Self-assembly of hard helices: a rich and unconventional polymorphism

Two-Step Chirality Transfer to Twisted Assemblies: Synergistic Interplay of Chiral and Aggregation Interactions

Chirality plays a pivotal role in both the origin of life and the self-assembly of materials. However, the governing principles behind chirality transfer in hierarchical self-assembly across multiple length scales remain elusive. Here, we propose a concise and versatile simulation strategy using the patchy particle chain model to investigate the self-assembly of rods interacting through chiral and aggregation interactions. We reveal that chiral interaction possessing an entropic nature, amplifies the fluctuations and twists in the alignment of rods, while aggregation interaction serves as a foundational platform for aggregation and assembly. When both interactions exhibit moderate absolute and relative values, their synergistic interplay facilitates the chirality transfer from rods to assemblies, resulting in the formation of chiral mesoscale ordered structures. Furthermore, we observe a two-step chirality transfer process by monitoring the formation kinetics of the twisted assemblies. This work not only provides a comprehensive insight into chirality transfer mechanisms, but also introduces a versatile mesoscale simulation framework for exploring the role of chirality in hierarchical self-assembly.

Stability of the chiral crystal phase and breakdown of the cholesteric phase in mixtures of active-passive chiral rods

In this study, we aim to explore the effect of chirality on the phase behavior of active helical particles driven by two-temperature scalar activity. We first calculate the equation of state of soft helical particles of various intrinsic chiralities using molecular dynamics (MD) simulation. In equilibrium, the emergence of various liquid crystal (LC) phases such as nematic (N), cholesteric , smectic (Sm) and crystal (K) crucially depends on the presence of walls that induce planar alignment. Next, we introduce activity through the two-temperature model: keep increasing the temperature of half of the helical particles (labeled as 'hot' particles) while maintaining the temperature of the other half at a lower value (labeled as 'cold' particles). Starting from a homogeneous isotropic (I) phase, we find the emergence of 2-TIPS: two temperature-induced phase separations between the hot and cold particles. We also observe that the cold particles undergo an ordering transition to various LC phases even in the absence of a wall. This observation reveals that the hot-cold interface in the active system plays the role of a wall in the equilibrium system by inducing an alignment direction for the cold particles. However, in the case of a cholesteric phase, we observe that activity destabilizes the phase by inducing smectic ordering in the cold zone while an isotropic structure in the hot zone. The smectic ordering in the cold zone eventually transforms to a chiral crystal phase with high enough activity.

Janus helices: From fully attractive to hard helices

The phase diagram of hard helices differs from its hard rods counterpart by the presence of chiral "screw" phases stemming from the characteristic helical shape, in addition to the conventional liquid crystal phases also found for rod-like particles. Using extensive Monte Carlo and Molecular Dynamics simulations, we study the effect of the addition of a short-range attractive tail representing solvent-induced interactions to a fraction of the sites forming the hard helices, ranging from a single-site attraction to fully attractive helices for a specific helical shape. Different temperature regimes exist for different fractions of the attractive sites, as assessed in terms of the relative Boyle temperatures, that are found to be rather insensitive to the specific shape of the helical particle. The temperature range probed by the present study is well above the corresponding Boyle temperatures, with the phase behaviour still mainly entropically dominated and with the existence and location of the various liquid crystal phases only marginally affected. The pressure in the equation of state is found to decrease upon increasing the fraction of attractive beads and/or on lowering the temperature at fixed volume fraction, as expected on physical grounds. All screw phases are found to be stable within the considered range of temperatures with the smectic phase becoming more stable on lowering the temperature. By contrast, the location of the transition lines do not display a simple dependence on the fraction of attractive beads in the considered range of temperatures.

Flow and assembly of cellulose nanocrystals (CNC): A bottom-up perspective - A review

Cellulosic sources, such as lignocellulose-rich biomass, can be mechanically or acid degraded to produce inclusions called cellulose nanocrystals (CNCs). They have several uses in the sectors of biomedicine, photonics, and material engineering because of their biodegradability, renewability, sustainability, and mechanical qualities. The processing and design of CNC-based products are inextricably linked to the rheological behaviour of CNC suspension or in combination with other chemicals, such as surfactants or polymers; in this context, rheology offers a significant link between microstructure and macro scale flow behaviour that is intricately linked to material response in applications. The flow behaviour of CNC items must be properly specified in order to produce goods with value-added characteristics. In this review article, we provide new research on the shear rheology of CNC dispersion and CNC-based hydrogels in the linear and nonlinear regime, with storage modulus values reported to range from ~10^-3 to 10³ Pa. Applications in technology and material science are also covered simultaneously. We carefully examined the effects of charge density, aspect ratio, concentration, persistence length, alignment, liquid crystal formation, the cause of chirality in CNCs, interfacial behaviour and interfacial rheology, linear and nonlinear viscoelasticity of CNC suspension in bulk and at the interface using the currently available literature.

Modeling the cholesteric pitch of apolar cellulose nanocrystal suspensions using a chiral hard-bundle model

Cellulose nanocrystals (CNCs) are naturally sourced elongated nanocolloids that form cholesteric phases in water and apolar solvents. It is well accepted that CNCs are made of bundles of crystalline microfibrils clustered side-by-side, and there is growing evidence that each individual microfibril is twisted. Yet, the origin of the chiral interactions between CNCs remains unclear. In this work, CNCs are described with a simple model of chiral hard splinters, enabling the prediction of the pitch using density functional theory and Monte Carlo simulations. The predicted pitch P compares well with experimental observations in cotton-based CNC dispersions in apolar solvents using surfactants but also with qualitative trends caused by fractionation or tip sonication in aqueous suspensions. These results suggest that the bundle shape induces an entropy-driven chiral interaction between CNCs, which is the missing link in explaining how chirality is transferred from the molecular scale of cellulose chains to the cholesteric order.

Shape-driven entropic self-assembly of an open, reconfigurable, binary host-guest colloidal crystal

Entropically driven self-assembly of hard anisotropic particles, where particle shape gives rise to emergent valencies, provides a useful perspective for the design of nanoparticle and colloidal systems. Hard particles self-assemble into a rich variety of crystal structures, ranging in complexity from simple close-packed structures to structures with 432 particles in the unit cell. Entropic crystallization of open structures, however, is missing from this landscape. Here, we report the self-assembly of a two-dimensional binary mixture of hard particles into an open host-guest structure, where nonconvex, triangular host particles form a honeycomb lattice that encapsulates smaller guest particles. Notably, this open structure forms in the absence of enthalpic interactions by effectively splitting the structure into low- and high-entropy sublattices. This is the first such structure to be reported in a two-dimensional athermal system. We discuss the observed compartmentalization of entropy in this system, and show that the effect of the size of the guest particle on the stability of the structure gives rise to a reentrant phase behavior. This reentrance suggests the possibility for a reconfigurable colloidal material, and we provide a proof-of-concept by showing the assembly behavior while changing the size of the guest particles in situ. Our findings provide a strategy for designing open colloidal crystals, as well as binary systems that exhibit co-crystallization, which have been elusive thus far.

Terpolymer Multicompartment Nanofibers as Templates for Hybrid Pt Double Helices

Hybrid inorganic/block copolymer (BCP) materials have become increasingly relevant for application in heterogeneous catalysis, microelectronics, and nanomedicine. While block copolymer templates are widely used for the formation of inorganic nanostructures, multicompartment templates could give access to more complex shapes and inner structures that are challenging to obtain with traditional processes. Here, we report the formation and characterization of hybrid platinum/polymer helices using multicompartment nanofibers (MCNFs) of polystyrene-block-polybutadiene-block-poly(tert-butyl methacrylate) (PS-b-PB-b-PT) triblock terpolymers as templates. Cross-linking of a PS-b-PB-b-PT helix-on-cylinder morphology resulted in uniform nanofibers with a diameter of 90 nm and a length of several micrometers, as well as an inner PB double helix (diameter 35 nm, pitch 25 nm, core 12 nm). The PB double helix served as template for the sol-gel reaction of H₂PtCl₆ into hybrid Pt double helices (Pt@MCNFs) as verified by STEM, electron tomography, AFM, and SEM. Carbonization of the Pt hybrids into Pt decorated carbon nanofibers (Pt@C) was followed in situ on a TEM heating state. Gradual heating from 25 to 1000 °C induced fusion of amorphous Pt NPs into larger crystalline Pt NP, which sheds light on the aging of Pt NPs in BCP scaffolds under high temperature conditions. The Pt@MCNFs were further sulfonated and incorporated into a filter to catalyze a model compound in a continuous flow process.

Biaxial, Twist-bend, and Splay-bend Nematic Phases of Banana-shaped Particles Revealed by Lifting the "Smectic Blanket"

We perform an extensive computational study on the phase behavior of hard banana-shaped particles, and show that biaxial, twist-bend, and splay-bend nematic phases are metastable with respect to a smectic phase for a system of hard bent spherocylinders. However, if the smectic phase is destabilized-either by polydispersity in the particle length or by curvature in the particle shape-stable biaxial, twist-bend, and splay-bend nematic phases are obtained. This provides a unified and consistent picture on the subtle role of particle shape on the phase behavior of bent rods.

A versatile simulation method for studying phase behavior and dynamics in colloidal rod and rod-polymer suspensions

Here, we present an implicit-solvent model for dynamic simulations of hard-rod and rod-polymer suspensions. Individual rods are represented by a rigid linear chain consisting of overlapping spheres which interact through a pseudohard-core potential based on the cut-and-shifted Mie (generalized Lennard-Jones) potential with exponents (50, 49). In the rod-polymer suspensions, the polymers are modeled as freely interpenetrable spheres with respect to each other, while there is the pseudohard-core repulsion between the polymer and rod spheres. Dynamic simulations with this model are carried out with a dissipative particle dynamics (DPD) thermostat-each sphere is put in a larger DPD sphere and thus interacts with others via additional pairwise frictional and random forces-which captures the effects of Brownian forces due to the solvent while conserving local momentum. The phase behavior of these models, obtained from continuous compression and expansion simulations, reproduces previous predictions based on theoretical calculations and Monte Carlo simulations. Our method is suited to study dynamic processes in these suspensions, including nucleation and self-assembly, and can be readily extended to colloidal particles of different shapes and chemistry.

Effect of Size Polydispersity on the Pitch of Nanorod Cholesterics

Many nanoparticle-based chiral liquid crystals are composed of polydisperse rod-shaped particles with considerable spread in size or shape, affecting the mesoscale chiral properties in, as yet, unknown ways. Using an algebraic interpretation of Onsager-Straley theory for twisted nematics, we investigate the role of length polydispersity on the pitch of nanorod-based cholesterics with a continuous length polydispersity, and find that polydispersity enhances the twist elastic modulus, K 2 , of the cholesteric material without affecting the effective helical amplitude, K t . In addition, for the infinitely large average aspect ratios considered here, the dependence of the pitch on the overall rod concentration is completely unaffected by polydispersity. For a given concentration, the increase in twist elastic modulus (and reduction of the helical twist) may be up to 50% for strong size polydispersity, irrespective of the shape of the unimodal length distribution. We also demonstrate that the twist reduction is reinforced in bimodal distributions, obtained by doping a polydisperse cholesteric with very long rods. Finally, we identify a subtle, non-monotonic change of the pitch across the isotropic-cholesteric biphasic region.

Self-Assembled Chiral Photonic Crystals from a Colloidal Helix Racemate

Chiral crystals consisting of microhelices have many optical properties, while presently available fabrication processes limit their large-scale applications in photonic devices. Here, by using a simplified simulation method, we investigate a bottom-up self-assembly route to build up helical crystals from the smectic monolayer of a colloidal helix racemate. With increasing the density, the system undergoes an entropy-driven cocrystallization by forming crystals of various symmetries with different helical shapes. In particular, we identify two crystals of helices arranged in binary honeycomb and square lattices, which are essentially composed of two sets of opposite-handed chiral crystals. Photonic calculations show that these chiral structures can have large complete photonic band gaps. In addition, in the self-assembled chiral square crystal, we also find dual polarization band gaps that selectively forbid the propagation of circularly polarized light of a specific handedness along the helical axis direction. The self-assembly process in our proposed system is robust, suggesting possibilities of using chiral colloids to assemble photonic metamaterials.

The Self-Assembly of Cellulose Nanocrystals: Hierarchical Design of Visual Appearance

By controlling the interaction of biological building blocks at the nanoscale, natural photonic nanostructures have been optimized to produce intense coloration. Inspired by such biological nanostructures, the possibility to design the visual appearance of a material by guiding the hierarchical self-assembly of its constituent components, ideally using natural materials, is an attractive route for rationally designed, sustainable manufacturing. Within the large variety of biological building blocks, cellulose nanocrystals are one of the most promising biosourced materials, primarily for their abundance, biocompatibility, and ability to readily organize into photonic structures. Here, the mechanisms underlying the formation of iridescent, vividly colored materials from colloidal liquid crystal suspensions of cellulose nanocrystals are reviewed and recent advances in structural control over the hierarchical assembly process are reported as a toolbox for the design of sophisticated optical materials.

Hard Competition: Stabilizing the Elusive Biaxial Nematic Phase in Suspensions of Colloidal Particles with Extreme Lengths

We use computer simulations to study the existence and stability of a biaxial nematic N_{b} phase in systems of hard polyhedral cuboids, triangular prisms, and rhombic platelets, characterized by a long (L), medium (M), and short (S) particle axis. For all three shape families, we find stable N_{b} states provided the shape is not only close to the so-called dual shape with M=sqrt[LS] but also sufficiently anisotropic with L/S>9,11,14,23 for rhombi, (two types of) triangular prisms, and cuboids, respectively, corresponding to anisotropies not considered before. Surprisingly, a direct isotropic-N_{b} transition does not occur in these systems due to a destabilization of N_{b} by a smectic (for cuboids and prisms) or a columnar (for platelets) phase at small L/S or by an intervening uniaxial nematic phase at large L/S. Our results are confirmed by a density functional theory provided the third virial coefficient is included and a continuous rather than a discrete (Zwanzig) set of particle orientations is taken into account.

Purely entropic self-assembly of the bicontinuous Ia3d gyroid phase in equilibrium hard-pear systems

We investigate a model of hard pear-shaped particles which forms the bicontinuous Ia[Formula: see text]d structure by entropic self-assembly, extending the previous observations of Barmes et al. (2003 Phys. Rev. E68, 021708. (doi:10.1103/PhysRevE.68.021708)) and Ellison et al. (2006 Phys. Rev. Lett.97, 237801. (doi:10.1103/PhysRevLett.97.237801)). We specifically provide the complete phase diagram of this system, with global density and particle shape as the two variable parameters, incorporating the gyroid phase as well as disordered isotropic, smectic and nematic phases. The phase diagram is obtained by two methods, one being a compression-decompression study and the other being a continuous change of the particle shape parameter at constant density. Additionally, we probe the mechanism by which interdigitating sheets of pears in these systems create surfaces with negative Gauss curvature, which is needed to form the gyroid minimal surface. This is achieved by the use of Voronoi tessellation, whereby both the shape and volume of Voronoi cells can be assessed in regard to the local Gauss curvature of the gyroid minimal surface. Through this, we show that the mechanisms prevalent in this entropy-driven system differ from those found in systems which form gyroid structures in nature (lipid bilayers) and from synthesized materials (di-block copolymers) and where the formation of the gyroid is enthalpically driven. We further argue that the gyroid phase formed in these systems is a realization of a modulated splay-bend phase in which the conventional nematic has been predicted to be destabilized at the mesoscale due to molecular-scale coupling of polar and orientational degrees of freedom.

The Shape of Things To Come: The Formation of Modulated Nematic Mesophases at Various Length Scales

The twist-bend nematic (NTB ) phase is a recently discovered liquid-crystalline phase that exhibits macroscopic chirality even when formed from achiral materials, and as such presents a unique testbed for studies concerning the spontaneous breaking of mirror symmetry in soft matter. It is primarily exhibited by materials for which the molecular structure is composed of two rigid aromatic units (such as biphenyl connected by a flexible spacer). The local structure of the NTB phase is nematic-like-with molecules having an average orientational order but no positional order-with a nanoscale helix in which the pitch (i.e., the repeat distance of the helix) is of the order of several nanometres. A helix is chiral, and so the bulk NTB phase-in the absence of a biasing chiral environment-spontaneously separates into macroscopic domains of opposite handedness. After discussing the structure of this mesophase and its elucidation, this concept article presents the molecular factors that determine its incidence. The apparent dependency primarily on molecular shape and bend angle rather than particular functional group combinations manifests in this mesophase being exhibited on length scales far beyond those of simple liquid-crystalline dimers, not only in oligomers and polymers, but also in aqueous suspensions of micron sized helical particles.

Twisting with a twist: supramolecular helix fluctuations in chiral nematics

Most theoretical descriptions of lyotropic cholesteric liquid crystals to date focus on homogeneous systems in which the rod concentration, as opposed to the rod orientation, is uniform. In this work, we build upon the Onsager-Straley theory for twisted nematics and study the effect of weak concentration gradients, generated by some external potential, on the cholesteric twist. We apply our theory to chiral nematics of nanohelices in which the supramolecular helix sense is known to spontaneously change sign upon variation of particle concentration, passing through a so-called compensation point at which the mesoscopic twist vanishes. We show that the imposed field offers exquisite control of the handedness and magnitude of the helicoidal director field, even at weak field strengths. Within the same framework we also quantify the director fluctuation spectrum and find evidence for a correlation length diverging at the compensation point.

Understanding the origin of liquid crystal ordering of ultrashort double-stranded DNA

Recent experiments have shown that short double-stranded DNA (dsDNA) fragments having six- to 20-base pairs exhibit various liquid crystalline phases. This violates the condition of minimum molecular shape anisotropy that analytical theories demand for liquid crystalline ordering. It has been hypothesized that the liquid crystalline ordering is the result of end-to-end stacking of dsDNA to form long supramolecular columns which satisfy the shape anisotropy criterion necessary for ordering. To probe the thermodynamic feasibility of this process, we perform molecular dynamics simulations on ultrashort (four base pair long) dsDNA fragments, quantify the strong end-to-end attraction between them, and demonstrate that the nematic ordering of the self-assembled stacked columns is retained for a large range of temperature and salt concentration.

Chiral self-assembly of helical particles

The shape of the building blocks plays a crucial role in directing self-assembly towards desired architectures. Out of the many different shapes, the helix has a unique position. Helical structures are ubiquitous in nature and a helical shape is exhibited by the most important biopolymers like polynucleotides, polypeptides and polysaccharides as well as by cellular organelles like flagella. Helical particles can self-assemble into chiral superstructures, which may have a variety of applications, e.g. as photonic (meta)materials. However, a clear and definite understanding of these structures has not been entirely achieved yet. We have recently undertaken an extensive investigation on the phase behaviour of hard helical particles, using numerical simulations and classical density functional theory. Here we present a detailed study of the phase diagram of hard helices as a function of their morphology. This includes a variety of liquid-crystal phases, with different degrees of orientational and positional ordering. We show how, by tuning the helix parameters, it is possible to control the organization of the system. Starting from slender helices, whose phase behaviour is similar to that of rodlike particles, an increase in curliness leads to the onset of azimuthal correlations between the particles and the formation of phases specific to helices. These phases feature a new kind of screw order, of which there is experimental evidence in colloidal suspensions of helical flagella.

Empty smectic liquid crystals of hard nanorings: Insights from a second-virial theory

Inspired by recent simulations on highly open liquid crystalline structures formed by rigid planar nanorings, we present a simple theoretical framework explaining the prevalence of smectic over nematic ordering in systems of ring-shaped objects. The key part of our study is a calculation of the excluded volume of such nonconvex particles in the limit of vanishing thickness to diameter ratio. Using a simple stability analysis we then show that dilute systems of ring-shaped particles have a strong propensity to order into smectic structures with an unusual antinematic order while solid disks of the same dimensions exhibit nematic order. Since our model rings have zero internal volume, these smectic structures are essentially empty, resembling the strongly porous structures found in simulation. We argue that the antinematic intralamellar order of the rings plays an essential role in stabilizing these smectic structures.

Liquid Crystal Ordering and Isotropic Gelation in Solutions of Four-Base-Long DNA Oligomers

Liquid crystal ordering is reported in aqueous solutions of the oligomer 5'-ATTAp-3' and of the oligomer 5'-GCCGp-3'. In both systems, we quantitatively interpret ordering as stemming from the chaining of molecules via a "running-bond" type of pairing, a self-assembly process distinct from the duplex aggregation previously reported for longer oligonucleotides. While concentrated solutions of 5'-ATTAp-3' show only a columnar liquid crystal phase, solutions of 5'-GCCGp-3' display a rich phase diagram, featuring a chiral nematic phase analogous to those observed in solutions of longer oligonucleotides and two unconventional phases, a columnar crystal and, at high concentration, an isotropic amorphous gel. The appearance of these phases, which can be interpreted on the basis of features of 5'-GCCGp-3'molecular structure, suggests distinctive assembly motifs specific to ultrashort oligonucleotides.

Continuum percolation of congruent overlapping spherocylinders

Continuum percolation of randomly orientated congruent overlapping spherocylinders (composed of cylinder of height H with semispheres of diameter D at the ends) with aspect ratio α=H/D in [0,∞) is studied. The percolation threshold ϕ_{c}, percolation transition width Δ, and correlation-length critical exponent ν for spherocylinders with α in [0, 200] are determined with a high degree of accuracy via extensive finite-size scaling analysis. A generalized excluded-volume approximation for percolation threshold with an exponent explicitly depending on both aspect ratio and excluded volume for arbitrary α values in [0,∞) is proposed and shown to yield accurate predictions of ϕ_{c} for an extremely wide range of α in [0, 2000] based on available numerical and experimental data. We find ϕ_{c} is a universal monotonic decreasing function of α and is independent of the effective particle size. Our study has implications in percolation theory for nonspherical particles and composite material design.

Simulating the pitch sensitivity of twisted nematics of patchy rods

Stiff, elongated biomolecules such as filamentous viruses, DNA or cellulose nanocrystals are known to form liquid crystals often exhibiting a helical supramolecular organization. Little is known about the microscopic origin, size and handedness of the helical pitch in these, so-called cholesteric phases. Experimental observations in chiral lyotropics suggest that long-ranged chiral forces of electrostatic origin acting between the mesogens are responsible for such organization. Using large-scale computer simulation we study the sensitivity of the pitch imparted by soft microscopic helices and confirm that the helical sense is sensitive to a change of packing fraction, magnitude of the molecular pitch and amplitude of the chiral interactions. In particular, we find evidence that the cholesteric helix sense may change spontaneously upon variation of particle density, at fixed molecular chirality. These pitch inversions have been reported in recent theoretical studies but simulation evidence remains elusive. We rationalize these sudden changes in the supramolecular helical symmetry on the basis of detailed measurements of the mean-torque generated by the twisting of the helices. The simulation methodology employed does not require confining the twisted nematic in a slab geometry and allows for a simultaneous measurement of the pitch and the twist elastic constant. We find that the twist elastic constant increases almost linearly with density suggesting that twisted nematic shows no signs of anomalous stiffening due to pre-smectic fluctuations at higher packing fraction.

Entropy-driven formation of chiral nematic phases by computer simulations

Predicting the macroscopic chiral behaviour of liquid crystals from the microscopic chirality of the particles is highly non-trivial, even when the chiral interactions are purely entropic in nature. Here we introduce a novel chiral hard-particle model, namely particles with a twisted polyhedral shape and obtain a stable fully entropy-driven cholesteric phase by computer simulations. By slightly modifying the triangular base of the particle, we are able to switch from a left-handed prolate (calamitic) to a right-handed oblate (discotic) cholesteric phase using the same right-handed twisted particle model. Furthermore, we show that not only prolate and oblate chiral nematic phases, but also other novel entropy-driven phases, namely chiral blue phases, chiral nematic phases featuring both twist and splay deformations, chiral biaxial nematic phases with one of the axes twisted, can be obtained by varying particle biaxiality and chirality. Our results allow to identify general guidelines for the stabilization of these phases.

Controlling Chirality of Entropic Crystals

Colloidal crystal structures with complexity and diversity rivaling atomic and molecular crystals have been predicted and obtained for hard particles by entropy maximization. However, thus far homochiral colloidal crystals, which are candidates for photonic metamaterials, are absent. Using Monte Carlo simulations we show that chiral polyhedra exhibiting weak directional entropic forces self-assemble either an achiral crystal or a chiral crystal with limited control over the crystal handedness. Building blocks with stronger faceting exhibit higher selectivity and assemble a chiral crystal with handedness uniquely determined by the particle chirality. Tuning the strength of directional entropic forces by means of particle rounding or the use of depletants allows for reconfiguration between achiral and homochiral crystals. We rationalize our findings by quantifying the chirality strength of each particle, both from particle geometry and potential of mean force and torque diagrams.