# softmatter:Brownian Dynamics Simulation (BD)

Brownian dynamics (BD) is a mesoscale simulation method commonly used to study solute-solvent systems without explicitly considering the solvent particles. When using BD it is assumed that solvent particles are small as compared to solute particles. To avoid explicitily calculating solvent interactions, the solvent particles are treated as a viscous medium. To account for the Brownian motion and dissipative losses that occur as a result of collisions with large numbers of solvent particles a stochastic force and a drag force are implemented into the simulation. The result is that larger systems and longer time scales are accesible to BD over traditional methods such as Molecular Dynamics Simulation.

## Method

Similar to Dissipative Particle Dynamics, BD obeys Newton’s equations of motion. The trajectory of each Brownian particle is governed by the Langevin equation[1, 2]:

$m_{i}\mathbf{\ddot{r}}_{I}(t)= \mathbf{F}_{i}^C(\mathbf{r}_{i}(t))+\mathbf{F}_{i}^R(t) -\gamma\mathbf{v}_{i}(t)$


where mi is the mass of bead i, ri, vi, and Fi are the postion, velocity, and force acting on bead i, respectively. γ represents the friction coefficient acting on the beads. The force acting on a particle can again be broken into three components, a conservative, random, and dissipative force respectively. It is assumed that there are no temporal or spatial correlations in the drag force and the random force is assumed to be stationary, Markovian, and Gaussian with zero mean. The variance of the random force obeys the fluctuation dissipation theorem and must satisfy the following conditions[3].

$\langle \mathbf{F}_{i}^R(t) \rangle = 0$

$\langle \mathbf{F}_{i}^R(t) \mathbf{F}_{j}^R(t') \rangle = 6\gamma k_{B}T\delta_{ij}\delta(t-t')$


The friction and noise terms couple the system to a heat bath where the friction term acts as a heat sink and the noise term as a heat source. Due to the stochastic nature of the Langevin equation, the equations of motion are stabilized and numerical errors that accumulate over long simulation times are minimized.

## Integrating the equations of motion

Typical time steps used to integrate the equations of motion in the BD simulations are ΔtBD = 0.01-0.02, two to four times larger than those used in standard Molecular Dynamics Simulations.

It is assumed that the random and drag forces are independent of their history, i.e. the appropriate autocorrelation functions decay rapidly compared with the simulation time step, and the friction coefficient is taken to be isotropic. The assumptions about the random force and the friction force are justified when one is concerned with studying systems in thermodynamic equilibrium irrespective of their path to equilibrium. The assumption that the friction coefficient is isotropic can result in an overestimate of the drag force on a anisotropic particle. When one is interested in cases where the system is in equilibrium and when the conservative force acting on a particle is large compared to the drag force, this assumption is justified. Further details of the BD method can be found in reference[1]. The main feature that distinguishes BD from DPD is that in BD the random and dissipative forces do not act in a pairwise fashion and therefore hydrodynamic interactions are not included and momentum is not conserved. More sophisticated treatments of BD exist that incorporate hydrodynamics, at the cost of much more detailed equations of motion and more computationally intensive calculations.

## Examples

BD has been successfully used to study a host of applications, including the simulation of Surfactants, Block Copolymers, Tethered Building Blocks and Patchy Particles.

## References

• [1] W. F. van Gunsteren; H. J. C. Berendsen and J. A. C. Rullmann, "Stochastic dynamics for molecules with constraints: Brownian dynamics of n-alkanes," Molecular Physics, 44, 66-95, (1981)
• [2] G. S. Grest; M. D. Lacasse; K. Kremer and A. M. Gupta, "Efficient continuum model for simulating polymer blends and copolymers," J. Chem. Phys., 105, 10583-10594, (1996)
• [3] R. kubo, "Fluctuation -dissipation theorem," Reports on Progress in Physics, 29, 255-284, (1966)
• [4] D. Frenkel and B. Smit, " Understanding molecular simulation : from algorithms to applications.," Academic,: San Diego, Calif. ; London : (2002)
• [5] M. P. Allen and D. J. Tildesley, " Computer Simulation of Liquids," Clarendon Press: Oxford (1987)