Confinement Reveals Hidden Splay-Bend Order in Twist-Bend Nematics

TL;DR

This study uses simulations to show how confinement in thin films reveals hidden splay-bend order in twist-bend nematic phases, leading to new phases and insights into molecular organization.

Contribution

It demonstrates that confinement enhances and reveals splay-bend order, uncovering new nematic and smectic phases not observable in bulk materials.

Findings

Confinement amplifies splay-bend order in twist-bend nematics.

A smectic splay-bend phase emerges near confining surfaces.

New phases appear during the transition from confined to bulk states.

Abstract

Using extensive Monte Carlo (MC) and molecular dynamics (MD) simulations, we investigate how spatial confinement affects molecular organization within thin films of the nematic twist-bend ($\mathrm{N_{TB}}$) phase. Our simulations show that confinement markedly amplifies the otherwise elusive splay-bend order, primarily by suppressing the intrinsic three-dimensional heliconical structure characteristic of bulk $\mathrm{N_{TB}}$. Remarkably, when the $\mathrm{N_{TB}}$ phase is confined between parallel walls imposing planar anchoring, and the bulk wave vector is oriented parallel to the walls, a smectic splay-bend ($\mathrm{S_{SB}}$) phase spontaneously emerges near the confining surfaces. This intermediate structure subsequently transforms into the bulk $\mathrm{N_{TB}}$ phase either directly via a smectic splay-bend-twist ($\mathrm{S_{SBT}}$) phase or through a sequence involving both the $\mathrm{S_{SBT}}$ and the nematic splay-bend-twist ($\mathrm{N_{SBT}}$) phases. Notably, the $\mathrm{N_{SBT}}$ phase becomes particularly pronounced as the molecular bend angle approaches its maximum attainable value in bulk $\mathrm{N_{TB}}$; this regime occurs in close proximity to the $\mathrm{N}\text{--}\mathrm{S_{A}}\text{--}\mathrm{S_{SB}}$ triple point on the bulk phase diagram. Our findings reveal a compelling and intricate interplay among chirality, confinement, and molecular ordering, further evidenced by the calculated elementary director distortions. Crucially, this study opens promising avenues for experimental exploration: confined thin-film geometries serve as powerful model systems for revealing and characterizing novel nematic and smectic liquid-crystal phases that remain elusive in, or currently inaccessible to, bulk experiments.

Equations

$$\tau (x)=\frac{1}{NT}\left|\sum _{t=1}^T\sum _{n=1}^N\Theta (\Delta -|x_n(t)-x|)e^{2\pi ik{z}_n(t)/Z}\right|,$$

$$\begin{array}{r}A(x,z)=\frac{1}{NT}\sum _{t=1}^T\sum _{n=1}^N{A}_n(t)\Theta ({\Delta }_x-|x_n(t)-x|))\\ \times \Theta ({\Delta }_z-|z_n(t)-z|),\end{array}$$

Full text

nematic twist–bend, Splay–bend order,thin layers, order at surfaces, Monte Carlo and Molecular Dynamics simulation

††preprint: APS/123-QED

Introduction.— Over the past decade, significant advances in liquid-crystal research have been driven by the discovery of novel polar nematic phases [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In these phases, molecules exhibit diverse forms of long-range orientational and polar order, while their centers of mass remain randomly distributed, as in isotropic fluids. Among the most striking examples are the twist-bend nematic ($\mathrm{N_{TB}}$) [1, 2, 3, 14, 15] and the splay-bend nematic ($\mathrm{N_{SB}}$) [10, 11] phases, realized in systems of chemically achiral bent-core molecules and colloids.

In the $N_{TB}$ phase, achiral bent-shaped mesogens form a heliconical, locally polar nematic (Fig. 1). The primary order parameter is a transverse vector polarization $\mathbf{p}(z)$, orthogonal to both the director $\mathbf{n}(z)$ and the helical axis (parallel to the wavevector $\mathbf{k}$). The texture combines twist–bend distortions of $\mathbf{n}$ with a co-precessing polarization. The local point symmetry is chiral, polar monoclinic $C_{2}$, with the twofold (polar) axis parallel to $\mathbf{p}$. The director maintains a constant tilt and, together with $\mathbf{p}$, precesses with a single pitch ($\frac{2\pi}{k}$), typically on the order of $\sim 10\,\mathrm{nm}$.

Formation of the $\mathrm{N_{TB}}$ phase requires no molecular chirality, yielding equally probable left- and right-handed heliconical domains. The weakly first-order transition from the uniaxial nematic ($\mathrm{N_{U}}$) or isotropic ($\mathrm{Iso}$) phase to $\mathrm{N_{TB}}$ constitutes spontaneous mirror-symmetry breaking in the absence of long-range translational order. By contrast, the nonchiral and globally nonpolar $\mathrm{N_{SB}}$ phase exhibits periodic splay–bend modulations of the director and polarization fields confined to a single plane, with the polarization usually either perpendicular to the local director or locally vanishing (Fig. 1).

From a theoretical perspective, Meyer [16] first proposed the $\mathrm{N_{TB}}$ and $\mathrm{N_{SB}}$ phases by linking shape-induced spontaneous polarization to splay or bend deformations. In 1990, we introduced a flexopolarization-induced coupling between the alignment tensor and polarization fields that yields twist–bend order within a generalized Landau–de Gennes (Ginzburg–Landau–type) framework [17], which has since been validated quantitatively for $\mathrm{N_{TB}}$-forming CB7CB-like mesogens [18].

A critical advance came in 2001 with Dozov’s work [19], which generalized the Oseen–Zocher–Frank elastic theory by proposing that molecular shapes favoring bend can reduce, and even invert, the nematic bend elastic constant $K_{33}$, thereby stabilizing either the $\mathrm{N_{TB}}$ or $\mathrm{N_{SB}}$ phase depending on the ratio of splay ($K_{11}$) to bend ($K_{33}$) elasticity. Specifically, if $K_{11}>2K_{33}$, the heliconical $\mathrm{N_{TB}}$ structure is favored, whereas if $K_{11}<2K_{33}$, the planar $\mathrm{N_{SB}}$ structure becomes more stable. Subsequent theoretical work [20, 4] showed that Dozov’s elastic theory can be recast as a flexopolarization mechanism underlying the inversion of $K_{33}$. These studies also predicted more complex phases featuring the coexistence of splay, bend, and twist deformations [21, 22, 23, 24, 25, 26, 27] (Fig. 1).

Experimentally, the $\mathrm{N_{TB}}$ phase has been widely observed across numerous thermotropic liquid-crystal systems [4, 15, 28], whereas the $\mathrm{N_{SB}}$ phase and other complex polar nematic phases remain rare, reported primarily in colloidal systems [10, 11] or under applied electric fields [29, 30]. This scarcity persists despite Dozov’s relatively broad—and, in principle, readily satisfied—elastic-constant criteria, underscoring the need for further theoretical and computational studies to elucidate the mechanisms governing the stability of polar nematic phases.

Motivated by these experimental findings and by the unresolved scarcity of a stable $\mathrm{N_{SB}}$ phase—contrasting with the widespread occurrence of its parent $\mathrm{N_{TB}}$ phase—we investigate whether confinement can stabilize $\mathrm{N_{SB}}$. We examine the molecular organization in thin films of a bulk-stable $\mathrm{N_{TB}}$ phase confined between parallel walls imposing planar anchoring. Studying such confinement can clarify the mechanisms governing the stability of splay–bend order and reveal phenomena relevant to both fundamental research and technological applications [31]. Our objective is to advance theoretical understanding and to inform future experimental studies of confined bent-core molecular systems.

Model.— To investigate the effects of confinement on bent-core nematics, we employed two closely related coarse-grained models that capture essential features of molecular ordering. In our MC simulations, performed in the constant-pressure ensemble, each molecule was modeled as a rigid assembly of eleven mutually tangent spheres (diameter $\sigma=1$) arranged equidistantly along a circular arc with a tunable bend angle $\chi$ ranging from $180^{\circ}$ (linear chain) to $0^{\circ}$ (semicircle) (see Fig. 2). For MD simulations, the hard-sphere repulsion of the MC model was replaced by the truncated and shifted repulsive part of the Lennard–Jones potential, i.e., the differentiable Weeks–Chandler–Andersen (WCA) interaction [32, 33]. The WCA sphere diameter was matched to its hard-sphere counterpart via the Heyes–Okumura formula [34], thereby ensuring quantitative consistency in phase behavior and observables across both models [27]. Greco and Ferrarini [35] first showed—using MD simulations and density-functional theory (DFT)—that packing entropy alone can stabilize the $\mathrm{N_{TB}}$ phase. Importantly, their molecular model was identical to the coarse-grained arc-of-spheres model defined above. Kubala, Tomczyk, and Cieśla extended this analysis by combining MC and MD simulations and mapping the bulk phase diagram as a function of bend angle and packing fraction, thereby identifying the stability regions of the $\mathrm{N}$, $\mathrm{N_{TB}}$, and smectic phases [27]. Building on these results (see Fig. 3), our present work focuses on the confinement-induced structural organization of the $\mathrm{N_{TB}}$ phase between two parallel walls.

Simulation Methods.— To investigate the effects of confinement on bent-core nematics, we employed two closely related coarse-grained models described above. MC simulations were performed in the isothermal–isobaric (isotension) ensemble using custom software developed by P.K. (see Code and datasets availability). MD simulations were carried out with LAMMPS [36], using both $NpT$ and $NVT$ ensembles.

We considered a confinement geometry in which a monodomain of the $\mathrm{N_{TB}}$ phase was placed between two parallel, structureless planar walls of finite extent. The walls were oriented parallel to the $y$–$z$ plane, and the helical wave vector of the confined $\mathrm{N_{TB}}$ domain was aligned with the $z$ axis. Periodic boundary conditions were applied along $y$ and $z$ (parallel to the walls). Initial configurations were prepared by equilibrating bulk samples (see Ref. [27]) and then introducing the confining walls. System sizes reached up to $N=12{,}000$ molecules (MC) and $N=24{,}000$ molecules (MD). Equilibration ran up to $3\times 10^{8}$ MC cycles and $5\times 10^{6}$ MD steps, followed by production runs of $3\times 10^{8}$ MC cycles and $6\times 10^{7}$ MD steps for ensemble averaging. Walls were planar and structureless; in MD simulations, wall–particle interactions were modeled with the WCA potential. Both approaches yielded quantitatively consistent results.

*Results.—*Detailed simulations were carried out along the blue lines in Fig. 3. In all cases, the equilibrium order observed at the walls is a smectic splay–bend ($\mathrm{S_{SB}}$) phase, in which director modulation is coupled to density modulation. Moving away from the walls toward the center of the sample—where the bulk $\mathrm{N_{TB}}$ phase is stable—the splay distortions and density modulations decay through a sequence of intermediate structures. Representative results for $\chi=110^{\circ}$ (MC) and $\chi=135^{\circ}$ (MD) are shown in Figs. 4–6.

Near the $\mathrm{S_{SB}}\!-\!\mathrm{S_{A}}$ coexistence wedge and close to the $\mathrm{N_{TB}}\!-\!\mathrm{S_{A}}$ boundary (Fig. 3), the intermediate structures are characterized by $\mathrm{N_{SBT}}$ ordering adjacent to $\mathrm{N_{TB}}$ regions. Upon approaching the walls, the $\mathrm{N_{SBT}}$ phase gradually transforms into $\mathrm{S_{SBT}}$, which ultimately converts into $\mathrm{S_{SB}}$ near the walls. This evolution of the $\mathrm{N_{TB}}$ phase is illustrated in the top panel of Fig. 5 for $\chi=135^{\circ}$ and packing fraction $\eta=0.335$.

Furthermore, as the packing fraction decreases, the local smectic order parameter $\tau$, defined as

[TABLE]

also decreases (Fig. 6). Here, the sums run over $N$ molecules and $T$ configuration snapshots; $z_{n}(t)$, $Z$, and $k$ denote the $z$-position of the $n$-th molecule at time $t$, the box length along $z$, and the number of density–modulation periods along the wave–vector direction, respectively. The Heaviside step function $\Theta$ restricts the average to molecules whose centers lie within a slab of half-width $\Delta=0.12\,L_{\mathrm{mol}}$ centered a distance $x$ from the nearest wall. For each $\eta$, the resulting $\tau(x)$ was fitted with $\tau(x)=\tau_{0}e^{-x/\lambda}$. The $\eta$-dependence of $\tau_{0}$ and $\lambda$ is shown in the bottom panel of Fig. 6. With decreasing packing fraction, the $\mathrm{S_{SB}}$ phase at the walls also weakens and eventually disappears as $\eta$ approaches values characteristic of the bulk nematic or isotropic phases.

Guided by the structural analysis above, we further quantify the interfacial fine structure of orientational order by decomposing the director–gradient field into the canonical Oseen–Frank modes (splay, twist, bend, and saddle–splay). This mode-resolved perspective provides, to our knowledge, the first direct bridge between particle-resolved simulations of confined $\mathrm{N_{TB}}$ and continuum elasticity, and it pinpoints where boundaries select distinct elastic responses—most notably how interfacial layers accommodate chirality and activate the saddle–splay channel. Experiments and modeling by Xia et al. demonstrate that suitably programmed surfaces can control symmetry via this channel [37]. In our system, the channel is activated differently: competition between the heliconical bulk $\mathrm{N_{TB}}$ texture and planar surface anchoring selects the observed sequence of interfacial ordering. Following Selinger’s geometric formulation, we monitor the saddle–splay interfacial density on the same footing as splay, twist, and bend [38].

For completeness, and to connect with our maps, for $i\in\{x,y,z\}$ we define $\mathrm{splay}_{i}=n_{i}\,(\nabla\!\cdot\!\mathbf{n})$ and $\mathrm{bend}_{i}=\big(\mathbf{n}\times(\nabla\times\mathbf{n})\big)_{i}$, while the pseudoscalar and scalar fields are $\mathrm{twist}=\mathbf{n}\!\cdot\!(\nabla\times\mathbf{n})$ and $\mathrm{saddle\text{–}splay}=-\,\nabla\!\cdot\!\Big(\mathbf{n}\,(\nabla\!\cdot\!\mathbf{n})+\mathbf{n}\times(\nabla\times\mathbf{n})\Big)$, respectively. We also evaluate the coarse-grained polarization vector field $\mathbf{p}(x,z)$, which, in our sterically driven model, is given by the local average of the molecular short axis ${\hat{\mathbf{b}}}$.

To compute any coarse-grained observable $A(x,z)$ from a microscopic quantity $A_{n}$, we use the same slab averaging employed for $\tau(x)$:

[TABLE]

where $\Delta_{x}=0.04\,L_{\mathrm{mol}}$ and $\Delta_{z}=0.03\,L_{\mathrm{mol}}$ are the slab half–widths. Choosing $A_{n}=\hat{\mathbf{b}}_{n}$ yields the polarization $\mathbf{p}(x,z)$. Choosing $A_{n}=\tfrac{3}{2}\,\hat{\mathbf{a}}_{n}(t)\otimes\hat{\mathbf{a}}_{n}(t)-\tfrac{1}{2}\,\mathbf{I}$ yields the alignment tensor $\mathbf{Q}(x,z)$. The local director $\mathbf{n}(x,z)$ is then defined as the normalized eigenvector of $\mathbf{Q}(x,z)$ corresponding to its largest-magnitude (nondegenerate) eigenvalue. Spatial derivatives are obtained by convolving the discretized director field with standard $3\times 3$ Sobel kernels to approximate first-order gradients [39, 40]. Results are shown in Fig. 7.

Guided by Fig. 7, we find a robust interfacial orientational pattern representative of the two blue simulation paths in Fig. 3. Near the center of each interfacial layer, the texture locks into a heliconical nematic twist–bend state ($\mathrm{pitch}=\frac{2\pi}{k}\approx 4.5L_{mol}$ in Fig. 7), and the deformation maps show the corresponding position–independent signatures. Along this mid–plane of the $\mathrm{N_{TB}}$ slab we find (up to numerical accuracy) $\mathrm{splay}=0$, $\mathrm{bend}_{z}=0$, and saddle–splay $=0$, while twist retains a fixed sign across the slab. The in–plane bend components $\mathrm{bend}_{x}$ and $\mathrm{bend}_{y}$ exhibit the same periodic modulation with the expected quarter–period phase shift along $z$. The $n_{z}$ component of the director is nearly constant and less than unity, indicating saturated tilt, while the polarization field is essentially collinear with bend: $p_{x}$, $p_{y}$, and $p_{z}$ stripes with the same wavelength and phase as $\mathrm{bend}_{x}$, $\mathrm{bend}_{y}$, and $\mathrm{bend}_{z}$, respectively. This is consistent with the bend–polarization relation $\mathbf{p}\sim\,\mathbf{n}\times(\nabla\times\mathbf{n})$, up to an overall scale and a sign set by the handedness. These features again confirm that the layer’s $\mathrm{N_{TB}}$ interior is consistent with bulk twist–bend ordering and acts as a phase–matching sub-layer between the two walls.

The boundary–driven structure—set by the competition between planar anchoring and the heliconical bulk—is confined to the near–wall regions $x/L_{\max}\!\lesssim\!2$ and $x/L_{\max}\!\gtrsim\!10$. There, $n_{z}$ displays the same axial wavelength and comparable amplitude at both walls, but the bright/dark bands are offset by almost half a period along $z$. This near–antiphase relation is the signature of the improper symmetry that relates the two interfacial skins (reflection about the mid–plane combined with a half–pitch translation along $z$). Small even–harmonic content in the boundary layers explains the slight, systematic misalignment of extrema (crests do not map exactly onto troughs).

Across the $S_{SBT}$ skins (splay–bend–twist), a weak, alternating twist localizes and coexists with alternating splay and saddle–splay; this is the entropic/elastic cost of steering the texture away from the purely planar bend favored by the walls. Directly at the walls ($S_{SB}$) the twist channel is suppressed, the splay components strengthen, and the saddle–splay shows sign–selective lobes that are phase–locked to the splay/bend bands—consistent with a saddle–like (negative Gaussian curvature) distortion producing surface torques that reinforce the interfacial splay–bend texture. While splay and saddle–splay are strongest in the smectic regions, they remain weak but finite in the $\mathrm{N_{SBT}}$ bands. Finally, the $y$-component of the bend field is nearly identical across all regions.

Concerning polarization at the walls, $p_{x}$ develops a finite mean component with opposite signs on the two sides—numerically $p_{x,L}\!\approx\!-0.17$ and $p_{x,R}\!\approx\!+0.17$. This antisymmetric offset is consistent with the flexo–splay contribution, proportional to $\mathbf{n}\,(\nabla\!\cdot\!\mathbf{n})$: the near–wall splay $(\nabla\!\cdot\!\mathbf{n})$ is odd under the mid–plane reflection $x\!\to\!L-x$, so the $x$-projection of $\mathbf{p}$ changes sign from one wall to the other. By contrast, $\mathrm{bend}_{x}$ at the walls is dominated by its oscillatory fundamental and has (nearly) zero mean, so it carries no comparable constant component. Consequently, the sample–averaged polarization along $x$ vanishes by symmetry, while the two walls host equal–and–opposite interfacial polarizations.

Discussion.— Understanding how periodically modulated polar nematics form and remain stable is pivotal for advancing liquid–crystal theory and enabling reconfigurable optical elements and display concepts. With planar anchoring and the wavevector parallel to the plates, confinement selects a robust wall–to–wall architecture: thin $\mathrm{S_{SB}}$ skins at the walls (as in strictly 2D flexible bend–core systems [41]), $\mathrm{S_{SBT}}$ buffers, and—near the $\mathrm{N_{TB}}$–smectic threshold—an interior $\mathrm{N_{SBT}}$ band, phase–matched by a heliconical $\mathrm{N_{TB}}$ core. Mode–resolved maps show families of distortions along $z$, strong splay and saddle–splay at boundaries, a fixed–sign twist across the slab, and a half–pitch antiphase of $n_{z}$ between the walls. The polarization $\mathbf{p}$ forms stripes in phase with bend in the $\mathrm{N_{TB}}$ interior, while equal–and–opposite mean $p_{x}$ develops at the two walls, so the sample–averaged polarization vanishes.

Across our representative sweep of the phase diagram we did not find a stable $\mathrm{N_{SB}}$ in three dimensions; to our knowledge it has only been stabilized in strictly 2D boomerang models [42]. This rationalizes the preference for twist-mediated $\mathrm{S_{SBT}}$–$\mathrm{N_{SBT}}$ pathways over a pure $\mathrm{N_{SB}}$ phase. Mechanistically, the saddle–splay density concentrates at walls and at $\mathrm{S_{SB}}\!\leftrightarrow\!\mathrm{S_{SBT}}$ crossovers; this boundary channel allows confinement to “program” symmetry and handedness, consistent with the surface–driven control reported by [37]. Our results, bridging particle simulations of confined $\mathrm{N_{TB}}$ with continuum elasticity, provide practical design rules: by tuning anchoring, geometry, and proximity to the $\mathrm{N_{TB}}$–smectic threshold, one can assemble prescribed $\mathrm{S_{SB}}/\mathrm{S_{SBT}}/\mathrm{N_{SBT}}/\mathrm{N_{TB}}$ stacks and program interfacial chirality and polarization for chiral photonics, polarization gratings, and low–power electro–optic devices.

Code and datasets availability— The source code of an original RAMPACK simulation package used to perform Monte Carlo sampling is available at https://github.com/PKua007/rampack. The input script for LAMMPS and RAMPACK along with the datasets generated during and/or analyzed during the current study are available from S.D. and P.K. upon reasonable request.