Theory and Simulation Studies of Self‐Assembly of Helical Particles

Abstract

While it is possible to control the enantioselective process by using depletion interactions and faceting of the bulding blocks [5], helices are among the natural objects to focus on when dealing with chirality. New functional materials [9] [10] can be produced by exploiting the intrinsic chirality of the helical structures, which are useful in catalysis and demixing of enantiomers [11] [12]. The importance of the helix in nature is unquestionable: proteins, polysaccharides, DNA and RNA, the so called molecules of life, have a helical structure. The helical shape is exhibited in nature also by microorganisms, like filamentous viruses, and cell organelles, like bacterial flagella. Filamentous viruses are formed by a DNA of RNA core, wrapped by a coating of helically arranged proteins. Well-known examples are Tobacco Mosaic Virus (TMV), the first discovered virus [13], and viruses related to filamentous phage fd, whose mutants are present in nature (M13, fd), while others can be obtained by genetic engineering. They have been widely investigated as models of highly anisotropic, colloidal systems, with the advantage of being essentially monodisperse and that their length, of the order of a micrometer, makes them suitable for imaging techniques, such as optical microscopy. Bacterial flagella are helical macromolecular structures assembled from a single protein (flagellin). Their helical shape can be tuned with high precision by regulating external parameters such as temperature or pH, and their large size, of the order of microns, makes them very handy for optical observations. Because of their shape anisotropy, helical biopolymers and colloidal particles may exhibit liquid crystal phases at high densities [14]. These phases are often tacitly assumed to be the same as those occurring in systems of rod-like particles. However, it cannot be taken for granted that at such high densities the intrinsic helicity of the shape can be neglected. To explore the effect of self-assembly of helical polymers and colloids, and in particular to discover whether there is anything special just determined by the helical shape, we have undertaken a comprehensive investigation of the phase behavior of hard helices [15][16][17][18][19], interacting through purely steric repulsions, using Monte Carlo simulations and an extension of Onsager theory [20], a density functional theory (DFT) that was originally proposed to explain the onset if nematic ordering in a system of hard rods. These studies have revealed an unexpectedly rich phase behavior, the most interesting result being the existence of special phases characterized by screw -like ordering. Such kind of organization had been proposed for DNA, based on theoretical considerations [21], and had been observed in dense suspensions of flagellar filaments [22]. Hard helices are an athermal system: phase transitions are controlled by density and are driven by the entropy gain on moving from the less to the more ordered phase. This is a minimalist model, possibly insufficient to account entir...

Perturbative density functional methods for cholesteric liquid crystals

“…We note that an approach involving the explicit third-order virial contribution to the excess free energy was also implemented with some suc-cess for such systems, albeit under stringent symmetry assumptions. 120,121 It would be interesting to investigate whether its generalisation to configurations of freelyrotating particles may lead to significant improvements over the MPL approximation in our case.…”

Self-Assembled Chiral Photonic Crystals from a Colloidal Helix Racemate

“…In this phase, particles are best packed when each helix interacts with its like-chiral neighbors through the same zero interaction phase, and the system stays in this in-plane liquid crystal state before transforming to crystalline states at very high pressure, as indicated by the increase of ψ 6 BO in Figure f. This chiral liquid crystal phase (also named as screw liquid crystal) has recently been explored extensively both experimentally and theoretically. − …”

“…The behavior of d ij ( z i / j ) is similar to d ij (ϕ i / j ); thus it is not shown. Because of such complex anisotropic interactions between the helices, general analytical overlap-checking algorithms between two helices are currently unavailable, and helices can only be modeled as connected hard spheres. − The computational cost thus scales up dramatically with the increase of coarse-grained degree and the length of helices. In the Methods section, we rigorously prove that for the specific case of parallel helices, as a result of screw axis symmetry of helices, d ij can be written as a periodic function of a single variable ΔΨ ij , which is defined as the interaction phase : where C i / j is the chirality sign of the helices i / j ( C R = 1, C L = −1).…”

“…This chiral liquid crystal phase (also named as screw liquid crystal) has recently been explored extensively both experimentally 56 and theoretically. [62][63][64] For all these self-assembly processes, we find the density jump in the equation of state associated with the sudden change of the corresponding crystalline order (Figure 4). This implies that the formations of these structures are all first-order phase transitions, which can be intuitively understood from the perspective of optimal packing.…”

“…Compared with top-down building of helical crystal whose complexity scales up with the number of helices, fabrication of free helical colloids or fibers with controllable chirality is more technically accessible using current nanotechniques, − chirality-transferring chemical synthesis methods, − or even biological helical structures such as DNA tubes, spiral bacteria, , bacterial flagella, spirulina algae, and plant vessels . Although there are several theoretical and simulation studies trying to elucidate the self-assembly mechanism of helical particles − and fibers, , no attempt has been made to connect the self-assembled structures with their optical properties.…”