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Light-scattering study of Fréedericksz transitions in nematic liquid crystals
P. Galatola
To cite this version:
P. Galatola. Light-scattering study of Fréedericksz transitions in nematic liquid crystals. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.1995-2010. �10.1051/jp2:1992247�. �jpa-00247784�
Classification
Physics Abstracts
61.30G 64.70M 78.35
Light-scattering study of Fr4edericksz transitions in nematic liquid crystals
P. Galatola
Dipartimento di Fisica, Politecnico di Torino, 10129 Torino, Italy, and Consorzio Interuniversi- tario Nazionale di Fisica della Materia, Unith di Torino Politecnico, Italy
(Received 30 December 1991, accepted in final form 5 August 1992)
Abstract We introduce a novel technique to study the static distortion of
a nematic liquid crystal in the neighbourhood of a Frdedericksz transition, based on the analysis of the power
spectral density of quasi-elastic light-scattering. Under suitable conditions the symmetry break- ing of the director profile due to the Frdedericksz transition induces a similar symmetry breaking in the angular dependence of the power spectral density, thus allowing a precise determination of the critical field. Experimental results are shown for the case of splay (twist) transition in-
duced by an electric (magnetic) field. The correlation of the scattered fields at different angles is analyzed.
1 Introduction.
One of the most interesting and studied phenomena in the physics of nematic liquid crystals
is the Frdedericksz transition, in which
a nematic sample uniformly aligned between parallel planes undergoes a transition to a deformed state due to the action of magnetic, electric
or optical fields [ii. In the most usual configurations the transition is second order: when the applied field exceeds a well defined critical value, the liquid crystal continuously distorts
breaking the original symmetry of the director profile. Other cases are known for which the transition can be first order: bend deformation in an electric field [2, 3], application of crossed electric and magnetic fields [4, 5], conducting nematics under the action of an electric field [6],
zerc-field distortions [7], presence of an optical field [8].
Apart from the practical applications, the interest in such transitions stems from the pos-
sibility of accurately determining the material constants (e,g. the elastic constants) from the
knowledge of the modification of the director profile under the influence of the applied fields,
and in particular from the critical fields; other interesting informations (e,g. the viscosity cc- efficients) can be obtained from the analysis of the dynamics of the transitions.
Various techniques have been employed so far to determine the director profile as, for in- stance, the measurement of the optical transmission of a liquid crystal cell placed between
crossed polarizers [9], the conoscopic measurements [10] and the capacitive measurements [11].
In this paper we describe a novel method based on the analysis of the modification of the
homodyne power spectrum in suitable light-scattering geometries.
As it is well known, the strong scattering of light in nematics is due to the thermally excited fluctuations of the director ii]; in homogeneously aligned samples, total and dilserential cross- section measurements allow the evaluation of the three elastic constants [12] and of the ratio between the elastic constants and the dielectric and magnetic anisotropies [13], whilst power spectrum measurements of the scattered intensity allow one to determine the ratio between the elastic and the viscous constants and between the elastic constants and the magnetic or
dielectric anisotropies [14-19]. Multiple scattering elsects and the minimisation of various
sources of errors under generic geometrical conditions have been investigated [20]. Light-
scattering techniques have been proposed to measure the surface anchoring energies [21, 22].
Here we show that under suitable geometric conditions it is also possible to study the static distortion of the director field: in particular we will exploit the symmetry breaking induced by
the Frdedericksz transition.
In section 2 ~ve analyze our experimental set-up and results in the case of a splay-type Frdedericksz transition induced by an electric field: the critical voltage of this second order transition is well marked by the opposite variations of the linewidths and amplitudes of the power spectra relative to two scattering directions symmetric with respect to the undistorted
director configuration. ~ife also report, for the first time to our knowledge, cross-spectrum mea-
surements for which we still lack a satisfying theoretical model able to explain all the features that we observed. In section 3 we show how the same technique can be modified in order to
detect the twist-type FrAedericksz transition in a magnetic field. Finally in section 4 we briefly
discuss our results with particular attention to the splay geometry, for which we introduce a very simple model that is nevertheless in reasonable agreement with the experimental data.
2. Splay ti~ausitioii in
au electi~ic field.
The experimental set-up employed to detect the electrically induced splay FrAedericksz transi- tion is schematically shown in figure I. The radiation emitted from a 10 mW cw He-Ne laser L (1
= 632.8 nm) is sent through the optical isolator composed by the linear polarizer Pi and the quarter-~vave plate Q; the circularly-polarized beam is then made divergent by the lens
Ll, passed through the spatial filter F and focused onto the planarly-oriented liquid crystal
cell LC by the converging lens L2: the beam waist on the sample is m 200 pm. The incident beam is normal to the cell and linearly polarized by P2 in a direction orthogonal to the direc- tor, corresponding to the ordinary polarization. The planarly-oriented liquid crystal cell was
prepared by Tecdis S.p.A. in a class-1000 clean room and consists of two glass plates coated with a conducting ITO electrode and a surfactant polyimide layer mechanically rubbed in one direction to obtain a strong planar anchoring. The empty cell thickness was interferometrically
measured and found to be (8.3+ 0.05) pm. The cell was then filled, by capillary action in a low pressure glass bell, with a liquid crystal having positive dielectric and magnetic anisotropies (4-cyanc-4'-n-pentylbiphenyl, known as 1(15, by BDH).
The scattered light is detected by two low-noise photodiodes (EG&G FFD-100 operated in the photoconductive regime) Dl and D2 laying on the plane of the director and of the normal to the LC cell and symmetrically located with respect to the transmitted beam. The two
polarizers Al and A2 in front of the photodiodes select the extraordinary-polarized component of the scattered light (parallel to the scattering plane). The current from the photodiodes is
amplified by two high-gain transresistance amplifiers and sent to a digital dual-channel FFT
DV PC SA
A2 D2
I ~2
I ~i
L pi Q Li F L2 P2 LC
Al Di
Fig. 1. Experimental set-up for the electrically induced splay Frdedericksz transition: L He-Ne laser;
P1 linear polarizer; Q quarter-wave plate; L1 diverging lens; F spatial filter; L2 converging lens; P2 linear polarizer; LC liquid crystal cell; D1, D2 photodiodes; A1, A2 analyzers; SA dual channel FFT
spectrum analyzer; PC computer; DV digital voltmeter. pi and fl2 are the two scattering angles.
signal analyzer (AND AD-3525) connected to a PC.
The splay Frdedericksz transition is induced by a sinusoidal I kHz zero mean value electric field normal to the glass plates obtained by a voltage, generated by a digital to analog converter in the PC, applied to the ITO electrodes. The precise RMS voltage applied to the cell is
measured by a digital voltmeter (HP 3456A). For small distortions in the neighbourhood of the Frdedericksz threshold the period of the driving field is much smaller than the cell response
time, and thus no peaks related to the frequency of the applied voltage are observed in the noise power spectra; on the other hand, far above the Frdedericksz threshold (typically twc- three times the threshold voltage), a sharp peak at twice the frequency of the applied field, corresponding to the quadratic dielectric response of the liquid crystal, is sometimes observed, accompanied also by higher harmonics and smaller components at the frequency of the field and its odd harmonics, presumably due to a flexoelectrical response. The presence and the
heights of the peaks depend on the scattering angles.
The position of the director, and thus the polarization of the incident and scattered light,
was adjusted by rotating the sample around the incident beam until extinction of the light
transmitted through the sample and a polarizer crossed with respect to the incident polarization
was found. Particular care was taken in order to reduce the scattering by stray reflections and surface defects, given the reduced thickness of the cell: this is the main reason for the elaborate optics between the laser and the sample.
For an unbounded nematic liquid crystal with homogeneous alignment and no external fields
applied, with our scattering geometry, the power spectrum of the scattered light intensity,
both under hoinodyne and heterodyne conditions, has a Lorentzian shape ii]; the width of the Lorentzian in the homodyne regime is twice that for the heterodyne situatiin. The effects of
the finite cell thickness have been considered in [23]: in our conditions they only amount to a
quantization of the allowed values of the longitudinal wavevectors of the director fluctuations,
an elsect that in our case, where high values of the longitudinal wavevectors are detected, can be neglected. To test whether we were working in the homodyne or in the heterodyne regime we
checked the statistics of the scattered light intensity: in fact, as the complex scattered electric field has Gaussiaii statistics independent of the phase, it is easily shown that the heterodyne signal must have a Gaussian statistics as well, whilst the homodyne signal has an exponential character. In our case the statistics of the detected light is always well fitted by an exponential,
thus meaning that we are working in a good homodyne regime.
When the cell is distorted, above the Frdedericksz threshold, one would expect the noise power spectra to be modified and to consist of a superposition of various Lorentzian, due to the non-uniform tilt of the nematic director in the cell. As a matter of fact, however, it turns out that the spread of the widths of the dominant Lorentzian contributions is smaller than the resolution allowed by the experimental errors, and thus we could extract only an average
linewidth by fitting the spectra with a single Lorentzian.
-3 6
-~~ cJ
~' -41 ~
ol
b
0 50 100 150 200
f (Hz)
Fig. 2. Homodyne power spectra S as a function of the frequency f for O-E scattering with no
applied voltage, normal incidence and scattering angles pi
=
-6° (Si,
curves a) and fl2
=
6° (52,
curves b). The solid lines are the experimental data, the dashed lines Lorentian least-squares fits.
Figure 2 shows the two power spectral densities Si, 52 of the scattered intensities as a function of the frequency f, obtained with no applied voltage and ordinary (extraordinary) polar12ation for the incident (scattered) beam-OE geometry-at room temperature (21°C)
and scattering angles pi " -6° (Si, curves a) and fl2 = 6° (52, curves b). The spectrum analyzer was set at 200 Hz full-scale with a spectral resolution of 400 lines, corresponding to a frame time of 2 s, and 500 time averages: thus the recording of each spectrum required
1000 s. The solid lines represent the experimental data while the dashed curves
are Lorentzian
least-squares fits
~~'~~~~
l +
)~'l,2)~
~~'~~
where the indices 1, 2 must be selected concurrently. For Si we get an amplitude Al
=
-37.3 dB and a linewidth (HWHM) ri " l12 Hz; for 52, instead, we have A2 = -38.9 dB and
r2 " l19 lI2. Tile siiiall differences between tile two spectra are due to several minor sources
of errors, more precisely:
I) residual parasitic static scatterings which increase the amplitude and reduce the linewidth of the spectra;
it) a small surface tilt angle of the director (by independently measuring the optical transmit- tance, as a function of the applied voltage, of the cell placed between crossed polarizers
making an angle of 45 degrees with respect to the incidence plane, in which the director lies,
we estimated that the mean tilt angle is less than 4 X 10~~ degrees [24]);
iii) difficulty in the symmetrical positioning of the two photodiodes due to the divergence of the transmitted beam and to the short distance between the photodiodes and the sample required by the low scattered intensity;
iv) a slight imbalance between the gains of the two photodetecting chains.
-33 2
(a) (b)
/~ #~
t
n 4
At f /
15
~ 0
~ 0 ~ ~ ~ ~ ~ ~ ~ ~ ~
~j ~ ~~
~_-38 $
j~ ~' ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
A A A ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
~ ~
~ ~ ~ ~ ~ ~
~ ~~ $ ~ i
~~
$ i~
~
%
5 o
o.o o. 5 o o.o o 5 1 o
V (Volt) V (Volt)
Fig. 3. Variation of the homodyne power spectra as a function of the RMS applied voltage V for O-E scattering with normal incidence and scattering angles pi " -6° (Ai, ri and fl2 " 6° (A2, r2).
a) Amplitudes Ai (o) and A2 (b); b) linewidths ri (o) and r2 (b). The data are obtained through
a Lorentzian least-squares fit of the experimental spectra.
We then applied and slowly increased by small steps the sinusoidal voltage to the cell and for each value of the voltage we recorded, under stationary conditions, the corresponding power
spectra for the two selected scattering angles. As previously discussed, each spectrum was
then fitted with a single Lorentzian: figure 3 shows the variation of the amplitude A and of the linewidth r of the fitted Lorentzians as a function of the RMS voltage V. We can see that the spectra remain basically unchanged until the Frdedericksz threshold voltage is reached:
from this point on one of the t>vo spectra tends to become wider and lower, whilst the other
gets narrower and higher. The quantity that most accurately locates the transition is the
ratio between the linewidths R = ri/r2 of the two spectra (see Fig. 4): the transition voltage
VF determined from this curve is 0.780 ~ 0.002V and coincides, within the bounds of the experimental errors, with the value that we obtained by independent fringe measurements [24].
The symmetry breaking of the two spectra, which occurs only for the O-E scattering, reflects the symmetry breaking of the director field induced by the Frdedericksz transition; in particular the variation of the linewidths is due to the asymmetrical change of the two scattering wavevectors as the director rotates: this immediately allows one to determine the direction of rotation of the director. When the applied field is increased far beyond the Frdedericksz threshold, the two spectra tend to become symmetrical again, as the cell tends to become homeotropically aligned away from two tiny surface layers.
JOURNAL DE PHYSIQUE II T 2, N' II, NOVEMBER 1992 75
3
o
o o o 5 io
V (Volt)
Fig. 4. Ratio R
= ri/r2 bet~veen the linewidths ri and r2 (see Fig. 3b) as a function of the RMS
voltage V.
b
uJ
a
-4
0 loo 200
f Hz)
Fig. 5. Same as figure 2 but for an applied RMS voltage of 0.85 V, beyond the Fr4edericksz threshold
VF " 0.780 V.
In figure 5 we display the two power spectra corresponding to a RMS voltage of 0.85 V: it
can be seen that even for a very distorted cell they preserve a good Lorentzian shape, thus
justifying our way to fit the experimental data.
We note that no field effects are observed below the transition, and in particular no critical
slowing down: this is due to the fact that the electric field sensitively influences only the splay-
bend fluctuation modes characterized by the longitudinal wavevector ~r/d, where d is the cell thickness, and transverse wavevectors close to zero. In our scattering geometry, instead, we
are detecting a single twist-bend fluctuation mode characterized by high values for both the transverse and the longitudinal wavevectors. The possibility of indirectly detecting the critical and quasi-critical modes, based on the detection of a large number of coherence zones, was
io zoo
~ ~
~~~~ 0 _
/~ ~ ~
a n a~ a a
7l
zg
~ ~ ~ ~~A ~ ~ n a
~ 0 -
~~~ ~
-3 0
-zoo
lo loo 1000
f (Hz)
Fig. 6. Modulus M (o, left-hand scale) and phase ~ (b, right-hand scale) of the cross spectrum for E-E homodyne scattering with incidence angle di
=
48.5° and scattering angles pi
"
-6° and
fl2 " 6°; the incident and scattered fields and the director are all in the same plane. The two signals
are equally correlated over all the frequencies (the phase ~ is equal to zero and the coherence, here not
shown, is constant). The sample is
a 26 ~m thick planar cell filled with BDH K15 liquid crystal with
no applied voltage. The modulus is normalized with respect to the geometric mean of the amplitudes
of the spectra of the two signals. The solid line is a Lorentzian least-squares fit of the modulus M having amplitude A
= -5.25dB and linewidth r
= 39.1Hz.
exploited in [25]: this approach, however, poses several problems, as discussed in [24].
With no applied voltage the intensities of the two scattered fields are entirely uncorrelated,
as shown by their cross-spectrum, and in particular by the phase of the cross-spectrum, which
turns out to be completely random. The same holds true for the same scattering geometry but
for extraordinary incident polarization and ordinary scattering polarization (E-O geometry)
and for different incident and scattering angles. The situation, instead, is completely different in the case of extraordinary polarization for both the incident and the scattered beam (E-E geometry): with this arrangement the intensities of the scattered fields, for suitably selected pairs of angles, contain a strong correlated part, as shown in figure 6. For this E-E geometry the contribution of the static surface defects to the scattered field was much greater than in the previous configurations: thus to get homodyne spectra we had to prepare another cell, having a thickness of 26 pm, made by two ITO-coated glass plates, evaporated with a Sioz layer at grazing incidence (50° from the normal) to obtain a strong planar anchoring, and filled
by capillary action with the same liquid crystal as in the thinner cell (K15). Even so we had to select a non-zero incidence angle in order to increase the E-E scattering cross-section.
To verify that the phases of the two scattered fields are also correlated, we measured the cross-spectrum in heterodyne conditions (see Fig. 7: here the data were obtained again with the 8.3 ~m cell, with the surface defects acting as local oscillator). In this case the linewidths of the spectra are halved and, depending on the relative phases between the local oscillators and the two signals, one can obtain both correlated or, as in the case shown, anticorrelated
signals. The degree of correlation, given by the coherence C-defined as the ratio between the square modulus of the cross-spectrum and the product of the two self-spectra-is constant over all the frequencies, indicating that the correlation is static and not dynamic. The value of C
depends on the number of coherence zones on the two detectors and, in the case of heterodyne
