frequency response of cantilever beams immersed in viscous
TRANSCRIPT
Frequency response of cantilever beams immersed
in viscous fluids
Cornelis Anthony van Eysden∗
NORDITARoslagstullsbacken 23, SE-10691 Stockholm, Sweden
John Elie Sader†
Department of Mathematics and StatisticsThe University of Melbourne, Victoria 3010, Australia
Dated: November 5, 2013
1 Introduction
The dynamic response of microcantilever beams is used in a broad range ofapplications, including ultra-sensitive mass measurements and imaging withmolecular and atomic scale resolution (Newell, 1968; Binnig et al., 1986; Bergeret al., 1997; Craighead, 2000; Sader, 2002; Lavrik et al., 2004; Ekinci & Roukes,2005). Importantly, many of these applications are performed in a fluid en-vironment (gas or liquid), which can significantly affect the dynamic responseof a microcantilever. To explain this behavior, theoretical models have beendeveloped that rigorously account for the effect of the surrounding fluid, whichhave been validated by detailed experimental measurements; models exist forflexural, torsional and extensional vibrational modes of cantilever beams (Sader,1998; Green & Sader, 2002; Paul & Cross, 2004; Maali et al., 2005; Clarke et al.,2005; Basak et al., 2006; Paul et al., 2006; van Eysden & Sader, 2007, 2009b;Brumley et al., 2010; Castille et al., 2010; Cox et al., 2012). These studies estab-lish that viscosity plays an essential role in the frequency response of cantileversof microscopic size (∼ 100µm in length), such as those used in the atomic forcemicroscope (AFM) and in micro-electromechanical systems (MEMS). This con-trasts to macro-scale cantilevers (∼ 1m in length) which are insensitive to theeffects of fluid viscosity (Chu, 1963; Lindholm et al., 1965).
∗Electronic address: [email protected]†Electronic address: [email protected]; Corresponding author
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The most common approach used to model the frequency response of a can-tilever beam immersed in a viscous fluid is to approximate the fluid motion by atwo-dimensional flow field due to local displacement of the beam (Sader, 1998).Significantly, for cantilever beams of high aspect ratio (length/width), such asthose used in the AFM, this approach has proven to be highly successful inpredicting the frequency response of the fundamental mode and the next fewhigher order modes. Of recent interest has been the use of higher order can-tilever modes (Maali et al., 2005; Braun et al., 2005; Ghatkesar et al., 2008),which enables operation at high quality factors in highly viscous fluids. How-ever, as mode order increases, the spatial wavelength of the modes decreases,ultimately leading to violation of the two-dimensional flow field approximationand breakdown in the validity of these models. This limitation is also present inthe well-known theory of Chu (1963) for the resonant frequencies of a cantileverbeam immersed in an inviscid fluid, which is valid for cases where fluid viscositycan be ignored. Models that account for three-dimensional effects, and thusare valid for all modes, have been developed for both invscid and viscous fluids(Elmer & Dreier, 1997; van Eysden & Sader, 2006a, 2007).
Operation at higher mode numbers not only increases the importance of(incompressible) three-dimensional effects discussed above, but fluid compress-ibility can also play a major role. The reason for this feature is that the acousticwavelength in the fluid decreases with increasing mode number and ultimatelybecomes comparable to and smaller than the dominant hydrodynamic lengthscale of the beam. Models that rigorously account for the three-dimensionaleffects of fluid compressibility are reported in (van Eysden & Sader, 2009b).
In this chapter, we present an overview of the above-mentioned theoreticalmodels for the fluid-structure interaction of cantilever beams immersed in fluid.The focus is on beams whose lengths greatly exceeds their widths, as is oftenencountered in practice. In §2.1, the two-dimensional theory for flexural modesat low mode number is presented, where fluid viscosity plays an essential role.Models for torsional, in-plane and extensional modes, also at low mode numbers,are summarized in §2.2, §2.3, §2.4, respectively. The general, three-dimensionaltheory for arbitrary mode order is presented in §3, with a focus on the flexuralmodes; models for torsional modes are reported in (van Eysden & Sader, 2007,2009b) and not detailed here. Both incompressible (§3.1) and compressible(§3.2) flows are discussed, accompanied by scaling analyses for determining theregimes where fluid compressibility is expected to be important.
2 Low order modes
2.1 Flexural oscillation
In many applications, such as those found in AFM and MEMS, the fundamentalmode and the next few modes are typically interrogated. In such cases, flowaround the cantilever can be well approximated by one that is two-dimensionaland explicit formulas for the frequency response derived. The theory pertaining
2
L
b
Figure 1: Schematic illustration showing the plan-view dimensions of a rectangular can-tilever. Origin of the coordinate system is at the centroid of the beam cross section at itsclamped end.
to a rectangular cantilever beam immersed in a viscous fluid was first derivedby Sader (1998). This model is summarised in this section, and contains thefollowing principal assumptions:
1. Cross section of the beam is uniform along its entire length and is rectan-gular in geometry;
2. Length of the beam L greatly exceeds its width b, which in turn greatlyexceeds it thickness h, see Figure 1;
3. Amplitude of oscillation is far smaller than any geometric length scale ofthe beam, which allows the Navier-Stokes equations to be linearised;
4. Internal (structural) dissipative effects are negligible in comparison withthose of the fluid; and
5. Fluid is incompressible in nature and is unbounded in space.
The governing equation for the elastic deformation of the beam executingflexural oscillations is (Landau & Lifshitz, 1970)
EI∂4w(x, t)
∂x4+ µ
∂2w(x, t)
dt2= F (x, t) , (1)
where w(x, t) is the deflection function of the beam in the z-direction, E isYoung’s modulus, I is the second moment of area (often referred to as themoment of inertia) of the beam cross section, µ is the mass per unit length of thebeam, F (x, t) is the external applied force per unit length in the z-direction, xis the spatial coordinate along the length of the beam, and t is time; see Figure1. For a cantilever beam of uniform rectangular cross section, I = bh3/12(Landau & Lifshitz, 1970) and µ = ρcbh , where ρc is the cantilever density.The corresponding boundary conditions for the (clamped-free) cantilever beam
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are [w(x, t) =
∂w(x, t)
∂x
]x=0
=
[∂2w(x, t)
∂x2=∂3w(x, t)
∂x3
]x=L
= 0 . (2)
To calculate the frequency response of the beam, we take the Fourier transformof the governing equation (1). Scaling the spatial variable, x, by the length Lgives
EI
L4
∂4w(x|ω)
∂x4− µω2w(x|ω) = F (x|ω) , (3)
where x now denotes the scaled length coordinate, and
X(ω) =
∫ ∞−∞
X(t)eiωtdt , (4)
is the Fourier transform of any function X of time and i is the usual imaginaryunit. For simplicity, we shall henceforth omit the superfluous “∼” notationin (4), while noting that all dependent variables refer to their Fourier spacecounterparts. Solving (3) in the absence of any applied load, leads to the well-known result for the vacuum radial resonant frequencies
ωvac,n =C2n
L2
√EI
µ, (5)
where n = 1, 2, ... is the mode order and Cn is the nth positive root of
1 + cosCn coshCn = 0 , (6)
which is well approximated by Cn ≈ (n− 1/2)π for n ≥ 2.We note in passing that while the present analysis focuses on cantilever
beams, it is equally applicable to clamped-clamped beams provided (i) the eigen-values Cn for clamped-clamped beams are used, and (ii) the deflection functionw(x, t), and corresponding homogeneous solutions φn(x), for a clamped-clampedbeam are implemented in subsequent analyses. These are obtained by applyingthe appropriate boundary conditions for a clamped-clamped beam in (1). Thisyields the eigenvalues Cn satisfying −1 + cosCn coshCn = 0, which are wellapproximately by Cn ≈ (n + 1/2)π for all mode numbers n ≥ 1 (Landau &Lifshitz, 1970). This feature also holds for other theoretical models, includingChon et al. (2000); Green & Sader (2002, 2005); van Eysden & Sader (2006a,2007, 2009a).
Next, we turn our attention to the analysis of the load F (x|ω) applied tothe cantilever. This load is decomposed into two components:
F (x|ω) = Fhydro(x|ω) + Fdrive(x|ω) , (7)
corresponding to the hydrodynamic load Fhydro(x|ω) exerted by the fluid on thecantilever beam due to its motion, and a driving force Fdrive(x|ω) that excitesthe cantilever. The hydrodynamic load is expressed generally as
Fhydro(x|ω) =π
4ρω2b2Γf (ω)w(x|ω) , (8)
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where ρ is the fluid density and Γf (ω) is the normalised hydrodynamic load,termed the hydrodynamic function. The superscript f refers to the flexuralmodes and will be used to distinguish between results for the torsional modes.It is important to emphasise that (8) is derived under the assumption that thelength of the cantilever greatly exceeds its width, and as such, is formally con-sistent with the underlying assumptions of the beam equation, (1). The hydro-dynamic function Γf (ω) is dimensionless and depends on the radial frequencyω through the dimensionless parameter
Re =ρωb2
η, (9)
where η is the fluid shear viscosity. The parameter Re is commonly termed theReynolds number1 and indicates the importance of inertial forces in the fluidrelative to viscous forces.
Cantilevers used in the AFM and MEMS applications are small enough tobe significantly excited by the thermal motion of molecules in the fluid in whichthey are immersed; they undergo Brownian motion, in accordance with thefluctuation-dissipation theorem (Landau & Lifshitz, 1969). Equipartition ofenergy dictates that an energy of kBT/2 is imparted to each mode of oscillation,where kB is Boltzmann’s constant and T is the absolute temperature. For thecase where Brownian motion of the fluid molecules excite the cantilever, thethermal noise spectrum of the squared magnitude of the displacement functionis (van Eysden & Sader, 2008, 2009b)
|w(x|ω)|2s =3πkBT
2k
ρb
ρch
C41
ω2vac,1
∞∑n=1
ωΓfi (ω)
|C4n −B4
n(ω)|2φ2n(x) . (10)
The corresponding result for the slope is∣∣∣∣∂w(x|ω)
∂x
∣∣∣∣2s
=3πkBT
2k
ρb
ρch
C41
ω2vac,1
∞∑n=1
ωΓfi (ω)
|C4n −B4
n(ω)|2
(dφn(x)
dx
)2
, (11)
where the subscript s refers to the spectral density, k is the normal springconstant of the cantilever beam (Roark, 1943), the subscript i refers to theimaginary component, and the function Bn(ω) is defined as
Bn(ω) = C1
(ω
ωvac,1
)1/2(1 +
πρb
4ρchΓf (ω)
)1/4
. (12)
In the limit of small dissipative effects, i.e., when the quality factor greatlyexceeds unity, the modes of the cantilever are approximately uncoupled and thefrequency response of each mode of the beam is well approximated by that of asimple harmonic oscillator (SHO). We obtain the following expressions for the
1The convention adopted for the Reynolds number conforms with Batchelor (1974). TheReynolds number is often associated with the nonlinear convective inertial term in the Navier-Stokes equation. This latter convention has not been adopted here.
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resonant frequency ωfR (defined as the undamped in-fluid natural frequency)and quality factor Qf for the nth mode of vibration (Sader, 1998):
ωR,nωvac,n
=
(1 +
πρb
4ρchΓfr (ωR,n)
)−1/2
, (13)
Qn =
4ρch
πρb+ Γfr (ωR,n)
Γfi (ωR,n), (14)
where the subscript r refers to the real component. Note that for a drivencantilever, the resonant frequency corresponds to the frequency at which thedriving force is 90 out of phase with the displacement of the cantilever; ingeneral this does not coincide with the frequency of maximum amplitude of eachresonance peak. We emphasize that the expressions in (13) and (14), which areformally derived in the limit of large quality factor, are also valid in the Stokeslimit (Re→ 0), where the added apparent mass and damping coefficient of thecantilever are frequency independent.
To use the results (10)–(14), the hydrodynamic function, Γf (ω), must bedetermined. This is obtained from solution of the linearised Navier Stokes equa-tion for a rigid beam of infinite length, with identical cross section to that of thecantilever beam, undergoing transverse oscillatory motion. For a cantilever ofthin rectangular cross section, this is well approximated by that of an infinitelythin blade:
Γfrect (ω) = Ω (ω) Γfcirc (ω) , (15)
where the real and imaginary parts of the correction Ω (ω) are given by
Ωr (ω) =(0.91324− 0.48274τ + 0.46842τ2 − 0.12886τ3
+0.044055τ4 − 0.0035117τ5 + 0.00069085τ6)
×(1− 0.56964τ + 0.48690τ2 − 0.13444τ3 + 0.045155τ4
−0.0035862τ5 + 0.00069085τ6)−1
,
Ωi (ω) =(−0.024134− 0.029256τ + 0.016294τ2 − 0.00010961τ3
+0.000064577τ4 − 0.000044510τ5)× (1− 0.59702τ
+0.55182τ2 − 0.18357τ3 + 0.079156τ4 − 0.014369τ5
+0.0028361τ6)−1
,
τ = log10 (Re/4) , (16)
and Γcirc(ω) is the hydrodynamic function for a circular cylinder:
Γfcirc (ω) = 1 +4iK1
(−i√iRe/4
)√iRe/4K0
(−i√iRe/4
) , (17)
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where Kn(x) are modified Bessel functions of the second kind (Abramowitz& Stegun, 1972), and the Reynolds number is defined in (9)2. Equation (15)is accurate to within 0.1% over the entire range 0.1 < Re < 1000 for bothreal and imaginary parts, and possesses the correct asymptotic forms Re → 0and Re → ∞. In the absence of any simple exact analytical formula for thehydrodynamic function of an infinitely thin rectangular beam, (15) serves as auseful, practical expression.
Using the expression for the hydrodynamic function (15), and the abovetheoretical model, the frequency response of the cantilever can be explored. Thefrequency response is completely characterised by two dimensionless parameters:
Re =ρωvac,1b
2
η, T =
ρb
ρch, (18)
which enable us to study cantilevers of arbitrary dimensions and material prop-erties, immersed in fluids of arbitrary density and viscosity. The first parameterRe is a normalised Reynolds number, which indicates the relative importanceof inertial forces in the fluid to viscous forces. If this parameter greatly exceedsunity, then viscous forces exert a relatively weak effect and the flow will bepredominantly inviscid in nature. The second parameter T is the ratio of theadded-apparent mass due to inertial forces in the fluid (in the absence of viscouseffects), to the total mass of the beam. This parameter indicates the relativestrength of fluid inertia and is used to distinguish between immersion in gas andliquid, as will be discussed below.
Note that values of T for beams immersed in gases and liquids typicallydiffer by three orders of magnitude. This is a direct result of the differencein densities of gases relative to those of liquids. In contrast, values for Re ingases and liquids differ by only one order of magnitude, since the kinematicviscosities of gases are typically one order of magnitude greater than those ofliquids. Consequently, we shall present separate results for gases and liquidsthat account for these differences.
In Figure 2 we present results for the peak frequency ωp (i.e., that causingthe peak response) and quality factor Q of the fundamental mode, as a functionof both Re and T . The peak frequency is numerically calculated from thefrequency response, (13), whereas the quality factor is obtained directly from(14). Consequently, results presented for Q give quantitative information aboutthe resonance peak provided Q 1, since the analogy with the frequencyresponse of a SHO is only valid in those cases. For Q ≤ O(1), however, no suchanalogy exists and Q only presents qualitative information about the resonancepeak. In particular, for Q ≤ O(1) one can conclude that substantial broadeningof the resonance peak is present, and that the modes are significantly coupled inthe frequency domain. A reduction in Q will then result in further broadeningof the peak and an increased coupling of the modes. Finally, it is interesting
2Note that the definition of Re in (9) differs from that used in Sader (1998) by a factorof 4. This is because (9) is the form that appears naturally in the exact solution to Γf (ω)around an infinitely flat blade; see van Eysden & Sader (2006b, 2009a). For consistency, thisdefinition is adopted throughout this chapter.
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0.001 0.01 0.1 1 10 100
0.003
0.01
0.03
0.1
0.3
1
Re__
ω
ωva
c,1
______
p
1−
T = 0.005, 0.015, 0.045__
T increasing__
0.01 0.1 1 10 100 1000
0.0001
0.001
0.01
0.1
Re__
ω
ωvac,1
______p
__
T = 5, 15, 45T increasing__
0.001 0.01 0.1 1
1
10 100
10
100
1000
Re
__
T = 0.005, 0.015, 0.045__
Q
T increasing__
0.01 0.1 1
1
10 100 1000
0.3
3
10
Re__
__
T = 5, 15, 45
Q
T increasing__
Figure 2: Peak frequency ωp of fundamental resonance relative to frequency in vacuumωvac,1, and quality factor Q = Q1, (14), for the fundamental mode. Upper row: T = 0.045(short-dashed line); T = 0.015 (dashed line); T = 0.005 (solid line). Lower row: T = 45(short-dashed line); T = 15 (dashed line); T = 5 (solid line). Results in the limit Re→∞ forT = 0.045, 0.015, 0.005 are (1− ωp/ωvac,1) = 0.0172, 0.00584, 0.00196, respectively. Resultsin the limit Re→∞ for T = 45, 15, 5 are ωp/ωvac,1 = 0.166, 0.280, 0.451, respectively.
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to note that a non-zero peak frequency is observed in all cases presented. Suchbehavior is not observed in a SHO model, where the peak frequency is found tobe identically zero for all quality factors Q ≤ 1/
√2. These results demonstrate
that for Q ≤ O(1) the frequency response of a cantilever beam is not analogousto that of a SHO.
2.2 Torsional oscillation
We now turn our attention to the torsional modes of oscillation, as presented byGreen & Sader (2002). This analysis follows along analogous lines to that pre-sented above for flexural oscillation. Due to this similarity, we only summarisethe key results of this model. To begin, the vacuum frequencies of a cantileverbeam executing torsional oscillations are given by
ωvac,n =Dn
L
√GK
ρcIp, (19)
Dn =π
2(2n− 1) , (20)
where G is the shear modulus of the cantilever and n = 1, 2, 3, ... . For a thinrectangular beam Ip = b3h/12 and K is the torsional constant of the cross-section, which in the case of a thin rectangle reduces to bh3/3 (Roark, 1943).
The total moment per unit lengthM(x|ω) applied to the beam is decomposedinto a term corresponding to the driving moment Mdrive(x|ω) and the momentdue to hydrodynamic loading of the cantilever Mhydro(x|ω), i.e.,
M(x|ω) = Mhydro(x|ω) +Mdrive(x|ω) . (21)
The hydrodynamic load is then expressed generally as
Mhydro(x|ω) = −π8ρω2b4Γt(ω)Φ(x|ω) , (22)
where Φ(x|ω) is the deflection angle about the longitudinal axis of the cantileverand Γt(ω) is the hydrodynamic function. The expression for the thermal noisespectrum of the cantilever is (van Eysden & Sader, 2008, 2009b)
|Φ(x|ω)|2s =6πkBT
kΦ
ρb
ρch
D21
ω2vac,1
∞∑n=1
ωΓti(ω)
|D2n −A2
n(ω)|2γ2n(x) , (23)
where the subscript s again refers to the spectral density, kφ is the torsionalspring constant (Roark, 1943) and
An(ω) =π
2
ω
ωvac,1
(1 +
3πρb
2ρchΓt(ω)
)1/4
. (24)
In the limit of small dissipative effects, the conditions for which are discussedabove, the resonant frequency ωR,n and quality factor Qn for the nth mode of
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vibration are given by
ωR,n
ωvac,n=
(1 +
3πρb
2ρchΓtr(ωR,n)
)−1/2
, (25)
Qn =
2ρch
3πρb+ Γtr(ωR,n)
Γfi (ωR,n). (26)
The hydrodynamic function Γt (ω) is obtained from the solution of the lin-earised Navier Stokes equation for flow around a rigid thin blade of infinitelength that is executing torsional oscillation; this is analogous to the flexuraloscillation case. The required result is given in Green & Sader (2002), and isderived using the boundary integral technique of Tuck (1969). An approximatenumerical formula is provided for this result, which has the correct asymptoticbehaviour as Re → 0 and Re → ∞, and is accurate to better than 0.1% overthe range 10−4 ≤ Re ≤ 104. For details the reader is referred to Green & Sader(2002).
2.3 In-plane flexural oscillation
The model in §2.1 can be directly applied to cantilever beams of arbitrary crosssection. All that is required is specification of the hydrodynamic function forthe geometry under consideration. Brumley et al. (2010) calculated the hy-drodynamic function for a rectangular cylinder of arbitrary width-to-thicknessratio. It was found that the hydrodynamic function is weakly affected by thebeam thickness, for width-to-thickness ratios greater than unity – this showsthat approximation of a beam of small, but finite thickness, by one that is in-finitely thin is a good approximation for out-of-plane flexural oscillation; see§2.1. The situation is very different when the beam width is smaller than itsthickness, corresponding to in-plane flexural oscillation. In this latter case, thewidth-to-thickness ratio can strongly affect the hydrodynamic load – this geo-metric ratio must be specified to yield accurate results, in general. To facilitatepractical application, tabulated data for the hydrodynamic function, spanninga wide range of width-to-thickness ratios, is presented in Brumley et al. (2010).Cox et al. (2012) also examined the frequency response of the in-plane flexuralmodes of a cantilever beam of rectangular cross section.
2.4 Extensional oscillation
Theoretical models for the extensional modes of cantilever beams and free-freebeams have also been derived (Pelton et al., 2009; Castille et al., 2010; Peltonet al., 2011; Chakraborty et al., 2013). These models build on the same princi-ple developed for flexural oscillations, and approximate the hydrodynamic loadat any position along the beam by that of a rigid cylinder. Since these modestypically operate at frequencies much higher than those generated by the fun-damental flexural mode, the viscous boundary layer near the surface is much
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smaller than the cantilever width. This, in turn, enables approximation of thehydrodynamic load by a local solution derived from Stokes’ 2nd problem for theoscillation of a half-space in a viscous fluid (Landau & Lifshitz, 1959). Thesemodels have been successfully compared to experimental measurements.
3 Arbitrary mode order
3.1 Incompressible flows
In the previous section, we examined the frequency response of cantilever beamsof high aspect ratio (length/width) operating at low mode order, where the flowfield is well approximated as being two-dimensional. As mode order increases,however, the spatial wavelength of the modes decreases which ultimately leads toviolation of this two-dimensional approximation and breakdown in these models.Due to the importance of higher order modes in AFM and MEMS applications,a rigorous analysis of the three-dimensional flow field around a cantilever isvital. This is relevant to applications that utilize the higher order modes ofcantilever beams (van Noort et al., 1999; Ulcinas & Snitka, 2001; Stark, 2004;Braun et al., 2005).
Finite-element fluid-structure models for cantilevers immersed in viscous flu-ids have been developed, e.g., see Paul & Cross (2004); Maali et al. (2005); Basaket al. (2006). Some of these models enable cantilever plates of arbitrary aspectratio (length/width) to be investigated; however, all such models rely on so-phisticated and computationally intensive numerical techniques. Since manycantilevers used in practice possess large aspect ratios, it is desirable to havetheoretical models of an analytical nature that can be readily implemented,and this forms one aim of the present chapter. In addition to developing suchmodels, we systematically investigate the effect of increasing mode number onthe general characteristics of the frequency response, which is relevant to thedesign and operation of cantilevers in practice. Analytical solutions for the hy-drodynamic functions and their use in development of analytical models for thefrequency response of cantilevers in fluid, at arbitrary mode order, are reportedin van Eysden & Sader (2006b, 2007); these models are summarized here.
We consider a beam of high aspect ratio (length/width) whose width greatlyexceeds its thickness, as before. The hydrodynamic function is given by thatof an infinitely long oscillating thin blade with deflection function w(x|ω) =Z0exp(iκx); see van Eysden & Sader (2007). This yields the exact analyticalsolution for an incompressible viscous fluid (van Eysden & Sader, 2006b):
Γf (ω, n) = 8a1 , (27)
where the coefficient a1 is obtain by solving the system of linear equations
M∑m=1
(Aκq,m +ARe
q,m
)am =
1 : q = 10 : q > 1 ,
(28)
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for 1 ≤ q ≤M and
Aκq,m = −42q−1
√πG21
13
(κ2
16
∣∣∣∣ 32
0 q +m− 1 q −m+ 1
), (29)
AReq,m = −κ2 24q−5
√πG21
13
(κ2 − iRe
16
∣∣∣∣ 12
0 q +m− 2 q −m
)−24q−1
√πG21
13
(κ2 − iRe
16
∣∣∣∣ 12
0 q +m− 1 q −m+ 1
), (30)
in terms of Meijer G-functions (Luke, 1969). The integer M is to be increaseduntil convergence in the solution is obtained; features of these solutions arediscussed by van Eysden & Sader (2006b). The hydrodynamic function Γf (ω, n)depends on the radial frequency ω through the Reynolds number (9), and onthe mode order n through the dimensionless parameter
κ = Cnb
L, (31)
where Cn is as defined in (6). This parameter gives the relative aspect ratio ofeach spatial mode, and as such, is referred to as the “normalised mode number.”The asymptotic result for Γf (ω, n) in the limit as κ → 0 is identical to (15).For the complementary limit where κ 1, the hydrodynamic function has theasymptotic form,
Γf (ω, n) =8
π|κ|
√κ2 − iRe√
κ2 − iRe− |κ|, κ→∞ . (32)
In the inviscid limit, the matrix elements Aq,m ≡ Aκq,m+AReq,m are given by (van
Eysden & Sader, 2009b)
Aq,m = −42q−1
√πG21
13
(κ2
16
∣∣∣∣ 32
0 q +m− 1 q −m
). (33)
The dependence on ω and mode number, n, in (29) and (30) is contained inthe argument of the Meijer G-function, and subsequently within the coefficientsan. Similar formulas exist for the torsional modes of oscillation (van Eysden &Sader, 2009a).
Importantly, the hydrodynamic function now has an explicit dependence onthe mode number n, and we make the replacements Γf (ω) → Γf (ω, n) andΓt(ω) → Γt(ω, n) in all equations in the previous section. Consequently, (8)now places no restriction on the mode order n and rigorously accounts for thethree-dimensional flow field around the beam (while ignoring end effects).
We first turn our attention to investigate the effect of increasing mode num-ber on the frequency response of cantilever beams immersed in fluid. Impor-tantly, the fundamental torsional resonant frequency greatly exceeds the fun-damental flexural resonant frequency (Green & Sader, 2002), as is commonly
12
observed in measurements (Green et al., 2004). Since the higher order torsionalmodes are rarely probed in practice, we focus our discussion exclusively on theflexural modes. Nonetheless, we note that the influence of higher order modenumber on the torsional frequency response will be very weak in comparisonwith the flexural modes, since the torsional hydrodynamic function Γt(ω, n)is much more weakly dependent on mode number, n, in comparison with thecomplementary dependence of the flexural hydrodynamic function Γt(ω, n) (vanEysden & Sader, 2006b). For further details regarding the torsional hydrody-namic function, the reader is referred to (van Eysden & Sader, 2006b).
To completely characterise the frequency response for higher order modes,the following dimensionless parameter is required in addition to Re and T de-fined in (18),
κ = C1b
L. (34)
This third parameter is a normalised aspect ratio and accounts for the inducedthree-dimensional hydrodynamic flow around the beam. In the singular limit ofκ→ 0, the low mode number theory of §2.1 is recovered.
In Figure 3, we present results for the thermal noise spectrum of the slopeof the cantilever at its end point (x = 1), as this is the quantity of interest inAFM measurements. Results are presented in gas (left-hand side) and liquid(right-hand side) for κ = 0, 0.125, 0.25, 0.5, which corresponds to a geometricaspect ratio of L/b = ∞, 15, 7.5, 3.75, respectively. In gases, we see that thefrequency response in the neighbourhood of the fundamental resonance peakis very weakly dependent on the aspect ratio of the cantilever; see left-handside of Figure 3. This is a result of the weak dependence of the hydrodynamicfunction on κ for the value of normalised Reynolds number considered; see Table1. However, as the mode number increases, the thermal noise spectra begin todeviate significantly from the κ = 0 solution, as seen in the lower left-handpanel of Figure 3. This can be understood in terms of the normalised modenumber κ, which is increasing [see (31)], resulting in a significant change in thehydrodynamic function Γf (ω, n) [see Table 1 and (32)], which is reflected in thefrequency response.
Comparing the left- and right-hand sides of Figure 3, we find that immersionin liquid has a dramatic effect in comparison with that of gas. The resonancepeaks broaden and shift significantly to lower frequencies. Note that the peakfrequency of the fundamental mode is now approximately four times lower thanthe resonant frequency in gas. Nonetheless, inspecting the upper right-handpanel of Figure 3 we find that the fundamental resonance peak is very weaklyaffected by changing the aspect ratio of the cantilever between L/b = 3.75 and∞. The same is not true of the next mode, however, where a significant effecton both the peak frequency and quality factor is observed for the largest valueof κ = 0.5 (corresponding to the smallest value of L/b = 3.75). Interestingly,it is found that as the normalised aspect ratio κ is increased (resulting in adecrease in L/b), the normalised peak frequencies shift to higher frequenciesand approach the vacuum frequencies. This effect increases with increasingmode number n. Importantly, even the smallest nonzero value of κ considered
13
1
10
100
0.7 0.8 0.9 1 1.1 1.2 1.3
ω
ωvac ,1
H
Increasing κ
Gas
T = 0.01
Re = 10
Gas
T = 0.01
Re = 10
10- 5
10- 3
10- 1
H
101
0 20 40 60 80 100
ω
ωvac ,1
Increasing κ
0 0.5 1 1.5 2 2.5 3 3.5
0.01
0.1
1Liquid
T = 10
Re = 100
ω
ωvac ,1
H
Increasing κ
0 5 10 15 20 25 30
1
10- 2
10- 4
10- 1
10- 3
10- 5
Liquid
T = 10
Re = 100
ω
ωvac ,1
H
Increasing κ
Figure 3: Normalised thermal noise spectrum (slope) H = |w′(x, ω)|2s kωvac,1/(kBT ), (11),of the flexural modes in gas. The ′ refers to the derivative with respect to x. Normalised modenumbers κ = 0, 0.125, 0.25, and 0.5 corresponding to L/b =∞, 15, 7.5, and 3.75, respectively.Solid line corresponds to κ = 0 result, and is identical to model in §2.1. Upper row: Firstmode (left), first and second modes (right); Lower row: First 6 modes. Results given for a gaswith Re = 10, T = 0.01 (left-hand side) and liquid with Re = 100, T = 10 (right-hand side).
14
Tab
le1:
Inco
mp
ress
ible
hyd
rod
yn
am
icfu
nct
ion
Γf
(ω,n
)fo
rfl
exu
ral
mod
esas
afu
nct
ion
of
Rey
nold
snu
mb
erR
ean
dn
orm
alise
dm
od
enu
mb
erκ
,acc
ura
teto
six
sign
ifica
nt
figu
res.
(a)
Rea
lco
mp
on
ent.
Re→∞
resu
lts
ob
tain
edu
sin
gin
vis
cid
theo
ry(3
3).
(b)
imagin
ary
com
pon
ent.
κ0
0.1
25
0.2
50.5
0.7
51
23
57
10
20
(a)lo
g10Re
−4
3919.4
159.3
906
22.4
062
9.1
3525
5.6
2175
4.0
5204
1.9
3036
1.2
764
0.7
64081
0.5
45683
0.3
81972
0.1
90986
−3.5
1531.9
59.3
861
22.4
061
9.1
3525
5.6
2175
4.0
5204
1.9
3036
1.2
764
0.7
64081
0.5
45683
0.3
81972
0.1
90986
−3
613.4
26
59.3
42
22.4
052
9.1
3523
5.6
2174
4.0
5204
1.9
3036
1.2
764
0.7
64081
0.5
45683
0.3
81972
0.1
90986
−2.5
253.1
09
58.9
094
22.3
962
9.1
3504
5.6
2172
4.0
5203
1.9
3036
1.2
764
0.7
64081
0.5
45683
0.3
81972
0.1
90986
−2
108.4
29
55.2
882
22.3
078
9.1
3319
5.6
2153
4.0
5199
1.9
3036
1.2
764
0.7
64081
0.5
45683
0.3
81972
0.1
90986
−1.5
48.6
978
40.7
883
21.5
187
9.1
1481
5.6
196
4.0
516
1.9
3035
1.2
764
0.7
64081
0.5
45683
0.3
81972
0.1
90986
−1
23.2
075
22.7
968
17.5
378
8.9
437
5.6
0057
4.0
4771
1.9
3027
1.2
7639
0.7
6408
0.5
45683
0.3
81972
0.1
90986
−0.5
11.8
958
11.9
511
11.0
719
7.8
9716
5.4
3378
4.0
1051
1.9
2942
1.2
7629
0.7
64074
0.5
45682
0.3
81972
0.1
90986
06.6
4352
6.6
4381
6.4
7227
5.6
5652
4.6
4017
3.7
46
1.9
2114
1.2
7536
0.7
64012
0.5
45671
0.3
8197
0.1
90986
0.5
4.0
7692
4.0
594
3.9
9256
3.7
2963
3.3
7543
3.0
0498
1.8
5532
1.2
6646
0.7
63397
0.5
45564
0.3
81953
0.1
90985
12.7
4983
2.7
3389
2.6
9368
2.5
639
2.3
9884
2.2
2434
1.6
1821
1.2
0592
0.7
57637
0.5
4452
0.3
81786
0.1
90981
1.5
2.0
2267
2.0
108
1.9
8331
1.9
004
1.7
9834
1.6
9086
1.3
1175
1.0
4626
0.7
21165
0.5
35593
0.3
8018
0.1
90932
21.6
063
1.5
9745
1.5
7723
1.5
169
1.4
4276
1.3
6416
1.0
8036
0.8
78177
0.6
35443
0.4
96169
0.3
68548
0.1
90459
2.5
1.3
623
1.3
5532
1.3
3934
1.2
9142
1.2
3203
1.1
6842
0.9
32812
0.7
59965
0.5
51349
0.4
35586
0.3
34279
0.1
86672
31.2
1727
1.2
1141
1.1
9792
1.1
5718
1.1
0624
1.0
5117
0.8
4292
0.6
86229
0.4
93924
0.3
87183
0.2
95972
0.1
72722
3.5
1.1
3038
1.1
2518
1.1
1316
1.0
7668
1.0
3073
0.9
80721
0.7
8879
0.6
41744
0.4
58699
0.3
56289
0.2
68907
0.1
5445
41.0
7814
1.0
7334
1.0
6221
1.0
2827
0.9
85314
0.9
38346
0.7
56309
0.6
15164
0.4
37743
0.3
37813
0.2
52327
0.1
40852
∞1
0.9
95787
0.9
85974
0.9
55852
0.9
17424
0.8
75076
0.7
08262
0.5
76362
0.4
07863
0.3
11839
0.2
29184
0.1
20958
(b)lo
g10Re
−4
27984.8
44628.5
55176.1
71754
86311.5
100062
152411
203623
305570
407436
560225
1069521
−3.5
9816.3
314113.1
17448.2
22690.6
27294.1
31642.3
48196.5
64391.4
96629.7
128843
177159
338212
−3
3482.4
74464.1
65517.7
27175.4
18631.1
510006.2
15241.1
20362.3
30557
40743.6
56022.5
106952
−2.5
1252.6
61415.4
21745.1
72269.0
92729.4
23164.2
34819.6
56439.1
49662.9
712884.3
17715.9
33821.2
−2
458.3
86
457.8
63
552.8
62
717.6
35
863.1
38
1000.6
31524.1
12036.2
33055.7
4074.3
65602.2
510695.2
−1.5
171.3
97
160.9
51
177.7
02
227.2
05
273.0
13
316.4
49
481.9
67
643.9
15
966.2
97
1288.4
31771.5
93382.1
2−
165.8
679
62.2
225
61.6
26
72.6
542
86.5
364
100.1
44
152.4
18
203.6
25
305.5
7407.4
36
560.2
25
1069.5
2−
0.5
26.2
106
25.2
124.1
432
24.7
484
27.9
459
31.8
957
48.2
199
64.3
973
96.6
308
128.8
43
177.1
59
338.2
12
010.8
983
10.6
158
10.1
909
9.7
009
9.9
1067
10.6
48
15.3
139
20.3
81
30.5
604
40.7
448
56.0
229
106.9
52
0.5
4.7
8389
4.6
9492
4.5
3952
4.2
4925
4.0
9701
4.0
9433
5.0
1844
6.4
9605
9.6
7379
12.8
879
17.7
171
33.8
214
12.2
3883
2.2
0681
2.1
4583
2.0
088
1.8
9659
1.8
2463
1.8
5993
2.1
7718
3.0
8849
4.0
8581
5.6
0598
10.6
956
1.5
1.1
2164
1.1
0851
1.0
8208
1.0
1654
0.9
53355
0.9
01676
0.8
1464
0.8
44519
1.0
4394
1.3
2116
1.7
8306
3.3
8349
20.5
96697
0.5
90686
0.5
78118
0.5
45082
0.5
10467
0.4
79247
0.4
03803
0.3
83595
0.4
09256
0.4
69688
0.5
89749
1.0
7377
2.5
0.3
32285
0.3
29276
0.3
2283
0.3
05262
0.2
85953
0.2
6763
0.2
16732
0.1
94409
0.1
86218
0.1
95634
0.2
21631
0.3
49855
30.1
91043
0.1
89434
0.1
85931
0.1
76166
0.1
65118
0.1
54323
0.1
22124
0.1
05573
0.0
938839
0.0
925686
0.0
9682
0.1
26835
3.5
0.1
12082
0.1
11181
0.1
09199
0.1
03595
0.0
971392
0.0
907188
0.0
707736
0.0
59728
0.0
505049
0.0
476557
0.0
471326
0.0
534759
40.0
665172
0.0
659974
0.0
648471
0.0
615627
0.0
577366
0.0
538889
0.0
416384
0.0
345727
0.0
282418
0.0
25856
0.0
24611
0.0
252877
15
2 4 6 8 10 12 14 16 18 200.003
0.01
0.03
0.1
n
Gas
κ = 0.0625
ωvac,n
ωR,n
1 −
2 4 6 8 10 12 14 16 18 20
0.003
0.01
0.03
n
Gas
κ = 0.25
ωvac,n
ωR,n
1 −
2 4 6 8 10 12 14 16 18 20
0.1
0.2
0.3
n
Liquid
κ = 0.0625
ωvac, n
ωR, n
2 4 6 8 10 12 14 16 18 20
0.1
0.2
0.3
0.4
0.5
n
Liquid
κ = 0.25
ωvac, n
ωR, n
Figure 4: Normalised resonant frequency, (13), as a function of mode number n. Linesgiven as a guide. Low mode number model (solid circle), arbitrary mode number model(open circles), and large κ asymptotic model (open rectangles). Upper row: κ = 0.0625corresponding to L/b = 30. Lower row: κ = 0.25 corresponding to L/b = 7.5. Results givenfor gas Re = 1, T = 0.01 (left-hand side) and liquid Re = 10, T = 10 (right-hand side).
(corresponding to the largest finite value of L/b = 15) has a significant effect, asis strikingly evident in Figure 3 (lower right-hand panel). From these results itis clear that the frequency response of the higher modes of practical cantileverscan strongly depend on the normalised aspect ratio κ. This finding for thehigher modes contrasts directly to the fundamental mode that is found to bevirtually independent of κ.
The effect of varying aspect ratio on the resonant frequencies and qualityfactors is presented in Figures 4 and 5, respectively, where results for the first20 modes are calculated using (i) the arbitrary mode number model, (ii) the lowmode number model presented in §2.1, and (iii) the large κ asymptotic model thatis obtained using the hydrodynamic function in (32). For immersion in gas, wefind that the resonant frequencies are very weakly affected by the surroundingfluid, in agreement with previous studies (Sader, 1998; Green & Sader, 2002;Paul & Cross, 2004; Maali et al., 2005; Basak et al., 2006; Paul et al., 2006; Chonet al., 2000); see Figure 4. We also observe that the arbitrary mode numbermodel matches the low mode number model in the limit of κ = 0 (correspondingto L/b → ∞), but can diverge significantly as the aspect ratio, L/b, of thecantilever is reduced. Nonetheless, we find that the low mode number model isalways valid for the fundamental mode for the range of normalised aspect ratiosκ considered; this was also found to be the case up to κ = 0.5, beyond whichthe underlying assumptions in all models must be drawn into question.
16
2 4 6 8 10 12 14 16 18 20
30
100
300
1000
n
Qn
Gas
κ = 0.0625
2 4 6 8 10 12 14 16 18 20
30
100
300
1000
n
Qn
Gas
κ = 0.25
2 4 6 8 10 12 14 16 18 20
1
3
10
30
n
Qn
Liquid
κ = 0.0625
2 4 6 8 10 12 14 16 18 20
1
2
3
5
10
Liquid
κ = 0.25
n
Qn
Figure 5: As for Figure 4, except results given for quality factor Qn, (14), as a function ofmode number n.
In Figure 4, the arbitrary mode number model is found to give identicalresults to the low mode number model for the fundamental mode then departsfrom this solution as the mode number n is increased, ultimately approachingthe large κ asymptotic model. Similar results are also obtained for the resonantfrequencies in liquid, although the deviation is much more pronounced due tothe dramatic effect of liquid on the frequency response in comparison with gas,cf. the left- and right-hand sides of Figure 4.
Complementary results to Figure 4 are presented for the quality factor Qnin Figure 5. For all panels it is evident that as the mode number n increases,the quality factor approaches infinity. This is a consequence of the diminishinghydrodynamic load at high mode number [see Table 1 and (32)], an effect whichis also present in Figure 4, where the resonant frequencies in fluid approachthe vacuum results. Again, we find that the low mode number model agreeswell with the arbitrary mode number model for the fundamental mode always,but as the mode number n is increased these results begin to depart from thelow mode number model and approach the large κ asymptotic model. It isinteresting to note, however, that the deviation from the low mode numbermodel for immersion in gas is much more pronounced than the deviation inliquid, cf. left- and right-hand panels of Figures 5. For all cases presented, itis found that the quality factor in liquid is well approximated by the low modenumber model, regardless of the mode number; see (van Eysden & Sader, 2007).
Finally, we comment on the validity of the above model for practical can-
17
tilevers. In the κ = 0 limit, the low mode number model of §2.1 is valid for largeL/b provided the viscous boundary layer, defined as b/
√Re, does not become
comparable to the length of the beam, at which point the flow field is directlyaffected by the three-dimensional nature of the cantilever geometry. For thehigher order modes, the relevant length scale is the spatial length scale of os-cillations L/Cn, and the requirement that this is much less than the viscousboundary layer thickness can be expressed as
Re > κ2 . (35)
Approximating the resonant frequency of the cantilever by that in vacuum, wefind that the condition (35) is independent of mode number and is equivalent toRe > κ2. This condition is typically satisfied by practical microcantilevers, sup-porting the validity of the low mode number model of §2.1, for the fundamentalmode. To examine the condition (35) carefully, we can evaluate the ratio
Re
κ2= h
ωR,n
ωvac,n
ρ
η
√E
12ρc, (36)
which must be greater than unity. It is interesting that this ratio is indepen-dent of the cantilever length and width and only depends on its thickness. Forcantilever materials and fluids typically used in AFM and MEMS applications(Binnig et al., 1986; Berger et al., 1997; Ho & Tai, 1998; Sader, 1998; Saderet al., 1999; Chon et al., 2000; Green & Sader, 2002; Paul & Cross, 2004; Lavriket al., 2004; Maali et al., 2005; Basak et al., 2006; Paul et al., 2006), (36) canbe estimated to give Re/κ2 ∼ 108hωR,n/ωvac,n, where h is the thickness inmetres. This indicates that the above ratio greatly exceeds unity for microcan-tilevers of practical relevance. The ratio ωR,n/ωvac,n can be calculated using(13). Only in cases of very heavy fluid loading, or very thin cantilevers, whereωR,n/ωvac,n 1 will the ratio in (36) equal unity. Thus, for most cantileversof practical relevance, the condition Re κ2 will be satisfied, indicating thearbitrary mode number model captures the true frequency response at reso-nance of these cantilever beams. Complete three-dimensional computationalfluid dynamics simulations of the cantilever dynamics are not essential in suchsituations, even for very small cantilevers. Nonetheless, by tuning the mate-rial properties of the cantilever and fluid, it may be possible to reach this limitpractically; see (36).
3.2 Compressible flows
In general, the flow around a cantilever can be considered incompressible pro-vided the acoustic wavelength greatly exceeds the dominant length scale of theflow (Morse & Ingard, 1987; van Eysden & Sader, 2009a). For practical micro-cantilevers with high aspect ratios (length/width) and width-to-thickness ratios,operating at low mode numbers (i.e., the fundamental flexural mode and thenext few modes), the flow is approximately two-dimensional and its dominant
18
length scale is the beam width, b. In such cases, the acoustic wavelength greatlyexceeds the dominant length scale, and hence, the incompressibility assumptionis easily satisfied (Sader, 1998). This has been well established by the experi-mental validation of numerous theoretical models based on incompressible flow,e.g., see Chon et al. (2000); Basak et al. (2006); Ghatkesar et al. (2008). Foroperation at higher order modes, however, it is important to reassess how thisincompressibility condition is affected.
To investigate the effects of fluid compressibility, the hydrodynamic functionΓf (ω, n) must be generalised to include compressibility effects. A solution canbe obtained using the model presented in the previous sections, where the in-compressibility condition (assumption 5) above is relaxed. The exact analyticalsolution for the flow around an infinitely long oscillating flat blade in a com-pressible medium is presented in van Eysden & Sader (2009a). The result isobtained from (27) under the following replacement for (29):
Aκq,m = −42q−1
√πG21
13
[1
16
(κ2 − ς2 Re
Re− 4/3iς2
)∣∣∣∣ 32
0 q +m− 1 q −m+ 1
],(37)
while (30) is unchanged. In the inviscid limit, (33) generalises to
Aq,m = −42q−1
√πG21
13
(κ2 − ς2
16
∣∣∣∣ 32
0 q +m− 1 q −m
). (38)
The hydrodynamic function Γf (ω, n) is dimensionless and depends on fluid com-pressibility through the dimensionless parameter
ς =ωb
c, (39)
where c is the sound speed of the fluid. Therefore, ς is a normalised wave numberfor the propagation of sound in the medium and is a measure of the fluid com-pressibility. Note that for the continuum approximation to be valid, we requireς Re (Landau & Lifshitz, 1959) and the argument of the Meijer G-function inthe first term of (37) reduces to (κ2 − ς2)/16 for cases of practical interest; seevan Eysden & Sader (2009a). In the limit as ς → 0, the incompressible resultfor Γf (ω, n) presented above is obtained.
3.2.1 Scaling analysis
The conditions for which fluid compressibility becomes important can be deter-mined using a scaling analysis. For an elastic beam, the dominant length scaleof the flow is the minimum of (i) the beam width b and (ii) the length scale ofspatial oscillations of the beam, given by
λbeam =2πL
Cn≈ 4L
2n− 1. (40)
Since Cn increases with increasing mode number n, the length scale of spatialoscillations reduces until it becomes smaller than the beam width. Thus, spatial
19
oscillations of the beam will eventually set the dominant hydrodynamic lengthscale, as the mode number is increased.
The wavelength of acoustic oscillations generated by the beam in a fluid isc/fn, where fn is the resonant frequency. For the purpose of a scaling analysis,the resonant frequency can be approximated by the corresponding result invacuum, i.e., (5). The wavelength of sound is then
λsound =
(C1
Cn
)2c
fvac,1≈ 1.425
(2n− 1)2
c
fvac,1, (41)
where ωvac,1 = 2πfvac,1. Therefore, the acoustic wavelength decreases in inverseproportion to the square of the mode number n, for high mode numbers. Crit-ically, this has a stronger dependence on mode number than the length scale ofspatial oscillations. Therefore, as the mode number increases, the acoustic wave-length eventually becomes comparable to the spatial wavelength of the beam,and compressibility can no longer be ignored. This so-called coincidence pointcorresponds to the onset of significant radiation damping of the beam, whereenergy is carried away in the form of sound waves (Morse & Ingard, 1987; vanEysden & Sader, 2009a).
Equating the expressions in (40) and (41), determines the critical modenumber nc at which coincidence is reached,
nc =0.178c
fvac,1L. (42)
For a cantilever 250µm in length with a vacuum frequency of 20 kHz, immersedin air, coincidence is reached at nc ≈ 12. In liquids, however, the speed ofsound is a factor of ∼ 5 larger than in air and the critical mode number isapproximately nc ≈ 60, which is unlikely to be probed in practice. The preciseregime for which the effects of fluid compressibility become manifest is clarifiedbelow using the numerical results of the above model.
3.2.2 Numerical results
To completely characterise the frequency response for higher order modes, thefollowing dimensionless parameter is required in addition to Re, T and κ definedin (18) and (34),
ς =bωvac,1
c. (43)
The parameter ς is the normalised wave number for the propagation of soundthrough the medium, and is a measure of the fluid compressibility. When theacoustic wavelength is large, this parameter approaches zero and the incom-pressible limit is attained.
Another dimensionless parameter of fundamental importance in compress-ible flow is ς/κ, which is the ratio of the beam length to the acoustic wavelength,for the fundamental mode. Since (42) can be written as
nc =0.596κ
ς, (44)
20
0 20 40 60 80 100
10- 2
H 10- 1
1
10
10- 3
ωω vac ,1
Liquid
Re = 10
T = 10
0 20 40 60 80 100 120 140
10- 5
H
10- 3
10- 1
10
ωω vac ,1
Gas
Re = 5
T = 0.01
Figure 6: Normalised thermal noise spectrum (slope) H = |w′(x, ω)|2s kωvac,1/(kBT ) forthe flexural modes, showing the effects of fluid compressibility. The ′ refers to the derivativewith respect to x. Results given for (left-hand panel) gas: Re = 5, T = 0.01, κ = 0.125 andς/κ = 0, 0.025, 0.05, 0.075, 0.1; (right-hand panel) liquid: Re = 10, T = 10, κ = 0.125 andς/κ = 0, 0.005, 0.01, 0.015, 0.02.
it then follows that ς/κ determines the mode number at which coincidence isreached. The dimensionless parameter ς/κ will be used in the following resultsto vary the effects of compressibility.
In Figure 6, we present the thermal noise spectrum for the slope (angle) atthe end of the cantilever for compressible flow. Results are given for practicalvalues of κ, ς, and Re in gases [Figure 6, left-hand panel] and liquids [Figure 6,right-hand panel], which clearly demonstrate the effect of increasing compress-ibility. Figure 6 (left-hand panel) shows that increasing the gas compressibilitybroadens the resonance peaks, which corresponds to enhanced dissipation. Thiseffect is due to the generation of sound waves that radiate from the cantilever,which manifests itself more strongly with increasing mode number. This en-hancement in dissipation is in direct contrast to the incompressible limit, wherethe effects of increasing mode number reduce dissipation (see Figure 4); this lat-ter effect is due purely to the increasing importance of three-dimensional flow.Compressibility therefore counteracts the effects of three-dimensional flow withincreasing mode number. This feature is evident in (37), where the variables κand ς appear in combination.
From the right-hand panel of Figure 6, we observe that compressibility hasvirtually no effect on the thermal noise spectrum for cantilevers operating in aliquid, as the spectra are indistinguishable from the incompressible result (all fivecurves coincide). This supports the above scaling argument that compressibilitydoes not become significant in liquids until very high mode numbers are attained;for typical microcantilevers, this corresponds to mode numbers in excess of 20.We therefore focus the following discussion on the behaviour of cantilevers ingas.
In Figure 7, the resonant frequencies and quality factors as a function ofmode number are presented. On the left-hand side, values of κ and ς correspondapproximately to those used in the above scaling analysis. On the right-handside a larger aspect ratio of L/b = 30, corresponding to κ = 0.0625, is presented.
21
2 4 6 8 10 12 14 16 18 20 22
0.0003
0.001
0.003
0.01
0.0001
ωvac,n
ωR,n
1 −
n
2 4 6 8 10 12 14 16 18 20 22
100100
300
10001000
3000
n
Q
κ = 0.125
ζ= 0.05κ
κ = 0.125
ζ= 0.05κ
ωvac,n
ωR,n
1 −
n
Qκ = 0.0625
ζ= 0.05κ
2 4 6 8 10 12 14 16 18 20 22 24
0.0003
0.001
0.003
0.01
n
2 4 6 8 10 12 14 16 18 20 22 24
100
300
1000
3000
10000
κ = 0.0625
ζ= 0.05κ
Figure 7: Upper row: Normalised resonant frequency 1− ωR,n/ωvac,n. Lower row: Qualityfactor Qn as a function of mode number n. Results given for a gas with Re = 5, T = 0.01,ς/κ = 0.05 and κ = 0.125 (left-hand side) and κ = 0.0625 (right-hand side). Lines are given asa guide only. Complete solution (solid circles), viscous incompressible solution (open circles),and inviscid compressible solution (open rectangles).
22
Three solutions are given: (i) the viscous incompressible solution (open circles)of §3.1, (ii) the inviscid compressible solution (open rectangles), (38), and (iii)the complete (viscous compressible) solution (closed circles), (37). For low modenumbers, compressibility effects are weak and the viscous incompressible andcomplete solutions coincide. However, as the mode number increases, the viscousincompressible solution begins to deviate from the complete solution. On theleft-hand side, coincidence is reached at mode number 14, where the inertialcomponent of the hydrodynamic load reaches a maximum, in agreement withpredictions from the scaling analysis. However, on the right-hand side, thecoincidence point now occurs at n ∼ 17, despite (44) predicting the same result;the discrepancy arising from the approximate nature of the scaling analysis,which neglects the fluid loading on the beam. Above coincidence, sound wavesare strongly generated in the medium, and a further increase in mode numberresults in rapid diminishing of this (inertial) hydrodynamic load. For largemode numbers, the complete solution coincides with inviscid compressible result,as the viscous penetration depth over which vorticity diffuses decreases withincreasing mode number.
Corresponding results for the quality factor in gas are given in the lowerrow of Figure 7. As for the frequency response, the quality factor is domi-nated by incompressible viscous effects at low mode numbers, and approachesthe inviscid compressible solution for large mode numbers, with the transitionoccuring approximately at coincidence. Importantly, above coincidence we findthat the quality factor reduces significantly with increasing mode number. Thisfinding differs greatly from the incompressible result and is due to the stronggeneration of acoustic waves that greatly enhance dissipation, i.e., radiationdamping. This non-monotonic behaviour of the quality factor with increasingmode number should be observable experimentally.
The above theoretical models for compressible flow have been derived usingthe hydrodynamic function for an infinitely long beam executing uniform oscil-lations with discrete spatial wave numbers. We conclude by commenting on thepractical implications of these models to cantilever beams of finite length. Inthe limit where the acoustic wavelength is much greater than the length of thecantilever beam, the flow is incompressible and the validity of these models iswell established (Elmer & Dreier, 1997; Chon et al., 2000; Green & Sader, 2002;Paul & Cross, 2004; Maali et al., 2005; Braun et al., 2005; Basak et al., 2006;Paul et al., 2006; van Eysden & Sader, 2006a, 2007; Ghatkesar et al., 2008);this is the practical case for operation in the low order modes. In the oppositesituation where the acoustic wavelength is smaller than the spatial wavelengthof beam oscillations, i.e., above coincidence, the effects of compressibility are lo-calised and hence unaffected by the finite length of the beam; this correspondsto the high mode number limit. Therefore, the cantilever beam model is strictlyvalid in the two limits of small and large acoustic wavelengths relative to thedominant hydrodynamic length scale. In the intermediate regime where theacoustic wavelength is larger than, but comparable to the spatial wavelength ofbeam oscillations, the flow probes a spectrum of spatial wave numbers due tofinite beam length; this has not been rigorously considered here. Nonetheless,
23
since the above solution holds rigorously in the bounding asymptotic limits ofsmall and large wavelengths, it is expected to yield a good approximation in theintermediate regime.
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