Two-dimensional Order Parameter Research Articles

Radial Alignment of Carbon Nanotubes Via Dead-End Filtration

Dead-end filtration is a facile method to globally align single wall carbon nanotubes (SWCNTs) in large area films with a two-dimensional order parameter, , approaching unity. Uniaxial alignment has been achieved using pristine and hot-embossed membranes but more sophisticated geometries have yet to be investigated. In this work, we create three different patterns with radial symmetry and an area of 3.8 cm². Two of these patterns are replicated by the filtered SWCNTs and values of » 0.85 are obtained. Each of the radially aligned SWCNT films are characterized by scanning cross-polarized microscopy in reflectance and laser imaging in transmittance with linear, radial and azimuthal polarized light fields. The former is used to define a novel indicator akin to the two-dimensional order parameter using Malu’s law, yielding 0.82 for the respective film. The films are then transferred to a flexible printed circuit board and terminal two-probe electrical measurements are conducted to explore the potential of those new alignment geometries. Figure 1

(Invited) The Impact of Carbon Nanotube Length and Diameter on Their Global Alignment by Dead-End Filtration

Dead-end filtration on polyvinylpyrrolidone-coated polycarbonate track-etched membranes has proven to be an effective method to prepare macroscopically (3.8 cm²) aligned thin films from solution-based single-wall carbon nanotubes (SWCNTs). However, to make this technique broadly applicable, the role of SWCNT length and diameter on the alignment process must be understood. To date, most groups report the alignment of unsorted large diameter (» 1.4 nm) SWCNTs, but systematic studies on their small diameter (» 0.78 nm) counterparts are rare and in the case of sorted SWCNT species, non-existent. In this work, films are prepared from two series of length sorted fractions comprised of small and large diameter SWCNTs, respectively. The degree of alignment is characterized by cross polarized microscopy, scanning electron microscopy and Raman spectroscopy. For the longest fractions, the two-dimensional order parameter, , values of » 0.66 and » 0.76 are reported for the small and large diameter SWCNTs, respectively, and Raman maps over an area of 2500 μm2 are presented. A comparison of Derjaguin, Landau, Verwey and Overbeek (DLVO) theory calculations with the aligned domain size is then used to propose a law identifying the required length of a carbon nanotube with a given diameter and zeta potential for alignment.

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice.

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two phase transitions when the density of plates is increased: the first from a disordered fluid phase to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen Cartesian direction, corresponding to a twofold (Z_{2}) symmetry breaking of translation symmetry along the layering direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two adjacent slabs have a lower density, thus breaking the symmetry between the three types of plates. We show that the slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, interslab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is twofold symmetry breaking of lattice translation symmetry along all three Cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with universality class of the three-dimensional O(3) model with cubic anisotropy, while the layered to sublattice transition is first-order in nature.

Dispersion and Tunable Alignment of Colloidal Silver Nanowires in a Nematic Liquid Crystal for Applications in Electric–Optic Devices

The dispersion and tunable alignment of colloidal nanomaterials is desirable for practical applications in electric-optic (E-O) devices; however, it remains challenging for large one-dimensional nanomaterials with a large aspect ratio. Here, we demonstrate a large-scale, simple, multi-microdomain, and noncontact photoalignment technology to align colloidal silver nanowires (AgNWs, length ∼4.5 μm, diameter ∼70.6 nm) in a liquid crystal (LC) with a high two-dimensional order parameter (about 0.9). The AgNWs are precisely self-assembled via photomasks with twisted nematic and planar alignment models in microdomain regions. The AgNW orientation is tuned with an electric field, through the rotation of an LC director n, which allows three-dimensional (3D) tunable orientation combined with photoalignment. The colloidal dispersions of AgNWs in the LC cell influenced the ion transfer, elastic constant, dielectric anisotropy, and near LC alignment, changing the E-O properties of the LC devices. The 3D tunable orientation of an AgNW by photoalignment and an electric field could provide a new way to assemble large colloidal nanomaterials and fabricate functional E-O devices.

Wetting behavior of aqueous 1-alkyl-3-methylimidazolium tetrafluoroborate {[Cn MIM][BF4] (n = 2, 4, 6)} on graphite surface

Herein, molecular dynamics simulations have been conducted to investigate the effect of cation chain length on the wetting and interfacial properties of aqueous imidazolium-based ionic liquids (ILs) consisting of 1-ethyl-3-methylimidazolium [EMIM], 1-butyl-3-methylimidazolium [BMIM], and 1-hexyl-3-methylimidazolium [HMIM] cations with common tetrafluoroborate [BF4] anions on a graphite surface in the IL concentration range of 10–50 wt%. The difference between the contact angles of aqueous [EMIM][BF4] and [BMIM][BF4] droplets is very small regardless of the IL content. However, a noticeable reduction in the contact angle is observed for the IL with a longer alkyl chain [HMIM][BF4] at high IL concentrations. Furthermore, interfacial tensions and works of adhesion are calculated to quantify the wettability of the studied ILs; as a result, lower works of adhesion are obtained for the longer alkyl chain compounds (Sali et al., 2017). Droplet characteristics near the surface are obtained from two-dimensional density contours and orientation order parameters. The majority of IL molecules are located near the solid–liquid interface in all the studied cases. However, the [BMIM][BF4] and [HMIM][BF4] ILs with longer alkyl chains are also detected at the liquid–vapor interface. Finally, IL interactions with water molecules through hydrogen bonding are investigated at various concentrations. The obtained interfacial tensions are in good agreement with the experimental values reported in the literature. The results of this study can help elucidate the wetting and interfacial properties of aqueous imidazolium-based ILs.

Banded spherulites and twisting lamellae in poly–ε–caprolactone

By crystallization of poly–e–caprolactone (PCL) at temperatures (55 ∘C ≥ Tc ≥ 50 ∘C) close to the melting point (Tm = 61 ∘C), banded spherulites form. These structures show a periodicity of P = 32 µm as determined by polarized optical microscopy (POM). In weakly-to-non-banded spherulites formed under similar conditions, a less sharp distribution of characteristic length scales is observed, that exhibits the same mean value. Within bright and dark regions of the banded spherulites, the two-dimensional molecular order parameter of different crystal directions of the lamellae is deduced from tightly focused (15 × 50 µm2) polarized infrared-spectroscopy measurements. From the oscillation of the order parameters of the crystalline $\underline {a}$ -, $\underline {b}$ -, and $\underline {c}$ -axes, banding in pure PCL is proven to result from lamellae growing in a helicoidal fashion along the spherulites radius. No deviation of lamellar growth and radius is determined within experimental uncertainty (± 4∘). Furthermore, spatially averaged IR-microscopy results in the same characteristic polarization dependence in banded and weakly-to-non-banded spherulites. In conjunction with the mentioned characteristic length scales, we conclude that the mechanism, which results in banding, is also active in non-banded spherulites of PCL.

Order parameters in the diffusion of rods through two- and three-dimensional fixed scatterers

Characteristic diffusion in a Lorentz gas system and its extensions are typically described through a scale-free order parameter. The relative validity of possible order parameters for the diffusion of a thick tracer rod through spherical scatterers in two and three dimensions as a function of the rod length $L$ and scatterer density $\ensuremath{\rho}$ has been examined by simulation. We find that the often-used two-dimensional order parameter, $\ensuremath{\rho}{L}^{2}$, is accurate. However, its naive generalization to three dimensions, changing the exponent according to dimensionality, does not fit the observed results. A revised order parameter can be generated through a theory based on the behavior of the mean-free path of the tracer. We find that this generalized order parameter captures the multidimensional diffusive behavior.

Ordering of ferromagnetic nanoparticles in nematic liquid crystals.

Dispersions of magnetic nanoparticles in a nematic liquid crystal were investigated as magnetic fields were applied in three different boundary condition geometries: (i) planar substrates and B⊥n, (ii) planar substrates and B‖n, and (iii) homeotropic substrates and B⊥n. Particle chaining is observed when a magnetic field is applied, with a periodicity perpendicular to the chains. Furthermore, linear chains are observed for the magnetic field applied perpendicular to the director, while zigzag chains are formed when the magnetic field direction is parallel to the director field. This is attributed to a change from a dipolar defect configuration around dispersed nanoparticles, to a quadrupolar one, i.e. the change from satellite to Saturn-ring defects. This effect is largely independent of the sample thickness. The dynamic development of the chain length, as well as their two-dimensional order parameter was studied in all cases. Chain lengths increased rapidly until saturation at approximately l = 30 μm after a time of about t = 10 s. Similarly, the chain order parameters increased until saturation between S = 0.8-0.9, independent of sample geometry.

Highly conductive single-walled carbon nanotube thin film preparation by direct alignment on substrates from water dispersions.

A safe, scalable method for producing highly conductive aligned films of single-walled carbon nanotubes (SWNTs) from water suspensions is presented. While microfluidic assembly of SWNTs has received significant attention, achieving desirable SWNT dispersion and morphology in fluids without an insulating surfactant or toxic superacid is challenging. We present a method that uniquely produces a noncorrosive ink that can be directly applied to a device in situ, which is different from previous fabrication techniques. Functionalized SWNTs (f-SWNTs) are dispersed in an aqueous urea solution to leverage binding between the amine group of urea and the carboxylic acid group of f-SWNTs and obtain urea-SWNT. Compared with SWNTs dispersed using conventional methods (e.g., superacid and surfactants), the dispersed urea-SWNT aggregates have a higher aspect ratio with a rodlike morphology as measured by light scattering. The Mayer rod technique is used to prepare urea-SWNT, highly aligned films (two-dimensional nematic order parameter of 0.6, 5 μm spot size, via polarized Raman) with resistance values as low as 15-1700 Ω/sq in a transmittance range of 2-80% at 550 nm. These values compete with the best literature values for conductivity of SWNT-enabled thin films. The findings offer promising opportunities for industrial applications relying on highly conductive thin SWNT films.

Phase Behavior of Bow-Shaped Hard Particles in Two Dimensions

The phase behavior of a two-dimensional hard-particle model is studied via Monte Carlo simulations using the grand canonical, the isobaric and the canonical ensembles. This model consists of a three-segmented line whose geometry resembles a bow shape. The model reduces to some limiting cases: hard needles and bent-core particles. Manipulating the molecular parameters, a variety of molecular shapes were generated. As a result, several liquid crystalline structures like nematic and tetratic were obtained. Additionally, there are some other regions where the molecules form curvilinear paths. As the density increases, the formation of clusters of two or more particles is observed, producing assemblies with different shapes depending on the particular values of the molecular parameters. One interesting example is when the clusters have chiral features despite the particles are achiral. The two-dimensional tetratic, nematic and polar order parameters as well as the orientational correlation functions g2(r∗) and g4(r∗) and the distribution functions g∥ and g⊥ were calculated to describe the resulting mesophases. Besides this, the Gibbs ensemble was used to investigate some cases where indications of first-order phase transitions appeared. The mesophases diagrams are also reported.

Single Molecule Tracking Studies of Flow-Aligned Mesoporous Silica Monoliths: Aging-Time Dependence of Pore Order

Single molecule tracking (SMT) methods are employed to characterize the in-plane alignment and order of cylindrical mesopores in flow-aligned surfactant-templated silica monoliths prepared within glass microfluidic channels. The majority of dye molecules observed in wide-field fluorescence videos of these samples exhibit one-dimensional (1D) diffusive motions. Orthogonal regression analysis of these motions provides a measure of the mesopore orientation distribution function, which in turn is used to quantify the mesopore order via a two-dimensional orientational order parameter, <P>. Mesopore organization is explored as a function of aging time between sol preparation and filling of the microfluidic channels. Channels filled well before gelation of the sol are shown to incorporate large monodomains having average pore alignment within a few degrees of the flow direction. These monodomains extend over several millimeters and yield aging-time-independent <P> values larger than ~0.80. In contrast, channels filled near the time of sol gelation yield monoliths with misaligned pores that are also more disordered, having <P> ≈ 0.35. The SMT results are compared to those from small-angle X-ray scattering anisotropy experiments; these data are consistent across the range of samples investigated. A model describing the aging-time dependence of sol organization is presented. These studies demonstrate that well-aligned mesoporous silica monoliths can be obtained by simple flow alignment procedures but that short sol aging times are required in order to achieve optimum pore organization.

Templating effects of lyotropic liquid crystals in the encapsulation of amyloid fibrils and their stimuli-responsive magnetic behavior

The relationship between the symmetry of inverted lyotropic liquid crystals (LLC), used as hosting complex fluid, and amyloid fibrils, confined within the LLC matrix, was studied with the aim of exploiting the LLC water reservoirs for encapsulation of large protein aggregates. We used β-lactoglobulin (βlg) fibrils as a model system for high aspect ratio amyloid fibrils and encapsulated them into three different types of LLC mesophases composed of glycerol monolinoleate, with or without linoleic acid and water, yielding respectively lamellar, inverse bicontinuous cubic and inverse columnar hexagonal symmetries. The impact of fibrils confinement within the LLC on their secondary structure and spatial organization was studied by combining small angle X-ray scattering (SAXS), attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR), and atomic force microscopy (AFM) techniques. FTIR indicated that the βlg fibrils were incorporated within the aqueous layers of the lamellar phase while in the cubic and hexagonal structures they were mostly located at the lipid–water interface, along the channels. Furthermore, each mesophase affected the assembly of the amyloid fibers in accordance with the nature of the space group of the LLC structure considered. The two-dimensional order parameter of the fibrils, which was calculated on the basis of AFM images, revealed mainly random orientation distribution of the fibrils when these were confined within the lamellar and cubic phases, which can be understood by the orientation degeneracy of the fibrils in a 2D confinement and by the isotropic nature of the cubic phases. On the contrary, the fibers exhibited nearly perfect orientation when confined within the columnar hexagonal phase as a consequence of the unidirectional orientation of the LLC and the high aspect ratio of the fibrils. These trends were further exploited to induce orientation of the LLC by decorating the encapsulated amyloid fibrils with magnetic nanoparticles capable to respond to an external magnetic field stimulus: a coupling of the orientation of the nanoparticle-decorated amyloid fibrils and that of the LLC mesophase was demonstrated by small angle neutron scattering under the application of a constant magnetic field.

Significant softening ofc66and hidden order parameter in the magnetic ordered state ofLa2−xCaxCoO4(0.4≤x≤0.7)

We have measured the longitudinal and transverse sound velocities of ${\text{La}}_{2\ensuremath{-}x}{\text{Ca}}_{x}{\text{CoO}}_{4}$ $(0.4\ensuremath{\le}x\ensuremath{\le}0.7)$ propagating along [100], [110], and [001] from 200 K down to 3 K with a frequency of 10 MHz. It was found that ${c}_{66}$ shows a remarkable softening below ${\text{Co}}^{2+}$ spin-ordering temperature ${T}_{N}$ in the whole doping range, while $\frac{{c}_{11}\ensuremath{-}{c}_{12}}{2}$ exhibits a clear hardening below ${\text{Co}}^{3+}$ spin-ordering temperature ${T}_{N2}$ only when $x=0.5$. To explain the observed anomaly, the Landau free energy for the ${\text{La}}_{2\ensuremath{-}x}{\text{Ca}}_{x}{\text{CoO}}_{4}$ system has been constructed, including the coupling energy between elastic strains and a two-dimensional order parameter $({Q}_{1},{Q}_{2})$, having a spatial modulation along [110] or $[\overline{1}10]$. We argued that this represents orbital order, and it is this order that leads to the softening of ${c}_{66}$.

From sea-urchins to starfishes: controlling the adsorption of star-branched polyelectrolytes on charged walls.

We demonstrate that star-branched polyelectrolytes are a versatile soft component forming a variety of well-characterised conformations upon electrostatic complexation on oppositely charged planar surfaces. We discover and characterise five complexation shapes, described by a two-dimensional morphological order parameter. These shapes are distinguished on the basis of their deviations from sphericity, the orientation of the polyelectrolyte arms, and the equilibrium centre-to-surface distance of the stars. Conformational transformations can be steered by changing the electrostatic surface-star coupling and/or the star functionality. For low functionalities and/or strong external fields, full collapse of the star onto the wall, forming a starfish configuration, is observed. Otherwise, partially adsorbed states with a fraction of chains pointing away from the surface and with the star core being detached from the wall are formed.

Nematic membranes: Shape instabilities of closed achiral vesicles

We consider the coupling between the local curvature tensor of a membrane and the local two-dimensional nematic order parameter, deriving it from a quasi-microscopic argument. This coupling makes the nematic director aligned along the lowest curvature eigenvector in a local metric. Local bending of a membrane may then generate nematic ordering. Alternatively, emerging nematic order leads to shape instabilities of closed vesicles. The theory is applied to a spherical isotropic vesicle, which turns into a prolate shape with two +1 disclinations on its poles as the nematic order sets in the membrane, described within the Landau-de Gennes continuum model.

Vortex dynamics in organic superconductor [formula omitted][formula omitted], [formula omitted

Organic superconductors κ -(ET) 2X (X=Cu(NCS) 2, Cu[N(CN) 2]Br) show anomalous elastic properties in both normal and superconducting states. We have measured the sound velocity of these salts near the superconducting transition temperature T c in the selected magnetic fields with different directions. Under the magnetic field, the sound velocity as a function of temperature shows remarkable elastic anomalies. It was mainly caused by interplay between the vortices and the elastic strain. The Δ C S - N , which is the difference of elastic constant C ( T ) in the superconducting state from the normal state, shows the softening toward lower temperature after following the jump at T c . This discontinuous jump at T c has been found on the longitudinal mode and on the transverse mode. It suggests that two-dimensional order parameter plays an important role in the superconductivity of κ -(ET) 2X system.

Multiscale coupling of mesoscopic- and atomistic-level lipid bilayer simulations

A multiscale method is presented to bridge between the atomistic and mesoscopic membrane systems. The atomistic model in this case is the united atom dimyristoylphosphatidylcholine membrane system, although the method is completely general. Atomistic molecular dynamics provides the expansion modulus which is used to parametrize a mesoscopic elastic membrane model. The resulting elastic membrane model, including explicit mesoscopic solvent, shows appropriate static and dynamic undulation behaviors. Large membranes of approximately 100 nm in length can then be easily simulated using the mesoscopic membrane system. The critical feedback from the mesoscopic system back down to the atomistic-scale system is accomplished by bridging the stress (or surface tension) of a small region in the mesoscopic membrane to the corresponding atomistic membrane system. Because of long length-scale modes of membranes such as undulation and buckling, the local tension responds differently from the frame tension, when subjected to external perturbations. The effect of these membrane modes is shown for the stress response of a local membrane region and therefore the atomistic membrane system. In addition, certain equilibrium static and dynamic properties of stand-alone and multiscale coupled systems are presented for several different membrane sizes. Although static properties such as two-dimensional pair-correlation function and order parameters show no noticeable discrepancy for the different systems, lipid self-diffusion and the rotational relaxation of lipid dipoles have a strong dependence on the membrane size (or long-wavelength membrane motions), which is properly modeled by the present multiscale method.

Solid/solid phase transitions in confined thin films: A zero temperature approach

We report a density functional theory study of confinement induced solid/solid phase transitions in a thin film (modeled as methane) at T=0. The solid film is confined by two graphite surfaces represented by a mean-field potential. As the wall separation is varied the structure of the confined film changes, which influences its density and the solvation force. Using the directly accessible grand canonical potential density we determine the stable phases and calculate the exact location of the phase transitions. We observe a series of phases having square and triangular symmetry. At low wall separations we find zig-zag buckling and an asymmetric buckled phase, whose structure is consistent with the strongest buckling instability of a triangular monolayer predicted by Chou and Nelson [Phys. Rev. E 48, 4611 (1993)] but, to our knowledge, has not been observed as a stable phase before. We find that the two-dimensional order parameters Psi(4) (square symmetry) and Psi(6) (triangular symmetry) show unphysical behavior in the transition region between square and triangular symmetry. Thus, in the present model they fail to predict the right location of the phase transitions.

Spin-Polarized Tunneling Spectroscopy and Shot Noise in Ferromagnet/f-Wave Superconductor Junctions**The project supported by National Natural Science Foundation of China under Grant No. 10347129 and the Natural Science Foundation of the “333” Project of Jiangsu Province of China

The tunneling spectroscopy and shot noise in ferromagnet/insulator/triplet-superconductor (FM/I/triplet-SC) structures are studied by taking into account the roughness interfacial barrier and exchange splitting in the FM. For the triplet-SC of Sr2RuO4, we consider two-dimensional f-wave order parameter symmetries having nodes within the RuO2 plane, which reasonably describe both thermodynamic and thermal conductivity data. It is shown that the ferromagnetic exchange splitting gives rise to a decrease in the differential conductance, the average current, and the shot noise power, while the noise power-to-current ratio is increased; the interface roughness is found to lead to a decrease in the differential conductance and the average current, and an increase in the noise power-to-current ratio.

Canonical and microcanonical partition functions in long-range systems with order parameter space of arbitrary dimension

We consider systems in which the canonical partition function can be expressed as the integral in an n-dimensional space (the order parameter space) of a function that also depends parametrically on the number N of degrees of freedom and on the inverse temperature β. We show how to compute, together with the canonical entropy and specific heat, also the corresponding microcanonical quantities, generalizing in this way some results already in the literature for the case n=1. From the expressions that are obtained it is possible to derive the necessary conditions for the equivalence of the canonical and of the microcanonical ensembles. We finally study a simple model, with a two-dimensional order parameter space, in which ensemble inequivalence is realized.