Sedimentation of the neutron rich isotope $^{22}$Ne may be an important source of gravitational energy during the cooling of white dwarf stars. This depends on the diffusion constant for $^{22}$Ne in strongly coupled plasma mixtures. We calculate self-diffusion constants $D_i$ from molecular dynamics simulations of carbon, oxygen, and neon mixtures. We find that $D_i$ in a mixture does not differ greatly from earlier one component plasma results. For strong coupling (coulomb parameter $Gamma>$ few), $D_i$ has a modest dependence on the charge $Z_i$ of the ion species, $D_i propto Z_i^{-2/3}$. However $D_i$ depends more strongly on $Z_i$ for weak coupling (smaller $Gamma$). We conclude that the self-diffusion constant $D_{rm Ne}$ for $^{22}$Ne in carbon, oxygen, and neon plasma mixtures is accurately known so that uncertainties in $D_{rm Ne}$ should be unimportant for simulations of white dwarf cooling.
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