Perturbation calculation of the distribution functions of mixtures of hard spheres

The distribution functions for a mixture of additive hard spheres of diameter σ_ii are calculated by means of an expansion in powers of σ _ij ⁿ - dⁿ , using an unperturbed single-component hard-sphere fluid of diameter d. Although the expansion converges only when the hard spheres in the mixture are nearly equal in size, it is useful because it is the only practical scheme available for obtaining the distribution functions of multicomponent hard-sphere mixtures.     

Surface tension compressibility relation

The generalized van der Waals result for the surface tension of a simple liquid-vapour interface is shown to follow from the exact statistical mechanical expression with the approximation c(r, z ₁, z ₂) = c(r, n(z¯)). Using the statistical mechanical equation for the derivative of the density profile we derive the relation between the surface tension and the isothermal compressibility, to lowest order in T-T_c .     

Dielectric properties of a mixture of dipolar hard spheres

A theory for the dielectric constant, ε, of a fluid mixture of dipolar hard spheres is formulated by generalizing the methods developed by Ramshaw and Wertheim for the pure fluid case. The resulting expression for ε depends on the pair distribution functions, g _αβ(r ₁, θ₁, r ₂, θ₂) for a dipolar mixture. Due to the unavailability of exact representations for these dipolar pair distribution functions, the results of the mean spherical approximation are employed in the formalism developed. Numerical results are given for ε as calculated from the pair distribution functions for a spherical volume of macroscopic dimensions. The compositional dependence of the ε obtained in this way for a specific mixture is compared with the corresponding properties of the well established theories of Clausius-Mossotti-Debye and Onsager. In addition, the relative importance of the dipole moment and size of the hard sphere parameters in determining ε for a dipolar mixture (the correlative behaviour of which is described by the mean spherical approximation) is evaluated. It is found that the differences in hard core diameters can be largely ignored, in that ε for an ‘effective’ single component fluid can be given to within 2–5 per cent relative error (at worst) of the mean spherical approximation's result. Such an ‘effective pure fluid’ is described as having the same polarization content as the actual mixture being considered. Thereby, the properties of the effective fluid are determined by the quantity y = 4πβ(m ₁ ² ρ₁ + m ₂ ² ρ₂)/9 where m_i and ρ _i are the dipole moment and number density of component i in the binary mixture, with β = (kT)^-1.     

Moments of the Percus-Yevick hard-sphere correlation function

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ . We compare exact moments with shell approximationsM _n [h ^s ] corresponding to integration fromr=0 tos+1 fors=3–8, and with hybrid approximationsM _n [h ^s +h ^a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to . For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a ] is even better thanM _n [h ⁸]     

Observation of instabilities in a Paul trap with higher-order anharmonicities

Systematic measurements of the relative ion number stored in a Paul trap within the stability diagram given by the solution of the equation of motion reveal many lines, where only few or no ions can be confined. The observations can be explained by the presence of perturbations from higher-order components in the trapping potential, which is a quadrupole potential in the ideal case. The resonances follow the equation (n _r /2) _r + (n _r /2) _z = 1,n _r +n _z =N, where 2N is the order of the perturbation,n _r ,n _z are integer and _r , _z are stability parameters of the trap. The experiments were performed on H⁺ and H ₂ ⁺ ions, which are detected after a storage time of 0.3 s by ejection from the trap.     

Permeability of mixed soft and hard granular material: Hydrogels as drainage modifiers

We measure the flow of water through mixed packings of glass spheres and soft swellable hydrogel grains, at constant sample volume. Permeability values are obtained at constant sample volume and at porosities smaller than random close packing, for different glass bead diameters D and for variable gel grain diameter d, as controlled by the salinity of the water. The gel content is also varied. We find that the permeability decays exponentially in n(D/d ) ^b , where n = N _gel/N _glass is the gel to glass bead number ratio and b is approximately 3. Therefore, flow properties are determined by the volume fraction of gel beads. A simple model based on the porosity of overlapping spheres is used to account for these observations.     

Simple one-centre calculation of breathing force constants and equilibrium internuclear distances for NH3, H2O and HF

Total molecular energies, breathing force constants and equilibrium internuclear distances are determined for the NH₃, H₂O and HF molecules using a single determinant wave function of the simplified one-centre form s ² s′² p _x ² p _y ² p _z ², where each of the five Slater orbitals, s, s′, p _x , p _y , p _z , is characterized by an effective orbital exponent ζ and an effective principal quantum number n. Five different calculations are performed for each molecule: in (a) the orbitals are centred on the heavy atom, the parameters n are taken to be integral, and the orbitals p _x , p _y and p _z are given the same ζ values (the spherical approximation); in (b) the orbitals p _x , p _y and p _z are allowed to have different ζ values (the ellipsoidal approximation); (c) and (d) are the same as (a) and (b) except that non-integral n values are allowed; (e) is the same as (d) except that the orbital centre also is taken to be a variational parameter. The values obtained are compared with experimental values (the agreement is surprisingly good) and with values from previous one-centre wave functions. The electronic densities for the various spherical approximations are tabulated.     

Anisotropy profiles of optical fibers and preforms

An experimental method to measure the optical anisotropy profiles of optical fibers and preforms is presented. The local values of n_z?n_r and n_z?n_?, (r,?,z) being a cylindrical coordinate system associated with the fiber or preform, are obtained.     

Viscoelastic behaviour of fluids with steeply repulsive potentials

We analyse the shear stress, C _s(t) and pressure or ‘bulk’, C _b(t) time-correlation functions for steeply repulsive inverse power fluids (SRP) in which the particles interact via a pair potential with the analytic form, φ(r) = ε(σ/r) ⁿ , in a new approach to the understanding of their viscoelastic properties. We show analytically, and confirm by molecular dynamics simulations, that close to the hard-sphere limit both these time-correlation functions have the analytic form, C _s(t)/C _s(0) and C _b(t)/C _b(0) = 1 – T*(nt*)²+ O((nt*)⁴), where T* = k _B T/ε, is the reduced temperature, k _B is Boltzmann's constant and t* = (ε/mσ²)^½ t is the reduced time. This leads to an alternative and much simpler derivation of formulae for the shear and bulk viscosities which for the limiting case of hard spheres are numerically very close to the traditional Enskog relations. These simple relations for the finite and continuous SRP interaction are in satisfactory agreement with the essentially exact molecular dynamics simulation results for ca. n ≥ 18.     

Quantum corrections to thermodynamics of polar hard sphere fluids

The quantum corrections to the thermodynamic properties of polar hard sphere fluids and fluid mixtures are estimated taking into account the influence of dipole and quadrupole moments. Expressions are given for the second virial coefficient, free energy and pressure and results are given for different values ofμ* andϑ*. The first order quantum correction arises due to the translational contribution only. The quantum effect increases with density,μ* andϑ*. Numerical results are also estimated for binary mixtures of (i) hard spheres and dipole hard spheres and (ii) hard spheres and quadrupole hard spheres. The ‘excess’ free energy for dipole hard sphere binary mixture is also reported. It is found that the ‘excess’ quantum effect depends on the concentration and the particle diameter ratio and increases with increase ofμ* andϑ*.     

广义相对论中负指数多层球的结构及稳定性

本文运用广义相对论讨论了满足负指数多方状态方程的流体多层球。经典引力理论与相对论性理论的差别由σ标志。σ是球体中心处的压力与密度之比。通过数值积分得到了n<0的相对论性Emden函数。在-1ed^*也增大。 关键词：     

The mean spherical model for charged hard spheres

The reduced mean electrostatic potential v(r) and the radial distribution functions gij(r) for a system of charged hard spheres of equal diameter are calculated from the solution of the mean spherical model equation given by Waisman and Lebowitz. An analytical solution is given for v(r) and the gij(r) are shown to be the sum of the Percus-Yevick uncharged hard-sphere distribution function and an electrostatic term. The correct qualitative behaviour of the mean potential is predicted at high concentrations but the radial distribution functions are only accurate for low valency electrolytes at high concentrations.     

Approximate solution of the Percus-Yevick equation for a binary mixture of hard spheres with non-additive diameters

We present an approximate solution of the Percus-Yevick integral equation for a binary mixture of hard spheres with non-additive diameters. Defining R_ij the distance of closest approach between particles of species i and j by R ₁₂ = ½(R ₁₁ + R ₂₂) + α, we obtain a closed set of equations for the direct correlation functions c_ij (r) when 0 < α ? min [½(R ₂₂ - R ₁₁), ½R ₁₁]. Our expressions for c_ii (r), and for c ₁₂(r) in the range 0 < r ? ½[R ₂₂ - R ₁₁] - α, agree with those previously obtained by Lebowitz and Zomick.     

Behaviour of F and Fz Centres Under Thermal Stimulation in KCl:Eu2☎ Irradiated with Ionizing and UV Radiation

The optical absorption (OA) of F and F_z, centres in beta-and UV-irradiated KCl: Eu^2? have been studied. The spectra measured at room temperature and during the subsequent heatings up to 600 K, were analyzed by a deconvolution procedure. The OA bands of the F and F_z centres are well described respectively by asymmetric and symmetric gaussiane with parameters that are linearly dependent on the temperature. The thermal stability of the F_z centres correlates with the low-temperature thermostimulated luminescence (TL) peaks. The stability of the F centres is determined at least by two processes. Part of the F centres is destroyed together with the F_z centres. The remaining ones are destroyed during further heating up to 470 K, correlating with the TL peak at 425 K (at heating rate of 0.1 K/s).     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

A mixed spectral and finite difference model to study baroclinic annulus waves

A mixed spectral and finite difference model to study finite amplitude baroclinic waves in a differentially heated rotating annulus is presented. The model consists of the full Navier-Stokes equations and the heat equation. The field variables f = f(r, φ z; t) are decomposed into zonally averaged components f_o(r, z; t) and eddy components f(r, φ, z; t), the latter being periodic in f and represented in terms of Fourier series. The unknowns f_o(r, z; t) and f^{c, s}(r, z; t), which are Fourier amplitudes of f′(r, φ, z; t) are governed by two-dimensional primitive equations with the addition of source terms. These equations are solved semi-implicitly by the alternating direction implicit method on variable grids.A simplified model with two Fourier components which permits self-interaction of the chosen wave and the interaction of the wave and the mean fields had been used to repeat a computation done by G. P. Williams, who used a fully three-dimensional finite difference algorithm. We can reproduce almost all of Williams' results in 1/20 of the computing time with the present model. It only requires 1/30 the additional computer storage of Williams' finite difference model over the axisymmetric problem.The potential of the present model for investigation of multiwave interaction as well as the advantages and disadvantages of the two different approaches is discussed.     

Perturbation theory of polar hard Gaussian overlap fluid mixtures

A perturbation theory of polar hard Gaussian overlap fluid mixture is discussed. Explicit analytic expressions for the second and third varial coefficients are given. Numerical results are estimated for the thermodynamic properties of quadrupolar hard Gaussian overlap fluid and fluid mixture. It is found that the excess free energy and internal energy depend on concentrationsc ₁,c ₂, molecular diameter ratioR, shape parameterK and the quadrupole momentsQ*₁,Q*₂.     

Mode-coupling theory for the dynamics of tunnelling systems in dielectrics

A mode-coupling theory (MCT) is presented for the spin-boson model with a spectral density which accounts for a heat bath made up of lattice vibrations of a dielectric solid (superohmic dissipation). A usual decoupling approximation provides a set of non-linear integral equations which are solved both numerically by iteration on a computer and analytically by means of a frequency dependent ansatz for the memory functions. There is a transition to incoherent motion at a temperatureT ^* where the bare two-level energy is equal to the damping rate, in contradiction to results obtained previously from a path integral formulation. The discrepancy arises since in the MCT the relevant self-energy function does not exhibit a 1/z-pole atz=0. For tunnelling systems in dielectrics this yields a new relaxation mechanism due to incoherent tunnelling: the present results might require to modify some of the basic assumptions of the standard tunnelling model for dielectric glasses.     

A GMSA calculation of the radial distribution functions of molten RbCl

Summary The generalized mean spherical approximation (GMSA) for a fluid of charged hard spheres of equal diameters is used for calculating the radial distribution functions of molten RbCl. The results are compared to neutron diffraction data and a good overall agreement is found between the experimental and the theoreticalg _ij (r). Further application of this same approach to the case of ionic melts with ionic species of different diameters is discussed. Work supported in part by Comitato Regionale Ricerche Nucleari and Struttura della Materia (CRRNSM), Sicily, Italy.