Dissipative Particle Dynamics: A Method to Simulate Soft Matter Systems in Equilibrium and Under Flow - Selected Topics of Computational and Experimental Fluid Mechanics - page 60

4 Selected Examples of DPD in Flow Simulations

4.1 Polymer Brushes with Soft Potentials

There are a number of applications in soft matter systems that require of surfaces
covered by polymers to improve factors such as colloidal stability, reduction of
interfacial tension for wetting purposes, or improvement of lubrication. The most
common procedure to accomplish this is by grafting polymer chains of varying
length to surfaces, changing the grafting density also. Within the context of com-
puter simulations, when one is dealing with systems that model colloidal dispersions
or surfaces with grafted polymers, to ensure that the confined system is in chemi-
cal, mechanical and thermal equilibrium with the non-confined fluid it is necessary
to use the so-called Grand Canonical ensemble, which is well-known in statistical
mechanics. It requires that the chemical potential, the volume of the system and the
temperature be fixed during the simulation. To do so, one needs to perform Monte
Carlo simulations, which are relatively demanding from the computational point of
view because the number of particles in the simulation box is fluctuating and, as the
concentration is increased, the probability of introducing new particles into the sys-
tem becomes exceedingly small. However, for moderate densities it has been shown
to be a very useful tool. Within the context of DPD there are now some works that
have explored the properties of fluids confined by surfaces covered with polymers,
using the Monte Carlo method in the Grand Canonical ensemble (MCGC). A com-
mon approximation consists of substituting the colloidal particles by planar surfaces,
given the disparity of sizes between them and the solvent molecules. Also, planar
surfaces are useful to model pores or mimic experimental arrangements such as that
of the atomic force microscope or the surface force apparatus (Israelachvili 2011 ).
These planar walls have been implemented in DPD through two methods: either by
fixing in space some layers of DPD particles, placed at the ends of the simulation box
(Goujon et al. 2004 ), or by introducing an effective wall potential that acts on parti-
cles close to the ends of the simulation box (Gama Goicochea 2007 ). Several models
for the wall potential have been proposed; the first one being a linearly decaying
force, in the spirit of the other DPD forces, namely (Gama Goicochea 2007 ):

a w , i 1

z
z c

F i ( z ) = − z dž . (13)

dž

z direc-
tion (perpendicular to the plane where the surfaces are placed), a w , i is the inter-
action strength of such force, and z c is a cutoff distance, beyond which the force
becomes identically zero. For obvious reasons, this model is usually called the “DPD
wall” force. Recently, a self-consistent surface force was obtained for DPD particles
(Gama Goicochea and Alarcón 2011 ). It considers an infinite planar wall made up of
regularly spaced, identical DPD particles that interact with other particles on the wall
and with particles in the fluid through the usual DPD conservative force (see Eq. 7

Equation ( 13 ) represents the force that acts on the i th-particle in the