APPENDIX 1: SPECTRAL METHODS FOR MEASURING THE NORMAL CANTILEVER SPRING CONSTANT k

Transcription

1 APPENDICES APPENDIX 1: SPECTRAL METHODS FOR MEASURING THE NORMAL CANTILEVER SPRING CONSTANT k User measurement of cantilever spring constants, both normal k and torsional k,isa rich subtopic in AFM. As is extensively reported in the literature, there are various spectral calibration methods wherein one examines either the externally or the thermally driven cantilever resonance spectrum. These methods vary in precision, ease of laboratory implementation, and simplicity of data reduction. It is not the purpose of this appendix to comparatively review different strategies or recap their historical progression. Several review-type articles have effectively served this purpose [1 7]. Instead, we describe three methods that have passed the scrutiny of different reviewers and seem by consensus to be quite accurate (within 1%, as with the reference cantilever method of Section 3.7) and relatively free of difficulties or pitfalls. Some of the following, follows from the method of Section 3.7. Recall that in this method the user collects a force Z curve on a compliant surface of known stiffness a tipless, single-material, rectangular reference cantilever and compares the contact slope (ds Z /dz) to that on a surface of (nearly) zero compliance, the chip to which the cantilever is attached. The determination of spring constant k of this purely rectangular reference cantilever proceeds via k ¼ Ewt 3 /4L 3 (Eq. 3.6), given Young s modulus E and cantilever dimensions w, t, and L for width, thickness, and length, respectively. Width and length can be microscopically measured to acceptable accuracy, whereas thickness generally cannot. But one can also Atomic Force Microscopy: Understanding Basic Modes and Advanced Applications, First Edition. Greg Haugstad. Ó 212 John Wiley & Sons, Inc. Published 212 by John Wiley & Sons, Inc. 437

2 438 APPENDICES develop an expression for k wherein thickness has been eliminated [4], k ¼ ð59:3þwf p L ffiffiffi 3 r p ffiffiffi ; ða1:1þ E thus requiring only the plan-view dimensions of length L and width w and the material constants of density r and Young s modulus E, plus a measurement of cantilever resonance frequency f in air or vacuum. This is how manufacturers compute spring constants (typically claiming three significant digits) for the above reference cantilevers, and thus how the user can check these values. But the reference cantilever method of Section 3.7 suffers one obvious drawback: the need to push the tip against surfaces other than the sample(s) of interest reference cantilever and rigid surface such as the cantilever chip with enough force to establish the force Z trend (slope) in contact. In some cases, the user may prefer to never touch a solid surface in this way, say if only studying DLVO or steric forces (and if measurements of the deflection sensitivity S are found to be adequately consistent from one cantilever mounting to the next, for the sought precision of force and distance calibration). Of course, one can wait until all measurements of interest have been completed before performing force curves on the reference cantilever surface, depending on successful calibration without damaging the tip or cantilever. But in some cases, one would prefer to obtain k with a no-touch method, whether in air or in a liquid medium of interest. This is one of the attractive features of the following spectral methods. A1.1 Plan-View/Resonance Frequency Method In this method, one applies the preceding formalism to the cantilever used for AFM operation, that is, assuming that the inclusion of the mass of the tip as well as small deviations from purely rectangular geometry (e.g., angled corners) have only a minor effect on the determined value of k [2]. Plan-view cantilever dimensions (i.e., X Y, viewed from above the sample) are usually specified to acceptable accuracy and precision by the cantilever manufacturer (i.e., within the goal of 5 1% overall calibration accuracy). Otherwise, the cantilever length and width can be determined using a good light microscope, certainly within 5% error and approaching 1% in the case of length. In air or vacuum, the determination of resonance frequency is most easily achieved within a frequency-sweep measurement ( cantilever tune ), as performed to select the driving frequency in dynamic mode. If, however, one is calibrating in a heavy damping medium of operational interest (e.g., water) with a relatively low-mass cantilever, the damped-driven oscillator (DDO) resonance frequency may be significantly different from what is truly sought, the intrinsic resonance frequency, as per Equation 2.18 (i.e., if b/m is not v ), v DDO ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v b 2 m : ða1:2þ

3 APPENDICES 439 In this latter case, it is better to determine the thermally driven that is, by k B T energy, meaning Brownian motion resonance frequency. To do this, one examines the power spectral density (PSD) of the cantilever s thermal fluctuations in deflection (a rapid succession of measurements versus time) and finds the dominant resonance peak. This is similar to the use of PSD to measure environmental vibrations, as discussed in Section 8.3.3, but examining much higher frequencies and with the tip away from the sample. In a heavy damping medium, these thermally driven motions are typically slow enough (for soft cantilevers with low resonance frequencies, as used for high force sensitivity in liquids) to characterize using standard AFM data acquisition systems (i.e., at sampling rates in the tens of kilosamples/s); indeed, for very soft cantilevers, the thermally driven motion may be slow enough even in air to characterize in this fashion. One can simply capture a deflection signal image out of tip sample engagement (zeroing Z-feedback gains) at the highest achievable scan rate, then perform a 1D PSD along the fast-scan (X) direction using the instrument software or a third-party AFM analysis program, as exemplified in Figure A1.1 for a nominal k ¼.2 N/m rectangular silicon cantilever. This averages all the 1D PSDs corresponding to each raw scan line of deflection data. Within such programs, it is typically the dominant spatial wavelength or wave number that the software reads out for a given image size. Multiplying by the scan velocity, one determines resonance frequency in cycles/s: ½Spatial wave number ðcycles=mmþš 2 ½Scan size ðmmþš ½Scan frequency ðs 1 ÞŠ: One must also note whether a nonzero overscan parameter (e.g., called Rounding in Bruker scanner calibration software) is used. If, for example, the overscan is 1% (Rounding.1), then the actual scan size to insert into the above equation is 1% greater than the displayed scan size. Alternatively, one may simply zero the overscan parameter prior to acquiring the image. The latter was done for the image in Figure A1.1a, resulting in the following calculation of resonance frequency: f ¼½188 cycles=mmš2 ½:5 mmš½61:s 1 Š¼11:4 khz: The plan-view dimensions provided by the manufacturer of this silicon cantilever were w ¼ 5 mm and L ¼ 45 mm; using a density r ¼ 2.33 g/cm 3 and Young s modulus E ¼ 179 GPa, k is determined by Equation A2.1, h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 3 ð59:3þ m 11; 5 s 1 ð Þ m 233 kg=m 3 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N=m 2 ¼ :11 N=m: This example of a user-determined k differing from the manufacturer s nominal value by nearly a factor of two is not uncommon.

4 44 APPENDICES FIGURE A1.1 (a) Cantilever deflection image taken out of contact at 61 lines/s, at a scan size of.5 mm with no overscan, following first-order plane fitting (for viewing clarity). (b) Resulting X PSD (along fast-scan direction) and identification of spatial wave number (cycles/mm). It is very important to recognize that for many applications in air or vacuum, the cantilever s thermal motion is too fast (resonance frequency too high) to track versus time with a standard AFM data acquisition system. If one nevertheless attempts to measure the power spectral density of cantilever deflection as described above, one often obtains a peak and with a Lorentzian lineshape akin to that in

5 APPENDICES 441 Figure A1.1b. But in this case, the cantilever motion is being aliased, meaning sampled at a rate below the rates of fluctuational motion in such a way as to produce a virtual periodic signal. To understand aliasing, think of each successive measurement moving slightly further in phase along a sine wave but skipping several periods between each measurement, thereby producing a virtual sine wave of much longer time period (lower frequency) than the actual motion being sampled. In such cases, a faster measurement device is needed, such as a spectrum analyzer or simply a fast data acquisition card (e.g., as part of a LabView system), and of course BNC access to the cantilever deflection signal. Some newest-generation AFM systems, at this writing, are standard-equipped with data acquisition cards extending into the megahertz regime. The plan-view/resonance frequency method of this section introduces sources of error that are absent, in one way or another, from the methods of the next two sections. These include the values of cantilever density and Young s modulus and/or the measurements of plan-view dimensions. A1.2 Sader Method This method is named after the author of several papers proposing and refining procedures whereby cantilever resonance frequency and damping are analyzed to determine spring constant [8,9]. Plan view-dimensional measurements are required as in the preceding section, but not the density and Young s modulus of the cantilever. What is additionally required is the density and viscosity of the immersion fluid, typically air, and strongly different cantilever dimensions L w t (but in practice L/w > 3 has been found to be acceptable [9]). The spring constant expression is with k ¼ 7:52 r f w 2 LQf 2 G iðreþ ða1:3þ Re ¼ 2pr f f w 2 4h f ; ða1:4þ where r f and h f are the mass density and viscosity of the fluid medium, Q is the oscillator quality factor, and G i is the imaginary component of the hydrodynamic function, which is in turn a function of the Reynolds number Re via Equation A1.4. The hydrodynamic function is shown graphically in reference [8] and at this writing can be evaluated via Sader s online calculator or downloadable Mathematica files [1]. One might attempt to measure values of f and Q from frequency sweep measurements (sometimes called cantilever tune ). As discussed in Section A, this is acceptable for f provided that the damping is not too high (the case in air) and the cantilever mass not to low (usually true). But the Q value is dependent on the shape

6 442 APPENDICES of the resonance peak. In Section 5.2.6, extraneous contributions to this shape for an externally driven cantilever were discussed, in terms of the frequency-dependent coupling of the driving signal to the cantilever base. A more reliable way to measure Q with high accuracy is to mathematically fit the peak in the PSD of the cantilever s thermally driven motion with a Lorentzian function plus background: A 1 Q A white þ A f =f þ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ða1:5þ 2 ðf =f Þ 2 þ Q 2 1 ðf =f Þ 2 where the constant A white is the white noise (frequency independent), A f is the 1/f noise, and A 1 Q is the peak height. Using this expression to fit the spectrum in Figure A1.1b produces Q ¼ 35 and f ¼ 11.5 khz. Applicable values of fluid density and viscosity for air were 1.17 kg/m 3 and kg/m/s. Together with the values of w and L from Section A, one obtains k ¼.94 N/m with the Sader method, within roughly 1% of the value computed using the method of Section A1.1, and consistently about half of the manufacturer s nominal value of.2 N/m. A1.3 Thermal Method The name for this method often includes an additional noun, for example, thermal tune, thermal k, or thermal noise. Alternatively, it is sometimes designated by the authors of the original paper, Hutter and Bechhoefer [11], and/or the authors of later papers refining the method (e.g., Butt and Jaschke [12]). Obviously, the measurement of thermal noise is not unique to this method. What is unique is the use of the amplitude of thermally driven cantilever motion as the gauge of spring constant. This requires, of course, that the magnitude of thermal fluctuations is significantly greater than the noise floor of the deflection (cantilever inclination) measurement electronics; thus, the method is often not applicable to the stiffest cantilevers such as those often used in dynamic AFM in air. The fluctuations are characterized across a range of frequencies via the power spectral density, but the measurement is dominated by thermal noise in the vicinity of the fundamental resonance frequency. The precise value of this frequency, however, does not enter the computation. The key relationship in the thermal method derives from a fundamental principle of physics, the equipartition theorem for a single oscillatory degree of freedom, which relates the cantilever s thermal motion to k B T, or ks2 z ¼ 2 k BT; k ¼ k BT : ða1:6þ s 2 z

7 APPENDICES 443 The quantity in brackets, the mean squared vertical displacement of the tip measured over a sufficiently long time, is approximately evaluated in the frequency domain by integrating under the power spectral density peak for the fundamental mode (i.e., subtracting the background) such as that in Figure A1.1b. For good accuracy of integration, this requires a high point density in the frequency domain, which is not the case in Figure A1.1b; it would result if sampling the data for longer time intervals. But there are further details that introduce corrective factors in the ultimate expression for k. One factor derives from the fact that not all the cantilever s kinetic energy is contained in the fundamental mode as if an ideal spring. Rather, approximately 3% of the energy is contained in higher eigenmodes as can be determined by solving a fourth-order differential equation within the context of beam theory (see Section 9.2.1). A second consideration is the fact that the actual linear tip displacement s z in length units is not being measured with the position-sensitive photodetector, rather, the cantilever s angle of inclination, not far from the tip end of the cantilever. The relation between this angle of inclination and the vertical tip displacement is not the same for a thermally driven, free cantilever as it is for a statically loaded cantilever (Section 3.2), whether the latter is in sample contact or experiencing some other equilibrium net force such as DLVO. To account for both of these corrective factors (which act in the same direction, to decrease the determined value of k), in the case of a rectangular cantilever, Equation A1.6 is replaced by k ¼ :817 k BT ; ða1:7þ where the asterisk indicates the measured displacement of the freely fluctuating cantilever, as calibrated via the sensitivity S discussed in Section 3.2, which is determined via the force Z slope in static loading measurements (force curve on rigid surface). For the case of a V-shaped cantilever, in comparison with a rectangular cantilever of the same length and leg width, it has been numerically shown that the prefactor in Equation A1.7 is replaced by.764, a 7% reduction [13]. In general, one anticipates a similar, slightly reduced value for other V-shaped compared to rectangular cantilevers of similar length and leg width. The above corrective factors can require further modification if the laser spot diameter is considerable relative to the length of the cantilever. In this case, a range of inclination angles may be contributing to the position of the reflected laser beam spot on the photodetector. This issue is only expected to matter for very small cantilevers, such as recently developed at this writing. s 2 z APPENDIX 2: DERIVATION OF VAN DER WAALS FORCE DISTANCE EXPRESSIONS The problem of deriving the distance dependence of van der Waals force between a tip of axisymmetric geometry and a flat sample (Section 2.1.1) is central to AFM modeling and simulations. First, one integrates over the dipole dipole interaction

8 444 APPENDICES between a single-point dipole (inside the tip), at a distance z from the sample surface, and the continuum of point dipoles of number density n 1 throughout the semi-infinite half-space of the sample, as shown in the lower portion of Figure A2.1. Assuming a nonretarded dipole dipole potential VðrÞ ¼ Cr 6 ; ða2:1þ FIGURE A2.1 Illustration of two spatial integration domains in cylindrical coordinates: first, an infinite half-space interacting with a point dipole on the principal axis of the tip, and second, over an axisymmetric tip with bottommost point, a distance D from the surface of the half-space.

9 APPENDICES 445 where C is a constant and r is distance, one may integrate over the semi-infinite half-space cylindrical coordinates, VðzÞ ¼ Z 1 Z 1 Cð2pr dz 1 n 1 Þdr 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 ða2:2þ r 2 1 þ ð z þ z 1Þ 2 and thus [14] Z 1 1 VðzÞ ¼ 2Cpn 1 4ðz þ z 1 Þ 4 dz 1 ¼ Cpn 1 6z 3 : ða2:3þ Second, one may integrate in cylindrical coordinates over all point dipoles of number density n 2 in the tip, extending the z limit to infinity by assuming negligible van der Waals interactions at large distances (and neglecting retardation effects). Given a distance D between the end of the tip (at z 2 ¼ ) and the sample surface, we have VðDÞ ¼ Cpn 1 6 Z 1 rðz2þ Z ð2pr dz 2 Þn 2 dr 2 2 ðd þ z 2 Þ 3 or VD ð Þ ¼ A H 6 Z 1 ½rðz 2 ÞŠ 2 dz 2 ðd þ z 2 Þ 3 ; ða2:4þ where the Hamaker constant is A H ¼ p 2 Cn 1 n 2, and rðz 2 Þ describes the functional shape of the tip. Here, we consider the model tip geometries rðz 2 Þ ¼ 8 pffiffiffiffiffiffiffiffiffi s2rz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z >< 2 2 a b þ 1 1 tanðbþðz 2 þ z Þ >: tanðbþz 2 ða2:5 A2:8Þ for the cases of paraboloid of revolution, hyperboloid of revolution (offset such that the end of tip is located at z 2 ¼ ), and truncated and full cones, respectively, as depicted in top portion of Figure A.2. (z is the distance of the end of the truncated cone from the mathematical end point of a virtual full cone.) For the purpose of consistent tip aspect ratios in the latter three models, we choose tan(b) ¼ a/b, b being the tip half-angle; for consistent radius of curvature in the first two models, we choose R ¼ a 2 /b.

10 446 APPENDICES For any tip geometry, one may evaluate force by taking the derivative with respect to D prior to evaluating the integral, given a smooth, continuous, and finite integrand over the integration interval in Equation A2.4. Thus, we have F ¼ dv dd ¼ A H 2 The four tip models r(z 2 ) produce integrals of the form Z 1 ½rðz 2 ÞŠ 2 dz 2 ðd þ z 2 Þ 4 : ða2:9þ Z 1 z n 2 I n ðdþ ¼ dz 2 ðd þ z 2 Þ 4 ða2:1þ with n ¼, 1, 2. The solutions are [14] I 1 ðdþ ¼ 1 I ðdþ ¼ 3ðD þ z 2 Þ 3 1 z 2 ¼ ¼ 1 3D 3 " # 1 2ðD þ z 2 Þ 2 þ D 3ðD þ z 2 Þ 3 " # I 2 ðdþ ¼ 1 D þ D þ z 2 ðd þ z 2 Þ 2 D 2 3ðD þ z 2 Þ 3 1 z 2 ¼ 1 z 2 ¼ ¼ 1 6D 2 ¼ 1 3D : ða2:11þ ða2:12þ ða2:13þ Thus, for the four tip geometries and using Equations A2.5 A2.13, we obtain the following expressions: Paraboloid of revolution Hyperboloid of revolution FD ð Þ ¼ A HR 6D 2 FD ð Þ ¼ A HR D 6D 2 b þ 1 ða2:14þ ða2:15þ Truncated cone FD ð Þ ¼ A H tan 2 b 1 þ z 6D D þ z2 D 2 ða2:16þ Full cone FD ð Þ ¼ A H tan 2 b : ða2:17þ 6D

11 APPENDICES 447 The magnitude of these four F(D) expressions are plotted on a log log scale in Figure 2.2, using A H ¼ J, R ¼ 1 nm, b ¼ 14 nm, z ¼ 5 nm, and b ¼ 15. We see that the paraboloid or hyperboloid models produce nearly identical distance dependences, whereas the cone and truncated cone produce force distance trends very different from the other two models. APPENDIX 3: DERIVATION OF ENERGY DISSIPATION EXPRESSION, RELATIONSHIP TO PHASE One can derive a mathematical expression relating the energy dissipated per tip sample interaction in dynamic AFM to the measured value of phase (Section 2.4.4). In dynamic equilibrium, the power imparted by the driving force at the base of cantilever, at any driving frequency, must be equal to the power loss due to the cantilever motion (viscous medium and any intrinsic cantilever anelasticity) plus the power loss due to tip sample interaction [15,16]: P driver ¼ P cantilever þ P tip-sample : ða3:1þ In the general case, one must integrate the energy imparted over a full driving cycle, multiplied by the number of driving cycles per second, and then subtract the energy loss due to cantilever damping over its oscillation cycle, multiplied by the number of cantilever oscillations per second, to solve for the tip sample power dissipation. For the driving expression, one first notes that the instantaneous power delivered is the force on the driver times its velocity [17], given by where and ks ½ z ðþ z t dr ðþ t Š dz dr dt s z ðþ¼a t cosðvt þ wþ z dr ðþ¼a t dr cosðvtþ are the time-dependent vertical positions of tip and cantilever base, respectively, and k is the cantilever spring constant. This assumes harmonic cantilever response, with negligible energy in higher or subharmonics; the tip sample interaction simply affects tip amplitude A as well as phase shift w between drive and response. Integrating this power over a time of one driving period, and multiplying by the driving frequency f ¼ v/2p, determines P driver, P driver ¼ v 2p Z2p=v ks ½ z ðþ z t dr ðþ t Š dz dr dt dt ¼ 1 ka 2 drav sin w: ða3:2þ

12 448 APPENDICES For the cantilever damping expression, one integrates the velocity-dependent resistive force b ds z dt ¼ k dsz Qv dt ; where it is assumed that the damping velocity is approximately equal to the tip velocity. Here, the point is to express the viscous damping coefficient b in terms of experimentally measurable quantities: the cantilever spring constant k, the free resonance frequency v, and quality factor Q of the freely oscillating cantilever at a distance just barely beyond tip sample interaction (see Section 5.2.3). This does not imply that the effective quality factor of the cantilever under dynamic tip sample interaction remains that of the free oscillator (certainly not the case) or that the operational drive frequency must be v. Integrating this damping over one cycle period determines P cantilever, P cantilever ¼ v 2p Z2p=v k 2 dsz k dt ¼ 1 A 2 v 2 : Qv dt 2 Qv ða3:3þ Taking the difference of the expressions for P driver (Eq. A3.2) and P cantilever (Eq. A3.3) and using the relationship A ¼ QA dr yields P tip-sample ¼ 1 2 ka 2 v Q A sinw v : A v The energy dissipation per cycle is the power times a time period of 2p/v or E tip-sample ¼ pka2 Q A sin w v : A v Typically, v v to a very good approximation, resulting in the following useful expressions, solving in terms of the energy dissipation per cycle normalized to the total oscillator energy during imaging, E tip-sample 1=2kA 2 ¼ 2p Q sin w 1 ; ða3:4þ A=A the right side importantly being independent of cantilever spring constant and dependent on the amplitude reduction A/A rather than the magnitude of A or A.A disadvantage of this expression is that it diverges as A/A goes to zero. A similar expression, which avoids such a divergence, normalizes the energy dissipation per cycle to the total oscillator energy when free of the surface, E tip-sample 1=2kA 2 ¼ 2p Q sin w A : ða3:5þ A

13 APPENDICES 449 One can also solve for the phase angle, w ¼ sin 1 QE tip-sample þ A ; ða3:6þ pkaa A with the inverse sine function providing two possible values of phase for a given value of energy dissipation (i.e., within each of the attractive and repulsive regimes). APPENDIX 4: RELATIONSHIPS IN MENISCUS GEOMETRY, CIRCULAR APPROXIMATION Figure A.4.1 describes the relationships between geometric parameters needed to describe a capillary meniscus in solid sphere flat geometry (Section 6.3.3), within FIGURE A4.1 Cross-sectional geometry of capillary meniscus between a sphere and a flat surface in the model known as the circular approximation. The curvature of the liquid vapor interface is described by two radii: r 1 is that of the shown circular arc, in a plane perpendicular to the flat surface, and r 2 is that of a circular cross section parallel to the flat surface (extending in and out of the diagram). Thus, the 3D meniscus geometry is a toroid. Geometric relationships allow one to express r 1 in terms of the filling angle c. A surface tension vector is shown at the contact circle (of radius r 2 ) between the meniscus and the sphere. This and the Laplace pressure over the horizontal meniscus cross section act to exert an attractive force between the sphere and the flat surface (Section 6.3).

14 45 APPENDICES the circular (toroidal) approximation of meniscus shape (referring to the circulararc liquid vapor interface shown in the plane of the diagram). These include three angles: the filling angle c and the contact angles between the meniscus and the solid surfaces of sphere, u s, and flat, u f ; three radii of curvature, the sphere radius R and the radii describing the circular meniscus cross sections perpendicular to the flat solid surface, r 1, and parallel to the flat solid surface, r 2 (cutting through the diagram), and the distance D between the lowest point of the solid sphere and the solid flat surface. The principal sought relationship is obtained by equating a vertical length, the sum of distance D and spherical cap height R(1 cos c), to the height of the meniscus r 1 cos(u s þ c) þ r 1 cos (u f ). (The latter requires identification of complementary angles.) A second needed expression is the vertical component of the surface tension (per unit length) around the contact circle between meniscus and solid sphere, which is the fraction sin(u s þ c) of the total force (per unit length), the latter directed as shown along the tangent of the liquid vapor interface at the three-phase line (i.e., the contact circle), the point of contact with solid sphere. REFERENCES [1] Burnham, N.A., et al., Comparison of calibration methods for atomic-force microscopy cantilevers. Nanotechnology, 23, 14: 1 6. [2] Gibson, C.T., D.A. Smith, and C.J. Roberts, Calibration of silicon atomic force microscope cantilevers. Nanotechnology, 25, 16: [3] Butt, H.-J., B. Cappella, and M. Kappl, Force measurements with the atomic force microscope: Technique, interpretation and applications. Surf. Sci. Rep., 25,59: [4] Clifford, C.A. and M.P. Seah, The determination of atomic force microscope cantilever spring constants via dimensional methods for nanomechanical analysis. Nanotechnology, 25, 16: [5] Cook, S.M., et al., Practical implementation of dynamic methods for measuring atomic force microscope cantilever spring constants. Nanotechnology, 26, 17: [6] Matei, G.A., et al., Precision and accuracy of thermal calibration of atomic force microscopy cantilevers. Rev. Sci. Instrum., 26, 77: [7] Ohler, B., Practical Advice on the Determination of Cantilever Spring Constants. Bruker Application Note #AN94. 27, p [8] Sader, J.E., J.W.M. Chon, and P. Mulvaney, Calibration of rectangular atomic force microscope cantilevers. Rev. Sci. Instrum., 1999, 7 (1): [9] Chon, J.W.M. and J.E. Sader, Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids. J. Appl. Phys., 2, 87 (8): [1] Sader Research Group, Atomic Force Microscope Cantilevers, edu.au/afm/index.html. [11] Hutter, J.L. and J. Bechhoefer, Calibration of atomic-force microscope tips. Rev. Sci. Instrum., 1993, 64:

15 APPENDICES 451 [12] Butt, H.-J. and M. Jaschke, Calculation of thermal noise in atomic force microscopy. Nanotechnology, 1995, 6: 1 7. [13] Stark, R.W., T. Drobek, and W.M. Heckl, Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy. Ultramicroscopy, 21, 86: [14] Beyer, W.H., Standard Mathematical Tables. 26th ed., 1981, Boca Raton, FL: CRC Press. [15] Cleveland, J.P., et al., Energy dissipation in tapping-mode atomic force microscopy. Appl. Phys. Lett., 1998, 72: [16] Garcia, R., et al., Phase contrast in tapping-mode scanning force microscopy. Appl. Phys. A, 1998, 66: S39 S312. [17] French, A.P., Vibrations and Waves. 1971, New York: W. W. Norton, 316.