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Breakdown of the Stokes-Einstein relation in two, three, and four dimensions
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10.1063/1.4792356
/content/aip/journal/jcp/138/12/10.1063/1.4792356
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/12/10.1063/1.4792356

Figures

Image of FIG. 1.
FIG. 1.

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).

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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 .)

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.)

Image of FIG. 8.
FIG. 8.

(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 ).

Image of FIG. 9.
FIG. 9.

The time dependences of the dynamical susceptibility χ4(t) and of the non-Gaussian parameter α2,A (t) in the 4DKA model. Peak values and extracted from such time dependence are described further below.

Image of FIG. 10.
FIG. 10.

(Left) T dependences of the peak height of the dynamical susceptibility χ4(t) in the 2D R10 model. (Right) T dependences of the peak height of the non-Gaussian parameter α2,A (t) in the 2D R10 model.

Image of FIG. 11.
FIG. 11.

(Left) T dependences of the peak height of the dynamical susceptibility χ4(t) in the 2DMKA model. (Right) T dependences of the peak height of the non-Gaussian parameter α2,A (t) in the 2DMKA model.

Image of FIG. 12.
FIG. 12.

(Left) T dependences of the peak height of the dynamical susceptibility χ4(t) in the 3DKA model. (Right) T dependences of the peak height of the non-Gaussian parameter α2,A (t) in the 3DKA model.

Image of FIG. 13.
FIG. 13.

(Left) T dependences of the peak height of the dynamical susceptibility χ4(t) in the 4DKA model. (Right) T dependences of the peak height of the non-Gaussian parameter α2,A (t) in the 4DKA model.

Image of FIG. 14.
FIG. 14.

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.

Image of FIG. 15.
FIG. 15.

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

Image of FIG. 16.
FIG. 16.

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

Image of FIG. 17.
FIG. 17.

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

Image of FIG. 18.
FIG. 18.

(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 ) = 105 (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.

Image of FIG. 19.
FIG. 19.

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.

Image of FIG. 20.
FIG. 20.

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

Generic image for table
Table I.

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. τα or plots; (b) ξ AG = ratio of slopes from AG plots using D A and τα; high T exponents are obtained from D A vs. τα or η/T plots.

Generic image for table
Table II.

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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/content/aip/journal/jcp/138/12/10.1063/1.4792356
2013-03-14
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Breakdown of the Stokes-Einstein relation in two, three, and four dimensions
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/12/10.1063/1.4792356
10.1063/1.4792356
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