^{1}, Jürgen Horbach

^{2}, Kurt Binder

^{2,a)}, Michael E. Fisher

^{3}and Jan V. Sengers

^{3}

### Abstract

A symmetrical binary, Lennard-Jones mixture is studied by a combination of semi-grand-canonical Monte Carlo (SGMC) and molecular dynamics (MD) methods near a liquid-liquid critical temperature . Choosing equal chemical potentials for the two species, the SGMC switches identities to generate well-equilibrated configurations of the system on the coexistence curve for and at the critical concentration, , for . A finite-size scaling analysis of the concentration susceptibility above and of the order parameter below is performed, varying the number of particles from to 12 800. The data are fully compatible with the expected critical exponents of the three-dimensional Ising universality class. The equilibrium configurations from the SGMC runs are used as initial states for microcanonical MD runs, from which transport coefficients are extracted. Self-diffusion coefficients are obtained from the Einstein relation, while the interdiffusion coefficient and the shear viscosity are estimated from Green-Kubo expressions. As expected, the self-diffusion constant does not display a detectable critical anomaly. With appropriate finite-size scaling analysis, we show that the simulation data for the shear viscosity and the mutual diffusion constant are quite consistent both with the theoretically predicted behavior, including the critical exponents and amplitudes, and with the most accurate experimental evidence.

Two of the authors, (M.E.F. and S.K.D.), are grateful for support from the National Science Foundation under Grant No. CHE 03-01101. S.K.D. also acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) via Grant No. Bi 314/18-2 and thanks Professor Kurt Binder and Dr. Jürgen Horbach for supporting his stay in the Johannes Gutenberg Universität Mainz, Germany, where all the simulations were carried out with their close collaboration. The authors appreciate correspondence with Professor Y. Shapir, Professor A. Onuki, and Professor A. Yethiraj.

I. INTRODUCTION

II. MODEL AND SIMULATION TECHNIQUES

III. STATIC CRITICAL PROPERTIES

IV. SELF-DIFFUSION COEFFICIENT AND SHEAR VISCOSITY NEAR CRITICALITY

V. INTERDIFFUSION: FINITE-SIZE SCALING

VI. SUMMARY

### Key Topics

- Critical point phenomena
- 21.0
- Self diffusion
- 13.0
- Shear rate dependent viscosity
- 12.0
- Monte Carlo methods
- 10.0
- Data analysis
- 9.0

## Figures

Probability distributions of the relative concentration of particles for and chemical potential difference at several temperatures (a) below and (b) above , respectively. For clarity many independent data points have been omitted. Also, for clarity, error bars are not shown in this and the following figures, if they are comparable to the size of the symbols.

Probability distributions of the relative concentration of particles for and chemical potential difference at several temperatures (a) below and (b) above , respectively. For clarity many independent data points have been omitted. Also, for clarity, error bars are not shown in this and the following figures, if they are comparable to the size of the symbols.

Coexistence curve of the symmetrical (truncated) Lennard-Jones binary fluid in the plane of temperature and concentration , for overall density , the precise choice of potentials being given in Eqs. (1)–(4). The open circles are the simulation results for a system of particles, while the broken curve is only a guide to the eye. The solid curve indicates a fit to Eq. (13) which yields as highlighted by the horizontal dot-dashed line.

Coexistence curve of the symmetrical (truncated) Lennard-Jones binary fluid in the plane of temperature and concentration , for overall density , the precise choice of potentials being given in Eqs. (1)–(4). The open circles are the simulation results for a system of particles, while the broken curve is only a guide to the eye. The solid curve indicates a fit to Eq. (13) which yields as highlighted by the horizontal dot-dashed line.

Finite-size scaling plots of the susceptibility for temperatures above using the trial values of marked in the figure. The Ising values and have been accepted and simulation results for at temperatures , 1.46, 1.48, 1.50, 1.52, and 1.55 are presented. Particle numbers from to are included, as indicated (while the linear dimensions of the simulation box are ). The dashed lines are guides to the eye: in light of the degree of data collapse and the expected scaling function behavior stated in Eq. (17), the estimates and 1.421 are quite acceptable.

Finite-size scaling plots of the susceptibility for temperatures above using the trial values of marked in the figure. The Ising values and have been accepted and simulation results for at temperatures , 1.46, 1.48, 1.50, 1.52, and 1.55 are presented. Particle numbers from to are included, as indicated (while the linear dimensions of the simulation box are ). The dashed lines are guides to the eye: in light of the degree of data collapse and the expected scaling function behavior stated in Eq. (17), the estimates and 1.421 are quite acceptable.

The fourth-order cumulant plotted vs for several system sizes, as indicated in the figure. The broken horizontal line indicates the value of the at for Ising-type systems. The vertical line at represents our preferred estimate of . The smooth curves in the enlarged plot (b) are fits to tanh functions.

The fourth-order cumulant plotted vs for several system sizes, as indicated in the figure. The broken horizontal line indicates the value of the at for Ising-type systems. The vertical line at represents our preferred estimate of . The smooth curves in the enlarged plot (b) are fits to tanh functions.

Plot of the structure factors (a) and (b) for various temperatures vs momentum . The various curves are shifted up by 0.2 relative to one another for clarity. All data refer to a system of particles. The inset in part (b) represents an Ornstein-Zernike plot which yields estimates for via Eq. (22).

Plot of the structure factors (a) and (b) for various temperatures vs momentum . The various curves are shifted up by 0.2 relative to one another for clarity. All data refer to a system of particles. The inset in part (b) represents an Ornstein-Zernike plot which yields estimates for via Eq. (22).

Plots of (a) the reduced susceptibility and (b) the correlation length vs . Part (c) shows the variation of with . The lines represent fits using the anticipated Ising exponents. All the data refer to systems of particles.

Plots of (a) the reduced susceptibility and (b) the correlation length vs . Part (c) shows the variation of with . The lines represent fits using the anticipated Ising exponents. All the data refer to systems of particles.

(a) Log-log plot of the mean square displacements of all the particles vs time with , for systems containing particles, at the critical concentration and the seven temperatures indicated. The plots for different are displaced by factors of 2. (b) Variation of the reduced self-diffusion constant with temperature.

(a) Log-log plot of the mean square displacements of all the particles vs time with , for systems containing particles, at the critical concentration and the seven temperatures indicated. The plots for different are displaced by factors of 2. (b) Variation of the reduced self-diffusion constant with temperature.

A log-log plot of the reduced shear viscosity vs temperature. The line represents a least squares fit to the theoretical form (28) with and , yielding an amplitude .

A log-log plot of the reduced shear viscosity vs temperature. The line represents a least squares fit to the theoretical form (28) with and , yielding an amplitude .

Plot of the Stokes-Einstein diameter as defined in Eq. (29) vs temperature. The dashed line serves as a guide to the eye.

Plot of the Stokes-Einstein diameter as defined in Eq. (29) vs temperature. The dashed line serves as a guide to the eye.

Plot of the interdiffusion coefficient vs time at three different temperatures for systems of particles. The knees visible at short times are due to the discrete integration time step .

Plot of the interdiffusion coefficient vs time at three different temperatures for systems of particles. The knees visible at short times are due to the discrete integration time step .

Log-log plot of the interdiffusion coefficient as calculated vs . The line is a fit to the power law which yields . The data correspond to . Note that we do not show the error bars in this figure. However, error bars are shown in the subsequent presentations of the interdiffusional Onsager coefficient.

Log-log plot of the interdiffusion coefficient as calculated vs . The line is a fit to the power law which yields . The data correspond to . Note that we do not show the error bars in this figure. However, error bars are shown in the subsequent presentations of the interdiffusional Onsager coefficient.

Plot of the reduced Onsager coefficient vs for a system of particles. Note the “background” contribution and the sharp rise as is approached. The four highest data points span the range from 1.9% to 4% above ; but the experiments (see Ref. 11) probe the range .

Plot of the reduced Onsager coefficient vs for a system of particles. Note the “background” contribution and the sharp rise as is approached. The four highest data points span the range from 1.9% to 4% above ; but the experiments (see Ref. 11) probe the range .

Finite-size scaling plots of for the critical part of the reduced interdiffusional Onsager coefficient , with , , and trial values for the effective background contribution . The approximate Ising value has been adopted and, for convenience, we have set in the abscissa variable, , that approaches unity when . The filled symbols represent data at for different system sizes of to 6400 particles and fixed density . The solid arrows on the right hand axis indicate the central theoretical estimate for the critical amplitude : see text.

Finite-size scaling plots of for the critical part of the reduced interdiffusional Onsager coefficient , with , , and trial values for the effective background contribution . The approximate Ising value has been adopted and, for convenience, we have set in the abscissa variable, , that approaches unity when . The filled symbols represent data at for different system sizes of to 6400 particles and fixed density . The solid arrows on the right hand axis indicate the central theoretical estimate for the critical amplitude : see text.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content