^{1,a)}, George Jackson

^{1,b)}, Felipe J. Blas

^{2}and Enrique de Miguel

^{2}

### Abstract

A novel test-area (TA) technique for the direct simulation of the interfacial tension of systems interacting through arbitrary intermolecular potentials is presented in this paper. The most commonly used method invokes the mechanical relation for the interfacial tension in terms of the tangential and normal components of the pressure tensor relative to the interface (the relation of Kirkwood and Buff [J. Chem. Phys.17, 338 (1949)]). For particles interacting through discontinuous intermolecular potentials (e.g., hard-core fluids) this involves the determination of functions which are impractical to evaluate, particularly in the case of nonspherical molecules. By contrast we employ a thermodynamic route to determine the surface tension from a free-energyperturbation due to a test change in the surface area. There are important distinctions between our test-area approach and the computation of a free-energy difference of two (or more) systems with different interfacial areas (the method of Bennett [J. Comput. Phys.22, 245 (1976)]), which can also be used to determine the surface tension. In order to demonstrate the adequacy of the method, the surface tension computed from test-area Monte Carlo (TAMC) simulations are compared with the data obtained with other techniques (e.g., mechanical and free-energy differences) for the vapor-liquid interface of Lennard-Jones and square-well fluids; the latter corresponds to a discontinuous potential which is difficult to treat with standard methods. Our thermodynamic test-area approach offers advantages over existing techniques of computational efficiency, ease of implementation, and generality. The TA method can easily be implemented within either Monte Carlo (TAMC) or molecular-dynamics (TAMD) algorithms for different types of interfaces (vapor-liquid, liquid-liquid, fluid-solid, etc.) of pure systems and mixtures consisting of complex polyatomic molecules.

We are grateful to Erich Müller, Jean-Pierre Hansen, and Ruth Lynden-Bell for useful discussions. One of the authors (G.J.G.) would like to thank BP Exploration for funding a studentship. We acknowledge further support from the EPSRC of the UK (GR/N03358, GR/N35991, and GR/R09497), the Joint Research Equipment Initiative (JREI) for computer hardware (GR/M94427), and the Royal Society-Wolfson Foundation for the award of a refurbishment grant. Two of the authors (F.J.B. and E.dM.) are also grateful for financial support from Project No. FIS2004-06227-C02-01 of the Spanish Dirección General de Investigación. One of the authors (E.dM.) also acknowledges financial support from Secretaría de Estado de Educación y Universidades (Spain) within the Programa de Movilidad de Profesorado.

I. INTRODUCTION

II. SIMULATION OF THE INTERFACIAL TENSION

A. Mechanical relation

B. Thermodynamic free-energy difference

C. Finite-size scaling

D. Thermodynamic free-energyperturbation: New test-area approach

III. RESULTS AND DISCUSSION

IV. CONCLUSION

### Key Topics

- Surface tension
- 134.0
- Free energy
- 110.0
- Monte Carlo methods
- 27.0
- Tensor methods
- 26.0
- Helmholtz free energy
- 21.0

## Figures

Comparison of the reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table I for details); the squares to preliminary TAMC data (Ref. 102); and the diamonds to the results of Trokhymchuk and Alejandre (Ref. 34) determined from the mechanical route. The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 104).

Comparison of the reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table I for details); the squares to preliminary TAMC data (Ref. 102); and the diamonds to the results of Trokhymchuk and Alejandre (Ref. 34) determined from the mechanical route. The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 104).

Comparison of the reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table II for details); the squares to preliminary TAMC data (Ref. 102); the triangles to the results of Trokhymchuk and Alejandre (Ref. 34) determined from the mechanical route; the asterisk to the result of Miyazaki *et al.* (Ref. 26) and the diamonds to the results of Salomons and Mareschal (Ref. 57) both obtained with the free-energy difference method of Bennett; the circles to the data of Potoff and Panagiotopoulos (Ref. 81), and the crosses to that of Hunter and Reinhardt (Ref. 80), both obtained using a finite-size scaling technique. The continuous curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 105). The dashed curve corresponds to the unconstrained correlation to the data with and .

Comparison of the reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table II for details); the squares to preliminary TAMC data (Ref. 102); the triangles to the results of Trokhymchuk and Alejandre (Ref. 34) determined from the mechanical route; the asterisk to the result of Miyazaki *et al.* (Ref. 26) and the diamonds to the results of Salomons and Mareschal (Ref. 57) both obtained with the free-energy difference method of Bennett; the circles to the data of Potoff and Panagiotopoulos (Ref. 81), and the crosses to that of Hunter and Reinhardt (Ref. 80), both obtained using a finite-size scaling technique. The continuous curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 105). The dashed curve corresponds to the unconstrained correlation to the data with and .

Comparison of the reduced surface tension for the vapor-liquid interface of the square-well system with a range of as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table III for details); and the squares to preliminary TAMC data (Refs. 16 and 102). The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 106).

Comparison of the reduced surface tension for the vapor-liquid interface of the square-well system with a range of as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table III for details); and the squares to preliminary TAMC data (Refs. 16 and 102). The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 106).

Comparison of the reduced surface tension for the vapor-liquid interface of the square-well system with a range of as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table IV for details); the squares to preliminary TAMC data (Refs. 16 and 102); the triangles to the results of Orea *et al.* (Ref. 45) from the mechanical route; and the asterisks to the data obtained by Singh *et al.* (Ref. 47) with a finite-size scaling approach. The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 106).

Comparison of the reduced surface tension for the vapor-liquid interface of the square-well system with a range of as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table IV for details); the squares to preliminary TAMC data (Refs. 16 and 102); the triangles to the results of Orea *et al.* (Ref. 45) from the mechanical route; and the asterisks to the data obtained by Singh *et al.* (Ref. 47) with a finite-size scaling approach. The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 106).

Comparison of the reduced surface tension for the vapor-liquid interface of the square-well system with a range of as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table V for details); the squares to preliminary TAMC data (Refs. 16 and 102); and the triangles to the data obtained by Singh *et al.* (Ref. 47) from a finite-size scaling technique. The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 106).

Comparison of the reduced surface tension for the vapor-liquid interface of the square-well system with a range of as a function of reduced temperature : the black circles correspond to the simulation results obtained in this work with the TAMC technique (see Table V for details); the squares to preliminary TAMC data (Refs. 16 and 102); and the triangles to the data obtained by Singh *et al.* (Ref. 47) from a finite-size scaling technique. The curve represents the Guggenheim corresponding-states law [cf. Eq. (65)] with , , and (Ref. 106).

## Tables

The reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) computed with the test-area Monte Carlo (TAMC) simulation method as a function of reduced temperature . The systems of LJ particles are equilibrated for MC cycles, and averages are accumulated over ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from ten subaverages each of .

The reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) computed with the test-area Monte Carlo (TAMC) simulation method as a function of reduced temperature . The systems of LJ particles are equilibrated for MC cycles, and averages are accumulated over ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from ten subaverages each of .

The reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) computed with the TAMC simulation method as a function of reduced temperature . The systems of LJ particles are equilibrated for MC cycles, and averages are accumulated over ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from ten subaverages each of .

The reduced surface tension for the vapor-liquid interface of the Lennard-Jones system with a cutoff (unshifted) computed with the TAMC simulation method as a function of reduced temperature . The systems of LJ particles are equilibrated for MC cycles, and averages are accumulated over ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from ten subaverages each of .

The reduced surface tension for the vapor-liquid interface of the square-well system with a range computed with the TAMC simulation method as a function of reduced temperature . The systems of SW particles are equilibrated for MC cycles, and averages are accumulated over ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from 40 subaverages each of .

The reduced surface tension for the vapor-liquid interface of the square-well system with a range computed with the TAMC simulation method as a function of reduced temperature . The systems of SW particles are equilibrated for MC cycles, and averages are accumulated over ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from 40 subaverages each of .

The reduced surface tension for the vapor-liquid interface of the square-well system with a range computed with the TAMC simulation method as a function of reduced temperature . The systems of SW particles are equilibrated for MC cycles, and averages are accumulated over a ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from 20 subaverages each of .

The reduced surface tension for the vapor-liquid interface of the square-well system with a range computed with the TAMC simulation method as a function of reduced temperature . The systems of SW particles are equilibrated for MC cycles, and averages are accumulated over a ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from 20 subaverages each of .

The reduced surface tension for the vapor-liquid interface of the square-well system with a range computed with the TAMC simulation method as a function of reduced temperature . The systems of SW particles are equilibrated for MC cycles, and averages are accumulated over a further ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from 20 subaverages each of .

The reduced surface tension for the vapor-liquid interface of the square-well system with a range computed with the TAMC simulation method as a function of reduced temperature . The systems of SW particles are equilibrated for MC cycles, and averages are accumulated over a further ; 1 cycle involves MC trial displacements. Test changes in the area of are made once every cycle. The errors are estimated from the standard deviation of the mean determined from 20 subaverages each of .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content