Shear Modulus and Shear-Stress Fluctuations in Polymer Glasses

Using molecular dynamics simulation of a standard coarse-grained polymer glass model, we investigate by means of the stress-fluctuation formalism the shear modulus $\mu$ as a function of temperature $T$ and sampling time $\mathrm{\Delta }t$. While the ensemble-averaged modulus $\mu (T)$ is found to decrease continuously for all $\mathrm{\Delta }t$ sampled, its standard deviation $\delta \mu (T)$ is nonmonotonic, with a striking peak at the glass transition. Confirming the effective time-translational invariance of our systems, $\mu (\mathrm{\Delta }t)$ can be understood using a weighted integral over the shear-stress relaxation modulus $G(t)$. While the crossover of $\mu (T)$ gets sharper with an increasing $\mathrm{\Delta }t$, the peak of $\delta \mu (T)$ becomes more singular. It is thus elusive to predict the modulus of a single configuration at the glass transition.

Figure 1

Key findings as a function of temperature $T$. The vertical dashed lines indicate the glass transition temperature $T_g\approx 0.38$. (Main panel) Shear modulus $\mu (T)$ for different sampling times $\mathrm{\Delta }t$ showing that the transition becomes more and more steplike with an increasing $\mathrm{\Delta }t$. (Inset) Corresponding standard deviation $\delta \mu (T)$ showing a peak at $T\approx T_g$ which becomes sharper with an increasing $\mathrm{\Delta }t$. Also included are the shear-stress relaxation modulus $G(t)$ and its standard deviation $\delta G(t)$ taken at a time $t={10}^4$ (the bold solid lines).

Figure 2

First moments ${\mu }_A$, ${\mu }_0$, ${\mu }_1$, ${\mu }_F$, and $\mu$ vs temperature $T$ using a half-logarithmic representation. The data are obtained for $\mathrm{\Delta }t=\mathrm{\Delta }{t}_{\max }={10}^5$. With decreasing temperature, $\mu \approx {\mu }_1$ increases rapidly at $T_g$, but it remains continuous. Interestingly, ${\mu }_0$ deviates from ${\mu }_A$ and ${\mu }_1$ from $\mu$ below $T_g$.

Figure 3

Standard deviations $\delta {\mu }_A$, $\delta {\mu }_0$, $\delta {\mu }_1$, $\delta {\mu }_F$, and $\delta \mu$ as a function of $T.\delta {\mu }_A$ is found to be small, and $\delta \mu \approx \delta {\mu }_F$ for all $T$’s. Below $T_g$, the standard deviations $\delta {\mu }_1$ and $\delta {\mu }_0$ become rapidly similar and orders of magnitude larger than $\delta {\mu }_F$. This confirms the presence of strong frozen shear stresses.

Figure 4

Shear modulus $\mu$ as a function of sampling time $\mathrm{\Delta }t$ for a broad range of $T$’s, as indicated in the figure. $\mu (\mathrm{\Delta }t)$ decreases continuously with $\mathrm{\Delta }t$. Note that a smaller temperature increment $\mathrm{\Delta }T=0.01$ is used around $T_g$ (the solid lines), where $\mu (\mathrm{\Delta }t;T)$ changes much more rapidly with $T$. The dashed-dotted lines are obtained using Eq. (4) by integrating the directly measured shear-stress relaxation modulus $G(t)$. The vertical lines mark the sampling times used in Fig. 1.

Figure 5

Stress relaxation modulus $G(t)$ for a broad range of $T$’s. The data are rather similar to the shear modulus $\mu (\mathrm{\Delta }t)$ presented in Fig. 4. The dashed vertical line marks the time $t={10}^4$ used for $G(t)$ and $\delta G(t)$ in Fig. 1.

Figure 6

(Main panel) Distribution $p(\overline{\mu })$ for $\mathrm{\Delta }t={10}^5$ for different $T$’s. (Inset) Difference ${\mu }_{\text{med}}-\mu$ of the median ${\mu }_{\text{med}}$ and the ensemble average $\mu$ vs $T$ for two sampling times. The difference has a peak slightly below $T_g$ corresponding to very lopsided distributions.