Frustration and helicity in the ordered phases of a discotic compound

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Frustration and helicity in the ordered phases of a discotic compound

Paul A. Heiney, Ernest Fontes, Wim H. de Jeu, Antoni Riera, Patrick Carroll, Amos B. Smith

To cite this version:

Paul A. Heiney, Ernest Fontes, Wim H. de Jeu, Antoni Riera, Patrick Carroll, et al.. Frustration and helicity in the ordered phases of a discotic compound. Journal de Physique, 1989, 50 (4), pp.461-483.

�10.1051/jphys:01989005004046100�. �jpa-00210929�

Frustration and helicity in the ordered phases ^{of a discotic} compound

Paul A. Heiney (1), Ernest Fontes (1), Wim H. de Jeu (2,*) , Antoni Riera (3),

Patrick Carroll (3) and Amos B. Smith, ^III (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 on the Structure of Matter, University ^of Pennsylvania, Philadelphia, ^PA 19104, ^{U. S . A.}

(Reçu ^{le 6} septembre ^1988, accepté le 17 octobre 1988)

Resume.

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 relative- ment petit.

Abstract.

We have used X-ray diffraction to study the discotic mesophases Dhd ^and Dho and the low temperature crystalline ^K phase ^of hexa(hexylthio)triphenylene. ^The high- temperature 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.

Classification

Physics ^Abstracts

64.70Md - 61.30Eb

61.65 + d

(*) ^{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

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 short- range 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.

In this paper ^we present X-ray results in a single-crystal geometry of the various phases ^of hexa(hexylthio)triphenylene [9] (HHTT, Fig. la). This discotic 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 order-}

disorder 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

Fig. ^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.

transitions involve several columnar phases. ^{The K} phase ^{in the} present compound ^{turns out}

to have a monoclinic 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.

2. Experimental.

The HHTT sample ^was synthesized by ^{Kohne et al. as} described elsewhere [9], ^{and was found}

to give ^the following phase sequence :

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 well-}

characterized instrumental resolution function and a well characterized and oriented strand

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 self-} supporting. 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 q 1 ^=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}

vertical resolution Aq,,,,

^0.27 -1, ^{and for} high ^vertical resolution qvert

^0.027 -1. In 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

by ^Gaussian lineshapes with widths corresponding ^to àqll ^and Aq, . The vertical resolution

was 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. Experimen- tally, ^the broadening ^was described by ^two mosaic widths. The polar misalignment ^{of the z-}

axis 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 ^1.5-

2°. As a result, ^{in our} analysis ^{the intrinsic} sample scattering ^{was modelled} by ^{a three-}

dimensional 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 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 instrumental longitudinal resolution. For example, ^the long ^{axis dimension} àq

^0.047 A-1

FWHM results from the sum in quadrature of the vertical resolution width Oq,,ert

0.027 Â -1 ¹ FWHM and the polar mosaic width

Figure ^{2b shows an} equal intensity ^contour plot ^{of raw} scattering data collected in the qZ

1.727 Â -1 ¹ plane (L

1). ^The lineshapes ^of peaks ^{in this} plane ^are determined by ^the

convolution of the vertical resolution function and mosaic broadening. ^{In this} sample ^the

Table 1.

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.}

strand axis was 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.

Fig. ^2. - (a) Equal intensity ^contour plot ^of X-ray scattering data collected in the 6z - Qx plane ^about

the (2, ^0, 0.379) peak position ; (b) equal intensity ^contour plot ^of X-ray scattering data collected in the q2

1.727 A -1 ¹ plane (L

1 ). ^The (001), (101) ^and (011) peak positions, and the line [x, x, 1 ], ^are

indicated. Contours are drawn at values of 4 642, 3 167, ² 154, 1 602, 1 000, 681, 464, 317, 215, 160, 100 and 68 counts/30 s. The six peaks ^with 1 1 1 ( ^{33 symmetry have} approximately equal intensity ^maxima

of 5 800 counts/30 s. Data are missing from the lower left quadrant ^{due to a} mechanical limitation of the diffractometer.

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

Fig. ^3.

Perspective schematic of diffraction pattern. qx and qz ^indicate equatorial and meridional components ^{of the} scattering vector, ^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.

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 vs.} T from the Dhd phase ^{to the constant value} measured in the

Dho phase ^{indicates that we} may be close ^{to a} second-order transition. which is however

Fig. ^4.

Scattering profiles ^{at three} temperatures upon heating through the coexistence region ^{at the} 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.

Fig. ^5.

Intercolumn distance d, ^as derived from the fundamental (100) peak, ^versus temperature.

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.

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

of the fundamental (100) ^vector is q 1

^{0.334 Â-1.}

Upon formation of the Dho phase, ^{a number of new} reflections appear, ^as indicated by ^the

open circles indicated in figure 3. All the reflections in the Dho phase ^are resolution-limited, 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

peaks consistently deviate from these commensurate positions by ^{at least one FWHM} [11].

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} 3 ^x 3 R ^30° superlattice, ^{i.e. the new} fundamental is indexed

as a 1 1 0 33 ³ ). ^{The new unit cell} therefore consists of three columns. The evolution of some of these superlattice peaks ^{at the} Dhd-Dho phase transition is shown in the right-hand ^{column of} figure ^4. The absence of any ^new peaks ^{in the} equatorial (HKO ) plane implies ^{that the}

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. ^Semi- empirical 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

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.

4. Structure of the K phase.

The unit cell was determined to be monoclinic :

From the systematic 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

Fig. ^6.

Structure of the K phase. (a) Stereoscopic ^{view of an} individual molecule. Ellipses ^indicate

r.m.s. amplitudes ^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.

helical ^structure with a pitch ^P

^{2 c. The ratio} a/b = 1.13 found for the unit cell is far away 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 ^semi- empirical conformational analyses of related triphenylene derivatives [16]. Alkyl ^chain tilting

will break the reflection symmetry ^{of an isolated} molecule, leading ^to D3 symmetry (one ^3-

fold 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.

We now discuss the models used for least-squares analysis of diffraction data in the

Dho phase. The molecules lie 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

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

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.

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.}

At this point the first problem encountered in the data analysis ^becomes apparent. ^The

structure factor is 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

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

w5 ^x 13 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

Ô ^x J3 superlattice order in the Dho phase is associated with groups of three columns 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 of generality, ^{we can take} 00

⁰ (or random), Ho ^{= 1 and} Zo

^0. (Of ^course, because of the 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

the nonzero intensity ^{of the} 111 ( ^{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

a second sublattice of molecules, ^{all at} height Zo

^{0. If we then fix} Zl

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 factor :

where

The calculated intensity ^{was obtained} by squaring ^{the total structure} factor and multiplying

it 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

with the angular widths determined by the mosaic and instrumental resolution, ^as discussed in section 2 :

Several different forms for the Debye-Waller ^{factor were} tested. While at first it might

seem 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-phase- preserving translations Z. An appropriate Debye-Waller factor is then

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

fluctuations. _Unm measures the extent to which the vertical and rotational motions ^are

coupled ; ^{it can} range from --t 2 1 u,, u,,, 1 (perfectly correlated or anticorrelated) ^{to zero.}

Putting everything together, the scattered intensity ^is given by :

where {GJ ^{spans the} V3 x V3 ^{R 300} superlattice 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 have an 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,

Table II. - Parameters giving ’rise to the fit in ^{table 1.}

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.

Also, ^{in some fits we} allowed the molecules to rotate a small amount randomly ^about

random axes 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.

The 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.

However, ^{the n}

^{3 and n}

⁶ planes should have the same intrinsic intensities as their

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 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 previously discussed, ^{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