**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#

**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 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:}
kirstein@mpikg.fta-berlin.de

b!_{Current address: Physical Chemistry 1, University of Lund, Sweden.}

1782

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

*gi*52*ma*

]2_{u}

]*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

*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,*

r*f*:

*ma*5*Cmmd*5*Cm*r*V,* ~2!

*where Cm*is the added mass coefficient. It can be shown, that

*for an ideal fluid, Cm*51.

17

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

*gV*52*CV*

]*u*

]*t*, ~3!

*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 g*5*gi*

1*gV* to the equation of motion of the body in vacuum. The

*problem can be completely solved if the two constants C _{m}*

*and C*are known.

_{V}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*5

*u*0

*exp(i*v

*t), and surrounded by an incompressible viscous*fluid of densityr

*f*, 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 D*0 which is filled by the fluid, as outlined in Fig. 1.

*Later we will consider the limit D*0*→*` 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 and*u. The time dependent spatial distribution
of the velocity potential is then described by the linearized
Navier–Stokes equation17,19
¹4_{c}_{2}1
n
]
]*t* ¹
2_{c}_{5}_{0,} _{~}_{4}_{!}
wheren is the kinematic viscosity of the fluid. The
compo-nents of the velocity field in cylindric coordinates are derived
from c:
*ur*52
1
*r*
]c
]u, *u*u5]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 r*5*D/2 the*
boundary conditions are

*u _{r}*5

*u*0 cosu exp~

*i*v

*t*!

*u*u52

*u*0 sin u exp~

*i*v

*t*!. ~6!

*At r*5

*R*5

*D*0/2 all velocity components have to vanish:

*ur*50, *u*u50. ~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*
*Rk*5
v
n
*D*2
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*5*D/2:*

*p*~*R,*u!52r*f*

]c~*R,*u!

]*t* . ~9!

All components along the whole surface are summed to ob-tain the resulting force:

*g*~*t*!52

### E

02p

*p*~*R,*u!*R cos*u*d*u. ~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:

*g*52*Cmmd*
*d*2*u*
*dt*22*CV*
*du*
*dt*
5*Mda*v@Re~*H*!sinv*t*1Im~*H*!cosv*t*#, ~11!

*which allows to extract the expressions for Cmand CV*from

*the real and imaginary part of a complex master function H:*

*Cm*5Re~*H*!, ~12!

*CV*52*md*vIm~*H*!, ~13!

*md*5r*f*p*R*2. ~14!

*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*0and 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

*H*5*D*0
2_{1}

*D*2

*D*_{0}22*D*2. ~15!

~2! If the ideal fluid is infinitely extended ~n50 and
*D*_{0}*→*`!*, H*5*1 and therefore C _{m}*5

*1 and C*50. In this case

_{V}*the added mass equals the displaced mass m*5

_{a}*m*5rp

_{d}*R*2

_{.}

~3! In an infinite viscous fluid ~*v*Þ*0 and D*_{0}*→*`!, the
*function H can be approximated as*

*H*511*4K*1~a!

a*K*0~a!, ~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 (Rk*5v*D*2/4n

@*1) and an infinite viscous fluid, H can be further simplified*
to

*H*511 4

### A

*2R*2

_{k}*i*4

### A

*2R*. ~17!

_{k}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 R _{k}* 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 CV*for 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* ]
4* _{u}*
]

*z*41~

*CS*1

*CV*! ]

*u*]

*t*1~

*m*1

*ma*! ]2

*]*

_{u}*t*250, ~18!

*where E is Young’s modulus, I is the moment of area*
*(I*5p*D*4*/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/*]*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 CV*5*ma*50.

The resonance frequencies of the damped cantilever are
the complex solutions of Eq. ~18! *using u(t)*5*u*_{0}*exp(i*v*t)*
and result in
v~*f*
*n*!_{5}

### A

_{k}

*n*4 v

*f 0*2

_{2}

_{l}2

_{1}

_{i}_{l}

_{,}

_{~}

_{19}

_{!}where v

*f 0*2

_{5}

*EI*~

*m*1

*ma*!

*L*4 , ~20! l

*f*5

*CV*1

*CS*2~

*m*1

*m*!, ~21!

_{a}*and L is the total length of the cantilever. The coefficients*k* _{n}*
are fixed by the boundary conditions as k

*n*

2
5*3.52, 22.4, 61.7,..., for n*51,2,3,..., respectively. Note,
that besides the factor k* _{n}*4, Eq.~19!is identical to the

*well-known 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.
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

*u*~v!5 *u*0

### A

~v*f 0*

2 _{2}_{v}2_{!}2_{1}_{4}_{l}2_{v}2. ~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 Rk*5v*D*

2

/4n for a cylindrical body vibrating in an infi-nitely extended viscous fluid.

*Qf*5
v0
2l5
v0~*m*1*ma*!
~*CS*1*CV*! '
v0
Dv, ~23!

where Dvis the full width of the line taken at 1/& of the

maximum value.

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*

5v*f*/2p compared to the frequencies measured in vacuum

*or air, f _{v}*, can now easily be deduced from Eq.~19!as

*ff*

*f*5

_{v}### A

*m*

*m*1

*ma*5 1

### A

11*Cm*r

*f*r

*b*, ~24!

wherer* _{b}*is 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*
*Qv*5
*CS*
*CS*1*CV*
*f _{v}*

*ff*5 1 11

*CV*

*CS*

*f*

_{v}*ff*, ~25! and

*CV*

*CS*5@2Im~

*H*!# r

*f*r

*b*

*ff*

*f*

_{v}*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

52p*f , which should be identical to the unknown frequency*
*f _{f}*. Therefore, a self-consistent calculation procedure must

*be performed: The resonance frequency measured in air f*is

_{v}*used as a first approach to calculate R*

_{k}*and the constant C*

_{m}*to obtain f*, etc. until self-consistency is approached. Two cycles give satis-fying results for most cases.

_{f}. This is used again for the evaluation of R_{k}**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 R _{k}* and the factors

*C*to take into account the new geometry.

_{m}and C_{V}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 W*5*T. Both conditions are fulfilled if the constants*
*Cm* *and CV* are multiplied by the dimensionless parameter

*W/T leading to the following transformation:*

*C _{m}*h5

*W*

*T* *Cm*, *CV*

h_{5}*W*

*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 R*5*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. *
There-fore, we define the Reynolds number as

*R _{k}*h5v

n *W*2. ~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
*E*56.531010*N/m, I*51

4p*r*

4 _{where r}_{5}_{62.5}_{3}_{10}_{2}6 _{m} _{~}_{half}
the diameter of the fiber!*, and m*5r*B*p*r*2, where r*B*

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 f _{v}, the quality factor Q_{v}*, the

*maximum amplitude a(*v

*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 CS*from the

relation

*CS*5

2p*f _{v}m*

*Q _{v}* . ~29!

The values are listed in Table I together with data
ob-tained from the third eigenmode. The internal or structural
*damping coefficient CS*shows 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 f _{v}*

*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 f*2of 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 f*1of 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
*n*51.

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 n*51
*to more than 20% for n*57. 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 (*k*nW/L)*

*where L is the length of the cantilever. In their paper the*
mode number k*n* was nameda*n. There, the function f was*

calculated numerically and has the asymptotic behavior
*f (x)*5*1/2x for x→*` *and f (x)*'*0.2 for x→*0. In our
*ap-proach the factor Cm*also depends on frequency and hence,

on the length and the mode number of the beam. The explicit
*expression of Cm*is
*Cm*511
4

### A

*2Rk*511 4 k

*nW*

*L*

### A

2v0 n . ~30!*It is clear that C _{m}*.

*1 for all values of R*

_{k}*and C*1 for

_{m}→*high values of R*asymptotic behavior explains the increasing discrepancy of

_{k}and thus for large x. This difference in the*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 Q _{v}*5500 for

*the vibration in air, we obtain Qf*55.9 in water which

cor-responds to a width of D*f*5700 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*

### 8

3*T, where W*

### 8

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 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 (r*f*

51 g/cm3_{), and bromoform (}_{r}

*f*52.82 g/cm3). The frequencies are given in

kHz, the density of the silicon cantilevers is r*b*52.33 g/cm3. The

experi-mental data and the calculated values of the third column are taken from
Elmer and Dreier~Ref. 13!*the frequencies ff*are obtained from our theory.

The data of the cantilever are: ~a! *W*544m*m, T*52.18mm; ~b! *W*

537m*m, T*56mm; ~c! *W*537m*m, T*55.75mm; ~d! *W*529m*m, 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:r*b*

54.22 g/cm3 ~weighted average of gold and Si3Ni4!*, W*580.4m*m, T*

50.58mm.
n *f*exp * _{f}*Chen

_{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

**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
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 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*which depends only on the diameter of the rod, the kinematic viscosity of the fluid, and the vibration frequency. Since the resonance frequency

_{k}*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.

**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 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

c~*r,*u!5*u*0

## S

*A*1

*D*2

*r* 1*A*2*r*1*A*3*DI*1~l*r*!

1*A*_{4}*DK*1~l*r*!

## D

sinu exp~*i*v

*t*!, ~A1!

*where I*1

*and K*1are the modified Bessel functions of the first order and first and second kind, respectively. We introduce the abbreviations l5

### A

*i*v n ~A2! and a5l

*D*2 , b5l

*D*0 2 , g5

*D*0

*D*, ~A3!

*to write the constants A*1*– A*4 which are determined by the
boundary conditions~6!and~7!:

*A*15$2a2@*I*0~a!*K*0~b!2*I*0~b!*K*0~a!#
12a@*I*1~a!*K*0~b!1*I*0~b!*K*1~a!#
22ag@*I*0~a!*K*1~b!1*I*1~b!*K*0~a!#
14g@*I*1~a!*K*1~b!2*I*1~b!*K*1~a!#%/D, ~A4!
*A*25$2ag@*I*1~b!*K*0~b!1*I*0~b!*K*1~b!#
1a2_{g}2_{@}_{I}_{0~}_{a}_{!}_{K}_{0~}_{b}_{!2}_{I}_{0~}_{b}_{!}_{K}_{0~}_{a}_{!#}
22ag2@*I*1~a!*K*0~b!1*I*0~b!*K*1~a!#%/D, ~A5!
*A*35$22a*K*0~b!24g*K*1~b!1g2@2a*K*0~a!
1*4K*1~a!#%/D, ~A6!
*A*45$22a*I*0~b!24g*I*1~b!1g2@2a*I*0~a!
1*4I*1~a!#%/D, ~A7!
and
D5a2_{~}_{1}_{2}_{g}2_{!@}_{I}_{0~}_{a}_{!}_{K}_{0~}_{b}_{!2}_{I}_{0~}_{b}_{!}_{K}_{0~}_{a}_{!#}
12ag@*I*0~a!*K*1~b!2*I*1~b!*K*0~b!1*I*1~b!*K*0~a!
2*I*0~b!*K*1~b!#12ag2@*I*0~b!*K*1~a!2*I*0~a!*K*1~a!
1*I*1~a!*K*0~b!2*I*1~a!*K*0~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

*H*5$2a2@*I*0~a!*K*0~b!2*I*0~b!*K*0~a!#24a@*I*1~a!*K*0~b!1*I*0~b!*K*1~a!#14ag@*I*0~a!*K*1~b!1*I*1~b!*K*0~a!#
28g@*I*1~a!*K*1~b!2*I*1~b!*K*1~a!#%/$a2_{~}_{1}_{2}_{g}2_{!@}_{I}_{0~}_{a}_{!}_{K}_{0~}_{b}_{!2}_{I}_{0~}_{b}_{!}_{K}_{0~}_{a}_{!#1}_{2}_{ag}_{@}_{I}_{0~}_{a}_{!}_{K}_{1~}_{b}_{!}
2*I*1~b!*K*0~b!1*I*1~b!*K*0~a!2*I*0~b!*K*1~b!#12ag2@*I*0~b!*K*1~a!2*I*0~a!*K*1~a!1*I*1~a!*K*0~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

s5*Eu*1m *du*

*dt*5*E*

## S

*u*1a

*du*

*dt*

## 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*EI*

## S

] 4 ]*z*41a ]

*u*]

*t*

## D

1*m*]2

*]*

_{u}*t*250, ~B2!

and the solutions are found from the dispersion relation

*k*45 *m*v
2
*EI*~11*i*av! '
*m*
*EI* ~v
2_{2}_{i}_{av}3_{!}_{,} _{~}_{B3}_{!}
*where k*45k* _{n}*4

*/L*4 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

*k*45*m*
*EI* ~v

2_{2}_{i2}_{l}_{v}_{!}_{.} _{~}_{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

v3 _{by}_{v}3_{5}_{v}

*n*

2 _{v}

. Under these circumstances, the solutions ~B3!and~B4!of Eqs.~18!and~B2!are identical.

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