Page 227 - Introduction to Colloid and Surface Chemistry

216 Colloid stability

With the exception of highly polar materials, London dispersion
forces account for nearly all of the van der Waals attraction which is
operative. The London attractive energy between two molecules is
very short-range, varying inversely with the sixth power of the
intermolecular distance. For an assembly of molecules, dispersion
forces are, to a first approximation, additive and the van der Waals
interaction energy between two particles can be computed by
summing the attractions between all interparticle molecule pairs.
The results of such summations predict that the London interaction
energy between collections of molecules (e.g. between colloidal
particles) decays much less rapidly than that between individual
molecules.
For the case of two spherical particles of radii a t and a 2, separated
in vacua by a shortest distance //, Hamaker 197 derived the following
expression for the London dispersion interaction energy, V A:

2
-_A ( X +xy + x }
A 2 'I *»»! sj I (8.8)
12 x + xy 4- x x + xy + x + y {x + xy + x + y )
where
tj
x — an(j y = a { / # 2
<*\+<*2
A is a constant, known as the Hamaker constant.
For equal spheres, with a 1 = a 2 = fl(i.e.jc = HI2a), equation (8.8)
takes the form

A_ 1 1
12 (8 9)
'
If a small interparticle separation is assumed, such that H <g a (i.e. x
<t 1), this rather awkward equation simplifies to

12 2x 12H
Values of V A calculated from this equation will be overestimated on
account of the above approximation.
Values of V A calculated from any of the above equations will be
overestimated at large distances (// > c. 10 nm) owing to a neglect of
the finite time required for propagation of electromagnetic radiation