COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVERGENCES ABOVE A SMECTIC-A TO SMECTIC-C PHASE TRANSITION OBSERVED BY RAYLEIGH SCATTERING

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COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVERGENCES ABOVE A SMECTIC-A TO SMECTIC-C PHASE TRANSITION

OBSERVED BY RAYLEIGH SCATTERING

M. Delaye

To cite this version:

M. Delaye. COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVER- GENCES ABOVE A SMECTIC-A TO SMECTIC-C PHASE TRANSITION OBSERVED BY RAYLEIGH SCATTERING. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-350-C3-355.

�10.1051/jphyscol:1979368�. �jpa-00218764�

COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVERGELYCES ABOVE A oSMECTIC-A TO SMECTIC-C PHASE TRANSITION OBSERVED

BY RAYLEIGH SCATTERING

M. DELAYE

Laboratoire de Physique des Solides (*), Universitk de Paris-Sud Biit. 5 10,9 1405 Orsay, France

RCsumC. - Nous analysons les fluctuations angulaires critiques des molCcules au-dessus d'une transition smectique-A smectique-C par une technique de spectromitrie par battements de photons.

La comparaison de diverses composantes de Fourier de ces fluctuations met en hidence la saturation de leur susceptibilitC et de leur temps de relaxation dans le domaine critique. On en dCduit une mesure de la longueur de coherence et de son anisotropie. Les valeurs trouvCes pour les exposants y et v confirment l'analogie faite par de Gennes avec la transition /Z de l'hilium-4.

Abstract. - Using light beating spectroscopy, we analyse the critical angular fluctuations of the molecules above a second order smectic-A to smectic-C transition. Comparing various Fourier components of these fluctuations we put into evidence the quenching of their susceptibility and relaxation time in the critical domain. We deduce an absolute evaluation of the coherence length and of its anisotropy. The results on y and v exponents confirm the de Gennes's analogy with the I transition of helium-4.

During the last five years spectacular progress in the understanding of phase transitions have been made.

The renormalization group theory has been very successful and has inspired a host of experiments.

In particular the smectic-A to smectic-C transition appears to be a good model-system for phase tran- sitions in liquid crystals [l]. For this system de Gen- nes [2] has predicted an analogy with the L-transition of helium-4. The aim of experiments will thus be to choose between his prediction and a mean-field's [3].

Indeed we know some smectic-A to smectic-C (SA ³ SC) transitions to be second order. Moreover, the direct coupling between the order parameter and light waves makes light scattering technique a perfectly adapted tool. We have already shown [4]

the existence of critical angular fluctuations above a second order SA -+ Sc transition. In this paper, using the same light beating spectroscopy technique, we present a wave-vector analysis of these fluctuations.

From the quenching of large wave-vector Fourier components of the fluctuations we deduce the critical behavior of the coherence length and its anisotropy.

Let us recall some basic notations. For the S, -+ S, transition the order parameter is the tilt angle $ of the moIecules compared to the normal to the smectic layers. Note that $ is a two-dimensional quantity [2].

In the S, phase the average tilt ( $ ) is kept to zero through the action of a restoring torque - B,

I

$

1,

with B, an elastic modulus. The fluctuations ( $2

>

however are non-zero : they are expected to diverge approaching T, (S, -+ S, transition temperature), as B, goes to zero. The $ fluctuations are correlated over coherence lengths

<,,

^and

<,

in the directions respectively parallel and perpendicular to the mole- cules. In the simplest Landau model [3], these fluctua- tions are purely damped, with a relaxation time

T, = ?/B, where is a viscosity coefficient. All these physical quantities - B,,

t l l , t,,

ij - are expected to vary as power laws of the relative departure from critical temperature E = ATIT ( T is the temperature and AT = T - T,). The power-law exponents are v for both

t I 1

^and

l, (C

⁼

to

^{E-' (1)) and y} for the tilt

$' ) ⁼

-

⁼

-

one hand the mean field theory assumes the constancy of the viscosity coefficient [3]. On the other the renormalization group [5] predicts its divergence with a very small exponent compared to y :

- y =

5,

(3) with A of the order of the unity and y

-

^{2 X 1 0 - ~} for a two dimensional order para- meter (l). In what follows, the divergence will

( l ) As predicted by Hohenberg and Halperin 151 the relaxation

rate varies as o+ cc

<-'

^{cc E+"' with z = 2}

+

^{c q ; c} is of the order of - 0.5 :col can be alternatively written as B,/?, thus o, cc if X? is the exponent associated to 6. From the comparison of the two expressions one deduces :

(*) Laboratoire associC au C.N.R.S. X$ = y - zv = (2 - q ) v - (2

+

^{q ) v = - qv(l}

+

c ) E - 0.5 q v .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979368

COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVERGENCES C3-351

thus be neglected. We then expect zc to vary as

The g m of our work is to measure y and v and to decide between mean field (v = 0.5 y = l ) and renormalization group theory (v = 0.66 y = 1.3).

Let us restrict our analysis to the smectic-A phase above Tc. The angular fluctuations modulate the refractive index giving rise to a characteristic light scattering of Rayleigh width

r

⁼ 117,. By choosing several transfer wave-vectors q in the light scattering experiment, we can analyze various Fourier compo- nents of the molecular fluctuations. If q is small enough so that the associated wavelength 2 n/q remains larger than t(T) in the whole experimental temperature range, the Rayleigh scattered light simply shows the divergence of the amplitude and the relaxa- tion time of the angular fluctuations. Intensity and correlation time are described by the following formulae :

1 - ( , p )

-E-&-Y

B* (5)

On the contrary, using a large transfer wave-vector Q we will reach a temperature where 2 n/Q and ( become of the same order of magnitude. Closer to T,, the scattering experiment analyzes smectic-C-like fluctuations inside the coherence length

5.

These fluctuations remain regular, leading to a quenching effect. For a relaxational system with no conservation laws - it is the case with the S, + S, transition -

theory predicts [5] that the quenching can be expressed, in a first approximation by Orstein Zernike forms :

The additional term B,

Q 2 t2

represents the nema- tic-like contribution to the elastic free energy (it can be also written as KQ where K is a Frank constant).

To be more precise the fluctuations can be analyzed in two orthogonal modes. Denoting by 1 the direction normal to the molecular axis Oz in the plane (Oz, Q) and by 2 the direction normal to both Q and Ox expression (5) should read :

where B is the compression modulus normal to the layers and K,, K,, K, the usual Frank elastic constants.

A convenient choice of the scattering geometry and

polarization [6] will allow us to select one of these two modes at will.

Our experiments have been performed on p-nonyl- oxy benzoate-p-butyloxyphenol

which displays the following phases :

S, SA N

5

isotropic

.

The smectic phases consist in monomolecular layers as checked by X-rays [7]. This compound was chosen for its good stability (temperature drift from 0.5 to 2 mK/hour) and for its very small latent heat peak (less than 10 cal/mole). In addition it is much less absorbing than the compound used in reference [4] : we can thus prevent the sample from increasing its temperature by more than 1 mK, while bringing the laser source power up to 500 pW.

The influence of defects on T, may be important.

The strains corresponding to an edge dislocation of one layer Burgers vector shifts Tc down by some 40 mK for a 500 C( thickness sample [S]. In order to reduce the volume affected by this effect we minimize the number of dislocations by adjusting the paral- lelism of the two boundaries glass-plates (the angle they make is

-

10-4). These glass-plates can be alternatively coated by silane [9], leading to an homeo- tropic sample, or SiO evaporated [10], leading to a planar sample. Despite these precautions, there is little doubt that defects are left in the sample (surface defects for example). These plates are glued on a sample-holder placed in an electronically regulated oven. To measure the oven's temperature we compare a Minco platinum resistor located just below the sample to a Leeds and Northrup standard resistor using a Guildline a.c. bridge ( # 9 975). The stability and the resolution obtained are better than 1 mK ;

the absolute accuracy is not better than 10 mK. For a 10 mK temperature jump and a precision of about 1 mK, time required to reach thermal equilibrium is of the order of five minutes.

The light source is a Spectra Physics Argon laser ( # 165). Its stability in intensity was tested using a spectrum analyzer and measured to be better than 0.1

%

in the range from 10 to 50 000 Hz. The pola- rization of the incident beam can be changed using a half-wave plate and is accurately defined by a Glan prism right at the bottom of the sample. On the scattered beam, another Glan prism is used as an analyzer. The laser beam is filtered through an Oriel spatial filter and focused on the sample making a 600 p diameter spot. The laser power is conti- nuously adjustable from a few pW to a few hundreds mW. A small fraction of the incident beam is deflected

to a cooled phototube which controls the incident intensity. We have built two parallel set-up to analyze simultaneously the light scattered at two different angles : this seems a reasonable method to avoid T, shifts when comparing two sets of experiments.

The geometrical constraints on the oven allow only 3 scattering angles for each sample configuration.

To determine the corresponding wave vector we have measured the ordinary and extraordinary indices (no = 1.61 and n, = 1.77) close to T,. For no we have used an Abbe refractometer ; the ratio n,/no was deduced from the characteristic S, diffraction pattern described in reference [ll]. We thus cover a wave vector range from 1.5 ^X 103 cm-' to 1 .? X 105 cm- l . For each set up, the light scattered over less than one coherence area is measured by a fourteen-stage cooled phototube. A digital clipped ATNE (') correlator computes in real time the auto- correlation function C(z) of the photopulses. We choose a scaled or a clipped mode [l21 at will.

The next problem will be to interpret the corre- lation function. In an homodyne case [l21 the expres- sion of C(z) is

C(z) = i:(l

+

^{a e- , (8)}

where is is the scattered intensity and a a factor intro- duced by the clipping procedure [13]. A Hewlett Packard calculator can compare this expression to the experimental data with a 3 parameters least squares fit and calculate a, is and 7,. In fact we never get an homodyne regime : the texture defects elasti- cally scatter the light acting as a local oscillator of intensity io. The ratio is/io varies from

-

0.5 (large Q close to T,) to less than lOP3 (small q far from T,).

This situation involves several difficulties : a) The true auto-correlation function is now :

C(z) = (io

+

^is)'

+

2 i, is e-'lTc

+

^{i: e- 2'1rc} (9) Unfortunately the fitting procedure is not able to distinguish between (6) and (7). We thus use the same calculator fit as for the homodyne regime and correct the deduced z, value using a suitable technique.

b) The clipping and scaling procedures distort C(T) but these distortions have only been calculated [l41 in two limiting cases :

1) Z (average photocounts/sample time) is very small with respect to 1. The function built by the corre- lator is then very close to the true correlation function.

2) In a strong heterodyne regime (is/io

<

1). We then proceed to specific corrections in is and z, [14].

In fact for small scattering angle and close to T, our data do not fulfil these conditions. The evaluation of z, is not greatly affected but the intensity measure- ment becomes quite unaccurate.

c) Another problem arises from phototube's after-

(=) ATNE : Applications des Techniques Nouvelles en Elec- tronique ; Zone Industrielle de Courtabceuf, 91405 Orsay, France.

pulses : at small q and for AT

-

200 mK their contri- bution in the microsecond range becomes of the same order than the studied signal, distorting com- pletely the correlation function. This limitation also explains why there would not be any real advantage in mixing an external local oscillator to the smectic-A signal.

Let us now present the experimental results. We will first identify the two orthogonal modes (5.1) and (5 .2).

On the one hand, mode 1 is isolated using ordinary polarization for both incident and outgoing beams.

Note that the choice of an incident ordinary pola- rization avoids any Kerr effect of the laser field.

We have checked this mode close to T, in conditions where

B e ; B:

Q:.

(10)

This condition is easily fulfiled since the compression modulus B does not diverge at the transition. The expression (5.1) then reduces to :

and the corresponding correlation time is

On the other, mode 2 is detected in the same geometry but between cross polarizers :

We have measured at the same time z,, and T,, for large Q and have plotted the difference l/z,, - llz,, versus temperature. As expected we have found a constant value leading to a reasonable evaluation of K, - KZ (2 X 10-' cgs). In the following we will restrict to the analysis of mode 2.

We choose to define T, by the sudden appearance at small q of a strong signal characteristic of the S, phase. The corresponding accuracy for a newly made sample is AT =

+

2 mK. On the contrary, for old samples, the overlap of the two signals (S, and SJ occurs over an extended temperature range (up to 40 mK for a very old sample).

We will first discuss the intensity data. At small wave-vector, I, diverges close to T,, as already shown in reference [4]. However, due to the mixing of S, and S, signals just above T,, and to various corrections, the intensity plots have a rather chaotic aspect (cf. Fig. la). If we proceed to a linear fit on a log-log scale we can find for different samples exponents spanning the range 1 to 1.3. These values include both theoretical predictions (mean field and renormalization group) and, in view of experimental uncertainties, it seems impossible to draw a firm conclusion. The intensity data for large wave-vector are more reproducible (see Fig. lb). As expected we do observe a quenching of the intensity divergence

COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVERGENCES C3-353

FIG. 1. - Static data : a) at small wave-vector q = 3 X 103 cm-' ; b) at large wave-vector Q , = 1.2 ^X 105 cm-' (U) ; c) at large

wave-vector Qll = 1.4 X 105 cm-' (0).

close to T,. If we fit the results with the following law :

-

I I cc BL(1

+ ^{Q 2}

^{t 2 ) cc Cte}

+

^{B,, E - Y}

,

supposing y = 2 v, we find that the exponent y lies in the range 1 .l to 1.35, with a maximum probability at about 1.27.

We will now try to deduce a more accurate conclu- sion from the dynamical results. Our typical data are plotted on figure 2 on a log-log scale. For the small wave vector q (Fig. 2a) we observe a linear dependence over the range 4 mK-700 mK. A linear least-square fit with 3 parameters (T,, 7j/Bo,, y ) gives an exponent y = 1.35 _+ 0.05, the value of which is very sensitive to the choice of T,. In the small q limit, planar and homeotropic samples give analogous results. For large wave-vectors, we observe, close to T,, the expected quenching (Fig. 2b for

Q , ⁼ 1.2 X 105 cm-'

in the homeotropic geometry; figure 2c for Q l l = 1.4 ^X 105 cm-' in the planar one). Far above T,

(Q<

p 1) the three curves merge. The data reveal that z, depends on the orientation of Q (at constant magnitude) characteristic of an anisotropy. These data can be analyzed in two independent ways.

First, assuming that y = 2 v, we can fit 1 1 with ~ ~ the law :

FIG. 2. - Dynamics data : a) at small wave-vector g,, = 3 X 103 cm-' .

The continuous line represents the best fit with y = 1.35 f 0.05.

b) At large wave-vector Q, parallel to the smectic layers (best fit y = 1.26 f 0.06). c) At large wave-vector QII perpendicular to the

layers (y = 1.22

+

^0.07).

with -

T o = 9

2 2'

BO,

Q to

Figure 2b then yields :

and figure 2c :

Note that T, is no more a crucial parameter of this fit. Secondly, we can compare the data for small and large wave-vectors and deduce { as

The computed values of

tII

^and

5 ,

are plotted on figure 3.

24

%(A)

00000

,104

*.

*=*, *.

F.//

'

. ..

^**'.

%L

*.* **

- 103 . *.

S..

. .

-3 102 *. *.

. .

10' 3x10' 10' AT(m K).

I I

FIG. 3. - Coherence lengths versus temperature.

c,,

^and 5, ^are

respectively parallel and perpendicular to the molecules.

A linear fit gives v = 0.67

+

^0.01

toll

⁼ 5.8

A

from figure 3b, v = 0.62 f 0.02

to,,

^{= 8.8}

A

^from

figure 3c. In fact, from one sample to the other, we find some dispersion on the results. For small wave vector, the absolute values for zc show a good repro- ductibility (even for an dld sample for which T, has shifted down to 55 OC). On the contrary, the absolute values of zQ are not exactly reproducible : on the same sample but at different places we can find a dispersion of 50

%

on the zQ values. Neverthe- less, the general features depicted on figure 2b remain identical and the dispersion on the values of y computed by the first fit procedure, is not too large.

In contrast, the second fit procedure will be affected by these deviations : the plots for small and large wave-vectors may no more merge at large AT, giving rise to a non-physical asymptotic behavior for far from Tc. In fine, this non-exact reproductibility imposes a statistical study on many different samples ; if the ageing of the sample does not seem to affect the critical exponents, the texture defects undoubtly produce a dispersion on their values. As an example the probability distribution obtained on y over 23 samples is plotted on figure 4a and the distribution obtained on v over 10 samples is shown on figure 4b.

We can now proceed to an interpretation of the results. If nothing had been known about the critical dependence of the viscosity ij, and if static results had been more reliable, a difference between the exponents found by static and dynamic measurements could have been interpreted as an exponent for Ij.

In fact, keeping in mind that any critical exponent of - q should be very small, if non zero, we are led to the conclusion that dynamical data are directly related

FIG. 4. - Statistical results : a ) number of experiments as a func- tion of y values ; b) number of experiments as a function of y values.

to the value of the exponent y. In fact, we have to take into account the regular dependence of 5j (with an activation energy of 0.5 eV for example). The correction just affects the last temperature decade (0.5 to 5 K ) and does shift the y value by more than 1 or 2 ^X 10-2. A statistical study over 42 samples gives as final result y = 1.2 (5)

+

0.01 over 3 tempe- rature decades. For the determination of v, the problem of an eventual divergence of I j has no meaning since I j vanishes in the computation of 5. We find v = 0.64

+

^{0.05 as} an average over 13 samples over 1 or 2 temperature decades. The behavior of the S,.-+ S, transition is thus helium-like. Finally we note the difference between the absolute values of

tll

and t,, which reflects the molecular anisotropy.

The comparison of the values

toll

⁼ 8.8

A

and

to,

^{= 5.8}

A

to the length and width of the molecules (respectively 28.6 and 6 A) could suggest that the cohe- rence properties extend over a larger number of molecules transversally than along the mean align- ment direction. This is compatible with the zig-zag model of reference [7], where longitudinal coherence can be lost more easily along melted end chains.

To conclude we have observed the critical behavior of the soft mode associated with the smectic-A to smectic-C transition : both the intensity and the relaxation time of the thermally excited angular

COHERENCE LENGTH AND ANGULAR SUSCEPTIBILITY DIVERGENCES C3-355

fluctuations diverge when approaching from above a SA + S, second order transition. For large wave- vectors, we have displayed for the first time, and with clear evidence, the quenching of both quantities when reaching the critical domain Q< > 1. The static study of the transition is some what disappointing due to defect induced problems (non-reproductibility) and to indirect intensity evaluation. This deprives us from viscosity measurement. Yet an independent evaluation of and its temperature dependence would be of interest ; unfortunately, this evaluation appears as highly non trivial in smectic phases. Within the frame of theoretical predictions on non-diverging

viscosity, the dynamical data give helium-like expo- nents y

-

^{1.2 (5) f 0.01 v}

-

^{0.64 f 0.05. Our data}

allow an absolute measurement of the coherence length versus temperature, and of its anisotropy.

To check the universality of our results, it would finally be of interest to reproduce the same kind of experiments on different compounds presenting also a second order SA + S, transition.

We thank J. Jacques and C . Germain for the synthesis of the compound and for DTA measure- ments, M. Boix for SiO evaporations, the Orsay Liquid Crystal Group for fruitful discussions and M. Gabay and T. Garel for theoretical enlightments.

References

[l] For a general review on Liquid Crystal refer to DE GEN- NES, P. G., The Physics of Liquid Crystals (Oxford, Univ.

Press, London) 1974.

[2] DE GENNES, P. G., C . R. Hebd. S a n . Acad. Sci. 274 (1972) 758.

[3] VAN HOVE, L., Phys. Rev. 93 (1954) 1374.

LANDAU, L. D. and KHALATNIKOV, I. M., Dokl. Akad. Nauk SSSR % (1954) 469; reprinted in Collected Papers of L. D. Landau (Pergamon, London) 1965.

DE DEL AYE, M. and KELLER, P., Phys. Rev. Lett. 37 (1976) 1065.

[5] HOHENBERG, P. C. and HALPERIN, B. I., Rev. Mol. Phys. 49 (1977) ,435.

[6] GROUPE D'ETUDE DES CRISTAUX LIQUIDES D'ORSAY, J. Chem.

Phys. 51 (1969) 816.

[7] BARTOLINO, R., DOUCET, J., DURAND, G., Ann. Phys. (Compte rendu de ((Physics and Applications of Smectic and Lyotropic Liquid Crystals )p. Madonna di Campiglio) 1978.

[8] RIBOTTA, R., MEYER, R. B., DURAND, G., J. Phys. Lett. 35 (1974) 161.

[9] KAHN, F. J., Appl. Phys. Lett. 22 (1973) 386.

1101 URBACH, W., BOIX, M., GWON, E., Appl. Phys. Lett. 25 (1974) 479.

[l l] RIBOTTA, R., DURAND, G., LITSTER, J. D., Sol. State Commun.

12 (1973) 27.

[l21 For a general review of light beating spectroscopy technique refer to Cummins &Pike, Photon Correlation spectroscopy and Velocimetry (Plenum Press, New York and London) 1977.

[l31 JAKEMAN, E., OLIVER, C. J., PIKE, E. R., J. Phys. A 4 (1971) 827.

[l41 JAKEMAN, E., J. Phys. A 3 (1970) 201.

[l51 GETHNER, J. S., FLYNN, G. W., Rev. Sci. Znslrwn. 46 (1975) 586.