Atomic theory of viscoelastic response and memory effects in metallic glasses

An atomic-scale theory of the viscoelastic response of metallic glasses is derived from first principles, using a Zwanzig-Caldeira-Leggett system-bath Hamiltonian as a starting point within the framework of nonaffine linear response to mechanical deformation. This approach provides a generalized Langevin equation (GLE) as the average equation of motion for an atom or ion in the material, from which non-Markovian nonaffine viscoelastic moduli are extracted. These can be evaluated using the vibrational density of states (DOS) as input, where the boson peak plays a prominent role in the mechanics. To compare with experimental data for binary ZrCu alloys, a numerical DOS was obtained from simulations of this system, which also take electronic degrees of freedom into account via the embedded-atom method for the interatomic potential. It is shown that the viscoelastic $\alpha$-relaxation, including the $\alpha$-wing asymmetry in the loss modulus, can be very well described by the theory if the memory kernel (the non-Markovian friction) in the GLE is taken to be a stretched-exponential decaying function of time. This finding directly implies strong memory effects in the atomic-scale dynamics and suggests that the $\alpha$-relaxation time is related to the characteristic time scale over which atoms retain memory of their previous collision history. This memory time grows dramatically below the glass transition.

Figure 1

Kohlrausch empirical fits (solid lines) of experimental data (symbols). From top to bottom the lines and symbols correspond to temperatures in the following order: 536, 603, 670 K $\left(T_g\right)$. Solid curves are Kohlrausch $\sigma \left(t\right)\sim exp-{(t/\tau )}^{\beta }$ empirical fittings used to calibrate results, where the parameters $\beta$ and $\tau$ were chosen to be 0.69 and 0.87 min, 0.55 and 4.03 min, and 0.55 and 14.87 min for $T_g,0.9T_g$, and $0.8T_g$ respectively.

Figure 2

Vibrational density of states (DOS) from a simulated ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ system. Solid, dashed, and dotted lines correspond to DOS at a 670, 603, and 536 K, respectively. The curves have been shifted upward in order to be distinguishable for the reader. The inset shows the DOS normalized by the Debye law ${\omega }_p^2$, which shows clear evidence of a strong boson peak.

Figure 3

Real part of the complex viscoelastic modulus. From right to left solid lines represent $E^{\prime }$ for $T_g,0.9T_g$, and $0.8T_g$ respectively, from the Kohlrausch best fitting of our experimental data. Symbols are calculated based on our theory. For $T_g,0.9T_g$, and $0.8T_g,b$ was chosen to be 0.72, 0.58, and 0.58; $r$ was taken to be $1.2\times {10}^{-6},7\times {10}^{-6}$, and $3.4\times {10}^{-6}$. $\nu \left(0\right)=0.137$ is the same for all temperatures. Rescaling constants have been taken to adjust the height.

Figure 4

Imaginary part of the complex viscoelastic modulus. From right to left solid lines represent $E^{\prime \prime }$ for $T_g,0.9T_g$, and $0.8T_g$, respectively, from empirical Kohlrausch fittings of the experimental data. Symbols are calculated from our theory. For $T_g,0.9T_g$, and $0.8T_g,b$ in the memory kernel of our theory was chosen to be 0.72, 0.58, and 0.58; $r$ was taken to be $1.2\times {10}^{-6},7\times {10}^{-6}$, and $3.4\times {10}^{-6}$. $\nu \left(0\right)=0.137$ is the same for all temperatures. Rescaling constants have been taken to adjust the height.