Transport and aging in glassy systems

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Transport and Aging in Glassy Systems

Transport und Altern in glasartigen Systemen

Dissertation

zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.)

an der Universit¨at Konstanz Fachbereich Physik

vorgelegt von

B ERND R INN

Dissertation der Universität Konstanz Datum der m ündlichen Pr üfung: 9.11.2001 Referenten: Priv. Doz. Dr. P. Maaß

Prof. Dr. P. Nielaba Prof. Dr. R. Schilling

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Special edition of the book with the ISBN: 3-89825- 366-X

dissertation.de - Verlag im Internet GmbH Pestalozzistr. 9

10 625 Berlin

URL: http://www.dissertation.de

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Kurzzusammenfassung

Thema dieser Arbeit ist die Untersuchung der Gleichgewichts– und Nichtgleichge- wichts–Dynamik, die von thermich aktivierten Sprungprozessen in ungeordneten Energielandschaften herr ¨uhrt. Modelle, welche von der Sprungbewegung eines “Zu- standspunktes” in einer ungeordneten komplexen Energielandschaft ausgehen, werden seit l¨angerem zur Beschreibung der spezifischen Eigenschaften von glasartigen Materialien verwendet (siehe z. B. [Gol69]).

Im ersten Teil der Arbeit wird eine Theorie des effektiven Mediums f ¨ur die Be- schreibung des dispersiven Transports in amorphen Materialien entwickelt, welche die Einbeziehung asymmetrischer ¨Ubergangsraten erlaubt. Die vorgeschlagene Effek- tive–Medien–Theorie wird auf Ein–Teilchen–Systeme und Systeme mit Harter–Kugel–

Abstoßung angewandt. Bei den Ein–Teilchen–Systemen werden beispielhaft die Dif- fusionskoeffizienten im Frequenzraum berechnet und mit numerischen Ergebnissen aus Monte–Carlo Simulationen verglichen. F ür verd ünnte Gittergase mit Harter–Ku- gel–Wechselwirkung berechnen wir die Leitfähigkeit f ür verschiedene Temperaturen und Konzentrationen und zeigen, daß aus der Effektiven–Medien–Theorie Skalenei- genschaften f ür die frequenzabhängige Leitfähigkeit folgen, wie sie f ür ionenleitende Gläser gemessen worden sind.

Im zweiten Teil der Arbeit untersuchen wir das Nichtgleichgewichts–Phänomen der physikalischen Alterung in glasartigen Materialien. Dieses wird bei vielen ver- schiedenartigen Systemen beobachtet, wenn diese auf experimentell zugänglichen Zeitskalen keinen Gleichgewichtszustand erreichen. Dabei gehen wir von einem asymptotisch exponentiellen Abfall der Energiezustandsdichte aus. Ein solcher Ab- fall, der das Auftreten einer Glas übergangstemperatur impliziert, wird nach der sta- tistischen Theorie der Extremalwerte f ür die Verteilung der tiefsten Energien in einer komplexen Energielandschaft erwartet und konnte auch in Molekulardynamiksimu- lationen von Gläsern nachgewiesen werden. In einer idealisierten Beschreibung be- trachten wir eine streng exponentielle Zustandsdichte. In Energielandschaften mit dieser Zustandsdichte untersuchen wir Sprungprozesse und das mit diesen verbun- dene Zerfallen der Korrelationen. Die Korrelationen zerfallen umso langsamer, je mehr Zeit verstrichen ist seit dem Abk ühlen des Systems unter die Glas übergangs- temperatur. Dieses Phänomen bezeichnet man als physikalisches Altern. Wir zeigen, daß das Modell ein überraschend reiches Szenario verschiedener Skaleneigenschaften wie sublineares Altern, ultra–langsames Altern, ein verallgemeinertes Skalenregime und Vielfachskalierung besitzt. Wir formulieren ein (allgemeiner anwendbares) “Kon- zept des partiellen Gleichgewichtes”, das wir f ür unser Modell mathematisch exakt behandeln k önnen und das es uns erlaubt, die Skaleneigenschaften des Modells kor- rekt vorherzusagen. Schließlich zeigen wir, daß entgegen der naiven Erwartung eine

“Mean–Field–Behandlung”, bei der die Energielandschaft zeitlich schnell fluktuiert, selbst in hochdimensionalen Energielandschaften zu qualitativ anderen Ergebnissen f ¨uhrt.

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D Continuous–Time Monte–Carlo simulations IX D.1 General description . . . IX D.2 Technical aspects . . . X D.3 Calculation of^D^{^} from simulations . . . X E Summary of formulae of the asymmetric EMA for special DOS XIII E.1 Formulae and results for a constant DOS . . . XIII E.1.1 Exact limites . . . XIII E.1.2 Formulae of the asymmetric EMA . . . XIII E.1.3 Temperature rescaling . . . XIV E.2 Equations and results for a Gamma DOS . . . XVI E.2.1 Exact limites . . . XVI E.2.2 Formulae of the asymmetric EMA . . . XVI E.2.3 Temperature rescaling . . . XVII E.3 Equations and results for a bimodal DOS . . . XVII E.3.1 Exact limites . . . XVII E.3.2 Formulae of the asymmetric EMA . . . XIX E.3.3 Temperature rescaling . . . XX E.4 Formulae and results for the dilute hard–core system . . . XX E.4.1 How to determine the fugacity . . . XX E.4.2 Limits of the self consistency equation . . . XXI E.4.3 ^D(!)obtained by averaging over the energy . . . XXII E.4.4 Low frequency limit: Relation between^(!)and^D(!) . . . XXIII

F Number of distinct visited states XXIX

F.1 Trap model (⁼⁰) . . . XXIX F.2 One dimension (⁰^<¹) . . . XXX F.3 General arguments . . . XXXI F.4 Monte–Carlo results . . . XXXI

G ^C(t;˜ ^S)in the limit^S^!¹ XXXIII

Bibliography XXXV

Index XL

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List of Figures

1.1 (a)Viscosityversus^T^g^=Tfor different glasses. (b)Correlation and relaxation times obtained from different measurements for the organic

glass o–terphenyl. . . 2

1.2 Typical relaxation spectrum of a glass for two temperatures^T¹^<^T². . . . 3

1.3 Dependence of the loss–part of the dielectric permeability of glycerol on the waiting time. . . 4

1.4 Torque when applying a torsion to polycarbonate. . . 5

1.5 Parametric plot of the relaxation versus the correlation function for a sequence of increasing waiting times for a mean–field spin glass model. 8 3.1 Illustration of the energy levels involved in a jump. . . 28

3.2 Tracer diffusion coefficient for a constant DOS in^d⁼¹. . . 32

3.3 Tracer diffusion coefficient for a constant DOS in^d⁼³. . . 33

3.4 Tracer diffusion coefficient for a Gamma DOS in^d⁼¹. . . 35

3.5 Tracer diffusion coefficient for a Gamma DOS in^d⁼³. . . 36

3.6 Tracer diffusion coefficient for a bimodal DOS in^d⁼³with^q⁼^0:7. . . . 38

3.7 Tracer diffusion coefficient for a bimodal DOS in^d⁼³with^q⁼^0:25. . . . 39

4.1 Effective scaling of ac–conductivity data for (AgI)^x(AgPO³)^1-x. . . 42

4.2 Effective scaling of ac–conductivity data for (Na²O)^x(B²O³)^1-x. . . 42

4.3 EMA result of the dilute hard–core lattice gas. . . 47

4.4 Scaling of the EMA diffusion coefficient for different temperatures and concentrations (“generic” version). . . 48

4.5 Scaling of the EMA diffusion coefficient for different temperatures and concentrations (BNN version). . . 48

4.6 Scaling of the EMA diffusion coefficient for different temperatures and concentrations (^T=cversion). . . 49

4.7 Mastercurve of (Na²O)^x(GeO²)^1-x glasses compared to the EMA mastercurve. . . 49

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6.1 Sketch of a rugged energy landscape of a glas with various metastable

minima. . . 64

6.2 Mean square displacement of the QJM for^d⁼¹⁰,⁼¹⁼³and⁼¹⁼⁴. . . 66

6.3 Aging function^C(t⁺^t^w^;^t^w⁾of the QJM from Monte–Carlo simulations for^(d;^;⁾⁼^(10;^1=4;³⁼⁸⁾and different waiting times^t^w. . . 68

7.1 Aging function (a)^C(t⁺^t^w^;^t^w⁾and (b)¹^-^C(t⁺^t^w^;^t^w⁾from eq. (7.20) (thin lines) and from simulations of the AJM. . . 73

8.1 Jumps out of a valley with energy^Eminafter time^t^w. . . 81

8.2 First scaling function for a parameter set with^>. . . 86

8.3 First scaling function for a parameter set with^<. . . 87

8.4 Scaling function^F²⁽²⁾for the same parameter set as in fig. 8.3. . . 88

8.5 ^C(t^w⁺^t;^t^w⁾ and¹^-^C(t^w⁺^t;^t^w⁾as a function of the scaling variable from Monte–Carlo simulations and the partial equilibrium formula. . . . 89

8.6 Ultra–slow aging behavior obtained from the PEC. . . 96

8.7 Illustration of the effect of a bimodal distribution ofvalues. . . 97

8.8 Aging functions when using a bimodal distribution ofvalues. . . 99

8.9 Three scaling regimes in the aging function when using a trimodal distribution ofvalues. . . 100

8.10 Two scaling regimes in the aging function when using a bimodal distribution ofvalues. . . 101 E.1 Tracer diffusion coefficient for a constant DOS in^d⁼². . . XV E.2 Tracer diffusion coefficient for a Gamma DOS in^d=². . . XVIII E.3 Tracer diffusion coefficient for a bimodal DOS in^d⁼¹with^q=^0:7. . . . XXIV E.4 Tracer diffusion coefficient for a bimodal DOS in^d⁼²with^q=^0:7. . . . XXV E.5 Tracer diffusion coefficient for a bimodal DOS in^d⁼¹with^q=^0:25. . . . XXVI E.6 Tracer diffusion coefficient for a bimodal DOS in^d⁼²with^q=^0:25. . . . XXVII F.1 Number of distinct visited states^S(t^w⁾for^(;⁾ ⁼ ^(1=4;³⁼⁸⁾and dif-

ferent dimensions^d. . . XXXII F.2 Number of distinct visited states^S(t^w⁾for^(;⁾ ⁼ ^(1=6;¹⁼⁴⁾and dif-

ferent dimensions^d. . . XXXII

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Notation

h i thermal average

disorder average

weighting parameter for the final state energy in the transition rate

a lattice constant

inverse temperature^1=T

0 unit of one over energy in Densities of State

b (d)

() frequency–dependent function in EMA self–consistency equation

(t

1

;t

2

) response function of an internal variable^m(t¹⁾to an external field^H(t²⁾

C(t

1

;t

2

) correlation function^hm(t¹^)m(t²⁾ⁱof some (generic) internal variable^m(t)

Æ

n;m K^RONECKER delta,¹forⁿ⁼^m,⁰otherwise

Æ() D^IRACdelta function, defined by

R

1

-1

dxÆ(x)f(x)=f(0)

d spatial dimension

D(t) time–dependent tracer diffusion coefficient 2d¹t

h x(t)-x(0)

2

i

D

0 diffusion coefficient for^t=⁰

D

1 diffusion coefficient for^t!¹

D(!) frequency–dependent tracer diffusion coefficient lim^!0^{D (}^{^} ^-^i!)

^

D(z) L^APLACE transform ^z²

2d

L[h x(t)-x(0)

2

i](z)

"() dielectric relaxation (permittivity) function

^

f(z) L^APLACE transform of a function^f(t),^L[f(t)](z)⁼

R

1

0 dte

-zt

f(t)

E

0 unit of energy in Densities of State (part I) or ground–state energy (part II)

E

n site energy on site^xⁿ

F() scaling function

F

n;k

(t) effective lattice G^REENmatrix of the EMA

2F1

() hypergeometric function

(z;E) Effective transition rate of the EMA

TR

(z;E) temperature rescaled effective transition rate of the EMA

() Gamma function

G

n;k

(t) lattice GREENmatrix

G (d)

00

() scaled lattice matrix element for^ddimensions

H(t) external field adjoint to the (generic) internal variable^m(t)

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H

n;k(t) symmetrized lattice G^REENmatrix

j

(st) stationary probability current

fugacity

scaling variable

L number of states (“lattice sites”) in one dimension (linear dimension)

m(t) internal (generic) variable, adjoint to the external field^H(t)

M(!) mobility

0 unit of hopping rate (thermally activated)

a attempt frequency

m;n,

(E

m

;E

n

) symmetrized transition rate^w^m;n^e^-E^m, write^(E^m⁾if rate does not depend on^Eⁿ

n occupation number on the lattice with components⁽ⁿ¹^;^:^:^:^;ⁿ^N⁾

N total number of lattice sites (part I) or total number of hops performed (part II)

! frequency variable

(t

1

;t

1

) relaxation function of an internal variable^m(t¹⁾when disturbed by an external field^H(t¹⁾

C(t

1

;t

2

) aging function

() waiting time distribution density

qc site percolation threshold

qEA E^DWARDS–A^NDERSON order parameter of some internal variable^m, equals ^hmi²

p

(^eq) equilibrium distribution on the lattice with components^(p⁽1^eq)

;:::;p

(^eq)

N )

() Density of States

dc long–time conductivity

(!) complex conductivity, equal to⁰^(!)⁺ⁱ⁰⁰^(!)

temperature divided by the glass transition temperature^T⁰

hopping time

x^(t^w⁾ characteristic relaxation time after^t^w

T hopping time (PARETOdistributed random variable)

t

w waiting time

T temperature in units of energy

T

0 glass transition temperature

T

g calorimetric glass transition temperature

w

m;n,

w(E

m

;E

n

) transition rate, write^w(E^m⁾if rate does not depend on^Eⁿ

!x crossover frequency

x

n lattice vector of lattice siteⁿ

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Notation

coordination number, equal to^2dfor a hypercubic lattice

z LAPLACE variable

z

E energy–dependent LAPLACEvariable^z^e^-E

Z partition sum

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1 Introduction

1.1 Glassy systems

Glassy materials are characterized by the absence of structural long–range translational order (while on short ranges the structure may be similar to a crystal) and the fact that their relaxation time becomes exceedingly long and diverges for any practical purpose when temperature is decreased below the glass transition temperature

T

g. Glassy materials are therefore never in equilibrium on laboratory or even on ge- ological time scales. For an overview see [Bin95, Jäc86, Won76]. The divergent relaxation time can be observed in a variety of different quantities as elastic viscosity, mechanical torque relaxation, conductivity spectra, dynamical light scattering or neu- tron scattering measurements. An especially well–known connection is given by the MÂXWELL relation,^'^G¹, where ^G¹ is the long–time shear modulus and the shear relaxation time. Examples are shown in fig. 1.1. As one can see, relaxation times as a function of temperature change over many orders of magnitude. This behavior can be described by a VÔGEL–TÂMMANN–FÛLCHERlaw

=

0 exp

A T

0

T-T

0

; (1.1)

where VOGEL–TAMMANN–FULCHERtemperatures^T⁰, which are determined from fits of experimental data, have been found to be smaller than^T^gand to be rather close to the K^AUZMANN temperature^T^K. (At^T⁼^T^Kthe entropy of the glass would be equal to the entropy of the crystal, if the difference of glass and crystal in the specific heats at^T^gwould be extrapolated to temperatures^T<^T^g^(min), see e. g. [Ang97].)

Glassy materials include traditional (anorganic) network–glasses like SiO²or BeF², organic glasses and polymers like polystyrene, metallic glasses, which are usually very fast cooled alloys like Sn–Cu, and systems where only a subsystem is in a glassy (dis- ordered) state like spin–glasses or orientational glasses. Strictly speaking only a material that has been prepared by cooling a melt below its glass transition temperature is called “glassy”, while other materials with intrinsic disorder are called “amorphous”.

But in the further text we will not make a strict distinction between these two classes of materials when talking about “glassy materials”.

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Figure 1.1: (a)Viscosityversus^T^g^=Tfor different glasses [Ang76]. ^T^ghas been defined by

(T

g )=10

13Poise.(b)Correlation and relaxation times obtained from different measurements for the organic glass o–terphenyl [Cah91]. The symbols show: NMR spin–lattice correlation time (), spin–spin correlation time (), dielectric relaxation time (⁺), dynamic Kerr effect (), light scattering () and dielectric relaxation ().

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1.2 Relaxation spectra of glasses

10

⁻⁶

10

⁻³

10

⁰

10

³

10

⁶

10

⁹

10

¹²

10

¹⁵

ν [Hz]

10

¹

10

⁻¹

10

²

10

⁻²

10

⁰

ε ’’

wing excess

β

fast

boson peak

α slow

T < T

₁ ₂

β [a. u.]

Figure 1.2:Typical relaxation spectrum of a glass for two temperatures^T¹^<^T².

1.2 Relaxation spectra of glasses

A typical dielectric relaxation spectrum^"⁰⁰⁽⁾ of a glassy material in double logarith- mic representation is shown in fig. 1.2. The spectrum reveals a number of peaks, that may be characterized by their peak–position and their width. Starting from low frequencies and approaching higher frequencies the contributions are calledpeak, slow

peak (JOHARI–GOLDSTEIN relaxations), excess wing, fastpeak and Boson peak.

For even higher frequencies, infrared excitations show up. The fastrelaxation can be described in the framework of the mode–coupling theory [G ¨ot91] as “rattling” in a dynamical cage. There is much evidence that the Boson peak is connected to the excess contribution to the Density of States of the phonons in the glass compared to the crystal, and a couple of theoretical explanations have been given, but none is yet commonly accepted. The slowrelaxations are not observed in all glassy materials.

In some cases they can be attributed to some known dynamical process like vibrations or rotations of special functional groups in the glass. It is known experimentally that they show thermally activated behavior.

The most outstanding feature of relaxation spectra of glasses is the large peak that evolves for higher frequencies into the excess wing. A theoretical explanation of its detailed form for a particular material is most challenging and has not yet been achieved. However, on a descriptive level there are some well established facts. The

peak is not symmetric, but the increasing wing is always steeper than the decreas- ing wing. In the time domain this peak corresponds to a non–exponential relaxation.

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There are different formulae in use for therelaxation, the most prominent being the KOHLRAUSCH–WILLIAMS–WATTSfunction (KWW)

(t) =

0 exp

"

-

t

KWW

#

(1.2) with a KWW exponentKWW^<¹. Furthermore it is known that thepeak of different materials for different temperatures can be scaled to a very good approximation on a mastercurve by the DIXON–NAGEL scaling¹: When is the peak position of the peak,^wthe full width at half maximum and^"the difference of the permeability at very low and at very high frequencies the spectra can be superimposed on a mastercurve with respect to therelaxation and the excess wing when plotting

w -1log

"

00

()

"

versus ^w^-1⁽¹⁺^w^-1⁾log

(D^IXON–N^AGELscaling)^:

1.3 Physical aging

Figure 1.3:Dependence of the loss–part of the dielectric permeability^"⁰⁰^(!)of glycerol on the waiting time^t^a[Leh98].

Fluids and crystals reach a stationary state after leaving the transient regime. There- fore, correlations ^hm(t²⁾^m(t¹⁾ⁱof some internal variable ^m(t) depend only on the time difference^t²^-^t¹ after the initial transient correlations have died out. The same

1When JOHARI–GOLDSTEINrelaxations are absent, both thepeak and the excess wing can be scaled by the DIXON–NAGELscaling.

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1.3 Physical aging

Figure 1.4:Torque when applying a torsion to polycarbonate [OCo97]. All curves collapse to one master curve when the time is rescaled with a proper time scalethat depends on the waiting time.

holds true for response functions^(t²^;^t¹⁾that describe the response of a variable at time^t²when at time^t¹a field^H⁰^Æ(t¹⁾has been applied.

The situation changes in general when glassy systems are concerned. It has been observed experimentally that response functions like the elastic modulus or the dielectric susceptibility of glassy samples depend on thewaiting time^t^w that has passed since the sample has been quenched below the glass transition temperature^T^g. Ex- amples of this dependence are shown in figs. 1.3 and 1.4. This phenomenon is called physical aging(oragingfor short) and the connected dynamics is referred to as “slow dynamics”.

The well known formulae for correlation and response functions usually depend on time–translational invariance and thus have to be generalized in order to describe aging in glassy materials. We will start with correlation functions of an internal variable^m(t),

C(t

w +t;t

w

) hm(t

w

+t)m(t

w

)i ; (1.3)

where denotes a disorder average. We assume that the sample cannot reach a stationary state for finite waiting times^t^w, in which case the correlation function can be written as the sum of a stationary part^Cst^(t)and a part^Cag^(t⁺^t^w^;^t^w⁾that contains the physical age of the system,

C(t

w +t;t

w

) = Cst^(t)⁺^Cag^(t^w⁺^t;^t^w⁾^: (1.4a) Typically the aging part^Cag^(t^+t^w^;^t^w⁾is governed for large waiting times by a scaling

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function^F^C ^t=x^(t^w⁾

with some characteristic time scalex^(t^w⁾,

Cag^(t^w⁺^t;^t^w⁾ ^F^C

t

x^(t^w⁾

for ^t^w ^!¹^: (1.4b)

Fig. 1.4 shows an experimental example, where this scaling has been successfully applied to a response function. Note that eq. (1.4b) is not the most general form: In prin- ciple it is possible that there is not only one characteristic time scale (and associated scaling regime), but that there are many characteristic time scalesx^;1^<x^;2^<^:^:^:<x^;n

and that different scaling regimes show up when the correlation function is plotted versus^t=x^;i for different x^;i. This behavior is called multi–scaling. In fact, multi–

scaling recently has been conjectured on the basis of analytical results for the Langevin dynamics of mean-field spin glass models [Cug94, Bou98].

Often the time scalexis proportional to the waiting time itself and^Cag^(t+t^w^;^t^w⁾

F

C (t=t

w

), which is callednormal aging. Another (observed) possibility issubaging;

in this casex increases slower than^t^w. In any casexwill increase with^t^w, leading to a slower and slower decrease of the correlations. Therefore, there is a subtle point when performing the limit^t ^! ¹ in the correlation function. For any finite waiting time this limit will vanish, leading to

lim

tw!1

lim

t!1 C(t

w +t;t

w ) = F

C

(1) = 0: (1.5a)

However, if eqs. (1.4) holds and the limits are taken in reversed order, we will get lim

t!1

lim

t

w

!1 C(t

w +t;t

w ) = F

0

(0) qEA^; (1.5b)

where the plateau value ^qEA is called E^DWARDS–A^NDERSON order parameter. The two limits eqs. (1.5) belong to different regimes of the two–time correlation function

C(t;t 0

): The stationary regime is reached, when ^t^w ^! ¹ is taken for some finite value of^t. In this limit the correlation function is described by the (time–translational invariant) part^Cst^(t), while from the aging part only the offset^qEA is present. The limit^t ^! ¹of the stationary regime is given by eq. (1.5b). In theaging regimethe limits^t;^t^w ^! ¹are taken for some fixed value of the scaling variable ^t=x^(t^w⁾. Here the correlation function is governed by the (non time–translational invariant) part^Cag^(t^w⁺^t;^t^w⁾, since^Cst^(t)has died away. The limit^!¹of the aging regime leads to eq. (1.5a). When considering large, but finite values of ^t^w and ^t, the two regimes can be distinguished most easily by the value of the correlation function: In the stationary regime the correlation function starts at^Cst⁽⁰⁾⁺^qEAand then decreases to^qEAwhile in the aging regime it starts at^F^C⁽⁰⁾ ⁼^qEAand decreases to^F^C⁽¹⁾⁼ ⁰. Thus, in comprehensive form we may write

C(t

w +t;t

w

) Cst^(t)⁺^qEA ^& ^qEA (stationary regime)^; (1.6a)

C(t

w +t;t

w ) F

C

t

x^(t^w⁾

. q

EA (aging regime)^: (1.6b)

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1.3 Physical aging

When formally setting

C(t

1

;t

2

) 0 for ^t² ^< ⁰^;

since at time ^t⁼ ⁰ the system has been quenched below ^T^g, we can generalize the LAPLACE transform to the two–time correlation function (and generally to two–time functions) by

^

C(z;t

w ) =

Z

1

0 dt

0

e -zt

0

C(t

w

;t

w -t

0

); (1.7)

thus leading to a waiting time dependent LAPLACE transform (and thereby the spec- tral density) of the correlation function. When eqs. (1.4) hold, this can asymptotically be written as

^

C(z;t

w )

^

Cst^(z)⁺

1

z G

C

zx^(t^w⁾

for ^t^w ^!¹^; (1.8)

where the scaling function^G^C^(zx⁾is determined by^F^C^(t=x⁾.

Note that sometimes the ansatz x⁼x^(t^w⁺^t)is used [Str78]. For normal aging this only affects the form of the aging function, since with^t=t^w

t

t+t

w

=

1+

;

while for the class of algebraic subaging behavior^t=t^w with⁰^<^<¹asymptotically for^t;^t^w ^!¹it holds

t

t+t

w

=

1+t -1+

w

;

leading to the same aging function for the two ans¨atze. Note that for practical pur- poses, where only finite waiting times can be reached, the two ans¨atze nevertheless can give different results.

When disturbing the system by some external field^H(t)and measuring the adjoint variable^hm(t)i, one gains information about theresponse function^(t;^t⁰⁾defined by

hm(t)i = Z

t

-1 dt

0

(t;t 0

)H(t 0

): (1.9)

When^H(t⁰⁾⁼^H⁰is chosen for⁰^<^t⁰^<^t^wand zero otherwise, the relaxation for times

t

w +t>t

wcan be measured,

h m(t

w

+t)i = H

0 Z

t

w

0 dt

0

(t

w +t;t

0

) (t

w +t;t

w )H

0

; (1.10)

where^(t;^t⁰⁾is calledrelaxation function. When applying a harmonic field instead of a constant field, the frequency–dependent susceptibility^(!;^t^w⁾(or the dielectric function ^"(!;^t^w⁾) is measured instead. To this end the harmonic field is switched on at time^t⁼⁰when the sample reaches the glassy phase; after a waiting time^t^w the

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adjoint variable^hm(t^w^+t)iis measured over a few periods and from this measurement the amplitude^m⁰^(!;^t^w⁾is determined and divided by the amplitude of the external field ^H⁰. From this rule of measurement it is clear that the considered frequencies need to have periods^2=! that are much shorter than^t^w. Formally the frequency–

dependent susceptibility can be obtained as a limit of the LAPLACE transform of the time–dependent susceptibility,

(!;t

w

) = lim

!0 +

^

(-i!;t

w

) = lim

!0 +

Z

1

0 dt

0

e (i!-)t

0

(t

w

;t

w -t

0

); (1.11) where, as in the case of the two–time correlation function,^(t¹^;^t²⁾⁰has been set for

t

2

<0. For both, the susceptibility^(!;^t^w⁾and the relaxation function^(t^w⁺^t;^t^w⁾, an ansatz of the form eq. (1.4) can be made in the limit of large waiting times,

(!;t

w

) st^(!)⁺

1

! G

!x^(t^w⁾

; (1.12)

(t

w +t;t

w

) st^(t)⁺^F

t

x^(t^w⁾

: (1.13)

C(t

w +t;t

w ) (tw

+t;tw +t)-(tw +t;tw

)

Figure 1.5:Parametric plot (with respect to^t) of the relaxation versus the correlation function for a sequence of increasing waiting times for a mean–field spin glass model [Bou98]. The dotted line is a limiting curve^t^w^!1.

For a system in the stationary regime the fluctuation–dissipation theorem (FDT)

st⁽⁰⁾^-st^(t) ⁼

1

T

Cst⁽⁰⁾^-^Cst^(t)

(1.14)

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1.4 Hopping models of glassy systems or in L^APLACEspace

^

st^(z) ⁼

1

T

Cst^(t⁼⁰⁾⁺^qEA^-^z^C^{^}st^(z)

(1.15) holds. In the aging regime, however, this is no longer true. When considering a parametric plot (with respect to ^t) of ^(t;^t^w⁾ ^(t^w⁺^t;^t^w⁺^t)^-^(t^w⁺^t;^t^w⁾ versus

C(t

w +t;t

w

) for some fixed (large but finite) ^t^w, the slope should be constant and equal to^-1=Tdue to the FDT. When doing this in aging systems, it turns out that this is the case for^C(t^w⁺^t;^t^w⁾ ^& ^qEA, i. e. the stationary regime. In the aging regime, i. e. for ^C(t^w⁺^t;^t^w⁾ ^. ^qEA, where the parameter of the plot can be expressed by the scaling variable and thus⁽⁾⁼st⁺^F⁽⁰⁾^-^F⁽⁾is plotted versus^F^C⁽⁾, the slope changes. This can be interpreted as some “effective dynamic temperature”

-1=Teff^(t^w^+t;^t^w⁾, where^Teffquite generally is higher than^T. An example for a mean–

field spin glass model from [Bou98] is shown in fig. 1.5.

1.4 Hopping models of glassy systems

Transport properties of glassy systems are often associated with a hopping motion in a random energy landscape. For example, for describing the dynamics of simple structural glass formers in the highly supercooled melt beyond the mode–coupling regime, hopping processes are expected to become dominant [Das86, G öt91]. A well established concept is to view the dynamics of the glass forming liquid as the motion of a point in a high–dimensional configuration space with many potential energy minima of varying depth. The first one to propose this idea was GÔLDSTEIN[Gol69], who de- veloped the picture of a potential energy landscapeÛ(whereÛ(r¹^;^:^:^:^;^r^N⁾is plotted versus the^3Nconfiguration variables) as a rugged energy landscape with many local minima. He justified the importance of the local minima by stating that:

Glasses at low temperatatures are mechanically stable against small deviations from their equilibrium positions and thus have to be near minima of the potential energy.

Experimentally a finite configuration entropy is found for glasses. Furthermore, the state of a glass depends strongly on the thermal treatment it has received in the transition regime to the glassy phase. GOLDSTEIN concludes that in the part of the potential energy landscape that represents the glassy phase, a large number of minima of different depth are accessible to the glassy system.

At the lowest temperatures transitions from one valley of the potential energy to another one have to be thermally activated (when the possibility of quantum–

mechanical tunneling is neglected).

G^OLDSTEINassumes that, although the potential energy is no local quantity, the transitions from one minimum to a near–by minimum in the potential energy landscape

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are “local” in the sense, that only particles in a small region of the sample are consider- ably displaced. He uses a consideration originally put forward by ZWANZIG[Zwa67]

regarding the vibrational spectrum to estimate the range of validity of the hopping approximation in glasses. He argues that only those vibrations can be excited in a visco–elastic sample whose period is small compared to the shear relaxation time.

Consequently the picture of a state point performing vibrations in a deep energy minimum should be justified if the main contributions to thermodynamic properties stem from frequencies obeying this condition. When assuming a DEBYEform for the spectrum with cut–off frequencymax^'¹⁰¹³Hz and taking into account that frequencies smaller thanmax⁼¹⁰contribute only a few tenths of a percent to the entropy, he concludes that (as a very rough estimate) for shear relaxation times¹⁰⁼max^'¹⁰^-12s the picture of vibrations in a deep minimum (and consequently the picture of a hopping motion between deep minima) makes sense. When comparing this to typical relaxation times^'¹⁰³s at^T^g, GOLDSTEIN concludes that the hopping approximation should remain valid even for a supercooled liquids above the calorimetric glass transition temperature. We like to note that the potential energy landscape itselfdoes notdepend on temperature, but that only the kinetic energy of the system is temperature dependent.

The hopping picture of glasses has been applied and extended by several au- thors (see e. g. [B¨as87, Ang95, Sti95]). Considerable work has been done to clarify the structural properties of the potential energy landscape. S^TILLINGER and W^EBER have shown [Sti82] that rather general arguments suggest an increase of the number of minima^(N)with the number of particle^Nas

(N) N!e N

(1.16) for a single–component system in the limit^N^!¹. Attempts have been made only recently to calculate the distribution functions of these energy minima (and other sta- tistical features) both from molecular dynamics simulations [Heu97, Don00, Ang00, Bro00] and theory [Sch98, Mez99] and to model relaxation dynamics in terms of such distributions [Die99].

For spin–glasses, hopping processes in ruggedfree energy landscapes have been proposed [Bou92, Sib97a, Sib98]. The free energy functionals are defined in a space ofpure states where cumulants vanish in the limit of large distances (^F[m¹^;^:^:^:^;^m^N^] with mean magnetizations^mⁱin the case of spin–glasses) instead of the configuration space of the particles. In these models the separation of the time scales between the time spent in a trap and the time spent to surmount a barrier can have its origin in energy barriers or in kinematic constraints created by the fact that parts of the state space are totally inaccessible (“entropic barriers”). Since free energy functionals depend on temperature, in these models the form of the rugged energy landscape itself is temperature dependent (contrary to the case of the potential energy landscape above).

Within such models, S^IBANIand H^OFFMANNhave proposed [Sib97a] that aging may be attributed to partial thermal equilibrium within one “valley” of a free energy landscape. Here theratioof the occurrence of two different states in a valley (or subspace) is assumed to have approximately the same value as in equilibrium, while theabsolute valuewill in general be very different.

Transport and aging in glassy systems