MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 105

path from a selected adhered state to a selected separated state. For a
supercritical contact the adhered state is obviously selected as the
quiescent point and the separated state is infinite separation. The work
of separation, represented by the hatched area in Fig. 4-15 for
the k S = 1.3 k 2 case, is thus easily determined. However, such work
obviously depends on the stiffness of the spring as can be observed by
comparing the areas under the curves in Fig. 4-10 : Less work is
expended in extending a stiffer spring during separation, giving rise to a
smaller value for the work of separation. The work of separation in this
context is clearly a system parameter and not a quantity that
characterizes the materials and their interactions that form the tip and
surface. In the limit of a rigid spring, the interaction force-separation
curve is recovered (approximated by the k S = 100 k 2 curve in Fig. 4-10 ,

say) and the integral gives the work as Φ 0 , the characteristic binding

potential. This quantity, quoted per unit area of surfaces separated is
often referred to (somewhat confusingly) as the “work of adhesion”
(strictly a work per unit area) or the (also confusingly named) “surface
energy.”
For a sub-critical contact, the work of separation can obviously be
determined in the same way by integration of the force-position response
from the quiescent point to infinite separation. (As work is a path
quantity, whether the path is reversible or irreversible is irrelevant,
providing the path is specified.) Once again, the stiffness of the spring
alters the force-position response and more compliant springs give rise to
greater work of separation as indicated in Fig. 4-13 . A measure of the
full hysteretic adhesion-separation process for sub-critical systems is the
area contained within the hysteresis loop, indicated by the cross-hatched
area in Fig. 4-15 . The work represented by this area is the energy lost by
the actuator in moving the system through the sequence of reversible and
irreversible segments of the contact cycle. Although still dependent on
the stiffness of the spring in the system, it is a good measure of the “work
of adhesion” as it represents both the adhesive and separation processes
during the contact. In the next sections we will consider a very simple
sub-critical system in which the pull-off force, work of adhesion, and
surface energy are all related through an assumed contact geometry.