Q.H.Zeng et al Prog.Polym.SeL 33(2008)191-269 In a uVT ensemble,one hypothesizes a new and time scales currently inaccessible by the classical configuration/by arbitrarily choosing one atom MD methods and proposing that it can be exchanged by an atom of a different kind. This procedure affects the 2.2.1.Brownian dynamics chemical composition of the system.Also,the move BD simulation is similar to MD simulations [20] is accepte with a cer pro ity.How However,it introduces a few new approximations the hange e energy c that allow one to pertorm simulations on the to the follo AU2o the move of sitional own up to a e e MD accepted.However.if AU>0.the move is acce pted d with an de with a certain probability which is given by △U are typically ignored,allowing a much larger time- Pi exp(- (13 step than that of MD.Therefore,BD is particularly useful for systems where there is a large gap of time where AU is the change in the sum of the mixing energy and the chemical potential of the mixture.If example,in polym the new configuration is rejected one counts the st m original configuration as a new of th ndes of the using some othe step.However.if the detailed motion of the solvent mer na ompos olecrhots M molecules is concerned they may he removed from the simulation and their effects on the polymer are rious factors represented by dissipative(-yp)and random ((t)) orce terms. the forces in the governing Eq.(1)is replaced by a Langevin equation 2.2.Microscale methods F0=F听-m+aa0 (14) The modeling and simulation at microscale aim to bridge molecular methods and continuum where F is the conservative force of particle j methods and avoid their shortcomings.Specifically acting on particle i,y and o are constants depending in nanoparticle-polymer systems. the study of on the system,P;the momentum of particle 1,and structural evolution (i.e.,dynamics of phase separa- (1)a Gaus con the the ine on st degree th interact that the macrosconic hehavior of the system will no be hydrodynamic.In addition.the effect of one solute molecule on another through the flow of methods.In contrast.the interactions betweer solvent molecules is neglected.Thus,BD can only components can be examined at an atomistic level reproduce the diffusion properties but not the but are usually not straightforward to incorporate hydrodynamic fle the simulation at the continuum level.Therefore,various s does not obey the Navier okes equations aluate and structure and particle by H BD.DPD.LB.time-depe I anda and Koel 11t n si hath and non-Newtonian fluids.including polymer melts methods.a polymer system is usually treated with and blends,on microscopic length and time scales. a field description or microscopic particles that Like MD and BD,DPD is a particle-based method. incorporate molecular details implicitly.Therefore However.its basic unit is not a single atom or they are able to simulate the phenomena on length molecule but a molecular assembly (i.e..a particle). In a mVT ensemble, one hypothesizes a new configuration j by arbitrarily choosing one atom and proposing that it can be exchanged by an atom of a different kind. This procedure affects the chemical composition of the system. Also, the move is accepted with a certain probability. However, one computes the energy change DU associated with the change in composition. The new configuration is examined according to the following rules. If DUo0, the move of compositional change is accepted. However, if DUX0, the move is accepted with a certain probability which is given by pi!j / exp DU kBT , (13) where DU is the change in the sum of the mixing energy and the chemical potential of the mixture. If the new configuration is rejected one counts the original configuration as a new one and repeats the process by using some other arbitrarily or system￾atically chosen atoms. In polymer nanocomposites, MC methods have been used to investigate the molecular structure at nanoparticle surface and evaluate the effects of various factors. 2.2. Microscale methods The modeling and simulation at microscale aim to bridge molecular methods and continuum methods and avoid their shortcomings. Specifically, in nanoparticle–polymer systems, the study of structural evolution (i.e., dynamics of phase separa￾tion) involves the description of bulk flow (i.e., hydrodynamic behavior) and the interactions be￾tween nanoparticle and polymer components. Note that hydrodynamic behavior is relatively straight￾forward to handle by continuum methods but is very difficult and expensive to treat by atomistic methods. In contrast, the interactions between components can be examined at an atomistic level but are usually not straightforward to incorporate at the continuum level. Therefore, various simula￾tion methods have been evaluated and extended to study the microscopic structure and phase separa￾tion of these polymer nanocomposites, including BD, DPD, LB, time-dependent Ginsburg–Landau (TDGL) theory, and dynamic DFT. In these methods, a polymer system is usually treated with a field description or microscopic particles that incorporate molecular details implicitly. Therefore, they are able to simulate the phenomena on length and time scales currently inaccessible by the classical MD methods. 2.2.1. Brownian dynamics BD simulation is similar to MD simulations [20]. However, it introduces a few new approximations that allow one to perform simulations on the microsecond timescale whereas MD simulation is known up to a few nanoseconds. In BD the explicit description of solvent molecules used in MD is replaced with an implicit continuum solvent de￾scription. Besides, the internal motions of molecules are typically ignored, allowing a much larger time￾step than that of MD. Therefore, BD is particularly useful for systems where there is a large gap of time scale governing the motion of different components. For example, in polymer–solvent mixture, a short time-step is required to resolve the fast motion of the solvent molecules, whereas the evolution of the slower modes of the system requires a larger time￾step. However, if the detailed motion of the solvent molecules is concerned, they may be removed from the simulation and their effects on the polymer are represented by dissipative (gp) and random (sz(t)) force terms. Thus, the forces in the governing Eq. (1) is replaced by a Langevin equation, FiðtÞ ¼ X jai F C ij gpi þ sziðtÞ, (14) where F C ij is the conservative force of particle j acting on particle i, g and s are constants depending on the system, pi the momentum of particle i, and z(t) a Gaussian random noise term. One conse￾quence of this approximation of the fast degrees of freedom by fluctuating forces is that the energy and momentum are no longer conserved, which implies that the macroscopic behavior of the system will not be hydrodynamic. In addition, the effect of one solute molecule on another through the flow of solvent molecules is neglected. Thus, BD can only reproduce the diffusion properties but not the hydrodynamic flow properties since the simulation does not obey the Navier–Stokes equations. 2.2.2. Dissipative particle dynamics DPD was originally developed by Hoogerbrugge and Koelman [21]. It can simulate both Newtonian and non-Newtonian fluids, including polymer melts and blends, on microscopic length and time scales. Like MD and BD, DPD is a particle-based method. However, its basic unit is not a single atom or molecule but a molecular assembly (i.e., a particle). ARTICLE IN PRESS Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 197