimensions of the cantilevers and colloidal probes were measured in an Environmental Sweep Electron Microscope (ESEM). Two cantilevers with silica spheres of different size were compared. The cantilever with the large sphere (radius 18 µm) had L0 = 90 µm, L1 = 11 µm, L2 = 36 µm, and kcal = 0.5 N/m, Lcal = 112 µm. The cantilever with the small sphere (radius 8 µm) had L0 = 113 µm, L1 = 5 µm, L2 = 16 µm, and kcal = 0.4 N/m, Lcal = 118 µm. The force and friction measurements were performed on mica in 0.1 mM NaCl in Milli-Q water. Immediately before use, the cantilevers were thoroughly rinsed with water and ethanol, dried under a flow of nitrogen, and cleaned with
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oxygen plasma (PDG-32G Plasma Cleaner, Harrick Scientific Corp., USA) on medium setting for 30 seconds. The friction coefficients were extracted from the difference between the extend and the retract constant compliance slopes, each of which represents an average value over the contact part of the force curve. The friction coefficients so obtained were then averaged over 10–15 force measurements, and the standard error on the mean obtained from these is reported below. For comparison, lateral friction measurements were made for the same set-ups as for the force measurements. Prior to the experiments, the lateral calibration was performed using a combination of the methods of Bogdanovic et al.[4] and of Feiler et al.[5] The cantilever to be calibrated was deflected and twisted by pressing it off-center with the tip of a tipped cantilever glued tip-up on the substrate, and this gave rise to a lateral deflection signal in the detector [4]. The ratio of the vertical and the lateral signals together with the lateral off-set distance was used to obtain the lateral calibration factor [5]. 3. Analysis Method Based upon the geometry shown in Fig. 1 and the analysis in Refs. [9, 10], the following constant are defined, a ≡ L1 cos θ0 + L2 sin θ0 2 c ≡ L30 cos θ0 + L20 a 3
b ≡ L1 sin θ0 − L2 cos θ0 d ≡ L20 cos θ0 + 2L0 a,
(2)
and also, 2 f ≡ L30 sin θ0 + L20 b, 3 i ≡ c cos θ0 + ad,
h ≡ L20 sin θ0 + 2L0 b, j ≡ f cos θ0 + ah.
(3)
In terms of these, the ratio of the horizontal axial motion of the probe on the substrate y2 to the vertical piezo-drive distance z2 , which is also equal to the ratio of the velocities, is c sin θ0 + bd y2 (t) y˙ 2 = = . z2 (t) z˙2 c cos θ0 + ad
(4)
If ∆V is the change in photodiode voltage, and if α± = ∆z2 /∆V± , (nm/V), are the measured slopes of the constant compliance lines on extension (+) and on retraction (−), then one defines e≡
−1 −1 α+ − α− −1 −1 . α+ + α−
(5)
The friction coefficient, which is the ratio of the horizontal friction force to the vertical surface force or load, µ = |Fy /Fz |, is given by µ=
ih − jd +
(ih − jd)2 + 4ihjde2 . 2jhe
(6)
Note that the spring constant of the cantilever was not required for this result. In this work we will not deal with surface forces directly. But for completeness the constant factor necessary to convert from the measured photo-diode voltage to surface force, ∆Fz = κ∆V , is 2kcal L3cal χ, (nN/V), (7) κ≡ 3d with the conversion factor for the change in angle of the cantilever being χ=
−1 −1 −1 + α− ) 2d(α+ , (nrad/V). i + µje
(8)
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
dFy/dFz
40 38 36 34 32 30 28 26 24 22 20
2
, (nm/V)
4. Results
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9 10 11 12 13 14 15
0.4 0 0
1 1
0.8
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50
5
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Load (nN)
100
9 10 11 12 13 14 15
Experiment Number
Experiment Number
Figure 2. Constant compliance factors for extension (filled symbols) and retraction (open symbols) for the large, (triangles, y˙2 = 1.4 µm/s), and for the small, (circles, y˙2 = 2.3 µm/s), colloid probes of Fig. 1. The small sphere data has been offset by -10 nm/V for clarity.
Figure 3. Friction coefficient. The average values are 0.27±0.01, (solid line, large probe, axial method), 0.29±0.03, (dotted line, small probe, axial method), and 0.31±0.01, (dashed line, large probe, lateral method, speed 2 µm/s, Fz ≥ 60 nN). The inset shows the local friction coefficient as a function of load obtained with the lateral method for the large (triangles) and small (circles) probes.
Figure 2 shows the measured constant compliance factors for the two probes in a number of experiments. The factor is larger for extension than for retraction. To put it another way, the rate of increase of detector voltage with piezo-drive distance, which is the slope, is less for extension than for retraction. This is because the torque due to the increasing friction force opposes the increasing cantilever bend on extension, whilst it enhances the bending on retraction. The relative difference between the measured constant compliances on each branch is on average 10% for the large probe and 4% for the small probe. That the effect of friction is greater for larger attached probes is consistent with the data graphically displayed in Fig. 1. Figure 3 shows the friction coefficient obtained from the constant compliance measurements, Eq. (6). The average value of the friction coefficient obtained in the axial measurements is in good agreement for the large and for the small probe, being 0.27±0.01 and 0.29±0.03, respectively. This is an important result since the diameter of the probe, the cantilever, and the placement of the probe on the cantilever were different for the two cases, as was the change in the slopes of the constant compliance lines, Fig. 2. That Eq. (6) properly accounts for these differences and yields a consistent value for the friction coefficient is strong evidence for the validity of the method. The mutual agreement between the value of the friction coefficient obtained with the axial method, (µ = 0.28), and that obtained with the lateral method, (µ = 0.31, large probe, Fz ≥ 60 nN, 20 friction loops, 5 loads), also confirms the validity of both methods. The lateral method shows a friction coefficient that increases with decreasing load at smaller loads for both the small and for the large sphere. This illustrates the importance of using the slope of the most linear part of the constant compliance curve to measure the friction coefficient. For the large probe in Fig. 3, the piezo-drive velocity was 1.8 µm/s and, from Eq. (4), the axial sliding velocity was 1.4 µm/s. For the small probe the piezo-drive velocity was 5.4 µm/s and the axial sliding velocity was 2.3 µm/s. The sliding speed in the lateral measurement was 2 µm/s. Based on this and unpublished data, the friction coefficient does not appear very sensitive to the speed in this regime. It is worth emphasizing one more conclusion from the data in Fig. 3: that the friction coefficient is here independent of the diameter of the colloid
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probe justifies by induction the science of nanotribology itself, since it validates measurements performed with colloidal-sized probes being extrapolated to macroscopic surfaces. 5. Discussion There are several restrictions on the present axial method for measuring the friction coefficient. The method is very sensitive to the difference in the measured slopes, and care must be taken to select the most linear part of the contact curve. Repeat measurements to average out noise and the effects of roughness are also necessary. One has ensure that the measured hysteresis between the extend and the retract curves is due to friction. Accordingly, the actual piezo-drive position has to be known accurately, preferably measured in situ with a position sensitive detector, either built in to the AFM, (as is done in the results reported here), or else as an added attachment [17, 18, 19, 20]. Similarly, the method cannot be used for viscoelastic materials where the hysteresis between extension and retraction is due to material relaxation rather than to friction [1, 19, 21]. (There does not appear to be any restriction on using the method for elastically deformable probes or substrates [1].) The relative change in slope is larger for larger probes, and so the method is less accurate or requires more repeat measurements for small colloid probes. Despite these restrictions, the present axial method has much to recommend it. It is quick, convenient, and requires no calibration. It can be performed during the course of a normal force measurement, and it can even be applied retrospectively to pre-existing data. The hysteresis is larger for larger probes, and the consequent ambiguity in the constant compliance slopes has in the past caused problems for normal force measurements. However, the present Eq. (7) now gives the correct calibration factor for this case, and in addition the friction data is commensurately more reliable for large probes because the difference in slopes is larger than the noise. The present results for silica sliding on mica showed reasonable agreement between the axial and lateral friction coefficients. It was also concluded that the friction coefficient was independent of the radius of curvature of the probe. This conclusion validates the application of nanotribological measurements to macroscopic surfaces. References
Attard P 2002 J. Adhesion Sci. Technol. 16 753 Ogletree D F, Carpick R W and Salmeron M 1996 Rev. Sci. Instrum. 67 3298 Meurk A, Larson I and Bergstrom L 1998 Mater. Res. Soc. Symp. Proc. 522 427 Bogdanovic G, Meurk A and Rutland M W 2000 Colloids Surfaces B 19 397 Feiler A, Attard P and Larson I 2000 Rev. Sci. Instrum. 71 2746 Ecke S, Raiteri R, Bonaccurso E, Reiner C, Deiseroth H J and Butt H J 2001 Rev. Sci. Instrum. 72 4164 Hoh J H and Engel A 1993 Langmuir 9 3310 Ruan J A and Bhushan B 1994 J. Tribol. Trans. ASME 116 378 Attard P, Carambassis A, and Rutland M W 1999 Langmuir 15 553 Stiernstedt J, Rutland M W, and Attard P 2005 Rev. Sci. Instrum. 76 083710 Attard P, Schulz J and Rutland M W 1998 Rev. Sci. Instrum. 69 3852 Cleveland J P, Manne S, Bocek D and Hansma P K 1993 Rev. Sci. Instrum. 64 403 Hutter J L and Bechhoefter J 1993 Rev. Sci. Instrum. 64 1868 Butt H J and Jaschke M 1995 Nanotechnology 6 1 Levy R and Maaloum M 2002 Nanotechnology 13 33 Sader J E, Chon J W M, and Mulvaney P 1999 Rev. Sci. Instrum. 70 3967 Hues S M, Draper C F, Lee K P and Colton R J 1994 Rev. Sci. Instrum. 65 1561 Koleske D D, Lee G U, Gans B I, Lee K P, DiLella D P, Wahl K J, Barger W R, Whitman L J and Colton R J 1995 Rev. Sci. Instrum. 66 4566 [19] Tyrrell J W G and Attard P 2003 Langmuir 19 5254 [20] Rutland M W, Tyrrell J W G and Attard P 2004 J. Adhesion Sci. Tech. 18 1199 [21] Gillies G S, Prestidge C A and Attard P 2002 Langmuir 18 1674
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
