On the interpretation of NMR data of non-rigid molecules in nematic phases: a serve test of the single conformation model

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On the interpretation of NMR data of non-rigid molecules in nematic phases: a serve test of the single

conformation model

D. Galland, F. Volino

To cite this version:

D. Galland, F. Volino. On the interpretation of NMR data of non-rigid molecules in nematic phases:

a serve test of the single conformation model. Journal de Physique II, EDP Sciences, 1991, 1 (2),

pp.209-223. �10.1051/jp2:1991156�. �jpa-00247508�

J

Phys

II1

(1991)

209-223 FtVRIER 1991, _PAGE 209

Classification

Physics

Abstracts

61 16N 61.30

On the interpretation of NMR data of non-rigid molecules in nematic phases

_{: a severe}

test of the single conformation model

D. Galland and F. Volino

(*)

Dkpartement

de Recherche Fondamentaie,

SPH/PCM,

C-E-N-G-, BP 85X, 38041 Grenoble Cedex, France

(Received 01 August1990, revved 29 October 1990,

accepted

31

October1990)

Abstract. The

large

sets of

dipolar

and

quadrupolar

NMR data of Celebre et al. [10, 11] for four

non-rigid

molecules baled _on the

ethoxybenzene

_moiety dissolved _{in a} nematic solvent _are

analyzed

in terms of _a model _in which the molecules _are assumed to exist

essentially

_{in one}

conformation It _is found

that

_{this model,}

in which the number of

adjustable

parameters is less than the number of

independent

data, _can describe very well the whole set of data for each of these molecules. The conformational parameters and the five elements of the order tensor are

unequivocally

deduced, with

high

_accuracy, for three molecules, and

only

estimated for the fourth

one The conformations and orders _are found _to be similar for the four molecules, and the relative values of the two order parameters in agreement ^{with the}

predictions

of _mean field theories.

These results _are contrasted with those of [10, 11] _in which many different conformations _are assumed to exist The

assumptions

_conceming the

possible

conformations and the _use of the ELS model [23] made _in this

approach

are criticized, and _{it is}

argued

^that our

(simpler) single

conformation model

probably provides

_{a more} realistic

descnption

of _a

liquid

at the molecular scale

Inboducdon.

The

descnption

of the Structure and

dynamic

behaviour of

intrinsically non-rigid

molecules _in

liquid phases

_{is an} involvbd and still controversial

problem

TWO

techniques

Which _are suitable

in this field _are the

scattering

of

X-ray

or neutron beams

[I]

and nuclear

magnetic

_resonance

(NMR)

in

liquid crystalline

media

[1, 2]

The scattering data contain

essentially

structural

information and detailed studies

[3]

suggest ^{that the} conformation and local

arrangement

in a

molecular

liquid

_{are very} similar to those

existing

in the solid

phase

at lower

temperature.

Moreover,

for mesogenic

compounds,

the local molecular

packing

_is found to be

«quite

similar _in nematic and isotropic

phases [4],

and for the

particular

case of

para-azoxyanisole

~PAA),

there _is «considerable short range

ordenng

of the

long

molecular _{axes in} both

phases

_»

[5].

These results suggest

that, despite

their

non-rigidity,

^the molecules _{in a}

liquid phase,

whether nematic _or

isotropic,

appear as rather hard

objects, existing

in

relatively

well

(*)

Member of C N R S

210 JOURNAL DE

PHYSIQUE

II lV° 2

defined conformations.

Clearly,

due to the fluid nature of _a

liquid,

fluctuations about _{a mean} conformation

certainly

_occur, but

they

_are

expected

to be of

relatively

small

amplitude

The

only large amplitude

intramolecular motions that _are allowed _are those which leave the molecular

shape essentially unchanged, namely symmetry operations

such _as

w-flip

of

phenyl

nngs,

methyl

_group rotation and

exchange

between dextro and laevo conformations. A model with these features will be called

single

conformation model »

Nuclear

magnetic

resonance

(NMR)

of molecules dissolved in nematic media _may be used to test these ideas

(that

_may be labelled _{as a «} hard point of

view).

The NMR method allows the

measuremqnt

^of _average

magnetic

interactions associated lklth _spins located at

specific

sites on the mblecules The

larger

the number of measured interactions, the _{more severe} will be the test. In _{a senes} of papers, the case of nematic PAA _was examined in detail

[6-9],

and it was shown that the whole set of existing proton ^and deutenum NMR

data

was consistent with the existence of _a

single conformation,

_{in agreement} with this _« hard point ^of view. In that

study,

thirteen interactions _were available for _a molecule

composed

of five

ngid

nioieties

linked

by

four

single

covalent bonds ^"

The _aim of this paper is to

perform

_a much _{more severe} test of this _view than with PAR.

For this purpose, we have chosen four molecules based _on the

ethoxybenzene'(EB)

_moiety.

This moiety ^is

composed

of

only

three

ngtd

units linked

by

two

single

covalent bonds

(Fig, I).

4 j

~

l0 YZ

,

~ ~_~ '

~ ~

i i

m

la) ₁₆₎

Fig. 1. Sketch of ethoxybenzene (EB) moiety

showing

the _vanous bond distances

(a)

and angles (b)

qied,

_in the

study,

_as well _as the ring frame Oxyz

for

molecules

iBd3,

_{4-fluoro EBd3 and 4-chloro}

kbd), _{spin_3}

is a proton, a fluonne and _a chlonne

reijectively,

_{and spins 8, 9,} 10

ark diuteronK Fir

molecule EBd10, all spins are deuterons

These molecules _are EBd3

(deuterated

_on the

methyl group), EBdIQ @erdeuierated _EB),~4-

fluoro EBd3 and 4-chloro

EBd3,

the two latter molecules

being

also deuterated _on the

riJdthyl grdup.

For all

tiese'molecules,

_{Celebre et al.}

_{[10, Ill}

measured _a

larje

_number of

magnetic

interactions, namely

fourteen

dipolar

interactions _in EBd3 and _in 4-fluoro EBd3

[11],

ten

dipolir

interactions in 4-chloro EBd3

[10]

^{and five}

quadrupolar interactions'in

_EBd10

[11],

when these molecules _are

dissolved,

at

300K,

in nematic solvent 152 These data _are

reproduced

in tables I to IV

lV° 2 ON THE INTERPRETATION OF NMR DATA 211

Table I.

-Dipolar

_D~~

couplings ofethoxybenzene-d3

dissolved _m 152 fit 300 K. The observed values _are taken

from ill].

The calculated values _are those

for

the

single conformation

model

discussed _in the _text.

D~~/Hz

i, j observed calculated

1, ^{2 2} 806.52 2 806 5

1; 3 358 78 359.5

1, 4 _1- 6 56 6 6

(1,

5

)

163.38 163 4

2,

3 ^' 69 35 69 4

1, 6 1202 17 1202.3

1, 8, 45 27 44 6

2,

6 241 77 242 6

2,

8 17 98 18.7

3,

6 168.48 166 5

3,

8 14,16 15,1

6,

7 4 020. 81 4 020.8

6,

8 137 85 137.8

Table II

-Dipolar couplings,

_D~~,

of4-fluoro ethoxybenzene-d3

dissolved _in 152 at 300 K The observed values are taken

from [11]

^{The calculated values} are those

for

the

single conformation

model discussed _m the text

D,~/Hz

i,

j

^{observed calculated}

1, ^{2 3 104.78 3 104 8}

1, 3 342.05 342.8

1, ^{4 10.23 10,I}

(1,

5

)

226.96 226 9

2,

3 -158 02 -1581

1, 6 334 82 334.8

1,

8 50.38 49 5

2,

6 268.99 270 5

2,

8 19.95 20 6

3,

6 -163 89 161.5

3,

8 13.91 14.5

6,

7 4 061 48 4 061.5

6,

8 lsl 61 lsl.6

JOURNAL DE PHYSIQUE II -T I, M 2, FiVRIER I991

212 JOURNAL DE

PHYSIQUE

II lV° 2

Table III

-Dipolar couplings,

_D~~,

of4-chloro ethoxybenzene-d3

dissolved _m 152 _at 300 K.

The observed values _are taken

from [10]

The

calculated

values _are those

for

the

single conformation

model discussed _in the _text.

D,~/Hz

I, _j observed calculated

1, 2 3 699 83 3 699.8

1, ^{4 27 21} 27, I

(1,

5

)

293 09 293, I

1, 6 574.06 574. I

1,

8 58 08 57.6

2,

6 313.78 313.2

2,

8 23.25 24

6,

7 4 438 25 4 438 2

6,

8 176,14 176,1

Table IV

Quadrupolar splittings ofperdeuterated ethoxybenzene (EBd10)

_m 152 _at 300 K.

The observed values _are taken

from [11]

The calculated values _are those

for

the

single conformation

model discussed _in the _text

Au,

, observed calculated

(Av1)

4 967 4 967

Av~

83 860 83 860

Av~

31 287 31 288

Av~

17 518 17 519

With fourteen interactions for

only

two

single

covalent

bonds,

the number of data _{is now}

larger

than the number of relevant parameters

(it

_was

equal

for PAR

[6-9]),

and the situation

is thus favorable for

such_a

test. We find that all these data can be _very well described

by

the

single

conformation model The conformations _are found to be

nearly

the _same for all

molecules,

the _main difference between the four molecules

lying

in the nature and

magnitude

of the onentational order.

Moreover,

_a distortion

predicted by

_quantum mechanical

calculations _is found to be necessary ^to obtain very

good

_agreement between

theory

and experiment ^In section

2,

_we describe the

ethoxybenzene

_moiety and discuss the values of the

geometncal

_{parameters as} well _as the _various internal motions The NMR data of

[lo,

I

I]

is

presented

and discussed _in section 3 The method of

analysis

_is

presented

in detail _in section 4 and the results for the four molecules _are described _in section 5 These results _as well _as the results obtained _in

[10, 11]

from _a

completely

different

approach

to the _same data _are

discussed and

compared

_in section 6

N 2 ON THE INTERPRETATION OF NMR DATA 213

2. The

ethoxybenzene moiety.

Figure la,

b shows _a sketch of the

ethoxybenzene

_moiety and define the

labelling

of the various spins from _{one to} ten. Interatomic distances and valence

angles

that have been assumed _in this

study

_are indicated

They

have been chosen

according

to the

literature, namely

quantum

chemistry

calculations

[10, 12], crystallographic

data of solid

phases

of

mesogenlc

dialkoxyazoxybenzene (nOAB)

molecules

[13-16]

Alteration of their values within reasonable ranges does not

change

the conclusions of the

study

_so this

point

will not be

discussed further The three

geometncal

_parameters that _are left to

fully

charactenze the conformation _are the

angles

_{pi, q~~} and

80

The

angles

_pi and _{q~~ are} torsional

angles

which characterize the conformation of the molecule and its

chirahty

The

angle 80 corresponds

to a

(small)

distortion _in the _ring

plane

which _is

predicted theoretically

for conformations of the

alkoxybenzene moiety

such that _{q~i is} smaller than 70°

[10, 12].

^Such distortion _is observed in the solid

phases

of nOAB molecules

[13-16]

where it _is found that pi is

always

small

(ring

and ether group

nearly coplanar).

The internal motions that _are assumed _in the model _are the

following

:

(i) ar-flJp

of the

phenyl

_ring which

exchange

_spins 1, 2 with

spins 5,

4.

(ii)

Rotational

jumps

of the

methyl

_group around its three fold axis which

exchange

_spins

8,

9 and 10 The

equihbnum position

of this group is fixed to that of the _mlnlmum internal

energy. There _is indeed _no known

examples

of solid

phases

of molecules

containing alkyl

chains where this is not also the case

[17]

(iii) Exchange

between dextro and laevo conformations

(called dynamical

racemlzation _in

[9])

^which is described

by

the simultaneous

change

in the _signs of

angles

_pi and

q~~ Thls

large amplitude

rotational

jump exchanges

_spins 6 and 7 The

exchange

between spins 9 and 10 _is

already

taken into account

by

the

methyl

_group rotation.

3. The NMR data of Celebre et al.

[10, iii.

For

EBd3,4-fluoro

EBd3 and 4-chloro

EBd3,

the set of fourteen

dipolar

interactions D~~

(Tabs. I,

II and

III)

_is

conveniently split

into three subsets

(i) Di~, Di~, Dj~, D15

₌

(D~~

+

D15)

and D~~ ^which concern the

phenyl

_ring

alone, (ii) _D~~

and D~~ which

2

concern the ethane unit alone and

(iii) Dj~, Dj~, D~~, D~~,

D~~, D~~ which _are interactions

between the _nng and the ethane unit. Note that _in subset

(i),

_we have

symmetnzed

the

problem by assuming

that

D~~

₌

Dj~

The difference between these two interactions _is very

weak,

and this

simplification

_is

equivalent

to _assume that the nng is

symmetric

with respect to Ox The deviation from this _{situation is} not

important

for _our purpose, and _we shall

systematically

_assume this symmetry, even for the

quadrupolar

data of EBd10 where _we have assunJed

Au1

=

(Au

j +

Av~)

In this latter _case, the measured difference _is indeed

easily

2

explained,

_as in PAA

[8], by slightly

different

inclinations,

_on the para-axis, of the

pnncipal

axis of the e-f g, ^tensor

(assumed

of

cylindrical symmetry)

_{acting on} the _various

phenyl

deuterons

Thus,

for the four

molecules,

there _is in fact _one datum less than mentioned

above, namely

thirteen for EBd3 and 4-fluoro

EBd3,

_nine for 4-chloro EBd3 and four for

EBd10. Note that for 4-chloro

EBd3,

the number of

dipolar

interactions _is reduced _since those

invo1vlng

_spin 3 do not exist The fact that the number of interactions _is

relatively small, namely

fourteen

dipolar

interactions for ten spins

(as

in EBd3 _or 4-fluoro

EBd3)

instead of

fourty five,

_is

easily explained by

the internal motions that render _some spins

magnetically

equivalent,

_as

explained

above.

214 JOURNAL DE

PHYSIQUE

II M 2

Th~s

property

^{allows the}

problem

of the

magnetic

interactions m these molecules to be treated w~th~n the framework of _a

completely ng~d

molecular

geometry,

^{to which} -a Cartesian frame _can be

attached,

and inside which the three kinds of

exchanges

mentioned _occur. The

moleculhr frame

Oxyz

that _we chose is the natural frame of the

phenyl

_ring, With Oz

along

the

para-ixis

and Ox _m the

plane

of the ring

(Fig

1)

4.

Interpretation

_{of the}

_magnetic

interactions.

4 BAsic EQUATIONS AND METHOD The

problem

for each molecule reduces _to

solving

_a

system ^of equations ^{which describe relations between the}

geometncal

_parameters of the molecule and the

dynamical

_parameters of its extemal motion,

namely

the five elements of ifs order matrix T. It _is conven~ent to define reduced

dipolar

interactions

D~)

^{and reduced}

quadrupolar splittings _Av~* by

the relations _:

hy,

_y~

D,~

=

D~)

4

«~

and

~ 3

e~ Qq

Vi~j~4Vi

~

where y~, y~ ^{are the} gyromagnetic ratios of _{spins i} and _j

(subscnpts

P, D _or F will

eventually

bj

^{used when}

^refernng

^more

specifically

to

proton,

^deuterium or fluorine

spins)

and

~

f~

_is the

quadrupolar courting

_{constant of spin I The numerical}

val~es if

interest m our

problem

_{are :}

hy(/4 «~

=

12012, hyp y~/4 ^«~ =18.439, hyp y~/4 gi~

=

l13

00, hy

~

y~/4 ^«~

= 17

.346,

, ~ ~

all

expressed

_m

kHz.l~

_{units, and} ^~

^~~

= 170 kHz _or 185 kHz for

aliphat~c

deuterons and h

aromatic

deuterons, respectively.

With these conventions, and _m the absence of

exihange,

the

equations

for the

dipolar

interactions can be written

[2, 18].

~

l~$

"

j _{(~2(lJ ))}

"

~j

~k

Tkf

_Uf

(I)

y y k, f _{x, y,z}

where

r,~ ^is the distance

between

spins i and _j and _{u an unit vector}

along

the _y direction The

equations for

the reduced

quadrupolar

interactions _are the _same, except ^{for the absence of} the factor

I/r(

and _u

along

the

pnncipal

_axis of the

e-f-g

tensor

acting

_{on spin i}

Equation'(I)

_may be rewritten _as

D,)

=

[P~(cos B)

_T~~ +

sm~

_B

cos 2 A

(T~~ T~~)

+

r 2 ~ .<

<J ₁

,

+ sin 2 B _sin A T~~ ^{+ sin 2 B cos A}

T~~.+ ^sm~

B sin 2 A

T~~] (2)

where B and A _are the

polar

and azimuthal

angles

of _{u m} the molecular frame

Oxyz

Wntten

m this way, this expression is suitable for

calculating

_averages,

symbolized~ by brackets,

_over

M 2 ON THE INTERPRETATION OF NMR DATA 215

all spin pairs that _are

magnetically equ~valent

because of the internal motions The thirteen

equations

^{take the form}

~ ^~

lP2

(C°S

B

~ l

sm~

B _cos 2 A

~ ~ sin 2 B _sin A

~

'J 3 ^zz +

j

₃ xx yy) ⁺

3 Yz +

~y ~y _~<j

sin 2 B _cos A

sin~

_{B sin 2 A}

+ T~~ + T~~

(3)

r( _r,(

and _can be rewritten _more

simply

_{as :}

Dil

₌

Ci

_{Tzz +}

C2 (Txx _Tyy)

+

C3

_{Tyz +}

C4

Tzx +

Cs

_Txy

(4)

For

computational

_{reasons, we} also introduce for the

analysis

of

dipolar

data the

following equation

(P~(M))

=

P~(cos BM)

_{T~~ +}

sin~ _BM

cos 2

AM

_T~~

_T~~)

₊ sin 2

BM

_sin

AM

_{T~~ +}

+ sin 2

BM

cos

AM

_T~~ +

sin~ _BM

sin 2

AM

_T~~

(5)

where

(P~(M)

represents ^the « order parameter » of the three fold _axis of the

methyl grout,

and

BM

^and

AM

are the

polar

and azimuthal

angles

of this _{axis m} the

Oxyz

frame

Assuming

that the behaviour of all these molecules _m the nematic solvent is close to that of

perdeuterated EBd10,

this parameter

(P~(M))

_may be

roughly

estimated from the

quadrupolar splitting Av~

^{measured with} EBd10

(Tab IV).

We have

Apt (P~(M)) (6)

The factor

1/3

_comes- from the averag~ng effect due to

methyl

_group rotation, assuming a

tetraedral

geometry

^{ThJs value of}

Apt

_is used _{as an} initial value _m the

fitting procedure

descnbed below As defined

by

these equations, ^which

separate clearly

the

purely

geometncal

_{parameters C~} from the

purely dynamical parameters T~i,

the

interpretation

^of the data that _we propose here

belongs

to the

general

framework that _was introduced

previously

for PAA

[8]

But at vanance with the

problem

of PAA where the available

expenmental

data _were insufficient _to separate

completely

the

geometrical

parameters ^from the

dynamical

parameters,

here,

the

high qual~ty

and abundance of the data allow a

complete

evaluation of both kinds of

parameters,

^{at least w~thm the framework of th~s}

model,

for molecules

EBd3,

4-fluoro EBd3 and 4-chloro EBd3

Th~s evaluation has been made _m three steps ^We use the fact that all the spins are located qn ^two

ng~d

units

only, namely

the

phenyl

_nng and the ethane unit, whose

geometries

are

assumed to be

perfectly

known. The

only geometrical

feature wh~ch _is not known _is the relative onentation of these two units The

geometncal parameters

^{that will be determ~ned} from the

analysis

are in fact of two

kinds, (i)

the nng distortion parameter ⁸

(see below)

and the bond distance d~ ^{that will be}

optim~zed

_m step

I,

and

(ii)

the

angles

_{oo> Pi} and q~~ that w~ll be

optimized

_{m step}

3, together

with the

dynamical parameter (P2(M)

We _now

detail the three step

procedure.

4 2 STEP I ANALYSIS OF DIPOLAR DATA RELATIVE To THE PHENYL RING Th~s

analysis

_is

rather

straightforward

It allows the determ~nation of the _two

diagonal

elements of matnx T

m the

Oxyz

frame and of the distortion parameter 8 defined

below,

and the

optim~zation

of the bond

length

d~ ^shown m

figure

la The five

equations corresponding

to interactions

216 JOURNAL DE

PHYSIQUE

II M 2

Dj~, Dj~, Dj~, Djs

and D~~ are all characterized

by

the fact that the distances

r,~ ^are constant, that _sin A and _sin 2A _{are zero} and that _cos A _is

averaged

to _zero

by

the

«-llJp.

The five

equations §imphfy,

_m

particular

the

off-diagonal

elements of T

disappear

The symmetry for the ring

implies

that

rjs = r~~ = rj~

/(1

+ 8

(7)

where 8 pictures a small deviation of the

ring

_geometry from the

perfect hexagonal

_symmetry

(contraction

_or dilation

along

the Ox

ax~s).

In the

appendix,

it is shown that for 8

«1,

the three equations ^for

Dj~, Djs

and

Dj~

_can be'written _as

("lD

is =

j j _{iTzz (Txx- Tyy)1 (9)}

YP r12

~

"/Dj~

= ~

l ₊ ~

8 T~~ 3

(1

~

8

(T~~- ~~)j

(10)

hyp

_{64 rj~} ^{4 4}

They

_are used to determine T~~,

T~~-

_T~~ and 8 The two last equations ^for

Dj~

and

D~~ ^which are of the form

~ _" ^~

D,~

=

(P2(cos ^B)

^{T~~ +}

^sm~

^{B cos 2 A}

^{(T~~ ~~)j (I I)}

Yi

Yj

_r,~ ²

are used to optim~ze ^the bond

length

_d~ in EBd3 and _m 4-fluoro EBd3.

4.3 STEP 2 _: ANALYSIS OF DIPOLAR DATA RELATIVE To THE ETHANE UNIT For any ^set of

values- of the four parameters

oo,

_{pi, q~~} and

(P~(M)),

it _is easy ^to fit the two internal interactions

D67

^and

D68

^and to

simultaneously satisfy

the equation ^for

(P~(M)). Indeed,

for each of these

three

equations, the

geometrical

coefficients of the type

Cj

to

Cs

can be

calculated,

and _since the

diagonal

elements of matnx Tare known from step

I,

these three equations constitute a linear systems where the three vanables _are the three

off-diagonal

elements T~~, T~~ and

T~~. We have.

'~~ D67 Ci

_Tzz

C2(Txx _Tyy)

=

c~

_{T~~ +}

c~

_{T~~ +}

cs

_T~~

(12)

~

~( ^{'~(~ D68}

^Di Tz=

D2 (Txx _Tyy)

₌

D~ T~~j D~ T~~+ Ds

_T~~

(13) (P2(M) Fj

_T~~

F2(T~~ _T~~)

=

F~

_T~~ +

F~

_{T~~ +}

Fs _T~~. (14)

Solving

this system allows the

4eterm~nation

_{of the three}

off-diagonal

elements of T

4 4 STEP 3 _{ANALYSIS OF DIPOLAR DATA CONNECTING PHENYL RING AND ETHANE FRAG-}

MENT. The equations that _remain to be solved _are those

corresponding

to interactions

Dj~, D~~, D~~, Dj~, D~~, D~~, namely

_six equations ^for EBd3 and 4-fluoro EBd3 but

only

four

equations

for 4-chlord

EBd3,

for four

adjustable

parameters,

namely _oo,

_{pi, q~~} and

(P~(M)).

For the three

molecules,

_{a mean square} fit

procedure involving

these _six

equations

as well _as

equations (12)-(14)

has allowed to find excellent solutions.

4.5 THE CASE OF PERDEUTERATED EB

(EBd10)

For

EBd10, only

four relevant

M 2 ON THE INTERPRETATION OF NMR DATA 217

quadrupolar

interactions _are

available,

_so that

assumptions

must be made to find _a

possible

solution Th~s molecule

being practically

the _{same as}

EBd3,

_{we can}

reasonably

_assume that

they

have the _same

geometry.

An additional

assumption

concerning the order _{is necessary}

since there _are

only

^{four data} to determ~ne the five _matrix elements of T. We have somewhat

arbitranly

assumed that the axial component of the T

pictured by

_So

(see below)

_is the _{same as} that of EBd3. The assignments and

signs

for the

quadrupolar

interactions _are the _{same as} those of

[11] Again

_{a quasi}

perfect,

but _now not

unique,

solution _is found

4 6 DIAGONALISATION _{OF TENSOR} T This calculation _is used to determJne the

principal

frame OXYZ of the order tensor and its two

eigenvalues,

the order parameters

Szz

and

Sxx Syy

The

following

conventions and notations w~ll be

adopted

_:

(i)

the X _axis is chosen such that

Sxx Syy

_is

positive

(ii)

the two

principal

values

Szz

and

Sxx Syy

can be

replaced,

without any loss of

generality, by

two the quantities So ^and x of _more direct

physical significance

defined

by.

~ZZ

" ~0

~2(CDS

_X

(15a)

Sxx Syy

=

So[I P~(cos X)1 (lsb)

Inversion of these formulae

yields

So

₌ 2

Syy (16a)

P2(cosx)

=

/ _(16b)

where l~s=

(Sxx Syy)/6jz

_is the amsotropy

parameter

^{of the} onentational order.

The

quantity

_{So represents} the axial component ^{of the order} tensor and the

angle

_X allows to

easily

visualize the

biaxiality. Indeed,

_{x can} be considered _as representing ^the

angular

fluctuation that would

perform

the molecule around its

pnncipal

axis O Y if th~s fluctuation _was described inside _a fictitious frame whose umaxial motion _is described

by

the

single

order parameter

6i

S. Results.

The calculated magnetic interactions using the best fitted values of the parameters are listed _m tables I _to IV for the four molecules. It _is observed that very

good _agreement

_is obtained _m all

cases Table V collects the

geometncal parameters oo,

_{pi, q~~,} 8 and

d~,

^{and table VI collects}

the

dynam~cal

_parameters Table VIa g~ves the five elements of the order matrix m the frame

Table V.- Geometrical parameters

of

the

four ethoxybenzene

derivatives molecules _con- sidered

m this work deduced

from

the

analysis of

the NMR data

of [10,

11] m terms

of

the

single conformation

model discussed in the _text Parameters

oo,

_{pi, q~ ~} ^and d~ are

pictured

_m

figure

I and

parameter

_{8 is}

defined

_m the text and _m the

appendix

Molecule 0

o'(°) _i (°) ₂ (°) ⁸ ~3

(/~)

EBd3 7 125 22.75 39.18 0 0036 _± 0 0004 1.096

4-fluoro EBd3 6 339 23 08 38.09 0 0067 _± 0 0004 288

4-chloro EBd3 6 813 21 54 37.09 _S 10-~

EBd10 7 125 22.75 39 18

218 JOURNAL DE

PHYSIQQE

-II M 2

Table VIa

Components,

_in the

Oxyz frame defined

_m

figure I, of

the order _tensor

of

the

four ethoxybenzene

derivatives molecules considered m'this

work,

deduced

from

the

analysis of

the NMR data

of [10, 11]

in terms

of

the.

single conformation

model discussed _in the text

Molecule

7~~~

(T~~ T~~)

T~= T~~ T~~

EBd3 0 35208 0.13677 0.12038

'0 i12447

_{0 057862}

4-fluoro EBd3

0.3895Q

0.099507 0 12971 0.0053374 0 056514

4-ihloro

_{EBd3 0.46415 0.082040 0 14408 0.033727 0 016288}

fBdl0

_not

exphcitely

calculated

Table VIb. Same

information

_{as m} table VIa,

expressed

_m

ternis _{of five} parameters of

_more

physical

and

geometrical sign#icance,

as

defined

_{in sections}

4,

5 6~ ^{is the axial} component

of

the order _{tensor, x} pictures ^{the biaxial disorder and the}

U,

are three Euler

angles (see

text

for

details

).

Molecule So _X (°)

Ui

_(°)

~2 f~ ~

~~~

EBd3 0 5690 28.35 30.92 11.55 14.03

4-fluoro EBd3 0 5803 25.80 28.53 11.37 8.206

4-chlorq

EBd3 0.6021 20. 31 20.61 1.13 15.85

E§d10

_{0.5690 33,10 38.55 9.011 13 15}

Table VIC.

-Prmcijal

values

of

the order _tensors. _

Molecule

Sxx

_SYY

_{~ZZ ~~~~ ~~~~}

EBd3 0.0921 0 2844 0.3765 0 1923

4-fluoro EBd3 0.1253 0 2901 0 4154 0.1648

4-chloro EBd3

0,1923'

_{0.3010 0 4933 0 1087}

EBd10 0.0300 0 2844 0 3144 0 2544

of the

phenyl

_ring whereas table VIb transforms this information into

physical

and

geometrical

quantities,

namely

the two parameters So ^and x defined

by

equations

(15)

and

(16)

and three

angles

which describe the _onentation of the nng frame

Oxyz

m the

pnncipal

frame OXYZ Instead of using the three conventional

[19]

Eulenan

angles Vi, V~

^and

V~, we

prefer

to give the three

angles Uj,

U~ ^and U~ related to

Vi,

V~ ^and V~

by

the relations

Uj

=

Vi

+

( ₍₁₇₎

'

U~ ₌

V~ (18)

U~ =

V~ f (19)

M 2 ON THE INTERPRETATION OF NMR DATA 219

which _are also Eulenan

angles,

but wh~ch

imply

_successive rotationi -around the

Oz, Ox,

Oz

axes instead of the

Oz, Oy,

Oz _axes, which _is the, _more usual convention

[19] Finally,

table VIC fists the

principal

values of the order tensor

From these tables, it appears that both the conformations and orders of the four molecules

are sim~lar

Figure

2 shows

stereoscopic

mews of the

predicted

conformation of the EBd3 molecule

projected along

the OX

(Fig 2a)

and

along

the OY

(Fig 2b) pnncipal

_axes It _is observed that the conformation is

elongated along

OZ and flattened _m the XOZ

plane.

It _is not necessary to present the

corresponding

_mews for the three other molecules because the

differences _are small and cannot

really

be

fully appreciated

_on these kinds of

figures.

z

~-+

x y

-~

z

x y

Fig

2

Stereoscopic

project~ons

along

the

principal

axis OX

(upper) jnd alonj

_the

_pnncipal

axis O Y

(lower)

of the conformation of EBd3 molecule,

predicted by

the

single

conformation model described _in the text The

pnncipal

ax~s OZ _is vertical _m both _{cases, as} md~cated

6. Discussion.

This

study

has

shown,

that the

single

conforrnation model _is

sufficiently good

to

fully

describe the

large

set of available NMR data for the four

non-rigid

molecules considered. The

important

improvement

with

respect

to _our prev~ous

study

with PAR

[6-9]

is that for two

molecules,

the number of

adjustable

_{parameters is} less than the number of

independent

data.

In

addition,

the results _{appear to} be very reasonable _:

(i)

The conformations of these four similar

molecules,

m the _same

solvent,

at the _same temperature, are found to be

similar,

_as

expected by

_{common sense}

(ii)

These conformations _are

elongated (cf Fig 2)

and _are found to be intermediate between conformations m the solid state, ^where the ether and ring

fragments

he

nearly

_m the

same

plane [13-16],

and those of the isolated molecules for which

quantum

chemical calculations

performed

_on

methoxybenzene _[12] predict

that the most stable conformation _is

for the ether

plane perpendicular

to the ring

plane

The _non-zero value found for

oo

is consistent w~th the

finding

that pi is rather

small,

of the order of 20°

(cf.

Tab.

V)

The

same intermediate situation was found for PAA A detailed discussion of th~s aspect ^{of the}

problem

_is made _m [9]

(iii)

The

principal

_axes of the order tensors, which have been deduced

completely

from the

calculations,

_are located where

they

_are

expected

to be The

long

_axis OZ is

along

the

220 JOURNAL DE PHYSIQUE II M 2

direction of maximum

elongation,

the OX _is m the

plane

_» of molecule

(the plane

in which the onentational disorder _is

maximum)

and OY _is

perpendicular

to th~s

plane

These features

can be

easily

be

appreciated

m

figures 2a,

b

using

_a

stereoscope.

(iv)

The order parameters _are also _as

expected

The overall order _is

larger

for the molecules of

larger

^molecular mass The _increase of the axial component ^of the order

(So)

_is

accompanied by

_a decrease of the biaxial disorder

(x),

_{as can} be

appreciated

_m

tables-VIb,

_c The difference found between EBd3 and EBd10 _is not

significant, essentially

due to the fact that for the latter

molecule,

the solution _is not unique. Th~s trend concerning the variations of the two order

parameters

is consistent w~th the

predictions

of _mean field

theories

[20-22]

(v)

Last but not

least,

the bond distances d~ are found to be _m agreement ^w~th

expected

values and the distortion 8 of the ring

along

Ox _is very small in all _cases

(cf.

Tab. V and the

Appendix).

All these results

support

the model

proposed.

In the next

section,

_{we compare our}

approach

with that

performed by

Celebre _et al _m

[10, 11]

7.

Comparison

with the

approach

of Celebre _et RI.

[10, iii.

In references

[10, 11],

the

analysis

of the _same data _is

performed

_assuming that the molecules

can

adopt

not

only

_one, but _a

large

number of

(very)

different conformations

(a

« soft »

point

of

view).

In order to avoid the determination of _a

virtually

infinite number of

adjustable

parameters,

namely

the

probability

of _occurrence and the five elements of the order tensor of each

conformation,

assumptions ^{about the}

possible

conformations and the onentational order

are made For the

conformat~ons, it~is

assumed that

they

differ

only by

the value of the

angle

q~2 The

angle

_{pi can} take the values 0 _{or «,} which _means that the

ring

and the ether _{group are}

assumed to be

always coplanar,

_{as m} solid

phases [13, 16]

The

probability

of the

conformation _{q~~ on} the other

hand,

is described

by

_a

potential

V

(q~~)

^which is

approx~mated

by

the _sum of the _seven first _terms of _its Fourier

expansion

The _seven

corresponding

coefficients _are the parameters used to

fully

charactenze the

probabilities

of the confor- mations. For the

ordering,

the so-called ELS model

[23]

_is

adopted

Th~s model _{is a mean} field

theory

for _nematic

phases

of

non-ngid molecules,

_m which the _main parameters are

coefficients in the

expression

of the _mean field

potential

In the

particular problem

treated

here,

the number of

independent

such coefficients _is reduced to

only

three. All the other

geometncal

parameters, ^such as bond

angles

and bond

lengths,

_are fixed to standard

values,

as m our

analysis

The total number of

adjustable parameters

_is thus ten,

exactly

_{as m our}

case.

And,

_as in _{our case,}

good

fits _are obtained for all

molecules,

the

quality

of _our fits

being

however

significantly better,

_{as can} be inferred

by

_{comparing our} tables I to IV w~th the

corresponding

tables _m

[10, 11]

At this

point

^{of the}

discussion,

it thus _appears that both models _can describe the _same

large

number of NMR

data, although they

_are

conceptually

different A similar situation _was found and discussed

recently by

_us

[24]

_m connection with the

analysis

of NMR data of d~rr~er model

compounds

of _nematic

polymers Essentially

the _same mathematical _reasons

why

such _a situation is

possible,

and the _same

physical

_{reasons m} favor of _our model _can be

given

here The reader is referred to

[24]

for details.

There _is however a difference between the _two situations.

Here,

two new arguments

against

the

approach

of

[10, 11]

_can be g~ven The first argument concerns the assumption _on the conformations. The

potential V(q~~)

_is found to be close to that of the isolated

molecule,

w~th

potential heights

between trans and

gauche

states of the order of _a few

kcal/mole [I I]

But m

the isolated

molecule,

the

potential height hindering

the rotation pi is also of _a few

kcal/mole,

at least in

PAR,

and the _most stable conformation is obtained for the nng

plane perpendicular

N 2 ON THE INTERPRETATION OF NMR DATA 221

to the ether

plane [12] Thus,

the rotational disorders for pi and _{q~~ are}

reasonably expected

to be of

comparable magnitude

whereas for th~s

model, they

_are

completely

different. The

assumption

concemmg _pi is

clearly

_in contradiction with the result found for _q~~. The second

argument

is _more fundamental and _concerns the

validity

the ELS

theory [23].

In th~s

theory,

the basic

assumption

is that the _mean field

potential

not

only

_governs the nature and

magnitude

of the

(long range)

onentational

order,

but also the

probability

of _occurrence of the various conformations This latter assumption is, m our opinion, a serious flaw of the

theory,

because _{it is}

equ~valent

to

identifying

the _mean field

potential

to the true intermolecular

potential.

In

fact,

the _mean field

potential

_is not a true

potential,

but

only

_an

effective _or

pseudo potential.

It charactenzes the

phase

and the

cleanng

transition It allows the calculation of

macroscopic

_quantities such _as

(long range)

order

parameters,

^heats of

transition, but

certainly

not molecular quantities such _as conformational

parameteri

The

cpnformations

_are determ~ned

by

the balance between the intramolecular

potential (as

calculated

for_ example by quantum

mechanical methods for isolated

molecules)

and the _true intermolecular

potential.

The true intermolecular

potential

does not vanish _m the

isotropic phase (it eventually

decreases

slightly,

_{m average,} due to thermal

expansion),

while the _mean field

potential

does.

Putting

the argument the other way

round,

if the _mean field

potential

were the true intermolecular

potential,

this would _mean that _m the isotropic

phase,

<the conformations _are determ~ned

by

the intramolecular

potent~al only,

which is

equivalent'to identify

the isotropic

phase

to a

(infinitely diluted)

_gas

phase

This _is

clearly

not

reasonable,

and _m contradiction with the scattenng experiments, as stated _in the introduction

The

ability

of _a molecule _to take very different conformations _is related to the free volume The

density

of _a

liquid being only slightly

smaller than that of _a

solid,

the free volume _is not

much

larger.

In the

solid,

there _is

only

one conformation

(m fact,

two the dextro and laevo conformations cf, the detailed discussions _m

[9]

^and

[24],

and

[13-16]).

This may

explain why

the _one conformation model

presented here,

_m which the

only

conformational

change

allowed is the

exchange

between these two quasi identical

conformations,

works _so well

8. Conclusion.

In th~s

work,

we have

produced

additional ev~dence that the

single

conformation model

[25]

_is

sufficient to descnbe the whole _set of NMR data of four

non-ng~d

molecules _{m a} fluid nematic

phase

ThJs evidence _is much _more conv~ncing than

previously presented [6-9]

_since the number of

adjustable

parameters is now

significantly

less than the number of

independent data,

_a situation which is not easy to achieve _m practice with

complex

molecules All the

predictions

of this model concerning the

conformation,

the nature and

magnitude

of the order for the four molecules considered _are

reasonable,

_{g~ving more}

support

to the _« hard _» point ^of view _as defined _m the introduction. The _« soft _» point ^{of view} cannot be excluded _on the basis of these results

only

but it _is not

possible

to prove ^it because _it

necessarily

contains _a priori ^a

(much) larger

number of

adjustable

_parameters This number _can

probably

be reduced _using

adequate approximations

or models For the

analysis

made _m

[10, 11],

we have

argued

that there _{is some} contradiction between the

hypothesis

concerning the conformation and the

results,

and

that,

_{m any case,} the ELS

theory

cannot be used to describe the

physical

situation.

Despite

these

arguments

m favor of _our

model,

the results

presented

here must not be taken too

l~tterally.

It _is well established

by optical

methods

(IR absorption

and Raman

scattering ^of

light)

that small

amplitude

fluctuations

certainly

exist around the

single

bonds.

The results of _our

analysis

_m which such fluctuations _are

neglected simply

_means that with the kind of NMR data considered _m this

work,

it is

impossible

to discriminate the static from the

dynamical

disorder around the bonds This _was

already pointed

out _m our previous work _on

222 JOURNAL DE PHYSIQUE II M 2

PAR

[8]. Thus,

the conformations found using

the'present

model must be

thought of,as average (over

these; small

amplitude fluctuations) conformations,

_{as seen}

by

NMR.

Consequently,

the numencal values given m the _various tables should not be considered'w~th all the

significant figures given. They

have been _given

however,

to allow interested readers to

repeat ^the calculations- with the

required

_accuracy. ^' _.~

To _summanze, the picture ^that emerges from th~s model _is

that, despite

the

large

number of

degrees

of freedom that it possesses

(i) large amplitude

internal motions described-

by

the symmetry

operations (« flips

^of _nng;

methyl

_group

rotation, dynam~cal racemization), (ii)

small

amplitude

fluctuations around the

bonds,

and

-(iii)

individual and collective translational and rotational _motions

(the

latter

being

described

by

the order

tensor),

a

non-ng~d

molecule in _a

liquid phase (whether

nematic _or

isotropic)

_{appears as a}

reasonably

well defined

object

This _is due _to the fact that _a

hqu~d

_is

practically

_as dense _{as a} solid

(m

which the conformations

are

perfectly defined)~

w~th very little free volume. Th~s situation _is

completely

different to that of a gas

phase

where the free volume is

virtually infinite, allow~ng

the existence of all the

conformations of the isolated molecule

Appendix.'

Here,

we derive the

e#uations

_for

_{Dj~, Dj~}

_and

_Djs

m

thi

presence of _a

smill'distortion

₈

(contraction

of

dilation)

of the hromatic

hexagbn along

Ox

Let _{x, y,} and _z the components of vector

ij

and _r its

length (r~

=

x~

₊

y~

₊

z~)

These three equations are of the form

~~/ _l~y

"

~l

_{~zz +}

~2(~xx

_~j<y)

j

with

~ ~

~

^~

'

Cj

~

~~~s~

~~~

~~

2~~

Let dx

= x8 the deformation of the _x component ^and

dD,~

^the

corresponding

variation of the-interaction

Taking

the

logarithmic

denvative of these two

coefficients,

_we obtain _:

dcj

₃

_8x~

₅

_z~ r~

~~ ~

~~2

₈

(2 ^r~

5

x~)

f /

₃

_{z~ r~ ~2 r~}

from which _we obtain the expressions of<the

perturbed

coefficients _:

'

ci+dci =~~~j~~ (1-~~~~~~~(~~(j

_2r

r 3z-r

and

C~+dC~=

^~

~

l+

~(2r~-5x~)

2~

r

Noting

that the

components

^of vectors

12,

14 and 15 _are

proportional

to

0,0,1,

/, _0,

_and

/, _0,

₀

respectively,

the

equations (8)-(10)

of the text _are

easily

derived These equations allow to calculate 1~~, T~~ _T~~ ^{and 8. In}

particular,

the expression of 8 _is

4

Dj~

128

Dj~

+ 36

/

_D

~ is

27

Dj~

384 D

j~ + 63

/ _Djs