polydispersity index. In such a way, we have tried to solve the non-linear algebraic system – involved in the PY result – *perturbatively in polydispersity*, starting from the monodisperse PY solution.

Baxter’s SHS model and PY solution
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The SHS model is defined as limiting case of  a particular square-well (SW) model [@Baxter68], based upon a potential including steeply repulsive core and short-ranged attractive tail, i.e.

$$\phi _{ij}^{\mathrm{Baxter\ SW}}\left( r\right) =\left\{ 
\begin{array}{ccc}
+\infty  &  & 0<r<\sigma _{ij}\equiv (\sigma _{i}+\sigma _{j})/2 \\ 
-\epsilon _{ij}^{\mathrm{Baxter~SW}} &  & \sigma _{ij}\leq r\leq
R_{ij}\equiv \sigma _{ij}+w_{ij} \\ 
0 &  & r>R_{ij}\text{ ,}%
\end{array}%
\right.   \label{eq0}$$

with$$\epsilon _{ij}^{\mathrm{Baxter~SW}}=k_{B}T\ \ln \left( 1+t_{ij}\frac{\sigma
_{ij}}{w_{ij}}\right) \text{ ,}$$where $\sigma _{i}$ is the HS diameter of species $i$, $\epsilon _{ij}^{%
\mathrm{Baxter~SW}} >0$ and $w_{ij}$ are the depth and the width of the well, respectively, $k_{B}$ denotes Boltzmann’s constant, $T$ the temperature. Moreover, $$t_{ij}=\frac{1}{12\tau _{ij}}\geq 0,$$where the conventional Baxter parameter $\tau _{ij}$ is an unspecified increasing function of $T$, while $\tau _{ij}^{-1}$ measures the strength of surface adhesion or ‘stickiness’ between particles of species $i$ and $j$.

The ‘sticky limit’ of $\phi _{ij}^{\mathrm{Baxter~SW}}\left( r\right) $ corresponds to taking $w_{ij}\rightarrow 0$. While the SW width goes to zero, its depth $\epsilon _{ij}^{\mathrm{Baxter~SW}}$ diverges, 