Nano-scale surface wrinkling in chiral liquid crystals and plant-based plywoods

Geometry-structure models for liquid crystal interfaces, drops and membranes: wrinkling, shape selection and dissipative shape evolution

We review our recent contributions to anisotropic soft matter models for liquid crystal interfaces, drops and membranes, emphasizing validations with experimental and biological data, and with related theory and simulation literature. The presentation aims to illustrate and characterize the rich output and future opportunities of using a methodology based on the liquid crystal-membrane shape equation applied to static and dynamic pattern formation phenomena. The geometry of static and kinetic shapes is usually described with dimensional curvatures that co-mingle shape and curvedness. In this review, we systematically show how the application of a novel decoupled shape-curvedness framework to practical and ubiquitous soft matter phenomena, such as the shape of drops and tactoids and bending of evolving membranes, leads to deeper quantitative insights than when using traditional dimensional mean and Gaussian curvatures. The review focuses only on (1) statics of wrinkling and shape selection in liquid crystal interfaces and membranes; (2) kinetics and dissipative dynamics of shape evolution in membranes; and (3) computational methods for shape selection and shape evolution; due to various limitations other important topics are excluded. Finally, the outlook follows a similar structure. The main results include: (1) single and multiple wavelength corrugations in liquid crystal interfaces appear naturally in the presence of surface splay and bend orientation distortions with scaling laws governed by ratios of anchoring-to-isotropic tension energy; adding membrane elasticity to liquid crystal anchoring generates multiple scales wrinkling as in tulips; drops of liquid crystals encapsulates in membranes can adopt, according to the ratios of anchoring/tension/bending, families of shapes as multilobal, tactoidal, and serrated as observed in biological cells. (2) Mapping the liquid crystal director to a membrane unit normal. The dissipative shape evolution model with irreversible thermodynamics for flows dominated by bending rates, yields new insights. The model explains the kinetic stability of cylinders, while spheres and saddles are attractors. The model also adds to the evolving understanding of outer hair cells in the inner ear. (3) Computational soft matter geometry includes solving shape equations, trajectories on energy and orientation landscapes, and shape-curvedness evolutions on entropy production landscape with efficient numerical methods and adaptive approaches.

Complex Nanowrinkling in Chiral Liquid Crystal Surfaces: From Shaping Mechanisms to Geometric Statistics

Surface wrinkling is closely linked to a significant number of surface functionalities such as wetting, structural colour, tribology, frictions, biological growth and more. Given its ubiquity in nature's surfaces and that most material formation processes are driven by self-assembly and self-organization and many are formed by fibrous composites or analogues of liquid crystals, in this work, we extend our previous theory and modeling work on in silico biomimicking nanowrinkling using chiral liquid crystal surface physics by including higher-order anisotropic surface tension nonlinearities. The modeling is based on a compact liquid crystal shape equation containing anisotropic capillary pressures, whose solution predicts a superposition of uniaxial, equibiaxial and biaxial egg carton surfaces with amplitudes dictated by material anchoring energy parameters and by the symmetry of the liquid crystal orientation field. The numerical solutions are validated by analytical solutions. The blending and interaction of egg carton surfaces create surface reliefs whose amplitudes depend on the highest nonlinearity and whose morphology depends on the anchoring coefficient ratio. Targeting specific wrinkling patterns is realized by selecting trajectories on an appropriate parametric space. Finally, given its importance in surface functionalities and applications, the geometric statistics of the patterns up to the fourth order are characterized and connected to the parametric anchoring energy space. We show how to minimize and/or maximize skewness and kurtosis by specific changes in the surface energy anisotropy. Taken together, this paper presents a theory and simulation platform for the design of nano-wrinkled surfaces with targeted surface roughness metrics generated by internal capillary pressures, of interest in the development of biomimetic multifunctional surfaces.

Post-buckling behaviors of thin-film soft-substrate bilayers with finite-thickness substrate

Surface buckling behaviors of thin-film soft-substrate bilayers have important research value. Recent research has focused on bilayers with infinite-thickness substrates. However, bilayers with finite-thickness substrates widely exist. To study this problem more comprehensively, we extended the stability theory of a beam on an elastic foundation to bilayers and then established a finite element method of bilayers using the neo-Hookean hyperelastic constitutive model. A self-contact analysis method was coupled to the finite element method so that the further buckling evolution of the film surface after folding could be simulated. Based on our analysis of various modulus ratios and thickness ratios, the evolution of the buckling path was significantly influenced by the thickness ratio. Without considering the situation of a prestressed substrate, four new buckling paths were found. Thus, we extended the single buckling path (under infinite thickness substrate) to five types. Our study also found that for path four, the substrate with a certain thickness exhibited a special final stable surface morphology. That is, regardless of the friction, a new periodic morphology after film folding appeared due to the contact slip of the film surface. Finally, further analysis showed that these five buckling paths were all dependent on different modulus ratios and thickness ratios.

Rate of Entropy Production in Evolving Interfaces and Membranes under Astigmatic Kinematics: Shape Evolution in Geometric-Dissipation Landscapes

This paper presents theory and simulation of viscous dissipation in evolving interfaces and membranes under kinematic conditions, known as astigmatic flow, ubiquitous during growth processes in nature. The essential aim is to characterize and explain the underlying connections between curvedness and shape evolution and the rate of entropy production due to viscous bending and torsion rates. The membrane dissipation model used here is known as the Boussinesq-Scriven fluid model. Since the standard approaches in morphological evolution are based on the average, Gaussian and deviatoric curvatures, which comingle shape with curvedness, this paper introduces a novel decoupled approach whereby shape is independent of curvedness. In this curvedness-shape landscape, the entropy production surface under constant homogeneous normal velocity decays with growth but oscillates with shape changes. Saddles and spheres are minima while cylindrical patches are maxima. The astigmatic flow trajectories on the entropy production surface, show that only cylinders and spheres grow under the constant shape. Small deviations from cylindrical shapes evolve towards spheres or saddles depending on the initial condition, where dissipation rates decrease. Taken together the results and analysis provide novel and significant relations between shape evolution and viscous dissipation in deforming viscous membrane and surfaces.

Mechanogeometry of nanowrinkling in cholesteric liquid crystal surfaces

Biological plywoods are multifunctional fibrous composites materials, ubiquitous in nature. The chiral fibrous organization is found in chitin (insects), cellulosics (plants), and collagen I (cornea and bone of mammals) and is a solid analog of that of cholesteric liquid crystals. The surface and interfaces of plywoods are distinguished by hierarchical topographies and nanowrinkling. In this paper, we present a theory to model the emergence of these surfaces and interfaces using liquid crystal-based shape equations that directly connect material properties with geometric wrinkling. The model applies to liquid crystal precursors of the plywood solid analoges. We focus on wrinkling geometry, wrinkling mechanics, and the mechanogeometry relationships that underlie multifunctionality ubiquitous in biological surfaces. Scaling wrinkling laws that connect mechanical pressures and stresses to folding and bending are formulated and quantified. A synthesis of the connections between mechanics and geometry is achieved using the topology of stress curves and curvature of the wrinkles. Taken together the results show that anchoring is a versatile surface morphing mechanism with a rich surface bending stress field, two ingredients behind many potential multifunctionalities.

Surface Anchoring Effects on the Formation of Two-Wavelength Surface Patterns in Chiral Liquid Crystals

We present a theoretical analysis and linear scaling of two-wavelength surface nanostructures formed at the free surface of cholesteric liquid crystals (CLC). An anchoring model based on the capillary shape equation with the high order interaction of anisotropic interfacial tension is derived to elucidate the formation of the surface wrinkling. We showed that the main pattern-formation mechanism is originated due to the interaction between lower and higher order anchoring modes. A general phase diagram of the surface morphologies is presented in a parametric space of anchoring coefficients, and a set of anchoring modes and critical lines are defined to categorize the different types of surface patterns. To analyze the origin of surface reliefs, the correlation between surface energy and surface nano-wrinkles is investigated, and the symmetry and similarity between the energy and surface profile are identified. It is found that the surface wrinkling is driven by the director pressure and is annihilated by two induced capillary pressures. Linear approximation for the cases with sufficient small values of anchoring coefficients is used to realize the intrinsic properties and relations between the surface curvature and the capillary pressures. The contributions of capillary pressures on surface nano-wrinkling and the relations between the capillary vectors are also systematically investigated. These new findings establish a new approach for characterizing two-length scale surface wrinkling in CLCs, and can inspire the design of novel functional surface structures with the potential optical, friction, and thermal applications.

Two-wavelength wrinkling patterns in helicoidal plywood surfaces: imprinting energy landscapes onto geometric landscapes

We present a model to investigate the formation of two-length scale surface patterns in biological and synthetic anisotropic soft matter materials through the high order interaction of anisotropic interfacial tension and capillarity at their free surfaces. The unique pattern-formation mechanism emerging from the presented model is based on the interaction between lower and higher order anchoring modes. Analytical and numerical solutions are used to shed light on why and how simple anisotropic anchoring generates two-lengthscale wrinkles whose amplitudes are given in terms of anchoring coefficients. The novel finding is that the surface energy landscape with its maxima and minima can be imprinted onto the surface geometric landscape. Symmetry relations and scaling laws are used to provide the explicit relations between the anchoring constants and surface profile of the two length scale wrinkles. These new findings establish a new paradigm for characterizing surface wrinkling in biological liquid crystals, and inspire the design of novel functional surface structures.

Multiple-wavelength surface patterns in models of biological chiral liquid crystal membranes

We present a model to investigate the formation of surface patterns in biological materials through the interaction of anisotropic interfacial tension, bending elasticity, and capillarity at their free surfaces. Focusing on the cholesteric liquid crystal (CLC) material model, the generalized shape equation for anisotropic interfaces using the Rapini-Papoular anchoring and Helfrich free energies is applied to understand the formation of multi-length scale patterns, such as those found in floral petals. The chiral liquid crystal-membrane model is shown to be analogous to a driven pendulum, a connection that enables generic pattern classification as a function of bending elasticity, liquid crystal chirality and anchoring strength. The unique pattern-formation mechanism emerging from the model here presented is based on the nonlinear interaction between bending-driven folding and anchoring-driven creasing. The predictions are shown to capture accurately the two-scale wrinkling of certain tulips. These new findings enable not only to establish a new paradigm for characterizing surface wrinkling in biological liquid crystals, but also to inspire the design of functional surface structures.

Recent Advances in Chiral Nematic Structure and Iridescent Color of Cellulose Nanocrystal Films

One unique property of cellulose nanocrystals (CNC) is their property of forming suspensions with chiral nematic order. This order can be preserved in films cast from the suspensions, raising the possibility of applications as photonic materials and templates. However, it has proved difficult to generate uniform, well-ordered chiral nematic materials from CNC. Recently, the importance of kinetic arrest due to gel formation in the later stages of evaporation has been recognized as a key step in film formation. In this brief review, recent developments regarding the structure of chiral nematic suspensions and films as monitored by polarized light microscopy are outlined, and attention is drawn to the importance of shear forces on the self-organization process.

Wetting of cholesteric liquid crystals

We investigate theoretically the wetting properties of cholesteric liquid crystals at a planar substrate. If the properties of substrate and of the interface are such that the cholesteric layers are not distorted, the wetting properties are similar to those of a nematic liquid crystal. If, on the other hand, the anchoring conditions force the distortion of the liquid crystal layers the wetting properties are altered, the free cholesteric-isotropic interface is non-planar and there is a layer of topological defects close to the substrate. These deformations can either promote or hinder the wetting of the substrate by a cholesteric, depending on the properties of the cholesteric liquid crystal.

Chiral Nematic Structure of Cellulose Nanocrystal Suspensions and Films; Polarized Light and Atomic Force Microscopy

Cellulosic liquid crystalline solutions and suspensions form chiral nematic phases that show a rich variety of optical textures in the liquid crystalline state. These ordered structures may be preserved in solid films prepared by evaporation of solvent or suspending medium. Film formation from aqueous suspensions of cellulose nanocrystals (CNC) was investigated by polarized light microscopy, optical profilometry and atomic force microscopy (AFM). An attempt is made to interpret qualitatively the observed textures in terms of the orientation of the cellulose nanocrystals in the suspensions and films, and the changes in orientation caused by the evaporative process. Mass transfer within the evaporating droplet resulted in the formation of raised rings whose magnitude depended on the degree of pinning of the receding contact line. AFM of dry films at short length scales showed a radial orientation of the CNC at the free surface of the film, along with a radial height variation with a period of approximately P/2, ascribed to the anisotropic shrinkage of the chiral nematic structure.