INDENTATION AND ADHESION AT SMALL LENGTH SCALES - Nanoindentation with Biological Applications - page 35

where m * is the effective mass of the oscillating element. (For closely
integrated tip-spring systems as in nanoindenters and AFMs, the
effective mass reflects the mass of the larger oscillating spring element
much more than the mass of the smaller tip; for AFM cantilevers,

m ∗ m cantilever / 4 , for example.) In the presence of contact forces, the tip

still behaves as a simple harmonic oscillator but with motion about a new
equilibrium position determined by the force balances discussed above
and with a resonant frequency determined by the effects of the probe
spring and the contact stiffness field. Typically, the behavior of the
contact stiffness field is not measured directly, but indirectly by the
changes in the resonant frequency and amplitude of vibration of the tip as
the probe position is changed. Such a determination can be made by a
dynamic measurement in which the tip is forced into oscillation by an
imposed oscillatory motion of the probe by the indenter and the
frequency of maximum vibration amplitude measured.
Determination can also be made by a “thermal” measurement, in
which the tip oscillates in response to imposed thermal equilibrium with
the indenter at temperature, T . The classical equipartition theorem
prescribes the average free amplitude of oscillation of the tip as a
function of temperature:

k S a 2 = k T (2-19)

where k is Boltzmann's constant and indicates the time-averaged
value of the quantity. Thus the free amplitude and frequency of
oscillation may be estimated from

a S =

( k T/k S ) 1/2 (2-20)

and

ω S = ( k S / m ∗ ) 1/2 . (2-21)
Taking T = 300 K and using the density of silicon to estimate the
effective mass of the k S 10 N m −1 AFM cantilever considered above

gives 340 kHz , an oscillatory behavior easily

measurable by good AFM photodiode detection systems. If the thickness
of the cantilever is reduced such that

a S 0.02 nm and

ω S

k S 0.01 N m −1 (as might be used