Stacked analyses of galaxy clusters at low-to-intermediate redshift show signatures attributable to dust, but the origin of this dust is uncertain. We test the hypothesis that the bulk of cluster dust derives from galaxy ejecta. To do so, we employ dust abundances obtained from detailed chemical evolution models of galaxies. We integrate the dust abundances over cluster luminosity functions (one-slope and two-slope Schechter functions). We consider both a hierarchical scenario of galaxy formation and an independent evolution of the three main galactic morphologies: elliptical/S0, spiral and irregular. We separate the dust residing within galaxies from the dust ejected in the intracluster medium. To the latter, we apply thermal sputtering. The model results are compared to low-to-intermediate redshift observations of dust masses. We find that in any of the considered scenarios, elliptical/S0 galaxies contribute negligibly to the present-time intracluster dust, despite producing the majority of gas-phase metals in galaxy clusters. Spiral galaxies, instead, provide both the bulk of the spatially-unresolved dust and of the dust ejected into the intracluster medium. The total dust-to-gas mass ratio in galaxy clusters amounts to $10^{-4}$, while the intracluster medium dust-to-gas mass ratio amounts to $10^{-6}$ at most. These dust abundances are consistent with the estimates of cluster observations at $0.2 < z <1$. We propose that galactic sources, spiral galaxies in particular, are the major contributors to the cluster dust budget.
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