Liquid Crystal Skyrmions as Elastic Multipoles

Understanding How Liquid-Crystal Skyrmions Behave Like Elastic Multipoles

Using electric fields and simplified mathematical models to explain how soft-matter solitons form chains and clusters

Introduction

Liquid crystals can support small, swirling structures known as skyrmions, which are localized regions where the orientation of the molecules twists into a stable pattern. Although these structures are formed from changes in a continuous material, they can move through the liquid crystal, interact with one another, and assemble into larger arrangements, so they often behave much like individual particles.

This behaviour has attracted attention because it may allow researchers to create materials whose internal structure can be rearranged using an electric field. Experiments have already shown that liquid-crystal skyrmions can gather into chains or spread into two-dimensional clusters when the field is repeatedly switched on and off. The difficulty lies in explaining how a change in the electric field produces such different forms of self-assembly.

A full simulation would need to track the orientation of the liquid crystal throughout the material as each skyrmion changes shape and interacts with its neighbours. That becomes computationally demanding once the system contains more than a small number of skyrmions. A basic particle model is much easier to simulate, although it may overlook the distortions in the liquid crystal that actually produce the forces between the skyrmions.

Teixeira et al. address this problem by describing each skyrmion through the distortion it creates in the surrounding liquid crystal. Far from the centre of the skyrmion, this distortion can be expressed using elastic multipoles, which play a role similar to multipoles in electrostatics. An electric charge produces one type of field, while a dipole or quadrupole produces a field with a different spatial pattern. In much the same way, the elastic dipole and quadrupole moments of a skyrmion describe how its distortion extends through the liquid crystal and influences nearby structures.

The orientation of the liquid-crystal molecules is represented by a director field, which gives the average direction in which the molecules point at each position. Within a skyrmion, this field twists away from the surrounding alignment while remaining stable as a whole. Once an external electric field is applied, the skyrmion deforms into a more asymmetric structure known as a bimeron. Its elastic multipole moments change along with its shape, altering both the direction and range of its interactions with other skyrmions.

The central idea of the study is that these changes in the multipole moments can explain why the same group of skyrmions may form chains under one set of electric-field conditions and clusters under another.

Method and Procedure Simplified

Teixeira et al. combined mathematical analysis with computer simulations, beginning with a model of the liquid crystal itself. They used the Frank–Oseen free energy, which describes how much energy is associated with distortions in the director field. Bending or twisting the molecular orientation raises the energy of the system, so this framework makes it possible to calculate which director configurations are stable and how they respond to external influences.

The model also accounted for the surfaces of the liquid-crystal cell. These surfaces impose an anchoring condition that encourages the molecules to point in a preferred direction. When an electric field is applied, it competes with this alignment and gradually deforms the skyrmion.

After obtaining the director-field configuration, the researchers examined the distortion outside the skyrmion’s central region. The distortion becomes weaker with distance, which makes it possible to describe the far field through a multipole expansion. From this expansion, they determined the elastic dipole and quadrupole moments associated with each skyrmion.

With no applied electric field, the skyrmion remains relatively symmetric and behaves mainly as an elastic dipole. Under an electric field, its shape becomes asymmetric and the quadrupole contribution becomes significant. These moments then provide a compact description of the effective interaction between neighbouring skyrmions.

The electric field used in experiments is often switched on and off, so the skyrmion does not immediately settle into a new shape whenever the field changes. Its multipole moments take time to adjust. The researchers represented this delayed response using exponential relaxation functions, which allowed the dipole and quadrupole moments to evolve gradually between their field-on and field-off values.

They then used the resulting interaction forces in particle-based molecular-dynamics simulations. Each skyrmion could be treated as a moving object whose interactions were determined by its changing elastic multipole moments, allowing the researchers to study much larger groups than would have been practical in a complete director-field simulation.

By varying the electric-field strength and the rate at which it was switched, they observed how the skyrmions moved over time and which collective structures eventually emerged.

Key Results Simplified

The simulations showed that an elastic-multipole description can reproduce the main forms of skyrmion self-assembly seen in experiments. A small number of multipole quantities was enough to capture how the interactions changed when the skyrmions were deformed by the electric field.

The shape of each skyrmion played a major role in determining the interaction. When the field was off, the skyrmion remained more symmetric and the elastic interaction was strongly screened, so its influence weakened quickly with distance. Once the field was applied, the skyrmion changed into a bimeron and produced a more directional distortion that extended farther through the liquid crystal. Nearby skyrmions could then influence one another over a larger distance, making organized structures more likely to develop.

The balance between the dipole and quadrupole contributions determined the form of that organization. A strong dipole interaction encouraged the skyrmions to align with one another, which led to the formation of elongated chains. When the quadrupole contribution became more important, the preferred arrangement spread across two dimensions and clusters emerged.

This relationship offers a physical explanation for the different structures reported in experiments. The transition between chains and clusters follows from changes in the elastic interaction created by the skyrmion’s shape, rather than being an accidental result of particle motion.

The electric-field strength affected this balance because stronger fields produced greater deformation and generally increased the relative influence of the dipole interaction. Under those conditions, chains were more likely to form. At lower or moderate field strengths, the quadrupole contribution could become more prominent, which favoured cluster formation.

The switching frequency mattered because the skyrmion needed time to adjust after the field changed. When switching occurred rapidly, the multipole moments could not fully reach their field-on or field-off values before the next change. At slower frequencies, the skyrmions had more time to relax toward their preferred shapes. The structure that formed therefore depended not only on the field strength but also on how the switching period compared with the relaxation time of the skyrmions.

This effect was particularly important at intermediate field strengths, where neither multipole contribution completely dominated. Small changes in the switching frequency could shift the balance enough to alter the final arrangement.

Across the range of conditions studied, the particle simulations reproduced the broad chain-to-cluster behaviour observed experimentally. The results suggest that the complicated collective motion of many skyrmions can be understood through the evolution of a relatively small set of elastic multipole moments.

What Is New?

Researchers had already observed that liquid-crystal skyrmions move and assemble under changing electric fields, although the connection between the microscopic director field and the resulting collective patterns was not fully understood.

Detailed numerical models could reproduce the liquid-crystal structure with considerable accuracy, but their computational cost limited the number of skyrmions that could be studied at once. Particle simulations could handle much larger systems, yet their interaction rules were often chosen without a direct derivation from the underlying liquid-crystal distortions.

The multipole approach developed by Teixeira et al. connects these two scales. The elastic dipole and quadrupole moments are obtained from the director field around a skyrmion, and those same moments are then used to calculate the forces in a many-particle simulation. The model remains manageable while retaining a clear physical connection to the liquid crystal.

The study also gives a more direct explanation of the chain-to-cluster transition. Chains appear when dipole-like interactions are strongest, while clusters emerge when quadrupole-like effects become more influential. Since the electric field changes the shape of the skyrmions, and their shape determines their multipole moments, the external field indirectly controls the way the skyrmions organise.

This framework makes it possible to think of liquid-crystal skyrmions as tunable elastic objects whose effective interactions change over time. It may therefore be useful for predicting how skyrmions will behave under field conditions that have not yet been tested experimentally.

Limitations and Issues

The model captures the main self-assembly behaviour, although several approximations limit how closely it represents a real liquid-crystal cell.

The analysis is mainly two-dimensional, while experimental systems often contain three-dimensional structures such as torons. A confined liquid crystal may vary across its thickness, and those variations can influence the stability and interaction of the skyrmions. The two-dimensional treatment captures the dominant behaviour more simply, but some features of the full structure are inevitably lost.

The multipole expansion was also restricted mainly to dipole and quadrupole terms. Higher-order contributions become weaker at long distances, which makes this approximation reasonable for separated skyrmions, although those terms may become important when two structures come close together or become strongly deformed.

The interaction model is largely pairwise, meaning that the force between two skyrmions is calculated and then combined with the forces from other pairs. In a real liquid crystal, one skyrmion may alter the director field around another and thereby change the second skyrmion’s multipole moments. These polarization and many-body effects are not fully included.

The authors suggest that such effects may be relevant to the repulsion observed between symmetric skyrmions, which the lowest-order multipole model does not completely reproduce.

The time dependence of the multipole moments is described through simple exponential relaxation. This gives the model a clear and practical way to represent delayed shape changes, but the actual dynamics may involve several relaxation processes and may respond differently under stronger or more rapidly changing fields.

A more complete model could incorporate three-dimensional director dynamics and allow the multipole moments to respond to the local arrangement of neighbouring skyrmions. Higher-order terms may also be needed when studying dense systems or short-range interactions.

Why Does It Matter?

The study gives researchers a practical way to connect the microscopic structure of a liquid crystal with the motion of a much larger group of skyrmions. That connection is important because useful material behaviour usually emerges from the collective motion of many structures, even though the underlying forces originate from molecular-scale distortions.

Liquid crystals are already used in display technology and optical devices because their properties respond to external fields. Skyrmions add another level of control, since they can move through the material and assemble into patterns that change when the field conditions are altered.

A material built around this behaviour could reorganize its internal structure without being physically rebuilt. Switching the electric field might move the system from a chain-like arrangement to a clustered one, changing how the material transmits light or responds mechanically.

This possibility fits into the broader field of programmable soft matter, where researchers aim to design materials whose properties can be adjusted after fabrication. Liquid-crystal skyrmions could act as mobile building blocks within such materials, provided their motion and interactions can be predicted reliably.

The elastic-multipole model helps with that prediction because it identifies the physical quantities that control self-assembly. Field strength changes the shape of the skyrmions, while switching frequency determines how completely they respond, and the resulting multipole moments govern the structures that appear.

The general approach may also be useful in other systems containing topological structures. Magnetic skyrmions, for example, interact through different physical mechanisms, although they also behave as localized objects whose collective organization depends on their surrounding fields. Describing such structures through effective moments may offer a useful way of connecting detailed field theories with larger-scale particle models.

Glossary:

Liquid crystal: A soft material that can flow while its molecules maintain a preferred orientation over part of the material.

Skyrmion: A stable, localized twisting pattern in a field. In this study, the twist occurs in the orientation of liquid-crystal molecules.

Director field: A mathematical description of the average direction in which nearby liquid-crystal molecules point at each position.

Elastic multipoles: Quantities that describe the spatial pattern of the distortion produced by a skyrmion in the surrounding liquid crystal.

Bimeron: The asymmetric structure formed when an electric field deforms a liquid-crystal skyrmion.

Dipole interaction: A directional elastic interaction that tends to align skyrmions and supports chain formation.

Quadrupole interaction: An elastic interaction with a more complex directional pattern that can favour two-dimensional clustering.

Frank–Oseen free energy: A mathematical expression used to calculate the energy associated with bending or twisting the director field.

Chain–cluster transition: The change between elongated chains and two-dimensional clusters as the relative importance of the multipole interactions changes.

Bibliography

Teixeira, A. W., Dias, C. S., and Tasinkevych, M. “Liquid Crystal Skyrmions as Elastic Multipoles.” Communications Physics, vol. 9, article 1, 2026. https://doi.org/10.1038/s42005-025-02400-x

Samarth Lamba | Writer | The STEM Review