Glenn T. Evans

The Onsager theory of the isotropic–nematic liquid crystal transition: Incorporation of the higher virial coefficients

In the Onsager theory for the phase transition from the isotropic fluid to the nematic liquid crystal phase, the Helmholtz free energy of a fluid of hard convex bodies (HCBs) is expressed as the sum of an entropy of a mixing‐like term and an energy‐like term (from the interaction of the HCBs). Whereas the Onsager theory expresses the interaction term in a virial expansion and determines the consequences of B2 alone, here we extend that treatment to incorporate B3 (with its attendant dependence on the mutual orientation of three HCBs). For HCBs (and specifically for D∞h ellipsoids) with large aspect ratios (5:1 or greater), the incorporation of B2 and B3 suffices to predict the variation of the order parameter 〈P2[cos(θ)]〉 with density in accord with the Monte Carlo (MC) results of Allen and Wilson. As the aspect ratio decreases (from 5:1) to more spherical molecules (say 3:1), virial coefficients of higher order than B3 contribute to the interaction term and their effect is represented in part by the y‐ex...

The isotropic to nematic liquid crystal transition for hard ellipsoids: an Onsager-like theory and computer simulations

The phase transition from an isotropic to a nematic phase for a classical fluid of hard ellipsoids has been studied using a version of a theory originally due to Onsager and by computer simulation. In the proposed form of the Onsager theory for the Helmholtz free energy, both the second and the third virial coefficients are treated exactly, but the fourth and higher virials are resummed in a manner consistent with the Carnahan-Starling equation of state for hard spheres. This same approach is applied to the calculation of the direct correlation function. A comparison of order parameters, transition densities and pressures calculated by simulation and by the resummed Onsager theory, suggests the following. (i) For 10 : 1 prolate hard ellipsoids, resumming the fourth and higher virial coefficients (rather than simply neglecting them) degrades the agreement by overestimating the importance of the higher virials. (ii) For 5 : 1 prolate and 1 : 5 oblate hard ellipsoids, the resummation yields a considerable im...

Aggregation of water molecules: Atmospheric implications

The equilibrium constants for water oligomers ranging from dimers to cyclic hexamers are determined using Wertheim’s theory of associating systems. In the present model for water, the pair potential has a spherical hard core, and tetrahedral hydrogen bonds which are represented by an energy parameter and an interaction volume. On the basis of the present theory, one predicts that in earth’s troposphere, water dimers and perhaps trimers may contribute to the absorption of solar radiation, but concentrations of higher oligomers are too low to influence the optical properties of the earth’s atmosphere.

Structure of the hard ellipsoid fluid

Molecular‐dynamics calculations are reported on fluids of hard ellipsoids over a range of densities and for several ellipsoidal aspect ratios. The pair correlation functions obtained from the simulations are expressed as functions of the minimum surface‐to‐surface separation, s, measured along the surface normal, s, and angles measured relative to the surface normal. Both isotropic and orientational correlations exhibit simpler behavior in the surface‐to‐surface than in the more customary center‐to‐center coordinate representation. For the hard‐ellipsoid fluid, the isotropic part of the pair correlation function, giso(s), behaves much like that of a hard‐sphere fluid. The surface‐to‐surface coordinates are well suited for studying pressure and collision rates because these properties depend upon surface contact distributions. They are also useful for studying the orientational order parameter, g2, because they enable one to readily identify a long‐range part and geometrical excluded volume contribution.

The orientational‐correlation time intercept in liquids

We propose a theory for the zero viscosity (η) or infinite temperature (T) intercept (τ0) in the much used quasihydrodynamic expression τ2=τ0+(const)(η/T) for the second rank orientational correlation time (τ2). Whereas we associate the (η/T) term with the integral over the torque–torque correlation function, we attribute the origin of τ0 to the integral over the cross‐correlation function between torques and kinetic or convective terms. We are unable to evaluate the time dependence of this cross‐correlation function exactly, but if a Gaussian form is assumed, a negative value is obtained for τ0. Experimentally, τ0 is sometimes positive and sometimes negative, which suggests that a monotonic form for the time dependence of the cross‐correlation function might be inadequate. Because we have not been able to evaluate this cross‐correlation function our explanation must still be considered a conjecture.

The Onsager theory of the isotropic-nematic liquid-crystal transition : biaxial particles in uniaxial phases

Dense fluids of biaxial hard convex bodies exhibit a phase transition from an isotropic to a nematic (I–N) liquid–crystalline state. In the Onsager theory, the I–N transition is attributed to the tendency of pairs of molecules to minimize their excluded volume (or second virial coefficient, B2 ). In the present work, the mutual orientation dependence of B2 is calculated for hard biaxial ellipsoids and is expressed in a Wigner function expansion. Orientational distribution functions are determined for the Onsager model and for the Lee model, which incorporates some density correlations beyond those in the Onsager theory. In the uniaxial phase of biaxial molecules, the two nonzero order parameters calculated on the basis of hard‐body models are in approximate accord with the findings of Maier–Saupe based theories. Introduction of biaxiality into a body has a pronounced effect: both the 〈P2〉 order parameter and the first orderness of the I–N transition are greatly reduced from that of comparable uniaxial bodies.

A simple kinetic theory model of reactive collisions of rigid nonspherical molecules

The classical kinetic theory for dilute gases of rigid convex molecules, as developed by Hoffman (1969), is now applied to the calculation on the bimolecular rate coefficient, the energy‐dependent reaction cross section σR(E), and the orientation‐dependent differential cross section, for general diatom–diatom reactions. Incorporated in the theory are the angular momentum and the convex shape of the colliding molecules, as well as the dependence of the barrier height upon mutual orientation. Several simple collision systems are considered, including that of two reactive ellipsoidal molecules. For atom–diatom scattering, it is found that, in the post‐threshold region (E≳E0), σR(E) has quadratic and higher‐order terms in E−E0 but no linear term. Like σR, the differential cross section depends sensitively upon the shape of the colliding molecules, as well as upon the angle‐dependent threshold energy. For the near‐spherical case, one obtains simple formulas that display explicitly the dependence of the cross s...

A van der Waals theory of nematic liquid crystals: a ‘convex peg in a round hole’ potential

The convex peg potential has been incorporated into a van der Waals theory of the (uniaxial) nematic to isotropic liquid crystal transition. In the convex peg model, molecules are envisioned to have a hard (here biaxial) core embedded in a spherically symmetric square well. Anisotropies in the potential are derived from both its repulsive and attractive regions. In accord with generalized van der Waals theories, the repulsive interactions are treated to all orders (albeit approximately using a resummation technique) and the attractive interactions are incorporated to lowest order. Global phase diagrams are determined and these predict the formation of one uniaxial nematic and two isotropic phases (vapour and liquid). From the phase diagram, one observes the coexistence of the nematic-vapour phases, the nematic-isotropic liquid phases, the vapour-liquid phases as well as a nematic-vapour-isotropic liquid triple point. By virtue of the effective anisotropic attractive forces, the first order character of th...

A kinetic theory calculation of the orientational correlation time of a rotorlike molecule in a dense fluid of spheres

The collective and single‐particle orientational correlation times are calculated using Boltzmann–Enskog kinetic theory for N rotorlike molecules in a bath of spheres. By assuming that the rotors and bath are hard convex bodies, the collision integrals can be reduced exactly to one‐ and two‐dimensional integrals, which are then readily amenable to numerical quadrature. The calculated correlation times depend on density and temperature in the same way as do the rough hard sphere results. However, the particle shape anisotropy plays the role of the roughness parameters. Comparison of the collective and single‐particle orientational correlation times with experiment (including molecular dynamics) indicates that the kinetic theory consistently neglects part of the frictional drag. In all cases studied, the calculated times were a factor of 2 to 4 below the experimental values.

Brownian dynamics simulation of alkane chain reorientation: A comparison of models

Orientational time correlation functions and correlation times are determined using a Brownian dynamics and rigid body formalism and are compared with the results obtained from our previous Mori approach. It is found that the Mori method equipped with a minimum set of primary variables approximately represents the freely rotating chain. The correlation functions obtained from Brownian dynamics were found to display a rapid initial decrease, followed by a slow nearly exponential fall‐off. First and second rank correlation times for pentane, hexane, nonane, and undecane are reported.