We study the equilibrium behaviour of a mixture of monodisperse hard spherecolloids and polydisperse non-adsorbing polymers at their $\theta$-point, usingthe Asakura-Oosawa model treated within the free-volume approximation. Ourfocus is the experimentally relevant scenario where the distribution of polymerchain lengths across the system is fixed. Phase diagrams are calculated usingthe moment free energy method, and we show that the mean polymer size $\xi_{\rmc}$ at which gas-liquid phase separation first occurs decreases with increasingpolymer polydispersity $\delta$. Correspondingly, at fixed mean polymer size,polydispersity favours gas-liquid coexistence but delays the onset offluid-solid separation. On the other hand, we find that systems with different$\delta$ but the same {\em mass-averaged} polymer chain length have nearlypolydispersity-independent phase diagrams. We conclude with a comparison toprevious calculations for a semi-grandcanonical scenario, where the polymerchemical potentials are imposed, which predicted that fluid-solid coexistencewas over gas-liquid in some areas of the phase diagram. Our results show thatthis somewhat counter-intuitive result arose because the actual polymer sizedistribution in the system is shifted to smaller sizes relative to the polymerreservoir distribution.
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