The pressure and temperature dependence of the orientational order in the nematic phase of 4-n-pentyl-d11-4'-cyanobiphenyl. A deuterium NMR study

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The pressure and temperature dependence of the orientational order in the nematic phase of

4-n-pentyl-d11-4’-cyanobiphenyl. A deuterium NMR study

J.W. Emsley, G.R. Luckhurst, B.A. Timimi

To cite this version:

J.W. Emsley, G.R. Luckhurst, B.A. Timimi. The pressure and temperature dependence of the ori- entational order in the nematic phase of 4-n-pentyl-d11-4’-cyanobiphenyl. A deuterium NMR study.

Journal de Physique, 1987, 48 (3), pp.473-483. �10.1051/jphys:01987004803047300�. �jpa-00210463�

The pressure and temperature dependence of the orientational order in the nematic phase ^of 4-n-pentyl-d11-4’-cyanobiphenyl. ^A deuterium NMR

study

J. W. Emsley, ^{G. R.} Luckhurst and B. A. Timimi

Department ^of Chemistry, ^The University, Southampton, ^{S09 5NH, U.K.}

(Requ ^{le 8} septembre 1986, accept6 le 18 novembre 1986)

Résumé.

On utilise la résonance magnétique du deutérium pour

la dépendance

température ^et

pression ^des dédoublements quadrupolaires ^de 4-n-pentyl-4’-cyanobiphényl ^{à chaînes} alkyl deutérées. Ces dédoublements donnent les paramètres ^d’ordre SiCD pour chaque position

^{la chaîne,} qu’on ^trouve indépendants ^de

la pression

^la ligne ^de transition nématique-isotrope. On utilise les résultats pour obtenir, par

équation ^d’état

de la phase nématique, la variation des SiCD

^la température à ^{volumes constants de} 0,243 dm3/mole-1 ^et 0,248 dm3/mole-1. On compare

deux groupes de SiCD

^les prédictions d’une théorie de champ moyen pour des molécules flexibles. On obtient ainsi la dépendance

densité des coefficients d’interaction de la théorie ; ^les

résultats suggèrent que le potentiel ^de couple moyen ^est dominé par des forces à courte portée.

Abstract.

Deuterium NMR spectroscopy has been used to measure the temperature and pressure dependence ^of quadrupolar splittings ⁱⁿ alkyl chain deuteriated 4-n-pentyl-4’-cyanobiphenyl. ^The quadrupolar splittings

^{used to}

obtain order parameters SiCD for ^each position ^{in the chain, and these}

^{found to be} independent of pressure ^at the

nematic-isotropic transition temperatures. ^{The data}

^used together ^with

equation ^{of state} for the nematic phase

to derive the variation of SiCD ^with temperature at constant volumes of 0.243 dm3 mol-1 and 0.248 dm3 mol-1. These two sets of SiCD

compared ^{with the} predictions ^of

^field theory of flexible molecules. The density dependences

obtained of the averaged interaction coefficients which appear in the theory, and the results suggest that relatively short range forces dominate the potential ^{of mean} torque.

Classification

Physics ^Abstracts

61.16N

61.30

1. Introduction.

The characteristic shared by ^all liquid crystal phases ^is

their long range orientational order ; this has been studied for many mesogens by ^{a wide} variety ^of techniques [1]. ^{The most} fundamental aim of such studies is to compare the experimental values with the

predictions of molecular theories, ^{or with the results}

from computer simulation experiments. ^{The orienta-}

tional order varies strongly ^with temperature ^{and most} studies concentrate on measuring ^this dependence.

However, ^{there is also a} _strong dependence ^on density

caused by changes in the radial distribution function so

that data collected from experiments ^on samples ^at

constant pressure but varying temperature ^{will also} contain an unknown contribution to the variation of orientational order which is caused by changes ⁱⁿ density. ^The theories and simulations usually ^{refer to} systems ^{at constant} density, ^so that it is clearly ^desirable

to obtain data which can be used to separate ^{the effects} of density from those of temperature.

There is no direct method for determining ^orienta-

tional order as a function of temperature at constant

density. To obtain this dependence ^{it is} necessary ^to

measure the orientational order as a function of both temperature and pressure and subsequently ^{to convert}

this data into the required dependence ^{of order on} density by using ^an experimentally determined equation

of state for the liquid crystal phase. ^There have, however, ^been only ^a very few studies of the relation-

ship ^between pressure, volume and temperature ^for

liquid crystal phases ^{and there are even fewer} examples

of combining this data with that on orientational order

[2-6]. The earliest measurements of orientational order

as a function of temperature and pressure [7, 8] ^were

made on 4,4’-dimethoxyazoxybenzene (PAA) ^before

an equation ^{of state for this} compound ^{had been} reported ; consequently ^the interpretation of the results

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004803047300

focussed on those aspects ^{which can} be discussed without the necessity ^of deriving ^the density depen-

dence of orientational order. Both Deloche et al. [7]

and McColl [8] ^used proton ^{NMR to measure orienta-} tidnal order by assuming ^{a direct} proportionality ^be-

tween AH, a separation ^{between two peaks in the}

spectra, ^and P2, an ^order parameter, which is for the molecular long axis about which the molecule is _sup-

posed ^{to be} axially symmetric. Deloche, ^{Cabane and}

Jerome [7] ^{found that} the order parameter ^{at the} nematic-isotropic coexistence line, P2I, ^{is a constant,}

which is in accord with the molecular field theory developed by ^{Maier and} Saupe [9] for nematics com-

posed ^of rigid, cylindrically symmetric particles. ^In

contrast, ^Horn and Faber [4] ^have reported ^{that for 4-} n-pentyl-4’-cyanobiphenyl (5CB) and 4-methoxyben- zylidene-4’-n-butylaniline (MBBA), P2I ^{varies with}

pressure. The values of the order parameter ^were obtained by measurements of refractive indices [3] ^but

the data for 5CB have been confirmed by measuring

AH in the proton ^NMR spectrum [5]. ^{Wallis and} Roy [5] ^studied the pressure and temperature dependence

of &H for ^{a series of} alkyl-cyanobiphenyls ^{and deduced}

that P2I decreases with increasing pressure for 5CB,

7CB ^{and 8CB, but is constant for 6CB.} They ^{also found}

P2 ^{to be} independent of pressure for the nematogen ^4-

methoxy-4’-cyanobiphenyl ^(10CB). Deviations from the prediction ^that P 2 ^{is a constant could be caused} by

a number of factors and perhaps ^{the most} important ^is

that the Maier-Saupe theory ^{refers to molecules which}

are cylindrically symmetric ^and rigid, ^{whereas none of}

the mesogens studied fulfill these criteria.

We should also note that the departure ^from cylindri-

cal symmetry ^{means that a} single ^order parameter provides ^an inadequate description ^of orientational

ordering ^{and that two order} parameters ^are necessary

[10] ; for flexible molecules the description ^of ordering

is even more complex [11, 12]. ^McColl [8] introduced a

thermodynamic coefficient T, ^{defined as}

which is a measure of the relative importance ^of

temperature ^and density ⁱⁿ establishing ^{nematic order.}

It can be obtained by measuring P2 ^{as a} function of T

and p ^and using ^an equation ^{of state to} determine V at

values of T and p. ^Alben [13] ^has suggested ^{that T is a}

sensitive test of molecular theories of liquid crystal- linity, being dependent ^{on the nature} of the forces

contributing ^{to the} potential ^{of mean} torque, U(I3),

which is defined via f (,B ), ^the singlet orientational distribution function,

Z is the orientational partition ^{function and the}

molecules have been assumed to be rigid ^and axially symmetric, ^{so that} 13 ^{is the} angle ^{between the} symmetry axis and the director. P2 is related to U(.8 ) by

so that P2 ^{is a constant when} U(,B )IkT ^{is constant. We}

may expand U(p ) ^{as an infinite sum of} Legendre polynomials PL(cos /3 ) ^{of rank} L,

The potential ^{of mean} torque vanishes in the isotropic phase and in the liquid crystal phase will increase as the orientational order increases, ^so that uL is expressed ^as

e L PL. ^The expansion coefficients EL will be dependent

on volume since they ^are averages of ^the pair potential

over the radial distribution function. If the drastic

assumption is made that only ^{the term with L}

²

should be retained in equation (4), ^then

With these assumptions ^{r is} determined by ^the

volume dependence ^of E2(V), ^and writing ^{this as}

identifies _y with T. Alben [13] noted that if dispersive

forces are responsible ^{for the} anisotropic potential ^then

y should be 2 whilst if repulsive ^{forces dominate then y}

will tend towards infinity, the value appropriate ^for

hard rods. For real nematogens ^the density dependence

of E2 (V ) ^{is probably more complex than the form given}

in equation (6), ^{but if} this form is retained then T will be found experimentally ^{to be} non-integral, ^{with a}

value dependent ^{on the} balance of forces contributing

to U(J3). Experimental values of F range from 1.9 in

4,4’-di-n-hexyloxyazoxybenzene [6] ^to 6.0 for 5CB [4].

These changes ^{seem to} suggest that there are major

differences in the forces contributing ^{to the} potential ^of

mean torque ^{in these} nematogens, ^{but this would be}

surprising bearing ^{in mind} the similarities in the struc- tures of the compounds. ^{It is} possible ^{that the} changes

in F are caused in part by differences in the degree ^of

molecular flexibility, ^{so that F reflects not} only ^an

average ^over different kinds of force, ^{but also over the} conformations adopted by the molecule.

The experimental characterisation of the orientation- al order of a flexible molecule is a formidable task.

Thus Al , ^the component along the director of some

second-rank property ^of the mesogen, is related to

ordering ^matrices Sas ^{for each molecular} conformation and assuming ^{these to be finite in number we obtain} [11]

Here pn ^{is the} statistical weight of the nth molecular

conformation, xyz ^{are axes} located in some rigid ^sub- unit, ^and A’,s ^are components ^{of the measured} quantity

referred to these axes. The scalar Ao may be obtained from measurements on the isotropic phase, ^{or in}

favourable cases is zero. For molecules lacking symmet- ry in all conformations, ^which is usual for mesogens, there are five independent, ^non-zero components ^of S" for each of N conformations, ^{which could be} determined only by measuring ^{5 N} independent ^values

of Ã1. ^In practice ^{this has} proved ^{to be} impracticable,

but deuterium NMR has allowed the measurement of order parameters SCD for each C-D bond in different

rigid ^{sub-units of the} mesogenic molecules. The availa-

bility ^{of these order} parameters provides ^a powerful

test of those few theories of orientational order which allow for molecular flexibility and indeed has prompted

their development [14-16]. ^We have shown in an earlier report [17] ^that good ^deuteron spectra ^can be obtained of liquid crystals ^{at elevated} pressures. ^{We now} describe

a detailed study ^{of the} pressure and temperature

variation of the orientational order of individual C-D bonds in the nematogen 5CB which has a fully

deuteriated alkyl ^{chain. This is one of the few} liquid crystals for which an equation ^{of state has been}

determined [3,18]. ^{We shall show, in section 3.1, how}

deuterium NMR can be used to determine ri ^{for each}

deuteriated site in the molecule ; ^{these values are then} compared ^{with the T value} obtained for 5CB by ^Horn

and Faber [4], ^{which is an} average for the whole molecule. We have also investigated ^the pressure

dependence ^of SCND, ^{the order} parameters ^{for individual} C-D bonds at the nematic-isotropic phase boundary,

and we shall discuss the results in sections 3.2 and 3.3.

Flexible molecules, ^{such as} 5CB, ^{also have the} interesting possibility ^{that the} statistical weights p. may be density dependent. ^{Such a} dependence ^{has been}

inferred for n-alkanes from studies of their vibrational spectra [19], although ^{this method of} studying ^confor-

mational distributions has been criticised [20]. ^In liquid crystal phases pn ^has contributions frorn ^the anisotropic

forces responsible ^for producing orientational order in addition to the internal forces which determine the conformational distribution in the isotropic phase.

Consequently, ^the density dependence ^{of the order}

parameters for the C-D bonds is determined by ^the changes ^with density of both pn and the potential ^of

mean torque. ^We describe, in section 3.4, ^an _attempt ^to determine the magnitude ^{of these two effects} by comparing ^the experimental SCD ^{with those} predicted by ^{a molecular} theory ^{which includes the effect of}

molecular flexibility explicitly [15].

2. Experimental.

The sample ^{was a mixture of} approximately ^ten parts ^of 5CB to one of 5CB-dll. ^{The NMR} measurements were

made on approximately 0.2 g contained in a thin-

walled, ^screw top container constructed from Teflon.

The pressure vessel has been described previously [17]

and was used with a Bruker CXP 200 spectrometer.

Each spectrum ^{was obtained} by ^Fourier transforming

the average ^{of 2} 000 free induction decays, ^{and a} typical

spectrum ^{is shown in} figure ^{1. The} peaks ^{have been} assigned ^{to deuteron} positions by Emsley ^{and Turner} [21] ^{as shown in} figure 1 with the molecular labelling given ⁱⁿ figure ^2. Spectra ^{were recorded at 21} temperat-

Fig. ^1.

^{30.7 MHz} deuteron NMR spectrum ^of 5CB-dll. ^The peaks

labelled with their assignment ^{to deuteron} positions.

Fig. ^2.

Structure and atomic labelling ^for 5CB-dll.

Fig. 3.

Quadrupolar splittings åiít for the first methylene

group in 5CB-dll

function of temperature, T, ^and pressure, p. The smooth

through ^{the data are} guides ^to

the eye and

used to interpolate between data points. ^The

data

collected at temperatures (K) of, ^{from left to} right,

295.3, 297.3, 299.2, 301.2, 303.0, 305.5, 308.1, 313.0, 318.2,

322.9, 328.2, 334.1, 339.5, 344.9, 350.2, 356.5, 362.4, 365.3,

366.9, 369.1, ^371.6.

ures between 26 °C and 90 °C and at pressures between 1 bar and 2.7 kbar. The highest pressure which could be used was determined by ^the freezing point ^{of the} pressurizing ^fluid [17]. ^At temperatures ^{above the} TNI, corresponding ^to atmospheric pressure, (TNI (1 )), ^a

spectrum ^was obtained first ^at the highest’ pressure ^at which the sample ^was still nematic and subsequent

spectra ^{were recorded at} lower pressures until the

sample ^became isotropic. ^Below TNI(L) spectra ^were recorded at increased pressure until the supercooled

nematic froze. Figure 3 shows the quadrupolar splittings å VI ^{1 obtained as a function of} temperature ^and

pressure ; similar data sets were obtained for each of the other four deuteriated alkyl ^chain positions.

3. Results and discussion.

The quadrupolar splittings ^{are related to} SCD ^{at each} position along ^the alkyl ^chain by

where qi ^{is the} quadrupolar coupling ^{constant for the}

ith C-D bond ; ^{we assume this to} equal ^{168 kHz} [22] ^for

all positions along the chain. Equation (8) ^{also assumes}

axial symmetry ^{of the} quadrupolar ^tensor about the C- D bond direction ; deviations from such symmetry ^are small and can be safely neglected.

There are several aspects ^{of the} data collected by ^us

that we wish to discuss. In section 3.1 we shall discuss the pressure dependence ^of SCN’ D ^{and after} discussing ⁱⁿ

section 3.2 the equations ^{of state} which have been obtained for 5CB we describe in section 3.3 how we

obtain site specific ^{r values.} Finally, in section 3.4 we

compare ^our variation of SCD ^with temperature ^at

constant volume with the predictions ^{of a molecular} theory.

3.1 THE PRESSURE DEPENDENCE OF SbD ^AT TNI.

The smallest value of v along ^{each curve of constant} temperature ^but varying pressure ⁱⁿ figure ^{3 does not} necessarily correspond ^{to the} value when nematic and

isotropic phases ^{coexist. We have} attempted, ^therefore,

to obtain accurate values of åVi ^at TNI ^{as a function of}

pressure by carrying ^{out a} careful search for the values of pressure ^{at constant} temperature ^where spectra ^from

both phases coexist. This biphasic region ^{arises in} part from inhomogeneties ⁱⁿ both T and p, but since the

sample ^{contains a small amount of unknown} impurities

we also expect ^{a smaller} region ^{where the two} phases

can coexist at constant T and p ^{and where the} quad- rupolar splittings ^are expected ^{to be} essentially ^constant [23]. This latter region proved ^{to occur} only very close to the point ^{where the} sample ^becomes entirely ^isot- ropic ^{and was} extremely time-consuming ^{to locate} accurately. ^{A careful} search for this region ^{was carried}

out at four of the temperatures ^{used in our} experiments

with the results for the splittings shown in table I. The

splittings ^at TNI obtained in _{this way} are essentially independent of pressure ^at each position ^{in the chain.}

This result contrasts with that of Horn and Faber [4], ^as

well as of Wallis and Roy [5] that there is a decrease of 15 % in orientational order of 5CB when the pressure is increased to the point ^where TNI ^{is 150 °C.}

Horn and Faber based their conclusion on the variation of refractive indices, ^{whilst Wallis and} Roy

used a measure of the width of the proton spectrum. ^In

both cases the assumed proportionality between these

experimental quantities ^and P2, ^{and the added} assump- tion that such proportionality ^is independent ^of density,

is less precise ^{than that} between å î1 ^and SICD, ^{and this}

may be the principle ^reason for the different results.

We return to a discussion of the significance ^{of this}

result in section 3.4.

3.2 THE EQUATION OF STATE.

To convert the data obtained as a function of temperature and pressure into

å îJ ^{as a} function of T and V requires ^a knowledge ^of

the relationship between p, T and V. An important

reason for choosing ^{5CB as the} liquid crystal ^most

suitable for study ^{was that an} equation ^{of state had}

been reported by ^Horn [3]. ^{However, Horn’s} equation

was subsequently ^{found to be} inconsistent with density

measurements on 5CB made by Dunmur and Miller

[24] ^and by Roy [25]. ^{Horn obtained} the volume at different values of T and p by measuring ^{the refractive}

index parallel, iij, , ^and perpendicular, nl, ^{to the}

nematic director. These measurements have been re-

peated ^{for 5CB} by Bunning [18, 26] ^{who also derived a}

new equation ^{of state.}

However, ^{this latter} equation ^{of state} predicts that,

close to the nematic-isotropic transition, ^{the molar}

Table I.

Quadrupolar splittings for ^the alkyl ^{deuterons in} 5CB-d11 ^{measured in the} biphasic region ^{close to} TI.

(*) Taken from reference [23].

Fig. ^4.

Variation of volume with pressure for 5CB ^at 101.8 °C

predicted by Bunning [18]. ^The straight line fitted to the points ^at high pressure is used ^to relate V to p ^{close to} the phase transition.

volume of 5CB should decrease with decreasing

pressure, ^as shown in figure ^4, whereas in practice ^an

increase is observed. Thus neither the equation ^{of state}

derived from Horn’s data, ^{nor from that of} Bunning

can be relied on to obtain molar volumes at all the

temperatures ^and pressures used ^{in our} experiments.

To overcome this difficulty ^{we have} adopted ^an empirical procedure ^{which we shall now describe and} justify. ^{This allows us to use the data} produced ^both by

Horn and by Bunning ^{in order to} predict ^{the densities}

measured by Roy [25], ^{and also to} give ^a qualitatively

correct dependence of the molar volume on pressure

near TNI. ^{We start with the} relationship ^{used to obtain}

the molar volume from nl ^and nl, namely

where

and VS ^{is a} scaling factor which Horn assumed to be

independent ^{of T} and p. ^A measurement of the density

of 5CB at 23 °C made by Gannon and Faber [27] ^was

used by ^{Horn to determine} VS to be ^{0.0843 dm3} mol- 1. The temperature and pressure dependences ^of iii ^and nl ^{were derived} by ^{Horn for the} entire range of the nematic phase ^for which data was collected. For each component ^the relationship ^{used was}

a

refers to either the parallel ^{or to the} perpendicular

component ^{and T* is} TNI (p ) ^{+ o.1. The variation of the} nematic-isotropic transition temperature, TNI (p ), ^with

pressure is calculated from

Combining equations (8)-(11) enables the molar vol-

ume to be predicted ^at any value of T and p ^{in the}

nematic range of 5CB. We have used this procedure ^to predict ^the temperature dependence ^{of V in} the range 22.5 °C to 35.3 °C for which 5CB exists in the nematic

phase ^at atmospheric pressure. Comparing these values of V with those measured by ^Dunmur and Miller [24]

revealed significant differences between the two sets of

data, ^{which can be} eliminated by allowing Vs in equation (8) ^to vary between 0.0817 dm3 mol-1 ^at

35.3 °C to 0.082 dm3 mol-1 ¹ at 22.5 °C. Roy [25] ^has

measured the molar volume of 5CB at four temperat-

ures between 36.5 °C and 84.2 °C and at each tempera-,

ture over a range of pressures extending ^{to almost}

2 kbar at the highest temperature. ^{This data was used} by ^us, together ^with equations (8-11), ^to establish both the temperature ^and pressure dependence ^of Vs. ^At

constant T and varying p the value of Vs ^is given b)

with Vo having ^{the linear} temperature dependence

With these values of V s it. is possible, using ^{Horn’s data,}

to fit the molar volume determined by Roy ^for tempera-

tures up ^to 62 °C to within 0.1 %. At temperatures higher ^{than 62 °C the} agreement ^between predicted ^and experimental values becomes substantially ^{worse. To}

overcome this failure we have used a similar approach

to calculate V for specific ^values of p ^and T, ^{but now} using the refractive index data collected on 5CB by Bunning [18]. ^{This data was} analysed by Bunning ^who

fitted it to the slightly ^different equation ^{for the}

pressure and temperature dependence ^of n-I ^and nl

Combining equations (8), (9), (11) ^and (14) ^{enabled us}

to predict the molar volumes at temperatures ^and

pressures for which Roy ^has measured the density ; again ^{there is a} considerable discrepancy between the measured and predicted values which can be eliminated

by allowing V s ^to vary with T and p. We have determined the variation of V s ^{to be}

but now Vo ^varies non-linearly ^{with T.} Drawing ^a

smooth curve through the variation of the volume

Vo ^with temperature ^{enabled us to} interpolate ^{values of}

Vo ^{and then} using equations (8), (9), (11), (14) ^and (15) ^{it was} possible ^{to fit the} density measurements of

Roy ^to within 0.1 % at temperatures ^above 45 °C, ^but

with considerably ^lower precision below this tempera-

ture. Our procedure, therefore, ^for determining ^{V at}

each value of T and p employed ^{in our} experiments ^{is to}

use equations (8), (9) ^and (11) ^{and to combine these}

with equations (10), (12) ^and (13) ^for temperatures up to 45 °C and above this temperature ^with equations (14)

and (15), together ^{with the} graphical interpolation procedure discussed for the value of Vo.

There still remains a problem ^of relating volume, temperature ^and pressure close ^{to the} nematic-isotropic

transition temperatures TNI(p), ^{because in this} region

the equation ^{of state derived from the data of} Bunning gives ^a qualitatively ^incorrect variation of V with p ^at

constant T. To overcome this problem ^{we first} plot

values of V against p ^as predicted by Bunning ^{for the} temperature ^{of interest. The} predicted value of volume varies in an essentially ^{linear manner} with pressure until

approximately ^{0.2 kbar from the} phase transition when it decreases dramatically, ^as illustrated in figure ^{4 for a}

temperature ^{of 101.8 °C. We make the} assumption ^that

the linear portion ^{of the} curve can be extrapolated

towards lower pressures and will represent the relation-

ship between p, V and T with greater precision ^than

that predicted by ^{the use of} Bunning’s relationship. ^In practice ^{our aim is to} obtain the values of pNI and

TNI ^{for a} particular ^{value of} V, ^{so that in} figure ^{4 we}

seek the value of pressure for which V is 0.238 dm3 mol-1 ¹ whilst the nematic-isotropic transition tempera-

ture is 101.8 °C. The departure ^from the linear relation-

ship ^{for this} temperature ^{occurs at} about 150 bars below the transition point. ^The measurement of

å fJ ^with good precision ^{close to} TNI ^{is also} difficult, ^as discussed in section 3.1 and hence we have not attemp-

ted to obtain the temperature variation of the quad- rupolar splittings ^{at constant volume} (see ^section 3.4)

closer than 2.5 °C from the transition, ^{but we use the}

observed independence ^to pressure of APi ^at TNI ^to

determine the values of the quadrupolar splittings ^at

the transition for the two values of molar volume.

3.3 SITE DEPENDENCE OF T. - We define a site

dependent Fi ^as

The volumes and temperatures corresponding ^{to con-}

stant SbD ^{are located as the} points ^of intersection of lines of constant Aiii (and ^{hence of constant} %o) ^with

smooth curves drawn through the variations of åVi ^with

pressure ^{at constant} temperature, ^as illustrated in

figure ³ for A rl. ^The equation ^{of state is then used to}

convert these points into values of T and V at constant

SbD’ ^{In this} way the variation of In V with In T at constant SbD ^was obtained ; typical ^{results are shown}

for position ^{1 in} figure ^{5. To a} good approximation

Fig. ^5.

^{In V} against ^{In T} at constant values of â fj 1.

such plots ^are linear and for each position ^have gradients ^{which are} independent ^of SbD ^to within about 5 %. The values of Fi ^obtained in this way ^are given ⁱⁿ

table II, ^{and we note that} they decrease in magnitude appreciably ^{as the distance of the C-D} bond from the aromatic core increases. The value of 6 found for T by

Horn and Faber [4] ^{for 5CB was derived from measure-}

ments of the birefringence. ^This anisotropic property ^is

an average for the whole molecule, ^{but has} probably ^a

dominant contribution from the aromatic part ^{since the} refractive index is related to the electric polarisability

which has a large contribution from the 7r-electrons.

Thus, ^we may attribute the Tvalue of 6 to being ^{that for}

the aromatic rings, ^which together ^{with the data on} ri in table I prompts ^{us to} propose ^{that site} specific ^r values are largest ^{for the} relatively rigid ^{aromatic cores}

and decrease steadily along alkyl chains. This is _sup-

ported by the observation that the average r values obtained for the homologous ^{series of} 4,4’-di-n-alkylox- yazoxybenzenes [6] increase from 1.9 for the n-hexyl compound ^to 4.2 for the methyl ^{member of the series.}

Clearly, however, the observation that Fi ^{is site} dependent ^{for 5CB casts doubt on the} usefulness of the average r values for testing ^molecular theories of orientational order. The availability ^{of the} Fi presents

a new challenge ^to theory ^{which we} approach ^here by comparing ^the density dependence ^of orientational order with a molecular theory ^{which considers} explicitly

the effect on the SbD ^of changes in molecular confor- mation.

Table II.

Dependence of r ^on position ^{in the} n-pentyl

chain of 5CB-d11..

3.4 MOLECULAR INTERPRETATION OF THE DENSITY DEPENDENCE OF ORIENTATIONAL ORDER. - The

theory has been described in detail elsewhere [15] ^and

here we summarise only the essential features. We

assume that the conformational distribution of the molecule can be approximated by ^{a discrete set of} conformers, ^{so that} å Vi ^is given by equation (7) ^with Al.=- AP ; ^note that for the quadrupolar interaction

Ao ^{is zero.} Thus, ^{if the} quadrupolar coupling ^constants

qi, which ^we have already ^{assumed to be site} indepen-

dent for the alkyl chain, ^{are also assumed to be} independent ^{of the molecular} conformation, ^{then com-} bining equation (7) ^with (8) ^{leads to}

where Sb) ^{is an order parameter for} the ith C-D bond in the nth conformation. To obtain Pn and Sf) ^{we write} U(n, (JJ), ^the single particle energy, ^as

where w represents the orientation of the molecule.

The conformers result from hindered rotations about C-C bonds in the alkyl chain, ^{and we assume that their} energies U;nt (n ) ^are given by the isomeric state model of Flory [28]. ^{In its} simplest ^{form this} gives

where Ng is the number of gauche segments ^{in the} alkyl

chain and Etg ^is the energy difference between gauche

and trans forms.

The potential ^{of mean} torque, Uext(n, co)7 ^{for a}

molecule in conformation n and at an orientation w with respect ^to the director is defined in terms of the

single particle orientational distribution function

f(n. o) by

the orientational partition function Z is

The symmetry ^{of each} conformation in the molecules of interest is less than C3, ^{which means} [10] ^{that the} ordering potential ^will depend ^{on the Euler} angles f3

and y of the director in the molecular reference frame.

We, therefore, ^write Ue., (n, w ) ^as

The 82, m ^are elements of the interaction tensor in irreducible form for the nth conformation expressed ⁱⁿ

its principal molecular frame. To obtain these tensor

components ^{we locate a common reference frame for} each molecular conformation, ^{and for 5CB we choose} this to be in the biphenyl group of the aromatic core, ^as shown in figure ^2.

The elements of En in this reference frame are

obtained as tensorial sums of contributions from rigid,

molecular sub-groups, ^{which are also assumed to be} cylindrically symmetric. Thus, the aromatic core contri- butes a term e c" = (3/2 )112 ^Xzz

^{Xa. Each Ci - 1_Ci}

link in the alkyl chain contributes E 0

Xee ^directed along ^{the C-C bond, and} similarly ^{for each C-D bond}

there is a contribution Ecd 2, 0

^Xed. Note that these.

group interaction ^{tensors are} assumed not to change

with the conformation and the conformational depen-

dence of En is contained entirely ^{in the functions of the}

orientations (a c, i ^{W ’CD, and W ’CD2 that the Ci -1_Ci,}

C‘-D (1 ) ^and Ci-D(2) ^bonds make with the reference frame in each conformation. Thus, ^the components ^of el in the reference frame are

where 0 is the DCD angle ^{in the} methyl group, k ^{is the}

number of carbon atoms in the alkyl ^{chain and} C2, . (W ) is the second modified spherical ^harmonic [29]. Diagonalisation ^{of En from} equation (23) gives ^the principal components ^{for use in} equation (22).

The four quantities Xa, X , Xcd ^and Etg ^{together with}

an assumed geometry [30] determine the magnitude ^of

the order parameters SICD, ^or equivalently ^{of the} quadrupolar splittings åVi. We may investigate ^there-

fore the temperature and volume dependence ^of Xa, Xcc, XCd ^and Etg ^{by obtaining} the values which best fit the observed SCD at ^different temperatures. ^{We have}

chosen to do this for two values of the molar volume ; figure ^{6 shows the} temperature ^variation of åV1 1 at

V m

^{0.238 dm3 mol-1 and} Vm

^{0.243 dm3 mol-1 and}

for comparison ^we show also the temperature ^variation

Of AP, at constant, atmospheric pressure.

Fig. ^6,

Temperature variation of 0 v at ^{constant volume} for V,,,, = ^0.243 dm3 mol-.l (0), ^{0.238 dm3 mol-1}

(O ) , ^and

atmospheric pressure (0).

Similar curves were obtained for each position along

the chain. The most striking difference between the three curves in figure 6 is the shifting along ^the temperature ^{axis for} constant å v l’ which is caused by the change ⁱⁿ TNl ^with density. Removing this differ-

ence between the curves by plotting å VI 1 against TNI - ^{T, as shown in} figure 7, ^{reveals a smaller but} appreciable density dependence ^of orientational order-

ing ^{at constant reduced} temperature. ^This density dependence ^of orientational order is also apparent when comparing ^{the variation} of AV, ¹ at constant pressure with that at constant volume ; in the former

case a change ⁱⁿ temperature ^also produces ^a change ⁱⁿ density ^{so that there are two factors} combining ^to give ^a

more rapid change ⁱⁿ orientational order and hence in the quadrupolar splitting.

Fig. ^7.

Variation of 0 v 1 with TNI - ^{T for constant molar}

volumes of 0.243 dm3 (0), ^{and 0.238 dM3} (0), ^{and for} atmospheric pressure (0).

Figure ^{8 shows the} temperature dependence ^of åVi ^{at all chain} positions ^{for the two molar volumes and}

this data was used to determine the parameters ^{in the} potential ^{of mean} torque ^at each value of T and V by ^a

least squares optimisation.

We note first that the quadrupolar splittings, ^and

hence SICD, ^{for each} position ^{in the} alkyl ^{chain are} independent ^of density ^at TNI, ^{but as} temperature decreases a difference develops between the å Vi ^values

at the two densities. This behaviour contrasts with that

predicted by ^the Maier-Saupe ^{mean field} theory ^for axially symmetric rigid molecules, ^{for which the} poten- tial of mean torque, U(Ø), ^{has the form of} equation (5). ^The single interaction constant, £2 (V ), ^is

determined by ^the magnitude ^of TNI (p ), ^{so that this} potential predicts ^a density independent ^{variation of} P2 with reduced temperature T/TNI(p). Introducing

additional terms into the potential, ^{to allow for either} higher rank interactions or lower symmetry, ^{or non-} rigidity, ^{will in} principle ^{introduce a} density dependence

into the variation of order parameters ^{with reduced} temperature, but there is no reason why ^{the order}

Fig. ^8.

^Variation of A i;i ^with TNI - ^{T for all} positions along

the alkyl ^{chain in} 5CB-dll ^at

^constant molar volume of 0.243 dM3 Mol- (8) ^{and 0.238 dm’ mol-’} (0).

parameters ^{must be} equal ^at TNI (p ). ^{In the case of the} potential given by equation (23), ^the equality ^{of the}

values of S’CND’ at ^{the two} densities would seem to imply

that Xa*

X ,,IkT, Xc*c

xcc/kT’ x£

XedlkT, ^and

Et*g

Etg/kT should all be independent ^of density.

This is, however, improbable, particularly ^for Et;.

Thus, changing the pressure from one bar, ^when TNI ^{is 308} K, ^{to 1.83} kbar, ^when TNI is 374.9 K would

imply ^{an increase in} Etg of 22 %. A change ⁱⁿ

Etg with pressure has been proposed for alkanes [19] ^on

the basis of the pressure dependence found of their Raman spectra, but much smaller in magnitude ^and negative. ^The interpretation of the Raman data has been criticized recently [20], ^but even so a 20 % increase of Et appears improbable ^{as a reason for the}

constancy ^of scb ^{which we observe. It is more probable,}

in fact, ^that Etg ^{has a} negligible change ^on increasing

the pressure ^to 2 kbar and hence in our analysis ^we

make this assumption, ^{and fix} Et, ^{to be 3.8 kJ} mol- 1, ^{the value found to fit best the order} parameters

at atmospheric pressure for both ^{5CB and} 8CB [30].

The values of Xa, Xee ^and Xed ^which give ^{the best} agreement between observed and calculated %D ^were

determined for each temperature ^{at the two} pressures, and are shown in figure ^{9. The} quality ^{of the} agreement

between observed and calculated values of i ^is

almost as good ^{as that} obtained when Etg rather than

Etg ^{is kept constant and is better than 1 % at all chain} positions. ^The temperature dependence ^of Xa, Xee ^and Xed is caused by ^their dependence ^{on the} degree ^of

orientational order, ^{and as} suggested by Counsell et al.

[30], ^{we write}

Fig. ^9.

Variation with TN, - ^{T at constant molar volumes of}

0.243 dM3 Mol- (filled symbols) ^{and 0.238 dm3 mol-} (open symbols) ^of Xa (0, 0), Xcc (D, .) ^and Xed (0, *) ^{for 5CB.}

Here PZ is the order parameter of the assumed symmetry axis of the aromatic core averaged ^{over all} conformations, whilst P2 ^and P 2 cd ^are

S" is the order parameter in the nth conformation of the Ci -Ci + 1 bond, The 6 a 13 ^{can be} interpreted ^{as radial}

averages of the anisotropic interactions between seg- ments

and j8 in different molecules ; they ^{are then} density dependent, whilst the temperature dependence

of the parameters Xa, X ^and Xed at ^{constant volume is}

caused by ^the temperature dependent order par- ameters. In principal ^{the six} E,,p parameters might ^be

obtained by fitting ^{the derived} values of the interaction and order parameters ^with equations (24)-(26). ^In practice, ^{the order} parameters P2, P’ ^and Pf ^{are, to a}

good approximation, proportional ^{to one} another in the temperature range that ^we have studied, ^{so that} equations (24)-(26) ^can be rewritten as

where A,, ^{is the} proportionality ^{constant between} P2 ^and P". These three relationships predict ^{the ratios} À cc

Xccl Xa ^and A cd

Xcd/Xa ^{to be} temperature

independent, and this is observed to be the _case, as

shown ⁱⁿ figure 10. The values of Àcc ^and Acd ^deter-

mined for constant pressure [30] ^are of similar mag-

Fig. ^10. - Àce (0,8) ^and A,d (0, +) against TNI - ^{T for} V m

^{0.243 dM3 Mol-} (open symbols) ^{and 0.238 dM3 mol-1}

(filled symbols).

nitude to those at constant volume, ^but they ^{have a}

small temperature dependence, increasing ^{in each case} by about 15 % on decreasing ^the temperature by ^ten degrees ^from TNI ; ^{at lower} temperatures ^{the ratios are}

essentially ^{constant. The} E,,p ^cannot be obtained

separately ^but only ^{as the three} combinations

The dependences ^of Xa, Xcc ^and Xcd ^on P2 ^are essentially ^{linear at the two} densities, ^{as shown in} figure 11, and from the slopes ^we obtain the values of

Ta, F, and £d given in table III. If the density depen-

dence of these three _average interaction parameters ^is

of the form E-,, V ya (see Eq. (6)) ^{we obtain the values}

of _ya shown in table III. It is not possible ^to give ^a

Fig. ^11.

Dependence ^of Xa (0, 0), X,, (0, 0) ^and Xcd (0, *) ^for V m

^{0.243 dm3 mol-1}

(open symbols) ^and

0.238 dM3 Mol- (filled symbols). ^{The lines}

the result of

fitting ^{the data to linear} relationships constrained to pass

through ^the origin.

Table III.

Average interaction coefficients Ea, Bc, -

and their density exponents 7a, Y c and yd for 5CB-d11.

(*) From reference [30].

precise interpretation ^{of the} origin ^{of the} magnitude ^of

these volume exponents ^{in terms of} particular ^{kinds of}

intermolecular forces, ^since they ^{refer to the} averages

£a rather than to the interaction constants between

pairs of molecular segments. However, ^{we can} specu- late that their relative magnitudes ^{reflects a} change ⁱⁿ

the balance of forces contributing ^to the individual EaP ^values. Thus, the value of Yc

yd of 10 suggests that short range repulsion plays ^a major ^{role in} determining ^the interactions between the alkyl ^chains

and the mean field from the surrounding ^molecules.

Such short range forces ^{are also} important in determin-

ing ^the interaction of the aromatic cores with the mean

field since _ya, although much smaller than yc and yd, is significantly larger ^{than the value of 2} predicted

for dispersion ^forces.

4. Conclusion.

The pressure and temperature dependence reported

here of the orientational order parameters %D ^for

different sites in a mesogen is the first such study ^{to be}

made. We have explored interpretations ^{of the data}

which are on the one hand model independent, ^{such as}

the r coefficient introduced by ^McColl [8], ^{and on the}

other hand are dependent ^on specific’models ^{for the} potential ^{of mean} torque ^{for a} particle ⁱⁿ the molecular field of its neighbours.

We have redefined r to be site dependent ^{and our}

results show that this coefficient tends to decrease as

the deuteron site is. moved _away from the aromatic

core. It would be tempting ^{to infer from this site} dependence ^{of r that} the interaction of the individual chain segments with the molecular field of all other molecules changes ^{in character as the} segment ^is

removed further from the aromatic core. However, ^the

r are complex averages and in principle ^are dependent

not only ^{on the nature of the} potential ^{of mean} torque, Uext (n , w ), but may also be influenced by ^the density

dependence of the internal energy, Ui.t (n), ^{via the} trans-gauche energy difference Etg. ^{However, our data}

show that Etg ^{has only a small and probably negligible}

density dependence for the small range of density changes ^studied. Any ^{effect on} orientational ordering produced by ^a change ⁱⁿ Etg ^{is far} outweighed by ^the

much larger change in the order parameters SCD resulting ^{from a} variation in the potential ^{of mean} torque. ^The temperature independent coefficients E,,p which appear in the potential ^{of mean} torque ^as parameterized by Counsell et al. [30], ^{and which are a}

measure of the average strength of interaction between

rigid sub-units a and /3 (core, ^{C-C and C-D} fragments),

could not be obtained individually ^{from our} data, ^but only ⁱⁿ certain linear combinations, £a. ^This is unfortu- nate because it prevents ^{us from} determining ^the density dependence ^{of the} segment interactions and hence of testing theories of their origin ^{in terms of} specific kinds of intermolecular forces. The Ëa depend

upon how segment

interacts with the _core, C-C and C-D segments in all other molecules, ^{and their} depen-

dence on density, ^{which we assumed to} be of the form V Ya, could be obtained from our data. The values of ya, yc and yd are all much greater than the value

expected ^for dispersive forces and suggest ^a strong influence of short-range forces in determining ^the magnitude ^{of the} interaction constants in the potential

of mean torque.

The results described here show that monitoring ^the

pressure and temperature dependence ^{of site} specific

order parameters by ^{deuteron NMR} provides ^{a much}

greater insight ^{into the} intermolecular forces responsi-

ble for producing orientational order than do studies which can measure only ^some single, conformationally averaged, ^order parameter. ^{A detailed} interpretation

of such data depends ^{on the} characterization of an

equation ^{of state for the} liquid crystalline phase. ^The paucity ^{of such} pVT data is the major ^factor limiting

the extension of these studies to other liquid crystals,

and in this respect ^{it would be most useful to} produce equations ^{of state for a} homologous ^{series, such as the} 4-n-alkyloxy-4’-cyanobiphenyls. ^{All the lower members}

of this series (n

^{1 to} 6) show nematic phases ⁱⁿ

temperature ranges ( ²⁰⁰ °C) ^{amenable to} study by

deuteron NMR.

Acknowledgments.

We are grateful ^{to Dr.} T. E. Faber for invaluable discussions concerning ^the equation ^{of state for 5CB.}

We also wish to thank the Science and Engineering

Research Council for a grant ^{towards the cost of the} Bruker CXP spectrometer ^and for the award of a

fellowship ^{to Dr. B. A. Timini.}