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8.6DISSIPATIVE PARTICLE DYNAMICSPep Espa˜nolDept. Física Fundamental, Universidad Nacional de Educaci´on a Distancia,Aptdo. 60141, E-28080 Madrid, Spain1. The Original DPD ModelIn order to simulate a complex fluid like a polymeric or colloidal fluid,a molecular dynamics simulation is not very useful. The long time and spacescales involved in the mesoscopic dynamics of large macromolecules orcolloidal particles as compared with molecular scales imply to follow anexceedingly large number of molecules during exceedingly large times. Onthe other hand, at these long scales, molecular details only show up in a rathercoarse form, and the question arises if it is possible to deal with coarse-grainedentities that reproduce the mesoscopic dynamics correctly. Dissipative particledynamics (DPD) is a fruitful modeling attempt in that direction.DPD is a stochastic particle model that was introduced originally as anoff-lattice version of Lattice gas automata (LGA) in order to avoid its latticeartifacts [1]. The method was put in a proper statistical mechanics contexta few years later [2] and the number of applications since then is growingsteadily. The original DPD model consists of a collection of soft repellingfrictional and noisy balls. From a physical point of view, each dissipative par-ticle is regarded not as a single molecule of the fl uid but rather as a collectionof molecules that move in a coherent fashion. In that respect, DPD can beunderstood as a coarse-graining of molecular dynamics. There are three typesof forces between dissipative particles. The first type is a conservative forcederiving from a soft potential that tries to capture the effects of the “pres-sure” between different particles. T he second type of force is a friction forcebetween the particles that wants to describe the viscous resistance in a realfluid. This force tries to reduce velocity differences between dissipative par-ticles. Finally, there is a stochastic force that describe the degrees of freedomthat have been eliminated from the description in the coarse-graining process.2503S. Yip (ed.),Handbook of Materials Modeling, 2503–2512.c! 2005 Springer. Printed in the Netherlands.2504 P. Espa˜nolThis stochastic force will be responsible for the Brownian motion of polymerand colloidal particles simulated with DPD.The postulated stochastic differential equations (SDEs) that define the DPDmodel are [2]dri= vidt (1)midvi=!j =/ iFCij(rij)dt − γ!j =/ iω(rij)(eij·vij)eijdt+ σ!j =/ iω1/2(rij)eijdWijHere, ri, viare the position and velocity of the dissipative particles, miis themass of particle i, FCijis the conservative repulsive force between dissipativeparticles i, j, rij= ri−rj, vij= vi−vj, and the unit vector from the j th particleto the ith particle is eij= (ri− rj)/rijwith rij= |ri− rj|. The friction coeffi-cient γ governs the overall magnitude of the dissipative force, and σ is a noiseamplitude that governs the intensity of the stochastic forces. The weight func-tion ω(r) provides the range of interaction for the dissipative particles andrenders the model local in the sense that the particles interact only with theirneighbors. A usual selection for the weight function in the DPD literature is alinear function with the shape of a Mexican hat, but there is no special reasonfor such a selection. Finally, dWij=dWjiare independent increments of theWiener process that satisfy the Itˆo calculus rule dWijdWi#j#= (δii#δjj#+ δij#δji#)dt. There are several remarkable features of the above SDEs. They are trans-lationally, rotationally and Galilean invariant. Most importantly, total momen-tum is conserved, d("ipi)/dt = 0, because the three types of forces satisfyNewton’s Third Law. Therefore, the DPD m odel captures the essentials ofmass and momentum conservation which are responsible for the hydrody-namic behavior of a fluid at large scales [3, 4]. Despite its appearance asLangevin equations, Eq. (2) is quite different from the ones used in BrownianDynamics simulations. In the Brownian Dynamics method, total momentumof the particles is not conserved and only mass diffusion can be studied.The above SDE are mathematically equivalent to a Fokker–Planck equa-tion (FPE) that governs the time-dependent probability distribution ρ(r,v; t)of positions and velocities of the particles. The explicit form of the FPE canbe found in Ref. [2]. Under the assumption that the noise amplitude andthe friction coefficient are related by the fluctuation–dissipation relation σ =(2kBT γ)1/2, the equilibrium distribution ρeqof the FPE has the familiar formρeq(r, v) =1Zexp#−1kBT$!imiv2i2+ V (r )%&(2)where V is the potential function that gives rise to the conservative forces FC,kBis Boltzmann’s constant, T is the equilibrium temperature and Z is thenormalizing partition function.Dissipative particle dynamics 25052. DPD Simulations of Complex FluidsOne of the most attractive features of the model is its enormous versatilityin order to construct simple models for complex fl uids. In DPD, the New-tonian fluid is made “complex” by adding additional interactions between thefluid particles. Just by changing the conservative interactions between the fluidparticles, one can easily construct polymers, colloids, amphiphiles, and mix-tures. Given the simplicity of modeling of mesostructures, DPD appears as acompetitive technique in the field of complex fluids. We review now some ofthe applications of DPD to the simulation of different complex fluids systems(see also Ref. [5]).Colloidal particles are constructed by freezing fl uid particles inside certainregion, typically spheres or ellipsoids, and m oving those particles as a rigidbody. The idea was pioneered by Koelman and Hoogerbrugge [6] and has beenexplored in more detail by Boek et al. [7]. The simulation results for shear thin-ning curves of spherical particles compare very well with experimental resultsfor volume fractions below 30%. At higher volume fractions somewhat incon-sistent results are obtained, which can be attributed to several factors. The col-loidal particles modeled in this way are to certain degree “soft balls” that caninterpenetrate leading to unphysical interactions. At high volume fractions sol-vent particles are expelled from the region in between two colloidal particles.Again, this misrepresents the hydrodynamic interaction, which is mostly dueto lubrication forces [8]. Depletion forces appear [9, 10] which are unphys-ical and

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