Hard Spheroids Research Articles

On Using the BMCSL Equation of State to Renormalize the Onsager Theory Approach to Modeling Hard Prolate Spheroidal Liquid Crystal Mixtures

Modifications to the traditional Onsager theory for modeling isotropic–nematic phase transitions in hard prolate spheroidal systems are presented. Pure component systems are used to identify the need to update the Lee–Parsons resummation term. The Lee–Parsons resummation term uses the Carnahan–Starling equation of state to approximate higher-order virial coefficients beyond the second virial coefficient employed in Onsager’s original theoretical approach. As more exact ways of calculating the excluded volume of two hard prolate spheroids of a given orientation are used, the division of the excluded volume by eight, which is an empirical correction used in the original Lee–Parsons resummation term, must be replaced by six to yield a better match between the theoretical and simulation results. These modifications are also extended to binary mixtures of hard prolate spheroids using the Boublík–Mansoori–Carnahan–Starling–Leland (BMCSL) equation of state.

A novel approach of high-voltage low-current electric energy input to synthesise cost-effective ultra-strong ductile material

ABSTRACT Adopting a unique electrical circuit design, here we treat a significant low-cost engineering material (eutectoid steel not containing costly alloying elements) with a high-voltage (100 kV) low-current (150 mA) energy input (energy level exceeding cohesive energy). A distinctive structural evolution is ascertained with treatment duration of only 5 min as an outcome of lamellar fragmentation and matrix supersaturation. This envisages an origin of dispersed nano-sized hard cementite spheroids embedded in nano-thick martensite crystals of stratified-tile-morphology along with distributed α-ferrite regions containing sub-microscopic cementite particles of various shapes. Apart from the conjoint effect of nano-scale dispersion strengthening and martensitic strengthening overhauling the effect of conventional lamellar strengthening on a gross scale; high dislocation density and systematically arranged dislocations of similar sign at incoherent cementite particle-matrix interface provides a unique combination of ultra-high strength (UTS ∼ 1.5 GPa), significantly high specific strength (188 MPa/g cm−3) and large ductility (%Elongation = 20). Therefore, in terms of the adopted synthesis route, structural evolution and mechanical property achieved, a new dimension is hereby added to the next-generation material development so as to meet the ever increasing demand for low-cost structural application. In turn, we elucidate a fundamental conceptualization for the first time which exemplifies disproportionate atomic migration at highly incoherent nano-sized cementite particle-martensite matrix interface in steel under high-voltage low-current energy input, resulting in accumulation of dislocations of similar sign so as to significantly enhance strength along with retention of substantial ductility.

Diffraction of a Gaussian Beam by a Strongly Elongated Spheroid

A high-frequency diffraction problem is considered for a Gaussian beam incident parallel to the axis of a strongly elongated spheroid. The parabolic equation method in spheroidal coordinates is used to construct the leading order term of the field asymptotics in the boundary layer near the surface in the form of an integral containing Whittaker functions. The field amplitudes on the surface of a perfectly hard spheroid are computed. High-frequency diffraction effects are discussed.

Acoustic scattering on spheroidal shapes near boundaries

A new expression for the Lame product of prolate spheroidal wave functions is presented in terms of a distribution of multipoles along the axis of the spheroid between its foci (generalizing a corresponding theorem for spheroidal harmonics). Such an “ultimate” singularity system can be effectively used for solving various linear boundary-value problems governed by the Helmholtz equation involving prolate spheroidal bodies near planar or other boundaries. The general methodology is formally demonstrated for the axisymmetric acoustic scattering problem of a rigid (hard) spheroid placed near a hard/soft wall or inside a cylindrical duct under an axial incidence of a plane acoustic wave.

Colloidal particles as elastic triads in nematic liquid crystals

ABSTRACTIn this short review we summarise already published results to manifest very important role of high order elastic terms in the formation of colloidal structures in nematic liquid crystals (NLC). We reveal that every colloidal particle in nematics can be effectively represented as a triad of nonzero elastic moments. Usually colloidal particles in NLCs are treated with their elastic dipole and/or quadrupole moments only. But we demonstrate that octupole, hexadecapole and even 64-pole moments play an important role as well and determine parameters of different 1D, 2D and even 3D colloidal crystals in NLCs. In general the triad of the first three nonzero elastic moments can describe almost all colloidal structures observed so far. Dipole particles should be considered as hard spheroids with a triad of the dipole, quadrupole and octupole moments. Quadrupole particles should be treated as hard spheres with a triad of quadrupole, hexadecapole and 64-pole elastic momentsPACS numbers: 61.30.Dk, 82.70.Dd, 64.70.M−

Structural Transition in a Fluid of Spheroids: A Low-Density Vestige of Jamming.

A thermodynamically equilibrated fluid of hard spheroids is a simple model of liquid matter. In this model, the coupling between the rotational degrees of freedom of the constituent particles and their translations may be switched off by a continuous deformation of a spheroid of aspect ratio t into a sphere (t=1). We demonstrate, by experiments, theory, and computer simulations, that dramatic nonanalytic changes in structure and thermodynamics of the fluids take place, as the coupling between rotations and translations is made to vanish. This nonanalyticity, reminiscent of a second-order liquid-liquid phase transition, is not a trivial consequence of the shape of an individual particle. Rather, free volume considerations relate the observed transition to a similar nonanalyticity at t=1 in structural properties of jammed granular ellipsoids. This observation suggests a deep connection to exist between the physics of jamming and the thermodynamics of simple fluids.

Phase behavior of shape-changing spheroids.

We introduce a simple model for a biaxial nematic liquid crystal. This consists of hard spheroids that can switch shape between prolate (rodlike) and oblate (platelike) subject to an energy penalty Δε. The spheroids are approximated as hard Gaussian overlap particles and are treated at the level of Onsager's second-virial description. We use both bifurcation analysis and a numerical minimization of the free energy to show that, for additive particle shapes, (i) there is no stable biaxial phase even for Δε=0 (although there is a metastable biaxial phase in the same density range as the stable uniaxial phase) and (ii) the isotropic-to-nematic transition is into either one of two degenerate uniaxial phases, rod rich or plate rich. We confirm that even a small amount of shape nonadditivity may stabilize the biaxial nematic phase.

High-frequency acoustic scattering from prolate spheroids with high aspect ratio.

The problems of high-frequency scattering by prolate soft and hard spheroids with high aspect ratio are studied. The asymptotics of the diffracted field and the far field amplitude are derived under the assumption that the spheroid is strongly elongated; that is, the ratio of its length measured in wavelengths to the square of its transverse wave size is on the order of unity. This ratio is presented as a parameter in all the asymptotic formulas, namely, in the asymptotics of the forward diffracted wave, the wave that is formed when forward wave "reflects" from the spheroid end, in the asymptotics of the far field amplitude, and effective cross-section. If this ratio tends to zero, the asymptotic formulas turn into classical. On the basis of the derived asymptotic formulas, the effects of the degree of elongation and the angle of incidence on the diffracted and scattered fields are analyzed. Comparison with some test examples allows one to expect good approximating properties of the presented asymptotic formulas.

Isotropic and nematic phases of the hard spheroid fluid: A virial approach

We use a high level virial expansion to investigate the properties of the isotropic and nematic phases of the hard spheroid fluid. We use the Monte Carlo techniques described previously to calculate the virial coefficients up to seventh order and we represent the dependence of these coefficients on particle orientations via a spherical harmonic expansion. The expansion coefficients are determined using Lebedev quadrature which carries out the angular integration required exactly. For fairly spherical spheroids (1/3<a/b<3, where a and b are the semiaxes of the spheroid) the virial expansion appears to be convergent in both the isotropic and nematic phases up to high density. For more aspherical particles, the predictions varied in a more erratic fashion upon increasing the number of virials. The use of a finite angular basis set to represent the orientationally dependent virial coefficients could be a contributory cause to this.

High-level virial theory of hard spheroids

We present a method for calculating high-order virial expansions of the isotropic-nematic phase transition which we apply here to hard spheroids. Studying a range of aspect ratios, for both oblate and prolate particles, we obtain equations of state, coexistence densities, and nematic order parameters, using expansions truncated at up to eighth virial level. For particles of large aspect ratios our results show rapid convergence, with truncation at sixth order sufficient to give excellent agreement with simulation data. For more spherical particles the convergence is less rapid, with results for up to eighth-order theory approaching but still not reaching simulation data. Our results indicate that high-order viral expansions are better suited to predicting equations of state than coexistence densities. We also test the validity of using the Onsager trial function to approximate the orientational distribution function, finding only small errors when making this approximation.

Calculation of direct correlation function for hard particles using a virial expansion

We have calculated the direct correlation function, c(1, 2), via a high-order virial expansion, for systems of hard spheres and spheroids, in both the isotropic and nematic phases. For hard spheres, we find that truncation at sixth order in density gives good agreement with simulation data. Close to freezing, the virial series still appears to converge to the simulation results, but there are significant discrepancies, particularly at very small separations. In the non-overlap region, the virial theory begins to capture the features found from simulation. We also calculate the pair distribution function, g(1, 2), from our estimates of c(1, 2), and find good agreement with simulation data at all densities up to freezing. For hard, prolate spheroids of aspect ratio 3 : 1, we calculate c(1, 2) and g(1, 2) in the isotropic phase, again finding good agreement with simulation data at moderate densities. Finally, we present the result of our calculations on c(1, 2) for 3 : 1 spheroids in the nematic phase.

Structure and stability of isotropic states of hard platelet fluids

We study the thermodynamics and the pair structure of hard, infinitely thin, circular platelets in the isotropic phase. Monte Carlo simulation results indicate a rich spatial structure of the spherical expansion components of the direct correlation function, including nonmonotonical variation of some of the components with density. Integral equation theory is shown to reproduce the main features observed in simulations. The hypernetted chain closure, as well as its extended versions that include the bridge function up to second and third order in density, perform better than both the Percus-Yevick closure and Verlet bridge function approximation. Using a recent fundamental measure density functional theory, an analytic expression for the direct correlation function is obtained as the sum of the Mayer bond and a term proportional to the density and the intersection length of two platelets. This is shown to give a reasonable estimate of the structure found in simulations, but to fail to capture the nonmonotonic variation with density. We also carry out a density functional stability analysis of the isotropic phase with respect to nematic ordering and show that the limiting density is consistent with that where the Kerr coefficient vanishes. As a reference system, we compare to simulation results for hard oblate spheroids with small, but nonzero elongations, demonstrating that the case of vanishingly thin platelets is approached smoothly.

Structure of molecular liquids: Closure relations for hard spheroids

We present the results of Monte Carlo simulations of hard spheroids of revolution of different elongations. Both prolate and oblate shapes are examined. A systematic study of the bridge function b(1,2), and direct comparison with the indirect correlation function gamma(1,2)=h(1,2)-c(1,2) at densities spanning the isotropic fluid range, allow us to evaluate the accuracy of various proposed closure relations for integral equations.

Structure of molecular liquids: Cavity and bridge functions of the hard spheroid fluid

We present methodologies for calculating the direct correlation function c(1,2), the cavity function y(1,2), and the bridge function b(1,2), for molecular liquids, from Monte Carlo simulations. As an example we present results for the isotropic hard spheroid fluid with elongation e = 3. The simulation data are compared with the results from integral equation theory. In particular, we solve the Percus-Yevick and hypernetted chain equations. In addition, we calculate the first two terms in the virial expansion of the bridge function and incorporate this into the closure. At low densities, the bridge functions calculated by theory and from simulation are in good agreement, lending support to the correctness of our numerical procedures. At higher densities, the hypernetted chain results are brought into closer agreement with simulation by incorporating the approximate bridge function, but significant discrepancies remain.

The equation of state of isotropic fluids of hard convex bodies from a high-level virial expansion

We have calculated virial coefficients up to seventh order for the isotropic phases of a variety of fluids composed of hard aspherical particles. The models studied were hard spheroids, hard spherocylinders, and truncated hard spheres, and results are obtained for a variety of length-to-width ratios. We compare the predicted virial equations of state with those determined by simulation. We also use our data to calculate the coefficients of the y expansion [B. Barboy and W. M. Gelbart, J. Chem. Phys. 71, 3053 (1979)] and to study its convergence properties. Finally, we use our data to estimate the radius of convergence of the virial series for these aspherical particles. For fairly spherical particles, we estimate the radius of convergence to be similar to that of the density of closest packing. For more anisotropic particles, however, the radius of convergence decreases with increased anisotropy and is considerably less than the close-packed density.

Transport tensors in perfectly aligned low-density fluids: self-diffusion and thermal conductivity.

The modified Taxman equation for the kinetic theory of low-density fluids composed of rigid aspherical molecules possessing internal degrees of freedom is generalized to obtain the transport tensors in a fluid of aligned molecules. The theory takes care of the shape of the particles exactly but the solution has been obtained only for the case of perfectly aligned hard spheroids within the framework of the first Sonine polynomial approximation. The expressions for the thermal-conductivity components have been obtained for the first time whereas the self-diffusion components obtained here turn out to be exactly the same as those derived by Kumar and Masters [Mol. Phys. 81, 491 (1994)] through the solution of the Lorentz-Boltzmann equation. All our expressions yield correct results in the hard-sphere limit.

Third virial coefficient of hard spheroids

The third virial coefficient B 3 of a fluid of spheroids is calculated by direct integration using the method of 2-point measures of convex body geometry. Calculations are performed for aspect ratios up to 10 for both prolate and oblate spheroids. The infinite set of 2-point measures has five indices. The three numerical ones, L, Lι and m, are standard quantities in the theory of the rotation group. These new calculations use truncation at the L = L = 2 level. The results show dramatic improvement over previous results at the L = Lι = 1 level and are in excellent agreement with existing Monte Carlo calculations of B 3.

Analytical Expressions for Calculating the Depletion Interaction Produced by Charged Spheres and Spheroids

Analytical expressions for predicting the depletion interaction in charged systems where long-range electrostatic effects are important are presented. The approach used was to model the nonadsorbing materials as hard particles but with an effective size representing the hard-core size and the effective size of the charged double layer. This effective size was then substituted into analytical expressions for predicting the depletion interaction between two hard spherical particles in solutions of nonadsorbing hard spheres or spheroids. We also found the effective thickness of the charged double layer providing the most accurate approximation to be the surface-to-surface separation distance between the particle and nonadsorbing species at which the electrostatic repulsion equals 0.5 kT.

Erratum: The application of some spherical reference theories to the hard spheroid fluid [J. Chem. Phys. 99, 6205 (1993)

Erratum: The application of some spherical reference theories to the hard spheroid fluid [J. Chem. Phys. <b>99</b>, 6205 (1993)

Transport coefficients of a simple and dilute gas of hard spheroids having C∞ ν symmetry

An uncorrelated collision (modified Enskog) kinetic theory has been applied to the derivations of formulae for the transport coefficients of a classical, simple and dilute gas of hard spheroidal molecules which have C ∞ ν symmetry. The only quantities appearing in these formulae which have not been evaluated explicitly are integrals over the orientational co-ordinates of two interacting molecules. The self-diffusion coefficient and shear viscosity formulae derived on the basis of this theory agree with those derived recently [R.G. Cole and G.T. Evans, Ann. Rev. Phys. Chem. 37 (1986) 105; G.T. Evans, Molec. Phys. 74 (1991) 775] by a Mori-generalised Langevin method. The volume viscosity and thermal conductivity expressions are new. The conductivity formula reduces to the Evans result if the effects of translational and rotational degrees of freedom are combined instead of being presented explicitly as in this work, and to the modified Eucken formula if suitable approximations are made. In the limit our formulae reduce to the corresponding transport coefficient expressions for a dilute gas of hard sphere (HS) molecules.