MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 91

The interaction between the two bodies, once again envisaged as a
surface and a probe, is described here by a Mie potential, Φ M ( z ): 2

A
z n

B
z m

(4-19) Φ M =− +

where A , B , n , m , z > 0. The simple, generalized form of Eq. 4-19 leads
to frequent use of the Mie potential in considerations of the behavior of
solids, in which both attractive and repulsive interactions exist between
atoms, molecules, and surfaces. There is, as we shall see, analytic
convenience in the fact that both terms are power-laws. The behavior of
Φ M is shown in Fig. 4-7 . z characterizes the separation of the surface and
the probe tip, although it will be shown that the term “separation” will
require some refinement. As before z is taken as positive directed
outward from the surface and z = 0 defines one boundary of the system.
Both terms in the Mie potential go to zero as z →∞, that is, there is no
potential for the interaction between the surface and the probe to perform
work when they are widely separated. The first ( A ) term describes an
attraction between the surface and the probe and may arise from
Coulomb interactions ( n = 1 to 3), fixed dipole interactions ( n = 2 to 4),
covalent bonding interactions ( n = 1 to 3), or fluctuating dipole (van der
Waals) interactions ( n = 6 to 8). The values of the amplitude term, A , and
the exponent, n , depend on the nature of the interaction and the geometry
of the probe tip and surface. The second ( B ) term describes repulsion
between the surface and the probe and may arise from Pauli exclusion or
antibonding interactions ( m = 9 to 11) or fluctuating dipole repulsion
( m = 12 to 14). (The repulsive terms B and m are largely empirical. The
Lennard-Jones potential for van der Waals interactions is a “6-12” Mie
potential.) As m > n , at large separations the long-range attraction
dominates and at small separations the short-range repulsion dominates;
the full potential passes through a minimum characterizing a stable
equilibrium at an intermediate separation. The force, F M ( z ), and
stiffness, k M ( z ) , fields associated with the Mie potential are

d Φ M
d z

nA
z n +1

mB
z m +1

(4-20) F M =− =− +

and