Calculation of local mechanical properties of filled polymers

A study is presented on the effects of smooth nanoparticles on the structure and elastic moduli of a polymer matrix. Structural changes between the unfilled polymer matrix and the nanocomposite give rise to the formation of a glassy layer that surrounds the nanoparticles. Results for the effects of particle size and concentration on the local and overall mechanical properties of the polymer are consistent with experimental macroscopic observations. At the molecular level, it is found that dispersed, attractive nanoparticles alter the nonaffine displacement fields that arise in the polymer glass upon deformation, thereby rendering the nanocomposite glass less fragile.

Figure 1

Filler-monomer pair distribution function $g_{12}\left(r\right)$ for the various types of interactions considered in this work. All results are shown as a function of the distance $r-R_f$ from the surface of the nanoparticle at $T=0.6$, $P=0.3$, and $N=32$. The layers are well defined and the effect is more pronounced for attractive particles.

Figure 2

Normalized filler-chain end pair distribution function $g_{12}^{\text{end}}\left(r\right)∕g_{12}\left(r\right)$ for both types of interaction studied (neutral, attractive) at $T=0.6$, $P=0.3$, and $N=32$ as a function of the distance $r-R_f$ from the surface of the nanoparticle. Chain end segregation to the surface is apparent for neutral interactions $R_f=2.0\sigma$, while for attractive fillers of the same size the chain end concentration is significantly smaller close to the particle. For small attractive particles, $R_f=1.0\sigma$, similar behavior to the neutral interactions for filler size $R_f=2.0\sigma$ is observed.

Figure 3

Mean second Legendre polynomial $P_2$ between the vector from the center of the nanoparticle to the monomer $i$ and the vector between $i$ and the next monomer in the same chain $i+1$ at $T=0.6$, $P=0.3$ and $N=32$. $⟨P_2⟩$ is given as a function $r-R_f$ from the surface of the particle. The preferential alignment of the chains parallel to the surface of the particle is observed.

Figure 4

Surface excess as a function of particle size at $T=0.6$, $P=0.3$, and $N=32$. A transition from enthalpic wetting $({\Gamma }_s>0)$ to entropic dewetting $({\Gamma }_s<0)$ is apparent at approximate $R_f=0.75$.

Figure 5

Distribution of the local shear modulus with respect to the distance $r$ from the center of the nanoparticle for attractive particles for two polymer chain lengths $(N=32,120)$ at $T=0.2$ and $P=0.3$.

Figure 6

(a) Distribution of the local shear modulus with respect to the distance $r$ from the center of the nanoparticle for attractive particles of different size at $T=0.2$, $P=0.3$, and $N=32$. (b) Glassy layer thickness with respect to particle size at $T=0.2$, $P=0.3$, and $N=32$.

Figure 7

Effect of volume fraction of nanoparticles of $R_f=2.0\sigma$ on shear modulus at $T=0.2$, $P=0.3$, and $N=32$.

Figure 8

(Color online) 3D distribution of domains of high positive shear modulus for the nanocomposite of particles of size $R_f=2.0\sigma$ and the pure polymer glasses at $T=0.2$, $P=0.3$, $\varphi \sim 0.07$, and $N=32$.

Figure 9

Investigation of the effect of surface area and volume fraction of the nanoparticles on the (a) shear modulus and (b) Young’s modulus of nanocomposites. The squares represent constant surface area results using Eqs. (10, 12) for the calculation of $G$ and $E$ for each particle size. (Constant volume fraction $\varphi =0.024$ and constant surface area $a=0.011{\sigma }^{-1}$ at $T=0.2$, $P=0.3$, and $N=32$.)

Figure 10

Effect of particle size on the distribution of the local shear modulus $G$ with respect to particle size keeping the volume fraction constant $(\varphi =0.023)$ at $T=0.2$, $P=0.3$, and $N=32$.

Figure 11

(Color online) Stress strain curve for both pure polymer and nanocomposite system with a strain rate of $5\times {10}^{-4}$/step. The nanocomposite curve is at all times higher than the pure polymer indicating a stronger material. The plastic events (drops in the stress) occur for the nanocomposite in higher strains than pure for the unfilled polymer suggesting a tougher material. The difference in total energy is plotted with respect to the strain. It is clear that the energy strain curve is not as sensitive to the plastic events.

Figure 12

Typical snapshots of the nonaffine displacement field $u\left(r\right)$ for the unfilled polymer in layers of thickness $1\sigma$ obtained after applying a uniaxial deformation of strain ${10}^{-5}$.

Figure 13

Typical snapshots of the nonaffine displacement field $u\left(r\right)$ for the nanocomposite system of volume fraction $\varphi \sim 0.07$ and particle size $R_f=2\sigma$ in layers of thickness $1\sigma$ obtained after applying a uniaxial deformation of strain ${10}^{-5}$.

Figure 14

(a) 3D plot of the 10% higher nonaffine displacements for the unfilled polymer. (b) 3D plot of the 10% higher nonaffine displacements for the nanocomposite. Nonaffine displacements obtained for a uniaxial deformation of strain ${10}^{-5}$.

Figure 15

Pair distribution of the nonaffine displacements. We refer as max to the 10% highest non-affine displacements. From the distribution of the highest nonaffine displacements it is clear that these displacements exhibit a propensity to occur near each other forming clusters of nonaffinely displaced segments. For the nanocomposite system this effect is more pronounced. Nonaffine displacements obtained for a uniaxial deformation of strain ${10}^{-5}$.

Figure 16

Participation ratio $P_r$ as a function of applied strain (solid lines) and residual plastic displacement field $⟨\parallel v\parallel ⟩$ (dashed lines) obtained by reverse transformation back to the original macroscopic shape. Residual fields below ${10}^{-8}$ are due to numerical inaccuracies and the field is considered as reversible. For the nanocomposite system the particle volume fraction is $\varphi \sim 0.07$ and particle size $R_f=2\sigma$.

Figure 17

Correlation function $C\left(r\right)$ of the nonaffine displacement field as a function between pairs of distance $r$ for the unfilled polymer and the nanocomposite system of volume fraction $\varphi \sim 0.07$ and particle size $R_f=2\sigma$. For comparison of the peaks of the correlation functions, the polymer segment pair distribution function $g\left(r\right)$ is plotted and the polymer correlation function not counting contributions between nonaffine displacements of segments belonging in the same chain. Inset: Magnification of long-range results.

Figure 18

(a) Correlation function $C\left(r\right)$ of the nonaffine displacements as a function between pairs of distance $r$ starting from polymer beads lying in a shell of thickness $1.5\sigma$ around the particle with the rest of the system on the opposite side of the particle separated by an imaginary surface passing as shown in the figure. Also the correlation starting from the filler is plotted and for comparison the total nanocomposite correlation function counting all contributions is given. Nonaffine displacements obtained for a uniaxial deformation of strain ${10}^{-5}$. (b) Distribution of magnitude of residual plastic displacements around a nanoparticle. Residual plastic displacements obtained from reverse deformation that compresses the systems back to their initial shape upon a uniaxial deformation of strain ${10}^{-5}$.