Frustrated junctions in interfacial networks
Abstract
Networks of interfaces in materials and living systems evolve through motion and rearrangement of interfaces and junctions. Local equilibrium at a junction requires interfacial force balance. We show that in three dimensions the classical Herring condition for triple lines is necessary but insufficient: four triple lines meeting at a quadruple node may each satisfy Herring equilibrium, while the four conditions remain mutually incompatible. Such a frustrated node has no admissible local equilibrium geometry. In the semi-isotropic limit, the six interfacial energies must form the edge lengths of a non-degenerate tetrahedron, with constructibility determined by the Cayley-Menger determinant. This hidden compatibility constraint arises from three-dimensional geometry and applies broadly to foams, polycrystals, tissues and other interfacial networks.
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BibTeX
@misc{hallberg2026frustrated,
title = {{Frustrated junctions in interfacial networks}},
author = {Håkan Hallberg and Vasily V. Bulatov and Bryan W. Reed and Mukul Kumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29030},
eprint = {2609.29030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29030},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Frustrated junctions in interfacial networks
AU - Håkan Hallberg
AU - Vasily V. Bulatov
AU - Bryan W. Reed
AU - Mukul Kumar
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29030
UR - https://arxiv.org/abs/2609.29030
AB - Networks of interfaces in materials and living systems evolve through motion and rearrangement of interfaces and junctions. Local equilibrium at a junction requires interfacial force balance. We show that in three dimensions the classical Herring condition for triple lines is necessary but insufficient: four triple lines meeting at a quadruple node may each satisfy Herring equilibrium, while the four conditions remain mutually incompatible. Such a frustrated node has no admissible local equilibrium geometry. In the semi-isotropic limit, the six interfacial energies must form the edge lengths of a non-degenerate tetrahedron, with constructibility determined by the Cayley-Menger determinant. This hidden compatibility constraint arises from three-dimensional geometry and applies broadly to foams, polycrystals, tissues and other interfacial networks.
ER -