Monomer,Dimer, Trimer…Macromolecules: in the topological world

As the famous popular science book “One Two Three... Infinity: Facts and Speculations of Science” describes how the world is built from the simplest units, we utilize liquid crystal systems to explore the possibilities of topological world based on the simplest units: skyrmion and point defects.
Monomer,Dimer, Trimer…Macromolecules: in the topological world
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Upon arriving at the Physics Department of CU Boulder, visitors are immediately captivated by the impressive tower named in honor of the renowned physicist, George Gamow (Fig. 1a). To reach the highest floor of this tower, the 11th floor, one must start from the hallway; which showcases George Gamow's famous science book "One Two Three ... Infinity" and then pass through each floor connected by stairs, ascending to the top floor. Interestingly, in the realm of topology, we find that the elementary units, skyrmions, can be connected with each other through another kind of fundamental topological structures known as topological point defects, creating dimers, trimers, and macromolecules, which we refer to as polyskyrmionomers.

Figure 1) a: Gamow Tower at CU-Boulder. b: Schematic of a 2D skyrmion with n(r)  shown by arrows according to the coloured sphere (bottom inset) representing the order parameter space of vectorized n(r). c: Schematic of a skyrmion tube. d. Schematic of a toron (left) and a hyperbolic point defect (right). e: A schematic shows analogies between nuclei and skyrmion, electron and point defect, atom and toron, molecule and twistion2, and macromolecule and polyskyrmionomer. f: Polarizing optical micrographs and brightfield transmission-mode optical images of polyskyrmionomers. Scale bars indicate 5 μm. g: Encoding of word “CU” using the ASCII binary table.

Figure 1) a: Gamow Tower at CU-Boulder. b: Schematic of a 2D skyrmion with n(r)  shown by arrows according to the coloured sphere (bottom inset) representing the order parameter space of vectorized n(r). c: Schematic of a skyrmion tube. d. Schematic of a toron (left) and a hyperbolic point defect (right). e: A schematic shows analogies between nuclei and skyrmion, electron and point defect, atom and toron, molecule and twistion2, and macromolecule and polyskyrmionomer. f: Polarizing optical micrographs and brightfield transmission-mode optical images of polyskyrmionomers. Scale bars indicate 5 μm. g: Encoding of word “CU” using the ASCII binary table.

Skyrmion was originally proposed as a model of the nucleon by Tony Skyrme and is a topologically stable field configuration in particle physics. In two-dimension,  skyrmions are often referred to as “baby skyrmions” (Fig. 1b). They serve as low-dimensional condensed matter analogues of Skyrme solitons in high-energy and nuclear physics, and also find extensive applications in race-track-memory data storage and other spintronics. When embedded in 3D samples, such as chiral liquid crystal and magnetic systems, the skyrmions exist in the form of translationally invariant 2D tube-like structures (Fig. 1c), which are topologically protected and cannot be eliminated from a uniformly oriented background without destroying the order. However, by introducing singular defects (Fig. 1d), the skyrmion tube terminates at a pair of opposite-charged point defects (Fig. 1d). This structure is referred to as toron1, which plays the role of macroscopic “atoms” (Fig. 1e). The interaction among torons is repulsive thus they cannot be bound into “molecules”. But in our chiral LC system, the introduction of additional opposite-charged point defects can bind the soliton “atoms” together, forming a stable structure akin to “covalent bonds” (Fig. 1e)2. In turn, these defects effectively play the role of shared “electrons” (Fig. 1e) and the skyrmions can be treated as the “nuclei”. In this way, the higher-order polymer-like structures, polyskyrmionomers, is analogues to the macromolecules in the polymeric world (Fig. 1e-f).

By carefully studying the structures of polyskyrmionomers, we find 2D skyrmion number computed for planar cross-sections varies at different sample depths along both the z and x directions, which shows the potential for data storage. Along the z direction, the skyrmion number Nsk(z) equals to unity at the midplanes but may be different near top and bottom confining surfaces. To classify the structural diversity of such solitonic assemblies, we use Nt and Nb as the superscripts and subscripts of “S” to denote a structure exhibiting Nsk(z)=Nt skyrmions at the top and Nsk(z)=Nb skyrmions at the bottom, where “S” stands for the “skyrmion”. Along the polyskyrmionomer’s chain extension direction, the x direction, by changing the number and positions of point defects, different meta-stable states of higher order polyskyrmionomers can be created, which feature different skyrmion number series. Within these series, the skyrmion number has only three values: -1, 0, 1; if we treat Nsk(x) = -1 as (0)2 and Nsk(x) = +1 as (1)2 in the binary data presentation form, with Nsk(x) = 0 as a space separating different bits, we can use a polyskyrmionomer of order 9 to store an 8-bit binary string by arranging the specific sequence of defect positions, and as an example we show a polyskyrmionomer-based encoding of “CU” in Fig. 1g. A standard English alphabet have been encoded by 26 different order-9 polyskyrmionomers, which can be a new language basis of “Skyrmioness”.

Moreover, the polyskyrmionomers exhibit rich and interesting dynamic properties under the action of oscillating voltage, which are dependent on their structures and symmetries. For example, there are three types of dimers: ,  and .  and  configurations rotate clockwise and counter-clockwise, respectively, and  structures remain static. This is consistent with their symmetry, as one can obtain  and  from each other by upside-down flipping, and flipping  transforms it into itself. Hence,  and  exhibit opposite rotation directions, while  does not have a rotation direction preference. We also observed that higher-order structures rotate slower than the lower-order ones and exhibit more complex dynamics, like bending, self-folding and waving, as also established for their semi-flexible chemical macromolecular analogues.

Figure 2) a: A dimer array. b: A dimer gas. Trajectories of the geometric centre are plotted versus time by means of the colour-coded scale displayed in (a). c: Star-shaped polyskyrmionomers.

Similar to atoms, where external fields can cause various vibrations reflecting interactions between the bound quasi-atoms, we observed these topological molecules exhibit over-damped vibrations caused by external electric field. For an order-N polyskyrmionomer, the behaviour is analogues to N strings connected in a series, where the effective damping coefficient and eigenfrequency of the macrosoliton’s vibrations decay as 1/N. When placing these topological molecules close to each other, they exhibit out-of-equilibrium inter-solitonic interactions, altering directional rotations of each other and forming an array or gas (Fig 2a,b). The polyskyrmionomers can also interact with other spatially localized topological director configurations, like torons1, hopfions3 and möbiusons4. These co-propagations reveal the polyskyrmionomer’s potential in implementing transportation of topological cargo.

In addition to the linearly shaped polyskyrmionomers, the polyskyrmionomer family also includes star- and ring-shaped ones (Fig. 2c), analogues to corresponding macromolecules. Various shapes of polyskyrmionomers can undergo inter-transformation beyond the range of stabilizing voltage: the ring-shaped polyskyrmionomer can shrink and transform into linear ones, the star-shaped polyskyrmionomers can transform into linear ones or monomers, and the linearly shaped polyskyrmionomers can change from higher-order to lower-order configurations, accompanied by a progressive decrease in length and discrete decrease in skyrmion number. These properties could be considered being analogues of electrolytic splitting of conventional molecules.

Much like in molecules comprising atoms bound by covalent bonds, while sharing electrons, polyskyrmionomers form via individual skyrmions sharing singular point defects. Such solitonic configurations can form dimer, trimer and even macromolecules. This may help establish solitonic structures in chiral LCs as model systems to study not only the topological solitons in various experimentally inaccessible physical systems, but also be useful for modelling the behaviour of polymers and both equilibrium and active matter systems made from semi-flexible chain-like building blocks.

References:

[1] Smalyukh, I. I., Lansac, Y., Clark, N. A., & Trivedi, R. P. Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids. Nat. Mater. 9, 139–145 (2010).

[2] Ackerman, P. J. & Smalyukh, I. I. Reversal of helicoidal twist handedness near point defects of confined chiral liquid crystals. Phys. Rev. E 93, 052702 (2016).

[3] Ackerman, P. J. & Smalyukh, I. I. Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids. Nat. Mater. 16, 426–432 (2017).

[4] Zhao, H., Tai, J.-S. B., Wu, J.-S. & Smalyukh, I. I. Liquid crystal defect structures with Möbius strip topology. Nat. Phys. 19, 451-459 (2023).

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