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Antiferroelectric surface layers in a liquid crystal as observedby synchrotron X-ray scattering

E. F. Gramsbergen, W. H. de Jeu (*)

Solid State Physics Laboratory, Melkweg 1, 9751 EP Groningen, The Netherlands

and J. Als-Nielsen

Risø National Laboratory, DK-4000 Roskilde, Denmark

(Rep le 28 octobre 1985, accepté le 10 dgcembre 1985)

Résumé. - Nous étudions la réflectivité aux rayons X de la surface d’un cristal liquide dont les molécules ontdes groupes terminaux polaires (cyano-); l’instrument est un spectromètre 3 axes à haute résolution combiné àune source de rayonnement synchroton. A la surface de la phase smectique Al, nous observons le développementde quelques doubles couches antiferroélectriques qui peuvent être distinguées de la structure en simples couchesdu matériau massif. Nous développons un modèle qui sépare la densité électronique en un facteur de forme molé-culaire et des facteurs de structures de mono- et bicouches. Avec un nombre restreint de paramètres ajustables,ce modèle rend compte de la courbe de réflexion observée, qui est fort complexe. Il permet de conclure que (i)la première couche moléculaire est disposée avec les queues plutôt que les têtes vers la surface, (ii) le paramètred’ordre smectique de la première mono- et bicouche est saturé, (iii) l’organisation antiferroélectrique disparaitabruptement et non pas exponentiellement.

Abstract. - The X-ray reflectivity form the surface of a liquid crystal with terminally polar (cyano substituted)molecules has been studied using a high-resolution triple-axis X-ray spectrometer in combination with a syn-chrotron source. It is demonstrated that at the surface of the smectic A1 phase a few antiferroelectric double layersdevelop that can be distinguished from the bulk single layer structure. A model is developed that separates theelectron density in a contribution from the molecular form factor, and from the structure factor of the mono-and the bilayers, respectively. With only a few adjustable parameters it accounts for the rather complex observedreflection curve. It shows that (i) the first molecular layer has tails up rather than heads up, (ii) the smectic orderparameter of the first mono- and bilayer is saturated, (iii) the antiferroelectric bilayering does decay rather abruptlyand not exponentially.

J. Physique 47 (1986) 711-718 AVRIL 1986, :

ClassificationPhysics Abstracts61.10 - 61.25 - 61.30 - 68.10

1. Introduction.

Since the first observation of a smectic A-smectic A

phase transition [1], many examples of SA polymor-phism have been reported. These cases always involvestrongly asymmetric elongated molecules with a

large terminal dipole moment (-CN or -N02 endsubstituents). Today at least four « exotic >> SA phasesare known [2]. The most common type of phase withthese compounds is the socalled SAd phase. Disregard-ing diffuse scattering it gives in X-ray experiments aBragg peak at Qo = (2 7C//’) n, where n is the director,

1 the molecular length and I l’ 2 L Apart from apossible reentrant nematic phase, with decreasingtemperature, the phases SAII Sx and SA2 may beobserved. The SA2 phase has long-range antiparallelcorrelation of the molecular orientation in neigh-bouring layers (Fig. la). It is an antiferroelectric phasethat gives rise to two Bragg peaks at Qo = (2 n/0 nand at approximately Qo/2. In the SA phase in additionthe density modulation shows phase fluctuations,leading to a regular lattice of « kinks » in the plane± n (Fig. lb). Consequently, the Bragg spot at Qo/2is split into two spots situated off axis [3]. The SAIphase shows a Bragg spot only at Qo. This can beinterpreted as a « classical >> SA phase in which theup-down distribution of the molecular orientation israndom. Alternatively it can be considered as a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704071100

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Fig. 1. - (a) Smectic layering with the orientation of thealiphatic tail-polar head molecules alternating from layerto layer. This smectic phase is denoted SA2. (b) In the Sxphase there is a regular domain pattern of up- and down-domains in each layer. (For the sake of clarity the picturesare drawn for perfect orientational and positional order).

variation on the SX structure, in which the lattice of« kinks » is not regular.

This paper is concerned with an X-ray study of thesurface of a liquid crystal around the nematic-smecticAl phase transition. It is shown that - contraryto the bulk - at the surface of the SAl phase a Braggpeak develops at Qo/2, which means that at the surfacea series of antiferroelectric double layers exist. Theresults are discussed using a model for the electrondensity profile that can be separated into contributionsof the molecular from-factor, and structure factorsof mono- and double-layers, respectively. The planof the paper is as follows : after a short descriptionof the experiment in the next section, in section 3 theresults are given. In section 4 the model is described,including the type of fits to the data that can beobtained. Section 5 contains a concluding discussion.

2. Experimental.

The compound studied is

which has the following sequence of phase transi-tions [4] :

A sample is prepared in the form of a droplet coveringf"Ott.I 2 cm2 of a glass plate treated such that n is perpen-dicular to the glass plate as well as to the upper liquidcrystal-air interface.The experiments have been carried out at HASY-

LAB (Hamburg), with a modified high-resolutiontriple-axis spectrometer that has been describedelsewhere [5, 6]. The geometry shown in figure 2 issuch that the liquid surface remains horizontal underall circumstances. The horizontal synchrotron beam

Fig. 2. - Scattering diagram and resolution volume for

specular reflection. The angular divergence, e91, of theincident radiation is determined by the source height(- 2 mm) and the source-to-sample distance (20 m) wherease62 is determined by the detector slit height (0.20 mm) andthe sample-to-detector distance (620 mm).

is deflected downwards by an angle 0, (spread in thevertical direction A01 ) by Bragg reflection from a tiltedGe monocrystal to produce the incident wave vectorkl. The height of the sample is controlled such that thebeam always strikes a fixed position of the sample.The detector is placed behind a narrow slit that

accepts radiation at angles 02 in the vertical directionand t/J2 in the horizontal (out of plane) direction, witha spread A02 and A4/2, respectively. As a consequenceof translational invariance in the horizontal plane,the scattering cross-section for X-rays that are eitherspecularly reflected from the flat surface or Braggscattered from the smectic layers parallel to the

surface, is proportional to 6(Q.,) 6(Qy) [7]. The reso-lution of the spectrometer tuned to this radiation isindicatetd by the parallelogram in figure 2, with

02 = 81 and t/J2 = 0. Scattering from the bulk materialcan be monitored by moving the detector in the x or zdirection such that QZ = 2 k, sin 0 does not intersectwith the resolution parallelogram.

3. Results.

The X-ray reflectivity of the surface has been measuredas a function of Q. with Qx = Qy = 0, for five tempe-ratures around the NSA1 phase transitions. The signalfrom the bulk, obtained by making the same scans witha slight offset in Q" was subtracted to ’obtain thesurface reflectivity R(6z) In order to facilitate com-parison with the theoretical reflectivity from the modelto be described below, e result has been devidedby the reflectivity calculated from Fresnel’s law,RF(Q.)- This quantity gi s the reflectivity for radia-tion incident on a plan r dielectric discontinuity,

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similar as commonly used in optics. As for X-raysthe dielectric permittivity of the material is slightlysmaller than 1 (s N 1 - 6 x 10- 6), Fresnel’s lawpredicts total reflection for scattering angles 0 smallerthan 0c 0.15°, and a steep fall-off - Qz 4 above thisangle. This Fresnel reflectivity can be calculatedprecisely with no adjustable parameters and is takenas the background on which the structure of a rea-listic surface (as contrasted with just a discontinuity)manifests itself The region of interest (0.50 0 1.5°)is well above 0c.The reflectivity divided by Fresnel’s law,

RZ(Qz)I RF(Qz) is depicted in figure 3. In this figure Qzis the value of the wavevector transfer inside the liquidcrystal, which is lower than the external value accord-ing to Q’in = Qi out - Q 2. The peak at 6,/6o = 1with Qo = 0.215 A-1 corresponds to the smectic

layers as a pretransitional effect in the nematic phasewhen the NSAL phase transition is approached. Thecorresponding periodicity is approximately equal tothe length of a stretched molecule. The layers extendto a depth çL(T) below the surface, with ÇL divergingat the transition temperature. The scan at the lowest

Fig. 3. - The measured reflectivity R(Q), relative to thecalculated Fresnel reflectivity RF(Q), versus wavevector

transfer Q = 2 k sin 0 relative to Qo = 2 x/L = 0.215 A-1,Notice the logarithmic scale on the ordinate. Data foreach temperature have been displaced by one decade. Thefull line is the best least squares fit of a model described inthe text.

temperature is below this point, and the peak becomesresolution limited. This behaviour is very similaras observed in the SAd phase of 80 CB, p, p’-octyloxy-cyanobiphenyl [6, 7].

In addition to the peak at Qo a second structure isobserved in the region 6z = Qo/2, determined byinterference of the specularly reflected wave with awave diffracted from a layer structure of double

periodicity. The interference is constructive for

Q.IQO 0.5 giving a peak around Qz/Q = 0.45, anddestructive for Q,,IQO > 0.5 leading to an « antipeak »at Q.IQO = 0.55. Since all peaks are asymmetric eachof them may be parameterized by its position, height,width and skewness. This would lead altogether to12 parameters to fit the data at each temperature.We shall in the following section describe a physicalmodel with much fewer adjustable parameters.

4. Model.

We shall first give a qualitative interpretation of thedata.At the interface between liquid and vapour there

will be a pronounced tendency for the molecules tobe oriented so the aliphatic tail is pointing towardsthe vapour, consequently there will be an orientedtop layer of molecules.The nematic phase is susceptible to smectic A

layering and since one layer is already formed at thetop, it is followed by another layer below, a layerbelow that, and so on. The penetration depth of suchlayers, ÇL’ increases as the temperature approachesthe nematic-smectic A transition temperature becausethe susceptibility for layering becomes more and morepronounced. This layering appears in the reflectioncurve R(Q)IRF(Q) as a peak around Q/Qo = 1 whereQo = 2 7c/(one layer thickness). The peak gets higherand sharper as ÇL increases in accordance with the dataaround Q/Qo = 1 in figure 3.However, the top layer is oriented. If there is a

tendency for molecules to stick their heads togetherrather than head-tail, then the second layer will beoriented opposite to the first layer. The two layerstogether form a bilayer and bilayering will penetrateinto the bulk with another penetration depth çp,where the index P indicates polarization, whereasindex L in ÇL indicates layering. Since there is no

particular response to polarization in the nematic

phase one must expect çp to be smaller than ÇL andalso to be roughly independent of temperature. Sincethe period of bilayering is twice as large that for thesmectic layering, it appears as a peak in the reflectedwave around Q/Qo = 0.5. Since this wave can interfereeither constructively or destructively with the ordinaryFresnel reflected wave, one can with the proper phasesaccount for a peak around Q/Qo 0.5 and an anti-peak » for Q/Qo > 0.5 as observed in the reflectioncurve.

We shall now cast this qualitative picture into aquantitative model. To this end we recall the basic

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relation between the reflectivity and the electron

density :

We shall first consider the electron density in a singlemolecule shown in figure 4a.

Fig. 4. - (a) Position of atoms in the molecule shown inthe plane of the molecular-axis and the two benzene ringsclosest to the polar head. The third benzene ring is perpen-dicular to the drawing plane and appears as a cross-hatchedstrip. The numbers coincide with the indices k in table I.

(b) The electron density around each atom is assumed tobe a Gaussian distribution of width a. The resulting densitydistribution along the molecular axis relative to the bulkdensity is shown.

From analogy with similar chemical bonds it seemsplausible that the two benzene rings closest to thepolar head are approximately co-planar, but the

plane of the third benzene ring is turned over aboutn/2. With bond length and bond angles taken fromspace-filling molecular models, this leads to a quanti-tative projection of the molecule as shown in figure 4a.However, the final results depend only very slightlyon the assumptions concerning the relative orienta-tions of the benzene rings.The electron density projected on the molecular

axis is approximated by a superposition of Gaussians,Zk( 2 c Q) -1 exp[ - ( - (k)2/2 U2], centred at thedifferent atomic positions (k in the molecule and witha prefactor Zk equal to the atomic number (6 for C,8 for CH2, 7 for N etc.). The width a of the Gaussian istaken to be 1 A. The values of (k and Zk are listed intable I. The resulting density/(0 with - L/2 C

Table I. - Atomic numbers Zk and coordinates Ck ofthe atoms and groups in the molecules investigated.The indices k coincide with the numbers in figure 4a.

L/2 normalized to an average value of unity, is shownin figure 4b.

It should be noted that this density also representsthe density of a fully polarized layer. We shall decom-pose/(0 into its average value of unity plus a sym-metric part f.(C) plus an antisymmetric partJ:a(C) :

where L1 is unity for - L/2 ( L/2 and zeroelsewhere. Such a decomposition is useful because thesymmetric part of h«() represents the density of arandomly oriented layer. It follows then that the anti-symmetric part of f.(C) must represent the differencebetween a completely polarized and a randomlyoriented layer. Since the master formula for thereflectivity is expressed in terms of the Fourier trans-form of the density p(z) it is intuitively clear that we

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shall need the Fourier transform of/(0 or rather of itssymmetric and antisymmetric parts :

The integration limits are between quotation marksbecause the Gaussian smearing implies that the limitsextend beyond - ! Land! L, respectively.

In evaluating C(Q) and S(Q) the Gaussian modelof the molecule is particular convenient. Explicitformulae are given below and the functions C(Q) andS(Q) are shown in figure 5. It is found that the shapeof S(Q) is quite sensitive for the position of the origoof the coordinates C

We shall now return to evaluation of the Fouriertransform O(Q), cf. equation (1c). Assuming complete

Fig. 5. - The deviation from unity of the normalizedmolecular electron distribution is decomposed into its

symmetric and antisymmetric parts. The correspondingcosine- and sine-transforms versus wavevector Q/Qo are

shown.

polarization one has (p(z) - p)/p = f(Q + f,(C) forthe top layer and (p(z) - P)fP = fs(C) - h(C) for thenext layer because all molecules have reversed direc-tions. Both the polarization and the layering decayas one goes further into the bulk with characteristicpenetration depths çp and ÇL’ respectively. The z-coordinate in equation (1) is decomposed into thelayer index n and intra-molecular coordinate C :

It is now assumed that the shape of the layer densityis invariant and equal tof,(C) + ( - 1)"f.(C) but withdecaying amplitudes for mono- and bi-layer structure :

We then find

The first term originates from the average density p(step-function), smeared by a Gaussian surface fuzzi-ness due to thermal vibrations and a phase factorexp[iQ,o] that allows for a displacement Co of thestep function relative to the origo of the molecularcoordinates.We shall now discuss a convenient mathematical

form of the amplitude functions As(n ; ÇL) and Aa(n ; çp).For clarity we shall for a moment omit the indiceson A and 03BE. The simplest form of A(n ; Q would be

However the data, in particular for the bi-layer cor-relation function, require a more sophisticated decaythan just a simple exponential. One reason is that theamplitude is at its saturation value of unity at the toplayer and it is therefore natural that the correlationfunction starts out with a horizontal tangent. Onepossibility is then the function,

or more general

The properties of this function and its Fourier trans-form are discussed in Appendix 1.

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With the ingredients of the model described abovefits to the experimental data can be obtained ofexcellent quality, as shown as full lines in figure 3.The values of the parameters used are given in table II.

Table II. - Least-squares fit values of the parametersin the model described in the text. The layer spacing inthe bulk SAt phase is Lo = 29.2 A. The value of 03BELlabelled with an asterix is in fact the resolution width.The index J of the decay function Yj(x) is given bothfor layering (JL) and polarization (Jp).

5. Discussion.

The first observations of smectic layering free surfacein liquid crystals in the nematic phase [6-8] involvedonly one layer spacing and in comparison to figure 3only data in the region 0.9 Q/Qo 1.1 were

important. These data were analysed by a purelyphenomenological model and the question of thedegree of orientational, or nematic order, at thesurface was not addressed. In the present work theknowledge of the structure of the molecules is incor-porated into the model of analysis. For comparisonwith earlier work let us for a moment neglect theantiferroelectric layering and thereby the third termin equation (7). The molecular modelling is given bythe cosine transform C(Q) in equation (7) and theamplitude function As(n ; ÇL) can then be interpreteddirectly. We find that A, =1 for n = 0, the first layer,which means that the nematic order parameter is

fully saturated at the surface. The first and secondterm in equation (7) gives rise to interference betweenthe ordinary Fresnel wave (first term) due to the dis-continuity in refractive index and the wave scatteredfrom the layered structure (second term). We havein the present analysis as well as in previous work [8]introduced a phenomenological phase factor

exp[iQCO] between these two waves. In previous workit turned out that the phase factor gave rise to astrongly asymmetric peak around Q = Qo with cons-tructive interference for Q Qo and destructive inter-ference for Q > Qo. The data for the present materialdoes not display this phenomenon around Q = Qo.

However, the spectrum around Q/Qo = 0.5 doesindeed exhibit strong interference effects arising fromthe antiferroelectric surface layers. The molecularmodel which has been introduced in the present workenters solely by its Fourier transforms given in

figure 5. With this model we find the physically verysimple and appealing result that the top layer hascomplete orientational order (all molecules havetails up) and in the next layer all heads are up. Thisantiferroelectric degree of order decays rather rapidlyas one goes further away from the surface. This decayis not exponential but is qualitatively rather like thecurve labelled J = 5 in figure 6. The penetrationdepth 03BEp is only weakly temperature dependent asshown in table II.

Fig. 6. - The penetration into the bulk of mono-layersand bi-layers is not a simple exponential but has the charac-teristic shape of a logistic curve which is modelled by thefunction Yj(x) shown versus x/J(J + 1). The definition ofYJ(x) is given in equation (A .1 ) in the text

As far as smectic layering, independent of orientationof the molecules within a layer, is concerned, thepenetration is much deeper and exhibits a strongtemperature dependence. In previous work [6] it wasfound that the penetration depth C;L coincides with thecorrelation range C; for smectic fluctuations in bulk.This is very likely to be true also in the present case,but we have not yet completed a study of the bulkproperties. In the bulk one can also observe fluctuationof antiferroelectric double layers. It will be interestingto compare the bulk correlation range with jp reportedhere.The r.m.s. values of the smearing parameters Us

for the surface and a for the molecular electrondensity are roughly equal. The values obtained, 4 Aand 4.5 A, respectively, seem quite reasonable.The value found for jo indicates, somewhat sur-

prisingly, that the actual surface is elevated about 5 Aabove the first smectic layer. Possible explanationsare : absorption of water on the surface, or the presenceof decomposition products of the molecules, that arelying on the surface, or a different molecular confor-mation in the top layer.

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One could speculate about the importance of theseobservations for molecular models of the various

types of smectic A phase. The presence of antiferro-electric bilayering indicates that the surface fieldinduces locally smectic A2 type of ordering in thesmectic Al phase. This is certainly not possible for a« classical >> smectic A phase composed of symmetricmolecules. It remains to be seen if such an effect is

present in single layer smectic A phases of weaklypolar molecules. If not, then the present results indi-cate that the SAl phase is closely related to the SA2phase, which is of interest in context with some recenttheoretical models [8]. Then the SAl phase should bedistinguished from a « classical >> SA phase. A possiblemodel is an SAZ structure with locally broken up-downsymmetry and in addition phase fluctuations similarlyas in the SA phase, but randomly distributed.

Finally it should be noted that the present obser-vation of surface double layers is also highly relevantfor the interpretation of X-ray results on free standingsmectic films [9]. It indicates that below a criticalnumber of smectic layers the two surfaces of the filmcan induce a phase transition.

Acknowledgment.

The authors wish to thank Prof. G. Heppke (TU,Berlin) for providing them with the substance. Theexcellent research conditions provided by Hasylabare gratefully acknowledged. This work form partof the research program of the « Stichting voorFundamenteel Onderzoek der Materie » (Foundationfor Fundamental Research on Matter, FOM) andwas made possible by financial support from the« Nederlandse Organisatie voor Zuiver Wetenschap-pelijk Onderzoek » (Netherlands Organization for theAdvancement of Pure Research, ZWO).

Appendix.

Consider the function Yj(x) defined by the finiteseries :

The derivatives of Yj(x) are :

Therefore, for J > 0, Yj(x) starts out with a horizontaltangent at x = 0 and decays with convex curvatureuntil at x = J the curvature becomes concave. With

increasing J higher and higher derivatives of Yj(x)vanishes at x = 0 so in terms of the variable x/(J + 1)the shape gets more and square-like. Indeed, the

limiting shape for J = oo is a square as immediatelyinferred from the identity e+x = E Xj/j ! 1

Let the decaying function A(n) be

For J = 0 A(n) is the simple exponential decayexp[ - njç] and for J > 0 the curvature changes fromconvex to concave around the j’th layer, so in physicalterms we denote ç to be the correlation range.We shall now consider the structure factor

where + applies for A = As and - applies for A = Aa.It is convenient to interchange the summation over jand n

with

in terms of the complex variable

where again + applies for A = As and - for A = Aa.The summation in equation (A. 5) is carried out for

increasing values of j starting with j = 0 :

By differentiating (A. 8) with respect to a on bothsides we obtain

or

By differentiating equation (A. 9) with respect to a oneobtains s2(Q), and so on. The explicit expressions for

Thus with the particular choice of A(n) given byequations (A. 3) and (A .1 ) the structure factor, whichin principle is a sum over an infinite number of layers,is evaluated as a finite sum of (J + 1) terms.

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