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Blistering is a phenomenon sometimes observed in sputtered-deposited thin films but seldom investigated in detail. Here, we consider the case of titania-doped germania (TGO)/silica multi-layers deposited by ion beam sputtering. TGO is a candidate as high refractive index material in the Bragg mirrors for the next iteration of gravitational waves detectors. It needs to be annealed at 600°C for 100h in order to reach the desired relaxation state. However under some growth conditions, in 52-layer TGO/silica stacks, blistering occurs upon annealing at a temperature near 500°C, which corresponds to the temperature where Ar desorbs from TGO. In order to better understand the blistering phenomenon, we measure the Ar transport in single layers of TGO and silica. In the case of <1 μm-thick TGO layers, the Ar desorption is mainly limited by detrapping. The transport model also correctly predicts the evolution of the total amount of Ar in a 8.5 μm stack of TGO and silica layers annealed at 450°C, but in that case, the process is mainly limited by diffusion. Since Ar diffusion is an order of magnitude slower in TGO compared to silica, we observe a correspondingly strong accumulation of Ar in TGO. The Ar transport model is used to explain some regimes of the blisters growth, and we find indications that Ar accumulation is a driver for their growth in general, but the blisters nucleation remains a complex phenomenon influenced by several other factors including stress, substrate roughness, and impurities.