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Register a new userThe relation between stretched-exponential relaxation and the vibrational density of states in glassy disordered systems
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Amorphous solids or glasses are known to exhibit stretched-exponential decay over broad time intervals in several of their macroscopic observables: intermediate scattering function, dielectric relaxation modulus, time-elastic modulus etc. This behaviour is prominent especially near the glass transition. In this Letter we show, on the example of dielectric relaxation, that stretched-exponential relaxation is intimately related to the peculiar lattice dynamics of glasses. By reformulating the Lorentz model of dielectric matter in a more general form, we express the dielectric response as a function of the vibrational density of states (DOS) for a random assembly of spherical particles interacting harmonically with their nearest-neighbours. Surprisingly we find that near the glass transition for this system (which coincides with the Maxwell rigidity transition), the dielectric relaxation is perfectly consistent with stretched-exponential behaviour with Kohlrausch exponents $0.56 < beta < 0.65$, which is the range where exponents are measured in most experimental systems. Crucially, the root cause of stretched-exponential relaxation can be traced back to soft modes (boson-peak) in the DOS.
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We use a microscopically motivated Generalized Langevin Equation (GLE) approach to link the vibrational density of states (VDOS) to the dielectric response of orientational glasses (OGs). The dielectric function calculated based on the GLE is compared with experimental data for the paradigmatic case of two OGs: Freon 112 and Freon 113, around and just above $T_g$. The memory function is related to the integral of the VDOS times a spectral coupling function $gamma(omega_p)$, which tells the degree of dynamical coupling between molecular degrees of freedom at different eigenfrequencies. The comparative analysis of the two Freons reveals that the appearance of a secondary $beta$ relaxation in Freon 112 is due to cooperative dynamical coupling in the regime of mesoscopic motions caused by stronger anharmonicity (absent in Freon 113), and is associated with comparatively lower boson peak in the VDOS. The proposed framework brings together all the key aspects of glassy physics (VDOS with boson peak, dynamical heterogeneity, dissipation, anharmonicity) into a single model.
We propose an atomistic model for correlated particle dynamics in liquids and glasses predicting both slow stretched-exponential relaxation (SER) and fast compressed-exponential relaxation (CER). The model is based on the key concept of elastically interacting local relaxation events. SER is related to slowing down of dynamics of local relaxation events as a result of this interaction, whereas CER is related to the avalanche-like dynamics in the low-temperature glass state. The model predicts temperature dependence of SER and CER seen experimentally and recovers the simple, Debye, exponential decay at high temperature. Finally, we reproduce SER to CER crossover across the glass transition recently observed in metallic glasses.
We study the effect of rapid quench to zero temperature in a model with competing interactions, evolving through conserved spin dynamics. In a certain regime of model parameters, we find that the model belongs to the broader class of kinetically constrained models, however, the dynamics is different from that of a glass. The system shows stretched exponential relaxation with the unusual feature that the relaxation time diverges as a power of the system size. Explicitly, we find that the spatial correlation function decays as $exp(-2r/sqrt{L})$ as a function of spatial separation $r$ in a system with $L$ sites in steady state, while the temporal auto-correlation function follows $exp(-(t/tau_L)^{1/2})$, where $t$ is the time and $tau_L$ proportional to $L$. In the coarsening regime, after time $t_w$, there are two growing length scales, namely $mathcal{L}(t_w) sim t_w^{1/2}$ and $mathcal{R}(t_w) sim t_w^{1/4}$; the spatial correlation function decays as $exp(-r/ mathcal{R}(t_w))$. Interestingly, the stretched exponential form of the auto-correlation function of a single typical sample in steady state differs markedly from that averaged over an ensemble of initial conditions resulting from different quenches; the latter shows a slow power law decay at large times.
This paper is concerned with the connection between the properties of dielectric relaxation and ac (alternating-current) conduction in disordered dielectrics. The discussion is divided between the classical linear-response theory and a self-consistent dynamical modeling. The key issues are, stretched exponential character of dielectric relaxation, power-law power spectral density, and anomalous dependence of ac conduction coefficient on frequency. We propose a self-consistent model of dielectric relaxation, in which the relaxations are described by a stretched exponential decay function. Mathematically, our study refers to the expanding area of fractional calculus and we propose a systematic derivation of the fractional relaxation and fractional diffusion equations from the property of ac universality.
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, that we take from numerical evaluation of a marginally-stable harmonic disordered lattice, featuring a strong boson peak (excess of soft modes over Debye $simomega_{p}^{2}$ law). The dielectric function calculated based on this ansatz is compared with experimental data for the paradigmatic case of glycerol at $Tlesssim T_{g}$. 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, especially in the reactive modulus. 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 in [Phys. Rev. Lett. 117, 035501 (2016)]. $alpha$-wing asymmetry and stretched-exponential behaviour 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.
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