Topology and Martensitic Phase Transformations

TL;DR

The work establishes a topology-based criterion for martensitic phase transformations by linking crystal Bravais lattices to triply periodic minimal surfaces (TPMS). It shows that end states of a martensitic path reside on TPMS of the same genus, reflecting diffusionless connectivity preservation and correspondence between lattice-neighbor graphs and TPMS skeletal graphs. The study demonstrates this through Cu/Al (non-martensitic genus change), Zr/Na (genus-conserving martensitic paths), and NiTi (B2 to B19′ with oPb TPMS) cases, highlighting how TPMS topology can predict transformation feasibility. The findings offer a framework for understanding transformation pathways and guiding the design of shape memory materials from a topological perspective, with implications for non-magnetic crystals and their TPMS realizations.

Abstract

Triply periodic minimal surfaces (TPMS) are discovered to conform to surfaces of given charge density distributions embedded in crystals [Z. Kristallogr. \textbf{170}, 138 (1985)]. Based on our previous work [Phys. Rev. Mater. \textbf{9}, 073802 (2025)], we discovered that crystals can have surfaces of a given charge density converging to TPMS. We also discovered that end states connected by a martensitic phase transformation should have their corresponding TPMS being topologically equivalent. In this work, we gave an explanation for the topological continuity of a martensitic phase transformation and studied how TPMS indicate whether a non-magnetic crystal can undergo a martensitic phase transformation or not.

Figures (11)

• Figure 1: Examples of TPMS. (a) Schwarz P surface (P), (b) Schoen F-RD surface (F-RF), (c) Schoen I-WP surface (I-WP), (d) Schoen H$^\prime$-T surface (H$^\prime$-T).
• Figure 2: Example of forming a skeletal graph. (a) I-WP surface. From (b) to (d) shows a continuous shrinking of the I-WP surface. (e) Ball-stick illustration of the skeletal graph of the I-WP surface. Spheres represent ends and sticks represent arms connecting two ends.
• Figure 3: From (a) to (d) are skeletal graphs of the P, F-RD, I-WP and H$^\prime$-T surfaces, respectively. Balls represent ends and sticks represent arms connecting two ends.
• Figure 4: Sketch of how lattice points locate at flat points on a TPMS. (a) NiTi lattice (simple cubic) on the P surface, (b) BCC lattice on the I-WP surface, (c) FCC lattice on the F-RD surface, (d) hexagonal lattice on the H$'$-T surface.
• Figure 5: (a) Sketch of a flat point on P crossing which the line connecting a Ni atom (red sphere) and a Ti atom (blue sphere) penetrating the P surface. (b) Sketch of all flat points obtained by connecting one atom at the conner to the one at the centre. (c) Sketch of the Bravais lattice of B2 NiTi, with the start point chosen to be the one given in (a). (d,e,f) are the the view of (a,b,c) form the $[\overline{1}10]$ direction, respectively.