Page 155 - Book Hosokawa Nanoparticle Technology Handbook

3.5 INTERACTIONS BETWEEN PARTICLES FUNDAMENTALS
Therefore, the limiting value of the external force F s However, it is widely used for the analysis of the force
is given by: because of the reasonable agreement with the actual
phenomena [18].
On the other hand, DMT theory [19], developed by
Ad
F (3.5.11) Derjaguin, Muller and Toporov, has no such an
s
16 z 2
unphysical situation, i.e. the attractive force does not
act in a line but a band. The contact area by DMT the-
As an external force F ( 0) is gradually applied to ory is reduced to zero as the separation force
separate the particle from the other particle or flat increases. Therefore the resulting pull-off force is the
surface, the radius of the contact area a decreases, same as that calculated by equation (3.5.1).
and the particle is separated at F F . The separa- Muller et al. [20, 21] introduced a Lennard–Jones
s
tion force F s gives the adhesive force, which is also potential (6–12 potential) into a model and showed
called ‘pull-off force’. The magnitude of F in equa- that the JKR theory is applicable to soft materials and
s
tion (3.5.11) is 3/4 of the magnitude of F in equa- the DMT theory is applicable to hard materials, and
v
tion (3.5.1). Substituting equation (3.5.11) into also referred to the intermediate materials. Table 3.5.3
equation (3.5.8) gives the radius of the contact area summarizes the difference of these theories [18].
a on the separation, i.e.: In the case that a small amount of liquid is held
s
between particles or between a particle and flat
3 kAd 2 surface, an attractive force arising from the liquid
a 3 (3.5.12) bridge acts between these bodies. The force is known
s
128 z 2
as “liquid bridge force” or “capillary force”. The
liquid bridge force is predominant in adhesive forces
One difficulty with the JKR theory is that the stress at between surfaces in gases. The formation of liquid
the edge of the contact area has an infinite value. bridge depends on the wettability of the surfaces.
Table 3.5.3
Theories for elastic deformation and adhesion between spherical particles [18].
Hertz JKR DMT Muller et al.

F
Force between
surfaces
D

Stress under
compressive
load

Shape under
compressive load

Shape under
zero load

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