Approximative “one particle” bridge function B
„1…
„r… for the theory
of simple fluids
Jean-Marc Bomont
a͒
and Jean-Louis Bretonnet
Laboratoire de Physique des Milieux Denses, Université Paul Verlaine, 1, Boulevard F. D. Arago,
57078 Metz Cedex 3, France
͑Received 18 January 2007; accepted 12 April 2007; published online 4 June 2007;
publisher error corrected 6 June 2007͒
New properties for the one particle bridge function B
͑1͒
͑r͒, which are necessary to the calculation of
the excess chemical potential

ex
, are derived for the hard sphere fluid. The method, which only
requires the knowledge of the bridge function B
͑2͒
͑r͒, is based on an investigation of the correlation
function dependence on the Kirkwood charging parameter. In this framework, the unavoidable
question of topological homotopy is addressed. As far as B
͑2͒
͑r͒ is considered as exact, this work
provides useful information on B
͑1͒
͑r͒ in the well identified dynamical regimes of the hard sphere
fluid. Signatures of the transitions between these regimes are identified on the trends of B
͑1͒
͑r͒. This
approach provides self-consistent results for

ex
that agree very well with simulation data. © 2007
American Institute of Physics. ͓DOI: 10.1063/1.2737046͔
I. INTRODUCTION
The primary goal of liquid theory is to predict the mac-
roscopic properties of classical fluids from the knowledge of
the interaction potential u͑r͒ between the constituent par-
ticles. For the past 50 years, integral equation ͑IE͒ theory for
homogeneous or inhomogeneous fluids has been applied
1,2
to
describe the local arrangement of particles through the pair
correlation function g͑r͒ and related thermodynamic proper-
ties. In the past, IE theory has been qualified
3
as a relatively
weak field method compared to other methods involving
analytical equation of state and exact simulation data, be-
cause it relies on the so-called bridge functions B
͑2͒
͑r͒ and
B
͑1͒
͑r͒͑a priori unknown correlation functions͒ that are in-
finite series of irreductible diagrams D
i
that, respectively,
read
4
B
͑2͒
͑r͒ =
2
1
2
D
1
͑r͒ +
3
͓D
2
͑r͒ + D
3
͑r͔͒ +
4
͓¯͔ + ¯ ͑1͒
and
B
͑1͒
͑r͒ =
1
3
B
͑2͒
͑r͒ +
3
͓
1
6
D
2
͑r͒ −
1
12
D
3
͑r͒
͔
+
4
͓¯͔ + ¯ ,
͑2͒
where
is the density of the fluid. Unfortunately, these series
cannot be summed up exactly. Contrary to B
͑1͒
͑r͒, the study
of B
͑2͒
͑r͒ and the development of new and better closure
relations for this function have been the subject of increasing
interest. As attested in the literature,
5
over the past two de-
cades several attempts have been made to improve upon
these closure relations and to extend the range of validity of
IE theory. Among these attempts, one finds closure relations
whose improvement consists either ͑i͒ in providing accurate
expressions
6–10
of B
͑2͒
͑r͒ or ͑ii͒ in calculating it in a self-
consistent manner.
11–14
From Eq. ͑2͒, it is obvious that the
accuracy of B
͑1͒
͑r͒ is conditioned by the accuracy of B
͑2͒
͑r͒.
Little is known about the one particle bridge function, except
that B
͑1͒
͑r͒Ϸ͑1/3͒B
͑2͒
͑r͒ for low densities,
15
but its expres-
sion remains unknown for higher densities. Moreover, no
method has been developed yet to extract B
͑1͒
͑r͒ from nu-
merical simulation.
To treat various problems for open systems such as
phase equilibria, obtaining the chemical potential

ex
is not
straightforward, especially at high densities, contrary to
usual correlation functions. It can be either obtained by com-
puter simulation or by analytical formulas. In the first case,
the test particle insertion method
16
based on the pair distri-
bution theorem
17
can be used. The chemical potential is then
calculated from the potential energy between a test particle
͑solute particle͒ and each of the particles in the fluid. In
contrast, this work focuses on the analytical approach, re-
quiring the knowledge of B
͑2͒
͑r͒ and B
͑1͒
͑r͒. Although recent
advances
18
have been achieved for B
͑2͒
͑r͒, neither study nor
improvements to B
͑1͒
͑r͒ have been made since the work of
Kiselyov and Martynov,
4
unfortunately. Nevertheless, B
͑1͒
͑r͒
͓that is not so common as the ordinary bridge function
B
͑2͒
͑r͔͒ still remains attractive because it is well suited to the
study of fluid entropic properties. So it is of interest to focus
on it. Particularly, this work extends the idea of Kirkwood
coupling parameter integration to obtain the chemical poten-
tial from correlation functions estimated by IE theories and
provides improvements to B
͑1͒
͑r͒.
The paper is organized as follows. In order to calculate
the excess chemical potential, an extension of the usual Kirk-
wood charging parameter integration is proposed in Sec. II.
An expression for B
͑1͒
͑r͒ is derived, allowing us to calculate

ex
in a self-consistent manner over the entire range of
densities. The results are displayed and discussed in Sec. III.
It is shown that calculating the chemical potential by analyti-
a͒
Author to whom correspondence should be addressed. Electronic mail:
bomont@univ-metz.fr
THE JOURNAL OF CHEMICAL PHYSICS 126, 214504 ͑2007͒
0021-9606/2007/126͑21͒/214504/7/$23.00 © 2007 American Institute of Physics126, 214504-1