(709a) Chain Conformations in Polymer Nanocomposites: A Field Theory-Inspired Monte Carlo Simulation Approach

Nanocomposite materials containing particles of size 1-10 nm dispersed at a volume fraction often lower than 10^-3 within a polymer matrix are characterized by particle number densities of ~10²⁶ m^-3,  interfacial areas per unit volume of ~10⁹ m^-1, and interparticle spacings that are commensurate with the particle dimensions and the radii of gyration of matrix chains. The mechanical and rheological properties of a polymer-matrix nanocomposite may depart dramatically from those of the pure matrix.¹ The quantitative relationships between composition and size of polymer chains and nanoparticles, processing conditions, degree of dispersion of the nanoparticles, dynamics of the matrix chains, and macroscopic properties are still elusive.

Molecular simulation holds great promise as a means for understanding and predicting these relationships. The wide spectra of length and time scales characterizing structure and motion in nanocomposites, however, necessitate careful design of multiscale modeling approaches for addressing their properties. This paper briefly reports on an ongoing effort in this direction for materials consisting of nanoparticles of roughly spherical shape within amorphous polymer matrices.  Particular emphasis is laid on changes in the conformation and spatial extent of polymer chains resulting from the presence of nanoparticles.  Simulation predictions are validated by comparing against small angle neutron scattering (SANS) measurements.

We have developed a Field Theory-inspired Monte Carlo (FTiMC) simulation approach for predicting the structure of polymer matrix nanocomposites.²  Polymer chains are represented in a coarse-grained sense as freely jointed sequences of Kuhn segments.  The configurational partition function of the system is expressed as a functional integral over the paths of all chains and as an integral over all nanoparticle positions and orientations. For spherical nanoparticles, orientations play a role only when grafted chains are present. The effective Hamiltonian embodies (a) interactions between different nanoparticles; (b) nonbonded interactions between segments of the free and grafted chains with the nanoparticles; (c) nonbonded interactions among segments of the polymer chains, whether free or grafted.  Contributions (a) and (b) are  computed from the center of mass positions and diameters of the nanoparticles and of the polymer segments and from the atomic densities within each nanoparticle and each polymer segment, as sums of Hamaker potentials. Contribution (c), on the other hand, is calculated by invoking a self-consistent field approximation.  Instead of expressing the nonbonded interaction energy as a sum of pairwise interactions between segments, as is typically done in molecular simulations, we assume it is a functional of the three-dimensional density distribution of polymer segments, ρ(r). To express the latter density distribution we partition the system into cells using a cubic grid of spacing ΔL. We then assign each polymer segment to the center of the cell to which it belongs. The volume of each cell is taken as (ΔL)³ minus the volume of any sections of nanoparticles that find themselves in the cell; it is computed via a fast analytical algorithm. Following early work by Helfand, our nonbonded polymer-polymer effective Hamiltonian punishes departures of the local density in each cell from the mean segment density ρ₀ in the melt under the temperature and pressure conditions of interest.  It is expressed as a sum of contributions of the form κ₀k_BT(ΔL)³/(2 ρ₀)( ρ_m- ρ₀)² from each cell, ρ_m being the density of polymer segments in cell m.  The prefactor κ₀ is related to the isothermal compressibility κ_Tof the melt via κ₀=1/(k_BT κ_T ρ₀).  Clearly, the FTiMC method cannot accurately resolve structural features and properties with characteristic length scales smaller than ΔL.  In practice, ΔL is chosen such that each simulation cell contains ρ₀(ΔL)³ =O(10) Kuhn segments.  Inputs to the model are the chemical composition of the chains and nanoparticles, the segment density ρ₀ and isothermal compressibility κ_T of the bulk melt, the Kuhn length b of the chains, as well as the Lennard-Jones interaction parameters for all pairs of atoms or chemical groups constituting the chains and the nanoparticles. The molar masses of free and grafted chains, the surface grafting density σ,  the radius R_n and volume fraction φ of the nanoparticles and their state of dispersion in the polymer constitute important molecular design parameters.

To start the FTiMC simulation, an initial configuration is generated by placing the nanoparticles within the simulation cell, with periodic boundary conditions. Grafting points are picked randomly on the surface of each nanoparticle, redistributed on that surface through a brief MC procedure employing a short-range repulsive potential, and then kept fixed in all subsequent simulations. The intention here is to model the more or less uniform distribution of initiators and linkers connecting grafted chains to the nanoparticle surface, observed experimentally. Both grafted and free chains are constructed as three-dimensional random walks, the former off of the grafting points already fixed on the nanoparticle surfaces. Inhomogeneities in the segment density distribution of the initial configuration thus created are reduced through a “zero temperature MC optimization” run, during which only moves leading to a more uniform distribution of segments are accepted.

Equilibration during a FTiMC run is achieved through MC scheme designed to sample the probability density associated with the configurational partition function discussed  above. The fact that chain-chain interactions are accounted for only approximately, through the nonbonded effective Hamiltonian, allows bold rearrangements of the chain conformations to be attempted with significant probability of success. Rigid displacements and rotations of chains, mirroring transformations through planes passing through the chain centers of mass, and exchanges of entire chains keeping their center of mass positions fixed, are used. In addition, the internal conformations of chains are sampled using flips of internal segments, end segment rotations, reptation and pivot moves. For the displacement of a nanoparticle a new configurational bias move has been designed, which involves rearrangement of all chain segments within a spherical region centered at the midpoint of the initial and final positions of the center of mass of the nanoparticle, thereby alleviating the severe overaps between nanoparticle and surrounding chains.

The FTiMC approach is particularly useful for exploring the structure of high-molar mass systems, comparable to those studied experimentally or considered for technological applications.

We have applied our FTiMC approach to nanocomposites consisting of tightly crosslinked 35 nm-radius PS nanoparticles within molten linear PS matrices of various molar masses, which have been studied experimentally with SANS by Mackay and coworkers.³ Our simulation results reveal swelling of the polymer melt coils with increasing nanoparticle volume fraction φ (0≤φ≤0.15). This swelling is most pronounced when the unperturbed root mean square radius of gyration of the melt chains is commensurate with the nanoparticle radius. These observations are in excellent agreement with the SANS results from ref 3.

We have also applied FTiMC to study systems consisting of SiO₂ nanoparticles of radius 3.6 up to 13 nm dispersed in monodisperse atactic PS matrices of molar mass 20 to 100 kg/mol at 450 K and 500 K.  The nanoparticles were either bare or carried monodisperse surface-grafted PS chains of length 20 to 100 kg/mol at grafting densities 0.2 to 0.7 nm^-2. Cubic simulation boxes of edge length 60-200 nm were used.  Detailed SANS investigations of grafted chain conformations in such systems have been carried out by Chevigny et al.⁴  At low φ, the profile of grafted chain segments ρ_g(r) far from the surface of a nanoparticles is found to decay with distance r from the particle center according to the scaling prediction⁵ρ_g(r) ~ σ ^1/2 (R_n/r).  The thickness h of the “brush” of grafted chains, as calculated from the second moment of the grafted chain density distribution, is in excellent agreement with the experimentally reported value⁴ of 6.2 nm.  Free chains of low molar mass are found able to penetrate into the grafted corona and swell it (“wet” corona), while free chains of high molar mass tend to be expelled from the corona.  Simulation results for the brush height h are consistent with the Daoud and Cotton prediction⁶h ~ N_g^1/2 σ ^1/4b, with N_g being the length of grafted chains. Predicted scattering curves from the grafted polymer corona are in excellent agreement with SANS measurements.⁴

            Well-equilibrated configurations of cis-1,4-polyisoprene melts filled with SiO₂ nanoparticles, sampled via the FTiMC approach, are used as a starting point for crosslinking into nanoparticle-filled rubbers.  In hierarchical fashion, the configurations obtained  are further coarse-grained into networks of polymer strands between crosslinks, entanglement points, chain ends, and points of attachment on the nanoparticle surfaces, to predict mechanical reinforcement of the rubbers brought about by the nanoparticles.

Acknowledgments.  This work was funded by the European Union under the FP7-NMP-2007 program NANOMODEL, Grant Agreement SL-208-211778, and the FP7-NMP-2011-EU-RUSSIA program  COMPNANOCOMP, Grant Agreement 295355.

References

(1)  S.K. Kumar, R. Krishnamoorthi R. Annu. Rev. Chem. Biomol. Eng.  1, 37 (2010).

(2) G.G. Vogiatzis, E. Voyiatzis, E.; D.N. Theodorou, Eur. Polym. J. 47,  699 (2011).

(3) A. Tuteja, P.M. Duxbury, M.E. Mackay, Phys.Rev.Lett. 100,  077801 (2008).

(4) C. Chevigny, J. Jestin, D. Gigmes,  D. Schweins, E. Di Cola, F. Dalmas, D. Bertin, F. Boué,  Macromolecules 43, 4833 (2010).

(5) C.M. Wijmans, E.B. Zhulina, Macromolecules 26, 7214 (1993).

(6) M. Daoud, J.P. Cotton, J. Phys. France 43, 531 (1982).