Ultrasonic study of the nematic-isotropic phase transition in PAA
Domain: Physics
We present an ultrasonic investigation of the nematic-isotropic phase transition in p-azoxyanisole (PAA). Our measurements were performed at several frequencies ranging from 0.8 to 5 MHz as a function of temperature and, in the nematic phase, as a function of the orientation of the liquid crystal with respect to the ultrasonic wave vector. In both phases, our results are within the ωτ ⪡ 1 regime where τ is the acoustical relaxation time. In the isotropic phase, the results may be quantitatively interpreted using the dynamic heat capacity theory which predicts for α, the ultrasonic absorption, a critical exponent of 1.5. In the nematic phase, we find a critical exponent of ∼ 1 for Δα(T), the attenuation anisotropy, and α(T). This result seems to show that the Landau-Khalatnikov mechanism is the dominant contribution in the temperature range investigated which corresponds to Tc— T values from 1 °C to 20 °C. For comparison purposes we also include some data for PCB that we published some time ago.
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Publié le : 29/06/2012
Langue : Français
Nombre de pages : 9
Type de la publication : Rapports et thèses
Thème :
Savoirs > Science de la nature
Source : Journal de Physique
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789
Ultrasonic
study
of
the
nematic-isotropic
phase
transition
in
PAA
(*)
Y.
Thiriet
and
P.
Martinoty
Laboratoire
d’Acoustique
Moléculaire
(**),
Université
Louis-Pasteur,
4,
rue
Blaise-Pascal,
Strasbourg,
France
(Reçu
le
Il
décembre
1978,
révisé
le
19
avril 1979,
accepté
le
26
avril 1979)
Résumé.
2014
Nous
avons
étudié
la
variation
thermique
de
l’absorption
et
de
la
vitesse
ultrasonore
dans
un
échantil-
lon
de
para-azoxyanisole
(PAA)
orienté
par
un
champ
magnétique,
pour
des
fréquences
comprises
entre
0,8
et
5
MHz.
Dans
le
domaine
des
températures
étudiées,
nos
mesures
correspondent
au
régime
03C903C4
~
1
où 03C4
est
le
temps
de
relaxation
acoustique.
Du
côté
isotrope
de
la
transition,
nos
résultats
peuvent
être
interprétés
par
la
théorie
de
la
chaleur
spécifique
dynamique
qui
prédit
que
le
coefficient
d’absorption
0 3 B 1 ( T )
diverge
avec
un
exposant
1,5.
Du
côté
nématique,
039403B1(T),
l’anisotropie
ultrasonore
et
a(T)
divergent
avec
un
exposant
1.
Ce
résultat
semble
montrer
que
la
partie
critique
de
l’absorption
résulte
de
la
relaxation
du
paramètre
d’ordre
lui-même
(mécanisme
de
Landau-
Khalatnikov)
et
que
les
fluctuations
jouent
dans
cette
phase
un
rôle
négligeable,
tout
au
moins
dans
le
domaine
des
températures
étudiées
(de
1
°C
à
20 °C
de
Tc).
Nous
comparons
ces
résultats
à
ceux
que
nous
avions
obtenus
antérieurement
dans
le
p-n-pentyl
p’-cyanobiphenyle
(PCB).
Abstract.
2014
We
present
an
ultrasonic
investigation
of
the
nematic-isotropic
phase
transition
in
p-azoxyanisole
(PAA).
Our
measurements
were
performed
at
several
frequencies
ranging
from
0.8
to
5
MHz
as
a
function
of
temperature
and,
in
the
nematic
phase,
as
a
function
of
the
orientation
of
the
liquid
crystal
with
respect
to
the
ultrasonic
wave
vector.
In
both
phases,
our
results
are
within
the
03C903C4
~
1
regime
where 03C4
is
the
acoustical
relaxation
time.
In
the
isotropic
phase,
the
results
may
be
quantitatively
interpreted
using
the
dynamic
heat
capacity
theory
which
predicts
for
03B1,
the
ultrasonic
absorption,
a
critical
exponent
of
1.5.
In
the
nematic
phase,
we
find
a
critical
exponent
of ~
1
for
039403B1(T),
the
attenuation
anisotropy,
and
0 3 B 1 ( T ) .
This
result
seems
to
show
that
the
Landau-
Khalatnikov
mechanism
is
the
dominant
contribution
in
the
temperature
range
investigated
which
corresponds
to
Tc2014
T
values
from
1
°C
to
20 °C.
For
comparison
purposes
we
also
include
some
data
for
PCB
that
we
published
some
time
ago.
LE
JOURNAL
DE
PHYSIQUE
TOME
40,
AOÛT
1979,
Classification
Physics
Abstracts
61.30 - 64.70E - 62.80
1.
Introduction.
-
Orientational
order
fluctuations
near
the
nematic-isotropic
phase
transition
have
been
widely
studied
by
ultrasonic
absorption,
and
the
occurrence
of
a
pronounced
maximum
in
the
atte-
nuation
and
a
minimum
in
the
velocity
is
well
known
[1-6].
However,
some
questions
related
to
the
temperature
dependence
of
the
attenuation
para-
meters
are
still
unresolved
on
either
side
of
the
transi-
tion.
The
situation
is
particularly
complex
on
the
nematic
side,
where
unexpected
exponents
of
0.4
to
0.5
have
been
found
for
the
absorption
coefficient
a( T)
and
for
the
relaxation
frequency
r-1(T)
[2,
5].
However,
the
experiments
were
performed
on
MBBA
[2-4]
and
PCB
[5],
compounds
which
present
an
(*)
Presented
at
the
7th
International
Liquid
Crystal
Conference
Bordeaux,
July
1-5
1978.
(**)
(E.R.A.
au
C.N.R.S.).
intramolecular
relaxation
(in
the
same
frequency
range
as
the
critical
relaxation)
that
can
affect
the
critical
parameters.
Furthermore,
most
of
the
measu-
rements
were
made
in
unaligned
samples
or
in
a
fre-
quency
range
not
wide
enough
to
obtain
accurate
relaxational
data.
On
the
isotropic
side
of
the
transi-
tion,
the
situation
is
also
confusing,
since
exponents
of
1
[2]
and
1.5
[3,
5]
for
a(T)
have
been
reported.
The
purpose
of
the
present
work
is
to
make
a
detailed
study
of
the
nematic-isotropic
phase
transi-
tion
in
p-azoxyanisole
(PAA),
a
compound
in
which
no
rotational
isomerism
can
occur
in
the
end
groups.
Therefore,
in
spite
of
the
experimental
problems
associated
with
the
high
transition
temperature,
PAA
appears
to
be
a
promising
compound
for
such
a
study.
Another
advantage
is
the
low
value
of
its
shear
viscosity,
which
leads
to
a
very
small
background
absorption.
Our
measurements
were
performed
at
frequencies
from
0.8
to
5
MHz
as
a
function
of
temperature
and,
Article published online by
EDP Sciences
and available at
http://dx.doi.org/10.1051/jphys:01979004008078900
790
in
the
nematic
phase,
as
a
function
of
the
orientation,
0,
relative
to
q,
the
ultrasonic
wave
vector.
In
the
temperature
range
investigated,
the
data
are
within
the
wi
1
regime.
In
the
isotropic
phase,
our
results
may
be
quantita-
tively
interpreted
using
the
dynamic
heat
capacity
theory
[7],
which
predicts
for
a(T)
a
critical
exponent
of
1.5.
In
the
nematic
phase,
we
find
a
critical
expo-
nent
of -
1
for
both
oc(T)
and
Aot(T).
This
result
seems
to
show
that,
in
the
temperature
range
investigated,
which
corresponds
to
Tc -
T
values
from
20,DC
to
1
OC,
the
dominant
contribution
to
the
sound
absorp-
tion
arises
from
the
Landau-Khalatnikov
mecha-
nism
[8].
The
plan
of
thé
paper
is
as
follows :
In
section
2,
we
describe
our
experimental
technique.
The
theore-
tical
background
is
reviewed
in
section
3.
Our
results
are
presented
in
section
4
and
compared
with
those
we
obtained
earlier
in
PCB.[5],
and
are
analysed
in
section
5.
2.
Experimental.
-
The
ultrasonic
measurements
were
made
in
the
frequency
range
0.8-5
MHz,
using
the
acoustic
resonator.
This
technique,
which
employs
standing
sound
waves
in
a
cylindrical
cavity,
allows
the
simultaneous
determination
ofvelocity
and
absorp-
tion
values
with
high
precision
from
the
frequency
position
and
from
the
3-dB
bandwith
of the
resonance
peaks
of
the
cavity.
On
the
other
hand,
this
technique
requires
only
small
liquid
samples
of
about
3
ml.
A
detailed
description
of
the
apparatus
and
of
the
experimental
procedure
has
been
given
in
ref.
[9],
so
we
shall
not
discuss
them
further.
However,
because
of
the
problems
associated
with
the
high
transition
temperature,
we
have
made
a
more
elaborate
version
of
our
previous
cell.
A
sketch
of
this
new
cell
is
given
in
figure
1.
The
temperature
during
the
experi-
ments
was
regulated
to
within
±
0.02
OC
by
oil
circulating
from
a
constant
temperature
bath
through
the
annular
space
of
the
double-walled
cell.
Tempera-
ture
fluctuations
in
the
cell
were
controlled
by
measur-
ing
the
position
of
a
resonance
peak
over
a
long
period
of
time.
One
sensitive
way
to
do
this
is
to
keep
the
frequency
of
one
3-dB
point
of
a
resonance
peak
constant
and
to
observe
the
relative
variations
in
the
output
amplitude,
which
are
related
to
the
tempera-
ture
variations.
The
quartz
transducers
are
3
MHz,
X-cut,
optically
polished
plates,
30
mm
in
diameter.
The
transducer
spacing
was
2.04
mm.
With
this
value,
our
cell
shows
resonances
at
approximately
330
kHz
intervals.
PAA,
which
is
known
to
be
more
stable
than
a
Schiff’ s
base,
such
as
MBBA,
was
obtained
from
Merck,
and
used
without
further
purification.
The
transition
temperature
Tc
was
135 °C.
To
prevent
oxidation,
the
sample
inside
the
resonant
cavity
had
no
contact
with
the
atmosphere-i.e.,
the
sample
completely
filled
the
cavity.
The
transition
tempera-
ultrasonic
resonator
cell
Fig.
1 .
-
High-temperature
ultrasonic
resonator.
The
following
elements
are
indicated :
(1)
viton
rings
and
metal
rings
pressing
the
viton
rings
against
the
X-cut
quartz
61
and
Q2 ;
(2)
parallelism
adjustment
viton
ring ;
(3)
filling
hole ;
(4)
parallelism
adjustment
blocks ;
(5)
annular
spaces
for
thermostat
liquid ;
(6)
thermostat
mantles ;
(7)
BNC
connectors
with
contact
wires ;
(8)
threaded
ring
to
fix
one
half
of
the
cell
to
the
holder ;
(9)
holder ;
(10)
adjust-
ment
screws;
(11)
parallelism
adjustment
screws.
ture
and
the
quality
of
the
compound
were
verified
by
differential
thermal
analysis
before
and
after
the
experiment
and
a
slight
shift,
of
0.1
OC,
was
found
in
Tc.
Because
of
the
existence
of
a
two-phase
region,
we
have
not
analysed
the
temperatures
closest
to
T,.
For
the
other
temperatures,
our
data
plotted
as
a
function
of !
T -
rj
1
are
not
very
sensitive
to
the
shift
in
Tc.
To
protect
ourselves
from
hysteresis
effects,
we
performed
the
experiments
by
heating
the
sample.
The
sample
was
aligned
by
a
magnetic
field
of
10
kG,
and
for
each
temperature,
the
measurements
were
made
for
e
equal
to
00,
45°,
and
90°.
With
increasing
temperature,
the
attenuation
peaks
became
broader
(as
a
result
of the
increase
of the
ultrasonic
absorption)
and,
in
the
1
°C
around
Tc,
those
for
frequencies
higher
than
2
MHz
disappeared.
The
transition
tem-
perature
Tc
was
defined,
for
our
measurements,
as
the
highest
temperature
at
which
a
non-zero
value
of
Da
was
obtained.
Near
Tc
there
was
an
intermediate
temperature
range
of -
0.3
OC
to
0.4
OC
in
which
a
typical
reso-
nance
peak
became
a
double-peak.
With
a
slight
increase
in
the
temperature,
the
magnitude
of
one
of
791
the
peaks
increased
while
the
other
vanished.
Since
the
frequency
shift
between
the
two
peaks
corresponds
approximately
to
the
difference
in
the
velocity
between
the
nematic
phase
and
the
isotropic
phase,
we
believe
that
this
effect
was
due
to
gravitational
separation
into
a
two-phase
region
separated
by
a
horizontal
interface
(1).
No
quantitative
measurements
were
made
in
this
region.
The
assumption
of
a
two-phase
region
is
also
supported
by
the
fact
that
a
change
in
one
of
the
peaks
occurred
when
the
magnetic
field
was
rotated,
whereas
the
other
peak
was
unchanged.
Such
a
coexistence
region
is
presumably
due
to
small
amounts
of
impurity
which
must
be
difficult
to
elimi-
nate,
since
results
in
MBBA
have
shown
that
the
two
phases
coexist
even
after
repeated
distillations
[10].
3.
Theoretical
background.
-
3.1
Té
ISOTROPIC
PHASE.
-
The
anomalous
behaviour
of
the
ultrasonic
absorption
and
velocity
on
the
isotropic
side
of
the
N-I
transition
have
been
interpreted
by
Imura
and
Okano
[7]
in
terms
of
a
frequency-dependent
specific
heat
on
the
basis
of
de
Gennes’
statistical
continuum
theory
[11].
This
approach
considers
the
interaction
of
the
temperature
variation
of
the
sound
wave
with
the
thermal
fluctuations
of
the
tensor
order
parameter
Qrz/l’
which
are
described
by
a
correlation
function
here,
k
is
Boltzmann’s
constant, q
is
the
wave
number,
and
A
and
L
are
coefficients
of
the
Landau
expansion
of
the
free
energy.
The
temperature
dependence
of
A
is
assumed
to
be
given
by
A(T)
=
a(T -
Ti)
where
Ti
is
the
virtual
second-order
transition
temperature.
The
presence
of
the
sound
wave
induces
periodic
changes
in
A(T)
and
G(q).
Near
the
transition,
the
fluctuations
of
the
order
parameter
have
a
strong
spatial
correlation,
and
G(q)
cannot
follow
the
tempe-
rature
variations
induced
by
the
sound
wave.
This
phase-lag
produces
a
frequency-dependent
heat
capa-
city.
Since
the
sound
velocity
depends
on
the
specific
heat
ratio,
one
obtains
a
complex
frequency-depen-
dent
sound
velocity
whose
imaginary
part
gives
rise
to
the
sound
absorption.
According
to
this
theory,
the
absorption
per
wavelength
aÂ,
the
ultrasonic
absorption
aJf 2,
and
the
velocity
V
are
written
as
follows
(’) :
(1)
Two-phase
separation
effects
have
also
been
reported
in
ref.
[6].
e)
In
the
derivation
of
eq.
(2),
the
temperature
and
the
frequency
dependences
of
the
ultrasonic
wave
velocity
are
assumed
to
be
negligible.
where
cp
and
c°
are
the
heat
capacity
at
constant
pressure
and
constant
volume,
respectively,
in
the
absence
of the
fluctuations
of
the
order
parameter,
and
Acp
is
the
excess
specific
heat
due
to
the
fluctuations.
fi(x)
and
f2(x)
are
the
functions
which
provide
the
theoretical
curves
for
the
frequency
dispersion :
where
cvo
=
A(T)IM
is
the
relaxation
frequency
of
the
longest
wavelength
mode
(q
=
0)
of
G(q)
and u
is
the
transport
coefficient
appearing
in
the
relaxation
equation
of
G(q).
Since y
is
regular
at
the
transition
Wo
goes
to
zero
as
T -
Tc*.
Eqs.
(1)-(3)
are
valid
for
x >
1
(i.e.,
w «
wo).
For
x «
1
(w >
wo),
where
fluctuations
with
wave
num-
bers
much
greater
than
,- 1
are
predominant,
one
expects
a
breakdown
in
the
theory
due
to
the
inade-
quacy
of
the
Omstein-Zemike
form
for
G(q)
[12].
In
the
low-frequency
limit
(x »
1),
eq.
(2)
reduces
to
a
simple
relaxation
law
of
the
form :
where
r -1
is
the
acoustical
relaxation
frequency.
This
quantity
is
related
to
the
relaxation
frequency
coo
of
the
longest
wavelength
mode
of
G(q)
by :
Since
Acp -(T-
T:)-0.5
and
Wo "-1
(T -
Tc*),
it
follows
that
rx/f2 "-1
(T -
T:)-1.5.
On
the
other
hand,
the
relaxation
frequency
of
the
longest
wave-
length
mode
of
Qas
is
wm
=
A(T)Iil,
where Pl
is
the
transport
coefficient
appearing
in
the
relaxation
equation
for
QaP.
To
a
first
approximation y -
yy/2,
and
therefore
c-’ -
8
im 1.
Recently,
another
formulation
has
been
proposed
by
Matsushita
[13]
using
Mori’s
statistical
mecha-
nical
theory
of
sound
attenuation
and
applying
Kawasaki’s
mode-coupling
theory
to
the
order
para-
meter
correlation
functions.
This
approach
leads
to
temperature
and
frequency
dependences
of
the
sound
attenuation
and
of
the
velocity
which
are
essentially
equivalent
to
those
obtained
by
Imura
and
Okano.
3.2
THE
NEMATIC
PHASE.
-
Two
different
critical
contributions
are
expected
in
the
nematic
phase :
one
from
the
relaxation
of
the
fluctuations
of
the
order
parameter
(as
discussed
above)
(this
is
symmetric
792
with
respect
to
Tc)
and
another
from
the
relaxation
of S,
the
mean
value
of
the
order
parameter
(this
is
due
to
the
Landau-Khalatnikov
mechanism).
In
the
following,
we
are
only
concemed
with
the
relaxation
of S.
The
hydrodynamic
theory
of
nematics
gives
the
following
for
the
attenuation
[14]
where
0
is
the
angle
between
the
ultrasonic
wave
vector
and
the
director
and
V
is
the
velocity.
Vl’
V2,
and
V3
are
friction
coefficients
and
v4 -
V2
and
vs
are
volume
viscosities.
It
follows
that
the
anisotropy
in
rx/.f2,
i.e.,
the
difference
between
the
00
and
900
values,
is
given
by :
As
shown
by
Jâhnig
[15,
16],
the
hydrodynamic
theory
can
be
generalized
to
extend
outside
the
hydrodynamic
frequency
regime
by
retaining
the
structure
of
the
hydrodynamic
equations
and
intro-
ducing
a
frequency
dependence
of
the
elastic
and
dissipative
parameters
of
the
system.
Because
of
the
anisotropic
properties
of
the
elastic
tensor,
the
strength
of
the
coupling
between
the
mechanical
variables
and
the
order
parameter
S
depends
on
the
different
tensor
components.
As
a
consequence,
the
relaxation
of
S
appears
in
the
attenuation
anisotropy.
Assuming
that
the
volume
viscosities
are
the
only
relaxing
quantities,
Jâhnig
showed
that :
where
â.E1
and
AE2
are
certain
elastic
parameters.
For
co-r. «
1,
eq.
(9)
reduces
to :
which
is
eq.
(8)
with V4
=
’tm
I1E1
and
vs
=
’tm
M2
and
v 1
=
0.
According
to
mean-field
theory,
Tm-1
(Tl -
T).
In
fact,
a
calculation
by
Kawamura et
al.
[3]
based
on
the
mean-field
theory
but
modified
for
the
weak
first-
order
nature
of
the
transitions
predicts
for
alf 2
and
im 1
an
essentially
temperature
independent
behaviour
in
a
narrow
range
near
Tc,
changing
to
a
power
law
(Tl -
T)
at
temperatures
farther
below
Tc.
The
temperature
Tl
is
defined
by
A
coupling
between
the
sound
waves
and
the
fluc-
tuations
of
the
director
via
Frank’s
elastic
constants
has
been
proposed
by
Nagai et
al.
[5],
extending
Imura
and
Okano’s
theory
for
the
isotropic
phase
to
the
nematic
phase.
The
theory
of
Nagai et
al.,
which
does
not
explain
our
results
(see
section
4),
will
not
be
reviewed
here.
However,
it
should
be
noted
that
an
error
in
their
calculation
of
the
complex
specific
heat
suggested
an
anisotropy
in
the
critical
damping.
The
exact
calculation
shows
that
the
contri-
bution
arising
from
the
fluctuations
of
the
director
is
in
fact
isotropic.
4.
Results.
-
Figure
2
shows
typical
data
of
a/12,
the
ultrasonic
absorption
as
a
function
of f,
the
frequency.
The
separate
curves
are
for
temperatures
which
differ
by
the
amounts
indicated
from
the
transition
temperature
Tc.
The
figure
shows
that
the
values
of
a/f2
are
frequency
independent
in
the
tem-
perature
range
investigated.
Therefore,
these
values
are
those
for
the
low-frequency
limit
wr
«
1.
Fig.
2.
-
The
absorption
coefficient
divided
by
the
square
of
the
frequency,
as
a
function
of
frequency.
Individual
curves
are
for
various
temperatures
above
and
below
the
transition.
The
data
indicate
that
rx/12
is
frequency
independent
over
the
range
of
frequency
investigated
in
contrast
to
the
results
for
PCB
which
are
shown
in
the
insert
for
comparison.
In
contrast,
nematics
at
room
temperature,
like
MBBA
or
PCB,
show
for
the
isotropic
phase
a
strong
dispersion
in
the
same
frequency
range
(see
for
example
the
insert),
which shows
that
the
frequency
relaxation
of
the
order
parameter
for
these
compounds
is
lower
than
that
for
PAA.
This
difference
is
easily
explained
since
the
relaxation
frequency
is
inversely
propor-
tional
to
the
shear
viscosity
and
the
latter
is
smaller
for
PAA
than
for
PCB.
Figure
3
shows
the
sharp
maximum
of
the
ultra-
sonic
absorption
at
the
transition.
In
the
nematic
phase,
the
data
are
for
0
=
90°.
They
are
in
good
agreement
with
the
limited
attenuation
results
(the
793
Fig.
3.
-
The
attenuation
peak
in
PAA.
Below
Tc
the
data
are
for 0
=
90°.
The
results
for
PCB
at
0.5
MHz
are
shown
for
compa-
rison.
triangles)
of Kempf and
Letcher
[17].
For
comparison,
we
have
also
reported,
in
the
same
figure,
the
PCB
data
at
a
frequency
of
0.5
MHz.
Although
the
results
for
the
two
compounds
are
qualitatively
similar,
they
differ
quantitatively,
as
follows.
In
the
isotropic
phase,
the
slow
relaxation
of
the
order
parameter
in
PCB
leads
to
an
increase
of
the
ultrasonic
absorption,
which
is
therefore
larger
than
that
in
PAA.
However,
within
a
given
range
of
T -
Tc,
the
ultrasonic
absorption
increases
by
the
same
factor
for
PAA
and
PCB.
Thus,
the
temperature
dependence
of
the
ultrasonic
absorption
obeys
a
law
which
should
be
identical
for
the
two
compounds.
In
the
nematic
phase,
the
contribution
from
rota-
tional
isomerism
in
the
end
chain
of the
PCB
moleçule
causes
the
ultrasonic
absorption
to
be
larger
with
this
compound
and
the
critical
increase
to
be
less
sharp
than
in
PAA.
Moreover,
the
ultrasonic
absorption
in
PCB
is
so
high
at
the
T
farthest
below
Tc
that
there
is
probably
another
relaxation
process.
In
fact,
it
has
been
recently
shown
[18]
that
PCB
is
a
rather
peculiar
compound
which
in
the
nematic
phase
has
a
pro-
nounced
local
order
of
the
smectic
type,
which
could
contribute
to
the
attenuation.
Figure
4
shows
the
temperature
dependence
of
the
sound
velocity.
Within
our
resolution,
no
frequency
dispersion
was
observed
between
1
and
5
MHz.
Given
the
results
of
the
absorption
measurements,
these
values
of the
velocity
are
those
at
zero
frequency.
Fig.
4.
-
Temperature
dependence
of
the
sound
velocity.
Between
1
and
5
MHz
no
angular
dependence
and
no
frequency
dispersion
were
observed
within
our
resolution.
5.
Comparison
with
theory
and
discussion. -
5.1
THE
ISOTROPIC
PHASE.
-
To
compare
the
data
with
the
theory,
one
must
account
for
the
second-
order
transition
temperature
Ti
and
substract
the
contribution
of
the
shear
viscosity
and
of
the
non-
relaxing
volume
viscosity.
The
contribution
of
the
shear
viscosity
to
rx,f¡2
is
about -
20
x
10-17
cm-1
S2
without
significant
temperature
change.
In
this
esti-
mate,
we
used
the
capillary
measurements
of
ref.
[19].
For
the
non-relaxing
volume
viscosity
’1v,
we
have
assumed ’1v ’"
4/3
’1s,
as
for
a
conventional
liquid.
Fig.
5.
-
Temperature
dependence
of
the
ultrasonic
absorption
in
the
isotropic
phase
of
PAA.
The
solid
line
has
a
slope
of
1.5.
The
result
for
PCB
is
shown
for
comparison
(from
ref.
[5]).
794
Anticipating
a
power
law
dependence
on
(T -
Tc*)
we
made
a
log-log
plot
of
the
î.l/f2
values
obtained
versus
T -
Tc*,
varying
Tc*
within
a
reasonable
range.
With
Tc*
=
Tr
one
obtains
a
distinctly
bent
curve
to
which
one
cannot
fit
a
straight
line
passing
through
the
points.
Taking
Tc -
Ty
=
3
OC,
one
obtains
a
curve
with
the
opposite
bend.
For
Tc -
Tc* ~
1.1
°C,
one
obtains
the
1.52
power
law
shown
in
figure
5,
which
confirms
the
1.5
power
law
that
we
obtained
earlier
for
PCB
[5]
at
0.5
MHz
for
T >
Tc
+
2
°C.
Using
the
result
of
the
fit
in
the
nematic
phase
(see
next
section)
we
find
Tl -
Tc* "-1
1
°C.
Therefore
Tl
is
about
0.1
°C
above
Tc
while
T*
is
about
1.1
°C
below
Tr.
According
to
eq.
(5)
we
may
deduce
the
ratio
of the
relaxation
frequencies
from
attenuation
measurements
in
PAA
and
PCB
provided
the
ratio
Acplc’
is
known.
Calorimetric
measurements
showed
that
àcplco
is
of
the
same
order
of
magnitude
for
the
two
compounds,
and
from
the
data
of
figure
5
we
find
that
Since
the
temperature
variation
of
-rPC1B
is
known
we
may
estimate
and
therefore
at
This
high
value
of
the
relaxation
frequency
explains
that
our
values
of
aJf2
are
frequency
independent
between
1
and
5
MHz
in
the
temperature
range
investigated.
However,
since
to
deduce
-r;¿
we
have
used
the
results
of
experiments
from
many
sources,
the
above
value
must
be
considered
only
an
estimate.
We
tum
now
to
the
relationship
between
the
relaxa-
tion
frequency
of
the
order
parameter
and
the
relaxa-
tion
frequency
which
is
measured
by
ultrasonic
expe-
riments.
Both
quantities
have
been
measured
for
PCB
and
are
reported
in
figure
6.
Curve
(a)
shows
the
temperature
dependence
of
the
acoustical
relaxa-
tion
frequency
-r-t
(from
ref.
[5])
and
curve
(b)
shows
that
of
the
relaxation
frequency
of
the
order
para-
meter
im 1
(from
ref.
[20]).
It
is
clear
that
the
acoustical
relaxation
frequency
is
one
order
of
magnitude
higher
than
the
relaxation
frequency
of
the
order
parameter.
A
similar
observation
can
be
made
for
MBBA
[2].
From
measurements
shown
in
figure
6
we
find
which
is
consistent
with
the
theoretical
expectation,
and
therefore y -
il/2.
The
same
conclusion
was
obtained
at
the
A-N
phase
transition
in
CBOOA
[9]
Fig.
6.
-
(a)
Temperature
dependence
of
the
acoustical
relaxa-
tion
frequency
-r-l/2
7r
for
PCB
(from
ref.
[5]).
(b)
Temperature
dependence
of
the
relaxation
frequency
of
the
order
parameter
T.’/2 n
for
PCB
(from
ref.
[23]).
The
data
indicate
thatr-’ -
8 -r;
1
(see
text).
and
appears
to
be
a
general
feature
of
phase
transitions
in
liquid
crystals.
Finally,
from
the
data
in
figure
4
we
estimate
the
velocity
dispersion,
v(oo) -
V(0)
=
180
m.
In
this
estimate,
V(oo)
was
obtained
by
extrapolating
the
velocity
in
the
isotropic
phase
to
Tc.
Thus
and,
using
eq.
(3),
àcpIco _
1.5.
This
value
is
consis-
tent
with
that
found
by
differential
scanning
calori-
metry
(D.S.C.),
àcplco -
2.2,
considering
that
our
estimate
of
V(oo) -
V(O)
is
an
underestimate
and
that
D.S.C.
gives
only
an
order
of
magnitude.
5.2
THE
NEMATIC
PHASE.
-
Figure
7
shows
the
temperature
dependence
of
the
critical
increase
of
the
volume
viscosities
V4 -
V2
and
vs.
These
viscosity
coefficients
were
determined
using
the
following
equations :
795
Fig.
7.
-
Our
best
estimate
of
the
temperature
dependence
of
the
critical
part
of
the
volume
viscosities
V4 - V2
and
vs.
The
solid
straight
line
has
a
slope
of
1.05.
from
which
the
non-critical
absorption
must
be
removed.
To
this
end,
we
took
for v2
and
V3
the
values
in
ref.
[14]
and,
assuming
as
for
a
conventional
liquid
that
the
residual
part
of
the
volume
viscosities
v4
and
v5
are
of
the
same
order
of
magnitude
as
the
shear
viscosity,
we
evaluated
the
non-critical
absorption
as
-
70
x
10-"
cm-1
s2.
Using
this
value
we
obtained
the
straight
lines
shown
in
figure
7,
with
a
slope
of
1.05
±
0.05
and
with
uncertainties
determined
by
the
x2
test.
In
fact,
the
non-critical
absorption
is
so
small
compared
to
the
critical
absorption
that
a
log-log
plot
of
the
raw
values
of
(11)
and
(12)
appears
linear
with
a
slope
of -
1.
As
mentioned
previously,
two
contributions
to
the
critical
sound
attenuation
are
Fig.
8.
-
Log-log
plot
of
the
attenuation
anisotropy
Aa
in
the
nematic
phase
of
PAA.
Aa -
(v5 -
v4
+
vl).
The
upper
curve
is
the
raw
data.
The
lower
curve
with
a
slope
of N
1
corresponds
to
the
critical
part
and
the
contribution
of
the
coefficient
vl
has
been
removed.
expected
in
the
nematic
phase,
one
from
the
relaxa-
tion
of
the
order
parameter
fluctuations
and
another
from
the
relaxation
of
the
mean
value
of
the
order
parameter
(the
Landau-Khalatnikov
mechanism).
These
two
contributions
do
not
have
the
same
theo-
retical
temperature
dependence.
Since
the
critical
exponent
is
1.5
for
the
fluctuation
mechanism
and
1
for
the
Landau-Khalatnikov
mechanism,
the
straight
line
with
a
slope
of
1.05
shown
in
figure
7
seems
to
prove
that
the
Landau-Khalatnikov
mechanism
is
the
dominant
contribution
in
the
temperature
range
investigated,
that
is,
Tc -
T Jit
1
°C.
We
tum
now
to
the
temperature
dependence
of
Aa/f2 ,
the
attenuation
anisotropy,
which
has
the
form
given
by
eq.
(8).
Note
that
this
quantity
does
not
contain
the
contribution
of
the
specific
heat
which,
according
to
eq.
(5),
is
isotropic.
To
obtain
the
critical
part
of
Aalf ’,
one
must
substract
the
shear
viscosity
vi.
In
principle,
this
coefficient
may
be
deduced
from
the
effective
shear
viscosity vl
+ v2 - 2
V3,
which
is
given
by
the
following
combination :
However,
this
shear
contribution
is
so
small
com-
pared
to
the
critical
attenuation
that
it
can
be
deduced
only
for
the
T farthest
below
T.,
say,
Tc -
T
& # x 3 E ;
15
°C.
In
this
temperature
range,
Vt
+
v2 - 2
V3 ~
0.18
P.
Taking v2
=
0.034
P
and
V3
=
0.024
P
[14],
we
may
estimate
v1.
We
find
v1 ~
0.21
P.
Assuming
for Vl
the
same
temperature
dependence
as
the
capillary
viscosity
and
assuming vl
of the
same
order
of magni-
tude
as
v5,
we
obtain
from
our
measurements
of
the
attenuation
anisotropy
the
linear variation
of (vs -
v4)
with
a
slope
of
1,
shown
in
figure
8.
Although
the
experimental
uncertainties
are
large
for
the
T
farthest
from
Tc,
the
result
shows
the
effectiveness
of
the
Landau-Khalatnikov
mechanism.
According
to
this
mechanism,
the
position
of
the
attenuation
peak
occurs
at
cor.
=
1,
and
it
shifts
to
lower
temperature
with
increasing
frequency.
From
the
relative
position
of
the
peaks
at
two
different
frequencies,
it
should
be
possible
to
deduce
the
temperature
dependence
of
Irm 1 .
Using
a
pulse
technique
with
a
fixed-path
cell,
which
is
the
resonant
cell
itself,
we
measured
the
absorption
coefficient
across
the
transition
at
a
fre-
quency
of
10
MHz
(the
highest
frequency
available
with
our
apparatus)
and
compared
these
measure-
ments
with
ones
made
simultaneously
at
1
MHz.
Unfortunately
it
was
not
possible
to
deduce
the
law
for
zm 1
because
the
data
show
that
the
peak
at
10
MHz
still lies
within
the
coexistence
region.
Since
the
region
is
about
0.4
OC
we
can
say
only
that
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