The Stochastic Dynamics of Rectangular and V-shaped Atomic Force Microscope Cantilevers in a Viscous Fluid and Near a Solid Boundary

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by  M. T. Clark

The Stochastic Dynamics of Rectangular and V-shaped Atomic Force Microscope
Cantilevers in a Viscous Fluid and Near a Solid Boundary
∗
M. T. Clark and M. R. Paul
Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
(Dated: February 2, 2008)
Usingathermodynamicapproachbaseduponthefluctuation-dissipationtheoremwequantifythe
stochasticdynamicsofrectangularandV-shapedmicroscalecantileversimmersedinaviscousfluid.
8 Weshowthatthestochastic cantileverdynamicsasmeasuredbythedisplacement ofthecantilever
0 tip or by the angle of the cantilever tip are different. We trace this difference to contributions
0 from the higher modes of the cantilever. We find that contributions from the higher modes are
2 significant in the dynamics of the cantilever tip-angle. For the V-shaped cantilever the resulting
flow field is three-dimensional and complex in contrast to what is found for a long and slender
n
rectangular cantilever. Despite this complexity the stochastic dynamics can be predicted using a
a
two-dimensional model with an appropriately chosen length scale. We also quantify the increased
J
fluid dissipation that results as a V-shaped cantilever is brought near a solid planar boundary.
5
1

I. INTRODUCTION and to suggest simplified analytical approaches to de-
]
l scribe the cantilever dynamics for these complex situa-
l
a tions.
h The stochasticdynamicsofmicronandnanoscalecan-
- tileversimmersedinaviscousfluidareofbroadscientific
s
and technological interest [1, 2]. Of particular impor-
e II. THERMODYNAMIC APPROACH –
m tanceistheoscillatingcantileverthatiscentraltoatomic FLUCTUATIONS FROM DISSIPATION
force microscopy [3, 4]. Significant theoretical progress
.
t hasbeenmadeusingsimplifiedmodelsinthelimitoflong
a Thestochasticdynamicsofmicronandnanoscalecan-
m and thin rectangular cantilevers [5, 6, 7]. In this case, a tilevers driven by thermal or Brownian motion can be
two-dimensionalapproximationis appropriate(therefore
- quantified using strictly deterministic calculations. This
d neglecting effects due to the tip of the cantilever) and
isaccomplishedusingthefluctuation-dissipationtheorem
n has yielded important insights. However, it is not cer-
since the cantilever remains near thermodynamic equi-
o tain how well these approximations work for many sit-
librium [15, 16]. We briefly review this approach for the
c uations of direct experimental interest. For example, a
[ case of determining the stochastic displacement of the
commonly used cantilever in atomic force microscopy is
cantilever tip and then extend it to the experimentally
2 V-shaped and a theoretical description of the dynamics
important case of determining the stochastic dynamics
v of these cantilevers in fluid is not available.
of the angle of the cantilever tip.
6
3 Furthermore, micron and nanoscale cantilevers are of- Theautocorrelationofequilibriumfluctuationsincan-
1 tenusedin closeproximitytoa solidboundaryeither by tilever displacement can be determined from the deter-
2 necessityoroutofexperimentalinterest. Itiswellknown ministic response of the cantilever to the removal of a
1. experimentally and theoretically that the presence of a stepforcefromthe tip ofthe cantilever(i.e. atransverse
0 solid boundary increases the fluid dissipation resulting pointforceremovedfromthedistalendofthecantilever).
8 in reducedquality factorsand reducedresonantfrequen- If this force f(t) is given by
0 cies [7, 8, 9, 10, 11, 12, 13, 14]. Again, theoretical de-
v: scriptionsareavailableinthe limitoflongandthinrect- f(t)= F0 for t<0 (1)
i angularcantileversanditisuncertainiftheseapproaches (cid:26) 0 for t≥0,
X
can be applied to these more complex geometries.
r wheretistimeandF0 isthemagnitudeoftheforce,then
a In this paper we use a powerful thermodynamic ap-
the autocorrelationofthe equilibriumfluctuationsinthe
proachtoquantifythestochasticdynamicsofcantilevers
displacement of the cantilever tip is given directly by
due to Brownian motion for experimentally relevant ge-
ometries for the precise conditions of experiment includ- U1(t)
ing the presence of a planar boundary. Our results are hu1(0)u1(t)i=kBT , (2)
F0
validforthe precisethree-dimensionalgeometryofinter-
est and include a complete description of the fluid-solid where kB is Boltzmann’s constant, T is the tempera-
interactions. Using these results we are able to compare ture, and hi is an equilibrium ensemble average. In our
with available theory to yield further physical insights notation lower case letters represent stochastic variables
(u1(t)isthestochasticdisplacementofthecantilevertip)
and upper case letters represent deterministic variables
(U1(t) representsthe deterministic ringdownofthe can-
∗Electronicaddress: [email protected] tilever tip due to the step force removal). The spectral
2
L(µm)w(µm)h(µm)k (N/m)kt (N-m/rad)f0 (kHz)
y
z (1)197 29 2 1.3 1.6×10−8 71
(2)140 15.6 0.6 0.1 8.9×10−10 38
L x
TABLEI:Summaryofthecantilevergeometriesandmaterial
properties. (1)Therectangularcantilever. (2)TheV-shaped
(a)
h cantilever used is the commercially available Veeco MLCT
w Type E microlever that is used in AFM [18]. The geometry
is given by the cantilever length L, width w, and height h.
FortheV-shapedcantileverthetotallengthbetweenthetwo
L armsatthebaseisb=161.64µm. Thecantileverspringcon-
stant k, torsional spring constant k , and resonant frequency
t
in vacuum f0 are determined using finite element numerical
θ
simulations. Thecantileversareimmersedinwaterwithden-
w
sity ρ = 997 kg/m3 and dynamic viscosity η = 8.59 × 10−4
l
kg/m-s.
b y x
z tip-angle θ(t) is given by
Θ1(t)
hθ1(0)θ1(t)i=kBT . (4)
τ0
(b)
Here Θ1(t) represents the deterministic ring down, as
measured by the tip-angle, resulting from the removal
of a step point-torque. Again, the Fourier transform of
FIG.1: Schematicsofthetwomicronscalecantilevergeome-
the autocorrelationyields the noise spectrum.
tries considered (not drawn to scale). Panel (a), A rectan-
Apowerfulaspectofthisapproachisthatitispossible
gular cantilever with aspect ratios L/h = 98.5, w/h = 14.5,
and L/w = 6.8. The cantilever is composed of silicon with to use deterministic numerical simulations to determine
density ρc = 2329 kg/m3 and Youngs Modulus E = 174 U1(t)andΘ1(t) forthe precisecantilevergeometriesand
GPa. Panel (b), A V-shaped cantilever with aspect ratios conditions of experiment. This includes the full three-
L/h = 233, w/h = 30, and L/w = 7.8. The total width dimensionality of the dynamics which are not accounted
between the two arms normalized by the width of a single for in available theoretical descriptions. The numerical
armisb/w=10.36. Thecantileverplanformisanequilateral results can be used to guide the development of more
triangle with θ = π/3. The cantilever is composed of silicon
accurate theoretical models.
nitride with ρ =3100kg/m3 and E =172GPa. The specific
c
dimensions for the rectangular and V-shaped cantilever are
given in Table I.
III. THE STOCHASTIC DYNAMICS OF
CANTILEVER TIP-DEFLECTION AND
TIP-ANGLE
properties of the stochastic dynamics are given by the
Fourier transform of the autocorrelation.
The stochastic dynamics of the cantilever tip-
The thermodynamic approach is valid for any conju-
gate pair of variables [15]. For example, it is common displacement u1(t) and that of the tip-angle θ1(t) yield
interesting differences. Using the thermodynamic ap-
in experiment to use optical techniques to measure the
proach, insight into these differences can be gained by
angle of the cantilever tip as a function of time [4]. It
performing a mode expansionof the cantilever using the
has also been proposed to use piezoresistive techniques
initial deflection required by the deterministic calcula-
to measure voltage as a function of time [17]. The ther-
tion. The two cases of a tip-force and a tip-torque re-
modynamic approach remains valid for these situations
sult in a significant difference in the mode expansion co-
by choosing the correct conjugate pair of variables.
efficients which can be directly related to the resulting
In this paper we also explore the stochastic dynamics
stochastic dynamics.
of the angle of the cantilever tip. In this case, the angle
For small deflections the dynamics of a cantilever
of the cantilever tip is conjugate to a step point-torque
with a non-varying cross section are given by the Euler-
applied to the cantilever tip. If this torque is given by
Bernoulli beam equation,
τ(t)= τ0 for t<0 (3) ∂2U ∂4U
(cid:26) 0 for t≥0, µ ∂t2 +EI ∂x4 =0, (5)
where τ0 is the magnitude of the step torque, then the where U(x,t) is the transverse beam deflection, µ is the
autocorrelation of equilibrium fluctuations in cantilever massperunitlength,E isYoung’smodulus,andI isthe
3
momentofinertia[19]. Forthecaseofacantileverwhere n bn (tip-force) bn (tip-torque)
a step force has been applied to the tip at some time in
the distantpastthe steady deflectionofthe cantileverat 1 0.97068 0.61308
t=0 is given by 2 0.02472 0.18830
3 0.00315 0.06473
F0 x3 2 4 0.00082 0.03309
U(x)=− −Lx , (6) 5 0.00030 0.02669
2EI (cid:18) 3 (cid:19)
where L is the length of the cantileverand the appropri-
ate boundary conditions are U(0) = U′(0) = U′′(L) = 0 TABLE II: The fraction of the total energy Eb contained in
the first five beam modes given by the coefficients b . The
′′′ n
and U (L) = −F0/EI. The prime denotes differentia- tip-force results are for a rectangular beam that has been
tion with respect to x. deflected by theapplication of a point force tothe cantilever
Similarly, the deflection of the same cantilever beam tip. The tip-torque results are for a rectangular beam that
duetotheapplicationofapoint-torqueatthecantilever- has been deflected by the application of a point torque to
tip is quadratic in axial distance and is given by the cantilever tip. The coefficients clearly show that the tip-
torque case has significantly more energy contained in the
τ0 2 higher modes.
U(x)= x , (7)
2EI
where the appropriate boundary conditions are U(0) =
U′(0) = U′′′(L) = 0 and U′′(L) = τ0/EI. The angle ˙u21¸1/2(nm) ˙θ12¸1/2(rad)
of the cantilever measured relative to the horizontal or
undisplaced cantilever is then given by tanΘ=U′(x). (1) 5.6 5.0×10−7
(2) 20 7.0×10−9
The mode shapes fora cantileveredbeam aregivenby
Φ (x) = −(cosκL+coshκL)(cosκx−coshκx)
n TABLE III: The magnitude of stochastic fluctuations in tip-
− (sinκL−sinhκL)(sinκx−sinhκx), (8) deflection and in tip-angle for the rectangular (1) and V-
shaped (2) cantilevers. These values were obtained from nu-
where n is the mode number, and the characteristic fre- merical simulations simulations of the beams in vacuum.
quencies are given by κ4 =ω2µ/EI. The mode numbers
κaresolutionsto1+cosκLcoshκL=0[19]. Theinitial
cantilever displacement givenby Eqs.(6) and (7) can be
expanded into the beam modes with mode number is more gradual. The fifth mode for
thetip-torquecasecontainsmoreenergythanthesecond
∞
mode for the tip-force case. Although we have only dis-
U(x)= a Φ (x), (9)
n n cussed a mode expansion for the rectangular cantilever,
nX=1 the V-shaped cantilever will exhibit similar trends since
the transverse mode shapes are similar to that of a rect-
with mode coefficients a . The total energy E of the
n b
angular beam [20].
deflected beam is given by
The variation in the energy distribution among the
EI L ′′ 2 modes required to describe the initial deflection of the
E = U (x) dx, (10)
b
2 Z0 cantilevercanbe immediately connectedto the resulting
stochastic dynamics. For the deterministic calculations
which is entirely composed of bending energy. The frac-
the initial displacement can be arbitrarily set to a small
tion of the total bending energy contained in an individ-
value. In this limit the modes of the cantilever beam
ual mode is given by
are not coupled throughthe fluid dynamics. As a result,
the stochastic dynamics of each mode can be treated as
b = EI L(a Φ′′(x))2dx. (11) the ring down of that mode from the initial deflection.
n 2Eb Z0 n n This indicates that the more energy that is distributed
amongst the higher modes initially the more significant
The coefficients b for the rectangular cantilever of Ta-
n the ring downand,using the fluctuation-dissipationthe-
bleIareshowninTableII. Forthecaseofaforceapplied
orem, the more significant the stochastic dynamics.
to the cantilever tip, 97% of the total bending energy
is contained in the fundamental mode and the energy The mode expansion clearly shows that the tip-torque
contained in the higher modes decays rapidly with less casehasmoreenergyinthe highermodes. This suggests
than 1% of the energy contained in mode three. When that stochastic measurements of the cantilever tip-angle
a point-torque is applied to the same beam it is clear willhaveastrongersignaturefromthehighermodesthan
thatasignificantportionofthe bendingenergyisspread measurements of cantilever tip-displacements. Using fi-
over the higher modes. Only 61% of the energy is con- nite element simulations for the precise geometriesof in-
tainedinthe fundamentalmode andthe decayin energy terest we quantitatively explore these predictions.
4
IV. THE STOCHASTIC DYNAMICS OF A noise spectra shown in Fig. 3. In our notation the sub-
RECTANGULAR CANTILEVER script of G indicates the variable over which the noise
spectrum is measured: G is the noise spectrum for tip-
θ
We have performed deterministic numerical simula- angleand Gu is the noise spectrumfor tip-displacement.
tions of the three-dimensional, time dependent, fluid- The equipartition theorem of energy yields,
solid interaction problem to quantify the stochastic dy-
∞
1 k T
namics of a rectangular cantilever immersed in water B
G (ω)dω = (12)
u
using the thermodynamic approach discussed in Sec- 2π Z0 k
∞
tionII. Thedeterministicnumericalsimulationsaredone 1 k T
B
G (ω)dω = (13)
using a finite element approach that is described else- 2π Z0 θ kt
where [21, 22].
The stochastic fluctuations in cantilever tip- where k and k are the transverse and torsional spring
t
displacement for a rectangular cantilever in water constants, respectively. The curves in Fig. 3 are normal-
have been described elsewhere [15, 16, 23, 24]. In the ized using the equipartition result to have a total area
following we compare these results with the stochastic of unity. Using this normalizationthe areaunder a peak
dynamics as determined by the fluctuations of the is an indication of the amount of energy contained in
cantilever tip-angle. The geometry of the the specific a particular mode. Figure 3 shows only the first two
micron scale cantilever we explore is given in Table I. modes, althoughthe numericalsimulations include allof
AsdiscussedinSectionIItheautocorrelationsinequi- the modes (within the numerical resolution of the finite
librium fluctuations follow immediately from the ring element simulation). The energy distribution across the
down of the cantilever due to the removal of a step firsttwomodesshowsthesignificanceofthesecondmode
force(toyieldhu1(0)u1(t)i)orsteppoint-torque(toyield for the tip-angle dynamics.
hθ1(0)θ1(t)i). The autocorrelations of the rectangular
cantilever are shown in Fig. 2. The magnitude of the
noise is quantified by the root mean squared tip-angle
and deflection which is listed in Table III.
10-5
A comparison of the autocorrelations yields some in-
teresting features. At short times hθ1(0)θ1(t)i shows the
presenceofhigherharmoniccontributions. Thisisshown
more clearly in the inset of Fig. 2. This further sug- 10-6
θ
G
gests that the angle autocorrelations are more sensitive
to higher mode dynamics as discussed in Section III. G,u
10-7
1
1 10-8
0 0.5 1 1.5 2 2.5 3 3.5
ω/ω
〉) 0.5 0.5 0
θ(t1
0) 0
θ(1 0 0.005 0.01 FIG. 3: The noise spectra of stochastic fluctuations in can-
〉〈), 0 tilever tip-angle (dashed) and tip-deflection (solid) for the
u(t1 rectangular cantilever. The curves are normalized to have
0) thesame area, however only thefirst two modes are shown.
u(1-0.5
〈
Using a simple harmonicoscillatorapproximationit is
straight forward to compute the peak frequency ω and
f
-1 qualityQ forthe cantileverin fluid. Using a singlemode
00 00..0055 00..11 00..1155 00..22
t(ms) approximation yields the values shown in Table IV. As
expectedthere is asignificantreductioninthe cantilever
frequency when compared with the resonant frequency
FIG. 2: The normalized autocorrelation of the rectangular invacuum ω0 and the quality factor is quite low because
cantilever for tip-deflection (solid) and tip-angle (dashed). of the strong fluid dissipation. The values of ωf and Q
(Inset) A detailed view of the autocorrelation at short time for tip-angle and tip-dispacement are nearly equal. This
differences to illustrate the influence of higher modes in the is expected since the displacements and angles are very
tip-angle measurements. small,resultinginnegligiblecouplingbetweenthemodes.
Any differences in ω and Q can be attributed to using
f
TheFouriertransformoftheautocorrelationsyieldthe a single mode approximation.
5
Itisusefultocomparetheseresultswiththecommonly
usedapproximationofanoscillating,infinitelylongcylin-
der with radius w/2 [5, 6, 16]. The cantilever used here
has an aspect ratio of L/w ≈ 7 and the infinite cylinder
theory is quite good at predicting of ω and Q.
f
ωf/ω0 Q
(1) 0.35 3.34
(2) 0.36 3.26
TABLE IV: The peak frequency and quality factor of the
fundamental mode of the rectangular cantilever determined
by finite element simulations using the thermodynamic ap-
proach. (1)iscomputedusingthecantilevertip-displacement
dueto theremoval of astep force. (2) is computedusing the
cantilevertip-angleduetotheremovalofapoint-torque. The
frequency result is normalized by the resonant frequency in
vacuum ω0. Using the infinite cylinder approximation with
a radius of w/2 the analytical predictions are Q = 3.24 and
ωf/ω0 =0.34.
V. THE STOCHASTIC DYNAMICS OF A
V-SHAPED CANTILEVER
WenowexplorethestochasticdynamicsofaV-shaped
cantilever in fluid. An integral component of any theo-
retical model is an analytical description of the result-
ing fluid flow field caused by the oscillating cantilever.
Thedeterministicfiniteelementsimulationsthatweper-
formed yield a quantitative picture of the resulting fluid FIG. 4: The fluid flow near the tip of the cantilever as il-
dynamics. Exploringthe flowfieldsfurtheryieldsinsight lustrated by the velocity vector field calculated from finite
into the dominant features that contribute to the can- element numerical simulations. A cross section of the x−y
plane at z = 0 is shown (see Fig. 1) that is a close-up view
tilever dynamics.
of the tip-region. The shaded region indicates the cantilever
As discussed earlier, for long and slender rectangular
(becauseofthesmall deflectionsusedinthesimulations that
cantilevers the flow field is often approximated by that
cantilever does not appear to be deflected). (top) The flow
of a cylinder of diameter w undergoing transverse oscil-
fieldnearthetipoftherectangularcantilever. Thisflowfield
lations. This approach assumes that the fluid flow is es- is at t=6µs and the magnitude of the largest velocity vector
sentially two-dimensionalinthe y−z plane andneglects shown is -0.3 nm/s. (bottom) The flow field near the tip of
any flow over the tip of the cantilever. Figure 4 (top) il- theV-shapedcantilever. Thisflowfieldisatt=7.2µsandthe
lustratesthistipflowfortherectangularcantileverusing magnitude of the largest velocity vector shown is -26 nm/s.
vectors of the fluid velocity in the x−y plane at z = 0. Theshadedregionindicatesthetipregionwherethetwosin-
The figure is a close-up view near the tip of the can- gle arms have merged. The open region to the left is where
thetwosingle arms haveseparated revealing theopen region
tilever. It is evident that the flow over the rectangular
in the interior of the V-shapedcantilever.
cantileverisnearlyuniforminthe axialdirectionleading
uptothe tip. However,nearthe tipthereis asignificant
tip flow that decays rapidly in the axial direction away
from the tip. The increasing significance of the tip flow thex−y planeatz =0. Theshadedregionindicatesthe
as the cantilever geometry becomes shorter (for exam- part of the cantilever where the two arms have merged.
ple, by simply decreasing L) is not certain and remains To the right of the shaded region illustrates flow off the
an interesting open question. However,for the geometry tip and to the left indicates flow that circulates back in
used here it is clear that this tip-flow is negligible based between the two individual arms.
upontheaccuracyoftheanalyticalpredictionsusingthe In order to illustrate the three-dimensional nature of
two-dimensional model. thisflow,theflowfieldinthey−z planeisshownattwo
Figure 4 (bottom) illustrates the tip flow for the V- axiallocationsinFig.5. Figure5(top)isataxiallocation
shaped cantilever, again by showing velocity vectors in x=77µm. Thetwoshadedregionsindicatethetwoarms
6
of the cantilever. Each arm is generating a flow with a
viscous boundary layer (Stokes layer) as expected from
previous work on rectangular cantilevers. However, the
Stokes layers interact in a complicated manner near the
center. It is expected that as one goes from the base of
thecantilevertothetipthatthesefluidstructureswould
transition from non-interacting to strongly-interacting.
Figure 5(bottom) illustrates the flow field at axial lo-
cation x=108.8µm, the axial location at which the two
arms of the cantilevermerge to form the tip region. The
length of the shaded region is therefore 36µm or twice
that of a single arm shown in Fig. 5(top). For this tip
regiontheflowfieldissimilartowhatwouldbeexpected
of a single rectangular cantilever of this width.
Overall,itisclearthatthefluidflowfieldismorecom-
plex for the V-shaped cantilever than for the long and
slender rectangular beam. For the V-shaped cantilever
the flow is three-dimensional near the tip region where
the two arms join together.
Central to the flow field dynamics are the interactions
of the two Stokes layers caused by the oscillating can-
tilever arms. The thickness of these Stokes layers are
expected to scale with the frequency of oscillation as
δ /a ∼ R −1/2 where a is the half-width of the can-
s ω
tilever and R = ωa2/ν is a frequency based Reynolds
ω
number (often called the frequency parameter). For the
relevant case of a cylinder of radius a oscillating at fre-
quency ω the solution to the unsteady Stokes equations
yields a distance of approximately5δ to capture 99%of
s
the fluid velocity in the viscous boundary layer [25]. For
a single arm of the V-shaped cantilever this distance is
nearly 10µm. In comparison, the total distance between
the two arms at the base is 125µm. This separation is
large enough such that the two Stokes layers have neg- FIG. 5: The fluid velocity vector field at two axial positions
ligible interactions near the base. However, as the arms along the V-shaped cantilever calculated from deterministic
finite element numerical simulations. Cross sections of the
approachone anotherwithaxialdistance the Stokes lay-
y−zplaneareshown(seeFig.1),theentiresimulationdomain
ers overlapand eventually merge at the tip.
is not shown and the shaded region indicates the cantilever.
Despite the complicated interactions of the three-
Bothimagesaretakenatt=7.2µsandthemaximumvelocity
dimensional flow caused by the cantilever tip and the
vector shown is -26 nm/s. (top) The y − z plane at x =
axialmergingofthetwoStokeslayers,theV-shapedcan- 77µm. The skewed width of a single arm of the cantilever
tilever behaves as a damped simple harmonic oscillator. in this cross-section is 18µm. The distance separating the
The autocorrelations in tip-angle and tip-displacement two cantilever arms is 36µm. (bottom) The y−z plane at
that are found using full finite element numerical sim- x=108.8µm. This is thepoint at which thetwo single arms
ulations are shown in Fig. 6. It is again clear that the join to make a continuouscross-section of width 2w.
tip-angle dynamics have significant contributions from
the higher modes, see the inset of Fig. 6. The area nor-
malized noise spectra are shown in Fig. 7. the V-shaped cantilever from full finite-element numeri-
Using a simple harmonicoscillatoranalogya peak fre- cal simulations. For the rectangular beam the equations
quency and a quality factor can be determined from the are well known (c.f. Ref. [16]) and yield a unique value
′ ′ ′
first mode in the noise spectra of Fig. 7. These values of length L, width w , and height h as shown below,
are given in the first two rows of Table VI. The qual-
ity of the cantilever is Q ≈ 2 and the peak frequency is 3EI Ew′h′3
k = = , (14)
reduced significantly, ωf/ω0 ≈0.2. compared to the res- L′3 4L′3
oflnuaidn.t frequency in the absence of a surrounding viscous Q = mfωf = π4ρρfchw′′ +Γ′(w′,ωf), (15)
It is insightful and of practical use to determine the γf Γ′′(w′,ωf)
geometry of the equivalent rectangular beam that would
yield the precise values of k, ω , and Q calculated for where the peak frequency is determined from the maxi-
f
7
mass, γ is the fluid damping, Γ is the hydrodynamic
f
′
function for an infinite cylinder, Γ is the real part of Γ,
1 ′′
and Γ is the imaginary part of Γ. Equations (14)-(15)
1
can be solved to yield values for the unknown geometry
′ ′ ′
of the equivalent rectangular beam L, w , and h which
θ〉0)(t)1 0.5 0.05 athreingnievre,nanindwTiadbelretVha.nTthheeVeq-suhivaapleedntcabnetailmeveisr.sIhmorptoerr-,
θ(1 0 0.02 0.04 tantly, the width of the equivalent beam is nearly twice
〉〈), 0 that of a single arm of the V-shaped cantilever.
u(t1 ′ ′ ′
0) L/L w/w h/h
u(1-0.5
〈
0.8 1.9 0.8
-1
00 00..11 00..22 00..33 TABLEV:Thegeometryoftheequivalentrectangularbeam
t(ms)
thatyieldstheexactvaluesofk,ω ,andQfortheV-shaped
f
cantileverthat havebeen determined from full finite-element
numerical simulations. The length, width, and height of the
FIG. 6: The normalized autocorrelation of equilibrium fluc- equivalent beam (L′,w′,h′) are calculated using Eqs. (14)-
tuations in the tip-deflection hu1(0)u1(t)i (solid lined) and (15) and are normalized by the values of (L,w,h) for the
in tip-angle hθ1(0)θ1(t)i (dashed-line) for the V-shaped can- V-shaped cantilevergiven in Table I.
tilever. The inset shows a close-up of the dynamics for short
timedifferencestoillustratetheinfluenceofthehighermodes These results suggest that the parallel beam approx-
in thetip-angle measurements. imation (PBA) [26, 27, 28, 29] commonly used to de-
termine the spring constant for a V-shaped cantilever
may also provide a useful geometry for determining the
dynamics of V-shaped cantilevers in fluid. In this ap-
proximation the V-shaped cantilever is replaced by an
10-5
equivalent rectangular beam of length L, width 2w, and
height h to yield a simple analytical expression for the
spring constant. This has been shown to be quite suc-
θ cessful for V-shaped cantilevers that have arms that are
G
,u10-6 not significantly skewed. The results of using the geom-
G etry of this approximation to determine ω and Q from
f
the two-dimensional cylinder approximation are shown
onthe thirdrowofTableVI. Itisclearthatthisis quite
accurate. It is expected that these results will remain
10-7
useful for cantilever geometries that do not deviate sig-
0 0.5 1 1.5 2 2.5 nificantly from that of an equilateral triangle as studied
ω/ω here. An exploration of the breakdown of this approx-
0
imation is possible using the methods described but is
beyond the scope of the current efforts.
FIG. 7: The noise spectra for the V-shaped cantilever as de-
termined from the tip-displacement G (solid line) and from
x
tip-angleGθ (dashedline). Thecurvesarenormalizedtohave VI. QUANTIFYING THE INCREASED
an area of unity,with only thefirst two modes shown. DISSIPATION DUE TO A PLANAR BOUNDARY
In practice, the cantilever is never placed in an un-
mum of the noise spectrum,
bounded fluid and the influence of nearby boundaries
4kBT 1 must be accounted for to provide a complete description
G = (16)
u k ω0 ofthedynamics. Inmanycasesthecantileverispurpose-
T0ω˜Γ′′(R0ω˜) fully brought near a surface out of experimental interest
× [(1−ω˜2(1+T0Γ′(R0ω˜)))2+(ω˜2T0Γ′′(R0ω˜))2]. in order to probe some interactionwith the cantileveror
to probe the surface itself. To specify our discussion we
In the above equations ω˜ = ω/ω0 is the normalized fre- will consider the situation depicted in Fig. 8 showing a
quency, α = 0.234 is a constant factor to determine an cantilever a distance s from a planar boundary. In the
equivalent lumped mass for a rectangular beam, m is followingwestudy the casewherethe cantileverexhibits
f
theequivalentmassofthecantileverplustheaddedfluid flexuraloscillations in the direction perpendicular to the
8
ωf/ω0 Q from what is expected to be a strong influence of the
wall to a negligible influence. Figure 9 clearly shows a
(1) 0.21 1.98 reduction in the peak frequency and a broadening of the
(2) 0.22 2.04 peak as the cantilever is broughtcloser to the boundary.
(L,2w,h) 0.19 1.98
Infact,forthesmallerseparationsthepeakisquitebroad
and the trend suggests that eventually the peak will be-
come annihilated as the cantilever is brought closer to
TABLEVI:Thepeakfrequencyandqualityfactorofthefun-
the boundary.
damental mode of the V-shaped cantilever determined by fi-
niteelementsimulations usingthethermodynamicapproach.
(1) is computed using the cantilever tip-displacement due to
the removal of a step force. (2) is computed using the can-
tilever tip-angle due to the removal of a point-torque. The 1 s/δ =2.42 (a)
third line represents theoretical predictions using the geom- 0.8 s
etry of an equivalent rectangular beam given by (L,2w,h). 0.6 s/δs=9.67
Thefrequencyresultisnormalizedbytheresonantfrequency
in vacuumω0. 0.4
u
G
0.2
boundary. However, we would like to emphasize that
our approach is general and can be used to explore ar-
bitrary cantilever orientations and oscillation directions
if desired. The fluid is assumed to be unbounded in all
otherdirections. Itiswellknownthatthepresenceofthe
boundary will influence the dynamics of the cantilever
[8, 9, 13]. The result is a reduction in the resonant fre- 0 0.1 0.2ω/ω0.3 0.4 0.5
0
quency and quality factor. This has been described the-
oretically for the case of a long and thin cantilever of
simple geometry where the fluid dynamics have been as- 1
(b)
sumed two-dimensional [7, 10, 11, 12]. 0.8 s/δ =3.63
0.6 s
s/δ =14.51
s
0.4
Gθ
0.2
h
s
0 0.1 0.2ω/ω0.3 0.4 0.5
0
FIG. 8: A schematic of a cantilever a distance s away from
a solid planar surface (not drawn to scale). The cantilever
FIG. 9: Panel (a) The noise spectra G of stochastic fluc-
u
undergoes flexural oscillations perpendicular to thesurface.
tuations in cantilever tip-deflection for separations s =
10,12,15,20,40µm. Panel(b)thenoisespectraG ofstochas-
θ
In the following we use the thermodynamic approach tic fluctuations in cantilever tip-angle for separations s =
withfiniteelementnumericalsimulationstoquantifythe 15,25,60µm. Thespectrahavebeennormalized bythemax-
dynamics of the V-shaped cantilever as a function of its imum value of Gu or Gθ. The smallest and largest values of
separationfroma boundary. We haveperformed8simu- separationarelabeledwithallothervaluesappearingsequen-
lationsoverarangeofseparationsfrom10to60µmusing tially.
both the tip-deflection and tip-angle formulations. The
noise spectra for these simulations are shown in Fig. 9. Using the noise spectra we compute a peak frequency
Using the insights from our simulations of the V-shaped andaqualityfactorforthe fundamentalmodeasafunc-
cantilever in an unbounded fluid we expect the relevant tion of separation from the boundary, which are plot-
length scale for the fluid dynamics to be twice the width ted in Fig. 10. The horizontal dashed line represents
of a single arm, 2w. Using the peak frequency of the the value of the peak frequency andquality factor in the
V-shaped cantilever in unbounded fluid yields a Stokes absence of bounding surfaces using the two-dimensional
lengthδ =4.14µm. ScalingtheseparationbytheStokes infinite cylinder approximation [16] where the cylinder
s
length yields. 2.5 . s/δ . 15 which covers the range width has been chosen to be 2w. It is clear from the
s
9
results that for separations greater than s/δ & 7 the
s
V-shaped cantilever is not significantly affected by the
0.22
presence of the boundary. However, as the separation
decreases below this value the peak frequency and qual-
0.2
ity factor decrease rapidly.
The triangles in Fig. 10 represent the theoretical
predictions of Green and Sader [11, 12] using a two- 0.18
dimensional approximation for a beam of uniform cross- ω0
section that accounts for the presence of the boundary. / 0.16
We have used a width of 2w in computing these theo- ωf
reticalpredictions for comparisonwith ournumericalre- 0.14
sults. Despitethe complexandthree-dimensionalnature
of the flow field the theory is able to accurately predict 0.12
(a)
the qualityfactoroverthe rangeofseparationsexplored.
The frequency of the peak for the V-shaped cantilever
0.1
0 5 10 15
shows some deviation from these predictions. s/δ
s
In general, an increase in the period of oscillation for
a submerged object can be attributed to the mass of
fluid entrained by the object [30]. The lower peak fre-
quency calculated for the V-shaped cantilever using a
two-dimensional solution indicates an over-prediction of 2
the mass loading. This can be attributed to the three-
dimensional flow around the tip being neglected for this
approach. It is reasonableto expect the cantilevertip to
1.5
carryasmalleramountoffluidthanasectionofthebeam
Q
bodymovingwiththesamevelocity,seeFig.4. Thequal-
ityfactorrelatesto theratioofthe massloadingandthe
viscous dissipation and is less sensitive to deviations in-
1
curredfromthetwo-dimensionalapproximation. Despite
neglecting three-dimensional flow around the cantilever
(b)
tip, the two-dimensional model for the fluid flow around
the V-shaped cantilever gives an accurate prediction of 0.5
0 5 10 15
the peak frequency and quality factor. s/δ
s
VII. CONCLUSIONS FIG.10: Thevariation ofthepeakfrequency(panel(a))and
quality(panel(b))ofthefundamentalmodeoftheV-shaped
We haveshownthatthe thermodynamicapproachisa cantilever in fluid as a function of separation from a nearby
wall. Resultscalculatedusingtip-deflectionarecircles,results
versatileandpowerfulmethodforpredictingthestochas-
using tip-angle are squares, and theoretical predictions using
tic dynamics of cantilevers in fluid for the precise con-
the results of Ref. [12] are triangles. The peak frequency
ditions of experiment including complex geometries and
andqualityfactorofthefundamentalmodeinanunbounded
the presence of nearby boundaries. Available analyti-
fluid are ω/ω ≈0.19 and Q≈2 and are represented by the
f
cal predictions are for idealized situations including sim- horizontal dashed line. The distance s is normalized by the
plegeometrieswherethethree-dimensionalflownearthe Stokeslength δ where a=w toyield δ =4.14µm.
s s
cantilevertiphasbeenneglected. Althoughthishaspro-
vided significant insight, many situations of experimen-
tal interest are more complicated. It is often required to The thermodynamic approachis generalinthat it can
have a quantitative base-line understanding of the can- beusedtocomputethestochasticdynamicsofanyconju-
tileverdynamics for the precise conditionsof experiment gatepairofvariables. Wehaveshownthatthestochastic
in order to make and interpret measurements in novel dynamics that are measured depend upon the choice of
situations and in the presence of other phenomena of in- measurement. This could be exploited in future exper-
terest. iments, for example, to minimize or maximize the sig-
Weemphasizethatbyusingthefluctuation-dissipation nificance of the higher mode dynamics by choosing to
theorem a single deterministic calculation is sufficient to measure tip-deflection or tip-angle, respectively.
predict the stochastic behavior for all frequencies. Fur- Our results also suggest that despite the complicated
thermore, the deterministic calculation is computation- three-dimensional nature of the flow field around a V-
ally inexpensive and does not require special computing shaped cantilever, analytical predictions based upon a
resources. two-dimensional description are surprisingly accurate if
10
the appropriate length scales are used. We anticipate grant no. FA9550-07-1-0222. Early work on this
thatthesefindingswillbeofimmediateuseastheatomic project was funded by a Virginia Tech ASPIRES grant.
force microscope continues to find further use in liquid We would acknowledge many useful interactions with
environments. Michael Roukes, Mike Cross, and Sergey Sekatski.
Acknowledgments: This researchwasfunded by AFOSR
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