The glass transition: dynamic and static scaling approach

HAL Id: jpa-00212416

https://hal.archives-ouvertes.fr/jpa-00212416

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

The glass transition: dynamic and static scaling

approach

J. Souletie

To cite this version:

The

_glass

transition:

_dynamic

and static

_scaling

_approach

J. Souletie

Centre de Recherches sur les Très Basses

_{Températures, C.N.R.S.,}

BP 166 X, 38042 Grenoble

Cedex, France

(Reçu

le 24

_juillet

1989, révisé le 15 décembre _1989,

_accepté

le 12

_janvier

₁₉₉₀₎

Résumé. 2014 Nous décrivons la viscosité

et la chaleur

_spécifique

au

_voisinage

de la transition vitreuse dans le cadre

_classique

_{des théories d’échelle élaborées pour les transitions de 2e ordre.} On définit la transition par la réalisation du

paradoxe

de Kauzmann : pour cette _transition, toute

la différence

_d’entropie

entre les

_{phases liquide}

et cristalline se retrouve au-dessus de

Tc

où la viscosité, aussi,

diverge

par l’effet du ralentissement

critique.

On retrouve le

comportement des verres «

fragiles

» et « forts » en

_changeant

la dimension

_d’espace.

Dans la

pratique,

il semble que

chaque système

évolue d’un _comportement «

_fragile

»

(caractérisé

_par

z03BD ~ 6 et _{~ 0 ~}

10-4 poises)

vers un _comportement « fort » caractérisé par zv ~ 20 et des valeurs de ~0 apparemment non

_physiques.

Abstract. 2014 We discuss the

viscosity

and the

_specific

heat of

_{glass forming}

systems near their

glass

transition in the framework of usual

_{scaling theory}

at a second order transition. We define this transition

_through

the Kauzmann

_{paradox :}

i.e. we assume that all the _entropy difference

between the same

_crystal

and the

_liquid

phase

is found over the critical temperature

Tc

where the

_viscosity

also would

_diverge

as a manifestation of critical

_slowing

down. We

_provide

expressions

for different « dimensionalities » between the lower critical and

higher

critical

dimensions which would be

_appropriate

to describe all situations between the «

_fragile

» and the « _strong » cases. In

_practice

it seems that each _system

undergoes

a cross-over

_(or

a

_high

order

liquid

to

_liquid

_transition)

from

_fragile

to _strong behaviour when the

_viscosity

is of the order of a

thousand

_poises.

Classification

Physics

Abstracts

64.60F - 64.60H - 64.70P

In this paper we

_analyse

the behaviour of the

_viscosity

and of the

_specific

heat of

_glass

forming

systems

in the

_spirit

of what we have leamt from the

_study

of the

_dynamical

properties

of

_{spin glasses.}

The

_latter,

at some

_time,

were also

_interpreted

with the

Vogel-Tamman-Fulcher

_empirical

law inherited from the WLF

_theory

of the

_glass

transition.

Although

this

_step

was useful and is still a basis for fruitful

_analogies

the real clarification

came when it was realised that the same data could also be

_interpreted

as critical

_slowing

down near a second order transition

[1].

The

_reality

of the

_spin-glass

transition has been

established

_{statically through}

the

_study

of the

_divergence

of the

_higher

order

_{susceptibilities}

of these materials. The

_exponents,

for this

transition,

are much

_larger

than those measured in

ferromagnets :

this is the result of the effect of frustration which tends to decrease

JOURNAL DE PHYSIQUE. - T. 51, N° 9, 1cr MAI 1990

884

7c

and

hereby

increase the

_{J/ Tc}

ratio to which the

_magnitudes

of the _exponents are

_closely

related

_[2].

Small

exponents

mean

_sharp

static

_properties

and a

_dynamic

which is affected

_only

in the very close

vicinity

of

_{Tc. Large}

_exponents

_by

contrast, induce « dumb » static

_properties

and the

_dynamic

which is now affected at _very

_large

distances from

_Tc

_appears as the most

dramatic feature. It is

_tempting

to check whether some of this new

_knowledge

could be useful in an

_attempt

to reconsider the

_{phenomenology}

of the

_glass

transition. We will hereafter

analyse

this transition as a classical continuous transition with

_dynamic

_consequences

_(the

divergence

of the

_viscosity)

as well as _{static consequences}

_(the

_specific

heat

_anomaly

and the Kauzmann

_paradox)

which all reflect the

_divergence

of a correlation

_length.

Dynamics.

The

_viscosity

of

_liquids

increases on

_decreasing

the

temperature :

in a number of

glass-forming

systems

like

silica,

which are classified as «

_{strong systems »}

by Angell

[3]

it is well described

_by

the Arrhenius law

An additional

_parameter

is needed to describe the «

_{fragile »}

_{systems :}

it was introduced in

the form of a reference

_temperature

_To

in the

_{empirical Vogel-Fulcher}

law

The

_divergence

of the

_viscosity

can be related to the

_divergence

of a relaxation time

T s

through

the

relation q =

T

s G 00

where we have introduced the

high-frequency

shear

modulus

_{G 00 (with Goo -}

_{10-9, no _}

_{10- 4 poises corresponds}

to _{T 0 --}

10-13

_s).

The theories which deal with the

_glass

transition tend to view

_To

where the

_divergence

would occur as the

signature

of some kind of

_instability

of the

_equilibrium

_high

_temperature

_{liquid phase.}

In

practice

it becomes

_impossible

to check the

_validity

of this

_equation

because we have either a

first order

_melting

transition to the

_crystalline

state at

_TM

or

_because,

in the undercooled

liquid

phase,

the

_viscosity

would become of the order of 1013

_poises

at the

_glass

_temperature

Tg

larger

than

_To

_{and the time necessary} to

_perform

the

_experiment

at

_equilibrium

would become of the order of 104 s and exceed the times for which we are

_ready

to wait.

An

_interesting

_{correlation has been noticed in many}

_systems

between

_To

and the Kauzmann

temperature

TK [4].

At

_TK,

the

_liquid

_entropy

curve

_extrapolated

below

_Tg

would intersect the

crystal

entropy

curve. There is no clear

evidence,

so

_far,

that any process would

_prevent

this

alarming

intersection from

_occuring

if the

_experiment

could be carried out at

_equilibrium.

So

far, however,

the process is

stopped

at

_Tg

where the excess

_entropy

_Sex

=

Sl,q

-

Scrystal

remains

positive.

All this is accounted for in the Adam and Gibbs

_theory

_[5]

which relates the

divergence

of the

_viscosity

to the

_collapse

of the excess

_entropy

_{by introducing}

a barrier

height

which increases like

_{C 1 Sexe}

With the

_resulting

_equation

equation (2)

is recovered

_exactly

if

_{O CP ~}

T-1. Viscosity

measurements are

_currently

performed

over 12 or 14 decades. To fit the data over such enormous _{ranges it}

is,

in

practice,

usually

necessary to

_accept

some evolution of the

parameters

of

_{equation (2)}

_{with the range} covered. A

_difficulty

of the above

_picture

is

_precisely

that

_TK

tends to _appear

_consistently

observed that in some cases the ratio

B/To in

_{equation (2)}

could be

_imposed

to be the same

for a whole

_family

of

_systems

_(12.7

in

_{polyalcohols) :}

this would

_require

_{very little} sacrifices in the accuracy of the fit and it would

permit

to introduce more

_consistency

between the values

of

_To

and

_TK.

In the

_spirit

of what we have said

_above,

we will now check the

_aptitude

of the

slowing

down

_equation

to describe the same results. This

_equation

traduces the

_dynamic

effect associated with a

second order transition. It tells that

_{T diverges}

like a _{power of} a coherence

_length

_{e which}

itself

increases like

_{(1 -}

_Tc/t)-v

and

_diverges

at the transition

temperature

Tc.

We will rather use

the differential form of

_equation

₍₄₎

where 0’ -

z v Tc

is _{a measure} of the interaction

[2].

The

advantage

of

equation (5)

is that

To has been

eliminated ;

a

_plot

_{of 0, (T)}

vs. T

_yields

a

_straight

line which intersects the

0,(T) = 0

axis at

_T,

and

_{the 0,(T) = 1}

axis at

_{Tc + o’}

so that the two

_parameter

7c

and o’ are determined at a

_glance.

If we use

_{equation (2),}

or more

_generally

we obtain instead that

so that for the Fulcher law where =

1,

03A6t(T) (or

_more

exactly

T¢ T ( T ))

is _a

parabola

which

is

_tangent

to the

_T~T(T)

= 0 axis _at

To.

Figure

1 shows _a

0 , (T)

plot typical

of

glass-forming

systems

between

_fragile

and

_strong

of _very different natures :

_ionic,

_organic,

metallic.

then

data in the metallic

_compound

_{Pd77.SCu6Si16.5 [7]}

can

_equally

well be

_represented

_by

a

_straight

line or

_by

the

_parabola

which admits this

_straight

line as a

tangent. This

means that

_equation

(2)

or

₍₄₎

would fit

_equally

well the data. In order to chose between the two laws we would

need

_precisely

these data closer to

_Tc

and

_To

which

_require

so much time to be obtained. We know

_however,

from the

_properties

of the

_parabola,

that the

_intercept

_Tc

of the

_tangent

with the

_03A6T(T)

= 0 axis would sit

halfway

between the abcissa of the _contact

point

which is _a

typical

temperature

of the

_experimental

data and the summit of the

_parabola

which is

To.

For

_{given experimental}

data therefore

_Tc

will

_always

sit closer than

_To

from the

experimental

results : in

moisi

systems

this

_{places Tc higher}

than

_TK

and we have no

Kauzmann

_paradox

_{anymore. To go further} on with this

_comparison,

we _may

_expand

the

slowing

down

_expression

_(4).

_By

_truncating

this

_expansion

to its first two terms we recover the

886

Fig.

1. - The

temperature

dependence

of the

_viscosity

of different

_{glass forming}

systems is

compared

with the

_predictions

of the

_slowing

down

_{équation 11}

=

q,(1 - TcIT)-ZJI.

For

_amorphous

_{Pd77.5CU6SiI6.5}

as measured in reference

_[7]

the

_figure

shows how

_T,

= 570 K and zv = 13.9 _are determined _as,

respectively,

the

_intercept

and the

_slope

of a

_straight

line which best fits the a In

_T/a

_{In 7? vs. T} data. A Fulcher law which would

_correspond

to a

_quasi

_parabola

in this

plot,

would also fit the data within the

experimental

error. From the

_properties

of the

_{parabola Tc}

sits half way between

TF ~

505 K and the

experimental

data. Note in many systems the cross-over to a situation described

by

a

_larger

_exponent

when the _temperature is decreased. For

_typical

_{strong systems} like

_Si02

all data would sit in the Arrhenius

_{regime, perhaps}

because it was

_impossible

to

_perform

the measurements

_{high enough}

in temperature.

This identifies the

_{parameter B}

of

_{equation (2)}

with

_{0’ - z PT,}

and the observation

_[6]

that

B/To

can be made a constant for a

_family

of

_systems

would mean that these

_systems

pertain

to

a

_given

class of

_universality

as

_they

exhibit the same

_dynamical

_exponent.

To illustrate our

_point

let us take the data of

_Birge

and

_Nagel

_[21]

on

_glycerol

around 210 K. The authors fit their results over 5 decades with a Fulcher law with

_To

= 128 K and

B = 2 500 K. We

claim,

from

above,

that _a similar fit would be obtained with a _{power law}

with

_{Tc = (210}

+ 128

_)/2

= 169 K and z v = 2 500/169 = 14.7. The authors obtain

Tc

= 169 K

and z v = 15 from a direct fit.

_Similarly

in

_{propylene glycol}

around 190 K

they

find

To

=114 K and B = 2 020 from which we would deduce

_Tc

= 152 K and zv

= 13,3

instead of

7c

= 148 K and z v = 14.6 which

they

obtain

directly.

Specific

heat.

In

_glycerol,

a

_typical

_system

intermediate between «

_strong

» and «

fragile », To

= 128 K is smaller than

_{TK ~}

135 K. But in our

_picture

the

_singularity

is at

_T,

= 169 K where the

paradox

anymore.

Despite

of that

observation,

we share with

_Angell

and Smith

_[3]

the

prejudice

that the

_proximity

of

_TK

and

_Tc

is not accidental. Near

_Tg,

besides,

in a _{range where}

the relaxation times become

_{exceedingly large}

it _{is very}

_likely

that the

_specific

heat and the

corresponding

entropy

are underestimated with the result that

_TK

also would be

underesti-mated. In any case, the

suggestion

from the data is that most of the

_entropy

available for this

transition would be found over

_Tc.

This is a feature which can be

_{easily granted}

within the framework of the same

_scaling

theory

whose

_dynamical

_{consequences have been consirered above. The coherence}

_{length e}

which we have introduced defines n =

V 1 çd

renormalized « correlated »

objects

of volume

03BE d

in dimension d. We write the Gibbs

_potential

G for a _{gaz of n such}

_particles

Fig.

2. - Plot of

long 17

vs.

_log

(1 -

_TIT)

in the

_amorphous

_system

_{Pd77.5CU6S’16.5}

with

_{Tc =}

568 K as determined in the

_{representation}

of

figure

1.

By

differentiating equation (9)

with

_respect

to the

_temperature

we obtain

_[2]

the

_{entropy S}

and the

_specific

_Cp(T)

where 8 =

888

Fig.

3. The

_predictions

of the

_{scaling theory}

for the

_{viscosity r?}

_{( T)}

and the _entropy

_{S ( T)}

and the

specific

heat

_{C P ( T )}

with the constraint that the excess _entropy

_Sex

cancels at

_Tc

_{[Kauzmann’s paradox].}

We have

_{TI /Tlo}

=

(1 -

Tc/T)- B’/Tc,

S(T’)/SeX

=

[1

₊

(8 -

rj/r]

(1 -

Tc/T){B/Tc)-1

1 and

CP(T)/SeX

=

T-2(1 -

Tc/T){B/TC>-2.

The _parameters

Tlo, 6,6’

and

Sex

_are fixed. The different curves

are labelled

_by

the

Tc/6

ratio which

permits

to describe all situations between « _strong » and «

_{fragile »}

entropy

variation is shared between the two

_regimes

T --

_T,

and T>

_Tc

and there is an

anomaly

on both sides of

_{Tc. By imposing}

A = 0 _we would describe a transition with all the

entropy

variation below the critical

_temperature

as in the mean field

_theory

of

superconduc-tors and

_{ferromagnets.}

_If,

_by

_contrast, we fix A

equal

to

_Sex,

the total

_entropy

excess of the

liquid phase

over the

_{crystal phase,}

we

_imply

that the

_entropy

variation is

_entirely

in the

T >

_{T, regime}

and cancels at

_T,.

This

feature,

which

_implies

the

_freezing

of all

_degrees

of freedom at finite

_{temperatures,}

is

_present

_{in the random energy model of B. Derrida}

_{[6] :}

this is our

_{interpretation}

of the Kauzmann

_paradox.

_{With n0}

=10- 4

_poise

and A =

Sex,

assuming

0’/ 0

=

z/d

is fixed

[which,

_we

believe,

_respects

the

spirit

of the definition of the

_exponent

zv

but,

admittedly,

may necessitate further

discussion]

the

equations (4, 10)

and

₍₁₁₎

describe the

_viscosity,

the

_entropy

and the

_specific

heat in terms of the

_temperature

with one

adjustable

parameter

which is the

_{Tc/ o}

ratio. These

_predictions

are shown

_figure

3 for

representative

values of

_{Tc/ 0}

_{in the range} 0 =====

_{T 0}

where it is

_possible

to obtain solutions of the form of

_equations

₍₁₀₎

and

_(11).

The solutions

_{7c > o}

would make the

_{entropy S}

diverge

and are not

_{acceptable thermodynamically.}

For

_{Tc -->}

0,

on another

hand,

d v

_diverges

like

_8/Tc

and the

_{equations (4)}

or

₍₁₁₎

_generate

essential

_{singularities}

_{[2] :}

By sweeping

the

_{Tc/ o}

ratio over the

_permitted

_range we therefore recover situations which can be described

by

_{power laws and} are intermediate between the

_strong

case

_(essential

singularities)

and a

fragile

limit

_{Tc -., 8}

which cannot be

_overpassed

for

_{thermodynamical}

reasons. So that we account

_{semiquantitatively}

for the main feature of the classification of

Angell.

If we fix the

_z/d

ratio as we did in the

_{representation}

of

_figure

3,

it becomes

_possible

to

fix

the

_glass

_{temperature Tg}

where the

_viscosity

would reach 1013

_poise

an

_represent

the data in terms of

_T/Tg

as is common

_practice

_among

_{experimentalists.}

The

_figure

3

_reproduces

the

_general

features

_qualitative

and

_quantitative

of the variations which are

_{experimentally}

observed. In

_particular

we

_find,

in

_agreement

with the

_experiment,

that the

_jump

in the

specific

heat which is observed near

_Tg

is the

_larger

for the most

_fragile

_systems.

It becomes almost unnoticeable for the

_{strong systems}

_{being squeezed by}

the

_exponential

term of the

Schottky-like

contribution.

The continuous variation of the

_{Tc/ 0}

_{ratio which sweeps all}

_permitted

solutions amounts to

a continuous variation of the

_exponents

as we would

_expect

if we were able to _vary

continuously

the space dimension between its

higher

critical and lower critical values. This raises a

_difficulty

as we are not

_prepared

to do so : on the

_contrary

the existence of a finite

number _{of space dimensions has} a _{well known consequence which is the existence of} a finite

number of

_{corresponding}

« classes of

universality

». We will now see that if we

_accept

the idea

of a cross-over it is

_possible

to account for the same data in better detail and restrict to a finite

value

_{(possibly two)}

the number of different

_universality

classes.

Cross-overs.

Closer examination of

_figure

1 and of the inserts of

_figures

4 to _{7 shows that in many}

_systems

890

Fig.

4. - Neither _a

unique

critical

_slowing

down

_equation

nor a

_unique

Fulcher law would describe the

two

_regimes

that the a In

_T/a

In T

diagram

shows in insert. In the

_plot

of

_{log v}

vs.

0’/ T

where o’ =

zv Tc

the

viscosity

of

a-phenyl-o-cresol

would

diverge

when

0’/ T =

z v . Notice the remarkable

stability

of the

_high

_temperature

_dynamic

_exponent zv for _{systems of very different} natures

_(Figs.

4 to 7 and Tab.

_I).

TR

from a

_regime

with a smaller

exponent

to a

_regime

with a

_larger

_exponent

when the

temperature

is decreased below a _{range where the}

_viscosity

is of the order of 1 000

_poises

(see

Fig.

8).

Either side of the cross-over

_regime

can be described within the

_experimental

error

_by

a _{power law} or

_by

a Fulcher

law,

as we have said

above,

even if the range covered extends to 7

or 8 decades. However neither law can describe both sides of the cross-over without

systematic

errors. This

_point

has been

_recognized

_by

Macedo and

_Napolitano

_{[9]. [The}

error

curve 7J theor

-

77 _exp vs. T which

_they

deduced in

_B203

has a characteristic N

_shape.

This N results from the fact that the

_{corresponding 0t (T)}

curve has two different

_segments

that a

parabola

(Fulcher law)

or a

_straight

line

_{(power law)}

intersects in two

_points.

There are two

points

therefore where the

_slope

a In

_Tla

In T is correct and there are two

_{corresponding}

extrema for the error

curve.] Similarly Laughlin

and Uhlmann

[8]

have observed that the best

Fulcher law which describes the

_high

temperature

range in

organic liquids

deviates after 5 to 6 decades and that a different law would be needed to describe the lower

temperatures.

Bondeau and Huck

_[10]

on their

_side,

advocate for a continuous

_spectrum

of

_parameters

whose values

_depend

on the observation window.

_Finally

Taborek et al. stress _{that in many}

liquids

the

_viscosity

first evoluates like a _{power law} and then deviates towards Arrhenius

behaviour

_[14].

We

_similarly

find two

_regimes

for our _{power laws in many} cases. In the

_high

_temperature

Fig.

5. - After a

_high

_temperature « universal »

_regime

(zv - 6-7)

where the

_viscosity

is enhanced

from 10-4 to 103

_poises

the _system deviates to a second

_regime

described

_by

a

_larger

_exponent and an

unrealistic qo

traducing possibly

some reduction of the

_{dimensionality}

of the system. The same

description

can be

_applied

to all _systems where like in this

_propylene

carbonate we have data

_covering

viscosities much over and much below 1

_poise

_(data

from Ref.

_[10]).

See also

_figures

4, 6 and 7.

system

(see

Figs. 1

and 3 to

_8,

Tab.

_I,

and the discussion about the method which is made in

the note at the end of the

_paper).

The

_{corresponding}

_{n o is of the order of}

10- 4

to

10- 5

_poise

which would

_correspond

to a _{characteristic time Tao -}

10-13

to

10-14

s. None of

these values would

_surprise

_people

familiar with the

_dynamic

_properties

of

_{spin glasses}

(zv -

6-8 _{and T 0 ~}

10-13

s at 4

_K).

More

_recently

_[15],

still

_larger

_exponents

have been measured which characterize other classes of

_{spin glasses}

_{(zv ~}

oo in

_FeMgCl2,

zv ~ 15 in

CdMnTe... )

so that the values of the

_exponents

_reported

in the lower

_temperature

_regime

of the

_{glass forming}

_{liquids (see}

Tab.

_I)

would not raise too _{much emotion among}

_spin-glass

specialists

either : the

_problem

is associated _{with the To values which appear}

_extremely

scattered and

_unphysically

small in this second

_regime.

Such a

situation,

_though,

is the

anavoidable _{consequence of the fact that the} state with

_higher

_exponents

is obtained

_through

a cross-over at finite

_temperature

as is shown in the

_figures

1 and 3 to 8. The two

_intersecting

straight

lines in

_{the ~t ( T)}

_{plot correspond,}

in a In q

vs. 11T plot,

to two curves with different

curvatures which are

_tangent

to each other at the cross-over

_{point ; they}

have

necessarily

different intersections with the

_{1 / T}

= 0 axis and

correspond

therefore to different

To values. If we

_{have ) =}

_{)o (1 - Tc/T )- y}

in the first

_regime

and

_if

_{)i(1 - Tc/T )-}

v

,

892

Fig.

6. -The

logarithm

of the

_viscosity

of

_{KN03-Ca(N03)2}

vs.

8’/T

(data

from Ref.

[11]).

Fig.

7. - The

_logarithm

of the

_viscosity

of

_{B203 (data}

from Ref.

_[9])

vs.

_{0’/ T.}

Notice the

_larger

Table 1. - We

give

for

different

systems :

the lower and

higher

temperature

Ti

and

Tf

where measurements are

_available,

the

position

_of

the cross-over

_TR

and then

_for

the

_higher

and the lower

_temperature

_range

_respectively

the

_{best parameters}

_{0’, T,}

zv, q 0 which

fit

the data

with the critical

_slowing

down

_equation

q =

qo(1 - Tc/T )- zv (0’

=

z v Tc ).

Notice the

unrealistic values

_{of no}

in the

_temperature

range below

TR

for

all systems

except

silica

contrasting

with the remarkable

_stability

_{o f all}

_parameters in the

_higher

_{temperature range when}

accessible.

and

_{e’ 0}

is bound to _appear

_unphysical

if

_eo

is

_physical.

In the case where we cross-over from a

lower to a

_higher

dimension such as at the 3d transition of

_quasi

Id or

_quasi

2d

_ferromagnets

[2]

the

_exponent

y’

of the

_{susceptibility}

is smaller below

_TR

than in the first

_regime

and this leads to

_giant

moments which are an

_accepted

_{consequence of the presence of short range}

order.

_By

contrast, the cross-over which is observed in

_glasses

_implies

an increase of the

exponents

and is

_suggestive

of a restriction of the

_{dimensionality}

such as the 3d to 2d

cross-over which is

_expected

when the correlation

_length

becomes of the order of the thickness of

the

system.

Here we _may

_speculate

that a

_comparable

restriction results from the

_slowing

down effect itself : some of the

_degrees

of freedom become

_quenched

at

_typical

values of the

relaxation time. For

_example

if a

_{particular species}

of

_permutations

or rotations of chains or

of atoms becomes

_hampered

the result could be described as the occurrence of some sort of

quenched

disorder and of a new situation with new

_properties

whôse

_description

would call

for a different hamiltonian

(e.g.

_quenched

vs.

annealed).

The

_higher

_temperature

liquid

1

phase,

if it is

_universal,

would not _{favor any}

_special

kind _{of short range} correlations. We would thus have

_{simultaneously}

_{correlations which favour say cubic and}

_hexagonal

_symmetry,

etc... This frustrated situation would lead to a

_complete

_blocking

of the

_degrees

of freedom at

Te.

We may suppose that in the

liquid

II

_phase,

at lower

_{temperatures,}

some

_particular

correlations are favoured :

presumably

those which determine the

_symmetry

of the

crystal

phase.

This

_point

of view has been very

recently

substantiated

_by

the observation made

_by

Dupuy

and Jal

_[22]

that,

in

LiCI,

6

_H20,

the

_temperature

_TR

which

_they

identified

_using

our

894

transformation in the

_{equilibrium phase diagram.}

It seems,

_altogether,

that the

_systems

react

so as to

_postpone

to a lower

_temperature

and even down to 0 K the crisis which would occur

when the

_entropy

cancels. In the same

_regime

mode

_coupling

theories

_predict

_[16]

and

neutron

_spin-echo

as well as

_optical

studies reveal

_[17]

unconventional stretched

_exponential

relaxations which seem the

_{temporal signature}

associated with this

_ambiguity

on the choice of

Tc.

The above discussion

_suggests

a scenario for the

_glass

transition which would account for

the main features of

_Angell’s

classification

_{by assuming}

the existence of

_only

two or three

universality

classes. The situation is summarized on the

_figure

8 where the different

temperatures

which are relevant to the discussion are ordered with

_respect

to each other. In

all

_systems

we would

_expect

a

_high

_temperature

_regime

characterized

_by

universal

_exponents

and

parameters

which have

_{physically acceptable}

values

_(e.g.

zv - 6-8

_{and no ~}

10- 4

_poise).

This

_liquid

_regime

is observed over the

_melting

temperature

TM

and is unmodified in the

overcooled

_regime

below

_TM.

It can be described within the classical framework of second

order

_phase

transitions which assumes the existence of correlated

_objects

whose

size e

increases and tends to

_diverge

when the

_temperature

_approaches

_T,.

At some

_temperature

TR > Tc

however,

where e = eR

is still

_finite,

the

system

undergoes

the

_equivalent

of a

reduction of its space

dimensionality,

the new

_{liquid being}

characterized

_{by larger}

exponents

(z v ~

15-20 or z v -

_{oo )}

and

_{corresponding « unphysical »}

parameters

(n0~

10- 20

_etc...).

It is the

_magnitude

of the

_{T RI Tc}

ratio which will decide of the situation of the

_experimental

window with

_respect

to

_TR.

This ratio

_{obviously depends}

on the

_{microscopic properties}

of the

system.

It is small in

_Be203

or in the

_polymers

and the

_{liquid regime}

with zv ~ 6 is observed down to 10 % of

_T,,.

The

_viscosity

variation of these

_systems,

is therefore almost

_entirely

determined

_by

the « small »

_exponent

z v and

_they

are identified as

_fragile.

In

_{Si02 by}

contrast

or in

_BeF2

or

_Ge02

the

TRI Tc

ratio is

_large

and the observation window is confined to the T

_{TR regime.}

Such

_systems

are well described

_by

an Arrhenius law. In

_BeF2

the fact that

n 0 10- 4 lets

it be

_suspected

that there is some sizeable range of

_temperatures

T>

_TR

where the

_system

_might

be observed with a smaller

_exponent

_provided

the

temperature

could be raised

_{high enough.}

A

_{strong system}

like

_Si02

where we have at the

same time an Arrhenius law and an almost reasonable

no - 10- 8 poise

would appear as

representative

of the

_{« strong}

limit » where the

T R/ Tc

ratio tends to

_infinity.

The same discussion should

_apply

also to the

_specific

heat. The

_figure

8 shows

_readily

that

equation (11)

with a - 0

_(i.e.

_{Tc/8 ’"}

_0.5)

accounts for the main features of the

_specific

heat

anomaly

of

_fragile

_systems

as

_they

are

_given

in a

_compilation

of data which is due to

_Angell

[3].

We have not

_attempted

a more detailed check of our

_{argumentation}

about the cross over

as it would necessitate data around

_TR,

denser and more accurate than those which we know :

equation (13)

with two sets of constants for both sides of

_TR

has more

_parameters

than is

necessary to fit the data in the _{restricted range of}

_temperatures

_Tg

T

_TM

where we _may

measure at the same time the contribution of the overcooled

_{liquid phase}

and that of the

crystal.

Our

_argument

however

_permits

to answer

_qualitative

_questions

that the

_experiments

raise

_(see

for

_example

the discussion

_{by Birge}

_[21]

of his data in

_glycerol).

We _{may have} a

_step

discontinuity

with an

_exponent

around a - 0 and at the same time avoid an

_entropy

catastrophe

at a lower

_temperature

in these

_systems

whose

viscosity

departs

to Arrhenius behaviour because in our model the

_specific

heat would then

correspondingly

depart

towards

Schottky-like

behaviour as shown

_figure

8c. The

_problem

is then

_simply

to match at

TR

the

_magnitudes

and

_slopes

of a

_high

_temperature

«

fragile »

contribution

(a - 0)

with a

«

_strong

» low

_temperature

tail. The

adjustment

which

implies

to redistribute the

_entropies

so

Fig.

8. - The

_figure

a shows a

_{compilation by Angell}

_[3]

of

_specific

heat anomalies in different intermediate _systems on the

_fragile

side. The dash dotted line

_{ACP ~}

T-2

for T>

_T,

is the

_prediction

of

our model for a = 0. The fit _assumes that the contribution of the

crystal

is

_essentially

constant over the range covered. The

figure

b shows the -

d In

_T/a

In _q vs. T

_diagram

for a _system on the _strong side

(T RI Tc

large,

case

_I)

and for a _system on the

_fragile

side

_{(case II).}

The

_figure

c shows the

_{corresponding}

specific

heat anomalies

_assuming

the data are

_{represented by}

a = 0 in the «

fragile » high

_temperature

regime

and

_by

a

_Schottky

tail in the _strong low _temperature

_regime.

Below the cross over at

TR

an areas rule should hold in a

_{(ACp) /T}

vs. T

_diagram

between the

_Schottky

contribution and the

extrapolation

of the

_high

_temperature

_regime

down to

_Tc.

T RITe

ratio which

_again

determines whether the

_system

will be seen as

_{« fragile »}

or

«

_strong

».

To summarize it seems that the

glass

transition could be described

_by

a series of

_liquid

to

liquid

cross overs from a

_high

_temperature

_{« fragile » regime}

to a

_{« stronger »}

low

temperature

regime

with

_larger

z v, smaller

_Tc

and

_unphysical

_{n o values. We} found indications

896

different classes of

_universality

characterized

_{by z v - 6 :!: 1, z v - 20 :L- 5}

and z v - oo. In some

cases, one

_exponent

can describe viscosities over _up to 7 or 8 decades. But two

_exponents

at

least are _necessary to describe variations over 12 decades or so.

Discussion.

We have shown that it is

_possible

to make a coherent

_description

_{of many of the effects which}

characterize the «

glass

transition » in the framework of the classical

_{scaling theory}

_{of second}

order

_phase

transitions. This

_point

of view does not _{introduce very}

_spectacular

differences

with the traditional

_point

of view.

_Essentially

Adam and Gibbs _{propose that} T ~

Ta exp

(W/T)

where W increases like

S-1

i.e.

like e

(1/v)- d when

we _propose,

_instead,

that

W/ T ~

z

_{In e.}

In

_{spin glasses}

Fisher and Huse

_[18]

have

_{recently proposed}

that W increases like a _power

of e

which amounts to a reformulation of the Adam and Gibbs result.

_By

contrast

Rammal and Benoit and

_Henley

_[19]

have

_given

numerical and theoretical evidences that W increases like

_{In e}

on

_percolating

clusters. So that the discussion _{which opposes Fulcher law}

vs. _{power law is also actual in}

spin glasses.

We have

_quoted

some

_examples

of the

rationalizations that the new

_point

of view makes

_{possible :}

one

_example

is the identification

of the

exponent

zv with the

B/Tc

ratio and the

justification

of the

_stability

of this ratio.

Another interest is the connection which is made with a

_theory

which _{is very}

_general

and whose consequences are well established. For

_example

we are led to conclude that

7c

is a real

_instability

of the

_liquid

_phase

_{and it should exist in any}

_liquid.

The

_{crystal phase by}

contrast is the result of a first order transition and it is not connected with the

_{liquid by}

fluctuations _{of any kind : the}

_temperature

_TM

therefore where the

_melting

transition will

occur is not

_predictable

from our

_{phenomenological}

_argument.

It

_depends

on

_{microscopical}

properties

of the

_systems

and it seems that

_appropriate

information can now be obtained in the framework of the

_density

functional

_theory

of

_freezing

_[20].

The

_question

then is what is

the situation of

_TM

with

_respect

to

_{Tc ?}

If

_TM

is much

_larger

than

_Tc

there is a

_large

_entropy

difference between the

_liquid

and the

_crystal

which is used under the form of a latent heat at

TM

where the transformation takes

_place.

However the fluctuations associated with the 2nd order transition at

_Tc

can

_explain

deviations in the

_specific

heat or an increase in the

viscosity

which are

_actually

observed in the

_{liquid phase}

over

_TM

and cannot be

_easily

understood in the framework of an

_approach

to a first order transition. If

_TM

becomes closer

to

_7c

the

_slowing

_{down associated with the second order transition may become}

_{large enough}

to

_hamper

the

_growth

_{of any seed of the}

_{crystal phase}

and the

_system

will

_easily

become a

glass.

The difference in

_entropy

which

_persists

at

_Tg

will be then at the

_origin

of the time effects which are

_classically

observed below the

_glass

transition.

Finally

we estimate from the above that the

_glass

transition is well described in the framework of usual

_scaling

theories as a succession of aborted second order transitions. We see no

_strong

motivation to introduce as is

_generally

_done,

the

_concept

of a

_{« dynamical}

transition » since there are

_huge

static effects visible on the

_specific

heat for

_example.

II

remains that the

_{really convincing}

evidence which would substantiate our

_point

of view would

be a direct determination of the correlation

_length

that our model presupposes.

Note about the method :

While we _{agree with Taborek et} al.

[14]

that there are two

_regimes

in the

relaxation,

we differ on the estimate of the « universal »

_exponent

in the

high

_temperature

regime

where we obtain

6

_typically

when

_they

obtain 2 sometimes for the same

_system.

One reason is that we use, over

T,,

the non linearized

_{equation 03BE/03BEo}

=

(1-

Tc/T)- v

when

they

_use the linearized

expression

« paramagnetic »

limit

_{provided 8}

=

v Tc is

finite

also

leads,

as we have _seen, to essential

singularities

in the «

_strong

limit » where

Tc

vanishes which is the same limit of the variable

Tc/T.

By

contrast the linearized

_expression

tends to T- ’ which is an absurd result. When one

tries to fit data far from

_Tc,

with the linearized

_expression

one finds

[2]

an effective

_exponent

which tends to decrease the

_higher

the

_temperature

range : it is about divided

by

2 at

2

_Tc

and cancels when

_Tc/T

vanishes in order to

_compensate

this built-in artefact of the

formula.

Besides,

Taborek et al. use a criterion

which,

unlike our

_equation

(5),

is not

objective

since

_they

determine

_Tc

which

_gives

the best

_linearity

in a

_plot

of

B/n

vs. T and

_they

naturally

obtain with this

_Tc

the

_exponent

2 which

_they

presupposed.

Our

criterion,

by

contrast, has no other

_prejudice

than the

assumption

that the non-linearized

scaling

equation

is valid. One should not

_forget

_{in any} case that the determination of

exponents

remains a

tricky

question

where one does the best with the available accuracy and his own

prejudices...

All

_{interpretations}

therefore,

including

ours, should be taken with a «

_grain

of salt » until an

agreement

on the method is obtained.

Acknowledgments.

The author is

_grateful

to J. Chevrier and M.

_Papoular

for many

illuminating

comments... He also wishes to thank the

_organisators

of a

_meeting

in

_Aspen

(in 1985)

where he

_got

acquainted

with many of the

problems

discussed in the

_present

_paper.

References

[1]

BONTEMPS N., RACHENBACH _J., CHAMBERLIN R. V. and ORBACH R.,

Phys.

Rev. B 30

₍₁₉₈₄₎

6514 ;

SOULETIE J. and THOLENCE J. L.,

Phys.

Rev. B 21

₍₁₉₈₅₎

516 ;

OGIELSKI A. T. and MORGENSTERN I.,

Phys.

Rev. Lett. 54

_(1985).

[2]

SOULETIE J., J.

_Phys.

France 49

₍₁₉₈₈₎

1211.

[3]

ANGELL C. A., in « Relaxations in

Complex Systems

_», K. L.

Ngai

and G. B.

Wright,

Eds.

(National

Technical Information Service of U.S.

Department

of Commerce,

1984)

pp. 3-11 and Nuclear

_Phys.

B

_{(Proc. Suppl.)}

5A

₍₁₉₈₈₎

_{69 ;}

ANGELL C. A. and SMITH D. L., J.

_Phys.

Chem. 86

₍₁₉₈₂₎

3845.

[4]

KAUZMANN W., Chem. Rev. 43

₍₁₉₄₈₎

219.

[5]

ADAM G. and GIBBS J. H., J. Chem.

_Phys.

43

₍₁₉₆₅₎

139.

[6]

DERRIDA _B.,

_Phys.

Rev. B

₍₁₉₈₁₎

2613.

[7]

CHEN H. S., J.

_Non-Cryst.

Sol. 27

₍₁₉₇₈₎

257.

[8]

LAUGHLIN W. T. and UHLMANN D. R., J.

_Phys.

Chem. 76

₍₁₉₇₂₎

2317.

[9]

MACEDO P. B. and NAPOLITANO A., J.

_Phys.

Chem. 49

₍₁₉₆₈₎

1887.

[10]

BONDEAU A. and HUCK J., J.

_Phys.

France 46

₍₁₉₈₆₎

1717.

[11]

WEILER R., BLASER S. and MACEDO P. B., J.

_Phys.

Chem. 73

₍₁₉₆₉₎

4147.

[12]

BRUCKNER R., J.

_Non-Cryst.

Sol. 6

₍₁₉₇₁₎

177 ;

FONTANA E. H. and PLUMMER W. H.,

Phys.

Chem. Glasses 7

₍₁₉₆₆₎

139.

[13]

NEMILOV S. V., Fiz. Tverd. Tela 6

₍₁₉₆₄₎

1375 ;

MOYNIHAN C. T. and CANTOR S., J. Chem.

_Phys.

48

₍₁₉₆₈₎

115.

[14]

TABOREK _P., KLEIMAN R. N. and BISHOP D. J.,

Phys.

Rev. B 34

₍₁₉₈₆₎

1885.

[15]

BERTRAND D., REDOULES J. P., FERRÉ J., POMMIER J. and SOULETIE J.,

Europhys.

Lett. 5

₍₁₉₈₈₎

271 ;

DEKKER C., ARTS A. F. M., DE WIJN H. W., VAN DUYNEVELDT A. J. and MYDOSH J. A.,

Phys.

Rev. Lett. 61

(1988)

1780 ;

898

[16]

GÖTZE W. and SJÖGREN L., J.

_Phys.

C. Solid State

_Phys.

20

(1987)

879.

[17]

MEZEI F., KNAAK W. and FARAGO B.,

Phys.

Rev. Lett. 58

₍₁₉₈₇₎

571.

[18]

FISHER D. S. and HUSE D. A.,

Phys.

Rev. Lett. 56

(1986)

1601.

[19]

RAMMAL R. and BENOIT A.,

Phys.

Rev. Lett. 55

₍₁₉₈₅₎

1949 ;

HENLEY C. L.,

Phys.

Rev. Lett. 54

₍₁₉₈₅₎

2030.

[20]

RAMAKRISHNAN T. V. and YUSSOUFF M.,

Phys.

Rev. B 19

₍₁₉₇₉₎

2775 ;

ALEXANDER _S.,

_Phys.

Chem.

_Liq.

18

₍₁₉₈₈₎

_{207 ;}

BRAMI B., JOLY F., BARRAT J. L. and HANSEN J. P.,

Phys.

Lett. A 132

₍₁₉₈₈₎

187.

[21]

BIRGE N. O and NAGEL S. R.,

Phys.

Rev. Lett. 54

₍₁₉₈₅₎

2674 ;