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MOLECULAR DYNAMICS SIMULATIONS OF THE ELASTIC MODULI OF POLYMER-CARBON NANOTUBE COMPOSITES Michael Griebel and Jan Hamaekers Department of Applied Mathematics, University of Bonn, Wegelerstr. 6, D-53115 Bonn, Germany. 1 Introduction Very long pure carbon tube-like structures were first re- ported by Iijima in 1991. 1 These nanotubes (NT ) can be used to reinforce polymer composites. Because the bend- ing stiffness of NTs is in the range of 0.4–4 TPa, 2 they pos- sess the potential for large increases in strength and stiffness when compared to typical carbon-fiber-reinforced polymer composites. In this work, we derive stress-strain curves from molecular dynamics (MD) simulations of polyethy- lene (PE) NT composites to predict their macroscopic elas- tic moduli and compare them to the rule-of-mixtures. In several earlier works, MD simulations have been suc- cessfully applied to predict elastic properties of PE/NT composites. 3,4 So far, the application of strain has been ac- complished by uniformly expanding the length of the simu- lation cell and the coordinates of the atoms in the direction of the deformation. Then, a MD simulation or a potential energy minimization is performed to equilibrate the sys- tem and to measure the corresponding stress. In the present work, we carry out the application of strain by employing a Parrinello-Rahman-Nos´ e Lagrangian to control stress and temperature. 5,6 We compute the stress-strain curves of three periodic model-systems, an infinite (10, 10) carbon NT, a capped (10, 10) carbon NT embedded in PE, and the PE matrix itself. To model the bonded interaction within these hydrocarbon systems, we use a many-body bond order po- tential (REBO) due to Brenner 7 with an additional van der Waals term 8 (model I ). Alternatively, we model the PE ma- trix by a united-atom potential 9 and just the NT by Bren- ner’s potential (model II ). In both models, the nonbonded interaction of the atoms is represented by a Lennard-Jones potential. We exploit the slopes of the stress-strain curves to derive different elastic moduli and constants. 2 Computational Methods To obtain an isothermal-isobaric ensemble (NPT ), we intro- duce additional degrees of freedom to an N -particle con- stant volume and constant energy ensemble (NVE) with cartesian coordinates ~x i , masses m i and a potential V . We define a time-dependent matrix ˆ h =[ ~a 1 ,~a 2 ,~a 3 ] consist- ing of the basis vectors of the simulation cell and re-scale the coordinates ˆ ~s i = ˆ h -1 ~x i . We also re-scale the time t by ¯ t = R t 0 γ (τ )dτ and obtain the velocities in the form ˙ ~x i ( ¯ t)= γ ˆ h ˙ ˆ ~s i (t). Then, we define the fictitious potentials P ext det ˆ h and N f k B T ln γ with the external pressure P ext and the target temperature T , the system’s number of de- grees of freedom N f and Boltzmann’s constant k B . Now a so-called Parrinello-Rahman-Nos´ e Lagrangian can be pos- tulated and an extended Hamiltonian H = 1 2 N X i=1 ~ p T ~ s i G~ p ~ s i m i + 1 2 tr(p T h p h ) M P + 1 2 p 2 γ M T + V (h,h~s 1 ,...,h~s N )+ P ext det h + N f k B Tη (1) with variables ~s i (t) := ˆ ~s i ( ¯ t), h(t) := ˆ h( ¯ t), G := h T h, and η(t) := ln γ ( ¯ t), can be derived. 5,6,10 Here, M P and M T are fictitious masses. The resulting equations of motion (EQM) read as ˙ ~s i = ~ p ~ s i m i , ˙ h = p h M P , ˙ η = p γ M T , (2) ˙ ~ p ~ s i = -h -1 ∇ ~x i V - G -1 ˙ Gp ~ s i - p γ M T p ~ s i , (3) ˙ p h = (Π int - diag (P ext )) h -T det h - p γ M T p h , (4) ˙ p γ = N X i=1 ~ p T ~ s i G~ p ~ s i m i + tr(p T h p h ) M P - N f k B T. (5) Here, the internal stress tensor Π int can be written as Π int = 1 det h ˆ N X i=1 m i h~s i ~s T i h T - d dh V ! h T . (6) To accomplish various tensile load cases, we use an addi- tional external stress tensor Π ext within the EQM (4) ˙ p h = (Π int - diag (P ext )+Π ext ) h -T det h - p γ M T p h . (7) Additionally, we apply a constraint to get a symmetric dis- placement matrix e = hh -1 equilibrated - 1, which then equals the linear strain tensor ε. For the numerical solution of the system of the ordinary differential equations (2)-(5), we employ the “predictor- corrector” time integration scheme based on Beeman’s ap- proach 11 and later modified by Refson 12 , to solve the diffi- culty of the velocity-dependent forces in the EQM (2)-(5). 3 Numerical Experiments All tensile load tests are carried out at normal conditions, i.e. for T = 273.15 K and for P ext =1.01325 · 10 -4 GPa. The fictitious masses are set to M T = 10.0u ˚ A 2 and M P = 10.0u. We use a timestep of 0.1 fs for model I and 0.2 fs for model II. We use a stress rate of 0.01 GPa/ps in all tensile load test cases and measure the induced stress π := -Π int and strain ε. Details of the studied systems are given in figure 1 and the derived elastic constants are summarized in tables 1-3.
