Mixtures Of Hard Rods Research Articles

Exploring anisotropic pressure and spatial correlations in strongly confined hard-disk fluids: Exact results.

This study examines the transverse and longitudinal properties of hard disks confined in narrow channels. Employing an exact mapping of the system onto a one-dimensional polydisperse, nonadditive mixture of hard rods with equal chemical potentials, we compute various thermodynamic properties, including the transverse and longitudinal equationsof state, along with their behaviors at both low and high densities. Structural properties are analyzed using the two-body correlation function and the radial distribution function, tailored for the highly anisotropic geometry of this system. The results are corroborated by computer simulations.

One dimensional lattice fluid mixture with nearest neighbour interactions

We present an exact derivation of the free energy functional of a fluid mixture of hard rods with arbitrary sizes on a one-dimensional lattice. Our approach is based on the Wertheim cluster theory which consists of mapping a system with finite range interactions to the system with pure hard-core interaction but with modified activities. We show that the free energy functional has the same form as the fundamental measure functional. The interactions part of the free energy has two contributions, one from the one-particle cavity restricted to the hard rod or hard-sphere diameter and a second from the two-particle cavity which includes the finite range of the interaction. In the limit of a one-component system, our results reduce to the one derived using the Markov chain approach. For vanishing interactions, the density functionals coincide exactly with the previously derived for the mixture of hard rods with pure hard-core interaction.

Structural properties of hard-disk fluids under single-file confinement.

The structural properties of confined single-file hard-disk fluids are studied analytically by means of a mapping of the original system onto a one-dimensional mixture of non-additive hard rods, the mapping being exact in the polydisperse limit. Standard statistical-mechanical results are used as a starting point to derive thermodynamic and structural properties of the one-dimensional mixture, where the condition that all particles have the same chemical potential must be taken into account. Analytical results are then provided for the nth neighbor probability distribution function, the radial distribution function, and the structure factor, a very good agreement being observed upon comparison with simulation data from the literature. Moreover, we have analyzed the scaling form for the disappearance of defects in the zigzag configuration for high pressure and have obtained the translational correlation length and the structural crossover in the oscillation frequency for asymptotically large distances.

Excluded volume interactions and phase stability in mixtures of hard spheres and hard rods.

In this paper we study excluded volume interactions, the free volume fraction available, and the phase behaviour, in mixtures of hard spheres (HS) and hard rods, modeled as spherocylinders. We use free volume theory (FVT) to predict various physical properties and compare to Monte Carlo computer simulations. FVT is used at two levels. We use the original FVT approach in which it is assumed that the correlations of the HS are not affected by the rods. This is compared to a recent, more rigorous, FVT approach which includes excluded volume interactions between the different components at all levels. We find that the novel rigorous FVT approach agrees well with computer simulation results at the level of free volume available, as well as for the phase stability. The FVT predictions show significant quantitative and qualitative deviations with respect to the original FVT approach. The phase transition curves are systematically at higher rod concentrations than previously predicted. Furthermore, the calculations revealed that a certain asphericity is required to induce isostructural fluid-fluid coexistence and the stability region is highly dependent on the size ratio between the rods and the spheres.

Demixing, surface nematization, and competing adsorption in binary mixtures of hard rods and hard spheres under confinement.

A molecular simulation study of binary mixtures of hard spherocylinders (HSCs) and hard spheres (HSs) confined between two structureless hard walls is presented. The principal aim of the work is to understand the effect of the presence of hard spheres on the entropically driven surface nematization of hard rod-like particles at surfaces. The mixtures are studied using a constant normal-pressure Monte Carlo algorithm. The surface adsorption at different compositions is examined in detail. At moderate hard-sphere concentrations, preferential adsorption of the spheres at the wall is found. However, at moderate to high pressure (density), we observe a crossover in the adsorption behavior with nematic layers of the rods forming at the walls leading to local demixing of the system. The presence of the spherical particles is seen to destabilize the surface nematization of the rods, and the degree of demixing increases on increasing the hard-sphere concentration.

Models of blockage-induced selectivity

We examine blockage-induced selectivity in a particulate stream flowing through a channel. Each component of the mixture is characterized by the transit time, τ, necessary to pass through the channel. The model is motivated by filtration and other processes involving blockage. The transit time distribution of exiting particles depends on the entering particle distribution, , the intensity, λ, of the entering stream, and the blocking rule. With the simple rule that a blockage occurs whenever two particles are present in the channel, the properties of the exiting stream are directly related to the Laplace transform of the entering distribution, . For any entering distribution, the exiting stream is enriched in faster moving components. The selectivity of a species in a binary mixture can be mapped to a thermodynamic system, namely a hard rod mixture at a given pressure and temperature that can model the adsorption of gas mixtures in nanopores. We also examine an alternative rule according to which blocking only occurs if a faster moving particle catches up to a slower one in the channel. The selectivity is quantitatively different compared to the simple blocking rule. In a binary mixture the majority component in the entering stream is further enhanced in the exiting stream, independently of the transit times.

Phase behaviour in complementary DNA-coated gold nanoparticles and fd-viruses mixtures: a numerical study.

A new gel-forming colloidal system based on a binary mixture of fd-viruses and gold nanoparticles functionalized with complementary DNA single strands has been recently introduced. Upon quenching below the DNA melt temperature, such a system results in a highly porous gel state, that may be developed in a new functional material of tunable porosity. In order to shed light on the gelation mechanism, we introduce a model closely mimicking the experimental one and we explore via Monte Carlo simulations its equilibrium phase diagram. Specifically, we model the system as a binary mixture of hard rods and hard spheres mutually interacting via a short-range square-well attractive potential. In the experimental conditions, we find evidence of a phase separation occurring either via nucleation-and-growth or via spinodal decomposition. The spinodal decomposition leads to the formation of small clusters of bonded rods and spheres whose further diffusion and aggregation leads to the formation of a percolating network in the system. Our results are consistent with the hypothesis that the mixture of DNA-coated fd-viruses and gold nanoparticles undergoes a non-equilibrium gelation via an arrested spinodal decomposition mechanism.

Sedimentation stacking diagrams of binary mixtures of thick and thin hard rods.

We use Onsager theory and the local density approximation to study sedimentation-diffusion equilibrium density profiles of binary mixtures of thick and thin hard rods. We construct stacking diagrams for three diameter ratios, and find that even a simple spindle-shaped phase diagram with only isotropic-nematic demixing can lead to counter-intuitive stacking sequences such as an isotropic phase sandwiched between two nematic phases. For the most complex phase diagram considered here, we find sixteen distinct stacking sequences, including several with five sedimented layers. By adding sedimentation paths to composition-pressure and density-density phase diagrams and calculating density and composition profiles, we show that conclusions about bulk phase diagrams of binary mixtures on the basis of sedimentation-diffusion equilibria should be drawn warily.

Exact density functional for hard-rod mixtures derived from Markov chain approach

Using a Markov chain approach we rederive the exact density functional for hard-rod mixtures on a one-dimensional lattice, which forms the basis of the lattice fundamental measure theory. The transition probability in the Markov chain depends on a set of occupation numbers, which reflects the property of a zero-dimensional cavity to hold at most one particle. For given mean occupation numbers (density profile), an exact expression for the equilibrium distribution of microstates is obtained, which means an expression for the unique external potential that generates the density profile in equilibrium. By considering the rod ends to fall onto lattice sites, the mixture is always additive.

Self-assembly of 2D membranes from mixtures of hard rods and depleting polymers().

We combine simulations and experiments to elucidate the molecular forces leading to the assembly of two dimensional membrane-like structures composed of a one rod-length thick monolayer of aligned rods from an immiscible suspension of hard rods and depleting polymers. Computer simulations predict that monolayer membranes are thermodynamically stable above a critical rod aspect ratio and below a critical depletion interaction length scale. Outside of these conditions alternative structures such as stacked smectic columns or nematic droplets are thermodynamically stable. These predictions are confirmed by subsequent experiments using a model system of virus rod-like molecules and non-adsorbing polymer. Our work demonstrates that collective molecular protrusion fluctuations alone are sufficient to stabilize membranes composed of homogenous rods with simple excluded volume interactions.

Modeling and Understanding Smectic-Phase Formation in Binary Mixtures of Rodlike Polysilanes: Comparison of Onsager Theory and Experiment

We study a binary mixture of short and long parallel hard rods in the smectic A phase to understand the role of length ratio (weight ratio) in the formation of three different types of ordering which were observed in binary mixtures of helical polysilanes. Using Onsager’s second virial theory of parallel hard rods, and modeling the short and long polymers as hard cylinders of different lengths (L1 ≠ L2) and same diameters (D), we show that the normal smectic A phase (S1) forms for length ratios (l = L2/L1) between 0.58 and 1, the microsegregated smectic structure (S2) takes place for 0.32 < l < 0.56 and 0 < l < 0.39 in short-rod-rich and long-rod-rich phases, respectively, and the two-in-one ordering (S3) is stable only in mixtures rich in long rods for 0.33 < l < 0.58. These results are in very good agreement with the available experimental data of binary mixtures of polysilanes [Okoshi et al. Macromolecules 2009, 42, 3443]. In addition, the theory predicts the existence of a partially microsegregated smectic structure (S4), in long-rod-rich mixtures, for 0.3 < l < 0.39. Our theoretical results show that two or even three smectic order parameters should be measured, for all types of smectic phases, in order to obtain the correct positional distribution functions from the X-ray diffraction pattern.

Surface and smectic layering transitions in binary mixtures of parallel hard rods

The surface phase behavior of binary mixtures of colloidal hard rods in contact with a solid substrate (hard wall) is studied, with special emphasis on the region of the phase diagram that includes the smectic A phase. The colloidal rods are modeled as hard cylinders of the same diameter and different lengths, in the approximation of perfect alignment. A fundamental-measure density functional is used to obtain equilibrium density profiles and thermodynamic properties such as surface tensions and adsorption coefficients. The bulk phase diagram exhibits nematic-smectic and smectic-smectic demixing, with smectic phases having different compositions; in some cases they are microfractionated. The calculated surface phase diagram of the wall-nematic interface shows a very rich phase behavior, including layering transitions and complete wetting at high pressures, whereby an infinitely thick smectic film grows at the wall via an infinite sequence of stepwise first-order layering transitions. For lower pressures complete wetting also obtains, but here the smectic film grows in a continuous fashion. Finally, at very low pressures, the wall-nematic interface exhibits critical adsorption by the smectic phase, due to the second-order character of the bulk nematic-smectic transition.

Phase diagram of colloid-rod system

The semigrand ensemble theory [H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren, Europhys. Lett. 20, 559 (1992)] in conjunction with the fundamental measure density functional theory [V. B. Warshavsky and X. Song, Phys. Rev. E 69, 061113 (2004)] are used to construct the Helmholtz free energy densities of a mixture of uncharged colloidal hard spheres and colloidal rods in its solid and liquid phases. Given these free energy density functions, we apply the free energy density minimization method [G. F. Wang and S. K. Lai, Phys. Rev. E 70, 051402 (2004)] to crosshatch the system's regions of phases in coexistence. The calculated results show that the triangular area bounded by gas-liquid, gas-solid, and liquid-solid coexisting two phases which has been called the coexistence region of gas-liquid-solid corresponds in fact to sets of two phases in coexistence. The phase boundaries which define our calculated coexistence domains compare very well with previous theoretical calculations. The relevance of the phase-diagram domains to three phases in coexistence will be discussed.

Do Multilayer Crystals Nucleate in Suspensions of Colloidal Rods?

We study the isotropic-to-crystal transformation in a mixture of colloidal hard rods and nonabsorbing polymer using computer simulations. We determine the height of the nucleation barrier and find that the critical cluster consists of a single crystalline layer growing laterally for all polymer fugacities considered. At lower supersaturation, the free energy of a single hexagonally packed layer increases monotonically with size, while the nucleation barrier of a second crystalline layer is extremely high. Hence, the nucleation of multilayer crystals is never observed. Multilayer crystals form only in the spinodal decomposition regime, either where, in an intermediate stage, single crystalline membranes coalesce into multilayer clusters or where, at higher polymer fugacity, smaller clusters of rods stack on top of each other to form long filaments. Eventually, these transient structures evolve into a thermodynamically stable bulk crystal phase.

Demixed and ordered phases in hard-rod mixtures

We analyse demixing and ordering transitions in systems of hard cylindrical particles. The second virial approximation of Onsager and a bifurcation analysis, as introduced by Koda and Kimura, are used to evaluate the free energies, pressures, and density distribution functions in mixtures of equally long but differently wide cylinders. The spatial density distribution along the one relevant coordinate is of particular importance as it provides more detailed information concerning the nature of the phase transition than the bare bifurcation diagnosis. Detailed results are given for the nematic–nematic spinodal and the nematic–smectic transitions. Allowing for the absence of an isotropic phase, our results are in good qualitative agreement with those for freely orienting rods reported previously, and indicate a complex sequence of phase diagrams as the diameter dissimilarity of the two components is increased, with upper and lower critical points bounding nematic and smectic demixing regions. However, experimental results on colloidal rods show that nematic demixing occurs at a diameter ratio much smaller than ours or those for freely rotating fluids, indicating that Onsager-type theories may be insufficient to reproduce this phenomenon in a quantitative manner and, consequently, that more sophisticated approaches, presumably incorporating particle flexibility and additional interactions, are required.

Fundamental-measure density functional for mixtures of parallel hard cylinders

We obtain a fundamental-measure density functional for mixtures of parallel hard cylinders. For this purpose we first generalize to multicomponent mixtures the fundamental-measure functional proposed by Tarazona and Rosenfeld for a one-component hard disk fluid, through a method alternative to the cavity formalism of those authors. We show the equivalence of both methods when applied to two-dimensional fluids. The density functional so obtained reduces to the exact density functional for one-dimensional mixtures of hard rods when applied to one-dimensional profiles. In a second step, we apply an idea put forward some time ago by two of us, based again on a dimensional reduction of the system, and derive a density functional for mixtures of parallel hard cylinders. We explore some features of this functional by determining the fluid-fluid demixing spinodals for a binary mixture of cylinders with the same volume, and by calculating the direct correlation functions.

Depletion potentials in colloidal mixtures of hard spheres and rods

The depletion potential between a hard sphere and a planar hard wall, or two hard spheres, imposed by suspended rigid spherocylindrical rods is computed by the acceptance ratio method through the application of Monte Carlo simulation. The accurate results and ideal-gas approximation results of the depletion potential are determined with the acceptance ratio method in our simulations. For comparison, the depletion potentials are also studied by using both the density functional theory and Derjaguin approximations. The density profile as a function of positions and orientations of rods, used in the density functional theory, is calculated by Monte Carlo simulation. The potential obtained by the acceptance ratio method is in good agreement with that of density functional theory under the ideal-gas approximation. The comparison between our results and those of other theories suggests that the acceptance ratio method is the only efficient method used to compute the depletion potential induced by nonspherical colloids with the volume fraction beyond the ideal-gas approximation.

Use of Parsons-Lee and Onsager theories to predict nematic and demixing behavior in binary mixtures of hard rods and hard spheres

Parsons-Lee and Onsager theories are formulated for the isotropic-nematic transition in a binary mixture of hard rods and hard spheres. Results for the phase coexistence and for the equation of state in both phases for mixtures with different relative sizes and composition are presented. The two theories explain correctly the general behavior observed in experiments and computer simulations for these fluids. In particular, the theory accounts for the destabilization of the nematic phase when spherical or globular macromolecules are added to a system of rodlike colloids, and the entrance of the system into a demixed regime at high volume fractions of the spherical particles. Upon demixing a nematic state rich in rods coexists in equilibrium with an isotropic state much more diluted in the rodlike component. Onsager theory fails on quantitative grounds for aspect ratios of the rodlike molecules smaller than 100, and in the cases where the molar fractions of spheres becomes close to unity. On the contrary, the Parsons-Lee approximation remains accurate down to aspect ratios as small as 5. The spinodal analysis indicates that the isotropic-isotropic and nematic-nematic coexistences become feasible for sufficiently large spheres and long rods, respectively. The latter type of coexistence interferes partially with the isotropic-nematic coexistence regime of interest to the present work. Overall, the study serves to rationalize and control key aspects of the behavior of these binary nematogenic colloidal systems, which can be tuned with an appropriate choice of the relative size and molar fractions of the particles.

Binary Mixtures of Hard Rods: A Short Account

Results of recent calculations aimed at predicting the liquid-crystalline phase behavior of binary mixtures of hard spherocylinders are shortly reviewed. These calculations have been made by adopting a simple, re-summed second-virial Onsager-type theory, and include, among the usual isotropic and nematic phases, smectic phases too. The paradigmatic hard spherocylinders with diameter D and length of the cylindrical part equal to 5D have been taken as reference component, to which hard spherocylinders of different geometry are added. The liquid-crystalline phase diagram of the bidisperse mixtures so generated have been traced in all range of composition as a function of pressure. Various phenomena, such as strong demixing, presence of triple, critical and azeotropic points, and smectic phases with different symmetries, have been observed.

Surface phase transitions in athermal mixtures of hard rods and excluded volume polymers investigated using a density functional approach

Using fundamental measures' density functional framework based on Wertheim's first order perturbation theory [J. Chem. Phys. 87, 7323 (1987)] we study the surface phase transitions in athermal polymer-needle mixtures, which demix in bulk into the isotropic polymer-rich (rod-poor) and polymer-poor (rod-rich) phases. We find that the polymer-rich (rod-poor) phase wets the hard wall at coexistence and the wetting transition is of first order. In the partial wetting regime we find a sequence of layerings but these transitions are gradually suppressed as the chain length increases. For long enough chains we detect the prewetting line. Rods exhibit pronounced ordering at the wall in the polymer-rich phases. Our results imply that experiments on the (isotropic) wetting transition for colloidal rod-polymer mixtures should be easier to carry out than those for the colloidal rod-sphere mixtures because the wetting transition occurs at lower rod densities. On the other hand, layerings in sphere-needle mixtures may turn out to be difficult to observe experimentally because some of them will be metastable with respect to the freezing transition, whereas the remaining ones are located very close to the binodal.