HIGH RESOLUTION X-RAY AND LIGHT SCATTERING STUDIES OF BILAYER SMECTIC A COMPOUNDS

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HIGH RESOLUTION X-RAY AND LIGHT

SCATTERING STUDIES OF BILAYER SMECTIC A COMPOUNDS

J. Litster, J. Als-Nielsen, R. Birgeneau, S. Dana, D. Davidov, F.

Garcia-Golding, M. Kaplan, C. Safinya, R. Schaetzing

To cite this version:

J. Litster, J. Als-Nielsen, R. Birgeneau, S. Dana, D. Davidov, et al.. HIGH RESOLUTION X-RAY AND LIGHT SCATTERING STUDIES OF BILAYER SMECTIC A COMPOUNDS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-339-C3-344. �10.1051/jphyscol:1979366�. �jpa-00218762�

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JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 4, Tome 40, Avril 1979, page C3-339

PHASE TRANSITIONS.

HIGH RESOLUTION X-RAY AND LIGHT SCATTERING STUDIES OF BILAYER SMECTIC A COMPOUNDS

J. D. LITSTER, J. ALS-NIELSEN (*), R. J. BIRGENEAU, S. S. DANA, D. DAVIDOV (**), F. GARCIA-GOLDING, M. KAPLAN, C. R. SAFINYA and R. SCHAETZING

Physics Department, Massachusetts Institute of Technology Cambridge, Massachusetts, 02139, U.S.A.

Resum&. - Nous prksentons les rksultats obtenus par diffraction des rayons-X A haute r6solution et diffusion de lumikre i la transition smectique A-nkmatique du cyanobenzylidkne-octyloxyaniline, octyloxy-cyanobiphBny1, et octyl-cyanobiphknyl. Les effets pr6transitionnels en phase nematique sont correctement dkcrits par analogie avec le modele hklium 4 proposk par de Gennes B condition de tenir compte des corrections trks fines likes 51 I'absence d'un vkritable ordre a longue distance en phase smectique. Les constantes Blastiques en phase smectique ont un comportement anormal proba- blement lib aux fluctuations de phase qui sont logarithmiquement divergentes.

Abstract. - We summarize the results of high resolution X-ray and light scattering studies of the smectic A-nematic transition in cyanobenzylidene-octyloxyaniline, octyloxy-cyanobiphenyl, and octyl-cyanobiphenyl. Pretransitional behavior in the nematic phase is essentially consistent with the He4 analogue proposed by de Gennes with subtle effects arising from the lack of true long range order in the smectic phase. Elastic constants in the smectic phase show anomalous behavior probably associated with the logarithmicalty divergent phase fluctuations,

1. Introduction. - Liquid crystals have been fasci- nating objects of study by those seeking to understand the various phases in which condensed matter can exist ; in recent years, smectic phases with orientational and several kinds of translational long range order intermediate between crystalline solid and isotropic liquid states have received special attention. The different smectic phases seem likely to provide theoreti- cally interesting and experimentally accessible systems in which to test and extend the concepts that have emerged from the application of renormalization group techniques to statistical mechanics [l]. The basic idea is that thermally excited fluctuations can exert a profound influence on phase transitions which are not strongly first order. The important factors are thought to be : (i) the number of degrees of freedom of the order parameter (the fluctuating quantity) which is determined by spatial symmetry, (ii) the wave vector dependent susceptibility which indicates how easily fluctuations are excited and is determined by the range and isotropy of the molecular interactions, and (iii) the volume in reciprocal space available to

(*) Permanent Address : Research Establishment Riso, Roskilde, Denmark.

(**) Permanent Address ^:Dept. of Physics, Hebrew University, Jerusalem, Israel.

these fluctuations, which is determined by the spatial dimensionality of the system.

If the number of spatial dimensions exceeds an upper marginal dimensionality d*, the fluctuations do not have a significant effect on the dhase transition.

One may still learn a great deal about the behaviour of the material if the fluctuations are probed by scattering spectroscopy, but the mean field approxi- mation is adequate for statistical mechanical calculations. When the spatial dimensionality is less than a lower marginal dimensionality do, thermal fluctuations are sufficiently important to prevent the phase transition and the establishment of long range order. The existence of d* and d o as well as their values for various systems can be understood by simple phenomenological arguments [2, 31, but calculations for intermediate dimensionality, where long range order occurs but is strongly affected by fluctuations, require the renormalization group techniques. Expe- rimentally, the use of scattering spectroscopy to probe the wave vector dependent susceptibility and the dynamics of the fluctuations is a powerful tool to test the applicability of these theoretical concepts.

Aside from the nematic-isotropic transition, which appears to be well understood on a phenomenological basis [4], the simplest liquid crystal transition these theoretical and experimental approaches have been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979366

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C3-340 J . D. LITSTER et al.

applied to is that between smectic A (SmA) and nematic (N) phases. It has been subjected to inves- tigation by several groups around the world, and much of the work up to 1976 is summarized in refe- rence [5].

Light scattering is an excellent probe of orientational fluctuations of liquid crystal molecules (e.g.

director modes) while X-rays couple to mass density fluctuations. Combined light scattering and X-ray scattering is therefore an ideal tool to study the SmA-N transition, which is the establishment of a one dimensional density wave in a three dimensional orienta- tionally ordered liquid. We have carried out such studies at M.I.T. over the past year and a half on three compounds which form bilayer [6] smectic A phases ; it is our aim in this conference presentation to summarize our results and to discuss their significance.

In what follows we present the results of measurements on p-cyanobenzylidene-p-n-octyloxyaniline (CBOOA), octyloxy-cyanobiphenyl (80 CB), and octylcyanobiphenyl(8 CB).

2. Theoretical background. - Various aspects of the model used to discuss the SrnA-N transition are due to McMillan [7], Kobayashi [8], aGd de Gennes [9], with extensions by Jahnig and Brochard [10]. We summarize here only aspects relevant to an inter- pretation of our experimental results. The SmA order parameter is the amplitude of a one dimensional density wave whose wave vector go is parallel to the nematic director. Thus the density may be written

where d = 2 n/qo is the smectic layer spacing and y5 =

I

$

I

eiq* is a complex two component order parameter. When the phase of

+

is written as go U , u corresponds to a displacement of the smectic layers in the z direction (parallel to the unperturbed nematic director no). The symmetry properties of

+

^enabled

de Gennes to propose a free energy density of the Landau-Ginsberg form

where K,, K,, K, are the splay, twist, and bend Frank elastic constants. In the SmA phase

( I * I >

⁼^*o ⁼^{( -}alP)"2;

if we take ( ( ^I)( ) to be spatially uniform then (2) becomes

Taking the Fourier transform of (3) and applying the equipartition theorem leads to expressions for the mean squared fluctuations in the director. If we choose the wave vector q to lie in the X-z plane we obtain

The coefficients B = q;/M, and D =

t+!J$

q;/M, physically give, respectively, the restoring forces for fluctuations in the phase of (i.e. layer thickness) and fluctuations in molecular orientation away from the normal to the layers. Eq. (4) with B = D = 0 des- cribes the usual director fluctuations in the nematic phase. The presence of fluctuations in SmA short range order modifies the Frank elastic constants. By carry- ing over the calculation of Schmid [l11 for fluctuation diamagnetism in superconductors, de Gennes predicted

where Kio is the normal nematic phase value of the Frank constant and

tll

= (2aMv)-'l2 and

c,

⁼(2 EM,)- ' l 2 are the longitudinal and transverse correlation lengths for Sm short range order.

The X-rays are scattered from fluctuations in

+

^and

thus the scattering cross-section is proportional to

The term

9:

in (6) is not predicted straightfor- wardly from (2), but we include it since it is necessary to fit our experimental results.

In a mean field theory a = ao(T - T,) and a second order transition with classical behaviour occurs at T,. However, application of the Ginsburg criterion to (2) shows [2] that d* ⁼4 for the SmA-N transition ; in that case one might expect (2) to provide an approximate description if the temperature dependence of a, M,, and M, are modified to correspond to the scaling laws. For example, B and D are analogous to the superfluid density in a superfluid and thus should depend on temperature as (

t+!J

)' $ [12].

Although SmA liquid crystals, superconductors, and superfluids are all systems with d* ⁼q ~ h e long coherence length in superconductors means that experimentally observed behaviour is mean field like.

The interaction range in smectics is sufficiently short that critical behaviour near the SmA-N transition should be closely analogous to that of superfluid He4.

There is however one fundamental difference in that the lower marginal dimensionality do is predicted to be three for SmA phases and two for He4. In addition,

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HIGH RESOLUTION X-RAY AND LIGHT SCATTERING STUDIES C3-341

coupling to director fluctuations is predicted to make the SmA-N transition weakly first order [13].

3. Experimental. - Experimental details of the light scattering measurements have been given in a previous publication [5] so we shall not discuss them further. A limited discussion of the high resolution X-ray set up was presented earlier [14], and we shall present a complete discussion in an article which is in preparation and will be submitted to the Physical Review.

The CBOOA was obtained from Eastman Organic Chemicals, while 80 CB and 8 CB were obtained from British Drug Houses. All materials were used as received from the manufacturer.

There is an experimental difficulty with the X-ray studies that we have not yet entirely solved. It was not always possible to make full use of the instrumental resolution because of mosaic spread arising from macroscopic variations in the orientation of the nematic director and smectic planes over the illu- minated region of the sample. The mosaic was measured from the limiting transverse linewidth at Tc.

Thus data were analyzed by convolving (6) with the measured instrumental resolution function [l41 and whatever mosaic corrections were found to be necessary. These were not usually significant, but did lead to substantial corrections for some measurements with t = TITc - 1 < 10-4. Whenever mosaic corrections were made, they are indicated on the figures displaying the data.

4. Analysis of results. - The X-ray scattering experiments were analyzed using eq. (6). A typical scan for all materials is like that shown for CBOOA at t = 0.005 2 in figure 1. The lower half of the figure shows longitudinal (q, = g, = 0) and transverse (q, = qI1 = 0) scans. The upper half of the figure shows clear systematic errors and an unacceptable 'X unless the q;2 term is included. This same anomalous q; term had to be included to fit data for all three compounds and thus seems to be a feature of the

SrnA-N transition in these bilayer compounds.

We first discuss the results for 80 CB in which qo = 0.197

A-'.

In figure 2 we show the values of and

c,

for SmA short range order in the nematic phase of 80 CB. Data are shown for two samples ; the amount of mosaic correction is shown by the line extending downward from the data points to the uncorrected values. The solid lines were obtained from a least squares fit to the data ; we found the data could be represented by power laws

5 -

^t-"

with the effective exponents

The susceptibility is also shown in the figure and diverges with an exponent y = 1.32. In figure 5 we show the coefficient c of eq. (6). Although c tends

CBOOA c = O , X ²=26 t = 0.0052 + c=.13, x2=1.5

-.40 -.30 720 -.l0 0 +.l0 t 2 0 +.30 +.40 WAVEVECTOR q/q,

FIG. 1. -X-ray scattering cross-section for longitudinal and transverse scans in CBOOA at t = 0.005 2. The upper half of the figure illustrates the necessity to include a q: term in the cross-section

( q , = 0.179 A-').

to zero as t -+ 0, the

4

contribution to the linewidth becomes increasingly important as

<,

^diverges.

The same figure also shows the ratio

511/<,

which evolves from 5 & 1 at t = 10-' to 9 f 1 at t = 10-4.

We also measured K; and K, in 80 CB by means of quasielastic light scattering. The intensity I of the scattered light was obtained from the zero intercept of the autocorrelation function and from the d.c. photo- current (identical results). We also measured E,, the anisotropy in the dielectric constant, and obtained K, and K, by dividing ^E: T by the intensity. To obtain absolute values of K, and K , we used our measu- rements of the relaxation time for the director fluctuations and the twist and bend viscosities reported in the literature [15]. The results are shown in figure 3.

The K, data for t < 5 X 10-, (where K! is not important) were fit to a power law and found to diverge as t- ^0.66,consistent with the divergence of

c,,

; the fit is shown as a solid line in the figure. (For both light scattering and X-ray scattering experiments, Tc was measured rather than taken as an adjustable parameter.) We know all of the quantities in (5) from the X-ray measurements and thus can calculate the divergent parts of K, and K, ; the calculated values are

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C3-342 J. D. LITSTER er al.

TC = 66.785 CENTIGRAOE

m m m m

d

~ ( a r b . units)

-I

FIG. 2. - The longitudinal and transverse correlation lengths for short range SmA order in the nematic phase of 80 CB. The open and filled circles are for two samples ; mosaic corrections (see text) are

indicated by vertical lines.

FIG. 3: - Elastic constants K, and K, detepnined by intensity of light scattering in the nematic phase of 80 CB. The dashed lines are the divergent contributions calculated from X-ray data and

eq. (5).

shown as dashed lines in figure 3. It is apparent that (5b) underestimates the divergent part of K, by a factor 2.5 ; we found a similar underestimate in 8 CB, an effect which is perhaps not too surprising since (5) is the result of a mean field calculation. It is also apparent from figure 3 that the contribution of SmA short range order fluctuations to K, is simply not evident for t > lOV4, a result of the fact that (<,/<11)2 4 1 in the bilayer smectics. It seems likely that K: also dominates in CBOOA, thus impossibly large background corrections are necessary and probably explain the widely differing divergences reported in the literature for K, in CBOOA.

From light scattering measurements in the SmA phase we determined the squares of the penetration depths, BIK, and DIK;. The former was obtained by the dependence of the spectrum of director mode fluctuations on q, [16], while D/K; came from intensity measurements normalized to the nematic phase. Both these quantities show the same anomalous behaviour as

in

CBOOA 1161, with B/K1 -- (- Q9, 9 = 0.30+0.05, and D/Ky

-

(-t)+", y)' = 0.50 $- 0.03, as shown by the solid lines in the figure 4.

The results of our measurements in 8 CB are similar to those for 80 CB, but there is a greater evolution in the ratio (11/<1 from 4 f 1 at t = 10-2 to 9 1 at t = 10-4. As a result the effective expo- nents are vll = 0.67

+

^0.03 ^{and v,}⁼^0.51

*

^0.03.

The data are shown in figure 6.

From light scattering measurements we obtained the divergence of K, in 8 CB. From (5b) we see that &:/I should scale as

tII +

⁽²⁴n/kTq@ K:. We

FIG. 4. - Elastic constants BIK, and D / K ~ obtained by light scattering measurements in the SmA phase of 80 CB. The solld lines are least squares fits with slopes of 0.30 (BlK,) and 0.50 (DIK;).

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HIGH RESOLUTION X-RAY AND LIGHT SCATTERING STUDIES C3-343

therefore plot &:/l in figure 7, normalized to give

tl,

⁼^{1 600}

A

^{at t}⁼lOP4, which is the X-ray result.

The solid line is a fit to a power law divergence (v = 0.62 f 0.03) with a small (40

A)

temperature independent background. Actually, comparison with published measurements for K, [l71 shows that (5b)

8 0 C B

Tc

-

6 6 . 7 8 5 C E N T I G R A D E SAMPLE M O S A I C I T Y - 1 . 6 0

TC = 6 5 . 9 2 5 C E N T I G R A D E SAMPLE M O S A I C I T Y - q . 9 2

FIG. 5. - The coefficient c of eq. (6) and the ratio t1,/5, for 80 CB.

Mosaic corrections are indicated by vertical hnes.

gives a divergent contribution to K, which is about four times too small. Smectic phase light scattering measurements also showed B/K, going to zero with an exponent = 0.26

+

^0.06.

Finally we turn to the X-ray measurements for CBOOA ; these have been published previously [l41 but were analyzed with an incorrect instrumental resolution function which made it appear there was a crossover to mean field behavior for t < I O - ~ . When the data are properly analyzed, see figure 8, one finds effective exponents v11 = 0.70

+

^0.04,

v, = 0.62 f 0.05, and y = 1.30 f 0.06. The ratio of

rll/(,

evolves from 5.5 ^f0.1 at t = 10-2 to 8 -+_ 1 at t = 10-4.

In all of the materials studied the different effective exponents for v,, and v, are simply the result of the ratio

tll/(l

changing by a factor from 1.4 to 2.2 over two decades in t. If there were truly different divergences for

t,,

and

t,,

then this factor would change from

-

^{2 to}^COin the last 30 mK above T,. Our data do not indicate this happens, nor do they exclude it ; thus we may speak only of effective exponents over the temperature range studied.

8 C B

TC- 3 3 . 6 8 0 C E N T I G R A D E El m

Cu $-0. 1 9 8 X'

FIG. 6. - Plot of tll and l, for 8 CB. Open and filled circles are for two samples and mosaic corrections are indicated by vertical

lines.

30

10-4 10-3 10-2

T / Tc - l

FIG. 7. - The divergence of K, in the nematic phase of. 8 CB;

see text for an explanation of the ordinate scale.

5. Conclusions. - The use of both X-ray and light scattering spectroscopy in the same laboratory has helped elucidate the nature of the SmA-N phase transition, which turns out to be an interesting one because it is at the lower marginal dimensionality. The effects of marginal dimensionality are subtle ones, with logarithmic corrections to mean field theory at d*, for example. The phase fluctuations which destroy long range order at do in smectics also diverge only as In (q, L ) where L is the sample dimension.

In the nematic phase for t > 10-S the short range order extends over distances less than 10 000

A.

^Thus

it is not surprising that nematic phase pretransitional

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c3-344 J. D. LITSTER et al.

FIG. 8. - Susceptibility,

cl(,

and

<,

from X-ray measurements in the nematic phase of CBOOA. These data correct the incorrect

results of ref. [14].

behaviour should be consistent with de Gennes' model and the helium analogue. Our exponents are probably the most accurate ones for the SrnA-N transition

to-date; they tend to be somewhat lower than the helium analogue (d = 3, n = 2) values [l81 but are consistent within accumulated errors. Thus the only unusual effect one sees is the

4

term which probably results from phase fluctuations within the regions of SmA short range order in the nematic phase. It is tempting to associate the evolution in

5,,/c,

with anisotropic-isotropic crossover predicted by Lubensky and Chen [20], but quantitative calculations are necessary to check this. This behaviour is also consistent with a feature of bilayer smectics, that

tI1/t,

^is

larger than the lengthlwidth ratio of the molecules.

Thus

5,

becomes comparable to the width of a molecule while is still considerably greater than the length. Since

tll

^at ^t⁼^{1 O P 3} ^{is 1 230}

A,

608

A,

and 383

A,

in CBOOA, 80 CB, and 8 CB, respectively, that would also explain why the evolution in

t ,1/5,

is not.seen in CBOOA and quite obvious in 8 CB over the decade from t = 10-3 to t = 10-2.

In the smectic phase, the lack of long range order has little effect on the heat capacity [21], but quite profoundly affects other quantities which are sen- sitive to the diverging phase fluctuations. We believe this lack of true long range order probably accounts for the anomalous behaviour of BIK, and DIK;.

Finally, if the SmA-N transition is first order, it is very weakly so. Any first order transition must occur within 10 mK of T, for all of the materials we studied.

We suggest that the data now accumulated for the SmA-N phase transition are sufficient to motivate a renewed attempt at a theory which includes the effects

of marginal dimensionality.

Acknowledgments. - This research was supported by the Joint Services Electronics Program (Contract No. DAAB07-76-C-1400) and the National Science Foundation (Grant No. DMR-76-18035).

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