such as the industrially important triblock polymer “pluronic” that consists

of three consecutive blocks ethylene oxide “ propylene oxide “ ethylene ox-

ide, e.g., EO13 -PO30 -EO13 . Spontaneous formation of lamellar, hexagonal,

bicubic and other structures has been observed, where the order remains lo-

cal and only very slowly extends to larger distances. When shear is applied,

ordering over longer distances is induced. See Fig. 10.1 for an example.3

3 Some of the relevant articles are Zvelindovski (1998a, 1998b), van Vlimmeren et al. (1999),

Maurits et al. (1998a, 1998b, 1999), Sevink et al. (1999) and Morozov et al. (2000).

10.3 The mean ¬eld approach to the chemical potential 303

a d

100 15,000

b e

1,000 20,000

c f

10,000 24,000

Figure 10.1 Six snapshots of the evolution of a diblock-copolymer melt of the type

A10 B10 in a mesoscopic continuum simulation at T = 300 K. At time 0 a homo-

geneous melt is subjected to a repulsive A“B interaction (χ = 0.8); it develops

a lamellar structure (a“c). After 10,000 time steps (c) a shear rate of 0.001 box

lengths per time step is imposed; the lamellar structure orients itself along the di-

rection of shear into a co-called perpendicular orientation (d“f). The dimensionless

density of A is shown as shades of gray only for values larger than its volume av-

eraged value (= 0.5). Figure courtesy of Dr Agur Sevink, Leiden University. See

also Zvelindovsky et al. (1998a, 1998b).

11

Dissipative particle dynamics

In this chapter we consider how continuum dynamics, described by contin-

uum equations that are themselves generalizations of systems of particles,

can be described by particles again. The particle description in this case is

not meant to be more precise than the continuum description and to repre-

sent the system in more detail, but is meant to provide an easier and more

physically appealing way to solve the continuum equations. There is the

additional advantage that multicomponent systems can be modeled, and

by varying the relative repulsion between di¬erent kinds of particles, phe-

nomena like mixing and spinodal decomposition can be simulated as well.

The particles represent lumps of ¬‚uid, rather than speci¬ed clusters of real

molecules, and their size depends primarily on the detail of the boundary

conditions in the ¬‚uid dynamics problem at hand. The size may vary from

superatomic or nanometer size, e.g., for colloidal systems, to macroscopic

size. Since usually many (millions of) particles are needed to ¬ll the required

volume with su¬cient detail, is it for e¬ciency reasons necessary that the

interactions are described in a simple way and act over short distances only

to keep the number of interactions low. Yet, the interactions should be su¬-

ciently versatile to allow independent parametrization of the main properties

of the ¬‚uid as density, compressibility and viscosity. Although dissipative

particle dynamics (DPD), which is meant to represent continuum mechan-

ics, di¬ers fundamentally from coarse-grained superatom models, which are

meant to represent realistic molecular systems in a simpli¬ed way, the dis-

tinction in practice is rather vague and the term DPD is often also used for

models of polymers that are closer to a superatom approach.

The origin of DPD can be identi¬ed as a paper by Hoogerbrugge and Koel-

man (1992),1 who described a rather intuitive way of treating ¬‚uid dynamics

1 See also Koelman and Hoogerbrugge (1993), who applied their method to the study of hard-

sphere suspensions under shear.

305

306 Dissipative particle dynamics

problems with particles. Essentially their model consists of particles with

very simple, short-ranged conservative interactions with additional friction

and noise terms that act pairwise and conserve momentum and average en-

ergy. The addition of friction and noise functions as a thermostat and allows

an extra parameter to in¬‚uence the viscosity of the model. But there are

predecessors: notably the scaled particle hydrodynamics (SPH), reviewed by

Monaghan (1988) with the aim to solve the equations of ¬‚uid dynamics by

the time evolution of a set of points. SPH was originally developed to solve

problems in astrophysics (Lucy, 1977). It is largely through the e¬orts of P.

Espa˜ol2 that DPD was placed on a ¬rm theoretical footing, and resulted in

n

a formulation where the equation of state (i.e., pressure and temperature as

functions of density and entropy or energy) and properties such as viscosity

and thermal conductivity can be used as input values in the model, rather

than being determined by the choice of interparticle interactions (Espa˜ol n

and Revenga, 2003). In Espa˜ol™s formulation each particle has four at-

n

tributes: position, momentum, mass and entropy, for which appropriate

stochastic equations of motion are de¬ned.3 Another model, originated by

Flekkøy and Coveney (1999),4 uses ¬‚uid “particles” based on Voronoi tesse-

lation that divides space systematically in polyhedral bodies attributed to

moving points in space.

We shall not describe these more complicated DPD implementations, but

rather give a short description of a popular and simple implementation of

DPD given by Groot and Warren (1997). This implementation is close to

the original model of Hoogerbrugge and Koelman (1992). One should be

aware that simplicity comes at a price: models of this kind have intrinsic

properties determined by the interaction functions and their parameters and

simulations are generally needed to set such properties to the desired values.

The model contains stochastic noise and friction, and represents therefore

a Langevin thermostat (see Chapter 6, page 196). Such a distributed ther-

mostat causes isothermal behavior rather than the adiabatic response that

is usually required in realistic ¬‚uid dynamics.

2 Espa˜ol (1995) derived hydrodynamic equations from DPD and evaluated the probability den-

n

sity from the Fokker“Planck equation corresponding to the stochastic equations of motions.

3 The formal derivation of thermodynamically consistent ¬‚uid particle models is based on the

GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) formalism

¨ ¨ ¨ ¨

of Ottinger (Grmela and Ottinger, 1997; Ottinger and Grmela, 1997; Ottinger, 1998). In this

formalism the change in a set of variables that characterize the state of a system is expressed in

terms of the dependence of energy and entropy on the state variables; this is done in such a way

that energy is conserved and entropy cannot decrease, while the ¬‚uctuation“dissipation theorem

is satis¬ed. See Espa˜ ol et al. (1999) for the application to hydrodynamic generalization.

n

4 See also Flekkøy et al. (2000) and Espa˜ ol (1998). Serrano and Espa˜ ol (2001) elaborated on

n n

this model and the two approaches were compared by Serrano et al. (2002).

11.1 Representing continuum equations by particles 307

11.1 Representing continuum equations by particles

The system consists of particles with mass mi , position r i and velocity v i .

Each particle represents a ¬‚uid element that moves coherently. The particles

interact pairwise though two types of forces: a potential-derived conservative

force and a dissipative friction force that depends on the velocity di¬erence

between two interacting particles. The energy dissipation due to the dissi-

pative force is balanced by a random force, so that the total average kinetic

energy from motion with respect to the local center of mass, excluding the

collective kinetic energy (the “temperature”), remains constant. Since all

forces act pairwise in the interparticle direction and are short-ranged, the

sum of forces is zero and both linear and angular momentum is conserved,

even on a local basis. Since mass, energy and momentum conservation are

the basis of the continuum equations of ¬‚uid dynamics, DPD dynamics will

follow these equations on length scales larger than the average particle sep-

aration and on time scales larger than the time step used for integration the

equations of motion.

The equations of motion are Newtonian: