Length-Scale Dependence of the Stokes-Einstein and Adam-Gibbs Relations in Model Glass Formers

Anshul D. S. Parmar, Shiladitya Sengupta, and Srikanth Sastry

Phys. Rev. Lett. 119, 056001 – Published 31 July 2017

Abstract

The Adam-Gibbs (AG) relation connects the dynamics of a glass-forming liquid to its thermodynamics via the configurational entropy and is of fundamental importance in descriptions of glassy behavior. The breakdown of the Stokes-Einstein relation (SEB) between the diffusion coefficient and the viscosity (or structural relaxation times) in glass formers raises the question as to which dynamical quantity the AG relation describes. By performing molecular dynamics simulations, we show that the AG relation is valid over the widest temperature range for the diffusion coefficient and not for the viscosity or relaxation times. Studying relaxation times defined at a given wavelength, we find that SEB and the deviation from the AG relation occur below a temperature at which the correlation length of dynamical heterogeneity equals the wavelength probed.

Received 6 July 2016

DOI:https://doi.org/10.1103/PhysRevLett.119.056001

Physics Subject Headings (PhySH)

Classical transport Glass transition

Glasses Glassy systems

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Anshul D. S. Parmar^1,2, Shiladitya Sengupta^1,3, and Srikanth Sastry^1,*

¹Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bengaluru 560064, India
²TIFR Center for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
³Department of Fundamental Engineering, Institute of Industrial Science, The University of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japan

^*sastry@jncasr.ac.in

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Vol. 119, Iss. 5 — 4 August 2017

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Figure 1
(a) Wave number ( $k$ ) dependence of the SE breakdown in the KA model. $k$ -dependent relaxation times are computed from $F_{s}^{A} (k, t)$ using the definition $F_{s}^{A} (k, τ^{'}) = 0.01$ (see Supplemental Material [44] for a comparison with the standard definition of $1 / e$ ). The SE breakdown using the shear viscosity is also shown. Lines are power law fits to $D_{A} = A X^{- ξ}$ , where $X$ is either $τ_{A}^{'}$ or $η / T$ . At high $k$ , high-temperature and low-temperature regimes can be clearly seen. The shear viscosity $η$ is coupled to $τ_{A}^{'} (k^{*})$ , i.e., $τ_{α, A}$ . (b) Testing the AG relation for $D_{A}$ , $η / T$ , and $τ_{A}^{'} (k)$ for different $k$ values for the KA model. The $τ_{A}^{'} (k)$ and $η / T$ have been scaled so that they coincide with $D_{A}^{- 1}$ at one high temperature ( $1 / T S_{c} \approx 1.49$ ). Lines are fits to the AG relation $X (T) = X_{o} \exp (A_{X} / T S_{c})$ , where $X = τ_{A}^{'} (k), η / T$ , or $D_{A}$ . The activation energy for $τ_{A}^{'} (k)$ and $η / T$ shows a change of slope as a consequence of the SEB [i.e., $A_{D} / A_{τ^{'} (k)} = ξ (k)$ ]. The AG relation holds for the $D_{A}$ in the considered temperature range, but there is a systematic deviation in $τ_{A}^{'} (k, T)$ from the AG relation due to the presence of SEB. However, the deviation observed in the KA model is not large enough to obtain a reliable length scale from the $k$ dependence of such a deviation.
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Figure 2
The inset shows $k$ dependence of the SE breakdown temperature $(T_{SEB})$ . See the text for the detailed procedure. The main panel shows a direct comparison of the length scale $l (T)$ obtained from the $k$ dependence of the SE breakdown temperature with the dynamic heterogeneity length scale $ξ_{4} (T)$ obtained from finite size scaling of Binder cumulant $T$ and shows that they are proportional in the KA model. DH length-scale data were obtained with permission from Ref. [22].
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Figure 3
(a) Wave number ( $k$ ) dependence of the Stokes-Einstein breakdown in the SqW model. $k^{*} = 2 π$ is the first peak of the static structure factor $S (k)$ for $A$ -type particles. As $k$ increases, gradually a fractional SE regime appears at low $T$ . The viscosity plotted against diffusivity shows that $τ_{α, A} \propto η / T$ is a good description of the data across the breakdown temperature. Lines are power law fits to $D_{A} = A X^{- ξ}$ , where $X$ is either $τ_{A}$ or $η / T$ . (b) Testing the AG relation for $D_{A}$ , $η / T$ , and relaxation times $τ (k)$ for different $k$ values for the SqW model. $τ_{A} (k)$ and $η / T$ have been scaled so that they coincide with $D_{A}^{- 1}$ at one high temperature ( $1 / T S_{c} \approx 0.65$ ). Lines are fits to the AG relation $X (T) = X_{o} \exp (A_{X} / T S_{c})$ , where $X = τ_{A} (k), η / T$ , or $D_{A}$ . The AG relation is valid for $D_{A}$ across the SEB temperature. The relaxation times and the viscosity show two regimes with different slopes for the activation energy $A_{τ, η / T}$ . In the presence of SEB, the ratio of activation energies for viscosity and $τ_{A} (k)$ to that of $D_{A}$ is given by the fractional SE exponent $ξ . T_{SEB} (k)$ is obtained as the temperature at which $τ_{A} (k)$ shows deviations from the AG relation. (c) Inset: $k$ dependence of the SEB exponent shows that the breakdown is very strong in the SqW model. Main: $k$ dependence of $T_{SEB}$ for the SqW model, obtained from the breakdown of both the SE and the AG relations. For every $T$ , there is a normal SE regime above a crossover length scale $l (T)$ and a breakdown of the SE relation below that length scale.
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Figure 4
(a) $k$ dependence of the four-point structure factor $S_{4} (k, τ_{4})$ for the SqW model. Lines are fits to Ornstein-Zernicke relation $\lim_{k \to 0} (1 / S_{4}) = A (1 + k^{2} ξ_{4}^{2})$ , from which the dynamical heterogeneity length scale $ξ_{4}$ is estimated. (b) Comparison of the DH length scale and the crossover length scale, showing that they are the same within numerical precision.
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