Abstract:The interaction forces between nanoscale colloidal particles coated with end-grafted Lennard-Jones homopolymers are calculated using off-lattice Monte Carlo simulations in the NVT ensemble. The focus of this work is on grafted polymers that are of approximately the same size as the nanoparticle, a regime intermediate to the star-polymer and Derjaguin limits. The effects of chain length (N), nanoparticle diameter (σ c ), grafting density (F a ), and colloidpolymer and polymer-polymer interaction energies ( cp and pp ) on the polymer-induced force between the nanoparticles are explored. The inclusion of attractive dispersion interactions between the particle and polymeric modifier results in either long-ranged attraction and shortranged repulsion or pure repulsion, depending on the molecular parameters. The polymerinduced attraction occurs even under good solvent conditions below a threshold grafting density (F a ) and chain length (N) and could be attributed to both bridging (colloid-polymer) and intersegmental (polymer-polymer) attraction. Above the threshold F a and N values, chain entropy and excluded volume effects begin to dominate and lead eventually to polymer-induced repulsion and, consequently, nanoparticle stabilization. These results point to the importance of considering dispersion attractions between grafted segments and the nanoparticle surface in modeling these high-curvature colloid interactions.

A key issue in nanoscale materials and chemical processing is the need for thermodynamic and kinetic models covering colloid-polymer systems over the mesoscopic length scale (approximately 1-100 nm). We have applied Monte Carlo simulations to attractive nanoscale colloid-polymer mixtures toward developing a molecular basis for models of these complex systems. The expanded ensemble Monte Carlo simulation method is applied to calculate colloid chemical potentials (micro(c)) and polymer adsorption (gamma) in the presence of freely adsorbing Lennard-Jones (LJ) homopolymers (surface modifiers). gamma and micro(c) are studied as a function of nanoparticle diameter (sigma(c)), modifier chain length (n) and concentration, and colloid-polymer attractive strength over 0.3 < Rg/sigma(c) < 6 (Rg is the polymer radius of gyration). In the attractive regime, nanocolloid chemical potential decreases and adsorbed amount increases as sigma(c), or n is increased. The scaling of gamma with n from the simulations agrees with the theory of Aubouy and Raphael (Macromolecules 1998, 31, 4357) in the extreme limits of Rg/sigma(c). When Rg/sigma(c) is large, the "colloid" approaches a molecular size and interacts only locally with a few polymer segments and gamma approximately n. When Rg/sigma(c) is small, the system approaches the conventional colloid-polymer size regime where multiple chains interact with a single particle, and gamma approximately sigma(c)2, independent of n. In contrast, adsorption in the mesoscopic range of Rg/sigma(c) investigated here is represented well by a power law gamma approximately n(p), with 0 < p < 1 depending on concentration and LJ attractive strength. Likewise, the chemical potential from our results is fitted well with micro(c) approximately n(q)sigma(c)3, where the cubic term results from the sigma(c) dependence of particle surface area (approximately sigma(c)2) and LJ attractive magnitude (approximately sigma(c)). The q-exponent for micro(c) (micro(c) approximately n(q)) varies with composition and LJ attractive strength but is always very close to the power exponent for gamma (gamma approximately n(p)). This result leads to the conclusion that in attractive systems, polymer adsorption (and thus polymer-colloid attraction) dominates the micro(c) dependence on n, providing a molecular interpretation of the effect of adsorbed organic layers on nanoparticle stability and self-assembly.

The force between two nanoscale colloidal particles dispersed in a solution of freely adsorbing Lennard-Jones homopolymer modifiers is calculated using the expanded grand canonical Monte Carlo simulation method. We investigate the effect of polymer chain length (N), nanoparticle diameter (sigma(c)), and colloid-polymer interaction energy (epsilon(cp)) on polymer adsorption (Gamma) and polymer-induced forces (F(P)(r)) between nanoparticles in the full thermodynamic equilibrium condition. There is a strong correlation between polymer adsorption and the polymer-mediated nanoparticle forces. When the polymer adsorption is weak, as in the case of smaller diameters and short polymer chain lengths (sigma(c) = 5, N = 10), the polymers do not have any significant effect on the bare nanoparticle interactions. The adsorbed amount increases with increasing particle diameter, polymer chain length, and colloid-polymer interaction energy. In general, for strong polymer-particle adsorption the polymer-governed force profiles between nanoparticles show short-range repulsion and long-ranged attraction, suggesting that homopolymers would not be ideal for achieving stabilization in nanoparticle dispersions. The attraction is likely due to bridging, as well as polymer segment-segment interactions. The location and magnitude of attractive minimum in the force profile can be controlled by varying N and epsilon(cp). The results show partial agreement and some marked differences with previous theoretical and experimental studies of forces in the limit of flat walls in an adsorbing polymer solution. The difference could be attributed to incorporation of long-ranged colloid-polymer potential in our simulations and the influence of the curvature of the nanoparticles.

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The expanded ensemble Monte Carlo (EEMC) simulation method has been applied to calculation of the chemical potential of nanocolloidal particles in the presence of polymeric surface modifiers. Two general classes of surface modifiers have been studied -nonadsorbing and freely-adsorbing. For both systems, the infinite dilution colloid chemical potential was calculated as a function of the colloid diameter and the modifier chain length. The colloid chemical potential was found to decrease with increasing modifier chain length for both types of modifiers, albeit for different reasons. Empirical power-law scaling relationships were found to represent the simulation results well. A physical interpretation was proposed for the power law exponents obtained in the case of adsorbing modifiers.