A statistical model of the discotic mesophase

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PramS.ha, Vol. 16, N o . 3, March 1981, pp. 201-209. © Printed in India.

A statistical model of the discotic mesophase

R L O C Q U E N E U X

Laboratoire de Physique th6orique, Universit6 de Lille, I Ba^timent P5, B.P. 36, 59650 Villeneuve D'asco, France

MS received 8 December 1979; revised 17 January 1981

Abstract. The evaluation of the thermodynamic properties of an ideal discotic mesophase needs the determination of a partition function, that depends upon two basic types of energy storage, translational and rotational. Here, we suppose we can substitute for the complex effective intermolecular potential, different mean potentials acting o n each individual molecule. The defining assumptions for a discotic mesophase include the stipulations, first--that each simple disk-like molecule is, during most of the time, confined to a cell, secondly--that each molecule has an external rotation which is more or less hindered by a periodic potential. The cells are stacked in columns and the columns form a regular hexagonal array, each molecule moves in a cell as in an infinite potential well due to the neighbouring molecules.

This model has allowed the general formulation of the mesophase-free energy.

F r o m this, we obtain the form of the coefficient of isothermal compressibility when the external rotations of the molecules are hindered and the vibrational energy is weak.

Keywords. Discotic mesophase; cell theory.

1. Introduction

Some recent studies (Chandrasekhar et al 1977; Billard et al 1978; Levelut 1979) have shown that compounds with disk-like molecules can give partially ordered states; these molecules assemble in a discotic mesophase. In some ideal discotic mesophase, owing to the molecular interactions, the disks are stacked one on top o f the other in columns that constitute a regular hexagonal array, and these disks re- main parallel to one another.

To a first approximation, we consider that the translational, rotational motions o f molecules are independent and uncoupled and their energies may be separated, and furthermore that the contribution of internal rotation is negligible.

2. Determination of the translational free energy

The examination of the mesophase leads us to think that it is possible to represent this mesophase by a model inspired by the cellular models of fluids (see Munster 1974 or Cruickshank 1971). To evaluate the translational energy of the phase, we consider that each molecule moves in a cell as in an infinite well created by the neighbouring molecules.

So we suppose that we can substitute for the complex effective intermolecular potential, different mean potentials acting on each individual molecule; and wo also

201

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202 R Loequeneux

suppose that the cell theory can be used to study the ideal discotic mesophase. In this theory each molecule is, during most o f the time, confined to a cell whose boundaries are determined by the potential due to the neighbouring molecules. We suppose that the cells are placed between parallel planes Ill, II~ ... IIi .... The distance d~ between two successive planes is the thickness o f cells (this distance is equal to the mean distance between two successive molecules in the same column). In a c o l u m n the cell centres are put in a line A~, and let a be the angle between a line and the n o r m a l to plane. Between two successive planes, the cells f o r m a regular hexagonal array.

We consider, in the bulk o f fluid, a system o f N cells in t h e r m a l equilibrium with the rest o f fluid; let T be the temperature o f the system, V=Nvc, the volume (where v~

is the volume o f a cell). We choose a system with N~ cellular strata, each o f them having N= cells ( N = N A N , ) . We separate the translation m o t i o n into two parts, one on a plane parallel to the planes IIl, the o t h e r o n a line parallel to the lines Aj.

2.1 Translation parallel to the planes Ill

T o a first approximation, we suppose that in a plane ( i i ~ ) , the hexagonal cells may be replaced by circular cells o f the same surface s¢ (let Pc be the radius o f these circles) and we suppose that the molecules are rigid disks. Let pa be the radius o f these disks and sd their surfaces.

The average surface accessible to the centre o f the molecule, the so-called free surface per molecule s F is

s F = ~(pc -- PdY,

because the centre o f the molecule cannot a p p r o a c h within pa o f the cell's boundary.

Some geometrical considerations allow us to write 2p~ = 1.05 3~

where 3c is the distance between the centres o f two neighbouring hexagonal cells.

T h e canonical partition function o f one o f this lamellar systems may be written as:

Z~ , - ~ , ( N . L 1 , ~ 1 J )

where S,~ ---- N,, sc is the surface o f the system.

T o account for a more important disorder in a pile o f molecules, we may consider that there are multiple overlaps so that each element o f surface is counted more than once, let s' c be the surface o f these cells

s~ : e2sc or s" = r/-z so, where V -1 = E a n d 0 < "q ~ 1.

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Statistical model of discotic mesophase

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It follows from this that the canonical partition function of the lamellar system is Z ~ = ( 2¢r

h 2mkT]N/

!~. e2 N,~kS~[1 --

"o(N1rsa]l/Z]~~ --~ ] J )"

The free energy of the lamellar system can be obtained from the partition function, we get

f~r = -- kT

log Z,~.

The system has Na identical lamellar systems, so their contributions to the translational free energy of the system is:

F~,=Naf,~.

F ~ = - - N k T l o g ( 2 ~ r m k T ] - - N k T l o g ~S"[1

(N'rr Sd~1/212 ~ h ~ / (N,,L -- ~ / \ - - ~ / J ) 2.2

Translation parallel to the axis Aj

There are Na linear cells on a line A j, these cells are considered as segments of a line, and let d j = (ddcos a) be the length of a segment. We suppose that the molecules are rigid segments of the line, and let d~ ----

(ddcos a)

be the length of a rigid segment where de is the thickness o f a molecule.

The average length accessible to the centre of the molecule, the so-called free length per molecule d~- is

a k = a " - a;.

The canonical partition function of one of these linear systems may be written as

where

Za = [ (21r ~ kT) 1/~ _ _

Da =Nadc,

cos a NA D,~ / '

and the free energy of the linear system may be written as

fa = -- kT

log Z6.

In the system considered, there are N~ identical linear systems so their contribution to the translational free energy of the system is:

FA -~N, rfa,

~'2 / cos a Na DA / "

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204 R Locqueneux

The translational free energy o f a discotic mesophase is:

F = Fa-i- F,~

So we seet hat the translational free energy o f the discotic mesophase is a function o f microscopic quantities which can be calculated f r o m x-ray data (Chandrasekhar et al 1977; Levelut 1979).

3. Expression of rotational partition function

(i) T h e internal rotations in the molecule originate essentially from the flexibility o f the lateral chains; so their contributions to the free energy o f the mesophase is negligible to a first approximation (the moments o f inertia o f these lateral chains are very small c o m p a r e d with the m o m e n t o f inertia o f the molecule).

(ii) Now, we consider the rotation o f the molecule as a whole; the central part o f the molecule is a rigid body but the lateral chains are flexible. The rigid part o f the molecule has an axis of symmetry 8R o f order 3 which passes t h r o u g h the centre o f mass and is perpendicular to the plane of the molecule. On account o f the flexibility o f the lateral chains, the m o m e n t of inertia o f the molecule does not have a constant value. Yet, to the first approximation, we can replace real molecules by mean molecules with lateral chains in fixed mean positions. T h e n the molecule as a whole a n d the rigid part o f the real molecule have the same symmetry. Let J be the m o m e n t o f inertia o f this m e a n molecule.

T h e rotation o f a molecule as a whole is m o r e or less hindered by the presence o f lateral chains, b o t h its own and those o f others. According to the physical conditions, the distance between the centres o f neighbouring molecules is more or less important. It is evident that if this distance is larger t h a n the diameter o f a molecule when it reaches its full size, the interactions o f the lateral chains are weak and con- sequently the rotation o f the molecule as a whole is little hindered but it is evident that if this distance is smaller than the diameter o f this full-sized molecule then the rotation is more hindered or forbidden. M o r e o v e r the interactions o f the lateral chains depend on the positions o f the molecules in the nearest cells; so interactions exist between the translational and rotational degrees o f f r e e d o m o f the neighbouring molecules that we do not take into account.

T o a first approximation, these interactions o f the lateral chains may be approxi- mated by the following sinusoidal potential

u = V__o [ 1 - c o s . 41, 2

where ~ is the rotation o f the molecule, a = 6 for the molecule o f benzene-hexa-n- alkanoates (Chandrasekhar et al 1977) and a = 3 f o r the molecule o f hexa alco- xyderivatives o f the triphenylene (Billard et al 1978). It is evident that the value o f U 0 depends on V/N and T.

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Statistical model o f discotic mesophase 205 F r o m this approximation, it follows that the wave equation m a y be written as

d2ff q- ( 1 - - c o s 0" ~) ~b = O, d~ z h 2 k

(see, for example Munster 1974 or Pacault 1963).

This equation gives us

,4,t,2____~ + (a + 2q cos 2x) ~b (x) = 0 dx 2

which is the Mathieu equation (see for example Wittacker and W a t s o n 1973) where q -- _ _

8.~.Ju0

_{, a - -}

0 "2 h 2

32~r2J E - - and x

O.2 /12

There is an infinite sequence o f discrete energy levels which are not degenerate.

It follows that the hindered rotational partition function is:

. ~ R = (ZR)N where Z R -= Z~ exp - - - - . Ei

k T

W h e n the external rotations o f the molecule are forbidden,

Uo~ kT,

a n d we m a y expand the approximate potential in a series and retain only the first t e r m in this series, we obtain

2 L2 '

which is a potential o f a simple harmonic oscillator.

T h e n the partition function Z R becomes a vibrational partition function.

O f course, in the two cases, the quantity U depends on the sizes o f cells and o f molecules. So the value o f Z R depends on T and on V/N as well. As for the weak vibrations, the correspondant free energy is negligible. We determine, in these conditions, some t h e r m o d y n a m i c properties o f the mesophase.

4. Some thermodynamic properties of the discotic mesophase

We restrict o u r study to the effects o f the lamellar or linear strains which produce a variation o f the size o f the cells but do not change the shape o f the cells.

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206 R Locqueneux 4.1 General study

We suppose that lamellar stress acting on the regular hexagonal array is

D S~ N=, T

then

t~ ^- N~ k T 1-- ^{_ _} ^F ~ - -

[N=

^sd~,2l-1 , S~ L k s ~ l J

a n d the linear stress acting on the columns is:

c3Da N a, T, then N~ k T (1 .N.~ d a y 1

t;~-- ^{~D~ \} ^~ D~ / "

F r o m this, it follows that the translational and rotational specific heat and the coefficients o f thermal expansion a and the isothermal compressibility ~ m a y be determined. The application o f this model to the study o f the coefficient o f isothermal compressibility presents some particularities, therefore we chose this study, to set an example.

I n this model, we suppose that the translations parallel to the axes A and the translations parallel to the planes II are independent and uncoupled, so it is not possible to consider the strains caused in the planes II by stresses parallel to the axes A and vice versa. So, it is necessary to consider only particular stresses when studying the isothermal compressibility of the ideal discotic mesophase.

(i) W h e n the ideal mesomorphic phase is exposed to homogeneous lamellar stresses t~, the (lamellar) coefficient of isothermal compressibility m a y be written:

L D S~r JN~r, T

u -

(ii) When the ideal mesomorphic phase is exposed to uniform linear stresses ta, the (linear) coefficient o f isothermal compressibility m a y be written:

= -k D~ t - - D~ J

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Statistical model of discotic mesophase 207 The signs o f lamellar and linear stresses have been chosen arbitrarily. F r o m this choice, it follows that the lamellar a n d linear coefficients K,, and ~x are positive.

Nevertheless, as in the case o f graphite, it is possible for the coefficient K x to be positive a n d coefficient K= to be negative (Riley 1945) (indeed the graphite, as the discotic mesophase, has a lamellar structure) then it would be necessary to write:

t = +

4.2 Coefficients o f isothermal compressibility for two hexa-alcoxy derivatives o f the triphenylene

A n actual calculation o f the coefficients K= and KA is now possible. F o r example, we propose to calculate the isothermal compressibilities at 85°C o f two hexa n-a!coxy- 2-3-6-7-10-11 derivatives o f the triphenylene, the lateral straight chains Of which are alkanes with 5 or 7 carbon atoms (Billard et a11978; Levelut 1979).

The mean distance dc between two successive molecules in the same c o l u m n is 3.59 A f o r the two c o m p o u n d s , and the thickness de o f the molecules can be chosen between 1 A a n d 1.5 A, so the coefficient Ka is between 0.5 1011 and 0.4 1011 N -1.

The distance 3c between the centres o f two neighbouring hexagonal cells is 18.94 A for the pentyl derivative a n d 21.94 A for the heptyl derivative. I f we suppose that the radius o f the disk which represents the molecule is equal to the radius o f central part o f this molecule, which is not realistic, we have pe = 5.02 A for the two c o m p o u n d s . But some other values o f Pe can be chosen; it is possible, for example, to choose this radius equal to the distance between the centre o f the molecule and one o f the c a r b o n a t o m s o f the lateral straight chains, when the molecule is fully extended.

Table 1 gives the calculated values o f K~. In this table, for the two derivatives, the values o f pe are equal to the distances between the centre o f the molecule a n d the antepenultimate, penultimate and last carbon atoms o f the lateral chains. A pressure p = ta s~ -1 corresponds to the linear stress ta a n d a pressure p ---- t~ d~ -1, to the lamellar stress t,~; as a result, it is possible to make a comparison between the

Table 1. tqrN -1 m for two hexa alcoxy-derivatives of the triphenilene at 85°C

pa A pentyl derivative heptyl derivative

(pc = 9-94 A) (Pc = 11,52 A)

1/ = 0"9 ~/ = 1 ~/ = 0 . 9 ~/ = 1

5-02 237 204 380 335

7"8 155 51 - - - -

9"35 25 4 - - - -

10"3 5 no some 54 18

11"7 - - - - 11 no some

12.7 - - - - 0.01 no sense

(not very signi- ficant value)

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208 R Locqueneux

coefficients Ka sc a n d K= dc a n d t h e u s u a l coefficient o f i s o t h e r m a l c o m p r e s s i b i l i t y , K.

T a b l e 2 gives t h e c a l c u l a t e d v a l u e s o f coefficients Ka sc a n d ~c, dc for t w o h e x a - a l c o x y d e r i v a t i v e s o f t h e t r i p h e n y l e n e .

A t p r e s e n t , w e c a n o n l y n o t e t h a t t h e v a l u e s o f t h e s e c o e f f i c i e n t s a n d t h e e x p e r i - m e n t a l v a l u e s o f t h e coefficients o f i s o t h e r m a l c o m p r e s s i b i l i t y f o r s o m e c o m p o u n d s c l o s e l y r e l a t e d t o s t u d i e d c o m p o u n d s a r e in t h e s a m e r e g i o n ; s o K = 5 10 -1° N - t m s f o r p a r a a z o x y a n i s o l e ( C h i n a n d N e f f 1975) a n d K-~5, 7 10 -a° N -1 m 2 f o r N - - ( p e t h o x y b e n z y l i d e n e ) p - n b u t y l a n i l i n e ( K u s s 1978). W e n o t e t h a t t h e coefficient ~a is l i n k e d t o t h e r e l a t i v e c h a n g e o f t h e a v e r a g e d i s t a n c e d , b e t w e e n t h e n e i g h b o u r i n g m o l e c u l e s in t h e c o l u m n f o r a c h a n g e A G o f l i n e a r stresses e x e r t e d o n t h e i d e a l d i s c o t i c m e s o p h a s e

a n d t h a t t h e coefficient K~ is l i n k e d to t h e r e l a t i v e c h a n g e o f t h e d i s t a n c e ~¢ b e t w e e n t h e c e n t r e s o f t w o n e i g h b o u r i n g h e x a g o n a l ceils f o r a c h a n g e At,, o f l a m e l l a r stresses e x e r t e d o n t h e i d e a l d i s c o t i c m e s o p h a s e

o, LAt~,JT

3¢

LA-T=JT

5. Conclusion

F r o m t h e s t r u c t u r e o f a d i s c o t i c m e s o p h a s e , w e h a v e b u i l t u p a m o d e l i n s p i r e d b y t h e cell t h e o r y o f l i q u i d s t o d e t e r m i n e t r a n s l a t i o n a l f r e e e n e r g y o f t h e m e s o p h a s e .

F r o m t h i s c e l l u l a r m o d e l , it is p o s s i b l e t o d e t e r m i n e t h e t h e r m o e l a s t i c coefficient o f t h e m e s o p h a s e a s a f u n c t i o n o f t h e size o f t h e m o l e c u l e a n d o f t h e m e a n d i s t a n c e

Table 2. K.101° N-am ~ (K.10 u dyne -1. cm ~)

/~. pentyl derivative heptyl derivative

KaSc da

1 15 21

1.5 12 17

K~dc pa ~/ = 0.9 ~ = 1 ~ = 0.9 r/ = 1

5"02 850 732 1364 1203

7"8 556 183 - - - -

9"35 90 14 - - - -

10.3 18 no sense 194 65

11.7 - - - - 39 no sense

12.7 - - - - 0.04 no sense

(not very signi- ficant value)

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Statistical m o d e l o f discotic mesophase 209 b e t w e e n t h e n e i g h b o u r i n g m o l e c u l e s , t h e s e m i c r o s c o p i c q u a n t i t i e s m a y b e d e t e r - m i n e d b y x - r a y s t u d i e s .

I t is still t o o s o o n t o j u d g e t h e a p p r o p r i a t e n e s s o f t h i s m o d e l t o a c c o u n t f o r t h e t h e r m o d y n a m i c p r o p e r t i e s o f t h e d i s c o t i c m e s o p h a s e . N e v e r t h e l e s s it s e e m s q u i t e o b v i o u s t h a t a c e l l u l a r m o d e l is m o r e c l o s e l y r e l a t e d t o t h e r e a l s y s t e m w h e n t h i s sys- t e m is a d i s c o t i c m e s o p h a s e t h a n w h e n it is a liquid.

Acknowledgements

W e a r e m u c h i n d e b t e d t o P r o f e s s o r J B i l l a r d f o r v a l u a b l e d i s c u s s i o n s .

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