Dissipative Particle Dynamics-1758825109000381

发布时间:2021-10-19 00:01:54

International Journal of Applied Mechanics Vol. 1, No. 4 (2009) 737–763 c Imperial College Press

DISSIPATIVE PARTICLE DYNAMICS: INTRODUCTION, METHODOLOGY AND COMPLEX FLUID APPLICATIONS — A REVIEW

E. MOEENDARBARY?,? , T. Y. NG? and M. ZANGENEH?

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

?Department

of Mechanical Engineering, University College London Torrington Place, London WC1E 7JE, UK

of Mechanical & Aerospace Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798, Singapore ?mtyng@ntu.edu.sg Received 15 March 2009 Accepted 3 June 2009 The dissipative particle dynamics (DPD) technique is a relatively new mesoscale technique which was initially developed to simulate hydrodynamic behavior in mesoscopic complex ?uids. It is essentially a particle technique in which molecules are clustered into the said particles, and this coarse graining is a very important aspect of the DPD as it allows signi?cant computational speed-up. This increased computational e?ciency, coupled with the recent advent of high performance computing, has subsequently enabled researchers to numerically study a host of complex ?uid applications at a re?ned level. In this review, we trace the developments of various important aspects of the DPD methodology since it was ?rst proposed in the in the early 1990’s. In addition, we review notable published works which employed DPD simulation for complex ?uid applications. Keywords : DPD review; mesoscopic simulation; coarse-graining; boundary models; complex ?uid; multiphase ?ows.

?School

1. Introduction To bridge the gap between atomistic simulations and macroscopic network simulations, and to overcome the inherent di?culties faced by conventional methods when applied to complex ?uid systems, we need an intermediary technique focused at a length scale larger than the atomistic scale, but smaller than the macroscopic connection scale, Groot and Warren [1997], please see Fig. 1. Mesoscopic simulations aim at identifying characteristic physical lengths and times in the system in order to use them for simpli?cation of complex models. In particular, for soft matter and polymeric systems, the hydrodynamic behaviors are captured easily using continuum methods while it is expensive to handle them at atomistic levels.
? Corresponding

author. 737

738

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Fig. 1. Dissipative Particle Dynamics: a mesoscale technique for bridging the gap between the micro- and macro-scales.

Conversely, the interaction between components at the atomistic scale can be taken into account by employing molecular methods, but is not usually straightforward to consider them through continuum methods. Thus, the developments of intermediate techniques, which have both features of micro- and macro-scales methods, are essential. Depending on the nature of the problem, several mesoscopic simulation methods have been devised, developed and re?ned over the years. In the area of mesoscopic complex ?uid simulation, several particle-based (o?-lattice) methods such as smoothed particle hydrodynamics (SPH), DPD, ?uid particle method (FPM), as well as other grid-based techniques such as lattice gas automata (LGA) and lattice Boltzmann (LB) are the more notable techniques found in the literature. For more details on some of these techniques and their applications, please refer to the reviews Monaghan [1992], Koumoutsakos [2005] and Chen and Doolen [1998]. The dissipative particle dynamics (DPD) is a potentially very powerful and simple mesoscopic approach, which facilitates the simulation of the statics and dynamics of complex ?uids and soft matter systems at physically interesting length and time scales. Since 1990, when the method was ?rst developed in Europe, DPD has been applied in the study of the dynamical properties of a wide variety of systems and applications, as we shall describe them subsequently in this review. DPD, as an o?-lattice technique, does not su?er from some of the restrictions imposed by the lattice as in the LGA or LB methods. We believe that presently, the DPD is arguably one of the best mesoscale simulation techniques, and in the near future, it has the potential to emerge as an even more widely used modeling and simulation technique for many complex ?uid systems. The aim of this review is to document the most representative DPD works over the years. This review is organized in the following manner. In the next section, we provide an introduction of the DPD algorithm and summary of the theoretical background, the time integration techniques, and the boundary models which have been used widely in DPD implementations, all in a succinct manner. In Sec. 3, we

Dissipative Particle Dynamics

739

extend this review to cover works on other complex ?uidic applications in which DPD was employed as the simulation technique. Finally, we provide some concluding remarks, and o?er some possible future directions for the implementation of DPD methodology. 2. Dissipative Particle Dynamics: A Coarse-Grained Technique The dissipative particle dynamics (DPD) technique is an alternative method for mesoscopic complex ?uid simulation, which was ?rst devised and developed for simulating hydrodynamic behaviour by Hoogerbrugge and Koelman [1992] and Koelman and Hoogerbrugge [1993]. It was subsequently modi?ed by Espa? nol and Warren [1995]. This method can be perceived as the clustering a number of molecules into single particle where the number of molecules per DPD particle is known as the nol et al., 1997; coarse-graining parameter and is usually denoted by Nm , [Espa? Flekk?y and Coveney, 1999; Flekk?y et al., 2000; Kinjo and Hyodo, 2007a]. This coarse graining parameter plays a vital role and has signi?cant impact on the speed of simulation, Backer et al. [2005a]. For the DPD method, deriving the coarse graining procedure can include ?nite discretization of the Navier-Stokes equations in a Lagrangian moving Voronoi grid (tessellation) which is relatively easy to implement in two spatial dimensions, [De Fabritiis and Coveney, 2003; De Fabritiis et al., 2002; Serrano et al., 2002; Serrano and Espa? nol, 2001]. Espa? nol [1996] began from a microscopic description of a harmonic chain and worked out the equations of motion for a coarse-grained chain constructed from groups of clustered atoms. Later, it was noted by Cubero and Yaliraki [2005a, b] that Markovian approximation was not applicable when sound propagation plays an important role in the coarse graining procedure. As indicated in Kinjo and Hyodo [2007b], coarse-graining techniques facilitate simulations of complex systems in larger time spans and with reduced computational work by eliminating some extra degrees of freedom which are deemed unessential in the phenomena of interest. Thus one of the most important issues is to ?nd the upper limit for the coarse-graining parameter by which we could model the physical properties in an e?cient manner, and this issue was investigated in signi?cant detail by Pivkin and Karniadakis [2006a]. In Groot and Rabone [2001], they rationalized that the total DPD simulation speed-up with respect to MD can be estimated by 8/3 1000 Nm for a given system volume. Thus, for example if Nm = 3 and 7, then the speed-up factors are roughly 2 × 104 and 2 × 105 , respectively. Apart from the idea of lumping several atoms together and replacing them by a single bead, the quality and softness of e?ective interactions between these newly devised clusters also play important function. More speci?cally, the inclusion of these extremely soft coarsegrained potentials instead of the hard-core Lennard-Jones (LJ) models improves the computational e?ciency of the DPD model, Pool and Bolhuis [2006]. The DPD algorithm is essentially a combination of MD, BD and lattice gas automata (LGA), and derives its static and dynamic properties according to the theory in statistical mechanics, Marsh et al. [1997b]. Similar to molecular dynamics, the

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

740

E. Moeendarbary, T. Y. Ng & M. Zangeneh

time evolution of each DPD particle, which represents a cluster of molecules/atoms, can be calculated by Newton’s second law dri = vi , dt dpi = dt Fij ,
j =i

(1)

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

where ri , vi and pi are respectively the position, velocity and momentum vectors of particle i, and Fij is the total interparticle force exerted on particle i by particle j . The original interparticle force is de?ned by three components that lie along their lines of centres and conserves linear and angular momentum, Espa? nol [1997b]. D R C Speci?cally, Fij = FC ij + Fij + Fij , where a purely repulsive conservative force Fij , a D dissipative or frictional force Fij which represents the e?ects of viscosity and slows down the particles motion with respect to each other, and a random (stochastic) force FR ij which represents the thermal or vibrational energy of system, are summed to obtain the total force, and these force components can be individually written as
C FC ij = w (rij )eij ,

(2) (3) (4)

FD ij

= ?γw (rij )[vij · eij ]eij ,
D

R FR ij = σw (rij )θij eij ,

where eij = rij /rij , rij = ri ? rj , rij = |ri ? rj | and vij = (vi ? vj ). wC , wD and wR are the conservative dissipative and random r dependent weight functions. The θij term is a Gaussian white noise function with symmetry property θij = θji to ensure the total conservation of momentum and has the following stochastic properties θij (t) = 0, θij (t)θkl (t ) = (δik δjl + δil δjk ) δ (t ? t ) (5)

All of the above forces are acting within a sphere of interaction or cut-o? radius rc , which is the length scale parameter of the system. The symbols γ and σ are the coe?cients of the dissipative and random forces, respectively. The extension of DPD forces to include coloured noise instead of Gaussian white noise was reported in Cotter and Reich [2003]. Similar to the ?uctuation-dissipation theorem of Kubo [1966] but adapted for DPD, Espa? nol and Warren [1995] obtained the detailed balance condition which is su?cient condition guaranteeing that the system has a Gibbsian equilibrium wD (r) = [wR (r)]2 , σ 2 = 2γkB T /m, (6)

where kB is the Boltzmann constant and T the equilibrium temperature. This condition makes the DPD equations act like as a thermostat and because the algorithm depends on relative velocities and the interactions between particles are symmetric, it is an isotropic Galilean invariant thermostat which preserves the hydrodynamics, [Allen and Schmid, 2007; Groot, 2006; Stoyanov and Groot, 2005]. Later the ergodicity of the DPD dynamics in one dimension was proved by Shardlow and Yan [2006]. Recently an extended version of the DPD interactions (transverse DPD

Dissipative Particle Dynamics

741

thermostat) which includes the damping of the perpendicular components of the relative velocity was reported by Junghans et al. [2007]. The conservative force weight function can be de?ned as a simple decaying function of distance as ? ? ?aij 1 ? rij rij ≤ rc C rc (7) w (rij ) = ? ? 0 rij > rc where aij is the repulsion parameter between particle i and particle j . De?ning the conservative weight function and especially the repulsion parameter is one the most important aspects of DPD simulations. The de?nition of soft repulsive potential facilitates accessibility of much larger length and time scales, [Klapp et al., 2004; Vattulainen et al., 2002]. For the same types of solvent particles, Groot and Warren [1997], Groot and Rabone [2001] and Keaveny et al. [2005] matched the compressibility condition and determined the repulsion parameters as a function of DPD number density ρ and system temperature. The central and most cited work in the DPD area is that of Groot and Warren [1997] who speci?ed the repulsion parameter between di?erent types of particles. In order to consider interaction between components in the solution, they related the DPD parameters to the χ-parameters in Flory-Huggins theory for polymers, and subsequently obtained the relation between the χ-parameters and the repulsion parameters for unequal particles, [Groot and Warren, 1997; Groot and Rabone, 2001; Keaveny et al., 2005]. In order to address the relation of bead-bead interaction parameters to bead-size, solubility parameter, and surface tension authors of Maiti and McGrother [2004] re-visited the work of Groot and Warren [1997]. Later, Wijmans et al. [2001] extended the methodology of Groot and Warren [1997] for Gibbs ensemble Monte Carlo simulations. To predict the lower critical solution temperature (LCST) and to model pressure-induced phase separation of polymer-solvent mixtures, Van Vliet et al. [2000, 2003] extended the Flory-Huggins theory and introduced a pressure and temperature dependent repulsion parameter for DPD simulations. Following this, the polymer branching e?ects and its in?uences on the LCST curve was investigated in Van Vliet et al. [2002]. The dissipative and random weight functions take the general form ? ? ? 1 ? rij D R 2 rc w (rij ) = [w (rij )] = ? ? 0
s

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

rij ≤ rc rij > rc ,

(8)

where the exponent s = 1 in the original DPD algorithm and other values for s can be chosen to adjust the ?uid viscosity, Fan et al. [2006], √ Symeonidis et al. [2006]. R = σw ( r ) ? / dteij , where ?ij represents The random force transforms to FR ij ij ij an independent increment in a stochastic process, which is represented by a uniform distribution of random numbers with zero and unit variance, and is chosen independently for di?erent pairs of particles at each time step.

742

E. Moeendarbary, T. Y. Ng & M. Zangeneh

2.1. Theoretical aspects Extracting the DPD Fokker-Planck equation, which is analogous to MD Liouville equation, demonstrates the theoretical connection between dissipative particle dynamics and continuum ?uid mechanics and is the essential formalism for the derivation of the kinetic and hydrodynamic equations, [Espa? nol, 1995; Coveney and Espa? nol, 1997]. The Fokker-Planck equation governs the N -particle distribution function that provides the probability density of the microscopic variables of the system, which are the positions and momentums of all the particles. It is related to DPD by the corresponding Langevin equations, which are the stochastic di?erential equations for the dynamics of the particles subjected to conservative, dissipative, and random forces. Using standard projection operator techniques, Espa? nol [1995] obtained the DPD macroscopic hydrodynamic equations. The derivation of more general equation of motion for coarse-grained particles was recently reported by Kinjo and Hyodo [2007b]. Espa? nol and Serrano [1999] and Serrano et al. [1999] investigated the behavior of the velocity autocorrelation function and found the importance of collective hydrodynamic e?ects for high values of friction. Marsh et al. [1997a, 1997b] formulated the kinetic theory for DPD such that the transport coe?cients, namely the self-di?usion coe?cient and shear viscosity, in the hydrodynamic equations, were related to the DPD model parameters, [Evans, 1999; Masters and Warren, 1999; Noguchi and Gompper, 2007; Satoh and Majima, 2005]. The equilibrium solution is not obtained by solving the Fokker-Planck equation directly, but it is a natural outcome of proving the H-theorem and demonstrating that the Gibbs distribution is the stationary solution of the Fokker-Planck equation. Proof of an H-theorem is the fundamental result in statistical mechanics for a dynamical system and shows that DPD particles tend to migrate towards the equilibrium state, which yields the Gibbs distribution as the equilibrium solution, [Coveney and Espa? nol, 1997; Marsh et al., 1997a, 1997b]. The H-theorem yields stability of the solution and guarantees that all states in phase space lead to equilibrium. Detailed balance condition for DPD, equation (6), which is the same condition as the one in conventional Brownian motion, is the basic requirement in deriving the H-theorem. If this condition is violated, the H-theorem cannot be derived, and the Gibbs distribution is not a stationary solution of the Fokker-Planck equation. In addition to central pairwise forces, Espa? nol [1997b, 1998] and Espa? nol and Revenga [2003] proposed the ?uid particle model (FPM) in order to consider the noncentral shear forces between dissipative particles. This model can be considered as both the thermodynamically consistent version of smoothed particle hydrodynamics (SPH), Espa? nol et al. [1999], and the generalized DPD method that includes torques and angular velocities of the particles. Furthermore, it is not straightforward to determine the transport coe?cients from the original DPD algorithm and the physical scales in DPD simulations are unde?ned, Espa? nol and Revenga [2003]. Therefore, the introduction of FMP improves some of these inherent de?ciencies in original DPD technique.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

743

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Willemsen et al. [1998] proposed the combination of DPD with the Monte Carlo technique to compute thermodynamic properties in a more e?cient manner. More recently, Goicochea [2007] predicted absorption and disjoining pressure isotherms of con?ned polymers using this combined model. Also, combining the basic elements of DPD with reaction Monte Carlo method enabled L? ?sal et al. [2006] to introduce a new method termed the reaction ensemble DPD in order to evaluate the reaction equilibria of polymer systems and to calculate the polydispersity under various conditions. Although the classical DPD model can successfully simulate the hydrodynamics, it is not able to reproduce the thermodynamic behaviour of a real system accurately. To overcome this limitation and to obtain the correct dynamics of nonideal ?uids, Pagonabarraga and Frenkel [2000, 2001] introduced the “many body” DPD approach. In this model, the amplitude of the soft repulsions is dependent on the local excess free energy, which provides a wider range of possibilities for the equation of state (a density dependent conservative force approach). Following this, Tro?mov et al. [2002] re?ned this model for strongly nonideal systems and generalized it to multicomponent mixtures, and later they extended it for a constant pressure ensemble in Tro?mov et al. [2005]. Warren [2003a] and Klapp et al. [2004] noted that the density dependent pair potentials and “many body” e?ects are very important considerations for achieving correct physical properties. This is especially so in applications with vapor-liquid coexistence, and other works involving many-body DPD include those by Warren [2003b] and Tiwari and Abraham [2006, 2008]. 2.2. Dissipative particle dynamics with energy conservation The existence of non-isothermal phenomena and heat ?ow will inevitably intensify the complication of simulating complex systems, as can be observed in the works of Han et al. [2008], He and Qiao [2008] and Qiao and He [2008]. As the original DPD model was isothermal and could not incorporate energy transport, Espa? nol [1997a] and Avalos and Mackie [1997] independently proposed a generalization of DPD algorithm that incorporated the conservation of the total energy in particle-particle interactions in addition to conserving the total momentum, such that the thermal conduction and temperature gradients could be modelled. The related detailed balance and H-theorem were proved for this energy conserving DPD algorithm (EDPD) by Marsh and Coveney [1998]. In Ripoll et al. [1998], heat conduction was simulated with this model and the correct equilibrium ?uctuations and reproduction of Fourier law were observed. Later, Ripoll et al. [2001] analytically calculated the wave number dependent transport properties of the DPD ?uid and more recently, Ripoll and Ernst [2005] extended the kinetic theory for the generalized hydrodynamic regime to incorporate the heat conduction. Mackie et al. [1999] re?ned the energy model of DPD for every time step and derived the macroscopic equilibrium probability distribution and equations of state. Subsequently, in Avalos and Mackie [1999] and Mackie et al. [1999], they analyzed

744

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

the transport properties of this model and conducted some heat transport and thermal convection simulations to con?rm the consistency of the model. To facilitate the study of complex ?uids in the (N,V,E) ensemble, Pastewka et al. [2006] proved that the energy conserving Peters thermostat, Peters [2004], is equivalent to the EDPD in the limit of vanishing time-step. In Willemsen et al. [2000b], a consistent boundary condition to model phase change of materials with EDPD was developed. More recently Qiao and He [2007] employed the EDPD method to model heat conduction in nanocomposites which can be interpreted as a model for thermal transport in heterogeneous materials. They were able to account for the interfacial thermal resistance, which is an atomistic phenomenon, by using the EDPD model. Moreover, they estimated the thermal conductivity of heterogeneous nanocomposite and it was found that the embedding of high thermal conductivity nanoparticles enhances the thermal conductivity of the matrix materials. Based on the DPD model with conserved energy, Stoltz [2006] proposed an interesting mesoscopic model for the simulation of shock waves and he successfully simulated the shock waves in a crystalline polymer. Later this reduced model was re?ned by Maillet et al. [2007] to handle chemical reactions. 2.3. Time integration schemes An extra degree of di?culty arises in the time integration of the DPD algorithm because unlike MD, the DPD equations are stochastic and there is thus no guarantee for time reversibility of the integration process. Moreover, the dissipative force depends on the velocity, which in turn depends on the force, and so there exists a nonlinear coupling. Typically the simple Euler and velocity-Verlet algorithms can be used to integrate the DPD equations, [Espa? nol and Warren, 1995; Groot and Warren, 1997; Warren, 1998; Gibson et al., 1999a; Hafskjold et al., 2004]. In order to follow the phase space path of the particles more accurately and for better representation of the stochastic di?erential equations, various ?nite time-step implementations of DPD which are often based on analogies to higher-order solvers for conservative systems have been proposed. These include the Verlet or leap-frog algorithms that would be expected to follow the evolution up to the second-order in the time-step. However, care must be taken in the implementation to take account of the stochastic nature of the underlying equations, and an analysis of these methods for DPD can be found in Novik and Coveney [1998]. Correspondingly, other complex integration schemes such as the Lowe’s approach, Lowe [1999], self consistent Verlet, [Besold et al., 2000; Pagonabarraga et al., 1998], and Shardlow’s splitting method, Shardlow [2003], have been proposed. A detailed comparison of the performance of these integrators is given in Nikunen et al. [2003] and Vattulainen et al. [2002]. Vattulainen et al. [2002] found that the integrators in which the velocities and dissipative forces are estimated self-consistently, as in Besold et al. [2000] and Pagonabarraga et al. [1998], demonstrated better performance. More recently, the applicability of the stochastic Trotter integration scheme (a common MD integrator)

Dissipative Particle Dynamics

745

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

as an alternative updating algorithm for DPD was investigated by De Fabritiis et al. [2006], Serrano et al. [2006] and Thalmann and Farago [2007]. The strong dependence of the DPD equilibrium kinetic temperature to the timestep was explored in Marsh and Yeomans [1997]. Based on Andersen’s thermostat, Andersen [1980], Lowe [1999] formulated an alternative DPD approach in which the interaction potential does not include dissipative or random forces. The relative velocities of the proposed Lowe-Andersen thermostat are taken from the Maxwellian distribution which maintains rigorous temperature control. More importantly, higher viscosities and consequently higher Schmidt numbers, which are the characteristics of liquid dynamics, can be achieved by adjusting related parameters in this scheme. Later, to control temperature ?uctuations, Den Otter and Clarke [2000, 2001] proposed another method in which the coe?cients of the random and dissipative forces are tuned according to the size of the time-step. Chen et al. [2005] employed the Lowe-Andersen thermostat to simulate polymeric systems, and in particular to examine the e?ciency of this method for modelling microphase separation of diblock copolymers. It should be pointed out that one most e?cient algorithms, based on both computational cost and degree of accuracy, is the modi?ed velocityVerlet scheme 1 2 t +dt t = rt rt i + dtvi + dt fi i 2 t+dt t vi = vi + λdtfit
+dt t+dt fit+dt = fi (rt , vi ) i

(9)

1 t+dt t = vi + dt(fit + fit+dt ), vi 2 where the actual velocity-Verlet algorithm would be recovered for λ = 1/2. This modi?ed integration scheme was ?rst devised by Groot and Warren [1997] and subsequently used by many researchers in the area of DPD simulations. Due to the stochastic nature of the process, the order of the algorithm becomes indistinct, and the variable factor λ, introduced empirically, appears to account for some of the additional e?ects of the stochastic interactions. In addition to the di?culties of ?nding appropriate time integrating scheme arising from the stochastic nature of the DPD algorithm, other problems may be encountered when simulating complex soft matter systems. In particular, the use of the Lennard-Jones (LJ) potential for each bead pair of polymers requires smaller time-step compared to soft repulsion of typical DPD particles, Soddemann et al. [2003]. Due to the presence of both soft and hard potentials, Symeonidis et al. [2005] and Symeonidis and Karniadakis [2006] proposed the use of time-staggered algorithms to study the polymeric physical quantities (such as end-to-end distance or radius of gyration) e?ciently. For DPD simulations of lipid bilayers in water, Jakobsen et al. [2005] investigated the sensitivity of the pressure pro?les and the kinetic bead temperatures to the artifacts caused by varying the integration time step and

746

E. Moeendarbary, T. Y. Ng & M. Zangeneh

the thermostat. Subsequently, to speed up the DPD simulations, Jakobsen et al. [2006] introduced a multiple time-step integrating scheme based on the velocityVerlet algorithm, whereby the solvent particles are updated at lower frequencies than the bounded interactions within the solute. In addition, Allen [2006] pointed out the necessity of examination and measurement of the con?gurational temperature besides the typical kinetic temperature, for determining the optimum time-step values in multicomponent systems. 2.4. Boundary models De?ning the correct boundary conditions, especially at solid boundaries, is one of the main issues in DPD simulations involving wall-bounded geometries. In DPD, it is possible to employ the general implementation of boundary conditions that is usually used in the lattice Boltzmann method (LBM) and molecular dynamics (MD) formulations. However, due to the soft repulsion between the DPD particles, the ?uid particles are not naturally prevented from penetrating solid boundaries. This is unlike the MD method, and appropriate mechanisms thus have to be developed and implemented at the walls. In addition to impenetrability, the boundary should impose the correct velocity pro?le through in the general ?ow ?eld and also at the boundaries (no slip condition) while having consistency to macroscopic system properties; i.e. the temperature and density pro?les should obey the thermal and continuum limits, especially near the boundaries. A full description and classi?cation of DPD boundary models can be found in Moeendarbary et al. [2008], and here we shall brie?y review most of works related to DPD boundary models. In Revenga et al. [1998], the boundary is modelled as layer of ?xed DPD particles, and taking the continuum limit of this layer, the dissipative and stochastic forces on the DPD ?uid particles are determined analytically. In Revenga et al. [1999], the e?ects of specular, Maxwellian, and bounce back re?ections on sticking boundary conditions and temperature distributions, were examined. A similar method to Revenga et al. [1998], (i.e. obtaining e?ective forces for planar geometries), but adapted for cylindrical and spherical geometries, was used by Colmenares and Rousse [2006] to obtain explicit expressions for the e?ective random and dissipative forces for a point DPD particle. For the DPD simulation of the shearing of a liquid drop on a solid surface, Jones et al. [1999] used similar densities for the solid and liquid, but with a strong repulsive interaction between both phases to keep them separated. Also, in an attempt to avoid slip at wall boundaries, they imposed a certain velocity on all particles within a close distance from the wall. Willemsen et al. [2000a] proposed a scheme which intrinsically imposes no-slip boundary conditions in DPD without having to arti?cially use high wall densities to achieve the same. By adjusting the distance between the layers of wall particles, a new implementation was reported by Duong-Hong et al. [2004], for achieving no slip boundary condition with low density distortion. In order to reduce the undesirable e?ects on macroscopic properties, Visser et al. [2005] developed a notable method whereby
Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

747

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

impenetrable ?at and cylindrical solid walls were devised via a wall construction technique that employs parallel twin systems which set up the wall through backto-back placement. In order to measure the viscosity of ?ow in a particle method like the DPD, Backer et al. [2005b] introduced a novel type of periodic boundary conditions to simulate counter-?owing Poiseuille ?ows without the use of explicit boundaries. Based on an equivalent force between wall and DPD particles, Pivkin and Karniadakis [2005] developed a method to obtain the no slip boundary condition. Subsequently, Pivkin and Karniadakis [2006b] also proposed a general adaptive method to prevent the density ?uctuations. Haber et al. [2006] employed DPD simulations in ?ow generated by two rotating concentric cylinders and used this model to examine some of the earlier proposed boundary conditions for analyzing the velocity, density and temperature pro?les. For treating DPD simulations with higher dissipation rates, simple modi?cations to the Lees-Edward periodic boundary condition was made in by Chatterjee [2007]. Employing a stochastic boundary forcing technique for a number of benchmark DPD problems, Altenho? et al. [2007] obtained the accurate no slip boundary condition while minimizing other spurious ?uctuations. Another method for the treatment of solid-liquid interfaces was reported recently in Henrich et al. [2007], in which the solid representation is that of an amorphous, thermally rough ensemble of particles. 3. Complex Fluid Applications Elliott and Windle [2000] constructed a model for the geometrical packing of irregularly shaped ?ller particles in composite materials within the DPD framework. Following this, Rahatekar et al. [2005] simulated the packed assemblies of oriented ?bres suspended in a viscous medium and the DPD assembled structures were used in Monte Carlo simulations to estimate the network impedance. More recently, the modi?cation of the molecular dynamics package (DL POLY) to handle mesoscale modelling of ?bre networks was reported by Elliott et al. [2006]. The well-known attributes of carbon nanotubes (CNTs) which include excellent mechanical, thermal, and electrical properties, ?nds promising applications in nanocomposites, and designing their dispersion within polymeric matrices is a very important aspect in this emerging area of nanotechnology. By improving the Flory-Huggins theory for DPD parameters, Maiti et al. [2005] were able to obtain the equilibrium morphology of the polymer-nanotube composites. Later, Wescott et al. [2007] mapped the DPD calculated mesoscale morphology of these composites onto a ?nite-element grid and the electrical conductivity of the ?lms was estimated. In order to predict the structure of polymer-clay nanocomposites (PCN), nanotube–polymer composites and polymer blends (PCN), Fermeglia and Pricl [2007] and Scocchi et al. [2007a, b] reported a hierarchical procedure which bridges the atomistic and mesoscopic simulations. They calculated the interaction parameters of mesoscopic DPD model by mapping the corresponding energy values obtained from MD simulations. Figure 2

748

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

(a)

(b)

Fig. 2. Morphology of the clay-nylon 6 nanocomposite system obtained via DPD simulation: (a) perspective view and (b) frontal view. Orange-clay; blue-surfactant head; green-surfactant tails; red-polymer. Reprinted with permission from Scocchi et al. [2007b]. Copyright 2006 American Chemical Society.

shows the morphology of clay–nylon 6 nanocomposites obtained from DPD simulation, which is in good agreement with earlier experimental and atomistic simulation results. Recent studies have claimed that the motor proteins can be a potential tool for powering future nano-bio-mechanical systems. With an aim to study the interaction between the protein motors and bio-?laments, and to investigate the e?ects of motor density and the length of microtubule on the microtubule motion, Chen et al. [2007] conducted DPD simulations of molecular motors attached to a microtubule. In Bedau et al. [2006] a simple form of ligation was investigated by a DPD model extended to include the dynamic making and breaking of strong bonds. In Gazzola et al. [2007] and Buchanan et al. [2008], it was shown that the incorporation of the chemical reactions into the DPD model enables the additional study of the interplay between chemical reactions and self-assembly processes. More interestingly, the coupled di?usion, self-assembly, and chemical reaction processes required to model a full life cycle of a protocell were reported by Fellermann et al. [2007] and Fellermann and Sol? e [2007], using a DPD framework. In addition to successful application of DPD in the areas discussed in the earlier sections on polymeric related ?elds, there are some other publications connected to DPD simulations, which are more general and related to various ?uidic systems. Thus, in the following sub-sections, we shall classify and discuss these speci?c categories, and review some of the most representative works in these categories. 3.1. Suspension ?ow and colloidal system Understanding the rheological properties of colloidal suspensions with particles of di?erent sizes, shapes and compositions which are suspended in di?erent mediums

Dissipative Particle Dynamics

749

of various conditions, is one of the key issues in its modelling and simulation. The industrial applications of these suspensions range from paints, coating ?uidized beds to oil recovery. In addition to continuum based approaches like Brownian dynamics (see Ball and Melrose [1997] for more details) or Stokesian dynamics, which are computationally expensive for high shear rates Boek et al. [1997], there are alternative mesoscopic techniques such as the Lattice Boltzmann method (LBM) and DPD. The DPD simulation of suspensions is a possible way to accommodate continuum hydrodynamics at large length scales, while it is also capable of capturing some degree of molecular details. In the following discussion, we review the most representative works which used DPD to simulate colloidal suspensions. Boek et al. [1997] investigated the rheological properties of suspension of large colloidal particles such as spheres, rods and disks in a liquid of interacting point particles. They found the DPD method highly e?cient for the calculation of hydrodynamic interactions compared to methods which use continuum models for the solvent. In Boek et al. [1996, 1997], the viscosity as a function of shear rate and volume fraction of the suspended particles was calculated using DPD, and the results for dilute suspensions of rods and disks were in excellent agreement with theoretical predictions. In addition, for the semidilute regime, they found similar “Doi-Edwards” scaling behaviour for the concentrated rod suspensions. In an exposition on the issues and complications of using DPD to simulate colloids, Whittle and Dickinson [2001] noted that an increase in the size of the colloidal particles compared with the ?uid particles is expected to improve the quality of the hydrodynamic interactions, but this also increases the total number of particles in the simulation and is thus computationally costly. An additional complication arises from the depletion e?ects caused by structuring of the small particles between adjacent larger particles. With regard to these ?nite size and resolution e?ects in the DPD simulation of colloidal systems, Boek and Van Der Schoot [1998] examined the ?uid ?ow through a periodic array of spheres, rheology of suspended spheres and aggregation of solid spheres caused by depletion ?occulation. They found that for the case of ?uid ?ow through arrays, the size of system does not a?ect dimensionless drag, while for higher solid volume fractions, it was necessary to increase the system size to avoid the ?nite size and resolution e?ects. Gibson et al. [1998, 1999b] used DPD to e?ectively simulate adsorption of colloidal particles onto a polymer-coated surface and their results achieved good agreement with the theoretical predictions, in that as the size of the polymers relative to the colloidal particle increases or similarly as density of the polymers increases, the adsorption of particles onto the surface would be less likely. Van Der Kooij et al. [2001] obtained good results for the intrinsic viscosity of hard plate suspension by studying the rheology of dilute suspensions of these hard plate-like colloids through combining rheological measurements on suspensions of sterically stabilized plate-like colloids with DPD modelling for disks.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

750

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Kim and Phillips [2004] studied ?ows around spheres and cylinders at ?nite Reynolds numbers with a ?nite ?uid inertia in DPD simulations. Flow around immobile objects and translation and rotation of mobile objects were investigated and it was shown that under computationally feasible conditions, DPD simulations are quantitatively accurate up to Reynolds numbers of 50?100, and the inaccuracies at higher Reynolds numbers can be attributed to compressibility e?ects. Later, Chen et al. [2006] estimated the drag force and torque of uniform and shear ?ow around a stationary sphere by employing DPD simulation, and they observed the importance of the dissipative coe?cient on the resulting values of the drag force and torque. In Darias et al. [2003], suspensions constricted in cylindrical geometries were investigated using DPD. In this DPD model, the suspended soft spheres interact via a conservative force while the continuum phases interact through DPD forces. Pryamitsyn and Ganesan [2005] devised a simple extension of the DPD approach to model the rheology of macroparticles in complex ?uid solvents, and in addition, they were also able to track the glass transition and related dynamical phenomena of their suspension model. Later, De Palma et al. [2006] applied the DPD method to simulate the ?ow driven by a peristaltic micropump which consists of several colloidal spheres, and they obtained a detailed description of local ?ow properties which were reasonably accurate when compared with corresponding experimental data. Martys [2005] performed a series of simulation tests in order to compare the DPD results of various suspension models with other theoretical predictions, and he found that the original DPD method recovers the ?ow of a suspension for volume fractions in the dilute to semidilute systems regime well. However, at higher volume fractions and Peclet numbers P e, he found that it was necessary to adjust the time step and explicitly include lubrication forces into the DPD algorithm in order to consider important phenomena that must be resolved at small time and length scales. Satoh and Chantrell [2006] ?rst modelled the interactions between dissipative and magnetic particles of two magnetic particle system using an idealised model potential, and then moved on to study ferromagnetic colloidal dispersion in a multimagnetic particle system. The DPD simulations were carried out to investigate particle aggregates and the pair correlation function along an applied magnetic ?eld direction, and the results were found to be in good agreement with those of Monte Carlo and Brownian dynamics simulations. For simulation of ordered colloidal structures, Dzwinel and Yuen [2000] and Dzwinel et al. [2002] used a Lennard-Jones-type potential to de?ne the colloidal particle system and DPD particles to mimic the solvent. The phase transition of particles and spontaneous creation of spherical or worm-like micelles and their crystallization in stable hexagonal or wormlike structures were observed. In addition, the strongly variable properties such as the viscosity and partial pressure of the DPD solvent were found to be crucial in determining the speed of crystallization.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

751

3.2. Phase separation of immiscible ?uids and mixtures The growth kinetics of immiscible ?uids and simulating the phase separation of these mixtures is another interesting issue in soft matter modelling. A variety of techniques such as cell dynamical systems with/without Oseen tensor hydrodynamics, time dependent Ginzburg-Landau models, lattice-gas automata (LGA) and LBM have been used to simulate these systems Coveney and Novik [1996]. However, DPD is proving to be an alternative technique which combines the best aspects of LGA and MD to investigate these systems Arce et al. [2006]. Employing DPD, Coveney and Novik [1996] and Novik and Coveney [1997] investigated the domain growth and phase separation in two-dimensional binary immiscible ?uids. They studied the scaling laws for the domain size growth R(t) ? tn for both symmetric and asymmetric quenches and found the two-dimensional growth exponents being n = 1/2 and n = 2/3 at early and late times, respectively, for symmetric quenches, while n =1/3 throughout for asymmetric quenches. Furthermore, Laplace’s law was validated by a series of simple bubble experiments and the existence of surface tension between the two phases was con?rmed. Subsequently, Novik and Coveney [2000] studied the domain growth and phase separation of hydrodynamically correct binary immiscible ?uids of di?ering and equal viscosity, as a function of minority phase concentration, in both two and three spatial dimensions. For the latter three-dimensional case, it was observed that the characteristic domain size scales as n =1/3 for simulations of di?ering and equal viscosity ?uids developing from symmetric and slightly o?-critical quenches, see Fig. 3. Later, based on di?erence in dynamic properties and particularly by adjustment of friction coe?cients of DPD model, Yaneva et al. [2005] proposed a model of a binary mixture. The e?ect of volume fraction, radius, and mass of nanoparticles on the dynamics and morphologies of phase separation in three-dimensional ?uids containing nanospheres was investigated by authors Laradji and Hore [2004] using DPD. At low to moderate volume fractions of nanoparticles, the growth law was found to be similar to pure binary mixtures. However, slower growth regimes were observed as the volume fraction of the nanospheres is increased or their radius decreased, which is associated with crystallization of the nanoparticles within the preferred component. More recently, for moderate volume fraction of nanoparticles, Hore and Laradji [2007] explored the microphase separation induced by interfacial segregation of isotropic nanoparticles interacting symmetrically with the two ?uids. Liu et al. [2007a] studied the in?uence of polymerization on the phase separation of binary immiscible mixtures via DPD in two dimensions. They observed that during polymerization the bulk viscosity increased which slows down the spinodal decomposition process and suppressed the speed of phase separation. 3.3. The thin ?lm evolution and dynamics of drop There are various types of industrial and biological systems associated with suspension of droplets, deformation of individual drops in multiphase ?ow environments,

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

752

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Fig. 3. Time evolution of ?ve sample simulations of two (left) and three (right) dimensional equal viscosity ?uid of di?erent minority phase fractions varying from φ = 0.1 to φ = 0.5. Reprinted with permission from Novik and Coveney [2000]. Copyright 2000 by the American Physical Society.

and wetting processes. Until now and probably due to certain theoretical and experimental inadequacies, the physics of the dynamics of the drop’s motion and its shape and stability have not been fully understood. Accurate and robust numerical simulation techniques which can handle mesoscopic time and length scales are key tools which can help us to extract and understand the underlying physics of these complex systems, and DPD seems to be the proper mesoscale method which has recently attracted the attention of many investigators in this area. Employing MD and/or DPD, Dzwinel and Yuen [1999] and Dzwinel et al. [2000] studied the gravitational fall of a ?uid ?lm positioned on the underside of a plate (Raleigh-Taylor ?ow), and they observed the short-time rupture of the thin ?lm, its break up into contracted droplets and spikes, and ?nally the formation of bubbles. Moreover, formation of ?ngering instability, rivulets and horseshoe patterns were also observed. Later, Kong and Yang [2006] used DPD to study the e?ects of substrate topology on contact angle hysteresis. They found that discontinuous solid substrates (DSS) have relatively larger contact angle hysteresis at lower temperature, while from dynamic wettabilty viewpoint, a continuous solid substrate (CSS) is more suitable for building an ultrahydrophobic or ultralyophobic surface. More recently, a method to model adhesive, solid boundaries for the treatment of

Dissipative Particle Dynamics

753

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

solid-liquid interfaces was introduced by Henrich et al. [2007], and it was applied in a DPD framework to examine forced wetting and behaviour of dynamic angles. The dynamics of a drop con?ned through a liquid, and attached on a solid surface under a shear ?eld, was studied by Jones et al. [1999]. Through their DPD simulations, they found that the shear ?eld induces contact angle hysteresis in the drop, and this hysteresis increased with the shear rate. Consequently, when the shear rate exceeds a critical value, the drop was inclined to lift o? the boundary, and upon elongation of the drop, necks are developed and ?nally the drop breaks up as the necks rupture upon thinning. Following this, Clark et al. [2000] investigated the shape pro?le and detachment process of pendant drops, and they found that the DPD simulated shape pro?le was in a complete agreement with the solution of Laplace equation, while the neck formation and detachment of the drop was consistent with the earlier experimental data. In addition, they also investigated the time evolution of the shape of the drop as it underwent the break up process in shear ?eld, and the calculated critical value of the capillary was in reasonable agreement with experimental ?gures. In Warren [2003b], a many-body DPD model was constructed to cater for vapour-liquid coexistence, and this was used for the simulation of pedant drops. Liew and Mikami [2003] implemented several coarsegraining procedures for soft-matter, which also resulted models able to reproduce liquid-vapour coexistence, and they used this ‘soft-attractive-and-repulsive’ pairpotential model in combination with DPD to investigate dynamic properties in complex ?uids. DPD simulations of the realignment of a nematic nanodroplet suspended in an isotropic ?uid following a switch in the direction of an applied external magnetic ?eld were carried out by Levine and Polimeno [2007], and the results showed signi?cant spatial inhomogeneities in the properties of the nanodroplet, consistent with its ?uid structure. 3.4. Multiphase ?ows There are several applications, such as liquid jet break up, drop collisions, drop break up, mixing and multiphase ?ows in mircochannels, in which the ?ow involves di?erent phases. The key issue in these multiphase ?ow applications is the interfacial physics between di?erent phases of liquid, gas and solid. The DPD method, due to its mesoscopic features, can be a very useful simulation tool in this area. Based on mean ?eld theory, Tiwari and Abraham [2006] proposed a DPD model for twophase ?ows involving gas and liquid phases. In the validation of their proposed model, they carried out simulations of problems with interfacial dynamics, such as the small-and large-amplitude oscillations of liquid cylinders and capillary waves. Visser [2006] proposed three methods to handle the friction factor for the interaction between particles of unlike ?uids in multi-viscosity systems. The capability of DPD in capturing the qualitative jetting e?ects which occur during ceramic injection was reported in by Heldele et al. [2006].

754

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Liu et al. [2006] proposed a combination of short-range repulsive and long-range attractive interactions for the DPD modelling of multiphase systems, and using this model they studied the behaviour of liquid drops in a gas. Later a similar model was used to simulate multiphase ?uid ?ow in microchannel networks Liu et al. [2007c], unsaturated fracture junctions Liu et al. [2007b] and porous media Liu et al. [2007d], which is complex due to interplay of viscous, capillary and gravitational forces, microchannel geometry, and the in?ow conditions. Varying injection rates and the ?uid-?uid and ?uid-wall interaction strengths, and external forces, di?erent modes of ?ow such as thin ?lm ?ow, wetting and non-wetting ?ows, were observed by Liu et al. [2007b] and this veri?es the viability of the use of DPD methodology in the area of multiphase simulations. 4. Concluding Remarks The DPD method is a relatively new mesoscopic simulation technique for the simulation of complex systems, and it is capable of addressing certain features where other similar scale methods are unable. However, there are several issues associated with the DPD that may require new development work or future re?nements. These include possible new directions in theoretical aspects as well as its applications. In the theoretical development area, some of the interesting challenges can be formulation of more appropriate DPD algorithms and parameters that are capable of handling variety of problems such as multiphase problems, complex heat transfer studies (such as conduction and convection in nano-materials), simulation of biological membranes or macromolecules subject to di?erent ?ows and geometries, morphology investigation of self-assemblies, etc. Depending on the application; huge computational cost may be required, while fast time-evolution algorithms necessary for the speed up of the simulations are always essential. There are vast opportunities for particle methods to decrease computational cost by improving cell algorithms (such as the Neighbor List method), as well as e?cient parallel implementations, [Dzwinel et al., 1999; Boryczko et al., 2002, 2003; Sims and Martys, 2004; Oh and Klein, 2006]. Finally, it would be a major milestone, and just a matter of time when innovative researchers concurrently couple DPD with other particle based methods like MD in order to handle multiscale problems. In the nano-scale regions or close to boundaries where DPD is unable to capture the details of microscale or even nanoscale interactions, more re?ned techniques can be used while DPD is being used to simulate the bulk region. Acknowledgements The ?rst author, E. Moeendarbary, would like to thank Ms Shima Archangi for her kind assistance in gathering the reference documentation. In addition, he is grateful to his sponsors, namely the Dorothy Hodgkins Postgraduate Award (DHPA) and Ebara Corporation of Japan.

Dissipative Particle Dynamics

755

References
Allen, M. P. [2006] “Con?gurational temperature in membrane simulations using dissipative particle dynamics,” Journal of Physical Chemistry B, 110(8): 3823–3830. Allen, M. P. and Schmid, F. [2007] “A thermostat for molecular dynamics of complex ?uids,” Molecular Simulation, 33(1–2): 21–26. Altenho?, A. M., Walther, J. H. and Koumoutsakos, P. [2007] “A stochastic boundary forcing for dissipative particle dynamics,” Journal of Computational Physics, 225(1): 1125–1136. Andersen, H. C. [1980] “Molecular dynamics simulations at constant pressure and/or temperature,” The Journal of Chemical Physics, 72: 2384. Arce, J. C., Schaadt, A. and Bart, H. J. [2006] “Extraction with spinodal decomposition Experiment and simulation,” Chemical Engineering and Technology, 29(4): 487–494. Avalos, J. B. and Mackie, A. D. [1997] “Dissipative particle dynamics with energy conservation,” Europhysics Letters, 40(2): 141–146. Avalos, J. B. and Mackie, A. D. [1999] “Dynamic and transport properties of dissipative particle dynamics with energy conservation,” Journal of Chemical Physics, 111(11): 5267–5276. Backer, J. A., Lowe, C. P., Hoefsloot, H. C. J. and Iedema, P. D. [2005a] “Combined length scales in dissipative particle dynamics,” Journal of Chemical Physics, 123(11): 1–10. Backer, J. A., Lowe, C. P., Hoefsloot, H. C. J. and Iedema, P. D. [2005b] “Poiseuille ?ow to measure the viscosity of particle model ?uids,” Journal of Chemical Physics, 122(15): 1–6. Ball, R. C. and Melrose, J. R. [1997] “A simulation technique for many spheres in quasistatic motion under frame-invariant pair drag and Brownian forces,” Physica A: Statistical Mechanics and its Applications, 247(1–4): 444–472. Bedau, M. A., Buchanan, A., Gazzola, G., Hanczyc, M., Maeke, T., McCaskill, J., Poli, I. and Packard, N. H. [2006]. Evolutionary design of a DDPD model of ligation. Lecture Notes in Computer Science (including subseries Lecture Notes in Arti?cial Intelligence and Lecture Notes in Bioinformatics). 3871 LNCS: 201–212. Besold, G., Vattulainen, I., Karttunen, M. and Polson, J. M. [2000] “Towards better integrators for dissipative particle dynamics simulations,” Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 62(6 B): R7611–R7614. Boek, E. S., Coveney, P. V. and Lekkerkerker, H. N. W. [1996] “Computer simulation of rheological phenomena in dense colloidal suspensions with dissipative particle dynamics,” Journal of Physics Condensed Matter, 8(47): 9509–9512. Boek, E. S., Coveney, P. V., Lekkerkerker, H. N. W. and Van Der Schoot, P. [1997] “Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics,” Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 55(3 SUPPL. B): 3124–3133. Boek, E. S. and Van Der Schoot, P. [1998] “Resolution e?ects in Dissipative particle dynamics simulations,” International Journal of Modern Physics C, 9(8): 1307–1318. Boryczko, K., Dzwinel, W. and Yuen, D. A. [2002] “Parallel implementation of the ?uid particle model for simulating complex ?uids in the mesoscale,” Concurrency Computation Practice and Experience, 14(2): 137–161. Boryczko, K., Dzwinel, W. and Yuen, D. A. [2003] “Clustering revealed in high-resolution simulations and visualization of multi-resolution features in ?uid-particle models,” Concurrency Computation Practice and Experience, 15(2): 101–116. Buchanan, A., Gazzola, G. and Bedau, M. A. [2008]. Chapter 4 Evolutionary Design of a Model of Self-Assembling Chemical Structures. Studies in Multidisciplinarity, 5: 79–100.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

756

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Chatterjee, A. [2007] “Modi?cation to Lees-Edwards periodic boundary condition for dissipative particle dynamics simulation with high dissipation rates,” Molecular Simulation, 33(15): 1233–1236. Chen, L. J., Lu, Z. Y., Qian, H. J., Li, Z. S. and Sun, C. C. [2005] “The e?ects of LoweAndersen temperature controlling method on the polymer properties in mesoscopic simulations,” Journal of Chemical Physics, 122(10): 1–8. Chen, Q., Li, D. Y. and Oiwa, K. [2007] “The coordination of protein motors and the kinetic behavior of microtubule - A computational study,” Biophysical Chemistry, 129(1): 60– 69. Chen, S. and Doolen, G. D. [1998]. Lattice Boltzmann method for ?uid ?ows. Annual Review of Fluid Mechanics, 30: 329–364. Chen, S., Phan-Thien, N., Khoo, B. C. and Fan, X. J. [2006] “Flow around spheres by dissipative particle dynamics,” Physics of Fluids, 18(10). Clark, A. T., Lal, M., Ruddock, J. N. and Warren, P. B. [2000] “Mesoscopic simulation of drops in gravitational and shear ?elds,” Langmuir, 16(15): 6342–6350. Colmenares, P. J. and Rousse, R. [2006] “E?ective boundary forces for several geometries in dissipative particle dynamics,” Physica A: Statistical Mechanics and its Applications, 367: 93–105. Cotter, C. J. and Reich, S. [2003] “An extended dissipative particle dynamics model,” Europhysics Letters, 64(6): 723–729. Coveney, P. V. and Espa? nol, P. [1997] “Dissipative particle dynamics for interacting multicomponent systems,” Journal of Physics A: Mathematical and General, 30(3): 779–784. Coveney, P. V. and Novik, K. E. [1996] “Computer simulations of domain growth and phase separation in two-dimensional binary immiscible ?uids using dissipative particle dynamics,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 54(5): 5134–5141. Cubero, D. and Yaliraki, S. N. [2005a] “Formal derivation of dissipative particle dynamics from ?rst principles,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 72(3): 1–4. Cubero, D. and Yaliraki, S. N. [2005b] “Inhomogeneous multiscale dynamics in harmonic lattices,” Journal of Chemical Physics, 122(3): 1–9. Darias, J. R., Quiroga, M., Medina, E., Colmenares, P. J. and Paredes, R. [2003] “Simulation of suspensions in constricted geometries by dissipative particle dynamics,” Molecular Simulation, 29(6–7): 443–449. De Fabritiis, G. and Coveney, P. V. [2003] “Dynamical geometry for multiscale dissipative particle dynamics,” Computer Physics Communications, 153(2): 209–226. De Fabritiis, G., Coveney, P. V. and Flekk?y, E. G. [2002] “Multiscale dissipative particle dynamics,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 360(1792): 317–331. De Fabritiis, G., Serrano, M., Espa? nol, P. and Coveney, P. V. [2006] “E?cient numerical integrators for stochastic models,” Physica A: Statistical Mechanics and its Applications, 361(2): 429–440. De Palma, P., Valentini, P. and Napolitano, M. [2006] “Dissipative particle dynamics simulation of a colloidal micropump,” Physics of Fluids, 18(2). Den Otter, W. K. and Clarke, J. H. R. [2000] “The temperature in dissipative particle dynamics,” International Journal of Modern Physics C, 11(6): 1179–1193. Den Otter, W. K. and Clarke, J. H. R. [2001] “A new algorithm for dissipative particle dynamics,” Europhysics Letters, 53(4): 426–431. Duong-Hong, D., Phan-Thien, N. and Fan, X. [2004] “An implementation of no-slip boundary conditions in DPD,” Computational Mechanics, 35(1): 24–29.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

757

Dzwinel, W., Alda, W., Kitowski, J. and Yuen, D. A. [2000] “Using discrete particles as a natural solver in simulating multiple-scale phenomena,” Molecular Simulation, 25(6): 361–384. Dzwinel, W., Alda, W. and Yuen, D. A. [1999] “Cross-scale numerical simulations using discrete particle models,” Molecular Simulation, 22(6): 397–418. Dzwinel, W. and Yuen, D. A. [1999] “Dissipative particle dynamics of the thin-?lm evolution in mesoscale,” Molecular Simulation, 22(6): 369–395. Dzwinel, W. and Yuen, D. A. [2000] “A two-level, discrete-particle approach for simulating ordered colloidal structures,” Journal of Colloid and Interface Science, 225(1): 179– 190. Dzwinel, W., Yuen, D. A. and Boryczko, K. [2002] “Mesoscopic dynamics of colloids simulated with dissipative particle dynamics and ?uid particle model,” Journal of Molecular Modeling, 8(1): 33–43. Elliott, J. A., Benedict, M. and Dutt, M. [2006] “Applications of DL POLY to modelling of mesoscopic particulate systems,” Molecular Simulation, 32(12–13): 1113–1121. Elliott, J. A. and Windle, A. H. [2000] “Dissipative particle dynamics method for modeling the geometrical packing of ?ller particles in polymer composites,” Journal of Chemical Physics, 113(22): 10367–10376. Espa? nol, P. [1995] “Hydrodynamics from dissipative particle dynamics,” Physical Review E, 52(2): 1734–1742. Espa? nol, P. [1996] “Dissipative particle dynamics for a harmonic chain: A ?rst-principles derivation,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 53(2): 1572–1578. Espa? nol, P. [1997a] “Dissipative particle dynamics with energy conservation,” Europhysics Letters, 40(6): 631–636. Espa? nol, P. [1997b] “Fluid particle dynamics: A synthesis of dissipative particle dynamics and smoothed particle dynamics,” Europhysics Letters, 39(6): 605–610. Espa? nol, P. [1998] “Fluid particle model,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 57(3 SUPPL. A): 2930–2964. Espa? nol, P. and Revenga, M. [2003] “Smoothed dissipative particle dynamics,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 67(2 2): 267051–2670512. Espa? nol, P. and Serrano, M. [1999] “Dynamical regimes in the dissipative particle dynamics model,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 59(6): 6340–6347. ¨ Espa? nol, P., Serrano, M. and Ottinger, H. C. [1999] “Thermodynamically Admissible Form for Discrete Hydrodynamics,” Physical Review Letters, 83(22): 4542–4545. Espa? nol, P., Serrano, M. and Zu? niga, I. [1997] “Coarse-graining of a ?uid and its relation with dissipative particle dynamics and smoothed particle dynamics,” International Journal of Modern Physics C, 8(4): 899–908. Espa? nol, P. and Warren, P. [1995] “Statistical mechanics of dissipative particle dynamics,” Europhysics Letters, 30: 191–191. Evans, G. T. [1999] “Dissipative particle dynamics: Transport coe?cients,” Journal of Chemical Physics, 110(3): 1338–1342. Fan, X., Phan-Thien, N., Chen, S., Wu, X. and Ng, T. Y. [2006] “Simulating ?ow of DNA suspension using dissipative particle dynamics,” Physics of Fluids, 18(6). Fellermann, H., Rasmussen, S., Ziock, H. J. and Sol? e, R. V. [2007] “Life cycle of a minimal protocell–a dissipative particle dynamics study,” Arti?cial Life, 13(4): 319–345. Fellermann, H. and Sol? e, R. V. [2007] “Minimal model of self-replicating nanocells: A physically embodied information-free scenario,” Philosophical Transactions of the Royal Society B: Biological Sciences, 362(1486): 1803–1811.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

758

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Fermeglia, M. and Pricl, S. [2007] “Multiscale modeling for polymer systems of industrial interest,” Progress in Organic Coatings, 58(2–3): 187–199. Flekk?y, E. G. and Coveney, P. V. [1999] “From molecular dynamics to dissipative particle dynamics,” Physical Review Letters, 83(9): 1775–1778. Flekk?y, E. G., Coveney, P. V. and De Fabritiis, G. [2000] “Foundations of dissipative particle dynamics,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 62(2 B): 2140–2157. Gazzola, G., Buchanan, A., Packard, N. and Bedau, M. [2007]. Catalysis by self-assembled structures in emergent reaction networks. Lecture Notes in Computer Science (including subseries Lecture Notes in Arti?cial Intelligence and Lecture Notes in Bioinformatics). 4648 LNAI: 876–885. Gibson, J. B., Chen, K. and Chynoweth, S. [1998] “Simulation of particle adsorption onto a polymer-coated surface using the dissipative particle dynamics method,” Journal of Colloid and Interface Science, 206(2): 464–474. Gibson, J. B., Chen, K. and Chynoweth, S. [1999a] “The equilibrium of a velocity-Verlet type algorithm for DPD with ?nite time steps,” International Journal of Modern Physics C, 10(1): 241–261. Gibson, J. B., Zhang, K., Chen, K., Chynoweth, S. and Manke, C. W. [1999b] “Simulation of colloid-polymer systems using Dissipative Particle Dynamics,” Molecular Simulation, 23(1): 1–41. Goicochea, A. G. [2007] “Adsorption and disjoining pressure isotherms of con?ned polymers using dissipative particle dynamics,” Langmuir, 23(23): 11656–11663. Groot, R. D. [2006] “A local Galilean invariant thermostat,” Journal of Chemical Theory and Computation, 2(3): 568–574. Groot, R. D. and Rabone, K. L. [2001] “Mesoscopic simulation of cell membrane damage, morphology change and rupture by nonionic surfactants,” Biophysical Journal, 81(2): 725–736. Groot, R. D. and Warren, P. B. [1997] “Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation,” Journal of Chemical Physics, 107(11): 4423–4435. Haber, S., Filipovic, N., Kojic, M. and Tsuda, A. [2006] “Dissipative particle dynamics simulation of ?ow generated by two rotating concentric cylinders: Boundary conditions,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 74(4). Hafskjold, B., Liew, C. C. and Shinoda, W. [2004] “Can such long time steps really be used in dissipative particle dynamics simulations?,” Molecular Simulation, 30(13–15): 879–885. Han, Z. H., Cao, F. Y. and Yang, B. [2008] “Synthesis and thermal characterization of phase-changeable indium/polyalphaole?n nano?uids,” Applied Physics Letters, 92(24). He, P. and Qiao, R. [2008] “Self-consistent ?uctuating hydrodynamics simulations of thermal transport in nanoparticle suspensions,” Journal of Applied Physics, 103(9). Heldele, R., Schulz, M., Kauzlaric, D., Korvink, J. G. and Hau?elt, J. [2006] “Micro powder injection molding: Process characterization and modeling,” Microsystem Technologies, 12(10–11): 941–946. Henrich, B., Cupelli, C., Moseler, M. and Santer, M. [2007] “An adhesive DPD wall model for dynamic wetting,” Europhysics Letters, 80(6). Hoogerbrugge, P. J. and Koelman, J. [1992] “Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics,” Europhys. Lett, 19(3): 155–160. Hore, M. J. A. and Laradji, M. [2007] “Microphase separation induced by interfacial segregation of isotropic, spherical nanoparticles,” Journal of Chemical Physics, 126(24).

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

759

Jakobsen, A. F., Besold, G. and Mouritsen, O. G. [2006] “Multiple time step update schemes for dissipative particle dynamics,” Journal of Chemical Physics, 124(9). Jakobsen, A. F., Mouritsen, O. G. and Besold, G. [2005] “Artifacts in dynamical simulations of coarse-grained model lipid bilayers,” Journal of Chemical Physics, 122(20): 1–11. Jones, J. L., Lal, M., Ruddock, J. N. and Spenley, N. A. [1999] “Dynamics of a drop at a liquid/solid interface in simple shear ?elds: A mesoscopic simulation study,” Faraday Discussions, 112: 129–142. Junghans, C., Praprotnik, M. and Kremer, K. [2007] “Transport properties controlled by a thermostat: An extended dissipative particle dynamics thermostat,” Soft Matter, 4(1): 156–161. Keaveny, E. E., Pivkin, I. V., Maxey, M. and Karniadakis, G. E. [2005] “A comparative study between dissipative particle dynamics and molecular dynamics for simple- and complex-geometry ?ows,” Journal of Chemical Physics, 123(10): 1–9. Kim, J. M. and Phillips, R. J. [2004] “Dissipative particle dynamics simulation of ?ow around spheres and cylinders at ?nite Reynolds numbers,” Chemical Engineering Science, 59(20): 4155–4168. Kinjo, T. and Hyodo, S. [2007a] “Linkage between atomistic and mesoscale coarse-grained simulation,” Molecular Simulation, 33(4–5): 417–420. Kinjo, T. and Hyodo, S. A. [2007b] “Equation of motion for coarse-grained simulation based on microscopic description,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 75(5). Klapp, S. H. L., Diestler, D. J. and Schoen, M. [2004] “Why are e?ective potentials ‘soft’ ?,” Journal of Physics Condensed Matter, 16(41): 7331–7352. Koelman, J. and Hoogerbrugge, P. J. [1993] “Dynamic simulations of hard-sphere suspensions under steady shear,” EUROPHYSICS LETTERS, 21: 363–363. Kong, B. and Yang, X. [2006] “Dissipative particle dynamics simulation of contact angle hysteresis on a patterned solid/air composite surface,” Langmuir, 22(5): 2065–2073. Koumoutsakos, P. [2005]. Multiscale ?ow simulations using particles. Annual Review of Fluid Mechanics. 37:457–487. Kubo, R. [1966] “The ?uctuation-dissipation theorem,” Reports on Progress in Physics, 29(1):255–284. Laradji, M. and Hore, M. J. A. [2004] “Nanospheres in phase-separating multicomponent ?uids: A three-dimensional dissipative particle dynamics simulation,” Journal of Chemical Physics, 121(21): 10641–10647. Levine, Y. K. and Polimeno, A. [2007] “A dissipative particle dynamics study of the realignment of a nanodroplet of a nematic in a weak external magnetic ?eld,” European Physical Journal E, 23(1): 13–23. Liew, C. C. and Mikami, M. [2003] “A coarse-grained model for particle dynamics simulations of complex ?uids,” Chemical Physics Letters, 368(3–4): 346–351. L? ?sal, M., Brennan, J. K. and Smith, W. R. [2006] “Mesoscale simulation of polymer reaction equilibrium: Combining dissipative particle dynamics with reaction ensemble Monte Carlo. I. Polydispersed polymer systems,” Journal of Chemical Physics, 125(16). Liu, H., Qian, H. J., Zhao, Y. and Lu, Z. Y. [2007a] “Dissipative particle dynamics simulation study on the binary mixture phase separation coupled with polymerization,” Journal of Chemical Physics, 127(14). Liu, M., Meakin, P. and Huang, H. [2006] “Dissipative particle dynamics with attractive and repulsive particle-particle interactions,” Physics of Fluids, 18(1).

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

760

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Liu, M., Meakin, P. and Huang, H. [2007b] “Dissipative particle dynamics simulation of ?uid motion through an unsaturated fracture and fracture junction,” Journal of Computational Physics, 222(1): 110–130. Liu, M., Meakin, P. and Huang, H. [2007c] “Dissipative particle dynamics simulation of multiphase ?uid ?ow in microchannels and microchannel networks,” Physics of Fluids, 19(3). Liu, M., Meakin, P. and Huang, H. [2007d] “Dissipative particle dynamics simulation of pore-scale multiphase ?uid ?ow,” Water Resources Research, 43(4). Lowe, C. P. [1999] “An alternative approach to dissipative particle dynamics,” Europhysics Letters, 47(2): 145–151. Mackie, A. D., Avalos, J. B. and Navas, V. [1999] “Dissipative particle dynamics with energy conservation: Modelling of heat ?ow,” Physical Chemistry Chemical Physics, 1(9): 2039–2049. Maillet, J. B., Soulard, L. and Stoltz, G. [2007] “A reduced model for shock and detonation waves. II. The reactive case,” Europhysics Letters, 78(6). Maiti, A. and McGrother, S. [2004] “Bead-bead interaction parameters in dissipative particle dynamics: Relation to bead-size, solubility parameter, and surface tension,” Journal of Chemical Physics, 120(3): 1594–1601. Maiti, A., Wescott, J. and Kung, P. [2005] “Nanotube - Poly mer composites: Insights from Flory-Huggins theory and mesoscale simulations,” Molecular Simulation, 31(2– 3): 143–149. Marsh, C. A., Backx, G. and Ernst, M. H. [1997a] “Fokker-Planck-Boltzmann equation for dissipative particle dynamics,” Europhysics Letters, 38(6): 411–415. Marsh, C. A., Backx, G. and Ernst, M. H. [1997b] “Static and dynamic properties of dissipative particle dynamics,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 56(2): 1676–1691. Marsh, C. A. and Coveney, P. V. [1998] “Detailed balance and H-theorems for dissipative particle dynamics,” Journal of Physics A: Mathematical and General, 31(31): 6561– 6568. Marsh, C. A. and Yeomans, J. M. [1997] “Dissipative particle dynamics: The equilibrium for ?nite time steps,” Europhysics Letters, 37(8): 511–516. Martys, N. S. [2005] “Study of a dissipative particle dynamics based approach for modeling suspensions,” Journal of Rheology, 49(2): 401–424. Masters, A. J. and Warren, P. B. [1999] “Kinetic theory for dissipative particle dynamics: The importance of collisions,” Europhysics Letters, 48(1): 1–7. Moeendarbary, E., Lam, K. Y. and Ng, T. Y. [2008]. “A new ”bounce-normal” boundary in DPD calculations for the reduction of density ?uctuations.” 2008 Proceedings of the ASME Micro/Nanoscale Heat Transfer International Conference, MNHT 2008. Monaghan, J. J. [1992] “Smoothed particle hydrodynamics,” Annual Review of Astronomy and Astrophysics, 30(1): 543–574. Nikunen, P., Karttunen, M. and Vattulainen, I. [2003] “How would you integrate the equations of motion in dissipative particle dynamics simulations?,” Computer Physics Communications, 153(3): 407–423. Noguchi, H. and Gompper, G. [2007] “Transport coe?cients of dissipative particle dynamics with ?nite time step,” Europhysics Letters, 79(3). Novik, K. E. and Coveney, P. V. [1997] “Using dissipative particle dynamics to model binary immiscible ?uids,” International Journal of Modern Physics C, 8(4): 909–918. Novik, K. E. and Coveney, P. V. [1998] “Finite-di?erence methods for simulation models incorporating nonconservative forces,” Journal of Chemical Physics, 109(18): 7667– 7677.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

761

Novik, K. E. and Coveney, P. V. [2000] “Spinodal decomposition of o?-critical quenches with a viscous phase using dissipative particle dynamics in two and three spatial dimensions,” Physical Review E — Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 61(1): 435–448. Oh, K. J. and Klein, M. L. [2006] “A general purpose parallel molecular dynamics simulation program,” Computer Physics Communications, 174(7): 560–568. Pagonabarraga, I. and Frenkel, D. [2000] “Non-ideal DPD ?uids,” Molecular Simulation, 25(3–4): 167–175. Pagonabarraga, I. and Frenkel, D. [2001] “Dissipative particle dynamics for interacting systems,” Journal of Chemical Physics, 115(11): 5015–5026. Pagonabarraga, I., Hagen, M. H. J. and Frenkel, D. [1998] “Self-consistent dissipative particle dynamics algorithm,” Europhysics Letters, 42(4): 377–382. Pastewka, L., Kauzlaric, D., Greiner, A. and Korvink, J. G. [2006] “Thermostat with a local heat-bath coupling for exact energy conservation in dissipative particle dynamics,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 73(3): 1–4. Peters, E. A. J. F. [2004] “Elimination of time step e?ects in DPD,” Europhysics Letters, 66(3): 311–317. Pivkin, I. V. and Karniadakis, G. E. [2005] “A new method to impose no-slip boundary conditions in dissipative particle dynamics,” Journal of Computational Physics, 207(1): 114–128. Pivkin, I. V. and Karniadakis, G. E. [2006a] “Coarse-graining limits in open and wallbounded dissipative particle dynamics systems,” Journal of Chemical Physics, 124(18). Pivkin, I. V. and Karniadakis, G. E. [2006b] “Controlling density ?uctuations in wallbounded dissipative particle dynamics systems,” Physical Review Letters, 96(20). Pool, R. and Bolhuis, P. G. [2006] “Can purely repulsive soft potentials predict micelle formation correctly?,” Physical Chemistry Chemical Physics, 8(8): 941–948. Pryamitsyn, V. and Ganesan, V. [2005] “A coarse-grained explicit solvent simulation of rheology of colloidal suspensions,” Journal of Chemical Physics, 122(10): 1–13. Qiao, R. and He, P. [2007] “Simulation of heat conduction in nanocomposite using energyconserving dissipative particle dynamics,” Molecular Simulation, 33(8): 677–683. Qiao, R. and He, P. [2008] “Mapping of dissipative particle dynamics in ?uctuating hydrodynamics simulations,” Journal of Chemical Physics, 128(12). Rahatekar, S. S., Hamm, M., Sha?er, M. S. P. and Elliott, J. A. [2005] “Mesoscale modeling of electrical percolation in ?ber-?lled systems,” Journal of Chemical Physics, 123(13): 1–5. Revenga, M., Z? un ?iga, I. and Espa? nol, P. [1999] “Boundary conditions in dissipative particle dynamics,” Computer Physics Communications, 121: 309–311. Revenga, M., Z? un ?iga, I., Espa? nol, P. and Pagonabarraga, I. [1998] “Boundary models in DPD,” International Journal of Modern Physics C, 9(8): 1319–1328. Ripoll, M. and Ernst, M. H. [2005] “Model system for classical ?uids out of equilibrium,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 71(4): 041104/ 1–041104/14. Ripoll, M., Ernst, M. H. and Espa? nol, P. [2001] “Large scale and mesoscopic hydrodynamics for dissipative particle dynamics,” Journal of Chemical Physics, 115(15): 7271–7284. Ripoll, M., Espa? nol, P. and Ernst, M. H. [1998] “Dissipative particle dynamics with energy conservation: Heat conduction,” International Journal of Modern Physics C, 9(8): 1329–1338. Satoh, A. and Chantrell, R. W. [2006] “Application of the dissipative particle dynamics method to magnetic colloidal dispersions,” Molecular Physics, 104(20–21): 3287–3302.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

762

E. Moeendarbary, T. Y. Ng & M. Zangeneh

Satoh, A. and Majima, T. [2005] “Comparison between theoretical values and simulation results of viscosity for the dissipative particle dynamics method,” Journal of Colloid and Interface Science, 283(1): 251–266. Scocchi, G., Posocco, P., Danani, A., Pricl, S. and Fermeglia, M. [2007a] “To the nanoscale, and beyond!. Multiscale molecular modeling of polymer-clay nanocomposites,” Fluid Phase Equilibria, 261(1–2): 366–374. Scocchi, G., Posocco, P., Fermeglia, M. and Pricl, S. [2007b] “Polymer — Clay nanocomposites: A multiscale molecular modeling approach,” Journal of Physical Chemistry B, 111(9): 2143–2151. Serrano, M., De Fabritiis, G., Espa? nol, P. and Coveney, P. V. [2006] “A stochastic Trotter integration scheme for dissipative particle dynamics,” Mathematics and Computers in Simulation, 72(2–6): 190–194. Serrano, M., De Fabritiis, G., Espa? nol, P., Flekk?y, E. G. and Coveney, P. V. [2002] “Mesoscopic dynamics of Voronoi ?uid particles,” Journal of Physics A: Mathematical and General, 35(7): 1605–1625. Serrano, M. and Espa? nol, P. [2001] “Thermodynamically consistent mesoscopic ?uid particle model,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 64(4 II): 461151–4611518. Serrano, M., Espa? nol, P. and Z? un ?iga, I. [1999] “Collective e?ects in dissipative particle dynamics,” Computer Physics Communications, 121: 306–308. Shardlow, T. [2003] “Splitting for dissipative particle dynamics,” SIAM Journal of Scienti?c Computing, 24(4): 1267–1282. Shardlow, T. and Yan, Y. [2006] “Geometric ergodicity for dissipative particle dynamics,” Stochastics and Dynamics, 6(1): 123–154. Sims, J. S. and Martys, N. [2004] “Simulation of sheared suspensions with a parallel implementation of QDPD,” Journal of Research of the National Institute of Standards and Technology, 109(2): 267–277. Soddemann, T., D¨ unweg, B. and Kremer, K. [2003] “Dissipative particle dynamics: A useful thermostat for equilibrium and nonequilibrium molecular dynamics simulations,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 68(4 2): 467021– 467028. Stoltz, G. [2006] “A reduced model for shock and detonation waves. I. The inert case,” Europhysics Letters, 76(5): 849–855. Stoyanov, S. D. and Groot, R. D. [2005] “From molecular dynamics to hydrodynamics: A novel Galilean invariant thermostat,” Journal of Chemical Physics, 122(11): 1–8. Symeonidis, V., Em Kamiadakis, G. and Caswell, B. [2005] “A seamless approach to multiscale complex ?uid simulation,” Computing in Science and Engineering, 7(3): 39–46. Symeonidis, V. and Karniadakis, G. E. [2006] “A family of time-staggered schemes for integrating hybrid DPD models for polymers: Algorithms and applications,” Journal of Computational Physics, 218(1): 82–101. Symeonidis, V., Karniadakis, G. E. and Caswell, B. [2006] “Schmidt number e?ects in dissipative particle dynamics simulation of polymers,” Journal of Chemical Physics, 125(18). Thalmann, F. and Farago, J. [2007] “Trotter derivation of algorithms for Brownian and dissipative particle dynamics,” Journal of Chemical Physics, 127(12). Tiwari, A. and Abraham, J. [2006] “Dissipative-particle-dynamics model for two-phase ?ows,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 74(5). Tiwari, A. and Abraham, J. [2008] “Dissipative particle dynamics simulations of liquid nanojet breakup,” Micro?uidics and Nano?uidics, 4(3): 227–235.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.

Dissipative Particle Dynamics

763

Tro?mov, S. Y., Nies, E. L. F. and Michels, M. A. J. [2002] “Thermodynamic consistency in dissipative particle dynamics simulations of strongly nonideal liquids and liquid mixtures,” Journal of Chemical Physics, 117(20): 9383–9394. Tro?mov, S. Y., Nies, E. L. F. and Michels, M. A. J. [2005] “Constant-pressure simulations with dissipative particle dynamics,” Journal of Chemical Physics, 123(14): 1–12. Van Der Kooij, F. M., Boek, E. S. and Philipse, A. P. [2001] “Rheology of dilute suspensions of hard platelike colloids,” Journal of Colloid and Interface Science, 235(2): 344–349. Van Vliet, R. E., Dreischor, M. W., Hoefsloot, H. C. J. and Iedema, P. D. [2002] “Dynamics of liquid-liquid demixing: Mesoscopic simulations of polymer solutions,” Fluid Phase Equilibria, 201(1): 67–78. Van Vliet, R. E., Hoefsloot, H. C. J., Hamersma, P. J. and Iedema, P. D. [2000] “Pressureinduced phase separation of polymer-solvent systems with dissipative particle dynamics,” Macromolecular Theory and Simulations, 9(9): 698–702. Van Vliet, R. E., Hoefsloot, H. C. J. and Iedema, P. D. [2003] “Mesoscopic simulation of polymer-solvent phase separation: Linear chain behaviour and branching e?ects,” Polymer, 44(5): 1757–1763. Vattulainen, I., Karttunen, M., Besold, G. and Polson, J. M. [2002] “Integration schemes for dissipative particle dynamics simulations: From softly interacting systems towards hybrid models,” Journal of Chemical Physics, 116(10): 3967–3979. Visser, D. C. [2006] “Modelling multi-viscosity systems with dissipative particle dynamics,” Journal of Computational Physics, 214(2): 491–504. Visser, D. C., Hoefsloot, H. C. J. and Iedema, P. D. [2005] “Comprehensive boundary method for solid walls in dissipative particle dynamics,” Journal of Computational Physics, 205(2): 626–639. Warren, P. B. [1998] “Dissipative particle dynamics,” Current Opinion in Colloid and Interface Science, 3(6): 620–624. Warren, P. B. [2003a] “A manifesto for one-body terms: The simplest of all many-body interactions?,” Journal of Physics Condensed Matter, 15(48 SPEC. ISS.): S3467-S3473. Warren, P. B. [2003b] “Vapor-liquid coexistence in many-body dissipative particle dynamics,” Physical Review E — Statistical, Nonlinear, and Soft Matter Physics, 68(6 2): 667021–667028. Wescott, J. T., Kung, P. and Maiti, A. [2007] “Conductivity of carbon nanotube polymer composites,” Applied Physics Letters, 90(3). Whittle, M. and Dickinson, E. [2001] “On simulating colloids by dissipative particle dynamics: Issues and complications,” Journal of Colloid and Interface Science, 242(1): 106– 109. Wijmans, C. M., Smit, B. and Groot, R. D. [2001] “Phase behavior of monomeric mixtures and polymer solutions with soft interaction potentials,” Journal of Chemical Physics, 114(17): 7644–7654. Willemsen, S. M., Hoefsloot, H. C. J. and Iedema, P. D. [2000a] “No-slip boundary condition in dissipative particle dynamics,” International Journal of Modern Physics C, 11(5): 881–890. Willemsen, S. M., Hoefsloot, H. C. J., Visser, D. C., Hamersma, P. J. and Iedema, P. D. [2000b] “Modelling Phase Change with Dissipative Particle Dynamics Using a Consistent Boundary Condition,” Journal of Computational Physics, 162(2): 385–394. Willemsen, S. M., Vlugt, T. J. H., Hoefsloot, H. C. J. and Smit, B. [1998] “Combining dissipative particle dynamics and Monte Carlo techniques,” Journal of Computational Physics, 147(2): 507–517. Yaneva, J., D¨ unweg, B. and Milchev, A. [2005] “Non-Fickian interdi?usion of dynamically asymmetric species: A molecular-dynamics study,” Journal of Chemical Physics, 122(20): 1–6.

Int. J. Appl. Mechanics 2009.01:737-763. Downloaded from www.worldscientific.com by XI'AN JIAOTONG UNIVERSITY on 08/16/13. For personal use only.


相关文档

  • dissipative particle dynamics
  • Dissipative particle dynamics simulation
  • Foundations of dissipative particle dynamics
  • Introduction to Dissipative Particle Dynamics
  • Thermostat in dissipative particle dynamics
  • Molecular dynamics and dissipative particle dynamics simulations for the
  • Dissipative Particle Dynamics Foundation DPD大牛4
  • Dissipative Particle Dynamics Foundation DPD大牛2
  • Kinetic theory for dissipative particle dynamics the importance of collisions
  • Dynamical geometry for multiscale dissipative particle dynamics
  • 猜你喜欢

    电脑版