A molecular dynamics simulation study of nanoparticle interactions in a model polymer-nanoparticle composite James S. Smitha, Dmitry Bedrova, Grant D. Smitha,b,* aDepartment of Materials Science and Engineering, University of Utah, 122 S. Central Campus Drive, Rm. 304, Salt Lake City, UT 84112, USA bDepartment of Chemical and Fuels Engineering, University of Utah, 122 S. Central Campus Drive, Rm. 304, Salt Lake City, UT 84112, USA Received 4 September 2002 received in revised form 8 November 2002 accepted 15 December 2002 Abstract Molecular dynamics (MD) simulations were performed on a model polymer���nanoparticle composite (PNPC) consisting of spherical nanoparticles in a bead-spring polymer melt. The polymer-mediated effective interaction (potential of mean force) between nanoparticles was determined as a function of polymer molecular weight and strength of the polymer���nanoparticle inter- action. For all polymer���nanoparticle interactions and polymer molecular weights investigated the range of the matrix-induced interaction was greater than the direct nanoparticle���nanoparticle interaction employed in the simulations. When the polymer��� nanoparticle interactions were relatively weak the polymer matrix promoted nanoparticle aggregation, an effect that increased with polymer molecular weight. Increasingly attractive nanoparticle���polymer interactions led to strong adsorption of the polymer chains on the surface of the nanoparticles and promoted dispersion of the nanoparticles. For PNPCs with strongly adsorbed chains the matrix-induced interaction between nanoparticles reflected the structure (layering) imposed on the melt by the nanoparticle surface and was independent of polymer molecular weight. The nanoparticle second virial coe���cient obtained from the potential of mean force was utilized as an indicator of dispersion or aggregation of the particles in the PNPC, and was found to be in qualitative agreement with the aggregation properties obtained from simulations of selected PNPCs with multiple nanoparticles. # 2003 Published by Elsevier Ltd. Keywords: A. Nanostructures A. Particle-reinforced composites A. Polymer-matrix composites (PMCs) B. Modeling C. Computational simulation 1. Introduction Particles are important additives for altering and enhancing the properties of polymers [1]. A well-known example is the addition of carbon black to rubbers that is responsible for increased strength and durability [2,3]. Because of their very high surface area to volume ratio, the effect of nanoscopic particles (nanoparticles) on the properties of a polymer matrix and the resulting prop- erties of the polymer���nanoparticle composite, or PNPC, can be much more dramatic than is observed in con- ventional polymer���particle composites. Such PNPCs exhibit promising properties for a wide variety of appli- cations [4���8]. The properties of PNPCs are strongly influenced by nanoparticle size and filler fraction, nanoparticle shape, nanoparticle distribution, polymer molecular weight and the nature of the interactions between the nanoparticle and polymer matrix. There is a great need for insight that can be provided by theory and simulation regarding factors controlling the disper- sion of nanoparticles and the properties of the PNPCs as a function of these parameters. The application of theory and simulation methods to PNPCs is much less mature than in the related field of colloidal suspensions. Theoretical efforts that have been successful for colloid-polymer solutions (e.g., [9���16]) have not been fully extended to the dense polymer melts typical of a PNPC. Furthermore, until quite recently [15] theoretical studies of colloid-polymer solutions have dealt almost exclusively with cases where the radius of the colloidal particle is large compared to the radius of gyration of the polymer. The number of 0266-3538/03/$ - see front matter # 2003 Published by Elsevier Ltd. doi:10.1016/S0266-3538(03)00061-7 Composites Science and Technology 63 (2003) 1599���1605 www.elsevier.com/locate/compscitech * Corresponding author. Fax: +1-801-581-4816. E-mail address: gsmith2@gibbon.msu.utah.edu (G.D. Smith).

molecular simulation studies [17���19] that have been performed on PNPCs in order to gain insight in their structure and dynamics is also quite limited. These simulations revealed that the presence of nanoparticles as well as the strength of nanoparticle���polymer inter- actions strongly influence the dynamics, viscosity, and dynamic shear modulus of the polymer matrix and PNPC. Balazs et al. have shown in a series of lattice Monte Carlo and self-consistent field simulations [20��� 24] of diblock copolymer/nanoparticle mixtures that nanoparticle-polymer interactions strongly influence the dispersion of nanoparticles. In the present work molecular dynamics (MD) simulations have been employed to examine the polymer-induced interac- tions between nanoparticles in a dense polymer matrix as a function of polymer molecular weight and the strength of the nanoparticle-polymer inter- action, and to correlate polymer matrix effects with the dispersion of nanoparticles in a model PNPC. Here we concentrate on the regime where the radius of the particle, the radius of gyration of the polymer and the statistical segment length of the polymer are comparable which is particularly di���cult to address theoretically [15,16]. 2. System description and simulation methodology 2.1. Coarse-grained polymer���nanoparticle composites MD simulations as described below were performed on PNPCs consisting of two or five nanoparticles in a melt of 400, 800 and 1600 bead-necklace chains of length 20, 10, or 5 beads, respectively. The systems with two nanoparticles were utilized to determine the poten- tial of mean force between the nanoparticles which was subsequently used to calculate second virial coe���cient. These results were qualitatively compared with the dis- persion/aggregation behavior of nanoparticles in the five nanoparticle systems. The polymer chains were modeled as bead necklace chains [25,26] with bead dia- meters of ', defining the reference length scale. Each bead corresponds to 4���7 monomer units in a real, flex- ible polymer chain. Bond lengths were constrained to ' using the standard shake algorithm [27]. The polymer beads interacted with other polymer beads by Lennard- Jones interactions with a well depth of one ("pp=1), defining the energy scale for the simulations. The radius of gyration, Rg, of the bead necklace chains in a pure melt were 2.24' (20 bead chains), 1.50' (10 bead chains), and 0.98' (5 bead chains). The nanoparticles were modeled as spheres of Len- nard���Jones radius Rn=2.5'. The interactions between nanoparticles, Unn, and between polymer beads and nanoparticles, Unp, were modeled using modified Len- nard���Jones functions which account for the excluded volume of the beads and particles by offsetting the interaction range by REV: Unn��r�� �� 1 r 4 REV �� 4:0' Unn��r�� �� 4"nn ' r REV 12 ' r REV 6 " # r REV �� 4:0' ��1a�� Unp��r�� �� 1 r 4 REV �� 2:0' Unp��r�� �� 4"np ' r REV 12 ' r REV 6 " # r REV �� 2:0' ��1b�� where r is the separation distance between two interact- ing sites and the choice of interaction parameters "nn and "np is described below. All interactions in the system were truncated and shifted so that the energy and force are zero at the separation r=REV+2.5' (REV=0.0 for polymer bead���polymer bead interactions). In our simulations the interparticle interaction para- meter "nn was fixed so that the excess nanoparticle sec- ond virial coe���cient, B2 EX, is zero in the absence of polymer. The excess second virial coe���cient is the sec- ond virial coe���cient for the nanoparticle gas less B2 HS, the second virial coe���cient for hard spheres of diameter REV=4.0' and was calculated using the relation [28]: B2 EX �� 1 2 �� 1 REV exp�� Unn��r������4%r2dr 1 �� ��2�� A value of B2 EX=0 was obtained for "nn=1.412which then was used in all simulations to represent nano- particle���nanoparticle interactions. For each system we have performed simulations for nanoparticle-polymer interactions "np=1.0, 2.0 and 3.0. 2.2. Simulation methodology MD simulations for the PNPCs described above were carried out using the simulation package Lucretius [29]. All simulations were carried out in a periodic cubic cell at temperature T*=1.33 in units of "pp/kB where kB is the Boltzmann constant. Initially the systems were simulated in the NPT ensemble using the extended ensemble method [30] for at least 20 polymer chain relaxation times (Rouse times) at pressure P=0 yielding equilibrium densities &* =0.70, 0.68, and 0.63 for 20, 10, and 5 bead chains, respectively. The PNPCs were subsequently equilibrated in the NVT ensemble for at least 100 polymer relaxation times. Production runs of t*=71.5 103 were carried out in the NVT ensemble 1600 J.S. Smith et al. / Composites Science and Technology 63 (2003) 1599���1605