**Principal Investigator**
Richard Sadus **Project **g36

School of Computer Science and Software Engineering,
**Machine **VP

Swinburne University of Technology

**Molecular Simulation of Phase Transitions**

Fluid-phase transitions are a direct consequence of intermolecular interactions. Historically, however, prediction of vapour-liquid equilibria has relied almost exclusively on approximate theoretical models or on empirical equations of state rather than on rigorous models for intermolecular interaction at high fluid densities. The advent of new molecular simulation techniques such as the Gibbs ensemble method provides an opportunity to apply directly our knowledge of intermolecular potentials to the prediction of phase equilibria of fluids. The advantage of molecular simulation over other predictive methods is that comparison of simulation results with experimental data provides an unambiguous test for the accuracy of theory, particularly for the intermolecular potential used in the simulation. For a one-component fluid, the only assumption made concerning molecular interactions is the choice of intermolecular potential.

It is commonly assumed that the outcome of molecular interactions can be adequately attributed to the effect of two-body interactions alone. Accurate two-body potentials have been developed for some of the noble gases and the influence of three- or more-body interactions has been incorporated as "density effects" in some models. Consequently, the role of three- or more-body interactions is inadequately documented. Calculations using three-body interactions are typically limited to those based on the triple-dipole dispersion term of Axilrod and Teller; those calculations commonly contribute 5-10% of the pairwise additive energy of the liquid phase. These results suggest that pairwise calculations alone cannot fully account for the effects of intermolecular interactions. Several simulation studies for phase coexistence of both one-component and two-component fluids using pairwise potentials but no work on deviations from pairwise additivity has been reported.

**What are the basic questions addressed?**

The purpose of the work on the Fujitsu VP2200 Supercomputer was to examine the role of three-body interactions on the vapour-liquid coexistence of simple fluids.

**What are the results to date and future of the
work?**

The Gibbs ensemble algorithm was successful implemented.
The vapour-liquid coexistence properties of argon were determined
using both two body and three-body potentials. The main conclusions
were:

l Three-body
repulsion must be included to determine accurately the contribution
of three-body interactions. Including only three-body dispersion
interactions (e.g., the Axilrod-Teller potential) can lead to
misleading results.

l Three-body
intermolecular repulsion accounts for approximately 45% of the
total three-body interaction of the fluid.

l The contributions from three-body attraction and three-body repulsions almost cancel. Because of this cancellation effect, two-body intermolecular potentials are a good approximation for intermolecular interaction in the vapour and liquid phases.

**What computational techniques are used and why
is a supercomputer required?**

The Lennard-Jones potential was used to calculate interactions between pairs of molecules. More accurate pair-potentials for argon are available. However, the Lennard-Jones potential is simple to implement and it can be used to predict accurately the coexistence curve of argon with the exception of the near-critical region.

Two contributions to three-body interactions were used in the simulations. The Axilrod-Teller term accounts for the contribution of three-body dispersion interactions. Sherwood et al. recognised that three-body repulsions are significant at small interatomic separations. They developed two approximate models for three-body repulsive interactions at small interatomic separations. Their electrostatic distortion model is particularly suitable for our simulations because both angle-dependent terms and the pair separations are already evaluated in calculation of the Axilrod-Teller term. The contribution of three-body repulsion to the non-additivity of the third virial coefficient may be equivalent to 45% of the magnitude of the Axilrod-Teller term

The NVT-Gibbs ensemble2 was used to simulate the coexistence of liquid and vapour phases. A total of 300 molecules were partitioned between two boxes to simulate the vapour and liquid phases. The temperature of the entire system is held constant and surface effects are avoided by placing each box at the centre of a periodic array of identical boxes. Equilibrium is achieved by attempting molecular displacements (for internal equilibrium), volume fluctuations (for mechanical equilibrium) and particle interchanges between the boxes (for material equilibrium). The Gibbs-ensemble method has recently been reviewed.

The simulations were performed in cycles with each cycle typically consisting of 300 attempted displacements, a single volume fluctuation and 10-2000 interchange attempts. The maximum molecular displacement and volume changes were adjusted to obtain, where possible, a 50% acceptance rate for the attempted move. The number of attempted particle interchanges depends on the achievement of a satisfactory acceptance rate (5-10%). Ensemble averages were accumulated only after the system had reached equilibrium. The equilibration period was 40000 cycles (approximately between 20 and 80 million configurations) and a further 40000 cycles were used to accumulate the averages. The calculations were truncated at intermolecular separations greater than half the box length and appropriate long-range corrections were used to obtain the full contribution of pair interactions to energy and pressure. The full (untruncated) three-body potential was calculated to avoid uncertainties that arise when calculating three-body long-range corrections from unknown pair-distribution functions. The configurational properties were updated after each successful move.

Ideally, the contributions of both two-body and three-body interactions to the configurational energy of the fluid should be updated for each attempted move. However, it is currently not computationally feasible to include three-body interactions in the acceptance criterion because of the very large increase in computing time required to recalculate accurately triplet interactions. Consequently, only changes to two-body interactions contributed to the acceptance criterion and the predicted phase coexistence curve is not affect by three-body forces. Instead, the effect of three-body interactions to the energy, chemical potential and the pressure were estimated by re-calculating their contribution periodically during the course of the simulation.

The purpose of the work on the Fujitsu VP2200 Supercomputer
was to examine the role of three-body interactions on the vapour-liquid
coexistence of simple fluids. The Gibbs ensemble was used to
simulate vapour-liquid coexistence for argon and the results are
compared with experimental data. Pairwise interactions were described
using the Lennard-Jones potential. Three-body dispersion interactions
were based on the Axilrod-Teller term; however, the contribution
of repulsive three-body interactions was also included using an
electrostatic distortion potential developed by Sherwood et al.