# Brownian dynamics of confined rigid bodies

## Abstract

We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowingmore »

- Authors:

- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 (United States)

- Publication Date:

- OSTI Identifier:
- 22489694

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 143; Journal Issue: 14; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; DIFFUSION; LANGEVIN EQUATION; MONTE CARLO METHOD; PARTICLES; SEDIMENTS

### Citation Formats

```
Delong, Steven, Balboa Usabiaga, Florencio, and Donev, Aleksandar, E-mail: donev@courant.nyu.edu.
```*Brownian dynamics of confined rigid bodies*. United States: N. p., 2015.
Web. doi:10.1063/1.4932062.

```
Delong, Steven, Balboa Usabiaga, Florencio, & Donev, Aleksandar, E-mail: donev@courant.nyu.edu.
```*Brownian dynamics of confined rigid bodies*. United States. doi:10.1063/1.4932062.

```
Delong, Steven, Balboa Usabiaga, Florencio, and Donev, Aleksandar, E-mail: donev@courant.nyu.edu. Wed .
"Brownian dynamics of confined rigid bodies". United States. doi:10.1063/1.4932062.
```

```
@article{osti_22489694,
```

title = {Brownian dynamics of confined rigid bodies},

author = {Delong, Steven and Balboa Usabiaga, Florencio and Donev, Aleksandar, E-mail: donev@courant.nyu.edu},

abstractNote = {We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general, such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long time scales.},

doi = {10.1063/1.4932062},

journal = {Journal of Chemical Physics},

number = 14,

volume = 143,

place = {United States},

year = {Wed Oct 14 00:00:00 EDT 2015},

month = {Wed Oct 14 00:00:00 EDT 2015}

}