# Dynamics and Elastic Properties of Glassy Metastable States

^{1}

^{2}

*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.

## References

- Shelby, J.E. Introduction to Glass Science and Technology, 2nd ed.; The Royal Society of Chemistry: Cambridge, UK, 2005. [Google Scholar]
- Bernu, B.; Hiwatari, Y.; Hansen, J.P. Soft-sphere model for the glass transition in binary alloys: Pair structure and self-diffusion. J. Phys. C
**1985**, 18, L371. [Google Scholar] [CrossRef] - Wahnström, G. Molecular-dynamics study of a supercooled two-component Lennard-Jones system. Phys. Rev. A
**1991**, 44, 3752. [Google Scholar] [CrossRef] - Kob, W.; Andersen, H.C. Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I: The van Hove correlation function. Phy. Rev. E
**1995**, 51, 4626. [Google Scholar] [CrossRef] [Green Version] - Sollich, P. Predicting phase equilibria in polydisperse systems. J. Phys. Condens. Matter
**2002**, 14, R79–R117. [Google Scholar] [CrossRef] - Brazhkin, V.V. Can high pressure experiments shed light on the puzzles of glass transition? The problem of extrapolation. J. Phys. Condens. Matter
**2008**, 20, 244102. [Google Scholar] [CrossRef] - Angell, C.A.; Borick, S. Specific heats C
_{p}, C_{v}, C_{conf}and energy landscapes of glass forming liquids. J. Non-Cryst. Solids**2002**, 307–310, 393–406. [Google Scholar] [CrossRef] [Green Version] - Aoki, K.M.; Fujiwara, S.; Sogo, K.; Ohnishi, S.; Yamamoto, T. One-, Two-, and Three-dimentional Hopping Dynamics. Crystals
**2013**, 3, 315–332. [Google Scholar] [CrossRef] - Aoki, K.M.; Fujiwara, S.; Sogo, K.; Ohnishi, S.; Yamamoto, T. Molecular Dynamics Simulations of One-, Two-, Three-dimensional Hopping Dynamics. JPS Conf. Proc.
**2014**, 1, 012038. [Google Scholar] - Aoki, K.M.; Yoneya, M. Order Parameter Discretization in Metastable States of Hexatic Smectic B Liquid Crystal. J. Phys. Soc. Jpn.
**2011**, 80, 124603. [Google Scholar] [CrossRef] - Aoki, K.M. Network Analysis of Free Energy Landscaper of Metastable States of Hexatic Smectic B Liquid Crystal. J. Phys. Soc. Jpn.
**2014**, 83, 104603. [Google Scholar] [CrossRef] - Weeks, J.D.; Chandler, D.; Andersen, H.C. Role of Repulsive Forces in Determinig the Equilibrium Structure of Simple Liquids. J. Chem. Phys.
**1971**, 54, 5237. [Google Scholar] [CrossRef] - Aoki, K.M. Symplectic Integrator Designed for Simulating Soft Matter. J. Phys. Soc. Jpn.
**2008**, 77, 044003. [Google Scholar] [CrossRef] - Aoki, K.M. Molecular Dynamics Simulation of Anisotropic Molecules as a Model of Liquid Crystals. Ph.D. Thesis, Keio University, Yokohama, Japan, 1993. [Google Scholar]
- Aoki, K.M.; Yoneya, M.; Yokoyama, H. Extended methods of molecular dynamics simulations under hydrostatic pressure and/or isostress. J. Chem. Phys.
**2003**, 118, 9926. [Google Scholar] [CrossRef] - Aoki, K.M.; Yoneya, M.; Yokoyama, H. Molecular dynamics simulation methods for anisotropic liquids. J. Chem. Phys.
**2004**, 120, 5576. [Google Scholar] [CrossRef] - Aoki, K.M.; Yoneya, M.; Yokoyama, H. Constant surface-tension molecular-dynamics simulation methods for anisotropic systems. J. Chem. Phys.
**2006**, 124, 064705. [Google Scholar] [CrossRef] - Aoki, K.M. Anisotropy in condensed matter - liquid crystals, glass, and phase coexistence. J. Phys. Conf. Ser.
**2019**, 1252, 012004. [Google Scholar] [CrossRef] [Green Version] - Abraham, S.; Harrowell, P. The origin of persistent shear stress in supercooled liquids. J. Chem. Phys.
**2012**, 137, 014506. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fuereder, I.; Ig, P.I. Influence of inherent structure shear stress of supercooled liquids on their shear moduli. J. Chem. Phys.
**2015**, 142, 144505. [Google Scholar] [CrossRef] [Green Version] - Petry, W.; Bartsch, E.; Fujara, F.; Kiebel, M.; Sillescu, H.; Frago, B. Dynamic anomaly in glass transition region of orthoterphenyl. Z. Phys. B Condens. Matter
**1991**, 83, 175–184. [Google Scholar] [CrossRef] - Buchenau, U.; Zorn, R. A Relation between Fast and Slow Motions in Glassy and Liquid Selenium. EuroPhys. Lett.
**1992**, 18, 523–528. [Google Scholar] [CrossRef] - Angell, C.A. Formation of Glasses from Liquids and Biopolymers. Science
**1995**, 267, 1924–1939. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Magazù, S.; Maisano, G.; Migliardo, F.; Mondelli, C. Mean-Square Displacement Relationship in Bioprotectant Systems by Elastic Neutron Scattering. Biophys. J.
**2004**, 86, 3241–3249. [Google Scholar] [CrossRef] [Green Version] - Niss, K.; Dalle-Ferrier, C.; Frick, B.; Russo, D.; Dyre, J.; Alba-Simionesco, C. Connection between slow and fast dynamics of molecular liquids around the glass transition. Phy. Rev. E
**2010**, 82, 021508. [Google Scholar] [CrossRef] [Green Version] - Capaccioli, S.; Ngai, K.L.; Ancherbak, S.; Paciaroni, A. Evidence of Coexistence of Change of Caged Dynamics at T
_{g}and the Dynamic Transion at T_{d}in Solvated Proteins. J. Phys. Chem. B**2012**, 116, 1745–1757. [Google Scholar] [CrossRef] [PubMed] - Vural, D.; Glyde, H.R. Intrinsic mean-square displacements in proteins. Phys. Rev. E
**2012**, 86, 011926. [Google Scholar] [CrossRef] [Green Version] - Kegel, W.K.; van Blaaderen, A. Direct Observation of Dynamical Heterogeneities in Colloidal Hard-Sphere Suspensions. Science
**2000**, 287, 290–293. [Google Scholar] [CrossRef] [Green Version] - Weeks, E.R.; Crocker, J.C.; Levitt, A.C.; Schofield, A.; Weitz, D.A. Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition. Science
**2000**, 287, 627–631. [Google Scholar] [CrossRef] [Green Version] - Weeks, E.R.; Weitz, D.A. Properties of Cage Rearrangements Observed near the Colloidal Glass Transition. Phys. Rev. Lett.
**2002**, 89, 095704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kaufman, L.J.; Weitz, D.A. Direct imaging of repulsive and attractive colloidal glasses. J. Chem. Phys.
**2006**, 125, 074716. [Google Scholar] [CrossRef] [PubMed] - Gokhale, S.; Sood, A.K.; Ganapathy, R. Deconstructing the glass transition through critical experiments on colloids. Adv. Phys.
**2016**, 65, 363. [Google Scholar] [CrossRef] - Rouxel, T. Elastic Properties and Short-to Medium-Range Order in Glasses. J. Am. Ceram. Soc.
**2007**, 90, 3019. [Google Scholar] [CrossRef] - Novikov, V.N.; Sokolov, A.P. Poisson’s ratio and the fragility of glass-forming liquids. Nature
**2004**, 431, 961. [Google Scholar] [CrossRef] - Yannopoulos, S.N.; Johari, G.P. Poisson’s ratio of glass and a liquid’s fragility? Nature
**2006**, 442, E7–E8. [Google Scholar] [CrossRef] [PubMed] - Mott, P.H.; Dorgan, J.R.; Roland, C.M. The bulk modulus and Poisson’s Ratio of “incompressible” materials. J. Sound Vib.
**2008**, 312, 572. [Google Scholar] [CrossRef] - Anderson, D.L. Theory of Earth; Blackwell Scientific Publications: Boston, MA, USA, 1989; Chapter 5. [Google Scholar]
- Ha, A.; Cohen, I.; Zhao, X.; Lee, M.; Fischer, T.; Strouse, M.J.; Kivelson, D. Supercooled Liquids and Polyamorphism. J. Phys. Chem.
**1996**, 100, 1–4. [Google Scholar] [CrossRef] - Cohen, I.; Ha, A.; Zhao, X.; Lee, M.; Fischer, T.; Strouse, M.J.; Kivelson, D. A Low-Temperature Amorphous Phase in a Fragile Glass-Forming Substance. J. Phys. Chem.
**1996**, 100, 8518–8526. [Google Scholar] [CrossRef] - Angell, C.A. The amorphous state equivalent of crystallization: New glass types by first order transition from liquids, crystals, and biopolymers. Solid State Sci.
**2000**, 2, 791–805. [Google Scholar] [CrossRef] - Kurita, R.; Tanaka, H. On the abundance and general nature of the liquid–liquid phase transition in molecular systems. J. Phys. Condens. Matter
**2005**, 17, L293–L302. [Google Scholar] [CrossRef] - Kobayashi, M.; Tanaka, H. The reversibility and first-order nature of liquid–liquid transition in a molecular liquid. Nat. Commun.
**2016**, 7, 13438. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Murata, K.; Tanaka, H. Link between molecular mobility and order parameter during liquid–liquid transition of a molecular liquid. Proc. Natl. Acad. Sci. USA
**2019**, 116, 7176–7185. [Google Scholar] [CrossRef] [Green Version] - Walton, F.; Bolling, J.; Farrell, A.; MacEwen, J.; Syme, C.D.; Jiménez, M.G.; Senn, H.M.; Wilson, C.; Cinque, G.; Wynne, K. Polyamorphism Mirrors Polymorphism in the Liquid-Liquid Transition of a Molecular Liquid. J. Am. Chem. Soc.
**2020**, 142, 7591–7597. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tanaka, H. Liquid-liquid transition and polyamorphism. J. Chem. Phys.
**2020**, 153, 130901. [Google Scholar] [CrossRef] [PubMed] - Helfand, E.; Rice, S.A. Principle of Corresponding States for Transport Properties. J. Chem. Phys.
**1960**, 32, 1642–1644. [Google Scholar] [CrossRef] - Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids; Claredon Press: Oxford, UK, 1987; Appendix B. [Google Scholar]
- Hoover, W.G.; Gray, S.G.; Johnson, K.W. Thermodynamic Properties of the Fluid and Solid Phases for Inverse Power Potentials. J. Chem. Phys.
**1971**, 55, 1128–1136. [Google Scholar] [CrossRef] - Aoki, K.M.; Yonezawa, F. Scaling properties of soft-core parallel spherocylinder near crystal-smectic-phase transition. Phys. Rev. E
**1993**, 48, 2025–2027. [Google Scholar] [CrossRef] - Aoki, K.M.; Yoneya, M.; Yokoyama, H. Entropy and heat capacity calculations of simulated crystal-hexatic smectic-B- smectic-A liquid-crystal phase transitions. Phys. Rev. E
**2010**, 81, 021701. [Google Scholar] [CrossRef] [Green Version] - Aoki, K.M. Structural transformation of smectic liquid crystals under surface tension. Mol. Cryst. Liq. Cryst.
**2017**, 647, 92–99. [Google Scholar] [CrossRef] [Green Version]

**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.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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