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Tilt of Atomic Force Microscope Cantilevers: Effect on Spring Constant and Adhesion Measurements Lars-Oliver Heim, Michael Kappl, and Hans-Ju¨rgen Butt* Max-Planck-Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany Received November 12, 2003. In Final Form: January 14, 2004 In atomic force microscopy, the cantilevers are mounted under a certain tilt angle R with respect to the sample surface. In this paper, we show that this increases the effective spring constant by typically 1020%. The effective spring constant of a rectangular cantilever of length L can be obtained by dividing the measured spring constant by cos2 R(1 - 2D tan R/L). Here, R is the tilt angle and D is the size of the tip. In colloidal probe experiments, D has to be replaced by the radius of the attached particle. To determine the effect of tilt experimentally, the adhesion force between spherical borosilicate particles and planar silicon oxide surfaces was measured at tilt angles between 0° and 35°. The experiments revealed a significant decrease of the mean apparent adhesion force with a tilt of typically 20-30% at R ) 20°. In addition, they demonstrate that the adhesion depends drastically on the precise position of contact on the particle surface.
Introduction Since its invention, the atomic force microscope (AFM)1 has gained widespread use and it has become a routine tool in surface analysis. One important application is the measurement of force curves.2 When analyzing force curves, it is always assumed that the cantilever is oriented horizontally with respect to the sample surface. In reality, this is not the case. Usually cantilevers are mounted under a certain tilt R (Figure 1). A tilt is necessary to ensure that the tip, and not the chip onto which the cantilever is attached, touches the sample first. In commercial AFMs, the tilt angle ranges from 7° to 20°. Knowledge of the spring constant is a prerequisite for any quantitative AFM force measurement. In this paper, we consider the increase of the effective spring constant with tilt angle. This increase can lead to a systematic error in AFM force measurements, such as for example adhesion measurements. Adhesion phenomena are important in everyday life and industrial applications such as coating technology and powder processing and handling. With the invention of the atomic force microscope, adhesion could for the first time be studied directly at the nanometer scale. In chemical force microscopy, the adhesion between surfaces of defined surface chemistry is analyzed.3 Furthermore, by attaching particles to the end of AFM cantilevers the interaction between particles of typically 2-30 µm diameter can be measured.4 In many publications, only the relative adhesion is determined, that is, the adhesion force at a certain position of a sample with respect to another position on the sample. In this way, the distribution of functional groups on a surface can be monitored. Relative adhesion experiments are not affected by cantilever tilt since the tilt would affect all adhesion values equally. In other applications, absolute values of the adhesion force are measured. In absolute adhesion
Figure 1. Schematic drawings of (a) a cantilever tilted by an angle R with respect to a horizontal surface and (b) a microfabricated tip of length D and a spherical particle of radius R attached to the end of the cantilever.
* To whom correspondence should be addressed. E-mail: butt@ mpip-mainz.mpg.de. Fax: +49-6131-379-310. Phone: +49-6131379-111.
experiments, a tilt of the cantilever can have a significant influence. In particular, when studying the adhesion between fine particles absolute values are required.5-7 This study was motivated by another question: Does the adhesion of a particle depend on the precise position of contact or is it the same irrespective of the precise location of contact on the particle surface? This is relevant with respect to the flow behavior of powders and granular media. If adhesion depends on the precise location of contact, the “high-affinity sites” on the particle surface are going to dominate the aggregation and flow behavior. By changing the tilt angle of the cantilever, the position of contact with a planar surface could be changed for single particles. In the theory section, we calculate the effect of cantilever tilt on the effective spring constant. For a tilted cantilever, only part of the force is detected because the force is not directed perpendicular to the cantilever. In addition, the
(1) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930-933. (2) Cappella, B.; Dietler, G. Surf. Sci. Rep. 1999, 34, 1-104. (3) Noy, A.; Vezenov, D. V.; Lieber, C. M. Annu. Rev. Mater. Sci. 1997, 27, 381-421. (4) Kappl, M.; Butt, H.-J. Part. Part. Syst. Charact. 2002, 19, 129143.
(5) Heim, L. O.; Blum, J.; Preuss, M.; Butt, H.-J. Phys. Rev. Lett. 1999, 83, 3328-3331. (6) Heim, L.; Ecke, S.; Preuss, M.; Butt, H.-J. J. Adhes. Sci. Technol. 2002, 16, 829-843. (7) Ecke, S.; Raiteri, R.; Bonaccurso, E.; Reiner, C.; Deiseroth, H. J.; Butt, H. J. Rev. Sci. Instrum. 2001, 72, 4164-4170.
10.1021/la036128m CCC: $27.50 © 2004 American Chemical Society Published on Web 02/13/2004
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force component directed parallel to the cantilever causes a torque. This torque leads to only a small change of deflection but a significant change of the inclination of the cantilever at the end. Since commercial instruments use the optical lever technique to detect cantilever deflection, which actually measures the inclination, the torque causes a change in the detected signal. In the Experimental Section, we describe results of adhesion (pull-off) forces measured under different tilt angles. The experiments were done with a custom-made setup in which the tilt of the cantilever can be changed. In this way, we measured the adhesion force versus tilt angle. Calculations We consider a cantilever of length L with a rectangular cross-section of thickness d and width w. It is tilted by an angle R with respect to the sample surface (Figure 1). We assume that in a force measurement the distance is varied in a direction normal to the sample surface. The shape of the cantilever is described by a function z(x), where z is the vertical and x the horizontal coordinate. We calculate the shape of the cantilever at a given force F acting at the end of the tip. Three cases are considered. (i) In the first case, the size of the tip is neglected and we describe the behavior of a bare cantilever. (ii) Second, we analyze the shape of a cantilever with a microfabricated tip, which has a pointlike end of length D. (iii) In the third case, a spherical particle of radius R is attached to the end of the cantilever as it is commonly used in the colloidal probe technique. (i) To calculate the cantilever shape, we use the fact that in equilibrium the sum of all torques at any part of the cantilever is zero. If we neglect the extension of the tip, we have to take two contributions into account: First, we consider the torque due to the applied force. At a given position x, the lever arm is L cos R - x. The force perpendicular to this lever is F, leading to torque F(L cos R - x). (Alternatively we could have chosen a coordinate along the cantilever axis leading to a lever L - x/cos R and a force F cos R. This results in the same torque.) This torque is compensated by the restoring elastic response of the cantilever, which is described by a torque EI d2z/dx2, where I ) wd3/12. Considering both contributions, we get the differential equation
d2z F(L cos R - x) ) EI 2 dx
(1)
Rearranging eq 1 leads to
F d2z ) (L cos R - x) 2 EI dx
(2)
After integration and using the boundary condition dz/dx(x ) 0) ) -tan R, we get
z′ )
F dz ) (2L cos R x - x2) - tan R dx 2EI
(3)
changes xm. To be precise, we would need to determine xm from the condition that the whole length of the cantilever is equal to L:
L)
dz ∫0x x1 + (dx )
2
m
dx
(5)
As a good approximation, we can use xm ≈ L cos R. Inserting this into eq 4 leads to
∆z′ )
FL2 2 cos R 2EI
(6)
The inclination is reduced by a factor cos2 R compared to a horizontal cantilever. The effect on the deflection is even higher. With z(x ) 0) ) 0, an integration of eq 3 leads to
z)
(
)
F x3 L cos R x2 - x tan R 2EI 3
(7)
The deflection at the end of the cantilever is
∆z )
FL3 cos3 R 3EI
(8)
It is even reduced by a factor cos3 R. What is the consequence for a force measurement? In most AFMs, the optical lever technique is used to detect cantilever deflection. With the optical lever technique, the inclination and not the deflection is detected. According to eqs 6 and 8, both are proportional:
∆z )
2L cos R ∆z′ 3
(9)
The inclination is reduced by a factor cos2 R. We can take this into account by dividing the spring constant of the corresponding horizontal cantilever, 3EI/L3, by cos2 R to get an effective spring constant. The spring constant is defined as K ) F/∆z. Compared to the spring constant of a horizontal cantilever, one could expect that the spring constant of a tilted cantilever is increased by a factor cos-3 R (eq 8). Due to the fact that the proportionality between inclination and deflection changes by another factor cos R (eq 9), the effective spring constant changes by cos-2 R. Spring constants of AFM cantilevers can be measured in different ways. The most common methods are (1) to measure the change in resonance frequency caused by an added mass,8 (2) to determine the resonance frequency and the quality factor and then calculate K,9 and (3) to determine the amplitude of thermal noise.10 With all three methods, the uncorrected spring constant is determined and we have to divide it by cos2 R to get the effective spring constant. Calibration with a reference cantilever11,12 and some less frequently used methods, such as using the gravitational force of an attached particle,13 lead to the correct effective spring constant.
The inclination at the end of the cantilever changes by
F (2L cos R xm - xm2) ∆z′ ) 2EI
(4)
when a force is applied. Here, xm is the x coordinate of the tip; the index “m” indicates that this is the maximal x-value. For a horizontal cantilever without force, xm is equal to L. For a tilted cantilever without force, it is given by xm ) L cos R. An applied force bends the cantilever and
(8) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403-405. (9) Sader, J. E.; Chou, J. W. M.; Mulvaney, P. Rev. Sci. Instrum. 1999, 70, 3967-3969. (10) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 18681873. (11) Torii, A.; Sasaki, M.; Hane, K.; Shigeru, O. Meas. Sci. Technol. 1996, 7, 179-184. (12) Gibson, C. T.; Watson, G. S.; Myhra, S. Nanotechnology 1996, 7, 259-262. (13) Senden, T. A.; Ducker, W. A. Langmuir 1994, 10, 1003-1004.
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Figure 2. Correction factor (i.e., inclination of a horizontal cantilever divided by the inclination of a tilted cantilever to which the same force is applied) versus the tilt angle R. For a cantilever without a tip, this factor is cos2 R (eq 6, continuous line). For a cantilever with a tip, we used a typical tip size of D ) 5 µm and cantilever lengths of L ) 100 µm (dashed) and L ) 200 µm (dotted); the correction factor plotted is cos2 R (1 - 2D tan R/L) (eq 12).
(ii) Now we take into account the effect of a protruding tip. A tip of length D mounted perpendicular to the cantilever direction reduces the effective lever of the force from L cos R - x to L cos R - D sin R - x. As a result, eq 1 has to be modified:
d2z F(L cos R - D sin R - x) ) EI 2 dx
(10)
(
)
x2 F Lx cos R - Dx sin R - tan R (11) EI 2
The inclination at the end changes by
∆z′ )
2D FL2 cos2 R 1 tan R 2EI L
(
)
(12)
Thus, when a force is applied the change in the detected signal is given by eq 12 rather than by ∆z′ ) FL2/2EI as for a horizontal cantilever. For typical values of L and D, the correction factor, cos2 R (1 - 2D tan R/L), is plotted in Figure 2. The detected signal is typically 10-20% lower than the signal expected for a horizontal cantilever. Accordingly, forces are underestimated by 10-20%. (iii) For a sphere attached to the end of the cantilever, the situation is different because the interacting area changes with the tilt (Figure 1b). The torque is F(L cos R - R tan R - x), which leads to the differential equation
F d2z ) (L cos R - R tan R - x) 2 EI dx
(13)
Integrations and inserting L cos R for x lead to
∆z′ )
2R tan R FL2 cos2 R 1 2EI L cos R
(
)
(14)
which for small angles (R < 20°) can be written as a good approximation as
∆z′ )
2R FL2 cos2 R 1 tan R 2EI L
(
This equation resembles eq 12; we just have to replace the length of the tip D by the radius of the particle R. One consequence of the increased effective spring constant is that all forces reported so far (including our own) were underestimated by typically 10-20%. This is certainly small enough that the main messages of most publications on AFM force measurements are still valid. For precise quantitative measurements, however, this correction should be taken into account. Since the parameters (L, D, or R) are usually known, the effective spring constant can easily be calculated by dividing the measured spring constant with the appropriate correction factor. Experimental Section
Rearranging and integrating once leads to
z′ )
Figure 3. Schematic of the setup used to measure adhesion forces at different tilt angles of the cantilever.
)
(15)
The setup used to measure adhesion forces at different tilt angles is a modification of the particle interaction apparatus described earlier.14 In this setup (Figure 3), the sample was fixed. A piezoelectric translator (Physik Instrumente, Waldbronn, Germany) moved the “head” up and down. The head contained the cantilever with its optical lever to detect cantilever deflection. It included the laser diode (3.0 mW, 670 nm wavelength), the adjustable cantilever holder, the mirror, and the two-dimensional position sensitive detector (PSD, SiTek, Sweden, active area of 2 × 2 cm2). The whole head could be tilted. This allowed an adjustment of the tilt between cantilever and surface in the range between zero and about 35°. Coarse positioning of the cantilever-sample distance is done by a motorized stage onto which the sample is mounted by micrometer screws (range, 0.9 cm). The piezoelectric translator has a range of 12 µm and a root-mean-square (rms) noise of 0.1 nm. The piezoelectric translator is equipped with an internal capacitance distance sensor to measure the actual position. An electronic controller monitors the signal of the capacitance distance sensor and regulates the voltage supplied to the piezo to adjust the position. Thus, hysteresis and creep of the translator were avoided. The chip with the cantilever was clamped to a holder above the sample stage. Light from the laser diode was coupled into an optical fiber and focused on the backside of the cantilever using a microfocusing optic to obtain a circular spot of 8 µm in diameter. The PSD, used to monitor the position of the reflected laser beam, provided an output current, which was amplified to voltage U. This voltage was proportional to the position of the laser spot with respect to the vertical direction. It was used to monitor the deflection of the cantilever in the normal direction. Particle, cantilever, and sample could be observed with an optical microscope during the experiment. The measurement itself is controlled by a personal computer. The software controls the stepper motor for coarse approach, generates the driving signal for the vertical piezoelectric translator, and reads U plus (14) Preuss, M.; Butt, H.-J. J. Colloid Interface Sci. 1998, 208, 468477.
Tilt of AFM Cantilevers
Figure 4. Scanning electron microscope images of borosilicate particles attached to the end of cantilevers. The scale bars are 5 µm. Particles 6, 8, 9, and 10 (cf. Table 1) are shown as typical examples.
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Figure 5. Typical force-versus-displacement curve measured with a borosilicate sphere of 11.7 µm diameter on a silicon wafer in air. The tilt angle was 32.5°. Positive forces are repulsive, and negative forces attractive. The adhesion force is indicated.
an auxiliary input channel. The software, developed in LabView (National Instruments Corp., Austin, TX), consisted of a realtime panel to operate the instrument plus off-line data handling and analysis procedures. D/A and A/D conversions are achieved by a PCI-6052E board (National Instruments) with 16-bit resolution for both input and output. Off-line software allows manipulating batches of data, calculating force-versus-distance curves, and performing basic analysis like fitting before and after contact regions, extracting pull-off force, jump-into-contact distance, and indentation depths. We used spherical borosilicate particles of roughly 5, 10, and 20 µm in diameter (Duke Scientific Corp., Palo Alto, CA). The particles were glued onto tipless cantilevers (Figure 4) using a small amount of epoxy resin (Epikote 1004, Shell) by the use of a micromanipulator (Narishige, model MMO 203, Japan) under optical control. The particles were positioned on a glass slide on top of a heating stage. Beside the particles the resin was placed. After heating the stage above the melting temperature of the resin, a small amount was taken with the tipless cantilever before touching the desired particle. The tipless silicon cantilevers (MikroMasch, Tallinn, Estonia) used had a length between 130 and 250 µm and a width of 36 µm. Each spring constant was determined with a reference cantilever as described before.15 Force curves were recorded as follows. After mounting the head with the cantilever, the laser beam was adjusted onto the backside on the cantilever. It was refocused to minimize interference effects resulting from the reflection of the cantilever and parallel of the silicon wafer.16 A silicon wafer was used as a reference surface to minimize the surface roughness. Before the measurements, the wafer was washed with ethanol and rinsed with water from a Milli-Q water system. To use the maximal range of the PSD, the laser spot was placed in its center by adjusting the inclination of the mirror. In the next step, the tilt angle was chosen and checked by optical microscopy. After coarse positioning of the particle above the wafer by the use of the z-motor and the manual stage, the piezo translator was set to a triangular periodic up-and-down movement. Then the software-controlled motion of the motorized stage was used to decrease the distance until contact between particle and surface was established in the lowest position of the movement. The unavoidable variation of the force applied to the contact before measuring the pull-off force (also called adhesion force in this article) has no influence on the result as shown in other experiments.6 This also holds for the humidity in the range present in our experiments (relative humidity of 35-45%). The frequency of the measure cycle was 1 Hz, resulting in a velocity of 24 µm/s between particle and reference. All particles (except particles 7 and 4) were inspected by scanning electron microscopy (Leo Gemini 1530) after the experiments. The microscope was operated at a voltage of 1-3
A typical force-versus-displacement curve recorded with a borosilicate microsphere on a silicon wafer is shown in Figure 5. From the retracting part, the adhesion force is obtained; it is the difference between the most negative force and the zero line. Adhesion forces were measured for at least 10 different positions on the sample. At each position, at least 5 force curves were recorded and the mean of all values was taken. The mean adhesion forces recorded at zero tilt angle together with the particle radii are listed in Table 1. For a better comparison, the adhesion forces were normalized by dividing them by the radius. The values range between 0.18 and 0.38 N/m with a total mean value of 0.24 N/m. According to the theory of Johnson, Kendall, and Roberts (JKR),17 the adhesion force Fa can be related to a work of cohesion W by Fa ) 3πRW/2, where R is the radius of the particle. Instead of W, usually a surface energy parameter γ is reported, where W ) 2γ. We find γ ) 0.025 N/m. This value agrees with previous results.5 Also, the variation of adhesion forces is of the same order as observed before by centrifuge and AFM experiments.5,6,18-20 Averaged adhesion forces were determined for tilt angles between 0° and 35° (Figure 6). We started with low tilt
(15) Preuss, M.; Butt, H.-J. Int. J. Miner. Process. 1999, 56, 99-115. (16) Jaschke, M.; Butt, H.-J. Rev. Sci. Instrum. 1995, 66, 12581259.
(17) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-313. (18) Polke, R. Bull. Soc. Special Chim. France 1969, A3241, 51-54.
Table 1. Number of Silicon Oxide Microspheres, Mean Adhesion Force Fa,a Spring Constant of the Cantilever K, Radius of the Particle R, and Normalized Adhesion Force Fa/R particle
Fa (nN)
R (µm)
K (N/m)
Fa/R (N/m)
1 2 3 4 5 6 7 8 9 10
1027 2097 505 869 2067 1366 440 1130 1112 2375
2.71 10.40 2.80 ≈5 9.24 5.83 ≈5 5.59 5.39 8.21
0.62 0.24 0.35 0.65 0.15 0.4 0.10 0.22 1.13
0.38 0.20 0.18 0.17 0.22 0.23 0.09 0.20 0.21 0.29
a Measured at 10 or more positions and averaged over at least 5 force-versus-displacement curves.
kV so that it was not necessary to coat the particles with a conducting layer.
Results and Discussion
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Figure 6. Mean adhesion forces versus tilt angle of the cantilever determined for 10 different microspheres. Each microsphere corresponds to one symbol. Adhesion forces are scaled for each particle by dividing each value with the mean adhesion force averaged over all tilt angles (and division by 1.4 to get a total mean value of roughly 100% at zero tilt). The error bars are the standard deviations of the adhesion force. The correction function calculated with eq 15 (R ) 5 µm, L ) 200 µm) is plotted as a continuous line.
angles going up to 35° and then reduced the tilt again in order to verify the results. To be able to compare adhesion forces, they were scaled by dividing with the mean (averaged over all angles) adhesion force. In this way, the large variation in absolute values was removed in Figure 6. Two phenomena were observed: First, there is a tendency for the apparent adhesion force to decrease with tilt angle. Second, for individual particles the adhesion force varied in an unpredictable although reproducible way from one tilt angle to the other. We discuss these two phenomena in the following. Decrease of Adhesion with Tilt Angle. The apparent adhesion force is expected to decrease proportional to cos2 R (1 - 2R tan R/L) according to eq 15. This was observed. Within the limit of our measurements, the results agree with the prediction. All adhesion forces determined by AFM reported so far should be corrected according to eq 12 or eq 15. Variation of Adhesion with Tilt. When changing the tilt angle, the precise location of the contact on the particle surface changes. For each degree change in tilt, the center (19) Polke, R.; Krupp, H.; Rumpf, H. Einflu¨ sse auf die Adha¨ sion von Feststoffteilchen; Stoffe, V. I. K. f. g., Ed.; Carl Hanser Verlag: Mu¨nchen, 1973. (20) Podczeck, F.; Newton, J. M.; James, M. B. Powder Technol. 1995, 83, 201-209.
Heim et al.
of the contact area shifts 175 nm assuming a particle radius of 10 µm. This change in the contact area caused a substantial fluctuation in the adhesion force. It could be much stronger than the general tendency described above. For example, for particle 9 (4 in Figure 6) the adhesion force was 1112 nN at R ) 0° and then decreased to 444 nN at R ) 10° and increased to 972 nN at R ) 18°. These values were reproducible and could be measured repeatedly. We could not correlate this unpredictable though reproducible variation with any structure observed with a scanning electron microscope (Figure 4). The underlying effect is probably the same one that leads to the variation in adhesion forces from particle to particle. We assume that it is caused by a heterogeneity of the particle surface. This might be a heterogeneity in the structure (namely, surface roughness), mechanical properties such as elasticity, chemical composition, or different crystalline faces at the surface. The subject of surface heterogeneity is thoroughly discussed for wetting phenomena.21,22 A relevant question is at what length scale the surface is heterogeneous and changes its properties. This length scale cannot be much smaller than the diameter of the contact area. Otherwise it would be averaged out and no variation of adhesion forces would be observed. Using JKR theory for a silicon oxide particle on a silicon oxide surface, we can estimate the radius a of the contact area:
a3 ≈
3R (F + 3πWR + x6πWRF + (3πWR)2) (16) 2E
Inserting E ) 7 × 1010 Pa, R ) 5 µm, and W ) 0.05 N/m, we get a typical contact radius of a ) 80 nm at zero force. For this reason, we conclude that surface heterogeneities which vary at length scales of the order of 100 nm cause the variation observed in adhesion experiments. Conclusions The apparent adhesion force decreases with the tilt angle of the cantilever by typically 20-30% at a tilt of 20°. To a large degree, this can be explained with the increase of the effective spring constant. In addition, the adhesion force varies with the precise position of the contact area on the particle surface. A typical lateral shift of the order of 100 nm could change the adhesion substantially. This is relevant with respect to understanding the flow behavior of powders. The “high energy” areas on the surface of particles will dominate it. LA036128M (21) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11-16. (22) Drelich, J.; Wilbur, J. L.; Miller, J. D.; Whitesides, G. M. Langmuir 1996, 12, 1913-1922.
