limitations. It will be shown that such a scaling downward of dimensions has the profound additional benefit of leading to a marked reduction in response time. This decrease arises primarily from the reduction in cantilever mass loading with the decreased dimensions.
A very rough estimate of the drag constant is possible by considering the drag on a sphere in low Reynolds number flow far from any surface. In this case, γeff =6πηa, where η=ρν is the shear viscosity of the solution and a is the radius of the sphere. For a
= 1µm, in water, the Nyquist formula yields S1/ 2F ~ 17 f N / Hz1/ 2.
cantilevers in the context of the atomic force microscope.2 Numerical evidence suggests that loading of a rectangular cantilever is well approximated by the loading of a circular cylinder of diameter equal to the width of the beam.8 The fluid loading of an infinite cylinder, first calculated by Stokes, is well known9 and can be written as an equivalent mass per unit length:
( )
2( )
4
=πρLw Γ ℜ
L f . 5
The prefactor is simply the volume displaced by the cylinder, while the function Γ which depends solely upon Reynolds number must be calculated from the motion of the fluid.
In this approximation, the fluidic forces at each frequency and on each section of the cantilever are proportional to the displacement at that point. For this case it can be shown that the structure of the cantilever modes is unchanged − only their frequency and damping is modified. The Stokes calculation for a cylinder yields
( ) ( )
( )
1 0
1 4iK i i i K i i
− ℜ ℜ = +
ℜ − ℜ
Γ
, 6 where Ko and K1 are modified Bessel functions. There are two important consequences
of this relation; first, 2πf Im{Γ} gives an effective, frequency-dependent, viscous force per unit length, Imℜ
{
Γ( )
ℜ}
πηu, where u is the velocity. The prefactor,{ ( ) }
ℜIm Γ ℜ , is of order 4 at ℜ=1 and is only a slowly varying function of ℜ. The similarity with the expression for the Stokes force 6πηau acting upon a sphere of radius a is apparent. However, unlike the case for the sphere, the dissipative drag coefficient for a cylinder does not asymptotically approach a constant value at low Reynolds numbers —
instead the prefactor decreases asymptotically as 8 / lnℜ at very small ℜ. For the fundamental mode of a rectangular cantilever, the fluidic damping term can be written as
{ }
2 Im
γeff ≅ α ⎡⎣ π f L ⎤⎦. 7
This is weakly frequency dependent, since the factor 2π f Im
{ }
Γ is not constant. The parameter α relates the mean square displacement along the beam to the displacement at its end. For the fundamental mode of a simple rectangular cantilever, α = 0.2436; for a cantilever that acts as a hinge (see Fig. 2.2.A), α = 0.333. Realistic cantilever geometries will be discussed in more detail in section 2.3, where the actual expressions used for α and K will be mentioned.The second consequence of fluidic loading is an increase in the effective mass per unit length given by Re{Γ}. This term becomes quite large at small ℜ. For the fundamental mode of a cantilever,
(
Re{ } )
eff C c
M ≅ α ρ V + L . 8
Here, Vc is the cantilever volume. Note that the fluid loading is determined by w2 and not wt; hence thin beams experience relatively large fluid loading. The value of Re{Γ}
is unity for large ℜ, is around 4 at ℜ= 1, and continues to increase as ℜ decreases.
Hence, for a silicon cantilever in water at a value of w/t = 2, the mass loading factor (defined as the ratio of fluid loading to inertial mass) is approximately 3 at ℜ= 1 and increases for proportionately thinner beams and lower Reynolds numbers.
can be estimated simply from the fluid properties as
{ }
{ }
Re ( )
~ ~ .
Im ( ) ω
γ
ℜ ℜ
eff eff
Q M Γ
Γ 9
Fig. 2.2.A Prototype silicon nanocantilevers.
The cantilevers extend over a fluidic “vias” (dark regions) formed by deep-etching the wafer through to its backside. The topmost electron micrograph shows the following geometrical parameters for this particular prototypical two-leg device: =15µm, w=2.5µm, b=0.58µm, and 1=4µm. The cantilever thickness is t=130nm, of which the top 30nm forms the conducting layer (with a boron doping density of 4x1019/cm3). From this top layer the transducer and its leads are patterned. The two electrical terminals are visible on the right. For this cantilever, the current path is along the <110> direction for which πL~4x10-10 Pa-1. 11,12 The two lower colorized images show other nanocantilevers above their respective fluidic vias (dark regions). The small gold pad visible at the cantilever tip is used for thiol-based biofunctionalization protocols.
0.2<Q<0.9 as ℜ changes from 10-3 to 1. As expected, this is many orders of magnitude smaller value than the Q’s obtained from semiconductor resonators in vacuum.1,10 Note that since Meff and γeff are frequency-dependent, this notion of Q is only approximate.
The displacement response function is given by the Fourier transform of equation 3:
( )
eff( )
1 2 eff( )
H f x
F K M f i π γf f
= =
− + . 10
(The average squared magnitude of H f
( )
was given in equation 4). We shall use this in the analysis below to relate effective sources of displacement noise back to the force domain (in electrical engineering parlance, “refer them to the input”) to enable evaluation of the practical force sensitivity attainable. The resultant motion of the cantilever tip from an applied force, F, is consequently dependent on the spring constant, K, which depends on the elastic properties of the material and device geometry and the frequency- dependent effective mass and damping which characterize the response of the beam in fluid. (This is analogous to the response function for a resonant device in vacuum, except that in the case of the latter the effective mass and damping are frequency-independent.) In Fig. 2.2.B we plot theoretical calculations of a normalized response function,( )
K H f , for three different cantilever geometries. At high frequencies (greater than 10% of the vacuum resonance frequency) the roll-off in device response due to fluid- induced effective stiffening (from fluid loading) is evident. At low frequencies (less than 1% of the vacuum resonant frequency) the effect of fluid on the cantilever response is
slight. It is in the intermediate region that a sharp resonance would be observed for a device in vacuum. In fluid there is a peak in responsivity (at least for cantilevers 1 and 2), but it is a very broad peak, greatly suppressed compared to the responance in vacuum.
The experiments of Viani et al.13 involving silicon nitride microcantilevers in water confirm this; a peak intensity response of order of twice the low frequency response is found.
Fig. 2.2.B Amplitude response functions for three prototypical fluid-loaded nanocantilevers.
The spring constant, K, is used here to plot a dimensionless, “normalized”, modulus of the response function (c.f. equation 10). The curves correspond to three examples whose properties are delineated in Table I. The Reynolds number for fluidic motion ranges from 5.0, to 0.19, to 0.07, respectively, and the response is seen to evolve from nearly critically damped to strongly overdamped.