Structural studies of the intercalated smectic C phases formed by thenon-symmetric a-(4-cyanobiphenyl-4º-yloxy)-x-(4-alkylaniline-benzylidene-4º-oxy) alkane dimers using EPR spectroscopy

P. J. Le Masurier¤ and G. R. Luckhurst*Department of Chemistry and Southampton L iquid Crystal Institute, University of Southampton,HighÐeld, Southampton, UK SO17 1BJ

Many examples of intercalated smectic C and A phases, exhibited by liquid crystal dimers, have been reported. While odd spacerdimers tend to form smectic C phases, even spacer dimers tend to give smectic A phases. The tendency to exhibit smectic Cphases appears to be related to the bent nature of the dimer, in which the mesogenic groups are inclined to each other for themajority of the spacer conformations. The non-symmetric a-(4-cyanobiphenyl-4@-yloxy)-u-(4-alkylanilinebenzylidene-4@-oxy)alkanes (CBOnO É m) do, however, exhibit intercalated smectic A and C phases, both formed by odd spacer dimers. As yet, nosatisfactory explanation as to the di†erence between the intercalated smectic A and C phases, for odd spacer dimers, has beendeveloped. Here, we report the results from angle-dependent EPR experiments and their analysis. Our results indicate that, withinthe intercalated smectic A phase, the mesogenic groups of the odd spacer dimers are, on average, parallel with the smectic layernormal. On entering the smectic C phase a tilt is observed which, for the CBO11O.6 dimer, is found to have a maximum value of18¡. A partial alignment of the tilt directions is also observed in the EPR magnetic Ðeld, which contrasts with the behaviour of thesmectic C phase formed by monomeric liquid crystals. The results are compared with XRD studies on an aligned sample of therelated dimer CBO9O.6.

1 Introduction

Liquid crystal dimers form a class of materials which has gen-erated considerable interest.1 Whereas the more familiarmonomer systems are generally comprised of a single rigidunit with a terminal Ñexible chain or chains, dimers have twomesogenic groups connected with a Ñexible spacer. Thissimple reversal of conventional molecular design allows thecreation of a whole class of liquid crystal materials thatpossess properties fundamentally di†erent to their monomercounterparts, properties that resemble to a greater extentthose of semi-Ñexible main-chain liquid crystal polymers.1 Themajority of dimer systems can be classed as symmetric, in thatthe two mesogenic groups are identical.2,3 However, a secondclass exists where the dimers have di†erent mesogenic groups,the so-called non-symmetric dimers.4h10 Here, we are con-cerned with the non-symmetric a-(4-cyanobiphenyl-4@-yloxy)-u-(4-alkylanilinebenzylidene-4@-oxy) alkanes (CBOnO É m),

which have been observed to form novel smectic phase struc-tures.4,9 At one extreme is the interdigitated arrangement, forwhich a layer spacing of ca. 1.8 times the molecular length isobserved, and at the other is the unusual intercalated phase,where the layer spacing is essentially half the molecular length.The intercalated phase is thought to result from favourablequadrupolar interactions between two unlike mesogenicgroups in the same layer, coupled with a random mixing ofspacers and chains.11 The non-symmetric, CBOnO É m exhibita smectic polymorphism, displaying SmA, SmC and SmIphases. The smectic phases adopted, in particular for the

¤ Present address : Department of Information Systems Engineer-ing, Osaka Sangyo University, Nakagaito 3-1-1, Daito-shi, Osaka 574,Japan.

dimers which form intercalated smectic phases, are seen to bestrongly inÑuenced by the parity of the Ñexible spacer. Fig. 1shows a comparison of the odd and even spacer dimers intheir all-trans conformation, and the proposed structure of theintercalated smectic phases which they form. Within the pro-posed structure of the intercalated smectic C phase the tiltdirection of the director alternates between layers whereas, fora conventional monolayer smectic C phase, the tilt directionchanges randomly from layer to layer. For the CBOnO É monly the odd spacer dimers form intercalated smectic Cphases. This tendency has also been observed for other inter-calated smectic phases exhibited by symmetric dimers, in eachcase the even spacer dimers form smectic A phases and theodd spacer dimers smectic C.12h15 Watanabe et al.12 suggestthat this parity-dependent phase formation is a consequence

Fig. 1 Odd and even spacer dimers in their all-trans conformation,and the resulting intercalated structure of (a) the smectic A phase and(b) the alternating smectic C phase

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of the degree of mesogenic group parallelism for the odd andeven spacer dimers, when the spacers adopt the all-trans con-formation. However, with the CBOnO É m there is an addedfeature. The phase structure for the odd spacer dimers is notlimited to a smectic C alone, all the intercalating dimers withodd spacers show BOTH smectic A and smectic C phases.The difficulty is then in drawing a distinction between thesetwo, when both are formed by dimers with mesogenic groupsinclined at an angle to each other. For the even spacer dimerthe structure is obviously smectic A, however, the odd spacerintercalated structure is more difficult to categorise. Attard etal.9 suggested a possible model to describe the intercalatedsmectic CÈintercalated smectic A transition, this involved abiasing of the distribution of the tilt directions of the mol-ecules resulting in a long-range correlation of the tilt angle.16There is, clearly, a need to test the proposed structures for theintercalated smectic C phase. We have, therefore, undertakenan EPR investigation of the structure of the intercalatedsmectic C of three non-symmetric a-(4-cyanobiphenyl-4@-yloxy)-u-(4-alkylanilinebenzylidene-4@-oxy)alkane dimers andreport the results of our study here. In the following sectionwe describe the application of EPR spectroscopy to study thestructure of smectic phases, in particular to the determinationof the tilt angle. The experimental details of the investigationare described in Section 3. The EPR results are given inSection 4 and their interpretation is discussed in Section 5.The relevance of these results to those of an X-ray study of thephases formed by an analogous non-symmetric dimer is dis-cussed in Section 6. Our conclusions are given in Section 7.

2 EPR spectroscopyEPR spectroscopy is a powerful tool in the study of liquidcrystal phases in general and for the smectic C and other tilted

phases in particular.17h19 When a paramagnetic spin probe isdissolved in a normal low-viscosity nematic the magnetic Ðeldin an X-band EPR spectrometer (typically 0.3 T) is sufficientlystrong to align the phase and form a monodomain. If thesample is then rotated about an axis orthogonal to the mag-netic Ðeld the phase is rapidly (of the order of s) realignedparallel to the magnetic Ðeld (provided However, the*s8 [ 0).situation is di†erent for a smectic phase, now its viscosity issufficiently large and the EPR Ðeld sufficiently small to allowthe sample to be rotated with no magnetically inducedrealignment of the director occurring. This change in directororientation can usually be followed by observing any mag-netic interaction associated with the spin probe. For example,the angle dependence of the hyperÐne interaction is given by

a6 (c) \[A3

M2g8

M2 ] (A3

A2 g8

A2 [ A3

M2g8

M2)cos2 c]1@2

g6 (c)(1)

where and are the components of the partiallyA3A

, A3M

, g8A

g8Maveraged tensors parallel and perpendicular to the director,

for the hyperÐne and g tensors, respectively. For a mono-domain smectic A phase the observed hyperÐne splittingshould vary with the angle between the Ðeld and the directoror layer normal, according to eqn. (1). Fig. 2(a) shows a plot ofthe predicted angular variation of the hyperÐne splitting for acholestane spin probe, with its single electron spin coupled tothat of the nitrogen nucleus. The situation for a normalsmectic C is a little more complicated and so more inter-esting.18 First, we must consider the situation when the layernormal is parallel to the magnetic Ðeld. The directors aretilted within individual smectic layers and the tilt angle is thesame for all smectic layers, however, for the macroscopicsample there will be no preferred tilt direction for the director.Summing over the layers, a cone of director tilt directions is,

Fig. 2 Theoretical angular variation of the hyperÐne interaction with the angle c between the magnetic Ðeld and the layer normal for amonodomain sample of (a) a smectic A phase and (b) a smectic C phase

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therefore, formed. When the orientation of the layer normal,or cone axis, is rotated away from the magnetic Ðeld, the tiltdirections are no longer energetically degenerate and a mag-netically preferred direction will result. This direction corre-sponds to the director being as close as possible to themagnetic Ðeld, thus minimising the magnetic energy. Theobserved hyperÐne interaction will, therefore, no longer be aminimum at a rotation angle, c, of 0¡ when the layer normal isparallel to the magnetic Ðeld. The hyperÐne spacing is nowgiven by

a6 (c)\[A3

M2g8

M2 ] (A3

A2g8

A2[ A3

M2g8

M2)cos2(c[ h)]1@2

g6 (c)(2)

where h is the tilt angle, and (c[ h) is used from 0¡ to 90¡ and(c] h) from 90¡ to 180¡. The minimum in now occursa6 (c)when the directors are parallel to the magnetic Ðeld, i.e. whenthe tilt angle equals the rotation angle. Fig. 2(b) also showsthe predicted angular variation in the hyperÐne splitting for asmectic C phase prepared in this way. As we can see, thehyperÐne interaction has a minimum, equal to when theA3

A,

rotation angle equals the tilt angle, that is the Ðeld pointsalong the edge of the cone. At the 90¡ orientation the hyper-Ðne interaction is smaller than for the monodomain smectic Aat the same orientation. The decrease in the hyperÐne inter-action when the monodomain is rotated through the 90¡ pointis again due to the director adopting the most favourable tiltdirection and so never being perpendicular to the magneticÐeld. Under certain conditions, a smectic C monodomain canbe created in which the tilt directions are the same and themagnetic Ðeld is unable to realign the director over the surfaceof the cone. Then the observed angular variation in the hyper-Ðne interaction is like that for a smectic A phase but shifted bythe tilt angle.17,19 Angle-dependent EPR spectroscopy, there-fore, provides a valuable tool in the study of tilted smecticphases.

3 Experimental

3.1 Synthesis

The synthetic route to the CBOnO É m (n \ 9, m\ 6 andn \ 11, m\ 6 and 10) has been described elsewhere8,9 andthis method was followed throughout. The purity of thedimers was checked by 1H NMR spectroscopy as well as fromtheir transition temperatures, determined using di†erentialscanning calorimetry (DSC) and optical microscopy ; theywere found to be in good agreement (^1 K) with previousstudies.9 The phases formed and the transition temperaturesare given in Table 1. These particular compounds were chosenbecause they all possess a nematic phase which can be alignedwith a magnetic Ðeld. The nematic phase is followed by anintercalated smectic A phase and so, on cooling the alignednematic, a monodomain smectic A phase is formed with thelayer normal parallel to the Ðeld. On cooling this into theintercalated smectic C phase, the layer normal remains paral-lel to the Ðeld but the director is tilted with respect to it.

Table 1 Transition temperatures for the non-symmetric a-(4-cyanobiphenyl-4@-yloxy)-u-(4-alkylanilinebenzylidene-4@-oxy)alkanesused in this study

CBOnO.m TCr/K TJSmI/K TSmISmC/K TSmCSmA/K TSmAN/K TNI/K

CBO9O.6 353 (335) 356 385 390 452CBO11O.6 366 (333) (359) 385 411 450CBO11O.10 373 (359) 381 408

3.2 Techniques

3.2.1 Low-Ðeld EPR experiments. The EPR experimentswere carried out using an ECS 106 Bruker EPR spectrometer,Ðtted with a goniometer which allowed rotation of the sampleabout an axis perpendicular to the magnetic Ðeld. The tem-perature was regulated using a temperature-control unit basedon a Eurotherm B-VT 2000 VTU. The paramagnetic spinprobe 3b-doxyl-5a-cholestane was used, since previous studieshave shown it to mimic the behaviour of the mesogenicgroups in the CBOnO É m.20 The spin probe was added to theliquid crystal in the minimum proportion necessary to elimi-nate induced chiral behaviour coming from the chiral probe,as well as to reduce the inÑuence of the probe on the tran-sition temperatures. This is easy to achieve, however, becauseof the high sensitivity of the EPR technique ; typically the con-centration of the spin probe is ca. 1 ] 10~4 M and has nosigniÐcant e†ect on the transition temperatures (^0.5 K). Thesolution of the spin probe in the liquid crystal was degassed(using a freezeÈthaw cycle under vacuum) Ðrst to reduceoxygen-induced line broadening and secondly to minimise thedecomposition of the spin probe at high temperatures. Amonodomain smectic A phase was prepared by Ðrst taking thesample into the nematic phase and then under the maximummagnetic Ðeld attainable (ca. 0.66 T) cooling it slowly (ca. 0.25K min~1) into the smectic A phase. In the resulting mono-domain the layer normal is aligned parallel with the Ðelddirection. Orientation-dependent studies were then under-taken on the smectic A phase, by rotating the sample tubeusing the computer-controlled goniometer accessory to theECS106 spectrometer. The microwave cavity was re-tunedautomatically after every rotation and spectra recorded at 10¡intervals from 0È180¡. The samples were then cooled, againunder a high magnetic Ðeld, into the smectic C phase. Angle-dependent studies were performed throughout the smectic Cphases at 2È3 K intervals allowing 10 min at each temperaturefor the samples to reach thermal equilibrium. For the rota-tional studies on the smectic C phase the Ðeld was increasedto 0.66 T before and during each rotation, in order to increasethe probability of forming a Ðeld-induced monodomain of tiltdirections.

3.2.2 High-Ðeld EPR experiments. The EPR experimentswere also carried out on a multi-band Bruker ESP 300E EPRspectrometer since this has a larger magnetic Ðeld of 1.8 T.The magnetic torque acting on the director is quadratic in themagnetic Ñux density and so the higher Ðeld increases thetorque by a factor of ca. 7.4 which increases signiÐcantly theprobability of creating a monodomain sample. The experi-mental details di†er only with respect to the Ðeld value aftereach rotation of the sample. In the high-Ðeld experiment theÐeld was increased to 1.8 T, for 1 min, after each rotation andbefore a spectrum was recorded. The high-Ðeld EPR measure-ments were performed at the EPSRC cw Multi-frequencyEPR Service Centre at the Chemistry Department of the Uni-versity of Manchester.

3.2.3 XRD studies. The X-ray experiments were performedusing a Marconi-Avionics GX20 rotating anode generatoroperated at 30 kV and 25 mA with a 0.1 mm focusing cap.The Cu-Ka radiation (j \ 1.542 was nickel-Ðltered andÓ)brought to a point focus using toroidal optics. The sampleswere contained in 1.5 mm diameter thin-walled tubes whichwere mounted horizontally in a purpose-built electricallyheated holder incorporating rare earth magnets, giving analigning Ðeld across the sample of ca. 1 T. The temperature ofthe sample was computer controlled (Microrobotics SkorpionK4) to ^0.2 K using a platinum sensor mounted in the

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Fig. 3 Stacked plots of the orientation-dependent EPR spectra for the cholestane spin probe dissolved in (a) the smectic A phase and (b) thesmectic C phase of CBO9O.6 at 385 and 362 K, respectively

sample holder close to the sample. The di†raction patternswere obtained using a purpose built optoelectronics X-raydetection system analogous to that described elsewhere.21 Theimages were processed using a modiÐed version of the TV4software kindly provided by Professor S. M. Gruner ofCornell University, USA. The calibration of the spacings mea-sured for the CBO9O.6 was checked using lauric acid andsilver behenate standard samples. The CBO9O.6 was cooledfrom the nematic at a rate of 1 K min~1. The XRD experi-ments were carried out at the Department of Chemistry atImperial College of Science, Technology and Medicine,London.

4 EPR results4.1 Orientation-dependent EPR results for CBO9O.6

Fig. 3 shows a stacked plot of the orientation-dependentspectra for the intercalated smectic A and C phases. For theintercalated smectic A phase, the spectra were recorded at 385K just before the transition to the intercalated smectic Cphase. The spectrum in the foreground is for the 0¡ orienta-tion with the director parallel to the magnetic Ðeld. The angleof rotation increases into the plot in 10¡ steps through 90¡ upto 180¡. The spectra have narrow linewidths and the stackedplot is symmetrical around the 90¡ spectrum, as required fromthe symmetry of the smectic A sample. If the recorded hyper-Ðne splittings are compared to those calculated using eqn. (1)for the orientation dependence of the hyperÐne interaction,the Ðt is good (see Fig. 4). These results show that a mono-domain sample of a smectic A phase has been created, whichis suitable for growing the aligned smectic C sample. Thissmectic A was then cooled into the smectic C phase, under

Fig. 4 Angle dependence of the nitrogen hyperÐne splitting for(=)the cholestane spin probe in the smectic A phase of CBO9O.6 com-pared with the values calculated from eqn. (1), (ÈÈ)

high magnetic Ðeld (ca. 0.66 T), and the rotation study repeat-ed at a temperature 23 K into the smectic C phase. The resultsare shown in Fig. 3(b) where the most obvious feature of thesestacked spectra is the departure from the normal three-linespectrum of the spin probe in the region between 0¡ and 90¡.The Ðve-line spectra found for these orientations are indicativeof a pronounced deviation of the director from a uniform dis-tribution which is quite unlike that for a conventional smecticC phase described in Section 2. Fig. 5 shows the individualspectra for the 0¡, 20¡, 40¡, 60¡ and 90¡ orientations (the 120¡,140¡, 160¡ and 180¡ being equivalent to these). The 0¡ spec-trum has only a slight asymmetry in the shapes of the threehyperÐne lines, which indicates an essentially uniform directordistribution with respect to the magnetic Ðeld, consistent withthat for a normal smectic C. However, when the layer normalis rotated away from the Ðeld direction the spectrum containsone sharp and four broad hyperÐne lines, showing that thedistribution of the director is non-uniform relative to the mag-netic Ðeld. The splitting of the low- and high-Ðeld lines is mostpronounced when the angle between the layer normal and theÐeld is 40¡. When the angle is 90¡ the normal three-line spec-trum is again observed and the lines are sharp, indicating that

Fig. 5 Selected EPR spectra of the cholestane spin probe taken fromthe stacked plot in Fig. 3(b), for the smectic C phase

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Fig. 6 Temperature dependence of the EPR spectra for the choles-tane spin probe dissolved in CBO9O.6 with the layer normal orientat-ed at 40¡ to the magnetic Ðeld

the director is uniformly aligned with respect to the magneticÐeld. The temperature dependence of this unusual smectic Cbehaviour was investigated by recording spectra at 3 K inter-vals from the end of the smectic A range to just inside thesmectic I phase. Fig. 6 shows the 40¡ spectra for selected tem-peratures. In the smectic A phase (385 K) a conventionalthree-line spectrum is observed, but as the sample is cooleddeeper into the smectic C phase the splitting of the high- andlow-Ðeld hyperÐne lines increases. This increase shows thatthere is a growing deviation in the director distribution from astate of uniform alignment on going further into the smecticC. Before we see how these unusual results might be under-stood we shall Ðrst consider the results for the other non-

Fig. 7 Low (ÈÈÈ) and high-Ðeld (É É É É É É É) EPR spectra for the chol-estane spin probe dissolved in the smectic C phase of CBO11O.6 at359 K, with c\ 40¡

symmetric dimers with intercalated phases, as well as those forthe experiments at the higher magnetic Ðeld.

4.2 Orientation-dependent EPR results for CBO11O.6 andCBO11O.10

The orientation-dependent spectra for the smectic A phase ofCBO11O.6 at 400 K, like those for CBO9O.6, were indicativeof a good monodomain and so the sample was cooled into thesmectic C phase. The observed orientation and temperaturedependence of the EPR spectra for the intercalated smectic Cphase were totally consistent with the CBO9O.6 results. Thisnon-uniformity of the director distribution could, however, bea result of our method of preparing the initial monodomain inthe smectic C phase. As a test, the following technique involv-ing pre-rotation of the smectic A monodomain was employedin an attempt to align the director uniformly in the smectic Cphase. While the sample was still in the smectic A phase justbefore the transition to the smectic C, the monodomain wasrotated by an angle considered to be close to the expected tiltangle, which in this case was 20¡. The sample was then cooledinto the smectic C phase with the magnetic Ðeld strength at itsmaximum value. For conventional smectic C phases this tech-nique of pre-rotation has the e†ect of producing a uniquedirector orientation, along the Ðeld direction. As a conse-quence, a monodomain of tilt directions, on the surface of thecone, is created. However, for the intercalated smectic C phaseof CBO11O.6 no such alignment of the tilt directions wasobserved.

The only member of the CBOnO.10 homologous series ofdimers displaying an intercalated smectic C phase isCBO11O.10. The results obtained for the smectic C phase ofthis dimer were again consistent with those for the other inter-calated dimers with the EPR spectra showing Ðve lines forintermediate angles. It would seem, therefore, that the inter-calated structure of the smectic C phase causes a distributionof tilt directions that cannot be aligned uniformly by themaximum magnetic Ðeld of the ECS 106 Bruker spectrometer.To see if this is a result of the relatively low value of the Ðeldthe experiments were repeated for CBO11O.6 using a highermagnetic Ðeld.

4.3 High-Ðeld EPR experiment

The Bruker spectrometer used for high-Ðeld experiments has amaximum Ðeld of 1.8 T which, as we have argued, shouldincrease the ordering of the tilt directions. At 359 K theorientation-dependent EPR spectra within the smectic Cphase are analogous to those for other non-symmetric liquidcrystal dimers which we have studied. However, the splittingsfor the high- and low-Ðeld lines observed at intermediateorientations (see e.g. Fig. 4) were found to be more apparent,because the linewidths for the sample produced at high Ðeldwere smaller. To illustrate this signiÐcant change in thespectra we show, in Fig. 7, the spectra obtained for the smecticC phase of CBO11O.6 prepared with low-(0.66 T) and high-(1.8 T) aligning Ðelds. In the following section we shall attemptto understand our results in terms of a model for the structureof the smectic C phase based on that shown in Fig. 1(b).

5 InterpretationThe key feature of our EPR studies of the intercalated smecticC phase formed by the non-symmetric dimers is the appear-ance of a Ðve-line spectrum when the magnetic Ðeld is inclinedwith respect to the layer normal, although at the extremeorientations of 0¡ and 90¡ the spectra contain just three hyper-Ðne lines. This behaviour is reminiscent of that observed byLuckhurst et al.17 for the smectic C phase of 4,4@-diheptyl-oxyazoxybenzene which had been prepared with two magneti-cally inequivalent sites created by producing the smectic C

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phase from a nematic with surfaces to align the smectic layersand a magnetic Ðeld parallel to the surfaces. The inequivalenceof the two director sites results from the tilt directions di†er-ing by 180¡. This is exactly the situation which is to beexpected for the proposed structure of the smectic C phase ofthe odd spacer dimers shown in Fig. 1(b). This is, of course, acaricature, because it shows the molecules as if they hadperfect conformational, translational and orientational order.However, what the structure is showing is that, in a mono-domain sample of the alternating, intercalated smectic Cphase, there are two director orientations, with tilt directionsdi†ering by 180¡, which alternate from layer to layer. If thesystem had the tilt directions uniformly aligned throughoutthe sample, as we have indicated, then the spectrum should, atsome optimal orientation of the layer normal with respect tothe magnetic Ðeld, consist of just two three-line spectra withthe central peaks superimposed. In practice a Ðve-line spec-trum is indeed observed although the linewidths are found tobe broader than for the individual spectra at the two extremeorientations. This, together with the observation that the spec-tral lines are sharpened when a strong magnetic Ðeld is usedto align the tilt directions, suggests that the spectra arebroadened by a distribution of the tilt directions with respectto the Ðeld.

To conÐrm this interpretation quantitatively it is necessaryto simulate the EPR spectra for the smectic C phase with dif-ferent orientations of the layer normal with respect to themagnetic Ðeld. Such simulations require the distribution func-tion for the directors with respect to the magnetic Ðeld, whichcan be related to the distribution function for the azimuthalangle, a. In the absence of the magnetic Ðeld all values of a areequally probable, but in the presence of the Ðeld there will bea bias resulting from the magnetic torque. To Ðnd the form forthis magnetic energy we Ðrst establish a laboratory-basedcoordinate system in which the z axis is parallel to the layernormal and the x axis is in the plane formed by the normaland the magnetic Ðeld ; the y axis is orthogonal to the othertwo axes (see Fig. 8). The polar angles for the two directorsare h and n [ h where h is the tilt angle ; it is assumed that theazimuthal angles for the directors are the same, which seemsreasonable given the basic molecular organisation in the inter-calated alternating smectic C phase [see Fig. 1(b)]. We furtherassume that the local diamagnetic susceptibility is cylin-drically symmetric about a director which, although notstrictly true for a smectic C phase, is also a reasonableapproximation. The anisotropic magnetic energy is then givenby

Umag \ [(*s8 B2/3k0)[(P2(nü 1 Æ BŒ )] P2(nü 2 Æ BŒ )] (3)

where is a unit vector parallel to the magnetic Ðeld and isBŒ k0the magnetic constant. Evaluating the two scalar products in

Fig. 8 Polar coordinates for the two director orientations in thealternating, intercalated smectic C phase

terms of the angles deÐning the director orientations and themagnetic Ðeld gives, after some algebra,

Umag\ [(*s8 B2/3k0)] [3(cos2 c cos2 h ] sin2 c sin2 h cos2 a) [ 1] (4)

Except for the trivial case when the magnetic Ðeld is parallelto the layer normal (c\ 0¡) the magnetic energy is a functionof the azimuthal angle for the directors. It is a minimum whenthe azimuthal angle is either 0¡ or 180¡ and a maximum whena is 90¡. The probability distribution function, p(a), is

p(a) \ Z~1 exp[[Umag(a)/kB T ] (5)

where the normalisation is

Z\P

exp[[Umag(a)/kB T ]da (6)

although this is a maximum when a is 0¡ or 180¡, the prob-ability adopts signiÐcant values for other azimuthal angles aswe can see from the results shown in Fig. 9. These were calcu-lated for a tilt angle of 18¡ and three di†erent orientations ofthe magnetic Ðeld (c\ 0¡, 45¡ and 90¡), with *s8 B2/3k0kBTequal to 1 and 5. As might be expected from the form of themagnetic energy, the probability distribution shows the great-est variation when the magnetic Ðeld is in the smectic plane(c\ 90¡). It is this distribution function for the azimuthalangle which controls the line broadening in the EPR spectrumand, to explore this quantitatively, it is necessary to simulatethe EPR spectra as a function of both the sample orientationand the Boltzmann factor To calculate the*s8 B2/3k0 kBT .form of a spectrum we need to know the angles andb1 b2between the magnetic Ðeld and the two directors, since it isthese angles for the assumed symmetry axes which determinethe hyperÐne spacing and the g tensor,

a6 (b) \[A3

M2g8

M2 ] (A3

A2g8

A2 [ A3

M2g8

M2)cos2 b]1@2

g6 (b)(7)

and

g6 (b) \ [g8M2] (g8

A2[ g8

M2)cos2 b]1@2 (8)

The two angles can be written in terms of the other angularvariables as

cos b1 \ cos c cos h ] sin c sin h cos a (9)

Fig. 9 Probability distribution function for Ðnding the director at anazimuthal angle a to the magnetic Ðeld, p(a), calculated for a tilt angleof 18¡, equal to (a) 1 and (b) 5, and c\ 0¡, 45¡ and 90¡*s8 B2/3k0 kB T

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and

cos b2 \ [cos c cos h ] sin c sin h cos a (10)

From these values of and we can calculate the reso-a6 (b) g6 (b)nance Ðelds for the nitrogen hyperÐne lines in the spectrum ata given director orientation, from these the spectrum isobtained by superimposing a lorentzian on each resonanceÐeld. The widths of the lorentzians associated with the individ-ual lines are given by

T 2~1(m)\ A] Bm] Cm2 (11)

where m is the nitrogen spin quantum number for the particu-lar transition. The linewidth coefficients A, B and C are foundto be orientation dependent and theory22,23 shows this depen-dence to be of the form

A\ A0 ] A2 P2(cos c)] A4 P4(cos c) (12)

where is an L th rank Legendre polynomial. It is dif-PL(cos c)

Ðcult to obtain all these angular linewidth coefficients for asmectic C phase and so we have adopted the following semi-empirical method. First, the widths of the three hyperÐne lineswere measured as a function of the director orientation in thepreceding smectic A phase and these results were smoothed byÐtting them to the form given in eqn. (12). The linewidthsobtained in this way were then converted to those appropriatefor the smectic C phase by scaling them individually to opti-mise the agreement between the observed and simulated EPRspectra for the complete range of sample orientations. Theindividual spectra were obtained by summing those associatedwith the range of azimuthal angles.

The spectra for the CBO11O.6 smectic C phase were simu-lated, starting with the spectra recorded at 359 K, where thesplitting of the high- and low-Ðeld lines had reached amaximum, and then repeating the simulations at 375 and 381K. The spectra when the layer normal was at 40¡ to the mag-netic Ðeld were simulated Ðrst because the Ðve-line characterof the spectrum is most apparent. This occurs because theorientation dependence of the hyperÐne splitting is greatest atthis angle. Fig. 10 shows the experimental and simulatedspectra obtained using the parameters listed in Table 2. Thesewere held Ðxed for the three temperatures and agreement wasobtained by varying the tilt angle. At 359 K the best Ðt wasfound when the tilt angle was set equal to 18¡ and the Bolt-zmann factor, was equal to 0.5. This suggests*s8 B2/3k0kBT ,that the Ðeld-induced alignment of the two directors over thesurface of the cone is relatively weak. There is seen to be goodagreement between the simulated and the experimentalspectra, which supports our model for the alternating, inter-calated smectic C phase in which a partial alignment isachieved by the magnetic Ðeld. The tilt angle was observed to

Table 2 Magnetic parameters used in the simulation of the EPRspectra of the cholestane probe dissolved in the non-symmetricCBO11O.6 dimer

parameter value

A3M/10~4 T 19.25

A3A/10~4 T 6.5

g8M

2.0058g8A

2.0062A0/10~4 T 2.47A2/10~4 T [0.34A4/10~4 T 0.42B0/10~4 T [0.33B2/10~4 T 0.06B4/10~4 T 0.23C0/10~4 T 1.49C2/10~4 T 1.17C4/10~4 T [1.74*s8 B2/3k0 kB T 0.5

Fig. 10 Simulation of the c\ 40¡ EPR spectrum for a cholestanespin probe dissolved in the smectic C phase of CBO11O.6 at 359, 375and 381 K; (ÈÈÈ) experimental, (É É É É É É É) simulation

decrease to 12¡ and 8¡ for the smectic C phase at the highertemperatures of 375 and 381 K, respectively. We again Ðndgood agreement between the experimental and simulatedspectra for these temperatures. Fig. 11 shows a plot of the tiltangle obtained from these simulations as a function of theshifted temperature the growth in h and its(TSmCSmA[ T ) ;form is in accord with previous EPR studies of conventionalsmectic C phases.18

To test the model further we have attempted to simulate thespectrum obtained from the sample prepared at high-Ðeld.Fig. 12 shows the simulated spectrum using the same param-eters as for the low-Ðeld experiment, the only di†erence beingin the Boltzmann factor, which was increased*s8 B2/3k0kBT ,

Fig. 11 Temperature dependence of the tilt angle, h, for the smecticC phase of CBO11O.6

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Fig. 12 Simulation of the c\ 40¡ EPR spectrum for the cholestanespin probe dissolved in the smectic C phase of CBO11O.6 at 359 K;(ÈÈÈ) experimental, (É É É É É É É) simulation, high-Ðeld experiment

from 0.5 to 4, in keeping with its quadratic dependence on themagnetic Ðeld strength. Such an increase has a major inÑuenceon the distribution function for the azimuthal angle which isdramatically steeper (see Fig. 9). This is reÑected by the nar-rowness of the spectral lines and hence the improvedresolution. The Ðt of the simulated to the experimental spec-trum, using the same parameters as for the low-Ðeld experi-ment, is not perfect, in particular the peak positions of thelow- and high-Ðeld lines do not agree well. This could be dueeither to an improved monodomain in the original smectic Afor the high-Ðeld experiment or an inaccuracy in the recordedtemperature, the improved monodomain being more likelywith the higher Ðeld. However, the important feature is theimproved spectral resolution, which is predicted by our modelfor the Ðeld-induced alignment of the directors and observedexperimentally.

6 XRD resultsIn order to support our detailed results for the tilt angle in thealternating, intercalated smectic C phase of CBO11O.6 wehave measured the X-ray scattering pattern of the alignedsmectic C of CBO9O.6. For comparison we have also deter-mined the scattering patterns for its other liquid crystalphases, namely nematic, smectic A, smectic I and crystal J. Allthe X-ray scattering patterns are shown in Fig. 13. The small-angle peaks are located on the meridian, which conÐrms thealignment of the director parallel to the magnetic Ðeld andshows that the layer normal is parallel to the Ðeld for thesmectic and the crystal phases. The distance between thepeaks on the meridian does not change to any signiÐcantextent with the structure of the phase. It corresponds to aperiodicity of 20.6 which is essentially one half of theÓ,molecular length, estimated from CPK models to be 41.8 Ó,and is in keeping with the intercalated structure of thesephases. It is also of considerable interest to note that theseparation between the di†use peaks on the meridian for thenematic phase corresponds to a distance of 21.3 whichÓ,shows that the nematic has a local intercalated structure.However, our major interest is in the tilt angle within thesmectic phase and this is obtained from the splitting of thewide angle arc.12 This splitting is most clearly apparent in theX-ray scattering pattern from the crystal J phase (see Fig. 13).The angle between the two lines joining the maxima to thecentre of the scattering pattern gives twice the tilt angle, whichis found to be 16¡. It is more difficult, but not impossible, tolocate these maxima for the two tilted smectic phases ; fromsuch measurements we estimate the tilt angle to be 13¡ in thesmectic I and 12¡ in the smectic C phase at 363 K. Although

Fig. 13 XRD patterns for an aligned sample of CBO9O.6 in thenematic, smectic A, smectic C, smectic I and crystal J phases

the error in these values is relatively high (ca. ^2¡) they are inaccord with the maximum value of 18¡ found using EPR spec-troscopy for the lowest point in the smectic C phase ofCBO11O.6, which was studied in detail. Finally, we shouldnote that for the smectic C there is a third di†use peak exactly

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on the equator, although this is not apparent from the XRDpattern shown in Fig. 13. This is analogous to that found forthe crystal J phase, which suggests that there is a local hex-agonal packing of the mesogenic groups in the intercalatedsmectic C phase analogous to that in the smectic I phase.

7 ConclusionsThe EPR spectra of the cholestane spin probe dissolved in theintercalated smectic C phase show that there are two distinctdirectors with tilt directions which di†er by 180¡. This struc-ture is consistent with the packing of the bent dimers, inwhich the direction of the mesogenic groups necessarily alter-nates from one layer to the next. The tilt angle is seen toincrease from 0¡ at the SmCÈSmA transition to a maximum of18¡ for CBO11O.6. This value for the tilt angle is consistentwith those determined by XRD for the uniformly alignedsmectic C, I and crystal J phases of CBO9O.6, where amaximum tilt angle of 16¡ was determined. This is in accordwith the bent geometry of a dimer in its all-trans conforma-tion in which a mesogenic group is estimated to make anangle of 15¡ with respect to the spacer. The extent to whichthere is a uniform alignment of the tilt direction is determinedby the orientation and magnitude of the magnetic Ðeld. Whenthe director is aligned by a Ðeld of 1.8 T in the smectic Cphase the tilt direction is observed to be quite uniform. In theintercalated smectic A phase exhibited by odd spacer dimersthe tilt directions for the mesogenic groups are uncorrelated,but at the transition to the smectic C phase long-range corre-lations between the tilt directions are established and the tiltangle grows until it reaches a limiting value dictated by themolecular geometry.

are grateful to the EPSRC for the award of a researchWestudentship to P. J. Le M. and for a grant (GR/H96904)towards the cost of the Bruker EPR spectrometer. We alsowish to thank Drs. R. H. Templar and J. M. Seddon (ImperialCollege of Science, Technology and Medicine, London) forallowing us to use the X-ray equipment and Mr. S. Moriartyfor helping us make the measurements. Miss P. Steans pro-vided valuable assistance in recording the EPR spectra and

we are grateful for this and to Dr. F. E. Mabbs for an alloca-tion of time on the EPSRC Multi-frequency cw EPR Serviceat the University of Manchester.

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Paper 7/09029C; Received 16th December, 1997

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