The stochastic response of nanoscale oscillators of arbitrary geometry
immersed in a viscous fluid is studied. Using the fluctuation-dissipation
theorem it is shown that deterministic calculations of the governing fluid and
solid equations can be used in a straightforward manner to directly calculate
the stochastic response that would be measured in experiment. We use this
approach to investigate the fluid coupled motion of single and multiple
cantilevers with experimentally motivated geometries.Comment: 5 pages, 5 figure

D. Chandler

L. Rosenhead

M. C. Cross

M. R. Paul

English

arXiv

The stochastic response of nanoscale oscillators of arbitrary geometry immersed in a viscous fluid is studied. Using the fluctuation-dissipation theorem, it is shown that deterministic calculations of the governing fluid and solid equations can be used in a straightforward manner to directly calculate the stochastic response that would be measured in experiment. We use this approach to investigate the fluid coupled motion of single and multiple cantilevers with experimentally motivated geometries

Paul, M. R.

Cross, M. C.

Caltech Authors - Main

P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23Stochastic Dynamics of Nanoscale Mechanical Oscillators Immersed in a Viscous Fluid
M. R. Paul* and M. C. Cross
Department of Physics, California Institute of Technology 114-36, Pasadena, California 91125, USA
(Received 5 January 2004; published 9 June 2004)235501-1The stochastic response of nanoscale oscillators of arbitrary geometry immersed in a viscous fluid is
studied. Using the fluctuation-dissipation theorem, it is shown that deterministic calculations of the
governing fluid and solid equations can be used in a straightforward manner to directly calculate the
stochastic response that would be measured in experiment. We use this approach to investigate the fluid
coupled motion of single and multiple cantilevers with experimentally motivated geometries.
DOI: 10.1103/PhysRevLett.92.235501 PACS numbers: 81.07.–b, 45.10.–b, 83.10.Mjthis will be a function of the microscopic phase space
variables consisting of 3N coordinates and conjugate
momenta of the cantilever, where N is the number of
particles in the cantilever. We investigate the situation
situation characteristic of fluid-damped nanoscale canti-
levers, since the spectral response is very broad, and a
large number of fixed-frequency simulations would be
needed to characterize this response. A second advantageSingle molecule force spectroscopy using nanoscale
cantilevers immersed in fluid is a tantalizing experimen-
tal possibility [1,2]. The precise manner in which nano-
mechanical devices will be utilized for single-molecule
force spectroscopy and sensing is currently under devel-
opment [3,4]; however, the detection system will rely
upon the change in response of the cantilever due to the
binding of target biomolecules. It is therefore important
to build a baseline understanding of the motion of a fluid-
loaded cantilever, or arrays of cantilevers, in the absence
of active molecules. This is the focus of this Letter.
The dynamics of the nanoscale structures considered
are dominated by Brownian fluctuations although the
mechanical structures are still large compared to the
molecular size of the fluid molecules. The elastic response
of a long and slender cantilever immersed in fluid has
been investigated previously in the context of atomic
force microscopy (see, for example, [5,6]). However, the
response of nanoscale cantilevers (very strong fluidic
damping) or short and wide cantilevers (where end effects
would be important) is not well understood.
We show here that the fluctuation-dissipation theorem
allows for the calculation of the equilibrium fluctuations
using standard deterministic numerical methods. This is
possible because the same molecular processes are re-
sponsible for the dissipation and the fluctuations. In the
case under consideration here, predominantly these are
the collisions of the fluid molecules with the cantilever,
although dissipative processes within the elastic material
of the cantilever could also be included.
The fluctuation-dissipation theorem comes in many
forms. For the fluid-damped motion of nanoscale canti-
levers, it is sufficient to use a classical formulation. The
most convenient form is the one originally discussed by
Callen and Greene [7] (see also [8]). Consider a dynami-
cal variable A. For the classical system of interest here,0031-9007=04=92(23)=235501(4)$22.50 where a force ft that couples to A is imposed. In this
case, the Hamiltonian of the system isH  H0  fA, and
we look at the linear response for very small f. It can then
be shown for the special case of a step function force
given by
ft 

F for t < 0
0 for t  0 (1)
that, in the linear response regime, the change in the
average value of a second dynamical quantity B (again
any function of the 3N coordinates and momenta) from
its equilibrium value in the absence of f is given by
hBti  	Fh
A0
Bti0, where 	  kBT1, kB is
Boltzmann’s constant, T is the absolute temperature,

A  A hAi0, 
B  B hBi0, and the subscript zero
on the average h i denotes the equilibrium average in the
absence of the force f. Thus, we can calculate a general
equilibrium cross-correlation function in terms of the
linear response as
h
A0
Bti0  kBThBtiF : (2)
There are no approximations involved in this result, ex-
cept that of assuming classical mechanics and linear
behavior. If in addition the dynamical variables are suffi-
ciently macroscopic that the mean hBti can be calculated
using deterministic, macroscopic equations, we have our
desired result.
There are a number of advantages to a formulation of
the fluctuation-dissipation theorem in time rather than
frequency for our purposes. First, the full correlation
function is given by a single (numerical) calculation,
the response to removing a step force. The spectral prop-
erties can be obtained by Fourier transform. This is
particularly advantageous for the low-quality factor (Q)2004 The American Physical Society 235501-1
lw
h
b
l1
Wall
y
x
z
FIG. 1. Schematic of the cantilever geometry (not drawn to
scale): l  3 m, w  100 nm, l1  0:6 m, b  33 nm.
The cantilever is silicon with a density s  2330 kg=m3,
Young’s modulus Es  125 GPa, and spring constant k 
8:7 mN=m [3].
(b)
ν (KHz)
G
x(ν
)(a
rb
.u
n
its
)
0 100 200 300
10-3
10-2
10-1
100
l
w
h
Wall
(a)
FIG. 2. (a) The noise spectrum from simulation (solid line)
and from theory (dashed line). For the theory calculation only
the fundamental mode has been considered. (b) A schematic of
the micron-scale cantilever used for validation: length l 
197 m, width w  29 m, and height h  2 m. The applied
step force is F1  26 nN.
P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23is that no expansion in modes of the oscillator is needed.
Although such an expansion is not too hard for a high-Q
oscillator where the dissipation has a negligible influence
on the mode shape, for small cantilevers in a fluid the
coupling to the fluid is large, and the motion of the fluid
complex, so that a mode analysis would be quite diffi-
cult. Finally, the expression Eq. (2) allows us to calcu-
late the correlation function and noise spectrum of
precisely the quantity measured in experiment, first by
tailoring the applied force to couple to one physical
variable measured in the experiment (A), and then by
determining the effect on the second physical variable
(B). This idea has been exploited in the very high-Q
situation of the oscillators used in gravitational wave
detectors [9], although there it was convenient to formu-
late the result in the frequency domain. In our case, for
example, if the displacement of the cantilever is mea-
sured through the strain of a piezoresistive layer near the
pivot point [3,4] of the cantilever, it is possible to tailor
the force F to couple to this distortion, and so determine
the ‘‘strain-strain’’ correlation function of one or more
cantilevers.
Our scheme consists of the following steps in a deter-
ministic simulation: (i) apply the appropriate force f,
constant in time, small enough so that the response
remains linear, and tailored to couple to the variable of
interest A, and allow the system to come to a steady
state; (ii) turn off the force at a time we label t  0;
(iii) measure some dynamical variable Bt (which might
be the same as A to yield an autocorrelation function, or
different) to yield the correlation function of the equilib-
rium fluctuations via Eq. (2). The fluid motion is calcu-
lated using the incompressible Navier-Stokes equations,
and the dynamics of the solid structures are computed
from the standard equations of elasticity. Using the so-
phisticated numerical tools developed for such calcula-
tions (algorithm described elsewhere [10,11]), we can find
accurate results for realistic experimental geometries that
may be quite complex, for example, the triangular canti-
lever design often used in commercial atomic force mi-
croscopy, or the reduced stiffness geometries currently
under investigation for use as detectors of single bio-
molecules as shown in Fig. 1.
For simplicity, we illustrate our approach by finding the
auto- and cross-correlation functions for the displace-
ments xit of the tips of one or two nanoscale cantilevers
with experimentally realistic geometries. To do this, we
calculate the deterministic response of the displacement
of each tip, which we call Xit i  1; 2, after switching
off at t  0 a small force applied to the tip of the first
cantilever, F1, given by Eq. (1). For this case, the equi-
librium auto- and cross-correlation functions for the
fluctuations x1 and x2 are precisely hx1tx10i
kBTX1t=F1 and hx2tx10i  kBTX2t=F1. The cosine
Fourier transform of the auto- and cross-correlation
functions yields the noise spectrum G11 and G12,235501-2respectively, which are the experimentally relevant
quantities.
For long (l * 100 m) and slender (l w) cantilevers,
the cantilever response can be well approximated as an
infinitely long oscillating cylinder [5,6]. We first validate
our numerical approach by investigating a cantilever in
this regime. We emphasize that for the experimentally
motivated nanoscale cantilevers of interest here an ap-
proximate theory is not available, yet our numerical
approach remains valid providing a means to gain valu-
able insight.
The micron-scale cantilever used for validation has the
simple beam geometry, as shown in Fig. 2(b) (see case c2
in [6]). For micron and nanoscale cantilevers immersed in
fluid, dissipation is dominated by the viscous motion of
the fluid driven by the cantilever vibrations. This can be
described by a Reynolds number based on the frequency
of oscillation ! as R  !w2=4, where  is the fluid235501-2
X1(t) X2(t)
F1
x
y (a)
s
(b) (c)
x1(t)
s
s
x2(t)
x1(t) x2(t)
F1F1
FIG. 3. Schematic showing various cantilever configurations.
In all configurations, the step force F1 is released at t  0,
resulting in the cantilever motion referred to by X1t. The
motion of the neighboring cantilever is X2t, and is driven
through the response of the fluid. (a) Two cantilevers with ends
facing. (b) Side-by-side cantilevers. (c) Cantilevers separated
along the direction of the oscillations.
P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23density and  is the viscosity. For the cantilevers of
interest here, the Reynolds numbers are typically 0:01 &
R & 1, indicating that this is in the low Reynolds number
regime. Small R corresponds to strong dissipation.
The noise spectrum, G11, is calculated from the
numerical results by taking the cosine Fourier transform
of the autocorrelations and is shown by the solid line in
Fig. 2(a). The two broad peaks can be identified with the
first two modes of the cantilever. The noise spectrum is
also calculated using the long cylinder analytic approxi-
mation and is shown by the dashed line. The analytical
result for the fundamental mode of the noise spectrum is
found in the following manner [5] (note that higher
harmonics could be included if desired). In Fourier space,
the equation of motion for the cantilever displacement is
m!2 	 kx^  F^f 	 F^B; (3)
where F^f is the force felt by the cantilever due to the fluid,
F^ f  me!2!x^; (4)
F^B is the fluctuating (Brownian) force, me is the effective
mass [12] of a fluid cylinder of radius w=2, m is the
effective mass of the cantilever, and ! is the hydro-
dynamic function (for an infinitely long cylinder of di-
ameter w oscillating in the x direction) which contains
both the fluid damping and fluid loading components
and is given by [13] !  1	 4iK1i

iR
p =
 iRp K0i iRp , where K1, K0 are Bessel functions
and i  1p . From the fluctuation-dissipation theorem,
the spectral density of the fluctuating force, GFB, can
be related to the dissipation due to the fluid and is given by
GFB  4kBTmeT0!i!; (5)
where !  2$. Solving for the spectral density of the
displacement fluctuations,Gx, from Eqs. (3)–(5) yields
Gx 4kBTk
1
!0
 ~!T0iR0 ~!f1 ~!21	T0rR0 ~!g2	 ~!2T0iR0 ~!2
;
(6)
where ~!  !=!o is the frequency relative to the vacuum
resonance frequency !0 

k=m
p
, R0 is the Reynolds
number based on !0, T0 is the ratio of the mass of fluid
contained in a cylindrical volume of radius w=2 to the
mass of the cantilever, and r and i are the real and
imaginary parts of , respectively. Sader’s analysis [5]
does not take into account the frequency dependence of
the noise force and assumes that the numerator is con-
stant. The frequency dependence is not large for R & 1;
however, the correction has been included in our analysis.
Equation (6) yields the analytical curve for G11 in
Fig. 2(a) and the agreement is excellent with our numeri-
cal results.235501-3A recent study of the fluid coupled motion of two
adjacent 1 m beads illustrates the importance of under-
standing the cross correlations in the fluctuations for use
in single-molecule force measurements [14]. The fluid
disturbance caused by an oscillating cantilever is long
range, producing motion of the other cantilevers through
the viscous drag. As a result, the stochastic motion of
multiple cantilevers will be correlated. However, the nu-
merical approach developed here remains valid for mul-
tiple fluid-loaded cantilevers of arbitrary geometry, and
our approach can be used to quantify the response of
multiple cantilevers with the precise complex geometries
used in experiment (as we show below) as well as to help
develop a better analytical understanding of idealized
geometries. Various possible cantilever configurations
are shown in Figs. 3(a)–3(c); here we will present results
for the end to end case and defer results for the other
geometries to a later paper.
We use our approach to calculate the behavior of the
experimentally motivated cantilever shown in Fig. 1. Full
three-dimensional simulations were performed for both
one cantilever and two cantilevers facing end to end in
fluid as shown in Fig. 3(a). Through the fluctuation-
dissipation theorem, the simulations yield results for the
cantilever autocorrelation function and the two cantilever
cross-correlation function shown in Figs. 4(b) and 4(c),
respectively. The value of hx10x10i is 0:471 nm2 indi-
cating that the deflection of the cantilever due to
Brownian motion in an experiment would be 0.686 nm
or about 2:3% of the thickness of the cantilever. The cross
correlation of the Brownian fluctuations of two canti-
levers is small compared to the individual fluctuations.
The largest magnitude of the cross correlation is
0:012 nm2 for s  h and 0:0029 nm2 for s  5h.
The noise spectra for both the one and two cantilever235501-3
F1
t = 0 t
F (a)
t (µs)
〈x 1
(0)
x 2(
t)〉
(nm
2 )
0 0.5 1 1.5 2
-0.01
-0.005
0
s = h
s = 5h (c)
t (µs)
〈x 1
(0)
x 1(
t)〉
(nm
2 )
0 0.5 1 1.5 20
0.2
0.4
(b)
FIG. 4. Absolute predictions of the auto- and cross-
correlation functions of the equilibrium fluctuations of the can-
tilevers shown in Figs. 1 and 3(a). (a) Schematic of the step
force boundary condition that is applied to the tip of the first
cantilever, F1  75 pN. (b) The autocorrelation and (c) the
cross correlation of the fluctuations (shown are five separations,
s  h; 2h; 3h; 4h; 5h, where only s  h; 5h are labeled, and the
remaining curves lie between these values in sequential order).
P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23fluctuations are shown in Figs. 5(a) and 5(b). Notice that
tuning the separation could be used to reduce the corre-
lated noise in some chosen frequency band.
The variation in the cross-correlation behavior with
cantilever separation as shown in Fig. 4(c) can be under-
stood as an inertial effect resulting from the nonzero
Reynolds number of the fluid flow, as follows. The flow
around the cantilever can be separated into a potentialν (MHz)
G
12
(ν)
(10
-
8
n
m
2
H
z)
0 0.5 1 1.5 2
-9
-6
-3
0
3 s = h
s = 5h
(b)
ν (MHz)
G
11
(ν)
(10
-
6
n
m
2
H
z)
0 1 2 3 4
0.4
0.8
1.2
(a)
FIG. 5. (a) The noise spectrum, G11. (b) The noise spec-
trum, G12, as a function of cantilever separation, s, for two
adjacent experimentally realistic cantilevers (see Fig. 1). Shown
are five separations s  h; 2h; 3h; 4h; 5h, where only s  h; 5h
are labeled, and the remaining curves lie between these values
in sequential order.
235501-4component, which is long range and propagates instanta-
neously in the incompressible fluid approximation, and a
vorticity containing component that propagates diffu-
sively with a diffusion constant given by the kinematic
viscosity . For step forcing, it takes a time 'v  s2= for
the vorticity to reach distance s. For small cantilever
separations, the viscous component dominates, for nearly
all times, and results in the anticorrelated response of the
adjacent cantilever in agreement with [14]. However, as s
increases, the amount of time where the adjacent cantile-
ver is subject only to the potential flow field increases,
resulting in the initial correlated behavior.
In closing, we would like to emphasize that the meth-
ods described here are applicable to atomic force micros-
copy, in general, and also to other nanostructure fluid
interaction problems which are of growing importance
as nanoelectromechanical systems technology advances.
This research was carried out within the Caltech
BioNEMS effort (M. L. Roukes, PI), supported by
DARPA/MTO Simbiosys under Grant No. F49620-02-1-
0085. We gratefully acknowledge extensive interactions
with this team. M. P. thanks the Burroughs Wellcome
Foundation ‘‘Interfaces’’ program for additional support.*Electronic address: mpaul@caltech.edu
URL: http://www.cmp.caltech.edu/~stchaos
[1] M. B. Viani, T. E. Scha¨ffer, and A. Chand, J. Appl. Phys.
86, 2258 (1999).
[2] M. L. Roukes, cond-mat/0008187.
[3] J. Arlett et al. (to be published).
[4] M. Roukes, S. Fraser, M. Cross, and J. Solomon,
U.S. Patent App. No. 60/224 109 (2000).
[5] J. E. Sader, J. Appl. Phys. 84, 64 (1998).
[6] J.W. M. Chon, P. Mulvaney, and J. Sader, J. Appl. Phys.
87, 3978 (2000).
[7] H. B. Callen and R. F. Greene, Phys. Rev. 86, 702 (1952).
[8] D. Chandler, Introduction to Modern Statistical
Mechanics (Oxford University Press, New York, 1987).
[9] Y. Levin, Phys. Rev. D 57, 659 (1998).
[10] H. Yang and V. Makhijani, AIAA-94-0179, 1994,
pp. 1–10.
[11] CFD Research Corporation, 215 Wynn Drive,
Huntsville, Alabama 35805, USA.
[12] The effective mass is defined such that the kinetic energy
based upon the displacement of the cantilever tip equals
the kinetic energy of the cantilever. For the fundamental
mode of a simple rectangular cantilever [see Fig. 2(b)]
me  0:243m.
[13] L. Rosenhead, Laminar Boundary Layers (Oxford
University Press, London, 1963).
[14] J.-C. Meiners and S. R. Quake, Phys. Rev. Lett. 82, 2211
(1999).235501-4


Stochastic dynamics of nanoscale mechanical oscillators immersed in a viscous fluid

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Stochastic Dynamics of Nanoscale Mechanical Oscillators Immersed in a Viscous Fluid

The stochastic response of nanoscale oscillators of arbitrary geometry immersed in a viscous fluid is studied. Using the fluctuation-dissipation theorem, it is shown that deterministic calculations of the governing fluid and solid equations can be used in a straightforward manner to directly calculate the stochastic response that would be measured in experiment. We use this approach to investigate the fluid coupled motion of single and multiple cantilevers with experimentally motivated geometries.Published versio

Paul, Mark R.

VTech Works (Virginia Tech)

Physical Review Letters

http://authors.library.caltech.edu/3452/1/PAUprl04a.pdf

The stochastic dynamics of nanoscale mechanical oscillators immersed in a viscous fluid

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