  We present a theory of effective electrostatic interactions in polydisperse
suspensions of charged macroions, generalizing to mixtures a theory previously
developed for monodisperse suspensions. Combining linear response theory with a
random phase approximation for microion correlations, we coarse-grain the
microion degrees of freedom to derive general expressions for effective
macroion-macroion pair potentials and a one-body volume energy. For model
mixtures of charged hard-sphere colloids, we give explicit analytical
expressions. The resulting effective pair potentials have the same general form
as predicted by linearized Poisson-Boltzmann theory, but consistently
incorporate dependence on macroion density and excluded volume via the Debye
screening constant. The volume energy, which depends on the average macroion
density, contributes to the free energy and so can influence thermodynamic
properties of deionized suspensions. To validate the theory, we compute radial
distribution functions of binary mixtures of oppositely charged colloidal
macroions from molecular dynamics simulations of the coarse-grained model (with
implicit microions), taking effective pair potentials as input. Our results
agree closely with corresponding results from more computationally intensive
Monte Carlo simulations of the primitive model (with explicit microions).
Simulations of a mixture with large size and charge asymmetries indicate that
charged nanoparticles can enhance electrostatic screening of charged colloids.
The theory presented here lays a foundation for future large-scale modeling of
complex mixtures of charged colloids, nanoparticles, and polyelectrolytes.
