Quantifying phase mixing and separation behaviors across length and time scales

M. Skolnick, S. Torquato

Research output: Contribution to journalArticlepeer-review


Phase mixing and separation phenomena abound in the formation of natural and synthetic material systems, including alloys, composites, granular media, geological media, complex fluids, and biological media. While characterizing phase mixing is critical to understanding material microstructure formation and physical properties, previous metrics that measure the degree of mixing in condensed phases often rely on empirical observation, are applicable only to certain microstructures, and/or fail to detect nontrivial changes in mixing with respect to length scale. To overcome these shortcomings, we introduce and study a mixing metric that is applicable to a broad spectrum of complex microstructures and leverages the hyperuniformity concept to provide a unifying framework to rank and classify mixing in materials across both length and time scales. Specifically, we employ the local phase volume-fraction/field-intensity variance σV/F2(R,t) associated with a spherical sampling window of radius R at time t as our metric. We demonstrate that σV/F2(R,t) provides a sensitive measure of mixing across both length and time scales of a structurally diverse set of real and simulated material microstructures: exotic disordered multihyperuniform avian retina photoreceptor cell patterns, time-evolving spinodal decomposition patterns, and a model material mixed via the baker's transformation. We also describe how this metric can be used to guide future work on fabricating materials whose bulk properties can be controlled by tuning their mixing characteristics across length and time scales.

Original languageEnglish (US)
Article number119774
JournalActa Materialia
StatePublished - Apr 15 2024

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Ceramics and Composites
  • Polymers and Plastics
  • Metals and Alloys


  • Alloying
  • Blending
  • Heterogeneous materials
  • Hyperuniformity
  • Mixing metric


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