Neutron spin echo study of the dynamics of undercooled selenium

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Neutron spin echo study of the dynamics of undercooled selenium

Ch. Simon, Gabriel Faivre, R. Zorn, F. Batallan, J. Legrand

To cite this version:

Ch. Simon, Gabriel Faivre, R. Zorn, F. Batallan, J. Legrand. Neutron spin echo study of the dy- namics of undercooled selenium. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.307-314.

�10.1051/jp1:1992144�. �jpa-00246485�

Classification Physics Abstracts

61.40 64.70 61.41

Neutron spin echo study of the dynamics of undercooled selenium

Ch. Simon

(I),

G. Faivre

(I),

R. Zom

(2),

F. Batallan

(3)

and J. F.

Legrand (4)

(~)

Groupe

de

Physique

des Solides, Universit£s Paris 6 _et 7, 75251 Paris Cedex 05, France (2)

Kemforschungsanlage

Jiilich, Institut fur

Feslkbrperforschung,

5170 Jiilicl~,

Germany

(3) Instituto de Ciencia de Materiales de Madrid, CSIC, Serrano 144, 28006 Madrid,

Spain

(4) ^Institut Laue

Langevin,

156X, 38042 Grenoble, France

(Received 16

July

1991, revised 28 October 1991, accepted 19 November I99I)

R4sumd.-Nous

pr6sentons

des r6sultats de _{mesures au}

spectrombtre

h £cho de

spin

sur la relaxation structurale du s61enium _en suffusion

prbs

de la transition vitreuse. Nous trouvons une

relaxation en deux

£tapes,

essentiellement conforme h celle trouv£e ant6rieurement dans d'autres

liquides,

h ceci

prbs

_que le temps

caract£ristique

_vi du stade lent varie _avec la

temp6rature

T

plus

lentement que ~(T)/T, off ~(T) est la viscosit£

macroscopique.

Nous

interpr6tons

cet dcart apparent ^{h la loi d'dchelle}

g6n£ralement

observde _comme la manifestation du fait que le s£16nium

liquide

est _un

polym~re d'6quilibre. L'analyse

des donn6es

sugg~re

d'autre part que l'exposant de Kohlrausch du stade lent d6pend de la

temp6rature.

Quant _au stade rapide de la relaxation, _nous

montrons

qu'il

_peut due

interpr£t£,

sans recourir h la th60rie du couplage de modes, _par

l'existence

d'h6t£rog6n6it6s

locales de la densit6 _au

voisinage

de la transition vitreuse.

Abstract.- We present neutron

spin

echo measurements _on the structural relaxation of undercooled

liquid

^selenium above

glass

transition. We find _a two-stage ^relaxation process essentially similar to that

generally

observed in

liquids,

_except for the fact that the

slow-stage

characteristic time _vj varies with the temperature ^T more

slowly

than ~(T)/T, where

~ (T) is the

macroscopic viscosity.

We

interpret

this apparent

discrepancy

with respect to the

usually

found

scaling

law _as the manifestation of the

equilibrium-polymer

nature of

liquid

selenium. On the other hand, the data suggest ^{that the} Kohlrausch exponent of the slow stage ^is temperature

dependent. Conceming

the

rapid

_stage of the relaxation, _we show that _a

possible

altemative _to the

mode-coupling interpretation

is to attribute it to the existence of local

inhomogeneities

of the

density

close to the

glass

transition.

Introduction.

First observations _on

glass forming liquids [1-3]

have shown

that, surprisingly,

at

long

^times

(t

up ^to

10ns)

and short

wavelengths (Q~~=a

_{the nearest}

_{neighbor distance),}

_the

_density

correlation function

W~(t)

seems to be

independent

of the

microscopic

structure,

exhibiting

the _same behaviour in _a salt

[I],

_a

polymer [2],

and _a molecular

liquid [3].

The relaxation is

clearly separated

in two processes. A certain fraction of

W~

^{is relaxed in a} very short time

308 JOURNAL DE PHYSIQUE ^I N° 3

even near the

glass

transition

T~ (conventionally

defined _as the temperature ^{where the}

macroscopic viscosity

_1~ reaches 1013

Poise).

The

amplitude

of this

rapid

_process varies from zero, below

T~,

^{to a constant value at some} temperature ^above

T~.,A

^slower process leads to the total relaxation of

~P~.

^{The Kohlrausch law}

(exp(- t/r~)fl

_can be fitted to it and it tums

out that the time _{constant vi} varies with temperature ^{in the} same way as

1~(T)/T,

where

1~(T)

is the

macroscopic viscosity

of the

liquid.

The _same behavior has been found in numerical simulations that

give ~P~(t)

^{in the} same time and _{wave vector} ranges

[4].

These results have been considered _as

strongly supporting

the mode

coupling theory.

This

theory [5, 6] predicts

the existence of _a critical temperature T~ ^{situated well above} T~ ^{and the}

separation

of the relaxation process in two distinct stages. ^Below

T~,

^{the relative}

amplitude Fq

^{of the slow process}

^(named

^{in this context the}

non-ergodicity parameter)

varies

as

(T-T~)'~,

_a law which indeed fits the

experimentally

measured variations of

F~,

^within

experimental

errors.

However,

the

viscosity

is

predicted

to grow without limit when T

approaches T~.

^{This last feature is of} course in clear

disagreement

with observation _: in real

liquids,

the

viscosity

is of the order of 100 Poise and its variation _rate with temperature is moderate in the

temperature

range in which T~ ^is

supposed

to lie.

Although

this

predicted

divergence

of the

viscosity

at T~

might

be avoided

by removing

some of the

approximations currently

done in the calculations

[7, 8],

_we cannot consider that the mode

coupling theory

is _a well established

interpretation

of the relaxation behaviour of the

liquids

near the

glass

transition. More

experimental

information _on the relaxation of

liquids

in the time and _wave

vector range

explored by

neutron diffraction is

clearly

desirable. Our choice of

liquid

selenium has been

guided by

the

following

considerations.

Liquid

selenium is

generally

considered _{as an}

equilibrium polymer [9, 10].

The viscoelastic relaxation spectrum ^{of standard}

(as opposed

to

equilibrium) polymers

at

large wavelengths

has been

extensively

studied and is _now well understood

[I I].

^A detailed neutron

scattering investigation

has been

performed

_on

polybutadiene by

Richter et al.

[2],

with the

following

results. The relaxation process in

polymers

is similar to that found in

non-polymeric liquids.

In

particular,

_vi varies with T _as

1~/T, which,

in the _context of the Rouse

theory

of the relaxation in

polymers,

means that _vi is

practically

identical with the so-called monomeric relaxation time To. This is not a trivial result

since, according

to the

generally accepted

vision of the Rouse

theory,

_To is _a

phenomenological quantity

with _no known

specific microscopic reality.

It was therefore

tempting

to

perform

the _same kind of

investigation

_on

liquid

selenium,

where the relation between _1~ and To is ^not

expected

to be the _{same as} in

ordinary organic polymers,

to check whether the _same

identity

between _vi and _To is found. Let _us be _a little _more

explicit

_on this

point.

In the

simplest

model of

equilibrium polymers,

one assumes that each atom is

covalently

bound to two

neighbors (thus,

the

liquid

is

mostly

_a linear

polymer),

but _one attributes _a finite average lifetime, function of temperature, to the covalent bonds.

Then,

it _can be shown

[10]

that the

liquid

has the _same viscoelastic

properties

as a standard

monodisperse polymer, except

^{that its effective} or

apparent degree

of

polymerization

^N

(T) depends

on temperature.

More

specifically,

the Rouse

regime

of standard

polymers

is described

by

the

dynamical

shear modulus

[11]

G

(t)

=

RT/vN

£ exp(- tq~/ro

_{N ~)}

,

~1 (l)

where N is the number of _monomers per

chain,

_To the

already

mentioned

phenomenological

monomeric time

(related

to the so-called monomeric friction

coefficient)

and _v the molar

volume per monomer. This formula is

explicitly supposed

to be valid

only

for times

substantially larger

^than _To,

corresponding

to

relaxing

_segments

long enough

to be

Gaussian,

and it is indeed observed that the

experimentally

measured relaxation modulus deviates from

equation (I)

^below about 10 _To.

By integrating

^formula

(I),

one obtains the

viscosity

_1~

1~ =

G(t)

dt

=

ar~/6RT/vNTo. (2)

This establishes

(for

low molecular

weight polymers,

but _a similar relation exists for

high polymers)

that the

temperature dependence

of

1~/T entirely

_comes ^from that of To. It is ^seen, from

equation (I),

that N and To ^are

independently

^obtained

by measuring G(t)

on a

sufficiently

wide interval of time. Such measurements have been

performed

_on

liquid

selenium between 40 and 70

°C,

I-e- below the

crystallisation

_gap of the

liquid,

and above its

glass

transition

(30°C) [11].

It _was found that N is

temperature dependent.

This _was

interpreted

in the way

explained above,

and it _was

predicted, by extrapolating

the low- temperature ^results

(therefore

with poor

quantitative accuracy), that,

in the temperature range above the

crystallisation

_gap, N

(T)

should be _a

decreasing

function of

T,

and

that,

_{as a}

consequence,

1~/T

should decrease _more

rapidly

than _To when T increases.

Experimental aspects.

Neutron

spin

echo spectrometry ^{delivers the} intermediate

scattering

function

I~(t)

^for times

up ^to 10 _ns. The measurements _were

performed

_on the INI I

spin

echo of the Institut Laue

Langevin [12].

The

incoming

neutron beam had _an average

wavelength

of 4.63

I

_with

a

distribution width of 20

9b,

which

corresponds

to _a maximum

experimental

time of 2 ns. We used the _« double echo _»

setup,

^which allows to extent the time range towards small times down to 2 ps

[13].

We chose

Q

~

l.9

i~

~, which is the maximum of the

scattering

function

I~(0)

^of the

liquid.

The

scattering angle

is then 90°. Since there is

only

one type ^of atoms in

selenium, I~ (t)

= ~P~

(t)/~P~ (0).

All the _{curves were} normalized

point by point

with respect

to the _curve measured _on the _same

sample

at 77 K. The

temperature

_range ^that can be

explored

is limited

by

the

glass

transition _at 303 K and

by

the

crystallization

of the

liquid

between about 350 K and 420 K. The

temperatures

^{under 350 K} were reached

by quenching

the

sample

from above the

crystal melting point (494 K)

in cold _water and then

heating (or cooling)

it

slowly

in the

cryofumace. Temperatures

above the

crystallization

_gap where

directly

reached

by quenching

from above 494 K. Below 494

K,

the

experiments

_were carried out until

crystallization eventually

manifests itself

by

an

abrupt slowing

down in the

dynamics.

The slow relaxation process.

The values of

I~(t)

^measured at different

temperatures

are

given

in

figure

I.

Figure

la presents ^{the data obtained above the}

crystallization

_{gap, as a} function of time.

Below the

crystallization

_gap,

only

_one value is

given

_at each

temperature (Fig, lb)

for it tums out that

I~(t)

does not

depend

on time between 2 ps and 2 ns : in this

temperature

range, the time constants of the slow relaxation process are much

larger

than 2 _ns. The

meaning

of the

corresponding

_constant values will be discussed in the next section.

Figure

^2a presents the _same data _as in

figure la,

reduced

according

to the usual

procedure

_:

the data

corresponding

to a

given

value of T _are shifted

along

the

logarithmic

time axis

by

a

factor

log (1~/T),

the values of

1~

(T)

in this temperature _range

being accurately

known

[14].

Inspection

of

figure

2a reveals _a

systematic

deviation from what whould be _a

satisfactory

reduction. The shift

by

the factor

1~/T

is

clearly

too

large,

in accordance with _our

expectation.

Another

systematic

deviation _can been _seen in

figure

2a _: the

higher

is the

temperature,

^the steeper ^{is the relaxation} curve.

310 JOURNAL DE PHYSIQUE I N° 3

(a) (b)

I

~ ~

())(

^0.95

1~0.6 i~.*

_~~~~ ^~~

^~~

f

8i~* * 480K

'~

_0.85

o.4 ~1"

(

. ~

*~~'f

_~

~ o.8

e~ _g

~ ^{* .}

°'~

'*

8 *

, o.75

~ *

o o.7

lo 100 lo00 290 300 310 320 330 340 350 360

time (ps) T (K)

Fig.

I.-a) Time

dependence

of the

density

correlation function

I~(t),

measured _at different temperatures, above the

crystallization

_gap of the

liquid.

b) Below 370 K,

I~(t)

^is

time-independent.

This constant value is

plotted

here _versus temperature.

(a> (b>

o.8 o.8

o.7 . * 480K o 7 f t

,+w ^* 500K . 605K

°'~ **#x ^{+ 510K o.6 m 575K}

o.5 *+#+x" x 540K

~~ ^x 540K

. 575K + 510K

a °.4 _{. .} 605K

'~

_{o.4 A} 500K

03 ⁺ 480K

o.2 o.2

O-I *

x o-1

~

o o

o.ol o.1 lo loo o.ol o-I I lo loo

10~^~t T/~ (s K/Poise) ^{t /~(T~}

(Cl

~'~

o.7 -fit

. 605K

~~ m 575K

x 540K

o.5

+ 510K

A 500K

O °'~

+ 480K

o.3

o.2

o-i

o

o. i ~ i o

( V«(T~)

Fig.

2. Reduction _curves of the data shown in

figure

la, obtained

by

various

procedures.

al Shifi

by

the factor _1~ (T)/T (~ _:

viscosity). b)

Shift with the characteristic time Ti of the Kohlrausch law _as the

only

fit parameter, c) as in b, with Ti and the exponent p as

independent

fit parameters.

In order to

analyse

these trends in _{a more}

quantitative

_way, we have fitted the data to the classical Kohlrausch law

I~ (t)

=

F~ exp(- t/r~

_)fl

,

(3)

where vi,

p

and

F~

_are three

parameters

to be

adjusted

for each

temperature. Figure

2b shows the reduction obtained when the fit is

performed

with

only

_vi let free _to vary with

temperature, p

and

F~ being

assumed to be temperature

independent (the

best fit values _are then

p

= 0.51 and

F~

=

0.85),

while

figure

2c presents ^{the reduction obtained with all the} parameters ^{free. The best fit values of} _vi ^and

p corresponding

to the latter _{case are}

given

in

figures

3 and

4,

_as functions of T. The _errors bars

correspond

to the 0.92 confidence

limit,

for each temperature. ^{The variations of the}

pre-exponential

factor are not

represented (the

average value is

F~

=

0.85).

We _are not

going

to

present

a formal discussion of the extent to which the trends visible in

figures

3 and 4 _are

statistically significant.

Such _a discussion would

essentially

aim _at

evaluating

the

validity

of the Kohlrausch law

itself,

_a

question

which is

beyond

the scope of this work. We _use here the Kohlrausch law

only

_{as a} convenient

algebraic

( a )

~~5

~ W

10 .

~ NT

~ o

$i ₁₀₀₀

W u

E~ ₁₀₀

~ ~

10 r

1

60

T(K)

( b 400

350

300

)

250

j~

200 #

~ lS0 ,

ioo

50

0

460 480 500 520 540 560 580 600

T(K)

Fig.

3. Variation with temperature ^of a) _T~, the Kohlrausch time, as obtained

by

the reduction

presented

in

figure

2c, and

Nro

calculated from

equation

(I) _; b) the ratio

Nrjri, representing

the effective chain

length

ⁱⁿ the

liquid

(see text).

312 JOURNAL DE PHYSIQUE I N° 3

~ ~

j

0.6

0.4

0.2

o

460 480 500 520 540 560 580 600

T(K)

Fig.

4.

Temperature

variation of the Kohlrausch exponent p, _as obtained by ^{the reduction}

presented

in figure 2c.

expression containing

a characteristic time

(vi)

and _a characteristic width of the time distribution related to

p

in the

following

_way. One

usually

defines the distribution P of the relaxation times of the system ^{in the} variable _p

= In

(r) by

exp(- t~)

=

P

(p ) dp exp(- t/r) (4)

Then, if A is the half-width _at half maximum of the distribution

[15]

p~~=

I ₊

A~ (5)

Let _{us now} tum to the discussion of the results

presented

in

figures

3 and 4.

The

quantity plotted

ⁱⁿ

figure

3b is the ratio NT

jr

i, where NT

o is calculated from

equation (I)

_as if the

liquid

were a standard

polymer (with

R/v

=

4.15

Pa/K [10]).

In the

equilibrium- polymer

vision of

selenium,

this ratio is the effective _«

dynamical

_» chain

length

N

(T).

The decrease

by

a factor 2.5 found between 460 and 600 K is in reasonable agreement with the temperature

dependence

of N

extrapolated

from the

low-temperature

viscoelastic results of reference

[10].

We therefore conclude that the _most natural

interpretation

of the present results is that

they

_support the

equilibrium-polymer interpretation

of

liquid

selenium _as well

as the

universality

of the

identity

of _vi with _To.

The increase of the Kohlrausch exponent

p

seen in

figure

4 _means,

according

to

equation (5),

that the half-width A of the time distribution decreases _as

temperature

^{increases. Such} a

decrease of A is

predicted by

mode

coupling

calculations

[5]

_as well _as

by

renormalization ideas

applied

to

glass

transition

[16].

In

figure 4,

A also shows _a

plateau

_as T

approaches T~.

^{It is not clear whether this feature is}

really significant,

but it may nevertheless be

interesting

to note that _a similar

temperature dependence (which

is not

predicted by

_any

model)

_was observed in molecular

glasses [17].

The

rapid

relaxation process.

Below 350 K, the

intensity

is found _{constant over} the

experimental

time range but smaller than the

intensity

measured at _zero time

(Fig. lb). Thus,

in this temperature range, the

characteristic time of the

rapid

movement is much shorter than 2 ps and

F~

=

1~ (t). Figure

5

presents ^the

corresponding

values of

F~.

^{The values of}

F~

obtained above the

crystallization

gap as the result of the fit described in the

previous section,

which have been found to be

temperature independent,

_are also

plotted

in

figure

5. It _can be _seen that the

parameter

F~ only

varies between

T~

^and

To,

a

temperature

^{located somewhere in the}

crystallization

gap. Such _a result _was

already

observed in the other

systems investigated by

neutron

diffraction

[1-3].

In

o-terphenyl [3],

the _{wave vector}

dependence

of

F~ (for

the incoherent

scattering)

has been measured and it _was shown that the results _can be fitted _to the

following law,

in which the temperature and the _wave vector

dependence

are

separated

_one from the

other _:

F~(T)

₌

fj

+

h~ ^{co(I T/T~)~'~,} (6)

where

fj

and

h~

_co are two temperature

independent parameters. Obviously,

this formula is also

compatible

with the data shown in

figure 5,

but cannot be

significantly

tested with them.

F d%

0.

~ ~" ~

~m

0.6 _-

n 0.4

m 0.2

m

0

0

T (K)

Fig. 5.

elaxation rocess

text).

Formula

(6) originates

from the mode

coupling theory [6]

and its

compatibility

with

experimental

data

(within experimental uncertainty)

_can be considered _{as a} support to this

theory.

However, it _can also be

given

_a different

meaning, suggested by

the

proportionality

between

fj

and

h~

co observed in

figure

6 of reference

[3]. Equation (6)

_can be written _as

F~(T)

₌

(i n(T))

+

n(T)J](pQ) ₍₇₎

where

Jo(x)

is the Bessel

spherical

function which appears in _a model of diffusive motion _{on a}

sphere

of radius _p

[18]. Equation (7) corresponds

to a model in which diffusion is confined within small

volumes,

which _can be visualized _as

regions

of lower

density.

The radius of these

inhomogeneities,

_p, does not

depend

_on temperature:

only

their

number,

hence the percentage ^of

moving

molecules

n(T),

is

temperature dependent. Applying

this model to the data in

figure 5,

and

assuming

that _n

= I above the

crystallization

_{gap, we} find _p

=

0.25

I.

The temperature

dependence

of _n obtained in this way is shown in

figure

5. At low temperatures, ^{there is} no diffusive motion and _n

=

0 _; between

T~

^and

To,

n varies from 0 to 1.

Conclusion.

Liquid

selenium

belongs

to _a very narrow class of

glass-forming liquids,

the

equilibrium

polymers.

The

present study

establishes that the two-stage relaxation _process

already

found in several other classes of

liquids

is also observed in

equilibrium polymers,

thus

reinforcing

the

314 JOURNAL DE PHYSIQUE I N° 3

presumption

of

universality

of this behaviour. The distribution of relaxation times centered

on ri is

reasonably

well

represented by

a Kohlrausch law.

From the _more

specific point

of view of

polymeric liquids,

_we also confirm that the characteristic time ri of the slow relaxation process is

practically

identical with the Rouse

phenomenological

time To. The Rouse time appears attached _{to a} distribution of relaxation

times, which, being

observed in

non-polymeric liquids

as well _as in

polymeric

_{ones, can} bear

no relation with the Rouse distribution of

equation (I). Thus,

_r~ ^identified with

vi, has

another,

_more fundamental

meaning

that the _one attributed to it in the Rouse

theory, meaning which, however,

_{we are not}

yet

^able to

specify.

Our last remark _concems the

glass

transition. What appears

specific

of the

vicinity

of the

glass

transition is the

rapid

relaxation process with _a temperature

dependent amplitude.

In

our

opinion,

the mode

coupling interpretation

of this process is _not

unambiguously supported by experimental

observation. The

rapid

_process

could, altematively,

be related to

spatial

inhomogeneities

of the

density,

the number

(but

not the

size)

of which increases _as the

temperature increases,

above the

glass

transition.

Acknowledgements.

We thank C.

Caroli,

D. Richter and U. Buchenau for many

helpful discussions,

J. Souletie and W.

Petty

for

communicating interesting preprints.

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