_{ij}\left( r\right) \right]
,$$since, for Baxter’s potential, the Mayer function, $f_{ij}(r)=\exp \left[
-\beta \phi _{ij}(r)\right] -1$, vanishes for $r>R_{ij}$.

The choice of $c_{ij}^{\mathrm{shrink}}(r)$ inside the well (region which ’shrinks’ in the sticky limit) defines one particular closure within the proposed class. Of course, $c_{ij}^{\mathrm{shrink}}(r)=c_{ij}^{\mathrm{PY}%
}(r)$ corresponds to the PY approximation. On the other hand, when $c_{ij}^{%
\mathrm{shrink}}(r)\neq c_{ij}^{\mathrm{PY}}(r)$, we are in the presence of *mixed closures,* which have frequently appeared in the literature [@Herrera91]. In order to define mixed closures simpler than the PY approximation, we consider the *density* expansion of the exact direct correlation function [@Barrat03], and denote as $C_{n}$ approximation a truncation of this series to order $O(\rho ^{n})$. The simplest two approximations are

$$\begin{array}{ccc}
\begin{array}{c}
c_{ij}^{\mathrm{shrink}}(r)=f_{ij}(r) \\ 
\end{array}
&  & 
\begin{array}{c}
\text{(}C_{0}\text{ closure)} \\ 
\end{array}
\\ 
c_{ij}^{\mathrm{shrink}}(r)=f_{ij}\left( r\right) \ \left[ 1+\left( \
\sum_{k}\rho _{k\ }f_{ik}\ast f_{kj}\right) \left( r\right) \right]  &  & 
\text{(}C_{1}\text{ closure),}%
\end{array}%$$

where $\rho _{k\ }$ is the number density of species $k$, while $\ast $ denotes convolutive integration [@Gazzillo04].

In the ‘sticky limit’ $R_{ij}\rightarrow \sigma _{ij}^{+}$ the well region shrinks, but a ‘memory’ of the approximation chosen for $c_{ij}^{\mathrm{%
shrink}}$ remains in the solution of the OZ integral equation. In fact, although all solutions 