ScienceDirect ELsevier Prog.Polym.Sai.33(2008)191-269 www.cl vier.com/locate/ppolys Multiscale modeling and simulation of polymer nanocomposites Q.H.Zeng,A.B.Yu.*G.Q.Lub ARC Centre of Excellence for Functional Nanomaterials.The University of Queensland.Brisbane.OLD 4072.Australia Polymer nanocor the reinforcement nd ti ng and simt that have been applied to polymer nanocomposites,covring from cale (e.g molecular dynamics,Monte Carlo) (e.g., rownian dynam dissipa tive particle dynamic lattice Boltzmann,time-dependent Ginzburg- polymer nanocomposites,including the thermodynamics and kinetics of formation,molecular structure and dynamics morphology.proc The present review aims to summariz the recent advance in the fundamental understandin of polymer nanocomposites 207ti,nanorube.day patelets)and stimate fuher ihsarea. Keywords:Polymer nanocomposite:Multiscale:Modeling:Simulation:Nanostructure:Nanoparticles Contents 18 2.1.Molecular scale methods 194 olecular dynamics Monte Carlo.......................................................... 196 Abbreviations:BD:Br nod:CHC:Cah onal theory D T629385442gRn60295956 E-mail address a.yu@unsw.eduau (A.B.Yu). 0079-6700/S-see front m er2007 Elsevier Ltd.All rights reserved. doi:10.1016/j-progpolymsci.2007.09.002
Prog. Polym. Sci. 33 (2008) 191–269 Multiscale modeling and simulation of polymer nanocomposites Q.H. Zenga , A.B. Yua,, G.Q. Lub a Centre for Simulation and Modeling of Particulate Systems and School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia b ARC Centre of Excellence for Functional Nanomaterials, The University of Queensland, Brisbane, QLD 4072, Australia Received 13 December 2006; received in revised form 5 September 2007; accepted 5 September 2007 Available online 3 December 2007 Abstract Polymer nanocomposites offer a wide range of promising applications because of their much enhanced properties arising from the reinforcement of nanoparticles. However, further development of such nanomaterials depends on the fundamental understanding of their hierarchical structures and behaviors which requires multiscale modeling and simulation strategies to provide seamless coupling among various length and time scales. In this review, we first introduce some computational methods that have been applied to polymer nanocomposites, covering from molecular scale (e.g., molecular dynamics, Monte Carlo), microscale (e.g., Brownian dynamics, dissipative particle dynamics, lattice Boltzmann, time-dependent Ginzburg–Landau method, dynamic density functional theory method) to mesoscale and macroscale (e.g., micromechanics, equivalent-continuum and self-similar approaches, finite element method). Then, we discuss in some detail their applications to various aspects of polymer nanocomposites, including the thermodynamics and kinetics of formation, molecular structure and dynamics, morphology, processing behaviors, and mechanical properties. Finally, we address the importance of multiscale simulation strategies in the understanding and predictive capabilities of polymer nanocomposites in which few studies have been reported. The present review aims to summarize the recent advances in the fundamental understanding of polymer nanocomposites reinforced by nanofillers (e.g., spherical nanoparticles, nanotubes, clay platelets) and stimulate further research in this area. r 2007 Elsevier Ltd. All rights reserved. Keywords: Polymer nanocomposite; Multiscale; Modeling; Simulation; Nanostructure; Nanoparticles Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 2. Modeling and simulation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 2.1. Molecular scale methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 2.1.1. Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 2.1.2. Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 ARTICLE IN PRESS www.elsevier.com/locate/ppolysci 0079-6700/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.progpolymsci.2007.09.002 Abbreviations: BD: Brownian dynamics; BEM: boundary element method; CDM: cell dynamic method; CHC: Cahn–Hilliard–Cook; CNT: carbon nanotube; DFT: density functional theory; DPD: dissipative particle dynamics; FEM: finite element method; LB: lattice Boltzmann; MC: Monte Carlo; MD: molecular dynamics; MM: molecular mechanics; NMR: nuclear magnetic resonance; ODT: order–disorder phase transition; PCL: poly (e-caprolactone); PMMA: polymethylmethacrylate; RVE: representative volume element; SCF: self-consistent field; SWCN: single-walled carbon nanotube; TDGL: time-dependent Ginsburg–Laudau Corresponding author. Tel.: +61 2 93854429; fax: +61 2 93855956. E-mail address: a.yu@unsw.edu.au (A.B. Yu).
192 O.H.Zeng et al.Prog.Polym.Sci.33 (2008)191-269 2.2 oscale method Brownian dynamics 187 Dissipative particle dynamics.. attice B ndent Cinburg-Landau method. 225 Dynamie DFT method 2.3. e and macroscale methods 132 self-similar ppr 2.3.3 Finite element method. 204 3.Modeling and simulation of polymer nanocomposites.................................... 32 Nanocomposite thermodynamics........................ Nanocomposite molecular structure and dynamic properties. 213 osite morphology 342 3 Nanocomposite rheological and processing behaviors. 222 6 al properties .......... 。。。，。。。。。。 36 Continuum models 3.6.3 Equivalent-continuum and self-similar models 42 Sequential and concurrent approaches 4.3 Current research status......................... 155 Ks. References 261 1.Introduction 2 the hierarchical characteristics of the structure and dynamics of polymer nanocomposites ran- Polymer materials reinforced with nanoparticles ging from molecular scale,microscale to mesos- (e.g..nanosphere,nanotube,clay platelet)have recently cale and macroscale,in particular,the molecular received tremendous attention in both scientific and structures and dynamics at the interface between industrial communit s due to their extraordinary nanoparticles and polymer matrix: 0J.However,fror point on the ion of nanopart fabrication of polym sites.Tpu mechan ment of such materials is still largely empirical and a isms of nanoparticles in polymer nanocomposites. finer degree of control of their properties cannot be achieved so far.Therefore.computer modeling and simulation will play an ever-increasing role in predict The purpose of this review is to discuss the application of modeling and simulation techniques to ing and designing material properties,and guiding suck polymer nanocomposites.This includes a broad subject covering methodologies at various length and time ntal is We les and many aspects o for th 1.the thermodynamics and kinetics of the forma an he ughly divided int tion of polymer nanocomposites:
2.2. Microscale methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 2.2.1. Brownian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 2.2.2. Dissipative particle dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 2.2.3. Lattice Boltzmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 2.2.4. Time-dependent Ginzburg–Landau method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 2.2.5. Dynamic DFT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 2.3. Mesoscale and macroscale methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 2.3.1. Micromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 2.3.2. Equivalent-continuum and self-similar approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 2.3.3. Finite element method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 3. Modeling and simulation of polymer nanocomposites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 3.1. Nanocomposite thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 3.2. Nanocomposite kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 3.3. Nanocomposite molecular structure and dynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.4. Nanocomposite morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3.4.1. Homopolymer nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3.4.2. Block copolymer nanocomposites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 3.5. Nanocomposite rheological and processing behaviors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.6. Nanocomposite mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.6.1. Molecular models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 3.6.2. Continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3.6.3. Equivalent-continuum and self-similar models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4. Multiscale strategies for modeling polymer nanocomposites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 4.1. Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 4.2. Sequential and concurrent approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 4.3. Current research status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5. Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Acknowledgment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 1. Introduction Polymer materials reinforced with nanoparticles (e.g., nanosphere, nanotube, clay platelet) have recently received tremendous attention in both scientific and industrial communities due to their extraordinary enhanced properties [1–10]. However, from the experimental point of view, it is a great challenge to characterize the structure and to manipulate the fabrication of polymer nanocomposites. The development of such materials is still largely empirical and a finer degree of control of their properties cannot be achieved so far. Therefore, computer modeling and simulation will play an ever-increasing role in predicting and designing material properties, and guiding such experimental work as synthesis and characterization. For polymer nanocomposites, computer modeling and simulation are especially useful in addressing the following fundamental issues: 1. the thermodynamics and kinetics of the formation of polymer nanocomposites; 2. the hierarchical characteristics of the structure and dynamics of polymer nanocomposites ranging from molecular scale, microscale to mesoscale and macroscale, in particular, the molecular structures and dynamics at the interface between nanoparticles and polymer matrix; 3. the dependence of polymer rheological behavior on the addition of nanoparticles, which is useful in optimizing processing conditions; and 4. the molecular origins of the reinforcement mechanisms of nanoparticles in polymer nanocomposites. The purpose of this review is to discuss the application of modeling and simulation techniques to polymer nanocomposites. This includes a broad subject covering methodologies at various length and time scales and many aspects of polymer nanocomposites. We organize the review as follows. In Section 2, we introduce briefly the computational methods used so far for the systems of polymer nanocomposites which can be roughly divided into three types: molecular scale methods (e.g., molecular dynamics (MD), Monte Carlo ARTICLE IN PRESS 192 Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269
Q.H.Zeng et al.Prog.Polym.Sci.33 (2008)191-269 193 Nomenclature S average compliance of polymer compo- site strain-concentration tensor of polymer T temperature composite U interaction potential A area of interface interaction △U change of potential energy acceleration of atom UGL()Ginzburg-Landu free energy B stress-concentration tensor of polymer UCDM free energy calculated in cell dynamical composite method O stiffness tensor of polymer composite UK kinetic energy in finite element model Cr elastic stiffness tensor of the effective Uv Hookian potential energy in finite ele- fiber in polymer composite ment model E total modulus of polymer composite Um total potential energy of molecule model E11 longitudinal modulus of polymer com- U total potential energy of truss model posite U Udh energies associated with covalent E22 transverse modulus of polymer compo- bonding stretching,bond-angle bending, site and van der Waals interactions,respec- Er modulus of filler in polymer composite tively Em modulus of matrix in polymer composite UU energies associated with truss ele- ei local particle velocity in lattice gas ments that represent covalent bonding automaton method,or lattice velocity stretching,bond-angle bending,and van vector in lattice Boltzmann method der Waals interactions,respectively force acting on the ith atom or particle displacement field in finite element model Fyc conservative force of particle j acting on V(r) potential functional particle i Vt volume of filler in polymer composite dissipative force of particle j acting on volume of matrix in polymer composite particle i friction coefficient FyR random forces of particle j acting on interface binding energy particle i E volume-average strain of polymer com- f(x,t)distribution function of the particles of posite component a with velocity e;along 可 volume-average strain of filler in poly- direction i at lattice x and time t mer composite (x,)equilibrium distribution function of the 高m volume-average strain of matrix in poly- particles of component a mer composite G chain gyration tensor infinitesimal strain tensor of filler in H Hamiltonian associated with a configura- polymer composite tion infinitesimal strain tensors of matrix in △H change of system Hamiltonian or free polymer composite energy random noise term with zero mean h,ho final and initial clay platelet separation (r,t) thermal noise term with zero mean kB Boltzmann constant Gaussian white noise M mobility of the order parameter field enthalpic interaction parameter Mp particle mobility 0 volume fraction of filler in polymer 171 atomic mass composite nx,t)particle occupation variable with direc- Um volume fraction of matrix in polymer tion i at distance x and time t composite P average segment orientation velocity of atom Pi momentum of particle 5 random number between 0 and I which is Pi-→j probability of accepting a new config- to determine the acceptance or rejection uration ( of a new configuration position of ith atom or particle 5 shape parameter depending on filler geometry and loading direction
ARTICLE IN PRESS Nomenclature A strain–concentration tensor of polymer composite Ai area of interface interaction a * acceleration of atom B stress–concentration tensor of polymer composite C stiffness tensor of polymer composite Cf elastic stiffness tensor of the effective fiber in polymer composite E total modulus of polymer composite E11 longitudinal modulus of polymer composite E22 transverse modulus of polymer composite Ef modulus of filler in polymer composite Em modulus of matrix in polymer composite ei local particle velocity in lattice gas automaton method, or lattice velocity vector in lattice Boltzmann method F ~i force acting on the ith atom or particle Fij C conservative force of particle j acting on particle i Fij D dissipative force of particle j acting on particle i Fij R random forces of particle j acting on particle i f s i ðx;tÞ distribution function of the particles of component s with velocity ei along direction i at lattice x and time t f s;eq i ðx; tÞ equilibrium distribution function of the particles of component s Gij chain gyration tensor H Hamiltonian associated with a configuration DH change of system Hamiltonian or free energy h, h0 final and initial clay platelet separation kB Boltzmann constant M mobility of the order parameter field Mp particle mobility mi atomic mass ni(x, t) particle occupation variable with direction i at distance x and time t P2 s average segment orientation pi momentum of particle pi-j probability of accepting a new configuration (j) ~ri position of ith atom or particle S average compliance of polymer composite T temperature U interaction potential DU change of potential energy UGL(f) Ginzburg–Landu free energy UCDM free energy calculated in cell dynamical method UK kinetic energy in finite element model UV Hookian potential energy in finite element model Um total potential energy of molecule model Ut total potential energy of truss model Ur , Uy , Uvdw energies associated with covalent bonding stretching, bond-angle bending, and van der Waals interactions, respectively Ua , Ub , Uc energies associated with truss elements that represent covalent bonding stretching, bond-angle bending, and van der Waals interactions, respectively u_ displacement field in finite element model V(r) potential functional Vf volume of filler in polymer composite Vm volume of matrix in polymer composite g friction coefficient gi interface binding energy ¯ volume–average strain of polymer composite ¯f volume–average strain of filler in polymer composite ¯m volume–average strain of matrix in polymer composite e f infinitesimal strain tensor of filler in polymer composite e m infinitesimal strain tensors of matrix in polymer composite zij random noise term with zero mean z(r, t) thermal noise term with zero mean Z Gaussian white noise l enthalpic interaction parameter vf volume fraction of filler in polymer composite vm volume fraction of matrix in polymer composite v * velocity of atom x random number between 0 and 1 which is to determine the acceptance or rejection of a new configuration xf shape parameter depending on filler geometry and loading direction Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 193
194 Q.H.Zeng et al.Prog.Polym.Sci.33 (2008)191-269 Pmax maximum stress along the tube axis dimensionless collision-relaxation time noise amplitude constant of component o of the fluid G volume-average stress of polymer com- x°=10/△t posite order parameter Gf volume-average stress of filler in polymer 中 value of the order parameter at particle composite surface Gm volume-average stress of matrix in poly- asymmetry of diblock copolymer mer composite 2 collision operation total stress of filler in polymer composite weight function for conservative force total stress of matrix in polymer compo- WD weight function for dissipative force site OR weight function for random force (MC)),microscale methods (e.g.,Brownian dynamics clusters as the basic units considered.The most (BD),dissipative particle dynamics (DPD),lattice popular methods include molecular mechanics Boltzmann (LB),time-dependent Ginzburg-Lanau (MM),MD and MC simulation.Modeling of method,dynamic density functional theory (DFT) polymer nanocomposites at this scale is predomi- method),and mesoscale and macroscale methods (e.g., nantly directed toward the thermodynamics and micromechanics,equivalent-continuum and self-similar kinetics of the formation,molecular structure and approaches,finite element method(FEM)).We do not interactions.The diagram in Fig.I describes the aim to provide a detailed description of each method equation of motion for each method and the typical but its basic principles,strengths and weaknesses,and properties predicted from each of them [11-16].We potential applications.Interesting readers can refer to introduce here the two widely used molecular scale relevant books,reviews and research articles for details. methods:MD and MC. In Section 3,we discuss in detail the applications of these modeling and simulation methods in some specific aspects of polymer nanocomposites,including 2.1.1.Molecular dynamics the thermodynamics and kinetics of the formation, MD is a computer simulation technique that allows molecular structure and dynamics,morphologies (i.e., one to predict the time evolution of a system of phase behaviors),rheological and processing behaviors, interacting particles (e.g.,atoms,molecules,granules, and mechanical properties.We pay more attention to etc.)and estimate the relevant physical properties clay-based polymer nanocomposites because of their [17,18].Specifically,it generates such information as importance in polymer nanocomposites and our own atomic positions,velocities and forces from which the research interests.Of course,we also refer to research macroscopic properties (e.g.,pressure,energy,heat activities in polymer systems reinforced by nanospheres capacities)can be derived by means of statistical and nanotubes,an area developed rapidly in the past mechanics.MD simulation usually consists of three few years.In Section 4,we highlight the importance of constituents:(i)a set of initial conditions (e.g.,initial multiscale strategies of modeling and simulation in positions and velocities of all particles in the system); understanding and predicting the hierarchical structure (ii)the interaction potentials to represent the forces and behaviors arising from the polymer and nanopar- among all the particles;(iii)the evolution of the ticle together with the properties observed at various system in time by solving a set of classical Newtonian scales.We discuss the current applications of some equations of motion for all particles in the system. multiscale methods in polymer nanocomposites.Final- The equation of motion is generally given by ly,we conclude the review by emphasizing the current d27 challenges and future research directions. 下(0=mdr (1) 2.Modeling and simulation techniques where F;is the force acting on the ith atom or particle at time t which is obtained as the negative gradient of 2.1.Molecular scale methods the interaction potential U,m;is the atomic mass and Fi the atomic position.A physical simulation involves The modeling and simulation methods at mole- the proper selection of interaction potentials,numer- cular level usually employ atoms,molecules or their ical integration,periodic boundary conditions,and
(MC)), microscale methods (e.g., Brownian dynamics (BD), dissipative particle dynamics (DPD), lattice Boltzmann (LB), time-dependent Ginzburg–Lanau method, dynamic density functional theory (DFT) method), and mesoscale and macroscale methods (e.g., micromechanics, equivalent-continuum and self-similar approaches, finite element method (FEM)). We do not aim to provide a detailed description of each method but its basic principles, strengths and weaknesses, and potential applications. Interesting readers can refer to relevant books, reviews and research articles for details. In Section 3, we discuss in detail the applications of these modeling and simulation methods in some specific aspects of polymer nanocomposites, including the thermodynamics and kinetics of the formation, molecular structure and dynamics, morphologies (i.e., phase behaviors), rheological and processing behaviors, and mechanical properties. We pay more attention to clay-based polymer nanocomposites because of their importance in polymer nanocomposites and our own research interests. Of course, we also refer to research activities in polymer systems reinforced by nanospheres and nanotubes, an area developed rapidly in the past few years. In Section 4, we highlight the importance of multiscale strategies of modeling and simulation in understanding and predicting the hierarchical structure and behaviors arising from the polymer and nanoparticle together with the properties observed at various scales. We discuss the current applications of some multiscale methods in polymer nanocomposites. Finally, we conclude the review by emphasizing the current challenges and future research directions. 2. Modeling and simulation techniques 2.1. Molecular scale methods The modeling and simulation methods at molecular level usually employ atoms, molecules or their clusters as the basic units considered. The most popular methods include molecular mechanics (MM), MD and MC simulation. Modeling of polymer nanocomposites at this scale is predominantly directed toward the thermodynamics and kinetics of the formation, molecular structure and interactions. The diagram in Fig. 1 describes the equation of motion for each method and the typical properties predicted from each of them [11–16]. We introduce here the two widely used molecular scale methods: MD and MC. 2.1.1. Molecular dynamics MD is a computer simulation technique that allows one to predict the time evolution of a system of interacting particles (e.g., atoms, molecules, granules, etc.) and estimate the relevant physical properties [17,18]. Specifically, it generates such information as atomic positions, velocities and forces from which the macroscopic properties (e.g., pressure, energy, heat capacities) can be derived by means of statistical mechanics. MD simulation usually consists of three constituents: (i) a set of initial conditions (e.g., initial positions and velocities of all particles in the system); (ii) the interaction potentials to represent the forces among all the particles; (iii) the evolution of the system in time by solving a set of classical Newtonian equations of motion for all particles in the system. The equation of motion is generally given by F ~iðtÞ ¼ mi d2 ~ri dt2 , (1) where F ~i is the force acting on the ith atom or particle at time t which is obtained as the negative gradient of the interaction potential U, mi is the atomic mass and ~ri the atomic position. A physical simulation involves the proper selection of interaction potentials, numerical integration, periodic boundary conditions, and ARTICLE IN PRESS rmax maximum stress along the tube axis s noise amplitude s¯ volume–average stress of polymer composite s¯f volume–average stress of filler in polymer composite s¯ m volume–average stress of matrix in polymer composite sf total stress of filler in polymer composite sm total stress of matrix in polymer composite t s dimensionless collision–relaxation time constant of component s of the fluid, ts ¼ ls =Dt f order parameter fs value of the order parameter at particle surface w asymmetry of diblock copolymer O collision operation oC weight function for conservative force oD weight function for dissipative force oR weight function for random force 194 Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269
Q.H.Zeng et al Prog.Polym.SeL 33(2008)191-269 9 QM MD MC otion of electron,nuclei motion of atoms motion of atom chroainger equat 70 ●3 ctronis distribution anoparticle surface Bonding and unbonding micdifasioncocfec hterealaioakineties Mechanical properties ethods and lo n of a We-VCYn The interaction potentials together with their parameters,i.e.,the so-called force field,describe in detail how the particles in a system interact with each other,i.e.,how the potential energy of a system depends on the particle coordinates.Such a force may be obt me (e.g., (2) me0ho order potential).The criteria for selec The first four terms represent bonded interac- field include the accuracy.transferability and tions,i.e.,bond stretching ond,bond-angle bend computational speed.A typical interaction potential fhreioanierqio and dihedral angle torsion and U may consist of a number of bonded and non- whil the last two bonded interaction terms interactions 1.e.,van de energy and el energy a U,2,,fN)= involved in a given interaction:NN and N. stand for the total numbers of these respective interactions in the simulated system; and linrersion uniquely specily an indi idual interaction of each type;i and j in the
the controls of pressure and temperature to mimic physically meaningful thermodynamic ensembles. The interaction potentials together with their parameters, i.e., the so-called force field, describe in detail how the particles in a system interact with each other, i.e., how the potential energy of a system depends on the particle coordinates. Such a force field may be obtained by quantum method (e.g., ab initio), empirical method (e.g., Lennard–Jones, Mores, Born-Mayer) or quantum-empirical method (e.g., embedded atom model, glue model, bondorder potential). The criteria for selecting a force field include the accuracy, transferability and computational speed. A typical interaction potential U may consist of a number of bonded and nonbonded interaction terms: Uð~r1;~r2; ... ;~rNÞ ¼ N Xbond ibond Ubond ðibond ;~ra;~rbÞ þ N X angle iangle Uangleðiangle;~ra;~rb;~rcÞ þ N Xtorsion itorsion Utorsionðitorsion;~ra;~rb;~rc;~rd Þ þ NXinversion iinversion Uinversionðiinversion;~ra;~rb;~rc;~rd Þ þ N X1 i¼1 X N j4i Uvdwði; j;~ra;~rbÞ þ N X1 i¼1 X N j4i Uelectrostaticði; j;~ra;~rbÞ. ð2Þ The first four terms represent bonded interactions, i.e., bond stretching Ubond, bond-angle bend Uangle and dihedral angle torsion Utorsion and inversion interaction Uinversion, while the last two terms are non-bonded interactions, i.e., van der Waals energy Uvdw and electrostatic energy Uelectrostatic. In the equation, ~ra;~rb;~rc, and ~rd are the positions of the atoms or particles specifically involved in a given interaction; Nbond, Nangle, Ntorsion and Ninversion stand for the total numbers of these respective interactions in the simulated system; ibond, iangle, itorsion and iinversion uniquely specify an individual interaction of each type; i and j in the van der Waals and electrostatic terms indicate the atoms involved in the interaction. ARTICLE IN PRESS Fig. 1. Molecular modeling and simulation methods commonly used for polymer nanocomposites [11–16]. Reproduced from (i) Lee, Baljon and Loring by permission of American Institute of Physics; (ii) Smith, Bedrov, Li and Byutner by permission of American Institute of Physics; (iii) Smith, Bedrov and Smith by permission of Elsevier Science Ltd.; (iv) Zeng, Yu, Lu and Standish by permission of American Chemical Society; (v) Vacatello by permission of Wiley-VCH; and (vi) Zeng, Yu and Lu by permission of Institute of Physics. Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 195