Crystallization dynamics on curved surfaces

Self-Assembly at Curved Biointerfaces

Most of the biological interfaces are curved. Understanding the organizational structures and interaction patterns at such curved biointerfaces is therefore crucial not only for deepening our comprehension of the principles that govern life processes but also for designing and developing targeted drugs aimed at diseased cells and tissues. Despite the considerable efforts dedicated to this area of research, our understanding of curved biological interfaces is still limited. Many aspects of these interfaces remain elusive, presenting both challenges and opportunities for further exploration. In this review, we summarize the structural characteristics of biological interfaces found in nature, the current research status of materials associated with curved biointerfaces, and the theoretical advancements achieved to date. Finally, we outline future trends and challenges in the theoretical and technological development of curved biointerfaces. By addressing these challenges, people could bridge the knowledge gap and unlock the full potential of curved biointerfaces for scientific and technological advancements, ultimately benefiting various fields and improving human health and well-being.

Mapping hyperbolic order in curved materials

Nature employs an impressive range of topologically complex ordered nanostructures that occur in various forms in both natural and synthetic materials. A particular class of these exhibits negative curvature and forms periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces like spherical systems, the understanding of patterning on such surfaces is highly complicated due to the structures being intrinsically intertwined in three dimensions. We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible "hyperbolic map" of the pattern. Specifically, we exemplify the method via the simplest triply periodic minimal surfaces: the Primitive, Diamond, and Gyroid in their universal cover along with decorations from a soft materials, whose structures involve decorations of soft matter on negatively curved surfaces, not necessarily minimal.

Elastic Instability of Cubic Blue Phase Nano Crystals in Curved Shells

Many crystallization processes, including biomineralization and ice-freezing, occur in small and curved volumes, where surface curvature can strain the crystal, leading to unusual configurations and defect formation. The role of curvature on crystallization, however, remains poorly understood. Here, we study the crystallization of blue phase (BP) liquid crystals under curved confinement, which provides insights into the mechanism by which BPs reconfigure their three-dimensional lattice structure to adapt to curvature. BPs are a three-dimensional assembly of high-chirality liquid crystal molecules arranged into body-centered (BPI) or simple cubic (BPII) symmetries. BPs with submicrometer cubic-crystalline lattices exhibit tunable Bragg reflection and submillisecond response time to external stimuli such as an electric field, making them attractive for advanced photonic materials. In this work, we have systematically studied BPs confined in spherical shells with well-defined curvature and boundary conditions. The optical behavior of shells has also been examined at room temperature, where the cholesteric structure forms. In the cholesteric phase, perpendicular anchoring generates focal conic domains on the shell's surface, which transition into stripe patterns as the degree of curvature increases. Our results demonstrate that both higher degrees of curvature and strong spatial confinement destabilize BPI and reconfigure that phase to adopt the structure and optical features of BPII. We also show that the coupling of curvature and confinement nucleates skyrmions at greater thicknesses than those observed for a flat geometry. These findings are particularly important for integrating BPs into miniaturized and curved/flexible devices, including flexible displays, wearable sensors, and smart fabrics.

A Lagrangian Thin-Shell Finite Element Method for Interacting Particles on Fluid Membranes

A recurring motif in soft matter and biophysics is modeling the mechanics of interacting particles on fluid membranes. One of the main outstanding challenges in these applications is the need to model the strong coupling between the substrate deformation and the particles' positions as the latter freely move on the former. This work presents a thin-shell finite element formulation based on subdivision surfaces to compute equilibrium configurations of a thin fluid shell with embedded particles. We use a variational Lagrangian framework to couple the mechanics of the particles and the substrate without having to resort to ad hoc constraints to anchor the particles to the surface. Unlike established methods for such systems, the particles are allowed to move between elements of the finite element mesh. This is achieved by parametrizing the particle locations on the reference configuration. Using the Helfrich-Canham energy as a model for fluid shells, we present the finite element method's implementation and an efficient search algorithm required to locate particles on the reference mesh. Several analyses with varying numbers of particles are finally presented reproducing symmetries observed in the classic Thomson problem and showcasing the coupling between interacting particles and deformable membranes.

Structure and assembly of the S-layer in C. difficile

Many bacteria and archaea possess a two-dimensional protein array, or S-layer, that covers the cell surface and plays crucial roles in cell physiology. Here, we report the crystal structure of SlpA, the main S-layer protein of the bacterial pathogen Clostridioides difficile, and use electron microscopy to study S-layer organisation and assembly. The SlpA crystal lattice mimics S-layer assembly in the cell, through tiling of triangular prisms above the cell wall, interlocked by distinct ridges facing the environment. Strikingly, the array is very compact, with pores of only ~10 Å in diameter, compared to other S-layers (30-100 Å). The surface-exposed flexible ridges are partially dispensable for overall structure and assembly, although a mutant lacking this region becomes susceptible to lysozyme, an important molecule in host defence. Thus, our work gives insights into S-layer organisation and provides a basis for development of C. difficile-specific therapeutics.

Numerical Simulation of Dendritic Pattern Formation in an Isotropic Crystal Growth Model on Curved Surfaces

In this paper, we present several numerical simulation results of dendritic pattern formation using an isotropic crystal growth model, which is based on phase-field modeling, on curved surfaces. An explicit time-stepping method is used and the direct computing method to the Laplace–Beltrami operator, which employs the point centered triangulation approximating Laplacian over the discretized surface with a triangular mesh, is adopted. Numerical simulations are performed not only on simple but also on complex surfaces with various curvatures, and the proposed method can simulate dendritic growth on complex surfaces. In particular, ice crystal growth simulation results on aircraft fuselage or metal bell-shaped curved surfaces are provided in order to demonstrate the practical relevance to our dendrite growth model. Furthermore, we perform several numerical parameter tests to obtain a best fitted set of parameters on simple surfaces. Finally, we apply this set of parameters to numerical simulation on complex surfaces.

Geometry and kinetics determine the microstructure in arrested coalescence of Pickering emulsion droplets

Arrested coalescence occurs in Pickering emulsions where colloidal particles adsorbed on the surface of the droplets become crowded and inhibit both relaxation of the droplet shape and further coalescence. The resulting droplets have a nonuniform distribution of curvature and, depending on the initial coverage, may incorporate a region with negative Gaussian curvature around the neck that bridges the two droplets. Here, we resolve the relative influence of the curvature and the kinetic process of arrest on the microstructure of the final state. In the quasistatic case, defects are induced and distributed to screen the Gaussian curvature. Conversely, if the rate of area change per particle exceeds the diffusion constant of the particles, the evolving surface induces local solidification reminiscent of jamming fronts observed in other colloidal systems. In this regime, the final structure is shown to be strongly affected by the compressive history just prior to arrest, which can be predicted from the extrinsic geometry of the sequence of surfaces in contrast to the intrinsic geometry that governs the static regime.

Topologically-guided continuous protein crystallization controls bacterial surface layer self-assembly

Many bacteria and most archaea possess a crystalline protein surface layer (S-layer), which surrounds their growing and topologically complicated outer surface. Constructing a macromolecular structure of this scale generally requires localized enzymatic machinery, but a regulatory framework for S-layer assembly has not been identified. By labeling, superresolution imaging, and tracking the S-layer protein (SLP) from C. crescentus, we show that 2D protein self-assembly is sufficient to build and maintain the S-layer in living cells by efficient protein crystal nucleation and growth. We propose a model supported by single-molecule tracking whereby randomly secreted SLP monomers diffuse on the lipopolysaccharide (LPS) outer membrane until incorporated at the edges of growing 2D S-layer crystals. Surface topology creates crystal defects and boundaries, thereby guiding S-layer assembly. Unsupervised assembly poses challenges for therapeutics targeting S-layers. However, protein crystallization as an evolutionary driver rationalizes S-layer diversity and raises the potential for biologically inspired self-assembling macromolecular nanomaterials.

Bacteria assemble the surface layer (S-layer), a crystalline protein coat surrounding the curved surface, using protein self-assembly. Here authors image native and purified RsaA, the S-layer protein from C. crescentus, and show that protein crystallization alone is sufficient to assemble and maintain the S-layer in vivo.

Crystallinity and Size Control of Colloidal Germanium Nanoparticles from Organogermanium Halide Reagents

Germanium (Ge) nanoparticles are gaining increasing interest due to their properties that arise in the quantum confinement regime, such as the development of the band structure with changing size. While promising materials, significant challenges still exist related to the development of synthetic schemes allowing for good control over size and morphology in a single step. Herein, we investigate a synthetic method that combines sulfur and primary amines to promote the reduction of organometallic Ge(IV) precursors to form Ge nanoparticles at relatively low temperatures (300 °C). We propose a reaction mechanism and examine the effects of solvents, sulfur concentration, and organogermanium halide precursors. Hydrosulfuric acid (H₂S) produced in situ acts as the primary reducing species, and we were able to increase the particle size more than 2-fold by tuning both the reaction time and quantity of sulfur added during the synthesis. We found that we are able to control the crystalline or amorphous nature of the resulting nanoparticles by choosing different solvents and propose a mechanism for this interaction. The reaction mechanism presented provides insight into how one can control the resulting particle size, crystallinity, and reaction kinetics. While we demonstrated the synthesis of Ge nanoparticles, this method can potentially be extended to other members of the group IV family.

Formation of cluster crystals in an ultra-soft potential model on a spherical surface

We investigate the formation of cluster crystals with multiply occupied lattice sites on a spherical surface in systems of ultra-soft particles interacting via repulsive, bounded pair potentials. Not all interactions of this kind lead to clustering: we generalize the criterion devised in C. N. Likos et al., Phys. Rev. E, 2001, 63, 031206 to spherical systems in order to distinguish between cluster-forming systems and fluids which display reentrant melting. We use both DFT and Monte Carlo simulations to characterize the behavior of the system, and obtain semi-quantitative agreement between the two. We find that the number of clusters is determined by the ratio between the size σ of the ultra-soft particles and the radius R of the sphere in such a way that each stable configuration spans a certain interval of σ/R. Furthermore, we study the effect of topological frustration on the system due to the sphere curvature by comparing the properties of disclinations, i.e., clusters with fewer than six neighbors, and non-defective clusters. Disclinations are shown to be less stable, contain fewer particles, and be closer to their neighbors than other lattice points: these properties are explained on the basis of geometric and energetic considerations.

Deformation and failure of curved colloidal crystal shells

Designing and controlling particle self-assembly into robust and reliable high-performance smart materials often involves crystalline ordering in curved spaces. Examples include carbon allotropes like graphene, synthetic materials such as colloidosomes, or biological systems like lipid membranes, solid domains on vesicles, or viral capsids. Despite the relevance of these structures, the irreversible deformation and failure of curved crystals is still mostly unexplored. Here, we report simulation results of the mechanical deformation of colloidal crystalline shells that illustrate the subtle role played by geometrically necessary topological defects in controlling plastic yielding and failure. We observe plastic deformation attributable to the migration and reorientation of grain boundary scars, a collective process assisted by the intermittent proliferation of disclination pairs or abrupt structural failure induced by crack nucleating at defects. Our results provide general guiding principles to optimize the structural and mechanical stability of curved colloidal crystals.

Smectic block copolymer thin films on corrugated substrates

In this work we study equilibrium and non-equilibrium structures of smectic block copolymer thin films deposited on a topographically patterned substrate. A Brazovskii free energy model is employed to analyze the coupling between the smectic texture and the local mean curvature of the substrate. The substrate's curvature produces out-of-plane deformations of the block copolymer such that equilibrium textures are modified and dictated by the underlying geometry. For weak curvatures it is shown that the free energy of the block copolymer film follows a Helfrich form, scaling with the square of the mean curvature, with a bending constant dependent on the local pattern orientation. On substrates of varying mean curvature simulations show that topological defects are rapidly expelled from regions with large curvature. These results compare well with available experimental data of poly(styrene)-co-poly(ethylene-alt-propylene) smectic thin films.

Defect formation and coarsening in hexagonal 2D curved crystals

In this work we study the processes of defect formation and coarsening of two-dimensional (2D) curved crystal structures. These processes are found to strongly deviate from their counterparts in flat systems. In curved backgrounds the process of defect formation is deeply affected by the curvature, and at the onset of a phase transition the early density of defects becomes highly inhomogeneous. We observe that even a single growing crystal can produce varying densities of defects depending on its initial position and local orientation with regard to the substrate. This process is completely different from flat space, where grain boundaries are formed due to the impingement of different propagating crystals. Quenching the liquid into the crystal phase leads to the formation of a curved polycrystalline structure, characterized by complex arrays of defects. During annealing, mechanisms of geodesic curvature-driven grain boundary motion and defect annihilation lead to increasing crystalline order. Linear arrays of defects diffuse to regions of high curvature, where they are absorbed by disclinations. At the early stage of coarsening the density of dislocations is insensitive to the geometry while the population of isolated disclinations is deeply affected by curvature. The regions with high curvature act as traps for the diffusion of different structures of defects, including disclinations and domain walls.