Attractive interactions between particles do not necessarily lead to these particles aggregating. This is the new finding from researchers at the Tokyo University of Science in Japan who say this behaviour comes from non-reciprocal interactions amongst particles of different sizes that exert unequal forces on each other. These interactions could serve as a general design principle for creating active matter with a structure that continuously reorganizes rather than simply relaxing towards a static state.

Nature abounds with examples of active matter, with some well-known examples being flocks of birds, fish and insects, sheets of biological cells and swarms of bacteria. Researchers have been trying to copy this collective behaviour and make synthetic active materials – such as self-propelled colloids and dense phases of mechanically driven biopolymers – in the laboratory. Before such applications see the light of day, however, they need to first understand the fundamental physics of these systems.

In their new study, a team led by Yutaka Sumino and Kiwamu Yoshii of Tokyo University of Science’s department of applied physics, came across a system in which colloidal particles interact through electrohydrodynamic flow. “We found that mixing particles of two different sizes in such a system completely changed the collective behaviour of these particles,” explains Sumino. “Instead of forming increasingly large crystal-like aggregates, as is the case for particles with the same size, the system remained highly dynamic.”

The physicists’ experiment involved suspending polystyrene colloidal particles with radii of 1 and 1.5 µm in water and confining them between transparent indium tin oxide-coated electrodes. This set up allowed them track more than 10,000 particles and quantitatively analyse their dynamics for more than an hour, as opposed to just a few minutes in previous such studies.

Newton’s third law holds

When the researchers applied an alternating electric field to the system, they observed electrohydrodynamic flows around the particles, the strength of which increased strongly as the size of the particles became bigger. The electrohydrodynamic-mediated attractive interactions therefore became asymmetric, they explain, something that causes larger particles to attract smaller ones more strongly. This imbalance leads to non-reciprocal interactions, meaning that a larger particle pushes a smaller one, but a smaller one doesn’t push back.

While the effective interaction between the particles appears to break Newton’s third law, it doesn’t because there is no violation of momentum conservation, explains Sumino: momentum is transferred to the surrounding fluid through the induced flows and is ultimately dissipated through friction with the substrate.

The researchers also observed that particles of different sizes spontaneously pair together to form asymmetric structures with a distinct front and tail. These pairs behave as self-propelled units and move through the suspension – even though individual particles cannot propel themselves. As more self-propelled pairs form, they assemble into larger clusters, but these clusters do not continue growing into large aggregates and instead repeatedly fragment, rearrange themselves and reform. Importantly, they note: “the larger particle tends to be at the front of these moving pairs. This head-heavy size asymmetry, together with excluded-volume interactions, promotes the fragmentation of larger clusters and thereby prevents continuous coarsening”.

Sumino and Yoshii combined their experiments with numerical simulations, thereby identifying non-reciprocal pair motion as the microscopic origin of this behaviour.

The results, they say, suggest that non-reciprocal interactions can serve as a general design principle for creating active materials with structures that continuously reorganize rather than simply relaxing toward a static state. “An interesting aspect,” explains Sumino, “is that self-propulsion does not need to be built into each individual particle: it can emerge collectively from non-reciprocal interactions between particles that do not self-propel on their own.”

If such interactions can be controlled externally, they could provide a way to design microscopic systems that collectively gather, transport, fragment or mix materials, he tells Physics World. “Possible directions include programmable active materials and microrobotic systems, although these applications are still some way off.”

Reporting their findings in Physical Review Letters, the researchers say they would now like to understand how general this non-reciprocal mechanism is and whether the same principle – collective activity emerging from non-reciprocal interactions – can be transferred to other experimental systems. In the system studied in this work, the strength of the non-reciprocity depends on parameters such as particle size, composition and the applied electric field. By systematically tuning these parameters, they aim to explore what other types of collective states can be generated and whether transitions between them can be controlled.

“A more detailed hydrodynamic analysis of the system is also needed to develop a quantitative understanding of the interactions and ultimately predict the collective motion of the colloids,” says Sumino.

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Nickel disulfide, NiS₂, is an unusual material. Its interior behaves as an electrical insulator, blocking the flow of current, while certain parts of its surface conduct electricity freely. This contradiction has puzzled physicists for decades, and a recent study offers a detailed explanation. The conduction, it turns out, is confined to tiny one-dimensional steps on the surface, and these narrow conducting channels are a direct consequence of how electrons are arranged inside the crystal.

Mott insulator

NiS₂ shares its crystal structure with pyrite, the mineral sometimes called fool’s gold. At ordinary conditions, its electrons do not move freely through the lattice: the strong electrical repulsion between them opens an energy gap that keeps the bulk of the material insulating. This behaviour defines what physicists call a Mott, or charge-transfer, insulator, a category of materials that simple band theory cannot explain. Band theory predicts that any material with a partially filled outer electron shell should conduct electricity, yet compounds like nickel oxide obstinately insulate instead. In 1949, the physicist Nevill Mott showed why: if two electrons crowding the same atomic site cost more energy than is available, the electrons stay locked in place and no current flows, regardless of what band theory predicts. NiS₂ behaves in much the same way.

Yet electrical-transport measurements had repeatedly found that crystals of NiS₂ conduct through their surfaces. Earlier scanning-tunnelling experiments helped explain part of the puzzle: the broad, flat surface remains insulating, but the electronic gap narrows sharply near atomic-scale steps, tiny cliffs just one or two atoms high where the surface abruptly changes level. A new study ¹ confirms this picture and examines the steps with much higher spatial resolution than before.

A not that smooth surface

A scanning tunnelling microscope (STM) works by bringing an extremely sharp needle close enough to a surface that electrons can leak across the gap between tip and sample. Measuring that leakage at different energies reveals both the surface’s shape and how easily electrons can enter or leave it. In NiS₂, these measurements uncovered electronic states concentrated tightly along the step edges: the states extend for a long distance parallel to a step but stay confined to roughly one to two nanometres across it. In other words, the electrons are effectively free to move in one direction while remaining trapped in the other two, forming genuine one-dimensional conduction channels.

Two distinct kinds of surface step were observed, depending on whether the outermost atomic layer exposes nickel or sulfur atoms. Both carry electronic states associated with their steps, but they are not equally important. The nickel-terminated steps host states that sit right at the energy where electrical conduction happens, making them the most likely explanation for the surface conductivity that transport experiments had measured for years. The sulfur-terminated steps also host states inside the insulating gap, but their energy is positioned further away, making them less directly responsible for carrying current.

The study also tested what happens when a magnetic field is applied perpendicular to the surface. The step-edge states survived fields as strong as 10 tesla, a field far stronger than any produced by a laboratory magnet used for everyday purposes. Their intensity weakened somewhat, particularly on the sulfur-terminated steps, but the states were never destroyed. This robustness matters because NiS₂ already displays complicated magnetic behaviour of its own, which might easily have disrupted a more fragile conducting state.

The deeper explanation lies in how electrons are distributed inside the crystal. In an ordinary insulator, the occupied electronic states can usually be pictured as clouds centred on the atoms themselves. NiS₂ is different: calculations show that part of its electron density is naturally centred at special positions between atoms, specifically near the midpoints of pairs of sulfur atoms, where no atom actually sits. Physicists call these obstructed charge centres, and materials built around them belong to a class known as obstructed atomic insulators, a concept that grew out of a broader effort over the past decade to classify materials by the geometry, and not just the energy, of their electron states.

Not ordinary metallic defects

This arrangement is not a chemical accident; it is protected by the symmetry of the crystal. When the crystal is cut to form a surface, though, the special positions that hosted this electron density can no longer fit together the way they did in the bulk. At a step edge, this mismatch creates what is known as a filling anomaly: an imbalance in how the available electronic states can be occupied, which leaves behind a state inside the insulating gap, localized right at the edge. Calculations of artificial step structures reproduce the main features seen under the microscope, including this localization, supporting the idea that the conducting channels are not ordinary metallic defects caused by broken chemical bonds, but a direct consequence of the crystal’s underlying electronic geometry.

The study also settles a long-standing question about NiS₂’s magnetism at low temperature. Below about 39 kelvin, the nickel magnetic moments arrange themselves into a non-collinear antiferromagnetic pattern, with neighbouring moments pointing in different directions that largely cancel out. Below about 30 kelvin, they rearrange again into a state that produces a small but measurable net magnetization. New neutron-diffraction measurements support a revised model of this low-temperature arrangement, resolving a debate about the material’s magnetic ground state that had lasted for roughly fifty years.

Taken together, the results give NiS₂ a significance well beyond its peculiar conducting surface. The material shows that strong repulsion between electrons, magnetism, and the geometry of electronic states can coexist within a single crystal. Such geometric protection is normally associated with topological materials, in which certain electronic states are shielded from disorder or defects by the crystal’s symmetry. The new results suggest that this protection can survive even when strong interactions push a material into being an insulator, making NiS₂ a rare setting for studying how the quantum-mechanical arrangement of electrons decides where electricity is allowed to flow within an otherwise insulating crystal.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research paper/s.

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