# Dynamics and Elastic Properties of Glassy Metastable States

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*Solids*2021)

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Properties

#### 2.1. Temperature Dependence

#### 2.2. State G

#### 2.3. State J

#### 2.4. State M

#### 2.5. State O

#### 2.6. Dynamical Heterogeneity at ${T}_{D}$

## 3. Elastic Properties

#### 3.1. Temperature Dependence of Modulus of Spontaneous Elastic Tension/Compression

#### 3.2. Temperature Dependence Spontaneous Strain Ratio

#### 3.3. Temperature Dependence of Spontaneous Bulk Modulus

## 4. Concluding Remarks

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Computer Simulation Methods

#### Appendix A.1. Symplectic Integrator for Soft Matter

#### Appendix A.2. Calculation of Metastable States

#### Appendix A.3. Model and Initial Configurations

#### Appendix A.4. Calculation of Transport Coefficient

#### Appendix A.5. Calculation of Elastic Properties

**Figure A1.**Spontaneous stress and strain of state G for time duration $\Delta t=1000$ at temperature $T=132$: (

**a**) longitudinal stress versus strain, (

**b**) transverse stress versus strain, (

**c**) bulk stress versus strain, and (

**d**) ratio between transverse and longitudinal strains, for system size $N=5376$. Red lines are the linear fit to all points.

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**Figure 1.**Temperature dependence of diffusion coefficient of states G (▼), J (□), M (×), O (+), and crystal (•), with that of supercooled liquid (∘) and thermodynamic equilibrium liquid (⨂) for system size $N=5376$.

**Figure 2.**Temperature dependence of fraction of mobile particles of states G (▼), J (□), M (×), O (blue +), and crystal (•), and supercooled liquid (∘) measured in time duration of $\Delta t=1000$ for system size $N=5376$.

**Figure 3.**Mean square displacement (MSD) of state G at temperatures $T=114$ (red), 122 (green), 126 (blue), 128 (magenta), 132 (cyan), and 136 (black) for system size $N=5376$.

**Figure 4.**Mean square displacement of state J at temperatures $T=99$ (red), 102 (green), 104 (blue), 106 (magenta), 112 (cyan), 114 (black), 116 (red), 118 (cyan), and 120 (blue) for system size $N=5376$. Lower values of MSD at lower temperatures when plotted with the same color. Inset show close up of $T=106$.

**Figure 5.**Displacement plot projected on the xy-plane of state J at $T=102$ at times (

**a**) $t=1.11\times {10}^{4}$, and (

**b**) $t=3.25\times {10}^{4}$, for system size $N=5376$.

**Figure 6.**Mean square displacement of state M at temperatures $T=80$ (green), 85 (blue), 90 (magenta), 95 (cyan), 98 (red), 100 (green), 102 (blue), 104 (magenta), and 105 (cyan) for system size $N=5376$. Lower values of MSD for lower temperatures when data plotted with the same color.

**Figure 7.**Mean square displacement of state O at temperatures (

**a**) $T=80$ (red), 100 (green), 116 (blue), 128 (magenta), 132 (cyan), (

**b**) $T=88$ (red), 96 (green), 108 (blue), 120 (magenta), 124 (cyan), for system size $N=5376$.

**Figure 8.**Self-van Hove correlation function of state O at (

**a**) $T=88$ for different time durations t = 11,000–19,000 (solid), 15,000–19,000 (dotted), 17,000–19,000 (dashed), 18,000–19,000 (long dashed), and 20,000–21,000 (red dot–dashed); (

**b**) $T=116$ for time t=16,700–19,700 (solid), 17,700–19,700 (dotted), 18,700–19,700 (dashed) and 20,000–21,000 (red dot–dashed), for system size $N=5376$.

**Figure 9.**Three dimensional non-Gaussian parameter near ${T}_{D}$ of states (

**a**) G at $T=126$, (

**b**) J at $T=112$, (

**c**) M at $T=102$, and (

**d**) O at $T=116$, for system size $N=5376$.

**Figure 10.**Temperature dependence of spontaneous elastic modulus of states G (▼), J (□), M (×), O (+), and crystal (•), in (

**a**) longitudinal ${E}_{\Vert}$ and (

**b**) transverse ${E}_{\perp}$ directions, for system size $N=5376$.

**Figure 11.**Temperature dependence of strain ratio ${\u03f5}_{\perp}/{\u03f5}_{\left|\right|}$ of states G (▼), J (□), M (×), O (+), and crystal (•), with supercooled liquid (∘) and thermodynamic equilibrium liquid (⊗), for system size $N=5376$.

**Figure 12.**Temperature dependence of bulk modulus of states G (▼), J (□), M (×), O (+), and crystal (•), with supercooled liquid (∘) and thermodynamic equilibrium liquid (⊗), for system size $N=5376$. Lines are fit to Equation (5) for crystal (red) and state M (green) with reference values ${B}_{0}$ and ${V}_{0}$ at $T=80$ of each state. Blue line is a linear fit to data of liquids.

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Aoki, K.M.
Dynamics and Elastic Properties of Glassy Metastable States. *Solids* **2021**, *2*, 249-264.
https://doi.org/10.3390/solids2020016

**AMA Style**

Aoki KM.
Dynamics and Elastic Properties of Glassy Metastable States. *Solids*. 2021; 2(2):249-264.
https://doi.org/10.3390/solids2020016

**Chicago/Turabian Style**

Aoki, Keiko M.
2021. "Dynamics and Elastic Properties of Glassy Metastable States" *Solids* 2, no. 2: 249-264.
https://doi.org/10.3390/solids2020016