# Relationship between thermodynamic perturbation and scaled particle theories
for fused dimers fluids

O. Bernard

(Received July 6, 2021, in final form August 2, 2021)

###### Abstract

Various approaches are reviewed that use scaled particle theories to describe
dumbbell fluids made of tangent or overlapped hard spheres. Expressions
encountered in the literature are written in a form similar to that presented
in the thermodynamic perturbation theory introduced by Wertheim for chains and
developed in statistical associating fluid theory (SAFT). Analogies and
differences observed in these two types of theoretical descriptions allow one
to propose alternative theoretical expressions to describe dumbbell fluids
with overlapping spheres.

Key words: hard dumbbells fluid, scaled particle theory, thermodynamic
perturbation theory

###### Abstract

Ó ðîáîò çðîáëåíî îãëÿä ðçíèõ ïäõîäâ äëÿ ïîáóäîâè òåîð¿ ìàñøòàáîâàíî¿ àñòèíêè ç
ìåòîþ îïèñó ïëèíâ ãàíòåëåïîäáíèõ àñòèíîê, óòâîðåíèõ ç òâåðäèõ ñôåð, ùî
ïåðåêðèâàþòüñÿ àáî òîðêàþòüñÿ îäíà îäíî¿. Íàäàíî íøå ïðåäñòàâëåííÿ âäîìèì ç
ëòåðàòóðè âèðàçàì ó ìàòåìàòèíé ôîðì, ïîäáíé äî ò¿, ÿêà âäîìà ç òåðìîäèíàìíî¿
òåîð¿ çáóðåíü, îòðèìàíî¿ Âåðòãàéìîì äëÿ ëàíöþæêâ óçàãàëüíåíî¿ ó ñòàòèñòèíé
òåîð¿ àñîöàòèâíèõ ðäèí. Àíàëîã¿ òà âäìííîñò ìæ öèìè äâîìà òåîðåòèíèìè ïäõîäàìè
äîçâîëèëè çàïðîïîíóâàòè àëüòåðíàòèâí âèðàçè äëÿ îïèñó ïëèíâ ãàíòåëåïîäáíèõ
àñòèíîê ç ñôåðàìè, ùî ïåðåêðèâàþòüñÿ.

Ключов слова: ïëèíè ãàíòåëåïîäáíèõ àñòèíîê ç òâåðäèìè ñåðöåâèíàìè, òåîðÿ
ìàñøòàáîâàíî¿ àñòèíêè, òåðìîäèíàìíà òåîðÿ çáóðåíü

##  1 Introduction

The description of thermodynamic properties of media containing non-spherical
objects is still important in many fields of fundamental or applied science.
It still presents difficulties, in particular in the description of phase
transitions and the various types of phases generated according to the
geometry of the objects. Convex objects were described previously using
extensions of scaled particle theory (SPT), which was initially introduced for
spheres [1]. These extensions, sometimes found using virial developments, have
given rise to various conformal theories [2, 3, 4, 5]. They have led in
particular to free volume theories [6] used to describe phase transitions, as
well as to some approaches of inhomogeneous fluids [7].

Furthermore, the chains of spheres linked together could also be described
within the thermodynamic perturbation theory (TPT) proposed by Wertheim [8,
9], which led to the statistical associated fluid theory (SAFT) [10] and its
extensions currently used to describe polymer fluids. This approach separates
the thermodynamic quantities considered (free energy or pressure) into a
reference contribution, related to the subunits in absence of connection, and
into a contribution induced by the connectivity between these subunits
necessary to form the chains. In addition, the spheres constituting the chains
are most often constrained to be tangent because the description of spheres
with overlap proved to be more difficult. However, in the extensions of the
TPT used to describe polymers, it is often useful to be able to describe
various subunits within chains as groups of spheres with overlap. Better
consideration of such subunits with the scaled particle theory could be useful
to improve the application of TPT to polymers.

On the other hand, chains of spheres cannot in principle be described with
scaled particle theory or its variants because these objects are non-convex.
However, the theoretical SPT models of spherocylinders have been adapted to
describe linear chains. In particular, this has provided a good description of
dumbbell fluids formed of two overlapping spheres. This is probably due to the
similarity in shape between these dumbbells and slightly elongated
spherocylinders. Another conformal theory, called scaled field particle (SFP)
theory and having similarities with SPT, has also been used to describe dimer
fluids. Then, in this work, these two versions of conformal theories were
reviewed and applied to describe thermodynamics of dimer fluids. This paper is
dedicated to Yu. V. Kalyuzhnyi, who has made significant contributions in many
different research areas related to statistical mechanics, especially in
integral equation theories.

The reference contribution in TPT is that of dissociated hard spheres. Then,
in the next section, the description of hard spheres and its use in TPT theory
is presented. A theoretical expression of the dimer pressure obtained using
integral equations is also given to later highlight links with the other
approaches presented. In the following section, various conformal or free
volume theories inspired by SPT are presented. Conformal theories describe the
thermodynamic quantities of fluids by using geometric parameters
characteristic of the bodies considered, such as their volumes, surface areas
and mean radii. The two types of conformal theories studied do not use exactly
the same set of parameters. In addition, for each type of conformal theory,
from the original result a corrective term has been added to obtain an
improved version. The corrective term depends also on the set of geometric
parameters. Since the two types of conformal theories do not use exactly the
same geometric parameters, the corrective term is different. So, for each
theory we have presented both the original version and the improved version to
be able to better study the influence of the corrective term.

At small separation between the spheres constituting the dimer, when the
larger sphere tends to completely overlap the smaller one, it is known that
the two types of conformal theories tend towards the same result. The
difference between the results given by the two approaches increases with the
elongation between the components of the dimer. Then, in the next section, in
order to highlight the differences between the two conformal approaches, these
theories were first applied to dimers made of tangent spheres. For this
particular configuration, all of the above-mentioned geometric parameters can
be expressed from the diameters of the hard spheres. Then, the thermodynamic
quantities coming from the various conformal theories, were rewritten as a
function of the diameters of the hard spheres instead of the aforementioned
geometric parameters. In particular, when the hard spheres have all the same
diameter, the pressure of the dimer fluid is described simply as a function of
the volume fraction. Moreover, these expressions were split into two part
according to the same dividing found in the TPT, namely a reference term of
dissociated hard spheres and a connectivity term due to the links between
spheres. This made it possible to better analyze the reference term and the
connectivity term in each of the theories. This made it possible to explain
the differences observed in the description of the thermodynamics deduced from
simulations.

After this analysis of previous works, in the following section alternative
relationships have been established to describe first dumbbells made of
tangent spheres, and then those with overlap. A comparison was given with the
results of simulations. In a last part we conclude this work and present the
prospects that it suggests.

##  2 From integral equations to perturbation theory for hard spheres and
tangent hard sphere dimers

###  2.1 Integral equation for dissociated hard spheres fluid

A proper account of the excluded volume of various species is essential to
describe thermodynamic properties such as the pressure $P$ of liquids and
concentrated solutions [11, 12, 13, 14]. Since the sphere is the simplest
object taking into account the repulsion at short distances, all the
constituents were first considered as hard spheres. In addition, molecules of
complex shape can be represented using collections of spheres linked together.
Therefore, it seems essential to first properly describe the properties of
these subunits constituting these assemblies. Numerical simulations make it
possible to precisely determine the structural and thermodynamic properties of
model systems. However, their implementation is cumbersome and does not lead
to explicit expressions of the studied quantities. As an alternative, studies
of the structural and thermodynamic properties of hard spheres were undertaken
using the Ornstein-Zernike (OZ) integral equations [15]. Among various closure
relations useful in solving these integral equations, the Percus Yevick (PY)
approximation [16] has proven particularly suitable to describe hard spheres.
In addition, it provides analytical expressions of some structural and
thermodynamic quantities. Notably, the PY approximation allows one to access
the radial distribution functions $g_{ij}(r)$ between all pairs of components
$i$ and $j$ present in the liquid [17, 18, 19]. Various thermodynamic
quantities can be calculated from the radial distribution functions
$g_{ij}(r)$ [20]. In particular, from the relation defining the
compressibility as a function of these $g_{ij}(r)$, an analytic expression of
the pressure has been deduced by integration [21, 22, 23]

$\beta P_{{\mbox{\sc\scriptsize
py}}}^{c}=\frac{6}{\piup}\left[\frac{\zeta_{0}}{1-\zeta_{3}}+\frac{3\,\zeta_{1}\zeta_{2}}{\left(1-\zeta_{3}\right)^{2}}+\frac{3\,\zeta_{2}^{3}}{\left(1-\zeta_{3}\right)^{3}}\right],$
(2.1)

where $\beta$ is equal to $1/k_{\rm B}T$ with $k_{\rm B}$ the Boltzmann
constant and $T$ the temperature. The terms $\zeta_{n}$ are given by

$\zeta_{n}=\frac{\piup}{6}\sum_{k}\rho_{k}\sigma_{k}^{n},$ (2.2)

where $\rho_{k}$ and $\sigma_{k}$ are the number density and the diameter of
component $k$, respectively. Subscript “PY”, under the symbol $P$, means that
the pressure was obtained in the PY approximation, and the superscript “$c$”
has been added to remind that this expression comes from the compressibility
relation. The virial relation also provides an analytical expression of the
pressure $P_{{\mbox{\sc\scriptsize py}}}^{v}$, which is however less precise
in comparison with the results deduced from numerical simulations. An
alternative expression of the pressure somewhat more accurate than that given
by equation (2.1), has been deduced by a combinaison of the pressure deduced
from the virial and the compressibility relations, first in the case of a
single component [24] and then for mixtures of several constituents [25, 26] :

$\beta P_{{\mbox{\sc\scriptsize
bm}}}=\frac{6}{\piup}\left[\frac{\zeta_{0}}{1-\zeta_{3}}+\frac{3\,\zeta_{1}\zeta_{2}}{\left(1-\zeta_{3}\right)^{2}}+\frac{3\,\zeta_{2}^{3}-\zeta_{2}^{3}\zeta_{3}}{\left(1-\zeta_{3}\right)^{3}}\right].$
(2.3)

The subscript “BM” means Boublík-Mansoori, to recall the initials of the two
first authors of the articles in which were established the corresponding
expression for mixtures of several constituents [25, 26]. The “BMCSL”
subscript (or superscript) is often used to denote this equation, to recall
the initials of all the authors who established it for mixture of components
[25, 26]. Here, we limited the number of letters in subscript to alleviate the
notation. Expressions giving an even better description of the pressure
obtained by simulation were proposed later. However, in this article we limit
ourselves to this level of refinement, compared to the expression (2.1)
determined using the PY approximation. Indeed, the passage of equation (2.1)
to (2.3) is simple enough to be used later in the context of expressions
describing non-spherical objects. Moreover, equation (2.3) is commonly used
both in perturbation theories and in applications to describe experimental
systems. Thereafter, from these expressions of the pressure, other quantities,
such as the free energy $A$ or the chemical potentials $\mu_{j}$ of various
$j$ constituents, can be deduced by integration or differentiation. They will
also be useful when applying perturbation theories which take the fluid of
dissociated hard spheres as a reference.

###  2.2 Integral equations for dimers made of tangent hard spheres

Model molecules, made of assemblies of hard spheres were considered. Dimers
made of hard tangent spheres are the simplest of these aggregates. In order to
describe dimers with integral equations, attractive sites were added on their
surfaces. These sites induce the formation of bonds between the spheres. If
there is only one site per sphere, only dimers can be formed. Since the sites
occupying a very small fraction of the surface of the spheres, the attractive
interactions are highly directional. Dimers are formed only when two
contacting spheres have their sites oriented towards each other. In the other
configurations, particles interact only with the purely radial potential (hard
spheres potential). The consideration of these highly directional potentials
permits using various integral equations to describe the structure and
interactions of these hard sphere assemblies [27, 28, 8, 9, 29, 30, 31, 32,
33]. In particular, Wertheim’s OZ equations (WOZ) [8, 9, 29, 30, 31, 32] were
applied to the description of dimers formed by tangent spheres [31, 34, 35,
36]. As an extension of the PY relation, a polymer Percus Yevick (PPY) closure
relation was used. Moreover, the density $\rho_{i}$ of various $i$ species, is
decomposed into the density of free $i$ species $\rho_{i}^{0}$, and dimers
$\rho_{{\mbox{\sc\footnotesize d}}}$. Considering, for simplicity only, two
species $1$ and $2$ which can link together to form dimers, we have the mass
conservation relations, for $i=1$ or $2$,

$\rho_{i}=\rho^{0}_{i}+\rho_{{\mbox{\sc\footnotesize d}}},$ (2.4)

with

$\rho_{{\mbox{\sc\footnotesize
d}}}=\rho_{1}^{0}\;\rho_{2}^{0}\;K^{0}\;g^{00}_{12}(\sigma_{12}),$ (2.5)

where $K^{0}$ is an association constant (which can be expressed as an
integral dependent on the attractive potential [31]), and
$g^{00}_{12}(\sigma_{12})$ is the pair distribution between unassociated
species $1$ and $2$ at a contact distance
$\sigma_{12}=(\sigma_{1}+\sigma_{2})/2$. The solution of the WOZ equations
allows one to accurately describe the interactions between these constituents.
As in the case of the dissociated hard spheres, analytic expressions of the
pressure, deduced from the virial and compressibility relations, were
obtained. Here again, the expression deduced from the compressibility is the
most efficient [31, 36].

$\displaystyle\beta P_{\mbox{\rm\tiny PPY}}^{c}$ $\displaystyle=\beta
P_{\mbox{\rm\tiny
PY}}^{c}-\rho^{0}_{1}\rho^{0}_{2}\;K^{0}\Bigl{[}g_{12}^{00}(\sigma_{12})\Bigr{]}^{2}$
(2.6a) $\displaystyle=\beta P_{\mbox{\rm\tiny
PY}}^{c}-\rho_{\mbox{\sc\footnotesize d}}\;g_{12}^{00}(\sigma_{12}).$ (2.6b)

The definition of $\rho_{{\mbox{\sc\footnotesize d}}}$, given by the equation
(2.5), has been used to go from equation (2.6a) to (2.6b). Furthermore, in the
PPY approximation, it was found that $g_{12}^{00}(\sigma_{12})$ is equal to
$g_{12}^{\mbox{\rm\tiny PY}}(\sigma_{12})$ [31, 35], which is the
corresponding pair distribution function at contact, obtained in the PY
approximation for hard spheres without attractive sites. Moreover,
$g_{12}^{\mbox{\rm\tiny PY}}(\sigma_{12})$ is given by [19]

$g_{12}^{\mbox{\rm\tiny
PY}}(\sigma_{12})=\frac{1}{1-\zeta_{3}}+\frac{3\,\zeta_{2}}{\left(1-\zeta_{3}\right)^{2}}\,\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}.$
(2.7)

Equation (2.6b) was also established within various integral equations devoted
to the description of properties of chain fluids [37, 38, 39, 40]. As a
generalisation in equation (2.6b), for chains of density $\rho_{c}$ containing
$m$ subunits, $\rho_{\mbox{\sc\footnotesize d}}$ is replaced by
$(m-1)\,\rho_{c}$. In the next section, when comparing the various approaches,
we come back to the explicit expression (2.6b) derived from the
compressibility relation within the PPY approximation.

###  2.3 Thermodynamic perturbation theory for dimers made of tangent hard
spheres

As an alternative, a thermodynamic perturbation theory (TPT) was also
developed to take into account the effect of the bonds formed between the
particles on the thermodynamic properties of the fluid [8, 9, 29, 30, 31, 32].
The generalization and application of this theoretical path in the SAFT led to
one of the most accurate equations of state for tangent hard sphere chains
[10, 41, 42, 43]. These approaches describe the thermodynamics in terms of the
reference fluid distribution functions regardless of the amount of bonding.
The residual Helmholtz free energy $A$ can be separated into two terms as
follows:

$\frac{\beta A_{{\mbox{\rm\tiny TPT}}}}{V}=\frac{\beta A^{\rm
ref}}{V}+\frac{\beta A^{\rm bond}}{V},$ (2.8)

where $A^{\rm ref}$ is the reference Helmholtz free energy contribution from
the dissociated free monomers, and $A^{\rm bond}$ is a contribution related to
the formation of bonds. In the 1st order perturbation theory, the term $A^{\rm
bond}$ can also be evaluated from the properties of the reference system
without association.

$\frac{\beta A_{{\mbox{\rm\tiny TPT}}}}{V}=\frac{\beta A^{\rm
ref}}{V}-(m-1)\,\rho_{c}\ln{g^{\rm ref}(\sigma_{12})}.$ (2.9)

In the second term of equation (2.9), related to the connectivity within the
chains, $m$ is the number of monomers, $\rho_{c}$ the number density of the
chain and $g^{\rm ref}(\sigma_{12})$ the pair distribution function between
two monomers at a distance of contact $\sigma_{12}$ in the reference fluid
without association. For simplicity, it is assumed here that all the monomers
are separated by the same contact distance in the chains. If the reference
fluid consists of hard spheres, the reference free energy can be described
using either the PY expression,

$\frac{\beta A^{c}_{{\mbox{\sc\scriptsize
py}}}}{V}=\frac{6}{\piup}\left[-\zeta_{0}\ln{(1-\zeta_{3})}+\frac{3\,\zeta_{1}\zeta_{2}}{1-\zeta_{3}}+\frac{3\,\zeta_{2}^{3}}{2\left(1-\zeta_{3}\right)^{2}}\right],$
(2.10)

corresponding to the expression of the pressure given by equation (2.1) or the
improved expression associated with the pressure given by (2.3),

$\frac{\beta A_{{\mbox{\sc\scriptsize
bm}}}}{V}=\frac{6}{\piup}\left[\left(\frac{\zeta_{2}^{3}}{\zeta_{3}^{2}}-\zeta_{0}\right)\ln{\left(1-\zeta_{3}\right)}+\frac{3\,\zeta_{1}\zeta_{2}}{1-\zeta_{3}}+\frac{\zeta_{2}^{3}}{\zeta_{3}\left(1-\zeta_{3}\right)^{2}}\right].$
(2.11)

This latter expression is the one generally used because it leads to the
results closest to those given by the simulations, in comparison with those
given by equation (2.10). However, we have also presented expression (2.10),
deduced from the PY theory, because it can also be recovered using other
theories presented below in this article. This property will be used later to
develop new improved expressions. In the second term of equation (2.9), the
pair distribution function at contact can be described either using an
expression deduced from the OZ equations, or using an expression corresponding
to the improved expression of pressure (2.3). The pressure $P_{{\mbox{\rm\tiny
TPT}}}=P^{\rm ref}+P^{\rm bond}$ can be deduced from equation (2.9) by
differentiation

$\beta P_{{\mbox{\rm\tiny TPT}}}=\beta P^{\rm
ref}-(m-1)\,\rho_{c}\left[1+\sum_{k}\rho_{k}\frac{\partial\ln{g^{\rm
ref}(\sigma_{12})}}{\partial\rho_{k}}\right].$ (2.12)

In the same way, when the reference fluid consists of hard spheres, the
reference pressure $P^{\rm ref}$ can be described either using the PY
expression (2.1), or the improved expression (2.3). It is recalled that in the
case where only dimers are formed, $m=2$, the concentration of chains
$\rho_{c}$ denotes the concentration of dimers $\rho_{{\mbox{\sc\footnotesize
d}}}$. Then, in equations (2.9) and (2.12), $(m-1)\rho_{c}$ can be simply
replaced by $\rho_{{\mbox{\sc\footnotesize d}}}$.

Attempts have been made to extend TPT to fluids of fused sphere chains [44,
45, 46, 47]. In equations (2.9) and (2.12), the distribution function $g^{\rm
ref}(\sigma_{12})$ is replaced by a cavity correlation function $y(l)$ where
$l\leqslant\sigma_{12}$, is the distance between the spheres linked together.
Moreover, an effective number of tangent monomers was used by requiring that
the second virial coefficient of the chain of tangent spheres should be equal
to that of the chain of fused spheres. In the case of fused spheres, this
second virial coefficient can be expressed in terms of parameters of non-
sphericity which account for the molecular shape of the chain. This approach
is based on the contribution of various conformal theories describing the
properties of objects of various shapes.

##  3 Conformal or free volume theories

One of the goals of conformal or free volume theories is to attempt to
describe individual chemical potentials and an equation of state of convex
body in terms of a relevant set of parameters such as the volume, surface area
and the mean radius of curvature which characterize the shape of all the
components. This approach was first performed in the scaled particle theory
(SPT).

###  3.1 Scaled particle theory

The SPT was first developed to describe the hard sphere fluids [1, 48].
Analytical expressions of the radial distribution functions at contact and of
thermodynamic quantities were obtained. It is noted, that the resulting
expression of the pressure is identical to that given in equation (2.1),
derived from the compressibility relation and the PY approximation. The SPT
was extended to the description of convex objects other than hard spheres,
such as ellipsoids and spherocylinders. In particular, the studies of prolate
spherocylinders led to an analytical description of the pressure [49, 50, 51]

$\beta P_{{\mbox{\rm\tiny
SPT}}}=\frac{\rho}{1-v}+\frac{rs}{\left(1-v\right)^{2}}+\frac{qs^{2}}{3\left(1-v\right)^{3}},$
(3.1)

where:

$\rho=\\!\sum_{i}\rho_{i},\;\;v=\\!\sum_{i}\rho_{i}V_{i},\;\;s=\\!\sum_{i}\rho_{i}S_{i},\;\;r=\\!\sum_{i}\rho_{i}R_{i}\;\;\mbox{and}\;\;q=\\!\sum_{i}\rho_{i}R_{i}^{2}.$

This result for $P$, is no longer only a function of the diameters of the
spheres, but is also expressed as a function of the volume $V_{i}$, of the
surface area $S_{i}$ and of the mean curvature $R_{i}$ of each $i$ object.
This formula is a generalization of that obtained within SPT for mixtures of
hard spheres. In this case, the geometric parameters are related to the
diameters $\sigma_{i}$: $R_{i}=\sigma_{i}/2$, $S_{i}=\piup\sigma_{i}^{2}$ and
$V_{i}=(\piup/6)\sigma_{i}^{3}$. Using these relations in equation (3.1), leds
to the relation (2.1), previously deduced from the compressibility in the PY
approximation.

An improved scale particle theory (ISPT) was developed as a generalization of
equation (2.3), and a more accurate expression of the pressure was obtained
[2]

$\beta P_{{\mbox{\rm\tiny
ISPT}}}=\frac{\rho}{1-v}+\frac{rs}{\left(1-v\right)^{2}}+\frac{qs^{2}\left(3-v\right)}{9\left(1-v\right)^{3}}.$
(3.2)

Here also, when all the constituents are spherical, this formula becomes
identical to that given by equation (2.3). In principle, SPT is only
applicable to convex objects. Dumbbells resemble prolate spherocylinders but
are not convex and the result of the SPT should not be used for these model
molecules. Nevertheless, the previous equation (3.2) was applied to fused
dimers [52] and the results agree well with the simulation data [53].
Otherwise, equation (3.1) leads to an exact second virial coefficient and to
an approximate third coefficient. On the other hand, equation (3.2) provides a
third virial coefficient closer to the exact value. It is noted that the
different values found for the third coefficient seems to be related to the
difference observed between the third terms to the right of the equality in
each of these two equations. Moreover, there are other theoretical
developments attempting to jointly describe the thermodynamics and the
structure of fluids of convex objects.

###  3.2 Scaled field particle theory

Scaled field particle (SFP) theory provides an explicit expression of direct
correlation functions in addition to the quantities also determined by SPT
[4]. The expressions of various thermodynamic quantities are very close to
those deduced within the SPT. In particular, the pressure is written with an
expression very close to equation (3.1), derived under the SPT:

$\beta P_{{\mbox{\rm\tiny
SFP}}}=\frac{\rho}{1-v}+\frac{rs}{\left(1-v\right)^{2}}+\frac{1}{12\piup}\frac{s^{3}}{\left(1-v\right)^{3}}.$
(3.3)

This formula differs only in the last term, where $qs^{2}$ is replaced by
$s^{3}/(4\pi)$.

In the same way, an improved scale field relation (ISFP) was proposed as a
generalization of that leading to equation (2.3). This relation can be easily
deduced from equation (3.2), by replacing, in the third term to the right of
the equality, $qs^{2}$ by $s^{3}/(4\piup)$.

$\beta P_{{\mbox{\rm\tiny
ISFP}}}=\frac{\rho}{1-v}+\frac{rs}{\left(1-v\right)^{2}}+\frac{s^{3}\left(3-v\right)}{36\piup\left(1-v\right)^{3}}.$
(3.4)

From the SPT and the SFP, we have introduced four possible expressions to
account for the pressure variations observed in a fluid of convex objects. In
the following sections, a comparison between these various relationships is
presented, when used to describe a fluid of tangent or fused hard dimers. New
expressions are also established based on the mathematical form of the TPT
equations (2.9) and (2.12).

##  4 Application of conformal theories to dumbbells

A comparison of the results obtained using various conformal theories
introduced in the previous section may be done, when the fluid is made of
dumbbells. Dumbbells made of two spheres, denoted by indices $1$ and $2$, of
diameters $\sigma_{1}$ and $\sigma_{2}$, separated by a distance:
$l\leqslant\sigma_{12}\equiv(\sigma_{1}+\sigma_{2})/2$, are considered. The
characteristic quantities $V_{12},S_{12}$ and $R_{12}$ of these objects are
[4, 46, 47]

$V_{12}=\frac{\piup}{12}\left[\sigma_{1}^{3}+\sigma_{2}^{3}+\frac{3}{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\;l+\frac{3}{16}\frac{\left(\sigma_{1}^{2}-\sigma_{2}^{2}\right)^{2}}{l}-\;l^{3}\right],$
(4.1)
$S_{12}=\frac{\piup}{2}\left[\sigma_{1}^{2}+\sigma_{2}^{2}+\left(\sigma_{1}+\sigma_{2}\right)\;\left(l+\frac{\left(\sigma_{1}-\sigma_{2}\right)^{2}}{4\;l}\right)\right],$
(4.2)
$R_{12}=\frac{1}{4}\left[\sigma_{1}+\sigma_{2}+\;l+\frac{\left(\sigma_{1}-\sigma_{2}\right)^{2}}{4\;l}\right].$
(4.3)

Assuming that $\sigma_{1}$ is greater than $\sigma_{2}$, one can first
determine the values taken by these quantities when
$l\leqslant(\sigma_{1}-\sigma_{2})/2$. In this case, sphere $2$ is completely
included in sphere $1$ and various geometric quantities defined above take the
values corresponding to sphere $1$ alone. Once these quantities are known,
when only dumbbells are present (with $\rho_{1}=\rho_{2}$), the pressure can
be determined. It is found that the expressions of $P_{{\mbox{\rm\tiny SPT}}}$
and $P_{{\mbox{\rm\tiny SFP}}}$ are identical to $P_{{\mbox{\sc\scriptsize
py}}}^{c}$, given by equation (2.1), with
$\zeta_{n}=(\piup/6)\rho_{1}\sigma_{1}^{n}$ and $n=0,1,2$ and $3$. In the same
way, the expressions of $P_{{\mbox{\rm\tiny ISPT}}}$ and $P_{{\mbox{\rm\tiny
ISFP}}}$ are identical to $P_{{\mbox{\sc\scriptsize bm}}}$, given by equation
(2.3). Then, when $l\leqslant(\sigma_{1}-\sigma_{2})/2$, SPT and SFP
expressions of the pressure merge to the PY compressibility relation for hard
spheres, and ISPT and ISFP expressions merge to the improved BMCSL for hard
spheres.

The limit where $l=\sigma_{12}$ is also very important. Again, in this case
the various geometric quantities can be explained in terms of the diameters of
the spheres only. The thermodynamics deduced from conformal theories leads to
relations similar to those obtained with TPT. These relationships are
presented in the following subsections.

###  4.1 Pressure for dimers made of tangent hard spheres

When the two spheres are in contact: $l=\sigma_{12}$, then
$V_{12}=(\piup/6)(\sigma_{1}^{3}+\sigma_{2}^{3})$ and
$S_{12}=\piup(\sigma_{1}^{2}+\sigma_{2}^{2})$, are the sum of the volumes and
surface areas of the two spheres, respectively. However, the quantity $R_{12}$
is different from the sum of the two separate radii, namely
$R_{12}=(\sigma_{1}+\sigma_{2})/2-\;\sigma_{1}\sigma_{2}/[2(\sigma_{1}+\sigma_{2})]$.

Now, in the case of a dimer formation equilibrium, let us denote by
$\rho^{0}_{1}$ and $\rho^{0}_{2}$, the densities of fully dissociated
particles $1$ and $2$, which can form dumbbells D, of number density
$\rho_{{\mbox{\sc\footnotesize d}}}$, with the equilibrium

$(1)+(2)\rightleftharpoons(\mbox{D}).$ (4.4)

With this chemical equilibrium, it is observed that the concentrations that
naturally occur are $\rho^{0}_{1}$, $\rho^{0}_{2}$ and
$\rho_{{\mbox{\sc\footnotesize d}}}$ and not directly $\rho_{1}$ and
$\rho_{2}$, the total concentrations (free or bound) of particles $1$ and $2$.
Then, from the SPT equations, the pressure $P_{\rm SPT}$ can be evaluated by
considering the fluid as a mixture of two kinds of hard spheres components, of
densities $\rho^{0}_{1}$ and $\rho^{0}_{2}$, and dimers, of density
$\rho_{{\mbox{\sc\footnotesize d}}}$. The geometric parameters, $R_{i}$,
$S_{i}$, $V_{i}$, of these two kinds of an object can be determined thanks to
the equations (4.1)–(4.3). The introduction of equilibrium makes it possible
to link the chemical potentials of free and linked species. In addition, the
quantities of the three species involved in equilibrium are related by the
mass conservation relations (2.4). It has been noted previously [54, 55] that
the thermodynamic relationships for systems with association can be described
equivalently, either with the densities of the various species in equilibrium
or with the total densities of species $1$ and $2$, considering
$\rho_{{\mbox{\sc\footnotesize d}}}$ the density of pairs as a function of
total densities. In the case of dumbbells made of tangent spheres, the
integral equations within the PPY approximation and the perturbation theory
lead to expressions which are functions of the total densities and of the
density of dimers. Then, from the conformal expressions (3.1)–(3.3), equations
are established which are also expressed as a function of the total and dimer
densities.

Thus, in the SPT equation (3.1) with three constituents, one can first replace
the densities $\rho^{0}_{1}$ and $\rho^{0}_{2}$ by
$\rho_{1}-\rho_{{\mbox{\sc\footnotesize d}}}$ and
$\rho_{2}-\rho_{{\mbox{\sc\footnotesize d}}}$, respectively, in the quantities
$\rho$, $r,s,q$ and $v$. One then obtains, when $l=\sigma_{12}$

$\beta P_{{\mbox{\rm\tiny SPT}}}=\beta P_{{\mbox{\sc\scriptsize
py}}}^{c}-\rho_{{\mbox{\sc\footnotesize
d}}}\;\left[g_{12}^{{\mbox{\sc\scriptsize
py}}}(\sigma_{12})-\frac{3\zeta_{2}^{2}}{\left(1-\zeta_{3}\right)^{3}}\;\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}\right],$
(4.5)

where $g_{12}^{{\mbox{\sc\scriptsize py}}}(\sigma_{12})$ is given by equation
(2.7). In the equation (4.5), the terms $\zeta_{n}$ ($n=0,\ldots,3$),
appearing in $\beta P^{c}_{{\mbox{\sc\scriptsize py}}}$ and in the second part
of the equation, are calculated by considering the total concentrations
$\rho_{1}$ and $\rho_{2}$ of the two constituents :

$\zeta_{n}=\frac{\piup}{6}\left(\rho_{1}\sigma_{1}^{n}+\rho_{2}\sigma_{2}^{n}\right).$
(4.6)

In the same way, for the improved SPT, equation (3.2) led to

$\beta P_{{\mbox{\rm\tiny ISPT}}}=\beta P_{{\mbox{\sc\scriptsize
bm}}}-\rho_{{\mbox{\sc\footnotesize d}}}\;\left[g_{12}^{{\mbox{\sc\scriptsize
py}}}(\sigma_{12})-\frac{\zeta_{2}^{2}\left(3-\zeta_{3}\right)}{\left(1-\zeta_{3}\right)^{3}}\;\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}\right].$
(4.7)

The same procedure can be applied to expressions deduced from the SFP
theories. When $l=\sigma_{12}$, equation (3.3) led to

$\beta P_{{\mbox{\rm\tiny SFP}}}=\beta P_{{\mbox{\sc\scriptsize
py}}}^{c}-\rho_{{\mbox{\sc\footnotesize d}}}\;g_{12}^{{\mbox{\sc\scriptsize
py}}}(\sigma_{12}).$ (4.8)

Similarly, from the equation for the improved SFP theory, one gets

$\beta P_{{\mbox{\rm\tiny ISFP}}}=\beta P_{{\mbox{\sc\scriptsize
bm}}}-\rho_{{\mbox{\sc\footnotesize d}}}\;g_{12}^{{\mbox{\sc\scriptsize
py}}}(\sigma_{12}).$ (4.9)

Note that the four equations (4.5), (4.7), (4.8), (4.9) have a mathematical
form similar to that of the TPT equation given by (2.12).

Figure 1: (Colour online) Equation of state for tangent hard sphere dumbbells.
Lines from top to bottom represent, respectively, $-\cdot\cdot\,-$ SPT
equation (4.5), $\cdot\cdot\cdot\,\cdot$ SFP equation (4.8), — ISPT equation
(4.7), \- - - ISFP equation (4.9). Diamonds are simulation results (from
[53]).

Indeed, they are all composed of a first term, related to the dissociated hard
spheres, and a second term due to the connectivity between the spheres. By
analogy with the equation (2.12), the first pressure term can be regarded as
$\beta P^{\rm ref}$, the reference pressure, and the second term as $\beta
P^{\rm bond}$, the contribution due to the formation of bonds. Apart from the
proportionality to $\rho_{\rm D}$, the connectivity term only depends on the
total densities of the hard spheres. In addition, the reference pressure is
given by the PY expression in the original SPT and SFP theories. As expected,
in the equations deduced from the improved theories, the reference pressure is
given by the BM expression. We notice that in SPT theories the term of
connectivity is different between the original version and the improved one.
On the contrary, the connectivity term is the same in the original SFP theory
and in the improved one. Moreover, the expression (4.8), deduced from the
original SFP theory, is identical to that given by (2.6b), which was
established using integral equations. Equation (4.9) was also proposed as an
alternative to equation (4.8) within the framework of chain theory [37].

The results of the calculations carried out with these various formulae, for
tangent homonuclear dimers (i.e., with a distance between its constituents:
$l=\sigma_{12}$), are shown in figure 1. As it has already been said before,
the curve deduced from the improved theory ISPT with equation (3.2) is closest
to the simulation results. Then, by comparing the initial expressions (3.1)
and (3.3) obtained with SPT and SFP respectively, it is seen that they give
values that are too high but that the latter is closer to the simulated
values. Finally, the improved version ISFP gives values that are too low.

###  4.2 Connection with TPT deduced from free energy

In the previous subsection, it was shown how, from the conformal equations
(3.1)–(3.3), applied to dumbbells formed by tangent spheres, the pressure can
be split into a term of hard spheres repulsion and a term of connectivity
between these spheres. This separation is similar to that encountered in the
expression (2.12) of pressure in the TPT theory. Now, to further the analogy
between the two kinds of description, we study the equations describing the
free energy in conformal theories. The excess free energy can be deduced from
the pressure by integrating the densities in each of the approximations
presented above. Thereby, starting from equation (3.1) for $P_{{\mbox{\rm\tiny
SPT}}}$, we find within the SPT

$\beta\frac{A_{{\mbox{\rm\tiny
SPT}}}}{V}=-\rho\ln{\left(1-v\right)}+\frac{rs}{1-v}+\frac{qs^{2}}{6\left(1-v\right)^{2}}.$
(4.10)

In the same way, the free energy associated with the $P_{\rm ISPT}$ pressure
is given by:

$\beta\frac{A_{{\mbox{\rm\tiny
ISPT}}}}{V}=\left(\frac{qs^{2}}{9v^{2}}-\rho\right)\ln{\left(1-v\right)}+\frac{rs}{1-v}+\frac{qs^{2}}{9v\left(1-v\right)^{2}}.$
(4.11)

As previously, the free energies $A_{{\mbox{\rm\tiny SFP}}}$, related to the
pressure $P_{{\mbox{\rm\tiny SFP}}}$, and $A_{{\mbox{\rm\tiny ISFP}}}$,
related to the pressure $P_{{\mbox{\rm\tiny ISFP}}}$, can be deduced from the
two preceding equations by replacing everywhere $qs^{2}$ by $s^{3}/(4\piup)$.

Now, the explicit relations of free energy, devoted to dimers formed of
tangent spheres are presented. The mixture of dimers of density
$\rho_{{\mbox{\sc\footnotesize d}}}$ and of the two dissociated monomers of
density $\rho^{0}_{1}$ and $\rho^{0}_{2}$ is consider again. As previously,
using mass conservation, $\rho^{0}_{1}$ and $\rho^{0}_{2}$ are expressed as a
function of the total concentrations $\rho_{1}$ and $\rho_{2}$ and of the
dimer concentration, namely,
$\rho^{0}_{1}=\rho_{1}-\rho_{{\mbox{\sc\footnotesize d}}}$ and
$\rho^{0}_{2}=\rho_{2}-\rho_{{\mbox{\sc\footnotesize d}}}$.

Equations (4.10) and (4.11) of the SPT theory are considered first. From the
original relation (4.10), when $l=\sigma_{12}$, the application of the mass
conservation leads to

$\beta\frac{A_{{\mbox{\rm\tiny
SPT}}}(l=\sigma_{12})}{V}=\beta\frac{A^{c}_{{\mbox{\sc\scriptsize
py}}}}{V}-\rho_{{\mbox{\sc\footnotesize d}}}\ln{{\cal G}_{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})},$ (4.12)

where $A^{c}_{{\mbox{\sc\scriptsize py}}}$ is given by the equation (2.10) and

$\ln{{\cal G}_{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})}=-\ln{(1-\zeta_{3})}+\frac{3\;\zeta_{2}}{1-\zeta_{3}}\;\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}-\frac{3\;\zeta_{2}^{2}}{2\left(1-\zeta_{3}\right)^{2}}\;\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}.$
(4.13)

As previously, in the expression of $A^{c}_{\rm PY}$ and $\ln{{\cal
G}_{{\mbox{\rm\tiny SPT}}}(\sigma_{12})}$, appearing in equations (4.12) and
(4.13), the terms $\zeta_{n}$ (with $n=0,\ldots,3$) are calculated with
equation (4.6) by considering only the total concentrations $\rho_{1}$ and
$\rho_{2}$ of the two constituents. A similarity can be seen between the form
of equation (4.12) and that of equation (2.9), describing $A_{{\mbox{\rm\tiny
TPT}}}$, the TPT excess free energy. The first term, to the right of the
equality in the equation (4.12), corresponds to the reference free energy
$A^{\rm ref}$ in the equation (2.9), when the PY approximation is chosen as
the reference. Then, the second term in equation (4.12), corresponds to the
contribution due to the connectivity between the hard spheres. We recall that
in the TPT equation, the second term, proportional to the density of dimers
$\rho_{{\mbox{\sc\footnotesize d}}}$, otherwise, depends only on the densities
of dissociated spheres. We observe that the second term of the equation
presents the same characteristics. If the analogy with the TPT equation is
relevant, then the second term depends on $\ln{g^{\rm ref}(\sigma_{12})}$. We
considered that the second term of the equation (4.12) had this meaning. That
is why we gave it the name in relation $\ln{{\cal G}(\sigma_{12})}$.
Furthermore, within the framework of the SPT approximation applied to hard
spheres, an explicit expression of the pair distribution function at contact,
that we named $g^{{\mbox{\rm\tiny SPT}}}_{12}(\sigma_{12})$, was deduced
previously [21].

$g_{12}^{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})=\frac{1}{1-\zeta_{3}}+\frac{3\,\zeta_{2}}{\left(1-\zeta_{3}\right)^{2}}\,\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}+\frac{3\,\zeta_{2}^{2}}{\left(1-\zeta_{3}\right)^{3}}\,\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}.$
(4.14)

Then, the logarithm of $g_{12}^{{\mbox{\rm\tiny SPT}}}(\sigma_{12})$ can be
written as

$\ln{\left[g_{12}^{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})\right]}=-\ln{(1-\zeta_{3})}+\ln{\left[1+\frac{3\,\zeta_{2}}{\left(1-\zeta_{3}\right)}\,\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}+\frac{3\,\zeta_{2}^{2}}{\left(1-\zeta_{3}\right)^{2}}\,\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}\right]}.$
(4.15)

It was found that equation (4.13) for $\ln{{\cal G}_{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})}$ can be recovered by taking a limited development to the
second order in $\zeta_{2}$ of this expression.

Next, by applying the same process to the free energy $A_{{\mbox{\rm\tiny
ISPT}}}$ of the improved SPT, when $l=\sigma_{12}$, it was found

$\beta\frac{A_{{\mbox{\rm\tiny
ISPT}}}(l=\sigma_{12})}{V}=\beta\frac{A_{{\mbox{\sc\scriptsize
bm}}}}{V}-\rho_{{\mbox{\sc\footnotesize d}}}\ln{{\cal G}_{{\mbox{\rm\tiny
ISPT}}}(\sigma_{12})}$ (4.16)

with

$\ln{{\cal G}_{{\mbox{\rm\tiny
ISPT}}}(\sigma_{12})}=-\ln{(1-\zeta_{3})}+\frac{3\;\zeta_{2}}{1-\zeta_{3}}\;\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}-\left(\ln{(1-\zeta_{3})}+\frac{\zeta_{3}}{\left(1-\zeta_{3}\right)^{2}}\right)\frac{\zeta_{2}^{2}}{\zeta_{3}^{2}}\;\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}.$
(4.17)

The first term, to the right of the equality in the equation (4.16),
corresponds to the reference free energy $A^{\rm ref}$ in the equation (2.9),
when the BM approximation is chosen as the reference.

As an alternative, the same process can be applied to the free energy
expressions of the SFP theory. From the original expression (4.10), by
replacing $qs^{2}$ by $s^{3}/(4\pi)$, when $l=\sigma_{12}$, one gets

$\beta\frac{A_{{\mbox{\rm\tiny
SFP}}}(l=\sigma_{12})}{V}=\beta\frac{A^{c}_{{\mbox{\sc\scriptsize
py}}}}{V}-\rho_{\rm D}\ln{{\cal G}_{{\mbox{\rm\tiny SFP}}}(\sigma_{12})},$
(4.18)

where $\ln{{\cal G}_{{\mbox{\rm\tiny SFP}}}(\sigma_{12})}$ is given by

$\ln{{\cal G}_{{\mbox{\rm\tiny
SFP}}}(\sigma_{12})}=-\ln{(1-\zeta_{3})}+\frac{3\;\zeta_{2}}{1-\zeta_{3}}\;\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}.$
(4.19)

Now, the same type of comparison can also be made between the original SFP and
the improved SFP equation. By following again the same process, starting from
equation (4.16) [but replacing $qs^{2}$ by $s^{3}/(4\piup)$], when
$l=\sigma_{12}$, it was found

$\beta\frac{A_{{\mbox{\rm\tiny
ISFP}}}(l=\sigma_{12})}{V}=\beta\frac{A_{{\mbox{\sc\scriptsize
bm}}}}{V}-\rho_{\rm D}\ln{{\cal G}_{{\mbox{\rm\tiny SFP}}}(\sigma_{12})}.$
(4.20)

As in the case of the SPT theory, during the transition from the original SFP
expression to the improved one, it is seen that, in the first term, the free
energy $A_{\rm PY}^{c}$ coming from the PY approximation is replaced by the
improved $A_{\rm BM}$ equation. On the other hand, contrary to what had been
found in the SPT theory, the second term remains unchanged during the
transition from the original SFP relation to the improved one. In view of the
numerical results presented in figure 1, the expressions (4.16) of the free
energy lead by differentiation to a value of the pressure $P_{\rm ISPT}$ that
is much better than the one deduced from equation (4.20) in the ISFP theory.
Since the first term $A_{\rm BM}$ due to the dissociated hard spheres is the
same, in equations (4.16) and (4.20), there is a reason to believe that the
difference observed in the resulting pressures comes from the connectivity
terms $\ln{{\cal G}}(\sigma_{12})$. A comparison between these various
expressions of $\ln{{\cal G}}(\sigma_{12})$ as a function of the volume
fraction is presented in table 1. It is observed that equations (4.13) and
(4.17) give results of the same order of magnitude for the term $\ln{{\cal
G}}(\sigma_{12})$, unlike the SFP equation (4.19). In addition, equations
(4.13) and (4.17), for $\ln{{\cal G}_{{\mbox{\rm\tiny SPT}}}(\sigma_{12})}$
and $\ln{{\cal G}_{{\mbox{\rm\tiny ISPT}}}(\sigma_{12})}$, respectively, give
the values very close to those for $\ln{g_{12}^{\rm SPT}}(\sigma_{12})$,
deduced from equation (4.14). Equation (4.17) gives the closest values. Then,
by comparing the improved ISPT expression to the original SPT equation, taking
into account the fact that the term $\ln{{\cal G}}(\sigma)$ related to
connectivity is of the same order of magnitude in the both cases, we deduce
that the main correction incorporated in equation (4.16), with respect to
equation (4.12), is made to the reference term of dissociated hard spheres.

Table 1:  Comparison between various expressions of $\ln{{\cal G}}(\sigma)$,
given by equations (4.13), (4.17) and (4.19), with
$\ln{g_{12}^{{\mbox{\rm\tiny SPT}}}}(\sigma_{12})$ deduced from (4.14), as a
function of the volume fraction $\zeta_{3}$, when $\sigma_{1}=\sigma_{2}$.
$\zeta_{3}$ | $\ln{{\cal G}_{{\mbox{\rm\tiny SPT}}}}(\sigma)$ | $\ln{{\cal G}_{{\mbox{\rm\tiny ISPT}}}}(\sigma)$ | $\ln{g^{{\mbox{\rm\tiny SPT}}}_{12}}(\sigma)$ | $\ln{{\cal G}_{{\mbox{\rm\tiny SFP}}}}(\sigma)$
---|---|---|---|---
0.05 | 0.1292 | 0.1292 | 0.1292 | 0.1302
0.1 | 0.2674 | 0.2675 | 0.2674 | 0.2720
0.15 | 0.4155 | 0.4160 | 0.4157 | 0.4272
0.2 | 0.5747 | 0.5758 | 0.5751 | 0.5981
0.25 | 0.7460 | 0.7485 | 0.7472 | 0.7877
0.3 | 0.9307 | 0.9356 | 0.9336 | 0.9995
0.35 | 1.1297 | 1.1391 | 1.1364 | 1.2385
0.4 | 1.3442 | 1.3608 | 1.3581 | 1.5108
0.45 | 1.5741 | 1.6027 | 1.6019 | 1.8251
0.5 | 1.8181 | 1.8664 | 1.8718 | 2.1931

##  5 Alternative relations

The results represented in the previous table suggest the possibility of
developing new expressions, allowing us to describe first dumbbells made of
tangent spheres, and then those with overlap. So far, relationships between
conformal and TPT theories have been presented. On the other hand, the contact
pair distribution function is known to be better described by the expression
deduced from the SPT than by that obtained with the Ornstein Zernike equation
and the PY approximation. Then, it was first looked for possible alternative
relationships between the expression of the pressure deduced from the WOZ
integral equations and the expression of the contact pair distribution
function deduced from the SPT.

###  5.1 Equations deduced from integral equations

In this framework, it was observed the equality of equation (4.8) of the SFP
theory with expression (2.6b) obtained independently with the WOZ integral
equations. Then, it seems also interesting to find possible relations between
the equations of the TPT and those deduced from integral equations. The
getting of the expression (2.6b) from the equation (2.6a), has been
reconsidered, in order to obtain an alternative expression, by assuming that
the contact pair distribution function is given by the SPT equation.
Previously, from equation (2.6a) the relation (2.5) defining the pair
concentration $\rho_{{\mbox{\sc\footnotesize d}}}$ was used to arrive at
equation (2.6b). The equality between $g^{00}_{12}(\sigma_{12})$ and
$g^{{\mbox{\sc\scriptsize py}}}_{12}(\sigma_{12})$ was also used. Now, it is
assumed that $g^{00}_{12}(\sigma_{12})$ is equal to $g^{{\mbox{\rm\tiny
SPT}}}_{12}(\sigma_{12})$ in the equation (2.5) defining
$\rho_{{\mbox{\sc\footnotesize d}}}$ the concentration of pairs:

$\rho_{{\mbox{\sc\footnotesize
d}}}=\rho_{1}^{0}\;\rho_{2}^{0}\;K^{0}\;g^{{\mbox{\rm\tiny
SPT}}}_{12}(\sigma_{12}).$ (5.1)

This definition is an improvement because the expression of the pair
distribution function at the contact given by SPT is known to be better than
that given by the PY approximation. On the other hand, the equality between
$g^{00}_{12}(\sigma_{12})$ and $g^{{\mbox{\sc\scriptsize
py}}}_{12}(\sigma_{12})$ is kept in equation (2.6a), because
$g^{00}_{12}(\sigma_{12})$ is effectively found to be given by the right-hand
part of the equation (2.7) in the framework of the WOZ equations. The use of
the new definition of the concentration of pair $\rho_{{\mbox{\sc\footnotesize
d}}}$, then leads, starting from equation (2.6a), by expressing
$\rho^{0}_{1}\rho^{0}_{2}K^{0}$ as a function of
$\rho_{{\mbox{\sc\footnotesize d}}}$, to the alternative expression

$\beta P=\beta P_{{\mbox{\sc\scriptsize
py}}}^{c}-\rho_{{\mbox{\sc\footnotesize
d}}}\;\frac{\Bigl{[}g_{12}^{{\mbox{\sc\scriptsize
py}}}(\sigma_{12})\Bigr{]}^{2}}{g_{12}^{{\mbox{\rm\tiny SPT}}}(\sigma_{12})}.$
(5.2)

Otherwise, inspired by equation (7.19) seen in [4], the following relation
between $\left[g^{{\mbox{\sc\scriptsize py}}}_{12}(\sigma_{12})\right]^{2}$,
$g^{{\mbox{\rm\tiny SPT}}}_{12}(\sigma_{12})$ and its derivatives was found

$g^{\mbox{\rm\tiny SPT}}_{12}(\sigma_{12})+\sum_{k}\rho_{k}\frac{\partial
g^{\mbox{\rm\tiny
SPT}}_{12}(\sigma_{12})}{\partial\rho_{k}}=\Bigl{[}g^{\mbox{\rm\tiny
PY}}_{12}(\sigma_{12})\Bigr{]}^{2}.$ (5.3)

Therefore, by multiplying by $\rho_{1}^{0}\;\rho_{2}^{0}\;K^{0}$ on both
sides, the following equality is obtained

$\rho_{{\mbox{\sc\footnotesize d}}}\;\frac{\Bigl{[}g^{{\mbox{\sc\scriptsize
py}}}_{12}(\sigma_{12})\Bigr{]}^{2}}{g_{12}^{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})}=\rho_{{\mbox{\sc\footnotesize
d}}}\;\left[1+\sum_{k}\rho_{k}\frac{\partial\ln{\left[g^{{\mbox{\rm\tiny
SPT}}}_{12}(\sigma_{12})\right]}}{\partial\rho_{k}}\right],$ (5.4)

with $\rho_{{\mbox{\sc\footnotesize d}}}$ given by (5.1). The left part of the
previous equation appears in equation (5.2) and, in view of equations (2.12),
the right part is the relation used in the TPT to calculate the connectivity
part of the pressure when $g^{\rm ref}(\sigma_{12})$, the reference pair
distribution function, is given by $g^{{\mbox{\rm\tiny
SPT}}}_{12}(\sigma_{12})$. Therefore, the relation (5.2), deduced from the WOZ
integral equations, can be obtained independently within TPT. Since the SPT
expression of the contact pair distribution function is better than that given
by the PY approximation, the equation (5.2) is probably an improvement over
(2.6b). However, the expression of the reference pair distribution at contact
with $g^{{\mbox{\rm\tiny SPT}}}_{12}(\sigma_{12})$ is known to be slightly
poorer than that given by the corresponding BM equation. Accordingly, the
equation (5.2) is probably somewhat less efficient than the corresponding TPT
equation obtained using the BM approximation.

###  5.2 Equations deduced from conformal theories

It has been seen in table 1, that, when $l=\sigma_{12}$, the connectivity term
given in the expression of the improved free energy ISPT is numerically close
to the one found for the initial free energy SPT. The improvement present in
the ISPT equations is provided by the first term describing the dissociated
hard spheres. The effect of connectivity is taken into account using a simpler
equation in the initial SPT theory.

In view of these observations, an alternative expression, providing a
correction to the SPT expression of the free energy (4.12), when
$l=\sigma_{12}$, was deduced. A modification can be made to the SPT expression
by only changing the term of dissociated hard spheres. Then, when
$l=\sigma_{12}$, a new equation for the free energy can be obtained:

$\beta\frac{A_{{\mbox{\rm\scriptsize
New}}}(l=\sigma_{12})}{V}=\beta\frac{A_{{\mbox{\sc\scriptsize
bm}}}}{V}-\rho_{{\mbox{\sc\footnotesize d}}}\ln{{\cal G}_{{\mbox{\rm\tiny
SPT}}}(\sigma_{12})}.$ (5.5)

Compared with equation (4.12), the first term $A^{c}_{{\mbox{\sc\scriptsize
py}}}$ is replaced by a more accurate term $A_{{\mbox{\sc\scriptsize bm}}}$,
and the second term remains unchanged because it leads to numerical values
close to that given by the corresponding term of the equation (4.16). The
corresponding expression of the pressure is obtained by differentiation of the
free energy:

$\beta P_{{\mbox{\rm\scriptsize New}}}=\beta P_{{\mbox{\sc\scriptsize
bm}}}-\rho_{{\mbox{\sc\footnotesize d}}}\;\left[g_{12}^{{\mbox{\sc\scriptsize
py}}}(\sigma_{12})-\frac{3\zeta_{2}^{2}}{\left(1-\zeta_{3}\right)^{3}}\;\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\left(\sigma_{1}+\sigma_{2}\right)^{2}}\right]\quad{\text{for:}}\quad
l=\sigma_{12}.$ (5.6)

This relation allows us to correctly describe the thermodynamics of dumbbells
made of tangent spheres. As such, it is of little use insofar as many
expressions are already available. Nevertheless, from this equation a more
general relation can be established to describe dimers with overlapping
spheres. Indeed, equations (5.5) and (5.6) are linear combinations of various
relations previously introduced

$\displaystyle\beta A_{{\mbox{\rm\scriptsize New}}}$ $\displaystyle=\beta
A_{{\mbox{\rm\tiny ISFP}}}+\beta\Bigl{[}A_{{\mbox{\rm\tiny
SPT}}}-A_{{\mbox{\rm\tiny SFP}}}\Bigr{]},$ (5.7a) $\displaystyle\beta
P_{{\mbox{\rm\scriptsize New}}}$ $\displaystyle=\beta P_{{\mbox{\rm\tiny
ISFP}}}+\beta\Bigl{[}P_{{\mbox{\rm\tiny SPT}}}-P_{{\mbox{\rm\tiny
SFP}}}\Bigr{]}.$ (5.7b)

Figure 2: (Colour online) Equation of state for overlapping hard sphere
dumbbells. From top to bottom, lines represent theoretical results for
different overlap distances: $\cdot\cdot\cdot\,\cdot$ New equation (5.8), —
ISPT equation (3.2). Symbols are the simulation results [53]. Diamonds and
black lines: $l=1$; Circles and red lines: $l=0.8$; Triangles and blue lines:
$l=0.6$, Squares and pink lines: $l=0.2$.

These new relationships add corrective terms to the previous ISFP equations.
Moreover, these latter expressions can also be used for dimers with
overlapping of the spheres by using the more general equations introduced
previously to describe them. Therefore, using equations (3.4), (3.1) and
(3.3), the equation (5.7b) has been evaluated for a dumbbell fluid where the
distance $l$ between sphere is less than or equal to $1$. The term in square
brackets, $\left[P_{{\mbox{\rm\tiny SPT}}}-P_{{\mbox{\rm\tiny SFP}}}\right]$
in the previous equation, can be simplified, giving

$\beta P_{{\mbox{\rm\scriptsize New}}}=\beta P_{{\mbox{\rm\tiny
ISFP}}}+\frac{s^{2}}{3\left(1-v\right)^{3}}\;\left(q-\frac{s}{4\piup}\right),$
(5.8)

where the difference $q-s/(4\pi)$, starting from equations (4.2) and (4.3),
can be evaluated according to

$\left(q-\frac{s}{4\piup}\right)=\frac{\rho_{{\mbox{\sc\footnotesize
d}}}}{16}\left[l-\frac{\left(\sigma_{1}-\sigma_{2}\right)^{2}}{4l}\right]^{2}.$
(5.9)

The results of the calculations performed with equation (5.8) are presented in
figure 2, for homonuclear dimers, for various separations of the constituents.
A comparison is also given with the results deduced from the ISPT equation
(3.2). It is observed that the new theoretical expression describes well the
values obtained by simulation. It provides values numerically very close to
those given by the ISPT equation. The new equation can be seen also as an
improvement of the improved SFP equation (3.4). The corrective term for
$P_{{\mbox{\rm\tiny ISFP}}}$, presented in equation (5.8), seems to correspond
numerically to the contribution required to recover the results obtained with
the expression of $P_{{\mbox{\rm\tiny ISPT}}}$.

##  6 Conclusion

In the previous sections, it has been shown how, from the conformal theories
applied to dumbbells made of tangent hard spheres, the pressure and the free
energy can be split into a term of repulsion of hard spheres and a term of
connectivity between these spheres. This separation is similar to that
encountered in the expression of pressure in the TPT theory. The analysis
presented in this work highlights the relative influence of these two
contributions in the different expressions presented in various conformal
theories studied. It has been found that for each class of conformal theory
the reference term is, for the original version that given by the PY theory,
and for the improved version that given by the BMCSL expression. Then, when
the scaled particle and the scaled field theory are compared, it was noted
that the terms of reference are the same, both for the original approach and
for the improved expressions. On the other hand, for each of the two classes
of conformal theory considered, the term of connectivity changes little when
one goes from the original expression to the improved one. In addition, the
term of connectivity is different between the scaled field and the scaled
particle theory. Connections and similarities have also been seen between the
expressions describing these two contributions and those found with the TPT
for dimers made of tangent spheres. Similarities were also found with some
relationships given in the context of integral equations. This work allows one
to explain differences found in the description of thermodynamic properties
deduced from simulations.

Furthermore, this analysis has allowed one to provide alternative expressions
of the connectivity terms, in particular for dimers with overlap. It could be
extended to fluids containing chains of a greater number of constituents. It
could also be useful to better represent subunits belonging to polymers
according to the group contribution formalism [56]. On the other hand, the
analysis of thermodynamic quantities such as free energy and pressure could be
extended to the activity coefficients of various species present in the fluids
of dimers [57]. Then, the study of the chemical equilibrium between the hard
spheres forming the overlapping dimers could be reviewed. Thus, this will
allow one to describe the background correlation in continuation of previous
studies on this subject [58, 59, 60].

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