Leonard-Jones chain mixtures: Variational theory and Monte Carlo simulation results

The Helmholtz free energy and pressure for binary Lennard-Jones chain mixtures are derived using a variational first-order perturbation theory. The reference system consisting of a binary mixture of freely jointed tangent hard sphere chains is solved in the Percus—Yevick approximation. The optimal diameter of segments in the hard sphere chain reference system is determined by minimizing the Helmholtz free energy for the chain mixture through the Gibbs—Bogoliubov inequality. The Lennard-Jones potential is introduced via a perturbation on the reference system and uses only the interchain radial distribution function, which is solved in the Percus—Yevick approximation. This simplification is compensated for by introducing a state-independent parameter κ_B for pure components to account for intrachain (three-body) effects included in the simulation but omitted from the theory. A simple mixing rule is proposed to calculate mixture integrals appearing in the perturbation term. New simulation data are presented for binary Lennard-Jones chain mixtures for a range of temperatures T*, chain lengths m and chain fraction, plus Lennard-Jones size (σ) and energy (ε) parameters. Six physical systems are studied: a 4-mer/4-mer mixture with ε₂₂/ε₁₁ = 1.2 and equal size parameters; a 4-mer/4-mer mixture with σ₂₂/σ₁₁ = 1.5 and equal energy parameters; a 4-mer/8-mer mixture with equal parameters; a 4-mer/8-mer mixture with ε₂₂/ε₁₁ = 1.2 and equal size parameters; an 8-mer/8-mer mixture with ε₂₂/ε₁₁ = 1.2 and equal size parameters; and an 8-mer/8-mer mixture with σ₂₂/σ₁₁ = 1.5 and equal energy parameters. All six systems are simulated at three segment densities (ρ = 0.2, 0.4 and 0.8), two reduced temperatures (T* = kT/ε₁₁ = 4 and 8) and six chain fractions (X ₁ = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0). Good agreement is obtained between simulation and theory over a range of parameters and conditions.