Length scale dependence of the Stokes-Einstein and Adam-Gibbs relations in model glass formers

TL;DR

This study investigates how the validity of the Adam-Gibbs and Stokes-Einstein relations depends on length scales in glass-forming liquids, revealing that the Adam-Gibbs relation holds best for diffusion over a wide temperature range.

Contribution

The paper demonstrates that the Adam-Gibbs relation is valid for diffusion across a broad temperature range and links the breakdown of the Stokes-Einstein relation to length scale-dependent dynamical heterogeneity.

Findings

Adam-Gibbs relation holds for diffusion over wide temperature range

Stokes-Einstein breakdown occurs below a certain temperature

Breakdown linked to dynamical heterogeneity length scales

Abstract

The Adam-Gibbs (AG) relation connects the dynamics of a glass-forming liquid to its the thermodynamics via. the configurational entropy, and is of fundamental importance in descriptions of glassy behaviour. The breakdown of the Stokes-Einstein (SEB) relation 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.

Equations

$${\eta }_{\alpha \beta }=\frac{V}{k_{B}T}{\int }_0^{\infty }dt⟨J_{\alpha \beta }(t)J_{\alpha \beta }(0)⟩.$$

$$J_{\alpha \beta }(t)=\frac{1}{V}\left(\sum _i^N\frac{p_{i\alpha }{p}_{i\beta }}{m}+\sum _i^N\sum _{j>i}^N{r}_{ij\alpha }{f}_{ij\beta }\right)$$

$${\eta }_{\alpha \beta }=\frac{1}{V{k}_{B}T}\underset{t\to \infty }{\lim }\frac{⟨(A_{\alpha \beta }(t)-A_{\alpha \beta }(0){)}^2⟩}{2t},$$

$$\frac{d{A}_{\alpha \beta }(t)}{dt}=J_{\alpha \beta }V,$$

$$A_{\alpha \beta }(t)-A_{\alpha \beta }(0)=V{\int }_0^{t}d{t}^{\prime }{J}_{\alpha \beta }(t^{\prime })$$

$$\Delta A_{\alpha \beta }(t)=m\sum _{\text{coll, s}}^t\Delta$$

$$G_s^g(r,t)={\left(\frac{3}{2\pi <r^2(t)>}\right)}^{\frac{3}{2}}exp\left(-\frac{3r^2}{2<r^2(t)>}\right)$$

Full text

The Adam-Gibbs (AG) relation connects the dynamics of a glass-forming liquid to its the thermodynamics via. the configurational entropy, and is of fundamental importance in descriptions of glassy behaviour. The breakdown of the Stokes-Einstein (SEB) relation 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.

Length scale dependence of the Stokes-Einstein and Adam-Gibbs relations in model glass formers

Anshul D. S. Parmar

Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur Campus, Bengaluru 560064, India

TIFR Center for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India

  

Shiladitya Sengupta

Dept. of Fundamental Engineering, Institute of Industrial Science, The University of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japan.

  

Srikanth Sastry

Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur Campus, Bengaluru 560064, India

It is now clear from extensive research over the last two decades that, as a liquid is gradually (super)cooled, its dynamics becomes spatially heterogeneous (“dynamical heterogeneity”) Silescu ; EdRev ; DHbook and the collective nature of the underlying relaxation processes can be quantified by various growing correlation length scales KDS-rev ; KDS-rev2 . Dynamics can be described by different measures - the translational diffusion coefficient ($D$), the shear viscosity ($\eta$) or the $\alpha$-relaxation time ($\tau_{\alpha}$). At high temperatures, where particle motions are diffusive, all these time scales are mutually coupled. $D$ and $\eta$ are related via the Stokes-Einstein (SE) relation HM ; LL : $D={mk_{B}\over c\pi}{T\over R\eta}$, ($m,R$ are respectively mass and radius of a diffusing particle, $T$ is the temperature of the liquid and the constant $c$ depends on stick or slip boundary condition). At high temperatures, owing to exponential decay of self-intermediate scattering function $F_{s}(k,t)=\exp(-Dk^{2}t)$ at all probe wave vectors $k$, $D$ also gets coupled to the relaxation time $\tau(k)$ measured from $F_{s}(k,t)$ : $Dk^{2}\tau(k)=constant$. Further, $\tau_{\alpha}$ is often used as a proxy for $\eta$ (or $\eta/T$) in the SE relation to save computational cost. At low temperatures in dense, viscous liquids, $D$ becomes much bigger than the value estimated from $\tau_{\alpha}$ ($\eta$) using the SE relation. This phenomenon is known as the breakdown of the SE relation (SEB) EdRev ; EdigerGr ; Kumar-etal ; Becker-etal ; Sengupta2013 ; SEBKA-4 ; SEB-Hopping . The regime showing SEB is often found to obey a fractional SE relation: $D\propto\tau_{\alpha}^{-\xi}$, where the SE exponent $\xi\in[0,1]$ measures the extent of SEB.

The question naturally arises as to what causes the SE breakdown. A commonly held point of view in the literature EdRev ; Silescu ; LL ; EdigerGr interprets the SEB as a consequence of the dynamical heterogeneity (DH) developing in the liquid upon cooling. DH simply means that there are populations of slow and fast particles which form transient clusters, making the dynamics spatially heterogeneous. The existence of DH leads to the expectation of a distribution of diffusion coefficients and relaxation times SEBKA-4 ; SEBKA-5 corresponding to populations of different mobility. The observed $D$ is dominated by the fast population, while the observed $\tau$ (or $\eta$) is governed mainly by the slow population, leading to the decoupling and the SEB. The observed decoupling (SEB) is an average effect in such a picture, which however, does not clarify the role played by a heterogeneity length scale.

The breakdown of the SE relation poses a puzzle with regard to the celebrated Adam-Gibbs (AG) relation, which describes AG ; AG-expt1 ; AG-expt2 the viscous slowdown upon cooling in terms of a parallel decrease in the configurational entropy ($S_{c}$) ScKA ; SWB of the liquid. Its rationalization is considered fundamental to understanding the glass transition problem AG-water ; AG-fragility ; KDS-PNAS ; AG-DH ; GET ; Freed ; AG-S2 ; RFOT-Rev-BB ; RFOT-WolynesGr ; RFOT-Rev-KT ; RFOT-fragility . The AG relation can be written as $X(T)=X_{0}\exp\left({A_{X}\over TS_{c}(T)}\right)$, where $X$ is an appropriate measure of dynamics i.e. $D,\eta$ or $\tau_{\alpha}$. It has been tested in many different systems and has proven to be an extremely useful, predictive relationship between dynamics and thermodynamics below the onset temperature AG-water ; AG-fragility ; KDS-PNAS ; AG-DH ; AG-S2 , despite ambiguity about the range of temperatures over which one should expect it to hold (SEB-Hopping ). The breakdown of the SE relation raises the question of which dynamical quantity should obey the Adam-Gibbs relation. The $k$ dependence of the breakdown of the SE relation implies a $k$ dependence of the AG relation itself which, to our knowledge, has not been tested before.

Here we study two different glass-formers - (a) the well-known Kob-Andersen (KA) binary mixture KAref and (b) the square well (SqW) model SqWdet ; SPZ ; ST ; SWB by performing NVT-MD simulations. For the KA model, we perform simulations for system size $N=1000$, number density $\rho=1.20$ and temperature range $T\in[0.46,5.00]$. Temperature is kept constant using the Brown and Clarke algorithm BC . To simulate the SqW model we carry out the event driven NVT-MD simulations. The SqW model is a 50:50 binary mixture of particles having hard core repulsion at contact as well as an attractive interaction. The ratio of diameters of the two types of particles is $1.20$, with $\sigma_{AA}=1.20\sigma_{BB}$. Further, for the AB interaction, the hard core diameter is additive, i.e. $\sigma_{AB}=(\sigma_{AA}+\sigma_{BB})/2$. The width of the attractive shell ($\Delta_{ij}$) is defined such as $\Delta_{ij}/(\sigma_{ij}+\Delta_{ij})=0.03$. System size is $N=5324$ particles, number density $\rho$= 0.77, and temperature range $T\in[0.31,0.7]$. Temperature is kept constant using the Lowe-Anderson thermostat Thermo1 ; Thermo2 . At each state point, runs are executed for duration $>100\tau_{\alpha}$, where $\tau_{\alpha}$ is the $\alpha-$ relaxation time. A length scale dependent relaxation time $\tau_{A}(k)$ is computed from the self part of the intermediate scattering function for $A$ type particles, (F^{A}_{s}(k,t)$$\equiv{1\over N_{A}}\sum_{i=1}^{N_{A}}\exp(i{\bf k}.({\bf r}_{i}(0)-{\bf r}_{i}(t)) where ${\bf{r}}_{i}(t)$ is the position of particle $i$). Configurational entropy is calculated by subtracting the vibrational component from the total entropy of the system, $S_{c}(T)=S_{total}(T)-S_{vib}(T)$ following ScKA ; SWB . For each model, we compute the $k$ dependence of the SE relation and its breakdown and extract a temperature dependent length scale from the said $k$ dependences. We show that this length scale compares very well with a dynamical heterogeneity length scale estimated independently using standard definitions. We also show that the breakdown of the SE relation corresponds to a deviation from the AG relation for the relaxation times at appropriate wavenumbers $k$. For diffusion coefficients, the AG relation holds for the entire temperature range but for the relaxation times and viscosity there is a deviation from the AG behaviour at low temperatures in a manner dictated by the SE breakdown.

The SEB in the KA model is well documented SEBKA-4 ; SEBKA-5 ; SEBKA-1 ; SEBKA-2 ; SEBKA-3 ; SEBKA-6 ; SEBKA-7 . In the KA model, for $k=k^{*}(\sim 7.25)$ (first peak of the static structure factor $S(k)$), the SE relation breaks down close to the onset temperature of slow dynamics. A fractional SE relation $D_{A}\propto\tau_{A}^{-\xi}$ is followed in the low temperature regime. Fig. 1A shows the diffusion coefficient (of species A) vs. the relaxation time data for different $k$ values. We have considered the conventional definition $F^{A}_{s}(k,\tau_{A})=1/e$ as well as a longer timescale $F^{A}_{s}(k,\tau_{A}^{\prime})=0.01$ Chong as measures of relaxation time and for subsequent analyses in the KA model, consider only the longer timescales $\tau_{A}^{\prime}$ (see Supplementary Material (SM)). We also measure the shear viscosity Leporini2001 ($\eta$) and find that $\eta/T\propto$ $\tau_{\alpha,A}$Sengupta2013 , i.e. the viscosity decouples from diffusivity. At low $k$ we see no SEB. At high $k$, data show two distinct power law regimes (fit lines) at high and at low $T$. To estimate the temperature of the SE breakdown $T_{SEB}(k)$, we study the product $D_{A}\tau_{A}^{\prime}(k)$ scaled by the corresponding value at a reference high T vs. the relaxation time for different $k$, deviation from $1$ will indicate SEB (see SM).

In Fig. 1B, we plot $D_{A}$, $\eta/T$ and $\tau_{A}^{\prime}(k)$ (shifted to coincide all data sets at a chosen temperature) as a function of $(TS_{c})^{-1}$. For diffusion coefficient and relaxation time for $k\rightarrow 0$ the AG relation is valid for the entire temperature range studied. However, as k increases, a change of slope SEBKA-3 in the data can be seen. The viscosity which is decoupled from $D_{A}$ but remains coupled to $\tau_{\alpha,A}$, also show similar behaviour. We consider this as a manifestation of the k-dependent SEB. However, the observed change in slope in the KA model is gradual, making it difficult to extract a reliable temperature of deviation from AG behaviour. We discuss the deviation from the AG relation in more detail in the context of the SqW model.

The SE breakdown temperature $T_{SEB}$ estimated from deviation from SE relation shows a systematic increase with increasing $k$ (inset Fig. 2). Thus the SE breakdown follows a scenario where both the breakdown exponent and the breakdown temperature are $k$ dependent. The dependence of $T_{SEB}$ on the probe wave number $k$ can be inverted to obtain a temperature dependent length scale $l(T)$, which we compare with an independently evaluated heterogeneity length. At any given temperature, particle displacements probed above this length scale will show diffusion coefficients and relaxation times to be coupled while probing below this length scale will reveal decoupling between timescales. Thus this length is similar to the non-Fickian to Fickian crossover lengthscale Chong ; Chong-Kob ; BCG ; Berthier , which however has been claimed to be distinct BCG from a heterogeneity length. In Fig. 2, we show the temperature dependences of the length scale of SEB and the length scale of dynamical heterogeneity $\xi_{4}(T)$ as computed in Ref. KDS-PNAS from the finite size scaling analysis of Binder cumulant. They are directly proportional to each other. This provides a new demonstration that the breakdown of the SE relation is strongly related to the emergence of dynamical heterogeneity with cooling.

Next, we study another glass-former, viz. the SqW model. As in the case of the KA model, we show diffusion coefficient and relaxation times for $A$ type particles. Fig. 3A, shows $D_{A}$ vs. $\tau_{A}(k)$ (In the SqW model, we measure $\tau_{A}(k)$ using $F^{A}_{s}(k,t)=1/e$). Like the KA model, the SqW model shows no SEB for $k\rightarrow 0$. At large $k$, there is a high T regime where the SE relation holds and a low T regime obeying a fractional SE relation. However, the fractional exponent reaches much lower values, implying that the SEB in the SqW model is stronger than in the KA model. We also measure shear viscosity from Helfand moments Alder ; haile ; HM (see SM) and find that, like the KA model, the SqW model shows $\eta/T\propto\tau_{\alpha},A$, and viscosity decouples from the diffusivity. Note that in the SqW model, the configurational entropy can be reliably estimated ScKA ; SWB in the temperature range where the SEB occurs (for all the $k$). Consequently, in the SqW model, the AG relation can be tested over a temperature range which shows both the normal SE relation as well as a $k$ dependent breakdown, making it possible to address for which quantity the AG relation is valid. If the AG relation is valid for $D_{A}$, $\tau_{A}(k)$ and $\eta/T$ then these timescales should overlap up to a constant in the normal SE regime or show deviations in the SEB or fractional SE regime, when plotted against $(TS_{c})^{-1}$. The change of slope between $D_{A}$ $vs$ $\tau_{A}(k)$ and $\eta/T$ should occur at the breakdown temperature $T_{SEB}$. In Fig. 3B, we show the AG plot for $D_{A}$, $\tau_{A}(k)$ and $\eta/T$ (shifted to match all data sets at a selected high temperature) vs. $(TS_{c})^{-1}$. For $D_{A}$, the AG relation is valid in the full range shown. For relaxation times $\tau_{A}(k)$ (and viscosity), two distinct regimes are obtained with a crossover at a $k$ dependent temperature.

The AG theory proposed a mechanism for structural relaxation via the concept of CRRs. Recent works have proposed cooperatively moving highly mobile particles, called “strings”, as candidates for CRR AG-DH ; Freed ; string2 ; string3 , in that the string length is found to be inversely proportional to the configurational entropy as envisaged in the AG theory. The string life time is found to be proportional to $t^{*}$, the time when the non-Gaussian parameter $\alpha_{2}$ is maximum AG-DH . In turn, $t^{*}$ (for type $A$) is proportional to $D_{A}^{-1}$, and decouples from $\tau_{\alpha,A}$ or $\eta$ in the presence of SEB (see SM). Another strong evidence comes from the observation that diffusion coefficients for finite segments of a system’s trajectory are related to the configurational entropy estimated for the same segment Nave2006 . These results strongly suggest that the time scale described by the AG relation is proportional to $D_{A}$ and not to $\tau_{\alpha,A}$ (or $\eta$). Theoretical investigations along the lines of AG-DH ; GET ; Freed ; Nave2006 therefore offer a promising way of investigating further and rationalising our results.

At high $k$, as the temperature decreases, there is a marked change of slope at the crossover temperature, which decreases as $k\rightarrow 0$. The $k$ dependence of this temperature ($T_{SEB}$) is shown in Fig. 3C. For comparison, we also show the $T_{SEB}$ obtained directly (from the temperature dependence of the product $D_{A}\tau_{A}(k)$ scaled by the corresponding value at a reference high temperature). There is a reasonably good agreement between the two estimates, implying that the breakdown of the AG relation for relaxation and viscosity is a manifestation of the SEB. We compute the length scale of dynamical heterogeneity $\xi_{4}(T)$ KDS-PNAS from the four-point dynamic structure factor $S_{4}(k,\tau_{4})$ ($\tau_{4}$ is the time at which $S_{4}(k=0,t)$ is maximum) (Fig. 4A) and in Fig. 4B, compare it to the length scale of SEB $l(T)$ obtained from the $k$ dependence of $T_{SEB}$. The data show that they are proportional. The strong SEB and large length scales suggest enhanced DH in the SqW model.The SqW model was originally a model for attractive colloids, and systematic studies suggest that the DH is more pronounced in attractive glass-formers JCP07 ; DHexpt ; DHsimu . At low temperatures and at the characteristic time scale, the peak values of dynamical susceptibility $\chi_{4}(t)$ and the non-Gaussian parameter $\alpha^{A}_{2}(t)$ are bigger in the SqW model, showing that the DH is stronger in the SqW model than in the KA model (see SM). Considering $A$ and $B$ particle diffusion coefficients SEBKA-6 , we find that they obey a fractional power law relationship over a wide temperature range, and thus the Adam-Gibbs relation hold for both consistently Sengupta2013 (see SM for details).

In summary, we have studied the wave number $k$ dependent relaxation times in two model liquids. We have shown that the breakdown of the SE relation occurs at $k$ dependent temperatures $T_{SEB}(k)$ that decrease with decreasing $k$. This allows us to identify a SEB length scale based on $T_{SEB}(k)$. We compare the length scale with an independently estimated dynamical heterogeneity length and demonstrate that they are the same up to an undetermined multiplicative constant. The fractional SE exponent varies continuously from a value close to $1$ at the lowest $k$ to values as low as $0.51$ for one of the models (SqW). For diffusion coefficients, the Adam-Gibbs relation is valid throughout the temperature range we describe. For relaxation times and viscosity, at higher temperature range where the SE relation is valid, the AG relation with the same activation free energy is valid; at temperatures below the crossover at the SEB temperature, a different AG relation is valid, given by the fractional SE exponent. In presence of the SEB, the AG relation cannot be valid for both diffusion and relaxation times. However the question of which quantity is better described by the AG relation hasn’t been addressed in the literature before. Why diffusion coefficient, rather than relaxation times or viscosity, should be better described by the AG relation is an interesting question that needs to be addressed further in future works.

Length scale dependence of the Stokes-Einstein and Adam-Gibbs relations in model glass formers (Supplementary Material)

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

Here we provide additional information regarding the following aspects of analysis of studied model glasses: Stokes-Einstein breakdown and viscosity data for ($i$) KA model, ($ii$) SqW model, ($iii$) Dynamical heterogeneity, ($iv$) Coupling of $t^{*}$ and diffusion time scales and ($v$) Stokes-Einstein and Adam-Gibbs relations for diffusion coefficients and relaxation times of $A$ and $B$ components in the KA and SqW binary mixtures.

I KA model

II SqW model

II.1 Runlength and finite size effect

Various system sizes ($N=$ $864$, $5324$ and $16384$) have been studied for temperature across the Stokes Einstein breakdown (SEB) temperature $T=0.33$ (defined form $1/e$ decay of self-intermediate scattering functions at wave vector $k^{*}=2\pi$). The SEB temperature and exponent do not change much across the considered system size ($N=5324$).

II.2 Shear Viscosity

The shear viscosity $\eta$ can be computed from the Green-Kubo and Einstein relation. The Green-Kubo relation for the shear viscosity is defined from the integral of the autocorrelation function of the stress tensor ($J_{\alpha\beta}(t)$) -

[TABLE]

Further, the stress tensor $J_{\alpha\beta}$ is defined as

[TABLE]

where, $[\alpha,\beta]$ are x, y and z components, $V$ is volume of the system of N particles, $m$ is mass of a particle, $p_{i}$ is the momentum of particle $i$, distance $r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$ and for the pair interaction $U(r_{ij})$ the force can be defined as $f_{ij}=-\frac{\partial U(r_{ij})}{\partial r}$.

The shear viscosity also can be evaluated using Einstein relation

[TABLE]

where, the Helfand moment $A_{\alpha\beta}(t)$ is described as

[TABLE]

also,

[TABLE]

The integral in the Eq. (5) has been estimated numerically, which is not affected by the periodic boundary conditions used in the MD simulations Leporini2001_s . In the large time limit, the linear behavior of $\left<(A_{\alpha\beta}(t)-A_{\alpha\beta}(0))^{2}\right>$ (where, $\alpha,\beta\in[x,y,z],\alpha\neq\beta$) provides the estimate of the shear viscosity.

In the Kob-Andersen model particles are interacting $via$ continues model and Helfand moment is computes from the stress tensor as defined in Eq. (2). In the Fig S1B, we show that computed viscosity is proportional to the $\tau^{\prime}(k=7.25)$.

But, in the SqW model, the particles interaction is discrete and hence particles travel with constant velocity between two collisions ($\Delta t$, time between successive collisions). Hence, Helfand moment can be rewritten as haile_s ; Alder_s ; HM_s

[TABLE]

where, the first term is the kinetic part, evaluated at time $s$ and multiplied by the time interval between two successive collisions, the second term represents the change in the velocity between a colliding pair of particles $i$ and $j$. Further, the shear viscosity $\eta_{\alpha\beta}$ can be computed using Eq. (3). For the SqW model, we find that the shear viscosity is proportional to the relaxation time $\tau(k=2\pi)$ [see Fig S6].

III Dynamical heterogeneity

III.1 Four point correlation ($\chi_{4}(t)$) and Non-Gaussian parameter $(\alpha^{A}_{2}(t))$

The non-Gaussian parameter peaks at a time $t^{*}$, corresponding to the crossover between the so-called cage regime and the diffusive regime of the MSD. The maximum value of the parameter $\alpha_{2}^{A}(t^{*})$ is a measure of the heterogeneous dynamics, noticeably this value is quite high compared to well studied KA model kob1997_s . Further to understand the DH, we study the particle displacement and the morphology of the mobile clusters.

III.2 The van-Hove function

The van Hove distribution function is a dynamical correlation function which characterizes the spatial and temporal correlation of a pair, i.e. the probability of finding a particle at time $t$ at distance $r$ from its position at time $t=0$.

We define a Gaussian approximation to the self part of the van Hove function as

[TABLE]

The deviation from the Gaussian approximation can be related to the presence of the dynamical heterogeneity. In this study van Hove function has been estimated for “$A$” type of particles.

III.2.1 Time evolution of the self Van Hove function at different temperatures

The van Hove function has been studied at various time, the range of the time is taken in the window of $F_{s}(k=2\pi,t)$. The distribution becomes bimodal for the low temperatures (especially below $T_{SEB}=0.33$).

III.3 Morphology of the active particles (SqW model)

At time $t^{*}$, top 10% of the fast moving particles are considered as mobile particles  AG-DH_s . These particles are considered to be in the same cluster if they are present in the first coordination shell ($\approx 1.7$). The cluster size of these particles increases at lower temperatures. The morphology of the clusters of these mobile particles has been studied at time $t^{*}$.

IV Coupling of $t^{*}$ and diffusion time scales

The present work provides the numerical evidence that the Adam Gibbs relation is valid for diffusion coefficient and not for relaxation times and viscosity but the precise, microscopic description of the mechanism(s) of structural relaxation in glass-forming liquids is currently lacking. In general, it is found that structural relaxation time $\tau_{\alpha,A}$ is proportional to viscosity and is not proportional to the translational diffusion time scale $D_{A}$. This decoupling is related to the emergence of mobile and immobile clusters upon cooling (DH). The AG theory proposed a mechanism for structural relaxation via the concept of cooperatively rearranging regions (CRR), which have been identified as clusters of highly mobile particles, named “strings”AG-DH_s ; Freed_s ; string2_s ; string3_s . In Ref. AG-DH_s it is shown that the life time of these strings is proportional to $t^{*}$, which is proportional to $D^{-1}$. Hence, in the presence of SEB, the AG relation is described for $D$ and not for decoupled quantities $\tau_{\alpha}$ and $\eta$. Fig. S12 shows that the diffusion coefficient and the $t^{*}$, for particles type A, are coupled in both the models considered in the present study.

V Stokes-Einstein and Adam-Gibbs relations for diffusion coefficients and relaxation times of $A$ and $B$ components in the KA and SqW binary mixtures

The studied model glass formers consist of two types of particles, with compositions of the KA and SqW models being $(80:20)$ and $(50:50)$ respectively. Changes in dynamics with temperature affect these components differently SEBKA-6_s . We study the relation of diffusion coefficients of the two components as a function of temperature for the two models, as well as relaxation times. Though the ratio of the diffusion coefficients (i.e. $D_{B}/D_{A}$) is observed to be temperature dependent, as seen in Fig. S13, we find that they have a fractional power law dependence

on each other over a large temperature range (Fig. S14), extending well beyond the regime where the SEB is observed for any quantity we study. The Stokes-Einstein relation breaks down for particles of type $B$ as well. Fig. S15 shows that the fractional SE relation $D_{A,B}\propto\tau_{A,B}^{-\xi}$ is observed for both components (when we plot the diffusion coefficient of a given particle type against relaxation times computed for the same particle type) at low temperatures, with very similar characteristics. Because of their fractional power law dependence, the Adam-Gibbs relation holds for diffusion coefficients of both particle types in the considered temperature range, with different activation energies.

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