Disentangling the Role of Chain Conformation on the Mechanics of Polymer Tethered Particle Materials

The linear elastic properties of isotropic materials of polymer tethered nanoparticles (NPs) are evaluated using noncontact Brillouin light spectroscopy. While the mechanical properties of dense brush materials follow predicted trends with NP composition, a surprising increase in elastic moduli is observed in the case of sparsely grafted particle systems at approximately equal NP filling ratio. Complementary molecular dynamics simulations reveal that the stiffening is caused by the coil-like conformations of the grafted chains, which lead to stronger polymer–polymer interactions compared to densely grafted NPs with short chains. Our results point to novel opportunities to enhance the physical properties of composite materials by the strategic design of the “molecular architecture” of constituents to benefit from synergistic effects relating to the organization of the polymer component.


Experimental Details
Materials: Particle brush synthesis was performed using surface-initiated atom transfer radical polymerization following a procedure described previously. 1 Table S1 gives an overview of the polystyrene (PS)-tethered silica materials used in the experiments.
Brillouin Light Scattering (BLS): Utilizing the photoelastic interactions between incident light and thermally activated phonons, BLS 3 records the spectra of inelastically Table S1: Parameters of the PS-tethered silica particles. The volume fraction of PS (φ PS ) is determined by using different methods. 2 The mass density (ρ) is determined by using the relation ρ = ρ PS φ PS + ρ silica (1 − φ PS ), where ρ PS = 1.05 g/cm 3 and ρ silica = 1.85 g/cm 3 .  with ρ being the density of self-assembled films.

Simulation Model and Methods
Simulation Model Nanoparticles (NPs) are modeled as spherical clusters of Lennard-Jones (LJ) particles. Each LJ particle has unit diameter a and unit mass m. The NPs are considered as uniform spheres of unit reduced density. The resulting interaction between two NPs at distance r is given by the Hamaker potential 4,5 where R is the radius of the NP, r c,NN is the cut-off distance of the potential, and A NN is where r is the distance between two monomers, r c,mm = 3a is the cut-off distance of the potential, and ε is the strength of the potential. The monomers are bonded via the finitely extensible nonlinear elastic (FENE) potential 7,8 Here, r 0 is the maximum bond extension which is set to r 0 = 1.5 a, and κ is the spring constant which is set to κ = 30ε/a 2 . These values prevent unphysical bond crossing.
The polymers are grafted to the NPs by rigidly attaching the first polymer bead to the NP surface. These grafting points are randomly distributed on the NP surface. Then the remainder of the chains is fastened to those grafting beads (see the schematic representation given in Fig. S1). The interaction between a monomer and an NP at a distance r apart is Figure S1: Schematic of (a) an NP with all the grafting beads, (b) a free polymer chain that will be grafted to the NP, and (c) the resulting NP with one grafted chain.
also described via the Hamaker potential 5 where r c,Nm is the cut-off distance of the potential, and A Nm is the Hamaker constant between an NP and a monomer.
The experimental systems are mapped to our MD simulations in the following way.  Table S2.
All our MD simulations are performed in the N V T ensemble using the HOOMD-blue is the unit of time. Initially, all grafted NPs are placed randomly in a cubic box that is sufficiently large to avoid any overlap between grafted chains of different NPs. At this stage, the monomer-monomer interactions are set to purely repulsive by truncating U mm at r c,mm = 2 1/6 a. Then the simulation box is gradually shrunken over 5 × 10 6 MD steps until the desired monomer density ρ is reached. The system is then simulated for another 5 × 10 6 MD steps. Finally, the attractive contribution of U mm is turned on by setting r c,mm = 3a, and the system is evolved for 2 × 10 7 time steps. Measurements were taken during the last 4×10 6 steps of this period. Figure S2 shows the pressure, P , the potential energy, E, and the polymer radius of gyration, R g , during this time. The data have been normalized by their mean values during this period. It clearly visible that all quantities have leveled off, which indicates that equilibrium has been reached. We repeated this procedure for selected state points using different starting configurations and did not observe any significant impact on the final structures and properties.
The bulk modulus is computed using the relation In order to determine the slope dP/dV , we ran multiple simulations where we isotropically Quantification of Brush Overlap: The overlap between the grafted polymers belonging to the same NP and the other NPs is quantified by δ, defined as 15 where ρ s is the monomer density of grafted polymers belonging to the same NP and ρ o is the monomer density of polymers belonging to other NPs.
Identification of Single and Double Kinks using Z1 Algorithm: Fig. S4 shows a schematic of different types of kinks formed by the grafted chains which are obtained by using the Z1 algorithm. [16][17][18] Here, point A represents a single kink, and points B and C represent double kinks. The double kink at point B is formed by the chains belong to the same NP (measure of Z s ), whereas point C demonstrates a double kink formed by the chains coming from two different NPs (measure of Z o ).
A B C Figure S4: Schematic representation of single and double kinks.   Persistence length: The persistence length p is calculated using the relation

Additional Experimental Results
where b is the average bond length and Θ ijk is the angle between two consecutive bond vectors b ij and b jk connecting monomers i, j and j, k, respectively. The monomers (and bonds) are numbered in an ascending order starting from the grafting point.

Coarse-Grained Model and its Treatment based on Integral Equation Theory:
The high-frequency shear modulus is computed using the Zwanzig-Mountain relation 20 where φ c = 1 − φ PS is the effective packing fraction of the hard NP core, U (r) is the effective potential between brush-coated spherical NPs, and g(r) is the corresponding radial distribution function. In this work, we employed the model developed by Rabani et al., 21,22 where the NP consists of a rigid core with radius R surrounded by a spherical polymer shell of thickness h (see Fig. S9). The effective interaction potential between NPs, U (r), can be estimated assuming uniform filling of the polymer shell with monomers of the grafted chains, which interact via the LJ potential. 21,22 The corresponding g(r) can be calculated from the non-local integral equation theory with hypernetted-chain closure. 21,22 Taking the values for φ c , h, and d from experiments, we can use eq. (8) to compute G ∞ . Figure