By expanding the exponent in Eq.(14) and keeping only terms up to third order in concentration (or ), we can retrieve the perturbation treatment outlined in Ref.:
Now we can write the profile in a density expansion
where the superscript (1)-(3) denote the order in bulk density or . From the very low density consideration, we know that the lowest first order solution must be:
which, used in the evaluation of , gives:
since any adjustment to this will be an correction to the right hand side of Eq.(16). Therefore, Eq.(16) will give the function correct to the second order in :
Similarly, we could now plough this function back to obtain the correction to the function :
which allows the evaluation of third order term :
Thus we could, in theory, extend this perturbation treatment to any order in concentration . In practice, this becomes impractical at high orders as it will become faster to solve the Eq.(14) directly as we do in the next section. Therefore, we shall limit our discussion to these first three orders.
Setting in Eq.(16), integrating out the variable , we obtain:
where the functions are:
with three dimensionless geometric functions, defined as :
with . Here we have taken the function to only depend on but not , due to the uniaxial symmetry imparted by the parallel plate geometry. (We shall assume this symmetry is unbroken, as discussed further below.) For the special case clearly , and are numerically found to be and 0 respectively with very high precision. This leads to:
and the resulting pressure is indeed identical to the second virial result of Onsager [18, 21].
The depletion force for general separation h is now given by equations (3, 23) as:
The force between spheres of radius R follows by the Derjaguin approximation, Eq.(1); the interaction energy between two such spheres is then given by a further integration:
where , and where can be found by two numerical integrations (with respect to h) of the functions defined previously (and themselves evaluated numerically). Fig.2 shows the three (dimensionless) functions of separation . Together these allow the third order depletion potential curve to be constructed for any parameters desired, Eq.(32). Depletion potential and force curves for some typical parameters are presented in Figs.3,4.