Restricted Orientation Model Research Articles

Stability, structure, compressibility factor, excess free energy and density profile of hard prolate pear shape fluids

ABSTRACT In this work, the Kerr coefficients of isotropic and uniaxial nematic liquid crystals consisting of hard ellipsoid and hard prolate pear-shaped molecules were theoretically studied using classical density functional theory. A new analytical approximate expression for the direct correlation function of hard pear- shaped molecules is applied. Using the expansion coefficients of this direct correlation function, we calculated the structural parameters, compressibility factor, and excess free energy of the hard pear-shaped liquid crystals. In addition, using the restricted orientation model, one particle density of a confined hard pear- shaped liquid crystal is calculated. The limiting stabilities and estimated transition densities are in qualitative agreement with available data. The results confirm the validity of this approximate direct correlation function.

Biaxial nematic stability in the rod-plate mixture with a dopant: The restricted-orientation model on the 3rd virial level

We analyze the effect of doping a binary mixture of uniaxial rod(R)- and plate(P)-like hard parallelepipeds having equal volumes and conjugated aspect ratios γR=1/γP=5. Uniaxial rods (L) or plates (T) with greater elongations γL=1/γT=10 were used as an admixture. The aim of the work is to examine this effect as a possible tool to stabilize the biaxial (Nb) phase in the P–R mixture, where Nb is known to be unstable with respect to demixing into two uniaxial nematics. The formalism to study orientational phase transitions is the off-lattice restricted-orientation (Zwanzig) model of a multi-component mixture of rectangular parallelepipeds with the EOS on the 3rd virial level. The phase biaxiality parameters of components and the second-order transition lines are located by direct minimization of the Gibbs energy. The results obtained by the examples of ternary systems show that introducing of small amounts of L or T into the reference P–R system promotes broadening of the Nb domain and may induce its stabilization. The case of L admixtures shows that an increase of the dopant particle volume results in a trend towards enhancement of the Nb phase relative stability.

The study of uniaxial–biaxial phase transition of confined hard ellipsoids using density functional theory

The density profiles and corresponding order parameters of the hard ellipsoids confined between two hard walls and also in contact with a single hard wall are studied using the density functional theory (DFT). The hypernetted-chain (HNC) approximation is used to write excess grand potential of the system with respect to the bulk value. To simplify the calculations, we use restricted orientation model (ROM) for the orientation of ellipsoids to find the density profiles and order parameters. DFT shows that there is a uniaxial–biaxial (U–B) phase transition near a single hard wall and also between two hard walls for a fluid consisting of uniaxial hard ellipsoidal particles with finite elongation.

Density and Polarization Profiles of Dipolar Hard Ellipsoids Confined between Hard Walls: A Density Functional Theory Approach

The density and polarization profiles of the dipolar hard ellipsoids confined between hard walls are studied using the density functional theory (DFT). The Hyper-Netted Chain (HNC) approximation is used to write excess grand potential of the system with respect to the bulk value. The number density is expanded up to zero and first order in polarization to find the results. For the zero order in polarization, the coupled integral equations for the directional densities are obtained. Then for the first order in polarization the coupled integral equations for the directional densities and polarization profiles are obtained. To simplify the calculations we use restricted orientation model (ROM) for the orientation of ellipsoids to find the density and polarization profiles. We also apply an electric field and write an expression for the excess grand potential of the system and obtain the coupled integral equations for the density and polarization profiles again. Finally we calculate the density and polarization profiles for different cases and compare the obtained results.

Solvation force in hard ellipsoid fluids with HNW interaction

In present work, using density functional theory and extended restricted orientation model, the one particle density of hard Gaussian overlap fluid near the colloid walls is calculated. The hard needle–wall interaction between molecules and colloids are considered. Using non-linear equation, proposed by Grimson–Rickyazen, the solvation force of hard ellipsoidal molecular fluid with hard Gaussian overlap interaction is calculated. We could not find the exact or simulation results for comparison. The results in the case k = 2.0 are compared with the solvation force of one-dimensional hard rod fluids. The results are corresponded, qualitatively.

Density functional theory of liquid crystals and surface anchoring: Hard Gaussian overlap-sphere and hard Gaussian overlap-surface potentials

This article applies the density functional theory to confined liquid crystals, comprised of ellipsoidal shaped particles interacting through the hard Gaussian overlap (HGO) potential. The extended restricted orientation model proposed by Moradi and co-workers [J. Phys.: Condens. Matter 17, 5625 (2005)] is used to study the surface anchoring. The excess free energy is calculated as a functional expansion of density around a reference homogeneous fluid. The pair direct correlation function (DCF) of a homogeneous HGO fluid is approximated, based on the optimized sum of Percus-Yevick and Roth DCF for hard spheres; the anisotropy introduced by means of the closest approach parameter, the expression proposed by Marko [Physica B 392, 242 (2007)] for DCF of HGO, and hard ellipsoids were used. In this study we extend an our previous work [Phys. Rev. E 72, 061706 (2005)] on the anchoring behavior of hard particle liquid crystal model, by studying the effect of changing the particle-substrate contact function instead of hard needle-wall potentials. We use the two particle-surface potentials: the HGO-sphere and the HGO-surface potentials. The average number density and order parameter profiles of a confined HGO fluid are obtained using the two particle-wall potentials. For bulk isotropic liquid, the results are in agreement with the Monte Carlo simulation of Barmes and Cleaver [Phys. Rev. E 71, 021705 (2005)]. Also, for the bulk nematic phase, the theory gives the correct density profile and order parameter between the walls.

STRUCTURE OF CONFINED HARD ELLIPSES USING MOLECULAR DENSITY FUNCTIONAL THEORY

Density functional (DF) theory is used to study confined two-dimensional molecular fluids. The approximate DF used here is based upon a functional Taylor series expansion of the excess grand potential about the density of a uniform molecular liquid. This formalism is applied to obtain the density profiles of the hard-ellipse fluid confined between two parallel hard walls. The required direct correlation function is obtained using Percus–Yevick and hypernetted chain approximations for low- and high-number density, respectively. Both the restricted orientation model and the extension of this model are used. Generally, we obtained that the number density of the hard ellipses with the major axes parallel to the wall is larger near the walls than the other directions. To check the results, we show that for isotropic hard-ellipse fluids the average number density in the middle of the wall is almost equal to the bulk density and as we expect the average number density of the ellipses at the wall is nearly equal to the amount of the reduced pressure, βP. We also perform Monte Carlo simulation and find reasonable agreement with our results.

Density functional theory of liquid crystals and surface anchoring

This paper applies the density functional theory to confined liquid crystals comprising ellipsoidal shaped particles interacting through the hard Gaussian overlap (HGO) potential. The restricted orientation model proposed by Rickayzen [Mol. Phys. 95, 393 (1998)] is extended to study the surface anchoring. The excess free energy is calculated as a functional expansion of density around a reference homogeneous fluid. The pair direct correlation function (DCF) of a homogeneous HGO fluid is approximated, based on the Percus-Yevick DCF for hard spheres; the anisotropy is introduced by means of the closest approach parameter. The average number density and orientational order parameter profiles of a HGO fluid confined in between planar walls are obtained using a hard needle-wall potential to represent the particle-wall interactions. For short and long needle lengths, the homeotropic and planar anchoring are observed, respectively. For the bulk isotropic phase the calculated density and order parameter profiles are in agreement with the Monte Carlo simulation of Barmes and Cleaver [Phys. Rev. E 69, 61705 (2004)]. However, for the bulk nematic phase the theory gives the correct density profile between the walls. The correct order parameters are obtained close to the walls whereas for the region in the middle of the walls, the agreement is less satisfactory.

Density profile and order parameter of a hard ellipsoidal fluid confined to a slit

The density profile and order parameter of a fluid of hard axially symmetric ellipsoidsconfined in between two parallel hard walls is obtained by using the density functionaltheory. The required input direct correlation function of the homogeneous fluid iscalculated by the variational method introduced by Marko (1989 Phys. Rev. 39 2050) andthe modified closest approach method proposed by Rickayzen (1998 Mol. Phys. 95 393).Here the restricted orientation model, ROM, is extended to study a fluid comprisingmolecules which can be aligned in more than six directions, making it more representativeof a normal fluid. The density profiles, the average number density and order parameterare obtained for different values of density and elongations. The results are inagreement with the previous theory and available Monte Carlo simulation results.

DENSITY PROFILES OF A HARD GAUSSIAN OVERLAP FLUID BETWEEN HARD WALLS

The density profiles of a hard Gaussian overlap (HGO) fluid confined in between hard walls and in contact with a hard wall are studied using the density functional theory. The hyper-netted chain (HNC) approximation is used to find the coupled integral equations for the density profiles. The restricted orientation model (ROM) is used. The required homogeneous direct correlation function (DCF) is obtained by solving Ornstein–Zernike (OZ) integral equation numerically, using the Precus–Yevick (PY) approximation and the procedure mentioned by Letz and Latz [Phys. Rev.E60, 5865 (1999)]. We also obtained the DCF of hard ellipsoidal (HE) fluid by using the modified closest approach introduced by Rickayzen [Mol. Phys.68, 903 (1989)]. For both HGO and HE, we calculate the density profiles of molecules parallel and perpendicular to the walls and we compare the results. The calculations are performed for various values of packing fractions of the fluid and various molecular elongations. For moderate elongations, k≤3, the results for HGO and HE are almost the same, especially for the density profile of the molecules parallel to the walls but for k=5 there are some discrepancies between the results, in particular for the density profiles of the molecules perpendicular to the walls.

Dimensional Crossover of the Fundamental-Measure Functional for Parallel Hard Cubes

We present a regularization of the recently proposed fundamental-measure functional for a mixture of parallel hard cubes. The regularized functional is shown to have correct dimensional crossovers to any smaller dimension, thus allowing its use to study highly inhomogeneous phases (such as the solid phase). Furthermore, it is shown how the functional of the slightly more-general model of parallel hard parallelepipeds can be obtained using the zero-dimensional functional as a generating functional. Extensions to the multicomponent system, a restricted-orientation model, and mixtures or prallel hard cylinders are also given.

A model for homogeneous and inhomogeneous hard molecular fluids: ellipsoidal fluids

The thermodynamic and structural properties of a fluid of hard axially symmetric ellipsoids are studied on the basis of two approximations, namely that (a) the orientations of the molecules can be restricted to a discrete number of directions and (b) the direct correlation function can be written as a superposition of three geometrical functions with weights which depend on the density of the fluid. The variational principle of Anderson and Chandler (1972) is used to determine the weights. The equation of state of the fluid is derived for a range of values of the ratio of axes (elongation) from 0.35 to 5.0. For comparison new simulations have been performed on the same systems. Except at the greatest elongation, the theoretical pressure agrees very well with the simulated pressure of the restricted orientation model as well as with that of a fluid of freely rotating ellipsoids. When the elongation is 5.0, the pressure of the fluid of freely rotating ellipsoids lies somewhat above the pressure of the restricted orientation model, although for this model the theory and simulation are still in accord. Thus far the theory has not revealed a transition to a nematic state although there is some evidence from simulation that when the elongation is 5.0, the isotropic state of the restricted orientation model is not stable. The direct correlation function of the homogeneous phase is used in conjunction with a density functional in order to investigate the density profile of the fluid confined to a slit. Except in the neighbourhood of a wall there is very good agreement between the theory and simulation. The discrepancy near the wall is to be expected from the simple form of functional used. Overall it appears that the approximation developed is adequate for describing the thermodynamic properties of the homogeneous and inhomogeneous isotropic state, but that it needs some improvement for studying the properties of the nematic state.

Phase transitions in molecular fluids

The structure of a confined fluid of hard circular cylinders confined to a slit is studied using a simple theory in the context of a restricted orientation model; previous work is extended to higher densities. It is shown that, as the density increases, the fluid within the central region of the slit undergoes phase transitions which depend on the ratio, l/d, of the length to the diameter of the cylinders, and these transitions are interpreted in terms of the bulk behaviour of the fluid. When l/d is 0·25, the bulk fluid first undergoes a first-order isotropic-nematic transition as the density increases and then makes a transition to a spatially ordered phase. On the other hand when l/d is 0·8, the fluid undergoes a transition to a spatially ordered phase and does not pass through a nematic phase at all. These results are qualitatively in agreement with results obtained by Frenkel and Mulder from the simulation of hard ellipsoids. The limitations of the model are discussed and suggestions made for modifications to improve upon the quantitative predictions of the model at high densities.

A density functional analysis of the restricted orientation model of liquid crystals and its implications for theories of orientational ordering

We compare the predictions of various density functional approximation schemes as applied to the restricted-orientation, hard, rectangular-parallelepiped model of liquid crystals. These calculations illustrate some features of density functional calculations which may aid in understanding other approximate, mean-field theories of orientational ordering phase transitions and their interfaces. The isotropic–nematic coexistence curve of this model is apparently well described by the so-called third-order y expansion, and the analytical expression for the nonideal free energy derived from the simple y expansion thus provides a benchmark against which to compare the density functional approximation schemes. Using standard relations applicable to inhomogeneous fluids, we show how expansions about the bulk isotropic phase compare more favorably with the ‘‘exact’’ y-expansion results when truncated at third order than do second-order truncations. Thus this model behaves somewhat differently than other hard-particle models of fluids, notably the hard-sphere one. We also examine the possibility of expanding the free energy about the ordered phase to obtain the properties of a disordered phase. An expansion about local values of the density leads to the widely used smoothed-density approximation and a hierarchy of systematic extensions. The latter appear more stable than conventional Taylor expansions about bulk disordered phases.

The structure of inhomogeneous molecular fluids: a restricted orientation model

It is suggested that the structural properties of an inhomogeneous fluid of very anisotropic molecules can be studied through a theoretical model in which the orientations of the molecules are restricted to certain discrete values. The general basis of the model is explained and then for illustrative purposes, it is applied to a fluid of cylindrical molecule confined between parallel planar hard structureless walls. Density functional theory and the analogue of the PY equation are used to obtain equations for the density profiles of the fluid which depend on the direct correlation function of the bulk fluid. As this function is unknown, it is approximated by a linear combination of geometrical functions. The equations are solved and the results presented for various values of the packing fraction of the fluid, the separation of the walls and the length to diameter ratio (l/d) of the molecules. A comparison with a recent Monte Carlo simulation of the fluid indicates that the approximations provide as good ...