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Structural and thermal confinement of Schwarz crystals

Dan Huang1,*, Huangliu Fu1,*, Meng Lyu2,*, Yuanwen Feng1, Ryoichi Kajimoto3, Mitsutaka Nakamura3, Takashi Honda4, Enke Liu2,†, Bing Li1,‡ et al.

Xiuyan Li1,5,§ and Ke Lu1,5

  • *These authors contributed equally to this work.
  • †Contact author: ekliu@iphy.ac.cn
  • ‡Contact author: bingli@imr.ac.cn
  • §Contact author: xyli@imr.ac.cn
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Phys. Rev. Research 8, 033017 – Published 6 July, 2026

DOI: https://doi.org/10.1103/lqy2-cshl

Abstract

As a nontrivial state of solid, Schwarz crystal manifests itself in extremely refined nanograins and triply periodic minimal surface (TPMS) grain boundaries, and therefore exceptional thermal and mechanical stability. However, it remains unexploited how the structural units interact and thermal transport is influenced by such complex atomic structures. Here, we investigate Schwarz crystals of Cu and Pt using neutron scattering and thermal transport measurements. Pair distribution function analysis reveals the coexistence of compressed grains and expanded TPMS grain-boundary regions, forming a spatially confined atomic environment. This leads to an unusual phonon hardening suggested by inelastic neutron scattering. As bulk materials, Schwarz crystals exhibit a quasilinear temperature dependence of thermal conductivity, which was previously only observed in low-dimensional materials. This contrast indicates that the spatial confinement of the Schwarz crystals plays a crucial role in determining the unique thermal conductivity, which also provides an emergent route to regulate thermal properties in bulk materials.

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Phys. Rev. Research 8, 033017– Published 6 July, 2026
  • Received 22 October 2025
  • Revised 23 April 2026
  • Accepted 9 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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