H. L. Frisch

Statistical Mechanics of Rigid Spheres

An equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid sphere molecules in contact with a rigid sphere solute of arbitrary size. A number of exact relations which describe rather fully the functional form of G(r) are derived. These are based on both geometrical considerations and the virial theorem. A knowledge of G(a) where a is the diameter of a rigid sphere enables one to arrive at the equation of state. The resulting analytical expression which is exact up to the third virial coefficient gives the fourth virial coefficient within 3% and the fifth, insofar as it is known, within 5%. Furthermore over the entire range of fluid density, the equation of state derived from theory agrees with that computed using machine methods. Theory also gives an expression for the surface tension of a hard sphere fluid in contact with a perfectly repelling wall. The dependence of surface tension on curvature is also given. ...

Aspects of the Statistical Thermodynamics of Real Fluids

By extending the ideas previously applied to the statistical mechanical theory of hard sphere fluids of Reiss, Frisch, and Lebowitz, an approximate expression has been determined for the work of creating a spherical cavity in a real fluid. In turn the knowledge of this entity permits an evaluation of properties such as the surface tension and the normal heats of vaporization of fluids and the Henrys law constants of fluid mixtures. The agreement between the calculated and experimental properties is satisfactory.

Double‐Quantum Light Scattering by Molecules

Double‐quantum light scattering by a system of molecules is discussed in this paper. Expressions have been obtained for the scattered light intensity considering both the coherent and incoherent contributions. In that coherent contributions are also considered in this treatment, it goes beyond the scope of previous studies. It is shown that, for molecules of low symmetry, elliptically polarized light must be used in order to determine five independent quadratic forms in the 18 symmetric components (βijk+βikj). According to the present results, the apparent discrepancy between the observed value of ⅓ for the depolarization ratio for CCl4 and the value to be expected from theory may be due to the fact that the coherent contribution had been neglected in previous theoretical considerations. In general, orientational correlation is essential if there is to be appreciable contribution from coherent scattering. For macromolecules, this constitutes a major difference between single‐ and double‐quantum scattering...

Percolation Processes and Related Topics

This paper is a general review of the current state of the theory of percolation processes and related topics. Besides presenting definitions and listing some of the principal available results and unsolved problems we have attempted to indicate a wide variety of examples drawn from physics, chemistry, biology, engineering and communication theory of problems which can profitably be considered special instances of a percolation process. The references, while not complete, should be sufficiently broad for the needs of most readers.

On the Stereoregularity of Vinyl Polymer Chains

We explore quantitatively the kind and extent of information concerning the distribution of configurational sequences in vinyl (and related) polymer chains obtainable from data such as presently available high‐resolution NMR spectroscopy measurements. After finding the number of observationally distinguishable types of sequences, containing n monomer units [n(ads)], we obtain the necessary consistency relations which must hold between the observed proportions of n(ads), explicitly for all n≤5. Geometrically, the possible regions accessible to the n(ad) proportions form certain simplexes in a space of D(n) dimensions. The extreme points always correspond to repeating polymers. The computed dimensionality number D(n) plays a central role in fixing the number of free parameters of the polymer propagation model determinable by an n(ad) analysis. We discuss in some detail certain Markovian and the non‐Markovian Coleman—Fox propagation model in relation to n(ad) analysis. We introduce the concept of a completel...

Statistical Mechanics of a Simple Entanglement

It is shown that the statistical mechanics of an entanglement produced by looping a polymer chain around an infinitely long straight bar can be treated exactly, essentially because successive turns of the chain around the bar can be considered as constituting a Markoff process. The elastic response of such a loop to a force exerted on one of its segments is nonlinear and surprisingly complex.

Scaled Particle Theory of Fluids

A statistical thermodynamic theory has been developed employing distance scaling as a coupling procedure. This is an extension to real fluids of the technique applied by Reiss, Frisch, and Lebowitz to rigid‐sphere systems. One considers molecules interacting with pair potential u(r), except for one particle which interacts with potential u(r/λ). This single particle, essentially a scaled version of a normal molecule, is termed a λ‐cule. It is convenient to restrict discussion to potentials with rigid cores at r=a and cutoffs at γa. Attention is focused on a function, θ(λ, ρ, T), which reduces to G of footnote reference 1 in the case of rigid spheres. The pressure, chemical potential, and work of expanding a λ‐cule are simply related to θ. One can write θ exactly for λ<1/2γ and simple connection conditions hold at λ=1/2γ. An integral condition and λ=∞ condition on θ also exist. While θ is not completely specified, the foregoing conditions determine much of its behavior.

Nonequilibrium Distribution Functions in a Fluid

The behavior of a nonequilibrium fluid is analyzed on a level intermediate between that of hydrodynamics, where microstructure is totally ignored, and a phase space description, where the complete N‐body problem must be solved. The study of the fluid at this level generally involves solving an appropriate transport equation. For liquids, the primary subject of this investigation, the Fokker‐Planck equation of Kirkwood is accepted as a working model and solutions are found by the methods of Chapman and Enskog and of Grad to terms linear in deviations from local equilibrium. (It is argued, however, for a different form of the pair space force than that suggested by Kirkwood and co‐workers.) The results are similar in form to distributions found with other kinetic models. Variational principles are also considered. It is shown that the one‐ and two‐ particle distribution functions have the property of maximizing the entropy subject to the constraints of given densities and fluxes. Alternatively, these distri...

Conditions Imposed by Gross Properties on the Intermolecular Potential

The conditions for, and implications of, the existence of a maximum in the second virial coefficient as a function of temperature are investigated. Brief comment is made on the use of the inversion of Laplace transforms, with respect to the reciprocal temperature, to obtain information about intermolecular potentials and cross sections from second virial coefficients and dilute gas transport coefficients.

Time Lag in Transport Theory

The time lag in the attainment of a stationary state or flow is defined for a class of linear and nonlinear transport equations. It is shown that the explicit calculation of the time lag can be reduced, for these equations, to the determination of the Greens function of a simpler, linear, time‐independent boundary‐value problem. For a linear transport equation knowledge of this Greens function enables one to recursively calculate certain time moments which afford an approximate representation of the solution. The possibility of an asymptotic exponential approach to the steady‐state solution of these linear transport equations is directly related to the convergence of the sequence of these time moments. An explicit example of a nonlinear diffusion in a finite system is used to illustrate the general theory. Certain properties of the time lag for the flux of this nonlinear diffusion problem are derived in the Appendix.