An algorithm for analysing NMR data of nematic liquid crystals applied to the aromatic core of para-azoxyanisole

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An algorithm for analysing NMR data of nematic liquid crystals applied to the aromatic core of

para-azoxyanisole

D. Galland, F. Volino

To cite this version:

D. Galland, F. Volino. An algorithm for analysing NMR data of nematic liquid crystals applied to the aromatic core of para-azoxyanisole. Journal de Physique, 1989, 50 (13), pp.1743-1757.

�10.1051/jphys:0198900500130174300�. �jpa-00211028�

An algorithm ^for analysing NMR data of nematic liquid crystals applied ^to the aromatic ^core of para-azoxyanisole

D. Galland and F. Volino (*)

Centre d’Etudes Nucléaires de Grenoble, DRF/SPh/PCM, 85X, 38041 Grenoble Cedex, ^France (Reçu le 12 décembre 1988, revisé le 16 mars 1989, accepté ^{le 17 mars} 1989)

Résumé.

On propose ^un algorithme pour analyser l’ensemble des données RMN, proton ^et deutérium, intéressant le c0153ur aromatique ^du para-azoxyanisole nématique (PAA). ^Cette analyse s’appuie ^{sur un} modèle dans lequel la non-rigidité ^de ce c0153ur aromatique ^{est décrite en}

termes de rotations internes par rapport ^{à une} conformation moyenne unique. ^{On montre} que ^cet ensemble de données n’est pas suffisant pour caractériser complètement ^cette conformation et les

paramètres ^d’ordre correspondants ; ^on peut ainsi estimer les incertitudes qui ^{affectent notre}

connaissance de la géométrie ^{et de la} dynamique ^{de ce} système ^{tant à} l’échelle des noyaux

aromatiques qu’à l’échelle du c0153ur mésogène. ^Il n’apparaît pas nécessaire de faire intervenir

plusieurs types ^de conformations ; par ^contre les effets de la biaxialité de l’ordre orientationnel moléculaire sont clairement mis en évidence.

Abstract. 2014 An algorithm ^is proposed ^to analyse ^{the whole set of} proton ^and deuterium NMR data associated with the aromatic core of nematic para-azoxyanisole (PAA) ^{in terms of a model in}

which the molecular non-rigidity is described as internal rotations within a single ^mean

conformation. It is shown that this set is not sufficient to completely determine this conformation and its order. Taking ^this feature into account, the actual uncertainties which affect ^our

knowledge of various conformation and motion parameters ^{at both} ring and molecular length

scales are estimated. There is no evidence of the existence of very different conformations in the nematic phase ^{and we} clearly prove that, ^{in all} cases, the molecular order matrix remains biaxial.

Classification

Physics ^Abstracts

61.16N - 61.30

1. Introduction.

At the molecular level, the orientational order of any rigid object embedded in a nematic is

usually ^described by ^the ordering matrix of any frame attached to this object [1]. This matrix is characterized by ^its principal ^frame Oxyz ^{and two} independent principal values, ^{or order} parameters SZZ ^and Sxx - Syy. Information about these quantities may be obtained by ^both macroscopic measurements and spectroscopic techniques. Among ^{the latter, NMR is} probably ^{the most suitable one} and it is widely ^{used to} characterize both the structure and orientational order of small molecules dissolved in nematic solvents [2].

For larger ^non rigid objects ^{such as} the molecules of neat nematic phases, ^the problem ^is

not fundamentaly different since a long molecule may be considered as a succession of rigid

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500130174300

fragments ; ^most of the NMR probes ^{are attached to one of these} fragments ^and supply ^some

information about the corresponding ^local ordering ^tensor. This is the case of the PAA molecule [3-6] ^{where two} distinguishable ^deuterium probes ^{are attached to} each of the two

rings of the molecule (cf. Fig. 1). ^{The new} difficulty ^{now is not related to the} flexibility

features of the molecule, ^{but to} the fact that phenyl rings ^no longer possess the perfect symmetry properties which characterize a small isolated molecule and that, ^as will be shown

later, ^a large part of the NMR information is devoted to describing the orientation of each

« local probe » inside the rigid fragment. ^{As a} consequence, the information actually ^{usable to} supply ^a clear response about local order parameters ^is dramatically ^reduced.

Fig. ^1. - The central aromatic core of PAA : lateral view showing the directions of interest ; (a) quadrupolar probes ; (b) dipolar probes. ^Note that these drawings ^are only approximate since the PAA molecule is non planar ^and distorted. The oxygen ^{atom on} the azoxy group is adjacent ^{to the better}

ordered ring ^A, according ^to [25].

A more serious problem ^is raised if « non local » probes ^{also are} observed. A typical example ^{of such a « molecular} probe » ^{is the} dipolar interaction between protons ^{attached to} distinct neighbour fragments. The situation is even more complex if the molecules in the nematic phase ^are thought ^as very ^« soft » objects ^{that can} adopt ^a large number of very different conformations as they would do in a gas phase. ^{This « soft »} point ^{of view seems to}

be presently dominating in the literature dealing with the NMR of liquid crystals [7-10].

However, ^{a nematic} phase being ^a condensed matter, ^{with a} density ^{close to that of a} solid,

the packing forces may restrict the possible conformations to a few conformations, presumably ^{close to that} existing in the solid phase [11]. ^{In section} 2, ^{we shall} explain that, ⁱⁿ

the framework of this « hard ^» point ^of view, ^{it is not} unreasonable to introduce two mean

« molecular » conformations and to define some Cartesian frame attached to each one as for a

rigid object ; ^{moreover, we shall} show that the ordering ^{of these two} frames with respect ^to the director can be described by ^{the same} ordering matrix with principal ^values SZZ ^and Sxx - Syy.

Using ^{such a} model, ^{it was shown in a} previous paper [5] ^{that a} large ^amount of deuterium NMR (DMR) ^{data on} partially ^and completely deuterated PAA could be explained ^self- consistently assuming ^that Sxx - Syy

^{0. Recently [16], the same model was used to interpret}

the proton ^NMR (PMR) spectrum ^{of PAAd6} (partially deuterated on the methyl groups).

Comparison ^with experiment ^{showed that a} good ^{fit could be obtained} only ^{if an} elongation ^of

the phenyl rings ^{of = 6.5 %} perpendicular ^to the para-axes ^was assumed. Such a large

deformation is not justified by ^{the structure in the solid} phase ^{and it was} argued [6] ^{that this}

result was an indication that the second-order parameter Sxx - Syy ^{may be non-zero in nematic}

PAA.

In this paper, ^we propose ^to discuss again, ^{in much more detail, the} problem ^{of the} biaxiality of the molecular ordering ⁱⁿ PAA, using ^{both DMR and PMR data for this} purpose.

The relationships between the structural and motional parameters ^{at the scale of the two} phenyl ^{A and B} rings ^{and at the scale of the} mesogenic ^{core are} explicitly ^{written and an} algorithm ^for solving ^{the set} of relevant equations ^is proposed. ^{We show} that the second molecular order parameter clearly controls the temperature dependence ^of quadrupolar splittings allowing ^{a rather accurate} estimate of its overall variation throughout ^{the nematic} phase. However, the number of equations ^{is too small for} complete calculation and the

magnitude ^{of this} parameter ^can only ^be roughly located within a rather wide range of

uncertainty.

2. The PAA molécule and the « hard » hypothesis.

The sketch of PAA has been widely displayed ^elsewere [5]. This molecule can be partially deuterated ; DMR and PMR experiments ^were performed using ^{PAAd8 and PAAd6} respectively. ^{In the} following, ^{we assume that} isotopic ^{effects are} negligible [5] ^{and we}

consider DMR and PMR data as both supplying information about the same physical system.

Only ^the mesogenic ^{core formed} by the central azoxy group of the ^two phenyl rings ^{A and B}

will be considered here (Fig. 1). ^{This core} being ^a three-body system, ^the only ^relevant

internal motions are the rotations of each phenyl ring ^with respect ^to the NNO group. The

hypothesis ^{we make} concerning ^the rotations, ^which corresponds ^{to the « hard »} point ^{of view}

mentioned above, ^{is that we can} classify ^these intemal rotations into rotations of large ^and

small amplitude.

The large amplitude rotations, ^{assumed to be} instantaneous, ^{are the} following : firstly, ^a phenyl ring ^can flip ^{about its} para-axis producing ^some averaging effect ; however, ^these

7r flips ^involve symmetry properties ^{of the} rings and leave the ordering matrix of the

mesogenic ^core unchanged. Secondly, the conformation can exchange with its mirror image

conformation (dynamical racemisation) by simultaneous rotations around the single ^bonds.

Such a motion is not a geometrical symmetry operation since it involves two physically distinguishable conformations but, ^for NMR, ^it actually belongs ^{to a} symmetry operation

since the nuclear spin Hamiltonian and thus the corresponding observable ordering ^{matrix are} again ^left unchanged. ^{These ir} flips and racemisation processes which leave the general

molecular shape unchanged ^{are the} only large amplitude intemal motions that we consider in

our model.

Now, ^{it is not} very realistic ^{to assume} the existence of these large amplitude ^motions only

because the potentials hindering ^{the rotations} around the single ^{bounds are not} infinitely high ; ^small amplitude fluctuations certainly ^{occur about the mean} positions (librations) ^which slightly ^{blur the} image ^{of the} single conformation. However, ^these librations are expected ^to

be much faster than the large amplitude ^{motions so that it seems} reasonable to introduce the concept ^{of mean} (over ^the librations) conformation to which the « molecular » frame is attached. Thus, from this « hard » point ^of view, ^the non-rigidity ^{of the aromatic core is} simply pictured ^as internal rotations (of large ^{and small} amplitude) ^{inside a} single

« molecular » frame attached to a dextro or levo conformation as described above, ^{and the} ordering ^{of these two} frames with respect ^to the director is described by ^{the same} ordering

matrix. In the following, the fluctuations of the orientation described by ^{this matrix will be}

referred to as external motions.

Another important consequence of the « hard » hypothesis ^{is the} decoupling ^between

internal and external motions. This decoupling results either from symmetry properties (large

amplitude rotational jumps) ^or from the fact that these internal motions again may be

assumed to be much faster than the extemal ^ones. This general hypothesis that external and

internal motions are uncoupled is fundamental if we wish to establish some relationships

between local order parameters (at the level of the rigid rings) ^{and order} parameters ^{at the}

level of the mesogenic ^core. Finally, ^{we assume, as in the} previous studies, ^{that the} geometry of the mean molecule is temperature independent. The mathematical consequences of this ^set of hypotheses ^are presented ⁱⁿ appendix ^1.

3. Quadrupolar ^data.

The quadrupolar ^data which will be used in this section are described in detail in [5]. ^There

are four deuterium probes ^{attached to each} ring ^A and B. The fast exchange process resulting

from 7r flips reduces this number to two distinguishable probes ^{which we} label in the following by the indices 1 and 2 (Fig. la). These data are resumed in figure ^2. They ^run through ^the temperature range from Ts

^{357.6 K to} Tk

406.2 K. Let uri ^and 4"Jri(= ^{0 ) be the polar and}

azimuthal angles ^of ring CDi ^bonds in frames with Oz along para-axis and Ox in the plane ^of

the ring and let T be the corresponding ordering matrix. The corresponding quadrupolar splitting ^is given by [12] :

where

Actually, ^for Ur. = 60°, P2 (cos ur.) = - ^1/8 and for all physically acceptable solutions, ^its magnitude ^is larger ^as that of the second term then :

Fig. ^2.

Normalized splittings 2/3 àvjlc, ^{versus absolute} temperature. Avi ^{are the} experimental

3

quadrupolar splittings ^and cr the quadrupolar coupling ^constant. The full lines correspond ^{to calculated}

values.

The aim of this section is to reduce this whole set of data to a few reliable parameters ^that

can be evaluated with sufficient accuracy and which summarize all the information (but only

this information) contained in these data. For this _purpose, we have used the following procedure : ^In the frame of the parameters defined in appendix I, equation (1) may be written :

where FZZ (T ) ^and Fn (T) ^{are two} normalized functions such that

where r and s are two parameters which describe the overall variation of Tzz ^and

n between Ts ^and Tk.

In view to reduce the number of adjustable parameters, ^{we transform} again equation (1)

into the form :

where y(T) ^{is an} empirical ^{function that we} have chosen to be of the form :

with

It depends ^{on two} arbitrary parameters y and À.

Our goal again ^{is to} reduce the number of adjustable parameters. ^{The main reason is that} the amount of experimental information is limited and it is not possible ^to accurately adjust ^a large ^{number of} parameters. ^{Thus r} has been assumed to be known and indeed it is easily

calculated from reported dipolar ^data (Fig. ^{13 of} [5]). ^{We have} used the value r

0.43 and it is the value which we have used again ⁱⁿ equation (8) ^of appendix ^{II. On the other hand, it}

was possible ^to impose ^{the same value y = 0.25 for both} Fzz ^and F Tl. ^{We show now that 6}

parameters only ^{can be} adjusted ^{and that two} of them have to be evaluated using ^another

source of information.

For each ring ^{we have :}

Then by ^addition and subtraction :

Equations (6), (7) ^and (8) will be used in appendix ^{II. Now we wish to} discuss the last one

and explain that it does not supply any additional information. After elimination of q it can be written :

The latter equation ^{would be} interesting ^{if it were able to} supply ^some information about

TZZ. Unfortunately, ^{due to the} experimental uncertainties, this information is negligible

because equation (10) is consistent with quadrupolar data for any TZZ ^{in a} wide range of values.

As a conclusion, equation (9) may be eliminated. It is clear that better information about

TZZ ^is supplied by dipolar ^{data on the condition} that the distance between the two nuclear spins

of interest is known. We have assumed that this distance is 2.47 Â ± 0.01 and have used

reported ^D-H dipolar interactions [5]. ^{We have thus obtained} Tzz

^{0.65 and have used this}

value to calculate S2 . C2 - S1. C ^{1 as a} function of the 3 adjustable parameters S2 + S1, S2 - S1 , S2. C2 + S1. C1.

Finally ^the fitting procedure ^was performed using only ¹⁰ adjustable parameters ; ^the result, ^{which is} equivalent ^{to the whole set of} quadrupolar ^data considered, ^{is the} following :

The mean error between the measured and calculated values 2 2 àli

is : The mean error between the measured and calculated values of

_3 . ^{is :}

3 Cr

Due to the fact that Fzz(T) ^and F Tl (T) ^are normalized, ^the dependence ^{of the best} Si ^and Ci parameters ^{on the} shape parameters ^À is very weak and this two sets of parameters

are largely independent.

These best numerical values obtained clearly emphasize ^the dissymmetry ^{of the PAA}

molecule. This dissymmetry ^was recognized very early ⁱⁿ previous ^{NMR studies of PAA} [13-

17]. ^Two phenomena may be proposed ^to explain this feature : either the internal ring

distortions or the mutual disorientation of the rings. ^This problem is discussed in the

subsequent ^sections.

4. Dipolar ^{PMR data.}

We now introduce some additional information supplied by ^PMR spectroscopy (Figs. ^{lb and} 3). The method of analysis ^used (exact simulation of the spectrum) ^{has been} reported

elsewhere [6]. It allows us to calculate two new dipole interactions :

-

the dipole interaction between two ring protons ^{in meta} positions (actually ^the

measured value belongs ^{to an} average ^over A and B) ;

-

the inter ring dipole interaction (i.e. the interaction involving ^{the 4} pairs 3-4, 3-6, ^4-5

and 5-6 averaged by the uncorrelated A and B ir flips).

At low temperature T,, the former can be estimated to be 8.85 % of the corresponding

mean ortho interaction. If we neglect here the small distortions of the rings, ^we may write :

where the angled ^brackets imply averaging ^over rings ^A and B. This relation reduces to :

For the inter ring interaction the best value obtained is about 11.3 % in the whole temperature range. These latter ^two values define the coefficients involved in equations (11.9)

and (11.10) ⁱⁿ appendix II.

Fig. ^3.

High ^field component of the PMR spectra of PAAd6. Dotted lines are experimental ; ^solid

lines are calculated spectra. The normalized frequency is à v / Tzz (T) > where à v is the experimental

frequency ^{scale and} (Tzz(T)) ^{is the} ring main order parameter averaged ^over ring ^{A and B.} (a) ^Low

temperature spectrum. (b) High temperature spectrum.

5. Some examples ^of application.

As explained ⁱⁿ appendix II, the number of equations ^is definitely smaller than the number of parameters introduced by ^our modelisation of the PAA properties. Then, ^{it is} possible ^to

describe several typical situations, all consistent with the experimental ^data. However, ^some parameters weakly depend ^{on the free} input ^{variables at} least for realistic solutions. These parameters ^{are :}

The reason is that these quantities only depend ^on TZZ according ^{to the} approximate equations :

The latter quantity represents the variation of the ring biaxiality parameter ^between T, ^and Tk. ^Other quantities, ⁱⁿ particular the absolute values of the biaxiality parameters, ^do depend ^{on the free} input variables ; ^{in the} following, ^{we consider as} examples ^three typical

situations and in each of these _cases, we distinguish ^a high ^{and a low} rigidity ^situation corresponding ^{to «} small » and « large » amplitude librations of the rings :

5.1 THE PARA-AXES A AND B ARE ASSUMED TO BE PARALLEL. - Then we assume that

UA= UB ^and U2 - UB. Furthermore, ^{we assume that} Ul angles ^{are zero} i.e. that the ring

para-axes lie within the principal ^{XOY molecular} plane. ^{In such a} situation, the observed A-B

dissymmetry ^{can be} interpreted ^{in terms of} ring distortions only. ^{For each} ring, the distortion will be characterized by ^the parameters

This situation is probably rather academic but it cannot be completely rejected ^{since the}

distortions of structure of the rings ^{that it} implies ^{are not} unrealistic. These distortions ^can also interpret the observed dipolar dissymmetry [5] ^since they ^{tend to} induce distances between _para nuclei smaller for A than for B.

5.2 THE PARA-AXES A AND B ARE NO LONGER PARALLEL BUT ut = UB

^0.

^{In such a}

situation, ^{the A-B} dissymmetry may also be explained ^{in terms} of molecular distortion. The chosen values of ut ^and UB ^are probably ^nearer the actual physical situation in view of the structure of the molecule in the solid phase [18].

5.3 SAME SITUATION BUT ASSUMING THAT UA

UB

’TT /2.

This condition no longer

describes a distortion within the mean molecular plane but rather a distortion perpendicular

to this plane (some distortion of this type ^is observed in the solid phase).

Table 1.

Some possible ring and molecular orientational parameters of nematic PAA.

Columns 1, ^{3 and 5} correspond ^to examples proposed in sections 5.1, 5.2 and 5.3 respectively involving high ^molecular rigidity. ^Columns 2, ^{4 and 6} correspond ^{to low} rigidity. ^The quantity

’Tl represents (Sxx - Syy)/ Szz ^{the molecular} asymmetry parameter ^{and the other} quantities ^are defined ^{in the text and} appendices.

The results of the computations ^are presented in table 1 and in figures ⁴ and 5 where

SZZ (T ) ^and Sxx(T) - Syy (T ) ^{are plotted.} These functions are calculated using ^the expressions :

where angled ^brackets imply ^an average ^over ring ^{A and} ring ^B.

The ring distortions calculated in these six situations appear ^to be of the same order of

magnitude ^as those observed in the solid phase ^and consequently it is difficult to rule out any of the proposed solutions. As a consequence, the question of the actual origin ^{of the A-B} dissymmetry ^cannot be answered. In the same footing, ^{NMR data} supply very poor information about the magnitude of the libration phenomena and then it is difficult to conclude about the character (static ^or dynamic) of the necessary dihedral angle ^{between A}

and B. However, despite the weak accuracy of ^our evaluation of the model parameters, ^the molecular biaxiality Sxx, - Sy ^{always appears to be positive for realistic values of} inclinations of the para-axes UA ^and U2’ ^{on the principal} molecular axis OZ.

6. Conclusion.

In conclusion, ^{we have} proposed ^a model of the mesogenic ^core of the PAA molecule taking

the ^« hard ^» point ^{of view as} defined in the introduction and have developed its mathematical

consequences in their full generality. ^{In this manner, it was} possible ^to accurately ^{define a}

Fig. 4. Fig. 5.

Fig. 4. - Molecular oriental parameters SZZ ^and Su - Syy of the central aromatic core of nematic PAA

versus temperature. (1), (3) ^and (5) belong ^to examples proposed in sections 5.1, 5.2 and 5.3

respectively involving high ^molecular rigidity.

Fig. ^5.

Molecular orientational parameters SZZ ^and Su - Syy of the central aromatic core of nematic PAA versus temperature. (2), (4) ^and (6) belong ^to examples proposed in sections 5.1, 5.2 and 5.3

respectively involving low molecular rigidity.

mathematical framework involving ²⁶ parameters ^linked by ⁸ equations. Quadrupolar ^and dipolar ^NMR data appear ^as supplying 10 additional equations ; ^the degree of indetermination of the problem ^{can thus be} accurately characterized. An algorithm ^is proposed ^to calculate 18 output parameters ^{as a function of 8} input variables considered as free within the range of their physically acceptable ^{values. This} approach is useful because it allows us to explore ^the

field of all the interpretations consistent with the NMR data and thus clearly characterizes the

questions left open. It supplies ^a reliable method for calculating the range of uncertainty ^of

the parameters ^{which are the most sensitive to} this indetermination. Among the latter are the various biaxiality parameters ^at the scale of the rings and of the molecule. However, ⁱⁿ spite

of our poor knowledge ^{about them, a} reliable conclusion emerges from our data analysis : ^the

molecular ordering ^{matrix is} always biaxial and its temperature dependence is consistent with the behaviour predicted by ^mean field theories [19-21] ^{and with recent} experimental ^results [22]. On the other hand, ^this analysis emphasizes ^the usefulness _{of any} source of information able to supply effective additional equations ^{and to decrease the} indetermination. This is the

case of PMR spectroscopy, especially ⁱⁿ systems involving phenyl rings : ^it supplies phenyl dipolar probes ^{far from the} magic angle orientation which is the characteristic feature (with ^its advantages ^{and its} disadvantages) ^{of the} phenyl quadrupolar probes. Moreover, ^{it is sensitive}

to interactions connecting ^two neighbour fragments ^and thus, it is well adapted ^for supplying

information of deeper ^molecular significance ; ^{but this} information can only be measured via

an exact simulation of the PMR spectrum ^or by sophisticated multiquanta experiments [23].

Last but not least, ^{the success of this « hard »} point ^{of view to describe the NMR data does}

not mean that the « soft » point of view is incorrect. Simply, the latter is much more

complicated ^{and we} have shown that there is no need to invoke the existence, in the nematic

phase, ^of several very different conformations. In fact, ^our finding that the PMR spectrum

can only ^be reproduced assuming relatively small librations of the rings ^{is in our} opinion ^a support ^{for the non} existence of conformations with large ^dihedral angles between the rings.

Conversely, ^we believe that there is so far no known example ^{of NMR} study ^of liquid crystals showing ^{that the « hard »} point of view is in contradiction with the data. In future papers, ^we shall consider some examples ^{of studies in terms of the « soft »} point ^{of view and contrast the}

results to those obtained taking ^{the « hard »} point ^{of view.}

Appendix ^I.

The following physical model which involves 26 unknown parameters ^linked by ⁸ equatioris,

has been used to interpret the various NMR data. Let us list these parameters.

P2 (cos ur1 A) ^{and P2 (cos} U r2 A) ^{(denoted p 1 A and P2} respectively) characterize the positions ^of

the two distinguishable quadrupolar probes lying ^inside ring A ; P1 B and P f ^{are the similar}

parameters ^{for the} ring ^{B. To each} ring is attached a local frame Oxyz ^{such as Oz is} parallel ^to

the para-axis ^and Oy ^is perpendicular ^{to the} ring plane. ^We show later how, taking ^into

account the symmetry properties of the nuclear spin Hamiltonian, ^{it is} possible ^{to define a} symmetry adapted ordering matrix and prove that this matrix is diagonal. Thus, ^{the local} order, ^at the level of the two rings, ^is completely characterised by TA, T.Axx - TAyy,

Tf, TxBx - T yy ^{at low} temperature Ts ^and by T Z ( 1 + rA ), (TAxx - T;Y)(l + rA)(l + SA),

TBzz ⁺ rB ), ( T.Bxx - TByy ) ( 1 ^{+ rB ) ( 1 + sB ), at high} temperature Tk. ^The corresponding ^order

tensors, ^at the molecular scale, will be defined by ^three eigendirections ^{OXYZ and} by ^the eigenvalues Szz, Sxx - Syy ^{at T}

^{Ts, and Szz (1 +} u ), (Sxx - Syy ) (1 ^{+ u ) (1 + v ) at T}

^{Tk. The}

actual conformation of the PAA molecule and the relative disorientation of the two rings ^A

and B are not accurately ^known. Thus, ⁶ parameters ^{have to} be introduced to define the molecular conformation and the molecular principal frame. These 6 parameters ^{are the 6} Eulerian angles ut, U2 , U3A , UB, UB, UB ^which characterize the orientation of each ring ^inside

the molecular OXI’Z frame [24]. Finally, ^we define the magnitude ^{of the libration motions} by 1/ (cos ² ’P A) , 1/ (cons ² Q B> , ^{at T}

Ts ^and by (1 ⁺ wA)/ cos ² ’Q A) , (1 ⁺ wB )/ cos ² ’PB)

at T

Tk. ^{We show now} that these 26 variables are linked by ⁸ equations. ^{Let us introduce}

the auxiliary variables :

We may notice that the determinant of matrix M is always ^{far from zero} for realistic values of

U2. Using ^these notations, ^{the 8} equations which define the physical ^{model can be written :}

at T

Ts ^{and the same} type ^of equations ^{for the} ring ^B.

At T

Tk, ^these equations ^{become :}

and the corresponding equations ^{for B.} (1.7) ^and (1.8)

These 8 equations ^{can be derived in the} following way : let S be the diagonal representation

of the molecular ordering ^{tensor in the} principal molecular frame OXYZ. Neglect ^{in a first}

instance the internal ring rotations. Then, the molecule may be considered ^{as a} rigid object.

In the local frame of a ring, the matrix representation of the molecular tensor become :

and it has lost its diagonal character. However, ^{let us} consider T as the local ordering ^matrix

at the level of the ring ^{A or} B and introduce again the various internal motions. Then, ^{as we}

show _now, the resulting ring matrix may be again considered as diagonal.

Firstly, ^{the ir} flips processes belong ^to the transformation :

Averaged ^{over the two} conformations, ^both Tyz ^{and Tzx} matrix elements vanishe.

Secondly, ^{let us} consider the transformation (for both A and B) :

It describes the exchange between the two equally possible dextro and levo conformations inside the same molecular frame. Actually, ^{due to the} fact that this latter process leaves

unchanged ^{the nuclear} spin Hamiltonian, ^{the NMR} spectroscopy ^{does not} supply any information about its probability ^at the time scale of an experiment ; thus, ^{it is neither}

necessary ^nor forbidden to take it into account when defining ^the ring ^{order tensors.}

However, ^the conceptual and formal advantages ^{to do so} may be advanced : the order ^tensors obtained now have the same symmetry properties ^as the nuclear spin Hamiltonian and their

representations ^become diagonal ( Txy vanishes). Finally, the libration motions are described

by :

and if _cp (t ) is defined in such a way ^as (sin 2 cp ) ^{= 0, the} resulting effect may be described by

the transformation :

Equations (1.1) ^to (1.8) ^{are so} easily ^{derived from} (1.9).

Appendix ^II.

We have to calculate 26 parameters ^linked by ¹⁸ equations :

-

the 8 equations which describe internal constraints of the physical ^model (Eqs. (1) ^to (8)

of appendix 1) ;

-

10 additional equations supplied by the various NMR data.

The 6 following equations belong ^to quadrupolar ^{data as} explained in section 3 :

and the same equations ^for B with the corresponding ^{data :}

Two dipolar ^{data are} supplied by ^DMR (Fig. ^{13 of} [15]) :

and two dipolar ^{data are} supplied by ^{PMR are} explained in section 4 :

The last equation ^{is the most} complicated ^{one because} it involves the vector defined by ^the

centers of the two rings, i.e. the distance A B and its orientation inside the molecular frame

(in ^the examples treated in section 5, this orientation corresponds ^{to a} polar angle equal ^to

zero or 12°). ^These geometrical data have to be added to the 6 Eulerian angles

UA, UB ^{in view to} completely obtain the expression of the inter rings dipole interaction measured by ^PMR and written here in the following symbolic ^{form :}

In this expression, ^{V is the vector defined} by ^{two nearest} neighbour protons belonging respectively ^{to A and B} rings, 0 ^{and 0 are the} polar and azimuthal angles ^{of V in the} principal

molecular frame, and the double brackets stand for averages ^{over 7T} flips ^{and libration}

motions. (For the latter motions, the average ^can be calculated assuming ^that çA(t) ^and

’P B (t) ^are either uncorrelated or not ; actually, the result weakly depends ^{on this} choice.) ^{As a} conclusion, ^{we have} only ¹⁸ equations ^{usable to} calculate 26 unknown parameters. ^{As a} consequence, the model appears ^as widely undetermined and 8 parameters may be freely

chosen. Among ^{these 26} variables, ^{we wish to} choose the 8 free parameters ^{so that the set of} points ^of this many dimensional space belongs ^{to a set of} hypothesis ^of physical ^{and molecular}

interest.

Let us now introduce two auxiliary parameters OA and 403A6B

Roughly, 0 A and -PB ^{are dihedral} angles ^between ring planes and the XOZ molecular plane.

For the sake of clarity, the frames ^are chosen such that cP A ^and 03A6B ^are positive quantities ⁱⁿ

which case e = 0 A + 03A6 B represents the dihedral angle ^{between the two} ring planes. ^{Such a} goal ^can be achieved using ^the following convention : the eigendirections ^{allow to define two}

frames (OXYZ)A ^and (OXYZ)B ^{in such a} way that the latter corresponds ^to the former by ^a

7T rotation around OY. That is possible since such a rotation leaves the ordering ^matrix unchanged. ^The advantage ^now is that the special ^case where the molecule has the

C2 symmetry corresponds ^to the very simple relations :

In the following, 4J A and 4JB ^will systematically replace ut ^and UB ^{as basic} parameters ^{of the} model.

We can now describe the method to solve the whole set of equations. ^{It is an iterative}

method involving actually ²⁸ parameters ^linked by ²⁰ equations ; ^for purely computational

reasons, it is useful to split the molecular parameters ^u and v defined in appendix ^{1 into}

uA, Me and VA, v]3 ^linked by ^{the two} additional equations :

As a consequence, ^u and v will be replaced by ^{UA and} vA ⁱⁿ equations (1.5) ^and (1.6) ^and by

UB and VB ⁱⁿ equations (1.7) ^and (1.8).

The variables which are introduced as input ^{data for} starting ^the computing procedure ^are

the following :

which always ^{will be} considered as free (at least inside a bounded range) ; then, ^{two variables} define the dihedral angle between the two rings :

one of these two variables may be used ^{as an} adjustable parameter ^to satisfy equation (11.10) ;

one of these two variables again may be considered ^{as an} adjustable parameter ^to satisfy equation (11.12) ; finally,

will be the adjustable parameter ^{used to} satisfy equation (II.11) (actually, ^the inequality

r A =1= rB is ^not directly observable and can be neglected ; ^{it is} only ^a theoretical constraint of the model).

Now, equation (11.8) completely define rA and rB and the calculation of the 16 residual

output ^{variables can} be achieved : by ^{addition of} equations (I.1) ^and (1.3), (1.2) ^and (1.4), ^and

taking ^{into account} (11.7) ^and (11.9), ^{we can obtain} Szz ^and Sxx - Syy. ^Using (1.1), (1.2), (1.3),

(1.4) again, ^{we obtain} TA, TA - TAyY, ^{Tf, TB -} TyBy ; equations (11.1), (11.2) ^and (11.3) ^for A,

(11.4), (11.5) ^and (11.6) ^{for B} easily supply P1, P2 and s for each ring. Then, ^{we can calculate}

UA and vA by (1.5) ^and (1.6), Ma and VB by (1.7) ^and (1.8). The last task now, is to verify ^that (II.10), (II.11) ^and (II.12) ^are satisfied. If it’s not the _case, we have to modify ^{the 3} adjustable parameters previously defined and perform again ^{all the same} procedure.

References

[1] ^{DE GENNES P.} G., ^The Physics ^of Liquid Crystals (Clarendon Press) ^1974.

[2] ^EMSLEY J. W. and LINDON J. C., ^NMR Spectroscopy using Liquid Crystal ^Solvents (Pergamon Press) ^1975.

[3] DIANOUX A. J. and VOLINO F., ^J. Phys. ^{France 41} (1980) ^1147.

[4] ^{VOLINO F.,} MARTINS A. F. and DIANOUX A. J., ^Mol. Cryst. Liq. Cryst. ⁶⁶ (1981) ^37.

[5] DIANOUX A. J., FERREIRA J. B., MARTINS A. F., GIROUD A. M. and VOLINO F., Mol. Cryst. Liq.

Cryst. ¹¹⁶ (1985) ^319.

[6] ^{FERREIRA J. B.,} MARTINS A. F., GALLAND D. and VOLINO F., Proc. of the Fifth European ^Winter

Conf. on Liquid Crystals (Borovetz, Bulgaria) ^Mol. Cryst. Liq. Cryst. ¹⁵¹ (1987) ^283.

[7] BURNELL E. E. and DE LANGE C. A., J. Mag. ^{Res. 39} (1980) ^461.

[8] EMSLEY J. W. and LUCKHURST G. R., ^Mol. Phys. ⁴¹ (1980) ^19.

[9] ^ZANNONI C., ^Nuclear Magnetic Resonance of Liquid Crystals, Ed. J. W. Emsley (Reidel, Dordrecht) 1985, p. 35.

[10] ^{DI BARI L., FORTE} C., VERACINI C. A. and ZANNONI C., ^Chem. Phys. lett. 143 (1988) ^263.

[11] ^{WALZ and HAASE W., Mol.} Cryst. Liq. Cryst. ^{Lett. 4} (1986) 53 and references therein.

[12] ^{DOANE J.} W., Magnetic Resonance of Phase Transitions, Eds. F. J. Owens, C. P. Poole Jr. and H.

A. Farach (Academic Press) ^1979, p. 171.

[13] ^{DIEHL P.} and TRACEY A. S., Mol. Phys. ³⁰ (1975) ^1917.

[14] ^{EMSLEY J. W.,} KHOO S. K. and LUCKHURST G. R., ^Mol. Phys. ³⁷ (1979) ^959.

[15] ^{DONG R. Y., TOMCHUK E. , WADE Chas. G.,} VISINTAINER J. J. and BOCK E., J. Chem. Phys. ⁶⁶

(1977) ^4121.

[16] ^{DONG R. Y., LEWIS J., TOMCHUK E. , WADE} Chas. G. and BOCK E., ^{J. Chem.} Phys. ⁷⁴ (1981) ^633.

[17] HAYAMIZU K. and YAMAMOTO O., ^J. Magn. ^{Res. 41} (1980) ^94.

[18] KRIGBAUM W. R., CHATANI Y. and BARBER P. G., ^Acta Crystallogr. ^{B 26} (1970) ^97.

[19] STRALEY J. P., Phys. ^{Rev. A 10} (1974) ^1881.

[20] LUCKHURST G. R., ^ZANNONI C., NORDIO, P. J. and SEGRE U. , ^Mol. Phys. ³⁰ (1975) ^1345.

[21] ^{BERGENSEN B.,} PALFFY-MUHORAY P. and DUNMUR D. A., ^Mol. Cryst. Liq. Cryst. ¹²⁹ (1985) ^375.

[22] ^{BAO-GANG WU,} ZIEMNICKA B. and DOANE J. W., ^{J. Chem.} Phys. ⁸⁸ (1988) ^1373.

[23] ^{SINTON S.} W., ^{ZAX D.} B., ^MURDOCH J. B. and PINES A., ^Mol. Phys. ⁵³ (1984) ^333.

[24] ^{TINKHAM M.,} Group theory ^{and Quantum Mechanics} (McGraw-Hill) ^1964.

[25] ^BODEN N., BUSHBY R. J. and CLARK L. D., J. Chem. Soc. Perkin Trans. 1 (1983) ^543.