P olymer-grafted nanoparticles (PGNs) have accumulated significant attention for their enhanced mechanical properties, attributed to high filler fractions and improved nanoparticle (NP) dispersion. [1][2][3][4][5][6] However, it is known that the increase in modulus due to the filler is generally accompanied by a significant reduction in material ductility, a key challenge limiting their broader application. 1,7,8 Recent studies show that PGN mechanical performance strongly depends on molecular design parameters like grafting density, chain molecular weight, and nanoparticle size. 4,9,10 PGNs achieve optimal cohesion and toughness at grafting densities of 0.2-0.4 [chains/nm 2 ], where entanglements per grafted chain are maximized. 11,12 At lower to intermediate grafting densities and higher molecular weights (N/N e ∼ 4 -6, where N e is the entanglement length), a transition to a crazegoverned ductile failure mechanism stabilizes crazing and reduces chain pull-out, mitigating brittle fracture. 12,13 However, it remains uncertain whether PGNs can exhibit toughness exceeding that of linear polymer assemblies of comparable chain length while retaining their mechanical enhancements, given the trade-oY between modulus and toughness governed by filler fraction and polymer-nanoparticle interactions.

This study uses coarse-grained molecular dynamics (CGMD) simulations of generic polymer and nanoparticle models based on established methodologies 14 to systematically investigate the influence of key design parameters on the material properties of PGNs. Specifically, this study examines the eYects of molecular weight (N), nanoparticle size (R NP ), and grafting density (ρ g ) on Young's modulus (E) and the work of failure referred to as toughness (Γ), of matrix-free PGNs below their glass transition temperature (T g ), as evaluated by tensile deformation simulations. The degree of interdigitation between PGNs is analyzed to investigate how the release of confinement in polymer coronas, driven by nanoparticle surface curvature and grafting density, aYects the mechanical properties of PGNs. Additionally, the entanglement behavior during deformation is examined to elucidate the mechanisms responsible for the observed enhancements in toughness.

increases chain rigidity and reduces entanglement length 15 while minimizing system size to control computational costs. Nanoparticles are modeled as hollow, spherical, rigid shells with surface beads as grafting sites and a core bead, following established simulation methods. 14 This captures the stiYness contrast between polymers (1-4 [GPa]) and nanoparticles (80-1000 [GPa]). 6,11,12,16 Nanoparticle mass is adjusted to match polymer glass density, and a shifted LJ potential enforces excluded volume constraints. 12,14 Initially, 32 PGNs are packed into a cubic simulation box in an FCC Bravais lattice, consistent with experiments on PGNs with similar design parameters. 10,[17][18][19] A series of NVT and NPT equilibration steps relax the PGN melts, following established protocols. 8,14,20,21 Relaxation is confirmed by a plateau in the grafted polymers' radius of gyration over time. To expedite equilibration, the bond swap algorithm in LAMMPS facilitates eTcient structural relaxation. 22,23 Before cooling, the FENE bond potential is replaced with a quartic bond potential, allowing bond breakage in highly deformed chains throughout the remaining simulations. Quartic potential parameters, adapted from glassy polymer mechanics studies, 24 match the FENE equilibrium bond length while enabling realistic bond rupture at the cutoY. This approach accurately approximates the covalent bond fracture force in polymers 12,[25][26][27] and is widely used in KG polymer deformation simulations. 24,[28][29][30][31][32] Systems are quenched from T = 1.0 to T = 0.4 (below T g ≈ 0.47) 33 and allowed to relax, removing residual stress.

Without prior aging, each PGN system undergoes uniaxial tensile deformation along the Z-axis at T = 0.4 with a time step of Δt = 0.01, and an engineering strain rate of dϵ/dt = 0.001, following established literature. 14,[34][35][36][37] The cross-sectional area remains constant during deformation, ensuring a consistent framework for analyzing fracture behavior. This methodology aligns with approaches used in studies of glassy polymer and epoxy fracture mechanics. 12,[38][39][40] PGNs are simulated across a broad design space, varying nanoparticle radii (R NP = 1, 2, 3, 5, 8 [σ]), grafted chain lengths (N = 20, 50, 100, 150, 200 beads), and grafting densities (ρ g = 0.5 and 0.2 [chains/σ 2 ] for dense and sparse systems, respectively). For statistical robustness, three trials are conducted for each system.

Stress along the Z-axis is monitored to evaluate key mechanical properties: Young's modulus (E), ultimate stress (σ max ), failure strain (ϵ f ), and toughness (Γ). E is derived from the initial linear stress-strain region (up to ϵ = 0.02), 14,41 while Γ is the area under the stress-strain curve. Simulations are performed using LAMMPS. 42 ' MECHANICAL RESPONSE Figure 1 illustrates the stress-strain behavior of selected PGN systems under uniaxial tensile deformation performed with a

Letter constant cross-sectional area, varying R NP , N, and ρ g . Figures 1a and 1b include linear polymer glasses with volumes comparable to the organic fractions of the corresponding PGN systems. While grafted chain lengths are directly compared to linear polymer glasses of matching lengths, higher molecular weight linear chains are also included, as prior studies suggest that the eYective chain length of linear chains should be approximately twice that of the grafted chains for a meaningful comparison. 40 Complete system curves are provided in Figure S1. PGN mechanical response generally follows a characteristic sequence: an initial elastic region, stress overshoot due to segmental orientation, 43,44 yielding, strain softening, a stress plateau, strain hardening in some cases, and eventual fracture. Yielding occurs at an average strain of ϵ z = 0.095 (standard deviation 0.01). Strain softening results from the glass undergoing mechanical rejuvenation, a process in which segmental mobility increases, leading to high structural fluctuations. 45 The stress plateau and the degree of strain hardening are strongly influenced by R NP , N, and ρ g . At constant ρ g and R NP , increasing N tends to increase the postyield σ max and ϵ f . Shorter grafts (N = 20 and 50) show minimal stress plateau and strain hardening. Lower ρ g leads to higher σ max at a constant R NP and N, consistent with prior findings in flat brush systems, where intermediate grafting densities enhanced the potential of mean force (PMF) which correlates with toughness. 11 Furthermore, at constant N and ρ g , ϵ f inversely correlates with R NP . Figure 2a presents Young's modulus (E) for PGNs as a function of R NP , N, and ρ g , alongside data for linear polymer glasses. PGNs with smaller R NP and shorter grafted chains exhibit larger standard deviations in E, likely due to spatial rearrangements during equilibration, as these systems tend to deviate significantly from their initial FCC configuration. This increased structural variability contributes to greater fluctuations in their early deformation response. 14 Although no obvious trends emerge, subtle parametric eYects on E are observed. Figure S2a shows Pearson correlation coeTcients (PCC) quantifying the linearity of relationships between R NP , N, ρ g , nanoparticle volume fraction (Φ NP ), and E. R NP has a moderate positive correlation with E, while N shows a moderate negative correlation. Φ NP , which encapsulates contributions from both R NP and N, exhibits the strongest positive correlation with E, consistent with previous findings linking shear modulus to Φ NP . 14,46 The linear relationship between Φ NP and E follows the Voigt model, which assumes uniform strain in both polymer and nanoparticle phases. 47 Since nanoparticles act as rigid inclusions, stress is primarily borne by the polymer matrix, leading to a proportional increase in E with Φ NP . As shown in Figure 2c, this relationship holds for ρ g = 0.5 [chains/σ 2 ], but deviates at ρ g = 0.2 [chains/σ 2 ], suggesting that while lower ρ g increases Φ NP , it may reduce E.

Figure 2b highlights the pronounced eYects of parametric variations on Γ (work of failure). PGNs with longer grafted chain lengths exhibit significantly higher values of Γ, with a strong positive correlation to N (Figure S2b), as longer chains allow greater extension before failure, enhancing energy absorption. While Φ NP shows a moderate negative correlation with Γ, its eYect on toughness is less significant than on E. Figure 2d shows an inverse relation between Γ and Φ NP for ρ g = 0.5 [chains/σ 2 ], but this trend weakens at ρ g = 0.2 [chains/ σ 2 ] suggesting that a higher Φ NP can enhance Γ. Toughness generally decreases in PGNs with larger nanoparticles when N remains constant, with R NP showing a moderate negative correlation with Γ. This is likely due to increased chain confinement near the surface of larger nanoparticles. As the nanoparticle surface curvature decreases, the concentrated polymer brush (CPB) region expands radially, 48,49 reducing interdigitation between PGNs. Free ends of chains grafted to adjacent PGNs struggle to penetrate the coronas of their neighbors, weakening the entanglement network and lowering toughness. The eYect of grafting density further reinforces this observation. PGNs with lower ρ g tend to exhibit higher toughness than their high-density counterparts, consistent with previous findings on PGN monolayers deformed above T g . 11,50 This eYect is particularly pronounced for larger R NP and longer N, where a denser CPB corona exacerbates confinement, further limiting interdigitation. Notably, the toughest PGN, In Figure 2a, PGNs with high N and low ρ g , which exhibit significant improvements in toughness compared to their densely grafted counterparts, show that E remains largely unaYected and, in some cases, is slightly enhanced. This eYect is attributed to sparsely grafted PGNs achieving a higher Φ NP , positively correlated with E. Interestingly, denser grafting leads to higher values of E for the shortest chain lengths, despite their lower Φ NP , particularly for low R NP . This finding is consistent with previous research indicating that star polymers with a higher number of arms and larger N exhibit the highest Young's modulus due to steric hindrance near the core, which promotes chain stretching and restricts segmental mobility. 40

The increased toughness observed in sparsely grafted PGN systems, particularly those with larger nanoparticles and longer chain lengths, is attributed to enhanced interdigitation. As ρ g decreases, grafted polymer chains with radii of gyration smaller than the grafting point spacing can relax into mushroom-like conformations on the nanoparticle surface, creating more space for neighboring PGN chains to intermingle. 48 For the linear glass with N = 20, R g is approximately 1.5 times the graft spacing for PGNs with ρ g = 0.2. While this suggests brush formation in the flat surface limit, surface curvature yields diminished confinement eYects as distance from the surface increases. At higher grafting densities, the reduced graft spacing leads to the formation of dense CPB structures, where the crowded polymer corona around each nanoparticle restricts interactions between neighboring PGN chains, thereby limiting interdigitation.

Figure 3 illustrates these eYects through the probability density of end-to-end lengths (R ee ) normalized by the average distance between surfaces of nearest-neighbor PGN particles (d NP ) (Figure 3a). Figure 3c-3d show the R ee /d NP for a sparsely (ρ g = 0.2 [chains/σ 2 ]) and densely (ρ g = 0.5 [chains/σ 2 ]) grafted system for N = 100 and R NP = 1, 5, and 8 [σ]. The data are fitted to a Gaussian (normal) distribution to determine the mean (μ) and skewness (γ). The extent to which chains extend past d NP is highly dependent on R NP , as evidenced by the high R ee /d NP and μ for small R NP . The observed decrease in normalized end-to-end lengths for larger nanoparticles is expected, as reduced nanoparticle surface curvature enhances polymer chain confinement near the nanoparticle surface. This forms a more extended CPB, creating greater resistance to interdigitation. 48,49 For small R NP and low ρ g , the normalized end-to-end length distribution of polymer chains has a lower γ. As ρ g increases, brush formation induces skewness in the distribution. For large R NP , where brush formation is already favored due to the low nanoparticle surface curvature, γ is larger at low ρ g , suggesting that a greater fraction of chains experience significant extension due to the increased available space and reduced steric hindrance of adjacent PGNs. Notably, γ remains positive in all PGNs, indicating that the grafted chain conformations tend to be skewed toward greater chain extension.

For each R NP , the breadth of the probability distributions of sparsely grafted PGNs is much greater than their densely grafted counterpart. Dense grafting increases steric hindrance near the nanoparticle surfaces, limiting the extension of neighboring PGN chains. In particular, larger nanoparticle radii and densely grafted PGNs prevent the free ends of grafted polymer chains from interacting with chains near the surfaces of adjacent nanoparticles. Previous work demonstrated that densely grafting long polymers onto the surface of large nanoparticles induces a free-end curling phenomenon. 14 These polymer chains' free ends, expected to primarily inhabit the interstitial pockets of the PGNs, adopt configurations akin to those observed in polymers under poor solvent conditions. Alternatively, sparsely grafted nanoparticles experience less confinement, enabling the free ends of the grafted polymer chains to more readily access the near-surface regions of neighboring PGNs. This release in nanoconfinement promotes greater interdigitation of polymer chain conformations. The enhanced chain interpenetration in sparsely grafted PGNs is strongly supported by the cohesive energy density (γ PGN ), which was found to be markedly improved in PGNs with lower grafting densities (Figure S3). Previous work on the microballistic impact of PGN thin films demonstrated that higher values of γ PGN are generally associated with increased specific penetration energy (E p *) and change in failure mechanisms. 21 This emphasizes sparser grafting densities foster stronger inter-PGN connectivity and yield higher toughness.

Deformation significantly alters the topological characteristics of entanglements within the PGN systems. The evolution of entanglements during deformation is analyzed using the Z1+ package 51 to assess failure mechanisms related to local connectivity through chain entanglements. This analysis provides insights into the role of entanglements, facilitated by high interdigitation, in sustaining mechanical integrity and their eYects on the fracture behavior of the PGNs. EYective inter-PGN entanglements, denoted as Z eff , refer to the kinks formed between grafted polymer chains of two distinct PGNs. Figure 4 illustrates the total number of entanglements normalized by the number of grafted chains in each system Z ( ) alongside the normalized number of eYective inter-PGN entanglements Z ( ) eff for both the initial undeformed state Z ( ) 0 and the ultimate stress state postyielding Z ( ) ult . This analysis encompasses four distinct PGNs, varying in nanoparticle size, chain length, and grafted density, providing insight into the dependence of entanglements on these parameters. Additionally, an entanglement analysis on a linear polymer glass of N = 200 is provided (volume comparable to ρ g = 0.2 [chains/σ 2 ], R NP = 1 [σ], and N = 200). Γ of each system, including the large volume linear glass comparable to the grafted length of N = 200, is plotted on the secondary y-axis. The results indicate a direct correlation between toughness and Z eff . Notably, tougher PGNs exhibit a significantly higher quantity of eYective entanglements in the undeformed state, as well as a greater retention of eYective entanglements under postyield ultimate stress. The low Z ̂observed in the short-chain system further supports the assertion that short grafted chains do not experience the same benefits of interdigitation evident by their lower number of entanglements.

The percent of eYective entanglements, = Z Z (%) / eff , is plotted in Figure S4. Tougher PGNs correlate with a higher percentage of eYective inter-PGN entanglements, which are more prominent in systems with smaller nanoparticle radii, longer polymer chains, and sparse grafting. Revisiting Figure 1, it can be concluded that strain hardening in PGNs is facilitated Nano Letters 2) subjected to a short equilibration. This system exhibited a significantly lower density of Z ef f and little to no strain hardening, compared to its well-equilibrated counterpart. Another significant observation is that the change in the normalized number of entanglements Z ( ) from the undeformed state to the postyielding ultimate stress state correlates directly with the toughness of each system (Figure S5). When subjected to large stresses, the eYective entanglements within PGNs begin to disentangle as chains are pulled out from the coronas of neighboring PGNs, and chains undergo scission. This process enables the material to absorb substantial energy through frictional dissipation and stress delocalization, associated with disentanglement and chain scission, respectively. Similarly, prior work indicates that, at high strain rates, PGNs fracture primarily via two mechanisms: segment pull-out and chain scission. 52 Figure S6 illustrates the normalized number of broken bonds (relative to the number of chains in each system) as a function of strain. All PGNs undergo a transition in their failure mechanism from disentanglement to chain scission at N = 100. Tougher PGNs, such as R NP = 5 [σ], N = 200, and ρ g = 0.2 [chains/σ 2 ], exhibit significantly more chain scission events at the point of fracture compared to their denser grafted counterparts which tend to be less tough. This finding suggests that a higher number of eYective inter-PGN entanglements can enhance fracture toughness by promoting not only progressive disentanglement but also prolonged fracture through globally dispersed cracking. As entangled chains break in concentrated regions, the loading is periodically redistributed, eTciently dissipating energy and delocalizing the stress, collectively contributing to an increase in toughness. Figure 5a illustrates how tougher PGNs utilize chain disentanglement and load redistribution to disperse stress across the many fibrils formed during deformation. The enhanced energy dissipation is primarily attributed to the improved stress distribution under large strain. In contrast, brittle PGNs exhibit much lower values of ΔZ ̂and stress localization along the impending fracture plane due to their insuTcient energy dissipation capability driven by disentanglement (Figure 5b).

This study provides a detailed analysis of how key design parameters (grafted chain length, grafting density, and nanoparticle size) modulate the mechanical properties of matrix-free glassy polymer-grafted nanoparticle (PGN) systems. High nanoparticle volume fractions generally enhance Young's modulus but reduce toughness. This trend deviates across grafting densities due to steric hindrance near the nanoparticle core, which induces chain stretching and limits segmental mobility. Topological analysis reveals that reducing grafting density alleviates nanoconfinement, promoting PGN interdigitation and toughness, while increasing nanoparticle volume fraction which can enhance modulus. Finally, entanglement evolution during deformation reveals that the primary mechanisms underlying PGN toughness are disentanglement and chain scission. These processes enable energy absorption through frictional dissipation and load redistribution, promoting stress delocalization and enhancing material toughness. These findings reveal the key factors that govern the balance between stiYness and toughness in PGN design, providing valuable guidance for the development of highperformance nanocomposites.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.5c00680.

Additional figures and discussion on the complete stress-strain curves, Pearson correlation coeTcients, cohesive energy density, entanglements, and broken bonds (PDF)

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AUTHOR INFORMATION Corresponding Author Sinan Keten -Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 60208-3109, United States; Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208-3109, United States; orcid.org/0000-0003-2203-1425; Email: s-keten@northwestern.edu Authors Arman Moussavi -Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 60208-3109, United States Nano Letters pubs.acs.org/NanoLett Letter