Active colloids are capable of locally converting chemical energy into directional motion offering the possibility to use such particles as autonomous micromotors in a range of new applications, from cargo transport at the microscale to the assembly of structures via autonomous local deposition of materials. Recently, a novel class of solitonic active matter has been realised experimentally, where the elementary building blocks are self-assembled topological solitons, dubbed “skyrmions”, in confined chiral liquid crystals (LCs) [1]. LC skyrmions are continuous, but topologically-protected localised configurations of the LC director field and are analogues of the Skyrme solitons in nuclear physics [2].

Skyrmions can “swim” under the action of weak oscillatory electric fields and exhibit rich collective dynamics with reconfigurable out-of-equilibrium skyrmion assemblies, which have no analogues in more traditional passive or active colloids [3]. By contrast to active colloids which are solid, LC skyrmions are soft as they lack physical interfaces and their motion is accompanied by the periodic expansion and contraction of topology-protected regions with large director twist [1], mimicking squirming motion of biological microorganisms. The skyrmion velocity sensitively depends on the strength and the modulation frequency of the applied electric field [1], and additionally, can be controlled by combining laser tweezer techniques and photo-patterning of the in-plane LC director [4].

Driven LC skyrmions offer a distinct paradigm of solitonic active particle-like structures without mass transport [1]. The experimental setups employed to stabilise and study active skyrmions are similar to those used in LC display technologies, which provides an opportunity for the development of reconfigurable electro-optic materials [5]. Numerical modelling based on the Frank-Oseen (FO) [1] and Landau-de Gennes (LdG) [6] free energies resolves the spatio-temporal structure of the LC order parameter field. However, these fine-grained models are computationally costly, which prohibits a comprehensive sampling of the vast space of model parameters.

Building upon the original idea of [7] we developed a collective variable model of driven motion of LC skyrmions in two-dimensional domains. In our model, the skyrmion configuration and its response to an external field is captured approximately by a few collective variables, which allowed us to address the computational challenges associated with the use of the fine-grained FO or LdG models. In particular, the collective variables include 1) the coordinates of the centre of the skyrmion core, 2) the far field nematic director n₀ , and 3) the width ξ of the twist wall around the skyrmion core. Figure 1(a) depicts a skyrmion trajectory x_s(t) upon switching the electric field on and off , and several snapshots of the approximate director configurations along the trajectory are shown in Figs. 1(b)–(e).

Our theoretical and numerical analysis demonstrates how the skyrmion velocity under periodic switching of the electric field on and off is related to the complex dynamics of n₀ and ξ. The latter provides a quantitative measure of the high twist region and changes in a non-reciprocal way within each on and off states of the field, resulting in the net skyrmion displacement over one period of the electric field. The analysis also revealed the necessary condition for the reversal of the skyrmion velocity with increasing field frequency. The existence of reversal depends on the ratio of the director relaxation times during the field-on and field-off states.

We are currently working on an extension of this method to many skyrmion systems, aiming to address out-of-equilibrium collective behaviour. It is worth noting that our method can be straightforwardly extended to magnetic skyrmions.

References

[1] P. J. Ackerman, T. Boyle, and I. I. Smalyukh, Nat. Commun. 8, 673 (2017).

[2] T. H. R. Skyrme,  Nuclear Physics 31, 556 (1962).

[3] H. R. Sohn, C. D. Liu, and I. I. Smalyukh, Nat. Commun. 10, 4744 (2019).

[4] H.R.O. Sohn, C.D. Liu, R. Voinescu, Z. Chen, I. I. Smalyukh, Optics Express 28, 6306 (2020).

[5] J.-S. Wu, and I. I. Smalyukh, Liq. Cryst. Rev. 10, 34–68 (2022).

[6] A. Duzgun, C. Nisoli, C. J. O. Reichhardt, and C. Reichhardt,  New J. Phys. 24, 033033 (2022).

[7] C. Long and J. V. Selinger, Soft Matter 17, 10437 (2021).