the parent phase. As a consequence, there is no a single coexistence line (‘binodal’) as in the one-component case, but one rather finds a *cloud curve*, representing the temperature-density dependence line of the low-density majority phase (’gas’), and a *shadow curve* representing the temperature-density dependence of the high-density minority phase (incipient ’liquid’). While for one-component systems these two curves are identical, for polydisperse systems in general they are not, with the exception of the critical point where they coincide by definition.

However, in order to apply the $C_{0}$ and $C_{1}$ approximations to the multi-component SHS model, we have to tackle a further important problem, that is the definition of the stickiness parameters $\tau _{ij}$.

Size-dependence of stickiness parameters
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In mixtures, $\tau _{ij}$ will depend on the particular pair $i,j$ considered, and reasonably should be expected to be related to the particles sizes. Assuming that we are dealing only with size-polydispersity, we can always decouple temperature and adhesion as $$\frac{1}{\tau _{ij}}=\frac{1}{\tau }\ \epsilon _{ij}=\frac{1}{\tau }\mathcal{%
F}\left( \sigma _{i},\sigma _{j}\right)$$where the last equality stems from the assumption of size-polydispersity and of a purely pairwise potential. Unfortunately, the exact form of the size-dependence of these stickiness parameters is still an open problem, 