On the applicability of the rotational diffusion model in liquid crystalline solvents. A nuclear magnetic relaxation study of toluene

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On the applicability of the rotational diffusion model in liquid crystalline solvents. A nuclear magnetic

relaxation study of toluene

J. Bulthuis, L. Plomp

To cite this version:

J. Bulthuis, L. Plomp. On the applicability of the rotational diffusion model in liquid crystalline

solvents. A nuclear magnetic relaxation study of toluene. Journal de Physique, 1990, 51 (22), pp.2581-

2593. �10.1051/jphys:0199000510220258100�. �jpa-00212555�

On the applicability of the rotational diffusion model in liquid crystalline solvents. A nuclear magnetic relaxation study

of toluene

J. Bulthuis and L. Plomp (*)

Department ^of Physical and Theoretical Chemistry, ^Chemical Laboratory, Vrije Universiteit, ^de Boelelaan 1083, ^{1081 HV} Amsterdam, The Netherlands

(Received ¹⁴ May 1990, revised 18 July 1990, accepted ²⁰ July 1990)

Abstract. 2014 The model of rotational diffusion in the presence of ^an orienting potential ^{has been} adapted ^{to the case of an} asymmetric top molecule in an orienting potential that is defined by ^two ordering parameters. ^It has been used to describe the relaxation by ^the quadrupolar mechanism of the deuterons in toluene, dissolved in

Phase V ». No set of diffusion constants could be found, which simultaneously reproduce ^all spectral densities that were derived from the measured relaxation rates. Only very approximate agreement could be obtained with sets of diffusion constants in which two constants are of the same order of magnitude and the diffusion about the axis in the plane of the benzene ring ^and perpendicular ^to the axis of the methylgroup, ^{is slower} by

an order of magnitude. ^The anisotropy ^{in the} viscosity ^of the solution has not explicitly ^{been taken}

into account, but it has been indicated how this might be done. At this stage, therefore, ^no definitive conclusions can be drawn, ^but clearly ^the application of the rotational diffusion model in liquid crystalline solutions should be considered with care.

Classification

Physics ^Abstracts

61.30 - 76.60

1. Introduction.

’

In previous ^nuclear magnetic relaxation studies of small probe molecules in liquid crystals, ^we

have laid emphasis ^on mechanisms that can explain ^the observed frequency dependent

relaxation rates [1, 2]. Recently, ^a commonly used reorientational model was tested, namely

fast rotational diffusion, under the influence of an orienting potential ^{that is} slowly fluctuating, by measuring ^2H relaxation times of a series of probe molecules of increasing ^size.

The advantage ^of using ^suitable probe molecules is that the reorientational motion is not

unduly complicated by ^internal degrees of freedom, ^{as will} mostly ^{be the case for} liquid crystal

molecules. This advantage ^{is offset} by ^{the smaller effects to be} expected. ^In addition, ^{use of a} simple rotational diffusion model for liquid crystal molecules, by assuming ^{that the diffusion}

tensor is axially symmetric, ^{may be} justified by ^the elongated shape ^{of these} molecules. For

small, non-axially symmetric probe molecules, ^{as for instance} toluene, ^{this is not a valid}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510220258100

assumption. This is the reason why for the deuterated probes toluene, p-xylene, ^and tolan, ^we

have concentrated on the _para and methyl deuterons. The relaxation of these nuclei is not, ⁱⁿ

a first approximation, ^affected by rotations about the longest symmetry axis, provided ^the

internal rotation of the methyl group is much faster, ^{so that it seems} reasonable to assume

effective axially symmetric diffusion for these particular nuclei. The results of the relaxation measurements for the ortho- and meta-deuterons can, however, ^{not be} explained ^{on the same}

basis. Therefore a full analysis ^{in terms} of rotational diffusion of a non-axially symmetric

molecule in an orienting potential is in order. Previously, Vold and Vold [3] worked into the

same direction, ^but they preferred ^to simplify ^the model of reorientational motion [4]. ^We

have not yet applied ^their model, primarily ^{because we wish to find out whether or not the} general ^and well-established rotation diffusion model is able to satisfactorily explain ^the results, ^{but also because in their model the} molecular motion is decomposed ^{in a rather} speculative way.

The measured relaxation rates can be expressed ^{in terms of} spectral densities, ^{which are the}

Fourier transforms of classical correlation functions describing the reorientational motions of the molecule studied. There are several strategies ^to calculate these correlation functions [5- 9]. Here, ^{as in} previous papers, ^we have used the stochastic molecular theory developed by

Freed and coworkers [5, 6]. ^{In the} following section this theory ^is briefly recapitulated ^and

extended for the case of fully anisotropic diffusion. In section 4 we apply the results to the

case of toluene as an illustrative example. Moreover, for this molecule data on the anisotropic

motion in ^an isotropic liquid ^are available. Preliminary calculations show that the results for

p-ylene ^{and tolan are} similar, except ^{that for} p-xylene ^a comparison ^{with the} para-deuteron ^is

of course not possible.

2. Theory.

The Hamiltonian of the interaction that gives ^{rise to} spin relaxation is written in spherical

tensor notation as

F represents the variables of the bath ; ^{A the} spin operators ^{of the} spin system studied. Since F is most conveniently ^defined in the molecular frame, ^a transformation to this frame is carried out, which, by ^{virtue of the} spherical ^tensor notation, ^can be written in terms of

Wigner rotation matrices

f2 represents ^{the Euler} angles ( a , /3, y ), defining ^the orientation of the molecular axes in the

laboratory system. ^The subscript t ^{of fl indicates} that f2 is time dependent.

Using ^standard relaxation theory, the relaxation rates can be related to spectral densities, which are the Fourier transforms of correlation functions associated with the interaction tensors F ; these correlation functions can be defined as

M and M’ refer to the space-fixed axes ; K and K’ to the molecular ^ones. To simplify ^the

notation, ^we replace il by il 0 and f2 t + r by fl.

The correlation functions of the Wigner rotation matrices in equation (3)

can be calculated if the relevant orientational probability ^{functions are known.}

The time evolution of the orientational probability P (fl , t ) is assumed to be adequately

described by ^the Smoluchowski equation [11-13]

We have assumed that the diffusion tensor is independent of the instantaneous molecular orientation with respect ^to the director of the liquid crystal. ^{This is a common} assumption, though ^not necessarily ^{the most realistic one, but one which can} hardly ^{be avoided in order to} keep the number of adjustable parameters ^within acceptable limits. The simplest way in which the angular dependence ^{of the diffusion tensor} might be taken into account, through ^the anisotropic viscosity ^{of the} medium, ^{seems to be} by ^{an extension to the} asymmetric ^{case of the} procedure recently suggested by Dozov and Kirov [14]. ^This straightforward approach yet complicates equation (5) considerably, ^{since L also acts on the diffusion tensor.}

U(n) ^{is the} orienting pseudopotential ^of the nematic medium, ^{which can} be written as a

series expansion ⁱⁿ Wigner rotation matrix elements

This potential is uniaxial and assumed to be cylindrically symmetric. ^{This means that averages}

over a vanish, ^{unless M}

0, ^{and that L must be even.} Often, only ^{the first term in} equation (6) is retained. Since we are dealing ^with probe ^molecules having C2 v-symmetry, ^{we have to}

include terms with K:o 0, ^{but all terms with} L >- 2 will be omitted. We note that this is mainly

based on practical considerations : terms with L

4 cannot be determined by NMR,

although (P 4) ^{can be obtained by} fluorescence depolarization [15]. Fortunately, ^{this term is} small, ^{if not} completely negligible. Therefore, ^we persist ⁱⁿ keeping only ^{terms with}

L

2 in equation (6). Using ^the terminology of Polnaszek and Freed [6],

the difference with Polnaszek and Freed being that here we have defined the angles

fl

(a, f3, y ) ^with respect ^{to the} laboratory frame rather than the molecular frame.

Moreover, ^{the same authors} evidently ^{take a} (y ^{in our} case) ^{relative to the} y-axis rather than to the x-axis. One should realize this difference when using their notation :

Values of A and _p were obtained from the measured orientation parameters SZZ ^and

SXx - Syy by solving ^the appropriate integral equations [1].

Several methods have been proposed ^{to solve} equation (5) ^{in order to} derive rotational correlation functions [5, 7-9]. We have extended the approach of Polnaszek et al. [5], ^to

accommodate for non-axially symmetric diffusion. First, ^the operator Fn ⁱⁿ equation (5) ^is

symmetrized

where P eq (n) ^{is the} equilibrium distribution function

The symmetrized ^diffusion operator ^{can then be written as}

where

In reference [6] the coefficients X have been given ^{for the case} of axially symmetric ^diffusion.

If the molecule has only sufficient symmetry ^{as to find the axis} system in which the diffusion

tensor diagonalizes, they ^are

For axial symmetry, ^these equations (13) ^{reduce to} equations (2.18) ^{of reference} [6] (with ^the exception ^of (2.18d), ^which obviously ^{contains a} typographical error).

The eigenfunctions ^of f n ^can be written as linear combinations of the eigenfunctions ^{of the} symmetric ^rotor

where, using ^Rose’s convention [16]

In equation (14) ^{no summation over m is needed, since the} orienting potential ^{is taken} axially

symmetric. Instead of using ^a perturbation approach ^to calculate the coefficients a in the

eigenfunction expansion ⁱⁿ equation (14), ^{for a} limited number of terms, the corresponding

matrix was diagonalized. ^Control calculations showed that contributions from terms with

n

3 and f

4 to the eigenfunctions (Eq. (14)) ^could safely ^be neglected ^{in most cases.} Only

in some instances was it necessary ^to include terms with n

4 and thus diagonalizing ^a

25 x 25 matrix. Those cases were obvious by giving ^sudden deviations from a smooth curve

(presumably ^caused by ^near degeneracies), ^when calculating spectral ^{densities for a series of} slowly varying input values of diffusion constant-ratios.

The conditional probability, ^{needed to calculate} the rotational correlation functions (4),

can be expanded ^{in the} eigenfunctions ^and eigenvalues ^{of the} operator ^{in the} Smoluchowski

equation (5). ^{If the} symmetrized operator, equation (11), ^{is used}

where p(noln, ^{T )} is related to p(noln, ^{T )} by

The equilibrium distribution is given ⁱⁿ equation (10).

Substitution of equations (17) ^and (10) ⁱⁿ equation (4) ^{leads to a term} (Prq (n 0) P eq (n) )1/2

which, by expanding ^the exponentials ⁱⁿ equation (10) ^{as a} Taylor-series, ^{can be written as}

Here, ^{ll = À} /3 ^{and R}

^p / J6; ^{C is a} normalization constant. In the calculations the

expansions ^{were carried to terms with} p, q

4, ^which appeared ^{to be} fully adequate ^{for the}

values of À and _p in the case of toluene (A

^{1.007 and} p = - 0.377).

The multiple products of rotation matrix elements occurring ⁱⁿ equation (18) ^{were reduced}

to binary products by successively applying Clebsch-Gordan series expansions. ^The remaining integrals, after substitution of (18) ⁱⁿ (4), ^were written in terms of 3 j symbols, ^{and calculated}

as such. Once the correlation functions (4) ^{have been} determined, ^the spectral ^{densities can}

be found by performing ^{a Fourier} transformation. The results were simplified by assuming

that the extreme narrowing ^{condition holds for each} successive term. According ^{to the}

definition used here, ^the components ^{of the} spectral density in the molecular frame do not include the elements of the interaction tensor F.

Because of the axial symmetry ^{about the} laboratory z-axis, ^{M’ - -} M, ^{and hence the} components ^{of the} spectral density ^{in the} laboratory ^{frame can be indicated} by ^a single ^index

M. According ^to equation (3), ^these components JM, ^{are related to} those in the molecular frame as

The tensor F, describing ^the interaction of the deuteron quadrupole ^moment with the local

electric field gradient is, ⁱⁿ general, ^not diagonal in the molecular frame. The components ⁱⁿ

the molecular frame, however, ^can simply be related to those in the local frame of the

deuteron of interest by ^an additional transformation. Then

where K and K’ refer to the molecular frame, ^{and I and I’ to the} local frame.

M

0, ^{1 or 2. For axial} symmetry in the local frame, ^which applies ^{if the} asymmetry ^{of the} efg

tensor is negligible ^{as is} usually ^{the case for} deuterium, only ^{terms with I}

^{l’ = 0 remain and}

one has

To perform ^the calculations, ^a computer program ^was written in which the steps ^mentioned above, ^{were built} in, ^the only difference being that the Euler angles ^{were defined with} respect

to the molecular frame, rather than the laboratory ^frame. The result should be invariant to this choice. We shall address this point again in section 5.

3. Experimental.

2H relaxation measurements were carried out on perdeuterated toluene, dissolved in nematic

liquid crystal ^N5 (formerly ^{known as « Phase} V » ; Merck, Darmstadt) ^{at a} temperature ^{of ca.}

30 °C and at different field strengths. The relaxation rates of quadrupolar - and Zeeman order

were obtained by combining results of Jeener-Broekaert -, or, where appropriate, ^selective

inversion recovery measurements [1, 2].

For further details of the experimental procedures ^{we refer to reference} [1].

4. Results and discussion.

From the toluene experiments, ^three pairs ^of spectral density components ^{can be} derived, ^i.e.

JI and J2 ^{for the} methyl deuterons, the para deuteron, and the ortho- and meta deuterons, respectively. ^{Since the} rotational diffusion of toluene _may be assumed to be in the extreme

narrowing limit for all measuring frequencies used, ^the spectral density ^is expected ^{to be} frequency independent. JI and, ^{to a minor} extent, J2, however, ^{contain a} frequency dependent contribution, ^{which in} previous papers [1, 2] ^was accounted for by ^director fluctuations, ^{and which vanish at} the highest measuring frequencies ^used. This makes it

possible to extract the frequency independent part ^{of the} spectral density components, ^which is assumed to be solely ^{due to} rotational diffusion.

To determine the principal elements of the diffusion tensor from the experimental spectral densities, ^{a number} of parameters ^{must be known.} First, ^the quadrupole couplings ^{of both the} methyl- ^and ring deuterons. These were obtained from other measurements in liquid crystals [19]. Secondly, ^{the molecular} geometry, ^{or rather} symmetry, should be known in order to

relate spectral density components ^{in the molecular frame to those} in the local frames of the

deuterons, ^via equation (19). ^{For our} purpose it ^was adequate ^to adopt hexagonal symmetry of the benzene ring. ^Small deviations, ^{as for} instance found by ^{Diehl et al.} [20], ^{do not affect}

our results significantly.

The constants A and p, defining the lowest order terms of the orienting potential, ^were

calculated from the orientation parameters SZZ ^and Sxx - Syy, ^{which, in turn, can be calculated}

either from the proton dipolar splittings, using the known geometry [20], ^{or from the}

deuterium spectra, using appropriate values of the qcc’s. ^In the latter case the relative signs ^of

the orientation parameters ^{can not} be determined. Only ^an elementary analysis ^{of the} proton

spectrum ^was sufficient, however, ^to show that the parameters ^have opposite signs. ^{This is in} agreement ^{with the} analysis previously ^made by ^{Diehl et al.} [20] ^{of the} proton spectrum ^of toluene in the nematic phase ^{of EBBA.} According ^{to the} definition of the orienting potential

in equation (8), ^{À and p} appear ^to have the same sign ^as SZZ ^and Sxx - Syy, respectively. ^The

values are : SZZ

0.139, Sxx-Syy = - 0.120 ; A

1.007, p - - 0.377. ^{The molecular axes}

are chosen ^as in figure ^1.

Fig. ^1. - Choice of molecular axes in toluene.

From relaxation measurements at several frequencies, ^the frequency independent part ^of the spectral density components JI ^and J2 for the three different deuterons in toluene could be estimated [1]. The results are collected in table 1.

Using the model of asymmetric ^rotation diffusion in an orienting potential, ^{as described} above, ^the problem ^{is to find a set} of diffusion constants which can reproduce ^the spectral

densities in this table. To this end, JI ^and J2 of the three deuterons were calculated for ^a large

range of relative diffusion constants D xX’ Dyy, ^{and DZZ. In figures 2} and 3 ratios

Table 1. - Frequency-independent contributions to spectral ^densities for ^{deuterons in toluene,} obtained from frequency-dependent relaxation measurements of deuterated toluene dissolved in

liquid crystal ^{Phase V} (see ^Ref. [1]). Spectral ^{densities are} given ⁱⁿ picoseconds. ^Estimated

(maximum) errors are in parentheses.

Fig. 2. - Plots of calculated ratios (JIIJ2),, (JIIJ2),,,@,,,, ^and (JI)o,m/ (JI)p ^versus Dxx/ Dyy for different ratios Dxx/ Dzz, ^{viz. (1) 0.2, (2) 0.4, (3) 1.0, (4) 2.0, (5) 4.0, (6)} 8.0. The dots indicate calculated values

through which smooth curves are drawn. These curves are also based on calculated points outside the

regions drawn. Deviations of the dots from a smooth curve reflect the uncertainties in the calculations

(see text). Experimental ^{values of} (.J../ J2)p, (JI/ J2)o, m’ ^and (JI)o, ml (JI)p ^{are indicated} by dashed lines.

Fig. 3. - Plots of calculated ratios (Jt/.J.2)p, (Jt/J2)o,m, ^and (JI).,,nl (JI), ^versus Dyy/Dzz for different

ratios Dxx/ Dyy, ^{viz. (1) 0.5, (2) 0.75, (3) 1.0, (4) 2.0, (5) 4.0, (6)} 8.0. See also figure ^2.

(JI /J2)p, (JI IJ2).,

^{and (JI} )0, ml (Jl )p ^are plotted against Dxx/ Dyy ^and Dyy/ Dzz, respectively,

for different ratios Dxx/ Dzz ^and DxX/Dyy.

The intersection point ^{of the curves in} figure ^{2a shows that} (JI/ J2)p ^is independent ^of Dzz ^if Dxx

D yy, ^{as it should be, since this} corresponds ^{with the case of axial} symmetry ^about the z-axis (see Fig. 1). ^If Dxx=F D yy, ^however, (Jl /J2 )p ^{will depend on Dzz.} This behaviour is also to be expected ^{in an} isotropic solution : because of the asymmetry, rotations about the z-

axis are not independent of those about the x- and y-axis any ^more. Nevertheless,

(JI/ J2)p is rather insensitive to changes ⁱⁿ DZZ. ^{In the same} way, (JI/ J2)o, m ^{does not strongly}

depend ^on D yy, ^{as can easily be seen} by combining figures ^{2b and 3b.}

Likewise, for axial symmetry ^{about the} x-axis, (JI )0, ml (JI)p ^is independent ^of Dxx ^Of

course, this does not apply ^{to the ratio} JI / J2 for either deuteron, ^{due to the} presence of the

orienting potential.

Inspection ^of figure ^2a shows, that in order to reproduce ^the experimental ^{value of}

(JI/ J2)p, DXJDYY should be smaller than unity. Contrary ^to this, it follows from figure ^{2c that}

the reverse should be true. Comparison of the other plots also makes clear that the

experimental ^results impose conflicting conditions on the model calculations. Taking very conservative estimates of the ^errors in the experimental ^{ratios of} spectral density components, these conflicting conditions can be summarized as : (i) Dxx/ Dvv S 1.0 ^{(Figs. 2a and 3a) ;} (ii)

From combined studies of depolarized Rayleigh scattering ^and 13C _NMR, Bauer et al. [21]

found 2Dxx _ DZZ, depending ^{on the} viscosity ^{of the} solution, ^and Dxx _ 2D yy. ^{In addition,}

Dxx ^{was almost} independent ^{of the} viscosity, in accordance with a hydrodynamic slip ^model.

As ^one _may expect ^{from the same} model, Dzz ^and Dyy ^{were shown to have a clear viscosity} dependence, which increases in this order. (Note ^{that our x- and} z-axes are interchanged compared with Ref. [21].) ^{For neat toluene at about 23} °C, ^{with a} viscosity ^{of 5.52 x} 10- 4 _{Pa s,} Bauer et al. [21] ] obtained the following values for the diffusion constants

Dxx= (12.8 ±3.0) ^{x 1010} s- 1, Dyy

^{(2.1 ± 0.3 ) x 1 0’o} s - 1, ^and Dzz

(5.3 ^± 0.5 ) ^x 101° s-1. These values of the diffusion constants could simply ^{be obtained} by using the relation Te

1/6 ^{D, since the} reported correlation times were defined in this _way.

On the basis of the results obtained by ^{Bauer et al.} [21], ^{which are in} agreement ^{with a} hydrodynamic point ^of view, the first condition mentioned above, Dxx/Dyy S ^{1.0, is very}

unlikely. Similarly, ^condition (iii), Dxxl Dzz ^{2.0, is at first} sight preferable ^over (iv), ^{at least}

in an isotropic ^solution. Previously [1], however, ^we have assumed the molecule to reorient as a prolate ellipsoid, ^with D zzl D xx

^{2, in agreement} with results obtained by Lambert et al.

[22], ^{and also} by ^Bluhm [23], ^from 13C-relaxation measurements. The latter author, ^who studied a series of methyl-substituted benzenes, ^{found values} of the rotational diffusion

constants for toluene and mesitylene ^{a factor of two} smaller than those found by ^{Bauer et al.}

[21]. Surprisingly, ^{he leaves a more} fundamental difference, namely DZZ being larger ^than Dxx ^{rather than} smaller, ^{as found} by ^{Bauer et} al., unnoticed. We note in passing ^{that Bluhm}

obtained nearly always physically meaningless ^{results when} applying ^a fully anisotropic

diffusion model (i.e. Dxx:o D yy, ^{Dz), and he ascribed} this result to the attainable experimen-

tal accuracy being totally inadequate [23]. ^{In the} present study ^{this is not the} problem : we just

do not obtain a reasonable fit to the experimental values, ^{not even when} allowing ^for large experimental ^errors.

If there were conclusive evidence that in an isotropic solution the condition D xxi D zz S 1.0

is unlikely indeed, ^{it still} might ^{be the} preferable condition in a nematic solution, ^{as a}

consequence of the anisotropic viscosity. ^{If one} accepts this, ^{the best} compromise between the

calculated and experimental spectral density ratios leads to nearly equal values for

DXx ^and Dzz, ^{and to a} D yy value that is smaller by ^{at least one order of} magnitude. ^{As an} example, ^if DxX/Dyy

^{40, and} Dxx/Dzz

^{0.8, the following} ratios of spectral ^{densities are}

found : (Jl /J2)p

^1.32, (Jl /J2 )o, m

^{1.06, and} (JI).@ml(JI)p = 0.46. The absolute values of the diffusion constants for this sample calculation are : Dxx

^{1.0 x 1010} s-1, Dyy

^{2.5 x}

10g s- l, ^and D zz

^{1.25 x 101°} s-1. The value of Dyy ^{is about two orders of magnitude smaller}

than for neat toluene [21], which is in agreement with the much more viscous solution

(’Tl ,..., 6 x 10- 2 Pa s [1] ). ^{The other two values differ} only ^{about an order of} magnitude. ^This

may, ^at least partly, be attributed to their much weaker viscosity dependence [21].

So far we have not considered the methyl deuterons. If the reasonable assumption ^{is made}

that the methylgroup undergoes fast internal rotation, ^fair agreement ^{with the} experimental

ratio (JI/J2)methyl is obtained for ratios Dzz/ Dxx ^and Dzz/ Dyy ^{larger than 500 or so. This is as}

good ^{as may be} expected ^{from the} questionable application ^{of the diffusion model to the case}

of fast internal rotation [23, 24].

5. Concluding ^remarks.

From our calculations it appears that ^no satisfactory agreement of the calculated spectral

densities with the experimental relaxation rates can be obtained _{with any} set of rotational diffusion constants. It is conspicuous ^{that at least a} reasonable agreement could be obtained if the relative sign ^of the orientation parameters ^was reversed, using ^the definition of Euler

angles in reference [6] (i.e. ^{a, or y,} depending ^on the direction of the transformation, ^taken

relative to the y-axis). ^This is, ^of course, ^not justified, unless this definition is used

consistently. ^We have scrutinized our calculations for _any inconsistent use of convention, which might ^{have the same effect as} sign reversal of one of the order parameters, ^{but we have} been unable to identify irregularities. Evidently, the relative signs ^{of the terms in}

A and _p in equations (13), ^are correct, because the result of the calculation is invariant to a

transformation of molecular _axes, and the concomitant transformation of À and _p. We also

carefully considered the choice of axis system in which the Euler angles ^are defined. In the outline of the theory, given above, ^these angles ^{were defined relative to the} laboratory ^axes,

whereas they ^{were defined relative to} the molecular axes when writing ^the computer program.

In either _case, the diffusion problem is formulated and solved in the molecular coordinate system. ^The correlation functions should be independent ^{of the sense of rotation} and from the

equations ^{that follow} using ^either definition, ^{it can be seen} that this is the case. Also, ^{from the}

form of the diffusion operator (Eq. ( 11 )), ^{it is} immediately clear that the sign ^of

(D xx - D yy) ^{is invariant to} the choice of definition, contrary ^{to what} has been stated

previously by ^Huntress [17]. Further evidence of the correctness of the calculations was

obtained by comparing the results for limiting cases, i.e. asymmetric rotational diffusion in an

isotropic liquid, ^and axially symmetric rotation in a single parameter orienting potential, ^with

those obtained by ^Huntress [17] ^and by Lin and Freed [18], respectively. ^{In both cases} perfect agreement ^{was obtained.}

Having excluded these more trivial sources of the discrepancy found, ^the following possible explanations emerge.

a. The neglect ^of higher than second order terms in the potential ^{of mean} orienting torque (Eq. (6)). ^{These terms are not} accessible by ^{the NMR} experiment, ^but they might ^be

estimated in the framework of the Maier-Saupe theory by extending ^the orienting potential by

the appropriate ^{4th order} spherical ^{harmonics and} recalculating ^{the order} parameters ^until consistency is reached. The value of (P 4), calculated from an orienting potential ^with

,À - 1 and p = 0, ^{is more than an order of} magnitude ^{smaller than} Szz (i.e. (P2». ^{This is in}

agreement ^with experimental ^values given ^{in the} literature (e.g. ^Ref. [15]). In view of this, ^{it is}

unlikely ^{that the} disagreement ^between theory ^and experiment may be attributed to the

neglect ^of higher ^{order terms in the} orienting potential.

b. The neglect ^{of the} anisotropy ^{in the} viscosity of the nematic solution. This is a

complication, ^{which is} very difficult to assess for two reasons. First, ^{no accurate} experimental

data is available on the anisotropy ^{in the} viscosity (not ^to mention the intricate problem ^of relating the measured viscosity components ^{to those of interest} here). Second, ^{the relation}

between the viscosity ^{and the different diffusion constants is not known. As an} approximation,

one could use a hydrodynamic model, ^with viscosity dependencies of the diffusion constants

reported by Bauer et al. [21].

To evaluate the effect of the anisotropic viscosity ^{in a} highly qualitative way, the angular dependence of the diffusion tensor might ^be introduced in the _way suggested by ^{Dozov and}

Kirov [14]. ^As already mentioned, ^this presents ^a complicated problem ^{which we have not}

started to tackle yet. ^{It is hard to see whether a better fit with the} experimental results could be obtained, but the effect of the viscous anisotropy ^{could at} least be made more explicit, allowing ^{a better} comparison ^{of the diffusion constants} with those obtained in isotropic

solutions. In the results presented ^{here, the effect} of the viscous anisotropy ^{is not} transparent, and could affect the diffusion constants implicitly, possibly leading ^to D zz being larger ^than D xX’ ^as suggested ^above.

c. One _may think of dynamical local ^{structure effects} (e.g. ^{described in terms of a «} Slowly Relaxing Local Structure »

SRLS - model by Freed et al. [6, 13]), ^{which in addition to}

director fluctuations play ^{a role in the} frequency dependent parts ^{of the} spectral ^densities JI and J2 of the para- and methyl-deuterons. However, ^{for a molecule like} toluene, ^{which is} only moderately ^{ordered in Phase} V, ^{these effects are estimated to be} small, and the ratios

JI/ J2 ^{will hardly be affected.}

Moro et al. [25, 26] ^{discussed the} coupling of rotational and translational motion, ^{which is}

of significance ^{at low} frequencies. ^{In smectic} phases ^{these lead to} relaxation effects similar to those ascribed to SRLS by Freed et al. [6, 13], mentioned above. For nematic phases, however, Moro et al. [26] argue that couplings between rotations and translations ^are ineffective.

Clearly, ^such suggestions challenge ^the correctness of the simple form of the Smoluchowski

equation given ⁱⁿ equation (5). ^Our calculations too indicate that this form is an oversimplifi- cation, ^{at least at a level where the} orientational dependence ^{of the diffusion tensor is not}

taken into account. On the basis of our relaxation data a definite conclusion as to the validity

of equation (5) ^{is therefore not warranted} until this deficiency ^{has been removed.}

Acknowledgement.

We acknowledge stimulating discussions with Dr. G. van der Zwan.

References

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