Parametrisation in Dissipative Particle Dynamics: Applications in Complex Fluids - Selected Topics of Computational and Experimental Fluid Mechanics - page 91

Fig. 2 Form of the
conservative force in the
DPD methodology

a

F ij = a ij ω C ( r ij ) e ij

F C

r c

r

representative monomers or sets of monomers) are joined by springs with a spring
constant k , so we have an extra conservative force of the form f ij =−

k r ij whenever

particle i is connected to particle j .
Dissipative forces account for the local viscosity of the medium, and are of
the form

r ij ) ˆ e ij · v ij ˆ F ij =− ʳˉ

D

( e ij , (7)

D

where v ij = v i − v j is the relative velocity, ʳ the dissipation constant, and ˉ ( r ij ) =

2 a dimensionless weight function.
Finally, the random (thermal) force disperses heat produced by the dissipative
force and invests it into Brownian motion in order to keep the temperature T locally
constant. It is of the form

( 1 − r ij / r c )

F ij =− ˃ˉ

R

( r ij ) ʾ ij ˆ e ij , (8)

/ √ ʴ t )

with ʾ ij , where ʴ t is the integration time-step and ʸ ij is a random

Gaussian number with zero mean and unit variance. A dimensionless weight function
ˉ

= ʸ ij ( 1

R

also appears.
Not all three forces are independent. The fact that the random force compensates
the energy dissipated in order to keep T constant means that it acts as a regulating
thermostat. The relation between the dissipative and random forces is

( r ij ) = ( 1 − r ij / r c )

2

2
2 k B T ,

˃

D R

ʳ = ˉ ( r ij ) = ˉ ( r ij ) , (9)

where k B is Boltzmann's constant. This expression is a consequence of the fulfillment
of the fluctuation-dissipation theorem (Español and Warren 1995 ).
When dealing with electrically charged species, such as polyelectrolytes, a prob-
lem with the DPD methodology, arising from the fact that the interactions are soft, is
the formation of ionic clusters which do not correspond to the real system. Electric
charges are usually treated as point charges whose potential diverges at their position
in space. In Groot ( 2003 ) and Gonzalez-Melchor et al. ( 2006 ) this problem is solved
by considering charge distributions over the DPD-particles. Suppose that we have a
system consisting of N particles, each one with a point charge q i and a position r i in