Natural Colloids and Nanoparticles in Aquatic and Terrestrial Environments - Environmental and Human Health Impacts of Nanotechnology - page 125

forces between colloidal particles and the direct experimental measurements of the
force as a function of surface separation carried out for particles immersed in a
liquid phase.

4.5.1 DLVO Theory

The abbreviation DLVO refers to the names of Derjaguin, Landau, Verwey and
Overbeek. They conducted the fi rst successful attempts to describe colloidal stabil-
ity interactions in Russia (Derjaguin and Landau, 1941) and Netherlands (Verwey
and Overbeek, 1948). The DLVO theory is based on the assumption that forces
between surfaces or colloidal particles can be regarded as the sum of two forces.
These are the short range, attractive van der Waals and the long range, repulsive
electrical double layer forces. The interplay between these two forces has many
important consequences on colloid stability and aggregation:

VVV
T

=+ (4.2)

AR

where V T is the total interaction energy, V A is the attractive van der Waals energy
and V R is the repulsive double layer energy.

4.5.1.1 Van der Waals Forces

Van der Waals forces are always short range attractive forces and arise from spon-
taneous electrical and magnetic polarizations, giving a fl uctuating electromagnetic
fi eld within the media and in the gap between surfaces or particles (Elimelech
et al. , 1995a). There are two approaches to calculate the van der Waals forces:
microscopic and macroscpic.
In the microscopic approach, the interaction force is the pairwise summation of
all relevant interatomic interactions (Hamaker, 1937) and can be described in terms
of geometrical parameters and a constant A, the “ Hamaker constant ”. For two
spheres of equal radius, R, at a surface to surface separation distance, h, apart along
the centre to centre axis, the total interaction energy can be given as (Liang et al. ,
2007 ):

2 2 2

AR
h

2 R

hR

2 R

hR

4

Vh

( ) =−

+ + ln 1 − (4.3)

A

2 2 2

6 + 4 h

(

+ 2

) (

+ 2

)

In the case of the interaction between a sphere and a plane at a distance, h, the
interaction energy can given as:

( )

AR
h

R
hR

h
hR

Vh

( ) =−

+ + ln (4.4)

A

6 + 2 +

The Hamaker constant depends on material properties such as density and
polarizability. The effective Hamaker constant depends also on the dispersion
medium. It is generally of the order of magnitude 10 − 20 - 10 − 21 J (Elimelech et al. ,
1995a ).

( )

2

AA (4.5)

≈ − A

eff particle medium