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We report multimodal scanning hard x-ray imaging with spatial resolution approaching 10 nm and its application to contemporary studies in the field of material science. The high spatial resolution is achieved by focusing hard x-rays with two crossed multilayer Laue lenses and raster-scanning a sample with respect to the nanofocusing optics. Various techniques are used to characterize and verify the achieved focus size and imaging resolution. The multimodal imaging is realized by utilizing simultaneously absorption-, phase-, and fluorescence-contrast mechanisms. The combination of high spatial resolution and multimodal imaging enables a comprehensive study of a sample on a very fine length scale. In this work, the unique multimodal imaging capability was used to investigate a mixed ionic-electronic conducting ceramic-based membrane material employed in solid oxide fuel cells and membrane separations …