Detailed measurements of the frequency responses of a series of rectangular atomic force microscope (AFM) cantilever beams, immersed in a range of fluids, have been performed to test the validity and accuracy of the recent theoretical model of Sader [J. Appl. Phys. 84, 64 (1998)]. This theoretical model gives the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Very good agreement between experimental measurements and theoretical calculations is found for all fluids considered. Furthermore, a critical assessment of the well-known inviscid model is presented, which demonstrates that this model is not applicable to AFM cantilever beams in general.

1.
T. R.
Albrecht
,
S.
Akamine
,
T. E.
Carver
, and
C. F.
Quate
,
J. Vac. Sci. Technol. A
8
,
3386
(
1990
).
2.
H.-J.
Butt
,
P.
Siedle
,
K.
Seifert
,
K.
Fendler
,
T.
Seeger
,
E.
Bamberg
,
A. L.
Weisenhorn
,
K.
Goldie
, and
A.
Engel
,
J. Microsc.
169
,
75
(
1993
).
3.
G. Y.
Chen
,
R. J.
Warmack
,
T.
Thundat
, and
D. P.
Allison
,
Rev. Sci. Instrum.
65
,
2532
(
1994
).
4.
J. E.
Sader
,
I.
Larson
,
P.
Mulvaney
, and
L. R.
White
,
Rev. Sci. Instrum.
66
,
3789
(
1995
).
5.
D. A.
Walters
,
J. P.
Cleveland
,
N. H.
Thomson
,
P. K.
Hansma
,
M. A.
Wendman
,
G.
Gurley
, and
V.
Elings
,
Rev. Sci. Instrum.
67
,
3583
(
1996
).
6.
T. E.
Schaffer
,
J. P.
Cleveland
,
F.
Ohnesorge
,
D. A.
Walters
, and
P. K.
Hansma
,
J. Appl. Phys.
80
,
3622
(
1996
).
7.
H.
Muramatsu
,
N.
Chiba
,
K.
Homma
,
K.
Nakajima
,
T.
Ataka
,
S.
Ohta
,
A.
Kusumi
, and
M.
Fujihira
,
Thin Solid Films
273
,
335
(
1996
).
8.
A.
Roters
and
D.
Johannsmann
,
J. Phys.: Condens. Matter
8
,
7561
(
1996
).
9.
T. E.
Schaffer
,
M.
Viani
,
D. A.
Walters
,
B.
Drake
,
E. K.
Runge
,
J. P.
Cleveland
,
M. A.
Wendman
, and
P. K.
Hansma
,
Proc. SPIE
3009
,
48
(
1997
).
10.
F.-J.
Elmer
and
M.
Dreier
,
J. Appl. Phys.
81
,
7709
(
1997
).
11.
M.
Tortonese
and
M.
Kirk
,
Proc. SPIE
3009
,
53
(
1997
).
12.
W.-H. Chu, Tech. Rep. No. 2, DTMB, Contract NObs-86396(X), Southwest Research Institute, San Antonio, TX (1963).
13.
U. S.
Lindholm
,
D. D.
Kana
,
W.-H.
Chu
, and
H. N.
Abramson
,
J. Ship Res.
9
,
11
(
1965
).
14.
D. G. Stephens and M. A. Scavullo, NASA TN D-1865 (April 1965).
15.
L.
Landweber
,
J. Ship Res.
15
,
97
(
1971
).
16.
G.
Muthuveerappan
,
N.
Ganesan
, and
M. A.
Veluswami
,
J. Sound Vib.
61
,
467
(
1978
).
17.
D. G.
Crighton
,
J. Sound Vib.
87
,
429
(
1983
).
18.
Y.
Fu
and
W. G.
Price
,
J. Sound Vib.
118
,
495
(
1987
).
19.
M. K.
Kwak
,
Trans. ASME, J. Appl. Mech.
63
,
110
(
1996
).
20.
R. E.
Hetrick
,
Sens. Actuators
18
,
131
(
1989
).
21.
J. E.
Sader
,
J. Appl. Phys.
84
,
64
(
1998
).
22.
For a composite beam, i.e., a beam composed of two or more layers, ρc is the average density of the beam.
23.
Park Scientific Instruments, 1171 Borregas Ave., Sunnyvale, CA 94089-1304.
24.
The theoretical models are also applicable to cantilever beams composed of crystalline materials, provided the crystal orientation is fixed over the length of the beam. The calibrated cantilevers satisfy this condition.
25.
Digital Instruments, 112 Robin Hill Road, Santa Barbara, CA 93117.
26.
Mathematica is a registered trademark of, and is available from Wolfram Research, Inc., 100 Trade Center Drive, Champaign, IL 61820–7237.
27.
The resonant frequency in vacuum of mode n can always be calculated from a knowledge of the fundamental resonant frequency ωvac,1 using the well-known formula ωvac,n=Cn2/C12ωvac,1, where Cn is the nth positive root of 1+cos Cncosh Cn=0.
28.
The shift in resonant frequency from vacuum to fluid is primarily accounted for in ωR,n, which neglects all dissipative effects in the fluid. Such dissipative effects are accounted for in the quality factor Qn, which introduces a comparatively small correction to the resonant frequency in fluid ωfluid.
29.
To establish the ultimate lower limit for L/b, for which the models are applicable, measurements need to be performed on cantilevers with aspect ratios smaller than those used in this study.
30.
These measurements were obtained by directing the AFM laser beam, which is normally used to measure the deflection of the cantilever, onto the cantilever substrate. The reflected signal was then processed in an identical manner to that described in the Appendix.
31.
AT-MIO-16E-1 board available from National Instruments, 6504 Bridge Point Parkway, Austin, TX 78730-5039.
32.
LabVIEW is a registered trademark of, and is available from National Instruments (see Ref. 31).
33.
R. C. Weast, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, 1985).
34.
J. P.
Cleveland
,
S.
Manne
,
D.
Bocek
, and
P. K.
Hansma
,
Rev. Sci. Instrum.
64
,
403
(
1993
).
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