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Next: Numerical Solutions of the Up: Theory of the Depletion Previous: Method

Perturbation Treatment

By expanding the exponent in Eq.(14) and keeping only terms up to third order in concentration tex2html_wrap_inline700 (or tex2html_wrap_inline714 ), we can retrieve the perturbation treatment outlined in Ref.[7]:

  equation148

Now we can write the profile tex2html_wrap_inline856 in a density expansion

equation155

where the superscript (1)-(3) denote the order in bulk density tex2html_wrap_inline700 or tex2html_wrap_inline714 . From the very low density consideration, we know that the lowest first order solution must be:

equation160

which, used in the evaluation of tex2html_wrap_inline886 , gives:

equation164

since any tex2html_wrap_inline922 adjustment to this will be an tex2html_wrap_inline924 correction to the right hand side of Eq.(16). Therefore, Eq.(16) will give the function tex2html_wrap_inline884 correct to the second order in tex2html_wrap_inline700 :

equation172

Similarly, we could now plough this function back to obtain the tex2html_wrap_inline922 correction to the function tex2html_wrap_inline932 :

equation178

which allows the evaluation of third order term tex2html_wrap_inline934 :

equation185

Thus we could, in theory, extend this perturbation treatment to any order in concentration tex2html_wrap_inline700 . 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 tex2html_wrap_inline902 in Eq.(16), integrating out the variable tex2html_wrap_inline790 , we obtain:

  equation195

where the functions tex2html_wrap_inline942 are:

equation205

equation208

equation212

with tex2html_wrap_inline944 three dimensionless geometric functions, defined as [7]:

equation217

equation221

eqnarray229

equation239

with tex2html_wrap_inline946 . Here we have taken the function tex2html_wrap_inline948 to only depend on tex2html_wrap_inline950 but not tex2html_wrap_inline952 , 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 tex2html_wrap_inline954 clearly tex2html_wrap_inline956 , and tex2html_wrap_inline958 are numerically found to be tex2html_wrap_inline960 and 0 respectively with very high precision. This leads to:

equation256

and the resulting pressure tex2html_wrap_inline964 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:

  equation262

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:

  equation268

where tex2html_wrap_inline970 , and where tex2html_wrap_inline972 can be found by two numerical integrations (with respect to h) of the functions tex2html_wrap_inline976 defined previously (and themselves evaluated numerically). Fig.2 shows the three (dimensionless) functions of separation tex2html_wrap_inline978 . 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.


next up previous
Next: Numerical Solutions of the Up: Theory of the Depletion Previous: Method

Yong Mao
Tue Sep 17 16:40:38 BST 1996