compound - W. de Jeu

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University of Pennsylvania, Philadelphia, PA 19104, U. S . A. ... published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys: .... iterative process the strand could ultimately be centered in the beam to within 20 itm and its.
J.

Phys.

France 50

(1989)

461-483

15 FÉVRIER

461

1989,

Classification

Physics Abstracts 64.70Md - 61.30Eb

-

61.65

Frustration and

+

d

helicity

in the ordered

phases of

a

discotic

compound Paul A. Heiney (1), Ernest Fontes (1), Wim H. de Jeu Patrick Carroll (3) and Amos B. Smith, III (3)

(2,*) ,

Antoni Riera

(3),

(1) Department of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. (2) FOM-Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands

(3) Department of Chemistry and Laboratory for Research University of Pennsylvania, Philadelphia, PA 19104, U. S . A. (Reçu

le 6

septembre 1988, accepté

le 17 octobre

on

the Structure of

Matter,

1988)

Nous avons étudié les mesophases Dhd et Dho et la phase cristalline K à basse température de l’hexa(hexylthio)triphenylene par diffraction de rayons X. La phase cristal liquide Dhd à haute température est formée d’un arrangement triangulaire à deux dimensions de colonnes avec un ordre entre colonnes à courte distance. La phase Dho à température intermédiaire est formée d’un arrangement triangulaire de colonnes ayant un ordre périodique de position et un ordre hélicoïdal incommensurable. En outre, on retrouve un super-réseau à trois colonnes qui peut résulter de la frustration imposée par l’interdigitation de la symétrie triangulaire. Nous suggérons que la transition Dhd ~ Dho. est déterminée par l’augmentation de rigidité et de longueur des chaînes hydrocarbonées due à la décroissance en température. La phase K est monoclinique. La transition K ~ Dho peut s’effectuer par un déplacement moléculaire relativement petit. Resume.

2014

X-ray diffraction to study the discotic mesophases Dhd and crystalline K phase of hexa(hexylthio)triphenylene. The hightemperature liquid crystalline Dhd phase is composed of a two-dimensional triangular array of columns, with short range intracolumn order. The intermediate-temperature Dho phase consists of a triangular array of columns with periodic positional and incommensurate helical intracolumn order. Furthermore, a three-column superlattice is found, which may result from the frustration imposed by molecular interdigitation in triangular symmetry. We suggest that the Dhd ~ Dho transition is driven by the increase in hydrocarbon tail stiffness and length with decreasing temperature. The K phase is found to be monoclinic. The K ~ Dho transition can be accomplished by a relatively minor displacement of the molecules. We have used Abstract. low and the temperature Dho 2014

(*)

Also at the

Open University,

P.O. Box

2690, 6401 DL Heerlen, The Netherlands.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005004046100

462

1. Introduction.

Liquid crystals [1] have provided us with interesting structures in which ordering takes place in less than three dimensions, and thus are unique model systems to study low-dimensional phase transitions [2]. Conventional rod-like mesogens may form a nematic phase, with shortrange positional and long-range orientational order, or various types of smectic phase, which have in common a well-defined density modulation in one dimension. The simplest type of smectic phase is smectic-A which has liquid-like order in the other two dimensions. Discotic liquid crystals [3] are composed of disc-shaped niolecules with rigid planar central cores and typically 6-8 flexible hydrocarbon chain tails. They form nematic phases, and also more highly ordered columnar mesophases consisting of two-dimensional arrays of columns. Columnar discotic phases have been classified according to the symmetry of the columns and the nature of the intracolumn order. The most commonly observed « hexagonal disordered » Dhd phase consists of a triangular array of columns with only fluid-like intracolumn order and consequently no long-range column-column correlation of the molecular heights. In cases where the molecules are tilted with respect to the long axis of the column, or where the 3-fold molecular symmetry is broken, a « rectangular disordered » Drd phase can also be found [4, 5]. Additionally, the molecules within each column may order to form the hexagonal ordered phase Dho. However, if true long-range intracolumn order develops then the columns must be correlated, since long-range order in one dimension is not possible. Ordered columnar phases may thus be analogs of higher smectic phases [6], which are three-dimensional crystals with local structures close to those of true smectic phases. There are theoretical indications [7, 8] that the phase transition from the crystalline phase to the Dhd phase may be second order ; however, this transition has not been studied experimentally. of the various phases of This discotic hexa(hexylthio)triphenylene [9] (HHTT, Fig. la). mesogen is unique in having both a Dho and a Dhd phase at temperatures intermediate between those of the crystalline (K) and isotropic (I) phases [10]. As far as we are aware this is the first example of a orderdisorder transition within the columns retaining the hexagonal columnar lattice intact. In other materials such a transition is usually accompanied by a change from a hexagonal to a rectangular lattice. Preliminary results on the Dhd-Dho phase transition have already been given elsewhere [10, 11]. In the Dho phase helical order develops within the columns, associated with the molecular orientations. The helical period is incommensurate with the intermolecular spacing. Additionally, a three-column superlattice develops, associated with both the helical phase and the vertical displacement of the three columns. A model will be presented in which this structure is a solution to the frustration imposed by both molecular interdigitation and rotational degrees of freedom in a triangular lattice. The Dhd phase shows an unusual negative thermal expansion coefficient, which is most likely associated with stiffening of the aliphatic tails of the molecules. Though the crystal structures of several mesogenic compounds have been determined [12], it has not been possible to establish general rules relating the structure of the crystalline phase to that of the mesophase that occurs upon melting. In many cases a considerable enthalpy change is associated with the first phase transition upon heating from the crystalline phase. This indicates important structural relaxations, and consequently little correspondence between the structures of the two phases can be expected. The situation may be different when the transition from the crystal to the isotropic liquid goes in a series of steps with enthalpy changes of the same order of magnitude. For rod-like molecules this is the case when several higher-ordered smectics occur, and similarly with disk-like molecules when the In this paper

we

present X-ray results in

a

single-crystal geometry

463

1. (a) Hexa(hexylthio)triphenylene (HHTT) ; (b) the pin is drawn slowly from the bulk material in the cup to produce a freely-suspended strand ; (c) the strand fixed axes and reciprocal lattice are defined with the column axis parallel to 6z and Qx along the (100) direction. With scattering vector q defined, ~ and X are used as azimuthal and polar angles, respectively.

Fig.

-

transitions involve several columnar phases. The K phase in the present compound turns out structure, but with a fundamental spacing that equals that in the Dho phase. Hence, the K-Dho transition can be accomplished by a relatively small displacement of the molecules. The plan of the paper is as follows. In section 2 the experimental methods are discussed, including some details of growing and characterizing discotic strands in the Dhd and Dho phases. Section 3 gives the X-ray results for the Dhd and Dho phases and a qualitative discussion of the structures and the phase transition. Section 4 contains the structure determination of the K phase. Using these ingredients in section 5 a model for the Dho phase is developed, followed by a brief discussion in section 6. to have a monoclinic

2.

Experimental.

The HHTT give the

to

sample was synthesized by Kohne et al. following phase sequence :

as

described elsewhere

[9],

and

was

found

Freely-suspended columnar phase strands with a single-crystal geometry were studied using a high-resolution 4-circle X-ray diffractometer. As discussed below, a sharp and wellcharacterized instrumental resolution function and

a

well characterized and oriented strand

464

permitted detailed analysis of the intrinsic peak positions and lineshapes. However, the anisotropic sample mosaic introduced uncertainties in the integrated peak intensities, and thus into the detailed analysis of the molecular conformations. Due to extreme supercooling, the evolution of the Dho phase was observed on cooling to well below 62 °C, and details of the Dho H K phase transition could not be studied. The unit cell structure of the K phase was determined from a single-crystal sample using an Enraf-Nonius CAD4 diffractometer. This technique is inappropriate for detailed lineshape measurements, but yields reliable integrated peak intensities for structural refinement. Freely-suspended strands of the Dhd and Dho phases were obtained using a technique first developed by Safinya et al. [5]. They were grown in situ in a two-stage temperature controlled oven [13]. Pictured schematically in figure lb, as a pin was inserted and then slowly (1 Um/min) pulled out of a cup reservoir, a fiber or strand of approximately 150-200 >m in diameter was drawn to a length of 1.5-2 mm. Retractable periscopes provided a microscopic view of the strand during its growth ; the best quality strands were uniform in thickness and optical clarity, while poor ones had opaque bulk-like regions. Strands were grown and annealed for 12-24 h at high temperatures in the Dhd phase. Good quality strands could not be drawn in the Dho phase, and strands in the nematic and isotropic phases were not selfsupporting. The column axes were approximately oriented along the direction of the strand pull, referred to as the strand z-axis. The orthogonal x-y plane is the plane of hexagonal symmetry of the column packing, and is referred to as the hexagonal basal plane. The reciprocal space basis is labeled Qx, Qy, and Qz ; Qb refers to an arbitrary direction in the hexagonal basal plane. qx, qy, qz and qb refer to measured diffraction components in the above directions (Fig. lc). In the discussion that follows below we will use a triangular reciprocal lattice to describe the Dhd and Dho phases : H and K are triangular Bragg indices in the basal plane (qx is oriented along the (H00 ) direction) and L is the index describing reflections oriented along the column axis. The temperature controlled sample oven was accurately positioned to within 10 >m of the center of a Huber 4-circle diffractometer. The X-ray windows defining the temperature stages were thin-walled beryllium cylinders coaxial with the oven long axis. The outermost window provided a sealed barrier against external thermal fluctuations and convection. The two internal windows were integral components of two concentric heating stages ; the inner stage was controlled at the desired sample temperature, and the second stage was held 1 °C lower. The temperature uniformity over the strand sample was balanced to better than 0.01 °C. A helium atmosphere was maintained inside the oven. The total attenuation of the X-ray beam due to the windows was 35 %. An Elliott GX-13 rotating anode generator with a fine-focus source diameter of 100 I£m was employed to collect the X-ray data on the Dhd and Dho - phase strands. A vertical focussing LiF(200) monochromator crystal provided an incident flux of approximately 107 CuKal photons/second into a 400 )JLm diameter circular spot at the sample position. Using a flat LiF(200) analysing crystal this configuration yielded an in-plane longitudinal resolution of AI q1 =0.005Â’ full width at half maximum (FWHM), and an in-plane transverse resolution of àq, = 0.0007 Â-1 FWHM. The vertical resolution was determined by slits placed after the monochromator and before the scintillation detector, adjusted such that the vertical angular convergence of the incident X-ray beam onto the sample matched the vertical angular acceptance of the detector. Two slit configurations were used, yielding for low 0.027 -1. In vertical resolution Aq,,,, 0.27 -1, and for high vertical resolution qvert the high vertical resolution mode, the total incident flux onto the sample was reduced by a factor of ten. The instrumental longitudinal and transverse resolution were modeled in our data analysis =

=

465

lineshapes with widths corresponding to àqll and Aq, . The vertical resolution triangular with an acceptance range of àqve,, as determined by the convolution of two square slit functions. In addition to instrumental broadening, the Bragg peaks were broadened by intrinsic sample mosaic. There were two possible sources of this broadening : (1) from a multicrystalline sample, the measured profiles are the sum of the intensity scattered by all domains in the region of the strand illuminated by the incident X-ray beam ; (2) in a single-crystal sample, broadening may result from strain fields inside the strand. Experimentally, the broadening was described by two mosaic widths. The polar misalignment of the zaxis of the strand could be described by a Gaussian distribution in polar angle with width of about 2. 2° . The azimuthal rotational distribution about the strand long axis was typically 1.52°. As a result, in our analysis the intrinsic sample scattering was modelled by a threedimensional Gaussian ellipsoid of revolution. The diffracted intensity was quite sensitive to both the position and the alignment of the strand in the beam. The alignment of the strand (i.e. the orientation of the reciprocal space axes) was determined by measuring the positions of the six primary (100) triangular peaks at regular intervals during an experiment. Likewise, the translational displacement of the strand was calculated and corrected by comparing the intensities of these six peaks. (Since the absorption of the X-ray beam by the sample was quite small, measurement of the beam attenuation by the sample was insufficient to accurately position the strand). Through an iterative process the strand could ultimately be centered in the beam to within 20 itm and its by

Gaussian

was

orientation determined to within 0.1 ° . The sample mosaic, strand position, X-ray spot-size and instrumental resolution coupled in a complicated way, resulting in systematic variations in measured intensity that were unrelated to the intrinsic scattering amplitude. For structural determinations, the quantity of interest is the integrated intensity under each Bragg peak. Peak intensities (which are listed in Tab. 1 for the Dho phase) were measured by two methods : 1) Strong peaks were characterized by diffraction profiles measured in three orthogonal directions, and 2) the intensities of weak peaks were determined by carefully measuring (i.e. counting for relatively long time periods) the scattering at the indexed Bragg position. In the second case, the integrated intensities were calculated from peak intensities using an analytic form derived from the convolution of the instrumental resolution with the sample mosaic. We estimate the overall uncertainty in our peak intensities to be ± 10-30 % for the strongest peaks and ± 0.5 (counts/s) for the weakest peaks. Nevertheless, since the peak intensities spanned a dynamic range of about 104, we can say a great deal about the structure of the Dhd and Dho phases, as will be discussed in detail later on. The effect of strand misorientation on peak positions was minimized by careful measurements of vector differences between peaks ; we estimate our overall precision in peak position at AI q 1 _«- 1.0 x 10- 3 A-1. The effects of instrumental resolution and sample mosaic on X-ray peak profiles are illustrated in figure 2. Figure 2a shows equal intensity contours of a peak in the qx - qz plane. The long axis of the rod-shaped contours results from the convolution of the instrumental vertical resolution with the polar mosaic, while the short axis results from the 0.047 A-1 instrumental longitudinal resolution. For example, the long axis dimension àq FWHM results from the sum in quadrature of the vertical resolution width Oq,,ert 0.027 Â -11 FWHM and the polar mosaic width =

=

shows an equal intensity contour plot of raw scattering data collected in the  -11 plane (L 1). The lineshapes of peaks in this plane are determined by the 1.727 qZ convolution of the vertical resolution function and mosaic broadening. In this sample the

Figure 2b =

=

466

Measured intensities (with estimated uncertainties) in counts per second, and calculated intensities for Bragg peaks in the Dho phase. Measured integrated intensities are derived from either peak intensities or orthogonal scans through Bragg peaks, as discussed in text. Calculated intensities are the result of a best fit to model III with adjustable parameters given in table II. Table 1.

-

misoriented from the oven long axis by a polar rotation of 1.75° about the [1, 1, 0] crystal direction, and was additionally broadened in this direction by approximately 2. 2° . Thus, the polar profiles of nominally equivalent peaks vary somewhat : for instance, the peaks along the [x, x, 1 ] line are broadened in the orthogonal direction by both mosaic and the instrumental resolution. The structure of the K phase was determined using a single crystal 0.04 x 0.15 x 0.05 mm3 sample which was recrystallized from hexane. The crystal was mounted on an Enraf-Nonius CAD4 automated diffractometer and studied with Ni-filtered CuKa radiation. A total of 5 280 reflections were measured by the w - 2 () scan technique, of which 2 656 with intensity 1 > 3 were used in the structure solution and refinement. strand axis

was

467

Fig. the

2. - (a) Equal intensity contour plot of X-ray scattering data collected in the 6z - Qx plane about (2, 0, 0.379) peak position ; (b) equal intensity contour plot of X-ray scattering data collected in the 1.727 A -11 plane (L 1 ). The (001), (101) and (011) peak positions, and the line [x, x, 1 ], are

q2 indicated. Contours =

and 68 counts/30

s.

of 5 800 counts/30 diffractometer.

s.

=

are

drawn at values of 4 642, 3 167, 2

The six

peaks with

Data

missing

are

1( 1 1

154, 1 602, 1 000, 681, 464, 317, 215, 160, 100 symmetry have approximately equal intensity maxima

33 from the lower left

quadrant due to

a

mechanical limitation of the

3. Columnar phases. The diffraction patterns observed in the Dhd and Dho phases are summarized in figure 3. In the Dhd phase only the reflections corresponding to the solid circles are found. The diffraction pattern in this phase is very similar to that previously observed in a hexa-substituted truxene mesogen [13]. In the equatorial plane, a triangular array of sharp (HKO )-type reflections is observed, corresponding to long-range (correlation length 03BE > 800 À) hexagonal ordering of the columns. A single diffuse meridional (001) peak at qz = 1.727 Â- 1 is due to intracolumn liquid-like ordering of the disks. The correlation length along the columns grows upon cooling from e 16 A at 89 °C to e 30 A at 76 °C, corresponding to from 4 to 8 stacked molecules. In the transverse directions, the (001) peak is diffuse over a spherical shell with polar width of approximately 35° FWHM. Additional diffuse scattering due to the hydrocarbon tails is centered 45° off the q, axis at a momentum transfer magnitude of 1.3 -1 ; it can be empirically described by a functional form =

=

468

3.

schematic of diffraction pattern. qx and qz indicate equatorial and meridional as discussed in text. Circle radii are proportional to the log of the peak intensity ; we measured 14 700 counts/s and 1.7 counts/s in the (100) and (300) peaks, respectively. Open circles indicate peaks observed in the Dho phase only ; solid circles those found in both D ho and D hd phases.

Fig.

-

Perspective

components of the scattering vector,

Such diffuse off-axis scattering has been previously observed in discotic mesophases [14], and may arise either from a preferred tendency of the tails to tilt up or down, resulting in an oriented tail-tail correlation function, or from short-range helical positional order. In any case, the diffuse scattering is most likely primarily due to the aliphatic tails, and corresponds to a liquid-like mean tail-tail separation of 4.8 À. No other peaks are observed, due to the absence of long-range intercolumn ordering of the molecules. All the reflections observed in the Dhd phase are also observed in the Dho phase. The appearance of the (100) and (001) peaks above and below the phase transition is shown in the left and middle columns of figure 4. The integrated intensities do not change appreciably through the transition, although the (001) peak sharpens to become resolution limited in the Dho phase. Figure 5 shows the intercolumnar distance d, calculated from the position of the triangular (100) peak, as a function of temperature. The dramatic negative thermal coefficient of expansion in the Dhd phase is most likely the consequence of the hydrocarbon tails becoming stiffer and therefore longer with decreasing temperature. Indeed, the same effect may provide the driving mechanism for the Dhd --+ Dh. transition ; as tail conformational degrees of freedom are « frozen out », steric hindrance between tails in adjacent columns should become more important, resulting in enhanced column-column coupling. This hypothesis is strengthened by the observation that the diffuse hydrocarbon-tail diffraction at 1.3 A -1 is weaker in the Dho phase than in the Dhd phase. (The diffuse maximum in the Dho phase is also essentially isotropic, rather than being centered off-axis ; it can be described empirically by a functional form

Extrapolation of d Dho phase indicates

vs.

that

T from the Dhd phase to the constant value measured in the we may be close to a second-order transition. which is however

469

Scattering profiles at three temperatures upon heating through the coexistence region at the Fig. 4. Dhd --+ Dho phase transition. The double-peaked (100) profiles indicate the progress of the transition (left). The sharp (001) peak of the Dh. phase weakens upon heating (center panel, top), and is replaced in the Dhd phase by a broadened diffuse peak (bottom center panel, note expanded qz axis). At right, scans along the qy direction with fixed qx 0, qz = 1.727 Â-show the transverse evolution of the (001) -

=

peak

and the

superlattice

and

peaks.

Intercolumn distance d, as derived from the fundamental (100) peak, versus temperature. 5. Closed circles indicate measurements made on warming, open circles measurements on cooling. Where two solid circles are shown at the same temperature, two distinct peaks were visible in the diffraction pattern, as shown in figure 4, indicating coexistence. Solid line is the result of a least-squares fit to the measured distances ; above 75 C it has a slope of - 0.037 À/C.

Fig.

-

470

preempted by a first-order jump. However, a superlattice peak intensity (see below) would be a better choice for an order parameter than d. All peak positions, including those of the new peaks discussed below, are essentially temperature-independent in the Dho phase ; the length 0.334 Â-1. of the fundamental (100) vector isq1 the formation of a of new reflections appear, as indicated by the number Dho phase, Upon in circles indicated 3. in the Dho phase are resolution-limited, All the reflections figure open indicating correlation lengths of at least 800 Â. We will continue to use the triangular (HKL ) Bragg indices of the Dhd phase to describe these reflections. New reflections in planes at qz = 0.654 Â -1, 1.309 A-1 and 1.727 Â -1 correspond to L 0.379, L 0.758 and =

=

=

L 1. The first two values are very close to L = 3/8 and L 6/8, but careful measurements of vector differences associated with reflection through the qz 0 plane have verified that the from these commensurate deviate positions by at least one FWHM [11]. peaks consistently There is thus some type of incommensurate modulation along the columnar axis. No new peaks are seen along the [0 0, L meridional axis. In the qx and qy directions, the new reflections correspond to a 30° superlattice, i.e. the new fundamental is indexed x =

=

=

3

as a

13313 ).0 The

new

3R

unit cell therefore consists of three columns. The evolution of some of

at the Dhd-Dho phase transition is shown in the right-hand column of 4. The of absence any new peaks in the equatorial (HKO ) plane implies that the figure electron density projections of the three columns onto the basal plane are identical, and therefore that the superlattice ordering does not entail any distortion of the underlying hexagonal mesh. The reflections in the L 0.379 and L 0.758 planes, and the absence of any peaks in these planes along the (OOL ) axis, are consistent with an incommensurate helical arrangement of molecules within the columns. Indeed, helical order has previously been observed in columnar phases of discotic and cone-shaped molecules [14, 15]. In most of these measurements the phase of the helix was uncorrelated from column to column, resulting in diffuse layer lines, though in one case [15] additional Bragg peaks were also reported. Semiempirical conformational analyses [16] of triphenylene derivatives similar to HHTT have indicated that adjacent molecules along a column minimize steric hindrance via a relative rotation of roughly 45°. Alternate tails around the circumference of each molecule tilt above and below the plane of the core, so that once the initial clockwise-counterclockwise symmetry is broken the helical order can extend to large distances. This analysis is supported by the results for the HHTT molecule as obtained from the structure in the K phase (see Sect. 4). The diffraction pattern for a group of atoms repeated along the z-axis by the operation of a non-integer screw is well-known [17] : diffraction maxima are found at

these

superlattice peaks

=

=

where m and n are integers, P is the pitch of the helix, and p is the rise per subunit. In the present case, the intracolumn molecular spacing is p 2 ’TT / (1.727 A-1 ) 3.638 A, and the pitch is given by P 6 ’TT / (0.654 A -1) 28.82 A 7.92 p, corresponding to a 45.50 rotation between adjacent molecules. This means that the minimum correlation length of 800 A mentioned earlier comprises at least 220 molecules or about 30 complete rotations. The superlattice order is associated with groups of three columns forming triangles. Within a group of three columns, we can assign to the molecules in each individual column an overall vertical displacement and helical phase. Without going into the details of the model to be discussed in section 5, it is clear that the superlattice ordering must involve both the helical phase differences and the vertical displacements, since superlattice peaks are seen in the (n 0 ; m = 1) plane as well as in the (n 3, 6 ; m 0) planes. =

=

=

=

=

=

=

=

471

4. Structure of the K

The unit cell

was

phase.

determined to be monoclinic :

absences HKL :H + K = 2 n + 1 and HOL : L 2 n + 1 the space group was either acentric Cc or centric C2/c. The structure was solved in the acentric space group Cc by the use of the MULTAN 11/82 program package which revealed the locations of 32 atoms. The remaining atoms were found from weighted Fourier syntheses. This structure revealed the presence of a molecular 2-fold axis, so structure refinement was performed in the centric space group C2/c by full-matrix least squares techniques. The hydrogen atoms were included in pre-calculated positions but were not refined. The refinement converged to R = 0.066 and RW = 0.077. Half of the unit cell is shown in figure 6b. Note that there are 4 molecules per unit cell, which are positioned in pairs on top of each other. These pairs are separated by c/2, and hence the molecules are regularly spaced in the direction along the c-axis. This situation is very different from that in the related hexa-alkyltriphenylenes where it has been found that two relatively close molecules are in turn rather widely spaced from the next pairs in the direction perpendicular to the discs [18]. In the HHTT K phase the two molecules in a pair are rotated by 180° with respect to each other, so that we can say that the columns have a From the

systematic

=

Structure of the K phase. (a) Stereoscopic view of an individual molecule. Ellipses indicate of anisotropic displacements ; (b) one half of a unit cell, viewed along the c axis. Atoms in the upper plane are indicated with closed circles, atoms in the lower plane with open circles.

Fig.

6.

r.m.s.

-

amplitudes

472

2 c. The ratio a/b = 1.13 found for the unit cell is far away helical structure with a pitch P from triangular symmetry which requires alb 0. However, the fundamental spacing between the molecular planes parallel to the diagonal of the unit cell can be calculated to be 18.8 Â, which exactly equals the spacing in the Dho phase. This explains the constancy through all three phases of the position of the low-angle diffraction peak reported for powder X-ray results [10]. Hence the K - Dho transition can be accomplished by a relatively small parallel displacement of molecules. In addition, of course, the angle {3 should then change from its value of 97.85° in the K-phase to 90°. A stereoscopic picture of the HHTT-molecule in the K-phase is shown in figure 6a. The structure of the 18 core carbon atoms is quite close to that of the simple triphenylene molecule [19]. Note that there is only one 2-fold axis in the plane of the molecule. The two pairs of alkyl chains close to this axis tilt out of the central molecular plane in opposite directions. The remaining two alkyl chains do not show this behavior. Root-mean-square (r.m.s.) motional amplitudes range from roughly 0.25 Â for the core atoms to almost 1.0 Â for the carbon atoms near the ends of the alkyl tails ; evidently, there is considerable thermal motion of the alkyl chains even in the K-phase. The HHTT molecule has an intrinsic D3 h symmetry. However, examination of simple space-filling models shows that adjacent tails interfere with each other. Thus, alternate alkyl tails are likely to tilt above and below of the plane of the triphenylene core, leading to a propellor-blade like molecule. This type of arrangement has also been found by semiempirical conformational analyses of related triphenylene derivatives [16]. Alkyl chain tilting will break the reflection symmetry of an isolated molecule, leading to D3 symmetry (one 3fold axis and three 2-fold axes). In the K phase, this symmetry is further broken due to the monoclinic environment, but at the K - Dho phase transition, the monoclinic environment of a molecule is replaced by a triangular environment. We expect that the conformation of the HHTT molecule will then revert to D3 symmetry ; a propellor-blade like structure can then lead to helical structure along the column. =

=

5. Model of the

Dho phase.

discuss the models used for least-squares analysis of diffraction data in the on a lattice defined by real-space triangular vectors a and b and columnar translation vector c. Following Cochran et al. [17], atoms are considered to lie along sets of concentric helices, each of which is defined by the equations x r cos (2 ’TT’Hz / P + cpo) and y=rsin(2’TT’Hz/P+CPo). H=:tl determines the sign of the helicity and CPo is an arbitrary helical phase. If a given set of atoms are spaced periodically along the 2-direction at positions Zk Zo + k x p, then the structure factor is given by the convolution of the transform of the continuous helix and the transform of a set of planes with spacing p. This gives We

now

Dho phase. The molecules lie

=

=

where m and n are integers, Jn is the n’th order Bessel function, and tan gi qy/qx. As discussed in section 3, the spacing is given by p = 3.638 Á, and the pitch is P = 28.82 À. We can model a single helical column of molecules by N coaxial discontinuous helices, where N is the number of atoms per molecule. If the position of the j’th atom in the molecule is given in cylindrical coordinates by (pl, Oj, zj), with associated atomic form factor fj (q), and we assume for generality a helical-phase preserving overall translation by =

473

Zo 2,

then the structure factor for

a

given (n ; m) reflection

is

given by

Note that Wo and Zo affect scattered phases, but not amplitudes. However, rotational and translational fluctuations will affect the experimental intensities differently. Also, if different columns have unequal heights or phases then interference can result in superlattice diffraction. The HHTT molecule (Fig. 7a) was assumed to retain its intrinsic D3 symmetry, as discussed in the previous section. The molecule can then be decomposed into three pairs « t » of units, t 0, 1, 2. The pairs are related by 2 ir/3 rotations, and we assume that the units are symmetric within each pair : relative to the 2-fold rotation axis bisecting the pair (0j, z) = (2 rt/3 ± 0,, ± Hzs), s = 1, 2, M, and there are M N/6 atoms in each unit. (This implies an alternating propellor-blade tilt of the tails, as discussed above. We assume that if the helicity of the column changes sign, then the helicity of the individual molecules =

...,

=

Fig. 7. Structure of the Dho phase. (a) Best fit to molecular conformation, drawn to scale. Atoms are numbered for reference in the text. 8cs is the angle between the sulphur atom Si and the first tail carbon atom ; Occ is the angle between adjacent carbons, assumed to be the same for all tail carbon atoms. Adjacent tails are tilted by --t Ot above and below of the plane of the paper ; (b) proposed structure of the 3-column superlattice (not to scale). Column 0 has the opposite helicity from columns 1 and 2, and is displaced by p/2 from those columns. -

474

also

does.) Thus,

the structure factor becomes

which factors into

The last term amounts to

a

selection rule : n must be

a

multiple

of 3.

point the first problem encountered in the data analysis becomes apparent. The given by a sum of 10 or more Bessel functions, each of which is an oscillatory function. In the traditional analysis of helical biological molecules such as proteins, the subunit giving rise to the helical layer lines typically comprises a small fraction of the total molecule, and can be considered to lie at a well-defined radius. By contrast, in the present case all atoms contribute more or less equally to the helical diffraction, and therefore all 10 Bessel functions, with different periodicities, contribute to the intensity (which is only measured at Bragg peaks). This results in a large number of local minima in a least squares fit, and considerable care is required to ensure that the final result is not only physically plausible At this

structure factor is

but also the best fit. As seen in equation (7), the helical sign and phase of a single column cannot be determined from diffraction measurements, since these are reflected only in scattered phase differences. However, if different columns have different phases and/or helicities, the interference of these can result in new superlattice peaks, and just such differences are implied by the x superlattice peaks seen for n # 0. An additional possibility, as discussed in an earlier paper [11], is that one or more columns might have random average helicity. The x superlattice order in the Dho phase is associated with groups of three columns

w5 13

Ô J3

k 0, 1, 2. Within each group of three columns, we assign the molecules in column k an overall rotational-phase-preserving displacement Zk along the column, a displacement Pk in the equatorial direction, and helical sign and phase Hk and ek : the equatorial displacement adds a factor exp (iqb - Pk) to the single-column structure factor phase in equation (7). The equatorial displacements are given by po 0, pl a, p2 b. Without loss 0. (Of course, because of the of generality, we can take 00 0 (or random), Ho = 1 and Zo 3-fold symmetry of the molecules, rotations are only defined modulo 120°). We further assume that the heights of columns 1 and 2 are symmetric with respect to column 0, i.e. Zl - Z = - Z2. Initial calculations were done with the Zk taken as adjustable variables. However, it was found that the fitted values of the Zk were quite insensitive to details of the molecular conformation. The nonzero intensity of the (001 ) peak rigorously excludes Z p /3, while =

=

=

=

=

=

=

475

111 (

intensity of the 33 peaks excludes Z = 0. Fits to the entire set of Bragg peak intensities consistently give 0.45 p , Z 10.55 p ; Z is probably exactly equal to p/2 and was fixed at this value in subsequent fits. This implies that columns 1 and 2 are at the same height, and form a sublattice of molecules all at the same height. Column 0 forms part of 0. If we then fix Zl a second sublattice of molecules, all at height Zo Z2 p /2, the terms involving Z reduce to factors of exp (i 2 7TmZ / p) = (- 1 )"’ in the appropriate places. Combining the phase factors for the different columns, after a certain amount of manipulation (see Appendix) we arrive at the following expression for the unit cell structure the

nonzero

=

=

-

=

factor :

where

The calculated intensity was obtained by squaring the total structure factor and multiplying by a normalization factor Io, a Lorentz-polarization factor H, and a Debye-Waller factor U. Taking into account the partial polarization of the beam by the monochromator and analyser crystals, and the anisotropic resolution function, an appropriate Lorentz-polarization factor was given by it

with the angular widths determined section 2 :

by the mosaic and instrumental resolution,

as

discussed in

Several different forms for the Debye-Waller factor were tested. While at first it might logical to use a factor of the form exp (- (q u )2 ), where the qi and ui are Cartesian coordinates, this neglects the fact that a simple translation of an individual molecule along a column breaks both translational and helical symmetry. A more natural choice is to use the same cylindrical coordinates used for the structure calculation, and to use helical-phasepreserving translations Z. An appropriate Debye-Waller factor is then seem

u, clearly measures the extent of random motion in the equatorial plane, which we assume to be isotropic. um measures the extent of phase-preserving vertical motion (we can think of the atoms « sliding along a helical wire »). u,, measures the extent of purely rotational

476

fluctuations. Unm

measures

the extent to which the vertical and rotational motions

coupled ; it can range from --t 2 1 u,, u,,,1 (perfectly correlated or anticorrelated) Putting everything together, the scattered intensity is given by :

V3 x V3 R 300 superlattice

are ’

to zero.

reciprocal lattice. The quantitative analysis of the helical and superlattice ordering was quite sensitive to estimates of the molecular conformations. However, the residual uncertainty in our integrated peak intensities and the lack of data beyond 2 Á -prevented a direct determination of detailed atomic coordinates in the Dho phase. In most cases we used bond distances derived from our crystallographic measurement of the K phase, averaged to reflect D3 symmetry. The following assumptions were made about the structure of the HHTT molecule in the Dho phase (Fig. 7a) : 1) Molecules in all three columns were assumed to have identical conformations, aside from possible reversals of helicity. 2) The inner core carbon and sulphur atoms (Cl, C2, C3 and SI) were assumed to have bond lengths given by the K-phase data ; a typical carbon-carbon distance is 1.45 Á. (It should be noted that if this bond distance was made a free parameter in the fits it typically converged to a somewhat smaller and unphysical value of 1.1 À.) The sulphur atom was assumed to lie in the plane of the triphenylene core, as in the K-phase structure. 3) The carbon atoms C4-C9 in each tail were considered to lie in a plane which was tilted out of the triphenylene core plane by an angle ± et. The angle between the sulphur-carbon bonds C3-Sl and S1 - C4 was taken to be a free parameter, Ocs. The carbon atoms were assumed to havean all-trans configuration in the chain, as shown in figure 7. The carbon-carbon bond distances were all fixed ait 1.485 Â ; the angles between adjacent carbon bonds along the tail were assumed to all be equal to a free parameter,

where {GJ

spans the

Table II. - Parameters

giving ’rise to the fit in

table 1.

477

Occ. This is clearly â somewhat limited model for the alkyl tails ; for example, it does not allow for the possibility of gauche conformations. However, the goal was to arrive at reasonable agreement between theory and data with the smallest number of adjustable parameters. Occ thus effectively controlled the length of the alkyl tail. Typical fits to the data gave 0, 10° ± 5, Ocs 80° ± 10 and Occ 110° ± 20. Note that this value for ecc is exactly what one would expect _ on the basis of a normal all-trans configuration. =

=

=

fits we allowed the molecules to rotate a small amount randomly about in the a-b plane, similar to the tilting motion which, if correlated, would result in a Drd phase [5]. Treated as a free parameter, this molecular tilt typically converged to a value of about 3.5° ; it was not strongly correlated with other parameters. We did not consider models in which the molecular tilt was correlated with the column helicity.

Also, in

random

some

axes

Debye-Waller terms typically converged to ur - 2 Â, um - 1.0 and un _ 0.15. The first two terms correspond to r.m.s. positional motion in both the basal and columnar directions of 1-2 Â : a fairly large amplitude motion compared with the 3.64 À molecular spacing along the columns. The third term corresponds to r.m.s. rotational fluctuations on the order of 5-10° about the z-axis. In this context it is interesting to note that apart from (n = 3, 6 ; m 0) no other helical peaks were observed, even though scattering in the planes (n = - 6 ; m = 1), (n = - 3 ; m = 1) and (n = 9 ; m = 0) planes (L = 0.253, 0.623 and 1.139) are allowed by symmetry. The absence of observed peaks in the n 9 plane is simply explained by the rapid decay in amplitude of the Jn Bessel function with increasing n. 3 and n 6 planes should have the same intrinsic intensities as their However, the n n 3. and n 6 counterparts. The absence of observed peaks in these planes is best explained by correlated fluctuations, i. e. a negative value for Unm. This most likely results from fluctuations in which the molecules, rather than moving vertically along the helical trajectory (un 0) or have a pure vertical motion, rotate opposite to the helical trajectory while moving vertically. This type of motion allows the tails of fluctuating molecules to move into the gaps The

=

=

=

-

=

-

=

=

=

between the tails of the molecules above

or

below them.

The formation of the three column unit cell in the Dho structure can be understood on the basis of steric frustration of the molecular tails. The intercolumn separation in the Dho phase is 5-15 % smaller than the diameter of the HHTT molecule with the tails fully extended. This implies that either the molecules must be at. different heights, forming an overlapping structure, or that if the molecules are at the same height, the tails must interlock in a gear-like fashion. Molecular interdigitation in a triangular lattice is frustrated in the sense that it is not possible to have all the neighbors of a molecule translated by ± p /2. (This situation is in fact isomorphous to the antiferromagnetic Ising model on a triangular lattice ; structures quite similar to the one described below have been observed [20] in linear-chain magnets where the chains make a hexagonal lattice and there is antiferromagnetic coupling in the basal plane.) Further frustration is introduced by the rotational degrees of freedom : while it is possible to construct an interlocking triangular mesh Qf gears, they cannot then rotate. This frustration is partially relieved by the formation of two sublattices at different heights. Once these sublattices have formed, it is clear that the behavior of molecules in column 0 may be quite different from that of molecules in the other two columns, and that these differences may result in a superlattice ordering of the helical phases. The superlattice ordering reflected in the n =1= 0 planes must result from some difference in helical signs or phases of the different columns. To account for all degrees of freedom in the unit cell, we must include for each of the three columns a helical sign Hk, helical phase CPk’ and vertical displacement Zk. As previouslydiscussed, we set Ho 1, eo 0, Zo 0 and Z1 = - Z2 = p/2, thus the remaining models have definite Hl, H2, 403A61 and 03A62. Additionally , it is possible that one or more columns rotate freely or have random =

=

=

478

average 0,1 and

phase [21]. 03A62:

We

systematically explored

the

following

models

as

a

function of

These comprise all the inequivalent models one can construct, assuming at most one random column. With optimized choices for molecular parameters and helical phases, all of the above models could be brought into rough agreement with the data. However, we found the best agreement with model III, in which columns 1 and 2 have a definite phase relationship with and opposite helicity to column 0. Specifically, we obtained the best fit for CP1 lP2 60°, i. e . with columns 1 and 2 identical. In an earlier paper [11], we reported good agreement with model IV. If fits are done using data only from the n 3 and n 6 helical planes, model III gives a goodness-of-fit parameter X 2 - 1.5 as opposed to X2 = 7.5 for 0 planes are included, both models give X 2 == 6 ; the fitted model IV. However, if the n molecular conformations in this case are somewhat different than if only the helical data are fitted. It therefore seems probable that the helical superlattice is best described by model III, but that some other important feature of the structure has been neglected in our analysis. For example, it is possible that the conformations of molecules in column 0 are quite different from those in columns 1 and 2. Also, in principal we should be able to discern between models III and IV by the presence of diffuse scattering sheets for n # 0 produced by columns with random helical phases ; however, in practice the sheets were either not present or had scattering intensities too low to be detected. A comparison of calculated and fitted intensities is given in table I. It can be seen that considerable agreement is found over several orders of magnitude in intensity, although the agreement is not perfect. Using model III discussed above, we can now directly calculate the electron density. We used the phases derived from our model and amplitudes obtained by taking the square root of the measured intensity (corrected for Lorentz and polarization factors) to generate the electron density profiles shown in figure 8. It should be emphasized that these profiles are to a large extent model-dependent, since the phases of the different peaks depend on the model chosen and on the fitted parameters. Nevertheless, a number of features become clear : the alternation of sheets of molecules, the helical rotation of molecules in columns 1 and 2, and even the « propellor » tilt of the tails by 5-10° above and below of the triphenylene core plane. =

=

=

=

=

6. Discussion.

the various phases and phase transitions it is clear that the absence of in the Dhd phase is related to the « melted » state of the alkyl chains. correlations intercolumn the from As seen negative thermal expansion coefficient in the Dhd phase, the Dhd--+ thus transition appears to be the consequence of increased intercolumn coupling due to Dh. the stiffening and lengthening of the tail chains with decreasing temperature. The transition to the K phase can then tentatively be associated with a further freezing-out of molecular motions. From the considerable supercooling observed, we concluded that probably there is in addition a considerable kinetic barrier before the structural rearrangement to the monoclinic K phase is accomplished.

Considering

479

Equal-density contours of electron density in the Dho phase, calculated from measured Fig. 8. amplitudes and phases resulting from model III. A type-0 column passes through the origin. (a)-(d) : x-y planes (normal to the column axes) at indicated values of z. Contours are given at 90, 80, 70, 60, 50, 40, 30, 20, and 10 % of the maximum density ; (e) : z-axis is the column axis, horizontal axis is along the line joining a type-0 and type-1 column. Contours are given at 80, 60, 40, and 20 % of the maximum density. -

While we have performed a true crystallographic determination of the structure of the K there is still some uncertainty about the detailed atomic positions in the Dho phase. However, an alternate tilting of the alkyl tails above and below of the plane of the core is strongly indicated by the accumulated weight of different kinds of evidence : the parameters from least-squares fits to theoretical models as discussed in section 5, the reconstructed electron density images, the structure of the K phase, and physical arguments from spacefilling models. Fits to the data also uniformly give a rather large value of Ocs, indicating that

phase,

480

adjacent pairs bend together to form roughly triangular structures, rather than extending radially from the core of the molecule (Fig. 7). It may at first seem surprising that an ordered phase with a net chirality can be generated by non-chiral molécules. However, this can be understood by assuming that the tails in a particular molecule form a propellor structure, thus breaking the clockwise-counterclockwise symmetry, and steric hindrance then forces this broken symmetry to propagate along the column. In a macroscopic sample with a structure described by model III we would expect to find a racemic mixture of regions with Ho = 1, Hl H2 = - 1 and regions with Ho - 1, Hl H2 = 1, i.e. neither sign of chirality would be preferred on the average. The distinction between models in which the helices in columns 1 and 2 have the same sign and phase (as in models I, III and IV), or are distinguishable (models II and V), corresponds to the distinction between double- and triple-q representations of the densities in a Landau theory for two-dimensional superlattice order [22]. However, the present case must be treated as a three-dimensional problem since superlattice formation is simultaneous with the development of long range intracolumnar positional and helical order. With the exception of magnetic systems [23], incommensurate helical order is relatively uncommon in condensed matter physics. It plays an important role, however, in many biological systems. DNA, which has the well-known double-helix structure [24], forms cholesteric nematic and smectic structures in solution [25]. Indeed, the ordering of DNA into a nematic rather than crystalline state was a crucial step [26] in deducing that it was indeed helical. An even closer parallel to the Dho phase of HHTT is provided by vertebrate striated muscle [27], which is built up from interpenetrating arrays of thick and thin filaments. Myosin cross-bridges on the thick filaments form incommensurate helices ; a J3 x J3 R 30° superlattice is produced by the relative phases of these helices. The actin monomers in the thin filaments also form a helical structure, and the relative incommensurability of the thick and thin helices is believed to play a central role in muscle contraction. We can therefore hope that the study of incommensurate helical order in discotic liquid crystals, which can be chemically redesigned in a relatively straightforward fashion, will prove useful as a model for biologically important structures as well as a challenge to theorists. the tails in

=

=

=

Acknowledgments.

samples were supplied by K. Praefcke and B. Kohne (TU, Berlin), for which we grateful. We thank A. R. McGhie for heat capacity measurements. We acknowledge useful conversations with J. K. Blasie, T. C. Lubensky and A. B. Harris. E. F. and P. A. H. were supported in part by National Science Foundation Grant No. DMR-83-51063, and in part by support from the Research Foundation. Acknowledgment is also made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. P. C. and A. B. S. were supported by the National Science The HHTT are

Foundation-Labôratory

for Research on the Structure of Matter (LRSM) Grant No. DMR85-19059 ; acknowledge LRSM support of the X-ray and calorimetry facilities. A. R. was supported by as NATO fellowship. W. H. dJ. was supportedvia the Stichting voor Fundamçnteel Onderzoek der Materie (Foundation for Fundamental Research on Matter, FOM) by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Research, NWO). we

also

Appendix. Diffraction from

a

superlattice

of helices.

The X-ray data on the Dho phase reveal à three column unit cell, therefore the modeled by a triangular iattice built from three sublattices of columns of types k

crystal is 0,1, 2.

=

481

Each columnar sublattice is composed of helical stacks of molecules having a helical sign Hk, helical phase Ok, vertical translation Zk (which preserves helical phase) and x-y basal plane position within the unit cell pk. The molecular structure factor Smol (Eq. (7)) is independent of the helical phase and displacement of the column, and since the cosine function is symmetric under change of sign of the argument it is also independent of the sign of the helicity. The total X-ray structure factor is therefore separable into contributions from the superlattice, unit cell basis, and molecular structure factor :

Super is

the structure factor for the

superlattice

f

R 30° superlattice reciprocal lattice. The molecular structure spans the f x factor Smol is derived in section 5. The column helical pitch P and the molecular z-spacing p determine qz 2 7r(n/P + m/p) for integer values of n and m. The unit cell is formed by three columns of types 0, 1 and 2 located on the vertices of an equilateral triangle with side1 al. We choose to calculate the unit cell structure factor Slat as a normalized sum over six columns of types 1 and 2 surrounding a column of type 0 (at 0). This procedure simplifies the algebra and (more importantly) makes clear the po underlying three-fold symmetry of the Dho structure. The same overall results can be obtained starting with the above three-column triangular unit cell, however the mathematical form is less tractable. Column 0 is treated separately, so that

where

{Gs}

=

=

where Sj is the phase factor for the column and displacements :

(from Eq. (7)) with arbitrary helical sign, phase,

and tan (03C8) q y/ qx. Slat is the sum of phase factors for column 0 and for three pairs of dissimilar columns of types 1 and 2. We first solve for the structure factor of a pair of dissimilar helices with signs, phases and positions (H1, (03A61, Zl, + d) and (H2, (fJ2, Z2, - d), located with respect to the x-axis by tan (f3) dy/dx : =

=

The models considered in the text have

and

Zl

=

-

Z2==

defining

Z

=

p /2,

so

that

482

yields

the structure factor for

a

dissimilar

To evaluate the unit-cell structure of dissimilar columns with /3

pairs

pair

of columns

factor, we add the structure factors of column 0 and three 0,2 7r/3, 4 7r/3 :

=

We evaluate So in two specific cases. (1) If the helical phase of column 0 is fixed, then without loss of generality we can set Ho 1, 03A60 0, and Zo 0, so that So exp [in(1/I’ + 7T /2)]. (2) If the helical phase of column 0 is random across the sample, then columns of this type produce only diffuse scattering planes for n =1= 0, and thus contribute negligible scattering intensity at the Bragg peak positions. In this case, the column 0 intensity results from the incoherent sum of scattering from N columns of type 0 in the sample, which is negligible from the 2 N other columns of compared to the coherent scattering (proportional to types 1 and 2 with fixed relative phase. For n 0 scattering, however, the relative helical signs and phases of the columns are irrelevant, and all three columns contribute equally to the coherent scattering from the full triangular lattice. (For this reason this original triangular reciprocal lattice is recovered ; Slat 0 extinguishes the superlattice peaks for n 0). With substitutions[ d1 = 1 a 1 cos (2 7r/3 - 03C8) cos (03C8 + 4 rr /3 ) and cos (4 7T /3 - 03C8) cos (03C8 + 2 7r/3), we conclude that the lattice structure factor for the unit cell is : =

=

=

=

N)

=

=

,

=

=

=

References

for example : VERTOGEN, G. and DE JEU, W. H., Thermotropic Liquid Crystals, Fundamentals (Springer Heidelberg) 1988. LITSTER, J. D. and BIRGENEAU, R. J., Phys. Today 35 (May 1982) 26. The properties of discotics are reviewed in : CHANDRASEKHAR, S., Phil. Trans. R. Soc. London A 309 (1983) 93 ; LEVELUT, A. M., J. Chim. Phys. 80 (1983) 149. LEVELUT, A. M., OSWALD, P., GHANEM, A. and MALTHETE, J., J. Phys. France 45 (1984) 745. SAFINYA, C. R., CLARK, N. A., LIANG, K. S., VARADY, W. A. and CHIANG, L. Y., Mol. Cryst. Liq. Cryst. 123 (1985) 205. See, e.g., BENATTAR, J. J., MoussA, F. and LAMBERT, M., J. Chim. Phys. 80 (1983) 99. KATS, E. I., JETP 48 (1978) 916. SUN, Y. and SWIFT, J., J. Phys. France 45 (1984) 1039. KOHNE, B., POULES, W. and PRAEFCKE, K., Chem. Zeit. 108 (1984) 113.

[1] See, [2] [3] [4] [5] [6] [7] [8] [9]

483

[10] GRAMSBERGEN, E. F., HOVING, H. J., DE JEU, W. H., PRAEFCKE, K. and KOHNE, B., Liq. Cryst. 1 (1986) 397. E., HEINEY, P. A and DE JEU, W. H., Phys. Rev. Lett. 61 (1988) 1202. FONTES, [11] [12] BERNAL, J. D. and CROWFOOT, D., Trans. Faraday Soc. 29 (1933), 1032 ; DOUCET, J., Ch. 14 in The Molecular Physics of Liquid Crystals, Eds. G. R. Luckhurst and G. W. Gray (Academic Press London) 1979. [13] FONTES, E., HEINEY, P. A., OHBA, M., HASELTINE, J. N. and SMITH, A. B. III, Phys. Rev. A 37 (1988) 1329. [14] LEVELUT, A. M., J. Phys. France 40 (1979) L81. [15] LEVELUT, A. M., MALTHETE, J. and COLLET, A., J. Phys. France 47 (1986) 351. [16] PESQUER, M. , COTRAIT, M. , MARSAU, P. and VOLPILHAC, V. ,J. Phys. France 41 (1980) 1039 ; COTRAIT, M., MARSAU, P. ,PESQUER, M. and VOLPILHAC, V., J. Phys. France 53 (1982) 355. [17] COCHRAN, W. , CRICK, F. H. C. and VAND, V., Acta. Cryst. 5 (1952) 581. [18] COTRAIT, M., MARSAU, P., DESTRADE, C. and MALTHETE, J., J. Phys. France 40 (1979) L-519. [19] AHMED, F. R. and TROTTER, J. , Acta. Cryst. 16 (1963) 503. [20] See, e.g., YELON, W. B., Cox, D. E. and EIBSCHÜTZ, M., Phys. Rev. B 12 (1975) 5007. [21] Models having a single column with a random helical phase are analogous to a superlattice structure found in the linear-chain magnet systems [20], in which two-thirds of the chains are antiferromagnetically ordered in the basal plane, while the remaining one-third are disordered. [22] DOMANY, E., SCHICK, M., WALKER, J. S. and GRIFFITHS, R. B., Phys. Rev. B 18 (1978) 2209. [23] See, e.g., GIBBS, D., MONCTON, D. E., D’AMICO, K. L., BOHR, J., and GRIER, B. H. , Phys. Rev. Lett. 55 (1985) 234 and references therein. [24] WATSON, J. D. and CRICK, F. H. C., Nature 171 (1953) 737. [25] STRZELECKA, T. E., DAVIDSON, M. W. and RILL, R. L., Nature 331 (1988) 457. [26] FRANKLIN, R. E. and GOSLING, R. G., Nature 171 (1953) 740. [27] For a recent review on the molecular basis of muscle contraction, see HUXLEY, H. E., Muscle and Nonmuscle Motility 1 (Academic Press, 1983), 1.