Communication: The criticality of self-assembled rigid rods on triangular lattices

The criticality of self-assembled rigid rods on triangular lattices is investigated using Monte Carlo simulation. We find a continuous transition between an ordered phase, where the rods are oriented along one of the three (equivalent) lattice directions, and a disordered one. We conclude that equilibrium polydispersity of the rod lengths does not affect the critical behavior, as we found that the criticality is the same as that of monodisperse rods on the same lattice, in contrast with the results of recently published work on similar models.     

Structure formation and fractionation in systems of polydisperse attractive rods

Fractionation effects and the formation of structured domains are investigated in polydisperse systems of attractive spherocylinders with the help of Monte Carlo simulations. For sufficiently high attractive interaction strength and pressure, the large rods in the system aggregate and form a highly ordered hexatic monolayer that coexists with an isotropic fluid of smaller rods. Fractionation diminishes with decreasing interaction strength but is still observed for hard rod systems, in which the large rods form a nematic droplet rather than a monolayer. Results for polydisperse systems are accompanied by phase diagrams for monodisperse systems of attractive spherocylinders. Here, the phase behavior is shown as a function of rod length and pressure.     

AFM studies on LB films of poly(n-hexyl isocyanate), poly(vinyl acetate), and their binary mixtures

LB films of rigid-rod-like poly(n-hexyl isocyanate) (PHIC), flexible poly(vinyl acetate) (PVAc), and binary mixtures of PHIC as well as of PHIC and PVAc transferred on a mica surface from the air-water interface were observed by AFM. The grain structure of three individual PHIC samples in the AFM images changed shape from a rigid rod to a coiled rod with an increase in the molecular weight due to changes in the chain rigidity of PHIC. On the other hand, the AFM image of PVAc was similar to that of a mica surface, indicating that PVAc forms a uniform and homogeneous film. For the binary mixtures of PHIC, the grain structure in the AFM image of the highest molecular weight PHIC was expanded with a similar shape after the addition of the smallest one, whereas it lost its shape after the addition of the middle one. Their peak-to-valley values in the AMF images were similar to those of the individual PHIC samples. For the binary mixtures of PHIC with the highest mass and PVAc, the grain in the AFM image of the PHIC lost its shape after the addition of PVAc and it changed shape from a connected partial lost coil to an extended bundle rod with an increase in the PVAc component.     

Towards Predicting Partitioning of Enzymes between Macromolecular Phases: Effects of Polydispersity on the Phase Behavior of Nonadditive Hard Spheres in Solution

The ability to separate enzymes, or cells or viruses, from a mixture is important and can be realized by the incorporation of the mixture into a macromolecular solution. This incorporation may lead to a spontaneous phase separation, with one phase containing the majority of one of the species of interest. Inspired by this phenomenon, we studied the theoretical phase behavior of a model system composed of an asymmetric binary mixture of hard spheres, of which the smaller component was monodisperse and the larger component was polydisperse. The interactions were modeled in terms of the second virial coefficient and could be additive hard sphere (HS) or nonadditive hard sphere (NAHS) interactions. The polydisperse component was subdivided into two subcomponents and had an average size ten or three times the size of the monodisperse component. We gave the set of equations that defined the phase diagram for mixtures with more than two components in a solvent. We calculated the theoretical liquid–liquid phase separation boundary for the two-phase separation (the binodal) and three-phase separation, the plait point, and the spinodal. We varied the distribution of the polydisperse component in skewness and polydispersity, and we varied the nonadditivity between the subcomponents as well as between the main components. We compared the phase behavior of the polydisperse mixtures with binary monodisperse mixtures for the same average size and binary monodisperse mixtures for the same effective interaction. We found that when the compatibility between the polydisperse subcomponents decreased, the three-phase separation became possible. The shape and position of the phase boundary was dependent on the nonadditivity between the subcomponents as well as their size distribution. We conclude that it is the phase enriched in the polydisperse component that demixes into an additional phase when the incompatibility between the subcomponents increases.     

Effects of Polydispersity on the Phase Behavior of Additive Hard Spheres in Solution

The ability to separate enzymes, nucleic acids, cells, and viruses is an important asset in life sciences. This can be realised by using their spontaneous asymmetric partitioning over two macromolecular aqueous phases in equilibrium with one another. Such phases can already form while mixing two different types of macromolecules in water. We investigate the effect of polydispersity of the macromolecules on the two-phase formation. We study theoretically the phase behavior of a model polydisperse system: an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modelled in terms of the second virial coefficient and are assumed to be additive hard sphere interactions. The polydisperse component is subdivided into sub-components and has an average size ten times the size of the monodisperse component. We calculate the theoretical liquid–liquid phase separation boundary (the binodal), the critical point, and the spinodal. We vary the distribution of the polydisperse component in terms of skewness, modality, polydispersity, and number of sub-components. We compare the phase behavior of the polydisperse mixtures with their concomittant monodisperse mixtures. We find that the largest species in the larger (polydisperse) component causes the largest shift in the position of the phase boundary, critical point, and spinodal compared to the binary monodisperse binary mixtures. The polydisperse component also shows fractionation. The smaller species of the polydisperse component favor the phase enriched in the smaller component. This phase also has a higher-volume fraction compared to the monodisperse mixture.     

Crystal nucleation in binary hard sphere mixtures: a Monte Carlo simulation study

We present calculations of the nucleation barrier during crystallization in binary hard sphere mixtures under moderate degrees of supercooling using Monte Carlo simulations in the isothermal-isobaric semigrand ensemble in conjunction with an umbrella sampling technique. We study both additive and negatively nonadditive binary hard sphere systems. The solid-fluid phase diagrams of such systems show a rich variety of behavior, ranging from simple spindle shapes to the appearance of azeotropes and eutectics to the appearance of substitutionally ordered solid phase compounds. We investigate the effect of these types of phase behavior upon the nucleation barrier and the structure of the critical nucleus. We find that the underlying phase diagram has a significant effect on the mechanism of crystal nucleation. Our calculations indicate that fractionation of the species upon crystallization increases the difficulty of crystallization of fluid mixtures and in the absence of fractionation (azeotropic conditions) the nucleation barrier is comparable to pure fluids. We also calculate the barrier to nucleation of a substitutionally ordered compound solid. In such systems, which also show solid-solid phase separation, we find that the phase that nucleates is the one whose equilibrium composition is closer to the composition of the fluid phase.     

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.     

Rotational diffusivity of rodlike molecules in amorphous polymeric matrices

In this note, scaling laws for rotational diffusivity of dilute monodisperse rigid-rod molecules (guest rods) in semidilute amorphous polymer solutions (host molecules) are derived. The coillike matrix molecules are modeled as a collection of flexibly connected rigid subunits. This allows an analogy with the Doi-Edwards theory for monodisperse rigid rods in semidilute solutions to be used in the analysis. Very strong dependencies are predicted for the rotational diffusivity of the rods on host polymer volume fraction and rod length. In semidilute polymer solutions the coils dramatically hinder the rotational freedom of the rods for r² ? ψ_p^?1, r being the rod aspect ratio and ψ_p the polymer volume fraction.     

Multiphase coexistence in polydisperse colloidal mixtures

The authors study the phase behavior of mixtures of monodisperse colloidal spheres with a depletion agent which can have arbitrary shape and can possess a polydisperse size or shape distribution. In the low concentration limit considered here, the authors can employ the free-volume theory and take the geometry of particles of the depletion agent into account within the framework of fundamental measure theory. The authors apply their approach to study the phase diagram of a mixture of (monodisperse) colloidal spheres and two polydisperse polymer components. By fine tuning the distribution of the polymer, it is possible to construct a complex phase diagram which exhibits two stable critical points.     

Simulation studies of self-assembly of end-tethered nanorods in solution and role of rod aspect ratio and tether length

We present temperature versus concentration phase diagrams for "shape amphiphiles" comprised of tethered moderate and low aspect ratio rods. Simulations of moderate aspect ratio rods (first reported by Horsch et al. [Phys. Rev. Lett. 95, 056105 (2005)]) predict their self-assembly into spherical micelles with bcc order, long micelles with nematic order, a racemic mixture of hexagonally ordered chiral cylinders, two perforated phases: one with tetragonal order and one with hexagonal order, and a smectic C lamellar phase. In contrast, we predict here that small aspect ratio tethered rods self-assemble into bcc ordered spherical micelles, hexagonally ordered cylinders, and a smectic C lamellar phase. We compare and contrast the phases obtained for the two aspect ratios and examine in further detail several unusual phases. Our simulations also reveal that for moderate aspect ratio rods there is a tendency toward phases with decreasing interfacial curvature with decreasing coil size, including a double gyroid phase. In addition, we investigate the role of tether length on the assembled structures. Our results are applicable to short rod-coil block copolymers and rodlike nanoparticles with polymer tethers, and to colloidal building blocks comprised of a flexible string of colloids tethered to a rigid string of colloids, with the interactions scaled appropriately.     

Compatibility of coils and rods bearing flexible side chains

We present a theoretical treatment of nematic-isotropic phase equilibria in mixtures which consist of random coils and comblike polymers, the latter components being composed of a rigid backbone and flexible side chains. The mixing partition function is evaluated by using the Flory lattice model. The comblike component is characterized by the axial ratio x_r of its rigid main chain and the number of flexible side chains z, each containing m segments. The coiled component is described by its number of segments x_c. The net exchange energy of mixing is assumed to be zero; i.e., we consider athermal solutions. It is shown that the flexible side chains attached to the rigid main chains markedly enhance the compatibility in the isotropic phase. If the ratio of the volume fraction of the side chains to the volume fraction of the main chains is high enough, there is even a finite range of concentration where the random coils mix homogeneously with the comblike component. This is in contrast to mixtures of rods and coils, which have been shown by Flory to be incompatible over nearly the full range of composition. These conclusions hold true only when ordered states are involved. For comblike polymers with flexible backbones mixed with random coils in isotropic melts, the resulting free energy of mixing is given by the familiar Flory-Huggins expression.     

The role of attractive interactions in rod-sphere mixtures

We present a computer simulation study of binary mixtures of prolate Gay-Berne particles and Lennard-Jones spheres. Results are presented for three such rod-sphere systems which differ from each other only in the interaction between unlike particles. Both the mixing-demixing behavior and the transitions between the isotropic and any liquid crystalline phases are studied for each system, as a function of temperature and concentration ratio. For systems which show macroscopic demixing, the rod-sphere interaction is shown to give direct control over interfacial anchoring properties, giving rise to the possibility of micellar phase formation in the case of homeotropic anchoring. Additionally, it is shown that on incorporating high concentrations of spheres into a system of rods with weak demixing properties, microphase-separated structures can be induced, including bicontinuous and lamellar arrangements.     

On the structure of binary mixtures of non-symmetric liquid crystal dimers: the alpha-(4-cyanobiphenyl-4-yloxy)-omega(4-alkylanilinebenzylidene-4-oxy) alkanes

Liquid crystal dimers, in which two mesogenic groups are linked by a flexible spacer, exhibit a rich smectic polymorphism for both symmetric and non-symmetric dimers which differ in the nature of the mesogenic groups. For example, smectic phases having monolayer, interdigitated and intercalated structures have been discovered. We have extended our studies of such systems to binary mixtures in an attempt to understand the origin of the different phase structures at the molecular level. The dimers studied include non-symmetric systems differing in the parity of the spacer and in the length of the terminal chains; for comparison we have also studied a mixture of symmetric dimers differing solely in the parity of the spacer. We have constructed the phase diagrams for the various mixtures and found that for certain systems the smectic phases exhibited by either one or both components can be destroyed. To investigate the local structure of the nematic phase for mixtures in which a smectic A phase is eliminated from the phase diagram we have determined their orientational order using NMR and ESR spectroscopy. To provide more direct information on the local structure an X-ray diffraction study was undertaken on certain of the mixtures.     

Structure, phase behavior, and inhomogeneous fluid properties of binary dendrimer mixtures

The effective pair potentials between different kinds of dendrimers in solution can be well approximated by appropriate Gaussian functions. We find that in binary dendrimer mixtures the range and strength of the effective interactions depend strongly upon the specific dendrimer architecture. We consider two different types of dendrimer mixtures, employing the Gaussian effective pair potentials, to determine the bulk fluid structure and phase behavior. Using a simple mean field density functional theory (DFT) we find good agreement between theory and simulation results for the bulk fluid structure. Depending on the mixture, we find bulk fluid-fluid phase separation (macrophase separation) or microphase separation, i.e., a transition to a state characterized by undamped periodic concentration fluctuations. We also determine the inhomogeneous fluid structure for confinement in spherical cavities. Again, we find good agreement between the DFT and simulation results. For the dendrimer mixture exhibiting microphase separation, we observe a rather striking pattern formation under confinement.     

Molecular alignment of rigid rods in nonrigid spherical pores

We have investigated the orientation ordering of two shish-kebab chains confined by spherically harmonic potentials through Monte Carlo simulations and asymptotic analysis. The rigid rod is modeled as shish-kebab chains consisting of tangent hard spheres aligned in the same axis, and the harmonic potential is chosen to model nonrigid cavities. We first show that the interactions between a rod and the spherically harmonic potential are independent of chain orientation, indicating that the alignment of two confined rods arises from the excluded volume interactions alone. In the strong fields, the order parameter of two confined rods converges to different values, depending on the parity of chain length. From asymptotic order parameters, we find that the rods of odd-number beads rotate more freely even under the limiting strong confinement. However, the two rods of even-number beads are essentially trapped in a configuration of perpendicular alignment through intercalation of their central grooves. We attribute the dependence of the parity of chain length to the different locations of the center-of-mass in a rod for these two cases. Furthermore, we compare the shish-kebab chains with different rod models in the simulations, and utilize these models to explore the effect of the local rod smoothness on molecular alignment. Our findings suggest that increasing local rod smoothness enhances the rotational degree of freedom for confined rods, and the effect of local rod roughness emerges under strong enough applied potentials.     

Polymer blends containing liquid crystal polymers

Polymer blends can be either composed of mixtures of flexible components, of a stiff chain and a flexible macromolecule, or of two stiff-chain polymers. All three cases may be dealt with in terms of the Flory lattice model. Special attention is paid to the influence of liquid crystalline order on the miscibility of the two polymers. For isotropic mixtures all three cases may be described in terms of the usual Flory–Huggins approximation. If a nematic phase is formed the miscibility of blends of rigid rods with flexible macromolecules (molecular composites) is strongly reduced because of entropic reasons. Highly ordered mixture of two stiff-chain polymers in melt can be described in terms of the regular solution theory leading to the same miscibility criterion as is valid for two flexible polymers. All deductions are compared to recent experimental work.     

The structure and phase transitions in polymer blends,diblock copolymers and liquid crystalline polymers: The Landau-Ginzburg approach

The polymer systems are discussed in the framework of the Landau-Ginzburg model. The model is derived from the mesoscopic Edwards Hamiltonian via the conditional partition function. We discuss flexible, semiflexible and rigid polymers. The following systems are studied: polymer blends, flexible diblock and multi-block copolymer melts, random copolymer melts, ring polymers, rigid-flexible diblock copolymer melts, mixtures of copolymers and homopolymers and mixtures of liquid crystalline polymers. Three methods are used to study the systems: mean-field model, self consistent one-loop approximation and self consistent field theory. The following problems are studied and discussed: the phase diagrams, scattering intensities and correlation functions, single chain statistics and behavior of single chains close to critical points, fluctuations induced shift of phase boundaries. In particular we shall discuss shrinking of the polymer chains close to the critical point in polymer blends, size of the Ginzburg region in polymer blends and shift of the critical temperature. In the rigid-flexible diblock copolymers we shall discuss the density nematic order parameter correlation function. The correlation functions in this system are found to oscillate with the characteristic period equal to the length of the rigid part of the diblock copolymer. The density and nematic order parameter measured along the given direction are anticorrelated. In the flexible diblock copolymer system we shall discuss various phases including the double diamond and gyroid structures. The single chain statistics in the disordered phase of a flexible diblock copolymer system is shown to deviate from the Gaussian statistics due to fluctuations. In the one loop approximation one shows that the diblock copolymer chain is stretched in the point where two incompatible blocks meet but also that each block shrinks close to the microphase separation transition. The stretching outweights shrinking and the net result is the increase of the radius of gyration above the Gaussian value. Certain properties of homopolymer/copolymer systems are discussed. Diblock copolymers solubilize two incompatible homopolymers by forming a monolayer interface between them. The interface has a positive saddle splay modulus which means that the interfaces in the disordered phase should be characterized by a negative Gaussian curvature. We also show that in such a mixture the Lifshitz tricritical point is encountered. The properties of this unusual point are presented. The Lifshitz, equimaxima and disorder lines are shown to provide a useful tool for studying local ordering in polymer mixtures. In the liquid crystalline mixtures the isotropic nematic phase transition is discussed. We concentrate on static, equilibrium properties of the polymer systems.     

Synthesis of poly(1.4.-benzamide) in solutions of randomly coiled polymers

The polycondensation of p-aminobenzoic acid by the use of triphenylphospine and hexachloroethane is described. One way to obtain a mixed solution of rigid rod and flexible coil polymers is the polycondensation of the rigid chains in a solution of a flexible polymer matrix. Results on matrix polycondensation of poly(p-benzamide) in solutions of poly(methyl methacrylate), poly(methyl methacrylate-co-styrene), polystyrene and polyacrylonitrile are reported. It is shown that there exists an interrelation between the phase behavior of the mixed polymer solutions and the influence of the matrix polymer on the synthesis of poly(1.4.-benzamide). The ternary phase diagrams of the rigid rod/flexible coil polymer solutions were determined.     

Re-entrant behavior in dynamics of binary mixtures of soft hybrid nanocolloids and homopolymers

We present results of measurements of temperature and wavevector dependent dynamics in binary mixtures of soft polymer grafted nanoparticles and linear homopolymers. We find evidence of melting of the dynamically arrested state of the soft nanocolloids with addition of linear polymers followed by a re-entrant slowing down of the dynamics with further increase in polymer density, depending on the size ratio, δ, of the polymers and the nanocolloids. For higher δ the re-entrant behavior is not observed, even for the highest added polymer density, explored here. Possible explanation of the observed dynamics in terms of the presence of a double-glass phase is provided.