^{1}and Jürgen Horbach

^{1,a)}

### Abstract

We present a hybrid method for the simulation of colloidal systems that combines molecular dynamics (MD) with the Lattice Boltzmann (LB) scheme. The LB method is used as a model for the solvent in order to take into account the hydrodynamic mass and momentum transport through the solvent. The colloidal particles are propagated via MD and they are coupled to the LB fluid by viscous forces. With respect to the LB fluid, the colloids are represented by uniformly distributed points on a sphere. Each such point [with a velocity at any off-lattice position ] is interacting with the neighboring eight LB nodes by a frictional force , with being a friction coefficient and being the velocity of the fluid at the position . Thermal fluctuations are introduced in the framework of fluctuating hydrodynamics. This coupling scheme has been proposed recently for polymer systems by Ahlrichs and Dünweg [J. Chem. Phys.111, 8225 (1999)]. We investigate several properties of a single colloidal particle in a LB fluid, namely, the effective Stokes friction and long-time tails in the autocorrelation functions for the translational and rotational velocity. Moreover, a charged colloidal system is considered consisting of a macroion, counterions, and coions that are coupled to a LB fluid. We study the behavior of the ions in a constant electric field. In particular, an estimate of the effective charge of the macroion is yielded from the number of counterions that move with the macroion in the direction of the electric field.

We thank Burkhard Dünweg, Vladimir Lobaskin, and Thomas Palberg for stimulating discussions. Moreover, we thank Norio Kikuchi for a critical reading of the manuscript and Hans Knoth for his help in the preparation of Figs. 9 and 11. We acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) under Grants No. HO 2231∕1 and No. HO 2231∕2, by the SFB 625 “Von einzelnen Molekülen zu nanoskopisch strukturierten Materialien,” and by the SFB TR6 “Colloidal Dispersions in External Fields.”

I. INTRODUCTION

II. LATTICE BOLTZMANN METHOD

III. THE COUPLING OF THE LB FLUID TO COLLOIDAL PARTICLES

IV. THE HYBRID MD∕LB SCHEME

V. RESULTS

A. Friction coefficient

B. Long-time tails

C. Charged colloids

1. Model and simulation details

2. A colloidal particle in an electric field

VI. CONCLUSIONS

### Key Topics

- Colloidal systems
- 57.0
- Friction
- 19.0
- Hydrodynamics
- 19.0
- Electric fields
- 17.0
- Navier Stokes equations
- 17.0

## Figures

Sketch of the model for a colloidal particle. The centers of the small spheres represent the points at which the colloidal particle interacts with the LB fluid. The cylinders that connect the spheres with each other are just guides to the eye.

Sketch of the model for a colloidal particle. The centers of the small spheres represent the points at which the colloidal particle interacts with the LB fluid. The cylinders that connect the spheres with each other are just guides to the eye.

for particle of radius as a function of for the indicated values of . The dashed lines are fits with Eq. (33) and the solid lines is a fit with Eq. (32) for , using the hydrodynamic radius as a fit parameter. The inset shows the variation of the hydrodynamic radius , normalized by the assigned sphere radius as a function of (see text).

for particle of radius as a function of for the indicated values of . The dashed lines are fits with Eq. (33) and the solid lines is a fit with Eq. (32) for , using the hydrodynamic radius as a fit parameter. The inset shows the variation of the hydrodynamic radius , normalized by the assigned sphere radius as a function of (see text).

as a function of the particle radius for , and the system sizes and .

as a function of the particle radius for , and the system sizes and .

Time-dependent velocity correlation function (solid lines) of the colloidal particle and the normalized average velocity of the LB fluid at the surface of the particle (dashed lines) for (a) , (b) , (c) , and (d) . The box length was set to , the radius of the particle to , and its mass to . The dotted lines are exponential functions that describe the short-time decay of (note that no fit parameter is involved).

Time-dependent velocity correlation function (solid lines) of the colloidal particle and the normalized average velocity of the LB fluid at the surface of the particle (dashed lines) for (a) , (b) , (c) , and (d) . The box length was set to , the radius of the particle to , and its mass to . The dotted lines are exponential functions that describe the short-time decay of (note that no fit parameter is involved).

for at the indicated system sizes. The dotted line shows the power law (see text).

for at the indicated system sizes. The dotted line shows the power law (see text).

The same as in Fig. 5, but now for the angular velocity correlation function and for . The dotted line shows the power law (see text).

The same as in Fig. 5, but now for the angular velocity correlation function and for . The dotted line shows the power law (see text).

Velocity autocorrelation function for and , as calculated for a kicked particle (solid line) and with thermal fluctuations (circles).

Velocity autocorrelation function for and , as calculated for a kicked particle (solid line) and with thermal fluctuations (circles).

The same as in Fig. 7, but now for the angular velocity correlation function.

The same as in Fig. 7, but now for the angular velocity correlation function.

Representative configuration of counterion and coion distribution around a macroion of charge . The macroion is the central large grey sphere, the 555 counterions are the light gray coloured small spheres, and the 300 coions are the black small spheres.

Representative configuration of counterion and coion distribution around a macroion of charge . The macroion is the central large grey sphere, the 555 counterions are the light gray coloured small spheres, and the 300 coions are the black small spheres.

Plot of the radial density distribution of counterions and coions around the macroion for the two indicated charges . The system with contains 555 counterions and 300 coions, whereas there are 471 counterions and 350 coions in the system with . The inset shows the fluctuation of the temperature of the counterions around its assigned value (solid line).

Plot of the radial density distribution of counterions and coions around the macroion for the two indicated charges . The system with contains 555 counterions and 300 coions, whereas there are 471 counterions and 350 coions in the system with . The inset shows the fluctuation of the temperature of the counterions around its assigned value (solid line).

Snapshot of the ionic distribution around the macroion (for the system with ) with an applied electric field of (top) and (bottom), respectively. The electric field is applied in the direction of the black arrow. The rest is similar to Fig. 9.

Snapshot of the ionic distribution around the macroion (for the system with ) with an applied electric field of (top) and (bottom), respectively. The electric field is applied in the direction of the black arrow. The rest is similar to Fig. 9.

as a function of the external electric field . is the average number of counterions that move with the particle. The inset shows the drift velocity as a function of the electric field .

as a function of the external electric field . is the average number of counterions that move with the particle. The inset shows the drift velocity as a function of the electric field .

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