INDENTATION AND ADHESION AT SMALL LENGTH SCALES - Nanoindentation with Biological Applications - page 32

Let us consider a typical indentation contact as measured by a
displacement-controlled AFM device. As before, we will consider
contact between a cylindrical tip and a flat surface. In this case, more
typical of AFM experiments, we will take the tip and the surface to the
same (isotropic) material with Young's modulus of 100 GPa and
Poisson's ratio of 0.25. In this case the indentation modulus is

53 GPa and, for the same assumed contact radius of 100 nm, the
contact stiffness is

M

10 4 N m −1 . The stiffness of the probe spring in
this case may be estimated by considering the spring as a simple
rectangular (clamped-fixed) cantilever, for which
k S = b S t 3 E ∗ /4 L 3 , (2-13)
where E ∗ is the plane-strain modulus of the cantilever material along its
length, and b S , t S , and L S are the width, thickness, and length of the
cantilever, respectively. (The flexibility in tuning AFM probe spring
stiffness even for this simple cantilever geometry is obvious given the
number of parameters in the above expression.) Taking E ∗ = 170 GPa
([110] silicon) and the cantilever to be 30 μm wide, 10 μm thick and 500
μ

k a 1 ×

10 N m −1 (which is at the stiffer end of the spectrum
of AFM cantilever stiffness values, 0.002 N m −1 to 200 N m −1

m long gives k S

, but would

be typical of that used for AFM indentation).
The above result reflects the usual conditions during AFM
indentation, the contact stiffness is much greater than the spring stiffness,

k S , and hence most of the displacement imposed on the system is
taken up by the cantilever spring, with negligible relative deformation of
the contact, i.e ., ws . Within the contact, the comparable (in this case,
identical) surface and probe indentation moduli,

k a

M surface M tip , lead to
contact displacement appearing equally as surface and probe
deformation. Within this limit, the cantilever displacement is given by

F
k S

s (2-14)

and taking a (moderate) AFM contact load of F = 10 μN gives s = 1 μm;
typical cantilever displacements are very small relative to the cantilever
thickness (and this enables a linear relationship to be approximated