Glassy interphases reinforce elastomeric nanocomposites by enhancing volume expansion under strain

Significance

By adding nanoparticles to elastomers, engineers turn materials with the compliance of rubber bands into composites strong enough to support a Boeing 747 on just a few tire patches. The physics driving this transformation has remained elusive for nearly a century. Using molecular simulations, we show that nanoparticles primarily reinforce elastomers by forcing them to expand in volume under strain, drawing on the polymer’s incompressibility to enhance its resistance to deformation. When nanoparticles are sticky, they cause surrounding elastomer layers to become viscous or even glassy, further amplifying this effect. These findings unify competing hypotheses for the mechanism into a cohesive model of how nanoparticles strengthen elastomers.

Abstract

For nearly a century, introduction of nanoparticles to elastomers has yielded extraordinarily tough nanocomposites that are critical to technologies from actuators to tires. The mechanisms by which this reinforcement occurs have nevertheless remained a central open question in material science. One widely debated hypothesis posits that strong interactions between polymer and particles induce “glassy bridges” that cement particles into a cohesive percolating network that resists elongation. Here, molecular dynamics simulations show that glassy particle shells do not primarily provide elongational cohesion. Instead, they amplify an underlying mechanism wherein competition between filler and elastomer networks causes the elastomer’s volume to increase on deformation. This induces contributions from the elastomer’s bulk modulus, which is of order 1,000 times larger than its Young’s modulus. These findings establish a unified understanding of low-strain reinforcement in filled elastomers as emanating from volumetric competition between coexisting particulate and elastomeric networks. This reframes and unifies our understanding of low-strain reinforcement, provides a clear-cut diagnostic for the presence of glassy bridging, and offers a design principle for tough elastomeric nanocomposites.

Engineers have long added nanofillers, such as carbon black or silica, to polymeric elastomers in order to make them stiffer and tougher (1) — enhancements that enable rubber to withstand millions of cycles of deformation without failing (e.g., in vehicle tires, vibration dampers, and actuators). The surfaces of the nanoparticles in these systems are often “sticky”—that is, they can attract and immobilize nearby polymer segments. This stickiness is believed to play a central role in the mechanical performance of these (2, 3) and other nanocomposites (4–6). However, the mechanism of this effect, and even the extent to which it genuinely drives nanoparticulate reinforcement of elastomers, remain unresolved. The central challenge is to disentangle multiple physical mechanisms that may arise in the presence of strong polymer–particle attractions (7). This represents a nearly 100-year problem in soft materials science: despite elastomeric nanocomposites’ role as the oldest and most economically important nanocomposite, we do not understand their fundamental behavior.

The large reinforcements realized in experimental nanocomposites (e.g., 10-fold or greater modulus enhancements) are now known to result from mechanical percolation of nanoparticles (5, 8–14). However, the nature of this percolation and its relationship to polymer–particle interactions remains an open question. Do particles percolate through direct particle–particle contacts (15–18)? In this case, what role do strong polymer–particle interactions play? Are particles instead interconnected via “bridges” of immobilized, glassy polymeric segments induced by strong polymer–particle interactions (19–24)? If so, how can we account for reinforcement in systems apparently lacking such effects (25–30)?

Here we employ molecular dynamics (MD) simulations of elastomeric nanocomposites incorporating strong polymer–particle attractions (strength controlled by forcefield parameter ${ϵ}_{\mathit{PF}}$) in order to answer these questions. To do this, we explore the synergistic effects of ${ϵ}_{\mathit{PF}}$ with filler loading ${\varphi }_F$ and structure $N_p$ on various mechanisms of reinforcement by measuring relaxation times, bulk moduli, Young’s moduli, and Poisson’s ratios for neat and filled elastomers. In doing so, we seek to reconcile several sets of facts. First, polymer dynamics can indeed be significantly slowed in the vicinity of attractive, rigid surfaces, reflecting a local increase in the glass transition temperature $T_g$ (11, 20, 31–50). Indeed, there appears to be a general tendency for interfaces to nucleate a relatively long-ranged dynamical interphase in polymers and other glass-forming liquids (31). This interphase consists of a gradient in local relaxation times over ranges considerably exceeding those of compositional or thermodynamic gradients. This makes the glassy bridge scenario highly plausible.

On the other hand, this proposal remains highly debated, with a distinct set of studies questioning its strength of evidence and relevance at typical use temperature far above $T_g$ (25–27, 51–54). Indeed, early work by Witten, Rubinstein, and Colby established a mechanism by which reinforcement could instead occur through direct particle–particle contacts, via resistance to lateral compression during deformation (18). Our recent simulation work has shown that a compression-dominated scenario reminiscent of this picture indeed governs reinforcement in the absence of glassy bridge effects (28–30, 54).

With this in mind, we seek to disentangle four possible mechanisms in which strong polymer–particle attractions could hypothetically augment mechanical response.

• M1.Strain localization: Strong attractions to particles could immobilize surrounding polymer, leading to strain localization and stress amplification in the remaining mobile elastomer domains. This “bound-rubber” mechanism was popular in early literature (51).
• M2.Glassy bridging: Regions of polymer between particles may vitrify to form glassy bridges, yielding a collective elongational cohesion of the nanoparticle network (5, 19–24, 32, 55–58).
• M3.Transient crosslinking: Arrested or dynamically slowed polymer regions around particles, or adhesions to the particles themselves, may act as long-lived physical crosslinks in the matrix. This could increase the effective crosslink density of the rubber, thus increasing the entropic elastic modulus of the polymer domains (9, 59–62).
• M4.Enhanced Poisson’s ratio mismatch effects: Finally, reduced-mobility polymer (glassy shells) around nanoparticles could enhance mechanical response by augmenting direct-contact reinforcement effects. Our recent work has shown that reinforcement in systems lacking glassy bridge effects emerges from a polymer–particle competition over the volume of the system under deformation (28–30, 54). Whereas elastomers tend to conserve volume under deformation (corresponding to a Poisson’s ratio of 0.5), jammed particle networks tend to increase in volume under deformation (corresponding to a Poisson’s ratio in the vicinity of 0.3). Because these networks share an outer boundary, they must compromise to exhibit an intermediate behavior at the mean composite level, leading to a growth in volume on deformation of the composite. This is resisted by the ($\mathcal{O}(\mathrm{GPa})$) bulk modulus of the elastomer matrix, dramatically enhancing resistance to deformation (28–30, 54). We expect that this effect will intensify in the presence of glassy shells that augment particle jamming behavior.

Our findings in this work show that while all of these mechanisms play a role in reinforcing the material, the primary impact of strong polymer–particle attraction is to enhance the Poisson’s ratio mismatch (M4) in the filled elastomer. In other words, the dominant reinforcement mechanism arises not from glassy shell-mediated elongational cohesion, but from a mismatch in Poisson’s ratio that draws a large bulk modulus contribution into the material stiffness. We also find that glassy shells around nanoparticles introduce a distinct ultra-low-strain (<2%) glassy response regime (M2). Our findings point to a diagnostic signature for glassy bridges based on the presence of this ultra-low-strain glassy response regime. This study establishes a unified and comprehensive framework for understanding nanoparticle-induced reinforcement in elastomers at low strains.

Results

Glassy Bridge and Shell Phenomenology in the Quiescent Limit.

As indicated by Fig. 1A, our systems can be divided into three classes based on the impact of filler particles on polymer segmental relaxation. When ${ϵ}_{\mathit{PF}}=1$, polymer dynamics are essentially unaffected by proximity to filler particles, indicating the absence of glassy shell or bridge effects. At intermediate interaction strengths (${ϵ}_{\mathit{PF}}=2$ or 3), polymer dynamics within two segment diameters of the nanoparticle surface are appreciably slower than the bulk but are not arrested on simulation timescales [i.e. over a relatively short range, as expected from the literature far above $T_g$ (34, 46, 63–69)]. That is, interfacial polymer segments are in an intermediate viscous shell regime in these systems. Finally, at strong interactions (${ϵ}_{\mathit{PF}}=5$), near-particle relaxation times exceed simulation timescales, indicating vitrification and the formation of glassy shells in near-surface polymer. In all cases, the relaxation time of polymers located between filler clusters (bridging regions, see Fig. 1B) is essentially the same as in near-surface shells. This indicates that glassy or viscous bridging, to the extent it is present, results from simple overlap of glassy shells rather than from uniquely strong dynamical suppression in interfiller bridging domains. Other variables (particle loading and structure) exert relatively minor influence on near-particle dynamics, such that ${ϵ}_{\mathit{PF}}$ is the dominant controlling parameter for shell classification (see section on relaxation times and SI Appendix, Fig. S1).

Fig. 1.

(A) Segmental relaxation time vs. polymer–filler interaction strength ${ϵ}_{\mathit{PF}}$ for the $N_p=7$ and 75 PHR filled elastomer system. Scatter points are included for “Interfiller” beads, as well as radial shells (1σ thick) around the nanoparticle surface. Relaxation time is determined as the time when the self-intermediate scattering function decays to a value of 0.2 at a wavenumber of 7.07. (B) Schematic illustration of interfiller glassy bridges and overlapping glassy shells. (C) Initial configuration for $N_p=7$, 75 PHR, and ${ϵ}_{\mathit{PF}}=1$. (D) Same as (C) but with ${ϵ}_{\mathit{PF}}=3$. Distinct filler clusters are shown in shades of gray; elastomer beads are color-mapped by relaxation time according to the color bar below (C) and (D).

The larger-scale implications of these findings can be seen in our initial configurations in Fig. 1C and D with polymer segments colored by their relaxation time. At ${ϵ}_{\mathit{PF}}=1$, regions near particles exhibit only marginally perturbed dynamics relative to bulk-like regions far from particles (panel C). In contrast, at high ${ϵ}_{\mathit{PF}}$, particles along with their glassy shells form a percolating network throughout the system (panel D), with bicontinuous domains of bulk elastomer and filler networks connected via immobilized polymer. While direct particle–particle contacts are difficult to visualize in these 2D projections, they are expected to be present at our filler loading and structure. Indeed, prior simulation and experimental studies suggest that the percolation threshold for carbon black-like aggregates is well below our lowest filler volume fraction (26.1%) (70, 71). Importantly, particle percolation via direct contacts is known to significantly influence mechanical properties even in the absence of frozen polymer shells (30), and we expect this influence to be further amplified by the presence of immobilized polymer, which enhances network connectivity and stiffness.

Emergence of Glassy Strain Softening and Yield at Low Strain in the Presence Of Strong Attractions.

As expected, increasing polymer–particle attraction enhances mechanical response (Fig. 2A–C). More importantly, it drives the emergence of a new feature in the low-strain response regime—a sharp yield at approximately 2% strain. While superficially this might be assigned to the Payne effect—a yield event in filled elastomers that is attributed to the breakdown of a physical filler network under deformation (1, 72)—in detail this does not appear to be the correct physical assignment. The Payne effect is a broad softening of the modulus that spans several decades in strain amplitude and is commonly observed most clearly in the 5 to 10% strain range (1, 72). In comparison, the yield observed in our strongly interacting systems is quite sharp and is uniformly at about 2%, suggesting a distinct yield mechanism.

Fig. 2.

Engineering stress response ($A-C$), species normal pressure multiplied by species volume fraction (filler is +ve and polymer is -ve) ($D-F$), and Poisson’s ratio ($G-I$), all plotted vs engineering strain. Data sets in each panel are identified by the legend in the bottom panel of its column. The left column corresponds to 100 PHR systems with various particle cluster sizes and polymer-particle interaction energies. The middle column corresponds to $N_p$ = 11 systems with various loadings and polymer-particle interaction energies. The right column corresponds to systems at 75 PHR and $N_p$ = 7, over a range of polymer-particle interaction energies. SE bars are generally smaller than the points and were computed from 100 thermal replicates each (SE bars typically smaller than data points).

Rather than emanating from particle network behavior, this yield event, and the ultra-high-modulus-regime that precedes it, must be understood as an unavoidable consequence of the presence of glassy bridges. This can be seen from a formal analysis of the connection between high-frequency glassy relaxation dynamics and stress–strain response via the generalized linear viscoelastic relations. The elongational stress σ is related to the relaxation modulus E via the equation (73),

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \sigma \left(t\right)=\dot{\epsilon }{\displaystyle \underset{0}{\overset{t}{\int }}}E\left(t-s\right)ds,\end{array}\end{array}$$ [1]

where t is the time since deformation began and $\dot{\epsilon }$ is the (constant) elongational strain rate.

For a glass-forming crosslinked polymer, $E(t)$ exhibits an initial glassy plateau, followed by an exponential decay, the Rouse regime, and then the rubbery plateau (see Fig. 3A and extended discussion in a dedicated section on detailed derivations in SI Appendix). We characterize the rheological response of the material by a Deborah number, defined as the product of its α relaxation time ${\tau }_0$ and strain rate $\dot{\epsilon }$ (where $\dot{\epsilon }{\tau }_0\ll 1$, $\dot{\epsilon }{\tau }_0\gg 1$, and $\dot{\epsilon }{\tau }_0\sim 1$ indicate fluid-like response, solid-like response, and intermediate response with comparable relaxation and deformation rates, respectively). Numerical solution of Eq. 1 with this form of $E(t)$ predicts a qualitative change in linear regime stress response as $\dot{\epsilon }{\tau }_0$ even loosely approaches unity. As shown by Fig. 3B, when $\dot{\epsilon }{\tau }_0$ approaches ${10}^{-3}$, a low-strain glassy modulus emerges in the stress response, followed by a softening by around 2% strain. This is not a genuine yield, but the result of integrating in the glassy and Rouse modes at low strain. This turnover will be further sharpened by actual nonlinear yield of glassy modes at similar strains (74–76).

Fig. 3.

(A) Relaxation modulus model (SI Appendix, Eq. S20 with 20 Kuhn segments, i.e., effective monomers) employed in computing anticipated linear response startup elongation modulus of an elastomer. (B) Predicted stress vs. strain for linear response in startup elongation, using the relaxation modulus in panel A, accounting for short-time integration of high frequency modes via Eq. 1.

With this in mind, we proceed by dividing the low-strain response into two regimes and considering them separately: 1) the rubbery modulus at strains beyond 2%, and 2) the ultra-low-strain transient glassy response that precedes it. At strains typically relevant to elastomers (ca. 5% or greater), stress response in the rubbery regime is most relevant; the impact of the ultra-low-strain glassy response regime is to generate a vertical offset in the stress response at higher (elastomer-relevant) strains.

Glassy Bridge Effects on the Rubbery Modulus.

As shown in Fig. 2, the rubbery response of systems with viscous and glassy shells exhibit the same mechanistic features of the Poisson’s ratio mismatch mechanism as systems without glassy shells (29, 30). Enhanced stress is underpinned by an enhanced internal normal stress balance between filler and polymer (Fig. 2D–F) and reduced Poisson’s ratio (Fig. 2G–I)—core signatures of the polymer/filler volume competition that underpins Poisson’s ratio mismatch reinforcement effects (29, 30).

To quantitatively assess the role of the Poisson’s ratio mismatch mechanism (M4), we compare its predictions for the rubbery moduli to measurements at 5.5% strain. The core prediction of M4 relates the composite modulus $E_c$ to the change in Poisson’s ratio in the composite relative to the neat elastomer, ${\nu }_c-{\nu }_e$ (29):

$$\begin{array}{c}\hfill E_c=f\left[E_e+2K_e\left({\nu }_e-{\nu }_c\right)\right],\end{array}$$ [2]

where f is a strain amplification factor (measured based on chain deformation rather than employed as a fit parameter, as described in SI Appendix section on quantification of strain localization) and $E_e$, $K_e$, and ${\nu }_e$ are the Young’s modulus, bulk modulus, and Poisson’s ratio of the neat elastomer measured at the same strain, respectively. This equation predicts that adding filler enhances the Young’s modulus by invoking a contribution from the elastomer’s bulk modulus, with a magnitude controlled by the suppression in composite Poisson’s ratio. We note that in using the elastomer’s bulk modulus in Eq. 2, we are making the assumption that most of the volumetric expansion is born by the polymer network. While we do not directly test this hypothesis, we argue that because the polymer is much less rigid than the filler, the volumetric strain localizes primarily in the polymer phase. Thus, using $K_e$ in Eq. 2 is an appropriate approximation to leading order. Moreover, we have found this approximation to be successful in prior work (see figure 11 in ref. 29).

In prior work, we reported that Eq. 2 predicts reinforcement across a wide range of nanoparticle loadings and structures (29), in the absence of glassy bridges. In contrast, as can be seen in Fig. 4A, for many of the systems with ${ϵ}_{\mathit{PF}}>1$, Eq. 2 overpredicts the degree of reinforcement when it is applied to the overall composite stress. However, as shown in Fig. 4B, it provides an excellent description of the elongational stress within the polymer component of the composite, a la

$$\begin{array}{c}\hfill {\overline{E}}_P=f\left[E_e+2K_e\left({\nu }_e-{\nu }_c\right)\right].\end{array}$$ [3]

Fig. 4.

(A) Normalized Young’s modulus $E_c/f$ and (B) normalized polymer Young’s modulus ${\overline{E}}_P/f$ vs. Poisson’s ratio mismatch $2\left({\nu }_e-{\nu }_c\right)$, all computed at a strain of 5.5%. These moduli are normalized by the strain amplification factor f, which is measured from simulation as described in SI Appendix and is not a fit parameter. All scatter point error bars are SE from 100 replicates and are typically smaller than data points. The dashed line denotes the slope predicted by the measured bulk modulus of the neat elastomer.

Here the overbar denotes a partial volumetric component stress within the composite, defined such that the total stress is given by a volume-fraction-weighted sum over partial volumetric component stresses ($\sigma ={\varphi }_P{\overline{\sigma }}_P+{\varphi }_F{\overline{\sigma }}_F$, $E={\varphi }_P{\overline{E}}_P+{\varphi }_F{\overline{E}}_F$, see Materials and Methods for formal definition). On the basis of the physical origin of the Poisson’s ratio mismatch theory, this is unsurprising: this relationship is the consequence of compelling the elastomer matrix to deform at a nonnative Poisson’s ratio, without considering direct contributions to elongational stress from the filler network. These contributions were negligible in the absence of glassy bridges and could be neglected to leading order; evidently they are significant here.

We can account for direct elongational contributions from the filler network by decomposing the composite modulus into direct contributions from polymer and particle components:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E_c-E_e={\varphi }_P\left({\overline{E}}_P-E_e\right)+{\varphi }_F\left({\overline{E}}_F-E_e\right).\end{array}\end{array}$$ [4]

Substituting Eq. 3 into Eq. 4 then yields a generalized prediction for the modulus of a composite, accounting for multiple distinct mechanisms of reinforcement, as in Eq. 5.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left(E_c-E_e\right)}_{\text{predicted}}=& \underset{\text{M1: Strain Localization}}{\underbrace{{\varphi }_P{E}_e(f-1)}}+\underset{\text{M2: Elongational Glassy Bridging}}{\underbrace{{\varphi }_F\left({\overline{E}}_F-E_e\right)}}\hfill \\ \hfill & +\underset{\text{M4:}\nu \text{Mismatch}}{\underbrace{2{\varphi }_{P}f{K}_e({\nu }_e-{\nu }_c)}}.\hfill \end{array}\end{array}$$ [5]

In Eq. 5, term 2 corresponds to mechanism 2—direct elongational stress contributions from the filler network that were missing in our original model. The first and third terms represent a decomposition of the direct polymer contribution into a term (term 1) quantifying strain localization effects from particles and nearby frozen elastomer (mechanism 1) and a term (term 3) quantifying the Poisson’s-ratio-mismatch mechanism (mechanism 4).

To test Eq. 5, we measure the direct filler elongational contribution (term 2) from simulation, we measure f and ν from simulations, and terms 1 and 3 then yield predicted values for the strain localization and ν mismatch mechanisms. As shown in Fig. 5, this equation provides a good leading order prediction of composite moduli in our simulated systems.

Fig. 5.

Plots of measured (MD simulation) and predicted (Eq. 5) difference in composite and neat Young’s moduli, $E_c-E_e$, along with the contributing terms in the right-hand-side of Eq. 5. All scatter point error bars are SEs from 100 replicates and are generally smaller than the data points. From left to right, the columns show data for the 100 PHR, $N_p=11$, and $N_p=$7 & 75 PHR systems, respectively.

Remarkably, as shown in Fig. 5, the direct filler contribution to composite stress, ${\varphi }_F\left({\overline{E}}_F-E_e\right)$, is in many cases negative. This seems surprising: hypothesized glassy bridging is predicated on the idea that glassy interparticle bridges lend elongational cohesion to the nanoparticulate network. However, the reason for this contrary outcome is simple. As discussed above, beyond 2% strain, the elongational glassy modes within the system have yielded or fully integrated in. Elongational glassy cohesive effects are thus necessarily lost beyond the ultra-low-strain regime and do not contribute to the rubbery elongational modulus. This essentially rules out the possibility of mechanism 2—direct elongational cohesion via glassy bridges—in the rubbery response regime.

Other mechanistic contributions to the stress response are evidently of second or higher order. Strain localization effects, as shown in Fig. 5, are of smaller magnitude than these effects and are not leading order. Moreover, where there are second-order deviations of the data from Eq. 5, they generally take the form of a modest underprediction. Returning to the list of postulated potential mechanisms in the introduction, we tentatively assign this second-order underprediction to mechanism 3—enhanced effective crosslink density due to tethering at the particle interface (9, 59–61). This would enter the model through an enhancement of $E_e$ relative to the neat elastomer value. Future work in the field should establish means of directly validating and quantifying this tentative assignment, but, in any case, the correction is at second order.

These results indicate that the Poisson’s ratio mismatch between nanoparticle network and elastomer matrix is the dominant mechanism of rubbery modulus reinforcement, including in the presence of glassy bridges. This implies that glassy shell and bridge effects primarily enhance the rubbery modulus beyond 2% strain by buttressing the particle network in the compressive direction, strengthening the Poisson’s ratio mismatch effect.

Ultra-Low-Strain Glassy Response Regime.

Finally, we consider the impact of glassy bridging in the ultra-low-strain regime prior to the <2% glassy softening. By extrapolating the line corresponding to the 5.5% tangent modulus back to a strain of zero, we extract the vertical offset $C_0$ in the rubbery response (at 5.5% strain) produced by the ultra-low-strain glassy response. As shown in SI Appendix, Fig. S3, $C_0$ is proportional to the tangent modulus of the system at 1.5% strain, which confirms that this offset reflects the degree of glassy stiffening in this ultra-low-strain regime.

The behavior of the ultra-low-strain regime, analyzed in this manner, resembles a percolating glassy network that is mechanically diluted by soft surrounding rubbery regions. As shown by Fig. 6, $C_0$ increases strongly with ${\epsilon }_{\mathit{PF}}$; this is consistent with the expected emergence of a glassy network as glassy shells emerge. It additionally increases with particle loading; this is consistent with a gel-like dilution model, akin to dilution of crosslinked elastomers by a solvent. In this case, however, the diluted network is glassy rather than rubbery. In contrast, $C_0$ is relatively insensitive to particle structure (Fig. 6A); this is reasonable for a percolated network where structure does not change the fraction of glassy material.

Fig. 6.

Extrapolation to the intercept of the stress–strain curve from the rubbery modulus at 5.5% strain ($C_0={\left[\sigma -E_c\epsilon \right]}_{\epsilon =5.5\%}$) vs. a) $N_p$ at 100 PHR, b) PHR loading at $N_p=11$, and c) ${ϵ}_{\mathit{PF}}$ at $N_p=7$ and 75 PHR. All scatter point error bars are SE from 100 replicates and are generally smaller than the data points.

Finally, we note that, in addition to explaining the presence of this ultra-low-strain glassy response, the linear viscoelastic theory analysis in SI Appendix sheds light on its expected strain rate dependence. The extent of this behavior is governed by the segmental Deborah number ${\text{De}}_s$, i.e., the ratio of the timescale of glassy relaxation to the timescale of observation or deformation. By definition, ${\text{De}}_s$ varies spatially in the presence of glassy bridging effects and is larger within bridges. Broadly, when ${\text{De}}_s\gg 1$ in interparticle regions, glassy bridges are present and the ultra-low-strain glassy response region emerges. At lower strain rates, when ${\text{De}}_s\ll 1$ in interparticle regions, glassy bridges are absent and the ultra-low-strain glassy regime is absent. This is confirmed by a small set of simulations at lower strain rates reported in SI Appendix, Fig. S4, which shows that the ultra-low-strain glassy response considerably weakens at a 10× lower strain rate, as predicted by our generalized linear viscoelastic analysis presented in SI Appendix and summarized above.

Discussion and Conclusions

This reasoning points to a clear-cut diagnostic for the presence of glassy bridge effects. Glassy bridges unavoidably introduce a glassy response at less than 2% strain, which then yields to the rubbery response. This is a physical consequence of introducing a glassy network induced by particle interfaces, diluted by rubbery content, and is formally required by the low-strain integration over the network’s glassy modes via Eq. 1. The presence of such a regime suggests the presence of glassy bridges; its absence unambiguously denotes their absence.

Taken together with our prior findings, these results suggest a unified understanding of the century-old question of the mechanism of low-strain reinforcement in filled elastomers. While earlier models emphasized bound rubber, glassy bridging, and transient crosslinking as causal mechanisms, our simulations demonstrate that these effects play quantitatively augmenting but not mechanistically causal roles. The causal mechanism of rubbery regime reinforcement is instead a competition between elastomer and particle networks over system volume under deformation, as reflected in their differing native Poisson’s ratios. This competition induces a bulk-modulus-mediated resistance to deformation. This mismatch is significantly amplified, but not caused, by glassy shells (when present).

This picture points to answers to the motivating questions posed in our introduction. Particles can percolate and provide reinforcement through either direct particle–particle contacts or via overlapping glassy shells. In either case, the dominant mechanism of reinforcement is volume expansion on deformation, driven by a Poisson’s ratio mismatch between filler and elastomer networks. This is why systems without glassy shells can exhibit large reinforcements. Strong polymer–particle interactions, when present, create glassy shells that buttress the particle network against compressive strain, biasing the composite toward more volumetric expansion and therefore toward greater contributions from the bulk modulus.

Finally, this study provides a framework for rational control of low-strain reinforcement in elastomeric nanocomposites via particle structure, loading, and interactions: the combination of these variables allows for a degree of independent control of the rubbery modulus vs. the ultra-low-strain glassy response. To illustrate this, in Fig. 7, we plot $C_0$ (which characterizes the strength of the ultra-low-strain glassy response) vs. the rubbery modulus of the composite at 5.5% strain across all studied systems. As can be seen here, introducing glassy shell and bridge effects (increasing ${ϵ}_{\mathit{PF}}$) tends to raise both the rubbery modulus and ultra-low-strain glassy modulus, as does increasing loading (especially when glassy bridges are present). In real-world systems, the glassy bridge effect strength can be controlled via chemical modifications to tune the strength of polymer/particle interactions, or via temperature [since glassy interface effects weaken on heating (34)]. In contrast, increasing particle structure tends to increase the rubbery modulus without strengthening the ultra-low-strain glassy response. In future work, it would further be of interest to probe additional molecular variables, such as filler stiffness, by which the Poisson’s ratio and shell effects might be modulated. This could both extend the validation of this theoretical framework to yet more diverse systems and provide additional microscopic control variables for tuning composite response.

Fig. 7.

Intercept extrapolated from the rubbery modulus at 5.5% strain vs. rubbery modulus at 5.5% strain. All scatter point error bars are SE from 100 replicates and are generally smaller than data points.

The combination of these variables thus offers the potential to control the shape of the stress response below 10% strain in a fairly facile way: filler cluster size and morphology can enhance the rubbery modulus without inducing stress softening related to glassy polymer layers; increasing filler loading and polymer–filler affinity enhances stiffness but introduces pronounced stress softening at low strains. These findings point the way to rational control of filler characteristics and polymer–filler interactions to target specific performance criteria in demanding industrial environments, such as automotive tires, vibration damping, and impact-resistant components.

Materials and Methods

We performed MD simulations of crosslinked bead-spring elastomeric nanocomposites at equilibrium and under uniaxial extension using the LAMMPS package (77–79). Two system types were studied: i) a neat elastomer network comprising 5000 polymer strands (20 beads each, 95% crosslinking) and ii) filled systems with the same elastomer network containing dispersed nanoparticle clusters, both prepared using Packmol (80) and a previously established equilibration MD workflow (30). Filler clusters were constructed from 3 to 11 primary particles and introduced at loadings of 75, 100, or 125 parts per hundred rubber (PHR), corresponding to volume fractions of 0.26 to 0.37. The strength parameter ${ϵ}_{\mathit{PF}}$ for the 12-6 Lennard-Jones (LJ) interactions between filler and particle beads was varied from 1 to 5 to probe the effects of interfacial dynamics (32, 46, 68, 81). All beads were modeled using the standard Kremer–Grest model, modified with crosslinking ends and breakable bonds (82–84). Simulations employed a time step of $1\times {10}^{-3}{\tau }_{\mathit{LJ}}$ (where ${\tau }_{\mathit{LJ}}$ is the LJ unit of time) and a constant engineering strain rate of $5\times {10}^{-5}/{\tau }_{\text{LJ}}$ [determined to be within the rubbery-plateau regime (28, 29)], with temperature ($T=1$ in LJ units) and the normal pressure ($P=0$) controlled via Nosé-Hoover thermostat and barostat, respectively. For each of the 15 distinct systems reported, SE were computed from an ensemble of 100 deformation trajectories per system, generated by thermally forking the equilibrated configuration and performing a brief equilibration prior to deformation. This yields, for each simulated condition, an ensemble of 100 simulations across which stress fluctuations are randomized and segmental positions are relaxed to provide better sampling. This produced a total of 1,500 deformation simulations (100 per system), corresponding to ∼15 y of aggregate compute time on our distributed resources. Periodic boundary conditions were enforced and the standard minimum image convention was employed in all distance computations.

Stress–strain response, Poisson’s ratio, bulk modulus, and polymer segmental relaxation times were computed to elucidate reinforcement mechanisms. Stress and modulus are defined in terms of engineering stress normalized by initial volume, with per-species and partial volumetric decompositions. Tangent moduli and Poisson’s ratio are computed from finite differences at 5.5% strain. Bulk modulus is computed via a fluctuation–dissipation theorem from a quiescent NPT simulation (85). Segmental relaxation times are obtained from self-intermediate scattering functions using a Kohlrausch–Williams–Watts fit (34, 45, 46, 68, 69, 86, 87). We used OVITO to visualize MD snapshots (88) and AMDAT to extract relaxation times from MD trajectories and to spatially decompose systems into interfacial, interfiller, and bulk regions (89, 90). Full details of the polymer and filler models, bonding potentials, annealing and crosslinking protocols, stress decomposition, and dynamic analysis are provided in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

3D trajectories, thermodynamic property CSV, and analysis codes, and figures have been deposited in USF Digital Commons (https://doi.org/10.17632/dg9jxp8sj6) (91).