Describing simple layers with a complex tensor

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While we tend to think of liquids as simple isotropic fluids, many commonplace liquids possess complex internal structures, such as layering. These include cosmetic skin creams, fabric softeners, and components in devices like the pixels in LCD screens. However, because they are liquid-like within each layer but the layers are ordered into stacks, modelling these so-called lamellar fluids is notoriously challenging.  

In our most recent article in Nature Communications, we introduce a theoretical framework for better modelling layered fluids, through an original, complex tensorial order parameter. 

We take an ideal stance: layering is a fundamental symmetry breaking. We argue that layering does not need to be seen as concurrent with local particle alignment – even though it typically is in smectics, which are beautiful but topologically complex layered liquid crystals. This approach allows us to consider a broader range of systems, from smectic liquid crystals to lipid bilayers, or even the layered ridges of fingerprints [1] to ``lasagna'' phases of nuclear pasta within neutron stars [2]. Our work seeks to view these systems at the mesoscale, not looking at particles or individual layers, but in enough detail to describe the complex defect structures that make these fluids so intriguing. 

While this rich landscape of defects makes these lamellar phases interesting, it is also what contrives to make them challenging to model. Previous theories have used a complex scalar order parameter and drawn an analogy with superconductors [3], but these face significant difficulties in the regions around defects [4]. The defects come in two types: either the layers can loop over themselves, or layers can be added or removed (shown in the below figure). Exciting recent work at the microscale (resolving layers at length scales comparable to individual molecules) has documented unique pairs of quarter-charge topological defects [5], and motivates a computationally practical theory for mesoscale and hydrodynamic modelling. 

A catalogue of defects in 2D smectics. (a) Single defects marked on a high-resolution photo of a fingerprint. +1/2 disclination-type defect, -1/2 disclination-type defect and edge dislocation marked by a red cross, yellow trilateral and red circle, respectively. (b) Schematics of (left) +1/2 disclination, (centre) -1/2 disclination and (right) dislocations type defects. (c-e) Simulations of defects in circular domains with boundary conditions requiring single defects. These match the schematics in (b) and show a visualisation of the layers from the eigendecomposition of the complex tensor order parameter. 

We achieve this by introducing a complex tensorial order parameter. Mirroring the construction of real tensorial order parameters for nematic liquid crystals in which particles tend to align, our novel tensor respects the symmetry of layered fluids, while being computationally versatile. Using a steepest gradient descent method on a free energy written in terms of this complex tensor, we can reproduce a standard catalogue of lamellar topological defects. These defects persist in simulations of bulk to form a kinetically arrested glassy configuration (figure 3 in the paper) and we study annihilation dynamics of both half and quarter-charge defect pairs. 

By combining local layer orientation and the extent of ordering into a single mathematical object, the complex tensor description is capable of accurately describing layers. We hope that being able to use this mesoscopic model to reliably describe defects will prove a valuable tool for modelling layered matter in more topologically interesting situations, whether biological, technological or even in everyday household products. 

  1. Glover, J.D., Sudderick, Z.R., Shih, B.B.J., Batho-Samblas, C., Charlton, L., Krause, A.L., Anderson, C., Riddell, J., Balic, A., Li, J. and Klika, V., 2023. The developmental basis of fingerprint pattern formation and variation. Cell 123, 1.
  2. Koren, M., 2018. The Pasta in Our Stars. The Atlantic. Available at: https://www.theatlantic.com/science/archive/2018/10/neutron-stars-nuclear-pasta/573166/
  3. de Gennes, P.G., 1972. An analogy between superconductors and smectics A. Solid State Communications 10, 753.
  4. Pevnyi, M.Y., Selinger, J.V. and Sluckin, T.J., 2014. Modeling smectic layers in confined geometries: Order parameter and defects. Physical Review E 90, 032507.
  5. Wittmann, R., Cortes, L.B., Löwen, H. and Aarts, D.G., 2021. Particle-resolved topological defects of smectic colloidal liquid crystals in extreme confinement. Nature Communications 12, 623.

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