and G. Vertogen

Notizen 1359

Oll the Odd-Even Effect in the Helix of Non-Sterol Cholesterogens

B. W. van der Meer

Solid S t a t e P h y s i c s L a b o r a t o r y a n d I n s t i t u t e for T h e o r e t i c a l P h y s i c s , U n i v e r s i t y of G r o n i n g e n , T h e N e t h e r l a n d s *

and G. Vertogen

I n s t i t u t e f o r T h e o r e t i c a l P h y s i c s , T o e r n o o i v e l d , C a t h o l i c U n i v e r s i t y of N i j m e g e n , T h e N e t h e r l a n d s Z . N a t u r f o r s c h . 3 4 a , 1 3 5 9 - 1 3 6 1 (1979);

r e c e i v e d S e p t e m b e r 21, 1979

A m o d e l is p r o p o s e d t o e x p l a i n t h e a l t e r n a t i o n of t h e screw-sense of t h e h e l i x w i t h t h e n u m b e r of b o n d s b e t w e e n t h e chiral c e n t r e a n d t h e r i n g s y s t e m in chiral c o m p o u n d s w i t h n e m a t i c - l i k e m o l e c u l a r s t r u c t u r e .

Non-sterol cholesterogens can be obtained from a nematogen through the replacement of a hydrogen atom of their alkyl or alkoxy chain by an alkyl group, thus creating a branching point at the chain and consequently a chiral centre at that point.

These systems exhibit an odd-even effect, i.e. the handedness of the helix alternates with the number of bonds between the chiral centre and the ring system [1, 2]. The purpose of this note is to explain this effect in terms of a simple model.

In these systems the handedness of the helix appears to depend on the position of the chiral centre at the chain as well as on the absolute configuration of the centre. A convenient definition of the configuration of a chiral centre at the chain is the following. Consider a molecule with a chiral centre having for instance a H-group and a CH3- group attached to it, see Figure l a . Next we proceed along the chain towards the ring system with the long axis of the molecule in a horizontal position and the H- and CH3-groups pointing obliquely upwards. Now the centre is labeled with a R (rectus, righthanded), if the CH3-group is to the right and with a S (sinister, lefthanded), if the CH3-group is to the left. This definition is in complete agreement with the convention of Cahn et al. [3]. The centres in Fig. 1 a and 1 b are therefore

R e p r i n t r e q u e s t s t o : D r . G. V e r t o g e n , I n s t i t u t e f o r Theo- r e t i c a l P h y s i c s , T o e r n o o i v e l d , C a t h o l i c U n i v e r s i t y of N i j m e g e n 6 5 2 5 - E D N i j m e g e n , N i e d e r l a n d e .

* M e l k w e g 1, 9718 E P G r o n i n g e n .

R-centres. The sign of the helix in a system of molecules with the chemical structure of Fig. 1 a can be reversed by changing the absolute configuration of the chiral centre (R -> S) or by shifting the branching point from a carbon atom n, counted from the ring system, to a carbon n — 1 (Figure 1 a and l b ) . Note that the latter case can be obtained by means of an inversion operation through the point P located halfway between n and n— 1 (Figure 1).

In the model we propose the spontaneous twist in these non-sterol systems is due to an orientational interaction between the substitutiongroup (CH3 in Fig. 1) and the neighbouring molecules. The sub- stitution group tends to align with the neighbouring molecule. This can be simply rephrased by stating that a certain axis of the substitution group, which in general deviates slightly from the symmetry axis of this group, tends to align with the long axes of neighbouring molecules. This additional inter- action can be described by a Maier-Saupe model

V' =-J

JP

{e

-a

) (1)

where eo is a unit vector along the axis of the group, P i is the second Legendre polynomial, P

(z) = | z

— a n d J is a coupling constant depending on the distance r'

= | rQi — s do | from the substitution group on molecule o to the long axis ai of molecule i; roi is the distance between the

M CMj

101 CD

H H

chain ring system

H H

161 OO

CH3 H up

F i g . 1. Chiral m o l e c u l e s c o n s i s t i n g of t w o p a r t s : t h e r i n g s y s t e m ( b e n z e n e r i n g s o r c y c l o h e x a n e r i n g s a n d l i n k a g e g r o u p s ) a n d t h e c h a i n (alkyl or a l k o x y c h a i n ) . T h e a s y m - m e t r i c c e n t r e is m a r k e d w i t h a n a s t e r i s k . T h e d i f f e r e n c e b e t w e e n t h e n u m b e r of b o n d s b e t w e e n t h e c e n t r e a n d t h e ring s y s t e m in (a) a n d (b) e q u a l s o n e . B o t h t h e u p p e r s i d e a n d d i r e c t i o n a r e i n d i c a t e d in o r d e r t o f a c i l i t a t e t h e d e f i n i t i o n of t h e a b s o l u t e c o n f i g u r a t i o n .

0340-4811 / 79 / 1100-1371 $ 01.00/0. — Please order a reprint rather t h a n making your o w n c o p y .

1360 Notizen F i g . 2. S c h e m a t i c r e p r e s e n t a t i o n of t w o i n t e r a c t i n g molecules. T h e s u b s t i t u t i o n g r o u p is d r a w n on o n l y o n e of t h e m o l e c u l e s .

centres of mass of the molecules; s denotes the distance between the substitution group and the long axis ao of molecule o: do is a unit vector point- ing from the long axis of o towards the group, do • «o = 0. This situation is schematically repre- sented in Fig. 2 (see also Figure 3). The summation in (1) runs over the nearest neighbours. Whether this interaction is due to steric or to dispersion forces is not relevant for the following argument.

Assuming t h a t the distance from the substitution group to the long axis is smaller t h a n the inter- molecular distance, we can approximate the inter- action (1) by

V _{= - J} (J _- sd

• not dJ/droi) P

(eo • «;) (2)

i

where «o» is a unit vector from o towards i. The unit vectors do and eo can be written as

do = dox fro + doy co ,

eo = eo

«o + eox fro + ßoy co , (3) where bo can Co denote unit vectors along the short

axes of molecule o with bo A co = ao. Next we assume t h a t the molecules rotate more or less uncorrected.

This means t h a t we are allowed to average over all orientations of 6o and co such t h a t

« 0 • fro =

bo • Co

Substitution of (3) into (2) and next making use of the following two relations valid for all vectors n and m , which are not affected by the rotation of molecule o (e.g. at and uot) [4]:

(1) (bo-n)(bo-m)

= (co • n) (co ~ m)

(4a)

= | (n • m) — i (» • ao) (m • ao),

(2) (b

•

(c

•

= | re

, (4b)

n-1

lb) — i o o \ 1

an X

Fig. 3. Schematic representation of a molecule with a chiral centre a t n (a) and at n — 1 (b).

where the average has been taken over bo and co, yields the following twist-producing interaction:

§ 2 * d J / d r0f [ ( o0 • c0) ( a0 A e0 • <f0)J i

• [(a

•

(a

u

)] , (5)

where we used the relation

EOZ

doy

eoy do

d

• ao = a

d

. (6) According to the continuum theory [5] the helical wavenumber q — 2 jr/pitch is given by

q = k

lk

2 , (") where k

is the elastic constant for twist and k

is

the elastic constant for spontaneous twist; k

is always positive, whereas k

can be positive or negative; ^ > 0 (&2>0) corresponds with a right- handed and <7<0 (Ä:2<0) corresponds with a left- handed helix. As follows from the molecular theory of the cholesteric phase [4, 6], the interaction (5) determines k

and therefore the sign of the helix.

The following proportionality relation holds

q ~ (a

- e

) ( a

A e

- d

) . (8) The odd-even effect in the twisting power becomes

obvious immediately. As explained above, a shift of the branching point from a carbon atom n to a neighbouring atom n— 1 (Fig. l a and l b ) , while keeping the same absolute configuration of the chiral centre, is equivalent to an interchange of the groups attached to carbon atom n and n — 1 by means of an inversion through the point P located halfway between n and n — 1. Under this inversion operation the vectors eo and do transform accord- ing to

d

-> — do , e

— e

. (9) Then we see from (8) that the helix changes sign.

Consequently we describe the odd-even effect. The transformation (9) is illustrated in Figure 3a and b).

This figure may also be helpful to visualize the odd-even effect. The orientational perturbation of the substitution group with axis eo is the strongest on the side where the group sticks out from the axis.

In Fig. 3a, therefore, the group induces a right- handed helix, whereas it gives rise to a lefthanded helix in the situation of Figure 3 b.

If the number of bonds between the chiral centre and the ring system increases, the factor

(a

• e

) (ao

e

• d

)

Notizen 1361

will decrease on the average due to an increase of the flexibility of that part of the alkyl chain. As a consequence the pitch will become larger with an

increasing number of bonds between the branching point and the ring system. This is observed experi- mentally [2].

[1] G . W . G r a y a n d D . G. M c D o n n e l l , Mol. C r y s t . L i q . C r y s t . (letters) 34, 211 (1977).

[2] G. H e p p k e a n d F . O e s t r e i c h e r , Z. N a t u r f o r s c h . 3 2 a , 899 (1977).

[3] R . A . C a h n , C. K . I n g o l d , a n d V. P r e l o g , E x p e r i e n t i a 12, 81 (1956).

[4] B . W . v a n d e r M e e r , G. V e r t o g e n , A. J . D e k k e r , a n d J . G. J . Y p m a , J . C h e m . P h y s . 65, 3 9 3 5 (1976).

[5] See e . g . F . C. F r a n k , Disc. F a r a d a y Soc. 25, 19 (1958);

J . N e h r i n g a n d A . S a u p e , J . C h e m . P h y s . 54, 337 (1971).

[6] B . W . v a n d e r M e e r a n d G. V e r t o g e n , P h y s . L e t t . A 71, 4 8 6 (1979).