Diminishing Interfacial Effects with Decreasing Nanoparticle Size in Polymer-Nanoparticle Composites

Using molecular simulations on model polymer nanocomposites at fixed filler loading, we show that interfacial polymer dynamics are affected less with decreasing nanoparticle (NP) size. However, the glass transition temperature $T_g$ changes substantially more for an extremely small NP. The reason for this apparent contradiction is that the mean NP spacing decreases with decreasing particle size. Thus, all polymers are effectively interfacial for sufficiently small NPs, resulting in relatively large $T_g$ shifts, even though the interfacial effects are smaller. For larger NPs, interfacial relaxations are substantially slower than the matrix for favorable NP-polymer interactions. The minority “bound” polymer dynamically decouples from the polymer matrix, and we only find small changes in $T_g$ relative to that of the bulk polymer for large NPs. These results are used to organize a large body of relevant experimental data, and we propose an apparent universal dependence on the ratio of the face-to-face distance between the NPs and the chain radius of gyration.

Figure 1

(a) The monomer relaxation time as a function of distance from the NP surface for different NP sizes determined by fitting $F_{\mathrm{self}}(q,t,r)$ using Eq. (2). For a weak NP interaction strength $\epsilon =0.25$ (black lines), the monomer relaxation is enhanced approaching the NP surface. For a strong NP interaction strength $\epsilon =2.0$ (red lines), the monomer relaxation is significantly slowed approaching the NP surface. (b) The relaxation time nearest to the NP surface as a function of interaction strength for different NP sizes. The data show that the effect of interactions on surface relaxation diminishes as NP size decreases.

Figure 2

(a) The intermediate scattering function for systems with different NP sizes at an identical NP concentration ($\varphi =0.042$). The primary and bound relaxation times show opposite dependences on particle size; specifically, the bound layer relaxation time decreases as the NP size decreases. For the smallest NP size ($d=3.3\sigma$), this bound layer does not affect the $T_g$. (b) $F_{\mathrm{self}}(q,t)$ for particle sizes $d=6.6\sigma$ and $d=3.3\sigma$ and NP concentration $\varphi =0.042$ [same data as (a), but with a linear scale on the ordinal axis]. As a result of interfacial layer overlap, the primary relaxation time of the $d=3.3\sigma$ system is significantly larger than that of the $d=6.6\sigma$ system. (c) $F_{\mathrm{self}}(q,t)$ of the pure polymer and the composites at small NP concentration; for small $\varphi$, the interfacial layers do not overlap for any case so that the effect of NP size on the relaxation time is less significant.

Figure 3

(a) The glass transition temperature $T_g$ for the overall composite for all NP sizes with identical NP concentration. The $T_g$ changes are larger for smaller NPs. For the smallest NP ($d=3.3$), $T_g$ increases roughly linearly with $ϵ$, qualitatively different from that for the larger NP; this is due to the overlap of the interfacial zones for the smallest NP. (b) $T_g$ for the polymer matrix for different NP sizes with identical NP concentration; data for the NP size $d=3.3$ are not included, since all polymers are interfacial for this case. The smaller NP has a larger effect on matrix $T_g$.

Figure 4

Schematics of the different regimes for interfacial effects of the NP. (a) The dilute regime, where the dynamics of interfacial polymer decouples from that of the polymer matrix when polymer-NP interactions exceed those of the matrix, leading to very little $T_g$ change. (b) The “bridging regime,” where the separation of the NPs allows the interfacial chains to bridge. The bridging of the interfacial polymers leads to a higher change in $T_g$ than in the dilute regime. (c) The interfacially dominated regime, where the concentration of NPs is high enough that interfacial zones overlap. In this regime, the $T_g$ has the largest change compared to the pure melt. The value of $r/R_g$ for $d=3.3\sigma$, $d=6.6\sigma$, and $d=10.0\sigma$ is $r/R_g=3.3/2.16=1.52$, $r/R_g=6.6/2.16=3.05$, and $r/R_g=10.0/2.16=4.63$, respectively. (d) The normalized change of $T_g$ as a function of $r/R_g$. The systems shown are composites of PMMA-silica (black circles), poly-2-vinyl pyridine (P2VP)-silica (red squares), OPAS-silica (green diamonds), and P2VP-alumina (blue triangles). These data are gathered from Refs. [16, 18, 19, 22, 29, 30]. To estimate the mean face-to-face separation $r$ between NPs, we use the expression $r/d=({\varphi }_{\max }/\varphi {)}^{1/3}-1$ [31], where ${\varphi }_{\max }$ is the maximum filling fraction (${\varphi }_{\max }\approx 0.7$), $d$ is the NP diameter, and $\varphi$ is the NP filling fraction.