Angular Pair Correlation Function Research Articles

Characterisation of bubble clusters in simulations of upward turbulent channel flow

AbstractThe paper reports on Direct Numerical Simulations of bubble‐laden flows performed to investigate the interaction of bubbles with turbulence in an upward flow between two parallel vertical plates. Two simulations are presented with uniform bubble diameter and a third one with bi‐disperse bubbles. Three different approaches are used to characterize bubble clustering in the investigated configuration. Horizontal pairing and tendency to form cluster has been found for all reported cases, slightly less pronounced in the bi‐disperse swarm. Among other results, a 60°‐alignment has been found by means of the angular pair correlation function for the mixed pairs in the bi‐disperse swarm. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the pair correlation function in a bubble swarm

Abstract The paper presents results on the dynamics of bubbles in a mono- and in a bi-disperse bubble swarm. The radial pair correlation function provides insight into the characteristic bubble clustering length. The angular pair correlation function shows that for small distances the most probable alignment of a bubble pair is horizontal.

Structure of hard ellipsoids: The Gaussian overlap model

The equilibrium properties of isotropic fluids composed of hard ellipsoids of revolution represented by a Gaussian overlap model are studied using the integral equation theories. The Percus–Yevick and the hypernetted-chain integral equations have been devised for calculating the angular pair-correlation function for fluids of hard ellipsoids of a revolution. This model is computationally simple and shears some similarities with the widely used hard ellipsoids. The methods used involve an expansion of angle-dependent functions appearing in the integral equations in terms of spherical harmonics and the harmonic coefficients are obtained by an iterative algorithm. We have considered all the terms of harmonic coefficients which involve l indices up to less than or equal to 6. The numerical accuracy of the results depends on the number of spherical harmonic coefficients considered for each orientation-dependent functions. Ellipsoids with length-to-width ratios of 2.0 and 3.0 are considered and the harmonic coefficients are compared with the molecular-dynamics results for prolate ellipsoids. We find that both the Percus–Yevick and hypernetted-chain theories are in reasonable agreement with the computer simulation results. The pressure obtained from the virial and compressibility routes of these fluids have also been compared with the computer simulation results which shows that these integral equations are thermodynamically inconsistent.

In search of temporal power laws in the orientational relaxation near isotropic-nematic phase transition in model nematogens.

Recent Kerr relaxation experiments by Gottke et al. have revealed the existence of a pronounced temporal power law decay in the orientational relaxation near the isotropic-nematic phase transition (INPT) of nematogens of rather small aspect ratio, kappa (kappa approximately 3-4). We have carried out very long (50 ns) molecular dynamics simulations of model (Gay-Berne) prolate ellipsoids with aspect ratio 3 in order to investigate the origin of this power law. The model chosen is known to undergo an isotropic to nematic phase transition for a range of density and temperature. The distance dependence of the calculated angular pair correlation function correctly shows the emergence of a long range correlation as the INPT is approached along the density axis. In the vicinity of INPT, the single particle second rank orientational time correlation function exhibits power law decay, (t(-alpha)) with exponent alpha approximately 2/3. More importantly, we find the sudden appearance of a pronounced power-law decay in the collective part of the second rank orientational time correlation function at short times when the density is very close to the transition density. The power law has an exponent close to unity, that is, the correlation function decays almost linearly with time. At long times, the decay is exponential-like, as predicted by Landau-de Gennes mean field theory. Since Kerr relaxation experiments measure the time derivative of the collective second rank orientational pair correlation function, the simulations recover the near independence of the signal on time observed in experiments. In order to capture the microscopic essence of the dynamics of pseudonematic domains inside the isotropic phase, we introduce and calculate a dynamic orientational pair correlation function (DOPCF) obtained from the coefficients in the expansion of the distinct part of orientational van Hove time correlation function in terms of spherical harmonics. The DOPCF exhibits power law relaxation when the pair separation length is below certain critical length. The orientational relaxation of a local director, defined in terms of the sum of unit vectors of all the ellipsoidal molecules, is also found to show slow power law relaxation over a long time scale. These results have been interpreted in terms of a newly developed mode coupling theory of orientational dynamics near the INPT. In the present case, the difference between the single particle and the collective orientational relaxation is huge which can be explained by the frequency dependence of the memory kernel, calculated from the mode coupling theory. The relationship of this power law with the one observed in a supercooled liquid near its glass transition temperature is explored.

A computer simulation of a potential derived from the gay-berne potential for lattice model

 (Received February 28, 2000; revised March 24, 2000) The lattice model of elongated molecules interacting via a potential derived from the Gay-Berne pair potential is proposed. We made a systematic study of the effect of varying the molecular elongation and intermolecular vector orientation dependence of the pair potential on the thermodynamic as well as the structural properties of liquid crystals. A Monte Carlo simulations of molecules placed at the site of a simple cubic lattice and interacting via the modified Gay-Berne potential with its nearest neighbours is performed. The internal energy, heat capacity, angular pair correlation function and scalar order parameter are obtained. The results are compared against predictions of molecular field theory, experimental results and that of other related simulations wherever possible. It is shown that for more elongated molecules the nematic-isotropic transition becomes stronger first order transition. For a given molecular elongation as the intermolecular vector orientation dependence becomes larger the nematic-isotropic transition becomes a stronger first order transition as measured by the rate of change of the order parameter and the divergence of the heat capacity. Scaling the potential well seems to have dramatic change on the effect of the potential well anisotropy on trends of nematic-isotropic transition temperature and divergence of the heat capacity. It is shown that the behaviour of many nematics can be described by proposed model with the elongation ratio of molecules and potential well anisotropy ranging from 3 to 5.Bull. Chem. Soc. Ethiop. 2000, 14(1), 69-90Bulletin of the Chemical Society of Ethiopia. ISSN: 1011-3924&nbsp

Angular distribution functions and specific local structures in liquid water

A method of structural analysis, based on restricted averages of the angular pair correlation function, is proposed and applied to liquid water. Minimum free energy paths for a molecule moving in an average local structure are obtained. Around a central molecule, accepting and donating neighbors form hydrogen bonds having the same energy minima but differing for the availability of low free energy states. Some bounded states have specific bridging functions and connect hydrogen-bond and transition states along negative free energy pathways, thus favoring local diffusion. Bridging and transition states play the role of structural defects and include bifurcated hydrogen-bond and interstitial water geometries. Our methods and results can be relevant to understand, from a structural point of view, the anomalous behavior of stressed and confined water.

Neutron diffraction and molecular dynamics investigations of the temperature dependence of the local ordering in liquid cyclopropane

The structure of the liquid cyclopropane and the evolution of the local ordering with temperature has been investigated along the liquid-vapour coexistence curve between the melting point (T m ≈ 146 K) and room temperature by neutron diffraction and molecular dynamics simulation. Experimentally, the intramolecular structure factor is well described using the molecular parameters close to those reported from spectroscopic calculations and is found to be almost temperature independent. In contrast, the intermolecular structure factor shows a strong variation in the translational and orientational ordering by cooling the liquid from room temperature to a temperature close to the melting point. Molecular dynamics simulations were performed using an atom-atom Lennard-Jones potential to model the intermolecular interaction between cyclopropane molecules. Excellent reproduction of thermodynamical and dynamical data reported for cyclopropane has allowed an assessment of the validity of the potential parameters used. Moreover, good agreement between the experimental and the simulated pair correlation function has been obtained, allowing the use of the simulated functions to discuss the evolution of the liquid structure with temperature. It is found that g CC(r) is the correlation function best able to describe the variation in the orientational order in the first shell of neighbours, from the relative evolution against temperature of the first two peaks observed at about 4·0 and 5·1 Å, respectively. A discussion of this evolution using angular pair correlation function and geometrical models based upon solid-phase data shows that, at low temperatures, the favoured configurations of cyclopropane molecules are those where, on average, their symmetry axes are perpendicular or parallel. When the temperature increases, configurations where the molecular axes are more randomly distributed tend to increase, thus leading to a more orientationally disordered liquid. It is emphasized that this trend is also consistent with the interpretation of light scattering experiments.

Comparisons of perturbation and integral equation theories for the angular pair correlation function in molecular fluids

We compare several recently proposed theories for the angular pair correlation function g( rω 1ω 2), including first- and second-order perturbation theory (the u-expansion)., a Padé approximant to this series, first-order f-expansion, the single superchain, generalized mean field, linearized hypernetted chain, and quadratic hypernetted chain approximations. Numerical results from these theories are compared with available computer simulation data for four model fluids whose intermolecular pair potential is of the form u 0+ u a, where u 0 is a hard-sphere or Lennard-Jones model, while u a is a dipole-dipole or quadrupole-quadrupole interaction; we refer to these model fluids as HS + μμ, HS+QQ, LJ+μμ, and LJ+QQ. Properties studied include the angular pair correlation function and its spherical harmonic components, the thermodynamic properties, and the angular correlation parameters G 1 and G 2 that are related to the dielectric and Kerr constants. The second-order perturbation theory is superior to the integral equation theories for the thermodynamic harmonics of g( rω 1ω 2) and for the thermodynamic properties themselves at moderate multipole strengths. For other harmonics and properties, the integral equation theories are better, with the quadratic hypermetted chain approximation being the best overall.

Perturbation theories for quadrupolar fluids

A new second-order perturbation expansion is derived for the angular pair correlation function of quadrupolar fluids, and calculations were made, to first order, for dense quadrupolar fluids with reduced quadrupole moments Q ∗ from 0.5 to 1.0. Both spherical and nonspherical references were used with the latter giving better results.

Spherical harmonic expansions of the angular pair correlation function in molecular fluids

Two spherical harmonic expansions of the angular pair correlation function, one with orientations referred to an intermolecular reference frame, the other with orientations referred to a space-fixed frame, have been studied by computer simulation. The convergence of the intermolecular frame expansion has also been investigated.

Second-order perturbation theory for the dielectric constant and kerr factor of fluids with dipoles and quadrupoles

Second-order perturbation theory is used to calculate the spherical harmonic coefficients of the angular pair correlation function and the dielectric constant and Kerr parameter of Stockmayer fluids with dipoles and quadrupoles. The results indicate that the dielectric constant and the Kerr parameter both decrease rapidly with increasing quadrupole moment for any given dipole moment. These results confirm the recent conclusions of Patey using integral equation methods. The effect of long-range forces is also investigated.

Generalized mean field theory of molecular liquids. II. Thermodynamic properties

The generalized mean field (GMF) theory for the angular pair correlation function g(rω1ω2) for a molecular fluid, derived in an earlier paper, is applied to the calculation of the fluid thermodynamic properties. Explicit expressions are derived for the contribution of the anisotropic intermolecular forces to the free energy, energy, and pressure of the fluid. Numerical results are presented for a fluid with Lennard-Jones plus quadrupole–quadrupole interaction, and are compared to computer simulation and to perturbation theory (Padé) results.

Second-order perturbation theory for the angular pair correlation function in molecular fluids

Second-order perturbation theory is used to calculate spherical harmonic coefficients of the angular pair correlation function g( rω 1ω 2) for a liquid in which the molecules interact with a pair potential that is the sum of Lennard-Jones and quadrupole-quadrupole parts. The theory is compared with both molecular dynamics results and with the predictions of the GMF ≡ LHNC, QHNC and first-order perturbation theories. Second-order perturbation theory gives excellent results for the harmonic coefficient g(224, r), but is poorer for g(222, r) and g(202, r).

Perturbation theory for molecular fluids: Third-order term in the Pople expansion

We derive the expression for the third-order perturbation term A3 in the Pople expansion of the Helmholtz free energy, for an arbitrary anisotropic intermolecular pair potential. The corresponding expression for the second-order perturbation term g2 in the expansion of the angular pair correlation function is also obtained. We then apply this expansion, and also a Padé approximation to the series, to the case in which the molecules interact with an anisotropic charge overlap model proposed by Pople. Comparison of the calculated internal energies with molecular dynamics results for this model show that the Padé gives good results in the range −0.2?δ?0.35, where δ is an anisotropic shape parameter (positive for prolate and negative for oblate spheroids). This range includes small polyatomic molecules (CO2, ethane, N2, etc.). We also apply the theory to binary mixtures in which one of the molecular species obeys a spherical Lennard-Jones potential while the other obeys spherical Lennard-Jones plus the Pople anisotropic overlap model. The anisotropic overlap forces have little effect on the phase diagram for such small polyatomics, in contrast to the large effect of electrostatic forces.

Perturbation theory of the pair correlation function in molecular fluids

We study the perturbation theory of the angular pair correlation function g(rω1ω2)in a molecular fluid. We consider an anisotropic pair potential of the form u = u0 + ua, where u0 is an isotropic 'reference' potential, and for simplicity in this paper we assume the perturbation potential ua to be 'multipole-like', i.e., to contain no l = 0 spherical harmonics. We expand g in powers of ua about g0, the radial distribution function appropriate to u0. This series is examined by expanding ha = h−h0 (where h = g−1) and its corresponding direct correlation function ca in spherical harmonic components. We consider approximate summations of the series in ua that automatically truncate the corresponding harmonic series, so that the Ornstein–Zernike (OZ) equation relating ha and ca can be solved in closed form. We first expand ca = c1 + c2 + … where cn includes all terms in ca of order (ua)n. Taking ua to be a quadrupole–quadrupole interaction, we find that a 'mean field' (MF) approximation ca = c1 gives rise to only three nonvanishing harmonic components in ha, so that OZ is solved explicitly in Fourier space. The MF solution for multipoles of general order is given in an appendix. Graphical methods are then used to identify the class of all terms in the ua series that are restricted to the harmonic space defined by MF. A portion of this class can be summed by solving OZ with the closure ca = −βg0ua + h0(ha−ca), where β = (kT)−1, h0 = g0−1 This system is designated as generalized MF (GMF), and solved by numerical iteration. Numerical results from MF and GMF are presented for quadrupolar ua, taking u0 to be a Lennard-Jones potential. Symmetries imposed by the restricted harmonic space are foreign to the full g, yet harmonics within this space are sufficient for evaluation of many macroscopic properties. The results are therefore evaluated in harmonic form by comparison with the corresponding harmonic components of the 'correct' g as evaluated by Monte Carlo simulation.

Calculation of the dielectric and Kerr constant angular correlation parameters

The angular correlation parameters G L involved in the dielectric and Kerr constants are written as averages over the angular pair correlation function. A perturbation theory for these parameters is developed, based on the expansion of Gubbins and Gray for the angular pair correlation function. Numerical calculations are presented showing the influence of polar and quadrupolar forces on these parameters.

Theory of surface tension for molecular fluids

The Kirkwood-Buff statistical mechanical theory of surface tension γ for monatomic fluids is extended to molecular fluids. A rigorous expression for γ is derived in terms of the angular pair distribution function f(z 1 R 12θ1θ2) of an equilibrium fluid-fluid system (liquid-gas, liquid-liquid, or gas-gas). The Fowler approximation is applied for the liquid-gas case, and a simple expression for γ is derived in terms of the bulk liquid angular pair correlation function g(R 12θ1θ2). Thermodynamic perturbation theory for g(R 12θ1θ2) is also used to calculate γ theoretically. The theoretical results are compared with experimental values.

Theory of inelastic neutron scattering from molecular fluids

The time-dependent angular pair correlation function is discussed and its use in the analysis of inelastic neutron scattering experiments from polyatomic fluids is described, including both the coherent and incoherent spectra. The set of formal results given here permits a systematic interpretation of neutron inelastic scattering spectra on simple molecular liquids. Neutron spectra second moments are reviewed, and a new result for the fourth moment is given for the incoherent spectrum. Numerical results for the moments are obtained. The fourth moment depends on the mean squared torque and the mean squared force acting on a molecule in the fluid, and may provide a means for studying intermolecular forces in dense fluids. In addition, a method of calculating the correlation function for weak anisotropic forces is outlined.

Perturbation theory for equilibrium properties of molecular fluids

Perturbation theory for the angular pair correlation function g(r12 ω 1 ω 2), using a fluid with isotropic intermolecular forces as the reference system, is applied to the calculation of a variety of macroscopic properties. Comparisons with experiment are made for methane, oxygen and nitrogen (and carbon monoxide for infra-red and Raman band moments) in the dense fluid and liquid states. Theoretical expressions are given and calculations made for thermodynamic properties (isothermal compressibility, pressure, configurational energy, entropy and specific heat) both along and away from the vapour-liquid co-existence curve, for infra-red and Raman band moments, and for neutron scattering cross sections. Excellent agreement with experiment is obtained for all properties, except for the infra-red and Raman band moments; this latter comparison is inconclusive because of large experimental uncertainties. The anisotropic intermolecular forces are found to have very little effect on the liquid isothermal compressibility, in agreement with the first-order theory. Molecular anisotropy has a relatively small effect on the configurational energy and on the Helmholtz free energy, but the effect is large for pressure and specific heat. The pressure is more sensitive to short-range anisotropic forces than the other properties, whereas the specific heat is particularly sensitive to the long-range anisotropic forces. Mean squared torques (derived from infra-red and Raman band moments) are very sensitive to the strengths of the anisotropic forces, and are more sensitive to higher terms in the multipole series than are the other properties. The structure factors for oxygen and nitrogen are found to be little affected by the anisotropic forces.

Perturbation theory for the angular correlation function

A perturbation expression for the angular pair correlation function g (2)(r 12, ω1, ω2) is derived for systems interacting via non central potentials based on the method developed by Gubbins and Gray [1]. The method uses the ‘correct’ (in the sense of Rushbrooke [3] and Cook and Rowlinson [4]) angle-averaged potential as the reference system about which the perturbation is made. A preliminary comparison between the original Gubbins-Gray expression for g (2)(r 12, ω1, ω2) and the present expression is made for a system of two-dimensional point dipoles.