Correlations in a smectic B plane

~ t i d State'Communications, Vol. 20, pp. 379-383, 1976.

Pergamon Press.

Printed in Great Britain

CORRELATIONS IN A SMECTIC B PLANE M. Descamps and G. Coulon Equipe de Dynamique des Cristaux Mol6culaires, Associ~e au C.N.R.S. (E.R).. 465), Universit~ de Lille I., B.P. 36, 59650 Vilieneuve d'Ascq, France

(Recetped 10 May 1976; in revised form 28 June 1976 by P.G. de Gennex) Several experimental studies have shown that, in all the smectic B phases, tilted or not tilted, the molecules rotate around their long axis. A herringbone type local order has been detected by means of X-ray scattering experiment. We are showing that a model, assuming discrete jumps of molecules with steric hindrance between neighbouring phenyl rings leads to a local order of this type. With this object we use a statistical method, based on a graphical series expansion, which gives the orientationai double probabilities. The theoretical X-ray scattering, calculated by means of the later ones, leads to focused peaks in exact coincidence with the experimentally observed ones. 1. INTRODUCTION WHETHER THEY are tilted or not, the ~mectic B meu> phases are such as the molecules are ananged in a two dimensional hexagonal (or approximatively hexagonal) array. According to Meyer and McMillan's theoryx the tilted smectic B phases (SH) are characterized by the alignment of the dipoles and the non-tilted ones by randomly oriented dipoles. Concerning these tilted phases, De Vriesz proposes a model in which the molecules are also stationary, with the molecular planes arranged in a parallel ordered packing for TBBA and in a herring-bone-type ordered packing in BBEA. But from the analyms of X-ray diffraction, Levelut et aLa have shown that the orthorhombic ceils which descn"oethe hexagonal lattice are centered (ABCD cell for example on Fig. 1) in each case and this is compatible with an orientational molecular disorder. Besides, NMR,4 Raman 5 and neutron scattering e data are consistent with an isotropic rotational motion of the molecules around their long axis in all tilted or not SB phases; in the last quoted study some partial orientational ordering of the molecules has been suspected. In a recent X-ray scattering study 7 Levelut has remarked the existence of twelve diffuse spots which can be grouped into three sets deriving from one another through a Ir/3 rotation, and this for several types of smectic B phases (tilted: TBBA, non-tilted: PBBA). These spots are located on extinction points of the reciprocal lattices of the three possible centered orthorhombic lattices. So, she suggests that their presence is related to the apparition of a local order of Pba2 spatial symmetry: i.e. the molecules can take randomly three orientations and arrange themselves locally in a

/

(a}

Fig. I. (a) Projection of the phenyl rings on the smectic plane. The molecules are arranged in a two-dimensional hexagonal array and can take randomly three orientations. Co) The three possible orthothombic cells. The local herring-bone-type order can develop itself along the three corresponding directior,.. herring-bone-type packing (Fig. 1). Then her conclusion is that the molecular rotational motions are geared. Assuming that the phenyl rings can take three orientations, the simple analysis of the lattice parameters, of the disposition of the molecular orientations and of the size of the phenyl rings shows clearly that steric hindrance makes impossible some sLmultaneonsconfigurations of neighbouring molecules. As we have already studied this kind of correlations and their influence on the local order in plastic crystalss'° our purpose here is to know how the steric hindrance by

379

380

CORRELATIONS IN A SMECTIC B PLANE

Vol. 20, No. 4

P (S..S”) 116

X

lorders

t

0 order

4 +

3:

n 4

#’ 2

3

4

5

iT

fb)

Fig. 2. (a) Localization of the nearest-neighbours of a given molecule up to the sixth ones. (b) Series expansion of the double probabilities P(&,,, S,). l/9 corresponds to the complete independence. The series truncated at the third order, leads to a slight negative value of P(&,,, S,) in the case of complete steric hindrance between the first neighbours. In most cases the convergence is rather slow, but the herring-bone local order is clearly shown. (c) Dependence on successive encountered molecules (n) of P(So, S,,) along the D direction: here, the correlations are of antiparallel type. The dotted line, corresponding to the fast order of the expansion, gives an idea of this dependence. itself can account for the local order between equilibrium positions. In this view, we use a series expansion method by means of a graphical technique which allows to evaluate the configurational partition function and the double probabilities (Section 2). Then, in Section 3 we study the corresponding theoretical X-ray scattering. 2. DESCRIPTION OF THE MODEL AND STUDY OF CORRELATIONS 2.1 We consider a plane hexagonal array of linear sticks (the lattice is triangular in the graph terminology). A stick corresponds to the projection of a phenyl ring. Its length is superior or equal to the lattice parameter. We assume, that on each site, a stick can take three equilibrium positions along the sides of the triangles. So two

neighbouring sticks would overlap each other should they get along the same triangular side. This model can also describe a six orientational disorder (by a n reorientation) if the same steric hindrance is to happen. We are going to study the local order induced by this steric effect. 2.2 Study of correlations by the weak-gmph method We have already studied the steric hindrance in plastic crystals9 by adjusting the graphical method used by Nagle for the hydrogen bonds problems.ro Here we are concerned with the corresponding problem in a plan The global description of correlations drives us to evaluate the partition function

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CORRELATIONS IN A SMECTIC B PLANE

381

,.°

Fig. 3. Theoretical isodiffusion pattern in the (hk0)* plane. First order maxima focused on (210), 0 / 2 , 3/2, 0) and (5/2, 1/2, 0). Second order maxima focused on (120), (5/2, 3/2, 0) and (7/2, 1/2, 0). in which N is the number of sites, the product is restricted to all nearest-neighbour pairs of sites; Si represents the orientation of the ith molecules. The def'mition of A(St, S;) is the following: A(Si, St) = 1 = 0

f i s t andS~ ate compatible ifnot

incompatibility imposes a -- x a- ~(Sl, Sj) is a normalisation factor. There are two kinds of compatibilities: compatibility 1 : no stick on (//) ~* ~(1) = x 2 _ y 2 compat~ility 2: only one stick on (ij) ~ ~(2) = x 2 + x y

(2)

The computing of (l) can be performed with a high accuracy using the weak graph method: then A(S~, Sj) has the following form:

Whatever the whole configuration there are 2 N ~ ( 2 ) factors and N ~ ( 1 ) factors. The definition of the Gj(Si) terms has been taken broadly enough in order to impose:

~. ctAs,) ~- o (4);

in which

A(s,, sj) = a - c~(s~)c/~(s~) ( s . sj)

Gj(St) = + x = --y

(a)

if the ith stick is on the edge (i/') if not.

(I) may be rewritten as series expansion based on the products of an increasing number of terms (CijCji). Each product o f p terms (CuC/i) correspon~ to a graph o f p edges drawn on the lattice. It is clear that the

thusx = 2y (4').

8i

This being imposed, only dosed weak graphs with no vertices of degree one have to be taken into account in

O).s. ~o

So, the series converges much faster and the zero order term ZoN (no graph) is the prevailing one:

(x2) ~N zoN = [ ~ 0 ) ] . [ ~ ( 2 ) ] ~ v

x (3)';

3N is the total number of edges.

382

CORRELATIONS IN A SMECTIC B PLANE

Thus the series has the following form: z ~

=

Zon[l + e(3) + e(4) +...]M ( ~ x 3 ) n [ 1 + 3 . 2 x lO_2+...] n =

(5)

in which eO) corresponds to the first closed graphs i.e. the triangles. Z is the effective number of orientations for a molecule. The description of the local order makesit necessary for us to know how the orientations on one site affect the orientations of the neighbouring sites, thus we must reckon the double probabilities: 1

P(S.,,Sp) = ~

~

s.s,..sp

I]' A(S,,81)

P(Sm,Sp) =

1 -- ~ z,

and focused scattering, the only effect of the thermal vibrations will be to decrease the scattering intensity for large angles. Consequently we do not take the thermal scattering into account in the calculation. In one point of the reciprocal space, the theoretical scattering power (in e.u.) is expressed by: (I(S)) = N 2 [(F)iaA(S)

+[(~.fif. exp(iS'rl.'?--l(F)la l (8)

- N E Y. Y.

(6)

in which the summation is over all the sites under the restriction that S,, and Sp are fixed orientations. The (4) condition being imposed, non-vanishing terms are, in addition to closed polygons, those lines consisting of successive pairs starting from the mth site and terminating at the pth site. Eventually: ~,uA.~(a)'Jv3t¢-' [..,~v . -~] (7)

w(l)w(=)

in which the summation is over an increasing number of edges. Ca is the contribution of all the n-edges closed graphs and the n-edges open graphs which join the ruth and pth sites. Here, we have calculated the double probabilities within the limit of the 6th neighbours of a given molecule. For the three first ones, (7) has been expanded up to third order. Concerning the outward molecules, the computing shows that the closed weak graphs with three or four edges are of no account. The first non-vanishing term expresses the more direct interactions between the m and p molecules. The results are given in the Fig. 2. Since the complete independence of the molecules corresponds to 1/9, the results concerning the first neighbours show that the most likely local order is a herring-bone one. The same trend occurs for more and more remote neighbours with a decay of the relative influences. Owing to the hexagonal symmetry this local order can develop itself along three directions apart (Fig. 1).

n/3

rmP Sm Sp

× [P(S..)e(s,,)

-

e(sm. sp)] exp (~s" rmp)

The fast term is the diffraction one. The second term represents the scattering due to the complete orientational disorder of the molecules. The summations are over all the atoms of one molecule. The averages are reckoned on the different equilibrium positions. This expression is a substitute for the classical one ((I F Is ) -- I( F ) Is) because, in a given molecule, the atomic rotational movements are performed solidarily. The last term expresses the correlation effect. Fm(S,n ) is the Fourier transform of the ruth molecule in the orientation. is the simple probability. We are here studying the scattering repartition in the reciprocal plane (hk0)*. (In all cases the unit cell in the direct lattice is an orthorhombic one.) Consequently, only the projections of the phenyl rings have to be considered. So the stick length and the distribution of the electronic densities are imposed. For the computing we consider an orthorhombic lattice with parameters a = 8.7 A and b = 5 A. The length of the sticks is about 6.4 A; it is superior to the nearest neighbours distance ('-" 5 A). Concerning the electronic densities only the carbon atoms are taken into account. Using the above calculated double probabilities, we fred that the theoretical scattering intensity is distributed as follows (Fig. 3). In a quarter of the (hkO)* plane the notable features are:

Sm

P(Sm)

0.305 forISl = ~ A - I 2n

3. THEORETICAL X-RAY DIFFUSE SCATTERING We have now to test the opportunity of this model and to see in what measure the single steric hindrance can explain the partial orientational ordering of molecules which has been experimentally detected. In this aim we calculate the X-ray diffuse scattering and compare it with the experimental ~ one. Here we only take interest in the scattering related to the orientational disorder of phenyl rings and to the above mentioned correlations. If the latter give a strong

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three intense scattering peaks about (210), (1/2, 3/2, O)

and (5/2, 1/2, 0)

0.425 forlSI = ~ A - I 27r

three slightly weaker scattering peaks about (120), (5/2,

3/2, 0) and (7/2, I/2, 0).

Then, for larger [ S I, other peaks, but less and less intense. The three peaks corresponding to the first order have exactly the same localization as those observed by

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CORRELATIONS IN A SMECTIC B PLANE

Levelut. The second order ones, which, experimentally, must be strongly weakened by the D.W., appear on some photographs, u Obviously, the higher order peaks do not show up on the photographs. Besides if we interpret the width of the theoretical peaks in the way of domain sizes, we find that they are ~ 20 A large, in good agreement with the experimental values. These last ones vary from 18 A in the tilted Sn to 60 A in the non-tided Ss.7 4. CONCLUSION Our model is based on the existence of an orientational disorder of the molecules about their long axis: they occupy randomly discrete positions at lr/3 apart. It assumes a complete steric hindrance between some orientations of neighbouring phenyl rings. This model is compatible with two types of rotating molecules: either the phenyl rings remain coplanar during the rotation - or rotate relatively to each other while keeping the same projection on the plane. The double probabilities, computed by means of a graphical method, prove the existence of a herring-bone type local order. Besides, we show that the theoretical scattering peaks, which result from this order induced by the only steric hindrance itself, superimpose, exactly,

the experimental ones. And it so happens whatever the experimentally studied mesophase might be, tilted or not. It must be noticed that, for example in TBBA, the experimental domain size remains almost always constant, whatever the temperature is. This fact suggests the existence of hit or miss interactions because of their slight responsivity to temperature. As a conclusion, although our hypothesis concerning the interactions do not leave out other sources of local correlations, dipolar for example, our calculations prove that steric hindrance between phenyl rings of neighbouring molecules is certainly of prime importance in the generation of the local order in the smectic B planes; this order characterizes the =hectic E phases (when they exist). Besides, these results show that, contrary to plastic crystals,s steric hindrance can lead to a rather focused X-ray scattering.

Acknowledgements - We are very indebted to Drs. A.M. Levelut and M. Lambert for acquainting us with their results before publication and for helpful discretions. Thanks to M. Le Cocq for reading the English manuscript and Mrs Foulon for typing it.

REFERENCES 1.

MEYER R J . & McMILLAN W.I.,, Phys. Rev. A9,899 (1974).

2.

DE VRIES A., J. Chem Phys. 61, 2367 (1974).

3.

LEVELUTA.M.,DOUCETJ.&LAMBERTM.,J. dePhys. 35,773 (1974).

4.

DELOCHE B., CHARVOLIN J., LIEBERT L. & STRZELECKI L.,J. de Phy~ ColI. 36, CI.21(1975).

5.

DVORJETSKI D., VOLTERRA V. & WIENER-AVNEAR E.,Phys. Rev. AI2, 681 (1975).

6.

VOLINO F., DIANOUX AJ. & HERVET H., Solid State Commun. 18,453 (1976).

7.

LEVELUTA.M.,J. dePhy~ CoIL 37, C.3-411 (1976).

8.

DESCAMPS M., Solid State Commun. 14, 77 (1974).

9.

DESCAMPS M., Chem. Phys. 10, 199 (1975).

10.

NAGLE J.F.,J. Math. Phys. 7, 1484 (1966).

11.

LEVELUT A.M. (private communication).

Plusieurs dtudes exp~rimentales ont rEvdld que, clans les phases smectiques B, inclindes ou non, les molecules tournent autour de leur grand axe. Un ordre local des molecules, en chevrons, a m~me ~tE mis en Evidence par diffusion X. Nous montrons qu'un module ~ sauts discrets avec ernl~chement stErique de certains cycles benzEniques voisins conduit ~ ce type d'ordre local. Pour ce faire, nous utilisons une rr~thode statistique de d~veloppement en gr=phes qui donne les probabilitEs doubles d'orientation. La diffusion X th~orique, calculEe & I'aide de ces derni~res, donne ceux observes exp~rimentalement.

383

des spots diffuses exactement superposables