Glasses and spin glasses: a parallel

HAL Id: jpa-00246440

https://hal.archives-ouvertes.fr/jpa-00246440

Submitted on 1 Jan 1991

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.

Glasses and spin glasses: a parallel

J. Souletie, D. Bertrand

To cite this version:

J. Souletie, D. Bertrand. Glasses and spin glasses: a parallel. Journal de Physique I, EDP Sciences,

1991, 1 (11), pp.1627-1637. �10.1051/jp1:1991230�. �jpa-00246440�

J. Phys. I France 1

(1991)

1627-1637 NOVEMBRE1991, _PAGE 1627

Classification

Physics

Abstracts

64.60F- 64.60H 64.70P 75.40G 75.10N

Glasses and spin glasses

_{: a}

parallel

J. Souletie

(I)

and D. Bertrand

f)

(I)

Centre de Recherches _sur )es Trds Basses

Tempkratures,

C-N-R-S-, Laboratoire associk h l'Universitk

Joseph

Fourier de Grenoble, B-P- 166 X, 38042 Grenoble Cedex 9, France

(2) Laboratoire de

Physique

des Solides, I-N-S-A-, Avenue de

Rangueil,

31077 Toulouse Cedex, France

(Received 21 May 1991, accepted in

final form

18

July

1991)

Rksulnk. Nous montrons que la classification entre verres « forts _» et verres «

fragiles qui

est

populaire

_pour les

liquides trempks

peut ^{dtre aussi}

appliquke

_{aux verres} de

spins.

Los caractdres

fragile,

interrnkdiaire _et fort

distinguent

^des classes d'universalitk diffkrentes dans

l'approche

habituelle _en lois d'kchelles. Alors _que la classe d'universalitk d'un _verre de

spins

refldte la

plus

_ou

moins

grande anisotropie

des interactions, _on observe dans les

liquides fragiles lorsqu'on

baisse la

tempkrature

des cross-overs vers des

rkgimes plus

_« forts

qui pourraient

dtre

interprdtks

_{comme une} restriction de la dimensionalitd

d'espace.

Abstract. We show that the classification of

glasses

^from strong » to _«

fragile

_{» can} also be

applied

to

spin glasses. Fragile,

interrnediate and strong ^{behaviours would} mean different

universality

classes and the strong ^{liInit would} correspond to a lower critical dimension in the

framework of the _current scaling

approach.

While the class of _a

spin

glass _seems to be attached _to the frozen in

anisotropy

of the interactions, there _are

experimental signs

in

glass-forming

systems that _a strong

regime

is the result of _{a cross-over} from _{a more}

fragile high-temperature

situation.

The _same fact _can also be

interpreted

as a cross-over towards _{a more}

anisotropic

state or as a

restriction of space

dimensionality.

I. Introduction.

In

figure la,

the

logarithm

of the

viscosity

_1~ Of _a number Of

glass-forming

systems

[1-7]

is

represented

_vs.

~

where

1~ 10~~

poises

at T~. ^The

viscosity

is related to the shear relaxation T

time _T

by

the

equation

_1~

=

G~

_T. The

high frequency

shear modulus

G~

is of the order of 10~.

Figure

I therefore is also _a

plot

of

log

_T vs.

T~/T

^where _T~

corresponds

to T 10~ _s. As 10~ _s is _a

long

time in

laboratory scale,

_T~

represents

a natural limit _to the range

where

experiments

have been

performed

at

equilibrium.

In _{some cases}

1~

(T~)

^{is obtained}

by

an

extrapolation

of the data at

temperatures

^T _{~ T~ with} one of the

expressions

which _we will discuss below. This

plot,

which is due to

Angell [I],

contrasts dilTerent behaviours. The

viscosity

data of the «strong» systems

align

_{on a}

straight

^line in

Angell's diagram

and therefore

obey

_an ^{Arrhenius law}

~/~o

_{= exp}

(@/n (1)

1628 JOURNAL

Dg PHYSIQUE

I bt I1

logic(n/P) logic(T/s)

~ ~

~~e~ylene carbonate ~

~~~$S~~i

+ Selenium + RbzcUorsceomF4

o Taluene o cdo 7Mno iTe

~~ ^a KN03ca(N03)1 a cdosMno4Te

D a-Phenyl,~eresal U Feo~MnosTiO3

W Salol ~

o CuMn4.6%

o Sioi EUo4SrosS

5

.

. _5

o -io

,5

,15

o

Tg/T

bt 11 GLASSES AND SPIN GLASSES _: A PARALLEL 1629

fragile side, sitting

between CuMn4atflv

[8]

and Mn aluminosilicate

[9].

However _recent

measurements

in

Feo~MgJ7Cl~ [10], Rb~cuo_78Coo,22F4 II Ii, Cdo_6Mno_4Te [12], Feo.5Mno_5Ti03 [13]

have filled the range between the

fragile

and the

strong

^{limit. The}

dynamical

data in

spin glasses

_can be fitted with the _same Fulcher law which _was introduced for the structural

glasses (Eq. (2)

with _«

=

1) [14]. However,

when the existence of _a

phase

transition _was established

on the basis of the

divergence

of the

higher

order

susceptibilities,

it became natural

[8, 15]

to use the

slowing

down

equation

T/To

=

(f/fo)~

=

(l TJT~~~~ (3)

which is the

dynamical counterpart

^{of the} same

scaling theory

which accounts for the static result. As

Castaing

and Souletie

[17]

have

shown,

the

dynamic scaling

follows in _a model which assumes that _a hierarchical

procedure

_governs the evolution from step n to step

n + I of Kadanofrs renormalisation. More

specifically,

if _we decide that _a cluster of order

n+ I is formed when _p clusters of order _{n are} in _a

given position

_we have T~~j =

2~ T~ = 2" _To when

simultaneously f~

~j ⁼

bf

~ ⁼

b~

to.

On

eliminating

_n between these two

equations

_one

justifies

the

dynamic scaling hypothesis T/To

₌

(f/fo)~

where _z

= p In

2/In

b

(4)

The

slowing

down

equation (3)

follows from the static

scaling hypothesis

which states that there is _{a power} law

divergence

at

J~

in the

temperature dependence

of the correlation

length

f

which is otherwise _an

analytic

function of

I/T

_at ~ll

temperatures larger

than

T~. ^We have therefore _:

f(T~/f0

"

(' (

^~

_('

~

~~j~'

=

(I-() _~Xp(i/T~.

Per definition

p(I In

remains finite for Tm T~ ^{and it} is assumed to be constant in the

approximation

which leads to

equation (3).

In the

high temperature

^{limit therefore}

t(n/to= (i

+

vTc/T+. )p(i/T~

where

vT~ = @' should remain finite _to insure that the

expression

remains

analytic (that

it _can be

expanded

in _terms

of-1/7~.

With this constraint

equation (3)

_generates essential

singularities [18]

when T~ ^cancels as v

diverges

like

@'In

^and one obtains

equation (I)

which would be

appropriate

for _{« a} lower critical

dimensionality

for which T~ = 0. The

exponents,

it is

expected,

should

depend only

on features which survive when

f diverges,

like the space

or the

spin

dimensions.

Systems

which share such features

belong

to the _same

universality

class. In _non frustrated systems the

universality

_even resists the introduction of _some disorder

[19]. Very

little however is known _on the elTect of frustration _so that the

principle

of the

universality

of _zv in

spin glasses

is _not evident.

One

difficulty

with

equation (3)

is that it has three

parameters.

^We propose in this _{case, to} take the

logarithm

and dilTerentiate _to obtain

[18]

ajnT

T-T~ T-T~

Py(T~"-@"

~~ ⁼ ~

(5)

which has

only

two

parameters.

It is then

possible, by dilTerentiating

the

experimental

data to

1630 JOURNAL DE

PHYSIQUE

I bt II

obtain _an

objective

criterion which determines these two

parameters

ⁱⁿ a

unique

step as the data of

P~(T)

_vs. T should

align

_{on a}

straight

line which intersects the T axis at T~ and the T=0 axis at

-I/zv.

The _same line intersects the

P~(T)=

I line _at

J~

+ where

= z@'

=

zvT~.

2.

Dynamic scaling

and the

glass

transition.

Figure

2a shows the

P~(T) plot

for

representative glass-forming

_systems between strong and

fragile [20].

The

following

features _are remarkable _:

a)

In many systems on the

fragile

side _we have _a

high temperature regime

where the

viscosity

is well

approximated by

_{a power} law with exponent zv 6.2 ±1.

b)

On

decreasing

the

temperature

to

TR

^{close to} T~, ^most of these

systems

_escape ^towards a

situation of lower

T[

hence of

higher (zv I'

and reach

eventually

Arrhenius

(strong)

behaviour

which

corresponds

to the T~ = 0 limit. We have found

(zvl'~15

in

KNO~Ca(NO~)~

and

Licl,

5.75

H~O [21], (zv1'

23 in

propylene

carbonate

[4], (zvl'

- oo in

B~O~

_or

a-Phenyl-

o-Cresol. In these last two systems where the data _{cover a}

particularly

wide range of decades

P~(T)

₌

-dlog(T)/dlog(q) Pr(T)

=

-dlog(T)/dlog(T)

o.2

o.05

o.1

T1t T1t

0

~1t

~~'~~

. Feo3M8o7Cb

1 _

" AlzMn3S130m

~~ +

_~ i ^{. Bz03}

m Propylene carbonate

-11

^{° ~~° ?'~°° 3~°}

. Selenium ,o i ^{° Cdo«Mno,Te}

I _

o Toluene ^° Feo smno sTi03

~~~

° KN03cajN03j9 ^{° CuMn4.6%}

-0.2 ^{* zuo4sro «s}

0 500 1000 0 lo 20 30

T/K T/K

Fig.

2. A

plot

of ^{~ ~~ ~} _vs. T shows that the data in the

high

temperature

regime

of several 3 In _~

different

fragile

systems

point

towards _an average P(~ value which is of the order of 0.16 and therefore _can be described with _a universal

dynaInic

exponent _zv of the order of 6.25. The

plot

of

figure

2a dramatizes the cross-over, at

T~,

^towards a stronger

regime

of lower T~ and

larger

zv.

Strong

systems, ^like silica, and interrnediate systems like

amorphous

selenium _can be

entirely

described with a

unique regime

where _{zv oo} and _zv ~15

respectively.

Notice that for selenium the

same power law accounts for the data _on both sides of _a

crystallisation

window. In

figure

2b Cumn, EUSrS _or Mn aluminosilicate _{are on} the

fragile

side _as

compared

to stronger

anisotropic

spin

glasses

like

Rb~cuo~~coo~~f~

_or

Feo~mgo~cl~

which exhibit _a

larger dynamical

_exponent. We do not know of any data which show _a

sign

of _{a cross-over} as observed in

glass forrning liquids.

But this is

perhaps

because

the data _cover in general _a much smaller range of decades.

bt 11 GLASSES AND SPIN GLASSES _: A PARALLEL 1631

one would

easily

be

tempted

to find three successive

regimes

with

increasing _exponents (zv ~6,

then _zv

~15,

and then _zv

-oc).

These

changes

of

regime, although they

are

particularly apparent

^{in the}

P~(T~ diagram

have also been noticed otherwise

[22]

and these systems are the _same where Arrhenius deviations to the Fulcher law have been

reported.

In

general

_no

expression

with 4 coefficients _seems

capable

to describe both

regimes

with _a

unique

set of

parameters.

c)

A number of

systems

^like Se

[23], glycerol [24],

Pd77_5Cu~Sij~_5

[16]

can be described

by

_a

unique

power law with _an intermediate _zv l 5. Silica _or

GeO~

_on another hand _can

entirely

be described

by

_an Arrhenius law

(zv

~ oc

). Perhaps

these

systems

are the

archetypes

for two other

(one intermediate,

_one

strong)

classes of

universality.

However it is not clear in these systems ^{that the}

temperature

^{has been raised}

high enough

to reach the « universal _»

fragile regime.

For

example,

in

silica,

_we have not yet ^{reached the} viscosities below I

poise

which

make the difference and exhibit

fragile

behaviour in the

fragile _systems.

d) Conversely

_one wonders if T~ ₌ ⁰

(zv

- oc

)

is not the ultimate limit for all

systems

^and whether in these

liquids where,

like

Licl,

5.75

H~O,

_an intermediate

(zvl'~15

to 25 is measured _one would _not reach still

larger _exponent

values if the data for

higher

viscosities

were available.

The discontinuities of the

slope

of the

P~(T) dependence

which _are observed at

TR ^where

f

is finite _can be

interpreted

_{as a cross-over} from _one to another

regime

which _can both be described

by

_{a power} law of the form of

equation (4). But,

_as

P~(T)

is

continuous,

these _two power laws should be

tangent

to _one another at the _cross-over

point.

The tangent itself intersects the

T~/T

₌ ^{0 axis in the Arrhenius}

plot (Fig. I)

at

T(

such that

[15]

In

(T(/To)

₌ zv

ln

l

~

+

~~ (6)

TR

TR Tc

The

prefactor

of any Arrhenius law fitted _to the data in the

regime

below the _cross-over would have this value and would appear

unphysically

small when

compared

with the

generally accepted

_To

^~10~l~

_s

(but T(

is the result of _an

extrapolation

to the

high

T limit of the

analytical

continuation of _a law which is

only

valid below

T~).

ln any case

T(

will appear

depressed

if

T[

is decreased and the effect will be stronger ^when

TR

is closer to T~. The existence of _a small To in some of the

strongest systems

^{like silica} thus

supports

^the

conjecture

that there exists _{a cross-over} from _{a more}

fragile

state at _a

higher

temperature.

Notice that the

continuity

of the

P~(T~

curve is _a

significant point

in _our

description equation (4)

defines

T as a power of

f

the static

scaling hypothesis

_on another hand defines the Gibbs

potential

_{as a power} of

f

_we have

G/T= f~

_in

zero field. Discontinuities of

P~ (T)

would _{mean a}

discontinuity

of the

entropy S( 7~

= d

(G/T)/@(I In

and _a first order transition.

The

suggestion

therefore is that in

supercooled liquids

the situation is well described at

high temperatures by

_{a power} law with _a finite T~ ^where a correlation

length

would

diverge,

but

that,

_on

decreasing

the temperature ^the

system

_escapes

by

_a

(succession on cross-over(s)

towards _a situation of lower T~. ^{The whole}

procedure

is

interrupted

at the

glass

transition _near T~ when the relaxation time becomes

large compared

to any reasonable

experimental

time

and _{we are no}

longer

able to observe the system ^at

equilibrium.

The _case of

B~O~

^is

particularly interesting

because this system _goes all the way

through

from _a

fragile

to a strong

situation. But in neither _{case even} in the

high-temperature fragile regime

it is described with _a

«reasonable» _To. This is

certainly

related to the fact

that,

in this system,

undamped

transverse sound _{waves are} observed still much _over the

melting point

and up ^to 300 K

[25].

In _our

interpretation

_we

certainly

would

expect something (another

_{cross over} to some

1632 JOURNAL DE

PHYSIQUE

I li° 11

different

situation)

to occur at _a still

higher temperature.

In other words it _seems that this is _a

liquid

which still has to melt.

In

figure

2b _we show that the

P~ 7~ plot

relative to the data of

figure

16. This

plot

calls the

following

comments _:

a')

Most

spin glasses

^should

point

_zv

~

8 _± 2

according

to the admitted truth based

mainly

on

early experiments

_on R.K.K.Y, systems ^{like Cumn and} on

systems

like EUSrS

[8, 15].

The data in

FeMnTiO~, Cdo,~Mno_4Te, Cdo_7Mno_4Te

or Mn aluminosilicate

point

_a

larger

zv ll _± 2. The criteria which

permit

to define the

T(7~ dependence

have been

improved

since the first

experiments

_were

performed [9-13]. Certainly

it would be useful to reconsider all the former data with the _new criteria. After this is done it is not

completely

clear if _two

classes _{or one} will be needed _to describe all these

fragile systems.

^{We had} no

difficulty

to

superimpose

the data _{on a}

single

_curve shown in

figure

3 with _{a zv} l I. The

superposition

of

the Cumn data _on the _{same curve} is

possible

if _we accept a To 10~ ^~° _s. We should stress that

the data for this

sample

_are of 3 distinct

origins

for 3 _very different ranges

(neutrons,

_a-c-,

d.c...)

and this

implies

further uncertainties.

b')

Feo_~Mgo_7Cl~ or

Rb~cuj _~Co~E4

^which

point respectively

_{zv co} and _zv 25 show _a different

(stronger)

behaviour. More

examples

would be needed in order to confirm that

logic(T/~°~

logic(~/no)

~°

iogiol?'T°)

~

~~

~°~~~~~~~~~

int©rm©i'~

o

m

15

# 15

~~~ong .

.

3teeag

: .

.

/

~ 10

fl

~ ~~ ~o 26.330 ~°

~~

20 ~~~~ ~

@/T "zUTe/T

°

0 ~'~~

/~ _$~~

~

5 5

o o

0 3 6 0 5 lo

0

IT

₌ zuT~

IT

0

IT

= zuT~

IT

Fig.

3. In the

plot

of

log

_~

/~o)

_vs.

HIT

shown in the insert of

figure

3a the data of

figure

la

display

three different

universality

behaviours _one

fragile

with _zv 6.2, _one intermediate with _zv ~15 and

one strong with

zv - oo. The curves representative of _{a pure case are} tangent to each other at their

common

origin

where _~

= ~o (see Tab.

I)

and the

positions

of the asymptotes ^where ~ would

diverge

is at _{= zv.} For _a mixed _case which _{crosses over} at T~ to a stronger

universality

class the contact

point

between two behaviours is at TR so that _To and the

position

of the _new 7~

depend

_on the T~ over T~ ratio. The main

figure

shows in _more detail that B~O~ ^which

belongs

to the _same

fagile

class _as

KNO~-Ca(NO~)~, propylene

carbonate, toluene,

a-phenyl-o-cresol,

salol _{crosses over} towards stronger behaviour _{at a} different

T~/7~

ratio. The _same

plot

for

spin glasses (Fig. 3b)

^shows similar features but

no indication for _{a cross over} from _one to another

regime.

tt II GLASSES AND SPIN GLASSES _: A PARALLEL 1633

these systems are evidence for _two other

(one

strong, one

intermediate)

classes ot'

universality.

Also in

Rb~cuo,78Coo_~~F4

^{variations of}

T~(w ⁾

over 16 decades in

frequency

have been obtained

by extrapolating

with the Cole-Cole law measurements which

actually

_{cover a}

more conventional range of about 5 decades. This method _assumes that the

phenomenological

Cole-Cole law is

actually

correct from the

high frequency

to the low

frequency

limit and that the data in these 2

regimes

are consistent with the _same values of the

parameters. Right

_or

wrong, this is still _a

hypothesis

which needs further confirmations and _can have _an incidence

on the

«

experimental

data which _are derived l'or

T(T).

c')

None of the

spin glasses

exhibits _a clear _{cross over} from _one to another

regime

_as

observed with the

glass forming liquids

and _a

unique

_exponent characterizes the whole

regime

down to the

«glass

transition» where

ergodicity

is lost. But the relative _range of I

/Twhere

_we have data is much smaller than for structural

glasses (Fig. 2).

In

Rb~cuj ~Co~E4

Table I. We show

for

the

d%ferent glass-forming

_systems

«experimental»

values

of

zv, T~ ^and ~o which _come

from

_a best

fit

_to the

P~(T) plot of figure

2a. The

corresponding

« universal _» values which

correspond

to the

fits

shown

figures

I and 3 have been obtained

by imposing

an average value

of

_zv. For _most

fragile

_systems where _zv is

of

the order

of

6.25 _a

cross-over at

TR

^towardY a stronger

regime

is observed when T~ is

approached.

The data

corresponding

_to this second

regime

when it exists are shown _on the lower line

for

each system.

T~ ^{is the} temperature ^{where the}

viscosity

becomes

of

the order

of10'~

_{P and}

_ergodicity

_{is lost.}

The

validity

_range

of

this

regime

is between

T~

and T~ ^{so that} ~o which is the T - oc limit

of

the

analytical

continuation

of

the law

appropriate

to this range is not

expected

_to

have any

simple,

direct

physical meaning (see Eq. (6)).

With average zv

imposed

Glasses _zv

/K

_~

o/P

_zv T~/K 0/K _~

o/P

of

6.65 618 4 110 1.06 6.25 632 1.19

oo 0 46 350 2 _x 10-24 [3)

5.31 173 921 1.9 _x 10-4 6.25 166 1.26 x 10-4 [4]

20.5 143 2 935 4.5 _x lo-I 15 148 6.4 _x 10-8

Selenium 13.26 285 3 785 8.8 _x 10-4 15 284 3 8.6 _x 10-~ [23]

oluene 4.76 130 620 3.4 _x 10-4 6.25 l17 731 2.2 _x 10-4 [5]

KN03 4.96 368 1825 3.4 _x 10-4 6.25 356 2 232 8.5 _x 10-5 [5]

Ca~lQ03)2 15.83 327 5 170 1.55 _x 10-1° 15 329 930 4.5 _x 10-1°

«-Phenyl

7.23 230 660 7.66 _x 10-6 241 6.25 233 455 2.3 _x 10-5 [2]

Cresol _oQ 0 33 200 2 _x 10-56

5.34 243 300 1.9 _x 10-5 [2]

oo 0 31 200 1.3 _x 10-51

oQ 0 62 353 1A _x 10-6 [6]

JOURNAL DE PHYSIQUE T I, M It, NOVEMBRE lwl 65

1634 JOURNAL DE

PHYSIQUE

I li° 11

moreover, a cross-over from

fragile

to intermediate state at _a

high temperature

^would

permit

us to account for the

unphysical

_To 10~ ^~~ _s which is found with _our

interpretation. Perhaps

such _{a cross-over} is

suggested by

the

experimental

^data

(Fig. I).

d')

In

spin glasses

the stronger

regime

is

clearly

associated with _a lower

dimensionality.

Thus

Feo_~MgJ_7Cl~

^is

reputed Ising

and two-dimensional.

We show in the insert of

figure

3a _a

representation

of In

(T/To)

_vs.

RI

T where

= zv T~ ^has

been defined in

equation (5).

We have found _{an average}

zv ~ 6.2 ± 2 in the

fragile regime

of the

fragile

_systems and determined T~ ^and To which best fit the data for this value

(see

Tabs. I and

II).

The fits _are shown in

figures

I and 3. Selenium is fitted with _{a zv} 13.5 and is

given

here _{as an}

example

of _an internlediate class which would also include e.g.

glycerol

_or Pdcusi.

Silica is shown _{as an}

example

of the

strong

systems ^{for which} T~ =

0 and _zv

- oc and which would also include

GeO~,

etc. In this

representation

all systems have the _same

slope

at their

common

high temperature origin

where _T

= To and

they

exhibit _a

divergence

at

~

= zv which T

defines T~. ^{The main}

figure emphazises

the situation of several

fragile _systems

which _cross

over to intermediate behaviour at _a

Tj/T~

ratio which is of the order of I.I in

o-terphenyl,

salol _or

propylene

carbonate and of the order of 1.2 in

B~O~.

Figure

3b is the

corresponding plot

for

spin glasses.

We have

imposed

_{an average}

zv~ll for the

«fragile» systems

^{shown and}

Rb~cuo_78Coo_~~F~

^and

Feo_~MgJ_7Cl~

are

proposed

_as illustrations for intermediate and

strong

^{behaviours with} zv~25 and

zv oc

respectively.

Table II. We

give for

the

d%ferent spin-glasses

the values

of

_{zv, T~} and _To which _{come out}

from

_a best

fit

_to the data

of figure

2b. The

corresponding

values obtained with _an average

zv =

I I

imposed correspond

to the

fits of figures

I and 3.

With average zv imposed

Spin

glasses _zv @/K

ro/s

=u T~/K @/K To/s of

oQ 0 56.72 1.48 _x 10 [10]

alurni- I1.06 2.95 32.74 2.56 _x 10-16 11 2.96 32.58 1.81 _x 10-16 [9]

no-silicate

10.1 6.34 64.2 3.16 _x 10-'2 6.53 11 6.29 9.62 _x 10-'3 [12]

26.33 3.02 79.54 5.58 _x 10-'9

10.84 12.26 133 4.89 _x 10-'4 11 12.25 135 3.66 _x 10-'4 [12]

12.87 20.21 260.1 10-'4 11 20.64 7.74 _x 10-'4 [13]

5.73 27.47 157.3 1.3 _x 10-'3 8 27.3 5.3 _x 10-'6 [8]

9.48 1.48 14.07 2.20 _x 10-" 1.53 11 1.46 16.04 4.18 _x 10-'2 [15]

li° II GLASSES AND SPIN GLASSES A PARALLEL 1635

3. Discussion.

We have shown that critical

slowing

down offers _an alternative to the

phenomenological

Fulcher law to fit the

viscosity

of

glasses

_or the relaxation times of

spin glasses.

Besides it introduces _some elements of rationalisation such _as the convergence

(shown

in

Fig. 2)

of the

P~(T)

lines towards _a limited number of « universal

exponent

^{values. Until} now,

however,

no conclusive evidence has been found in favour of either law from the strict

point

of view of the

quality

of the fits. The _reason is apparent ⁱⁿ

figure

2. With

equation (2)

_we have

so that _we expect a power law rather than its tangent ^{which would}

correspond

to

equation (5)

but the slow

dynamics

forbid the

region

_near

J~

where the difference would be conclusive, The _case however of selenium deserves

special

consideration. The measurements are

interrupted

_on

lowering

the

temperature

^{when the nucleation of}

crystallites

becomes _too considerable but it is

possible

to obtain another set of data

by melting

the

quenched amorphous

state in _a range near T~ ^where the

growth

of the

crystallites

can be maintained

sufficiently

slow. In

figure 2,

_{we see} that _a

unique

_exponent is necessary when two distinct Fulcher laws _were needed. This stresses the

continuity

of the

physics

which describes both sides of the

crystallisation

window and its relevance to critical

slowing

down rather than to

activated

dynamics.

Another

question

is relative _to frustration which is believed to be

responsible

for the slow

dynamics

in

spin glasses.

The

question

then is what would be the

origin

of frustration in

glasses

? A

key

to the _answer could be the observation which is made of _a

highly

universal

regime

at

high temperatures

ⁱⁿ the

fragile systems.

^The _presence of the _{same zv} 6 in

systems

a

priori

as different _as Salol and

KNO~Ca(NO~)~

reflects the fact that there is little _energy difference between different

crystalline

structures which _may

eventually

set-in. Hence the

suggestion

of _a

multivalleyed phase

_space with

nearly equivalent

minima _: hence the presence of frustration _as each structure has _no

compatibility

with the others. In contrast with the

spin- glass

_case where there is _one response for _one

situation, fragile liquids

_seem

capable

to

adapt

their

criticallity through

_a succession of

liquid-to-liquid

transitions towards

stronger

situations.

There _{are many}

general

_reasons

by

which _one would

justify

that _a

phase

_space becomes

stronger (or

_more

anisotropic

if _we

interpret

the situation at the

light

of the

spin glass data).

This may be that _{one or} several

particular valleys

are favoured

(e.g.

at temperatures close to the

melting point

_a

deeper valley ought

to

correspond

to the

crystalline

structure of the

system)

_{; on} the contrary,

valleys

which have _some

probability

at short range become

hampered

when

f

_grows

(e.g.

the

periodic

structures which fill the 3d space are favoured with

respect

to 5-fold

symmetries

which do

not).

We would expect a similar situation in _a

spin glass layer

where

f

would become of the order of the thickness and the correlations which

initially developed

in 3 dimensions would become restricted to the

plane. Castaing

and Souletie

[17]

observe that the model of activated

dynamics

describes critical

dynamics

in _a

phase

_space

which shrinks _as

f

_grows. Indeed

equation (7)

_can be

interpreted

_as

featuring

_a continuous evolution towards

stronger

^situations as the

slope

of the

P~(7~

_curves decreases and cancels when T~ is

approached.

The

problem

is that in actual

liquids

this evolution is realized

by steps

and _a succession of cross-overs is _more

appropriate

to describe the successive tiers of this

hierarchy.

This is

particularly

clear on the

P~ (T) plot

which _{we are}

tempted

to describe with a

succession of segments ^{rather than with the continuous} curve or the

hierarchy

of tiers that the activated

dynamic equation

would

imply (see Fig. 2).

It would be of _course very

interesting

to

check whether each tier _can be related to

particular physics

that _a

particular system

^would

1636 JOURNAL DE

PHYSIQUE

I bt 11

exemplify (e.g,

whether _{a zv} ~15 would _{mean a}

polymeric

^situation as is observed in

selenium2~).

It _seems

logical,

in the framework of the above

picture,

to consider that the fluctuations of

the ordered

amorphous phase

_are

complicated

intermixtures of the different _pure

phases

which constitute the different

valleys. Among

these fluctuations _pure

crystallites

of these

phases

would _occur with _a certain

probability and,

_among those

crystallites,

those which have the

appropriate

_symmetry would act as the _germs which would motivate the _occurrence of the first order

melting

transition at TM. ^The

TM/J~

ratio is _an

important

_parameter in this

picture.

If

J~

^is much smaller than TM the

liquid

becomes _a

crystal

at TM ^but the model

explains viscosity

increases that _a first order transition would not

justify.

If T~ ^{is close} to TM, ^then

high

viscosities _are reached in _{one or} the other of the successive

regimes

which _we have described above. Pure

crystallites

of

only

_one

phase

become

unlikely

at

large f

and

they

would _{grow very}

slowly. High viscosity gradients

which _can reach orders of

magnitudes

in _a few

degrees

affect the transport ⁱⁿ the

liquid

towards the

liquid

_to solid

interface _: the definition of what is

liquid

and what is solid is in

question;

it becomes

frequency dependent

The ultimate consequences are known: the

freezing

becomes

homogeneous

^and we have _a

glass.

An intermediate _case would be that of Se where _a

crystallisation

window

hampers viscosity

measurements in _an intermediate _range of

tempera-

tures.

4. Conclusion.

We have found that

spin glasses

show similar features from

fragile

to strong as usual

glass forming liquids.

These features _can be accounted for in the framework of usual

scaling theory

where the strong limit would

correspond

to _a lower critical

dimensionality.

Data _on real

glasses

of the

fragile

side

suggest

^{that the} strong

regime

^could result from _{a cross-over} from _a universal

high temperature fragile

situation

occurring

at

T~

which

depends

_on the

microscopic properties

of the system. ^{It is the}

magnitude

of the

Tj/J~

ratio which decides whether the

system

^will appear

stronger

or more

fragile

in the fixed

experimental

window. Another mark

of the _cross-over would be the presence of _an

unphysical

_To value in the

regime

below the

cross-over. Before _we finish _we should mention that a power law behaviour for shear

viscosity

is also

predicted by mode-coupling theory

which has been rather successful in

explaining

_some

data _on hard system

glasses

and

polymeric

_{ones as} well

[27].

It _seems premature ^{to decide} whether the

present analysis

is _or is not

contradictory

with these calculations which do not introduce any

prerequisite

that there should exist _a

diverging length f

in the system. ^In any

case the

present description

is

phenomenological.

The existence of _a

phenomenology,

_even

successful,

does not exempt us of

seeking

for _a

microscopical theory

and _we believe the

reverse also should be true,

specially

with

mode,coupling theory

whose calculations _are

by

_no

means transparent.

Acknowledgements.

We _are

grateful

to P. Chieux for several

illuminating

discussions and to G. Faivre and Ch.

Simon who

pointed

out to us the

interesting

situation of Se. Part of this work _was motivated

by

the

workshop

_{on «}

Glasses,

biomolecules and evolution in

May

1990 at the Santa Fe

Institute that _one of the authors

(J.S.)

wants to thank for its

hospitality.

bt 11 GLASSES AND SPIN GLASSES A PARALLEL 1637

References

[I]

ANGELL C. A., Relaxations in

Complex Systems,

K. L.

Ngai

and G. B.

Wright

Eds.

~National

Technical Inforrnation Service of U.S.

Department

of Commerce, 1984) _pp. 3-11, Nucl.

Phys.

B (Proc.

Suppl.)

sA (1988) 69.

[2] ^{LAUGHLIN W. T. and UHLMANN D.} R., J. Phys. Chem. 76

(1972)

2317.

[3] ^{MACEDO P. B. and NAPOLITANO} A., J.

Phys.

Chem. 49

(1968)

1887.

[4] BONDEAU A. and HUCK J., J.

Phys.

France 46

(1986)

1717.

[5] ^WEILER R., BLASER S. and MACEDO P. B., J.

Phys.

Chem. 73

(1969)

4147 ; WEILER R., BOSE R. and MACEDO P. B., J. Chem.

Phys.

s3

(1970)

1258.

[6] BRUCKNER R., J. Non-Cryst. Sol. 6

(1971)

177 _;

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

Phys.

Chem. Glasses 7

(1966)

139.

[7] ^MOYNIHAN C. T., BALITACTAC N., BOONE L. and LITOVITz T. A., J. Chem.

Phys.

ss

(1971)

3013.

[8] SOULETIE J. and THOLENCE J. L.,

Phys.

Rev. B 21

(1985)

516.

[9] BEAUVILLAIN P., RENARD J. P., MATECKI M. and PRtJEAN J. J.,

Europhys.

Lent. 2

(1987)

128 _; PRtJEAN J. J., «Chance and Matter, edited by J. Souletie, J. Vannimenus, and R. Stora Eds.

~North-Holland,

Amsterdam,

1986)

_p. 557.

[10]

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

Europhys.

Lent. s

(1988)

271.

[I

I]

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

Phys.

Rev. Lett. 61

(1988)

1780.

[12] GESCHWiND S., OGiELSKi A. T., DEVLIN G., HEGARTY J. and BRiDENBAUGH P., J.

Appl. Phys.

63

(1988)

3291 _;

ZHOU Y., RIGAUX C., MYCIELSKI A., MENANT M, and BONTEMPS N.,

Phys.

Rev. B 40 (1989) 81II.

[13] ARUGA H., TOKORO T. and ITO A., J.

Phys.

Sac. Jpn s7

(1988)

261.

[14]

THOLENCE J. L., Solid State Commun. 3s

(1980)

l13.

[15]

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

Phys.

Rev. B 30

(1984)

6514.

[16]

CHEN H. S., J. Non-Cryst. So/. 27

(1978)

257.

[17j CASTAING B. and SOULETIE J., J. Phys. I France, 403.

[18] SOULETIE J., J.

Phys.

France 49 (1988) 1211.

[19] HARRIS A. B., J. Phys. C 7

(1974)

1671.

[20] SOULETIE J., J. Phys. France sl

(1990)

883 in _« Relaxation in Complex Systems ^and Related

Topics

_», I. A.

Campbell

and C. Giovanella Eds.

~Plenum

Press, New York,

1990)

_p. 231.

[21]

DUPUY J., JAL J. F., CARMONA P., AOUIzERAT A., CARBY E., CHIEUX P., Relaxation in

Complex Systems

and Related

Topics,

_op, cit.,

p.175.

[22]

ROSSLER E.,

Phys.

Rev. Lett. 6s

(1990)

1595.

[23]

UBERREITER K. and ORTHMANN H. J., Kolloid Z. 123

(1951)

84 _; PERRON J. C., RABBIT J. and RIALLAUD J. F., Phil. Mag. 46

(1982)

321.

[24]

BiRGE N. O. and NAGEL S. R.,

Phys.

Rev. Lent. s4

(1985)

2674.

[25]

GRiMSDiTCH M., BHADRA R. and TORELL L. M.,

Phys.

Rev. Lent. 62

(1989)

2616.

[26]

FAiVRE G. and GARDiSSAT J. L., Macromolecules 19

(1986)

1988.

[27] LEUTHEUSSER E.,

Phys.

Rev. A 29

(1984)

2765 _;

BENGTzELiUS U., GOTzE W. and SJOLANDER A., J. Phys. C17

(1984)

5915

and _a review

by

GOTzE W.,

Liquids, Freezing

and the Glass Transition, J. P. Hansen, D. Levesque and J. Zinn-Justin Eds.

~North-Holland, Amsterdam)

in press.