STATIC AND DYNAMIC BEHAVIOR NEAR THE ORDER DISORDER TRANSITION OF NEMATIC LIQUID CRYSTALS

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STATIC AND DYNAMIC BEHAVIOR NEAR THE ORDER DISORDER TRANSITION OF NEMATIC

LIQUID CRYSTALS

T. Stinson, J. Litster, N. Clark

To cite this version:

T. Stinson, J. Litster, N. Clark. STATIC AND DYNAMIC BEHAVIOR NEAR THE ORDER DIS-

ORDER TRANSITION OF NEMATIC LIQUID CRYSTALS. Journal de Physique Colloques, 1972,

33 (C1), pp.C1-69-C1-75. �10.1051/jphyscol:1972113�. �jpa-00214903�

JOURNAL DE PHYSIQUE

Colloque C1, suppldment au no 2-3, Tome 33, Fkvrier-Mars 1972, page C1-69

STATIC AND DYNAMIC BEHAVIOR NEAR THE ORDER DISORDER TRANSITION OF NEMATIC LIQUID CRYSTALS (*)

T. W. STINSON and J. D. LITSTER

Department of Physics and Center for Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Mass. 02139

N. A. CLARK

(**)

Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts, 02138

Rbumk. -

Nous avons mesure le spectre et la polarisation de la lurnikre ddiffusee par les fluc- tuations de la phase isotrope du cristal liquide nkmatique p-methoxybenzol p-n-butylaniline. La birkfringence magnetique et l'intensit6 de la lumikre diffusee divergent cornrne (T- T$)-1 oh T,* est 1 OC de moins que la temperature de la transition de phase. La lumiere diffusk polariske parallelement

^?

la lumikre incidente a

ⁱ un

spectre Lorentzian de largeur qui varie comme (T- T$)/v. Nous avons determine la valeur du coefficient du transport

v

et trouvons qu'il a la m6me dkpendance avec la temperature que la viscositk de cisaillement. Le spectre de la lumiitl-e diffusee dkpolarisCe montre le couplage entre le pararnktre d'ordre et les ondes transversales excittes thermiquement. Nous avons determine le coefficient de couplage

^p

et avons trouve qu'il augmente anormalement pendant les

5

OC au-dessus de la transition de phase. La theorre phenomenologique de de Gennes est compatible avec nos resultats et nous donnons les valeurs num6riques des para- metres de sa thkorie.

Abstract.

- We have measured the spectrum and polarization of light scattered by fluctuations in the isotropic phase of the nematic liquid crystal p-methoxy benzylidene p-n-butylanilene. The magnetic birefringence and the intensity of the scattered light both diverge as (T-

T,*)-1

where Tc* is

1

OK below the phase transition temperature. The scattered light polarized parallel to the incident light has a Lorentzian spectrum of width

^T

that varies as (T- T,*)/v. We have determined the transport coefficient

^v

and find it has the same temperature dependence as the shear viscosity.

The spectrum of the depolarized scattered light shows evidence of coupling between the order parameter and thermally excited shear waves. We determine the coupling coefficient

p

and find it shows an anomalous increase in the

5

OC above the phase transition temperature. The phenome- nological theory of de Gennes is consistent with our results and we give numerical values for the parameters in his theory.

I. Introduction.

-

We report here the results of experiments we have carried out using light scattering techniques to study the static and dynamic properties of the isotropic phase of the nematic liquid crystal p-methoxy benzylidene p-n-butylanilene (MBBA).

In the nematic phase of this material the anisotropic molecules exhibit a long range orientational order with their long axes tending to align parallel, and they are randomly oriented in the isotropic phase.

The centers of mass of the molecules are randomly located (as in a normal liquid) in both phases and both phases have the usual hydrodynamic properties of liquids. The nematic-isotropic phase transition is similar to the order-disorder phase transitions that occur at the liquid gas critical point or magnetic Curie point. One prominent difference is that the nematic-isotropic transition is first order (has a latent

(*)

Supported

by

Advanced Research Projects Agency under contract

DAHC 15-67-C-0222.

heat) while critical point phase transitions are second order. The first order transition in liquid crystals is a result of the symmetry of the ordered phase [I].

Our measurements of the static properties of the isotropic liquid crystal phase show a divergence of the magnetic birefringence (analogous to the diver- ging susceptibility near the magnetic Curie point) as the phase transition is approached as well as a diverging intensity of scattered light (like the critical opalescence in a pure fluid). From the spectrum of the scattered light we obtain information on the dynamic properties of the liquid crystal. We observe a critical slowing of the fluctuations in order parameter and a coupling between the order parameter and thermally excited shear waves in the isotropic phase. De Gennes [2] has proposed a phenomenological model for the nematic-isotropic phase transition. We find that this model is consistent with our measurements and analyze our results to obtain numerical values for

(**)

Supported

by

ARPA

:

SD-88. the parameters of the model.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972113

el-70 T. W. STINSON AND J. D. LITSTER

11. Static properties. - We shall first present the

predictions of de Gennes' model for the static behavior in the vicinity of the nematic-isotropic transition and compare them with the results of our measurements.

In a later section we will discuss dynamic properties.

To describe the order-disorder phase transition, it is necessary to define an order parameter. Most liquid crystals possess dipole moments, but all expe- rimental evidence [3] is that these do not order and that the nematic phase has uniaxial quadrupolar symmetry. The order may then be specified by a symmetric traceless tensor [2] whose microscopic definition would be

Q

a@ = -

i < 3 Ia I,

^-

ha, > (1) where I,,

[,

are the Cartesian components of the symmetry axis of the molecules and the average is carried out over a small but macroscopic volume. For a biaxial nematic Qap would have five independent components, but in a uniaxial crystal we may write

where nu, np are the components of a unit vector parallel to the local optic axis (usually called the

<t

director

>))

and the scalar order parameter S is

given by [4]

s = * < 3 c 0 s 2 e - 1. (3) with 8 the angle between a molecular symmetry axis and the optic axis. Rotational invariance in the iso- tropic phase requires that the anisotropy in all other quantities be a scalar multiple of Quo. Therefore, we may write

for the diamagnetic susceptibility and dielectric constant tensors, respectively. The quantities AX and A& are then the anisotropies in xaB and

E,,

for a completely ordered liquid crystal and equations (4) and (5) provide a macroscopic definition of the order parameter.

In de Gennes' model the free energy per unit volume in the vicinity of the phase transition is expanded in a Landau type power series in temperature and the order parameter. Let us write this as

(")

A B

q

⁼

v ~ ( P , T) + 2

Qap Qpa

- 7 Q*P QPY Qya +

(*) In general there are two fourth power invariants, but for a a traceless tensor one can show

e = ~ epA ey6 e6.

^{= r1/21}

(ap

^Q,,)~

.

The notation is a, = alax,.

For a phase transition to occur it is necessary that the coefficient A go through zero [l]

;

the simplest form is A

=

a(T

- T:).

We may use (6) to calculate the desired static properties of the liquid crystal.

Let us first discuss the order-disorder phase tran- sition for a spatially uniform sample in zero field.

This is most easily done by rewriting (6) using the scalar order parameter

The equilibrium value of S minimizes the free energy and so is the root of

which corresponds to the lowest free energy. In the isotropic phase this is S

=

0, but for A < 25 B2/108 C a second minimum appears in

(7)

for a finite value of S.

As a becomes smaller ( T

+

T,*) the free energy at the second minimum becomes lovier, equalling the free energy for S

=

0 when A

=

50 ~'1243 C. Therefore at a temperature TK

⁼

T: + 50 B2/243 aC a first order phase transition to an ordered state occurs.

From (7) we predict the discontinuity in S to be SK

=

20 B/27 C and the latent heat is (314) aT, s;.

For values of S as large as S, more than just three terms in a power series, as in eq. (7), are likely required.

Nonetheless this simplified analysis shows that the first order nature of the phase transition is imposed by the cubic term in (7). This cubic term reflects the physical fact that positive anti negative values of S are not equivalent. This is because S > 0 represents positive birefringence and S

^<:

0 represents negative birefringence

;

the two situations correspond to quite different physical ordering of the molecules. (For negative S the molecules tend to order with their axes confined to a plane but randomly oriented within the plane.)

We have seen that the symmetry of the ordered state results in a mathematical form for the free energy that generally requires a first order phase transition.

However, if the coefficient B were zero the free energy

expression would be identical to that in the Landau

model for second order phase transitions (critical

points). If we are far enough above T: that the

quadratic terms in (6) dominate we might well expect

the liquid crystal to exhibit pretransitional effects

similar to those in the vicinity of a critical point. For

example, let us consider the effect of an applied

magnetic field on a uniform sample. Minimizing the

free energy (6) and keeping lowest order terms we

find

STATIC AND DYNAMIC BEHAVIOR NEAR THE ORDER DISORDER TRANSITION C1-71

If we take H to lie along z the resulting birefringence is found from (5) to be

The diverging magnetically induced birefringence is analogous to the diverging susceptibility near a magnetic critical point or diverging compressibility at the critical point of a pure fluid. This divergence was first observed in para-azoxyanisoie ( P A A ) by Zadoc-Kahn [5] and has also been studied by Tsvetkov [6]. Quite precise measurements on MBBA

In (11) Po is the power in the incident light beam which traverses a path length I in the medium, o, is the frequency of the incident light,

c

the speed of light in vacuum, dP,,/dQ the power scattered per unit solid angle, and 6gif(q) is the Fourier component of the dielectric constant that couples incident light polarized along i to scattered light polarized along S.

We use the equipartition theorem in conjunction with (6) to obtain the fluctuations in the order para- meter tensor and combine this with (5) to obtain

< >. The change in free energy associated with fluctuations in volume V is obtained from (6) as have been made by Stinson and Litster 171

;

their

smv=

/

results are shown in figure 1 for two samples with v d3r AQaj(r) Q ~ a ( r ) + L I aaQpy(r) aaQsy(r) +

slightly different transition temperatures. The data

in figure 1 clearly demonstrate the divergence predicted + L2 daQay(r) apQsy(r) ) . (12) by (10) and a least square fit to either sample yields

the same value for a. With

fi =

1.605 at 6 328 A [8], The equilibrium values of all Qas are zero in the we obtain a

⁼

(3.1 0.03)

^x

1012 AE ~ ~ e r ~ c m - ~ OK-l. isotropic phase and we decompose the fluctuations into statistically independent Fourier components as follows

1 1 1

j d3r ei(q-qT).r -

-

v h ( q

-

q') .

v

( 1 3 4

We choose q to lie along the z direction and subs- titute into (12). Keeping in mind that Q,# is traceless, we obtain

40 4 4 48 5 2 5 6 6 0

TEMPERATURE P C )

FIG. 1. - Temperature dependence of the magnetic birefrin- gence in MBBA. Data are shown for two samples with slightly

A

different phase transitions. The solid lines are a fit to eq. (10).

+ 1 (1 + 5:

q 2 )

[Qxx(q) - Qyy(n)12

+ 2 A(1 + 5:. q 2 ) Q,(q) Let us now turn to a discussion of the intensity

of light scattered by thermal fluctuations in the order

parameter. Light will be scattered in a given direction _-

_-

+ 2 ~ ( 1 + t: q 2 + t 5: q2) l ~ ; , ( q ) + Q ; = ( ~ ) I )- ⁽¹⁴⁾

only by a particular spatial Fourier component of

fluctuations in the dielectric constant whose vector q Applying the equipartition theorem to the various conserves momentum between the wave vectors of terms in (14), we obtain the fluctuations in the order the incident and scattered light. This conditions is parameter. For example, we find

kf

=

k , + q or else, for scattering by angle 0,

q

=

2 k sin

(8/2),

where ki and kf are the wave vectors 2 kT

< Q h ( 4 >

⁼

+ i 2 2

of the incident and scattered light and each has

_I

q + 5 <: ^q2>

^'

magnitude k. If the illuminated volume is V, then the

fraction of power scattered per unit length per unit H,,,, following de Gennes [2], we have defined solid angle is given by the Rayleigh ratio [9] two correlation lengths

L

1 2 L2

=

- and

5 2 =

7 .

A

C1-72 T. W. STINSON AND J. D. LITSTER

We may use these results to evaluate the fluctuations iv and fv normal to the scattering plane (determined in eif(q). Let us take the incident, i, and scattered, by k i and kf) and components iH and

^fH

in the scatter- f, polarization vectors of the light to have components ring plane. Then we obtain the results

We have carried out measurements of the absolute intensity of light scattered at 0

=

900 in the isotropic phase of MBBA. Our measurements were made with

i, =

1 and iH

=

0, and we accepted all polarization states of scattered light. Absolute calibration was made by comparison with a known concentration of 910 A dia. polystyrene spheres in water for which the Rayleigh ratio can be accurately calculated [lo].

For our experimental conditions (15) and (11) give a Rayleigh ratio

Measurements of the angular dependence of the intensity of the scattered light showed t2

^{q 2}

terms to contribute less than 1 % and so we have dropped them in obtaining (16). The results of our measurement are shown in figure 2, where we plot the absolute temperature divided by R, and are seen to confirm

FIG. 2. -The temperature (degrees Kelvin) divided by the Rayleigh ratio for MBBA. The solid line is a fit to eq. (16).

the prediction of (16). A least squares fit of a straight line to figure 2 gives us the result

a

=

4.8 x 10' Ae2 erg cm-3 OK-' .

The experimental uncertainty in this value comes for the absolute calibration procedure and could be as much as 10 %.

We see that de Gennes' rnodel correctly predicts the observed temperature dependence for the magnetic birefringence and the intensity of the scattered light.

Comparison of our numerical results for both measu- rements yields AxIAE

=

1.55 x in c. g. s. units.

The consistency of this result can be checked by estimating

Ae

and AX from a comparison of measure- ments made in the isotrop~c and nematic phases.

The refractive indices of MBBA have been measured by the I. B. M. Group [8]

;

from their data we estimate AE to be 1.1, with an uncertainty of about 10 %.

The I. B. M. Group has very recently measured the diamagnetic susceptibility in both the nematic and isotropic phases of MBBA [ll]. From their prelimi- nary results we estimate the anisotropy in the volume susceptibility to be AX

=

2 x with an uncer- tainty of perhaps 20 %. These data correspond to AxlAe

=

(1.8 + 0.3) x lo-'' which is in satisfactory agreement with our experiments. The coefficient a has the value 6 x lo5 erg c n ~ - ~ OK-

^l

with an uncer- tainty of about 30 %

;

a better determination of Ae and AX would reduce this uncertainty considerably.

111. Dynamic properties.

--

We have seen that de

Gennes' model of the phase: transition is confirmed

by our measurements of static properties in the iso-

tropic phase of MBBA. We should now like to discuss

this model applied to the dynamics of the isotropic

phase and compare its predictions with our measu-

rements of dynamic behavior. The time dependence

of the fluctuating quantities can be obtained if we

know their equations of motion [12]. The motion of

the fluid is governed by the usual equations of hydro-

dynamics. If we consider small fluctuations in density,

Sp, in temperature, ST, and in velocity, v, about the

STATIC AND DYNAMIC BEHAVIOR NE AR THE ORDER DISORDER TRANSITION C1-73

equilibrium state the linearized hydrodynamic equa- tions of motion are

In these equations pT

⁼

- (lip) (ap/aTjp is the volume thermal expansion,

Cp

and

C ,

are the specific heats per unit mass at constant pressure and volume, respectively, while

y = Cp/C,

is their ratio. The speed of sound is

Co =

[y(aP/ap)T]112, 1 is the thermal conductivity, and up, the hydrodynamic stress tensor.

In addition to equations [17] it is necessary to have an equation of motion for the order parameter, and de Gennes derives one in the following way. The force conjugate to Qap is, from (6),

-

a(p/aQ,,. A pheno- menological damping force can be included in the from of a term

V Q , ~ ,

where

is the motion of Qap relative to background rotation of the fluid wap

= (+)

(aavp - ]apva): (For small velocities due to thermal fluctuations QaB

=

aQap/at).

The observation of a diverging flow birefringence [6], [13] in the isotropic phase of PAA indicates that Qap is coupled to velocity gradients and this can be des- cribed phenomenologically by mezans of a force term y(vap - 3 v,, Sap) in the equation of motion. Expe- rimental studies of the ordered phase of PAA [14]

show that inertial effects do not play a significant role in fluctuations in Q,@, and the results we shall give presently confirm this for the isotropic phase of MBBA. Combining all of these forces, de Gennes arrives at the following equation of motion for the order parameter

:

where vap

=

aavp + dpva.

The steady state solution of (18) leads to a diverging flow birefringence QUp,

⁼

( p / A ) (vaB - 3 v y y Sap) in agreement with the observations of Tsvetkov and Tolstoi [13]. The equations of motion (17) and (18) provide a theory very similar to that used by Vol- terra [I51 fto discuss light scattered from ordinary liquids. Volterra's theory is essentially the same as an earlier theory of Leontovich [16] and differs physically from de Gennes' model of the liquid crystal by cou- pling the order parameter to shear strains rather than velocities.

The entropy production in the fluid is given by

From this we see that the Onsager relations [18] require a term in the stress tensor oa@ of the form 2 The stress tensor in (17b) is then given by

Here q is the shear viscosity of the fluid and y, is a volume viscosity. There is a slight coupling between diagonal fluctuations in Qnp and density fluctuations in the fluid. This effect is very small and affects the spectrum only at frequencies where the spectral density of fluctuations in Qap has fallen to a very low value.

We shall therefore make the approximation that the fluid is incompressible. Results for a compressible fluid can be obtained by following Volterra's treat- ment [15].

The easiest way to obtain the spectrum of the fluctuations from the equations of motion is that used

oy Mountain [17]. For example, consider fluctuations in Q,,. The Laplace transform of the equation of motion for the Fourier component of wave vector q can be obtained1from (18) as

Multiplying by Q,,(q, 0 ) and averaging we find (where r

⁼

Alv)

The spectrum < ~ : , ( q , o ) > is the Fourier time transform of < ^Q,(q, t ) Q,,(q, 0 ) > and this is equal to twice the real part of the right hand side of (21b) after replacing

s

by io. The spectrum of light scattered by fluctuations in QXy will then be proportional to

where o is the frequency relative to the incident frequency oo and < ^Q:,(~) > is obtained from (14) and the equipartition theorem. Assuming that the correlation lengths can be neglected, we apply this method to obtain

where b

=

q/p and

c =

(y/p) (1 - 2 y2/yv)

We carried out our experimedts with the incident light polarized normal to the scattering plane (i,

=

1).

The scattering angle was 6.550, corresponding to

q

=

1.824 x lo4 cm". Eq. (22) then gives the spec-

trum of the light polarized in the same direction

(fv

=

1) as the incident light and (23) gives the spec-

C1-74 T. W. STINSON AND J. D. LITSTER

trum of the depolarized (f,

=

1) scattered light. The ratio of the integrated intensities, zvH/Ivv, was 0.756 + 0.008, in excellent agreement with the theore- tical prediction of 3

:

4. The spectra were measured with a spherical Fabry-Perot interferometer that had a bandwidth of about 3 MHz. A rotating polarizer was placed in front of the interferometer and the photoelectrons counted by different counters for alternate 900 segments of the polarizer. In this way we measured both the polarized spectrum (22) and its ratio to the depolarized spectrum (23) as we swept the frequency of the interferometer. This method was necessary in order to make a reasonably precise measu- rement of the small difference between the two spectra in the presence of slight frequency instabilities in the laser. We used a Spectra Physics model 119 laser that had been running for more than 24 hours

;

it was not possible to use the frequency stabilization servo of the laser because it introduced about 4 MHz of frequency modulation broadening.

We found the polarized spectrum, Ivv(o), to be accurately represented by a Lorentzian line of width

r

⁼

^{a(T - T:)/v,} as reported by Litster and Stin- son [19]. Since we know the coefficient a from our measurements of the magnetic birefringence and inten- sity of scattered light, we may deduce the value of the transport coefficient v from the linewidth of the pola- rued spectrum. The values we obtained with

a

=

6 x lo5 erg cm-3 OK-'

are shown in figure 3. A least squares fit, shown as a solid line in the figure, gives v

=

3.54 x exp (3 570/T) poise, where T is in OK. Also shown in figure 3 are the data of Martinoty and Candau [20]

for the shear viscosity y. The solid line is a least squares fit, and has the equation y

=

1.77 x exp (3 680/T) poise. It is clear that the two transport

FIG. 3.

-

Transport coefficients for MBBA. The solid points are values of v deduced from our measurements of the linewidth of the scattered light. The circles are the measurements of Martinoty (ref. 20). Arrows indicate the ordering tempera- tures of the samples used for the measurements. The solid lines are both least square fits to expressions of the form

v = v o exp(W/T) as given in the text.

coefficients have the same temperature dependence within our experimental error. The ordering tempe- ratures of the two samples studied are indicated by arrows on the figure. It is apparent that no detectable pre-transitional effects occur in the temperature dependence of these transport coefficients in the isotropic phase.

We now discuss the spectrum of the depolarized light. The ratio of the depolarized (23) to polarized (22) intensities at the laser frequency ( o

=

0) is

(314) (1 - 2 P~IVV) -

The departure of this ratio fiom 3

:

4 enabled us to evaluate ,u2/qv at each temperature. To check the result we then used this value of p2/rp, our measure- ment of r, and Martinoty's values [20] of

y

to calculate

IvH/GV as a function of

w

(subject to the constraint that ratio of the integrated intensities be 3

:

49. Over most of the temperature range we found excellent agreement between the calculated and measured ratios as can be seen in the top half of figure 4. At the highest temperature we found slight systematic departures from the theoretically predicted shape of IvH/Ivv. These deviations are only slightly greater than a conservative estimate of our experimental errors (the worst case was for the highest temperature and is shown in the bottom half of the figure), but are an indication the theory may be breaking down at high temperatures.

This procedure enabled us to determine p2/qv with an uncertainty of about 15 % and the results are shown in figure 5. It is interesting to note that, while neither q nor v exhibit any pre-transitional effects in

12 6 FREQUENCY 0 (MHz) 6 12

u

^I

6 0 3 0 0 30 6 0

I

I

FREQUENCY (MHz)

FIG. 4. - Ratio of the depolarized (ZVH) to depolarized (ZVV) spectral power densities of light scattered in the isotropic phase of MBBA. The solid lines are fits to equations (22) and (23) of the text and the dots are experimental measurements. The upper curve is for a temperature 10.54 OC above the phase transition and

the lower curve is for T'

+

^{24.14 O C .}

STATIC AND DYNAMIC BEHAVIOR NEAR THE ORDER DISORDER TRANSITION C1-75

the isotropic phase,

p

shows an anomalous increase above its background value in the 5 OC immediately above the phase transition temperature.

FIG. 5.

- Values of

jiZ/qv

as a function of temperature for

MBBA;

the points were determined as discussed in the text.

IV. Conclusion.

-

We find the predictions of de Gennes' phenomenological model to be consistent with the divergences we have observed in the static properties (magnetic birefringence and intensity of light scattered) of the isotropic phase of MBBA near its ordering temperature. The model is also consistent with the dynamic behavior we have observed a critical slowing of the fluctuations and a coupling t o over- damped thermally excited shear waves. We have determined both the transport coefficient which serves as a viscosity for the damping of fluctuations in order parameter and the coupling constant

,u

between velo- city shear and the order parameter

;

only the latter shows any pre-transitional effects as the ordering temperature is approached.

References

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Collected Papers of L. D . Landau,

edited by TER HAAR (D.), (Gordon and Breach, New York, 1965), 193.

121 DE GENNES (P. G.),

Phys. Letters,

1969, 30A, 454 and

Mol. Cryst. and Liq. Cryst.,

1971, 12, 193.

[3] A recent review article is by BROWN

(G. H.),

DOANE

(J.

W.) and NEFF (V. D.),

CRC Critical Reviews in Solid State Sciences,

1970,

1,

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[4] MAIER (W.) and SAUPE (A.),

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151 ZADOC-KAHN (J.),

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[7] STINSON (T.

W.),

and LITSTER (J. D.),

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Letters,

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[8] HALLER (I.), HUGGINS (H. A.) and FREISER (M. J.),

Mol. Cryst. and Liq. Cryst.,

to be published.

[9] LANDAU (L. D.) and LIFSHITZ (E. M.),

Electrodyna- mics of Continuous Media

(Pergamon, New York, 1960), chap. XIV.

[lo] DENMON (H.), PAGONIS

(W.)

and HELLER (H.),

Angular Scattering Functions for Spheres

(Wayne State University Press, Detroit, 1966).

[ l l ] HALLER (I.), private communication.

[12] LANDAU (L. D.) and LIFSHITZ (E. M.),

Fluid Mecha- nics

(Pergamon, London, 1959), Chap. XVII.

[13] TOLSTOI

(N.

A,) and FEDOTOV

(L.

N.),

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[14] ORSAY LIQUID CRYSTAL GROUP,

Phys. Rev. Letters,

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[I51 VOLTERRA (V.),

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[16] LEONTOVICH (M. A.),

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[17] MOUNTAIN (R. D.),

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[l8] Ref. 12, paragraph. 58.

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(T.

W.),

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1970,41,996.

[20] MARTINOTY (P.) and CANDAU (S.),

cc

Determination

of Viscosity Coefficients of a Nematic Liquid

Crystal

11, Mol. Cryst. and Liq. Cryst., to be

published.