Terms Of Harmonic Coefficients Research Articles

A study of isotropic–nematic transition of quadrupolar Gay–Berne fluid using density-functional theory approach

The effects of quadrupole moments on the isotropic–nematic (IN) phase transitions are studied using the density-functional theory (DFT) for a Gay–Berne (GB) fluid for a range of length-to-breadth parameters in the reduced temperature range . The pair-correlation functions of the isotropic phase, which enter into the DFT as input parameters are found by solving the Percus–Yevick integral equation theory. The method used involves 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. All the terms of harmonic coefficients which involve l indices up to less than or equal to 6 are considered. The numerical accuracy of the results depends on the number of spherical harmonic coefficients considered for each orientation-dependent function. As the length-to-breadth ratio of quadrupolar GB molecules is increased, the IN transition is seen to move to lower density (and pressure) at a given temperature. It has been observed that the DFT is good to study the IN transitions in such fluids. The theoretical results have also been compared with the computer simulation results wherever they are available.

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.

The Percus–Yevick approximation for quadrupolar molecular fluids

The Percus–Yevick integral equation theory has been solved to study the equilibriumand structural properties of quadrupolar Gay–Berne fluids. The method usedinvolves an expansion of angle-dependent functions appearing in the integralequations in terms of spherical harmonics and the harmonic coefficients are obtainedby an iterative algorithm. All the terms of harmonic coefficients which involvel indices up to less than or equal to 6 have been considered. Molecules withlength-to-breadth ratios 3.0 and 4.0 have been considered and results are reported fordifferent densities, temperatures, and quadrupole moments. The values of paircorrelation functions have been compared with the available computer simulationresults.

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.