In the CR (Renazzo‐like) chondrite group, many chondrules have successive igneous rim (IR) layers, with an outer layer that contains a silica mineral and/or silica‐rich glass (silica‐rich igneous rims, SIRs). Models for SIR formation include (1) accretion of Si‐rich dust onto solid chondrule surfaces, followed by heating and cooling and (2) condensation of SiO(gas) onto the surface of partially molten chondrules. We evaluate these models, based on a petrographic study of five Antarctic CR chondrites that have undergone minimal secondary alteration. We obtained electron microprobe analyses of minerals and glass with quantitative wavelength‐dispersive spectroscopy mapping, and identified silica polymorphs with Raman spectroscopy. Common SIRs contain silica, low‐Ca pyroxene, Ca‐rich pyroxene, Fe,Ni metal, ± glass ± plagioclase ± rare olivine. We also describe near‐monomineralic SIRs where a narrow zone of cristobalite occurs at the outer edge of the chondrule. All crystalline silica is cristobalite, except for one SIR that consists of tridymite. Some rims contain silica‐rich glass (>80 wt% SiO2) but no silica mineral. Features such as sharp interfaces and compositional boundaries between chondrules and SIRs indicate that SIRs were formed from solid precursors. Consideration of the stability fields of silica polymorphs and computed liquidus temperatures indicates that SIRs were heated to >1500°C for limited time periods, followed by rapid cooling, similar to conditions for chondrule formation. We infer that in the CR chondrule formation region, the same heating mechanism was repeated multiple times while the chemical composition of the nebular gas evolved to highly fractionated silica‐rich compositions.