Orientational Anchors in Liquid Crystals

• Orientational anchors are boundary-imposed directions that define the easy axes in liquid crystals, setting preferred molecular alignments.
• They emerge from microscopic interfacial interactions, substrate treatments, and patterned geometries, which can induce mono- or bi-stable anchoring states.
• Studying anchoring energetics and transitions offers practical insights into defect control, phase switching, and the design of advanced liquid-crystalline materials.

Orientational anchors are preferred directions imposed on an orientationally ordered system by a boundary, interface, or prescribed geometric constraint. In liquid-crystalline systems, they are usually identified with easy axes: favored orientations of the director close to an interface, together with an anchoring strength that quantifies the energetic penalty for deviations from that orientation. Across recent work, orientational anchors appear as molecularly defined interfacial directions set by adsorbed dipoles or alkyl chains, as planar or homeotropic surface states selected by substrate treatment, as spatially patterned easy axes written by microstructure or photoalignment, and as hard topological constraints imposed at boundaries in systems with one or several coupled orientational order parameters (Lacaze et al., 2015, Nazarenko et al., 2010, Anquetil-Deck et al., 2013, Paik et al., 12 Mar 2026).

1. Definitions and physical scope

In the liquid-crystal literature, the easy axis is the orientation at an interface toward which the liquid-crystal director tends to align. If one easy axis only is defined for one given interface, the bulk director orients along or close to this axis. Two major effects compete to impose the anchoring directions of a liquid crystal: the interactions between molecules and the interface, and the substrate roughness (Lacaze et al., 2015).

The concrete realization of an orientational anchor depends on the material class. In lyotropic chromonic liquid crystals (LCLCs), plank-like molecules self-assemble in aqueous solution into polydisperse stacks or aggregates, and confinement between solid substrates produces surface anchoring that can be planar (tangential) or homeotropic, with self-assembled aggregates perpendicular to the substrate (Nazarenko et al., 2010). In thermotropic systems such as nCB films on MoS$_2$, ordered interfaces with well-defined orientations of adsorbed dipoles induce planar anchoring locked along the adsorbed dipoles or the alkyl chains, which then play the role of easy axes (Lacaze et al., 2015).

The term also extends beyond a single director field. In phases with two types of two-dimensional orientational order, orientational anchors can be implemented as boundary conditions that force the orientation of the order parameter at the system boundary to a prescribed direction, as in a flat disk with strong radial anchoring (Paik et al., 12 Mar 2026). This broader usage preserves the central idea: an orientational anchor is a boundary-imposed directional constraint that selects admissible equilibrium textures, defect content, or both.

2. Microscopic origin of easy axes

At the microscopic level, orientational anchors emerge from ordered interfacial motifs. In nCB/MoS$_2$ systems, Scanning Tunneling Microscopy reveals specific molecular arrangements at the interface, and X-Ray Diffraction together with Optical Microscopy shows how bulk alignment follows those interfacial directions. In 11CB/MoS$_2$, two alternating orientations of adsorbed cyanobiphenyl groups differ by about $20^\circ$, and each orientation acts as an easy axis. In 10CB/MoS$_2$, the double-row structure contains cyanobiphenyl groups and alkyl chains oriented at about $30^\circ$ to one another, and both directions can act as anchors. In 5CB/MoS$_2$, two orientations of adsorbed CB groups are again observed, but the macroscopic anchoring is monostable rather than bistable (Lacaze et al., 2015).

These observations establish that orientational anchors need not be unique or chemically equivalent. Adsorbed dipoles can define one or two easy axes; alkyl chains can provide an alternative easy axis; and the equivalence or non-equivalence of these microscopic axes can determine whether the macroscopic anchoring is bistable or monostable. This suggests that anchoring cannot be reduced to a purely geometric normal-versus-tangent classification: interfacial molecular order is often the decisive variable.

In LCLCs, the microscopic picture is different but leads to the same phenomenology. Traditionally, lyotropic liquid crystals favored planar anchoring due to excluded volume effects, aligning the long axes of aggregates parallel to the substrate. However, DMOAP treatment of glass plates can produce homeotropic alignment, and the observed transition from an initially homeotropic to a later planar state indicates that dynamic processes during self-assembly can select between competing interfacial orientations (Nazarenko et al., 2010).

3. Anchoring energetics and continuum descriptions

The standard phenomenology for a single easy axis is the Rapini–Papoular form,

$E = W \sin^2(\theta-\theta_0),$

where $W$ is the anchoring strength, $\theta$ is the director angle, and $_2$0 is the easy axis (Lacaze et al., 2015). In confined chromonic droplets, an equivalent planar substrate anchoring density is written as

$_2$1

while tangential anchoring at the nematic–isotropic interface is described by

$_2$2

Here $_2$3 is the planar anchoring strength, $_2$4 is the substrate-imposed easy axis, $_2$5 is the isotropic surface tension, $_2$6 is an anisotropy parameter, and $_2$7 is the interface normal (Paparini et al., 2023).

For systems with competing planar and homeotropic states, a higher-order surface free energy is required. In nematic LCLCs, a homeotropic surface can be modeled by

$_2$8

or, in angular form,

$_2$9

A similar expression applies to a planar substrate, and the full hybrid-cell free energy contains both anchoring and bulk elastic terms: $_2$0 The two minima correspond to $_2$1 and $_2$2, and the relative stability depends on the signs and magnitudes of $_2$3 and $_2$4 (Nazarenko et al., 2010).

Dual easy axes likewise require going beyond second order. For two easy axes at $_2$5 and $_2$6, the phenomenological potential

$_2$7

admits either two minima or one, depending on the sign of $_2$8 and the ratio $_2$9. The classical Rapini–Papoular model is not suitable for dual easy axes because it only permits a single energy minimum (Lacaze et al., 2015).

A distinct, but related, refinement concerns the modeling of strong anchoring. In Landau–de Gennes theory, finite anchoring is captured by a Rapini–Papoular surface term,

$20^\circ$0

with extrapolation length

$20^\circ$1

Retaining finite surface energy yields Robin-type conditions,

$20^\circ$2

rather than rigid Dirichlet conditions. In the large-domain limit, this produces an $20^\circ$3 correction to the director field near boundaries, whereas Dirichlet conditions predict only an $20^\circ$4 correction (Rajamanickam, 2 Jan 2026). A common misconception is therefore that “strong anchoring” is equivalent to fixed boundary values; the revised asymptotics show that finite anchoring energetics can remain leading-order relevant near boundaries and defects.

4. Anchoring transitions, bistability, and metastability

A central feature of orientational anchors is that they can support multiple competing minima and discontinuous transitions between them. In nematic LCLCs, both planar and homeotropic alignments are observed. The homeotropic state is stable for a finite period of $20^\circ$5–$20^\circ$6 hours, then relaxes into the planar state through nucleation and expansion of planar domains inside the homeotropic matrix. Optical retardation and director orientation change abruptly, and hybrid wedge cells exhibit a critical thickness $20^\circ$7 at which the system switches abruptly from uniform to hybrid director configurations. These are signatures of a first-order anchoring transition and of large nucleation barriers associated with a double-well surface potential (Nazarenko et al., 2010).

Ordered interfaces with dual easy axes generate a second class of discontinuous behavior. In 11CB/MoS$20^\circ$8, two adsorbed cyanobiphenyl orientations separated by about $20^\circ$9 lead to anchoring bi-stability. In 10CB/MoS$_2$0, domains align either with the cyanobiphenyl group orientation or with the alkyl chains, again resulting in bi-stability. In 5CB/MoS$_2$1, only monostable anchoring is observed although two adsorbed CB orientations are seen microscopically. The transition from bi-stable anchoring in 11CB/MoS$_2$2 to monostable anchoring in 5CB/MoS$_2$3 is interpreted as a first-order anchoring transition, tied to changes in the anchoring-potential parameters and to the ordering or non-equivalence of the two axes (Lacaze et al., 2015).

Temperature- and field-driven anchoring transitions illustrate a third route. In an organo-siloxane tetrapode nematic, homeotropic cells show a dark texture above a critical temperature $_2$4, while below $_2$5 the texture becomes birefringent, indicating that the director tilts away from the normal. The dependence of $_2$6 on cell thickness $_2$7, with $_2$8 decreasing as $_2$9 increases, identifies the effect as a surface anchoring transition rather than a bulk phase transition. A vertical electric field restores the dark homeotropic texture even below $30^\circ$0, showing that the birefringent state does not require a bulk biaxial nematic phase (Kim et al., 2013). This directly addresses a recurrent interpretive error: apparent optical biaxiality can arise from surface-induced reorientation.

Electrically tunable anchoring can also be produced by ion adsorption. In a flexoelectric nematic cell, one surface is strongly homeotropic, whereas the other surface adsorbs positive ions that act as an orienting surfactant. The surface free energy is

$30^\circ$1

with $30^\circ$2 and $30^\circ$3 determined by the surface density of adsorbed ions. The theory predicts threshold voltages for transitions between the homeotropic and hybrid homeotropic–planar states and establishes critical anchoring parameters beyond which the orientational transitions do not take place (Tarnavskyy et al., 2021).

5. Patterned anchors, geometry, and defect selection

Orientational anchors are not restricted to chemically uniform flat substrates. Patterning can split polar and azimuthal control and can generate several far-field textures from the same local anchoring rule. A periodic array of rectangles that alternately promote vertical and planar alignment provides one example. Monte Carlo simulation, experiment, and continuum theory show that the easy axis and effective anchoring energy can be controlled by the pattern design. For strong anchoring, for rectangle ratios $30^\circ$4 the nematic aligns in the direction of the long edge of the rectangles; in weak anchoring scenarios the preferential anchoring is degenerate between the two rectangle diagonals; and for intermediate combinations of anchoring coefficient and system length-scale, bistability between diagonally aligned and edge-aligned arrangements is predicted (Anquetil-Deck et al., 2013).

Patterned topography provides another route. For a periodic array of parallel grooves on a chemically homogeneous substrate with local homeotropic anchoring, the modified Frank–Oseen analysis shows that if the substrate periodicity is much larger than the extrapolation length, different nematic textures with distinct far-field orientations, and the anchoring transitions between them, are associated with the presence of topological defects either on or close to the substrate. Sawtooth and sinusoidal substrates exhibit a homeotropic-to-planar anchoring transition as the substrate roughness is increased, whereas crenellated substrates exhibit a homeotropic-to-oblique transition with a phase diagram that depends in a complex way on roughness and anchoring strength (Rojas-Gomez et al., 2016).

Photoimposed director fields turn orientational anchors into spatially varying elastic templates. In planar nematic cells with

$30^\circ$5

the director field contains one-dimensionally modulated regions of maximal splay and maximal bend. Colloidal spheres with perpendicular surface anchoring are driven into regions of maximum splay, while spheres with tangential surface anchoring settle into regions of bend (Peng et al., 2016). Here the “anchor” is not a single easy axis but a patterned orientational landscape.

Boundary anchors can also enforce topological charge. In a flat disk with strong radial anchoring, the boundary condition

$30^\circ$6

forces the orientation at each boundary site to point directly outward and imposes a total topological charge of $30^\circ$7 inside the disk. On the sphere, no external easy axis is imposed, but topology fixes the total charge. In coupled $30^\circ$8-atic and $30^\circ$9-atic order, weak coupling yields diffuse wall networks connecting defects, whereas strong coupling causes higher-order defects to merge into lower-order defects and produce stretched defect cores (Paik et al., 12 Mar 2026). A plausible implication is that orientational anchors should be understood as generators of admissible topological sectors, not only as local alignment rules.

The same logic extends to chiral nematics, where the strength of perpendicular surface boundary conditions controls whether localized structures appear as torons, skyrmions, twisted walls, fingers, or hybrids. The effective anchoring is characterized by $_2$0, or equivalently by the extrapolation length $_2$1. Strong anchoring favors defect-capped structures such as torons and CF-3 fingers, whereas weak anchoring permits defect-free skyrmions and twisted walls (Tai et al., 2019).

6. Measurement, inference, and contemporary extensions

Orientational anchors are experimentally accessible through both direct imaging and indirect parameter inference. In LCLCs, LC PolScope retardation mapping provides quantitative, spatially resolved measurements of birefringence as a proxy for local director orientation, and abrupt changes in $_2$2 reveal discontinuous reorientation. Fluorescence Confocal Polarizing Microscopy enables three-dimensional imaging of the director field across the cell, including sharp and sometimes tilted boundaries between homeotropic and planar domains (Nazarenko et al., 2010). In ordered nCB/MoS$_2$3 interfaces, STM identifies the interfacial molecular orientations, XRD determines how bulk alignment relates to interface structure, and optical microscopy visualizes anchoring domains (Lacaze et al., 2015).

Anchoring strength can also be extracted geometrically. For chromonic nematic droplets confined in thin cells with planar-aligning plates, the equilibrium droplet shape follows from a variational free-boundary problem with fixed area. The relevant dimensionless parameters are

$_2$4

and the observed aspect ratio and contour of batonnets, discoids, or tactoids can be matched to the free-energy minimum. The anchoring strength then follows from

$_2$5

The same analysis predicts shape bistability for sufficiently small droplets, where tactoid and discoid branches exchange stability as the area varies (Paparini et al., 2023).

Contemporary theory extends anchoring beyond passive single-director descriptions. At nematic–isotropic interfaces in active systems, extensile activity drives tangential or planar active anchoring and contractile activity drives perpendicular or homeotropic active anchoring. When elastic anisotropy and active anchoring compete, simulations show that active anchoring dominates except at very low activities, while elastic anisotropy mainly renders the active length anisotropic (Coelho et al., 2020). In a $_2$6-tensor description of tactoids, anisotropic anchoring encoded through higher-order elastic terms causes interface thickness to depend on the relative orientation of the director at the interface, makes interfaces biaxial for tangential alignment when anisotropy is introduced, sharpens surface defects in negative tactoids as elastic anisotropy increases, and favors semi-bipolar director configurations in positive tactoids (Schimming et al., 2022).

Taken together, these results define orientational anchors as a unifying concept for interfacial direction selection, defect organization, and multistability in ordered media. The most robust conclusion of the recent literature is not merely that surfaces prefer a given orientation, but that the form, multiplicity, and finiteness of anchoring energetics control which textures exist, which transitions are discontinuous, and which topological structures remain stable under confinement, patterning, activity, and coupling between distinct orientational order parameters.