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 Scho¨nhoffb) Max-Planck-Institut fu¨r Kolloid und Grenzfla¨chenforschung, 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. © 1998 American Institute of Physics. @S0021-8979~98!07916-X#
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 so-called tapping mode.1,2In this method, the tip–sample dis-tance is recorded dynamically: the cantilever is excited to oscillate in one of its eigenmodes and the tip amplitude pro-vides 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 ampli-tude 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.5,6It 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 anti-bodies bound to fragments of cell membranes.9
Most biological systems however must be investigated within their native liquid environment. Both methods, the
tapping mode and the shear force detection, have already been successfully applied to take images of soft samples un-der 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 vibra-tion, which is characterized by the quality factor of the can-tilever. 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,12,13use numerical analysis,14or estimate the effect analogous to a moving sphere.14–16All these theories need further parameters like an effective radius of a sphere or a geometrical factor which have to be extracted from experi-mental 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 broad-ening 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 hydrody-namic 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 fi-ber measured in air and in distilled water to the theoretically derived data and we discuss the behavior of AFM cantilevers
a!Author to whom correspondence should be addressed; electronic mail: email@example.com
b!Current address: Physical Chemistry 1, University of Lund, Sweden.
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 de-scription of an infinite cylinder vibrating in a viscous fluid. This is a typical engineering problem and the detailed solu-tion together with many applicasolu-tions 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 vari-able velocity ]u/]t, even under the condition of potential flow. In this case, not only the body itself has to be acceler-ated 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 of fluid was attached to it. This results in an additional inertial force
where ma 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 propor-tional to the displaced mass of the fluid, md, which is given
by the volume of the body, V, and the density of the fluid,
where Cmis the added mass coefficient. It can be shown, that
for an ideal fluid, Cm51.
In a viscous fluid, the response of all the fluid elements is not necessarily instantaneous. Instead, a phase shift be-tween 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
where CV 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 g5gi
1gV to the equation of motion of the body in vacuum. The
problem can be completely solved if the two constants Cm and CV are known.
In the following we will restrict ourselves to the problem of a freely vibrating cylindrical cantilever. We first calculate the constants Cmand CV for an infinite cylinder with a mass per unit length m oscillating with a small amplitude u 5u0 exp(ivt), and surrounded by an incompressible viscous fluid of densityrf, and viscosityh. To simplify the
bound-ary conditions we assume that the cantilever with diameter D is concentrically surrounded by a cylindrical vessel of diam-eter D0 which is filled by the fluid, as outlined in Fig. 1.
Later we will consider the limit D0→` to obtain the solu-tions for large liquid cells. With these assumpsolu-tions, the prob-lem can be treated as two dimensional where the fluid is described by a scalar velocity potentialc(r,u) in cylindrical coordinates r andu. The time dependent spatial distribution of the velocity potential is then described by the linearized Navier–Stokes equation17,19 ¹4c21 n ] ]t ¹ 2c50, ~4! wheren is the kinematic viscosity of the fluid. The compo-nents of the velocity field in cylindric coordinates are derived from c: ur52 1 r ]c ]u, uu5]c]r. ~5!
If the inner cylinder is oscillating with constant frequencyv and amplitudes u then the fluid on the surface of the cylinder must exactly follow this movement. Therefore, at r5D/2 the boundary conditions are
ur5u0 cosu exp~ivt! uu52u0 sin u exp~ivt!. ~6! At r5R5D0/2 all velocity components have to vanish:
ur50, uu50. ~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/D0 and a dimensionless parameter, the kinetic Reynolds number, defined as Rk5 v n D2 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 R5D/2:
]t . ~9!
All components along the whole surface are summed to ob-tain the resulting force:
p~R,u!R cosudu. ~10!
FIG. 1. A cylindrical cantilever vibrating in a viscous fluid that is enclosed by a cylindrical tube.
For steady-state oscillations the calculation yields a re-markably simple result:
g52Cmmd d2u dt22CV du dt 5Mdav@Re~H!sinvt1Im~H!cosvt#, ~11!
which allows to extract the expressions for Cmand CVfrom
the real and imaginary part of a complex master function H:
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/D0and the kinetic Rey-nolds number Rk. For many geometrical arrangements
which are of practical relevance, more simple expressions for H can be derived:
~1!For an ideal fluid,n50, and hence
~2! If the ideal fluid is infinitely extended ~n50 and D0→`!, H51 and therefore Cm51 and CV50. In this case the added mass equals the displaced mass ma5md5rpR2.
~3! In an infinite viscous fluid ~vÞ0 and D0→`!, the function H can be approximated as
where K0 and K1 are the modified Bessel functions of the zeroth, first order, and second kind, respectively.
~4! For large kinetic Reynolds numbers (Rk5vD2/4n
@1) and an infinite viscous fluid, H can be further simplified to
For most practical applications, Eqs.~16!and~17! pro-vide results of satisfying accuracy. The real and imaginary part of H is plotted in Fig. 2 versus Rk using the formula ~16!. The approximate expressions ~17! are also shown for large values of Rk. 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 Cmand CVfor the added mass and the
viscous damping are known, the problem of a vibrating can-tilever totally immersed into a viscous fluid is easily solved. In this case the equation of motion is given by
EI ] 4u ]z41~CS1CV! ]u ]t1~m1ma! ]2u ]t250, ~18!
where E is Young’s modulus, I is the moment of area (I5pD4/64), CS 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 bend-ing of the cantilever and are responsible for the dampbend-ing in vacuum. In a more accurate description they have to be con-sidered proportional to (]4]u/]z4]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 CV5ma50.
The resonance frequencies of the damped cantilever are the complex solutions of Eq. ~18! using u(t)5u0exp(ivt) and result in v~f n!5
Ak n 4 vf 0 2 2l21il, ~19! where vf 0 2 5 EI ~m1ma!L4 , ~20! lf5 CV1CS 2~m1ma!, ~21!
and L is the total length of the cantilever. The coefficientskn are fixed by the boundary conditions as kn
2 53.52, 22.4, 61.7,..., for n51,2,3,..., respectively. Note, that besides the factor kn4, Eq.~19!is identical to the well-known result of a damped harmonic oscillator with EI/L4 acting as the force constant.
In any scanning probe application the cantilever is driven by an external force with variable frequency. Analo-gous to the case of a harmonic oscillator the frequency de-pendence of the amplitude u(v) of the cantilever is de-scribed by
2 2v2!214l2v2. ~22! The width of the resonance band is characterized by the quality factor Q, defined as
FIG. 2. Real and imaginary part of the complex function H~16!vs kinetic Reynolds number Rk5vD
/4n for a cylindrical body vibrating in an infi-nitely extended viscous fluid.
Qf5 v0 2l5 v0~m1ma! ~CS1CV! ' v0 Dv, ~23!
where Dvis the full width of the line taken at 1/& of the
For smalllthe frequency shift due to this damping fac-tor can be neglected. This is justified even for very low qual-ity factors of less then 10.11 The ratio of the resonance fre-quencies of the cantilever immersed into the fluid ff
5vf/2p compared to the frequencies measured in vacuum
or air, fv, can now easily be deduced from Eq.~19!as ff fv5
Am m1ma5 1
A11Cm rf rb , ~24!
whererbis the mass density per unit length of the cantilever. Here we neglect the damping term l. Similarly, the ratio of the quality factors is obtained from Eq. ~23!as
Qf Qv5 CS CS1CV fv ff 5 1 11CV CS fv ff , ~25! and CV CS 5@2Im~H!# rf rb ff fv Qv. ~26!
It is important to note that the key number for the cal-culation of the resonance frequency and the quality factor of the oscillation in a viscous fluid is the kinetic Reynolds num-ber Rk. The latter depends on the vibration frequency v
52pf , which should be identical to the unknown frequency ff. Therefore, a self-consistent calculation procedure must be performed: The resonance frequency measured in air fv is used as a first approach to calculate Rk and the constant Cm to obtain ff. This is used again for the evaluation of Rk, etc. until self-consistency is approached. Two cycles give satis-fying results for most cases.
C. Cantilever with rectangular cross section
The simple expressions~11!and~17!were explicitly de-rived 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 coor-dinates are used. These solutions are then applicable to rect-angular shapes, at least if they are very flat. Since it is be-yond the scope of this paper to solve the Navier–Stokes Eq. ~4! in elliptical coordinates, we present the following ap-proach. The solutions of the cylindric problem are used, but we modify the kinetic Reynolds number Rk and the factors Cmand CV 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 gi, and the damping
term gV, are related to the volume of fluid 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
cyl-inder if W5T. Both conditions are fulfilled if the constants Cm and CV are multiplied by the dimensionless parameter
W/T leading to the following transformation:
T Cm, CV
T CV. ~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 R5D/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. There-fore, we define the Reynolds number as
n W2. ~28!
We would like to note that expressions~27!and~28!are not necessarily exact but only based on reasonable assump-tions. However, as we will show later, they lead to remark-ably 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 ampli-fier~EG&G, 5302!. For detection of the movement, the beam of a laser diode~Scha¨fter and Kirchhoff, 670 nm emission!is
FIG. 3. Principal sketch of a rectangular ~a! and a V-shaped ~b! AFM cantilever.
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.
focused on the tip of the fiber and the shadow of the beam is detected by a two-segmented photodiode ~LASER Compo-nents!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 inves-tigations 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 measure-ments the cell is filled until the tip is totally immersed into the liquid. A more detailed description of the whole instru-ment 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 234.5 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 E56.531010N/m, I51
4 where r562.531026 m ~half the diameter of the fiber!, and m5rBpr2, where rB
52.2 g/cm3 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 52.5 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
arbitrari-ness 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 frequen-cies. Nevertheless, the spectrum of Fig. 5~a!is a clear indi-cation that the fiber is vibrating in its eigenmodes.
When the tip is immersed into water the frequency spec-trum usually is superimposed by other resonances as is shown in Fig. 5~b!. Due to the damping of the fiber oscilla-tion caused by the viscosity the dither amplitude of the pi-ezotube 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 re-corded in air, also with high piezodrive voltage. The result-ing 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 fv, the quality factor Qv, the maximum amplitude a(vn), and an additional offset a0 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 CS from the
Qv . ~29!
The values are listed in Table I together with data ob-tained from the third eigenmode. The internal or structural damping coefficient CSshows frequency dependency. This is
unexpected for modal damping20 and may be caused by the approximation that the internal damping is proportional to the velocity, as is explained in the Appendix.
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 sensi-tivity range of the lock-in amplifier.
We use values for fv and CS 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 h51.031023 Pa s; as a result, the kinetic Reynolds number Rk 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 Cm and the viscous damping
coeffi-cient CV for each eigenmode. From these factors we deduce
the new frequencies ffand factors Qf 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 rectan-gular cross section was tested with data from the literature. We refer to the measured values of the frequency shift pre-sented by Elmer and Dreier13 for rectangular AFM cantile-vers 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 can-tilevers operating in air, water, and bromoform. The experi-mental 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 canti-levers, which holds for T!W. Additionally, they assumed an infinitely extended beam with periodic boundary condi-tions and they have neglected the viscosity of the medium.
FIG. 7. Second resonance maximum f2of 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.
FIG. 6. First resonance maximum f1of 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 maxi-mum of the curve and an additional offset were adjusted to the experimental data in order to correct instrumental influences.
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 fv Qv Rk CS Cm CV ff Qf
1 15.53 146.7 312 0.018 1.160 0.157 12.59 ~12.73! 18.9 ~18.7!
2 91.24 333.4 1812 0.046 1.066 0.379 74.88 ~75.0! 44.4 ~¯!
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 systemati-cally overestimated with an increasing error up to 30% for n51.
The opposite behavior occurs with our theory. It pro-vides 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 n51 to more than 20% for n57. This deviation of the calculated values from experimental data is not observed for the cylin-drical 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 ob-served 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 Cm is replaced by the factor 4 f (knW/L)
where L is the length of the cantilever. In their paper the mode number kn was namedan. There, the function f was
calculated numerically and has the asymptotic behavior f (x)51/2x for x→` and f (x)'0.2 for x→0. In our ap-proach the factor Cmalso depends on frequency and hence,
on the length and the mode number of the beam. The explicit expression of Cmis Cm511 4
A2Rk 511 4 knW L
A2v0 n . ~30!
It is clear that Cm.1 for all values of Rk and Cm→1 for high values of Rkand 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 de-scription. It is always above the maximum value of 0.8 given by Elmer and therefore leads to smaller values of the fre-quency.
The width of the resonance line in water was estimated for the first sample ~a! in Table II. Assuming Qv5500 for the vibration in air, we obtain Qf55.9 in water which
cor-responds to a width of Df5700 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 oper-ated 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 ad-justed 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
83T, where W
8is 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 experi-mental data. The differences increase for increasing kine-matic 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 param-eters. 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.
TABLE II. Data of rectangular AFM cantilever operated in air, water (rf
51 g/cm3), and bromoform (r
f52.82 g/cm3). The frequencies are given in
kHz, the density of the silicon cantilevers is rb52.33 g/cm3. The
experi-mental data and the calculated values of the third column are taken from Elmer and Dreier~Ref. 13!the frequencies ffare obtained from our theory.
The data of the cantilever are: ~a! W544mm, T52.18mm; ~b! W
537mm, T56mm; ~c! W537mm, T55.75mm; ~d! W529mm, T 53mm. n Air fv exp ff exp Fluid ff Elmer ff ~a! water 1 15.1 4.2 5.4 4.17 2 94.9 30 34.7 28.7 3 266.2 95 99.5 82.4 ] ] ] ] ] 7 1799 745 753.5 570 ~b! 1 169 85 94.5 85.3 2 1048 531 608 541 3 2862 1534 1764 1485 ~c! bromoform 1 158.4 44.5 59.7 51 2 984.5 322 386.8 326 3 2702 928 1136 901 ~d! 1 311.4 99.6 99.8 83.2 2 1920 563 666.8 528
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 n. The data of columns 1–3 are taken from Chen et al.~Ref. 14!. We have calculated ff with the following data:rb
54.22 g/cm3 ~weighted average of gold and Si3Ni4!, W580.4mm, T
50.58mm. n fexp fChen f f Air 0 18.25 ¯ ¯ Hexane 0.5 3.2 3.0 3.6 Water 1.0 2.2 2.2 2.8 Ethanol 1.5 2.3 2.4 3.1 Hexadecane 4.3 1.8 1.8 2.9
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 inter-actions of the cantilever with the liquid are summarized by two constants: the added mass coefficient Cm, which
ac-counts for the frequency shift, and the viscous damping co-efficient CV, 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 Rkwhich 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 Rk, a
self-consistent 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 Elmer13 and Chen.14The agreement was again with an error of less than 1% for rectangular cantilevers operated in water. Larger de-viations 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.
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 Hu¨ls AG, which we gratefully acknowledge. We thank Professor Mo¨hwald 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
Dsinu exp~ivt!, ~A1! where I1and K1are the modified Bessel functions of the first order and first and second kind, respectively. We introduce the abbreviations l5
Ai v n ~A2! and a5l D 2 , b5l D0 2 , g5 D0 D , ~A3!
to write the constants A1– A4 which are determined by the boundary conditions~6!and~7!:
A15$2a2@I0~a!K0~b!2I0~b!K0~a!# 12a@I1~a!K0~b!1I0~b!K1~a!# 22ag@I0~a!K1~b!1I1~b!K0~a!# 14g@I1~a!K1~b!2I1~b!K1~a!#%/D, ~A4! A25$2ag@I1~b!K0~b!1I0~b!K1~b!# 1a2g2@I0~a!K0~b!2I0~b!K0~a!# 22ag2@I1~a!K0~b!1I0~b!K1~a!#%/D, ~A5! A35$22aK0~b!24gK1~b!1g2@2aK0~a! 14K1~a!#%/D, ~A6! A45$22aI0~b!24gI1~b!1g2@2aI0~a! 14I1~a!#%/D, ~A7! and D5a2~12g2!@I0~a!K0~b!2I0~b!K0~a!# 12ag@I0~a!K1~b!2I1~b!K0~b!1I1~b!K0~a! 2I0~b!K1~b!#12ag2@I0~b!K1~a!2I0~a!K1~a! 1I1~a!K0~b!2I1~a!K0~a!#. ~A8! Here again, I and K are the modified Bessel functions of the second kind. From this the explicit form of the master func-tion H(a,b,g) is derived as
H5$2a2@I0~a!K0~b!2I0~b!K0~a!#24a@I1~a!K0~b!1I0~b!K1~a!#14ag@I0~a!K1~b!1I1~b!K0~a!# 28g@I1~a!K1~b!2I1~b!K1~a!#%/$a2~12g2!@I0~a!K0~b!2I0~b!K0~a!#12ag@I0~a!K1~b! 2I1~b!K0~b!1I1~b!K0~a!2I0~b!K1~b!#12ag2@I0~b!K1~a!2I0~a!K1~a!1I1~a!K0~b!
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
D, ~B1! wheresis the stress,mis the internal frictional coefficient, anda is the intrinsic loss factor. Therefore, the equation of motion of the cantilever Eq. ~18!is modified to
S] 4 ]z41a ]u ]t
D1m ]2u ]t250, ~B2!
and the solutions are found from the dispersion relation
k45 mv 2 EI~11iav! ' m EI ~v 22iav3!, ~B3! where k45kn4/L4 for the case of one end fixed and the other end freely vibrating and the approximation holds for very small a. In real systems av!1 and therefore the approxi-mate 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
k45m EI ~v
22i2lv!. ~B4! This expression is identical to Eq.~B3!ifa52l/v2. Since we are only interested in the frequency dependence in the vicinity of the eigenmodes, it is reasonable to approximate
. Under these circumstances, the solutions ~B3!and~B4!of Eqs.~18!and~B2!are identical.
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