Environmental Engineering Reference
In-Depth Information
1 Introduction
Dissipative Particle Dynamics (DPD) is already a well established simulation method
to study particle systems (Frenkel and Smit
2002
). In the seminal work by Hooger-
brugge and Koelman (
1992
) the motivation of developing DPD was to study the
hydrodynamic behavior with a particle based method and an integration scheme
very similar to that of Molecular Dynamics (MD). While hydrodynamic conditions
could be studied theoretically with MD, in practice, performing a simulation with the
typical time step of MD, for a large number of particles, so that the Reynolds number
of a fluid can be varied considerably, would be prohibitively expensive in computer
time, even with current computer systems (Murtola et al.
2009
). Additionally, most
of the thermostats, which would allow for an isothermal simulation in presence of
flow, will perturbe the hydrodynamic correlation between particles (Hünenberger
2005
). DPD tackled these two drawbacks to allow particle-based simulations in a
scheme very similar to MD (Frenkel and Smit
2002
; Allen and Tildesley
1987
).
As was originally deviced by Hoogerbrugge and Koelman (
1992
), Español and
Warren (
1995
), and Groot and Warren (
1997
), DPD allows for a higher time step
than usual MD simulations. This was achieved by defining the so-called “soft poten-
tials” as the interaction model between particles in the original DPD scheme. These
quadratic potentials, have linear and small derivatives (repulsive forces), as compared
to a typical interaction potential of MD simulation, such as the Lennard-Jones poten-
tial. The equations of motion of these “fluid particles” interacting with soft potentials
can be integrated with a time step with a factor of 10-100 times higher than that used
for the Lennard-Jones potential. This speed-up allows for the simulation of larger
systems, and therefore higher Reynolds numbers. The other important contribution
of DPD to the simulation of hydrodynamic phenomena is the use of a thermostat that
conserves locally linear and angular momentum, which is one of the assumptions of
the hydrodynamic continuum formulation of the equations of motion. Either locally
or globally, most of the widely used thermostats in MD, such as the Andersen, Nosé-
Hoover (Frenkel and Smit
2002
) or the Langevin (Hünenberger
2005
) thermostats,
violate Galilean invariance and momentum conservation, giving rise to screening of
hydrodynamic correlations (Soddemann et al.
2003
; Dünweg
2006
).
The DPD method solves this by establishing a thermostatic process similar to
Brownian Dynamics, in which a dissipative and a viscous force are applied over the
particles of the fluid. The difference is, however, that in the DPD method these forces
are applied in a pair-wise form, and in the direction of the line that connects a pair of
particles (see Fig.
1
). In this way, the total “external force” on the particles is zero,
resulting in local momentum conservation. Additional details will be given in Sect.
2
.
We can mention other two appealing features of DPD. It can be implemented from
a straight-forward modification of a working MD program, from which efficient
parallelization strategies have been deviced (Plimpton
1995
; Brown et al.
2012
).
Additionally, it has a great versatility to study complex fluids or other soft matter
systems in the hydrodynamics context by adding interactions among particles, for
example, harmonic or other spring-like interactions (bonded), to describe polymers,
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