Perspective: Dissipative Particle Dynamics

Pep Espanol1 and Patrick B. Warren2

1Dept. Fısica Fundamental, Universidad Nacional de Educacion a Distancia, Aptdo. 60141 E-28080, Madrid, Spain2Unilever R&D Port Sunlight, Quarry Road East, Bebington, Wirral, CH63 3JW, UK.

(Dated: December 13, 2016)

Dissipative particle dynamics (DPD) belongs to a class of models and computational algorithmsdeveloped to address mesoscale problems in complex fluids and soft matter in general. It is basedon the notion of particles that represent coarse-grained portions of the system under study andallow, therefore, to reach time and length scales that would be otherwise unreachable from micro-scopic simulations. The method has been conceptually refined since its introduction almost twentyfive years ago. This perspective surveys the major conceptual improvements in the original DPDmodel, along with its microscopic foundation, and discusses outstanding challenges in the field. Wesummarize some recent advances and suggests avenues for future developments.

I. INTRODUCTION

The behaviour of complex fluids and soft matter ingeneral is characterized by the presence of a large rangeof different time and space scales. Any attempt to re-solve simultaneously several time scales in a single sim-ulation scheme is confronted by the problem of takinga prohibitively large number of sufficiently small timesteps. Typically one proceeds hierarchically [1], by de-vising models and algorithms appropriate to the lengthand time scales one is interested in. Leaving aside quan-tum effects negligible for soft matter, at the bottom ofthe hierarchy we have Hamilton’s equations, with accu-rate albeit approximate potential energy functions, whichare solved numerically with molecular dynamics (MD).Nowadays some research teams can simulate billions ofparticles for hundreds of nanoseconds [2]. This opensup the possibility to study very detailed, highly realis-tic molecular models that capture essentially all the mi-croscopic details of the system. This is, of course, notenough in many situations encountered in soft matterand life sciences [3]. One can always think of a problemwell beyond computational capabilities: from the fold-ing of large proteins, to the replication of DNA, or thesimulation of an eukariotic cell, or the simulation of amammal, including its brain. While we are still very farfrom even well-posing some of these problems, it is ob-vious that science is pushing strongly towards more andmore complex systems.

Instead of using atoms moving with Hamilton’s equa-tions to describe matter, one can take a continuum ap-proach in which fields take the role of the basic variables.Navier-Stokes-Fourier hydrodynamics, or elasticity, ormany of the different continuum theories for complexfluid systems are examples of this approach [4]. Thesecontinuum theories are, in fact, coarse-grained versionsof the atomic system that rely on two key related con-cepts: 1) the continuum limit—i. e. a “point” of spaceon which the field is defined is, in fact, a volume elementcontaining a large number of atoms [5], and 2) the lo-cal equilibrium assumption—i. e. these volumes are largeenough to reproduce the thermodynamic behaviour of

the whole system [6]. The quantities from one volume el-ement to its neighbour are assumed to change little andthis allows the powerful machinery of partial differen-tial equations to describe mathematically the system atthe largest scales, allowing even to find analytical solu-tions for many situations. Nevertheless, the continuumequations are usually non-linear and analytical solutionsare not always possible. One resorts then to numeri-cal methods to solve the equations. Computational fluiddynamics (CFD) has evolved into a sophisticated field innumerical analysis with a solid mathematical foundation.

The length scales that can be addressed by contin-uum theories range from microns to parsecs. Remark-ably, the same equations (with the same thermodynam-ics and transport coefficients) can be used at very dif-ferent scales. Many of the interesting phenomena thatoccur in complex fluids occur at the mesoscale. Themesoscale can be roughly defined as the spatio-temporalscales ranging from 10–104 nm and 1–106 ns. These scalesrequire a number of atoms that make the simulation withMD readily unfeasible. On the other hand, it was shownin the early days of computer simulations by Alder andWainwright [7] that hydrodynamics is valid at surpris-ingly small scales. Therefore, there is a chance to use con-tinuum theory down to the mesoscale. However, at theseshort length scales the molecular discreteness of the fluidstarts to manifest itself. For example, a colloidal particleof submicron size experiences Brownian motion which isnegligible for macroscopic bodies like submarine ships.In order to address these small scales one needs to equipfield theories like hydrodynamics with fluctuating terms,as pioneered by Landau and Lifshitz [8]. The result-ing equations of fluctuating hydrodynamics also receivethe name of Landau-Lifshitz-Navier-Stokes (LLNS) equa-tion. There is much effort in the physics/mathematicalcommunities to formulate numerical algorithms with thestandards of usual CFD for the solution of stochasticpartial differential equations modeling complex fluids atmesoscales [9–17].

While the use of fluctuating hydrodynamics may be ap-propriate at the mesoscale, there are many systems forwhich a continuum hydrodynamic description is not ap-

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plicable (or it is simply unknown). Proteins, membranes,assembled objects, polymer systems et c. may require un-accessible computational resources to be addressed withfull microscopic detail but a continuum theory may notexist. In these mesoscale situations, the strategy to re-tain some chemical specificity is to use coarse-graineddescriptions in which groups of atoms are treated as aunit [18]. While the details of how to do this are verysystem specific, and an area of intense active research(see reviews in Refs. [19–21]), it is good to know thatthere is a well defined and sounded procedure for theconstruction of coarse-grained descriptions [22, 23] thatis known under the names of non-equilibrium statisticalmechanics, Mori-Zwanzig theory, or the theory of coarse-graining [4, 24–26]. Simulating everything, everywhere,with molecular detail can be not only very expensive butalso unnecessary. In particular, water is very expensiveto simulate and sometimes its effect is just to propagatehydrodynamics. Hence there is an impetus to develop atleast coarse-grained solvent models, but retain enoughsolute molecular detail to render chemical specificity.

At the end of the 20th century the simulation of themesoscale was attacked from a computational point ofview with a physicist intuitive, quick and dirty, approach.Dissipative particle dynamics (DPD) was one of the prod-ucts, among others [27–32], of this approach. DPD is apoint particle minimal model constructed to address thesimulation of fluid and complex systems at the mesoscale,when hydrodynamics and thermal fluctuations play arole. The popularity of the model stems from its al-gorithmic simplicity and its enormous versatility. Justby varying at will the conservative forces between thedissipative particles one can readily model complex flu-ids like polymers, colloids, amphiphiles and surfactants,membranes, vesicles, phase separating fluids, et c. Dueto its simple formulation in terms of symmetry principles(Galilean, translational, and rotational invariances) it isa very useful tool to explore generic or universal features(scaling laws, for example) of systems that do not dependon molecular specificity but only on these general prin-ciples. However, detailed information highly relevant forindustrial and technological processes requires the inclu-sion of chemical detail in order to go beyond qualitativedescriptions.

DPD, as originally formulated, does not include thischemical specificity. This is not a drawback of DPD perse, as the model is regarded as a coarse-grained versionof the system. Any coarse-graining process eliminatesdetails from the description and keeps only the relevantones associated to the length and time scales of the levelof description under scrutiny. However, as it will be ap-parent, the original DPD model could be regarded asbeing too simplistic and one can formulate models thatcapture more accurate information of the system withcomparable computational efficiency.

Since its initial introduction, the question “What dothe dissipative particles represent?” has lingered in theliterature, with intuitively appealing but certainly vague

answers like “groups of atoms moving coherently”. In thepresent Perspective we aim at answering this question byreviewing the efforts that have been taken in this direc-tion. We offer a necessarily brief overview of applications,and discuss some open questions and unsolved problems,both of fundamental and applied nature. Since the ini-tial formulation of the DPD model a number of excellentreviews [26, 33–39], and dedicated workshops [40, 41],have kept the pace of the developments. We hope thatthe present perspective complements these reviews witha balanced view about the more recent advances in thefield. We also provide a route map through the differentDPD variant models and their underlying motivation. Inthis doing, we hope to highlight a unifying conceptualview for the DPD model and its connection with the mi-croscopic and continuum levels of description.

This Perspective is organized as follows. In Sec. IIwe consider the original DPD model with its virtues andlimitations. In Sec. III we review models that have beenformulated in order to avoid the limitations of the originalDPD model. The SDPD model, which is the culminationof the previous models that links directly to the macro-scopic level of description (Navier-Stokes) is consideredin Sec. IV. The microscopic foundation of the DPD modelis presented in Sec. V. Finally, we present some selectedapplications in Sec. VI and conclude in Sec. VII.

II. THE ORIGINAL DPD MODEL

The original DPD model was introduced by Hooger-brugge and Koelman [42], and was formulated by thepresent authors as a proper statistical mechanics modelshortly after [43]. The DPD model consists on a set ofpoint particles that move off-lattice interacting with eachother with three types of forces: a conservative force de-riving from a potential, a dissipative force that tries toreduce radial velocity differences between the particles,and a further stochastic force directed along the line join-ing the center of the particles. The last two forces canbe termed as a “pair-wise Brownian dashpot” which, asopposed to ordinary Langevin or Brownian dynamics, ismomentum conserving. The Brownian dashpot is a min-imal model for representing viscous forces and thermalnoise between the “groups of atoms” represented by thedissipative particles. Because of momentum conservationthe behaviour of the system is hydrodynamic at suffi-ciently large scales [44–46].

The stochastic differential equations of motion for thedissipative particles are [43]

ri =vi

mivi =− ∂V

∂ri−∑j

γωD(rij)(vij · eij)eij

+∑j

σωR(rij)dWij

dteij (1)

3

FIG. 1: Dissipative particles interact pair-wise with a conser-vative linear repulsive force, and a Brownian dashpot made ofa friction force that reduces the relative velocity between theparticles and a stochastic force that gives kicks of equal sizeand opposite directions to the particles. These forces vanishesbeyond a cutoff radius rc.

where rij = |ri−rj | is the distance between particles i, j,vij = vi − vj is the relative velocity and eij = rij/rij isthe unit vector joining particles i and j. dWij is an inde-pendent increment of the Wiener process. In Eq. (1), γ isa friction coefficient and ωD(rij), ω

R(rij) are bell-shapedfunctions with a finite support that render the dissipativeinteractions local. Validity of the fluctuation-dissipationtheorem requires [43] σ and γ to be linked by the relationσ2 = 2γkBT and also ωD(rij) = [ωR(rij)]

2. Here kB isBoltzmann’s constant and T is the system temperature.As a result, the stationary probability distribution of theDPD model is given by the Gibbs canonical ensemble

ρ({r,v}) =1

Zexp

{−β

N∑i

miv2i

2− βV ({r})

}(2)

The potential energy V ({r}) is a suitable function of thepositions of the dissipative particles that is translation-ally and rotationally invariant in order to ensure linearand angular momentum conservation. In the original for-mulation the form of the potential function was taken ofthe simplest possible form

V ({r}) =1

2

∑ij

aij(1− rij/rc)2 (3)

where aij is a particle interaction constant and rc is acutoff radius. This potential produces a linear force withthe form of a Mexican hat function of finite range. With-out any other guidance, the weight function ωR(r) in thedissipative and random forces is given by the same linearfunctional form. Complex fluids can be modeled throughmesostructures constructed by adding additional inter-actions (springs and/or attractive or repulsive potentialsbetween certain particles) to the particles. Groot andWarren [47] offered a practical route to select the param-eters in a DPD simulation by matching compressibilityand solubility parameters of the model to real systems.

The soft nature of the weight functions in DPD al-lows for large time steps, as compared with MD thatneeds to deal with steep repulsive potentials. Howevertoo large time steps lead to numerical errors that dependstrongly on the numerical algorithm used. The area ofnumerical integrators for the stochastic differential equa-tions of DPD has received attention during the years with

increasingly sophisticated methods. Starting from thevelocity Verlet implementation of Ref. [47] and the self-consistent reversible scheme of Pagonabarraga and Hagen[48], the field has evolved towards splitting schemes [49–54]. A Shardlow [49] scheme has been recommended aftercomparison between different integrators [50], but thereare also other recent more efficient proposals [55–57].

Because of momentum conservation, the original DPDmodel in Eqs. (1)–(3) can be regarded as a (toy) model forthe simulation of fluctuating hydrodynamics of a simplefluid. As a model for a Newtonian fluid at mesoscales theDPD model has been used for the simulation of hydro-dynamics flows in several situations [58–64]. It shouldbe obvious, though, that the fact that DPD conservesmomentum does not makes it the preferred method forsolving hydrodynamics. MD is also momentum conserv-ing and can be used to solve hydrodynamics; for a re-cent review see Kadau et al. [65]. However, in termsof computational efficiency hydrodynamic problems arebest addressed with CFD methods with, perhaps, inclu-sion of thermal fluctuations.

In addition, the original DPD model suffers from sev-eral limitations that downgrade its utility as a LLNSsolver. The first one is the thermodynamic behaviour ofthe model. Taken as a particle method, the DPD modelhas an equation of state that is fixed by the conservativeinteractions. The linear conservative forces of the origi-nal DPD model produce an unrealistic equation of statethat is quadratic in the density [47]. The quadratic equa-tion of state in DPD seems to be a general property ofsoft sphere fluids at high overlap density. A well-knownexemplar is the Gaussian core model [66]. These systemshave been termed mean-field fluids and this includes thelinear DPD potential in Eq. (3). Many thermodynamicproperties for the linear DPD potential can be obtainedby using standard liquid state theory and it has been ourexperience that the HNC integral equation closure worksexceptionally well in describing the behaviour of DPD inthe density regime of interest [67–69]. Note that whileit is possible to fit the compressibility (related to secondderivatives of the free energy) to that of water, for exam-ple, the pressure (related to first derivatives) turns outto be unrealistic. The conservative forces of the originalmodel are not flexible enough to specify the thermody-namic behaviour as an input of the simulation code [70].

A second limitation is due to too simplistic frictionforces. The central friction force in Eq. (1) implies thatwhen a dissipative particle passes a second, reference par-ticle, it will not exert any force on the reference particleunless there is a radial component to the velocity [71, 72].Nevertheless, on simple physical grounds one would ex-pect that the passing dissipative particle would drag insome way the reference particle due to shear forces. Ofcourse, if many DPD particles are involved simultane-ously in between the two particles, this will result in aneffective drag. The same is true for a purely conserva-tive molecular dynamics simulation. It would be nice,though, to have this effect captured directly in terms of

4

modified friction forces in a way that a smaller numberof particles need to be used to reproduce large scale hy-drodynamics. Note that the viscosity of the DPD modelcannot be specified beforehand, and only after a recourseto the methods of kinetic theory can one estimate thefriction coefficient to be imposed in order to obtain agiven viscosity [44, 45, 73, 74]. As we will see, inclusionof more sophisticated shear forces allows for a more directconnection with Navier-Stokes hydrodynamics.

A third limitation of DPD as a mesoscale hydrody-namic solver is the fact that the DPD model (in an iden-tical manner as MD) is hardwired to the scale. Whatwe mean with this is that given a physical problem,with a characteristic length scale, we may always puta given number of dissipative particles and parametrizethe model in order to recover some macroscopic informa-tion (typically, compressibilities and viscosity). However,if one uses a different number of particles for exactlythe same physical situation, one should start over andreparametrize the system again. This is certainly verydifferent from what one would expect from a Navier-Stokes solver, that specifies the equation of state andviscosity irrespective of the scale, and one simply wor-ries about having a sufficiently large number of pointsto resolve the characteristic length scales of the flow. Inother words, in DPD there is no notion of resolution,grid refinement, and convergence as in CFD. There havebeen attempts to restore a scale free property for DPD[72, 75, 76], even for bonded interactions [77]. To get thisproperty, the parameters in the model need to depend onthe level of coarse-graining, but this is not specified in theoriginal model. Closely related to this lack of scaling isthe fact that there is no mechanism in the model to switchoff thermal fluctuations depending on the scale at whichthe model is operating. On general statistical mechan-ics grounds, thermal fluctuations should scale as 1/

√N

where N is the number of degrees of freedom coarse-grained into one coarse-grained (CG) particle. As thedissipative particles represent larger and larger volumeelements, they should display smaller and smaller fluctu-ations. But there is no explicit volume or size associatedto a dissipative particle. This problem is crucial, for in-stance, in the case of suspended colloidal particles or inmicrofluidics applications where flow conditions and thephysical dimensions of the suspended objects or physicaldimensions of the operating device determine whetherand, more importantly, to what extent thermal fluctua-tions come into play.

Finally, another limitation of the DPD model is thatit cannot sustain temperature gradients. Energy in thesystem is dissipated and not conserved, and the Browniandashpot forces of DPD act as a thermostat.

III. MDPD, EDPD, FPM

During the years, the DPD model has been enriched inseveral directions in order to deal with all the above lim-

itations. In this Section we briefly review these enrichedDPD models.

The many-body (or multi-body) dissipative particledynamics (MDPD) method stands for a modification ofthe original DPD model in which the purely repulsiveconservative forces of the classic DPD model are replacedby forces deriving from a many-body potential; thus thescheme is still covered by Eqs. (1)–(2), but a many-bodyV ({r}) is substituted for Eq. (3). The MDPD methodwas originally introduced by Pagonabarraga and Frenkel[78], Warren [79], and independently by Groot, [80] andsubsequently modified and improved by Trofimov et al.[70], reaching a level of maturity [38, 76, 81–85]. Thekey innovation of the MDPD is the introduction of adensity variable di =

∑j 6=iW (rij), as well as a free en-

ergy ψ(di) associated to each dissipative particle. HereW (r) is a normalized bell-shaped weight function thatensures that the density di is high if many particles areaccumulated near the i-th particle. The potential of in-teraction of these particles is assumed to be of the formV =

∑i ψ(di) [86]. This is a many-body potential of

a form similar to the embedded atom potential in MDsimulations [87, 88]. For multi-component mixtures themany-body potential may be generalized to depend onpartial local densities.

Despite its many-body character, the resulting forcesare still pair-wise, and implementation is straightfor-ward. However, not all pair-wise force laws correspondto a many-body potential. Indeed the existence of suchseverely constrains the nature of the force laws, and someerrors have propagated into the literature (see discussionin Ref. [84]). In Appendix A we explore how the force lawis constrained by the weight function W (r). The messageis: if in doubt, always work from V ({r}).

MDPD escapes the straitjacket of mean-field fluid be-haviour by modulating the thermodynamic behaviour ofthe system directly at the interaction level between theparticles. This allows for more general equations of statethan in the original DPD model, which is a special casewhere the one-body terms are linear in local densities.Indeed, one can easily engineer a van der Waals loop inthe equation of state to accommodates vapor-liquid co-existence. But this in itself is not enough to stabilize avapour-liquid interface, since one should additionally en-sure that the cohesive contribution is longer-ranged thanthe repulsive contribution. This can be achieved for ex-ample by using different ranges for the attractive andrepulsive forces [79, 81], or modelling the square gradientterm in the free energy [89].

The energy-conserving dissipative particle dynamicsmodel (EDPD) was introduced simultaneously and in-dependently by Bonet Avalos and Espanol [90, 91] as away to extend the DPD model to non-isothermal situ-ation. In this case, the key ingredient is an additionalinternal energy variable associated to the particles. Thebehaviour of the model was subsequently studied [92–95].The method has been compared with standard flow sim-ulations [96, 97], and recently a number of interesting ap-

5

plications have emerged [98, 99], including heat transferin nanocomposites [100], shock detonations [101], phasechange materials for energy storage [102], shock load-ing of a phospholipid bilayer [103], chemically reactingexothermic fluids [104, 105], thermoresponsive polymers[106], and water solidification [107].

The fluid particle model (FPM) was devised as a wayto overcome the limitation of the simplistic friction forcesin DPD [71, 72]. The method introduced, in addition toradial friction forces, shear forces that depend not onlyon the approaching velocity but also on the velocity dif-ferences directly. Shear forces have been reconsideredrecently [108]. The resulting forces are non-central anddo not conserve angular momentum. In order to restoreangular momentum conservation a spin variable is intro-duced. Heuristically, the spin variable is understood asthe angular momentum relative to the center of mass ofthe fluid particle. The model has been used successfullyby Pryamitsyn and Ganesan [109] in the simulation ofcolloidal suspensions, where each colloid is representedby just one larger dissipative particle, an approach alsoused by Pan et al. [110].

IV. DPD FROM TOP-DOWN: THE SDPDMODEL

While MDPD is still isothermal and EDPD still usesconservative forces too limited to reproduce arbitrarythermodynamics, the two enrichments of a density vari-able and an internal energy variable introduced by thesemodels suggest a view of the dissipative particles as trulythermodynamic subsystems of the whole system, consis-tently with the local equilibrium assumption in contin-uum hydrodynamics. There have been a number of workstrying to formalize this view of “moving fluid particles”in terms of Voronoi cells of points moving with the flowfield [111]. Flekkøy et al. formulated a (semi) bottom-up approach for constructing a model of fluid particleswith the Voronoi tessellation [112, 113]. A thermody-namically consistent Lagrangian finite volume discretiza-tion of LLNS using the Voronoi tessellation was presentedby Serrano and Espanol [114] and compared favourably[115] with the models in Refs. [112] and [113]. Whilethis top-down modeling based on the Voronoi tessellationis grounded in a solid theoretical framework, it has notfound much application due, perhaps, to the computa-tional complexity of a Lagrangian update of the Voronoitessellation [116].

In an attempt to simplify the Lagrangian finite Voronoivolume discretization model, the smoothed dissipativeparticle dynamics (SDPD) model was introduced shortlyafter [117], based on its precursor [118]. SDPD is athermodynamically consistent particle model based ona particular version of smoothed particle hydrodynam-ics (SPH) that includes thermal fluctuations. SPH is amesh-free Lagrangian discretization of the Navier-Stokesequations (NSE) differing from finite volumes, elements,

or differences in that a simple smooth kernel is used forthe discretization of space derivatives. This leads to amodel of moving interacting point particles whose simu-lation is very similar to MD. SPH was introduced in anastrophysical context for the simulation of cosmic matterat large scales [119, 120], but has been applied since thento viscous and thermal flows [121, 122], including multi-phasic flow [123]. An excellent recent critical review onSPH is given by Violeau and Rogers [124].

In the particular SPH discretization given by SDPD ofthe viscous terms in the NSE, the resulting forces havethe same structure of the shear friction forces in the FPM.By casting the model within the universal thermodynam-ically consistent generic framework [4], thermal fluctu-ations are introduced consistently in SDPD by respectingan exact fluctuation-dissipation theorem at the discretelevel. Therefore, SDPD (as opposed to SPH) can addressthe mesoscopic realm where thermal fluctuations are im-portant.

The SDPD model consists on N point particles char-acterized by their positions and velocities ri,vi and, inaddition, a thermal variable like the entropy Si (by asimple change of variables, one can also use alterna-tively the internal energy εi or the temperature Ti).Each particle is understood as a thermodynamic sys-tem with a volume Vi given by the inverse of the density

di =∑N

i W (rij), a fixed constant mass mi, and an inter-nal energy εi = E(Si,mi,Vi) which is a function of theentropy of the particle, its mass (i. e. number of moles),and volume. The functional form of E(S,M,V) is as-sumed, through the local equilibrium assumption, to bethe same function that gives the global thermodynamicbehaviour of the fluid system (but see below). The equa-tions of motion of the independent variables are [117]

dri =vidt

mdvi =∑j

[Pi

d2i+Pj

d2j

]Fijrijdt

− 5η

3

∑j

Fij

didj(vij + eijeij ·vij) dt+mdvi

TidSi =5η

6

∑j

Fij

didj

(v2ij + (eij ·vij)

2)dt

− 2κ∑j

Fij

didjTijdt+ TidSi (4)

Here, Pi, Ti are the pressure and temperature of thefluid particle i, which are functions of di, Si through theequilibrium equations of state, derived from E(S,M,V)through partial differentiation. Because the volume ofa particle depends on the positions of the neighbours,the internal energy function plays the role of the po-tential energy V in the original DPD model. In addi-tion, vij = vi − vj , and Tij = Ti − Tj . The functionF (r) is defined in terms of the weight function W (r) as

∇W (r) = −rF (r). Finally, dvi, dSi are linear combina-tions of independent Wiener processes whose amplitude

6

is dictated by the exact fluctuation-dissipation theorem[125].

It is easily shown that the above model conserves mass,linear momentum and energy, and that the total entropyis a non-decreasing function of time thus respecting thesecond law of thermodynamics. The equilibrium distri-bution function is given by Einstein expression in thepresence of dynamic invariants [126]. As the number ofparticles increases, the resulting flow converges towardsthe solution of the Navier-Stokes equations, by construc-tion.

SDPD can be considered as the general version of thethree models MDPD, EDPD, FPM, discussed in Sec. III,incorporating all their benefits and none of its limita-tions. For example, the pressure and any other thermo-dynamic information is introduced as an input, as in theMDPD model. The conservative forces of the originalmodel become physically sounded pressure forces. En-ergy is conserved and we can study transport of energyin the system as in EDPD. The transport coefficients areinput of the model (though, see below). The range func-tions of DPD have now very specific forms, and one canuse the large body of knowledge generated in the SPHcommunity to improve on the more adequate shape forthe weight function W (r) [122]. The particles have aphysical size given by its physical volume and it is pos-sible to specify the physical scale being simulated. Oneshould understand the density number of particles as away of controlling the resolution of the simulation, offer-ing a systematic ‘grid’ refinement strategy. In the SDPDmodel, the amplitude of thermal fluctuations scales withthe size of the fluid particles: large fluid particles dis-play smaller thermal fluctuations, in accordance with theusual notions of equilibrium statistical mechanics. Whilethe fluctuations scale with the size of the fluid particles,the resultant stochastic forces on suspended bodies areindependent of the size of the fluid particles and only de-pend on the overall size of the object [127], as it should.

The SDPD model does not conserves angular momen-tum because the friction forces are non-central. This maybe remedied by including an extra spin variable as in theFPM as has been done by Muller et al. [128]. This spinvariable is expected to relax rapidly, more and more so asthe size of the fluid particles decreases. For high enoughresolution the spin variable is slaved by vorticity. Theauthors of Ref. [128] have shown, though, that the inclu-sion of the spin variable may be crucial in some prob-lems where ensuring angular momentum conservation isimportant [129].

In summary, SDPD can be understood as MDPD fornon-isothermal situations, including more realistic fric-tion forces. The SDPD model has a similar simplicity asthe original DPD model and its enriched versions MDPD,EDPD, FPM. It has been remarked [130] that SDPDdoes not suffer from some of the issues encountered inEulerian methods for the solution of the LLNS equa-tions. The SDPD model is applicable for the simulationof complex fluid simulations for which a Newtonian sol-

vent exists. The number of studies using SDPD is nowgrowing steadily and range from microfluidics,[131] andnanofluidics [130], colloidal suspensions [132, 133], blood[134, 135], tethered DNA [136], and dilute polymeric so-lutions [55, 137, 138]. Also, it has also been used for thesimulation of fluid mixtures [139–142], and viscoelasticflows [143].

Once SDPD is understood as a particle method forthe numerical solution of the LLNS equations of fluctu-ating hydrodynamics, the issue of boundary conditionsemerge. While there is an extensive literature in theformulation of boundary conditions in the deterministicSPH [121], and in DPD [144–154], the consideration ofboundary conditions in SDPD has been addressed onlyrecently [142, 155, 156].

In SDPD, what you put is almost what you get. Theinput information is the internal energy of the fluid par-ticles as a function of density and entropy (or tempera-ture), and the viscosity. However, only in the high resolu-tion limit, for a large number of particles it is ensured theconvergence towards the continuum equations. There-fore, for a finite number of particles there will be alwaysdifferences between the input viscosity and the actual vis-cosity of the fluid and, possibly, between the input ther-modynamic behaviour of the fluid particle and the bulksystem. These differences could be attributed to numeri-cal “artifacts” of the particle model, similar to discretisa-tion errors that arise in CFD. Often the worst effects ofthese artifacts can be eliminated by using renormalizedtransport coefficients from calibration simulations. Thisis similar, for instance, to the way that discretisation er-rors in lattice Boltzmann are commandeered to representphysics, improving the numerical accuracy of the scheme[157]. In this context the availability of a systematic gridrefinement strategy for SDPD is clearly highly beneficial.

A. Internal variables

The SDPD model is obtained from the discretizationof the continuum Navier-Stokes equations. Of course,any other continuum equations traditionally used forthe description of complex fluids can also be discretizedwith the same methodology. In general, these contin-uum models for complex fluids typically involve addi-tional structural or internal variables, usually represent-ing mesostructures, that are coupled with the conven-tional hydrodynamic variables [4, 158]. The coupling ofhydrodynamics with these additional variables rendersthe behaviour of the fluid non-Newtonian and complex.For example, polymer melts are characterized by addi-tional conformation tensors, colloidal suspensions can bedescribed by further concentration fields, mixtures arecharacterized by several density fields (one for each chem-ical species), emulsions are described with the amountand orientation of interface, et c.

All these continuum models rely on the hypothesis oflocal equilibrium and, therefore, the fluid particles are re-

7

garded as thermodynamic subsystems. Once the contin-uum equations are discretized in terms of fluid particles(Lagrangian nodes) with associated additional structuralor order parameter variables, the resulting fluid parti-cles are “large” portions of the fluid. The scale of thesefluid particles is supra-molecular. This allows one tostudy larger length and time scales than the less coarse-grained models where the mesostructures are representedexplicitly through additional interactions between par-ticles (i. e. chains for representing polymers, sphericalsolid particles to represent colloid, different types of par-ticles to represent mixtures). The price, of course, isthe need for a deep understanding of the physics at thismore coarse-grained level, which should be adequatelycaptured by the continuum equations.

For example, in order to describe polymer solutions, wemay take a level of coarse graining in which every fluidparticle contains already many polymer molecules. Thisis a more coarse-grained model than describing viscoelas-ticity by joining dissipative particles with springs [159].The state of the polymer molecules within a fluid parti-cle may be described either with the average end-to-endvector of the molecules [160, 161], or with a conformationtensor [143]. In this latter case, the continuum limit ofthe model leads to the Olroyd-B model of polymer rhe-ology. Another example where the strategy of internalvariables is successful is in the simulation of mixtures.Instead of modeling a mixture with two types of dissipa-tive particles as it is usually done in DPD, one may takea thermodynamically consistent view in which each fluidparticle contains the concentration of one of the species,for example [139, 140, 142, 162]. Chemical reactions canbe implemented by including as an internal degree of free-dom an extent of reaction variable [104].

V. DPD FROM BOTTOM-UP

The SDPD model [117], or the Voronoi fluid particlemodel [114], are top-down models which are, essentially,Lagrangian discretizations of fluctuating hydrodynamics.These models are the bona fide connection of the originalDPD model with continuum hydrodynamics. However,the connection of the model with the microscopic level ofdescription is less clear. Ideally, one would like to fulfillthe program of coarse-graining, in which starting fromHamilton’s equations for the atoms in the system, onederives closed equations for a set of CG variables thatrepresent the system in a fuzzy impressionistic way.

Coarse graining of a molecular system requires a cleardefinition of the mapping between the microscopic andCG degrees of freedom. This mapping is usually well de-fined when the atoms are bonded, as happens inside com-plex molecules like proteins and other polymer molecules,or in solid systems. In this case, one can choose groupsof atoms and look at, for example, the center of massof each group as CG variables. For unbonded atoms asthose occurring in a fluid system, the main problem is

that grouping atoms in a system where the atoms maydiffuse away from each other is a tricky issue. We discussseparately the strategies that have been followed in or-der to tackle the coarse-graining of both, unbonded andbonded atoms.

A. DPD for unbonded atoms

The derivation of the equations of hydrodynamics fromthe underlying Hamiltonian dynamics of the atoms is awell studied problem that dates back to Boltzmann andthe origins of kinetic theory [24, 163]. It is a problem thatstill deserves attention for discrete versions of hydrody-namics [164–167], which is what we need in order to sim-ulate hydrodynamics in a computer. These latter worksshow how an Eulerian description of hydrodynamics canbe derived from the Hamiltonian dynamics of the under-lying atoms, by defining mass, momentum, and energy ofcells which surround certain points fixed in space. How-ever, Lagrangian descriptions in which the cells “movefollowing the flow”, are much more tricky to deal with.Typically, two types of groupings of fluid molecules havebeen considered, based on the Voronoi tessellation or onspherical blobs.

An early attempt to construct a Voronoi fluid parti-cle out from the microscopic level was made by Espanolet al. [111]. The Voronoi centers were moved accordingto the forces felt by the molecules inside the cell in theunderlying MD simulation. An effective excluded vol-ume potential was obtained from the radial distributionfunction of the Voronoi centers. The method was re-visited by Eriksson et al. [168] who observed “molecularunspecificity” of the Voronoi projection, in the sense thatvery different microscopic models give rise to essentiallythe same dynamics of the cells. In earlier work [169], aforce covariance method, essentially the Einstein-Helfandroute to compute the Green-Kubo coefficients [170], wasintroduced in order to compute the friction forces underthe DPD ansatz. The results are disappointing as theseauthors showed that the dynamics of the CG particleswith the forces of the DPD model measured from MDfor a Lennard-Jones system were not consistent with theMD results themselves.

More recently, Hadley and McCabe [171] propose togroup water molecules into beads through the K-meansalgorithm [172]. The algorithm considers a number ofbeads with initially given positions and construct theirVoronoi tessellation. The water molecules inside eachVoronoi cell have a center of mass that does not coin-cide with the bead position. The bead position is thentranslated on top of the center of mass and a retessella-tion is made again, with a possibly different set of watermolecules constituting the new bead. The procedure isrepeated until convergence. At the end, one has cen-troidal Voronoi cells in which the bead position and thecenter of mass of the water molecules inside the Voronoicell coincide. The K-means algorithm gives for every

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microstate (coordinates of water molecules) the value ofthe macrostate (coordinates of the beads) and, therefore,provides a rule-based CG mapping. Unfortunately, thereis no analytic function that captures this mapping and,therefore, it is not possible to use the theory of coarse-graining to rigorously derive the evolution of the beads.The strategy by Hadley and McCabe is to construct theradial distribution function and infer from it the pairpotential. Recently, Izvekov and Rice [173] have alsoconsidered this procedure in order to compute both, theconservative force and the friction force between beadsby extracting this information from force and velocitycorrelations between Voronoi cells. They find that veryfew molecules per cell are sufficient to obtain Markovianbehaviour.

Instead of using Voronoi based fluid particles, Vothand co-workers consider a sphere (termed a blob) andmove the sphere according to the forces experienced bythe center of mass of the molecules inside it [174]. Thedynamics of the blob is then modeled in order to repro-duce the time correlations of the blob. Subsequently asystem of N Brownian blobs is constructed in order toreproduce the above correlations.

Recently, another attempt to obtain DPD from theunderlying MD has been undertaken by Lei et al. [175]by using the rigorous approach of the theory of coarse-graining. However, in order to construct the “fluid par-ticles” these authors constraint a collection of Lennard-Jones atoms to move bonded, by maintaining a specifiedradius of gyration. The fluid no longer is a simple atomicfluid but rather a fluid made of complex “molecules” (theatomic clusters constrained to have a radius of gyration)whose rheology is necessarily complex.

Our impression is that we still have not solved satis-factorily the problem of deriving from the microscopicdynamics the dynamics of CG particles that capture thebehaviour of a simple fluid made of unbonded atoms.Work remains to be done in order to define the properCG mapping for a fully satisfactory bottom-up modelfor Lagrangian fluid particles representing a set of fewunbonded atoms or molecules “moving coherently”.

B. DPD for bonded atoms

When the atoms are bonded and belong to definitegroups where the atoms do not diffuse away from eachother, the CG mapping is well defined, usually throughthe center of mass variables. In Fig. 2 we show a starpolymer melt in which each molecule is coarse-grained byits center of mass, leading to a blob or bead description[176]. The important question is how are the CG interac-tions between the blobs. Two CG approaches, static anddynamic, have been pursued, depending on the questionsone wishes to answer.

Static CG is concerned with approximations to the ex-act potential of mean force that gives, formally, the equi-librium distribution function of all the CG degrees of free-

FIG. 2: Star polymer molecules (in different colors) in a meltare coarse-grained at the level of their centers of mass. Theresulting model is a blob model of the DPD type [176].

dom. Radial distributions, equations of state, et c. arethe concern of static coarse graining. There is a vast lit-erature in the construction of the potential of mean forcefor CG representations of complex fluids [18, 177, 178],and complex molecules [19, 21, 179]. Despite these ef-forts, there is still much room for improvement in thethermodynamic consistency for the modeling of the po-tentials of mean force [180]. If one uses the CG potentialfor the motion of the CG degrees of freedom, the result-ing dynamics is unrealistically fast, although this may bein some cases convenient computationally.

Dynamic CG, on the other hand, focuses on obtain-ing, in addition to CG potentials, approximations to thefriction forces between CG degrees of freedom. Withinthe theoretical framework of Mori-Zwanzig approach, itis possible to obtain in general the dynamics of the CGdegrees of freedom from the underlying Hamiltonian dy-namics. The first attempt to derive the DPD modelfrom the underlying microscopic dynamics was given byEspanol for the simple case of a one dimensional har-monic lattice [181]. The center of mass of groups ofatoms were taken as the CG variables and Mori’s pro-jection method was used. Because this system is analyt-ically soluble, a flaw in the original derivation could bedetected, and an interesting discussion emerged on the is-sue of non-Markovian effects in solid systems [182–186].

By following Schweizer [187], Kinjo and Hyodo [188]obtained a formal equation for the centers of mass ofgroups of atoms. The momentum equation contains threeforces, a conservative force deriving from the exact po-tential of mean force, a friction force and a random force.By modeling the random forces the authors of Ref. [188]showed that this equation encompass both, the BD andDPD equations. However, to consider the procedure inRef. [188] a derivation of DPD, it is necessary to spec-ify the conditions under which one obtains BD insteadof DPD (or vice versa). This was not stated by Kinjoand Hyodo. The crucial insight is that BD appears when

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“solvent” is eliminated from the description, this is, some(the majority) of the atoms are not grouped and are in-stead described as a passive thermal bath (or implicitsolvent). The friction force in this case is proportionalto the velocity of the particles, and the momentum ofthe CG blobs is not conserved. On the other hand, aDPD description appears when all the atoms are par-titioned into disjoint groups. In this case, the conser-vation of momentum induced by Newton’s third law atthe microscopic level leads to a structure of the frictionforces depending on relative velocities of the particles. Aderivation of the equations of DPD from first principlestaking into account linear momentum conservation waspresented by Hijon et al. [176]. The position-dependentfriction coefficient was given in terms of a Green-Kuboexpression that could be evaluated, under certain simpli-fying assumptions, directly from MD simulations, withinthe same spirit of an early derivation of Brownian Dy-namics for a dimer representation (non-momentum con-serving) of a polymer by Akkermans and Briels [189].The general approach was preliminarly tested for a sys-tem of star polymers (as those in Fig. 2). A subsequentthorough study of this star polymer problem by Karni-adakis and co-workers [190] has shown that the intro-duction of an intrinsic spin variable for each polymermolecule seems to be necessary at low concentrations inorder to have an accurate representation of the MD re-sults. The approach in Ref. [176] has been labeled byLi et al. [190] as the MZ-DPD approach, standing forMori-Zwanzig dissipative particle dynamics. Other com-plex molecules (neopentane, tetrachloromethane, cyclo-hexane, and n-hexane) have been also considered [191]within the MZ-DPD approach with interesting discussionon the validity of non-Markovian behaviour (more on thislater). A slightly more general approach for the deriva-tion of MZ-DPD equations has been given by Izvekov[173]. Very recently, Espanol et al. [192] have formulatedfrom first principles the dynamic equations for an en-ergy conserving CG representation of complex molecules.This work gives the microscopic foundation of the EDPDmodel for complex molecules (involving bonded atomsonly).

C. Non-Markov effects

The rigorous coarse-graining in which centers of massof groups of atoms are used as CG variables relies on abasic and fundamental hypothesis, which is the separa-tion of time scales of the evolution of the CG variablesand “the rest” of variables in the system. More accu-rately, the separation of time scales refers to the exis-tence, in the evolution of the CG variables themselves oftwo well-defined scales, a large amplitude slow compo-nent, and a small high frequency component that can bemodeled in terms of white noise. The dynamics of theCG variables can then be approximately described by anon-linear diffusion equation in the space spanned by the

CG variables [22, 23]. This separation of time scales doesnot always exist, either because the groups of atoms aresmall and the centers of mass momenta evolve in the sametime scales as the forces (due to collisions with atoms ofother groups) [191], or because of the existence of coupledslow processes not captured by the selected CG variables.When this happens, one strategy is to tweak the frictionand simply fit frictions to recover the time scales. Gaoand Fang used this approach in order to coarse grain awater molecule to one site-CG particle [193]. Anotherstrategy is to enlarge the set of CG variables with thehope that the new set will be Markovian. Briels [194] ad-dresses specifically the problem of CG in polymers andintroduces transient forces to recover a Markovian de-scription. Davtyan, Voth, and Anderson [195] have con-sidered the introduction of “fictitious particles” in orderto recover the CG dynamics observed from MD. The ficti-tious particles are just a simple and elegant way to modelthe memory kernel in a particularly intuitive way. If thestrategy to increase the dimension of the CG state spacedoes not work yet, it is still possible to formulate frommicroscopic principles formal non-Markovian models andto extract information about the memory kernel fromMD [196]. However, in the absence of separation of timescales, the computational effort required to get from MDthe memory kernel makes the whole strategy of bottom-up coarse graining inefficient. Note that the advantageof a bottom-up strategy for coarse graining is that oneneeds to run relatively short MD simulations to get theinformation (Green-Kubo coefficients) that is used in thedynamic equations governing much larger time scales. Ifone needs to run long MD simulation of the microscopicsystem to get the CG information, we have already solvedthe problem by brute force in the first place!

D. Electrostatic interactions

In many situations, one is interested in the consequen-tial effects of charge separation. This is particularly sofor aqueous systems where the relatively high dielectricpermittivity of water means that ion dissociation read-ily occurs. The relevance lies not only in the structuraland thermodynamic properties of ionic surfactants andpolyelectrolytes et c, [197, 198] but is also motivated bya burgeoning interest of electrokinetic phenomena [199–202].

An important point to make is that some relevant elec-trostatic effects simply cannot be captured in a short-range interaction (DPD-like or otherwise). For examplethe bare electrostatic energy of a charged spherical mi-celle of aggregation number N scales as N2/R ∼ N5/3,where R ∼ N1/3 is the micelle radius. This electrostaticenergy cannot be captured in either a volume (∼ N) orsurface (∼ N2/3) term. To be faithful to this physicstherefore, one has in some way to incorporate long-rangeCoulomb interactions explicitly into the DPD model.This area was pioneered by Groot, who used a field-

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based method [203]. Since then, more standard Ewaldmethods have also been used [67, 204], and in princi-ple any fast electrostatics solver developed for MD couldbe taken over into the DPD domain. One importantcaveat is that with soft particles (ie no hard core repul-sion) the singularity of Coulomb interactions needs tobe tamed through the use of smoothed charge models[67, 68, 203, 204]. Adding electrostatics is certainly com-putationally expensive, often halving the speed of DPDcodes. This drastic slow-down offsets the advantage ofthe DPD methods when compared to more traditionalcoarse-graining methods.

Related to the problem of charge separation is thequestion of dielectric inhomogeneities (image chargeset c), such as encountered in oil/water mixtures and atinterfaces. In field-based methods this can be resolvedby having a local-density-dependent dielectric permittiv-ity [203]. Alternatively one can introduce an explicitlypolarisable solvent model [205, 206]. Note that an elec-tric field in the presence of a dielectric inhomogeneityinduces reaction forces on the uncharged particles [203].In a field-based method, this is quite complicated to in-corporate in a rigorous way. For an explicitly polarisablesolvent model, these induced reaction forces of coursearise naturally and are automatically captured.

VI. SYSTEMS STUDIED WITH DPD

The number of systems and problems that have beenaddressed with DPD or its variants is enormous and wedo not pretend to review the extensive literature on thesubject. Nevertheless, to illustrate the range and vari-ety of different applications of DPD we give a necessarilybrief survey of the field. A general trend observed inthe application side, is the shift from the original DPDmodel, of “balls and springs” models, towards more spe-cific atomistic detail, in the line of MZ-DPD, or semi-bottom-up DPD (with structure based CG potentials andfitted friction).

Colloids: A recent review on the simulation of col-loidal suspensions with particle methods, including DPD,can be found in [207]. The first application of DPDto a complex fluid was the simulation of colloidal rhe-ology by Koelman and Hooggerbruge [208]. Since then,a large number of works have addressed the simulationof colloidal suspensions, with a variety of approaches torepresent the solute. Typically, a colloidal particle isconstructed out of dissipative particles that are movedrigidly [208, 209], or connected with springs [210, 211].Arbitrary shapes may be considered in this way [212], aswell as confinement due to walls [209, 213]. As a way tobypass the need to update the relatively large number ofsolid particles, some approaches represent each colloidalparticle with a single dissipative particle [109, 110, 214],leading to minimal spherical blob models for the colloids.These simplified models for the solute require the intro-duction of shear forces of the FPM type. Representing

a colloidal particle with a point particle is a strategyalso used in minimal blob models in Eulerian CFD meth-ods for fluctuating hydrodynamics [215]. A core can beadded in order to represent hard spheres with finite radii,supplemented with a dissipative surface to mimic bound-ary conditions [216], and still retain the one-particle-per-colloid scheme. Although general features show semi-quantitative agreement with experimental results [216],other simulation techniques like Stokesian Dynamics, andtheoretical work, it is clear that getting more detailedphysics of colloid-colloid interactions and colloid-solventinteraction (either through a MZ-DPD approach or byphenomenologically including boundary layers and top-down parametrization) may be beneficial to the field.

Blood: A colloidal system of obvious biological interestis blood. Blood has been simulated with DPD [217], andmore recently with SDPD [134, 135, 218]. Two recentreviews [219, 220] discuss the modeling of blood withparticle methods. Multi-scale modeling (i. e. MZ-DPD)seems to be crucial to capture platelet activation andthrombogenesis [221].

Polymers: An excellent recent review on coarse-graining of polymers is given by Padding and Briels [222].Below the entanglement threshold Rouse dynamics holdsand this is well satisfied in a DPD polymer melt [223].Above the threshold, entanglements are a necessary in-gredient in polymer melts. Because the structure basedCG potential between the blobs are very soft, it is neces-sary to include a mechanism for entanglement explicitly.This is one example in which the usual simple schemesto treat the many-body potential (through pair-wise in-teractions) fails dramatically. There are several methodsto include entanglements: Padding and Briels [224, 225]introduced the elastic band method for coarse-grainedsimulations of polyethylene. Another alternative to rep-resent entanglements is to use the Kumar and Larsonmethod [226–228] in which a repulsive potential betweenbonds linking consecutive blobs is introduced. Finally,entanglements can be enforced in a simpler way by hardexcluded volume LJ interactions [229], or through suit-able criterion on the stretching of two bonds and theamount of impenetrability of them [230].

Beyond scaling properties, effort has been directed to-wards a chemistry detailed MZ-DPD methodology, byusing structure based CG effective potentials and ei-ther fitting the friction coefficient [231–234], or obtain-ing the dissipative forces from Green-Kubo expressions[235]. In general, one can take advantage of systematicstatic coarse-graining approaches, like those for heptaneand toluene [236], to be directly incorporated to DPD.Very recently, new Bayesian methods for obtaining theCG potential and friction are being considered [237, 238](on pentane). The ultimate goal of all these microscopi-cally informed approaches is to predict rheological prop-erties as a function of chemical nature of the polymersystem with a small computational cost. As mentionedearlier, whatever improvement in the construction of CGpotentials will be highly beneficial also for the construc-

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tion of dynamic CG models. In this respect, the workon analytical integral equation approach of Guenza andco-workers [239] for obtaining the CG potential in poly-mer systems that ensures both, structural properties andthermodynamic behaviour seems to be very promising.

We perceive a powerful trend towards more micro-scopically informed DPD able to express faithfully thechemistry of the system. This trend is important whenconsidering hierarchical multi-scale methods in whichMD information is transferred to a dynamic CG DPDmodel, the DPD model is evolved in order to get topol-ogy and equilibrium states much faster than MD, andthen a back-mapping fine grained procedure recovers mi-croscopic states able to be evolved again with MD [240–242].

Other complex fluid systems involving polymers havebeen considered. An early work is the study of adsorptionof colloidal particles onto a polymer coated surface [213].Polymer brushes are reviewed by Kreer [243]. Self as-sembly of giant amphiphiles made of a nanoparticle withtethered polymer tail has been considered recently [244].Polymer membranes for fuel cells have been considered byDorenbos [245]. Polymer solutions simulated with DPDobey Zimm theory that includes hydrodynamic interac-tions [246]. Polymer solutions have also been studiedwith SDPD observing Zimm dynamics [137].

Phase separating fluids: In polymer mixtures, the χ-parameter mapping introduced by Groot and Madden[247] has been phenomenally popular because it links tolong-established polymer physical chemistry (there aretables of χ-parameters for instance, and a large litera-ture devoted to calculating χ-parameters ab initio). Thishas helped incorporate chemical specificity in DPD fromsolubility parameters [248, 249]. It is also known that χ-parameters can be composition dependent (PEO in wa-ter is the notorious example). This can be accommodatedwithin the MDPD approach. Akkermans [250] presents afirst principles coarse-graining method that allows to cal-culate the excess free energy of mixing and Flory-Hugginsχ-parameter. A related effort is given by Goel et al. [251].

DPD has been very successful in identifying mech-anisms in phase separation: Linear diblock copolymerspontaneously form a mesocopically ordered structure(lamellar, perforated lamellar, hexagonal rods, micelles)[247]. DPD is capable to predict the dynamical pathwaytowards equilibrium structures and it is observed thathydrodynamic interactions play an important role in theevolution of the mesophases [252]. Domain growth andphase separation of binary immiscible fluids of differingviscosity was studied in [253]. New mechanisms via in-ertial hydrodynamic bubble collapse for late-stage coars-ening in off-critical vapor-liquid phase separation havebeen identified [79]. The effect of nanospheres in themechanisms for domain growth in phase separating bi-nary mixture has been considered by Laradji and Hore[210].

Drop dynamics: A particular case of phase separatingfluids is given by liquid-vapour coexistence giving rise to

droplets. Surface-confined drops in a simple shear wasstudied in an early work [254]. Pendant drops have beenstudied with MDPD [81], while oscillating drops [255],and drops on superhydrophobic substrates [256], havealso been considered.

Amphiphilic systems: An early review of computermodeling of surfactant systems is by Shelley and Shel-ley [257]. A more recent review on the modeling ofpure membranes and lipid-water membranes with DPDis given by Guigas et al. [36]. Coarsening dynamics ofsmectic mesophase of amphiphilic species for a minimalamphiphile model was studied by Jury et al. [258] andmesophase formation in pure surfactant and solvent byPrinsen et al. [259]. More microscopic detail has been in-cluded by Ayton and Voth [260] with DPD model for CGlipid molecules that self assembly, a problem also consid-ered by Kranenburg and Venturoli [261]. Effort towardsmore realistic parametrization for lipid bilayers was givenby Gao et al. [262]. Prior to this Li et al. [263] formulateda conservative force derived from a bond-angle dependentpotential that allowed to consider different types of micel-lar structures. Microfluidic synthesis of nanovesicles wasconsidered by Zhang et al. [264]. Simulations of micelle-forming systems have also been reported [265, 266].

Oil industry: DPD simulations have also addressedproblems in the oil industry, from oil-water-surfactantdynamics [267], and water-benzene-caprolactam systems[268], to aggregate behavior of asphaltenes in heavy crudeoil [269], or the orientation of asphaltene molecules at theoil-water interface [270].

Biological membranes: A review of mesoscopic model-ing of biological membranes was given by Venturoli et al.[271]. Groot and Rabone [272] presented one of the firstapplications of DPD to the modeling of biological mem-branes and its disruption due to nonionic surfactants.Sevink and Fraaije [273] devised a coarse-graining of amembrane into a DPD model in which the solvent wastreated implicitly. Amphiphilic polymer coated nanopar-ticles for assisted drug delivery through cell membraneshas been recently studied [274, 275]. The diffusion ofmembrane proteins has been considered iby Guigas andWeiss [276].

Biomolecular modeling: The CG modeling of complexbiomolecules with a focus on static properties has beenaddressed in the excellent review by Noid [19]. Pivkinet al. [206] have modeled proteins with DPD force fields,which competes with the Martini force field [277].

Inorganic materials: DPD has also been used forthe CG modeling of solid inorganic materials. Coarse-grained representation of graphene turns out to be es-sential for study of large scale resonator technology[170, 278].

VII. CONCLUSIONS

The DPD model is a tool for simulating the mesoscale.The model has evolved since its initial formulation to-

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wards enriched models that, while retaining the initialsimplicity of the original, are now linked strongly to ei-ther the microscopic scale or the macroscopic continuumscale. In many respects, the original DPD model of Fig. 1is a toy model and one can do much better by using theserefined models. In this Perspective, we wish to conveythe message that DPD has a dual role in modeling themesoscale. It has been used as a way to simulate, on onehand, coarse-grained (CG) versions of complex molecularobjects and, on the other hand, fluctuating fluids. Whilethe first type of application, involving atoms bonded bytheir interactions, has a solid ground on the theory ofcoarse graining, there is no such a microscopic basis forDPD as a fluid solver. The best we can do today isto descend from the continuum theory and to formulateDPD as a Lagrangian discretization of fluctuating hydro-dynamics, leading to the SDPD model.

Therefore as DPD simulators we are faced with threealternative strategies:

#1 Bottom-up MZ-DPD: When dealing with molecu-lar objects made of bonded atoms, we may formulate anappropriate CG mapping and construct the DPD equa-tions of motion with momentum conserving forces [176].These equations contain the potential of mean force gen-erating conservative forces and position-dependent fric-tion coefficient, with explicit microscopic formulae: thepotential of mean force is given by the configuration de-pendent free energy function, and the position dependentfriction coefficient tensor is given by Green-Kubo expres-sions. Both quantities are given in terms of expectationsconditional on the CG variables and are, therefore, many-body functions. These are not, in general, directly com-putable due to the curse of dimensionality. One needs toformulate simple and approximate models (usually pair-wise with, perhaps, bond-angle and torsion effects) inorder to represent the complex functional dependence ofthese quantities. Together with the initial selection ofthe CG mapping, finding suitable functional forms is themost delicate part of the problem. Once this simple func-tional models are selected, constrained MD simulations[176, 189], or optimization methods [19–21, 237], may beused to obtain the CG potential.

The existence of a framework to derive dynamic CGmodels from bottom-up is a highly rewarding intellec-tual experience with a high practical value because 1) itprovides the structure of the dynamic equations, and 2)signals at the crucial points where approximations arerequired. The MZ-DPD approach is, in our view, animportant breakthrough in the field, as it connects thewell established world of static coarse-graining with theDPD world [19]. In this way, it provides a frameworkfor accurately addressing the CG dynamics. However,the usefulness to follow the program by the book is notalways obvious due to the large effort in obtaining theobjects form MD. In this case, one would go to the nextstrategy.

#2 Parametrization of DPD: We may insist on a par-ticularly simple form of linear repulsive forces and simple

friction coefficients (like the ones in the original/cartoonDPD model) and fit the parameters to whatever propertyof the system one wants to correctly describe (for exam-ple, the compressibility). Nowadays, we advise cautionwith this simple approach because, usually, many otherproperties of the system go wrong. The simple DPDlinear forces are not flexible enough in many situations.However, from what we have already learned from micro-scopically informed MZ-DPD in the previous #1 strat-egy, we may give ourselves more freedom in selecting thefunctional forms (as in MDPD) for conservative and fric-tion forces and have more free parameters to play with.Once it is realized that the potential between beads orblobs in DPD is, in fact, the potential of mean force, onecan use semi-bottom-up approaches in which the poten-tial of mean force is obtained from first principles, whilethe DPD friction forces are fitted to obtain the correcttime scales [231, 232, 279]. Although this strategy is lessrigorous, it may be more practical in some cases.

The #1 bottom-up MZ-DPD strategy above has notbeen yet successful when the interactions of atoms ormolecules in the system are unbonded, allowing twomolecules that are initially close together to diffuse awayfrom each other. These are the kind of interactionspresent in a fluid system. The main difficulty seems to bein the Lagrangian nature of a fluid particle that makesthe CG mapping not obvious. Although some attemptshave been taken in order to derive DPD for fluid systemswith unbonded interactions, we believe that the problemis not yet solved. However, for these systems one may re-gard the dissipative particles as truly fluid particles (i. e.small thermodynamic systems that move with the flow).We are lead to the third strategy.

#3 Top-down DPD: Assume that we know that a par-ticular field theory describes the complex fluid of interestat a macroscopic scale (Navier-Stokes for a Newtonianfluid, for example). Then one may discretize the the-ory on moving Lagrangian points according to the SPHmesh-free methodology. The Lagrangian points may beinterpreted as fluid particles. If we perform this dis-cretization within a thermodynamically consistent frame-work like generic [4], thermal fluctuations are automat-ically determined correctly [142], allowing to address themesoscale. This strategy leads to enriched DPD mod-els (SDPD is an example corresponding to Navier-Stokeshydrodynamics). The functional forms of conservativeand friction forces in this DPD models are dictated bythe mesh-free discretization, as well as the input infor-mation of the field theory itself. We have the impressionthat SDPD or its isothermal counterpart MDPD are un-derappreciated and underused. Although these meth-ods are appropriate for fluid systems, we foresee the useof MDPD many-body potentials of the embedded atomform also for CG potentials for bonded atom systems.While CG potentials depending on the global density arepotentially a trap [280, 281], the inclusion of many-bodyfunctional forms of the embedded atom kind dependingon local density is a promissing route to have more trans-

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ferable CG potentials [282], valid for different thermo-dynamic points. This expectation, though, needs to besubstantiated by further research. In particular, liquidstate theory for MDPD may need to be further devel-oped [82, 239].

This Perspective on DPD also points to several openmethodological questions.

We have already mentioned the open problem of de-riving from microscopic principles the dynamics of La-grangian fluid particles made of unbonded atoms. Oncethis problem is solved, we will need to face the next prob-lem of deriving from first principles the coupling of CGdescriptions of bonded and unbonded atoms (a proteinin a membrane surrounded by a solvent, for example). Aderivation from bottom-up of this kind of coupling in adiscrete Eulerian setting has been given recently in [167].

For the simulation of fluids, standard CFD methodsequipped with thermal fluctuations are readily catch-ing up with the mesoscale [9–17]. Methods for cou-pling solvents and suspended structures are being devised[167, 215], and therefore one may well ask what is the ad-vantage of a Lagrangian solver based on the relatively in-accurate SPH discretization over these high quality CFDmethods. Note that CFD methods allow for the rigoroustreatment of limits (incompressibility, inertia-less, et c)that may imply large computer savings, and which aredifficult to consider in SPH based methods. We believe(see Meakin and Xu [283] for a defense of particle meth-ods) that fluid particle models may still compete in sit-uations where biomolecules and other complex molecu-lar structures move in solvent environments, because onedoes not need to change paradigm: only particles forboth, solvent and beads, are used, with the correspond-ing simplicity in the codes to be used. Nevertheless, a faircomparison between Eulerian and Lagrangian method-ologies is still missing.

As SDPD is just SPH plus thermal fluctuations it in-herits the shortcomings of SPH itself. SPH is still facingsome challenges in both, foundations (boundary condi-tions) and computational efficiency [123, 124]. In thisrespect, a Voronoi fluid particle model [114], understoodas a Lagrangian finite volume solver may be an interest-ing possibility both in terms of computational efficiencyand simplicity of implementation of boundary conditions.Serrano compared SDPD and a particular implementa-tion in 2D of Voronoi fluid particles [284]. In termsof computational efficiency, both methods are compara-ble because the extra cost in computing the tessellationis compensated by the small number of neighbours re-quired, six on average, while in SDPD one needs 20-30neighbours.

Another interesting area of research is that of multi-scale modeling. In CFD, one way to reduce the compu-tational burden is to increase the resolution of the meshonly in those places where strong flow variations occur, orinteresting molecular physics requiring small scale reso-lution is taking place. An early attempt within DPDwas given by Backer et al. [285]. We envisage that meth-

ods for multi-resolution SDPD will be increasingly usedin the future [130, 142, 155, 286]. Multi-resolution isa problem of active research also in the SPH commu-nity [124]. Eventually, one would like to hand-shake theparticle method of SDPD with MD as the resolution isdecreased [287]. Note, however, that as the fluid parti-cles become small (say “four atoms per particle”) it isexpected that the Markovian property breaks down andone needs to account for viscoelasticity [288, 289], eitherwith additional internal variables [143, 160, 161], or with“fictitious particles” [195].

Finally, a very interesting research avenue is given bythe thermodynamically consistent (i. e., able to deal withnon-isothermal situations) Mori-Zwanzig EDPD intro-duced theoretically by Espanol et al. [192]. Up to now,CG representations of complex molecules have only in-cluded the location and velocity of the CG beads or blobs(sometimes its spin [190]), completely forgeting its inter-nal energy content. Given the fundamental importanceof the principle of energy conservation, it seems that inorder to have thermodynamically consistent and moretransferable potentials, we may need to start looking atthese slightly more complex CG representations.

VIII. ACKNOWLEDGMENTS

We acknowledge A. Donev for useful comments on themanuscript. PE thanks the Ministerio de Economıa yCompetitividad for support under grant FIS2013-47350-C5-3-R.

Appendix A: MDPD consistency

In MDPD the potential takes the form described inthe main text where V ({r}) =

∑i ψ(di), and di =∑

j 6=iW (rij). From this it is easy to show that the forcesremain pairwise, with

Fij = −[ψ′(di) + ψ′(dj)]W′(rij) eij . (A1)

Note that the weight function here is W ′(r). However,to our knowledge, there does not exist in the literaturea proof of the converse, namely that this relationshipbetween the weight functions is a necessary condition toensure the existence of V ({r}). We present here such aproof, following the line of argument in Ref. [84].

We start with a generalised MDPD pairwise force law,with an (as yet) arbitrary weight function ωc(r),

Fij = A(di, dj)ωc(rij) rij . (A2)

We assume the amplitude function A(di, dj) is symmetricsince otherwise Fij 6= −Fji. Let us denote partial deriva-tives with respect to the first and second density argu-ments by A[1,0] and A[0,1]. The symmetry of A(di, dj)then implies A[1,0](di, dj) = A[0,1](dj , di).

14

A generic radial force law can always be integrated,so we cannot deduce anything useful just by consider-ing pairs of particles. Instead, following Ref. [84], letus consider three isolated, collinear particles, at posi-tions xi (i = 1 . . . 3) such that x1 ≤ x2 ≤ x3. For thisconfiguration the densities are d1 = W (x12) + W (x13),d2 = W (x12) + W (x23), and d3 = W (x13) + W (x23).The pairwise forces are F12 = A(d1, d2)ωc(x12), F23 =A(d2, d3)ωc(x23), and F13 = A(d1, d3)ωc(x13). Finally,the summed forces on the particles are F1 = F12 + F13,F2 = −F12 + F23, and F3 = −F13 − F23.

The existence of a potential implies integrability con-straints like ∂F1/∂x2 − ∂F2/∂x1 = 0. Imposing thesegives rise to an expression which can be simplified (byconsideration of special cases) to a set of requirementsfor which the representative case is

ωc(x12)W ′(x23)A[1,0](d1 + d3, d1)

− ωc(x23)W ′(x12)A[1,0](d1 + d3, d3) = 0.(A3)

The symmetry relation between A[0,1] and A[1,0] has beenused. If we are allowed to cancel the A[1,0] functionswe are home and dry, since this implies ωc(x)W ′(y) =ωc(y)W ′(x) (for arbitrary arguments x and y), and thiscan only be true if ωc(x) ∝ W ′(x). However, the A[1,0]

functions only cancel if A[1,0](x+ y, x) = A[1,0](x+ y, y)(for arbitrary arguments x, y). A little thought shows

that a sufficient condition for this to be true is thatA(di, dj) = f(di) + f(dj). This is precisely the formthe force-law takes in Eq. (A1). The conclusion is thatin this case ωc(x) ∝ W ′(x) is a necessary condition forthe existence of the many-body potential V ({r}). It isalso sufficient, since we can absorb the proportionalityconstant into the definitions of di and ψ(d), and thenexplicitly V ({r}) =

∑i ψ(di). This proves the claimed

result above.For another example, we might be tempted to consider

A(di, dj) = f(di + dj), but retaining the weight functionωc(x) ∝ W ′(x). For this choice A[1,0](x, y) = f ′(x + y)and Eq. (A3) reduces to f ′(2x + y) = f ′(x + 2y). Thisis true for arbitrary x and y if only if f(x) is linear, andtherefore the force law is de facto of the form shown inEq. (A1). Thus, a non-linear function f(x) would be abad choice. For a further case study, see Ref. [84].

If we fail to satisfy Eq. (A3) then the potential doesnot exist. If the potential does not exist, we lose theunderpinning theory that the stationary probability dis-tribution is given by Eq. (2). Without this foundationwe are in uncharted waters, and there is no link to estab-lished statistical mechanics and thermodynamics.

In our opinion, in MDPD the burden rests on the userto display the V ({r}) which gives rise to the chosen forcelaw. The absence of an explicitly displayed potentialleads only to unwarranted complications.

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