John D. McCoy

Density functional theory of nonuniform polyatomic systems. I. General formulation

We extend the density functional theory of nonuniform fluids to the cases of systems composed of polyatomic species. By the method of Legendre transforms, one demonstrates the existence of a free energy density functional where the densities refer to the locations of interaction sites (not full molecular coordinates). A variational principle for the free energy is derived. The methodology retains nearly all the mathematical simplicity of the traditional theory of atomic fluids. Thus, it may provide a practical route to deriving mean field theories of assembly and phase transitions in complex systems. Certain nonlinearities intrinsic to polyatomic systems and absent in simple fluids become apparent in our analysis. These features are associated with the entropy density functional for systems with bonding constraints. They must be carefully assessed in accurate applications.

Density functional theory of nonuniform polyatomic systems. II. Rational closures for integral equations

With the density functional theory outlined in paper I, we address and formally solve the nonlinear inversion problem associated with identifying the entropydensity functional for systems with bonding constraints. With this development, we derive a nonlinear integral equation for the average site density fields of a polyatomic system. When external potential fields are set to zero, the integral equation represents a mean field theory for symmetry breaking and thus phase transformations of polyatomic systems. In the united atom limit where the intramolecular interaction sites become coincident, the mean field theory becomes identical to that developed for simple atomic systems by Ramakrishnan, Yussouff, and others. When the external potential fields are particle producing fields (in the sense introduced long ago by Percus), the integral equation represents a theory for the solvation of a simple spherical solute by a polyatomic solvent. In the united atom limit for the solvent, the theory reduces to the hypernetted chain (HNC) integral equation. This reduction is not found with the so‐called ‘‘extended’’ RISM equation; indeed, the extended RISM equation—the theory in which the HNC closure of simple systems is inserted directly into the Chandler–Andersen (i.e., RISM or SSOZ) equation—behaves poorly in the united atom limit. The integral equation derived herein with the density functional approach however suggests a rational closure of the RISM equation which does pass over to the HNC theory in the united atom limit. The new integral equation for pair correlation functions arising from this suggested closure is presented and discussed.

Local structure of polyethylene melts

Polymer‐RISM (Reference‐interaction‐site‐model) theory is used to examine the local structure of a dense polyethylene melt near the freezing point. Predictions for the static structure factor are found to be in near quantitative agreement with new x‐ray diffraction data obtained at 430 K and 1 atm.

A density functional treatment of the hard dumbbell freezing transition

We present the first implementation of our density functional theory [J. Chem. Phys. 85, 5971, 5977 (1986)] to investigate a fluid–solid phase transition. In this theory, designed specifically for polyatomic systems, the entropy functional with bonding constraints is treated exactly, and approximations are generated by truncating expansions of the intermolecular interaction part of the free‐energy density functional. We examine the theory resulting from the quadratic truncation of the interaction free energy, and determine the resulting phase diagram for hard dumbbell molecules. The results for short bond lengths are in accord with known trends from experiment and simulation. However, the theory predicts no plastic crystal transition for hard dumbbells with a bond length that might characterize nitrogen, for which the experimental β phase is a plastic crystal. Reasons for this behavior are discussed.

Conjectures on the glass transition of polymers in confined geometries

We hypothesize that the shift of the glass transition temperature of polymers in confined geometries can be largely attributed to the inhomogeneous density profile of the liquid. Accordingly, we assume that the glass temperature in the inhomogeneous state can be approximated by the Tg of a corresponding homogeneous, bulk polymer, but at a density equal to the average density of the inhomogeneous system. Simple models based on this hypothesis give results which are in agreement with experimental measurements of the glass transition of confined liquids.

Density functional theory of simple polymers in a slit pore. I. Theory and efficient algorithm

Previous applications of density functional (DF) theory required a single chain Monte Carlo simulation to be performed within a self-consistent loop. In the current work, a methodology is developed which permits the simulation to be taken out of the iterative loop. Consequently, the calculation of the self-consistent, medium-induced potential, or field, is decoupled from the simulation. This approach permits different densities, different forms of UM(r), and different wall–polymer interactions to be investigated from a single Monte Carlo simulation. The increase in computational efficiency is immense.Previous applications of density functional (DF) theory required a single chain Monte Carlo simulation to be performed within a self-consistent loop. In the current work, a methodology is developed which permits the simulation to be taken out of the iterative loop. Consequently, the calculation of the self-consistent, medium-induced potential, or field, is decoupled from the simulation. This approach permits different densities, different forms of UM(r), and different wall–polymer interactions to be investigated from a single Monte Carlo simulation. The increase in computational efficiency is immense.

A DENSITY FUNCTIONAL THEORY FOR PAIR CORRELATION FUNCTIONS IN MOLECULAR LIQUIDS

We employ density functional methods to derive an integral equation for the two‐point intermolecular correlation function in molecular liquids. This radial distribution function is expressed as a two molecule average over a Boltzmann factor involving a ‘‘bare’’ site–site interaction, plus a pairwise additive, intermolecular, medium induced potential which mimics the remaining molecules in the system. This theory is formally exact in the low density limit. While the theory is valid in general for large molecule and polymer liquids, we demonstrate its use here for the case of the simple diatomic liquid. In this application, good agreement is found at all densities for the radial distribution function and equation‐of‐state when compared with computer simulations. Furthermore, the theory appears to give pressures that are more thermodynamically consistent than those obtained with reference interaction site model (RISM) theory.

Microscopic equations of state of polyethylene: Hard-chain contribution to the pressure

The athermal contribution to the pressure of polyethylene is investigated via integral equations and mean field generalized Flory‐type theories. The molecules are modeled as fused‐hard‐sphere chains with fixed bond lengths and bond angles; torsional rotations are treated via the rotational isomeric state approximation with literature values for the trans–gauche energies. The hard sphere diameter is obtained by matching structure factor predictions of the polymer reference interaction site model (PRISM) theory for hard chains to data from wide‐angle scattering experiments. In all, five hard chain equations of state are investigated: three via different thermodynamic routes in the PRISM theory, and two via different extensions (to fused‐sphere chains) of the generalized Flory‐dimer (GFD) theory. The integral equation approaches consist of a free energy ‘‘charging’’ route, the compressibility route, and the ‘‘wall’’ route (where the pressure is obtained from the density profile of the fluid at a hard wall). ...

Density functional theory of freezing: Analysis of crystal density

The density functional theory of freezing is used to study the liquid to crystal phase transition in the hardsphere and Lennard‐Jones systems. An important step in the calculation is the parametrization of the solid phase average single particle density ρ(r). In this work two popular parametrizations are compared. The first method is a general Fourier decomposition of the periodic solid density in which the amplitude of each (non‐symmetry‐related) Fourier component is treated as an independent parameter. The second parametrization, which is more restrictive but easier to implement, approximates the solid density as a sum of Gaussian peaks centered at the sites of a periodic lattice. The two methods give essentially identical results for the phase diagrams for the two systems studied, but the crystal density predicted by the Fourier method exhibits significant anisotropies which are excluded from the Gaussian representation by construction.

The structure of a rotational isomeric state alkane melt near a hard wall

Polyatomic density functional theory was used to model tridecane chains near a hard wall under melt conditions. Polymer reference interaction site model (PRISM) liquid state theory provided the bulk structure input for the density functional. The density profile, the fractional distribution of sites, and the variation of the end‐to‐end separation of the chains as a function of distance from wall contact were calculated, and excellent agreement with the results of full multichain simulation was found.