A molecular dynamics simulation study to investigate the effect of filler size on elastic properties of polymer nanocomposites

Transcription

1 Composites Science and Technology 67 (27) COMPOSITES SCIENCE AND TECHNOLOGY A molecular dynamics simulation study to investigate the effect of filler size on elastic properties of polymer nanocomposites Ashfaq Adnan a, C.T. Sun a, *, Hassan Mahfuz b a School of Aeronautics and Astronautics, Purdue University, 35 North Grant St, West Lafayette, IN , USA b Department of Ocean Engineering, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 3343, USA Received 26 April 26; received in revised form 7 August 26; accepted 7 September 26 Available online 3 November 26 Abstract The influence of filler size on elastic properties of nanoparticle reinforced polymer composites is investigated using molecular dynamics (MD) simulations. Molecular models for a system of nanocomposites are developed by embedding a fullerene bucky-ball of various sizes into an amorphous polyethylene matrix. In all cases, bucky-balls are modeled as non-deformable solid inclusions and infused in the matrix with a fixed volume fraction. The interaction between polymer and the nanoparticle is prescribed by the Lennard-Jones nonbonded potential. The mechanical properties for neat polymer and nanocomposites are evaluated by simulating a series of unidirectional and hydrostatic tests, both in tension and compression. Simulation results show that the elastic properties of nanocomposites are significantly enhanced with the reduction of bucky-ball size. An examination at the atomic level reveals that densification of polymer matrix near the nanoparticle as well as the filler-matrix interaction energy play the major role in completing the size effect. Ó 26 Published by Elsevier Ltd. Keywords: A. Particle-reinforced composites; A. Nanostructures; A. Polymers; B. Molecular dynamics simulation; C. Elastic properties. Introduction * Corresponding author. Tel.: ; fax: address: sun@purdue.edu (C.T. Sun). Recently, nanoparticle reinforced polymeric composites have generated considerable attentions among materials scientists and engineers due to their potential of producing superior physical, chemical, and mechanical properties in composites structures. It appears from current experimental investigations that improvements in mechanical properties, specifically the elastic properties, of nanocomposites depends on a set of factors including nanoparticle size, shape, volume fraction, degree of dispersion, characteristics of polymer matrix and interactions between filler and matrix at their interface []. The influence of nano-scale filler size on elastic properties is, however, somewhat surprising because this type of effect is not seen in conventional composite materials. It is also not possible to explain such effect from continuum mechanics [2 4]. Moreover, it is difficult to find plausible correlations between filler size and mechanical properties from the reported experimental results. Cho et al. [5], for example, conducted a systematic experimental and numerical study to address the effect of particle size on elastic modulus, tensile strength and particle/matrix debonding fracture toughness of particulate composites. They varied the size of particles from macro (.5 mm) to nano (5 nm) scale and found that particle size at micro scale impart no influence on Young s modulus of the composite. However, the Young s modulus increased as the size of the particles decreased at nano-scale. Similar results were obtained by Chisholm et al. [6] who infused micro and nano sized SiC particles in epoxy matrix. They claimed that higher elastic modulus in nanocomposites was attained due to higher surface energy of nanoparticles. While it apparently shows that reduction in size would yield stiffer nanocomposites, reports are available where it is demonstrated that decreasing filler size has no effect [7] or some cases has negative effect [8] on elastic properties /$ - see front matter Ó 26 Published by Elsevier Ltd. doi:.6/j.compscitech

2 A. Adnan et al. / Composites Science and Technology 67 (27) It was reported that poor nanoparticle dispersion and weak filler-polymer interface adhesion were the reasons why nanocomposites could not perform over microcomposites [8]. The size issue appears to be further complicated from the work of Vollenberg et al. [9,] where they have claimed that in the case of polymer infused with surfacetreated filler, the elastic modulus is independent of the size of the inclusions. However, the modulus seems to be significantly dependent on the particle size when reinforced by untreated particle. It appears that when size of filler reduces to nano-scale, it is difficult to identify any definite mechanism that could portray how size influence elastic properties of nanocomposites. This is partly because of challenges in processing nanocomposites, and unavailability of sophisticated tools to inspect closely their in situ manufacturing as well as characterization process. In this regard, several researchers have attempted to perform molecular level computations to visualize the deep insight of mechanics of nanocomposites [ 4]. While there has been quite a few computational works done in the field of polymer nanocomposites, there is yet to address the effect of filler size on mechanical properties of nanocomposites. The purpose of this study is to investigate the effect of nanoparticle size on elastic properties of polymeric nanocomposites using MD simulations. For this, molecular models of a nanocomposite were constructed by reinforcing amorphous polyethylene (PE) matrix with nano sized buckminister fullerene bucky-ball (or simply bucky-ball). Bucky-balls of three different diameters (.7,.2 and.7 nm, respectively) were utilized to incorporate size effect in the nanocomposites. To represent them as a generic nanoparticle system, all bucky-balls were configured as rigid body. This is necessary because a bucky-ball embedded inside the polymer matrix may deform excessively depending on its size and may overshadow the composite mechanical properties attributed to filler size. The assumption of rigid bucky-ball will ensure that the shape of filler does not contribute to variation in elastic properties. The assumption may be unrealistic for large diameter buckyballs, it is a reasonable assumption for small bucky-balls and solid nanoparticles. In addition to this shape constraint, the volume fraction of the filler, matrix characteristics (density, molecular weight, molecular weight distribution, branch content, degree of crystallinity, etc.) and their force interaction with the nanoparticle were kept constant in all nanocomposites. Molecular models of the neat PE matrix were also developed for comparison. Elastic properties of the neat and nanocomposite systems were then evaluated using four different modes of deformation, namely, unidirectional tension and compression, and hydrostatic tension and compression, respectively. 2. Molecular dynamics simulations 2.. Molecular models Molecular models of nanocomposites were developed by symmetrically placing a spherical fullerene bucky-ball in the PE matrix, as shown schematically in Fig. a. The dashed box in Figs. a and b indicates the periodic cell or unit cell that was simulated by MD. Three types of bucky-balls, C 6,C 8 and C 32 (subscripts denote number of carbon atoms), were used to incorporate the size effect. All bucky-balls were infused in matrix by approximately 4.5 vol%. Periodic boundary conditions were employed to replicate the unit cells in three dimensions. In nanocomposites, the PE matrix was represented by united atom (UA) CH 2 units. The initial structure of the matrix was constructed by positioning the bucky-ball at the center of the unit cell and by randomly generating PE chain(s) on a tetrahedron lattice surrounding the bucky-ball. The method can be considered as a modified version of the conventional self-avoiding random walk (SARW) technique [5]. Three types of polymer chains were thus constructed for different nanocomposites. The characteristic features of the polymer chains are shown in Table. It can be observed from Table that the lengths and thereby the molecular weights (MW) of the chains in different unit cells are almost constant. The reason of maintaining such a structure was to establish consistent matrix properties [6,7]. The same chain structures were used to describe polymer in three different neat systems Force field Once the molecular structures were developed, corresponding molecular mechanics force fields were then Polymer Bucky-ball a Periodic Cell b Fig.. (a) Schematic diagram of polymer nanocomposites, (b) periodic cells used for MD simulations.

3 35 A. Adnan et al. / Composites Science and Technology 67 (27) Table Characteristics of polyethylene chains in various nanocomposites and neat systems PE-I PE-II PE-III Number of chains in PE 3 6 Number of UA units Molecular weight (g/mol) defined. In this study, the PE chains were described by appropriate bond stretching, angle bending and dihedral potentials between CH 2 units [8]. The non-bonded van der Waals, henceforth referred as VDW, interactions within or between PE chains were modeled using Lennard-Jones (LJ) potential [8,9]. The functional form and parameters of the force field are shown in Table 2. As mentioned earlier, the bucky-ball was modeled as a non-deformable solid inclusion. Such features were achieved by expressing them as a set of Frozen atoms during MD simulation [2,2]. From the MD point of view, the frozen atoms are defined as a set of atoms that are completely immobilized (i.e. their original coordinates remain unaltered throughout the entire simulation). As all atoms of the bucky-ball were defined as frozen, no force field parameters were required to calculate energies due to interaction between frozen atoms. The non-bonded interactions between PE and frozen bucky-balls were, however, modeled and evaluated by the LJ potential [8,9] Simulation details The MD simulations were performed using DL-POLY (version 2.4) simulation package obtained from Daresbury Laboratory [22]. All simulations were carried out at a temperature of 3 K with.5 fs time steps. The elastic properties of neat polymer and nanocomposites were evaluated by performing simulations in two major steps. In the first step, the equilibrium state of the molecular model was obtained. In the next step, the model was subjected to different strain fields and then re-equilibrated Equilibrium state The first step in establishing stress-strain relationship for a neat or nanocomposite material is to obtain the equilibrium state. From MD point of view, attainment of such a state requires fulfillment of two major criteria, i.e., to achieve energy stabilized state at a prescribed temperature, and to obtain the minimum initial stress state for the periodic box. In the current study, the stable state was achieved by consecutively subjecting the initial model to canonical (constant-nvt) and microcanonical (constant-nve) ensembles for 5, and, steps, respectively. In the second step, the unit cell size was adjusted to minimize the initial stresses using isothermal isobaric (constant- NPT) ensembles for several thousand steps depending on the molecular model and cell size. Subsequently, the system was further equilibrated by NVE for, steps. At the end of these steps, the molecular model was believed to be relaxed at 3 K with minimum initial stress. The final periodic cells of both neat and nanocomposites models are shown in Fig. 2. Detailed descriptions and connectivity of the generic ensembles mentioned above can be found elsewhere [5]. After the forgoing procedures, polymer densities for all models were calculated. The radial distribution function (RDF) g(r) was also calculated for all non-bonded VDW atom pairs which comprise atoms that contribute to nonbonded energy. These g(r) functions were then utilized to estimate VDW separation distance or gap (h) between polymer and bucky-ball for various nanocomposite models. The volume V vdw occupied by h,henceforth referred here as VDW Zone was also calculated for comparison. It is worthwhile to provide a brief description about the approach we followed in evaluating physical quantities (temperature, energy, stress etc.) from our MD simulations. It is known that any MD run contains two phases, equilibration and production. The former phase allows the molecular structure to settle at the desired thermodynamic state, whereas the latter phase enables one to calculate the desired thermodynamic parameters (already stabilized during equilibration phase) over a specified period of time. Duration of the production phase can be very Table 2 Functional form and parameters for force field model of polyethylene and bucky-ball Interaction type Potential type Functional form Parameters Bond Harmonic UðrÞ ¼ 2 k rðr r Þ 2 k r = 7 kcal/mol r =.53 Å Angle Harmonic cosine UðhÞ ¼ 2 k h½cosðhþ cosðh ÞŠ 2 k h = 2.5 kcal/mol h = 9.47 Dihedral Cosine U(/) =k / [ + cos(m/)] k / =. kcal/mol m = 3. PE PE: e =.3266 kcal/mol ( h Non-bonded Lennard-Jones UðrÞ ¼ 4e r r 2 i r 6 r r < r cut r P r cut r = 4.28 Å PE-Bucky: e =.729 kcal/mol r = Å r cut =.7 Å

4 A. Adnan et al. / Composites Science and Technology 67 (27) Fig. 2. Periodic box of different neat and nanocomposites model used for simulation. Note the size of the box obtained after complete equilibration. short (5 ps) when thermal fluctuations are small. The length should be quite long (as long as several nanoseconds) when data are subjected to high thermal noises. The sole purpose of the longer production time is to minimize noise-induced error in the time-averaged data. In our work, we monitored our production phase data for a short period of time ( 2 ps) and then identified the dominant frequency(s) and wavelength(s) of all simulation data points using Fast Fourier Transform (FFT). We then refined the length of production phase as integral multiple of the corresponding wavelengths. We also ensured that the production phase is long enough to provide statistically converged data. Compared to exceedingly long production runs, we found this technique efficient without compromising accuracy. This technique works well when noise contains few dominant frequencies Stress strain relation For each model at the equilibrated state, a uniform strain field (.5%) along the required directions was applied by proportionately scaling the corresponding unit cell and the atomic positions of PE chains. The positions of the bucky-ball atoms were, however, kept unchanged in order to maintain the undeformed shape of the bucky-ball. Using this technique, each of the periodic cells was strained to demonstrate unidirectional tension, compression, hydrostatic tension and compression, as shown schematically in Fig. 3. For unidirectional tension or compression, the applied strain was along one direction at a time (Fig. 3a and b). The procedure was then repeated for other two directions with respect to same un-deformed structure. In case of hydrostatic tension or compression, a.5% strain field was employed to all normal directions simultaneously, as seen in Figs. 3c and d. All systems were then re-equilibrated for 3, steps using NVE. In the next step, various stress components resulting from applied strains were recorded. The definition of stress is, however, different from continuum mechanics framework. At the atomic level, stress can be defined in the form of virial stress as,! r ij ¼ X M a v a i V va j þ X F ab i r ab j ðþ 2 a b6¼a where V is the volume of MD unit cell and V ¼ P a V a ; V a being the atomic volume of atom a; v a i is the i-component of the velocity of atom a, v a j is the j-component of the velocity of atom a, F ab i is the i-component of the force between atom a and b, and r ab j is the j-component of the separation distance between atoms a and b. It can be seen that Eq. () represents average atomic stresses for the volume of the periodic box. Here, the first term is associated with the contribution from kinetic energy due to thermal vibration and the second term is related to change in potential energy due to applied deformation. The negative sign is used to express tensile stress as a positive quantity (in MD, compression is generally expressed as positive). 3. Elastic constants 3.. Linear elastic stress strain relation The elastic behavior of the molecular systems is described using continuum mechanics. It is assumed that both neat polymer and nanocomposite posses isotropic material symmetry and linear elastic stress strain relation

5 352 A. Adnan et al. / Composites Science and Technology 67 (27) x 2 a b x x 3 c d Fig. 3. Schematic diagram of deformed (dashed line) and un-deformed (solid line) shape of unit cell showing: (a) unidirectional tension, (b) unidirectional compression, (c) hydrostatic tension, (d) hydrostatic compression. prevails. Under these assumptions the generalized constitutive relation of the equivalent continuum can be reduced to, >< r >= C C 2 C 2 >< e >= r 22 >: >; ¼ C 2 C C 2 5 e 22 ð2þ >: >; r 33 C 2 C 2 C e 33 where r ij, C ij, e ij are the stress, elastic constant, and strain components, respectively [23]. In the current study, strain is defined as engineering strain. All stress components are expressed as engineering virial stress in place of conventional virial stress definition shown in Eq. (). This is done by considering V in place of V in all calculations Unidirectional tension/compression For unidirectional tension or compression,.5% strain is applied along one direction. Hence for this constrained deformation, when e 22 = e 33 =. and e 5., Eq. (2) becomes >< r >= C C 2 C 2 >< e >= r 22 >: >; ¼ C 2 C C 2 5 : ð3þ >: >; r 33 C 2 C 2 C : Expanding the first row only, r ¼ C e ð4þ Similar expressions can be evaluated for stretching in 2 and 3 directions, respectively. It can be visualized from Eq. (4) that for each simulated tension or compression, three different C may be obtained. Now, as the applied strain is uniform for all direction, i.e. e = e 22 = e 33, the elastic constant C can be further averaged. It is, however, evident that the accuracy of these constants lies on the initial boundary condition which requires the minimum initial stress state for the un-deformed body. It was discussed earlier that such condition was obtained by adjusting the periodic cells using NPT ensemble. The process actually yielded approximately zero bulk stress in the unit cell. This criterion does not guarantee vanishing individual normal stress components as intended initially. To correct the effect of these initial stresses, the intrinsic stress components were calculated by subtracting the non-zero initial stresses, if existed, from the stresses recorded after the simulated deformation Hydrostatic tension and compression For hydrostatic tension and compression, the elastic bulk modulus K can be calculated as,

6 K ¼ r þ r 22 þ r 33 ð5þ 3 e þ e 22 þ e 33 It is described earlier that.5% uniform strain was applied in all three directions to simulate hydrostatic tension or compression. To obtain bulk modulus in tension and compression, all normal components of the intrinsic stress were recorded and then substituted in Eq. (5). It is known that both K and C may be expressed in terms of Young s modulus, E and Poisson s ratio m as ð mþe C ¼ ð6þ ð þ mþð 2mÞ E K ¼ ð7þ 3ð 2mÞ For a given m, Eqs. (6) and (7) may be used to see how the E obtained from C is correlated to the E from K. For neat PE, this transformed E would also provide a base for comparison with other reported results. 4. Results and discussion The linear elastic stress-strain relations of neat PE and bucky-ball reinforced nanocomposites obtained from MD simulations are presented in Tables 3 6. All results, either for neat PE or nanocomposites, represent the average of three independent samples, i.e., the entire simulation procedure is repeated three times with uncorrelated starting polymer structures. This approach provides more accurate statistical average to noisy data, which is unavoidable in MD simulation. For unidirectional tension or compression, the C for each sample is also average of all three Table 3 Evaluation of bulk modulus K for neat polyethylene by hydrostatic tension and compression Simulation type Bulk modulus, K (GPa) % Variation PE-I PE-II PE-III Average from mean Hydrostatic compression Hydrostatic tension Table 4 Evaluation of elastic constant C for neat polyethylene by unidirectional tension and compression Simulation type Linear elastic constant, C (GPa) % Variation PE-I PE-II PE-III Average from mean Unidirectional compression Unidirectional tension A. Adnan et al. / Composites Science and Technology 67 (27) Table 5 Evaluation of bulk modulus K for various nanocomposites by hydrostatic tension and compression System type Hydrostatic compression Hydrostatic tension K (GPa) % Gain/loss K (GPa) % Gain/loss C 6 -PE C 8 -PE C 32 -PE Neat-PE Table 6 Evaluation of elastic constant C for various nanocomposites by unidirectional tension and compression System type Unidirectional compression Unidirectional tension C (GPa) % Gain/loss C (GPa) % Gain/loss C 6 -PE C 8 -PE C 32 -PE Neat-PE normal directions. It is evident that C for any models is the average of nine data points and K is the average of three. In Tables 3 and 4, the elastic properties of neat polyethylene from various unit cells are compared. Appears that properties of a neat PE are not affected appreciably with the change in periodic box size. In most cases, the deviation is within % from mean. Since different cell sizes induce varying degrees of fluctuation in the noisy data, special care must be taken to minimize the effect of noiseinduced error in averaging the data. It is seen from Table 3 that the compressive bulk modulus of neat PE is higher than that in tension. This trend also prevails in simulated unidirectional tests, as shown in Table 4. The C of PE in tension is comparable with reported simulation results [3]. In order to see whether there exists appropriate correlation between K and C of PE as well as to verify our result with experiments, the average value of these parameters were extracted from Tables 3 and 4, and then utilized to evaluate E using Eqs. (6) and (7). By using m =.4527, which is based on some sample calculations from current simulations, the magnitude of E in tension is found as.784 and.87 GPa from Eqs. (6) and (7), respectively. In a similar manner, the E in compression is estimated as.8526 and.853 GPa. As pure amorphous polyethylene is not available in practice, it is not possible to verify the numerical results with experiments. Commercially available polyethylene is semi-crystalline and exists in various categories. Linear low density polyethylene (LLDPE) closely matches with our simulated amorphous polyethylene. The reported elastic modulus of LLDPE lies between 262 and 896 MPa [6] and our result falls within the range. These numbers also confirm that a satisfactory correlation exist between simulated unidirectional and hydrostatic tests

7 354 A. Adnan et al. / Composites Science and Technology 67 (27) Table 7 Bulk densities of polyethylene in neat PE and in nanocomposites System Model Density, (gm/cm 3 ) Neat Polymer PE-I.857 PE-II.833 PE-III.867 Nanocomposite C 6 -PE.8285 C 8 -PE.8252 C 32 -PE.83 for neat polyethylene. In view of this, it is also reasonable to conclude that the applied strain field (.5%) was within linear elastic range. It is evident from Tables 5 and 6 that elastic properties of nanocomposites are improved appreciably with the infusion of bucky-balls in PE matrix. The result also shows that size of the reinforcing filler has significant effect on C and K of nanocomposites, either in tension or compression. The trend shows that with the increase in filler size, the extent of enhancement in elastic properties is gradually reduced. The result is somewhat surprising because in all cases the volume fraction was maintained constant (4.5%). Moreover, initial matrix structure and interface characteristics were also consistent. In conventional composites, systems with equal volume fraction usually possess identical elastic properties [24]. The result implies that there exists some particle size dependent effect on the stress-strain relations. In order to investigate the issue further, bulk density of polymer, VDW gap and their volume were calculated and compared. In Table 7, the densities of polyethylene in neat polymer and nanocomposite systems are listed. It can be observed that the density of neat polyethylene is quite consistent among different unit cells. The average of these is found to be.825 gm/cm 3, which is close to the reported value for amorphous polyethylene [25]. However, it is interesting to note that the density of polyethylene in nanocomposites is somewhat higher than their neat counterpart. A decreasing trend in bulk density is also observed with the increase of filler size. It can be concluded from this observation that size of the filler has considerable influence on polymer density even with non-bonded inter-molecular interactions between polymer and nanoparticle. The effect can be well understood from the radial density distribution of PE for both neat and nanocomposites, as shown in Fig. 4 in which the distribution is constructed by measuring local densities of PE at various radial distances starting from the center to the half-length of the periodic box. It is interesting to find that local densities are not constant along the radial distance. A 2 25% increase in polymer density exists for all nanocomposites at a distance close to the nanoparticle. At further distances, the distribution fluctuates in a similar manner as in the neat polymer system. The fluctuating character is inherent because mass needs to be conserved [2]. The collective contributions of these factors yielded a decreasing trend in polymer bulk density with the increment of filler size. It appears from the analysis that polymer density distribution plays the foremost role in size effect. However, it is not elucidated why size difference influence polymer density. The discernible contribution from filler size can be realized from radial distribution plot as shown in Fig. 5. It is known that the radial distribution function for any atom pairs gives a measure on how corresponding atoms are distributed in three-dimensional space due to VDW interactions [5]. Hence, g(r) PE-Bucky refers to radial distribution of PE atoms with respect to Bucky-ball atoms. As atomic position of all bucky-balls were fixed, a plot of g(r) PE-Bucky would thus provide information about the polymer distribution due to interaction with a nanoparticle. Fig. 5 reveals that the size of Bucky-ball has strong influence on the g(r) plot. It is observed that the value of g(r) assumed zero from to 3.4 Å for all nanocomposites, then increases with radial distance. The zero value refers to the VDW thickness h. It is also evident that h does not depend on filler size. It is quite expected because parameters describing LJ potentials are identical for all nanocomposites and the nature of the h is known to be governed by such interactions between nanoparticle and polymer [9]. However, the relative distribution of polymer atoms towards the nanoparticle, as indicated by the variation in g(r) at a particular radial distance, is quite different with the change in filler size. It is obvious from Fig. 5 that more atoms are tending to disseminate across the polymer nanoparticle interface as the size of buckyball decreases. This is an indication that even though the parameters describing the non-bonded interaction between nanoparticle and polymer are identical for all nanocomposites, the radius of the filler might have some role on such atom dissemination. In order to see this, polymer nanoparticle VDW energy for various equilibrated nanocomposites were computed and shown in Table 8. Here negative sign implies attractive VDW force between polymer and nanoparticles. The energy represents the total pair-wise energy between nanoparticle and polymer atoms within cut-off radius (.7 Å for all nanocomposites). It is seen from Table 8 that nanocomposites with larger nanoparticles produce higher VDW energy. To see how much attractive force is offered by an individual bucky-ball atom to polymer atoms, all energies were normalized with respect to number of bucky-ball atoms. It is apparent from Table 8 that the smallest bucky-ball offers maximum attraction to polymer matrix as well as higher VDW zone volume fraction. It is evident from above results that with the reduction of filler size, an increased interfacial interaction takes place between polymer and nanoparticle. It appears from the above discussion that with the reduction in filler size, the bulk density of polymer and the attractive interaction energy between polymer and nanoparticle at the interface increase substantially. Enhancements of these parameters are then translated to improved elastic moduli.

8 A. Adnan et al. / Composites Science and Technology 67 (27) Density, ρ (gm/cm 3 ).4.2 PE-I Density, ρ ( gm/cm 3 ) C 6 VDW Gap C 6 -PE Density, ρ (gm/cm 3 ).4.2 PE-II Density, ρ ( gm/cm 3 ) C 8 VDW Gap C 8 -PE Density, ρ (gm/cm 3 ).4.2 PE-III Density, ρ ( gm/cm 3 ) C 32 VDW Gap C 32 -PE Fig. 4. Radial density distribution of various: (a) neat PE and (b) nanocomposite models. Space occupied by nanoparticles is schematically shown by the quarter circles. Radial Distribution Function, g(r)pe-bucky C 6 -PE C 8 -PE C 32 -PE Fig. 5. PE-Bucky radial distribution functions (RDF) of various nanocomposite models. Table 8 Polymer nanoparticle VDW energy and volume for various nanocomposites Model 5. Summary U VDW PE Buckyball kcal/mol Normalized U VDW PE Buckyball kcal/mol-atom C 6 -PE C 8 -PE C 32 -PE VDW Zone volume fraction, V vdw /V unit-cell In this study, the effect of filler size on elastic properties of bucky-ball reinforced polyethylene is studied using molecular dynamics simulations. In this technique, equilibrated structure for both neat and nanocomposites models are obtained first, then.5% strain field is applied to all

9 356 A. Adnan et al. / Composites Science and Technology 67 (27) models to simulate a series of tests including unidirectional tension, unidirectional compression, hydrostatic tension and hydrostatic compression. Linear Elastic Constant C and bulk modulus K are evaluated by assuming isotropic materials symmetry for both neat and nanocomposites. A good agreement is found between the neat polyethylene C and K indicating linear elasticity might prevail. Simulation result demonstrates that elastic properties of nanocomposites are significantly improved with the reduction of bucky-ball size. It is concluded that increased in polymer bulk density and polymer bucky-ball interface attractive energy per bucky-ball atom played dominant role in gaining improved properties with reduced filler size. Acknowledgement This work was supported by a National Science Foundation Grant No. HRD References [] Jordan J, Jackob KI, Tannenbaum R, Sharaf MA, Jasiuk I. Experimental trends in polymer nanocomposites a review. Mater Sci Eng A 25;393:. [2] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 973;2:57 4. [3] Benveniste Y. A new approach to the application of Mori-Tanaka s theory in composite materials. Mech Mater 987;6: [4] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc Roy Soc A 957;24: [5] Cho J, Joshi MS, Sun CT. Effect of inclusion size on mechanical properties of polymeric composites with micro and nanoparticles. Compos Sci Technol 26;66(3): [6] Chisolm N, Mahfuz H, Rangari VK, Adnan A, Jeelani S. Fabrication and mechanical characterization of carbon/sic-epoxy nanocomposites. Compos Struct 25;67():5 24. [7] Reynaud E, Jouen T, Gauthier C, Vigier G, Varlet J. Nanofillers in polymeric matrix: a study on silica reinforced PA6. Polymer 2;42(2): [8] Jiang L, Lam YC, Tam KC, Chua TH, Sim GW, Ang LS. Strengthening acrylonitrile nutadiene styrene (ABS) with nano-sized and micron-sized calcium carbonate. Polymer 25;46(): [9] Vollenburg PHT, De Haan JW, Van de Ven LJM, Heikens TD. Particle size dependence of the Young s modulus of filled polymers: 2. Annealing and solid-state nuclear magnetic resonance experiments. Polymer 989;3(9): [] Vollenburg PHT, Heikens TD. Particle size dependence of the Young s modulus of filled polymers:. Preliminarily experiments. Polymer 989;3(9): [] Smith JS, Bedrov D, Smith GD. A molecular dynamics simulation study of nanoparticle interactions in a model polymer nanoparticle composites. Compos Sci Technol 23;63: [2] Odegard GM, Clancy TC, Gates TS. Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 25;46(2): [3] Frankland SJV, Harik VM, Odegard GM, Brenner DW, Gates TS. The stress-strain behavior of polymer nanotube composites from molecular dynamics simulation. Compos Sci Technol 23;63: [4] Vacatello M. Monte carlo simulations of polymer melts filled with solid nanoparticles. Macromolecules 2;34: [5] Allen MP, Tildesley DJ. Computer simulations of liquids. New York (USA): Oxford University Press; 987. [6] Peacock AJ. Handbook of polyethylene: structures, properties and applications. New York (USA): Marcel Dekker Inc; 2. [7] Nunes R, Martin JR, Johnson JF. Influence of molecular weight and molecular weight distribution on mechanical properties of polymers. Polym Eng Sci 982;22(4): [8] Mayo SL, Olafson BD, Goddard III WA. DREIDING: A generic force field for molecular simulations. J Phys Chem 99;94: [9] Binder K. Monte carlo and molecular dynamics simulations in polymer science. New York (USA): Oxford University Press; 995. [2] Smith W, Rodger PM. Pressure in systems with frozen atoms; 22. Internet Source: wsmith22.pdf. [2] Smith W, Forester TR. DLPOLY-2.4 manual; 24. Internet Source: ALS/USRMAN2.pdf. [22] Smith W, Forester TR. DL_POLY_2.: A general-purpose parallel molecular dynamics simulation package. J Mol Graph 996;4:36 4. [23] Lai WM, Rubin D, Krempl E. Introduction to continuum mechanics. 3rd ed. USA: Butterworth-Heinemann Ltd; 999. [24] Mallick PK. Fiber-reinforced composites: materials, manufacturing, and design. New York (USA): Marcel Dekker, Inc; 993. [25] Brown D, Clarke JHR. Molecular dynamics simulation of an amorphous polymer under tension.. Phenomenology. Macromolecules 99;24: