The Stochastic Dynamics of Rectangular and V-shaped Atomic Force Microscope Cantilevers in a Viscous Fluid and Near a Solid Boundary
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n The Stochastic Dynamics of Rectangular and V-shaped Atomic Force MicroscopeCantilevers 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: November 4, 2018)Using a thermodynamic approach based upon the fluctuation-dissipation theorem we quantify thestochastic dynamics of rectangular and V-shaped microscale cantilevers immersed in a viscous fluid.We show that the stochastic cantilever dynamics as measured by the displacement of the cantilevertip or by the angle of the cantilever tip are different. We trace this difference to contributionsfrom the higher modes of the cantilever. We find that contributions from the higher modes aresignificant in the dynamics of the cantilever tip-angle. For the V-shaped cantilever the resultingflow field is three-dimensional and complex in contrast to what is found for a long and slenderrectangular cantilever. Despite this complexity the stochastic dynamics can be predicted using atwo-dimensional model with an appropriately chosen length scale. We also quantify the increasedfluid dissipation that results as a V-shaped cantilever is brought near a solid planar boundary.
I. INTRODUCTION
The stochastic dynamics of micron and nanoscale can-tilevers immersed in a viscous fluid are of broad scientificand technological interest [1, 2]. Of particular impor-tance is the oscillating cantilever that is central to atomicforce microscopy [3, 4]. Significant theoretical progresshas been made using simplified models in the limit of longand thin rectangular cantilevers [5, 6, 7]. In this case, atwo-dimensional approximation is appropriate (thereforeneglecting effects due to the tip of the cantilever) andhas yielded important insights. However, it is not cer-tain how well these approximations work for many sit-uations of direct experimental interest. For example, acommonly used cantilever in atomic force microscopy isV-shaped and a theoretical description of the dynamicsof these cantilevers in fluid is not available.Furthermore, micron and nanoscale cantilevers are of-ten used in close proximity to a solid boundary either bynecessity or out of experimental interest. It is well knownexperimentally and theoretically that the presence of asolid boundary increases the fluid dissipation resultingin reduced quality factors and reduced resonant frequen-cies [7, 8, 9, 10, 11, 12, 13, 14]. Again, theoretical de-scriptions are available in the limit of long and thin rect-angular cantilevers and it is uncertain if these approachescan be applied to these more complex geometries.In this paper we use a powerful thermodynamic ap-proach to quantify the stochastic dynamics of cantileversdue to Brownian motion for experimentally relevant ge-ometries for the precise conditions of experiment includ-ing the presence of a planar boundary. Our results arevalid for the precise three-dimensional geometry of inter-est and include a complete description of the fluid-solidinteractions. Using these results we are able to comparewith available theory to yield further physical insights ∗ Electronic address: [email protected] and to suggest simplified analytical approaches to de-scribe the cantilever dynamics for these complex situa-tions.
II. THERMODYNAMIC APPROACH –FLUCTUATIONS FROM DISSIPATION
The stochastic dynamics of micron and nanoscale can-tilevers driven by thermal or Brownian motion can bequantified using strictly deterministic calculations. Thisis accomplished using the fluctuation-dissipation theoremsince the cantilever remains near thermodynamic equi-librium [15, 16]. We briefly review this approach for thecase of determining the stochastic displacement of thecantilever tip and then extend it to the experimentallyimportant case of determining the stochastic dynamicsof the angle of the cantilever tip.The autocorrelation of equilibrium fluctuations in can-tilever displacement can be determined from the deter-ministic response of the cantilever to the removal of astep force from the tip of the cantilever (i.e. a transversepoint force removed from the distal end of the cantilever).If this force f ( t ) is given by f ( t ) = (cid:26) F for t <
00 for t ≥ , (1)where t is time and F is the magnitude of the force, thenthe autocorrelation of the equilibrium fluctuations in thedisplacement of the cantilever tip is given directly by h u (0) u ( t ) i = k B T U ( t ) F , (2)where k B is Boltzmann’s constant, T is the tempera-ture, and hi is an equilibrium ensemble average. In ournotation lower case letters represent stochastic variables( u ( t ) is the stochastic displacement of the cantilever tip)and upper case letters represent deterministic variables( U ( t ) represents the deterministic ring down of the can-tilever tip due to the step force removal). The spectral h Lw (a) yxz Lw θ xzy (b)b FIG. 1: Schematics of the two micron scale cantilever geome-tries considered (not drawn to scale). Panel (a), A rectan-gular cantilever with aspect ratios
L/h = 98 . w/h = 14 . L/w = 6 .
8. The cantilever is composed of silicon withdensity ρ c = 2329 kg/m and Youngs Modulus E = 174GPa. Panel (b), A V-shaped cantilever with aspect ratios L/h = 233, w/h = 30, and
L/w = 7 .
8. The total widthbetween the two arms normalized by the width of a singlearm is b/w = 10 .
36. The cantilever planform is an equilateraltriangle with θ = π/
3. The cantilever is composed of siliconnitride with ρ c = 3100kg/m and E = 172GPa. The specificdimensions for the rectangular and V-shaped cantilever aregiven in Table I. properties of the stochastic dynamics are given by theFourier transform of the autocorrelation.The thermodynamic approach is valid for any conju-gate pair of variables [15]. For example, it is commonin experiment to use optical techniques to measure theangle of the cantilever tip as a function of time [4]. Ithas also been proposed to use piezoresistive techniquesto measure voltage as a function of time [17]. The ther-modynamic approach remains valid for these situationsby choosing the correct conjugate pair of variables.In this paper we also explore the stochastic dynamicsof the angle of the cantilever tip. In this case, the angleof the cantilever tip is conjugate to a step point-torqueapplied to the cantilever tip. If this torque is given by τ ( t ) = (cid:26) τ for t <
00 for t ≥ , (3)where τ is the magnitude of the step torque, then theautocorrelation of equilibrium fluctuations in cantilever L ( µ m) w ( µ m) h ( µ m) k (N/m) k t (N-m/rad) f (kHz)(1) 197 29 2 1.3 1 . × − . × − L , width w , and height h .For the V-shaped cantilever the total length between the twoarms at the base is b = 161 . µ m. The cantilever spring con-stant k , torsional spring constant k t , and resonant frequencyin vacuum f are determined using finite element numericalsimulations. The cantilevers are immersed in water with den-sity ρ l = 997 kg/m and dynamic viscosity η = 8.59 × − kg/m-s. tip-angle θ ( t ) is given by h θ (0) θ ( t ) i = k B T Θ ( t ) τ . (4)Here Θ ( t ) represents the deterministic ring down, asmeasured by the tip-angle, resulting from the removalof a step point-torque. Again, the Fourier transform ofthe autocorrelation yields the noise spectrum.A powerful aspect of this approach is that it is possibleto use deterministic numerical simulations to determine U ( t ) and Θ ( t ) for the precise cantilever geometries andconditions of experiment. This includes the full three-dimensionality of the dynamics which are not accountedfor in available theoretical descriptions. The numericalresults can be used to guide the development of moreaccurate theoretical models. III. THE STOCHASTIC DYNAMICS OFCANTILEVER TIP-DEFLECTION ANDTIP-ANGLE
The stochastic dynamics of the cantilever tip-displacement u ( t ) and that of the tip-angle θ ( t ) yieldinteresting differences. Using the thermodynamic ap-proach, insight into these differences can be gained byperforming a mode expansion of the cantilever using theinitial deflection required by the deterministic calcula-tion. The two cases of a tip-force and a tip-torque re-sult in a significant difference in the mode expansion co-efficients which can be directly related to the resultingstochastic dynamics.For small deflections the dynamics of a cantileverwith a non-varying cross section are given by the Euler-Bernoulli beam equation, µ ∂ U∂t + EI ∂ U∂x = 0 , (5)where U ( x, t ) is the transverse beam deflection, µ is themass per unit length, E is Young’s modulus, and I is themoment of inertia [19]. For the case of a cantilever wherea step force has been applied to the tip at some time inthe distant past the steady deflection of the cantilever at t = 0 is given by U ( x ) = − F EI (cid:18) x − Lx (cid:19) , (6)where L is the length of the cantilever and the appropri-ate boundary conditions are U (0) = U ′ (0) = U ′′ ( L ) = 0and U ′′′ ( L ) = − F /EI . The prime denotes differentia-tion with respect to x .Similarly, the deflection of the same cantilever beamdue to the application of a point-torque at the cantilever-tip is quadratic in axial distance and is given by U ( x ) = τ EI x , (7)where the appropriate boundary conditions are U (0) = U ′ (0) = U ′′′ ( L ) = 0 and U ′′ ( L ) = τ /EI . The angleof the cantilever measured relative to the horizontal orundisplaced cantilever is then given by tan Θ = U ′ ( x ).The mode shapes for a cantilevered beam are given byΦ n ( x ) = − (cos κL + cosh κL ) (cos κx − cosh κx ) − (sin κL − sinh κL ) (sin κx − sinh κx ) , (8)where n is the mode number, and the characteristic fre-quencies are given by κ = ω µ/EI . The mode numbers κ are solutions to 1 + cos κL cosh κL = 0 [19]. The initialcantilever displacement given by Eqs. (6) and (7) can beexpanded into the beam modes U ( x ) = ∞ X n =1 a n Φ n ( x ) , (9)with mode coefficients a n . The total energy E b of thedeflected beam is given by E b = EI Z L U ′′ ( x ) dx, (10)which is entirely composed of bending energy. The frac-tion of the total bending energy contained in an individ-ual mode is given by b n = EI E b Z L ( a n Φ ′′ n ( x )) dx. (11)The coefficients b n for the rectangular cantilever of Ta-ble I are shown in Table II. For the case of a force appliedto the cantilever tip, 97% of the total bending energyis contained in the fundamental mode and the energycontained in the higher modes decays rapidly with lessthan 1% of the energy contained in mode three. Whena point-torque is applied to the same beam it is clearthat a significant portion of the bending energy is spreadover the higher modes. Only 61% of the energy is con-tained in the fundamental mode and the decay in energy n b n (tip-force) b n (tip-torque)1 0.97068 0.613082 0.02472 0.188303 0.00315 0.064734 0.00082 0.033095 0.00030 0.02669TABLE II: The fraction of the total energy E b contained inthe first five beam modes given by the coefficients b n . Thetip-force results are for a rectangular beam that has beendeflected by the application of a point force to the cantilevertip. The tip-torque results are for a rectangular beam thathas been deflected by the application of a point torque tothe cantilever tip. The coefficients clearly show that the tip-torque case has significantly more energy contained in thehigher modes. ˙ u ¸ / (nm) ˙ θ ¸ / (rad)(1) 5 . . × − (2) 20 7 . × − TABLE III: The magnitude of stochastic fluctuations in tip-deflection and in tip-angle for the rectangular (1) and V-shaped (2) cantilevers. These values were obtained from nu-merical simulations simulations of the beams in vacuum. with mode number is more gradual. The fifth mode forthe tip-torque case contains more energy than the secondmode for the tip-force case. Although we have only dis-cussed a mode expansion for the rectangular cantilever,the V-shaped cantilever will exhibit similar trends sincethe transverse mode shapes are similar to that of a rect-angular beam [20].The variation in the energy distribution among themodes required to describe the initial deflection of thecantilever can be immediately connected to the resultingstochastic dynamics. For the deterministic calculationsthe initial displacement can be arbitrarily set to a smallvalue. In this limit the modes of the cantilever beamare not coupled through the fluid dynamics. As a result,the stochastic dynamics of each mode can be treated asthe ring down of that mode from the initial deflection.This indicates that the more energy that is distributedamongst the higher modes initially the more significantthe ring down and, using the fluctuation-dissipation the-orem, the more significant the stochastic dynamics.The mode expansion clearly shows that the tip-torquecase has more energy in the higher modes. This suggeststhat stochastic measurements of the cantilever tip-anglewill have a stronger signature from the higher modes thanmeasurements of cantilever tip-displacements. Using fi-nite element simulations for the precise geometries of in-terest we quantitatively explore these predictions.
IV. THE STOCHASTIC DYNAMICS OF ARECTANGULAR CANTILEVER
We have performed deterministic numerical simula-tions of the three-dimensional, time dependent, fluid-solid interaction problem to quantify the stochastic dy-namics of a rectangular cantilever immersed in waterusing the thermodynamic approach discussed in Sec-tion II. The deterministic numerical simulations are doneusing a finite element approach that is described else-where [21, 22].The stochastic fluctuations in cantilever tip-displacement for a rectangular cantilever in waterhave been described elsewhere [15, 16, 23, 24]. In thefollowing we compare these results with the stochasticdynamics as determined by the fluctuations of thecantilever tip-angle. The geometry of the the specificmicron scale cantilever we explore is given in Table I.As discussed in Section II the autocorrelations in equi-librium fluctuations follow immediately from the ringdown of the cantilever due to the removal of a stepforce (to yield h u (0) u ( t ) i ) or step point-torque (to yield h θ (0) θ ( t ) i ). The autocorrelations of the rectangularcantilever are shown in Fig. 2. The magnitude of thenoise is quantified by the root mean squared tip-angleand deflection which is listed in Table III.A comparison of the autocorrelations yields some in-teresting features. At short times h θ (0) θ ( t ) i shows thepresence of higher harmonic contributions. This is shownmore clearly in the inset of Fig. 2. This further sug-gests that the angle autocorrelations are more sensitiveto higher mode dynamics as discussed in Section III. t (ms) 〈 u ( ) u ( t ) 〉 , 〈 θ ( ) θ ( t ) 〉 FIG. 2: The normalized autocorrelation of the rectangularcantilever for tip-deflection (solid) and tip-angle (dashed).(Inset) A detailed view of the autocorrelation at short timedifferences to illustrate the influence of higher modes in thetip-angle measurements.
The Fourier transform of the autocorrelations yield the noise spectra shown in Fig. 3. In our notation the sub-script of G indicates the variable over which the noisespectrum is measured: G θ is the noise spectrum for tip-angle and G u is the noise spectrum for tip-displacement.The equipartition theorem of energy yields,12 π Z ∞ G u ( ω ) dω = k B Tk (12)12 π Z ∞ G θ ( ω ) dω = k B Tk t (13)where k and k t are the transverse and torsional springconstants, respectively. The curves in Fig. 3 are normal-ized using the equipartition result to have a total areaof unity. Using this normalization the area under a peakis an indication of the amount of energy contained ina particular mode. Figure 3 shows only the first twomodes, although the numerical simulations include all ofthe modes (within the numerical resolution of the finiteelement simulation). The energy distribution across thefirst two modes shows the significance of the second modefor the tip-angle dynamics. ω / ω G u , G θ -8 -7 -6 -5 FIG. 3: The noise spectra of stochastic fluctuations in can-tilever tip-angle (dashed) and tip-deflection (solid) for therectangular cantilever. The curves are normalized to havethe same area, however only the first two modes are shown.
Using a simple harmonic oscillator approximation it isstraight forward to compute the peak frequency ω f andquality Q for the cantilever in fluid. Using a single modeapproximation yields the values shown in Table IV. Asexpected there is a significant reduction in the cantileverfrequency when compared with the resonant frequencyin vacuum ω and the quality factor is quite low becauseof the strong fluid dissipation. The values of ω f and Q for tip-angle and tip-dispacement are nearly equal. Thisis expected since the displacements and angles are verysmall, resulting in negligible coupling between the modes.Any differences in ω f and Q can be attributed to usinga single mode approximation.It is useful to compare these results with the commonlyused approximation of an oscillating, infinitely long cylin-der with radius w/ L/w ≈ ω f and Q . ω f /ω Q (1) 0.35 3.34(2) 0.36 3.26TABLE IV: The peak frequency and quality factor of thefundamental mode of the rectangular cantilever determinedby finite element simulations using the thermodynamic ap-proach. (1) is computed using the cantilever tip-displacementdue to the removal of a step force. (2) is computed using thecantilever tip-angle due to the removal of a point-torque. Thefrequency result is normalized by the resonant frequency invacuum ω . Using the infinite cylinder approximation witha radius of w/ Q = 3 .
24 and ω f /ω = 0 . V. THE STOCHASTIC DYNAMICS OF AV-SHAPED CANTILEVER
We now explore the stochastic dynamics of a V-shapedcantilever 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.The deterministic finite element simulations that we per-formed yield a quantitative picture of the resulting fluiddynamics. Exploring the flow fields further yields insightinto the dominant features that contribute to the can-tilever dynamics.As discussed earlier, for long and slender rectangularcantilevers the flow field is often approximated by thatof a cylinder of diameter w undergoing transverse oscil-lations. This approach assumes that the fluid flow is es-sentially two-dimensional in the y − z plane and neglectsany flow over the tip of the cantilever. Figure 4 (top) il-lustrates this tip flow for the rectangular cantilever usingvectors of the fluid velocity in the x − y plane at z = 0.The figure is a close-up view near the tip of the can-tilever. It is evident that the flow over the rectangularcantilever is nearly uniform in the axial direction leadingup to the tip. However, near the tip there is a significanttip flow that decays rapidly in the axial direction awayfrom the tip. The increasing significance of the tip flowas the cantilever geometry becomes shorter (for exam-ple, by simply decreasing L ) is not certain and remainsan interesting open question. However, for the geometryused here it is clear that this tip-flow is negligible basedupon the accuracy of the analytical predictions using thetwo-dimensional model.Figure 4 (bottom) illustrates the tip flow for the V-shaped cantilever, again by showing velocity vectors in FIG. 4: The fluid flow near the tip of the cantilever as il-lustrated by the velocity vector field calculated from finiteelement numerical simulations. A cross section of the x − y plane at z = 0 is shown (see Fig. 1) that is a close-up viewof the tip-region. The shaded region indicates the cantilever(because of the small deflections used in the simulations thatcantilever does not appear to be deflected). (top) The flowfield near the tip of the rectangular cantilever. This flow fieldis at t=6 µ s and the magnitude of the largest velocity vectorshown is -0.3 nm/s. (bottom) The flow field near the tip ofthe V-shaped cantilever. This flow field is at t=7.2 µ s and themagnitude of the largest velocity vector shown is -26 nm/s.The shaded region indicates the tip region where the two sin-gle arms have merged. The open region to the left is wherethe two single arms have separated revealing the open regionin the interior of the V-shaped cantilever. the x − y plane at z = 0. The shaded region indicates thepart of the cantilever where the two arms have merged.To the right of the shaded region illustrates flow off thetip and to the left indicates flow that circulates back inbetween the two individual arms.In order to illustrate the three-dimensional nature ofthis flow, the flow field in the y − z plane is shown at twoaxial locations in Fig. 5. Figure 5(top) is at axial location x = 77 µ m. The two shaded regions indicate the two armsof the cantilever. Each arm is generating a flow with aviscous boundary layer (Stokes layer) as expected fromprevious work on rectangular cantilevers. However, theStokes layers interact in a complicated manner near thecenter. It is expected that as one goes from the base ofthe cantilever to the tip that these fluid structures wouldtransition from non-interacting to strongly-interacting.Figure 5(bottom) illustrates the flow field at axial lo-cation x = 108 . µ m, the axial location at which the twoarms of the cantilever merge to form the tip region. Thelength of the shaded region is therefore 36 µ m or twicethat of a single arm shown in Fig. 5(top). For this tipregion the flow field is similar to what would be expectedof a single rectangular cantilever of this width.Overall, it is clear that the fluid flow field is more com-plex for the V-shaped cantilever than for the long andslender rectangular beam. For the V-shaped cantileverthe flow is three-dimensional near the tip region wherethe two arms join together.Central to the flow field dynamics are the interactionsof the two Stokes layers caused by the oscillating can-tilever arms. The thickness of these Stokes layers areexpected to scale with the frequency of oscillation as δ s /a ∼ R ω − / where a is the half-width of the can-tilever and R ω = ωa /ν is a frequency based Reynoldsnumber (often called the frequency parameter). For therelevant case of a cylinder of radius a oscillating at fre-quency ω the solution to the unsteady Stokes equationsyields a distance of approximately 5 δ s to capture 99% ofthe fluid velocity in the viscous boundary layer [25]. Fora single arm of the V-shaped cantilever this distance isnearly 10 µ m. In comparison, the total distance betweenthe two arms at the base is 125 µ m. This separation islarge enough such that the two Stokes layers have neg-ligible interactions near the base. However, as the armsapproach one another with axial distance the Stokes lay-ers overlap and eventually merge at the tip.Despite the complicated interactions of the three-dimensional flow caused by the cantilever tip and theaxial merging of the two Stokes layers, the V-shaped can-tilever behaves as a damped simple harmonic oscillator.The autocorrelations in tip-angle and tip-displacementthat are found using full finite element numerical sim-ulations are shown in Fig. 6. It is again clear that thetip-angle dynamics have significant contributions fromthe higher modes, see the inset of Fig. 6. The area nor-malized noise spectra are shown in Fig. 7.Using a simple harmonic oscillator analogy a peak fre-quency and a quality factor can be determined from thefirst mode in the noise spectra of Fig. 7. These valuesare given in the first two rows of Table VI. The qual-ity of the cantilever is Q ≈ ω f /ω ≈ .
2. compared to the res-onant frequency in the absence of a surrounding viscousfluid.It is insightful and of practical use to determine thegeometry of the equivalent rectangular beam that wouldyield the precise values of k , ω f , and Q calculated for FIG. 5: The fluid velocity vector field at two axial positionsalong the V-shaped cantilever calculated from deterministicfinite element numerical simulations. Cross sections of the y − z plane are shown (see Fig. 1), the entire simulation domainis not shown and the shaded region indicates the cantilever.Both images are taken at t=7.2 µ s and the maximum velocityvector shown is -26 nm/s. (top) The y − z plane at x =77 µ m. The skewed width of a single arm of the cantileverin this cross-section is 18 µ m. The distance separating thetwo cantilever arms is 36 µ m. (bottom) The y − z plane at x = 108 . µ m. This is the point at which the two single armsjoin to make a continuous cross-section of width 2 w . the V-shaped cantilever from full finite-element numeri-cal simulations. For the rectangular beam the equationsare well known (c.f. Ref. [16]) and yield a unique valueof length L ′ , width w ′ , and height h ′ as shown below, k = 3 EIL ′ = Ew ′ h ′ L ′ , (14) Q = m f ω f γ f = ρ c h ′ πρ f w ′ + Γ ′ ( w ′ , ω f )Γ ′′ ( w ′ , ω f ) , (15)where the peak frequency is determined from the maxi- t (ms) 〈 u ( ) u ( t ) 〉 , 〈 θ ( ) θ ( t ) 〉 FIG. 6: The normalized autocorrelation of equilibrium fluc-tuations in the tip-deflection h u (0) u ( t ) i (solid lined) andin tip-angle h θ (0) θ ( t ) i (dashed-line) for the V-shaped can-tilever. The inset shows a close-up of the dynamics for shorttime differences to illustrate the influence of the higher modesin the tip-angle measurements. ω / ω G u , G θ -7 -6 -5 FIG. 7: The noise spectra for the V-shaped cantilever as de-termined from the tip-displacement G x (solid line) and fromtip-angle G θ (dashed line). The curves are normalized to havean area of unity, with only the first two modes shown. mum of the noise spectrum, G u = 4 k B Tk ω (16) × T ˜ ω Γ ′′ ( R ˜ ω )[(1 − ˜ ω (1 + T Γ ′ ( R ˜ ω ))) + (˜ ω T Γ ′′ ( R ˜ ω )) ] . In the above equations ˜ ω = ω/ω is the normalized fre-quency, α = 0 .
234 is a constant factor to determine anequivalent lumped mass for a rectangular beam, m f isthe equivalent mass of the cantilever plus the added fluid mass, γ f is the fluid damping, Γ is the hydrodynamicfunction for an infinite cylinder, Γ ′ is the real part of Γ,and Γ ′′ is the imaginary part of Γ. Equations (14)-(15)can be solved to yield values for the unknown geometryof the equivalent rectangular beam L ′ , w ′ , and h ′ whichare given in Table V. The equivalent beam is shorter,thinner, and wider than the V-shaped cantilever. Impor-tantly, the width of the equivalent beam is nearly twicethat of a single arm of the V-shaped cantilever. L ′ /L w ′ /w h/h ′ k , ω f , and Q for the V-shapedcantilever that have been determined from full finite-elementnumerical simulations. The length, width, and height of theequivalent beam ( L ′ , w ′ , h ′ ) are calculated using Eqs. (14)-(15) and are normalized by the values of ( L, w, h ) for theV-shaped cantilever given in Table I.
These results suggest that the parallel beam approx-imation (PBA) [26, 27, 28, 29] commonly used to de-termine the spring constant for a V-shaped cantilevermay also provide a useful geometry for determining thedynamics of V-shaped cantilevers in fluid. In this ap-proximation the V-shaped cantilever is replaced by anequivalent rectangular beam of length L , width 2 w , andheight h to yield a simple analytical expression for thespring constant. This has been shown to be quite suc-cessful for V-shaped cantilevers that have arms that arenot significantly skewed. The results of using the geom-etry of this approximation to determine ω f and Q fromthe two-dimensional cylinder approximation are shownon the third row of Table VI. It is clear that this is quiteaccurate. It is expected that these results will remainuseful for cantilever geometries that do not deviate sig-nificantly from that of an equilateral triangle as studiedhere. An exploration of the breakdown of this approx-imation is possible using the methods described but isbeyond the scope of the current efforts. VI. QUANTIFYING THE INCREASEDDISSIPATION DUE TO A PLANAR BOUNDARY
In practice, the cantilever is never placed in an un-bounded fluid and the influence of nearby boundariesmust be accounted for to provide a complete descriptionof the dynamics. In many cases the cantilever is purpose-fully brought near a surface out of experimental interestin order to probe some interaction with the cantilever orto probe the surface itself. To specify our discussion wewill consider the situation depicted in Fig. 8 showing acantilever a distance s from a planar boundary. In thefollowing we study the case where the cantilever exhibitsflexural oscillations in the direction perpendicular to the ω f /ω Q (1) 0.21 1.98(2) 0.22 2.04(L,2w,h) 0.19 1.98TABLE VI: The peak frequency and quality factor of the fun-damental mode of the V-shaped cantilever determined by fi-nite element simulations using the thermodynamic approach.(1) is computed using the cantilever tip-displacement due tothe removal of a step force. (2) is computed using the can-tilever tip-angle due to the removal of a point-torque. Thethird line represents theoretical predictions using the geom-etry of an equivalent rectangular beam given by ( L, w, h ).The frequency result is normalized by the resonant frequencyin vacuum ω . boundary. However, we would like to emphasize thatour approach is general and can be used to explore ar-bitrary cantilever orientations and oscillation directionsif desired. The fluid is assumed to be unbounded in allother directions. It is well known that the presence of theboundary will influence the dynamics of the cantilever[8, 9, 13]. The result is a reduction in the resonant fre-quency and quality factor. This has been described the-oretically for the case of a long and thin cantilever ofsimple geometry where the fluid dynamics have been as-sumed two-dimensional [7, 10, 11, 12]. sh FIG. 8: A schematic of a cantilever a distance s away froma solid planar surface (not drawn to scale). The cantileverundergoes flexural oscillations perpendicular to the surface. In the following we use the thermodynamic approachwith finite element numerical simulations to quantify thedynamics of the V-shaped cantilever as a function of itsseparation from a boundary. We have performed 8 simu-lations over a range of separations from 10 to 60 µ m usingboth the tip-deflection and tip-angle formulations. Thenoise spectra for these simulations are shown in Fig. 9.Using the insights from our simulations of the V-shapedcantilever in an unbounded fluid we expect the relevantlength scale for the fluid dynamics to be twice the widthof a single arm, 2 w . Using the peak frequency of theV-shaped cantilever in unbounded fluid yields a Stokeslength δ s = 4 . µ m. Scaling the separation by the Stokeslength yields. 2 . . s/δ s .
15 which covers the range from what is expected to be a strong influence of thewall to a negligible influence. Figure 9 clearly shows areduction in the peak frequency and a broadening of thepeak as the cantilever is brought closer to the boundary.In fact, for the smaller separations the peak is quite broadand the trend suggests that eventually the peak will be-come annihilated as the cantilever is brought closer tothe boundary. ω / ω G u s / δ s = 9.67s / δ s = 2.42 (a) ω / ω G θ s/ δ s = 14.51s/ δ s = 3.63 (b) FIG. 9: Panel (a) The noise spectra G u of stochastic fluc-tuations in cantilever tip-deflection for separations s =10 , , , , µ m. Panel (b) the noise spectra G θ of stochas-tic fluctuations in cantilever tip-angle for separations s =15 , , µ m. The spectra have been normalized by the max-imum value of G u or G θ . The smallest and largest values ofseparation are labeled with all other values appearing sequen-tially. Using the noise spectra we compute a peak frequencyand a quality factor for the fundamental mode as a func-tion of separation from the boundary, which are plot-ted in Fig. 10. The horizontal dashed line representsthe value of the peak frequency and quality factor in theabsence of bounding surfaces using the two-dimensionalinfinite cylinder approximation [16] where the cylinderwidth has been chosen to be 2 w . It is clear from theresults that for separations greater than s/δ s & w in computing these theo-retical predictions for comparison with our numerical re-sults. Despite the complex and three-dimensional natureof the flow field the theory is able to accurately predictthe quality factor over the range of separations explored.The frequency of the peak for the V-shaped cantilevershows some deviation from these predictions.In general, an increase in the period of oscillation fora submerged object can be attributed to the mass offluid entrained by the object [30]. The lower peak fre-quency calculated for the V-shaped cantilever using atwo-dimensional solution indicates an over-prediction ofthe mass loading. This can be attributed to the three-dimensional flow around the tip being neglected for thisapproach. It is reasonable to expect the cantilever tip tocarry a smaller amount of fluid than a section of the beambody moving with the same velocity, see Fig. 4. The qual-ity factor relates to the ratio of the mass loading and theviscous dissipation and is less sensitive to deviations in-curred from the two-dimensional approximation. Despiteneglecting three-dimensional flow around the cantilevertip, the two-dimensional model for the fluid flow aroundthe V-shaped cantilever gives an accurate prediction ofthe peak frequency and quality factor. VII. CONCLUSIONS
We have shown that the thermodynamic approach is aversatile and powerful method for predicting the stochas-tic dynamics of cantilevers in fluid for the precise con-ditions of experiment including complex geometries andthe presence of nearby boundaries. Available analyti-cal predictions are for idealized situations including sim-ple geometries where the three-dimensional flow near thecantilever tip has been neglected. Although this has pro-vided significant insight, many situations of experimen-tal interest are more complicated. It is often required tohave a quantitative base-line understanding of the can-tilever dynamics for the precise conditions of experimentin order to make and interpret measurements in novelsituations and in the presence of other phenomena of in-terest.We emphasize that by using the fluctuation-dissipationtheorem a single deterministic calculation is sufficient topredict the stochastic behavior for all frequencies. Fur-thermore, the deterministic calculation is computation-ally inexpensive and does not require special computingresources. s / δ s ω f / ω (a) s / δ s Q (b) FIG. 10: The variation of the peak frequency (panel (a)) andquality (panel (b)) of the fundamental mode of the V-shapedcantilever in fluid as a function of separation from a nearbywall. Results calculated using tip-deflection are circles, resultsusing tip-angle are squares, and theoretical predictions usingthe results of Ref. [12] are triangles. The peak frequencyand quality factor of the fundamental mode in an unboundedfluid are ω/ω f ≈ .
19 and Q ≈ s is normalized by theStokes length δ s where a = w to yield δ s = 4 . µ m. The thermodynamic approach is general in that it canbe used to compute the stochastic dynamics of any conju-gate pair of variables. We have shown that the stochasticdynamics that are measured depend upon the choice ofmeasurement. This could be exploited in future exper-iments, for example, to minimize or maximize the sig-nificance of the higher mode dynamics by choosing tomeasure tip-deflection or tip-angle, respectively.Our results also suggest that despite the complicatedthree-dimensional nature of the flow field around a V-shaped cantilever, analytical predictions based upon atwo-dimensional description are surprisingly accurate if0the appropriate length scales are used. We anticipatethat these findings will be of immediate use as the atomicforce microscope continues to find further use in liquidenvironments.Acknowledgments: This research was funded by AFOSR grant no. FA9550-07-1-0222. Early work on thisproject was funded by a Virginia Tech ASPIRES grant.We would acknowledge many useful interactions withMichael Roukes, Mike Cross, and Sergey Sekatski. [1] H.-J. Butt and M. Jaschke, Nanotech. , 1 (1995).[2] K. L. Ekinci and M. L. Roukes, Review of Scientific In-struments (2005).[3] G. Binnig, C. F. Quate, and C. Gerber, Phys. Rev. Lett. , 930 (1986).[4] R. Garcia and R. Perez, Surface Science Reports pp. 197–301 (2002).[5] E. O. Tuck, J. Eng. Math , 29 (1969).[6] J. E. Sader, J. Appl. Phys. , 64 (1998).[7] R. J. Clarke, S. M. Cox, P. M. Williams, and O. E.Jensen, J. Fluid Mech. , 397 (2005).[8] F. Benmouna and D. Johannsmann, Eur. Phys. J. E ,435 (2002).[9] I. Nnebe and J. W. Schneider, Langmuir , 3195 (2004).[10] R. J. Clarke, O. Jensen, J. Billingham, A. Pearson, andP. Williams, Phys. Rev. Lett. , 050801 (2006).[11] C. P. Green and J. E. Sader, Phys. Fluids , 073102(2005).[12] C. P. Green and J. E. Sader, J. Appl. Phys. , 114913(2005).[13] C. Harrison, E. Tavernier, O. Vancauwenberghe,E. Donzier, K. Hsu, A. Goodwin, F. Marty, andB. Mercier, Sensor and Actuators A , 414 (2007).[14] R. J. Clarke, O. Jensen, J. Billingham, and P. Williams,Proc. R. Soc. A , 913 (2006).[15] M. R. Paul and M. C. Cross, Physical Review Letters , 235501 (2004). [16] M. R. Paul, M. Clark, and M. C. Cross, Nanotechnology , 4502 (2006).[17] J. L. Arlett, J. R. Maloney, B. Gudlewski, M. Muluneh,and M. L. Roukes, Nano Lett. Theory of elasticity (Butterworth-Heinemann, 1959).[20] R. Stark, T. Drobek, and W. Heckl, Ultramicroscopy ,207 (2001).[21] H. Q. Yang and V. B. Makhijani, AIAA-94-0179 pp. 1–10(1994).[22] ESI CFD Headquarters, Huntsville AL 25806. We use theCFD-ACE+ solver.[23] J. W. M. Chon, M. P., and J. E. Sader, Journal of AppliedPhysics , 3978 (2000).[24] M. T. Clark and M. R. Paul, Int. J. Nonlin. Mech. (2006).[25] C. Carvajal and M. R. Paul, unpublished.[26] T. R. Albrecht, S. Akamine, and C. F. Carver, T. S.amd Quate, J. Vac. Sci. Technol. A , 3386 (1990).[27] J. E. Sader, Rev. Sci. Instrum. , 4583 (1995).[28] J. E. Sader and L. White, J. Appl. Phys , 1 (1993).[29] H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger,E. Bamberg, A. Weisenhorn, K. Goldie, and A. Engel,J. Microsc. , 75 (1993).[30] S. G. Stokes, Trans. Camb. Phil. Soc.9