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.

From theoretical aspects to practical food Pickering emulsions: Formation, stabilization, and complexities linked to the use of colloidal food particles

We noticed that in literature, the term Pickering emulsion (PE) is used as soon as ingredients contain particles, and in this review, we ask ourselves if that is done rightfully so. The basic behavior taking place in particle-stabilized emulsions leads to the conclusion that the desorption energy of particles is generally high making particles highly suited to physically stabilize emulsions. Exceptions are particles with extreme contact angles or systems with very low interfacial tension. Particles used in food and biobased applications are soft, can deform when adsorbed, and most probably have molecules extending into both phases thus increasing desorption energy. Besides, surface-active components will be present either in the ingredients or generated by the emulsification process used, which will reduce the energy of desorption, either by reduced interfacial tension, or changes in the contact angle. In this paper, we describe the relative relevance of these aspects, and how to distinguish them in practice. Practical food emulsions may derive part of their stability from the presence of particles, but most likely have mixed interfaces, and are thus not PEs. Especially when small particles are used to stabilize (sub)micrometer droplets, emulsions may become unstable upon receiving a heat treatment. Stability can be enhanced by connecting the particles or creating network that spans the product, albeit this goes beyond classical Pickering stabilization. Through the architecture of PEs, special functionalities can be created, such as reduction of lipid oxidation, and controlled release features.

Dumbbell Impurities in 2D Crystals of Repulsive Colloidal Spheres Trap Dislocations

Impurity-induced defects play a crucial role for the properties of crystals, but little is known about impurities with anisotropic shape. Here, we study how colloidal dumbbells distort and interact with a hexagonal crystal of charged colloidal spheres at a fluid interface. We find that subtle differences in the dumbbell length determine whether it induces a local distortion of the lattice or traps a dislocation, and determine how the dumbbell moves inside the repulsive hexagonal lattice. Our results provide new routes toward controlling material properties and understanding fundamental questions in phase transitions through particle-bound dislocations.

Thermal preconditioning of membrane stress to control the shapes of ultrathin crystals

We employ the phospholipid bilayer membranes of giant unilamellar vesicles as a free-standing environment for the growth of membrane-integrated ultrathin phospholipid crystals possessing a variety of shapes with 6-fold symmetry. Crystal growth within vesicle membranes, where more elaborate shapes grow on larger vesicles is dominated by the bending energy of the membrane itself, creating a means to manipulate crystal morphology. Here we demonstrate how cooling rate preconditions the membrane tension before nucleation, in turn regulating nucleation and growth, and directing the morphology of crystals by the time they are large enough to be visualized. The crystals retain their shapes during further growth through the two phase region. Experiments demonstrate this behavior for single crystals growing within the membrane of each vesicle, ultimately comprising up to 13% of the vesicle area and length scales of up to 50 microns. A model for stress evolution, employing only physical property data, reveals how the competition between thermal membrane contraction and water diffusion from tensed vesicles produces a size- and time-dependence of the membrane tension as a result of cooling history. The tension, critical in the contribution of bending energy in the fluid membrane regions, in turn selects for crystal shape for vesicles of a given size. The model reveals unanticipated behaviors including a low steady state tension on small vesicles that allows compact domains to develop, rapid tension development on large vesicles producing flower-shaped domains, and a stress relaxation through water diffusion across the membrane with a time constant scaling as the square of the vesicle radius, consistent with measurable tensions only in the largest vesicles.

Nanostructured Surfaces Enhance Nucleation Rate of Calcium Carbonate

Nucleation and growth of calcium carbonate on surfaces is of broad importance in nature and technology, being essential to the calcification of organisms, while negatively impacting energy conversion through crystallization fouling, also called scale formation. Previous work studied how confinements, surface energies, and functionalizations affect nucleation and polymorph formation, with surface-water interactions and ion mobility playing important roles. However, the influence of surface nanostructures with nanocurvature-through pit and bump morphologies-on scale formation is unknown, limiting the development of scalephobic surfaces. Here, it is shown that nanoengineered surfaces enhance the nucleation rate by orders of magnitude, despite expected inhibition through effects like induced lattice strain through surface nanocurvature. Interfacial and holographic microscopy is used to quantify crystallite growth and find that nanoengineered interfaces experience slower individual growth rates while collectively the surface has 18% more deposited mass. Reconstructions through nanoscale cross-section imaging of surfaces coupled with classical nucleation theory-utilizing local nanocurvature effects-show the collective enhancement of nano-pits.

Guest-guided anchoring patterns of cyclodextrin supramolecular microcrystals on droplet surfaces

The growth of cyclodextrin inclusion complexes (ICs) on oil/water interfaces represents a beautiful example of spontaneous pattern formation in nature. How the supramolecules evolve remains a challenge because surface confinement can frustrate microcrystal growth and give rise to unusual phase transitions. Here we investigate the self-assembly of ICs on droplet surfaces using microfluidics, which allows directly visualizing packing, wetting and ordering of the microcrystals anchored on the surface. The oil guests of distinct molecular structures can direct the assembly of the ICs and largely affect anchoring dynamics of the ICs microcrystals, leading to a range of behaviors including orientating, slipping, buckling, jamming, or merging. We discuss the behaviors observed in terms of the flexibility of the building blocks, which offers a new degree of freedom through which to tailor their properties and gives rise to a striking feature of anchoring patterns that have no counterpart in normal colloidal crystals.

Self-limiting stacks of curvature-frustrated colloidal plates: Roles of intraparticle versus interparticle deformations

In geometrically frustrated assemblies local intersubunit misfits propagate to intra-assembly strain gradients, giving rise to anomalous self-limiting assembly thermodynamics. Here we use theory and coarse-grained simulation to study a recently developed class of "curvamer" particles, flexible shell-like particles that exhibit self-limiting assembly due to the build up of curvature deformation in cohesive stacks. To address a generic, yet poorly understood aspect of frustrated assembly, we introduce a model of curvamer assembly that incorporates both intraparticle shape deformation as well as compliance of interparticle cohesive gaps, an effect we can attribute to a finite range of attraction between particles. We show that the ratio of intraparticle (bending elasticity) to interparticle stiffness not only controls the regimes of self-limitation but also the nature of frustration propagation through curvamer stacks. We find a transition from uniformly bound, curvature-focusing stacks at small size to gap opened, uniformly curved stacks at large size is controlled by a dimensionless measure of inter- versus intracurvamer stiffness. The finite range of interparticle attraction determines the range of cohesion in stacks that are self-limiting, a prediction which is in strong agreement with numerical studies of our coarse-grained colloidal model. These predictions provide critical guidance for experimental realizations of frustrated particle systems designed to exhibit self-limitation at especially large multiparticle scales.

Icosahedral supracrystal assembly from polymer-grafted nanoparticles via interplay of interfacial energy and confinement effect

Self-assembly of nanoparticles (NPs) in drying emulsion droplets paves the way for intricate three-dimensional (3D) superstructures, given the myriad of control parameters for fine-tuning assembly conditions. With their substantial energetic dynamics that are acutely responsive to emulsion confinements, polymeric ligands incorporated into a system can enrich its structural diversity. Here, we demonstrate the assembly of soft polymer-grafted NPs into Mackay icosahedrons beyond spherical body-centered cubic (BCC) packing structures commonly observed for these soft spheres. This behavior is governed by the free energy minimization within emulsions through the interplay of the oil-water interfacial energy and confinement effect as demonstrated by the experimental observations of structural transitions between icosahedrons and BCC crystals and by corresponding free energy calculations. The anisotropic surface of the icosahedral supracrystals provides the capability of guiding the position of a secondary constituent, creating unique hybrid patchy icosahedrons with the potential to develop into multifunctional 3D clusters that combine the benefits of both polymers and conventional colloids.

Limits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra

We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies-in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g., periodicity)-is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.

Flower-shaped 2D crystals grown in curved fluid vesicle membranes

The morphologies of two-dimensional (2D) crystals, nucleated, grown, and integrated within 2D elastic fluids, for instance in giant vesicle membranes, are dictated by an interplay of mechanics, permeability, and thermal contraction. Mitigation of solid strain drives the formation of crystals with vanishing Gaussian curvature (i.e., developable domain shapes) and, correspondingly, enhanced Gaussian curvature in the surrounding 2D fluid. However, upon cooling to grow the crystals, large vesicles sustain greater inflation and tension because their small area-to-volume ratio slows water permeation. As a result, more elaborate shapes, for instance, flowers with bendable but inextensible petals, form on large vesicles despite their more gradual curvature, while small vesicles harbor compact planar crystals. This size dependence runs counter to the known cumulative growth of strain energy of 2D colloidal crystals on rigid spherical templates. This interplay of intra-membrane mechanics and processing points to the scalable production of flexible molecular crystals of controllable complex shape.

The cooperative migration dynamics of particles correlates to the nature of hexatic-isotropic phase transition in 2D systems of corner-rounded hexagons

In the two-dimensional (2D) melting transition of colloidal systems, the hexatic-isotropic (H-I) transition can be either first-order or continuous. However, how particle dynamics differs at the single-particle level during these two different melting transitions remains to be disclosed. In this work, by Brownian dynamics (BD) simulations, we have systematically studied the dynamic behavior of corner-rounded hexagons during the H-I transition, for a range of corner-roundness ζ = 0.40 to 0.99 that covers the crossover from the continuous to first-order nature of H-I transition. The results show that hexagons with ζ ≤ 0.5 display a continuous H-I transition, whereas those with ζ ≥ 0.6 demonstrate a first-order H-I transition. Dynamic analysis shows different evolution pathways of the dominant cluster formed by migrating particles, which results in a droplet-like cluster structure for ζ = 0.40 hexagons and a tree-like cluster structure for ζ = 0.99 hexagons. Further investigations on the hopping activities of particles suggest a cooperative origin of migrating clusters. Our work provides a new aspect to understand the dependence of the nature of H-I transition on the roundness of hexagons through particle dynamic behavior.

Programmable Potentials Choreograph Defects in a Colloidal Crystal Shell

Crystallization on spherical surfaces is obliged by topology to induce lattice defects. But controlling the organization of such defects remains a great challenge due to the long-range constraints of the curved geometry. Here, we report on DNA-coated colloids whose programmable interaction potentials can be used to regulate the arrangement of defects and even achieve perfect icosahedral order on a sphere. Combined simulations and theoretical analysis show how the potential can be tuned by changing the temperature, thereby controlling the number of defects. An explicit expression for the effective potential is derived, allowing us to distinguish the effects of entropic repulsion and enthalpic attraction. Altogether, the present findings provide insights into the physics of crystallization on curved spaces and may be used for designing desired crystal geometries.

High yield and wide lateral size growth of α-Mo2C: exploring the boundaries of CVD growth of bare MXene analogues

Synthesis of Mo2C bare MXenes, without surface terminations groups, via chemical vapor deposition (CVD) on metal foils is scientifically a very intriguing crystal growth process, and there are still challenges and limited fundamental understanding to overcome to obtain high yield and wide crystal size lateral growth. Achieving large area coverage via direct growth is scientifically vital to utilize the full potential of their unique properties in different applications. In this study, we sought to expand the boundaries of the current CVD growth approach for Mo2C MXenes and gain insights into the possibilities and limitations of large area growth, with a particular focus on controlling Mo concentration. We report a facile modification of their typical CVD growth protocol and show its influence on the Mo2C synthesis, with growth times spanning up to 3 h. Specifically, prior to initiating the CVD growth process, we introduced a holding step in temperature at 1095 °C. This proved to be beneficial in increasing the Mo concentration on the liquid Cu growth surface. We achieved an average Mo2C crystals coverage of approximately 50% of the growth substrate area, increased tendency of coalescence and merging of individual flakes, and lateral flake sizes up to 170μm wide. To gain deeper understanding into their CVD growth behavior, we conducted a systematic investigation of the effect of several factors, including (i) a holding step time on Mo diffusion rate through molten Cu, (ii) the Cu foil thickness over the Mo foil, and (iii) the CVD growth time. Phase, chemical and microstructural characterization by x-ray diffraction, x-ray photon spectroscopy, SEM and scanning/transmission electron microscopy revealed that the grown crystals are single phaseα-Mo2C. Furthermore, insights gained from this study sheds light on crucial factors and inherent limitations that are essential to consider and may help guide future research progress in CVD growth of bare MXenes.

Thermal stability and secondary aggregation of self-limiting, geometrically frustrated assemblies: Chain assembly of incommensurate polybricks

In geometrically frustrated assemblies, equilibrium self-limitation manifests in the form of a minimum in the free energy per subunit at a finite, multisubunit size which results from the competition between the elastic costs of frustration within an assembly and the surface energy at its boundaries. Physical realizations-from ill-fitting particle assemblies to self-twisting protein superstructures-are capable of multiple mechanisms of escaping the cumulative costs of frustration, resulting in unlimited equilibrium assembly, including elastic modes of "shape flattening" and the formation of weak, defective bonds that screen intra-assembly stresses. Here we study a model of one-dimensional chain assembly of incommensurate "polybricks" and determine its equilibrium assembly as a function of temperature, concentration, degree of shape frustration, elasticity, and interparticle binding, notably focusing on how weakly cohesive, defective bonds give rise to strongly temperature-dependent assembly. Complex assembly behavior derives from the competition between multiple distinct local minima in the free-energy landscape, including self-limiting chains, weakly bound aggregates of self-limiting chains, and strongly bound, elastically defrustrated assemblies. We show that this scenario, in general, gives rise to anomalous multiple aggregation behavior, in which disperse subunits (stable at low concentration and high temperature) first exhibit a primary aggregation transition to self-limiting chains (at intermediate concentration and temperature) which are ultimately unstable to condensation into unlimited assembly of finite-chains through weak binding beyond a secondary aggregation transition (at low temperature and high concentration). We show that window of stable self-limitation is determined both by the elastic costs of frustration in the assembly as well as energetic and entropic features of intersubunit binding.

Geometric frustration of hard-disk packings on cones

Conical surfaces pose an interesting challenge to crystal growth: A crystal growing on a cone can wrap around and meet itself at different radii. We use a disk-packing algorithm to investigate how this closure constraint can geometrically frustrate the growth of single crystals on cones with small opening angles. By varying the crystal seed orientation and cone angle, we find that-except at special commensurate cone angles-crystals typically form a seam that runs along the axial direction of the cone, while near the tip, a disordered particle packing forms. We show that the onset of disorder results from a finite-size effect that depends strongly on the circumference and not on the seed orientation or cone angle. This finite-size effect occurs also on cylinders, and we present evidence that on both cylinders and cones, the defect density increases exponentially as circumference decreases. We introduce a simple model for particle attachment at the seam that explains the dependence on the circumference. Our findings suggest that the growth of single crystals can become frustrated even very far from the tip when the cone has a small opening angle. These results may provide insights into the observed geometry of conical crystals in biological and materials applications.

The entropy-controlled strategy in self-assembling systems

Self-assembly of various building blocks has been considered as a powerful approach to generate novel materials with tailorable structures and optimal properties. Understanding physicochemical interactions and mechanisms related to structural formation and transitions is of essential importance for this approach. Although it is well-known that diverse forces and energies can significantly contribute to the structures and properties of self-assembling systems, the potential entropic contribution remains less well understood. The past few years have witnessed rapid progress in addressing the entropic effects on the structures, responses, and functions in the self-assembling systems, and many breakthroughs have been achieved. This review provides a framework regarding the entropy-controlled strategy of self-assembly, through which the structures and properties can be tailored by effectively tuning the entropic contribution and its interplay with the enthalpic counterpart. First, we focus on the fundamentals of entropy in thermodynamics and the entropy types that can be explored for self-assembly. Second, we discuss the rules of entropy in regulating the structural organization in self-assembly and delineate the entropic force and superentropic effect. Third, we introduce the basic principles, significance and approaches of the entropy-controlled strategy in self-assembly. Finally, we present the applications where this strategy has been employed in fields like colloids, macromolecular systems and nonequilibrium assembly. This review concludes with a discussion on future directions and future research opportunities for developing and applying the entropy-controlled strategy in complex self-assembling systems.

Polyhedral Colloidal Clusters Assembled from Amphiphilic Nanoparticles in Deformable Droplets

Polyhedral colloidal clusters assembled from functional inorganic nanoparticles have attracted great interest in both scientific research and applications. However, the spontaneous assembly of colloidal nanoparticles into polyhedral clusters with regular shape and tunable structures remains a grand challenges. Here, we successfully construct Mackay icosahedral and regular tetrahedral colloidal clusters assembled from gold nanoparticles grafted with a mixture of polystyrene (PS) and poly(2-vinylpyridine) (P2VP) homopolymers by precisely tuning the interfacial interaction between the nanoparticles and the oil/water interface. By increasing the proportion of hydrophilic P2VP ligands on the surface of gold nanoparticles, the Mackay icosahedral clusters can transform into regular tetrahedral clusters in order to maximize the surface area of the polyhedral assembly. Furthermore, we reveal the formation mechanism of these regular polyhedral colloidal clusters. The formation of polyhedral colloidal clusters is not only dependent on the entropy but also determined by the interfacial free energy. This finding demonstrates an effective approach to organize nanoparticles into polyhedral colloidal clusters with potential applications in various fields.

Early-stage bifurcation of crystallization in a sphere

Bifurcations in kinetic pathways decide the evolution of a system. An example is crystallization, in which the thermodynamically stable polymorph may not form due to kinetic hindrance. Here, we use confined self-assembly to investigate the interplay of thermodynamics and kinetics in the crystallization pathways of finite clusters. We report the observation of decahedral clusters from colloidal particles in emulsion droplets and show that these decahedral clusters can be thermodynamically stable, just like icosahedral clusters. Our hard sphere simulations reveal how the development of the early nucleus shape passes through a bifurcation that decides the cluster symmetry. A geometric argument explains why decahedral clusters are kinetically hindered and why icosahedral clusters can be dominant even if they are not in the thermodynamic ground state.

Delamination from an adhesive sphere: Curvature-induced dewetting versus buckling

Everyday experience confirms the tendency of adhesive films to detach from spheroidal regions of rigid substrates-what is a petty frustration when placing a sticky band aid onto a knee is a more serious matter in the coating and painting industries. Irrespective of their resistance to bending, a key driver of such phenomena is Gauss' Theorema Egregium, which implies that naturally flat sheets cannot conform to doubly curved surfaces without developing a strain whose magnitude grows sharply with the curved area. Previous attempts to characterize the onset of curvature-induced delamination, and the complex patterns it gives rise to, assumed a dewetting-like mechanism in which the propensity of two materials to form contact through interfacial energy is modified by an elastic energy penalty. We show that this approach may characterize moderately bendable sheets but fails qualitatively to describe the curvature-induced delamination of ultrathin films, whose mechanics is governed by their propensity to buckle and delaminate partially, under minute levels of compression. Combining mechanical and geometrical considerations, we introduce a minimal model for curvature-induced delamination accounting for the two buckling motifs that underlie partial delamination: shallow "rucks" and localized "folds". We predict nontrivial scaling rules for the onset of curvature-induced delamination and various features of the emerging patterns, which compare well with experiments. Beyond gaining control on the use of ultrathin adhesives in cutting-edge technologies such as stretchable electronics, our analysis is a significant step toward quantifying the multiscale morphology that emerges upon imposing geometrical and mechanical constraints on highly bendable solid objects.

Building blocks of non-Euclidean ribbons: size-controlled self-assembly via discrete frustrated particles

Geometric frustration offers a pathway to soft matter self-assembly with controllable finite sizes. While the understanding of frustration in soft matter assembly derives almost exclusively from continuum elastic descriptions, a current challenge is to understand the connection between microscopic physical properties of misfitting "building blocks" and emergent assembly behavior at the mesoscale. We present and analyze a particle-based description of what is arguably the best studied example for frustrated soft matter assembly, negative-curvature ribbon assembly, observed in both assemblies of chiral surfactants and shape-frustrated nanoparticles. Based on our particle model, known as saddle wedge monomers, we numerically test the connection between microscopic shape and interactions of the misfitting subunits and the emergent behavior at the supra-particle scale, specifically focussing on the propagation and relaxation of inter-particle strains, the emergent role of extrinsic shape on frustrated ribbons and the equilibrium regime of finite width selection. Beyond the intuitive role of shape misfit, we show that self-limitation is critically dependent on the finite range of cohesive interactions, with larger size finite assemblies requiring increasing short-range interparticle forces. Additionally, we demonstrate that non-linearities arising from discrete particle interactions alter self-limiting behavior due to both strain-softening in shape-flattened assembly and partial yielding of highly strained bonds, which in turn may give rise to states of hierarchical, multidomain assembly. Tracing the regimes of frustration-limited assembly to the specific microscopic features of misfitting particle shapes and interactions provides necessary guidance for translating the theory of size-programmable assembly into design of intentionally-frustrated colloidal particles.

Retrieving Grain Boundaries in 2D Materials

The coalescence of randomly distributed grains with different crystallographic orientations can result in pervasive grain boundaries (GBs) in 2D materials during their chemical synthesis. GBs not only are the inherent structural imperfection that causes influential impacts on structures and properties of 2D materials, but also have emerged as a platform for exploring unusual physics and functionalities stemming from dramatic changes in local atomic organization and even chemical makeup. Here, recent advances in studying the formation mechanism, atomic structures, and functional properties of GBs in a range of 2D materials are reviewed. By analyzing the growth mechanism and the competition between far-field strain and local chemical energies of dislocation cores, a complete understanding of the rich GB morphologies as well as their dependence on lattice misorientations and chemical compositions is presented. Mechanical, electronic, and chemical properties tied to GBs in different materials are then discussed, towards raising the concept of using GBs as a robust atomic-scale scaffold for realizing tailored functionalities, such as magnetism, luminescence, and catalysis. Finally, the future opportunities in retrieving GBs for making functional devices and the major challenges in the controlled formation of GB structures for designed applications are commented.

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.

Entropic control of nanoparticle self-assembly through confinement

Entropy can be the sole driving force for the construction and regulation of ordered structures of soft matter systems. Specifically, under confinement, the entropic penalty could induce enhanced entropic effects which potentially generate visually ordered structures. Therefore, spatial confinement or a crowding environment offers an important approach to control entropy effects in these systems. Here, we review how spatial confinement-mediated entropic effects accurately and even dynamically control the self-assembly of nanoscale objects into ordered structures, focusing on our efforts towards computer simulations and theoretical analysis. First, we introduce the basic principle of entropic ordering through confinement. We then introduce the applications of this concept to various systems containing nanoparticles, including polymer nanocomposites, biological macromolecular systems and macromolecular colloids. Finally, the future directions and challenges for tailoring nanoparticle organization through spatial confinement-mediated entropic effects are detailed. We expect that this review could stimulate further efforts in the fundamental research on the relationship between confinement and entropy and in the applications of this concept for designer nanomaterials.

Direct Measurements of Surface Strain-Mediated Lateral Interactions between Adsorbates in Colloidal Heteroepitaxy

Surface strain can alter the dynamics of adsorbates, and often, the adsorbates themselves induce and interact via their surface strain fields. In epitaxy, such strain-mediated effects get further compounded when a misfit strain exists due to lattice mismatch between the growing film and substrate. Here, we carry out particle-resolved imaging of heteroepitaxial growth of multilayer colloidal films where the particles interact via a short-range attraction. Surprisingly, although the misfit strain relaxed systematically with film thickness, the adcolloid diffusivity was nonmonotonic. We show that this nonmonotonicity stems from the competition between the spatial extent of self-induced in-layer strain and the short interaction range. Importantly, we provide direct evidence for long-ranged strain-mediated interactions between adsorbates and show that it alters the growing film's morphology.

Observation of two-step melting on a sphere

Melting in two-dimensional flat space is typically two-step and via the hexatic phase. How melting proceeds on a curved surface, however, is not known. Topology mandates that crystalline particle assemblies on these surfaces harbor a finite density of defects, which itself can be ordered, like the icosahedral ordering of 5-coordinated disclination defects on a sphere. Thus, melting even on a sphere, the simplest closed surface, involves the loss of both crystalline and defect order. Probing the interplay of these two forms of order, however, requires a system in which melting can be performed in situ, and this has not been achieved hitherto. Here, by tuning interparticle interactions in situ, we report an observation of an intermediate hexatic phase during the melting of colloidal crystals on a sphere. Remarkably, we observed a precipitous drop in icosahedral defect order in the hexatic phase where the shear modulus is expected to vanish. Furthermore, unlike in flat space, where disorder can fundamentally alter the nature of the melting process, on the sphere, we observed the signature characteristics of ideal melting. Our findings have profound implications for understanding, for instance, the self-assembly and maturation dynamics of viral capsids and also phase transitions on curved surfaces.

Structural properties and ring defect formation in discotic liquid crystal nanodroplets

In this work, we performedNpTMonte Carlo simulations of a Gay-Berne discotic liquid crystal confined in a spherical droplet under face-on anchoring and fixed pressure. We find that, in contrast to the unbounded system, a plot of the order parameter as function of temperature does not show a clear evidence of a first-order isotropic-nematic transition. We also find that the impossibility of simultaneously satisfy the uniform director field requirement of a nematic phase with the radial boundary conditions, results in the appearance of a ring disclination line as a stress release mechanism in the interior of the droplet. Under further cooling, a columnar phase appears at the center of the droplet.

Coloration in Supraparticles Assembled from Polyhedral Metal-Organic Framework Particles

Supraparticles are spherical colloidal crystals prepared by confined self‐assembly processes. A particularly appealing property of these microscale structures is the structural color arising from interference of light with their building blocks. Here, we assemble supraparticles with high structural order that exhibit coloration from uniform, polyhedral metal–organic framework (MOF) particles. We analyse the structural coloration as a function of the size of these anisotropic building blocks and their internal structure. We attribute the angle‐dependent coloration of the MOF supraparticles to the presence of ordered, onion‐like layers at the outermost regions. Surprisingly, even though different shapes of the MOF particles have different propensities to form these onion layers, all supraparticle dispersions show well‐visible macroscopic coloration, indicating that local ordering is sufficient to generate interference effects.

Cumulative geometric frustration and superextensive energy scaling in a nonlinear classical XY-spin model

Geometric frustration results from a discrepancy between the locally favored arrangement of the constituents of a system and the geometry of the embedding space. Geometric frustration can be either noncumulative, which implies an extensive energy growth, or cumulative, which implies superextensive energy scaling and highly cooperative ground-state configurations which may depend on the dimensions of the system. Cumulative geometric frustration was identified in a variety of continuous systems including liquid crystals, filament bundles, and molecular crystals. However, a spin-lattice model which clearly demonstrates cumulative geometric frustration was lacking. In this paper we describe a nonlinear variation of the XY-spin model on a triangular lattice that displays cumulative geometric frustration. The model is studied numerically and analyzed in three distinct parameter regimes, which are associated with different energy minimizing configurations. We show that, despite the difference in the ground-state structure in the different regimes, in all cases the superextensive power-law growth of the frustration energy for small domains grows with the same universal exponent that is predicted from the structure of the underlying compatibility condition.

Curvature-Mediated Forces on Elastic Inclusions in Fluid Interfaces

Heterogeneous fluid interfaces often include two-dimensional solid domains that mechanically respond to changes in interfacial curvature. While this response is well-characterized for rigid inclusions, the influence of solid-like elasticity remains essentially unexplored. Here, we show that an initially flat, elastic inclusion embedded in a curved, fluid interface will exhibit qualitatively distinct behavior depending on its size and stiffness. Small, stiff inclusions are limited by bending and experience forces directed up gradients of Gaussian curvature, in keeping with prior findings for rigid discoids. By contrast, larger and softer inclusions are driven down gradients of squared Gaussian curvature in order to minimize the elastic penalty for stretching. Our calculations of the force on a solid inclusion are shown to collapse onto a universal curve spanning the bending- and stretching-limited regimes. From these results, we make predictions for the curvature-directed motion of deformable solids embedded within a model interface of variable Gaussian curvature.

Theorem on the Compatibility of Spherical Kirigami Tessellations

We present a theorem on the compatibility upon deployment of kirigami tessellations restricted on a spherical surface with patterned slits forming freeform quadrilateral meshes. We show that the spherical kirigami tessellations have either one or two compatible states, i.e., there are at most two isolated strain-free configurations along the deployment path. The theorem further reveals that the rigid-to-floppy transition from spherical to planar kirigami tessellations is possible if and only if the slits form parallelogram voids along with vanishing Gaussian curvature, which is also confirmed by an energy analysis and simulations. On the application side, we show a design of bistable spherical domelike structure based on the theorem. Our study provides new insights into the rational design of morphable structures based on Euclidean and non-Euclidean geometries.

Two-dimensional crystalization on spheres: Crystals grow cracked

Here we study how curvature affects the structure of two-dimensional crystals growing on spheres. The mechanism of crystal growth is described by means of a Landau model in curved space that accounts for the excess of strain on crystal bonds caused by the substrate's curvature (packing frustration). In curved space elastic energy penalization strongly dictates the geometry of growing crystals. While compact faceted crystals are observed when elastic energy contribution can be neglected, cracked crystals with ribbonlike forms appear as the main mechanisms to reduce elastic frustration for highly curved systems.

Cumulative geometric frustration in physical assemblies

Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that cannot be realized globally. Recently, such frustrated assemblies were shown to exhibit filamentation, size limitation, large morphological variations and other exotic response properties. While these unique characteristics can be shown to be a direct outcome of the geometric frustration, some geometrically frustrated systems do not exhibit any of the above phenomena. In this work we exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies. The framework highlights the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly. We show that the structure of the compatibility conditions determines the behavior of small assemblies and in particular predicts their superextensive energy growth exponent. We illustrate the use of this framework to several well-known frustrated assemblies.

Shaping in the Third Direction; Synthesis of Patterned Colloidal Crystals by Polyester Fabric-Guided Self-Assembly

A polyester fabric with rectangular openings was used as a sacrificial template for the guiding of a sub-micron sphere (polystyrene (PS) and silica) aqueous colloid self-assembly process during evaporation as a patterned colloidal crystal (PCC). This simple process is also a robust one, being less sensitive to external parameters (ambient pressure, temperature, humidity, vibrations). The most interesting feature of the concave-shape-pattern unit cell (350 μm × 400 μm × 3 μm) of this crystal is the presence of triangular prisms at its border, each prism having a one-dimensional sphere array at its top edge. The high-quality ordered single layer found inside of each unit cell presents the super-prism effect and left-handed behavior. Wider yet elongated deposits with ordered walls and disordered top surfaces were formed under the fabric knots. Rectangular patterning was obtained even for 20 μm PS spheres. Polyester fabrics with other opening geometries and sizes (~300–1000 μm) or with higher fiber elasticity also allowed the formation of similar PCCs, some having curved prismatic walls. A higher colloid concentration (10–20%) induces the formation of thicker walls with fiber-negative replica morphology. Additionally, thick-wall PCCs (~100 μm) with semi-cylindrical morphology were obtained using SiO₂ sub-microspheres and a wavy fabric. The colloidal pattern was used as a lithographic mask for natural lithography and as a template for the synthesis of triangular-prism-shaped inverted opals.

Curvature-assisted self-assembly of Brownian squares on cylindrical surfaces

HYPOTHESIS

We hypothesize that curved surfaces, including cylindrical surfaces, which go beyond prior experiments using flat surfaces, can significantly influence and alter the phase behavior and self-assembly of dense two-dimensional systems of Brownian colloids.

EXPERIMENTS

Here, we report a first experimental study regarding the self-assembly of Brownian square platelets with an edge length L = 2.3 μm on cylindrical surfaces having different curvatures; these platelets are subjected to a depletion attraction in order to form a monolayer above the cylindrical surface, yet have nearly hard interactions within the monolayer. Simulations are also performed to confirm and explain the experimental observations.

FINDINGS

Phase diagrams as a function of curvature are determined experimentally. Interestingly, hexagonal rotator crystal structures are observed for tubes having radii > 10.9L, but a tetratic phase is seen instead for the 10.9L tube at the corresponding platelet area fractions. We show that this transition is caused by the curvature-induced orientation-dependence of the depletion attraction between the squares and the underlying cylindrical surface. Brownian dynamics simulation results confirm the experimental observations and also illustrate helical structures formed by squares packing on cylinders. Our results demonstrate a way towards control over the self-assembly of anisotropic particles through curvature and depletion-attraction-induced orientational confinement.

Curved colloidal crystals of discoids at near-critical liquid-liquid interface

The assembly of disc-shaped particles at curved liquid-liquid interfaces was studied by using confocal microscopy. The interface is formed by a phase-separating critical liquid mixture of 2,6-lutidine and heavy water, where the colloids spontaneously assembled forming a dome. The novelty of this system is three-fold. First, the domes can be constructed and annihilated remotely and reversibly, which allows dynamic control of the colloidal assembly. Second, the effect of curvature can be investigated by analyzing domes of different radii ranging from 5 μm to 125 μm. Third, the slow dynamics due to hydrodynamic interaction among the particles can be utilized to investigate the time-evolution of defect morphology. Unlike the widely studied repulsive colloids, the interparticle potential near the critical point has an attractive component. I contrasted the packing and defects morphology of a solid-like and liquid-like dome differing in particle number density. In the solid-like dome, a chain of 5- and 7-fold coordinated particles was observed. The analysis of trajectories showed that particles were bound in a potential well of a depth of about ten times the thermal energy, which matched well with the calculation of the pair-potential by considering the attractive critical Casimir force among the particles. In the liquid-like dome, 6-fold particles separated by clusters of 5- and 7-coordinated particles were observed, which is suggestive of liquid-solid coexistence. The uniqueness of this system will open up a new research avenue to investigate the effect of varying curvature on the crystallization, defects, and phase diagram of colloidal assemblies.

Reflectance Confocal Microscopy: A Powerful Tool for Large Scale Characterization of Ordered/Disordered Morphology in Colloidal Photonic Structures

The development of appropriate methods to correlate the structure and optical properties of colloidal photonic structures is still a challenge. Structural information is mostly obtained by electron, X-ray, or optical microscopy methods and X-ray diffraction, while bulk spectroscopic methods and low resolution bright-field microscopy are used for optical characterization. Here, we describe the use of reflectance confocal microscopy as a simple and intuitive technique to provide a direct correlation between the ordered/disordered structural morphology of colloidal crystals and glasses, and their corresponding optical properties.

Statistical Mechanics and Thermodynamics of Liquids and Crystals

Thermodynamic phases are the most prominent manifestation of emergent behavior [...].

Predicting the characteristics of defect transitions on curved surfaces

The energetically optimal position of lattice defects on intrinsically curved surfaces is a complex function of shape parameters. For open surfaces, a simple condition predicts the critical size for which a central disclination yields lower energy than a boundary disclination. In practice, this transition is modified by activation energies or more favorable intermediate defect positions. Here it is shown that these transition characteristics (continuous or discontinuous, first or second order) can also be inferred from analytical, general criteria evaluated from the surface shape. A universal scale of activation energy is found, and the criteria are generalized to predict transition order as surface shape symmetry is broken. The results give practical insight into structural transitions to disorder in many cellular materials of technological and biological importance.

Effects of the curvature gradient on the distribution and diffusion of colloids confined to surfaces

The properties and behavior of colloids confined to move on curved surfaces offer a fertile ground for analysis since the geometric constraints induce specific features that are not available in flat spaces. Given their pertinence for biological and physicochemical processes, both with potential useful applications, the development of the concepts and methodology necessary for a deeper understanding of these unconventional systems is indeed an essential pursuit. The present study discusses a general and rigorous algorithm for the implementation of Brownian dynamics simulations that solves underlying difficulties and shortcomings inherent to conventional first-order schemes. Still based on the Ermak-McCammon recipe, our approach complements it with the higher-order geodesical projections of the elementary jumps generated on the associated tangent plane. This strategy, which warrants the locally isotropic propagation of non-interacting particles, is tested with a model system of colloidal particles interacting through a screened Coulomb potential while confined to move on ellipsoidal surfaces. This allows us to measure the effects prompted by the curvature gradient on the static and dynamic properties of this system. The varying curvature thus induces energetically favorable configurations in which the particles maximize their Euclidean distancing by crowding the regions with the largest Gaussian curvature, while withdrawing from those with the lowest. In turn, these inhomogeneous distributions provoke the anisotropic self-diffusion of the confined colloids, which is examined by exploiting the pertinent geodesic radial coordinates. The proficient methods under consideration thus allows dealing with the rich and remarkable new phenomena generated by any distinctive surface geometry.

Stress buffering in cyclodextrin-based membranes coated on emulsion droplet surfaces

Surface instability of membranes not only plays a critical role in the morphological evolution observed in natural and biological systems, but also underpins a promising way for the bottom-up fabrication of novel functional materials. There is an urgent need for the design of novel building blocks into membranes, and the understanding of the abilities of the membranes to cope with mechanical stress is therefore of considerable importance. Here, we design membranes built with cyclodextrin-oil inclusion complexes, which are formed spontaneously at the oil/water interface by a self-assembly process. We select the oil phases of distinct molecular structures, namely, branched triglyceride oil and straight-chain n-dodecane, and examine the patterns in which the membranes adopt morphological transitions to buffer stress. We discuss two possible buffering scenarios for the behaviors observed in view of structural arrest and interfacial rheology, which are most closely linked to the rigidity of the membranes. The membranes represent fascinating models and shed some light on the origin of arrested stress relaxation.

Single-cell yolk-shell nanoencapsulation for long-term viability with size-dependent permeability and molecular recognition

Like nanomaterials, bacteria have been unknowingly used for centuries. They hold significant economic potential for fuel and medicinal compound production. Their full exploitation, however, is impeded by low biological activity and stability in industrial reactors. Though cellular encapsulation addresses these limitations, cell survival is usually compromised due to shell-to-cell contacts and low permeability. Here, we report ordered packing of silica nanocolloids with organized, uniform and tunable nanoporosities for single cyanobacterium nanoencapsulation using protamine as an electrostatic template. A space between the capsule shell and the cell is created by controlled internalization of protamine, resulting in a highly ordered porous shell-void-cell structure formation. These unique yolk-shell nanostructures provide long-term cell viability with superior photosynthetic activities and resistance in harsh environments. In addition, engineering the colloidal packing allows tunable shell-pore diameter for size-dependent permeability and introduction of new functionalities for specific molecular recognition. Our strategy could significantly enhance the activity and stability of cyanobacteria for various nanobiotechnological applications.

Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (e.g. macromolecules, particles) into ordered multi-unit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits, and the other, a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving non-covalent interactions between identical subunits "measure" and select the size of assembled structures? In this review, we begin with an introduction to the basic statistical mechanical framework for assembly thermodynamics, and use this to highlight the key physical ingredients that ensure equilibrium assembly will terminate at finite dimensions. Then, we introduce examples of self-limiting assembly systems, and classify them within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter - micellization of amphiphiles and shell/tubule formation of tapered subunits - as well as less widely known classes of assemblies, such as short-range attractive/long-range repulsive systems and geometrically-frustrated assemblies. For each of these self-limiting mechanisms, we describe the physical mechanisms that select equilibrium assembly size, as well as potential limitations of finite-size selection. Finally, we discuss alternative mechanisms for finite-size assemblies, and draw contrasts with the size-control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.

Protein crystallisation with air bubble templates: case of gas–liquid–solid interfaces

Crystal formation on air bubble–liquid interface, as soft template to efficiently prompt nucleation of proteins.

Defects and defect engineering in Soft Matter

Soft matter covers a wide range of materials based on linear or branched polymers, gels and rubbers, amphiphilic (macro)molecules, colloids, and self-assembled structures. These materials have applications in various industries, all highly important for our daily life, and they control all biological functions; therefore, controlling and tailoring their properties is crucial. One way to approach this target is defect engineering, which aims to control defects in the material's structure, and/or to purposely add defects into it to trigger specific functions. While this approach has been a striking success story in crystalline inorganic hard matter, both for mechanical and electronic properties, and has also been applied to organic hard materials, defect engineering is rarely used in soft matter design. In this review, we present a survey on investigations on defects and/or defect engineering in nine classes of soft matter composed of liquid crystals, colloids, linear polymers with moderate degree of branching, hyperbranched polymers and dendrimers, conjugated polymers, polymeric networks, self-assembled amphiphiles and proteins, block copolymers and supramolecular polymers. This overview proposes a promising role of this approach for tuning the properties of soft matter.

Self-Limiting Polymerization of DNA Origami Subunits with Strain Accumulation

Biology demonstrates how a near infinite array of complex systems and structures at many scales can originate from the self-assembly of component parts on the nanoscale. But to fully exploit the benefits of self-assembly for nanotechnology, a crucial challenge remains: How do we rationally encode well-defined global architectures in subunits that are much smaller than their assemblies? Strain accumulation via geometric frustration is one mechanism that has been used to explain the self-assembly of global architectures in diverse and complex systems a posteriori. Here we take the next step and use strain accumulation as a rational design principle to control the length distributions of self-assembling polymers. We use the DNA origami method to design and synthesize a molecular subunit known as the PolyBrick, which perturbs its shape in response to local interactions via flexible allosteric blocking domains. These perturbations accumulate at the ends of polymers during growth, until the deformation becomes incompatible with further extension. We demonstrate that the key thermodynamic factors for controlling length distributions are the intersubunit binding free energy and the fundamental strain free energy, both which can be rationally encoded in a PolyBrick subunit. While passive polymerization yields geometrical distributions, which have the highest statistical length uncertainty for a given mean, the PolyBrick yields polymers that approach Gaussian length distributions whose variance is entirely determined by the strain free energy. We also show how strain accumulation can in principle yield length distributions that become tighter with increasing subunit affinity and approach distributions with uniform polymer lengths. Finally, coarse-grained molecular dynamics and Monte Carlo simulations delineate and quantify the dominant forces influencing strain accumulation in a molecular system. This study constitutes a fundamental investigation of the use of strain accumulation as a rational design principle in molecular self-assembly.

Confined Crystal Melting in Edgeless Poly(l-lactic acid) Crystalsomes

Polymer single crystals tend to be quasi two-dimensional (2D) lamellae and their small lateral surfaces are the starting points of lamella melting and thickening. However, the recently discovered crystalsomes, which are defined for hollow single crystal-like spherical shells, are edgeless, self-confined, and incommensurate with translational symmetry. This work concerns the structure and melting behavior of these edgeless crystalsomes. Poly(l-lactic acid) crystalsomes were grown using a miniemulsion solution crystallization method. Differential scanning calorimetry and in situ wide-angle X-ray diffraction were used to follow the structural evolution of the crystalsomes upon heating. Our results demonstrated that the structure and melting behavior of crystalsomes are curvature-dependent and significantly different from their flat crystal counterpart.