Stefan Kirstein, a) Michael Mertesdorf, and Monika Schönhoff b) Max-Planck-Institut für Kolloid und Grenzflächenforschung, Berlin, Germany

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 4 15 AUGUST 1998 The influence of a viscous fluid on the vibration dynamics of scanning near-field optical microscopy fiber probes and atomic force microscopy cantilevers Stefan Kirstein, a) Michael Mertesdorf, and Monika Schönhoff b) Max-Planck-Institut für Kolloid und Grenzflächenforschung, Berlin, Germany Received 15 December 1997; accepted for publication 18 May 1998 The influence of a viscous fluid on the dynamic behavior of a vibrating scanning near-field optical microscopy fiber tip is investigated both theoretically and experimentally. A continuum mechanical description of a cylindric cantilever is used to calculate the resonance frequencies and the widths of the resonance bands. The linearized Naviers Stokes equations are analytically solved and describe the interaction of the beam with the viscous fluid. The contribution of the liquid to the shift and the broadening of the resonance lines is summarized by two constants that can be derived from a master function and the kinetic Reynolds number. The theoretical values are compared with experimental data collected from an optical fiber which is used as a probe in a scanning near-field microscope. Agreement, with a relative error of less than 1%, is achieved. The theory is further developed for the application to atomic force microscopy cantilevers with a rectangular cross section. Experimental data taken from literature are in good agreement with the theory American Institute of Physics. S X I. INTRODUCTION Scanning probe techniques such as scanning near-field optical microscopy SNOM and atomic force microscopy AFM have attracted increasing interest for the investigation of biological systems since images can be taken from soft materials. This is well established for AFM using the socalled tapping mode. 1,2 In this method, the tip sample distance is recorded dynamically: the cantilever is excited to oscillate in one of its eigenmodes and the tip amplitude provides a feedback signal. The oscillation prevents the tip from sticking at the sample surface by the short contact time and the back-driving force of the cantilever due to the bending. The situation is very similar in the case of a SNOM setup consisting of a tapered optical fiber where the so-called shear-force detection scheme is used for distance control. 3,4 The fiber oscillates at its resonance frequency and the amplitude is damped when the tip approaches the surface. The only difference between this mode and the tapping mode is the geometrical arrangement: the AFM cantilever vibrates in a direction normal to the surface, whereas the SNOM fiber oscillates parallel to the scanning plane. The true damping mechanism is less clear in the latter case. 5,6 It is known, that the damping forces experienced by the tip are small enough not to damage the surface of soft materials. For example, near-field optical microscopy has been used to image polymer/dye complexes, 7 J aggregates in monomolecular Langmuir Blodgett films, 8 and to localize dye labeled antibodies bound to fragments of cell membranes. 9 Most biological systems however must be investigated within their native liquid environment. Both methods, the a Author to whom correspondence should be addressed; electronic mail: kirstein@mpikg.fta-berlin.de b Current address: Physical Chemistry 1, University of Lund, Sweden. tapping mode and the shear force detection, have already been successfully applied to take images of soft samples under water. 2,10,11 However, the success of these methods is based on the increased sensitivity of the tip sample distance control due to the resonance enhancement of the tip vibration, which is characterized by the quality factor of the cantilever. This value is substantially reduced by the viscosity of the liquid environment and accompanied by a large shift of the resonance frequency. Various theoretical descriptions of the dynamic behavior of AFM cantilever and optical fiber probes are given in the literature. However, most of them are not very accurate, since they either neglect the viscous damping due to a liquid medium, 12,13 use numerical analysis, 14 or estimate the effect analogous to a moving sphere All these theories need further parameters like an effective radius of a sphere or a geometrical factor which have to be extracted from experimental resonance curves measured in a liquid environment. The aim of this work is to give an illustrative, but also exact, description of both the frequency shift and the broadening of the resonance band of a scanning probe due to the presence of a viscous fluid. It will be demonstrated that both values can be calculated a priori from the knowledge of the resonance curve measured in air, the viscosity and density of the liquid, and the diameter and mass density of the fiber. First, we present a full analytical solution of the hydrodynamic problem of a vibrating cantilever with cylindric cross section as is the case for a SNOM fiber. Second, a simple approach towards an extension of this theory to describe the dynamic behavior of AFM cantilevers is presented. Third, we compare experimental resonance curves of an optical fiber measured in air and in distilled water to the theoretically derived data and we discuss the behavior of AFM cantilevers /98/84(4)/1782/9/$ American Institute of Physics

2 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff 1783 operated in tapping mode with the help of literature data taken from Refs. 13 and 14. II. THEORETICAL DESCRIPTION A. Hydrodynamic problem for cylindric cross sections For a better understanding of the formulas used in Sec. IV we present a brief outline of the full hydrodynamic description of an infinite cylinder vibrating in a viscous fluid. This is a typical engineering problem and the detailed solution together with many applications is presented in Refs. 17 and 18. It is well known that a rigid body experiences resistance when it is moving in an ideal, incompressible fluid at variable velocity u/ t, even under the condition of potential flow. In this case, not only the body itself has to be accelerated but also a specific amount of the fluid which is sticking to the surface of the body and is moving with the same velocity. Hence, the structure component behaves as though an added mass oluid was attached to it. This results in an additional inertial force 2 u g i m a t 2, 1 where m a is referred to as the added mass. This force is in phase with the acceleration of the body, because every fluid element, even if it is far away from the surface, is accelerated instantaneously with the body. The added mass is proportional to the displaced mass of the fluid, m d, which is given by the volume of the body, V, and the density of the fluid, f : m a C m m d C m V, 2 where C m is the added mass coefficient. It can be shown, that for an ideal fluid, C m In a viscous fluid, the response of all the fluid elements is not necessarily instantaneous. Instead, a phase shift between the structure and the fluid motion has to be taken into account. This results in a second force, the fluid damping force, which is expressed by u g V C V t, 3 where C V is the fluid damping coefficient. The equation of motion of an arbitrarily shaped body moving in a quiescent viscous fluid is then obtained by adding the force g g i g V to the equation of motion of the body in vacuum. The problem can be completely solved if the two constants C m and C V are known. In the following we will restrict ourselves to the problem of a freely vibrating cylindrical cantilever. We first calculate the constants C m and C V for an infinite cylinder with a mass per unit length m oscillating with a small amplitude u u 0 exp(i t), and surrounded by an incompressible viscous fluid of density f, and viscosity. To simplify the boundary conditions we assume that the cantilever with diameter D is concentrically surrounded by a cylindrical vessel of diameter D 0 which is filled by the fluid, as outlined in Fig. 1. FIG. 1. A cylindrical cantilever vibrating in a viscous fluid that is enclosed by a cylindrical tube. Later we will consider the limit D 0 to obtain the solutions for large liquid cells. With these assumptions, the problem can be treated as two dimensional where the fluid is described by a scalar velocity potential (r, ) in cylindrical coordinates r and. The time dependent spatial distribution of the velocity potential is then described by the linearized Navier Stokes equation 17, t 2 0, 4 where is the kinematic viscosity of the fluid. The components of the velocity field in cylindric coordinates are derived from : u r 1 r, u r. If the inner cylinder is oscillating with constant frequency and amplitudes u then the fluid on the surface of the cylinder must exactly follow this movement. Therefore, at r D/2 the boundary conditions are u r u 0 cos exp i t u u 0 sin exp i t. 6 At r R D 0 /2 all velocity components have to vanish: u r 0, u 0. 7 Equations 4 7 are the complete mathematical description of the reaction of the fluid to the oscillation of the rigid rod. The solution of Eq. 4 is a complex expression of Bessel functions. It is listed for completeness in Appendix A. In principle, the solution only depends on the diameter ratio D/D 0 and a dimensionless parameter, the kinetic Reynolds number, defined as D 2 R k 4. 8 The total force g per unit length that acts on the cylinder is calculated from the pressure p on the cylinder surface, i.e., at R D/2: R, p R, f. 9 t All components along the whole surface are summed to obtain the resulting force: 2 g t p R, R cos d

3 1784 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff For steady-state oscillations the calculation yields a remarkably simple result: d 2 u du g C m m d dt 2 C V dt M d a Re H sin t Im H cos t, 11 which allows to extract the expressions for C m and C V from the real and imaginary part of a complex master function H: C m Re H, 12 C V m d Im H, 13 m d f R The function H is a complicated rational function of Bessel functions and is given in the Appendix. It does not depend on time but on the diameter ratio D/D 0 and the kinetic Reynolds number R k. For many geometrical arrangements which are of practical relevance, more simple expressions for H can be derived: 1 For an ideal fluid, 0, and hence H D 0 2 D 2 D 2 0 D If the ideal fluid is infinitely extended 0 and D 0, H 1 and therefore C m 1 and C V 0. In this case the added mass equals the displaced mass m a m d R 2. 3 In an infinite viscous fluid v 0 and D 0, the function H can be approximated as H 1 4K 1 K 0, 16 where K 0 and K 1 are the modified Bessel functions of the zeroth, first order, and second kind, respectively. 4 For large kinetic Reynolds numbers (R k D 2 /4 1) and an infinite viscous fluid, H can be further simplified to H i. 17 2R k 2R k For most practical applications, Eqs. 16 and 17 provide results of satisfying accuracy. The real and imaginary part of H is plotted in Fig. 2 versus R k using the formula 16. The approximate expressions 17 are also shown for large values of R k. We want to emphasize here, that this theory is only valid for amplitudes of oscillations that are small compared to typical dimensions of the problem, such as the diameter of the rod or the distance between the surface of the rod and the limit of the surrounding liquid. B. Vibration dynamics of a cantilever If the coefficients C m and C V for the added mass and the viscous damping are known, the problem of a vibrating cantilever totally immersed into a viscous fluid is easily solved. In this case the equation of motion is given by EI 4 u z 4 C S C V u t m m a 2 u t 2 0, 18 FIG. 2. Real and imaginary part of the complex function H 16 vs kinetic Reynolds number R k D 2 /4 for a cylindrical body vibrating in an infinitely extended viscous fluid. where E is Young s modulus, I is the moment of area (I D 4 /64), C S is a structural damping coefficient which describes internal losses, and m is the mass per unit length. The z coordinate is along the main axis of the cantilever. The internal losses are due to dissipation of energy by the bending of the cantilever and are responsible for the damping in vacuum. In a more accurate description they have to be considered proportional to ( 4 u/ z 4 t) in Eq. 18. However, it is shown in the Appendix that this description is identical to ours for the case of small internal losses. The motion of the beam in vacuum is described by setting C V m a 0. The resonance frequencies of the damped cantilever are the complex solutions of Eq. 18 using u(t) u 0 exp(i t) and result in n f 4 n 2 f 0 2 i, 19 where 2 f 0 EI m m a L 4, 20 f C V C S 2 m m a, 21 and L is the total length of the cantilever. The coefficients n 2 are fixed by the boundary conditions as n 3.52, 22.4, 61.7,..., for n 1,2,3,..., respectively. Note, that besides the factor 4 n, Eq. 19 is identical to the wellknown result of a damped harmonic oscillator with EI/L 4 acting as the force constant. In any scanning probe application the cantilever is driven by an external force with variable frequency. Analogous to the case of a harmonic oscillator the frequency dependence of the amplitude u( ) of the cantilever is described by u 0 u 2 f The width of the resonance band is characterized by the quality factor Q, defined as

4 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff 1785 Q f m m a C S C V 0, 23 where is the full width of the line taken at 1/& of the maximum value. For small the frequency shift due to this damping factor can be neglected. This is justified even for very low quality factors of less then The ratio of the resonance frequencies of the cantilever immersed into the fluid f /2 compared to the frequencies measured in vacuum or air, f v, can now easily be deduced from Eq. 19 as f v m m m a 1 1 C m f b, 24 where b is the mass density per unit length of the cantilever. Here we neglect the damping term. Similarly, the ratio of the quality factors is obtained from Eq. 23 as Q f C S f v 1 Q v C S C V 1 C V C S f v, 25 and C V Im H f Q C S b f v. 26 v It is important to note that the key number for the calculation of the resonance frequency and the quality factor of the oscillation in a viscous fluid is the kinetic Reynolds number R k. The latter depends on the vibration frequency 2 f, which should be identical to the unknown frequency. Therefore, a self-consistent calculation procedure must be performed: The resonance frequency measured in air f v is used as a first approach to calculate R k and the constant C m to obtain. This is used again for the evaluation of R k, etc. until self-consistency is approached. Two cycles give satisfying results for most cases. C. Cantilever with rectangular cross section FIG. 3. Principal sketch of a rectangular a and a V-shaped b AFM cantilever. The simple expressions 11 and 17 were explicitly derived for the cylindric cross section of the cantilever. It is obvious however that similar solutions can be derived for cantilevers with elliptical cross sections when elliptical coordinates are used. These solutions are then applicable to rectangular shapes, at least if they are very flat. Since it is beyond the scope of this paper to solve the Navier Stokes Eq. 4 in elliptical coordinates, we present the following approach. The solutions of the cylindric problem are used, but we modify the kinetic Reynolds number R k and the factors C m and C V to take into account the new geometry. Usually the AFM cantilever is oscillating in a direction perpendicular to the width W as indicated in Fig. 3. Both hydrodynamic forces, the inertial term g i, and the damping term g V, are related to the volume oluid that is moving together with the solid structure. It is obvious that this mass is increasing with the width of the cantilever. Therefore we assume that the forces are proportional to W. On the other hand, the solutions should be identical to the case of a cylinder if W T. Both conditions are fulfilled if the constants C m and C V are multiplied by the dimensionless parameter W/T leading to the following transformation: C m W T C m, C V W T C V. 27 The kinetic Reynolds number depends on the square of a typical size of the moving body. For the case of a cylinder this is the radius R D/2. In the case of an ellipsoid or a rectangle, this should be the dimension perpendicular to the direction of movement, which again is the width W. Therefore, we define the Reynolds number as R k W2. 28 We would like to note that expressions 27 and 28 are not necessarily exact but only based on reasonable assumptions. However, as we will show later, they lead to remarkably good results. III. EXPERIMENTAL SETUP The experiments are performed using a home-built SNOM setup as outlined schematically in Fig. 4. A tapered optical quartz fiber is fixed in a piezotube such that the tip protrudes 0.5 cm towards the sample. A sinusoidal voltage is applied to the piezo from the oscillator of a lock-in amplifier EG&G, For detection of the movement, the beam of a laser diode Schäfter and Kirchhoff, 670 nm emission is FIG. 4. Schematical drawing of the SNOM setup. Only the optical fiber, the dither piezo, the liquid cell, and the light path of the optical amplitude detection system are sketched.

5 1786 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff FIG. 5. Dither amplitude of a SNOM fiber tip probe measured in air with a low excitation amplitude a and in air and in water with high excitation amplitude b. The most intensive resonance lines are clipped by the sensitivity range of the lock-in amplifier. focused on the tip of the fiber and the shadow of the beam is detected by a two-segmented photodiode LASER Components via the lock-in amplifier. Amplitudes below 1 nm can be detected by this method. A rectangular glass cavity is mounted on top of a glass substrate to form a cell for investigations within a liquid environment. The walls of the cell are precisely parallel to each other and do not essentially disturb the optical light path of the laser diode. To avoid any complications from varying liquid levels during the measurements the cell is filled until the tip is totally immersed into the liquid. A more detailed description of the whole instrument is given in Ref. 11. IV. RESULTS AND DISCUSSION A. Vibration spectra of an optical fiber in water In Fig. 5 a the tip vibration amplitude versus excitation frequency is shown for a typical tip. For the case of ambient air, three very sharp and distinct resonance peaks are visible together with a broad background signal which emerges above 250 khz. The resonance frequencies are located at 15.4, 91.5, and khz. These can be well described by the eigenmodes of a cylindric cantilever fixed at one end and freely vibrating at the other with Eq. 20 and the parameters E N/m, I 4 r 1 4 where r m half the diameter of the fiber, and m B r 2, where B 2.2 g/cm 3 is the mass density of quartz glass. Since the length of the tip is known only with an accuracy of 0.2 mm, we use it as an adjustable parameter. With a value of L 2.5 mm the first mode is calculated correctly, while the second and third amount to 96.6 and 271 khz, respectively. The discrepancies in the data are explained by the arbitrariness of the value chosen for the length L and neglecting the tapered region of the tip. In Ref. 12 it was shown that the tip shape has a significant influence on the resonance frequencies. Nevertheless, the spectrum of Fig. 5 a is a clear indication that the fiber is vibrating in its eigenmodes. When the tip is immersed into water the frequency spectrum usually is superimposed by other resonances as is shown in Fig. 5 b. Due to the damping of the fiber oscillation caused by the viscosity the dither amplitude of the piezotube has to be increased significantly in order to obtain the same tip amplitude. In our case, the drive voltage applied to the piezoelectrodes was increased by a factor of more than 5. Under these conditions, many well-pronounced resonance features appear and it is not at all obvious which of them belong to the resonant vibration modes of the fiber. This holds especially for the high frequency range above 200 khz. The situation can be clarified if the tip amplitude is recorded in air, also with high piezodrive voltage. The resulting spectrum is shown in Fig. 5 b by the dotted curve. From comparison of this curve with that one taken in water in the frequency range below 200 khz, one can see that some lines are not affected by the fluid. In contrast, the fiber modes in water are shifted towards significantly lower frequencies than those in air. The shift, as well as the broadening of the lines, must be due to the liquid environment of the tip. The lines which do not shift in frequency are due to resonances of either the piezotube or the instrumental setup. They strongly depend on the actual construction of the instrument. Above 250 khz many resonance lines appear in water that are not present in air, even at high piezodrive voltage. These resonances lead to very high amplitudes, which are clipped by the lock-in amplifier. Since in this frequency range the wavelength of sound is of the order of the size of our liquid cell, standing acoustic waves can be excited within the fluid. This subject will not be discussed here. For the following, we will concentrate on the first two eigenmodes of the fiber. B. Resonance line analysis of optical fiber In Figs. 6 and 7 the first and second resonance lines of the SNOM fiber tip are shown in a magnified view. In both cases, Eq. 22 has been used to fit the data measured in air. The resonance frequency f v, the quality factor Q v, the maximum amplitude a( n ), and an additional offset a 0 are used as free fitting parameters. The best fit is indicated in Figs. 6 and 7 by the straight line. From the Q value we deduce the internal or structural damping factor C S from the relation C S 2 f vm. 29 Q v The values are listed in Table I together with data obtained from the third eigenmode. The internal or structural damping coefficient C S shows frequency dependency. This is unexpected for modal damping 20 and may be caused by the approximation that the internal damping is proportional to the velocity, as is explained in the Appendix.

6 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff 1787 FIG. 6. First resonance maximum f 1 of dither amplitude of the fiber tip. Top: data measured in air together with a best fit according to the amplitude function 22 straight line ; bottom: data measured with fiber immersed into distilled water together with calculated amplitude function. Only the maximum of the curve and an additional offset were adjusted to the experimental data in order to correct instrumental influences. FIG. 7. Second resonance maximum f 2 of the fiber tip. As in Fig. 5 the top curves show experimental data taken in air together with a best fit, the bottom curve shows the data measured in distilled water together with the calculated resonance line. We use values for f v and C S to calculate the resonance lines in water. Since the dimensions of the liquid cell are large compared to the diameter of the tip, we can apply the approximation of an infinite viscous fluid to our problem. The viscosity of water at 20 C is Pa s; as a result, the kinetic Reynolds number R k is above 300 for the lowest eigenfrequency and increases for the higher modes see Table I. This allows us to use Eq. 17 to calculate the added mass coefficient C m and the viscous damping coefficient C V for each eigenmode. From these factors we deduce the new frequencies and factors Q f in the fluid using Eqs. 24 and 25 as listed in Table I. Again we have calculated the frequency dependence of the tip amplitude in water using Eq. 22. The result is shown together with the experimental data in Figs. 6 and 7. Only the offset and the amplitude were adjusted to the experimental data. In principal, the amplitude should also be exactly given by the theory. However, the experimental conditions are different when measuring through the liquid cell. Additionally, a best fit of Eq. 22 to the experimental curves was made. The results are listed in brackets in Table I. As can be seen from the graphs and the data, the frequency shift, as well as the damping and hence the broadening of the resonance lines, are described to an accuracy of 1%. C. AFM cantilever The extension of our theory to cantilevers with rectangular cross section was tested with data from the literature. We refer to the measured values of the frequency shift presented by Elmer and Dreier 13 for rectangular AFM cantilevers and the data presented by Chen et al. 14 for V-shaped cantilevers. Unfortunately, no data were presented for the Q values of the cantilevers since they only derived theories for the resonance frequency. The first group has been using four different silicon cantilevers operating in air, water, and bromoform. The experimental data together with values calculated with their theory and our approach are listed in Table II. In their paper they have derived a theory for thin cantilevers, which holds for T W. Additionally, they assumed an infinitely extended beam with periodic boundary conditions and they have neglected the viscosity of the medium. TABLE I. Data of the first three eigenmodes of a SNOM fiber tip operated in air and in distilled water. The Q values and resonance frequencies noted in brackets are obtained from fits. The Q values of the second and third mode could not be measured unambiguously due to the background signal. n f v Q v R k C S C m C V Q f

7 1788 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff TABLE II. Data of rectangular AFM cantilever operated in air, water ( f 1 g/cm 3 ), and bromoform ( f 2.82 g/cm 3 ). The frequencies are given in khz, the density of the silicon cantilevers is b 2.33 g/cm 3. The experimental data and the calculated values of the third column are taken from Elmer and Dreier Ref. 13 the frequencies are obtained from our theory. The data of the cantilever are: a W 44 m, T 2.18 m; b W 37 m, T 6 m; c W 37 m, T 5.75 m; d W 29 m, T 3 m. n Air f v exp exp Fluid Elmer a water ] ] ] ] ] b c bromoform d As a consequence of the periodic boundary conditions their theory provides only good values for large mode numbers n. For decreasing n the resonance frequencies are systematically overestimated with an increasing error up to 30% for n 1. The opposite behavior occurs with our theory. It provides excellent results for very low n but the relative error increases systematically with increasing n, e.g., for sample a in Table II, the error ranges from less than 1% for n 1 to more than 20% for n 7. This deviation of the calculated values from experimental data is not observed for the cylindrical cantilever. Much larger deviations can be seen for samples d and c which were operated in bromoform, but at least for sample d we believe that this is partially caused by the inaccuracy in the values of W and T. Otherwise it would be difficult to explain why the frequency is overestimated in sample c and underestimated in sample d although the ratio W/T is not much different. The same tendency is observed for the values given by Elmer. It is interesting to note that the theory of Elmer and Dreier gives an expression for the frequency ratio which is identical to Eq. 24, if the modified values of Eqs. 27 and 28 are used and C m is replaced by the factor 4 f ( n W/L) where L is the length of the cantilever. In their paper the mode number n was named n. There, the function f was calculated numerically and has the asymptotic behavior f (x) 1/2x for x and f (x) 0.2 for x 0. In our approach the factor C m also depends on frequency and hence, on the length and the mode number of the beam. The explicit expression of C m is TABLE III. Summary of experimental data and calculated values of the resonance frequency of V-shaped AFM cantilevers operated in media with different kinematic viscosity. The data of columns 1 3 are taken from Chen et al. Ref. 14. We have calculated with the following data: b 4.22 g/cm 3 weighted average of gold and Si 3 Ni 4, W 80.4 m, T 0.58 m. C m 1 4 2R k 1 f exp f Chen Air Hexane Water Ethanol Hexadecane n W 2 0 L. 30 It is clear that C m 1 for all values of R k and C m 1 for high values of R k and thus for large x. This difference in the asymptotic behavior explains the increasing discrepancy of the results with increasing mode number n. At low mode numbers our expression obviously gives a more realistic description. It is always above the maximum value of 0.8 given by Elmer and therefore leads to smaller values of the frequency. The width of the resonance line in water was estimated for the first sample a in Table II. Assuming Q v 500 for the vibration in air, we obtain Q f 5.9 in water which corresponds to a width of f 700 Hz, which is a reasonable value. The most crude approximation made here was the treatment of the kinetic Reynolds number in Eq. 28. In order to examine this assumption we tried to reproduce the data of Chen et al. obtained for V-shaped cantilevers operated in media with different viscosities. The data and the calculated values are listed in Table III. The frequency shift was calculated by Chen using the model of a moving sphere. The radius and an additional geometrical factor were adjusted to the observed effect in water. In the case of V-shaped cantilevers the deviations from the cylinder geometry are even stronger than in the case of the rectangular cantilevers. We have approximated the cross section of the cantilever by a single rectangle of size 2W T, where W is the width of one side of the triangle see Fig. 3. Obviously, the calculated resonance frequencies for the different liquids show large deviations from the experimental data. The differences increase for increasing kinematic viscosities. However, this could also be caused by the wrong approximation of the geometry of the cantilever. We would like to emphasize, that our data are obtained from the geometry of the cantilever and the known properties of the fluids without any further adjustment of additional parameters. With respect to that, the deviations are in an acceptable range. Therefore, even for V-shaped cantilevers the theory outlined here can be used to calculate an approximate value of the resonance frequency and the resonance broadening when it is operated in an viscous fluid.

8 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff 1789 V. CONCLUSION The eigenfrequencies of a cylindrical fiber probe of a SNOM or an AFM cantilever are significantly lowered in a viscous medium compared to ambient air. Furthermore, the viscosity leads to broadening of the resonance peak in the frequency spectrum and hence to lowering of the Q value. Here we present a full analytical description for the behavior of cylindrical cantilevers, as was found in Ref. 17. All interactions of the cantilever with the liquid are summarized by two constants: the added mass coefficient C m, which accounts for the frequency shift, and the viscous damping coefficient C V, which explains the broadening of the resonance line in the presence of a viscous medium. These constants can be easily evaluated with the complex function H Fig. 2 from the kinetic Reynolds number R k which depends only on the diameter of the rod, the kinematic viscosity of the fluid, and the vibration frequency. Since the resonance frequency in water must be used for the evaluation of R k, a selfconsistent solution must be found. As was shown by comparison with experimental data this provides a simple method to determine quantitatively the fundamental parameters of the resonance behavior of SNOM fiber tips in liquids. The shift as well as the broadening was obtained to within a relative error of less than 1%. The theory can also be applied to rectangular and V- shaped AFM cantilevers, but with reasonable accuracy only for low eigenmodes. This was demonstrated by comparison with experimental data taken from the work of Elmer 13 and Chen. 14 The agreement was again with an error of less than 1% for rectangular cantilevers operated in water. Larger deviations were observed for bromoform and other liquids. However, even for the V-shaped cantilevers the description provides reasonable values for the frequency shifts which are nearly in the range of one order of magnitude. ACKNOWLEDGMENTS The idea to use the SNOM technique for imaging under water was mainly initiated by Hubert Motschmann within the framework of a joined project with Hüls AG, which we gratefully acknowledge. We thank Professor Möhwald and the Max-Planck-Gesellschaft for supporting this work. APPENDIX A: COMPLETE SOLUTION OF EQ. 4 The complete analytical solution of the Navier Stokes Eq. 4 reads as r, u 0 A 1 D 2 r A 2r A 3 DI 1 r A 4 DK 1 r sin exp i t, A1 where I 1 and K 1 are the modified Bessel functions of the first order and first and second kind, respectively. We introduce the abbreviations and i A2 D 2, D 0 2, D 0 D, A3 to write the constants A 1 A 4 which are determined by the boundary conditions 6 and 7 : A 1 2 I 0 K 0 I 0 K 0 and 2 I 1 K 0 I 0 K 1 2 I 0 K 1 I 1 K 0 4 I 1 K 1 I 1 K 1 /, A 2 2 I 1 K 0 I 0 K I 0 K 0 I 0 K I 1 K 0 I 0 K 1 /, A 3 2 K 0 4 K K 0 4K 1 /, A 4 2 I 0 4 I I 0 4I 1 /, I 0 K 0 I 0 K 0 2 I 0 K 1 I 1 K 0 I 1 K 0 A4 A5 A6 A7 I 0 K I 0 K 1 I 0 K 1 I 1 K 0 I 1 K 0. A8 Here again, I and K are the modified Bessel functions of the second kind. From this the explicit form of the master function H(,, ) is derived as H 2 2 I 0 K 0 I 0 K 0 4 I 1 K 0 I 0 K 1 4 I 0 K 1 I 1 K 0 8 I 1 K 1 I 1 K 1 / I 0 K 0 I 0 K 0 2 I 0 K 1 I 1 K 0 I 1 K 0 I 0 K I 0 K 1 I 0 K 1 I 1 K 0 I 1 K 0 1. A9

9 1790 J. Appl. Phys., Vol. 84, No. 4, 15 August 1998 Kirstein, Mertesdorf, and Schönhoff APPENDIX B: INTERNAL LOSSES The intrinsic loss of a vibrating beam is usually referred to as internal friction or imperfect elasticity. 12,21 It is taken into account by a velocity dependent part in the stress strain relation Eu du dt E u du B1 dt, where is the stress, is the internal frictional coefficient, and is the intrinsic loss factor. Therefore, the equation of motion of the cantilever Eq. 18 is modified to EI 4 z 4 u t m 2 u t 2 0, B2 and the solutions are found from the dispersion relation m 2 k 4 EI 1 i m EI 2 i 3, B3 where k 4 n 4 /L 4 for the case of one end fixed and the other end freely vibrating and the approximation holds for very small. In real systems 1 and therefore the approximate solution on the very right-hand side is applicable. The latter relation must be compared to the dispersion relation obtained from Eq. 18 which reads as k 4 m EI 2 i2. B4 This expression is identical to Eq. B3 if 2 / 2. Since we are only interested in the frequency dependence in the vicinity of the eigenmodes, it is reasonable to approximate 3 by 3 n 2. Under these circumstances, the solutions B3 and B4 of Eqs. 18 and B2 are identical. 1 M. Radmacher, R. Tillmann, M. Fritz, and H. Gaub, Science 257, P. Hansma et al., Appl. Phys. Lett. 64, R. Toledo-Crow, P. Yang, Y. Chen, and M. Vaez-Iravani, Appl. Phys. Lett. 60, E. Betzig, P. Finn, and J. Weiner, Appl. Phys. Lett. 60, M. Gregor, P. Blome, J. Schöfer, and R. Ulbrich, Appl. Phys. Lett. 68, F. Froelich and T. Milster, Appl. Phys. Lett. 70, P. J. Reid, D. A. Higgins, and P. F. Barbara, J. Phys. Chem. 100, K. Kajikawa, H. Hara, H. Sasabe, and W. Knoll, Colloids Surf., A 126, R. Dunn, G. Mets, L. Holtom, and X. Xie, J. Phys. Chem. 98, P. Moyer and S. Krämer, Appl. Phys. Lett. 68, M. Mertesdorf, M. S. Schönhoff, F. Lohr, and S. Kirstein, Surf. Interface Anal. 25, Y. Yang, D. Hehe, P. K. Wei, W. S. Famm, H. H. Gray, and J. W. P. Hsu, J. Appl. Phys. 81, F.-J. Elmer and M. Dreier, J. Appl. Phys. 81, G. Chen, R. Warmack, T. Thundat, and D. Allison, Rev. Sci. Instrum. 65, G. Chen, R. Warmack, A. Huang, and T. Thundat, J. Appl. Phys. 78, S. Inaba, K. Akaishi, T. Mori, and K. Hane, J. Appl. Phys. 73, S.-S. Chen, Flow-Induced Vibration of Circular Cylindrical Structures Hemisphere, Washington, New York, London, S. Chen, M. Wambsganss, and J. Jendrzejczyk, Trans. ASME, J. Appl. Mech. 43, L. Landau and E. Lifshitz, Lehrbuch der Theoretischen Physik, Band VI, Hydrodynamik Akademie, Berlin, L. Landau and E. Lifshitz, Lehrbuch der Theoretischen Physik, Band I, Klassische Mechanik Akademie, Berlin, L. Landau and E. Lifshitz, Lehrbuch der Theoretischen Physik, Band VII, Elstizitätstheorie Akademie, Berlin, 1991.