Ultrasonic study of the nematic-isotropic phase transition in PAA

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Ultrasonic study of the nematic-isotropic phase transition in PAA

Y. Thiriet, P. Martinoty

To cite this version:

Y. Thiriet, P. Martinoty. Ultrasonic study of the nematic-isotropic phase transition in PAA. Journal

de Physique, 1979, 40 (8), pp.789-797. �10.1051/jphys:01979004008078900�. �jpa-00209164�

Ultrasonic study ^{of the} nematic-isotropic phase transition in PAA (*)

Y. Thiriet and P. Martinoty

Laboratoire d’Acoustique Moléculaire (**), Université Louis-Pasteur, 4,

Blaise-Pascal, Strasbourg, ^France (Reçu ^{le Il décembre} 1978, révisé le 19 avril 1979, accepté ^{le 26} avril 1979)

Résumé.

Nous

étudié la variation thermique ^de l’absorption ^{et de la} vitesse ultrasonore dans

échantil- lon de para-azoxyanisole (PAA) orienté par

champ magnétique, ^{pour des} fréquences comprises ^entre 0,8 ^et

5 MHz. Dans le domaine des températures étudiées,

^mesures correspondent

régime

~ 1 où 03C4 ^est le temps de relaxation acoustique. ^{Du côté} isotrope ^{de la} transition,

^résultats peuvent ^être interprétés par la théorie de la chaleur spécifique dynamique qui prédit que le coefficient d’absorption 03B1(T) diverge

exposant 1,5. ^{Du côté} nématique, 039403B1(T), l’anisotropie ultrasonore et a(T) divergent ^avec

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

^rôle négligeable, ^tout

moins dans le domaine des températures ^étudiées (de 1 °C à 20 °C de Tc). Nous comparons

résultats à ceux que ^nous avions obtenus antérieurement dans le p-n-pentyl p’-cyanobiphenyle (PCB).

Abstract.

We present

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}

function of temperature and, ^{in the nematic} phase,

^{function of} the orientation of the liquid crystal ^with respect ^{to the} ultrasonic wave vector. In both phases, ^{our results are within the}

^{~ 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,

^critical exponent of 1.5. In the nematic phase, ^{we find}

^critical exponent of ~ 1 for 039403B1(T), the attenuation anisotropy, ^and 03B1(T). This result

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

also include some data for PCB that

published

^{time ago.}

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.

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

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}

half of the cell to the holder ; (9) holder ; (10) adjust-

ment screws; (11) parallelism adjustment

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}

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

velocity

^{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,

function of frequency. Individual

for various temperatures above and below the transition. The data indicate that rx/12 ^is frequency independent

^{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

Fig. ^3.

The attenuation peak in PAA. Below Tc ^{the data}

for 0

90°. The results for PCB at 0.5 MHz

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

angular dependence ^and

frequency dispersion

were

observed within

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

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

^{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;

(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 :

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}

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

is the

data. The lower

with

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 > 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

796

This high ^{value for} m 1 confirms the fact that our

measurements between 1 and 5 MHz correspond ^{to the}

co,r « 1 regime. ^{On the other hand,} this value for ’tm-1

appears ^to be slightly higher ^{than that deduced from}

the data in the isotropic phase

It is clear that it would be of interest to make measu- rements at frequencies higher ^{than 10} MHz, ^{but such}

measurements are difficult because of the enormous

absorption ^at the transition.

Because of the unèxpected exponents found ⁱⁿ

MBBA and PCB, ^{it was} generally thought ^{that the}

critical behaviour in the nematic side of the transition

was more complicated than that observed here, ^and

tentative interpretations ^{have been} proposed [3, 5].

For example, ^the theory discussed in ref. [5], ^which

assumes a coupling between the sound wave and the fluctuations of the director through ^the temperature variation of Frank’s elastic constants, predicts ^{for the}

relaxation frequency ^{a critical} exponent ^{of 0.5 in} agreement with the results in MBBA (0.4) ^{and PCB} (0.5). Obviously, ^this theory ^{fails to} explain the critical exponent ^{of 1} in PAA. The unexpected exponents ⁱⁿ MBBA and PCB are probably because the intramole- cular component ^{has not been} separated ^{from the}

critical component ^{and the} experiments ^were perform-

ed in unoriented samples. ^{In this} respect, it should be noted that a study of the attenuation anisotropy

in MBBA has been recently published by ^Castro

et al. [6]. ^{In that} paper the authors separated ^the

contribution due to rotational isomerism in the butyl

end _groups of the molecule, ^{from the} contribution due to the critical relaxation of the order parameter.

From this analysis they ^{deduced an} expopent ^{of - 1} for Da(T ), ^which is consistent with our direct determi- nation in PAA. However, ^on the basis of a reported exponent ^{of 1 in the} isotropic phase, they ^concluded

that the critical relaxation observed in Da is due to a

fluctuation mechanism similar to that observed above

T,r , that conclusion contrasts with ours, which states that da is mainly govemed by the Landau-Khalat- nikov mechanism. As for PAA, the value for tm-1

in the nematic phase ^{of MBBA}

appears ^to be one order of magnitude higher ^than

the corresponding value in the isotropic phase

Irm 1 - ¹⁰⁸ àT ^{s - 1 from the} li gh t-scatterin g ^{data of}

ref. [21 ]).

6. Conclusion.

We find that ^our ultrasonic mea- surements in PAA are those for the wr « 1 regime ⁱⁿ

the temperature range investigated, ^which corresponds

to T - Tc ^{values from - 0.5 °C to - 15 OC and to} Tc - ^T values from - 1 °C to - 20 °C. We estimate to within - 0.3 °C-0.4 °C the width of the transition

(i.e., the coexistence region ^{of the two} phases) ^and

we have not made quantitative measurements in this region.

In the isotropic phase, ^{the critical} component ^of a(T) ^{is due to} fluctuations in the order parameter, ^and the dynamic ^heat capacity theory, ^{with an} exponent ^of 1.5 for a(T), ^{describes our} experimental ^{data well.}

In the nematic phase, ^{we observe a} divergence ^of Da(T) ^{with a critical} exponent ^{of -} 1, ^{which we} attribute to the relaxation of the order parameter itselr (Landau-Khalatnikov mechanism). Since within

experimental ^error rx(T) diverges ^{in the same} way ^as

Aoc(r), this mechanism seems to be the dominant one

at least in the temperature range investigated (up ^to

1 °C from Tc). ^{Of course} very ^near Tc ^both the relaxa- tions of the fluctuations and the relaxation of the order parameter should contribute to the ultrasonic

absorption and, in this temperature range, it is likely

that a(T) ^behaves differently ^with respect ^to tempera-

ture (and frequency) ^than Aa(T).,

In both phases ^{we find an} exponent ^{of 1 for the} relaxation frequency, and in the nematic phase ^this

result eliminates _one of the suggested models for ultrasonic absorption ^{in this} phase.

There is a différence of one order of magnitude

between the relaxation frequency of the order para- meter and the relaxation frequency associated with the fluctuations of the order parameter.

A self-consistency ^is beginning ^to emerge within a

subset of ultrasonic results in the nematic and the

isotropic phases. ^{In the} isotropic phase, ^{studies on}

PAA and PCB seem to be consistent with a diverging

as (T - Tc*)-1.5. ^In the nematic phase, this work and ref. [6] ^are consistent with Aoe diverging ^{with an} exponent ^of 1, and the results given ^{in refs.} [6], [22]

and [23] Support ^the assumption ^{that rotational}

isomerism in the butyl ^end groups is responsible ^for

the normal (as opposed ^to critical) relaxation _process.

Liquid crystals present ^a great variety ^{of more or}

less second-order phase transitions. In our view, ^the dynamic ^heat capacity theory ^{is of} great interest in the

study ^{of these} phase transitions since it allows the

magnitude of the anomalies in the velocity ^{and in the} absorption per wavelength ^{to be} predicted ^{from the}

measurement of the specific heat. This point ^{is illus-}

trated in ref. [24] for the A-N transition in CBOOA and TBBA. Of course, in the vicinity ^{of the} transition,

more complicated behaviour is expected ^{due, for} example, ^to mode-coupling between the order _para- meter and the hydrodynamic shear mode [25], ^{or to}

fluctuation effects whose wave numbers are much greater than ç -1 [12].

Acknowledgments.

Part of this work was done

at the Laboratoire de Physique ^des Solides of the Université Louis-Pasteur. It is a pleasure ^{to thank}

everyone there for their kind hospitality. ^{We are also}

grateful ^{to Mme} Pouyet, of the Centre de Recherches

sur les Macromolécules, Strasbourg, ^for doing ^the

calorimetric experiments.

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