Hexagonal Liquid Crystal Research Articles

Synergistic inhibition properties and microstructures of self-assembled imidazoline and phosphate ester mixture for carbon steel corrosion in the CO2 brine solution

The synergistic effect of oleic imidazoline (OIM) and alkyl phosphate ester (APE) for corrosion inhibition of carbon steel in CO2-saturated brine solution was investigated by electrochemistry and weight loss measurements. The effectiveness of corrosion inhibition is correlated to their relative compact film of OIM and APE adsorbed on metal, which is further correlated to the self-assembled structure in solution and the adhesion parameters on the surface. The optimal ratio of OIM and APE for synergistic corrosion inhibition is equal to one. The adsorption of mixed inhibitors on the metal follows the Langmuir isotherm. OIM and APE with dominant hexagonal liquid crystals (H1) in the solution correlate to the incompact monolayer and bilayer on the steel surface. OIM/APE mixed corrosion inhibitor forms catanionic complexes with lamellar structure in solution, which correlates to compact film coverage on the metal surface and corrosion inhibition. The mechanism is further studied and correlated with their critical micelle concentration (CMC), surface tension, contact angle(θ), adhesion tension, and adhesion work (WA) in diluted solution and on the metal surface. The studies demonstrate that a correlation of corrosion inhibition to the self-assembly structure in solution and on the surface exists when the mixed surfactants are used as corrosion inhibitors.

The effect of alkyl chain length of the imidazole ionic liquid surfactants on the improving lubricating properties of the hexagonal lyotropic liquid crystal composed of the [C16imCn]Br/hydrotalcite/H2O

The hydrotalcites were dispersed in the hexagonal lyotropic liquid crystals (LLCs) composed of 1-hexadecyl-3-alkyl imidazolium bromide ([C16imCn]Br, n = 1, 2 and 3) and H2O. The effects of the hydrotalcite concentration and the alkyl chain length of the [C16imCn]Br on the properties and the lubricant performance of the [C16imCn]Br/hydrotalcite hexagonal LLCs were studied by small angle X-ray diffraction measurements, rheology and tribology. Compared with the pure hexagonal LLCs, the incorporation of hydrotalcite into the [C16imCn]Br hexagonal LLCs can increase the interlayer distances and the viscoelasticity of the hybrid hexagonal LLCs gradually. However, the friction coefficient and wear volume decrease firstly and then increase with the increasing of hydrotalcite content. With the increase of the alkyl chain length of the [C16imCn]Br, the viscoelasticity, the interlayer distance and the water permeability of the mixed LLCs gradually increase which decreases the friction coefficient and wear volume at the moderate hydrotalcite content. The mechanism of the effective improvement of lubrication performance of the hybrid hexagonal LLCs is the tribofilm formed by the physical adsorption and tribochemical reaction of [C16imCn]Br/hydrotalcite LLCs on the steel surface.

Larvicidal formulation containing N-tosylindole: A viable alternative to chemical control of Aedes aegypti

Aedes aegypti is currently a major public health problem. This mosquito is responsible for the spread of infectious diseases that have been causing epidemics worldwide. Surfactant-stabilized systems, such as microemulsions, liquid-crystalline precursors and liquid crystals, are promising sustained delivery formulations of hydrophilic and hydrophobic substances. These systems are biocompatible water-soluble reservoirs for N-tosylindole exhibiting biological activity against Aedes aegypti Linn. (Diptera: Culicidae) larvae. The ternary diagram displayed four regions: microemulsion (ME), liquid crystal (LC), emulsion (EM) and phase separation (PS). PLM and SAXS distinguished microemulsions, lamellar and hexagonal phase liquid crystals. The system had a lethal concentration of 50% (LC50 = 0.1 ppm, 0.36 µM) lower than pure N-tosylindole (0.24 ppm, 0.88 µM), which has limitations in aqueous media. Furthermore, the formulation displayed no toxicity to Artemia sp., a non-target organism. The system exhibited excellent larvicidal activity as an alternative to commercial larvicides that have shown resistance and toxicity to the environment by Ae. aegypti larvae due to prolonged use. In addition, a two-fold increase in potency was observed.

Fabrication of ballpoint-ink via encapsulating inorganic pigments in microemulsion gels.

The fabrication of ballpoint-ink might open up a new perspective on physico-chemical solubility thermodynamics. In this report, we present a method to encapsulate inorganic pigments, such as Fe(CNS)3 (red), Fe2Fe(CN)6 (blue), CdS (yellow), and CuS (black) into w/o microemulsion gels. The area of w/o microemulsions was first determined by titrating surfactants Tween-60 into the given composition of water and cyclohexane in the pseudo-three phase diagram. Three prosperous phase areas were successfully mapped using this method, namely: microemulsion (w/o, and o/w) or (μE), lamellar liquid crystal (Lα), and hexagonal liquid crystal (Hα), respectively. The results show that inorganic pigments were well soluble in the w/o microemulsion gel of the Tween-60/cyclohexane/water system. The highest solubility of inorganic pigments in the microemulsion gel is 3.63 ± 0.05 mg g−1 for the red pigment of Fe(CNS)3, and the lowest is 2.92 ± 0.05 mg g−1 for the yellow pigment of CdS. Hence, the solubility limit distribution for all pigments is 2.9 ± 0.05 <>3.63 ± 0.05 mg g−1. The cation and anion size strongly affected the inorganic pigments' solubility in the w/o-microemulsion system. Some quantity of the ink-made of inorganic pigments encapsulated in the microemulsion gel has been inserted into empty ballpoint sleeves as prototypes. The resulting self-made inks demonstrated that the physical appearances of the ink could mimic the factory-made inks. Nevertheless, the self-made ink should be investigated further for long lifespan use, especially for long-term stability and corrosion resistance.

Chemical deposition and exfoliation from liquid crystal template: Nickel/nickel (II) hydroxide nanoflakes electrocatalyst for a non-enzymatic glucose oxidation reaction

This work reports the synthesis of nickel/nickel hydroxides nanoflakes (Ni/Ni(OH) 2 -NFs) at room temperature via a novel chemical deposition and exfoliation from a liquid crystal template mixture. The nickel ions dissolved in the interstitial aqueous domain of the Brij®78 hexagonal liquid crystal template were deposited by a reducing agent of sodium borohydride that concurrently reduces the nickel ions and generates extreme hydrogen gas bubbles, that exfoliated the nickel/nickel hydroxide layers. The Ni/Ni(OH) 2 -NFs crystal structure, morphology, and surface area characterizations revealed the formation of semi-crystalline α-Ni(OH) 2 nanoflakes with a thickness of approximately 10 nm and a specific surface area of about 135 m 2 /g. The electrochemical measurements of cyclic voltammetry, chronoamperometry, and impedance analysis showed that the Ni/Ni(OH) 2 -NFs exhibited significant performance for the glucose non-enzymatic oxidation in an alkaline solution in comparison to the bare -nickel hydroxide ( bare -Ni(OH) 2 ) deposited without surfactant. The Ni/Ni(OH) 2 -NFs electrode showed superior glucose oxidation activity over the bare -Ni(OH) 2 catalyst with a sensitivity of 1.078 mA mM −1 cm −2 with a linear concentration dependency range from 0.2 to 60 mM and a detection limit of 0.2 mM (S/N = 3). The enhanced electrochemical active surface area and mesoporosity of the 2D nanoflakes make the Ni/Ni(OH) 2 -NFs a promising catalyst in the application of glucose non-enzymatic sensing.

Nanostructure and anisotropy of 3D printed lyotropic liquid crystals studied by scattering and birefringence imaging

Extrusion-based 3D printing of hexagonal and lamellar lyotropic liquid crystals is a powerful technique to produce hierarchical materials with well-defined anisotropic structure. Tailoring the properties of 3D printed objects requires a precise control of the nanostructure; however, a sufficiently high degree of anisotropy is often not achieved. In this study, scanning small angle X-ray scattering was performed in situ at the exit of the needle during 3D printing. We study the induced anisotropy and nanostructure in hexagonal and lamellar lyotropic liquid crystals. Mapping of extruded filaments during printing revealed that narrower nozzle diameters (370 µm) resulted in less anisotropic structures with a wider distribution of orientation angles across the cross section, while larger nozzle diameters (550 µm) resulted in more anisotropic structures with an overall higher degree of orientation. The apparent wall shear rate is higher for the narrower nozzle, which produces wall slip, resulting in a highly anisotropic shell, and a less aligned filament core. Further examination of the filaments revealed phase transitions due to solvent evaporation. The time scales were of 10–20 min of exposure to atmospheric conditions. Simultaneously, a loss in the macroscopic anisotropy of the hexagonal self-assembled structure was observed. These processes occur during and after extrusion-based 3D printing of liquid crystals and limit the fine control of the final structure. The variability of structures achieved for our different systems highlights the importance of structural characterization during and after extrusion to guarantee high anisotropy and well-defined structures.

Towards a High-Flux Separation Layer from Hexagonal Lyotropic Liquid Crystals for Thin-Film Composite Membranes.

Hexagonal lyotropic liquid crystals (HLLC) with uniform pore size in the range of 1~5 nm are highly sought after as promising active separation layers of thin-film composite (TFC) membranes, which have been confirmed to be efficient for water purification. The potential interaction between an amphiphile-based HLLC layer and the substrate surface, however, has not been fully explored. In this research, hydrophilic and hydrophobic microporous polyvinylidene fluoride (PVDF) substrates were chosen, respectively, to prepare TFC membranes with the active layers templated from HLLC, consisting of dodecyl trimethylammonium bromide, water, and a mixture of poly (ethylene glycol) diacrylate and 2-hydroxyethyl methacrylate. The pore size of the active layer was found to decrease by about 1.6 Å compared to that of the free-standing HLLC after polymerization, but no significant difference was observable by using either hydrophilic or hydrophobic substrates (26.9 Å vs. 27.1 Å). The water flux of the TFC membrane with the hydrophobic substrate, however, was higher than that with the hydrophilic one. A further investigation confirmed that the increase in water flux originated from a much higher porosity was due to the synergistic effect of the hydrophilic HLLC nanoporous material and the hydrophobic substrate.

Deconstruction of technical grade diglycerol isostearate enables the controlled preparation of hexosomes and liposomes

Glycerol-based technical grade surfactants of industrial interest generally contain a mixture of components with diverse composition. Consequently, their use cannot be generalized since the properties or the technical grade product will likely depend on batches and source. Herein, we address this issue by deconvoluting the role of individual components and pondering their influence on the phase behavior of mixtures. Technical grade diglycerol-monoisostearate (C41) was separated into different fractions based on polarity differences. Three fractions are obtained, which contain mainly unreacted diglycerol, the monoester and higher esterification degree amphiphiles. The formation of a single-phase reverse hexagonal liquid crystal (H2) in the water/C41 system is hindered by the presence of unreacted diglycerol, which dehydrates the hydrophilic channels of the H2 phase. On the other hand, the neat monoester forms a thermotropic lamellar liquid crystalline phase that melts at c.a. 40 °C, as confirmed by X-ray scattering (SAXS) and differential calorimetry (DSC). The formation of the lamellar phase in the neat monoester is ascribed to branching in the alkyl chain. The monoester also forms lamellar liquid crystals mixed with water and remains stable up to high water contents. Moreover, the addition of the higher esterification degree amphiphiles to the monoester/water mixtures promotes a transition from the lamellar phase to a reverse hexagonal one, confirmed by polarized optical microscopy and SAXS. Such transitions can be somehow predicted by calculations of the molecular packing parameters of mixtures. Similar to the lamellar phase, the reverse hexagonal liquid crystal remains stable in excess water. Finally, the coexistence between lamellar or hexagonal phases in excess water is harnessed to produce unilamellar vesicles and hexosomes, as confirmed by dynamic light scattering, SAXS, and cryo-electron microscopy.

Double-Chain Cationic Surfactants: Swelling, Structure, Phase Transitions and Additive Effects.

Double-chain amphiphilic compounds, including surfactants and lipids, have broad significance in applications like personal care and biology. A study on the phase structures and their transitions focusing on dioctadecyldimethylammonium chloride (DODAC), used inter alia in hair conditioners, is presented. The phase behaviour is dominated by two bilayer lamellar phases, Lβ and Lα, with “solid” and “melted” alkyl chains, respectively. In particular, the study is focused on the effect of additives of different polarity on the phase transitions and structures. The main techniques used for investigation were differential scanning calorimetry (DSC) and small- and wide-angle X-ray scattering (SAXS and WAXS). From the WAXS reflections, the distance between the alkyl chains in the bilayers was obtained, and from SAXS, the thicknesses of the surfactant and water layers. The Lα phase was found to have a bilayer structure, generally found for most surfactants; a Lβ phase made up of bilayers with considerable chain tilting and interdigitation was also identified. Depending mainly on the polarity of the additives, their effects on the phase stabilities and structure vary. Compounds like urea have no significant effect, while fatty acids and fatty alcohols have significant effects, but which are quite different depending on the nonpolar part. In most cases, Lβ and Lα phases exist over wide composition ranges; certain additives induce transitions to other phases, which include cubic, reversed hexagonal liquid crystals and bicontinuous liquid phases. For a system containing additives, which induce a significant lowering of the Lβ–Lα transition, we identified the possibility of a triggered phase transition via dilution with water.

Electric-Field-Induced Chirality in Columnar Liquid Crystals.

We describe a novel class of tetraphenylbenzene-based discotic molecules with exceptional self-assembling properties. Absorption and fluorescence studies confirmed the formation of J-type aggregates in solution. The discotic mesogens also show an enhancement of the emission upon aggregation. Interestingly, these discotic molecules displayed enantiotropic hexagonal columnar liquid crystal (LC) phases that can be switched into a helical columnar organization by application of an electric field. The helical columns arise from the electric-field-induced tilt of the polar fluorobenzene ring that directs all of the peripheral phenyl groups into a propeller-like conformation with respect to the central benzene core. A cooperative assembly process of these propeller-shaped molecules resolves into a helical columnar organization, in which the preferred helical sense is obtained from the stereogenic center proximate to the polar carbon-fluorine bond. The ease of inducing chirality in columnar LCs by an electric field presents opportunities to create next-generation chiral materials for a variety of applications.

Lubrication and Dynamically Controlled Drug Release Properties of Tween 85/Tween 80/H2O Lamellar Liquid Crystals

Lamellar liquid crystals have amazing lubricating and drug-solubilizing properties. Hence, the combination of drug molecules with lamellar liquid crystals is expected to be used in joint lubrication and treatment. In this study, the partial phase diagram of the Tween 85/Tween 80/H2O three-component system was determined. The phase structure of the system changed from a hexagonal liquid crystal to a lamellar liquid crystal with the increase of Tween 85 content. The lamellar liquid crystals showed superior lubricating properties due to their unique lamellar structure. Furthermore, the model of drug release during friction was established for the first time. It was found that the order of the lamellar liquid crystals increased with the increase of the mass ratio of Tween 85/Tween 80, leading to the decrease of the ibuprofen release rate. In addition, the release rate of ibuprofen increased progressively with the increase of the friction frequency, but the load had little effect on it. Therefore, the lamellar liquid crystals consisting of nonionic surfactants with good biocompatibility have potential application prospects for joint lubrication and treatment of arthritis.

Chromonic liquid crystals and packing configurations of bacteriophage viruses.

We study equilibrium configurations of hexagonal columnar liquid crystals in the context of characterizing packing structures of bacteriophage viruses in a protein capsid. These are viruses that infect bacteria and are currently the focus of intense research efforts, with the goal of finding new therapies for bacteria-resistant antibiotics. The energy that we propose consists of the Oseen-Frank free energy of nematic liquid crystals that penalizes bending of the columnar directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure; we also consider the isotropic contribution of the core and the energy of the unknown interface between the outer ordered region of the capsid and the inner disordered core. The problem becomes of free boundary type, with constraints. We show that the concentric, azimuthal, spool-like configuration is the absolute minimizer. Moreover, we present examples of toroidal structures formed by DNA in free solution and compare them with the analogous ones occurring in experiments with other types of lyotropic liquid crystals, such as food dyes and additives. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

Characterization of hexagonal lyotropic liquid crystal microstructure: Effects of vitamin E molecules

The effects of vitamin E (VE) on the liquid crystal behavior and structure of a lyotropic liquid crystal (LLC) system based on nonionic surfactant C12EO10/H2O are reported. Polarized optical microscopy, X-ray diffraction and FTIR combined with fluorescent analysis were utilized to study structural properties and the interactions between the vitamin E and hexagonal mesophase. It was revealed that up to 20 wt% VE is solubilized within the direct hexagonal mesophase. As a result, the lattice parameter and the melting point increased as compared with binary LLC C12EO10/H2O. The presence in the system C12EO10/H2O of VE in the amount above 30% (up to 60 wt%) leads to a structural transformation of the mesophase from the direct to reverse hexagonal mesophase. The structural changes of hydrophilic and hydrophobic domains in the presence of VE in LLC were characterized.

Using Deposition Rate and Substrate Temperature to Manipulate Liquid Crystal-Like Order in a Vapor-Deposited Hexagonal Columnar Glass.

We investigate vapor-deposited glasses of a phenanthroperylene ester, known to form an equilibrium hexagonal columnar phase, and show that liquid crystal-like order can be manipulated by the choice of deposition rate and substrate temperature during deposition. We find that rate-temperature superposition (RTS)-the equivalence of lowering the deposition rate and increasing the substrate temperature-can be used to predict and control the molecular orientation in vapor-deposited glasses over a wide range of substrate temperatures (0.75-1.0 Tg). This work extends RTS to a new structural motif, hexagonal columnar liquid crystal order, which is being explored for organic electronic applications. By several metrics, including the apparent average face-to-face nearest-neighbor distance, physical vapor deposition (PVD) glasses of the phenanthroperylene ester are as ordered as the glass prepared by cooling the equilibrium liquid crystal. By other measures, the PVD glasses are less ordered than the cooled liquid crystal. We explain the difference in the maximum attainable order with the existence of a gradient in molecular mobility at the free surface of a liquid crystal and its impact upon different mechanisms of structural rearrangement. This free surface equilibration mechanism explains the success of the RTS principle and provides guidance regarding the types of order most readily enhanced by vapor deposition. This work extends the applicability of RTS to include molecular systems with a diverse range of higher-order liquid-crystalline morphologies that could be useful for new organic electronic applications.

Spacer length effect on the aggregation behaviours of gemini surfactants in EAN

As the symbol of gemini surfactants, the spacer group connects two surfactant monomers and plays an important role in the self-assembling of gemini surfactants. Recent attention has been paid to the effects of spacer nature and length in aqueous systems. In this study, the aggregation behaviours of quaternary ammonium gemini surfactants 12-s-12 with different spacer length (s = 2, 3, 4, 5, 6, 8, 10, 12) have been investigated in the protic ionic liquid EAN. The micellization behaviour was characterized by surface tension measurement. With the help of small-angle X-ray scattering (SAXS), the size and shape of aggregates were verified. The spacer length effect could be reflected by the changes in the micellization and phase behaviours of 12-s-12/EAN systems. The critical packing parameter of gemini molecules would decrease monotonically with the spacer length, which is supported by SAXS results. Due to the charge screening effect of EAN, the aggregation behaviours of 12-s-12 would be mainly influenced by spacer length. Our results would be the important supplement to the spacer length effect of gemini surfactants in the nonaqueous systems. With the increase of spacer chain length, the reverse hexagonal lyotropic liquid crystal phase would transform into the micellar phase.

Self-assembly of an imidazolium surfactant in aprotic ionic liquids. 2. More than solvents.

As tailorable solvents, the physiochemical properties of ionic liquids can be tuned by the structure of ions. Herein, we investigate the structural effects of ILs on the self-assembly of surfactants. It has been confirmed that the cationic surfactant 1-hexadecyl-3-methylimidazolium bromide (C16mimBr) can self-assemble into micellar and lamellar lyotropic liquid crystal phases in the aprotic ionic liquid (AIL) 1-ethyl-3-methylimidazolium tetrafluoroborate ([Emim]BF4). In this work, we explore the aggregation behaviours in AILs with different alkyl chains on the imidazolium group, i.e., 1-propyl-3-methylimidazolium tetrafluoroborate ([Pmim]BF4), 1-butyl-3-methylimidazolium tetrafluoroborate ([Bmim]BF4), 1-hexyl-3-methylimidazolium tetrafluoroborate ([Hmim]BF4) and 1-octyl-3-methylimidazolium tetrafluoroborate ([Omim]BF4). With the increase of the cation chain length, AILs have better solubility of the solvophobic part of the surfactants and hence a weaker driving force for self-assembly. Therefore, the critical micellization concentration of C16mimBr in AILs increases as confirmed by the surface tension and small angle X-ray scattering characterizations. More interesting things happen to the phase behaviours. Besides the micellar and lamellar lyotropic liquid crystal phases, a hexagonal lyotropic liquid crystal phase is formed in [Pmim]BF4 while hexagonal and bicontinuous cubic lyotropic liquid crystal phases are formed in [Bmim]BF4, [Hmim]BF4 and [Omim]BF4. It is surprising to observe richer phase behaviours in solvents of lower cohesive energy. The detailed structural information of various aggregates has been obtained by small-angle X-ray scattering. It is demonstrated that AILs work as not only solvents but also co-surfactants.

Micellar Nanocarriers from Dendritic Macromolecules Containing Fluorescent Coumarin Moieties.

The design of efficient drug-delivery vehicles remains a big challenge in materials science. Herein, we describe a novel class of amphiphilic hybrid dendrimers that consist of a poly(amidoamine) (PAMAM) dendritic core functionalized with bisMPA dendrons bearing cholesterol and coumarin moieties. Their self-assembly behavior both in bulk and in water was investigated. All dendrimers exhibited smectic A or hexagonal columnar liquid crystal organizations, depending on the generation of the dendrimer. In water, these dendrimers self-assembled to form stable spherical micelles that could encapsulate Nile Red, a hydrophobic model compound. The cell viability in vitro of the micelles was studied in HeLa cell line, and proved to be non-toxic up to 72 h of incubation. Therefore, these spherical micelles allow the encapsulation of hydrophobic molecules, and at the same time provided fluorescent traceability due to the presence of coumarin units in their chemical structure, demonstrating the potential of these dendrimers as nanocarriers for drug-delivery applications.

Highly Ordered Organic Piezoresponsive Materials Obtained by Cross-linking Electroresponsive Columnar Liquid Crystal Compounds

Abstract An N,N′-diphenylurea derivative possessing six oleyloxy groups exhibited hexagonal columnar liquid crystal phases, and the columnar molecular aggregates were cross-linked with applying an electric field to produce a highly-ordered thin film which has a polarity in the thickness direction. The resulting aromatic polyurea film showed a relatively large vertical piezoelectric constant (d33 of 6.1 pm/V).

Construction of a liquid crystalline double helix supramolecular structure and its electro-responsive behaviour

ABSTRACT A chiral urea compound, N,N’-bis(4-(3,4,5-trialkoxyphenylmethyl)phenyl)urea (1a), possessing a long core and six chiral alkyl chains ((S)-citronellyl groups) was designed to construct a double helix supramolecular structure in a liquid crystal phase. As a result, the urea compounds exhibited a hexagonal columnar liquid crystal phase, and the molecules were self-organised into a double helix structure by intermolecular interactions (dipole-dipole interactions, hydrogen bonds, and π-π interactions). Furthermore, the compound’s responsive behaviour to an applied external electric field was investigated, and its phase transition behaviour, column alignment, columnar structures, and electro-responsiveness were compared with those of an achiral urea compound (1b) possessing six n-decyl groups to investigate the effect on introducing the chirality. These results indicated that higher-order structures such as double helices are effective in suppressing thermal motion and delaying polarisation relaxation.

Fine structure of viral dsDNA encapsidation.

Unraveling the mechanisms of packing of DNA inside viral capsids is of fundamental importance to understanding the spread of viruses. It could also help develop new applications to targeted drug delivery devices for a large range of therapies. In this article, we present a robust, predictive mathematical model and its numerical implementation to aid the study and design of bacteriophage viruses for application purposes. Exploiting the analogies between the columnar hexagonal chromonic phases of encapsidated viral DNA and chromonic aggregates formed by plank-shaped molecular compounds, we develop a first-principles effective mechanical model of DNA packing in a viral capsid. The proposed expression of the packing energy, which combines relevant aspects of the liquid crystal theory, is developed from the model of hexagonal columnar phases, together with that describing configurations of polymeric liquid crystals. The method also outlines a parameter selection strategy that uses available data for a collection of viruses, aimed at applications to viral design. The outcome of the work is a mathematical model and its numerical algorithm, based on the method of finite elements, and computer simulations to identify and label the ordered and disordered regions of the capsid and calculate the inner pressure. It also presents the tools for the local reconstruction of the DNA "scaffolding" and the center curve of the filament within the capsid.