MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 79

The force, F A ( z ) , imposed by the surface potential on the tip is given

by

d Φ A

d z

A
z

F A =− =− , (4-3)

and the stiffness field, k A ( z ) , associated with this force is given by

d F A
d z

A
z 2 ,

k A =− =− (4-4)

where it is noted that for this example the positive z -direction is taken
along the outward normal of the surface, towards the probe. The force,
F S ( z ) , imposed by the spring on the tip is

d Φ S

d z

F S =− =− k S ( z − s ) (4-5)

and of course the stiffness of this force is the invariant spring stiffness,
k S . The behavior of the force and stiffness fields is also shown in
Fig. 4-2 .
Consideration of the small- and large- z limits of the total force on the
tip, F = F A + F S , shows that F must pass through a maximum for all
values of s . The condition for this maximum is that the net stiffness
experienced by the tip vanish. Using the stiffness relations above and
setting the net stiffness to zero, k = k A + k S = 0, provides the tip-surface
separation at which the tip experiences a maximum force:

1/2

k

A

z = . (4-6)

max

S

This separation is indeed invariant with respect to s . Using the force
relations above gives the maximum total force on the tip at this
separation:

2( Ak S ) 1/2 . (4-7)
Both the magnitude and sign of this maximum force depend on the
indenter-imposed position s , which thus determines the state of the
system relative to the adhesion sequence described above: F max

F max = k S s −

> 0,

stable separated state ( s large); F max = 0 , unstable transition state ( s at a