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Orientational diffusivities in a disk-like lyotropic nematic phase : a strong backflow effect
M.B. Lacerda Santos, Y. Galerne, Geoffroy Durand
To cite this version:
M.B. Lacerda Santos, Y. Galerne, Geoffroy Durand. Orientational diffusivities in a disk-like ly- otropic nematic phase : a strong backflow effect. Journal de Physique, 1985, 46 (6), pp.933-937.
�10.1051/jphys:01985004606093300�. �jpa-00210039�
Orientational diffusivities in a disk-like lyotropic nematic phase :
a strong backflow effect
M. B. Lacerda Santos (+), Y. Galerne and G. Durand
Laboratoire de Physique des Solides, Université de Paris-Sud, Bât. 510,91405 Orsay Cedex, France
(Reçu le 15 novembre 1984, accepté le 31 janvier 1985)
Résumé. 2014 Nous avons étudié la phase nématique discotique d’un mélange lyotrope (laurate de potassium/déca- nol/D2O) par diffusion Rayleigh, en utilisant la technique de battements de photons. Les mesures du temps d’amor- tissement des fluctuations thermiques du directeur sont faites en fonction du vecteur d’onde q. La géométrie employée nous permet de déterminer les diffusivités associées aux déformations de type « torsion » et « éventail ».
Le fort rapport observé entre ces deux quantités s’interprète par une forte anisotropie de viscosité due aux écoule- ments induits (backflow). Cette anisotropie éventail-torsion trouvée dans un nématique discotique (lyotrope) est l’equivalent de l’anisotropie flexion-torsion des nématiques classiques (thermotropes), constitués de molécules en
bâtonnet.
Abstract. 2014 We have studied the disk-like lyotropic nematic phase of a K-Laurate/Decanol/D2O mixture by Ray- leigh scattering. The decay time of the director thermal fluctuations, obtained by light-beating techniques, is mea-
sured as a function of the wave vector. The employed geometrical configuration allows to determine the splay and
twist diffusivities. The observed large ratio between these two quantities is interpreted in terms of the anisotropy
between their associated viscosities, due to the backflow. This splay-twist anisotropy for disk-like (lyotropic)
nematics appears similar to the well-known bend-twist anisotropy for rod-like (thermotropic) nematics.
Classification
Physics Abstracts
61.30 - 62.10 - 51.20
Amphiphilic molecules (molecules having a hydro- philic head and a hydrophobic tail) in water tend to aggregate forming anisotropic micelles [1]. Under
proper temperature - concentration conditions [2]
these micelles exhibit long range orientational order characteristic of nematics. Three types of lyotropic
nematic phases have been found, exhausting the sym- metry possibilities [3]. Two of them are uniaxial phases [4] : a « disk-like » nematic phase (ND) and a
« cylindrical» nematic phase (Nc), with the uniaxial director n respectively parallel and perpendicular to
the average direction of alignment of the amphiphilic
molecules [5]. Finally, a biaxial nematic phase (NBx)
has been found [6], as an intermediate phase between
the uniaxial ones. In spite of the different nature of the constituent objects, lyotropic and thermotropic nema-
tics have been shown to be quite analogous in their
main physical properties. For instance, elastic cons-
tant data available up to now for lyotropic nematics [7]
are of the same order of magnitude as those for ther-
( + ) Permanent address : Universidade Federal de Minas
Gerais, Dept. de Fisica, 30000 Belo Horizonte MG, Brazil.
Partially supported by CNPq, Brazil.
motropic nematics. Dynamical properties of lyotropic
nematics are less known. Indeed, a recent light scatter- ing experiment [8] confirmed that the basic features known for the dynamics of thermotropic nematics [9]
extend quite well to lyotropic nematics. Namely, the scattering of light is essentially due to the thermal fluctuations of the director n. These fluctuations are
described by the two collective normal modes of relaxational type, with the relaxation rates given by [9]
Where i = 1(2) denotes the mode of fluctuations 6n
parallel (normal) to the (n, q) plane. Here, K1, K2, and K 3 are the Frank elastic constants characterizing splay, twist, and bend deformation, and f11 (q) and f12(q) are effective viscosities associated to mode 1 and mode 2. q and ql denote, respectively, the parallel
and the perpendicular components of the wave vector with respect to n. Then equation (1) allows in principle
for the determination of the orientational diffusivities,
K K
i.e., the pure deformation ratios K1 and K2 for
nsplay f1 twist,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004606093300
934
q 1 n, , and K3 for q/./n. In this paper we present Nbend
relaxation rate data for a disk-like lyotropic nematic (ND) phase, obtained by Rayleigh light scattering
in a geometry very close to q 1 n, allowing the first
determination of splay and twist diffusivities in these materials. We find a rather large deviation between these two quantities, while they are known to be comparable in classical thermotropic nematics [9].
We argue that this behaviour arises from a strong backflow effect experienced by the disk-like micelles dispersed in the water, with consequent reduction of the splay viscosity.
Our sample is a mixture of K-laurate, 1-decanol,
and D20, in proportions of 25.15/6.32/68.53 wt. %, respectively. The concentrations were chosen with reference to the Yu and Saupe phase diagram [6], looking for a large temperature range ND phase,
without the Nbx phase, in order to avoid complications
from pretransitional effects [10]. Our ND phase is
limited by a lower isotropic phase, at about 10 °C and a higher isotropic phase, at about 40 °C. The
compound was enclosed in a 1 mm thick Hellma cell, hermetically sealed to prevent concentration drifts.
Indeed, we keep here only data in a limited time range of 3 days. Untreated glass plates can orient sponta- neously the ND phase with n normal to them (homeo- tropic alignment). We clean the cell walls with a strong
detergent and rinse them carefully with distilled water.
It results in homeotropic samples. To improve the alignment we transfer the sample to a magnetic field.
Recalling that xa 0 [6], the procedure consists of
turning the sample few times in the field (Which is equivalent to putting the sample in a rotating field.)
We did this whenever the visual inspection between
crossed polarizers showed an excessive misalignment
of the sample. To observe the fluctuations of n we use
(as in Ref. [8]) the light beating spectroscopy tech-
nique. A 50 mW He-Ne laser (Ao = 6.328 A) beam, polarized following the extraordinary (e) or ordinary (o) directions, is sent across the horizontally hold sample cell along the k; direction, as shown in figure 1.
The scattered light is e or o analysed and collected
along the k direction by a photomultiplier and its photo-current analysed by a digital correlator. The choice of the k, direction is such that the vertical z-axis bisects the scattering angle 0. Also the horizontality
of the sample is controlled to 10- 3 rd It follows that q
is perpendicular to n. This statement is valid even for depolarized scattering, because of the weak birefrin- gence of lyotropic nematics (An - 2 x 10-3) [5]. 0 is
calculated from the external angle using the mean
refractive index n = 1.38, determined with an Abbe refractometer. Then, by varying 0 we can vary the q
modulus, q = 2 ko n sin 0/2, where ko = 2 nlao. This
q ± n scattering geometry is convenient [9] to observe
pure twist deformations in the depolarized configu- rations, (e, o), and (o, e). It forbids in principle the
observation of pure splay deformations in the (e, e)
Fig. 1. - Scattering geometry : k; = incoming wave vec- tor ; kl = scattering wave vector, q = k; - ks ; n = director along its equilibrium direction, the (vertical) z-axis. 03B4n1, bn2 Pictorial indication of the normal modes of fluctuations of n : Mode 1 is in the plane (n, q) and mode 2 normal to it
configuration ((o, o) is always forbidden). In practice
we have observed that the slight misalignment of the sample, when kept outside the rotating magnetic field, suffices to provide a manageable (e, e) signal.
We have checked independently that this signal is extinguished in a freshly realigned sample. In the working conditions, we estimate that the small mis-
alignment of n around the vertical axis z is E N 0.1 rd,
This implies that the qn term in equation (1), which is of E2 order, can be neglected. With these approximations equation (1) reduces, for each mode, simply to :
where l’l (== 11 twist) [9] is the rotational viscosity
coefficient.
To obtain the diffusivities K 1/plsplay and K21YI we
proceeded to measure the relaxation rates rsplay, in
the polarized configuration, and Ftwist, in the depola-
rized one, for different scattering angles 0 (always keeping q 1 z). At the small angles we use (0 7°) we
always get the heterodyne regime. Thus the T’s are simply the inverse of the characteristic times of the
single exponential form of our signals. The relaxation rates r are plotted in figure 2 against q2, for’both polarized and depolarized configurations. These mea-
surements were performed at room temperature I
T ~ 19 °C which is far from the transition tempera-
tures. During the whole duration of the experiment, temperature variations of ± 1 °C were registered
Fig. 2. - Relaxation rates (F) versus the square of the wave
vector (q2). Open circles (0) : Polarized scattering associated
to splay deformations (mode 1). Dots (0) : Depolarized scattering associated to twist deformations (Mode 2). Full straight lines denote the best fits to these data.
However they were not found to cause any systematic effect, within our experimental accuracy. Figure 2
shows also the least square straight line fits for the
splay and the twist modes, weighted for constant
relative errors. These straight lines fit reasonably well
the data and, furthermore, they intercept the origin.
This confirms the diffusive character of these modes, as expressed by equations (2), and (3). From the slope of
these lines we find the splay diffusivity DSplay = Kl/ fIsplay = 1.15 x 10-7 cm2/s and the twist diffusi-
vity Dtwist = K21YI = 0.16 x 10 - 7 cm2 js.
To start the discussion of the results, let us note the large ratio between the two orientational diffusivities
Dsplay and Dtwist. This denotes a strong anisotropy
effect. In table I we display this data together with
similar ratios for classical (rod-like molecules) ther- motropic nematics, for comparison. We note, then,
that a large ratio for thermotropics only occurs for Dbend/Dtwist, the D5playIDtW;sc ratios being of order of 1.
Let us recall that two factors are involved on orienta- tional diffusivities, namely, elastic constants and visco-
sities. Elastic constants in lyotropic nematics have been studied [7] by quasi-static means(’) in DACI), a system
(1) In principle elastic constants may be independently
determined by light scattering absolute intensity measure-
ments. Indeed, this possibility doesn’t apply to our almost
forbidden (e, e) signal.
Table I. - Orientational diffusivity ratios for the disk phase (ND) of our K-laurate, D20, decanol mixture,
obtained by Rayleigh scattering. For comparison, typi- cal values for thermotropic nematics (obtained from Ref. [9]) are shown. Frames enclose the largest value for each nematic system.
closely related to ours. These elastic constants are of
the same order of magnitude as in thermotropics.
Furthermore, there are no large differences between
them, roughly a factor of 2, except for special pre- transitional effects. In fact there is no reason in prin- ciple for a different result in our system. Therefore, in
all that follows we shall suppose K1 N K2, so that DaplaylDcwigt N YlI f/splay. . Let us concentrate then the discussion on the viscosities. We notice that the rota- tional viscosity coefficient y, is large compared to the splay viscosity f/splay. This can be discussed in two
steps. First, about the absolute magnitude of the visco- sities : if one visualizes the ND phase as oriented disks (thickness a, diameter b) dispersed in a liquid of averag-
ed viscosity q, one can evaluate [ 11 ] Yl ’" il bb3a
which may be compared to the value Yl ’" q -ab2 found by Helfrich [12] for a nematic phase constituted of cylindrical objects. Neutron [13] and X-ray [14]
measurements give an estimate of the form factor
bla - 2-2.5. We deduce then Yl 1’-1 10 q where we
can estimate q - 0.1 poise using the data from refe-
rence [7]. The second step concerns the anisotropy
of viscosities caused by the backflow. To recall this effect on a simple physical ground, let us consider a
disk micelle (thickness a, diameter b) rotating in a liquid of viscosity with an angular velocity co. This
disk rotation induces a flow vortex around the indi- vidual micelle. In a steady state, this vortex must have the same angular velocity m as the disk for the torque
density acting on the fluid to vanish. In this regime,
the director is « frozen » in the fluid vortex As shown in figure 3, the vortex can be decomposed in two pure shear flows, respectively normal and parallel to the disk plane. Depending on the direction of q, compared to n,
these local shear flows can interfere and modify the
effective viscosity of the distortion mode. When the interference is constructive, the macroscopic backflow
induced by each individual micelle accompanies the
individual rotation, and thus decreases the dissipation,
936
Fig. 3. - Rotating disk (I ( 6n bt = m ) and its induced flow.
The flow vortex can be decomposed into pure shear flows, parallel (a) and perpendicular(b) to the disk plane.
by replacing the large relative friction between micelles and the bulk by the smaller friction of the fluid on
itself: in the backflow. On the other hand, when the
interference is destructive, there is no backflow and the dissipation totally due to the friction of the micelles in the bulk, is maximum. This situation corresponds
to the pure twist case, simply leading to I1twist = Y1.
In the case of splay (q 1 n) (Fig. 4a) the interference of individual vortices is not complete. The shear flows
parallel to the disks cancel by interference, but the
shear flows perpendicular to the disks interfere cons-
tructively to give a macroscopic backflow. There is no
longer relative motion of the micelles compared to
this backflow. The only dissipation comes from the
local shear parallel to the disk plane. As the order of
magnitude of Vv is wb/b = co, the resulting splay viscosity is I1spla ’" N. The backflow reduces the
viscosity from ?lb la - 10 N down to q. In the case of bend (Fig. 4b), on the other hand, the backflow sup- presses only the friction from the shear flow parallel
to the disk faces. The viscosity reduction must be of
the order of 11 compared to 71 i.e. we expect the bend
viscosity to be comparable to the twist one.
This discussion of the compared viscosities of pure
twist, splay and bend distortions in a discotic nematic
can be motivated by the classical phenomenology
of nematodynamics.
Fig. 4. - Interference between the individual vortices in the
cases : a) q 1 n (splay) b) qin (bend). The built-up macro- scopic backflow assists the rotation of the micelles and there- fore reduces the dissipation. This reduction is strong for splay and weak for bend (N, phase).
In a uniaxial nematic [9], we have :
where the a’s and il’s are viscosity coefficients written in the same notations as in reference [9]. In the last two expressions, the backflow effect is accounted for by
the negative terms. These backflow terms can be
evaluated by considering the two Miesowicz configu-
rations « b » and « c » (Fig. 5) in which the nematic is oriented along the z and x directions, respectively,
and exhibits a pure velocity gradient Axz = ax avax -= w.
In these two cases, the stress tensor applied on the
sample is (9) : 0
In a first approximation we assume that n is frozen
in the flow. A non-zero velocity gradient (Axz = w)
can then exist without applying a stress (a’ x y = 0)
on the nematic, corresponding to hx a) and nz = w in the configurations b and c, respectively. We deduce Nb ~ a3 and nc - a2. In configuration c, the shear is parallel to the disk. Its associated viscosity
ilc is therefore of order of 11. It results (a2 - 11. From the relation y, = a3 - a2 [9], we also derive tlb - a3 - y, - 11. These estimates of the viscosity coeffi-
cients a2, a3, 11b and 11e allow us to calculate the b k-
flow terms in the cases of pure splay and bend They
are respectively l’t - 11 and 11. The viscosities of the
Fig. 5. - Miesowicz configurations « a », « b » and « c ».
pure splay and bend distortions are then deduced : 17splay N, 17bend 7r As 17 is much smaller than 71, the backflow reduces strongly the splay dissipation in
agreement with our experimental results. Similarly,
one can predict that the backflow reduction must be very weak on the bend distortion. Unfortunately, our
geometry did not allow us to measure the bend damp- ing time.
We can now compare this backflow effect with that
already observed in rod-like (thermotropic) nematics
where it is the bend viscosity which is strongly reduc-
ed [15]. This inversion is indeed expected because,
while in the thermotropic case the director n o lies »
along the long side of the « hydrodynamic objects » (rigid rods), in the disk-like lyotropic n is perpendicular
to the large dimension of the micelles. In a two-dimen- sional picture we could say that, as far as we are con- cerned with viscosities, a bend mode for rods is equiva-
lent to a splay mode for disks.
To conclude, we have opserved the dynamics of
director fluctuations of orientation in the disk-like nematic phase (ND) of the lyotropic system K-laurate/
1-decanol/D20 using light beating spectroscopy. We
measure the diffusivities of twist and splay director
reorientations. With the additional assumption about
the approximate equality of elastic constants we can
estimate the splay and twist viscosities of our lyotropic
nematic material. The main result is the small value of Nsplay" almost an order of magnitude lower than ~twist.
This result must be compared with the behaviour of rod-like (thermotropics) nematics, where ~bend is much lower than ~twist. In both cases, these large variations
of viscosities are the effects of backflow, which reduces
strongly the dissipation for the splay distortion of disk-like nematics, as well as for the bend distortion of rod-like nematics. For disk-like nematics this back- flow effect should be much weaker on the bend visco-
sity, which remains to be observed in lyotropic nema-
tics.
Acknowledgments.
’
We wish to thank L. Liebert for providing us purified samples of potassium laurate. We are also indebted to J. P. Marcerou for valuable phase diagram infor-
mation concerning large temperature range lyotropic
discotic phases.
References
[1] TANFORD, C., J. Phys. Chem. 78 (1974) 2469.
[2] HENDRIKX, Y. and CHARVOLIN, J., J. Physique 42 (1981) 1427.
[3] FREISER, M. J., Phys. Rev. Lett. 24 (1970) 1041.
[4] LAWSON, K. D. and FLAUTT, T. J., J. Am. Chem. Soc.
89 (1967) 5489;
RADLEY, K., REEVES, L. W., CHARVOLIN, J., LEVELUT, A. M. and SAMULSKI, E. T., J. Physique Lett. 40 (1979) L-587.
[5] GALERNE, Y. and MARCEROU, J. P., Phys. Rev. Lett.
51 (1983) 2109.
[61 Yu, L. J. and SAUPE, A., Phys. Rev. Lett. 45 (1980) 1000.
[7] HAVEN, T., ARMITAGE, D. and SAUPE, A., J. Chem.
Phys. 75 (1981) 352 ;
SAUPE, A., Il Nuovo Cimento D 3 (1984) 16 (Special
issue : Meeting on Lyotropics and Related Fields, Rende-Italy, Sept. 1982).
[8] KUMAR, S., YU, L. J. and LITSTER, J. D., Phys. Rev.
Lett. 50 (1983) 1672.
[9] DE GENNES, P. G., The Physics of Liquid Crystals,
Oxford (1974).
[10] LACERDA SANTOS, M. B., GALERNE, Y. and DURAND, G., Phys. Rev. Lett. 53 (1984) 787.
[11] LACERDA SANTOS, M. B., GALERNE, Y., DURAND, G.,
to be published.
[12] HELFRICH, W., J. Chem. Phys. 50 (1969) 100.
[13] HENDRIKX, Y., CHARVOLIN, J., RAWISO, M., LIEBERT, L.
and HOLMES, M. C., J. Phys. Chem. 87 (1983) 3991.
[14] FIGUEIREDO NETO, A. M., GALERNE, Y., LEVELUT, A. M.
and LIEBERT, L., to be published.
[15] Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst.
13 (1971) 187 ;
LEGER-QUERCY, L., 3e Cycle, Thesis, Orsay (1970).
