Kirkwood Integral Equation Research Articles

Structure and thermodynamics of polymeric liquids: Polymer kirkwood integral equation method

AbstractA statistical mechanical theory is employed to predict the structural and thermodynamic properties of polymeric liquids. The theory consists of a coupled set of self‐consistent integral equations for the pair correlation functions which relates the polymer structure to molecular interaction potentials. In this article, the essentials of the method are reviewed and the results are shown in comparison with other theories and simulation data. Quantitative agreements are found between the results of the theory and those of the simulations for a wide range of density and temperature.

Polymer kirkwood integral equations: Structure and equation of state of polymeric liquids

AbstractA self‐consistent theory of chain molecular liquids from polymer Kirkwood integral equations is presented. The theory predicts the intramolecular chain distributions and the intermolecular pair correlation functions from which thermodynamic functions can be correctly predicted. The utility of the theory is demonstrated with the chain distributions from self‐consistent equations at zero density for swollen and collapsed states of the chains and numerical results for the structure and equation of state of athermal chain molecular liquids at various chain lengths and densities. Structural and thermodynamic predictions of the theory are compared with Monte Carlo simulation results. The theory agrees satisfactorily with simulation data for monomer packing fractions up to 0.25. The investigation of the chain length dependence shows that a plateau value is reached for the equation of state as the number of monomer units reaches a value between 20 and 50.

On integral equations for the pair correlation function

The Kirkwood integral equations for triplet or higher-order correlation functions are examined with a view to obtain some useful approximate sets of integral equations for correlation functions. Based on a set where the potential energies are eliminated from the integral equations for triplet or higher-order correlation functions, the Percus-Yevick integral equation is obtained. The derivation is rather simple but requires a pair of assumptions. This derivation implies that the Percus-Yevick equation is essentially founded on the Kirkwood superposition approximation which is one of the assumptions. A couple of plausible extensions of the Percus-Yevick equation are then suggested for pair correlation function and their consequences are examined in the light of some exact relations for pair correlation function. One of such generalized equations is numerically solved to assess its accuracy.

Kinetic theory and irreversible thermodynamics of dense fluids subject to an external field

A formal theory of dense fluids subject to an external electric field is developed as an extension of the kinetic theory of dense fluids reported previously. The present formulation also refines and modifies the previous theory wherever deemed necessary and especially in the part related to the renormalization. The extended Gibbs relation and a theory of irreversible thermodynamics are developed. A generalized Kirkwood integral equation for correlation functions and a generalized equation for local nonequilibrium chemical potentials are also derived for nonequilibrium mixtures.

Modified Poisson–Boltzmann equation in electric double layer theory based on the Bogoliubov–Born–Green–Yvon integral equations

The lowest order equation of the Bogoliubov–Born–Green–Yvon system of integral equations is applied to the electric double layer to obtain a modified Poisson-Boltzmann equation. Charge oscillations are predicted for κ0a > 1.720 (κ0 bulk Debye–Hückel constant, a ionic diameter) and this critical value of κ0a is shown to be related to that derived by Martuynov. Comparison is also made with the modified Poisson–Boltzmann equation based on the Kirkwood integral equation.

Reformulation of the nth order poisson equation in strong electrolyte solution theory using the kirkwood integral equations

A Poisson equation for the nth order mean potential has previously been derived in the case of the primitive model of an electrolyte solution using the Kirkwood integral equations. We generalise this analysis to the case of an arbitrary short range pair potential between the ions.

Reformulation of the nth order poisson equation in strong electrolyte solution theory using the kirkwood integral equations

Reformulation of the nth order poisson equation in strong electrolyte solution theory using the kirkwood integral equations

Distribution Function Theory of Nematic Liquid Crystals

The method of distribution functions—governed by Kirkwood integral equations—is applied to anisotropic fluids of the nematic type. It is shown how to obtain the self-consistent-field theory of Maier and Saupe (assuming their anisotropic dispersion force) with the help of some approximations. When the repulsive forces are included an additional term is present in lnρ(1)(θ)[ρ(1)(θ) ≡ orientational distribution function]; estimating this term we conclude that the repulsive force cannot be neglected if l/d〉2 (l and d=molecular length and diameter) and the molecules are not too flexible. Also, the first few terms of Onsager's virial expansion for the free energy are derived.

Extremum principles for a nonlinear kirkwood integral equation

.ho(t) = t ~ 1, It] 1 . Here g(r) is the radia l d is t r ibut ion funct ion which is a nonnegat ive even funct ion of r and is such tha t (4) g(r)-> 1 as r -~ ~ . Numer ica l solutions of this k ind of equa t ion have been obta ined by KIRKWOOD, MAUN and ALD]~R (1). In this paper complenwnta ry var ia t iona l principles for in tegra l equat ions (2,a) are used to find approx ima te analyt ic solutions for equat ions of tile form (1).

Variational solution of a kirkwood integral equation

Variational solution of a kirkwood integral equation

Coupling-Parameter Expansion in the Kirkwood Integral Equation for Dense Fluids

A new treatment of the coupling-parameter dependence of terms in the Kirkwood integral equation is proposed. The effect of partial coupling ξ of molecule 1 on the pair correlation function g(13, ξ) = exp[− ξφ(13)]Y(13, ξ) is divided into two parts, the direct coupling term exp[− ξφ(13)] and the indirect coupling term Y(13, ξ). We argue, on physical grounds, that the indirect coupling term Y(13, ξ) is a relatively slowly varying function of ξ as compared with the direct coupling term over almost all the range 0 < ξ ≤ 1 except in a very small region near ξ ≤ 0. In the kernel of the Kirkwood integral equation, Y(13, ξ) is expanded in a Taylor series in the coupling parameter about ξ = 1. The first term of this series gives an integral equation which resembles both the Percus–Yevick and hypernetted chain equations, if we assume our approximation to be valid for the entire range 0 ≤ ξ ≤ 1. An interative procedure for higher-order terms in a resummation of the Taylor series in the coupling parameter is found which, in principle, permits calculations to any desired order in the coupling. However, for a strongly repulsive potential the coupling parameter representation encounters difficulty near ξ = 0. By subdividing the range of variation of ξ a new expansion is suggested. Preliminary numerical calculations indicate that this scheme gives a good representation of g(1, 3).

Theory of Moderately Concentrated Polymer Solutions

The transition between the theories of the thermodynamic properties of dilute and concentrated polymer solutions is investigated using the random-flight model with explicit consideration of segment—segment interactions. It begins with a qualitative discussion of the transition. Supposing the individual segments, instead of the entire polymer molecules, to be solute molecules, a general expression for the osmotic pressure π is derived by the method of Green. Then the apparent second virial coefficient Γ defined by Γ=π/RTc2−1/Mc, where c is the concentration and M the polymer molecular weight, is evaluated by a coupling-parameter method. The evaluation consists of solving differential equations which can be derived approximately from the Kirkwood integral equations for the distribution functions. The result is expressed as a function of c and α, where α is the linear expansion factor of the polymer molecule at finite concentrations. Thus, α is also evaluated as a function of c, though the expression is incomplete at very high concentrations. For good solvent systems, it is predicted that as α is decreased to unity with increasing concentration and then becomes independent of concentration, Γ increases first rapidly and then gradually, passing through the transition region. This prediction is in fairly good agreement with experiment for polyisobutylene in cyclohexane. For poor solvent systems, the theory is invalid except at low concentrations.

Virial Coefficients and the Classical Theory of Fluids for the Square-Well Potential

The term g2(r) proportional to the square of the density in the expansion of the radial distribution function in powers of the density has been evaluated for a square-well potential for the Born—Green—Yvon integral equation, the Kirkwood integral equation, the Percus—Yevick integral equation, the hyperchain approximation, and an integral equation derived by Meeron. The square-well potential used has a hard-core diameter of σ and attractive diameter of 2σ. Each g2(r) is then used to calculate Dp, the fourth virial co-efficient calculated from the pressure equation, and Dc, that calculated from the compressibility equation, for the above five liquid theories. This allows a comparison over the entire temperature range, rather than at infinite temperature, the only value given by the previously used hard sphere potential. It is shown that the Percus—Yevick equation yields virial coefficients in best agreement with the exact value over a large temperature range. All the results presented are low-order polynomials in f=exp(ε/kT—1), where ε is the depth of the well, and reduce to the hard-sphere results when f goes to zero.

The Percus-Yevick equation for the radial distribution function of a fluid

Synopsis The Percus-Yevick (PY) equation is shown to be associated with an exact equation for the radial distribution function g(r) of a classical fluid. The exact equation contains an infinite series that provides a systematic means of improving the Percus-Yevick equation. It is also shown that the Born-Green-Yvon and Kirkwood integral equations can be similarly embedded in exact equations by using the appropriate form of an exact relationship between the three-particle distribution function and g(r). An iterative procedure that generates the solution of the PY equation by a partial summation of the graphs associated with g(r) is considered. This partial summation, called by us the series-union summation, is similar to the series-parallel summation associated with the hypernetted chain equation, but is simpler. The significance of this partial summation is discussed in terms of the range of the direct correlation function. It is noted that in the hard-sphere case the series-union approximation is equivalent to assuming that the direct correlation function is identically zero beyond the hard-sphere diameter. This assumption is exact in one dimension. On the basis of the series-union approximation, the 5th virial coefficient for hard spheres has been computed by means of the Ornstein-Zernike compressibility equation. Its value is 0.121 b4 where b is four times the hard-sphere volume. The Monte-Carlo value of this coefficient is (0.115 ± 0.005) b4.

Virial Expansion of the Solution of the Kirkwood Integral Equation for the Radial Distribution Function of a Fluid

The solution gK(r) of the Kirkwood integral equation for the radial distribution function of a fluid is expressed as a series in the density. This series is compared with the virial expansion of the exact radial distribution function g(r). It is shown that the term g2(r) proportional to the square of the density in the expansion of gK(r) differs from the corresponding term in the expansion of g(r). For the special case of hard-sphere particles g2K(r) is computed and used to obtain two approximations to the fourth virial coefficient of the equation of state. These results are compared with corresponding results for an approximate Kirkwood equation and for the Born-Green-Yvon and Percus-Yevick integral equations. The best agreement with exact theory is obtained from the Percus-Yevick equation; the poorest from the Kirkwood equation and the approximate Kirkwood equation.

On the Kirkwood Superposition Approximation

The Kirkwood superpositIOn approximation is investigated by.making use of the expan­ sion theorem for the potential of average force and its correction term is found in a form of the expansion in powers of particle number density p. The lowest order term of this correction is calculated for a special configuration of three particles in a fluid consisting of hard spheres and the Kirkwood approximation is shown to overestimate the distribution function of triplets in this case. The Kirkwood integral equation with correction is solved by expanding the radial distribution function in powers of p. The solution thus obtained is shown to be exact up to the order of p2, while the Kirkwood approximation itself does not yield the correct term of the order of p2. It is shown that the distribution function of triplets can, in principle, be expanded in terms of radial distribtution functions and a few terms of this expansion are calculated explicitly, of which the first term just corresponds to the Kirkwood superposition approximation.

The Gaussian Approximation for the Probability Density of a Molecule in a Cell

Kirkwood's integral equation of the free volume theory of liquids has been used to test the Gaussian approximation for the probability density F(r) of a molecule in a cell. The approximation has been found to be good, even up to the critical point.

The Radial Distribution Function in Liquid Argon

The radial distribution function in liquid argon which is obtainable as a solution of Kirkwood's integral equation is calculated numerically, making use of the approximation as to the coupling parameter. The result obtained is compared with experimental radial distribution functions.