^{1,2}, Smarajit Karmakar

^{2}, Chandan Dasgupta

^{3}and Srikanth Sastry

^{1,2}

### Abstract

The breakdown of the Stokes-Einstein (SE) relation between diffusivity and viscosity at low temperatures is considered to be one of the hallmarks of glassy dynamics in liquids. Theoretical analyses relate this breakdown with the presence of heterogeneous dynamics, and by extension, with the fragility of glass formers. We perform an investigation of the breakdown of the SE relation in 2, 3, and 4 dimensions in order to understand these interrelations. Results from simulations of model glass formers show that the degree of the breakdown of the SE relation decreases with increasing spatial dimensionality. The breakdown itself can be rationalized *via* the difference between the activation free energies for diffusivity and viscosity (or relaxation times) in the Adam-Gibbs relation in three and four dimensions. The behavior in two dimensions also can be understood in terms of a generalized Adam-Gibbs relation that is observed in previous work. We calculate various measures of heterogeneity of dynamics and find that the degree of the SE breakdown and measures of heterogeneity of dynamics are generally well correlated but with some exceptions. The two-dimensional systems we study show deviations from the pattern of behavior of the three- and four-dimensional systems both at high and low temperatures. The fragility of the studied liquids is found to increase with spatial dimensionality, contrary to the expectation based on the association of fragility with heterogeneous dynamics.

We thank Vishwas V. Vasisht, Jack F. Douglas, and Francis W. Starr for useful discussions, and the CCMS, JNCASR for computational facilities. S.S.G. acknowledges financial support from the CSIR, India.

I. INTRODUCTION

II. STOKES-EINSTEIN BREAKDOWN, DYNAMICAL HETEROGENEITY, AND FRAGILITY

III. SIMULATION DETAILS

IV. RESULTS

A. The dimension dependence of the SE breakdown

B. Dynamical heterogeneity and the breakdown of the Stokes-Einstein relation

C. Dependence of the fragility on spatial dimensions

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Relaxation times
- 34.0
- Spatial dimensions
- 19.0
- Diffusion
- 18.0
- Viscosity
- 13.0
- Entropy
- 11.0

## Figures

α relaxation times τ obtained from (i) the overlap function (τ_{ qt }), (ii) the self-part (τ_{ FsAkt }) of the intermediate scattering function *F* _{ sA }(*k**, *t*) of one type (*A*) of particles, and (iii) from the full (τ_{ Fkt }) intermediate scattering function *F*(*k**, *t*) plotted against viscosity showing that τ ∝ η/*T* is a good description of data at low *T* in the 3D KA model. Systematic deviations are seen at high *T*. *k** is at the first peak of the partial structure factor *S* _{ AA }(*k*).

α relaxation times τ obtained from (i) the overlap function (τ_{ qt }), (ii) the self-part (τ_{ FsAkt }) of the intermediate scattering function *F* _{ sA }(*k**, *t*) of one type (*A*) of particles, and (iii) from the full (τ_{ Fkt }) intermediate scattering function *F*(*k**, *t*) plotted against viscosity showing that τ ∝ η/*T* is a good description of data at low *T* in the 3D KA model. Systematic deviations are seen at high *T*. *k** is at the first peak of the partial structure factor *S* _{ AA }(*k*).

Plots showing the breakdown of the SE relation in the 2D KA model. (Top) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Bottom) *T* dependence of *D* _{ A }τ. The data shown follow a fractional SE relation.

Plots showing the breakdown of the SE relation in the 2D KA model. (Top) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Bottom) *T* dependence of *D* _{ A }τ. The data shown follow a fractional SE relation.

Plots showing the breakdown of the SE relation in the 2DMKA model. (Left) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Right) *T* dependence of *D* _{ A }τ. The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in *D* _{ A } vs*.* τ plot, although *the high T exponent is bigger than 1*. The change of slope occurs at a temperature *T* _{ SEB }, which is close to *T* _{ onset }. (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the onset temperature of slow dynamics.) Also shown is the dependence of *D* on τ according to the generalized Adam-Gibbs relation discussed in the text.

Plots showing the breakdown of the SE relation in the 2DMKA model. (Left) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Right) *T* dependence of *D* _{ A }τ. The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in *D* _{ A } vs*.* τ plot, although *the high T exponent is bigger than 1*. The change of slope occurs at a temperature *T* _{ SEB }, which is close to *T* _{ onset }. (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the onset temperature of slow dynamics.) Also shown is the dependence of *D* on τ according to the generalized Adam-Gibbs relation discussed in the text.

Plots showing the breakdown of the SE relation in the 2DR10 model. (Left) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Right) *T* dependence of *D* _{ A }τ. The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in *D* _{ A } vs. τ plot, although *the high T exponent is bigger than 1*. The change of slope occurs at a temperature *T* _{ SEB }, which is close to *T* _{ onset }. (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the onset temperature of slow dynamics.) Also shown is the dependence of *D* on τ according to the generalized Adam-Gibbs relation discussed in the text.

Plots showing the breakdown of the SE relation in the 2DR10 model. (Left) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Right) *T* dependence of *D* _{ A }τ. The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in *D* _{ A } vs. τ plot, although *the high T exponent is bigger than 1*. The change of slope occurs at a temperature *T* _{ SEB }, which is close to *T* _{ onset }. (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the onset temperature of slow dynamics.) Also shown is the dependence of *D* on τ according to the generalized Adam-Gibbs relation discussed in the text.

Plots showing the breakdown of the SE relation in the 3DKA model. The α relaxation times τ are computed from the (i) overlap function (τ_{ qt }) (ii) *F*(*k*, *t*) at the peak of *S*(*k*) (τ_{ Fkt }). These two measures are mutually proportional and are used interchangeably. (Top) *D* _{ A } vs. τ and plot. τ_{ qt }, τ_{ Fkt }(*k**) are multiplied by constant factors to match all data sets at low temperature. (Bottom) *T* dependence of *D* _{ A }τ and . The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in the *D* _{ A } vs. τ plot. The high T exponent (= −1) expected from the SE relation is obtained from the *D* _{ A } vs. plot. The SE breakdown occurs at a temperature *T* _{ SEB } closer to the onset temperature *T* _{ onset } than *T* _{ c }. ^{ 17 } (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the Arrhenius to non-Arrhenius cross-over temperature; *T* _{ c } is the mode coupling transition temperature. All data points shown here are at *T* > *T* _{ c }.)

Plots showing the breakdown of the SE relation in the 3DKA model. The α relaxation times τ are computed from the (i) overlap function (τ_{ qt }) (ii) *F*(*k*, *t*) at the peak of *S*(*k*) (τ_{ Fkt }). These two measures are mutually proportional and are used interchangeably. (Top) *D* _{ A } vs. τ and plot. τ_{ qt }, τ_{ Fkt }(*k**) are multiplied by constant factors to match all data sets at low temperature. (Bottom) *T* dependence of *D* _{ A }τ and . The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in the *D* _{ A } vs. τ plot. The high T exponent (= −1) expected from the SE relation is obtained from the *D* _{ A } vs. plot. The SE breakdown occurs at a temperature *T* _{ SEB } closer to the onset temperature *T* _{ onset } than *T* _{ c }. ^{ 17 } (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the Arrhenius to non-Arrhenius cross-over temperature; *T* _{ c } is the mode coupling transition temperature. All data points shown here are at *T* > *T* _{ c }.)

Plots showing the breakdown of the SE relation in the 3D R10 model. (Top) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Bottom) *T* dependence of *D* _{ A }τ. The data shown follow a fractional SE relation.

Plots showing the breakdown of the SE relation in the 3D R10 model. (Top) *D* _{ A } vs. τ plot (here, τ is the α relaxation time from the overlap function *q*(*t*)). (Bottom) *T* dependence of *D* _{ A }τ. The data shown follow a fractional SE relation.

Plots showing the breakdown of the SE relation in the 4DKA model. (Left) *D* _{ A } vs. τ plot (τ here is the α relaxation time from the overlap function *q*(*t*)). (Right) *T* dependence of *D* _{ A }τ. The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in *D* _{ A } vs. τ plot. The high T exponent (= −1) expected for a homogeneous (Gaussian distribution of particle displacements) system is obtained from *D* _{ A } vs. τ plot. The SE breakdown occurs at a temperature *T* _{ SEB } closer to the onset temperature *T* _{ onset } than *T* _{ c }. (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the Arrhenius to non-Arrhenius cross-over temperature.)

Plots showing the breakdown of the SE relation in the 4DKA model. (Left) *D* _{ A } vs. τ plot (τ here is the α relaxation time from the overlap function *q*(*t*)). (Right) *T* dependence of *D* _{ A }τ. The low T data follow a fractional SE relation. A clear change of exponent occurs at high T in *D* _{ A } vs. τ plot. The high T exponent (= −1) expected for a homogeneous (Gaussian distribution of particle displacements) system is obtained from *D* _{ A } vs. τ plot. The SE breakdown occurs at a temperature *T* _{ SEB } closer to the onset temperature *T* _{ onset } than *T* _{ c }. (*T* _{ SEB } estimated as the point of intersection of high T and low T fits; *T* _{ onset } is the Arrhenius to non-Arrhenius cross-over temperature.)

(Left) The Adam-Gibbs (AG) relation in the 3DKA model using as the dynamical quantities *D* _{ A }, τ. Here, τ is the α relaxation time from the overlap function *q*(*t*). (Right) The AG relation in the 4DKA model using as the dynamical quantities *D* _{ A }, τ. The slopes are different for *D* _{ A } and τ indicating that the diffusion coefficient has a *different* dependence on the configuration entropy than the α relaxation time. The fractional SE exponent at low *T* can be interpreted as the ratio of the slopes (Table I ).

(Left) The Adam-Gibbs (AG) relation in the 3DKA model using as the dynamical quantities *D* _{ A }, τ. Here, τ is the α relaxation time from the overlap function *q*(*t*). (Right) The AG relation in the 4DKA model using as the dynamical quantities *D* _{ A }, τ. The slopes are different for *D* _{ A } and τ indicating that the diffusion coefficient has a *different* dependence on the configuration entropy than the α relaxation time. The fractional SE exponent at low *T* can be interpreted as the ratio of the slopes (Table I ).

The time dependences of the dynamical susceptibility χ_{4}(*t*) and of the non-Gaussian parameter α_{2,A }(*t*) in the 4DKA model. Peak values and extracted from such time dependence are described further below.

The time dependences of the dynamical susceptibility χ_{4}(*t*) and of the non-Gaussian parameter α_{2,A }(*t*) in the 4DKA model. Peak values and extracted from such time dependence are described further below.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 2D R10 model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 2D R10 model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 2D R10 model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 2D R10 model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 2DMKA model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 2DMKA model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 2DMKA model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 2DMKA model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 3DKA model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 3DKA model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 3DKA model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 3DKA model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 4DKA model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 4DKA model.

(Left) *T* dependences of the peak height of the dynamical susceptibility χ_{4}(*t*) in the 4DKA model. (Right) *T* dependences of the peak height of the non-Gaussian parameter α_{2,A }(*t*) in the 4DKA model.

*T* dependence of *F* _{ sAkt } in 2DR10, 2DMKA, 3DKA, and 4DKA models. Also shown are the 4-parameter fit curves of the form with 0 ⩽ β_{ KWW } ⩽ 1. See Ref. ^{ 81 } for details of the fitting procedure.

*T* dependence of *F* _{ sAkt } in 2DR10, 2DMKA, 3DKA, and 4DKA models. Also shown are the 4-parameter fit curves of the form with 0 ⩽ β_{ KWW } ⩽ 1. See Ref. ^{ 81 } for details of the fitting procedure.

Comparison of the *degree of heterogeneity in different dimensions* using the as a measure. Vertical lines correspond to *T* _{ SEB }.

Comparison of the *degree of heterogeneity in different dimensions* using the as a measure. Vertical lines correspond to *T* _{ SEB }.

Comparison of the *degree of heterogeneity in different dimensions* using as a measure.

Comparison of the *degree of heterogeneity in different dimensions* using as a measure.

Comparison of the *degree of heterogeneity in different dimensions* using the β_{ KWW } as a measure.

Comparison of the *degree of heterogeneity in different dimensions* using the β_{ KWW } as a measure.

(Top) Fragility plot of relaxation times for five models in 2,3,4 spatial dimensions. VFT fits to relaxation times shown are used to obtain the kinetic fragility *K* _{ VFT } (see discussion). The fragility plot employs a “simulation glass transition temperature” *T* _{ g } defined as τ(*T* _{ g }) = 10^{5} (reduced unit) to scale temperatures. The *K* _{ VFT } values are listed in Table II . (Bottom) Fragility plot for four models in 2,3,4 spatial dimensions using the inverse of the diffusion coefficient. *The plots show that systems at higher dimensions are more fragile.*

(Top) Fragility plot of relaxation times for five models in 2,3,4 spatial dimensions. VFT fits to relaxation times shown are used to obtain the kinetic fragility *K* _{ VFT } (see discussion). The fragility plot employs a “simulation glass transition temperature” *T* _{ g } defined as τ(*T* _{ g }) = 10^{5} (reduced unit) to scale temperatures. The *K* _{ VFT } values are listed in Table II . (Bottom) Fragility plot for four models in 2,3,4 spatial dimensions using the inverse of the diffusion coefficient. *The plots show that systems at higher dimensions are more fragile.*

*T* dependence of *TS* _{ c } of five models in 2,3,4 spatial dimensions plotted as *TS* _{ c }vs. *T*/*T* _{ K } so that the slope is an estimate of the thermodynamic fragility *K* _{ T } (listed in Table II ). The plot shows that the thermodynamic fragility increases with increasing spatial dimensionality.

*T* dependence of *TS* _{ c } of five models in 2,3,4 spatial dimensions plotted as *TS* _{ c }vs. *T*/*T* _{ K } so that the slope is an estimate of the thermodynamic fragility *K* _{ T } (listed in Table II ). The plot shows that the thermodynamic fragility increases with increasing spatial dimensionality.

The configurational entropy density *S* _{ c }(*e* _{ IS }) of the inherent structures plotted vs. their energy shifted by its minimum possible value *e* _{ ISmin } defined from *S* _{ c }(*e* _{ ISmin }) = 0. The lines are fits to the parabolic form: where is the IS energy at the peak of the distribution. The distribution is broader for higher dimensions, which partially explains the increase of the thermodynamic fragility with increasing spatial dimension.

The configurational entropy density *S* _{ c }(*e* _{ IS }) of the inherent structures plotted vs. their energy shifted by its minimum possible value *e* _{ ISmin } defined from *S* _{ c }(*e* _{ ISmin }) = 0. The lines are fits to the parabolic form: where is the IS energy at the peak of the distribution. The distribution is broader for higher dimensions, which partially explains the increase of the thermodynamic fragility with increasing spatial dimension.

## Tables

Estimates of the magnitude of the SE breakdown exponents in different spatial dimensions *D*. (Notations) (a) ξ^{ SE } = SE breakdown exponent obtained from *D* _{ A } vs. τ_{α} or plots; (b) ξ^{ AG } = ratio of slopes from AG plots using *D* _{ A } and τ_{α}; high T exponents are obtained from *D* _{ A } vs. τ_{α} or η/*T* plots.

Estimates of the magnitude of the SE breakdown exponents in different spatial dimensions *D*. (Notations) (a) ξ^{ SE } = SE breakdown exponent obtained from *D* _{ A } vs. τ_{α} or plots; (b) ξ^{ AG } = ratio of slopes from AG plots using *D* _{ A } and τ_{α}; high T exponents are obtained from *D* _{ A } vs. τ_{α} or η/*T* plots.

Fragility-related parameters for the models in different dimensions. *T* _{ K } is the Kauzmann temperature obtained from extrapolating *TS* _{ c } to zero. *K* _{ VFT } is the kinetic fragility obtained from VFT fits to relaxation times. *K* _{ T } is the thermodynamic fragility obtained from the *T*-dependence of *TSc*. *A* is the Adam-Gibbs coefficient. *K* _{ AG } = *K* _{ T }/*A* is the kinetic fragility estimated from the AG relation.

Fragility-related parameters for the models in different dimensions. *T* _{ K } is the Kauzmann temperature obtained from extrapolating *TS* _{ c } to zero. *K* _{ VFT } is the kinetic fragility obtained from VFT fits to relaxation times. *K* _{ T } is the thermodynamic fragility obtained from the *T*-dependence of *TSc*. *A* is the Adam-Gibbs coefficient. *K* _{ AG } = *K* _{ T }/*A* is the kinetic fragility estimated from the AG relation.

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