Q.H.Zeng et al Prog.Polym.SeL 33(2008)191-269 199 represents the lattice velocity direction.fct)is the distribution function of the particles of component 器a8 mh with velocity e along direction iat latticex and is minimized to simulate a temperature quench from time t,and;(x,/is the corresponding equili- the miscible region of the phase diagram to the brium distribution function. odynamic behavior immiscible region.Thus,the resulting time-dependent In the LB methoc the blend md equatio a d (23) this method to where UGL()is the Ginzburg-Landu free energy, For an incompressible binary polymer blend,the equation of motion can be written as follows: and U)is a potential that describes the coupling interaction between the ith particle and the surrounding fluid, D=Mi】+. (26 oo(r.t) UGL()=dr( (24) Here()sa thermal represen For a binary k>0 are phe ara ()describes a single homo neous phase while al=「drU】+F 27 k。T k。T <0,it yields two-or multiphases coexistence. where UFH()is the Flory-Huggins free energy of U.()=d山rVoe-r-s(p)-s》}, (25) mixing.given by UFH( KRI 2=忌n+“-+x1-9 the of th %>0 (28) ents its coup ginlCnaci8nand Here x is the enthalpic interaction parameter An important advantage of the LB method is that are the the two polymer componen V4 and N microscopic physical interactions of the fluid numbe n polymer Aa particles can be conveniently incorporated into the ely Th 8 with /2 numerical model. Compared the er Stokes LB me handle the system is ce the critical co s am centration whereas if and re it is immiscible. To simplify TDGL,Oono and co-workers pro- and posed cell dynamic method (CDM)which wa level.However.its main disadvanta is that it i the TDG typically not guaranteed to be numerically stable and aplacian term is rep by it may lead to physically unreasonable results,for nd.the free instance,in the case of high forcing rate or high iven by interparticle interaction strength UCDM[(r)] 2.2.4.Time-dependent Ginzhu TDGL is a microscale method for simulating the -4 In(cosh structural evolution of phase-separation in polymer 29) blends and block copolymers.It is based on the Cahn-Hilliard-Cook (CHC) nonlinear diffusion This model reproduces the growth kinetics of the equation for a binary blend and falls under the more TDGL model,demonstrating that such quantities represents the lattice velocity direction, f s i ðx; tÞ is the distribution function of the particles of component s with velocity ei along direction i at lattice x and time t, and f s;eq i ðx; tÞ is the corresponding equili￾brium distribution function. In the LB method, the thermodynamic behavior can be described by the following free-energy functional: UðfÞ ¼ UGLðfÞ þX N i¼1 UiðfÞ, (23) where UGL(f) is the Ginzburg–Landu free energy, and Ui(f) is a potential that describes the coupling interaction between the ith particle and the surrounding fluid, UGLðfÞ ¼ Z dr t 2 f2 þ u 4 f4 þ k 2 jrfj 2 , (24) where the term k 2jrfj 2 represents the free-energy cost of forming fluid–fluid interface, and u40 and k40 are phenomenological parameters. If t40, UGL(f) describes a single homogeneous phase, while to0, it yields two- or multiphases coexistence. UiðfÞ ¼ Z dr V0eðjrsij=r0Þ ðfðrÞ fðsiÞÞ2 , (25) where si is the position of ith particle and N the total number of particles. The constant V040 charac￾terizes the strength of the coupling interaction and r0 represents its range. An important advantage of the LB method is that microscopic physical interactions of the fluid particles can be conveniently incorporated into the numerical model. Compared with the Navier– Stokes equations, the LB method can handle the interactions among fluid particles and reproduce the microscale mechanism of hydrodynamic behavior. Therefore it belongs to the MD in nature and bridges the gap between the molecular level and macroscopic level. However, its main disadvantage is that it is typically not guaranteed to be numerically stable and may lead to physically unreasonable results, for instance, in the case of high forcing rate or high interparticle interaction strength. 2.2.4. Time-dependent Ginzburg– Landau method TDGL is a microscale method for simulating the structural evolution of phase-separation in polymer blends and block copolymers. It is based on the Cahn–Hilliard–Cook (CHC) nonlinear diffusion equation for a binary blend and falls under the more general phase-field and reaction-diffusion models [26–28]. In the TDGL method, a free-energy function is minimized to simulate a temperature quench from the miscible region of the phase diagram to the immiscible region. Thus, the resulting time-dependent structural evolution of the polymer blend can be investigated by solving the TDGL/CHC equation for the time dependence of the local blend concentration. Glotzer and co-workers have discussed and applied this method to polymer blends and particle-filled polymer systems [29]. For an incompressible binary polymer blend, the equation of motion can be written as follows: qfðr;tÞ qt ¼r Mr dU½fðr;tÞ dfðr;tÞ þ zðr;tÞ. (26) Here z(r, t) is a thermal noise term with zero mean, U is the free energy, and the mobility M may depend on the order parameter f. For a binary polymer blend, one typically takes U½fðrÞ kBT ¼ Z dr UFH ½fðrÞ kBT þ kðfÞjrfðrÞj2  , (27) where UFH(f) is the Flory–Huggins free energy of mixing, given by UFH ðfÞ kBT ¼ f NA ln f þ ð1 fÞ NB lnð1 fÞ þ wfð1 fÞ. (28) Here w is the enthalpic interaction parameter between the two polymer components. NA and NB are the number of segments in polymer A and polymer B, respectively. The Flory–Huggins model has a critical point at fc ¼ N1=2 B =ðN1=2 A þ N1=2 B Þ and wc ¼ ðN1=2 A þ N1=2 B Þ 2 =2NANB. If wowc the system is miscible at the critical concentration whereas if w4wc it is immiscible. To simplify TDGL, Oono and co-workers pro￾posed cell dynamic method (CDM) which was derived from the discretized TDGL equation [30,31]. Here, the Laplacian term is replaced by its isotropic discretized counterpart. And, the free￾energy functional in Eq. (27) is given by UCDM½fðrÞ kBT ¼ Z dr A lnðcosh fÞ þ 1 2 f2 þ D 2 jrfðrÞj2  . ð29Þ This model reproduces the growth kinetics of the TDGL model, demonstrating that such quantities ARTICLE IN PRESS Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 199