Phase behavior of hard colloidal platelets using free energy calculations

We investigate the phase behavior of a model for colloidal hard platelets and rigid discotic molecules: oblate hard spherocylinders (OHSC). We perform free energy calculations using Monte Carlo simulations to map out the phase diagram as a function of the aspect ratio L∕D of the particles. The phase diagram displays a stable isotropic phase, a nematic liquid crystal phase for L∕D≤0.12, a columnar phase for L∕D?0.3, a tilted crystal phase for L?0.45, and an aligned crystal phase for L∕D?0.45. We compare the results to the known phase diagram of hard cut spheres. Thin cut spheres are almost cylinder-shaped, while the interactions between real discotic mesogens and colloidal platelets are more consistent with the toroidal rims of the OHSC. Since the shapes of the OHSC and the cut spheres are otherwise similar, the phase diagrams of the two types of particles are quite akin. However, the tilted crystal phase for OHSC, which is of a crystal type that is frequently found in experiments on disklike molecules, has not been found for hard cut spheres. Furthermore, although we have found a cubatic phase, it was shown to be definitely unstable, whereas the stability of the cubatic phase of cut spheres is still disputed. Finally, we also show that the phase boundaries differ significantly from those for cut spheres. These are remarkable consequences of a subtle change in particle shape, which show that for a detailed comparison with the phase behavior of experimental particles, the OHSC should be used as a model particle.     

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     

Phase equilibrium of poly(n-butyl acrylate) and E7

The experimental equilibrium phase diagram of mixtures of linear poly(n-butyl acrylate) of molecular mass M_w = 112000 g mol^-1 and the low molecular mass LC mixture E7 has been established using polarized optical microscopy and light scattering techniques. The diagram is found to be reminiscent of an upper critical solution temperature system. Two independent series of samples with the same composition were studied, yielding consistent results. A region of nematic and isotropic coexisting phases and a region of a single isotropic phase were identified in the composition-temperature phase diagram. The results were analysed within a theoretical model combining the Flory-Huggins lattice theory for isotropic mixing and the Maier-Saupe theory for nematic ordering. Interestingly, no region of isotropic coexisting phases was observed in our experiments. This is probably due to the fact that the nematic interaction overwhelms the isotropic interaction in the region where (I + I) coexisting phases could appear. A preferential solubility of certain constituents of the LC mixture in the polymer could possibly be a reason for this behaviour.     

Phase equilibrium of poly(n-butyl acrylate) and E7

The experimental equilibrium phase diagram of mixtures of linear poly(n-butyl acrylate) of molecular mass M_w = 112000 g mol^-1 and the low molecular mass LC mixture E7 has been established using polarized optical microscopy and light scattering techniques. The diagram is found to be reminiscent of an upper critical solution temperature system. Two independent series of samples with the same composition were studied, yielding consistent results. A region of nematic and isotropic coexisting phases and a region of a single isotropic phase were identified in the composition-temperature phase diagram. The results were analysed within a theoretical model combining the Flory-Huggins lattice theory for isotropic mixing and the Maier-Saupe theory for nematic ordering. Interestingly, no region of isotropic coexisting phases was observed in our experiments. This is probably due to the fact that the nematic interaction overwhelms the isotropic interaction in the region where (I + I) coexisting phases could appear. A preferential solubility of certain constituents of the LC mixture in the polymer could possibly be a reason for this behaviour.     

Nematic-nematic demixing in polydisperse thermotropic liquid crystals

We consider the effects of polydispersity on isotropic-nematic phase equilibria in thermotropic liquid crystals, using a Maier-Saupe theory [Z. Naturforsch. A 13A, 564 (1958)] with factorized interactions. A sufficient spread (approximately 50%) in the interaction strengths of the particles leads to phase separation into two or more nematic phases, which can in addition coexist with an isotropic phase. The isotropic-nematic coexistence region widens dramatically as polydispersity is increased, leading to reentrant isotropic-nematic phase separation in some regions of the phase diagram. We show that similar phenomena will occur also for nonfactorized interactions as long as the interaction strength between any two particle species is lower than the mean of the intraspecies interactions.     

Phase transitions in mixtures of a side-on-side chain liquid crystalline polymer and low molar mass nematic liquid crystals

Miscibility phase diagrams of mixtures of side-on side chain liquid crystalline polymers (s-SCLCP) and low molar mass liquid crystals (E48 and E44) have been established by means of polarized optical microscopy and light scattering. E48 and E44 are cyanobiphenyl-based eutectic nematic liquid crystal (LC) mixtures with nematic-isotropic transition temperatures of 93 and 105 C, respectively. The phase diagram of the s-SCLCP/E48 system reveals the coexistence of an isotropic nematic region and a single nematic phase in order of descending temperature. The single nematic phase suggests that the pair is miscible in the nematic region. On the other hand, the s-SCLCP/E44 mixture shows liquid liquid and nematic nematic coexistence phases, suggestive of the immiscibility character of the pair. These nematic phase diagrams of the s-SCLCP/E48 and s-SCLCP/E44 have been analysed in the context of the combined Flory-Huggins (FH) free energy for isotropic mixing and the Maier-Saupe (MS) free energy for nematic ordering of the mesogens. This combined FH/MS theory is capable of predicting the observed nematic phase diagrams consisting of liquid liquid, liquid nematic, nematic nematic, and the pure nematic regions. The change of colour accompanying the appearance and disappearance of the inversion walls may be attributed to the temperature dependence of birefringence.     

Phase equilibria and photopolymerisation-induced phase transitions of mesogenic diacrylate monomer and low molecular mass liquid crystal mixture

Experimental phase diagrams of binary mesogenic mixtures of reactive mesogenic diacrylate (RM257) monomer and low molar mass liquid crystals (E7) were determined by means of differential scanning calorimetry and optical microscopy. The combined free energy densities of Flory–Huggins for liquid–liquid demixing, Maier–Saupe for nematic ordering, and phase field free energy for crystal solidification was proposed to describe the phase diagrams of the starting E7/RM257 mixtures. The phase diagram thus constructed is an ideal mixing type, exhibiting a narrow loop of isotropic + nematic (I + N) coexistence region followed by the crystal + nematic (Cr1 + N) region in descending order of temperature. Of particular interest is the permanent fixation of the mesophase structures upon photopolymerisation of neat RM257 in the corresponding nematic and crystalline phases. Upon photopolymerisation of a low RM257 content mixture in both isotropic and nematic states, the nematic–isotropic transition of E7 was found to persist. The permanent structural anchoring is seen upon photo-curing of the 90/10 RM257/E7 mixture in the crystalline state.     

Many-body interactions between particles in a polydisperse polymer fluid

We use a continuum chain model and develop an analytical theory for the interaction between many spheres immersed in a fluid of ideal polydisperse polymers. Assuming local spherical symmetry of the polymer field about each particle, combined with a local approximation, compact expressions are derived for the many-body interaction between the spheres. We use a mean-field approximation to investigate the fluid-fluid phase diagram for the mixture.     

Phase and orientational ordering of low molecular weight rod molecules in a quenched liquid crystalline polymer matrix with mobile side chains

We study the phase diagram and orientational ordering of guest liquid crystalline (LC) rods immersed in a quenched host made of a liquid crystalline polymer (LCP) matrix with mobile side chains. The LCP matrix lies below the glass transition of the polymer backbone. The side chains are mobile and can align to the guest rod molecules in a plane normal to the local LCP chain contour. A field theoretic formulation for this system is proposed and the effects of the LCP matrix on LC ordering are determined numerically. We obtain simple analytical equations for the nematic/isotropic phase diagram boundaries. Our calculation show a nematic-nematic (N/N) first order transition from a guest stabilized to a guest-host stabilized region and the possibility of a reentrant transition from a guest stabilized nematic region to a host only stabilized regime separated by an isotropic phase. A detailed study of thermodynamic variables and interactions on orientational ordering and phases is carried out and the relevance of our predictions to experiments and computer simulations is presented.     

On the phase behavior of blends of polymers and nematic liquid crystals

The phase behavior of mixtures of polymers and nematic liquid crystals (LC) is investigated. Two types of systems are examined. The first one deals with blends in which the polymer is made of linear chains. In this case, a systematic study of the effects of various parameters on the phase diagrams is performed. In particular, it is shown how increasing the polymer size and/or the LC molecule size increases the miscibility gap of the mixture. It also reduces the region where a single nematic phase is observed in the presence of a tiny amount of polymer. Likewise, the relative effects of the isotropic and the nematic interaction parameters on the phase diagrams are examined. The second part of this investigation deals with blends involving crosslinked polymers. Here, substantial differences are observed as compared to the case where the polymer components are made of linear chains. These differences are illustrated by showing the phase diagrams in similar conditions for both blends. Unlike the case of a linear polymer matrix, it is observed that the single nematic phase and the nematic-isotropic spinodal branches are absent from the phase diagram of crosslinked polymers. This results into significant distortions of the phase diagram. In order to highlight all these effects, examples representing hypothetical blends are considered. These examples are chosen for illustration of the results in which the choice of numerical parameters is made consistently with the existing values in the literature which makes comparison with published data possible.     

Influence of non-mesogenic impurities on a nematic to isotropic phase transition

Non-mesomorphic solutes depress the normal nematic-isotropic transition temperature in liquid crystals. When non-mesomorphic solutes are added to a nematic liquid crystal, the nematic-isotropic transition temperature is depressed and a two phase region is formed due to the presence of impurities of the solutes. The present paper explains the formation of this two phase region by the Landau-de Gennes phenomenological theory, which agrees fairly well with the experimental observations. We also note that this two phase region indicates the tricritical behaviour of the nematic-isotropic phase transition and the phase diagram near the tricritical point is also obtained.     

Phase behavior of colloidal hard tetragonal parallelepipeds (cuboids): a Monte Carlo simulation study

The impact of particle geometry on the phase behavior of hard colloidal tetragonal parallelepipeds (TPs) was studied by using Monte Carlo simulations in continuum space. TPs or "cuboids" of aspect ratios varying from 0.25 to 8 were simulated by approximating their shapes with multisite objects, i.e., via rigid clusters of hard spheres. Using equation of state curves, order parameters, radial distribution functions, particle distribution functions along three directions, and visual analysis of configurations, an approximate phase diagram for the TPs was mapped out as a function of aspect ratio (r) and volume fraction. For r > 3 and intermediate concentrations, the behavior of the TPs was similar to that of spherocylinders, exhibiting similar liquid crystalline mesophases (e.g., nematic and smectic phases). For r = 1, a cubatic phase occurs with orientational order along the three axes but with little translational order. For 1 < r < 4, the TPs exhibit a cubatic-like mesophase with a high degree of order along three axes where the major axes of the particles are not all aligned in the same direction. For r < 1, the TPs exhibit a smectic-like phase where the particles have rotational freedom in each layer but form stacks with tetratic order. The equation of state for perfect hard cubes (r = 1) was also simulated and found to be consistent with that of the rounded-edge r = 1 TPs, except for its lack of discontinuity at the cubatic-solid transition.     

Mixtures of liquid-crystalline polymers

A density-functional expansion method is used to derive the free energy of a polymer mixture. The expression obtained includes the entropy of mixing, the entropy of configuration of the chains and the interactions (both isotropic and anisotropic ones). The chains are modelled as interacting elastic lines (bend curvature). The method is very general, and we only focus our attention on binary mixtures. The phase diagram and the order parameters are calculated. We show some results for two types of mixtures: a nematic polymer in a non-mesomorphic particle (polymer or solvent) and in another nematic liquid crystal (small-molecule or polymer). We discuss the influence of the molecular weights, the persistence length and the interactions on the phase separation.     

Orientational ordering in hard rectangles: The role of three-body correlations

We investigate the effect of three-body correlations on the phase behavior of hard rectangle two-dimensional fluids. The third virial coefficient B3 is incorporated via an equation of state that recovers scaled particle theory for parallel hard rectangles. This coefficient, a functional of the orientational distribution function, is calculated by Monte Carlo integration, using an accurate parametrized distribution function, for various particle aspect ratios in the range of 1-25. A bifurcation analysis of the free energy calculated from the obtained equation of state is applied to find the isotropic (I)-uniaxial nematic (N(u)) and isotropic-tetratic nematic (N(t)) spinodals and to study the order of these phase transitions. We find that the relative stability of the N(t) phase with respect to the isotropic phase is enhanced by the introduction of B3. Finally, we have calculated the complete phase diagram using a variational procedure and compared the results with those obtained from scaled particle theory and with Monte Carlo simulations carried out for hard rectangles with various aspect ratios. The predictions of our proposed equation of state as regards the transition densities between the isotropic and orientationally ordered phases for small aspect ratios are in fair agreement with simulations. Also, the critical aspect ratio below which the N(t) phase becomes stable is predicted to increase due to three-body correlations, although the corresponding value is underestimated with respect to simulation.     

The phase diagram of the lyotropic nematic mesophase in the TTAB/NaBr/water system

The phase diagram of the nematic mesophase present in the tetradecyltrimethylammonium bromide/sodium bromide/water ternary system was determined. A calamitic nematic mesophase (N_C) was observed which extends to very high concentrations of electrolyte. The order parameters of the surfactant head group in the mesophases were studied by the NMR quadrupolar splitting of the deuterated surfactant. On increasing the temperature of nematic mesophases with low electrolyte concentrations, a phase separation occurs with the formation of a more highly ordered hexagonal phase and an isotropic phase. Diffusion measurements of the isotropic micellar solution by the NMR PFG method were used to estimate hydrodynamic radii at low surfactant concentrations and to study micelle diffusion as the concentration of the surfactant was increased to the liquid crystalline region. At higher surfactant concentrations, the diffusion coefficient reached a limiting value. The calamitic nematic mesophase in this surfactant/electrolyte/water system appears to be formed by long wormlike micelles.     

Sol/gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 1. Features of the i/n transition

The phase behavior of a natural nontronite clay was studied for size-selected particles by combining osmotic pressure measurements, visual observations under polarized light, and rheological experiments. In parallel, the positional and orientational correlations of the particles were analyzed by small-angle X-ray scattering. Aqueous suspensions of nontronite exhibit a true isotropic/nematic (I/N) transition that occurs before the sol/gel transition, for ionic strengths below 10(-3) M/L. In this region of the phase diagrams, the system appears to be purely repulsive. The I/N transition shifts toward lower volume fractions for increasing particle anisotropy, and its position in the phase diagram agrees well with the theoretical predictions for platelets. SAXS measurements reveal the presence of characteristic interparticular distances in the isotropic, nematic, and gel phases. The swelling law (separation distance vs swelling law) exhibits two regimes. For high volume fractions, the swelling law is one-dimensional as in layered systems and reveals the presence of isolated platelets. At lower volume fraction, distances scale as phi(-1/3), indicating isotropic volumic swelling. Finally, the experimental osmotic pressure curves can be satisfactorily reproduced by considering the interparticle distances between two charged planes whose effective charge is around 10% of the structural charge.     

Rigid rod molecules as liquid crystal thermosets. II. Rigid rod esters

Nine rigid rod ester monomers endcapped with maleimide, nadimide, and methylnadimide groups were prepared and studied by DSC and hot stage polarized light microscopy. All of the monomers showed thermotropic nematic liquid crystalline phases and could be thermally crosslinked in the nematic phase. The nematic texture was maintained in the crosslinked solid state. The nematic phase range was enlarged by B-staging the monomeric compounds. Heating the monomers for a short period of time in the nematic phase lowered the crystal to nematic transition temperatures and increased the nematic to isotropic transition temperatures. A nonequilibrium phase diagram was proposed to explain the melting behavior of these reactive liquid crystal thermoset materials.     

Interplay between cubic and hexagonal phases in block copolymer solutions

The phase behavior of a symmetric styrene-isoprene (SI) diblock copolymer in a styrene-selective solvent, diethylphthalate, was investigated by in situ small-angle X-ray scattering on isotropic and shear-oriented solutions and by rheology and birefringence. A remarkable new feature in this phase diagram is the coexistence of both body-centered cubic (bcc) and hexagonally close-packed (hcp) sphere phases, in a region between close-packed spheres (cps) and hexagonally packed cylinders (hex) over the concentration range phi approximately 0.33-0.45. By focusing on the transitions among these various ordered phases during heating and cooling cycles, we observed a strong hysteresis: supercooled cylinders persisted upon cooling. The stability of these supercooled cylinders is quite dependent on concentration, and for phi > or = 0.40, the supercooled cylinders do not revert to spheres even after quiescent annealing for 1 month. The spontaneous formation of spheres due to the dissociation of cylinders is kinetically hindered in this case, and the system is apparently not amenable to any pretransitional fluctuations of cylinders prior to the cylinder-to-sphere transition. This contrasts with the case of cylinders transforming to spheres upon heating in the melt. The application of large amplitude shear to the supercooled cylinders is effective in restoring the equilibrium sphere phases.     

Nematic-isotropic interfaces under shear: a molecular-dynamics simulation

We present a large-scale molecular-dynamics study of nematic-paranematic interfaces under shear. We use a model of soft repulsive ellipsoidal particles with well-known equilibrium properties, and consider interfaces which are oriented normal to the direction of the shear gradient (common stress case). The director at the interface is oriented parallel to the interface (planar). A fixed average shear rate is imposed with moving periodic boundary conditions, and the heat is dissipated with a profile-unbiased thermostat. First, we study the properties of the interface at one particular shear rate in detail. The local interfacial profiles and the capillary wave fluctuations of the interfaces are calculated and compared with those of the corresponding equilibrium interface. Under shear, the interfacial width broadens and the capillary wave amplitudes at large wavelengths increase. The strain is distributed inhomogeneously in the system (shear banding), the local shear rate in the nematic region being distinctly higher than in the paranematic region. Surprisingly, we also observe (symmetry-breaking) flow in the vorticity direction, with opposite direction in the nematic and the paranematic state. Finally, we investigate the stability of the interface for other shear rates and construct a nonequilibrium phase diagram.     

Landau theory for uniaxial nematic,biaxial nematic,uniaxial smectic-A,and biaxial smectic-A phases

We use the Landau theory of phase transitions to obtain the global phase diagram concerning the uniaxial nematic, biaxial nematic, uniaxial smectic-A and biaxial smectic-A phases. The transition between the biaxial nematic and biaxial smectic is continuous as well as the transition between the nematic phases and the transition between the smectic phases. The transition from uniaxial nematic and uniaxial smectic is continuous with a tricritical point. The tricritical point may be absent and the entire transition becomes continuous. The four phases meet at a tetracritical point.