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A Method for the Calibration of Force Microscopy Cantilevers via Hydrodynamic Drag Nobuo Maeda and Tim J. Senden* Department of Applied Mathematics Research School of Physical Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia Received November 16, 1999. In Final Form: July 31, 2000 A new method for the in situ and nondestructive calibration of cantilevers used in force microscopy employing hydrodynamics is described. Using the notion that the viscous drag on objects is scale invariant for similar Reynolds numbers, the drag on a 500:1 scale model of a typical cantilever was compared with that of the actual cantilever drag within the Stokes regime of laminar flow. The spring constant of the model cantilever was determined via static end loading with known masses, and the distributed load due to viscous drag was carried out in neat glycerol. A master curve of the ratio of distributed load to point load was determined as a function of distance from a horizontal surface. Similar curves were then obtained for actual cantilevers in water, the comparison of which to the master curve provided an estimate of the spring constant.
Introduction Since the invention of the force microscope and subsequent application to the field of surface force measurement over the past decade, it has always been of fundamental importance to know the spring constant of the cantilevers. Without precise knowledge of spring constants no quantitative force measurement is possible. Thus, it is not surprising that a number of methods have been developed to measure this simple quantity. Each has its advantages and limitations, depending upon the specific configuration of the cantilever used. Discussion and development of the various methods, outlined below, will no doubt continue for some time. The resonant frequency method1,2 and the gravitational field method3 require attachment of a known mass to the cantilever. This delicate process often renders the cantilever unusable for subsequent measurements, and given the large size of the mass the exact point of loading can be highly uncertain. The determination of the angular dependence of deflection on point loading works well only where the material moduli are known.4,5 A few comprehensive theoretical studies 6,7 have helped to redress some of these practical difficulties. One particular solution is to ensure a cantilever of simple geometry (typically a beam) exists alongside the cantilever used for the actual measurement. In this configuration the product of the Young’s modulus to the thickness cubed can be determined for the simpler geometry. This automatically reduces the problem of explicitly determining the material stiffness to knowing the geometric relationship between the more complicated cantilever and the simple beam.6,8 There are a few approaches which are model independent. The static ex situ loading of the cantilever with a * Corresponding author. E-mail:
[email protected]. (1) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64. (2) Walters, D. A.; Cleveland, J. P.; Thomson, N. H.; Hansma, P. K.; Wendman, M. A.; Gurley, G.; Elings, V. Rev. Sci. Instrum. 1996, 67, 3583-3590. (3) T. J. Senden and W. A. Ducker, Langmuir 1994, 10, 1003-1004. (4) C. J. Drummond and T. J. Senden, Mater. Sci. Forum 1995, 189190, 107-114. (5) Ogletree, D. F.; Carpick, R. W.; Salmeron, M. Rev. Sci. Instrum. 1996, 67, 3298-3306. (6) Sader, J. E.; Larson, I.; Mulvaney, P.; White, L. R. Rev. Sci. Instrum. 1995, 66, 3789-3798. (7) Sader, J. E. J. Appl. Phys. 1998, 84, 64-76.
precalibrated cantilever9,10 or a pendulum 1 are convenient but require the construction of a test rig. Care must be taken not to deflect the cantilever more than the deflections found under usual tip excursions and to maintain the surface cleanliness of the tip if precalibration is required. A heterodyne interferometry method12 is both accurate and ingenious but requires expensive equipment not usually available. Methods based on the measurement of the cantilever’s intrinsic thermal noise13,14 are elegant and require little in the way of equipment. Sources of extraneous noise can complicate the direct determination of the spring constant, as can assumptions concerning bending modes. These issues are currently being addressed and will undoubtedly stimulate the broader use of this method.15 The surface stress developed along the reflective coating of a cantilever under an applied electrochemical potential can also be employed to estimate the spring constant. As cunning as this approach is, it is very model dependent and requires extreme care in the calibration of the optical sensitivity.16 Nonetheless, the method might be valuable for certain in situ measurements where differential stress across the two surfaces of the cantilever will give rise to a bending moment. If a rectangular cantilever is available, measurements of the resonant frequency and quality factor give access to the spring constant.17 We have developed a simple, semi-empirical relationship between the deflection of a cantilever under a point load and the same cantilever under laminar fluid flow. The aim here is to demonstrate that given the right conditions, the hydrodynamic contribution to cantilever deflection is (8) Neumeister, J. M.; Ducker, W. A. Rev. Sci. Instrum. 1994, 65, 2527-31. (9) Scholl, D.; Everson, M. P.; Jaklevic, R. C. Rev. Sci. Instrum. 1994, 65, 2255-2257. (10) Gibson, C. T.; Watson, G. S.; Myhra, S. Nanotechnology 1996, 7, 259-262. (11) Butt, H.-J.; Siedle, P.; Seifert, K.; Fendler, K.; Seeger, T.; Bamberg, E.; Weisenhorn, A. L.; Goldie, K.; Engel, A. J. Microsc. 1993, 169, 75-84 . (12) Torii, A.; Sasaki, M.; Hane, K.; Okuma, S. Meas. Sci. Technol. 1996, 7, 179-184. (13) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 18681873. (14) Butt, H.-J.; Jaschke, M. Nanotechnology 1995, 6, 1-7. (15) Hutter, J. L.; Bechhoefer, J. submitted for a publication. (16) Miyatani, T.; Fujihara, M. J. Appl. Phys. 1997, 81, 7099-7115. (17) Sader, J. E.; Chon, J. W. M.; Mulvaney, P. Rev. Sci. Instrum. 1999, 70, 3967.
10.1021/la9914965 CCC: $19.00 © 2000 American Chemical Society Published on Web 10/13/2000
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linear with speed, independent of tip size and position, and is scale invariant. It has long been believed that Reynolds’ law of similitude holds for two flows of the same Reynolds number.18 Comparison of different systems is made by determining the Reynolds number, Re, for each system:
Re ) aνF/η
a
(1)
where a is the length scale and ν is the velocity of the object and F is the density and η is the viscosity of the fluid. In this way scale models of ungainly objects can be tested on laboratory scales by adjusting the properties of the test fluid. In addition to the numeric comparison afforded by Re, several regime in fluid flow can be defined approximately by their range in values of Re. Thus, the Stokes regime is typically for Re , 1 and denotes fluid behavior characterized by an absence of boundary layer effects and turbulence. This regime is the simplest where pure laminar flow can be expected; that is, the force on an object is proportional to its speed, and the fluid viscosity.
b
Method A 500:1 scale model of the long thin cantilever from the standard silicon nitride Nanoprobe set (Digital Instruments) was constructed from 0.12-mm-thick blue spring steel shim (McMaster-Carr: flat stock, not off a roll). The dimensions of the actual cantilever (Figure 1a) were measured by scanning electron microscopy and transferred to the CAD system of an electric discharge wire cutter. This technique does not stress the material being cut and leaves no burring. The model cantilever was clamped at 10° to the surface in a rigid jig made from 12-mmthick plate aluminum welded into a form which closely mimics the local geometry of surfaces around the actual cantilever when mounted in a commercial fluid cell (Digital Instruments); see Figure 1b. A 2.3 mm-high, square-pyramidal aluminum tip with a “rare-earth” magnet glued inside a cavity was used to test the effect of tip placement along the cantilever. For convenience the light lever method typically used in force microscopy was also employed to monitor cantilever deflection, but on a comparatively larger scale. By analogy the encoded vertical travel of the mill bed served as the “piezo-electric stage” in the model system, with reproducibility of (5 µm and range of ∼200 mm. The bed of the mill (Deckel FP4M) could be driven at rates ranging from 8 mm min-1 to 100 mm min-1, depending on gearing ratios, to within 1%. Most importantly, the mill bed could easily translate the halved 44-gal drum containing 100 kg of glycerol (food grade). At the bottom of the container a smooth glass plate acted in place of the “sample surface”. The cantilever jig, which included the light lever components and is equivalent to the “head” of the real microscope, was fixed rigidly in the tool chuck of the mill. From an acrylic box constructed to simulate the top surface of the fluid cell, the light from a simple 3-mW laser pointer was reflected off the polished surface of the model cantilever, onto a front silvered mirror, and then down a ∼19-m path across the workshop onto a screen. A video camera recorded the position of the reflected laser light, in addition to marking measurements directly onto the screen during experimental runs. The optical sensitivity (compliance) of the system was determined by displacing the model cantilever a known distance using the encoded vertical movement of the mill bed and comparing the projected deflection on the screen. The optical gain of the system, the ratio of the real deflection to the projected one, was in the range 1000-1100. The spring constant of the model cantilever was measured by placing three tiny, preweighed rare-earth magnets, 186 mg in total, one at a time at the very tip of the cantilever and monitoring the deflection at the screen. The spring constant of the model was thus determined to be 2.26 ( 0.02 N m-1. The glycerol was poured into the container and left to equilibrate for 24 h prior to commencing the experiment. This (18) Laudau, L. D.; Lifshitz, E. M. Fluid Mechanics, 2nd ed; Pergamon Press: Oxford, UK, 1987.
Figure 1. (a) Scanning electron micrograph of a long-thin cantilever (Nanoprobe, Digital Instruments). (b) A schematic showing the configuration of the mill, the model cantilever, the halved 44-gal drum of neat glycerine, and the light lever optics. The inset, lower left, shows a detail of the light lever optics. The proximity of the walls and lower surface closely approximate the configuration of the commercial fluid cell. The cantilever is inclined at 10° with respect to the lower surface, as is found in the actual fluid cell. Care was taken to construct the model fluid cell such that the optical path length was not affected by the fluid level, again as in the actual fluid cell. process removes the bubbles formed during pouring, which might have otherwise scattered the laser light. Since the viscosity of the glycerol is a strong function of water concentration and due to its hydroscopic nature, we took several samples from around the cantilever during the course of the experiment. The surface of the liquid was covered with thin polyethylene film (Gladwrap) to help reduce uptake of atmospheric water. The viscosity was determined by using Cannon-Fenske viscometers and the density determined by an Anton Paar DMA602 oscillating densimeter at the same temperature as the experiment: 16.6 ( 0.2 °C. No significant change in either the density or the viscosity was observed over the 2 days in which the experiment ran. Values are reported in Table 1. After calibrating the system, the model cantilever was driven at a constant speed for ≈5.37 µm × 500 ≈ 2.7 mm in glycerol, first one way and then back again. Total deflections were maintained below a few percent in comparison to the total length of the cantilever. This was done to ensure that the systems were well within the elastic limits for the materials used. In addition, the low speeds used ensured that Stokes-like flow could be safely assumed. It is important to note at this point that as the lightlever method can only detect a change in deflection, it is necessary to measure the total deflection in both directions and divide by 2 to give an “average”. The effect of drift on the measured deflection will be minimized if the average is used, rather than attempting to determine the difference between static and deflected positions for a single direction. Of course, the drainage of the fluid between the cantilever and sample will be different
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Maeda and Senden Table 1.
system
length (m)
speed range, (m s-1)
density, (kg m-3)
viscosity, (kg m-1 s-1)
Reynolds’ number
actual cantilever in water at 22 °C model cantilever in glycerol at 16.5 °C
1.87 × 10-4
(2.3 × 10-5)-(2.1 × 10-4)
998
0.000 955
0.0043-0.047
0.940
(1.4 ×
2.000
0.0086-0.027
10-4)-(4.3
in the two directions, but if the same approach is used for both model and actual cantilever, then average deflections should scale similarly. In general, the piezo extension range should be small with respect to the tip-sample separation, D. This condition works well when the ratio of the piezo extension to the separation is sufficiently small. Experiments with two different piezos (maximum range 5.37 and 2.45 µm, respectively) gave similar spring constants within error, suggesting that 5.37 µm is already small enough when compared with D ) 18 µm to use the concept of scale invariance. In addition, the results suggest that, within error, the deflection has reached a maximum after 2.45 µm. As water is the most common liquid used in force microscopy we developed the procedure based around the properties of water under typical laboratory conditions, that is, 22 °C. The same procedure used in the model system was followed for the actual cantilever. That is, the compliance was determined, followed by a series of deflection measurements at several speeds, at several tip-sample separations. In the actual system the accurate calibration of the piezo belies the overall accuracy of the hydrodynamic data. Although, the error in the etched silicon height standard (NT-MDT, Russia) used was less than 1%, we assessed the precision of the piezo to be around 3%. This therefore sets the error in each of the components; the piezo displacement (hence speed), the cantilever deflection (via the compliance), and the tip-sample separation (via the stepper motor). Milli-Q water was injected into the fluid cell and allowed to equilibrate for 20 min. The sample surface was freshly cleaved mica. Following equilibration, the cantilever was cycled at a constant rate, first 5.37 µm toward the surface and then 5.37 µm back, for a given tip-sample separation. The separation of the surfaces was achieved by using the stepper motor (calibrated by the piezo), taking backlash into account. Five cantilevers from three different batches were measured for comparison. The analysis relies on the semi-empirical relationship formed between an end-loaded cantilever and one where the load is distributed due to laminar flow. For any point loaded cantilever, Hooke’s law holds for deflections within the elastic limits of the material:
Fp ) kx
(2)
where Fp is point-loaded force, k is spring constant, and x is deflection of the cantilever. At constant velocity and for laminar flow, Stokes’ law provides a measure of the distributed load due to fluid flow acting over the cantilever:
Fu ) sηνa
(3)
where Fu is uniformly distributed force, s is shape factor, η is viscosity, ν is flow speed, and a is the size of the cantilever. We can measure x, η, ν, and a in both model and actual systems and we can obtain k for a model cantilever. The shape factor, s, is common in both systems and in principle we can determine k for the actual cantilever as
kr ) km(xmarνrηr/xramνmηm)
(4)
where subscripts r refer to real or actual system and m refer to model system. In practice, however, the distributed hydrodynamic load due to drag depends on the tip-sample separation from the surface, and therefore instead of using eq 4 it is convenient to define another factor, R.
R ≡ sFp/Fu ) (kx/νaη)
(5)
This is a ratio of point-loaded force given by Hooke’s law to a distributed force given by Stokes’ law, excluding shape factor,
×
10-4)
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Figure 2. Deflection due to hydrodynamic drag on the model cantilever in glycerol as a function of speed, for different tipsurface separations (in mm). The inset shows two sets of data collected at much higher speeds to demonstrate the linearity of the change in deflection with speed. The surface separation is shown in brackets, measured by using the mill encoder (( 5 µm). and has no dimension. Strictly, this ratio is a universal constant for a given cantilever shape and for a laminar flow perpendicular to the plane of the cantilever. However, the proximity of walls, including the sample surface, will alter the flow field and R will become separation dependent. In most fluids cells it is not convenient to separate the tip and surface by more than 50 µm. In the specific case of the nanoscope, it is the placement of the O-ring which limits maximum separation due to potential fluid leakage. Thus, we define R as a function of tip-sample separation, R(D). Here D represents the maximum tip-surface separation in a cycle of travel, as is the case in the nanoscope.
Results Figure 2 shows deflection of a model cantilever in glycerol as a function of speed, at different tip-sample separations above the surface. The linear relationship between deflection and speed is a good indication that we are within the Stokes regime, and the plot of R(D) versus tip-sample separation above the surface for the model cantilever system is shown in Figure 3. Without any attempt to describe the actual hydrodynamic flow in the system we have chosen a power law to empirically fit the data. It is probable that more than a single exponent would be required to model the data; however, given the short range in values this would be unwarranted. The fit becomes our master curve to which all actual cantilever data will be compared. The factor that enables the actual data to overlay the model fit is the spring constant. The use of a removable tip allowed the effect due to the presence or absence of the tip to be studied. Within experimental variation, and at separations greater than 4 times the height of the tip (>18 µm), there was no contribution to the overall cantilever hydrodynamics from the tip interacting with the sample. This will not be the case for large colloidal probes attached to the cantilever where the separations are smaller than a factor of 4
Calibration of Force Microscopy Cantilevers
Figure 3. Plot of R(D) versus tip-sample separation for the model cantilever system. Without any attempt to describe the system mathematically we have chosen a power law to empirically fit the data. D in the equation is calculated in meters.
Figure 4. Deflection versus tip-sample separation at different speeds for an actual cantilever and at two depths. Open symbols denote data recorded at a tip-sample separation of 18 µm; closed symbols are for D ) 46 µm. The horizontal dashed lines show the limits where the total cantilever deflection during one cycle was estimated for a given cycle rate. The total deflection is halved to give the mean deflection for a given speed. Note the vertical deflection scale is halved for every doubling of speed. In this way each cycle appears to have a similar average deflection, emphasising the linearity of the hydrodynamic conditions. The tip-sample separation reduces as the tip travels from right to left.
compared with the probe diameters. Special consideration is required in these cases.19 Figure 4 shows an example of a set of force curves measured in the force microscope in water at tip-surface separations of 18 (open circles) and 46 µm (filled circles). Here 18 µm is the minimum separation at which the effect of tip is negligible and 46 µm is the maximum separation (19) Craig, V. S. J. submitted for publication.
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Figure 5. Deflection of an actual cantilever in water as a function of speed is shown for different tip-sample separations above the surface. The separation was adjusted by using the stepper motor normally employed to move the head. Care was taken to avoid backlash by only changing the separation in one direction.
without risk of leaking the water. Arrows indicate where the total deflection was estimated and, like the model system, represent the point when the cantilever has reached constant velocity. In a fashion similar to Figure 2, Figure 5 shows the deflection of an actual cantilever in water as a function of speed and for different tip-sample separations. Again, the linear relationship between deflection and speed confirmed the assumption of the Stokes regime. In the final step of the analysis a plot of R(D) versus tip-sample separation is shown in Figure 6, where the tip-sample separation from the model fit has been rescaled by a factor of 500. For each cantilever the spring constant was adjusted until the least-squares difference between the actual data and the model fit for all specified tip-sample separations was collectively minimized. The spring constants thus determined are listed in Table 2. To implement this method the following steps should be followed: (1) Bring the cantilever into contact with a noncompliant surface and determine the optical sensitivity (compliance). Ensure both piezo and stepper motors are well calibrated prior to this. (2) Using the stepper motor separate the tip from the sample to a known distance of >20 µm. (3) Cycle the piezo over its maximum extension range, recording the difference between the final deflection values for extension and retraction curves. Scale the deflection by the optical sensitivity. Do this at various speeds, for example, 2, 4, 8, and 16 cycles s-1. Plot half the deflection difference as a function of speed in SI units and fit a straight line to the data. Record the slopes measured for the four speeds. (4) Repeat step 3 for another tip-sample separation and then again out to about 50 µm. (5) To convert the slopes (in seconds) to R-factors defined in eq 5, divide each slope by the cantilever length (0.0002 m for long thin cantilever) and by the viscosity (0.000 89 kg/m/s for water at 22 °C) and multiply by a nominal spring constant. (6) Plot the R-factors as a function of tip-sample separation.
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Figure 6. Plot of R(D) versus tip-sample separation above the surface for the actual cantilevers investigated. The spring constant was used to drive a least-squares fit to the model data. All variance in the actual systems lies between the 5% error limits of the model system. Note that only data from the actual cantilevers which fell within the range of the model data was used in the least-squares fit. Although the deflection data collected at 9.2-µm separation lies on the power law fit to the model data, it was not used in the least-squares minimization to obtain the spring constant. Despite this the close correspondence with the power law fit demonstrates that the model assumptions are well-behaved in the regime to which they have been applied. Table 2. batch no.
spring constant, (N m-1)
1 1 1 2 3
0.0181 0.0222 0.0172 0.0409 0.0265
(7) Find the corresponding model R-factors for each tipsample separation, using the equation defined in Figure 3. Convert the tip-sample separation to meters before substituting into the equation. (8) Finally, adjust the nominal spring constant so that the least-squares difference between the R values obtained in step 7 and 8 is collectively minimized. The value of the spring constant is given in N m-1. Discussion It is apparent from Table 2 that the spring constant of cantilevers varies significantly not only across from different wafers but also between those from within the same wafer. Indeed, the interference colors seen in the silicon nitride film of the batch no. 1 varies from one side to the other, indicating a variation in thickness within the wafer. This is not the case for all wafers and visual inspection will tell if the thickness is uniform across the wafer. An attempt was made to measure the thickness of the cantilevers via electron microscopy, but because of the practical difficulty in aligning the cantilever perpendicular, values were not considered accurate. A better method might be to pinch off a cantilever onto a cleaved mica surface and image the detached cantilever with the force microscope. Even with this measurement, one still has the difficulty of considering the mechanical contribution of the gold layer to the cantilever rigidity. The nominal spring constant provided by Digital Instruments for the type of cantilever used here (“long
Maeda and Senden
thin”) is 0.06 N m-1. According to the manufacturer the variation in thickness may range from 0.4 to 0.7 µm. Since the spring constant is proportional to the cube of the thickness, the concomitant variation in spring constant is from 0.017 to 0.095 N m-1. (This serves only as a guide as variations in the material properties of silicon nitride preclude any direct comparison.4) Our values in Table 2 fall within this range, but largely in the lower half. The method reported here provides an estimate of the lowest bound for spring constant one might expect from any given cantilever. This results from having determined the spring constant for a truly end-loaded cantilever in the model system. The stiffness of the cantilever varies approximately as the cube of the distance from the base of the cantilever. Therefore the actual position of the tip, or colloid probe, would have to be determined in order to correct the spring constant determined from the hydrodynamic method. For example, a tip may be as far as 10 µm from the cantilever end. For a 200-µm-long cantilever the spring constant would be around 15% stiffer than estimated from the hydrodynamic method. In the absence of a calibration “gold standard” we are reluctant to compare the spring constants obtained from this method to those from other methods. However, using the method of Cleveland et al.1 we can see that our results are within error range of this resonance method.20 Other types of cantilevers are generally too stiff for hydrodynamic drag to produce significant deflection. Given their large spring constants and low sensitivity, they are not often used for the quantitative force measurement. However, for some special cases one might require knowledge of the spring constant of a cantilever other than long-thin. Our method is capable of providing the spring constants for the remaining cantilevers on the same chip in such a case, though the method is no longer in situ. The thickness and material properties of all the cantilevers on a single chip should be almost constant. Thus given the spring constant of the long-thin type, the effective Young’s modulus for the long-thin type is readily found, and the spring constants of the other cantilevers can be found once their dimensions are measured (see, for example, part 4 of ref 17). Acknowledgment. We thank the helpful assistance of Tony Beasley in providing access to accurate viscosity measurements and similarly Jack Derlacki for his density measurements. Anthony Hyde and Tim Sawkins provided technical assistance and advice, which is always of great practical value. Numerous discussions with Vince Craig helped refine the approach used, and John Sader is gratefully thanked for his invaluable comments. Ron Cruickshank gave open access to his workshop and in particular the Deckel mill, the key component in this project. T.J.S. is a grateful recipient of an Australian Research Council funded Postdoctoral Fellowship. LA9914965 (20) In Dr. Craig’s recently submitted work,19 he has employed the Cleveland et al. method on the same batch of cantilevers we have used in our study (batch #1). Craig used a different sized cantilever to the one we have used. The method of comparison, is now commonly used (see ref 8), works around difficulties in determining E or t separately for different cantilevers on the same chip. Using this approach and the difference in the point of loading, Craig’s data would estimate our spring constant to be 0.022 N m-1, which is in reasonable agreement with our reported value.
