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Order and micellar density fluctuations in the biaxial, uniaxial (Nc and ND), and isotropic phases of a
lyotropic nematic liquid crystal studied by light beating spectroscopy
M. B. Lacerda Santos, Geoffroy Durand
To cite this version:
M. B. Lacerda Santos, Geoffroy Durand. Order and micellar density fluctuations in the biaxial, uniaxial (Nc and ND), and isotropic phases of a lyotropic nematic liquid crystal studied by light beating spectroscopy. Journal de Physique, 1986, 47 (3), pp.529-547. �10.1051/jphys:01986004703052900�.
�jpa-00210233�
Order and micellar density fluctuations in the biaxial, uniaxial (Nc and ND),
and isotropic phases of a lyotropic nematic liquid crystal studied by light beating spectroscopy
M. B. Lacerda Santos (*) and G. Durand
Laboratoire de Physique des Solides, Bât. 510, Université de Paris-Sud, 91405 Orsay Cedex, France
(Reçu le 19 aout 1985, accepte le 15 novembre 1985)
Résumé. 2014 Nous avons réalisé des mesures de diffusion de lumière (Rayleigh) pour étudier la dynamique des
fluctuations thermiques dans les phases nématiques uniaxe et biaxe d’un cristal liquide lyotrope. Le paramètre
d’ordre nématique biaxe a été défini par de Gennes comme un tenseur d’ordre 2, symétrique, à trace nulle. La dynamique des fluctuations du paramètre d’ordre prédit 5 modes normaux : 2 sont associés aux fluctuations de module des paramètres d’ordre uniaxe et biaxe (modes scalaires) : 3 sont associés aux petites rotations rigides de l’ellipsoide représentant le tenseur autour des axes principaux (modes orientationnels). Nous avons estimé, en
utilisant le formalisme de Landau-Khalatnikov, les dépendances en température des taux de relaxation associés aux
5 modes, pour les phases uniaxe et biaxe. Nous présentons des données du taux de relaxation obtenues en lumière diffusée polarisée et dépolarisée dans les phases biaxe, uniaxe Nc, uniaxe ND (Nc 2014 biréfringence négative; ND biréfringence positive), et isotrope du système lyotrope nématique K-laurate, décanol, D2O. Nous avons identifié
chacun des 5 modes prévus associés aux fluctuations du paramètre d’ordre. Notamment, nous présentons la démons-
tration expérimentale du nouveau mode orientationnel associé aux rotations du directeur biaxe et l’observation des fluctuations associées au module du paramètre d’ordre uniaxe et biaxe dans les phases ordonnées. En plus des
7 modes, nous avons trouvé un nouveau mode scalaire en diffusion polarisée. Ce mode lent est attribué provi-
soirement aux fluctuations de densité micellaire.
Abstract. 2014 We have performed Rayleigh scattering measurements to study thermally excited fluctuations in a
lyotropic nematic liquid crystal system exhibiting uniaxial and biaxial nematic phases. The biaxial nematic order parameter has been defined by de Gennes as a symmetric, traceless, 2nd rank tensor. The dynamics of the order parameter fluctuations predicts 5 normal modes : 2 are associated to the fluctuations in the (uniaxial and biaxial) magnitudes of the order (scalar modes); 3 are associated to the small rigid rotations of the order parameter ellipsoid,
around its principal axes (orientational modes). Using a Landau-Khalatnikov formalism we have estimated the temperature dependence of the relaxation rates for these 5 modes, valid for the uniaxial and biaxial phases. We
present relaxation rate data obtained from polarized and depolarized Rayleigh scattering in the biaxial, uniaxial Nc, uniaxial ND (Nc-negative birefringence; ND-positive birefringence), and isotropic phases of the K-laurate, decanol, D2O lyotropic nematic system. We have identified the 5 expected modes arising from the order parameter fluctuations. In particular, we present the experimental demonstration of the new orientational mode associated to the rotations of the biaxial « director », and the observation, in ordered phases, of the magnitude fluctuations of the uniaxial and biaxial order parameter. In addition to the 5 modes, we have found a new scalar mode in pola-
rized scattering. This slow mode is tentatively attributed to fluctuations in micellar concentration.
Classification
Physics Abstracts
64.70E - 61.30E - 78.35
1. Introduction.
Liquid crystals are intermediate states of condensed matter that may occur between a solid crystal and
a completely isotropic liquid Among them, the nematic state is the simplest one, as it exibits orien-
tational order only. This order shows no polar cha-
racter. As pointed out by de Gennes [1], the natural way to describe the macroscopic order in a liquid crystal is via a 2nd rank symmetric traceless tensor.
In many cases of interest, the liquid crystal order is subjected to important fluctuations originated from
thermal excitation. Rayleigh light scattering has
shown to be a useful tool to study fluctuations in
liquid crystals, as the light waves couple directly
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703052900
with the order parameter fluctuations via the optical
dielectric tensor. Such studies in thermotropic liquid crystals started more than fifteen years ago [1].
More recently, these studies have been focused on new materials presenting liquid crystal order, e.g.,
polymer mesophases [2], and lyotropic liquid crystals [3, 4], with which we are concerned here.
Differently from thermotropics, which are consti-
tuted of one single kind of molecule (more or less rigid, anisotropic), lyotropics are made by dispersing
surfactant (soap) molecules in water. Such molecules have an amphiphilic character, that is, they show one hydrophiphilic polar head and one hydrophobic (flexible carbon chain) part. When brought in contact
with water the amphiphilic molecules tend to form aggregates that accomplish the spatial segregation [5]
between polar (water) and non-polar (paraffinic)
media. The topological properties of the (polar head built) interface [6] between these two media are
strongly influenced by the amount of water available, resulting in the wide variety of phases displayed by lyotropic systems. Though the phase diagrams
differ for each particular system, the following general
features are commonly observed [7] : in the amphi- philic-concentrated region we find the viscous bire- fringent phases that result from the formation of infinite aggregates, e.g, lamellae (or bilayer), etc.
On the other hand, in the dilute region we find the isotropic solutions of micelles, that is, aggregates of limited sizes. Bilayers, micelles, etc. have in common
the shorter dimension of approximately twice the length of the amphiphile [8]. Anisotropic solutions
without long range positional order, i.e., lyotropic
nematics [9], can occur in the concentration region
intermediate to these extremes, if non-spherical
micelles are present. Magnetic [10] and structural studies [11] have established the existence of two types of uniaxial nematic phases classified as « cylindrical » (Nc) and « discotic » (ND), according to whether
the optical axis (or « director »), n orients parallel
or perpendicular to the external magnetic field H.
Birefringence studies [12] indicate that the amphi- philic molecules tend to align parallel to the optical
axis in the Nc phase (An > 0) while they align rather perpendicular to the optical axis in the ND phase (An 0). In between these negative and positive
uniaxial phases, Yu and Saupe [13] discovered a
biaxial nematic phase (Nbx) in the K-laurate, decanol, D20 lyotropic system. The observed normal sequence of phases apart from reentrance phenomena [13]
is the following (arrows in the sense of increasing temperature) : Isotropic (I) - ND --+ Nbx -+ Nc I.
Recent high-resolution synchrotron X-ray studies [14]
have shown that the ND -+ Nbx -+ Nc 2nd order [15,16]
phase transitions can be understood as being pure order-disorder phase transitions of biaxial shaped objects. According to this view, no important change
in the shape of the micelles should happen at these
transitions.
Pretransitional phenomena near an order-disorder
phase transition may include diverging fluctuations that induce a diverging intensity of scattered light (analogous to the critical opalescence in a pure fluid [17]) as well as, from the dynamical point of view, a critical slowing down of the thermal fluctua- tions. In the case of the nematic to isotropic (N-I) phase transition such effects are associated to the fluctuations of the (uniaxial) order parameter around its equilibrium zero value, and have been studied
long ago [18] in the isotropic phase of thermotropic
nematics. Recently the divergence of the scattered
intensity from uniaxial fluctuations has been observed in the isotropic phase of a lyotropic system in the vicinity of the ND phase [3, 19]. However, due to the
first order character of the N-I transition, these critical fluctuations do not really diverge. In close analogy,
there should also exist [20] biaxiality fluctuations,
in uniaxial nematic phases. Such fluctuations tend to
destroy the revolution symmetry around the director.
They have never been observed in thermotropics.
The occurrence of the biaxial nematic phase in lyo- tropics opened the possibility [21] to observe biaxiality
fluctuations as a pretransitional effect in one neighbour
uniaxial phase, « disordered » with respect to the biaxial ordered phase. The first experimental evidence of biaxiality fluctuations has been the observation [22]
of a critical opalescence near the ND -> Nbx and Nbx ->
Nc phase transitions. The associated critical slowing
down of biaxiality fluctuations has been recently
demonstrated by Lacerda Santos, Galeme, and
Durand [4] in the ND phase near the Nbx phase of the K-laurate, decanol, D20 system. In this paper we present the first Rayleigh scattering study focusing
on both the uniaxial and the biaxial phases, in the
same system. To analyse the data we recall that the tensorial order parameter introduced by de Gennes [23]
takes five independent variables to be completely specified [24], yielding to five normal modes in the
fluctuating part of the free energy. We present here the first estimate of the temperature dependence
of the intensity associated to the five fluctuation modes in presence of the biaxial phase. We do this
in a simplified way by working out the fluctuations
near the equilibrium defined by a specific Landau-
model [15, 16] allowing for the biaxial phase. Next, using a Landau-Khalatnikov [25] formalism we derive
the temperature dependence of the relaxation rates
associated to the 5 modes. Among the five modes,
two are scalar » modes, associated to the fluctuations in the magnitude of, respectively the uniaxial and the biaxial order parameters. The other three are orienta- tional » modes associated to the small rigid rotations
of the order parameter ellipsoid Among the latter
three modes, two are the well known [1] ] uniaxial
« director » fluctuation modes, recently extended to lyotropics [3, 27]. Here we present the first observa- tion of the mode due to the rotations of the « biaxial director ».
To our knowledge, scalar modes have never been
reported in the ordered (non-isotropic) phases of
nematic systems, except in the previous work, refe-
rence [4]. Here we present relaxation rate data, that
for the first time show fluctuations in the magnitude
of the uniaxial order parameter in the nematic side.
We also present preliminary relaxation data in the
isotropic phase of our lyotropic system. These measure-
ments could be obtained from low frequency light beating techniques because of the exceptionally weak
first-order character of the N-I transition, which is
known to be caused [26] by the existence of the biaxial
phase. Completing the characterization of the scalar
ordering fluctuation modes we present the critical
slowing down effect on the biaxiality fluctuation as
the Nbx phase is approached from the Nc (first pre-
sentation) and ND sides as well as the data of this mode in the biaxial phase itself.
Finally, an unpredicted supplementary mode has
been found This low relaxation rate mode can be
explained by the fluctuation in the density of the
micelles at constant density of amphiphiles, that is,
the mechanism of creation and destruction of micelles [42]. These micella density fluctuations also affect the anisotropic part of the dielectric tensor
(the « norm » of the order parameter), although they give no effect on the isotropic part
The plan of the article is the following. In section 2 .1
a qualitative discussion of the expected order para- meter five modes is given, as well as a brief recall of the light beating methods for measuring relaxation
rates. The Landau model describing the equilibrium properties of nematic systems allowing for a biaxial phase [15, 16] is reviewed in section 2.2. Next, in section 2.3, we present simplified calculations of the
fluctuations, using the Landau-Khalatnikov forma- lism in deriving the dynamics.
This allows us to estimate the temperature depen-
dence of the expected 5 modes. In section 3 we describe
the experimental techniques. Finally, in section 4
we present and discuss our light scattering data in
addition with the simple model proposed to explain
the extra (6th) mode from micella density fluctuations.
2. TheoreticaL
2.1 INTRODUCTION. - It is well known [28] that
when light is propagating with a wave-vector k;
through a homogeneous dense medium, the scattering
of light in the direction specified by k f is caused by
the thermally excited fluctuations of the dielectric tensor E which have spatial dependence as exp(iq . r),
where q = kf - k;, r defining the spatial position.
For a nematic liquid crystal we can [23] decomposf
into a sum of two terms :
Here the first term contains the isotropic part of the dielectric tensor while all its anisotropic parts are
in the second term. Fluctuations of 81
= 1/3
1 t E are associated to density fluctuations [28] and are oflittle interest in liquid crystals [1]. The important piece in the (anisotropic) second term of equation (1)
is the traceless tensor Q, called the tensor order para- meter. In fact Q can be defined [23] macroscopically by equation (1), the coefficient E. being related to
the anisotropies for a completely ordered sample [24, 18]. As pointed out by de Gennes [23] this definition of the order parameter covers both uniaxial and biaxial nematics. For an oriented sample the undistorted state can be represented, in the appropriate coordinate
system x, y, z, by the diagonal form of Q :
Here P is a measure of the degree of biaxiality of the sample. In the uniaxial nematic phases P = 0, and the order is described by a single number, e.g., S, in case that the optical axis is chosen along the z axis.
S should be proportional to the conventional order parameter [1] defined on molecular basis (see next section), provided the molecular anisotropies super- pose in a simple additive way. In the case of lyotropics
the weak birefringence ( ~ 2 x 10- 3) allows us to
use the dielectric tensor in this context [29].
In the presence of fluctuations, Q can always be diagonalized in a reference frame rotated with respect
to the laboratory axes. Keeping the laboratory frame representation, at a given time, must now be written
in a general non-diagonal form. Note that a biaxial
(symmetric) traceless tensor takes five independent
variables to be completely specified [24]. It follows that, in a system with a tendency to undergo a biaxial phase transition, the order parameter fluctuations must split into five normal modes. To learn more
about these modes, at thermal equilibrium, we write
down the Landau-de Gennes [23] expansion of the
free energy density in terms of the order parameter invariants, in the general form,
where summation over repeated indices is implied, and 0 .. = ð/ðxa. From equation (3), the change in
free energy associated with fluctuations in volume V is obtained as
Note that by taking the Fourier transforms of the Q elements,
6F can be expressed as a sum of q components, which is useful in the presence of gradient terms. The analysis of the fluctuating modes is simpler in the isotropic phase where all the equilibrium values
vanish and the terms of orders higher than Q2 can be neglected Let us take q, along the z-axis. It results that the following set of five independent variables,
__
diagonalizes the Landau-de Gennes free energy in the isotropic phase, as shown by Stinson, Litster, and Clark [18]. A similar set of normal modes appears in a microscopic (mean field) model by Blinc, Lugomer,
and Zeks [20], when applied to one uniaxial nematic
phase oriented along z: As is usual [1], let us specify
the preferred direction of the uniaxial nematic sample by a unit vector n, called « director ». Then, the first
two modes from (6) (sketched in Fig. 1 (a, b)) represent only changes in the magnitude of the order parameter (i.e., its eigenvalues) and are thus called « scalar
Fig. 1. - Scalar modes ðO (biaxiality fluctuations) and 65 (uniaxiality fluctuations) in the uniaxial nematic phases;
Nc (a and b), and ND (c and d), oriented accordingly to the
text. (Obs : the sign of An was not taken into account in the
diagrams).
modes : QZZ represents a change in the magnitude
of the z axis of the order parameter ellipsoid, which
is compensated by opposite changes in the x and y
axes. It is a « uniaxial » mode. 6(Qxx - Qyy), on the
other hand, represents a change in the plane trans-
verse to the z-axis, the magnitude of the latter being kept constant. This mode tends to destroy the sym- metry of revolution around z and thus is called
« biaxial » mode. The other three modes are asso-
ciated to the off-diagonal tensor elements. In the example, 6Qx,, and ðQvz are dominated by rigid rota-
tions of the ellipsoid axis, or, equivalently, of the
director n, respectively around the x and y axes (Fig. 2(a)), which cost few energy [1]. These « orienta-
tional » modes of uniaxial nematics have been known for a long time for thermotropics [30, 1], and more recently also for lyotropics [3, 4, 27]. The fluctuations
on Qxy (see Fig.1(a)), on the other hand, are equivalent
to those (scalar) on (Qxx - Qyy), being related by a
450 rotation of the x and y axes in the transverse plane.
These biaxiality >> fluctuations become important
when the biaxial phase is approached (pre-transitional effect), as it has been theoretically predicted [21] and experimentally verified [4]. Now, what happens with bqxy when the biaxial phase is reached ? Assuming
that the biaxial phase is oriented along the principal
axes, it is natural to expect that bQxy becomes domi-
Fig. 2. - Orientational modes in uniaxial (a : Nc ; c : ND)
and biaxial nematic phases (b, d). The two representations
of the biaxial orientational mode refer’to how it appears both (b) near the Nbx - N, and (d) near the Nbx - ND phase transitions.
nated, in its turn, by the new orientational mode
arising from the rigid rotations of the ellipsoid
around z (Fig. 2(b)). As the revolution symmetry in the x-y plane is now broken, let us introduce a new
director m taken along one of the principal directions
in this plane, say y. The « biaxial orientational mode »
can now be associated to the small in-plane rotations
of m (Obs : the triad of orthogonal vectors cha- racterizing the undistorted state of the biaxial nematic
can be completed accordingly by, I = m x n [31]).
Obviously, the three orientational modes correspond,
at a given time, to the rotations which bring the laboratory frame axes along the fluctuating eigen-axes.
Once the normal modes are found, the equipar-
tition theorem can be applied to calculate the mean
squared fluctuations at thermal equilibrium, as done
in reference [18] for the isotropic phase. We shall
present the extension of this calculation for the biaxial
phase in section (2. C). As Q and E are linked by equation (1), light waves are directly coupled to the
order parameter. The fluctuations
o i -W
will thenproduce Rayleigh scattering in the material. As stated,
when the sample is illuminated with an incident laser been along ki the light scattered in a given direc- tion, along kf, results from the particular Fourier
component bsif(q) of the fluctuations in the dielectric tensor whose wave-vector q = k f - k;. Here, the subscripts i and f denote the unit vectors i and f specifying respectively, the polarizations of the incident
and the scattered wave fronts. That is, the fluctuating
component of the dielectric tensor,
couples the incident optical electric held, Ei i, to the
scattered field, Ef f. To observe the resulting temporal
fluctuations in Ef the scattered beam is focused on
the photocathode of a photomultiplier (PM) tube (a square-law detector). The output current i(t)
of the PM is then sent to a correlator that analyses
the corresponding fluctuations.
Two regimes of detection have to be distinguished, leading to different interpretations of the measured autocorrelation function. We briefly review the main results concerning these two regimes, referring the reader, for more details, to the specialized litera-
ture [32, 33]. In the homodyne (or self-beat) regime only the scattered light arising from the fluctuating
processes impinges on the photocathode. Calling E.
this « signal » scattered field, the autocorrelation function ( i(O) i(t) > ( ... > meaning time average), describing the fluctuations in the photocurrent i(t),
will be proportional to the « homodyne correlation function » [32],
that is,
where is = 1 Es{t) 12 ), and a is a constant related
to the efficiency of the PM tube. In the « heterodyne » regime, on the other hand, a small portion of the
unshifted laser light is mixed with the fluctuating
field signal Es, providing a local oscillator field E10
that beats with E. on the photocathode surface.
Only the slow time varying (and dc.) terms contribute
for the photocurrent autocorrelation function, which is now given by [32]
where i10 = I E10 12), and gl(t), that intervenes
in the light beating term, is the « heterodyne corre-
lation function » given by,
Where the proportionality coefficient I, groups the
physical parameters characterizing the light scattering experiment, all being time independent quantities.
For t = 0,
is the scattered intensity.
Note that when the amplitude of the local oscillator is much greater than the signal amplitude, i.e., when 1 E10 I >> 1 Es I, the homodyne contribution is g2 in
equation (9) can be neglected We mention that [32],
as far as Es can be considered as a random Gaussian
variable, the following relation between the cor-
relation functions gl and g2 holds :
The main consequence of (12) is that, for a one- exponential relaxation process, the homodyne cor-
relation function is still exponential but with a half
relaxation time compared with the heterodyne cor-
relation function.
Considering again the connection between c and Q,
we note from equation (1) that, whenever the fluc- tuations on 81 (isotropic part) can be neglected, the
whole time dependence of the scattering amplitude beif(q, t) comes from the order parameter fluctuations,
that is,
Coming back to equation (7), two types of light scattering are to be retained : one « depolarized »
with il f, arising from fluctuations in the off-diagonal
components ofit The other, « polarized » scattering,
can be made only due to the diagonal fluctuations
of E if we put *1//i. With these arrangements, from equation (13) it can be seen that orientational modes will appear only on depolarized scattering while
