Isotropic-nematic phase transition of nonaqueous suspensions of natural clay rods

A novel model system for studying the behavior of hard colloidal rods is presented, consisting of sterically stabilized particles of natural sepiolite clay. Electron microscopy and scattering results confirmed that the organophilic clay particles were individual, rigid rods when dispersed in organic solvents. With a length-to-diameter ratio of approximately 27, the particles showed nematic ordering for volume fractions phi > 0.06. Polarizing microscopy revealed that the phase separation process involved nucleation, growth, and coalescence of nematic domains. The phase volumes and particle concentrations in the coexisting phases were determined. The dependence of these quantities on the total concentration of the suspension agrees well with Onsager's [Ann. N. Y. Acad. Sci. 51, 627 (1949)] isotropic-nematic phase transition theory extended to bidisperse and polydisperse rod systems, and with previous experimental results for rigid rodlike particles. Particle size distributions were obtained by analyzing transmission electron microscopy images. A significant fractionation with respect to rod length (but not diameter) was observed in the coexisting isotropic and nematic phases. The relative polydispersity of both daughter phases was distinctly smaller than that of the parent suspension. The phase behavior of these daughter fractions agrees well with the predictions for hard spherocylinders of corresponding aspect ratios. An isotropic-nematic-nematic phase equilibrium was seen to develop in phase separated samples after 1 month standing and is ascribed to the effect of polydispersity and possibly gravity. The second nematic phase appearing is dominated by very long rods.     

Performance of a rigid rod statistical mechanical treatment to predict monolayer ordering: a study of chain interactions and comparison with molecular dynamics simulation

A statistical mechanical model that treats hydrocarbon self-assembled monolayer (SAM) chains as rigid rods is examined to interrogate the mechanisms involved in monolayer ordering. The statistical mechanical predictions are compared to fully atomistic molecular dynamics simulations of SAMs with different packing densities. The monolayer chain order is examined as a function of surface coverage, chain-surface interactions, and chain–chain interactions. Reasonable interaction potentials are deduced from ab initio electronic structure calculations of small model systems. It is found that the chain-surface interaction is the most important parameter in formation of flat-lying monolayer phases, while formation of standing phase monolayers is driven most importantly by increased density of molecules at the surface. A brief discussion of the utility and validity of the rigid rod treatment is given in light of the molecular dynamics results.     

Rigid rod anchored to infinite membrane

We investigate the shape deformation of an infinite membrane anchored by a rigid rod. The density profile of the rod is calculated by the self-consistent-field theory and the shape of the membrane is predicted by the Helfrich membrane elasticity theory [W. Helfrich, Z. Naturforsch. 28c, 693 (1973)]. It is found that the membrane bends away from the rigid rod when the interaction between the rod and the membrane is repulsive or weakly attractive (adsorption). However, the pulled height of the membrane at first increases and then decreases with the increase of the adsorption strength. Compared to a Gaussian chain with the same length, the rigid rod covers much larger area of the membrane, whereas exerts less local entropic pressure on the membrane. An evident gap is found between the membrane and the rigid rod because the membrane's curvature has to be continuous. These behaviors are compared with that of the flexible-polymer-anchored membranes studied by previous Monte Carlo simulations and theoretical analysis. It is straightforward to extend this method to more complicated and real biological systems, such as infinite membrane/multiple chains, protein inclusion, or systems with phase separation.     

Numerical study of the phase behavior of rods in solution: Warner–Flory model with polydisperse system of rods

Phase equilibria of polydisperse rods in solution are studied in terms of the Warner–Flory model. Numerical calculations are made with a Gaussian-like asymmetric distribution of rod lengths. The aim of these studies is to investigate (i) the dependence of the phase diagram on the shape of the polydispersity distribution function and (ii) the isotropic–anisotropic phase separation within the coexistence range. It is shown that, as a consequence of polydispersity, even for a narrow distribution, the phase diagram is very simple, with only one type of anisotropic phase present. However, within the biphasic range, the properties of the anisotropic phase vary significantly with temperature. At high temperatures, they are essentially governed by steric effects and at low temperatures by both steric effects and soft anisotropic interactions between the rods.     

Isotropic–nematic phase separation and demixing in mixtures of spherical nanoparticles with length‐polydisperse nanorods

An extension of Onsager theory is developed to simulate isotropic–nematic phase separation in a mixture of spheres with length‐polydisperse system of rods. This work is motivated by recent experimental data on nanorod liquid crystals. Prior theoretical investigations indicate that both polydispersity and the presence of spheres should increase the biphasic–nematic phase transition, that is, the nematic cloud point. Results indicate that the phase diagrams undergo drastic changes depending upon both particle geometry and rod length polydispersity. The key geometric factor is the ratio between the sphere diameter and the rod diameter. In general, length fractionation is enhanced by the addition of spheres, which may be experimentally advantageous for separating short nanorods from a polydisperse population. Simulation results also indicate that the nematic cloud and shadow curves may cross one another because of the scarcity of spheres in the shadow phase. In general, these results do indicate that the nematic cloud point increases as a function of sphere loading; however, in certain areas of phase space, this relationship is nonmonotonic such that the nematic cloud point may actually decrease with the addition of spheres. This work has application to a wide range of nanoparticle systems, including mixtures of spherical nanoparticles with nanorods or nanotubes. Additionally, a number of nonspherical particles and structures may behave as spheres, including crumpled graphene and tightly coiled polymers. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

The effects of shape and flexibility on bio-engineered fd-virus suspensions

We present a theoretical model to describe binary mixtures of semi-flexible rods, applied here to fd-virus suspensions. We investigate the effects of rod stiffness on both monodisperse and binary systems, studying thick-thin and long-short mixtures. For monodisperse systems, we find that fd-virus particles have to be made extremely stiff to even approach the behavior of rigid rods. For thick-thin mixtures, we find increasingly rich phase behavior as the rods are either made more flexible or if their diameter ratio is increased. For long-short rod mixtures we find that the phase behavior is controlled by the relative stiffness of the rods, with increasing the stiffness of the long rods or decreasing that of the short rods resulting in richer phase behavior. We also calculate the state point dependent effective shape of the rods. The flexible rods studied here always behave as shorter, thicker rigid rods, but with an effective shape that varies widely throughout the phase diagrams, and plays a key role in determining phase behavior.     

Conformational and dynamic effects of pendant-PEK chains on rigid-rod polymers by molecular dynamics simulation

Molecular dynamics simulations were performed with models of a single-component ‘hairy-rod’ molecular composite composed of flexible meta poly(aryl ether ketone) (mPEK) chains (the ‘hairs’) grafted to a poly(p-phenylene benzobisthiazole) (PBZT) rigid-rod backbone. The molecular-composite concept, conceived to circumvent solubility problems and improve compressive strength of PBZT, relies on an even distribution of rods in a coil-like matrix. Two-molecule simulations show that the pendant mPEK chains associate with the rods causing large bends in the rod backbone. Simulations of bulk systems imply that the mPEK chains greatly reduce the correlation of rod orientation without altering rod spacing. Radial distribution and orientation correlation functions as well as correlation volumes are computed.     

Debye-Hückel theory for mixtures of rigid rodlike ions and salt

Like-charged surfaces are able to attract each other if they are embedded in an electrolyte solution of multivalent rodlike ions, even if the rods are long. To reproduce this ability the Poisson-Boltzmann model has recently been extended so as to account for the rodlike structure of the mobile ions. Our model properly accounts for intraionic correlations but still neglects correlations between different rodlike ions. For sufficiently long rods, the model shows excellent agreement with Monte Carlo simulations and exhibits two minima - a depletion and a bridging minimum - in the interaction free energy. In the present work, we generalize the Poisson-Boltzmann model to systems with polydisperse rod lengths and arbitrary charge distributions along the rods, including the presence of salt. On the level of the linearized Debye-Hu?ckel model we derive a general criterion for whether an electrolyte with given distribution of rodlike ions is able to mediate attraction between like-charged surfaces. We numerically analyze two special cases, namely the influence of salt on symmetric and asymmetric mixtures of monodisperse rodlike ions. The symmetric mixture is characterized by the presence of both negatively and positively charged (but otherwise identical) rodlike ions. For the asymmetric mixture, the system contains rodlike ions of only one type. We demonstrate that the addition of salt retains the depletion minimum but tends to eliminate the bridging minimum.     

Isotropic-nematic transition of hard rods immersed in random sphere matrices

Using replica density functional theory and Monte Carlo computer simulations we investigate a system of annealed hard spherocylinders adsorbed in a matrix of quenched hard spheres. Theoretical predictions for the partition coefficient, defined as the ratio of density of rods in the matrix and that in a reservoir, agree well with simulation results. Theory predicts the isotropic-nematic transition to remain first order upon increasing sphere packing fraction, and to shift towards lower rod densities. This scenario is consistent with our simulation results that clearly show a jump in the nematic order parameter upon increasing the rod density at constant matrix packing fraction, corresponding to the isotropic-nematic transition, even for sphere matrix packing fractions < or approximately equal to 0.3.     

Osmotic pressure of a nematic solution of polydisperse rod-like macromolecules

For a nematic solution of polydisperse rigid rods the osmotic pressure is proved to be independent of the length distribution and proportional simply to the total number density. This rule holds in the gaussian approximation to the Onsager theory. Its accuracy is demonstrated for bidisperse systems.     

Large-scale structure in gels of attractive block copolymer micelles

We have used ultra-small-angle scattering (USANS) and fluorescence microscopy to demonstrate the existence of a nonfractal large-scale structure in attractive micellar gels of poly(styrene)-poly(acrylic acid) block copolymers, which have some characteristics of attractive colloidal glasses. The nature of the large-scale structure appears to depend systematically on the strength of attraction. Our systems display scattering that follows I approximately q(x) in the low q regime, with x varying from approximately -3 to -4 as the strength of attraction is decreased. This scattering behavior appears to be the result of surface scattering from large, highly polydisperse aggregates with rough interfaces.     

Computation of the nonhomogeneous equilibrium states of a rigid-rod solution

The nonhomogeneous equilibrium phase behavior of a solution of rigid rods is analyzed for a periodic one-dimensional system. Stable and unstable equilibrium solutions for the distribution function are computed as extrema of the free energy of the system expressed by the nonhomogeneous generalization of Onsager's [Ann. N.Y. Acad. Sci. 51, 627 (1949)] theory, which models interaction between rods on the scale of a single rod length. Biaxial equilibrium solutions are computed in a periodic system by discretizing the Euler-Lagrange nonlinear integral equation by the finite-element method and using Newton's method to solve the resulting set of nonlinear equations. Stable states for isotropic-nematic coexistence are computed in a periodic system rather than the semi-infinite system used in previous calculations. The density and order parameter profiles evolve monotically from the isotropic phase to the nematic phase. Unstable, nonhomogeneous, equilibrium states are also computed for concentrations of rods that exceed the value for spinodal decomposition. These nonhomogeneous states are characterized by combinations of bend, twist, and splay distortions in physical space and correspond to unstable attractors in the dynamic process of isotropic-nematic spinodal decomposition. For large systems, the nonhomogeneous states develop wide, bulklike nematic regions separated by thin regions with sharp gradients in orientation. The free energy formulation was also used to compute the accurate neutral stability curve; this curve shows the limits of applicability of the low-wave-number approximations frequently used in the study of spinodal decomposition.     

Isotropic-nematic interfacial tension of hard and soft rods: application of advanced grand canonical biased-sampling techniques

Coexistence between the isotropic and the nematic phase in suspensions of rods is studied using grand canonical Monte Carlo simulations with a bias on the nematic order parameter. The biasing scheme makes it possible to estimate the interfacial tension gamma(IN) in systems of hard and soft rods. For hard rods with LD=15, we obtain gammaIN approximately 1.4kBT/L2, with L the rod length, D the rod diameter, T the temperature, and kB the Boltzmann constant. This estimate is in good agreement with theoretical predictions, and the order of magnitude is consistent with experiments.     

Liquid-crystal phase diagrams of binary mixtures of hard spherocylinders

We have built the liquid crystal phase diagram of several binary mixtures of freely rotating hard spherocylinders employing a second-order virial density functional theory with Parsons scaling, suitably generalized to deal with mixtures and smectic phases. The components have the same diameter and aspect ratio of moderate value, typical of many mesogens. Attention has been paid to smectic-smectic demixing and the types of arrangement that rods can adopt in layered phases. Results are shown to depend on the aspect ratio of the individual components and on the ratio of their lengths. Smectic phases are seen not to easily mix together at sufficiently high pressures. Layered phases where the longer rods are the majority component have a smectic-A structure. In the opposite case, a smectic-A(2) phase is obtained where the shorter particles populate the layers and the longer ones prefer to stay parallel to the latter in the interlayer region.     

Depletion induced isotropic-isotropic phase separation in suspensions of rod-like colloids

When non-adsorbing polymers are added to an isotropic suspension of rod-like colloids, the colloids effectively attract each other via depletion forces. We performed Monte Carlo simulations to study the phase diagram of such rod-polymer mixture. The colloidal rods were modeled as hard spherocylinders; the polymers were described as spheres of the same diameter as the rods. The polymers may overlap with no energy cost, while the overlap of polymers and rods is forbidden. Large amounts of depletant cause phase separation of the mixture. We estimated the phase boundaries of isotropic-isotropic coexistence both in the bulk and in confinement. To determine the phase boundaries we applied the grand canonical ensemble using successive umbrella sampling [J. Chem. Phys. 120, 10925 (2004)], and we performed a finite size scaling analysis to estimate the location of the critical point. The results are compared with predictions of the free volume theory developed by Lekkerkerker and Stroobants [Nuovo Cimento D 16, 949 (1994)]. We also give estimates for the interfacial tension between the coexisting isotropic phases and analyze its power-law behavior on the approach of the critical point.     

Phase behavior of polarizable spherocylinders in external fields

Applied electric fields are known to induce significant changes in the properties of systems of polarizable molecules or particles. For rod-shaped molecules, the field-induced behavior can be rather surprising, as in the case of the negative electric birefringence of concentrated solutions of rodlike polyelectrolytes. We have investigated the interplay of shape anisotropy and field-induced anisotropy in molecular dynamics simulations of systems of polarizable soft spherocylinders in an electric field, in the limit of infinitely anisotropic polarizability, taking full account of mutual induction effects. We find a novel crystalline structure (K(2)) in the high-field limit, whose formation is driven by interactions between induced dipoles. For high pressures, the phase diagram exhibits a polar nematic phase between the hexagonal close-packed crystal phase and the K(2) phase. We also compare this system with an analogous system of spherocylinders with permanent electric dipoles and find that qualitatively similar behavior is obtained in the limit of strong coupling of the permanent dipoles to the external field.     

Percolation, phase separation, and gelation in fluids and mixtures of spheres and rods

The relationship between kinetic arrest, connectivity percolation, structure and phase separation in protein, nanoparticle, and colloidal suspensions is a rich and complex problem. Using a combination of integral equation theory, connectivity percolation methods, nai?ve mode coupling theory, and the activated dynamics nonlinear Langevin equation approach, we study this problem for isotropic one-component fluids of spheres and variable aspect ratio rigid rods, and also percolation in rod-sphere mixtures. The key control parameters are interparticle attraction strength and its (short) spatial range, total packing fraction, and mixture composition. For spherical particles, formation of a homogeneous one-phase kinetically stable and percolated physical gel is predicted to be possible, but depends on non-universal factors. On the other hand, the dynamic crossover to activated dynamics and physical bond formation, which signals discrete cluster formation below the percolation threshold, almost always occurs in the one phase region. Rods more easily gel in the homogeneous isotropic regime, but whether a percolation or kinetic arrest boundary is reached first upon increasing interparticle attraction depends sensitively on packing fraction, rod aspect ratio and attraction range. Overall, the connectivity percolation threshold is much more sensitive to attraction range than either the kinetic arrest or phase separation boundaries. Our results appear to be qualitatively consistent with recent experiments on polymer-colloid depletion systems and brush mediated attractive nanoparticle suspensions.     

Influence of perpendicular external magnetic field on microstructures of monolayer composed of ferromagnetic particles: analysis by means of quasi-two-dimensional Monte Carlo simulation

We investigated the influences of the magnetic field strength and particle areal density on the microstructure of a quasi-two-dimensional monolayer composed of ferromagnetic particles by means of a Monte Carlo simulation. The magnetic field was applied along a direction perpendicular to the plane of the monolayer. Microstructures of the monolayer obtained in the simulations were analyzed in terms of radial distribution and orientational distribution functions. Formation of the microstructures is discussed from the perspective of particle-particle interaction energy and the perpendicular magnetic susceptibility of the monolayer was calculated from simulated magnetization curves. The obtained results are summarized as follows. For small areal density of particles, formation of chain-like structures is prevented by the repulsive magnetic interaction between particles due to orientations of the magnetic moments in the particles along the magnetic field direction. For intermediate areal density of particles, the chain-like structures remain even when a relatively strong magnetic field is applied, because contributions of the attractive magnetic interactions increase. For large areal density of particles, mixtures of chain-like and locally ordered structures appear due to the anisotropic attractive magnetic interactions in the absence of the magnetic field. However, when a sufficiently strong magnetic field is applied, the magnetic interactions between particles change to isotropic repulsive interactions, which results in the short-range repulsive steric interactions between particles becoming dominant with the appearance of hexagonal close packed structures.     

Linear quadrupoles with added octopole fields

Two methods of adding relatively small octopole fields to the main quadrupole field of quadrupoles and linear ion traps with cylindrical rods are investigated. The first, 'stretching' the quadrupole by moving two rods out from the axis, produces a combination of higher order fields with similar magnitudes in which the octopole field is not necessarily the greatest. The quadrupole field strength is changed significantly and a large potential appears on the axis. The second method uses rod pairs of different diameters. It adds octopole components of up to several percent while all other higher order fields remain small. An axis potential is also added, but it is only a few percent of the radio-frequency (RF) voltage and approximately equal to the strength of the octopole field. The axis potential can be removed by moving the larger rod pair out from the axis or applying unbalanced RF to the electrodes.