200 O.H.Zeng et al.Prog.Polym.Sci.33 (2008)191-269 are in is the calar order parameter. the di PA-PB ee-energy term. d p investigate the phase-separation of er nand M is the mobility for the order p Variable describes the asymmetry of the diblock. nanoparticles [32-36]. is the noise field.and parameter I determines the thickness of the lamellar domains which is related to 22 5 Dynamic DET method the length of the diblock.The free energy U()is Dynamic DFT method is usually used to model given by the local free energy U and polymer- the dynamic behavior of polymer systems and has particle coupling interaction term Up: U(6)=U:++Ucp (31 m th with TDGL mo el U=d/(,》+号v,2 (32) d However.in contrast to traditional phenomenolo gical free-e nergy expansion methods employed in Up=Cdr∑Vv-R(,)-中，)， (33) the TDGL approach,the free energy is not truncated at a certain level,and instead retains the =1 is the value of the full polymer path integral numerically. At the enging comput un expen The motion of particle is described by the me mply Hug Langevin equation addition.viscoelasticity.which is no =M,,-a0)+n (34 OR included in TDGL approaches,is included at the level of the Gaussian chains.A similar DFT th approach has been developed by Doi and co workers [38,39]and forms the basis represe ts a pa e no their nev software tool Simulation Utilities for Soft and Hard Inter 2.3.Mesoscale and macroscale methods at s40 Despite the importance of understanding the The of dyn nic DeT method is that the molecular structure and nature of materials,their instantaneous unique conformation distribution ca behavior can be homogenized with respect to be obtained from the off-equilibrium density profile different ects which can be at different scales by coupling a fictitious external potential to the the observed macrosc behavior Is Hamiltonian.Once such distribution is knowr the the energy is ther calculated by standar The orce me.The continuur m rial is th ned to gra ent of t fur an a subiected to the pody forces such as a ity and surface forces equations for both nolymer and particle in the Generally speaking.the macroscale methods(or diblock polymer-particle composites 1351 alled cont uum methods hereafter)obey the The dynamics of microphase separation in a melt undamental ws of:(1)continuity. derived from of diblocks is described through the following onservatio of ma equation [341: 8r,0 chang (30) an arbitrary oint is equal to the resultant momentare insensitive to the precise form of the double-well potential of the bulk free-energy term. The TDGL and CDM methods have recently been used to investigate the phase-separation of polymer nano￾composites and polymer blends in the presence of nanoparticles [32–36]. 2.2.5. Dynamic DFT method Dynamic DFT method is usually used to model the dynamic behavior of polymer systems and has been implemented in the software package Meso￾dynTM from Accelrys [37]. The DFT models the behavior of polymer fluids by combining Gaussian mean-field statistics with a TDGL model for the time evolution of conserved order parameters. However, in contrast to traditional phenomenolo￾gical free-energy expansion methods employed in the TDGL approach, the free energy is not truncated at a certain level, and instead retains the full polymer path integral numerically. At the expense of a more challenging computation, this allows detailed information about a specific poly￾mer system beyond simply the Flory–Huggins parameter and mobilities to be included in the simulation. In addition, viscoelasticity, which is not included in TDGL approaches, is included at the level of the Gaussian chains. A similar DFT approach has been developed by Doi and co￾workers [38,39] and forms the basis for their new software tool Simulation Utilities for Soft and Hard Interfaces (SUSHI), one of a suite of molecular and mesoscale modeling tools (called OCTA) developed for the simulation of polymer materials [40]. The essence of dynamic DFT method is that the instantaneous unique conformation distribution can be obtained from the off-equilibrium density profile by coupling a fictitious external potential to the Hamiltonian. Once such distribution is known, the free energy is then calculated by standard statistical thermodynamics. The driving force for diffusion is obtained from the spatial gradient of the first functional derivative of the free energy with respect to the density. Here, we describe briefly the equations for both polymer and particle in the diblock polymer–particle composites [35]. The dynamics of microphase separation in a melt of diblocks is described through the following equation [34]: qfðr;tÞ qt ¼ Mr2 dU½fðr;tÞ dfðr;tÞ G½fðr;tÞ l þ xðr;tÞ, (30) where f is the scalar order parameter, f ¼ rArB, i.e., the difference of the respective local densities of the A and B segments of AB diblocks. The constant M is the mobility for the order parameter field. Variable l describes the asymmetry of the diblock. x is the noise field, and parameter G determines the thickness of the lamellar domains which is related to the length of the diblock. The free energy U(f) is given by the local free energy Ul and polymer– particle coupling interaction term Ucpl: UðfÞ ¼ Ul þ þUcpl, (31) Ul ¼ Z dr f 1ðfðr; tÞÞ þ D 2 ðrfðr; tÞÞ2  , (32) Ucpl ¼ C Z dr X i Vðr RiÞðfðr;tÞ fsÞ 2 , (33) where fs ¼ 1 is the value of the order parameter at the particle surface, and V(r) is the potential functional. The motion of particle is described by the Langevin equation R_i ¼ Mp f i qUðfÞ qRi þ Zi, (34) where Mp is the particle mobility, fi is the force acting on the particle from other particles, and Z represents a Gaussian white noise. 2.3. Mesoscale and macroscale methods Despite the importance of understanding the molecular structure and nature of materials, their behavior can be homogenized with respect to different aspects which can be at different scales. Typically, the observed macroscopic behavior is usually explained by ignoring the discrete atomic and molecular structure and assuming that the material is continuously distributed throughout its volume. The continuum material is thus assumed to have an average density and can be subjected to body forces such as gravity and surface forces. Generally speaking, the macroscale methods (or called continuum methods hereafter) obey the fundamental laws of: (i) continuity, derived from the conservation of mass; (ii) equilibrium, derived from momentum considerations and Newton’s second law; (iii) the moment of momentum princi￾ple, based on the model that the time rate of change of angular momentum with respect to an arbitrary point is equal to the resultant moment; ARTICLE IN PRESS 200 Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269