MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 80

critical value); and F max 0 , stable adhered state ( s small). Setting

F max = 0 provides the critical indenter-imposed position, s 0 , at which the
system is in the transition state between the two stable states:

<

1/2

k

A

s = 2 = 2 z , (4-8)

0 max

S

where the 0 subscript indicates zero force on the tip. The indenter-
imposed position s 0 is a characteristic length scale of the system and is
determined by a balance between the attraction of the surface force
interaction and the resistance to solid (spring) deformation.
The condition F = 0 of course defines equilibrium and hence for the

indenter-imposed condition of s = s 0 the spring is extended towards the
surface by s 0 / 2 such that the tip is resting in unstable equilibrium at a
separation from the surface of z ∗ s 0 / 2 , where the asterisk indicates an

equilibrium state. The characteristic force scale for the system is thus the
force exerted by the spring on the tip in this unstable equilibrium
state, F S ( s

=

( Ak S ) 1/2 . The characteristic energy scale
for the system is the deformational energy contained in the spring at this
state,

= s 0 /2) = ( k S s 0 /2) =

Φ= ( ss / 2) = ks ( / 2)

2

/ 2 = A / 2 . The characteristic stiffness of

S 0 S 0

the system, k S , is evident.
Using the characteristic scales of the system allows the potential,
force, and stiffness fields experienced by the tip to be specified in terms
of the internal variable of the tip position:

Φ=( A /2) 2ln( z / z A ) + 4( z / s 0 − s / s 0 ) 2 ,

[ ]

1/2

F k = ( ) − s z zsss / 2 ( − / − / ) , and (4-9)

S 0 0 0

− s 2 /4 z 2 k = k S + 1 ,

in which the central role of the indenter-imposed probe position, s ,
relative to the critical position, s 0 , in determining system behavior is
obvious. Figures 4-3, 4-4, and 4-5 show the variations of Φ( z ), F ( z ), and
k ( z ), and their components for s / s 0 = 1.5 , s / s 0 = 1, and s / s 0 = 0.5 ,
respectively. The invariant position of the maximum in the force at s 0 /2
and variation of the sign of the maximum force with s / s 0 is clear.