G. Stell

Thermo-dynamic perturbation theory for simple polar fluids:partI

The study of polar fluids begun in a previous paper is continued. Calculations for the Stockmayer potential are extended to include the term of order μ6, where μ is the dipole moment. The effects of higher-order terms are then approximated by means of a simple Pade extrapolation procedure, and the liquid-gas coexistence curve is located in this approximation. An orientation-independent but temperature-dependent potential that is thermodynamically equivalent to an arbitrary orientation-dependent potential is introduced and used to assess the lowest-order thermodynamic effects that result from the presence of quadrupole and octupole terms in the pair-potential. Several values of quadrupole and octupole moments representative of a dipolar molecule (HCl) as well as linear molecules (N2, O2 and CO2, for which μ=0) are considered.

Theory of polar liquids

Two recent contributions to the statistical theory of polar fluids, namely the perturbation theory of Stell, Rasaiah and Narang (SRN) and the meanspherical-approximation (MSA) results of Wertheim, and of Nienhuis and Deutch, are compared and contrasted for the conceptually simple model of hard spheres, diameter R, with central point dipoles, of strength μ (dipolar hard spheres). It is shown that the MSA approach replaces correlation functions which enter correctly into the SRN theory by their low-density limits : to this extent it is unsatisfactory. On the other hand the MSA work does suggest reasons why the naive Pade approximant featuring in SRN theory may be expected to do reasonable justice to the physics of the problem. Numerical comparisons of the excess free-energy (as compared with non-polar hard spheres) as a function of reduced density, ρ* = ρR 3, are given at two temperatures, T* = 2 and T* = 0·25, where T* = kTR 3/μ2. Similar curves, for T* = 1 and T* = 0·5, are available from the authors. The...

Microstructure of two‐phase random media. I. The n‐point probability functions

The microstructure of a two‐phase random medium can be characterized by a set of general n‐point probability functions, which give the probability of finding a certain subset of n‐points in the matrix phase and the remainder in the particle phase. A new expression for these n‐point functions is derived in terms of the n‐point matrix probability functions which give the probability of finding all n points in the matrix phase. Certain bounds and limiting values of the Sn follow: the geometrical interpretation of the Sn and their relationship with n‐point correlation functions associated with fluctuating bulk properties is also noted. For a bed or suspension of spheres in a uniform matrix we derive a new hierarchy of equations, giving the Sn in terms of the s‐body distribution functions ρs associated with a statistically inhomogeneous distribution PN of spheres in the matrix, generalizing expressions of Weissberg and Prager for S2 and S3. It is noted that canonical ensemble of mutually impenetrable spheres a...

Polydisperse systems: Statistical thermodynamics, with applications to several models including hard and permeable spheres

The statistical thermodynamics of polydisperse systems of particles is investigated. A Gibbs–Duhem relation is obtained and the equilibrium conditions relevant to a two‐phase system are derived. Systems of hard spheres, and hard spheres with Kac tails, are treated as illustrative examples with analytic results given in the context of scaled‐particle (Percus–Yevick) theory as well as the polydisperse generalization of the thermodynamic approximation of Mansoori et al. An exact treatment of the analogous one‐dimensional systems is also given. Quantitative results using a Schultz distribution of diameters are presented. A model of interpenetrable particles introduced previously by one of us—the permeable‐sphere model—is also considered. Its thermodynamics and pair distribution functions are shown to be exactly obtainable in the context of the Percus–Yevick approximation. For this model, polydispersivity in both particle size and particle impenetrability is considered analytically. The pair potential for this...

Thermodynamics of the MSA for simple fluids

An analytic study is made of the thermodynamic properties of the mean spherical approximation for a simple fluid and simple fluid mixtures. It immediately suggests several improved approximations.

Comment on: Fluid distributions in two‐phase random media: Arbitrary matrices

It is pointed out that the Ornstein–Zernike equations recently used by Madden in treating quenched‐annealed mixtures are approximate. The exact equations are given and briefly discussed.

Criticality and phase transitions in ionic fluids

Recent experimental investigations of criticality and phase separation in ionic fluids have revealed behavior of great theoretical interest. In seeking to understand the experiments, some of which appear to exhibit argonlike criticality and some of which exhibit “classical” (mean-field) criticality, a convenient starting point is the restricted primitive model (RPM) of symmetrically charged hard spheres, all of equal diameter σ, each sphere bearing a positive or negative charge of magnitudeq. There is overall charge neutrality, so that the expected number densities of the anions and cations are equal,ρ+=ρ-. Studies of RPM charge-charge and density-density correlation functions indicate that the fluctuation-suppressing mechanism that yields mean-field critical behavior in nonionic systems with long-range interparticle potentials is not operative in the RPM. On the basis of plausible assumptions, Ising-like behavior is instead expected. The above work is summarized. New work of Zhang and the author is outlined, showing that when one loses the RPM symmetry (through, e.g., different valence, diameter, or dipole moment of anions and cations) a strong coupling between charge-charge and density-density correlation ensues. The way in which this can be expected to give rise to mean-field or mean-field-like behavior is noted. Other new observations concern the mean-field analogy found by Høye and the author between the parameter 2/(d−2) (d is the dimensionality) in that model and the monomer number in high polymers, with respect to the coexistence-curve shape dependence on those parameters.

Microstructure of two‐phase random media. III. The n‐point matrix probability functions for fully penetrable spheres

We examine the two‐ and three‐point matrix probability functions for a two‐phase random and homogeneous system of impenetrable spheres. For such a system, we give an exact analytical expression for the two‐point matrix function S2 through second order in the number density of particles. Moreover, the two‐point matrix function is evaluated, for the first time, for a very wide range of densities. We also discuss the evaluation of the three‐point matrix function S3 for an impenetrable‐sphere system and provide new expressions that may be used to estimate it.

Thermodynamic perturbation theory for multipolar and ionic liquids

We extend earlier work of ours on the use of Pade approximants in the theory of multipolar and ionic potentials. The new features are (i) extension of our work to mixed multipole terms and inclusion of polarizability, (ii) formulation and implementation of a systematic means of getting analytic approximations for all the two-body and three-body terms appearing in the theory, (iii) assessment of the ionic Pade results in the low-concentration region important in ionic-solution applications, (iv) use of the Pade in a mixed perturbation theory of improved accuracy in that low-concentration regime. The results of (iii) and (iv) are used to study the remarkable lowdensity charged-sphere critical point recently discovered by Stell, Wu, and Larsen.

Kinetics of polymer gelation

The kinetics of polymer growth and especially gelation are discussed for a system of reacting f‐functional monomeric units. The gelation models of Flory and Stockmayer are examined, and their implications concerning sol–gel interaction after gelation has occurred are clarified. A new model, in which the gel does not cross link, is considered. A mathematical analysis shows that the onset of gelation is related to the formation of a shock‐wave solution of the equation satisfied by the generation function.