4. Mechanics of the AFM cantilever in a fluid

The AFM cantilever operates quite differently in fluid. In addition to the obvious viscous damping, the cantilever must move fluid with it, leading to a greatly increased effective mass and a substantial reduction of its resonant frequency (Butt, Siedle et al. 1993).

O’Shea and Welland (O'Shea, Lantz et al. 1998) estimate the Reynolds number for low amplitude DFM from

(9)

where r_{L and h are the fluid density and viscosity respectively and X is a characteristic cantilever dimension (e.g., width). Taking values for water (rL=1000kg/m^{3, h=10-3Pa s),a lever width of 50mm and frequencies in the hundreds of Hz to tens of kHz gives Re>1, so that inertial motion is important. This is experimentally evident in the lowering of the resonant frequency upon immersion. The same authors give the following approximate expression for this effect:}}

(10)

where L is the cantilever length, b its width, t its thickness and rc its density (the expression was derived for rectangular cantilevers). For a silicon cantilever in water, rL/rc=1/2.3 and taking L=250mm, b=50mm and t=2mm gives or about a factor 4 reduction in resonance frequency in water, in reasonable agreement with observations.

The cantilever damping, g, scales roughly linearly with the viscosity of the medium and the dimensions of the cantilever. The damping is often parameterized by the mechanical Q-factor. This is the ratio of the amplitude of motion at resonance to that at low frequency (i.e., the driving amplitude if the cantilever is driven by direct displacement). It is also (approximately) the ratio of the resonant frequency to the full-width at half height of the resonance peak (plotted as amplitude vs. frequency). For a simple harmonic oscillator, the damping force, Fd, and Q are given by

, . (11)

is the cantilever velocity and m is the effective mass of the cantilever (it includes the mass of displaced fluid as discussed above). The Q-factor, therefore, decreases in water and also decreases for larger (i.e. softer cantilevers) (O'Shea, Lantz et al. 1998). This is because such cantilevers have a lower resonant frequency and larger dimensions.

For cantilevers of the dimensions discussed above (i.e., stiffness, k, of about 1 N/m) the Q factor in water is about 3 or 4. In air, the same cantilevers would have a Q of over 100.