Connectivity, Not Density, Dictates Percolation in Nematic Liquid Crystals of Slender Nanoparticles

Structural Color from Cellulose Nanocrystals or Chitin Nanocrystals: Self-Assembly, Optics, and Applications

Widespread concerns over the impact of human activity on the environment have resulted in a desire to replace artificial functional materials with naturally derived alternatives. As such, polysaccharides are drawing increasing attention due to offering a renewable, biodegradable, and biocompatible feedstock for functional nanomaterials. In particular, nanocrystals of cellulose and chitin have emerged as versatile and sustainable building blocks for diverse applications, ranging from mechanical reinforcement to structural coloration. Much of this interest arises from the tendency of these colloidally stable nanoparticles to self-organize in water into a lyotropic cholesteric liquid crystal, which can be readily manipulated in terms of its periodicity, structure, and geometry. Importantly, this helicoidal ordering can be retained into the solid-state, offering an accessible route to complex nanostructured films, coatings, and particles. In this review, the process of forming iridescent, structurally colored films from suspensions of cellulose nanocrystals (CNCs) is summarized and the mechanisms underlying the chemical and physical phenomena at each stage in the process explored. Analogy is then drawn with chitin nanocrystals (ChNCs), allowing for key differences to be critically assessed and strategies toward structural coloration to be presented. Importantly, the progress toward translating this technology from academia to industry is summarized, with unresolved scientific and technical questions put forward as challenges to the community.

Universal Critical Behavior of Percolation in Orientationally Ordered Janus Particles and Other Anisotropic Systems

We combine percolation theory and Monte Carlo simulation to study in two dimensions the connectivity of an equilibrium lattice model of interacting Janus disks which self-assemble into an orientationally ordered stripe phase at low temperature. As the patch size is increased or the temperature is lowered, clusters of patch-connected disks grow, and a percolating cluster emerges at a threshold. In the stripe phase, the critical clusters extend longer in the direction parallel to the stripes than in the perpendicular direction, and percolation is thus anisotropic. It is found that the critical behavior of percolation in the Janus system is consistent with that of standard isotropic percolation, when an appropriate spatial rescaling is made. The rescaling procedure can be applied to understand other anisotropic systems, such as the percolation of aligned rigid rods and of the q-state Potts model with anisotropic interactions.

Continuum percolation in colloidal dispersions of hard nanorods in external axial and planar fields

We present a theoretical study on continuum percolation of rod-like colloidal particles in the presence of axial and planar quadrupole fields. Our work is based on a self-consistent numerical treatment of the connectedness Ornstein-Zernike equation, in conjunction with the Onsager equation that describes the orientational distribution function of particles interacting via a hard-core repulsive potential. Our results show that axial and planar quadrupole fields both in principle increase the percolation threshold. By how much depends on a combination of the field strength, the concentration, the aspect ratio of the particles, and percolation criterion. We find that the percolated state can form and break down multiple times with increasing concentration, i.e., it exhibits re-entrance behaviour. Finally, we show that planar fields may induce a high degree of triaxiality in the shape of particle clusters that in actual materials may give rise to highly anisotropic conductivity properties.

Effect of Nanorod Physical Roughness on the Aggregation and Percolation of Nanorods in Polymer Nanocomposites

Using molecular dynamics simulations, we elucidate the effect of nanorod roughness on nanorod aggregation, dispersion, and percolation in polymer nanocomposites (PNCs). By choosing coarse-grained models that enable systematic variation of the nanorod roughness and by selecting purely repulsive pairwise interactions for nanorods and polymer chains, we show how nanorod roughness affects the entropic driving forces for various PNC morphologies. At this entropically driven limit, we find that increasing nanorod roughness hinders nanorod aggregation and promotes nanorod percolation in the polymer melt. As nanorod roughness increases, the nanorod volume fraction needed to induce nanorod aggregation also increases. Increasing nanorod roughness increases the configurational entropy of the polymer chains and lowers the entropically induced depletion attraction between nanorods.

Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and perfectly oriented deposition and removal

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (particles occupying k consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θ_{c,k}[ϑ_{c,k}], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k}[ϑ_{c,k}] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θ_{c,k}^{i}[ϑ_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[ϑ_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, ϑ_{c,k}, θ_{c,k}^{i}, and ϑ_{c,k}^{i} are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θ_{c,k}[θ_{c,k}^{i}] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θ_{c,k→∞}=0.500(2) [θ_{c,k→∞}^{i}=0.500(1)]; (2)ϑ_{c,k}[ϑ_{c,k}^{i}] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑ_{c,k→∞}=0.5334(6) [ϑ_{c,k→∞}^{i}=0.4666(6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(ϑ)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{c,k}^{i}=1) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the θ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.

From Equilibrium Liquid Crystal Formation and Kinetic Arrest to Photonic Bandgap Films Using Suspensions of Cellulose Nanocrystals

The lyotropic cholesteric liquid crystal phase developed by suspensions of cellulose nanocrystals (CNCs) has come increasingly into focus from numerous directions over the last few years. In part, this is because CNC suspensions are sustainably produced aqueous suspensions of a fully bio-derived nanomaterial with attractive properties. Equally important is the interesting and useful behavior exhibited by solid CNC films, created by drying a cholesteric-forming suspension. However, the pathway along which these films are realized, starting from a CNC suspension that may have low enough concentration to be fully isotropic, is more complex than often appreciated, leading to reproducibility problems and confusion. Addressing a broad audience of physicists, chemists, materials scientists and engineers, this Review focuses primarily on the physics and physical chemistry of CNC suspensions and the process of drying them. The ambition is to explain rather than to repeat, hence we spend more time than usual on the meanings and relevance of the key colloid and liquid crystal science concepts that must be mastered in order to understand the behavior of CNC suspensions, and we present some interesting analyses, arguments and data for the first time. We go through the development of cholesteric nuclei (tactoids) from the isotropic phase and their potential impact on the final dry films; the spontaneous CNC fractionation that takes place in the phase coexistence window; the kinetic arrest that sets in when the CNC mass fraction reaches ∼10 wt.%, preserving the cholesteric helical order until the film has dried; the ’coffee-ring effect’ active prior to kinetic arrest, often ruining the uniformity in the produced films; and the compression of the helix during the final water evaporation, giving rise to visible structural color in the films.

Shape, geometric percolation, and electrical conductivity of clusters in suspensions of hard platelets

Using Monte Carlo simulations, we investigate how geometric percolation and electrical conductivity in suspensions of hard conducting platelets are affected by the addition of platelets and their degree of spontaneous alignment. In our simulation results for aspect ratios 10, 25, and 50, we consistently observe a monotonically decreasing percolation threshold as a function of volume fraction, i.e., the addition of particles always aids percolation. In the nematic phase, the distribution of particles inside the percolating clusters becomes less spherically symmetric and the aspect ratio of the clusters increases. However, the clusters are also anisotropically shaped in the isotropic phase, although their aspect ratio remains constant as a function of volume fraction and is only weakly dependent on the particle aspect ratio. Mapping the percolating clusters of platelets to linear resistor networks, and assigning unit conductance to all connections, we find a constant conductivity both across the isotropic-nematic transition and in the respective stable phases. This behavior is consistent with the other observed topological properties of the networks, namely, the average path length, average number of contacts per particle, and the Kirchhoff index, which all remain constant and unaffected by both the addition of particles and the degree of alignment of their suspension. In contrast, using an anisotropic conductance model that explicitly accounts for the relative orientation of the particles, the network conductivity decreases with increasing volume fraction in the isotropic, and further diminishes at the onset of the nematic while preserving the same trend deep in the nematic. Hence, our observations consistently suggest that, unlike for rodlike fillers, the network structures that arise from platelet suspensions are not very sensitive to the particle aspect ratio or to alignment. Hence platelets are not as versatile as fillers for dispersion in conductive composite materials as rods.

Geometric percolation of hard nanorods: The interplay of spontaneous and externally induced uniaxial particle alignment

We present a numerical study on geometric percolation in liquid dispersions of hard slender colloidal particles subject to an external orienting field. In the formulation and liquid-state processing of nanocomposite materials, particle alignment by external fields such as electric, magnetic, or flow fields is practically inevitable and often works against the emergence of large nanoparticle networks. Using continuum percolation theory in conjunction with Onsager theory, we investigate how the interplay between externally induced alignment and the spontaneous symmetry breaking of the uniaxial nematic phase affects cluster formation in nanoparticle dispersions. It is known that particle alignment by means of a density increase or by an external field may result in a breakdown of an already percolating network. As a result, percolation can be limited to a small region of the phase diagram only. Here, we demonstrate that the existence and shape of such a "percolation island" in the phase diagram crucially depends on the connectivity length-a critical distance defining direct connections between neighboring particles. For some values of the connectivity range, we observe unusual re-entrance effects, in which a system-spanning network forms and breaks down multiple times with increasing particle density.

Unusual geometric percolation of hard nanorods in the uniaxial nematic liquid crystalline phase

We investigate by means of continuum percolation theory and Monte Carlo simulations how spontaneous uniaxial symmetry breaking affects geometric percolation in dispersions of hard rodlike particles. If the particle aspect ratio exceeds about 20, percolation in the nematic phase can be lost upon adding particles to the dispersion. This contrasts with percolation in the isotropic phase, where a minimum particle loading is always required to obtain system-spanning clusters. For sufficiently short rods, percolation in the uniaxial nematic mimics that of the isotropic phase, where the addition of particles always aids percolation. For aspect ratios between 20 and infinity, but not including infinity, we find reentrance behavior: percolation in the low-density nematic may be lost upon increasing the amount of nanofillers but can be regained by the addition of even more particles to the suspension. Our simulation results for aspect ratios of 5, 10, 20, 50, and 100 strongly support our theoretical predictions, with almost quantitative agreement. We show that a different closure of the connectedness Ornstein-Zernike equation, inspired by scaled particle theory, is as least as accurate in predicting the percolation threshold as the Parsons-Lee closure, which effectively describes the impact of many-body direct contacts.

Percolation of aligned rigid rods on two-dimensional triangular lattices

The percolation behavior of aligned rigid rods of length k (k-mers) on two-dimensional triangular lattices has been studied by numerical simulations and finite-size scaling analysis. The k-mers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The connectivity analysis was carried out by following the probability R_{L,k}(p) that a lattice composed of L×L sites percolates at a concentration p of sites occupied by particles of size k. The results, obtained for k ranging from 2 to 80, showed that the percolation threshold p_{c}(k) exhibits a increasing function when it is plotted as a function of the k-mer size. The dependence of p_{c}(k) was determined, being p_{c}(k)=A+B/(C+sqrt[k]), where A=p_{c}(k→∞)=0.582(9) is the value of the percolation threshold by infinitely long k-mers, B=-0.47(0.21), and C=5.79(2.18). This behavior is completely different from that observed for square lattices, where the percolation threshold decreases with k. In addition, the effect of the anisotropy on the properties of the percolating phase was investigated. The results revealed that, while for finite systems the anisotropy of the deposited layer favors the percolation along the parallel direction to the alignment axis, in the thermodynamic limit, the value of the percolation threshold is the same in both parallel and transversal directions. Finally, an exhaustive study of critical exponents and universality was carried out, showing that the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered.