Yuri Yamada

Fluid–fluid Transition and Negative Expansion in 2 Step-function Molecules System by Statistical Mechanics

In order to perform the statistical mechanical calculations, we adopt the periodic cubic system with two molecules in the unit cell. Our model potential function consists of a step-function and the hard sphere wall. We assume the minimum image convention and get the canonical partition function. We find the fluid–fluid phase transition and the negative thermal expansion in the system. We discuss the thermodynamic properties vs. temperature plots under the constant volume. We show some results of the Monte Carlo (MC) simulations on the system with the same potential function under the periodic cubic boundary condition for a comparison. The theoretical results on pVT relations are in agreement with the MC simulations on the two-molecule system. The results of 108-molecule system with the MC simulations are expected to have the similar phase transition near the state in the case of the two-molecule system.

Monte Carlo simulation on the free energy of homogeneous nucleation in the supersaturated Lennard–Jones vapor phase

The free energy of homogeneous nucleation is estimated by the Monte Carlo simulations on the supersaturated Lennard-Jones (LJ) vapor phase. The calculation is performed on the each system with a fixed number of particles N (1 < N < 109). The periodic boundary condition is assumed. The volume per a particle is 43.2 in units of LJ system. No carrier gas particles are included. The range of temperature is from 0.01 to 1.00 in units of Lennerd-Jones system. The initial configuration is distorted cubic lattice, and the sample is cooled to low temperature, where all molecules become a large cluster. In the second stage, it is heated to vaporize, where the monomer phase is obtained. In the third run, the low temperature sample is heated with a condition that the cluster is not allowed to decompose in the Monte Carlo (MC) simulations. The free energies of the cluster phase and the monomer one are obtained by the thermodynamic integration. The obtained critical size is about 30-40 at temperature 0.67 in units of LJ system, which is comparable to the molecular dynamics (MD) simulation with the carrier gas particles.

Van der Waals type equation of state for Lennard-Jones fluid and the fluctuation of the potential energy by molecular dynamics simulations

Molecular dynamics (MD) simulations of a Lennard-Jones system are performed to obtain the pVT and UVT relations. An extended van der Waals equation of state (EOS) is derived by statistical mechanics on the perturbation approximation. A hard sphere system is used as the reference system. The Ree–Hoover EOS is assumed for the hard sphere system. The attraction energy term in the canonical ensemble partition function is extended by a cluster expansion. The new EOS includes three parameters, two of which are the interaction parameters in the Lennard-Jones interaction. The last parameter is the effective volume of the hard sphere system as the reference. The extended van der Waals EOS reproduces the pVT and UVT relations, at least qualitatively, whereas the original van der Waals EOS can explain only the pVT relation. The present EOS can also explain the fluctuations of the potential energy as calculated by MD simulations with 1000 molecules in the unit cell. In this sense, the present EOS gives better heat capacities at constant volume C v, even at lower temperatures than the original van der Waals EOS, which includes the attraction part of the equation, with no temperature dependence in the internal energy U.

The phase diagram of the step-function system by molecular simulations

The pressure–volume–temperature (pVT) relations for intermolecular interactions described by a repulsive step-function are obtained by molecular dynamics (MD) and Monte Carlo (MC) simulations. The system is modeled as a cubic unit cell with periodic boundary condition, where the unit cell contains 108 molecules. The pressure is obtained by the virial equation, and the phase diagram is estimated by Maxwell construction. The self-diffusion coefficients are calculated by MD simulation and used to assign phases as solid or fluid. High- and low-density solid and fluid phases are identified at low temperatures, and negative expansion is observed in the fluid phase near the low-density solid.