Elasticity and plasticity in stiff and flexible oligomeric glasses

In this paper we focus on the mechanical properties of oligomeric glasses (waxes), employing a microscopic model that provides, via numerical simulations, information about the shear modulus of such materials, the failure mechanism via plastic instabilities, and the geometric responses of the oligomers themselves to a mechanical load. We present a microscopic theory that explains the numerically observed phenomena, including an exact theory of the shear modulus and of the plastic instabilities, both local and system spanning. In addition we present a model to explain the geometric changes in the oligomeric chains under increasing strains.

Figure 1

The intermediate scattering function vs time (for various temperatures) and the relaxation time vs inverse temperature. Upper panel: $\kappa =0$; lower panel: $\kappa =2$.

Figure 2

A typical stress vs strain curve obtained by AQS straining of 256 polymers of chain length 20 with stiffness parameter $\kappa =2$. Smooth (linear) increases in the strain are punctuated with sharp drops. The trajectory of stress vs strain is reversible only until the first drop. The sharp drops are plastic events as explained in the text. Inset: Blow-up of the first few plastic drops.

Figure 3

Log-log plots of the average magnitude of the stress drops $\Delta \sigma$ and energy drops $\Delta U$ in the steady state (after the yield strain had been passed) as a function of the number of particles in the system.

Figure 4

Raw and rescaled pdf's for the energy drops in the steady state. The data collapse in the tails of the distributions supports the scaling laws presented in Eqs. (9) and (10).

Figure 5

Upper panel: Stress vs strain for different $\kappa$ for the stiff polymer $\left(\alpha =1\right)$. The stress reaches to the same steady state for the finite $\kappa$. Bottom panel: The variation of total internal potential energy with strain. The data averaged over 40 independent realizations are shown.

Figure 6

Variation of the three individual contributions to the total potential energy arising from different interactions of each with strain. Upper panel: $U_{\mathrm{LJ}}$ vs strain for different $\kappa$. The potential energy due to the LJ interaction increases on increase of $\kappa$. Middle panel: $U_{\mathrm{FENE}}$ vs strain for different $\kappa$. Bottom panel: $U_{\mathrm{angle}}$ vs strain. The potential energy $U_{\mathrm{angle}}$ decreases on increase of the stiffness of the chain.

Figure 7

Variation of end to end length $R_{\mathit{ee}}$ of the stiff polymers with the applied strain $\gamma$ as a function of $\kappa$.

Figure 8

Snapshot of the polymer in the box for $\kappa =0$ (upper panel) and $\kappa =10$ (lower panel). From left to right: (a) $\gamma =0$, (b) $\gamma =1$, and (c) $\gamma =5$. For $\kappa =0$ the polymers are coiled for zero strain, and they stretch on average upon increasing of the strain. The opposite occurs for $\kappa =10$.

Figure 9

The average angular measure $\langle cos\theta \rangle$ (upper panel), the average equilibrium end-to-end length $R_{ee}\left(0\right)$ (middle panel,) and the average persistence length $\left({\ell }_p\right)$ (lower panel) as a function of $\kappa$ at zero strain $\left(\gamma =0\right)$. The red dots with error bars are the data obtained from numerical simulations, and black curves are the theoretical estimates obtained using the theory discussed in Sec. 3d.

Figure 10

The variation of the smallest eigenvalue ${\lambda }_P$ as $\gamma$ is increased, In the upper panel we see the eigenvalue dips to zero, then recovers after the instability is over, and again dips to zero at the next instability. In the lower panel we choose to blow up the region of the first instability to demonstrate the approach of the eigenvalue to zero with a square-root singularity [Eq. (23)].

Figure 11

The eigenvector ${\Psi }_P$ and the nonaffine displacement field associated with the first plastic instability as $\lambda \to 0$. The length units are normalized to the full length of the box.

Figure 12

Log-log plot of the eigen value of the plastic mode ${\lambda }_P$ vs ${\gamma }_P-\gamma$ for $\kappa =2$ (top panel) and for $\kappa$=8 (bottom panel) near the first elementary plastic event for semiflexible $\left(\alpha =2\right)$ polymer. The exponent is approximately $0.5$.

Figure 13

The eigenfunction ${\Psi }_P$ and the nonaffine displacement field associated with a shear localizing plastic instability as $\lambda \to 0$. Note the global connection between the series of quadrupoles arranged along the line, such that the displacement field is pointing right above and left below the line. This IS the phenomenon of shear localization. The length units are normalized to the box size.

Figure 14

Comparison of the theoretically calculated and the numerically estimated shear modulus for various values of $\kappa$. The relatively large error bars stem from the relative smallness of the system, in which different realizations give a spread of values of the shear modulus. Nevertheless the agreement between theory and simulations is quite satisfactory.