MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 84

For s / s 0 1, there are two equilibrium tip-surface separations, z ∗

( Fig. 4-3 ) . Using the force equations above and setting the total force to
zero generates a quadratic equation in z that can be solved to give the z ∗
values generally in terms of the indenter-imposed position:

>

s ± ( s 2 − s 2 ) 1/2

2

z ∗ = . (4-10)

Taking the + sign above gives the larger, stable equilibrium z ∗ value
associated with probe approach to the surface. Taking the - sign gives
the smaller unstable equilibrium value that is not accessed until the
critical value of s = s 0 is reached, at which point the two solutions merge
into a single metastable value z ∗ = s 0 /2.

Experimentally, the indenter-imposed position, s , is the independent,
controlled variable, and the displacement of the tip from the imposed
position, Δ z = z ∗ − s , is a dependent, measured-response variable (using

the + sign):

Δ z = ( s 0 /2) ( s 2 / s 2 − 1) 1/2 − s / s 0 . (4-11)

Often this displacement is represented as the force exerted by the
spring, k S Δ z =− F S :

− F S = ( k S s 0 /2) ( s 2 / s 2 − 1) 1/2 − s / s 0 (4-12)

Noting that equilibrium requires F A =− F S this equation enables the
surface force to be determined from position-displacement, s -Δ z ,
measurements, although as a function of imposed probe position, not
tip-surface separation.
The net stiffness experienced by the tip, k , is another dependent
variable, given by (again taking the + sign)

− 2

( )

1/2

s 2 / s 2 k = k S 1 − s / s 0 + − 1 (4-13)

As noted in Chapter 2 , the behavior of the stiffness field is usually not
measured directly, but indirectly by changes in the resonant frequency
and amplitude of vibration of the tip. In the presence of the surface force
field, the net stiffness experienced by the tip changes with the
equilibrium separation from the surface and hence with imposed probe