Chapter 6 – Forces  233

parabolic shape. Whereas the local stiffness of the sample is often difficult to determine dir­

ectly, the well-​defined geometry and mechanical properties of the cantilever enable a more

accurate estimation of its stiffness to be made.

For standard mode force sensing, the AFM cantilever oscillations can be approximated

as those of a solid beam flexing about its point of attachment, namely, that of a beam-​like

structure that has one relatively compliant axis (normal to the cantilever flat surface) in terms

of mechanical stiffness, with the two remaining orthogonal axes having comparatively high

stiffness values. The beam bending equation of classical mechanics, for which one end of the

beam is fixed while the other is allowed to undergo deflections, can be used to estimate the

bending stiffness parallel to the z-​axis, kz as

(6.20)

k

Yw

w

w

z

y

y

x

=



^



^

4

3

where

wi is the width of the cantilever beam parallel to the ith axis (Figure 6.6c)

Y is the Young’s modulus

This beam–​cantilever system also has a resonance frequency ν0 given by

(6.21)

v

k

m

w

w

Y

z

z

x

0

0

2

1

2

0 162

=

=

π

ρ

.

where

m0 is the effective total mass of the cantilever and AFM tip

ρ is the cantilever density

Measurement of the resonance frequency far from the surface (such that the potential energy

function is dominated solely by the mechanical elasticity of the cantilever with negligible

contributions from surface-​related forces) can be used to estimate kz. However, this relies

on accurate knowledge of the material properties of the cantilever, which are often not easy

to obtain due to the variability from cantilever to cantilever in a given manufactured batch.

An alternative method involves measuring the vertical mean squared displacement (

)

z²

far away from the sample surface, which requires only an accurate method of measuring z.

Deflections of the cantilever can be very accurately detected usually involving focusing a laser

onto the reflecting back of the polished metal cantilever and imaging the reflected image onto

a split photodiode detector. The voltage response is converted ultimately to a corresponding

distance displacement of the tip, provided that the cantilever stiffness has been determined.

We can then use the equipartition theorem in a similar way as for quantifying the stiffness

of optical and magnetic tweezers to estimate k

k T

z

z

B

=

〈

〉

/

2 . At a room temperature of

~20°C, kBT is ~4.1 pN⋅nm (see Chapter 2). Thus, if the voltage output V from a photodiode

results in a volts per nm of cantilever vertical deflection, the z stiffness is given roughly by

4.1a²/​〈V²〉 in units of pN/​nm. Typical cantilever stiffness values used for probing biological

material in AFM are ~0.1 pN/​nm (see Worked Case Example 6.2).

The interaction forces experienced by the tip include electrostatic and chemical forces as

well as van der Waals (vdW) forces. The Morse potential is a good qualitative model for the

chemical potential, in characterizing the potential energy due to the separation, z, of two

atoms that can form a chemical bond to create a diatomic molecule when they approach each

other to within ~0.1 nm, as might occur between atoms of the sample and the approaching

tip. The shape of the potential energy curve has a minimum corresponding to the equilibrium

atom separation, σ, which is ~0.2 nm, so the term potential energy well is appropriate. The

functional form of the Morse potential is

(6.22)

U

E

Morse

bond

exp

(z

)

exp

(z

)]])

= −

−

−

−

[

[ −

−

(2

2

κ

σ

κ

σ