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Next: Perturbation Treatment Up: Theory of the Depletion Previous: Introduction

Method

Our calculation method generalizes that of Ref.[7], which we follow closely. We first find the depletion force, tex2html_wrap_inline718 per unit area between two infinite parallel plates, at separation h; the force tex2html_wrap_inline722 between two large spheres of radius tex2html_wrap_inline704 then follows by the Derjaguin approximation [14]:

  equation36

from which the depletion potential tex2html_wrap_inline726 can be obtained:

  equation39

To find tex2html_wrap_inline728 we invoke the ``pressure sum rule" [15], which expresses p, the (osmotic) pressure exerted on a hard wall by a solution of thin rods, as tex2html_wrap_inline732 where tex2html_wrap_inline734 , calculated below, is the density of rod ends in contact with the wall. For hard rod particles, this relation is exact to all orders in the virial expansion; it is the analogue of Henderson's formula for the pressure in terms of the contact density for the hard sphere case [16]. It may be proved rigorously (for rods in vacuo) by elementary kinetic theory (see Appendix).

For two hard parallel plates immersed at separation h in a solution of depletant, the force per unit area on one of the plates is simply the differential pressure on its two sides:

  equation45

Here tex2html_wrap_inline738 , the density of rod ends in contact with a plate, separated by h from another, is given by:

  equation48

where tex2html_wrap_inline742 denotes the number per unit volume of midpoints of rods with orientation tex2html_wrap_inline744 , evaluated on the plane of contact at distance tex2html_wrap_inline746 from the boundary.

Now we present an argument that will enable us to obtain tex2html_wrap_inline738 : we use essentially the same idea as lies behind the potential distribution theorem due to Widom [17]. Suppose we put our system of rod particles and parallel plates in contact with a hypothetical reservoir, in which rods are exempt from mutual excluded-volume interactions. We first calculate the relation between the particle density in the reservoir tex2html_wrap_inline750 and the bulk density tex2html_wrap_inline700 . Imagine an infinitesimal volume centered at position r in the bulk solution, in which we consider placing (the midpoint of) a rod of orientation tex2html_wrap_inline756 . This would exclude from a volume tex2html_wrap_inline758 the midpoint of another rod with relative orientation tex2html_wrap_inline760 defined with respect to coordinate axes in the first rod. We define our angular coordinates so that tex2html_wrap_inline762 (hence tex2html_wrap_inline764 ); the number density of rods in the reservoir with orientation in the range ( tex2html_wrap_inline766 ) is then tex2html_wrap_inline768 .

The presence of our proposed rod at r is permitted only if the volume tex2html_wrap_inline772 is empty for all orientations tex2html_wrap_inline774 ; this probability is simply the product of the probabilities of all sub-volumes tex2html_wrap_inline776 being empty:

  equation58

which can be multiplied out to give terms of different orders in concentration, cf. Ref.[5]:

  equation65

here tex2html_wrap_inline778 denotes the volume excluded by rod 1 that is otherwise available to rod 3, given the presence of (hence the exclusion due to) the rod 2. Now the picture of rod 3 touching and revolving around rod 2, which touchs rod 1, allows us to deduce that the chance of rod 1 and 2 excluding rod 3 simultaneously is smaller than that of rod 1 alone excluding 3 by roughly a factor of D/L, which is very small for thin rods. In other words, very thin rods are unlikely to form any loops; or mathematically, m mutually touching thin rods are likely to have just m-1 touching points. So we make the mean field approximation of tex2html_wrap_inline786 , which, in the bulk, is essentially the Onsager approximation, valid for tex2html_wrap_inline680 . (Notice that although, as we confirm below, the Onsager theory leads to truncation of the virial expansion for the pressure at second order [18], the same does not apply to the depletion force.)

In the limit of large aspect ratio, eq.(6) then becomes:

  equation79

equation84

(This is, of course, independent of the orientation tex2html_wrap_inline790 of the rod we are considering at r.) To find the density of rod centres at r (which must equal tex2html_wrap_inline700 , since r is an arbitrary point in the bulk) we argue that, if a rod is allowed there (probability tex2html_wrap_inline800 ) the density is tex2html_wrap_inline750 ; otherwise it is zero. Hence tex2html_wrap_inline804 where tex2html_wrap_inline708 is the reduced density. Inverting this relation gives:

  equation92

This completes our derivation of the relation between tex2html_wrap_inline750 and tex2html_wrap_inline700 . Of course, equating the chemical potential in the bulk (with mutual excluded volume interaction between two rods) tex2html_wrap_inline812 for tex2html_wrap_inline814 , with that in the reservoir tex2html_wrap_inline816 (without such interaction), gives the same result. However, the argument given above is directly generalizable to the depletion problem as we now show.

For use in Eq.(3), we need to calculate tex2html_wrap_inline738 , the end density contacting a confining plate separated by h from another. Consider a rod touching the plate with orientation tex2html_wrap_inline790 and a second rod touching this rod, with relative orientation tex2html_wrap_inline774 . (See Fig.1.) The height from the wall of the two ends of the first rod are respectively tex2html_wrap_inline826 and tex2html_wrap_inline828 ; those of the second rod, tex2html_wrap_inline830 and tex2html_wrap_inline832 ; tex2html_wrap_inline834 and tex2html_wrap_inline836 locate the position of the contact point of the rods as shown. The tex2html_wrap_inline838 variables can be expressed in terms of the others (using elementary geometry) as tex2html_wrap_inline840 , tex2html_wrap_inline842 and tex2html_wrap_inline844 . The infinitesimal volume excluded by the first rod to the (midpoint of the) second rod, both at fixed orientations tex2html_wrap_inline846 and with their contact point specified in the range ( tex2html_wrap_inline834 to tex2html_wrap_inline850 , tex2html_wrap_inline836 to tex2html_wrap_inline854 ), is [18]:

  equation101

We can now write tex2html_wrap_inline856 , the density of rods with orientation tex2html_wrap_inline790 at distance tex2html_wrap_inline826 away from the wall, as:

  equation105

with tex2html_wrap_inline862 where H's, the Heaviside unit step functions, are needed to eliminate all configurations forbidden by the hard interaction with the confining walls. tex2html_wrap_inline866 is the prior probability that the volume excluded by such a rod, at a distance tex2html_wrap_inline826 away from the wall, contains no other rods. So Eq.(11) states that all positions and orientations that are allowed to a rod are equally populated with an occupation of tex2html_wrap_inline870 , which is the same rod density per unit solid angle as in the reservoir; the probability of a chosen position and orientation being allowed is simply tex2html_wrap_inline872 .

As before (cf.Eqs.(5-7)), the probability tex2html_wrap_inline866 can be expressed as the product of the probabilities of all sub-volumes tex2html_wrap_inline876 being empty:

  equation114

The orientation tex2html_wrap_inline878 denotes that of the second rod in a coordinate system relative to the plates rather than that of the first rod (with respect to which tex2html_wrap_inline774 is defined). Making the same approximation of ignoring loop configurations as earlier, tex2html_wrap_inline866 is then given by:

eqnarray119

  equation122

This, together with Eqs.(9,11), leads to the following self-consistent integral equation for the density distribution function tex2html_wrap_inline884 :

  equation127

with tex2html_wrap_inline886 defined by Eq. (13) and (10) as:

  equation134

Notice the density tex2html_wrap_inline856 is exactly the same if we reverse the direction of rod 1 (this swaps the value of tex2html_wrap_inline826 and tex2html_wrap_inline828 and change tex2html_wrap_inline756 to tex2html_wrap_inline896 ). We can therefore choose tex2html_wrap_inline898 in the calculation of the function tex2html_wrap_inline856 without loss of generality. Setting tex2html_wrap_inline902 , we have, by definition, the contact density tex2html_wrap_inline904 , which will lead to the depletion force via Eq.(4). Note that, if we write down the grand thermodynamic potential of the system as a functional of the density distribution tex2html_wrap_inline884 , then the minimization of this potential with respect to the function tex2html_wrap_inline884 gives rise to exactly the same self-consistent integral equation (14). This was pointed out (for a single plate) in Ref.[19], though the resulting equation was not solved there. An approximate solution for a single plate is provided in Ref.[20]. In this paper, we numerically solve the equation with high precision for the geometry of two parallel plates.

The generalization of Eq.(14) to include an arbitrary enthalpic interaction between rods is rather straightforward: we simply need to insert a term with an appropriate Boltzmann factor in Eq.(11). In this way we could, in principle, treat cases with any pairwise interaction potential between rods; however we do not pursue this here.


next up previous
Next: Perturbation Treatment Up: Theory of the Depletion Previous: Introduction

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