section 1.5
As we wish to use the concept of the potential of mean force later on for the description of polymer colloid mixtures it is instructive to derive it first for colloidal particles dispersed in a solvent. This treatment is largely taken from Lekkerkerker et al [17].
We consider a suspension of N colloidal particles dispersed in a
solvent in a volume V which is in equilibrium with a large reservoir solely
containing the solvent. In the reservoir the solvent has a (hydrostatic) pressure 
and a fugacity 
where
is the chemical potential of the solvent and 
is
the inverse temperature.
Since we are imposing the fugacity of the solvent, the colloidal suspension can be
regarded an open system and the proper partition function for the
system is then
where 
is the canonical configurational partition
function for N colloidal particles and m solvent molecules at constant
V and T
where 
is the interaction potential of the configuration
given by the positions r of the N colloids and the m solvent molecules.
We define the potential of mean force 
by writing the grand
canonical partition function as
Combination of last three equation shows that the potential of mean force is given by
All possible solvent configurations contribute to 
. It
is the average potential the colloidal particles feel due to all
interactions of and with the solvent. If one takes the derivative of
the 
with respect to a particle's position 
it
becomes clear why the 
is called the potential of mean force.
This is the average force on the colloidal particle due to the collisions with the solvent particles. One can connect the partition function to the thermodynamic grand potential by the familiar equation
From this potential all the thermodynamical quantities can be derived. For example, the osmotic pressure, which is the difference between the pressure in the suspension and the reservoir pressure is
McMillan and Mayer also showed how the potential of mean force approach can be used to obtain the virial expansion of the osmotic pressure [16]. Because of the far-reaching analogy, it is possible to apply theories that were designed for simple liquids to the phase behavior of colloidal dispersions, and what is more important for this thesis, it is possible to conduct computer simulations of colloidal suspensions, as if the colloids were large atoms interacting through the potential of mean force. These kind of simulations can be used to test the assumptions made in the theories, they can provide reference data that can be compared both to experiments and theory, and they may provide new insights in the usually complicated phase behavior of colloidal dispersions.
section 1.5