Frank H. Stillinger

General Restriction on the Distribution of Ions in Electrolytes

The rigorous second‐moment condition previously derived for “primitive‐model” electrolyte ion atmospheres in equilibrium is generalized to arbitrary mixtures of electrolytes of unrestricted charge species. No special assumptions regarding the nature of solvent dielectric behavior are required, and the condition remains valid even in the presence of specific chemical interactions that lead to complex ion formation.

Ion‐Pair Theory of Concentrated Electrolytes. I. Basic Concepts

The statistical thermodynamics of symmetrical “primitive‐model” electrolytes is formulated in such a way that all ions are uniquely paired. The behavior of the resulting fluid of “polar molecules” may conveniently be described by a wavelength‐dependent dielectric constant e(k). A rigorous formula of the Kirkwood type for e(k) is derived. Since ion‐atmosphere mean charge densities may be obtained from e(k), this dielectric function is utilized in construction of an electrolyte free‐energy expression [Eq. (50)], as well as to establish an exact second‐moment condition on the ion atmospheres [Eq. (73)]. From the latter it is demonstrated that for rigid spherical ions of diameter a, the ion atmospheres necessarily each have nonuniform charge sign when κa > 61 / 2 (κ− 1 = Debye length).

Interfacial Solutions of the Poisson‐Boltzmann Equation

The linearized Poisson‐Boltzmann equation is considered for boundary conditions corresponding to a fixed point‐charge ion near the planar boundary between an electrolytic solution and a dielectric substrate. Use of the Fourier expansion for this fixed charge density allows the mean potential to be synthesized in the form of a simple quadrature. Subsequently, it is possible to compute the reversible work necessary to displace the ion atmosphere surrounding this charge into one hemisphere as the charge is brought up to the interface from the interior of the electrolyte phase. Implications for ionic adsorption processes are discussed. The result of a similar analysis for the mean potential surrounding a point dipole oriented normally to the boundary is also presented.

Ground‐State Energy of Two‐Electron Atoms

We examine the ground‐state energy e(λ) for two electrons bound to an infinite‐mass point nucleus regarded as a function of the complex coupling constant λ for the interelectron interaction. In the units used, the ground states of the homologous series H−, He, Li+, Be++, ···, correspond, respectively, to λ=1, ½, ⅓, ¼, ···, so the λ power‐series expansion of e(λ) is equivalent to the familiar expansion in inverse nuclear charge Z−1. It is argued that the power series has a finite radius of convergence imposed by the existence of a branch point on the positive real axis at λ=λ*≅1.1184 with exponent approximately 6/5. Furthermore, e(λ*) apparently lies above the continuum limit, but still corresponds to a localized wavefunction.

Ion‐Pair Theory of Concentrated Electrolytes. II. Approximate Dielectric Response Calculation

The theoretical program initiated in the preceding paper is continued. The wavelength‐dependent dielectric response function for the “primitive‐model” electrolyte, e(k), is evaluated by means of a self‐consistent torque calculation for ion pairs in an external field. The result is used to obtain the longestranged component of the ion‐atmosphere charge density. The rigorous second‐moment condition on the ion atmosphere then is used to compute crudely the short‐ranged components. Besides the usual lowconcentration terms, the implied free energy [Eq. (76)] and activity coefficient [Eq. (77)] contain negative terms varying with the four‐thirds power of concentration.

Analytic Approach to Electron Correlation in Atoms

A novel perturbative treatment of electron correlation in N‐electron atoms is devised. The unperturbed starting point is a central‐force “hydrogenic” problem in the full dN‐dimensional configuration space (d = dimensionality). The central potential in this solvable “hydrogenic” problem is obtained by averaging the actual electron–electron and electron–nucleus potentials over all dN − 1 hyperspherical polar angles in the configuration space. The relevant projected Greens functions are computed for the ground states of the model one‐dimensional two‐electron atom (with delta function interactions), as well as for the real three‐dimensional helium isoelectronic sequence. The corresponding first‐order wavefunctions exhibit weakly singular logarithmic behavior (at three‐particle confluence) of the type first advocated by Fock. Second‐order energies are evaluated for both of these two‐electron problems. The basic ingredients of our hyperspherical coordinate method for three‐electron atoms are displayed, in prep...

Rigid Disks at High Density

As shown in a previous publication, the Helmholtz free energy per particle (divided by kBT) for a rigid disk system has the following asymptotic form in the limit of the reduced density θ=A0/A→1:F/NkBT∼2ln(λ/a)−2ln(θ−1−1)+C+D(θ−1−1)+···, in which C and D are appropriate numerical constants, λ is the mean thermal de Broglie wavelength, a is the disk diameter, and A0 is the close‐packed value of the area, A, of a system of N particles. Moreover, a technique for estimating the value of C was developed. This technique was based upon a product representation for the partition function which considered a sequence of correlated motion of larger and larger sets of contiguous particles. An approximate value of C was previously obtained from all contributions for four or fewer correlated disks. This article extends these calculations to fifth order (contributions of all sets of five particles) with the summary result: C=0.14384–0.013857+0.014322–0.0073211–0.004222+···=0.14384–0.011078+···, in which the first number...

Compressibility of Simple Fused Salts

It is pointed out that the isothermal compressibility of monatomic molten salts should nearly equal that for a hypothetical fluid of uncharged ion cores at the same over‐all particle density as the salt. The validity of this relation is equivalent to verification of conjectures concerning the number‐average correlation function for ion pairs in the melt. When the ion core forces are treated as those acting between rigid spheres, it is readily possible to predict the distance of closest approach for an anion‐cation pair from experimentally determined compressibilities. The corresponding distances obtained in this manner for a series of molten halides are consistent with known ion sizes, and furthermore exhibit the proper homologous series trends.

Elasticity in Rigid‐Disk and ‐Sphere Crystals

Our previously developed product representation for the partition function of rigid molecules under high compression is generalized to include distorted reference lattices. The resulting strain component variations of free energy then permit extraction of elastic constants, both linear and nonlinear. The high‐compression elastic behavior of the two‐dimensional rigid‐disk crystal is explicitly evaluated through triplet cluster contributions. For rigid spheres in three dimensions in the face‐centered‐cubic lattice, singlet and pair cluster contributions have been evaluated, and the sound velocity ratio is reported for propagation along the 〈100〉 directions. Consideration of singlet and pair contributions for hexagonal packing suggests not only that this structure is more stable than the face‐centered‐cubic case, but that it exhibits a spontaneous small contraction along the c axis.

Self‐Diffusion in Dense Fluids

The time‐dependent solutions to the low‐order BBGKY hierarchy are examined for the case of inter‐diffusion of two mechanically identical isotopes. In order to effect a closure directed at evaluation of the pair‐distribution‐function perturbations and the self‐diffusion constant D, we invoke a dynamical superposition approximation, and a truncated expansion in inverse powers of the initial composition fluctuation wavelength. The potential feasibility of this general approach to calculation of dense‐fluid transport properties is illustrated by explicit numerical calculations for the dense fluid of rigid spheres, using in addition a pair‐space local equilibrium assumption. Although the resulting pair‐distribution‐function perturbations seem to be in accord with physical intuition, cumulative errors in the approximation sequence render the self‐diffusion coefficients predicted at variance with other calculations. Systematic improvements of the present scheme are outlined.