An exactly solvable model for a colloid–polymer ... - bristol.ac.uk

290 J.M. Brader, R. Evans / Physica A 306 (2002) 287 – 300
truncation at the pair term is exact only for size ratio q¡0:1547, the geometry in 1D
is much more forgiving andtruncation at the pair potential is exact for q ≡ a=b ¡ 1.
Once the problem has been reduced to one of calculating the partition function of a
one-component system with nearest-neighbour interactions, which is the case for q¡1,
standardtechniques can be appliedwhich make the solution possible. Quite how one
gives a physical interpretation to the 1D-pair ‘depletion’ potential is not entirely clear;
there is no longer an intuitive picture of the polymer being pushedout from the
region between the colloids. The eective pair potential must now be regarded as one
of combinatorial origin which simply reects the fact that more weight is given to
particular congurations of the colloids as a result of the number of ways in which
the polymer can be positionedaroundthem. Regardless of interpretation, the mapping
to an eective one-component system with only zero, one andtwo-body potentials is
formally exact for q¡1 andmay be regardedas a useful trick for calculation.
In order to eect the integrating out we treat a homogeneous mixture in the semigrand-canonical
(N c ;L;z p ;T) ensemble and write the corresponding thermodynamic
potential F in one-component form
exp[ − F]= 1 ∫ ∫
dX 1 ··· dX
N c ! Nc
Nc exp[ − (H cc + )] ; (3)
where all integrals are over the colloidcoordinates, X i , on the line of length L. Following
the treatment of the model in 3D [7] the grand potential of ideal polymer coils
in the external eldof a xedconguration of N c colloids with coordinates {X i }, with
i =1; 2;:::;N c can now be expanded in terms of the colloid–polymer Mayer f function
f ij ≡ f(X i ;x j ) = exp[ − cp (X i − x j )] − 1:
∫
∑N c ∫
∑N c ∫
− = z p dx j + z p dx j f ij + z p dx j f ij f kj + ··· ; (4)
i=1
where =(k B T) −1 and z p = −1
p exp( p ) is the fugacity of the reservoir of polymer.
is the thermal de Broglie wavelength of species with = c; p. The expansion
can be written as a sum of contributions n which are classiedaccording to the
number n =0; 1; 2;:::;N c of colloids which interact simultaneously with the ‘sea’ of
ideal polymer, i.e.,
− = − 0 − 1 − 2 −···
i¡k
∑N c
= z p L − z p N c (a + b) − (X ij )+··· : (5)
i¡j
The zero and one-body terms are independent of the colloid rod coordinates and have
the following interpretation: 0 is the grand potential of ideal polymer rods in a length
L at fugacity z p since z p = Pp r = r p, where Pp r is the pressure and r p the number
density of the ideal polymer in the reservoir, and 1 =N c is the grandpotential
requiredto insert a single colloidin the sea of ideal polymer. The thirdterm is the
sum of pairwise potentials with the eective pair potential (X ) given by a simple