Direct link between boson-peak modes and dielectric $\alpha$-relaxation in glasses

We compute the dielectric response of glasses starting from a microscopic system-bath Hamiltonian of the Zwanzig-Caldeira-Leggett type and using an ansatz from kinetic theory for the memory function in the resulting generalized Langevin equation. The resulting framework requires the knowledge of the vibrational density of states (DOS) as input, which we take from numerical evaluation of a marginally stable harmonic disordered lattice, featuring a strong boson peak (excess of soft modes over Debye $\sim {\omega }_p^2$ law). The dielectric function calculated based on this ansatz is compared with experimental data for the paradigmatic case of glycerol at $T\lesssim T_g$. Good agreement is found for both the reactive (real) part of the response and for the $\alpha$-relaxation peak in the imaginary part, with a significant improvement over earlier theoretical approaches. On the low-frequency side of the $\alpha$ peak, the fitting supports the presence of $\sim {\omega }_p^4$ modes at vanishing eigenfrequency as recently shown [E. Lerner, G. During, and E. Bouchbinder, Phys. Rev. Lett. 117, 035501 (2016)]. $\alpha$-wing asymmetry and stretched-exponential behavior are recovered by our framework, which shows that these features are, to a large extent, caused by the soft boson-peak modes in the DOS.

Figure 1

Density of states (DOS) with respect to eigenfrequency ${\omega }_p$ at $Z=6.1$ (solid line), i.e., close to the marginal stability limit $Z=6$ that we identify here as the solid-liquid (glass) transition; plots of the DOS at $Z=7, Z=8$, and $Z=9$ are also shown and are marked as dashed, dot-dashed, and dotted lines, respectively.

Figure 2

The DOS normalized by Debye's ${\omega }_p^2$ law, for (from bottom to top) $Z=9, Z=8, Z=7$, and $Z=6$, which gives evidence of the boson peak at low ${\omega }_p$. The eigenfrequency of the boson peak scales as ${\omega }_p^{\text{BP}}\sim (Z-6)$, as known from work for disordered systems with cental-force interactions [18, 21, 37].

Figure 3

Real part of the dielectric function as a function of the frequency of the applied field. Symbols are experimental data of the real part of the dielectric function of glycerol at $T=184$ K from Ref. [39]. The solid line is our theoretical calculation for the Markovian-friction case, i.e., $\nu =\text{const}$ in Eq. (15). The dot-dashed line is the real part of the Fourier transform when we consider the best-fitting (empirical) stretched-exponential function with $\beta =0.65$. We have taken $C=10, \nu /m=1620$, and $A_1=0.039$. For the empirical fitting, $\beta =0.65$ and $\tau =6555$. Rescaling constants are used to adjust the height of the curves.

Figure 4

Dielectric loss modulus as a function of the frequency of the applied field. Symbols are experimental data of the imaginary part of the dielectric function of glycerol at $T=184$ K from Ref. [39]. The solid line is our calculation for the Markovian friction case, i.e., $\nu =\text{const}$ in Eq. (16). The dot-dashed line is the imaginary part of the Fourier transform when we consider the best-fitting (empirical) stretched-exponential (Kohlrausch) function with $\beta =0.65$. In our calculation, we have taken $C=10, \nu /m=1620$, and $A_2=0.0437$. For the empirical fitting, $\beta =0.65$ and $\tau =6555$. Rescaling constants are used to adjust the height of the curves.

Figure 5

Dielectric loss modulus as a function of the frequency of the applied field. Symbols are experimental data of the imaginary part of the dielectric function of glycerol at $T=184$ K from Ref. [39]. The solid line is the theoretical description presented in this work for the non-Markovian-friction case; i.e., $\tilde{\nu }\left(\omega \right)$ in Eq. (16) is the Fourier transform of Eq. (8). The dot-dashed line is the imaginary part of the Fourier transform when we consider the best-fitting (empirical) Kohlrausch relaxation function $\epsilon \left(t\right)\sim exp{(-t/\tau )}^{\beta }$ with $\beta =0.65$. In our calculation, we have taken $C=555, A_2=120,{\nu }_0=6\times {10}^6$, and $b=0.3$ (for the physical justification of the latter value, see Sec. 2c). For the empirical Kohlrausch fitting, $\beta =0.65$ and $\tau =6555$, as before. Rescaling constants are used to adjust the height of the curves.

Figure 6

Time-dependent dielectric response. The solid line is calculated using our Eq. (19) with physical parameters calibrated in the fitting of Fig. 3. The dashed line represents the stretched-exponential Kohlrausch function that closely approximates our prediction, calculated using the parameters $\beta =0.56$ and $\tau =5655$. Rescaling constants are used to adjust the height of the curves.