Resonant Cantilevers

The material properties and geometry of the cantilever determine its spring constant and ultimately its resonant frequency. Because they are long and thin objects of width w for which / t, the usual approximate expression for the spring constant K is 

The deposition of mass AM results in decrease of the resonant frequency according to 

The geometrically dependent correction factor nc has value 0.24 for the most common rectangular cantilevers. Equation (4.48) is valid only if the spring constant K 

Because the vibration is spontaneous (i.e., thermally actuated) and the readout is optical (Fig. 4.23), there are no electrical connections required for their operation even in conducting electrolyte solutions. This greatly simplifies the packaging and encapsulation for biosensing, for example. 

Inherent in the oscillation of any object is the existence of different modes of oscillations and of higher harmonics. In addition to the lateral modes there is also the torsional mode. All modes can be detected in the x-y-directions by the PSD (Fig. 4.22). In the case of resonant cantilevers, the different modes have different mass sensitivity (Kim et al., 2001). They are well separated on the amplitude spectrum (Fig. 4.24), and show different sensitivities to absorption of vapors into the selective coating applied to the cantilever (Fig. 4.25). 

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Fig. 1.14 The amplitude, phase, and powder dissipation of the resonating cantilever measured as a function from the sample surface (d) according to Cleveland et al. The sample first was approached and then withdrawn. (Data obtained using Si cantilever/tip tapping on Si wafer.) Reproduced with permission from [17]...

Fig. 3.7 (a) Frequency shifts of a PEUT-coated cantilever (/ =353 kHz) upon alternate exposure to different concentrations of toluene and air. (b) Frequency shifts of a resonant cantilever as a function of analyte concentration for various organic volatiles from left to right, toluene, -octane, ethyl acetate, and ethanol (PEUT layer thickness 3.7 p.m) (Reprinted with permission from Hierlemann et al. (2000). Copyright 2000 Elsevier)... 

Rg. 3.17 Measured and modeled (solid lines) n-octane sensitivity vs polymer thickness for different cantilever temperatures. AT refers to ambient temperature (gas phase and sensor chip). The 150-pm-long resonant cantilever was fabricated using CMOS technology (Reprinted with permission from Lange et al. (2002). Copyright 2002 American Chemical Society)... 

More sophisticated mass transducers were proposed by using resonant cantilevers similar to those adopted in atomic force microscopy In spite of the claimed properties, these sensors were never demonstrated in practical applications. 

Modification of an AFM to operate in a dynamic mode aids the study of soft biological materials [58]. Here a stiff cantilever is oscillated near its resonant frequency with an amplitude of about 0.5 nm forces are detected as a shift to a new frequency... 

Most NC-AFMs use a frequency modulation (FM) teclmique where the cantilever is mounted on a piezo and serves as the resonant element in an oscillator circuit [101. 102]. The frequency of the oscillator output is instantaneously modulated by variations in the force gradient acting between the cantilever tip and the sample. This teclmique typically employs oscillation amplitudes in excess of 20 mn peak to peak. Associated with this teclmique, two different imaging methods are currently in use namely, fixed excitation and fixed amplitude. 

Pulsed-force mode AFM (PFM-AFM) is a method introduced for fast mapping of local stiffness and adliesion with lower required data storage than recording force-distance curves at each point on the x-y plane [115]. A sinusoidal or triangular modulation is applied between the tip and sample (either via lever or sample piezo) at a lower frequency than that of either the piezo or cantilever resonance frequency. Tip and sample then come... 

The most crucial component of an AFM is the cantilever. The deflection should be sufficiently large for ultra low forces (0.1 nN). Therefore, the spring constant should be as low as possible (lower than 1 N/m). On the other hand, the resonance frequency of the cantilever must be high enough (10 to 100 kHz) to minimize the sensitivity to mechanical vibrations (e.g., vibrational noise from the building —100 Hz, frequency of the corrugation signal up to a few kHz). The... 

One point, which is often disregarded when nsing AFM, is that accurate cantilever stiffness calibration is essential, in order to calculate accurate pull-off forces from measured displacements. Althongh many researchers take values quoted by cantilever manufacturers, which are usually calculated from approximate dimensions, more accurate methods include direct measurement with known springs [31], thermal resonant frequency curve fitting [32], temporary addition of known masses [33], and finite element analysis [34]. 

The dynamic imaging mode can be further classified into two subcategories intermittent contact mode (also known as tapping mode) and noncontact mode. In both techniques, the AFM tip is attached to the end of an oscillating cantilever. For the intermittent contact technique, the cantilever is vibrated near its resonance frequency. The amplitude of the oscillation is typically 100-200 nm with the tip intermittently contacting... 

All the STM results from our group presented in this chapter employed the variable temperature STM, with tips made by electrochemical etching of tungsten wire. For noncontact AFM (NC-AFM), we employ commercial conducting silicon cantilevers with force constants of approximately 2-14 rn 1 and resonant frequencies of approximately 60-350kHz (Nanosensors and Mikromasch). The NC-AFM images we present here were recorded in collaboration with Professor Onishi at Kobe University and employed a UHV JEOL (JSPM-4500A) microscope. 

Equation 28 states, that co is proportional to t. The effect of the thickness of the gel film on the frequency of the first resonance mode has been investigated. When the buoyancy is taken into account, the experimental results have quantitatively followed Eq. 28. It has been found that the buoyancy plays an important role in the occurrence of the electric field-associated vibration of gel film. The vibration of the gel film in an electric field has thus roughly analyzed as a mechanical bending vibration of a uniform cantilever beam. 

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