^{1,2}, Heidi Perry

^{3}, Peter Harrowell

^{2,a)}and David R. Reichman

^{3}

### Abstract

Using computer simulations, we show that the localized low frequency normal modes of a configuration in a supercooled liquid are strongly correlated with the irreversible structural reorganization of the particles within that configuration. Establishing this correlation constitutes the identification of the aspect of a configuration that determines the heterogeneity of the subsequent motion. We demonstrate that the spatial distribution of the summation over the soft local modes can persist in spite of particle reorganization that produces significant changes in individual modes. Along with spatial localization, the persistent influence of soft modes in particle relaxation results in anisotropy in the displacements of mobile particles over the time scale referred to as -relaxation.

We would like to thank L. Berthier, G. Biroli, J. P. Bouchaud, A. Heuer, C. O’Hern, and L. O. Hedges for useful discussions. H.P. and D.R.R. would like to thank P. Verrocchio for providing the equilibrated 3D configurations and the NSF for financial support. A.W. and P.H. acknowledge the support of the Australian Research Council.

I. INTRODUCTION

II. BACKGROUND

A. The isoconfigurational ensemble and short time heterogeneities

B. Irreversibility

C. Instantaneous and quenched normal modes in supercooled liquids

III. MODEL AND ANALYSIS

IV. RESULTS

A. The spatial distribution of irreversible relaxation

B. The spatial distribution of quasilocalized normal modes

C. Does directional motion have a structural origin?

D. The spatial correlation between irreversible reorganization and quasilocalized modes

E. The evolution of the spatial distribution of quasilocalized modes

F. The overlap of low frequency quenched modes and imaginary instantaneous modes

V. CONCLUSIONS

### Key Topics

- Normal modes
- 26.0
- Glass transitions
- 8.0
- Polarization
- 8.0
- Spatial dimensions
- 5.0
- Spatial scaling
- 5.0

## Figures

The probability distributions for the minimum number of “lost” initial neighbors after having first lost neighbors. Note that after losing neighbors, the probability of recovering all of the initial neighbors (i.e., having ) is essentially zero. The distributions show peaks at suggesting that particles typically recover two of their initial neighbors. These calculations where carried out over a time interval of at .

The probability distributions for the minimum number of “lost” initial neighbors after having first lost neighbors. Note that after losing neighbors, the probability of recovering all of the initial neighbors (i.e., having ) is essentially zero. The distributions show peaks at suggesting that particles typically recover two of their initial neighbors. These calculations where carried out over a time interval of at .

Distribution of times at which particles first lose their fourth nearest neighbor (data averaged over isoconfigurational ensembles of 100 runs for a total of ten initial configurations of particles at ).

Distribution of times at which particles first lose their fourth nearest neighbor (data averaged over isoconfigurational ensembles of 100 runs for a total of ten initial configurations of particles at ).

Contour plots of the probability of a particle losing four original neighbors, the criterion for IR, over 100 isoconfigurational runs for six different initial configurations.

Contour plots of the probability of a particle losing four original neighbors, the criterion for IR, over 100 isoconfigurational runs for six different initial configurations.

Contour plots of the participation fraction summed over the 30 lowest frequency modes for the quenched initial configurations of the same six configurations used in Fig. 3.

Contour plots of the participation fraction summed over the 30 lowest frequency modes for the quenched initial configurations of the same six configurations used in Fig. 3.

Maps of the participation fractions for the 11 lowest frequency normal modes of the local potential energy minimum associated with a single initial configuration. Modes with a participation ratio are colored red while the more delocalized modes are colored blue. The intensity of the color increases with the magnitude of the squared amplitude. The corresponding irreversibility map for this configuration is provided for comparison.

Maps of the participation fractions for the 11 lowest frequency normal modes of the local potential energy minimum associated with a single initial configuration. Modes with a participation ratio are colored red while the more delocalized modes are colored blue. The intensity of the color increases with the magnitude of the squared amplitude. The corresponding irreversibility map for this configuration is provided for comparison.

A comparison of the particle displacements averaged over the isoconfigurational ensemble and the displacements associated with the four lowest frequency normal modes. The first column labeled “displacements” contains the same displacement map repeated for ease of comparison. The vectors are proportional to the isoconfigurational average of the particle displacement over . The middle column labeled “Mode Displacements” shows the 4 lowest frequency normal modes of the initial configuration. The vectors are proportional to the mode polarization vector on each particle. Those vectors colored blue are correlated with displacement map vectors with an angle of less than while the red regions are anticorrelated with angle greater than It should be noted that each eigenvector provides us with lines in space along which particle moves with the actual forward or backward direction depending on the (arbitrary) phase of the vibration. The third column labeled “comparison” shows plots of the scalar product of the mode polarization vector on each particle with the average displacement vector. Darker colors signify greater magnitude of the scalar product, with blue (red) signifying positive (negative) signs of the scalar product.

A comparison of the particle displacements averaged over the isoconfigurational ensemble and the displacements associated with the four lowest frequency normal modes. The first column labeled “displacements” contains the same displacement map repeated for ease of comparison. The vectors are proportional to the isoconfigurational average of the particle displacement over . The middle column labeled “Mode Displacements” shows the 4 lowest frequency normal modes of the initial configuration. The vectors are proportional to the mode polarization vector on each particle. Those vectors colored blue are correlated with displacement map vectors with an angle of less than while the red regions are anticorrelated with angle greater than It should be noted that each eigenvector provides us with lines in space along which particle moves with the actual forward or backward direction depending on the (arbitrary) phase of the vibration. The third column labeled “comparison” shows plots of the scalar product of the mode polarization vector on each particle with the average displacement vector. Darker colors signify greater magnitude of the scalar product, with blue (red) signifying positive (negative) signs of the scalar product.

Contour plots of the low frequency mode participation (as in Fig. 3), overlaid with the location of particles (white circles) whose isoconfigurational probability of losing four initial nearest neighbors within is greater than or equal to 0.01.

Contour plots of the low frequency mode participation (as in Fig. 3), overlaid with the location of particles (white circles) whose isoconfigurational probability of losing four initial nearest neighbors within is greater than or equal to 0.01.

Plots of the participation fraction in the 30 lowest frequency normal modes for quenched configurations taken every along a trajectory from the initial configuration. The color code is the same as previous figures. While there are clearly variations occurring in the distribution of modes (and hence in the identity of the quenched minimum or inherent structure) over , substantial elements of the mode distribution persist. The top left map is the IR map for the initial configuration, included for comparison.

Plots of the participation fraction in the 30 lowest frequency normal modes for quenched configurations taken every along a trajectory from the initial configuration. The color code is the same as previous figures. While there are clearly variations occurring in the distribution of modes (and hence in the identity of the quenched minimum or inherent structure) over , substantial elements of the mode distribution persist. The top left map is the IR map for the initial configuration, included for comparison.

(a) Contour plot of the participation fraction summed over the 30 lowest frequency modes for a quenched configuration. (b) Contour plot of the maximum value of the participation fraction observed over five runs starting from the configuration in Fig. 10(a). (c) Particles whose isoconfigurational probability of losing four initial nearest neighbors within is greater than or equal to 0.01 (white circles) overlaid on the participation fraction map for the initial configuration. d) As in (c) except that the overlay is over the map of the maximum participation fraction shown in (b).

(a) Contour plot of the participation fraction summed over the 30 lowest frequency modes for a quenched configuration. (b) Contour plot of the maximum value of the participation fraction observed over five runs starting from the configuration in Fig. 10(a). (c) Particles whose isoconfigurational probability of losing four initial nearest neighbors within is greater than or equal to 0.01 (white circles) overlaid on the participation fraction map for the initial configuration. d) As in (c) except that the overlay is over the map of the maximum participation fraction shown in (b).

A comparison of the sum of the participation fractions of the normal modes with imaginary frequency with the sum of participation fraction of the same number of quenched normal modes, along with the map of the local DW factor and a map with vectors representing the motion of each particle from the instantaneous configuration to the IS.

A comparison of the sum of the participation fractions of the normal modes with imaginary frequency with the sum of participation fraction of the same number of quenched normal modes, along with the map of the local DW factor and a map with vectors representing the motion of each particle from the instantaneous configuration to the IS.

For the 2D system, the overlap of the 10% of particles with the largest DW factors with (black dots) the top 10% of clustered particles with the largest amplitudes in the sum of the lowest 30 normal modes and with (red squares) a random selection of 10% of the particles, all maps have had clusters of less than eliminated.

For the 2D system, the overlap of the 10% of particles with the largest DW factors with (black dots) the top 10% of clustered particles with the largest amplitudes in the sum of the lowest 30 normal modes and with (red squares) a random selection of 10% of the particles, all maps have had clusters of less than eliminated.

Similar to Fig. S6 for the 3D system, in this dimension the minimum cluster size is .

Similar to Fig. S6 for the 3D system, in this dimension the minimum cluster size is .

## Tables

Comparison of the degree of similarity in the sum of low-frequency quenched normal modes and imaginary frequency instantaneous normal mode maps for ten configurations. The left two columns give the percentage of particles with a value of the amplitude of the sum of the mode polarization vectors on each particle greater than 25% of the minimum value (the “active portions”). The overlap column lists the percentages of those particles that are presented in both the left and middle columns.

Comparison of the degree of similarity in the sum of low-frequency quenched normal modes and imaginary frequency instantaneous normal mode maps for ten configurations. The left two columns give the percentage of particles with a value of the amplitude of the sum of the mode polarization vectors on each particle greater than 25% of the minimum value (the “active portions”). The overlap column lists the percentages of those particles that are presented in both the left and middle columns.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content