B. Widom

Some Topics in the Theory of Fluids

It is shown how certain thermodynamic functions, and also the radial distribution function, can be expressed in terms of the potential energy distribution in a fluid. A miscellany of results is derived from this unified point of view. (i) With g(r) the radial distribution function and Φ(r) the pair potential, it is shown that g exp (Φ/kT) may be written as a Fourier integral, or as a power series in r2 the terms of which alternate in sign. (ii) A potential‐energy distribution which is independent of the temperature implies an equation of state which is a generalization of a number of well‐known approximations. (iii) The grand partition function of the one‐dimensional lattice gas is obtained from thermodynamic arguments without evaluating a sum over states. (iv) If in a two‐dimensional honeycomb (three‐coordinates) lattice gas fr(r=0, 1, 2, 3) is the fraction of all the empty sites which at equilibrium are neighbored by exactly r filled sites, then at the critical density the values of all four of the fs ...

Surface Tension and Molecular Correlations near the Critical Point

The van der Waals, Cahn—Hilliard theory of interfacial tension is reformulated for a fluid in the neighborhood of its critical point. The reformulated theory becomes equivalent to the Ornstein—Zernike, Debye theory of molecular correlations when the interface thickness is identified with the correlation length. When account is taken of the nonclassical behavior of the compressibility and coexistence curve, the theory is found to be in good agreement with independently known facts in three‐dimensional systems, yet slightly but unambiguously wrong in two dimensions. When one of the hypotheses of the original theory is replaced by an alternative hypothesis, the resulting theory is found to be correct in both two and three dimensions.

Random Sequential Addition of Hard Spheres to a Volume

The configurations produced by placing spheres one at a time in a volume, subject to no overlap of the spheres but otherwise placing each at a point chosen at random, differ in a fundamental way from the configurations characteristic of a hard‐sphere system of the same density which is in thermodynamic equilibrium. The difference is illustrated in four examples. The density at which the system becomes jammed is found in each case. The distribution of gaps produced by the random sequential placement of spheres on an infinite line is found for all densities up to that of jamming.

Structure and Free Energy of the Interface between Fluid Phases in Equilibrium near the Critical Point

The theory of van der Waals and of Cahn and Hilliard, which yields the density or concentration profile through an interface and the associated interfacial tension, is generalized by replacing the equation of state assumed originally by one that reproduces more accurately the known thermodynamic singularities at the critical point. Though the correct equation of state is not known, its presumed homogeneity of form is alone sufficient to allow most features of the interface to be determined explicitly. In particular, the maximum density gradient in the interface, the asymptotic behavior of the interface profile at large distances from the position of its maximum gradient, and the surface tension, are each obtained up to a dimensionless, and presumably universal, proportionality constant that reflects only the functional form of the equation of state. Numerical evaluation of the surface tension requires a knowledge of the limits approached by the correlation length and compressibility in the homogeneous flu...

A rigid sphere model for the melting of argon

A model is proposed for liquid and solid argon, in which the repulsive forces are represented by rigid cores and the attractive forces by a uniform background potential which depends on density but not on temperature. Thermodynamic consistency implies the equation of state: where p 0 is the pressure exerted by rigid spheres at the same number density n and temperature T, and a is independent of n and T. When p 0 is taken from the calculations of Alder and Wainwright it becomes possible to make an absolute calculation of the values of several dimensionless thermodynamic quantities characteristic of the triple point. All of these except certain second derivatives of the free energy agree very well with experiment, suggesting that the structure of liquids and solids near the triple point is mainly determined by the repulsive forces, the attractive forces serving merely to keep the molecules together.

Lattice model of microemulsions

A lattice model of microemulsions is proposed. It proves to be equivalent to a spin‐1/2 Ising model in a magnetic field, with ferromagnetic nearest‐neighbor, antiferromagnetic next‐nearest‐neighbor (next‐nearest defined as two lattice steps, regardless of the metrical distance), and three‐spin interactions. The respective interaction constants H, J, M, and L in the Ising model are related to the ratios zBB/zAA and zAB/(zAAzBB)1/2 of the activities of the oil (AA), water (BB), and surfactant (AB), to the surfactant‐film‐curvature energy (surfactant–surfactant interaction energy) K, and to the curvature‐bias parameter (Bancroft parameter) λ, in the microemulsion model. A table of translations is given. In mean‐field approximation the symmetrical version of the model, in which H=L=0 (or zBB/zAA =1 and λ=0 in microemulsion language), is equivalent also to the ANNNI (anisotropic, or axial, next‐nearest‐neighbor Ising) model. The analog of the three‐phase (Winsor III) equilibrium of surfactant solutions is iden...

Degree of the Critical Isotherm

An argument is presented which indicates that if g is the degree of the critical isotherm, d the degree of the coexistence curve, and f the power of ∥ Tc—T ∥—1 with which the isothermal compressibility becomes infinite as the critical point is approached, then g=1+fd. The relation of this result to other theoretical and experimental facts is discussed.

The hydrophobic effect

The thermodynamics of the hydrophobic effect, as measured primarily through the temperature dependence of solubility, is reviewed, and then a class of models that incorporate the basic mechanism of hydrophobicity is described. These models predict a quantitative relation between the free energy of hydrophobic hydration and the strength of the solvent-mediated attraction between pairs of solute molecules. It is remarked that the free energy of attraction being just of the order of the thermal energy kT may be important for the effective operation of the hydrophobic effect in proteins. Deviations from pairwise additivity of hydrophobic forces are also briefly discussed.

A model microemulsion

The microemulsion model of de Gennes et al., which is a modified version of the earlier model of Talmon and Prager, is modified further by imposing a microscopic cell‐size cutoff. Two free‐energy minima compete and lead to a composite free‐energy surface consisting of two intersecting sheets. The microemulsion phase arises from one sheet and the oil‐ and water‐rich phases with which it may be in equilibrium arise from the other. The Schulman condition, according to which the surfactant‐film pressure equals the oil–water interfacial tension, is found to hold to good approximation in the middle‐phase microemulsion that contains comparable amounts of oil and water. The oil‐ and water‐filled domains in that microemulsion are found to be about 75–80 A across. That phase is of such high osmotic compressibility that it would be opalescent or turbid due to fluctuations of composition. The model shows a range of phase equilibria like that seen in experiment, including two‐ and three‐phase equilibria, critical poin...

Biochemistry at 100°C: Explosive Secretory Discharge of Bombardier Beetles (Brachinus)

The defensive chemical spray of bombardier beetles is ejected at 100�C, with a heat content of about 0.2 calorie per milligram.