The dynamics of a small model glass former as viewed from its potential energy landscape [Elektronische Ressource] / vorgelegt von Burkhard Doliwa

Description

The Dynamics ofa Small Model Glass Formeras Viewed fromIts Potential Energy LandscapeDissertationzur Erlangung des GradesDoktor der Naturwissenschaften“”am Fachbereich Physikder Johannes Gutenberg–Universit atin Mainzvorgelegt vonBurkhard Doliwageboren in HadamarMainz 2002Tag der mundlicÄ hen Prufung:Ä 25.2.2003ContentsIntroduction 1Chapter 1 Supercooled Liquids and the Glass Transition 51.1 Phenomenology. . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Theories Describing the Viscous Slowing Down. . . . . . . . . 10Chapter 2 On the Simulations 152.1 Interaction Potentials.. . . . . . . . . . . . . . . . . . . . . . . 152.2 Equation of Motion and its Integration. . . . . . . . . . . . . . 202.3 Units.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Chapter 3 The Energy Landscape Approach 253.1 Statistics of Minima. . . . . . . . . . . . . . . . . . . . . . . . 283.2 Total Number of Minima. . . . . . . . . . . . . . . . . . . . . . 373.3 Configurational Entropy. . . . . . . . . . . . . . . . . . . . . . 47Chapter 4 Dynamics: From Hopping to Diffusion 534.1 The Long-Time Diffusion Coefficient. . . . . . . . . . . . . . . 544.2 Metabasin Hopping. . . . . . . . . . . . . . . . . . . . . . . . . 594.3 From Hopping to Diffusion. . . . . . . . . . . . . . . . . . . . . 644.4 Waiting Time Distributions. . . . . . . . . . . . . . . . . . . . 674.5 All this does not work with single basins. . . . . . . . . . . . .

Sujets

• Physik
• Astronomie

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2002
• Nombre de lectures: 7
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

The Dynamics of
a Small Model Glass Former
as Viewed from
Its Potential Energy Landscape
Dissertation
zur Erlangung des Grades
Doktor der Naturwissenschaften“
”
am Fachbereich Physik
der Johannes Gutenberg–Universit at
in Mainz
vorgelegt von
Burkhard Doliwa
geboren in Hadamar
Mainz 2002Tag der mundlic? hen Prufung:? 25.2.2003Contents
Introduction 1
Chapter 1 Supercooled Liquids and the Glass Transition 5
1.1 Phenomenology. . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Theories Describing the Viscous Slowing Down. . . . . . . . . 10
Chapter 2 On the Simulations 15
2.1 Interaction Potentials.. . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Equation of Motion and its Integration. . . . . . . . . . . . . . 20
2.3 Units.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 3 The Energy Landscape Approach 25
3.1 Statistics of Minima. . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Total Number of Minima. . . . . . . . . . . . . . . . . . . . . . 37
3.3 Configurational Entropy. . . . . . . . . . . . . . . . . . . . . . 47
Chapter 4 Dynamics: From Hopping to Diffusion 53
4.1 The Long-Time Diffusion Coefficient. . . . . . . . . . . . . . . 54
4.2 Metabasin Hopping. . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 From Hopping to Diffusion. . . . . . . . . . . . . . . . . . . . . 64
4.4 Waiting Time Distributions. . . . . . . . . . . . . . . . . . . . 67
4.5 All this does not work with single basins. . . . . . . . . . . . . 72
Chapter 5 From PEL Structure to Hopping 77
5.1 Activation Energies from Metabasin Lifetimes. . . . . . . . . . 80
5.2 Non-Local Ridge Method for Finding Transition States. . . . . 87
5.3 Energy Barriers from PEL Topology. . . . . . . . . . . . . . . 90
5.4 Barrier Crossing. . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Where is the PEL ground? . . . . . . . . . . . . . . . . . . . . 103
5.6 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
iii CONTENTS
Chapter 6 Finite-Size Effects 113
6.1 Static properties. . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Dynamic properties. . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Chapter 7 The Molecular Slowing Down 123
7.1 Liquid-like and Solid-like Regions on the PEL. . . . . . . . . . 125
7.2 From PEL Structure to Diffusion. . . . . . . . . . . . . . . . . 128
7.3 Relation to Existing Work. . . . . . . . . . . . . . . . . . . . . 131
7.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
AppendixANotation 141
A.1 Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2 Symbols and Acronyms. . . . . . . . . . . . . . . . . . . . . . 142
AppendixBSystem Specifications 143
B.1 Calculation of the Hesse Matrix. . . . . . . . . . . . . . . . . . 143
B.2 Details of the BMLJ Potential. . . . . . . . . . . . . . . . . . . 144
B.3 Simulation Runs. . . . . . . . . . . . . . . . . . . . . . . . . . 145
B.4 Details of the BMSS70 Potential. . . . . . . . . . . . . . . . . 146
AppendixCMiscellaneous 149
C.1 Configurational Partition Function of an Harmonic Oscillator. 149
C.2 A Comment on Entropy. . . . . . . . . . . . . . . . . . . . . . 149
C.3 Derivation of Eq. 4.5. . . . . . . . . . . . . . . . . . . . . . . . 150
C.4 Useful Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
C.5 Configurational Entropy from Gaussian Distributions. . . . . . 153
AppendixDBouchaud’s Trap Model 155
D.1 Single System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D.2 Combination of Two Independent Systems. . . . . . . . . . . . 157
Appendix EWaiting Times of a System Composed of Two Indepen-
dent Subsystems 161
E.1 Distribution of Waiting Times. . . . . . . . . . . . . . . . . . . 161
E.2 Mean Waiting Time of the Combined System. . . . . . . . . . 163
Bibliography 165Introduction
In our every-day life, we are surrounded by amorphous solids. This does not only
cover the ordinary window glass or the optical fiber which we use for short and
long-range communications. Among other things, most of the engineering plastics
and also –honey– belong to this kind of material.
Although the technical processing of glasses has been known for millennia now,
thephysicalprinciplesbehindglassformationarestillnotfullyunderstood. Whatis
clearis that the melt, uponsufficientlyrapidcooling, remains in the liquid statein-
steadofcrystallizing. Duetothelowtemperature,then,molecularmotionbecomes
very sluggish, e.g. happens on the time scale of seconds rather than picoseconds.
For this reason, the liquid is extremely viscous, thus acting very much like a solid.
Although the underlying crystal is the thermodynamically stable state, the system
is too slow to reach it within the experimental time scale; the liquid is said to be
metastable or supercooled.
On a phenomenological basis, one defines the glass transition temperature (T )g
2as the point where characteristic molecular relaxation times reach 10 s, or alter-
13natively, where the shear viscosity exceeds · = 10 poise. To get an impression
of the order of magnitude, consider a cubic centimeter of a substance with exactly
this viscosity, where a shear force of 100 N is constantly applied. After one day, the
deformation of the probe is ca. 1 mm.
For the development of glasses and their industrial application, it is highly de-
sirable to understand why certain compositions of materials readily form glasses on
cooling a melt while others have a strong tendency to crystallize. This remains
oneof thegreatunsolvedmysteriesofglassscience, althoughempiricaldescriptions
have been developed which successfully account for the glass-forming ability in cer-
tain specific cases. Nevertheless, choosing the right composition for a glass former
with some desired properties is still in an ’alchemy’ stage. Of course, the question
of the ease of glass formation on cooling a melt is intimately related to the problem
of how do glasses form.
One puzzling observation is that the dramatic slowdown of dynamics is accom-
12 Introduction
panied by no more than subtle changes in structure. One also says that the glass
transition is a kinetic phenomenon. Moreover, the mobility of particles can vary by
orders of magnitude between different regions of the same sample, which is referred
to as dynamic heterogeneity. These, and other aspects of glass formation, like the
jump in specific heat at T , are not fully understood. The major goal would beg
to predict dynamic (and also thermodynamic) properties of the system in question
from first principles, i.e. from the microscopic interaction potentials. The prob-
lem is that the lacking periodicity of the inter-molecular structure complicates the
application of the classical tools of theoretical physics.
The physics of glass-forming liquids is a complex many-body problem. This
shows up, e.g., in the non-exponential decay of dynamic response functions or
in the non-Arrhenius behavior of viscosities in many glass formers. Apart from
phenomenological models as that of (Adam and Gibbs, 1965) and analytical the-
ories like mode-coupling theory (G otze and Sjogren, 1992) or the theory of spin
glasses (Mezard et al., 1987), computer simulations are having an increasing im-
pactonthisfieldofresearch. Inparticular,thepotentialenergylandscapeapproach
(PEL) proposed by Goldstein over thirty years ago (Goldstein, 1969) could be im-
plemented numerically with considerable success (Stillinger and Weber, 1982). The
idea is to consider the high-dimensional vector of all particle coordinates as a point
moving on the surface of the total potential energy. At low temperatures, the sys-
temstaysnearthelocalminimaofthePEL,theso-calledinherentstructures. Using
only their local properties one is then able to predict all thermodynamic quantities.
At high temperatures, this attempt breaks down since the system is no longer con-
fined to the very vicinity of the local minima. In recent years, important pieces of
information have been gained about the PEL of different glass-forming systems via
extended computer simulations (Sastry, 2001; Sastry et al., 1998; Sciortino et al.,
2000; Sciortino et al., 1999). It has turned out that the PEL description starts to
work when cooling below T … 2T , where T is the critical temperature of mode-c c
coupling theory (MCT). In this regime the equation of state could be expressed
completely in terms of a few parameters that characterize the statistical properties
of PEL minima (La Nave et al., 2002a).
The current understanding of dynamics in terms of the PEL structure is much
less satisfying. However, it is felt that the molecular slowing down, as expressed,
e.g., by the diffusion coefficient D(T), should be related to the hopping over PEL
barrierswhichseparatetheminima. Thequantificationofthisideawillbethemain
subject of this thesis. The first question that arises is
† Is there a simple relation between D(T) and the hopping between minima?
We shall demonstrate in chapter 4 that there is, if one considers whole superstruc-3
turesofmanyPELminima(metabasins), ratherthansingleminima. Theexistence
of metabasins has been hypothesized some years ago (Stillinger, 1995) and recently
been demonstrated within simulations (Buc?