Smectic order on arbitrary curved substrate can be described by a differential form of rank one (1-form), whose geometric meaning is the differential of the local phase field of the density modulation. The exterior derivative of 1-form is the local dislocation density. Elastic deformations are described by superposition of exact differential forms. We use the formalism of differential forms to systematically classify and characterize all low energy smectic states on torus as well as on sphere. A two dimensional smectic order confined on either manifold exhibits many topologically distinct low energy states. Different states are not accessible from each other by local fluctuations. The total number of low energy states scales as the square root of the system area. We also address the energetics of 2D smectic on a curved substrate and calculate the mean field phase diagram of smectic on a thin torus. Finally, we discuss the motion of disclinations for spherical smectics as low energy excitations, and illustrate the interesting connection between spherical smectic and the theory of elliptic functions.
Smectic orders on curved substrates can be described by differential forms of rank one (1-forms), whose geometric meaning is the differential of the local phase field of density modulation. The exterior derivative of 1-form is the local dislocation density. Elastic deformations are described by superposition of exact differential forms. Applying this formalism to study smectic order on torus as well as on sphere, we find that both systems exhibit many topologically distinct low energy states, that can be characterized by two integer topological charges. The total number of low energy states sc…
We study smectic-liquid-crystal order in a cell with a heterogeneous substrate imposing surface random positional and orientational pinnings. Proposing a minimal random elastic model, we demonstrate that, for a thick cell, the smectic state without a rubbed substrate is always unstable at long scales and, for weak random pinning, is replaced by a smectic glass state. We compute the statistics of the associated substrate-driven distortions and the characteristic smectic domain size on the heterogeneous substrate and in the bulk. We find that for weak disorder, the system exhibits a three-dimens…
We explore the smectic liquid crystal order in a cell with a "dirty" substrate imposing random pinnings. Within harmonic elasticity we find a subtle three-dimensional surface disorder-driven transition into a pinned smectic-glass, controlled by a three-dimensional Cardy-Ostlund-like fixed line, akin to a super-rough phase of a two-dimensional xy model. We compute the associated random substrate-driven distortions of the smectic-glass state, identify the characteristic length scales on the heterogeneous substrate and in the bulk, and discuss a variety of experimental signatures.
Ordered phases on curved substrates experience a complex interplay of ordering and intrinsic curvature, commonly producing frustration and singularities. This is an especially important issue in crystals as ever-smaller scale materials are grown on real surfaces; eventually, surface imperfections are on the same scale as the lattice constant. Here, we gain insights into this general problem by studying two-dimensional smectic order on substrates with highly localized intrinsic curvature, constructed from cones and their intersections with planes. In doing so we take advantage of fully tractabl…
In a 2D liquid crystal, each topological defect has a topological charge and a characteristic orientation, and hence can be regarded as an oriented particle. Theories predict that the trajectories of annihilating defects depend on their relative orientation. Recently, these predictions have been tested in experiments on smectic-C films. Those experiments find curved trajectories that are similar to the predictions, but the detailed relationship between the defect orientations and the far-field director is different. To understand this difference, we extend the previous theories by adding the e…
Self-assembly of ordered nanometer-scale patterns is interesting in itself, but its practical value depends on the ability to predict and control pattern formation. In this paper we demonstrate theoretically and numerically that engineering of extrinsic as well as intrinsic substrate geometry may provide such a controllable ordering mechanism for block copolymers films. We develop an effective two-dimensional model of thin films of striped-phase diblock copolymers on general curved substrates. The model is obtained as an expansion in the film thickness and thus takes the third dimension into a…
Surface-induced profiles of both nematic and smectic order parameters in a nematic liquid crystal, ranging from an orienting substrate to "infinity", were evaluated numerically on base of an extended Landau theory. In order to obtain a smooth behavior of the solutions at "infinity" a boundary energy functional was derived by linearizing the Landau energy around its equilibrium solutions. We find that the intrinsic wave number of the smectic structure, which plays the role of a coupling between nematic and smectic order, strongly influences the director reorientation. Whereas the smectic order …
The liquid crystal octylcyanobiphenyl (8CB) was doped with the chiral agent CB15 and spin-coated onto a substrate treated for planar alignment of the director, resulting in a film of thickness several hundred nm in the smectic-A phase. In both doped and undoped samples, the competing boundary conditions - planar alignment at the substrate and vertical alignment at the free surface - cause the liquid crystal to break into a series of flattened hemicylinders to satisfy the boundary conditions. When viewed under an optical microscope with crossed polarizers, this structure results in a series of …
We construct a hydrodynamic theory of noisy, apolar active smectics, in bulk suspension or on a substrate. Our predictions include: quasi-long-ranged smectic order in dimension d = 2, and long- ranged in d = 3, extending previously published results to all dynamical regimes; Kosterlitz-Thouless melting to an active nematic at high and low concentrations in d = 2; nonzero second-sound speed parallel to the layers; the suppression of giant number fluctuations by smectic elasticity; instability to spontaneous undulation and flow in bulk contractile smectics; a layer spacing instability, possibly …
We analyze the transverse intersubstrate pseudo-Casimir force, arising as a result of thermal fluctuations of the liquid crystalline layers of a smectic-A film confined between two planar substrates in a bookshelf geometry, in which the equidistant smectic layers are placed perpendicular to the bounding surfaces. We discuss the variation of the interaction force as a function of the intersubstrate separation in the presence of surface anchoring to the substrates, showing that the force induced by confined fluctuations is attractive and depends on the penetration length as well as the layer spa…
Focal conic domains (FCDs) in smectic-A liquid crystals have drawn much attention both for their exquisitely structured internal form and for their ability to direct the assembly of micro- and nanomaterials in a variety of patterns. A key to directing FCD assembly is control over the eccentricity of the domain. Here, we demonstrate a new paradigm for creating spatially varying FCD eccentricity by confining a hybrid-aligned smectic with curved interfaces. In particular, we manipulate interface behavior with colloidal particles in order to experimentally produce two examples of what has recently…
We investigate the phase behaviour of a two-dimensional colloidal model system of ultra-soft particles on a substrate which varies periodically along one spatial direction. Our calculations are based on mean-field density functional theory for a system of particles interacting via an ultra-soft potential, that is, the generalized exponential model with index four (\mbox{GEM-4}). For suitable substrate periodicities (with commensurability parameter $p=2$), we find a succession of phase transitions from a modulated liquid to a locked smectic and then to a locked floating solid phase. The appeara…
We study the adsorption of semiflexible polymers such as polyelectrolytes or DNA on planar and curved substrates, e.g., spheres or washboard substrates via short-range potentials using extensive Monte-Carlo simulations, scaling arguments, and analytical transfer matrix techniques. We show that the adsorption threshold of stiff or semiflexible polymers on a planar substrate can be controlled by polymer stiffness: adsorption requires the highest potential strength if the persistence length of the polymer matches the range of the adsorption potential. On curved substrates, i.e., an adsorbing sphe…
We study thin self-assembled columns constrained to lie on a curved, rigid substrate. The curvature presents no local obstruction to equally spaced columns in contrast to curved crystals for which the crystalline bonds are frustrated. Instead, the vanishing compressional strain of the columns implies that their normals lie on geodesics which converge (diverge) in regions of positive (negative) Gaussian curvature, in analogy to the focussing of light rays by a lens. We show that the out of plane bending of the cylinders acts as an effective ordering field.
We use Monte Carlo simulations to study the finite temperature behavior of vortices in the XY- model for tangent vector order on curved backgrounds. Contrary to naive expectations, we show that the underlying geometry does not affect the proliferation of vortices with temperature respect to what is observed on a flat surface. Long-range order in these systems is analyzed by using the classical two-point correlation functions. As expected, in the case of slightly curved substrates these correlations behave similarly to the plane. However, for high curvatures, the presence of geometry-induced un…
We argue that for complete wetting at a curved substrate (wall) the wall-fluid surface tension is non-analytic in $R_i^{-1}$, the curvature of the wall and that the density profile of the fluid near the wall acquires a contribution proportional to the gas-liquid surface tension $\times R_i^{-1}$ plus higher-order contributions which are non-analytic in $R_i^{-1}$. These predictions are confirmed by results of density functional calculations for the square-well model of a liquid adsorbed on a hard sphere and on a hard cylinder where complete wetting by gas (drying) occurs. The implications of o…
The Gaussian (saddle splay) rigidity of fluid membranes controls their equilibrium topology but is notoriously difficult to measure. In lipid mixtures, typical of living cells, linear interfaces separate liquid ordered (LO) from liquid disordered (LD) bilayer phases at subcritical temperatures. Here we consider such membranes supported by curved supports that thereby control the membrane curvatures. We show how spectral analysis of the fluctuations of the LO-LD interface provides a novel way of measuring the difference in Gaussian rigidity between the two phases. We provide a number of conditi…
Motivated by recent experimental work on multicomponent lipid membranes supported by colloidal scaffolds, we report an exhaustive theoretical investigation of the equilibrium configurations of binary mixtures on curved substrates. Starting from the Jülicher-Lipowsky generalization of the Canham-Helfrich free energy to multicomponent membranes, we derive a number of exact relations governing the structure of an interface separating two lipid phases on arbitrarily shaped substrates and its stability. We then restrict our analysis to four classes of surfaces of both applied and conceptual interes…
We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler-Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or globa…
Flaps can be detached from a thin film glued on a solid substrate by tearing and peeling. For flat substrates, it has been shown that these flaps spontaneously narrow and collapse in pointy triangular shapes. Here we show that various shapes, triangular, elliptic, acuminate or spatulate, can be observed for the tears by adjusting the curvature of the substrate. From combined experiments and theoretical models, we show that the flap morphology is governed by simple geometric rules.
Eukaryotic cells adhere to extracellular matrix during the normal development of the organism, forming static adhesion as well as during cell motility. We study this process by considering a simplified coarse-grained model of a vesicle that has uniform adhesion energy with a flat substrate, mobile curved membrane proteins and active forces. We find that a high concentration of curved proteins alone increases the spreading of the vesicle, by the self-organization of the curved proteins at the high curvature vesicle-substrate contact line, thereby reducing the bending energy penalty at the vesic…
When thermal energies are weak, two dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two distinct physical effects are identified as key factors that contribute to the interaction between the shape of the underlying surface and the resulting lamellar morphology. The first is a local ordering fi…
It is widely recognized that the shape of epithelial cells is determined by the tension generated by the actomyosin cortex and the adhesion of cells to the substrate and to each other. To account for these biological and structural contributions to cell shape, different physical models have been proposed. However, an experimental procedure that would allow a validation of a minimal physical model for the shape of epithelial cells in 3D has not yet been proposed. In this study, we cultured MDCK epithelial cells on substrates with a sinusoidal profile, allowing us to measure the shape of the cel…
A new set of mathematical identities is presented for axi-symmetric sessile drops on flat and curved substrates. The geometrical parameters, including the apex curvature and height, and the contact radius, are related by the identities. The validity of the identities are checked by various numerical solutions both for flat and curved substrates.
We study the global influence of curvature on the free energy landscape of two-dimensional binary mixtures confined on closed surfaces. Starting from a generic effective free energy, constructed on the basis of symmetry considerations and conservation laws, we identify several model-independent phenomena, such as a curvature-dependent line tension and local shifts in the binodal concentrations. To shed light on the origin of the phenomenological parameters appearing in the effective free energy, we further construct a lattice-gas model of binary mixtures on non-trivial substrates, based on the…
The adhesion of a stiff film onto a curved substrate often generates elastic stresses in the film that eventually give rise to its delamination. Here we predict that delamination of very thin films can be dramatically suppressed through tiny, smooth deformations of the substrate, dubbed here "wrinklogami", that barely affect the macroscale topography. This "pro-lamination" effect reflects a surprising capability of smooth wrinkles to suppress compression in elastic films even when spherical or other doubly-curved topography is imposed, in a similar fashion to origami folds that enable construc…
We formulate a straightforward scheme of statistical mechanics for inhomogeneous systems that includes the virial series in powers of the activity for the grand free energy and density distributions. There, cluster integrals formulated for inhomogeneous systems play a main role. We center on second order terms that were analyzed in the case of hard-wall confinement, focusing in planar, spherical and cylindrical walls. Further analysis was devoted to the Lennard-Jones system and its generalization the 2k-k potential. For this interaction potentials the second cluster integral was evaluated anal…
In this work I derive analytic expressions for the curvature dependent fluid-substrate surface tension of a hard sphere fluid on a hard curved wall. In a first step, the curvature thermodynamic properties are found as truncated power series in the activity in terms of the known second and third order cluster integral of the hard-sphere fluid near spherical and cylindrical walls. These results are then expressed as packing fraction power series and transformed to different reference regions which is equivalent to consider different positions of the dividing surface. All the obtained series expa…
The adhesive contact between elastic solids with randomly rough, self affine fractal surfaces is studied by molecular dynamics (MD) simulations. The interfacial binding energy obtained from the simulations of nominally flat and curved surfaces is compared with the predictions of the contact mechanics theory by Persson. Theoretical and simulation results agree rather well, and most of the differences observed can be attributed to finite size effects and to the long-range nature of the interaction between the atoms in the block and the substrate in the MD model, as compared to the analytical the…
Determining the equilibrium configuration and shape of curved two-dimensional films with (generalized) liquid crystalline (LC) order is a difficult infinite dimensional problem of direct relevance to the study of generalized polymersomes, soft matter and the fascinating problem of understanding the origin and formation of shape (morphogenesis). The symmetry of the free energy of the LC film being considered and the topology of the surface to be determined often requires that the equilibrium configuration possesses singular structures in the form of topological defects such as disclinations for…
