Programming emergent symmetries with saddle-splay elasticity

Abstract

The director field adopted by a confined liquid crystal is controlled by a balance between the externally imposed interactions and the liquid's internal orientational elasticity. While the latter is usually considered to resist all deformations, liquid crystals actually have an intrinsic propensity to adopt saddle-splay arrangements, characterised by the elastic constant [Formula: see text]. In most realisations, dominant surface anchoring treatments suppress such deformations, rendering [Formula: see text] immeasurable. Here we identify regimes where more subtle, patterned surfaces enable saddle-splay effects to be both observed and exploited. Utilising theory and continuum calculations, we determine experimental regimes where generic, achiral liquid crystals exhibit spontaneously broken surface symmetries. These provide a new route to measuring [Formula: see text]. We further demonstrate a multistable device in which weak, but directional, fields switch between saddle-splay-motivated, spontaneously-polar surface states. Generalising beyond simple confinement, our highly scalable approach offers exciting opportunities for low-field, fast-switching optoelectronic devices which go beyond current technologies.

Fig. 1

Liquid crystals on circular posts. a Distorted director field with saddle-splay rich region indicated in red. b Schematic of director reorientation over the top of a post. Liquid crystal is increasingly free to reorient with post height above the surface. c–j Stationary liquid crystal configurations with horizontal sections through the director field at the top of the post (c–f), simulated polarized optical microscope images (insets) and three dimensional reconstructions (g–j) colored by energy density. Post height is $h=0.2\mathrm{\mu }\mathrm{m}$ for (c–e), and $h=2\mathrm{\mu }\mathrm{m}$ for (f). k, l Experimentally observed pinwheel and boojum structures in 5CB (4-cyano-4′-pentylbiphenyl), respectively

Fig. 2

Simulated molecular alignment configurations. Calculated structures for 5CB on circular SU8 posts, colored by energy density of each elastic deformation mode. Post height is $h=0.2\mathrm{\mu }\mathrm{m}$ for pinwheel, radial, and boojum structures, and $h=2\mathrm{\mu }\mathrm{m}$ for uniform structure

Fig. 3

Spontaneous chiral symmetry breaking for 8CB on circular posts. POM images of 8CB on a 0.2 μm and b 2.0 μm posts at various temperatures. Both samples were slowly heated (1 °C/min) from smectic phase to nematic phase. c Schematic of the smectic layers adopted by 8CB on top of a SU8 post, forming a toroidal focal conic domain (TFCD). Within a TFCD, radial in-plane LC director is strongly imposed. d Frequency histograms of quantitative chirality measurements for 8CB at 34 °C in the nematic phase over 40 posts. The magnitude of the chirality is influenced by the post height, such that short posts induce greater chiral symmetry breaking. e–g Simulated POM images of 8CB alignment at 34 °C as a function of saddle-splay constant ($K_{24}∕K_{22}$) and SU8 surface anchoring strength. Blue dash boxes indicate structures consistent with experiment as described in the main text; the intersection of these three regions is highlighted in red

Fig. 4

Control of defect location. a Polarizing microscope image of 5CB in contact with trefoil posts. b Section through a calculated structure with defect located at the center of the trefoil; c a structure with defect centered on one of the disks. Insets: Simulated POM images for each structure

Fig. 5

Periodic annular posts. a Schematic of annuli pattern, showing several periods. b Experimental observation and c simulation of 5CB on an array of touching annuli. An extended array of −1/2 defects forms and breaks the pattern symmetry. d–l Calculated director profiles (d, g, j) at the top of post, (e, h, k) simulated POM images and (f, i, l) director profiles throughout the the cell for W = 8 μJ/m² and d–f $K_{24}∕K_{22}=0.7$, g–i$K_{24}∕K_{22}=1$ (the value that best agrees with experiment) and j–l $K_{24}∕K_{22}=1.9$. m Simulated microscope images of 5CB on periodic annular posts as a function of saddle-splay constant $K_{24}∕K_{22}$ and surface anchoring strength. The parameter pairing that best agrees with experiment is highlighted with a red box

Fig. 6

Demonstration of a multistable device. a Device schematic showing electrodes and region of annuli. b–e Set of stable, cardinal states that may be switched into by the application of a small (0.5 V/$\mathrm{\mu }$m) transverse, directional electric field as indicated by each yellow arrow. Each of these states remains stable upon removal of the field and any starting state can be used as a precursor to any chosen new state. f By careful tuning of the applied field, the system can also access the four ordinal states, in which the defects reside along system diagonals. All scale bars are $20\mathrm{\mu }\mathrm{m}$. Insets show the simulation configuration of Fig. 3e, with the distorted regions highlighted