196 O.H.Zeng et al.Prog.Polym.Sci.33 (2008)191-269 There are many algorithms ation of mo don using f difference interface and also toth an commonly used in MDsi tions [17.All algorithms assume that the atomic 2.1.2.Monte Carlo position,velocities and accelerations a can be MC te also called me olis r approximated by a Taylor series expansion: don +80)=0+0t+5a0r2+…, numbers to generate a sample population of the (3) system from which one can calculate the properties of interest.A MC simulation usually consists of iu+0)=i(0+a0t+5b0)6r+, thre (4) e typical steps.In the first step.the physical problem unde mo In th at+80)=a)+bu)8t+. emodel is so Generally speaking.a good integration algorithm ical should conserve the total energy and momentum ined dat In the d h and be time- statistical methods.MCp ovides only the informa eversible.It should lso be easy to tion on equilibrium pronerties (e g free enerov step.The vene phase equilibrium).different from Md which gives mit a bly出 gorithn and s the positions-8t)from the pre ious sten t-t to calculate the new positions r(t+8r)at t+8t. at lly mo ing one ator According to Taylor series expansion of Eq.(3). we have 7u+60=70+t081+5a0ǒr2+…, △H=HU-H), (11) where H()and the Hamiltonian associated original an gurat 7u-60)=70-(01+5a062- (7) s.If△H Adding Eqs.(6)and(7)together gives cording to the ould g rule t Tu+80)=270-7t-6)+au)ǒ2 of lower energy Hence the move ent is immedi (8) ately accented and the displaced atom remains in its The velocities at time t and t+t can be new position.If AH0,the move is accepted only respectively estimated with a certain probability pwhich is given by 元(0=[T(u+8)-7t-6/28, (9) ep) (+58)=+0-0/8 (10) where is the Bot al between 0 and I and determine the new configuration according to the following rule: (uVT),microcanonical (NVE),canonical (NVT) and isothermal-isobaric (NPT).The constant tem m(←) the move is accepted perature and pressure can be controlled by adding over an and b the move is not acepied. respe tively. MD int allows us to in configuration is rejected.one the o and peats the polymer structure and dynamics in the vicinity ofThere are many algorithms for integrating the equation of motion using finite difference methods. The algorithms of verlet, velocity verlet, leap-frog and Beeman, are commonly used in MD simula￾tions [17]. All algorithms assume that the atomic position ~r, velocities v * and accelerations a * can be approximated by a Taylor series expansion: r *ðt þ dtÞ ¼ r *ðtÞ þ v *ðtÞdt þ 1 2 a *ðtÞdt 2 þ , (3) v *ðt þ dtÞ ¼ v *ðtÞ þ a *ðtÞdt þ 1 2 bðtÞdt 2 þ , (4) a *ðt þ dtÞ ¼ a *ðtÞ þ bðtÞdt þ . (5) Generally speaking, a good integration algorithm should conserve the total energy and momentum and be time-reversible. It should also be easy to implement and computationally efficient, and per￾mit a relatively long time-step. The verlet algorithm is probably the most widely used method. It uses the positions ~rðtÞ and accelerations a *ðtÞ at time t, and the positions~rðt dtÞ from the previous step tdt to calculate the new positions ~rðt þ dtÞ at t+dt. According to Taylor series expansion of Eq. (3), we have r *ðt þ dtÞ ¼ r *ðtÞ þ v *ðtÞdt þ 1 2 a *ðtÞdt 2 þ , (6) r *ðt dtÞ ¼ r *ðtÞ v *ðtÞdt þ 1 2 a *ðtÞdt 2 . (7) Adding Eqs. (6) and (7) together gives r *ðt þ dtÞ ¼ 2 r *ðtÞ r *ðt dtÞ þ a *ðtÞdt 2 . (8) The velocities at time t and t þ 1 2dt can be respectively estimated v *ðtÞ¼½r *ðt þ dtÞ r *ðt dtÞ=2dt, (9) v * t þ 1 2 dt ¼ ½r *ðt þ dtÞ r *ðtÞ=dt. (10) MD simulations can be performed in many different ensembles, such as grand canonical (mVT), microcanonical (NVE), canonical (NVT) and isothermal–isobaric (NPT). The constant tem￾perature and pressure can be controlled by adding an appropriate thermostat (e.g., Berendsen, Nose, Nose–Hoover and Nose–Poincare) and barostat (e.g., Andersen, Hoover and Berendsen), respec￾tively. Applying MD into polymer composites allows us to investigate into the effects of fillers on polymer structure and dynamics in the vicinity of polymer–filler interface and also to probe the effects of polymer–filler interactions on the materials properties. This will be discussed in Section 3.3. 2.1.2. Monte Carlo MC technique, also called Metropolis method [19], is a stochastic method that uses random numbers to generate a sample population of the system from which one can calculate the properties of interest. A MC simulation usually consists of three typical steps. In the first step, the physical problem under investigation is translated into an analogous probabilistic or statistical model. In the second step, the probabilistic model is solved by a numerical stochastic sampling experiment. In the third step, the obtained data are analyzed by using statistical methods. MC provides only the informa￾tion on equilibrium properties (e.g., free energy, phase equilibrium), different from MD which gives non-equilibrium as well as equilibrium properties. In a NVT ensemble with N atoms, one hypothe￾sizes a new configuration by arbitrarily or system￾atically moving one atom from position i-j. Due to such atomic movement, one can compute the change in the system Hamiltonian DH: DH ¼ HðjÞ HðiÞ, (11) where H(i) and H(j) are the Hamiltonian associated with the original and new configuration, respec￾tively. This new configuration is then evaluated according to the following rules. If DHo0, then the atomic movement would bring the system to a state of lower energy. Hence, the movement is immedi￾ately accepted and the displaced atom remains in its new position. If DHX0, the move is accepted only with a certain probability pi-j which is given by pi!j / exp DH kBT , (12) where kB is the Boltzmann constant. According to Metropolis et al. [19] one can generate a random number x between 0 and 1 and determine the new configuration according to the following rule: xp exp DH kBT ; the move is accepted; x4exp DH kBT ; the move is not accepted: If the new configuration is rejected, one counts the original position as a new one and repeats the process by using other arbitrarily chosen atoms. ARTICLE IN PRESS 196 Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269