JOURNAL OF APPLIED PHYSICSVOLUME 90, NUMBER 7

1 OCTOBER 2001

Ionic contribution to the dielectric properties of a nematic liquid crystal
in thin cells
Surajit Dhara and N. V. Madhusudanaa)
Raman Research Institute, C. V. Raman Avenue, Bangalore 560 080, India

共Received 9 January 2001; accepted for publication 1 June 2001兲
We report measurements on dielectric properties of a nematic liquid crystal in planar aligned cellswith two different thicknesses 共1.2 and 6.7 ␮m兲. The nematic has a wide thermal range and thetemperature variation of the effective dielectric constant ( ⑀⬜ ) of the thick cell reflects that of theorientational order parameter. On the other hand, the thin cell exhibits a higher value of ⑀⬜ whichis essentially temperature independent. We have conducted detailed dielectric measurements in thefrequency range of 10 Hz–1.1 kHz on both the cells and analyzed the data using a simplified model
which takes into account the ionic contribution. It leads to an estimation of the temperaturedependencies of the number density and diffusion coefficient of the impurity ions. © 2001
American Institute of Physics. 关DOI: 10.1063/1.1388163兴

I. INTRODUCTION

Dielectric spectroscopy is used to characterize the electrical properties of condensed matter. The influence of ionson the real and imaginary parts of the frequency dependent
dielectric constants of both solid and liquid electrolytes has
been studied both experimentally and theoretically for a longtime.7–9 The theory has since been extended for polymer
melts,10 which has been subsequently simplified and used todescribe the effect of ions on the dielectric response in theisotropic 共I兲 phase of liquid crystals.11,12 The latter studies
were made on cells with a thickness of about 5 ␮m or larger.
In view of the emerging interest in using much thinner cells
in LCDs, as well as an intrinsic interest in the problem, it
would be useful to conduct experiments in much thinner
cells, with thickness ⬍2 ␮m. In this article, we report frequency dependence of dielectric properties on a nematogenwhich exhibits the nematic phase over a very wide temperature range. The measurements have been made on cells withtwo different thicknesses and indeed the thinner 共⯝1.2 ␮m兲cell shows a pronounced influence due to the ionic contribution. In Sec. II, we give the experimental technique, anddescribe the results and discuss the latter on the basis of an
appropriate theoretical model in Sec. III. We make some concluding remarks Sec. IV.

Nematic 共N兲 liquid crystals are orientationally ordered
liquids, made of rod-like organic molecules.1 The averageorientation direction of the molecules is specified by a unit
vector n̂ called the director which is apolar in nature. The
medium combines anisotropies in dielectric, optical, andother properties with a facility to flow and hence exhibits
pronounced electro-optic effects, which are exploited in flat
panel liquid crystal display 共LCD兲 devices.2 The chemicalcompound exhibiting the nematic itself is usually a good
insulator. However, unless special precautions are taken, thematerial has ionic impurities, giving rise to an electrical conductivity ␴ ⬃10⫺10 (⍀ cm) ⫺1 . The conductivity has positive
anisotropy ⌬ ␴ ⫽ ␴ 储 ⫺ ␴⬜ , in which the subscripts refer todirections in relation to n̂. Indeed in materials with negative
dielectric anisotropy ⌬⑀, a dc or low frequency ac electric
voltage beyond a threshold value can generate electrohydrodynamic 共EHD兲 instabilities, due to the ionic conductivity.1
The display devices make use of electric field 共E兲 effects inmaterials with positive ⌬⑀, which favors an alignment of n̂
parallel to E. Since a nematic is a liquid, its response isrelatively slow,1 and the response time ␶ ⯝ ␩ d 2 / ␲ 2 K, where␩ is an effective viscosity, K an effective curvature elastic
constant and d the sample thickness. A typical LCD cell hasa thickness ⯝5 ␮m. The response time can be reduced considerably by reducing the thickness. Furthermore, in the surface stabilized ferroelectric devices,3 as well as the bistable
twisted nematic displays4,5 which are being investigated inrecent years, the thickness has to be quite small, ⯝2 ␮m or
lower. When the thickness of the sample is reduced, at any
given applied voltage the field increases and the influence ofions on the electrical properties of the cell becomes significant. The ionic contribution has been discussed in detail onlyin the case of EHD instabilities, and on the dc switchingprocess in a boundary layer bistable nematic display.6

II. EXPERIMENT

The main aim of this investigation is to look for the
influence of ions on dielectric properties of liquid crystals. In
the display devices, the liquid crystal is aligned with its director parallel to the glass plates. The dielectric anisotropy
共⌬⑀兲 of the material is positive so that the nematic shows arealignment of the director above a threshold voltage2
共Freedericksz transition兲. We have chosen a material withnegative dielectric anisotropy to avoid this problem. However, such a material can give rise to EHD instabilities beyond a threshold of about ⯝5 V or more, depending on thefrequency.1 We conduct our experiments at 1 V which is well
below the threshold value. The compound studied is 2-cyano
4-heptylphenyl-4⬘-pentyl-4-biphenyl carboxylate 关7共CN兲5a兲Electronic mail: nvmadhu@rri.res.in

0021-8979/2001/90(7)/3483/6/$18.00

3483© 2001 American Institute of Physics

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3484

J. Appl. Phys., Vol. 90, No. 7, 1 October 2001

FIG. 1. Schematic diagram of the planar aligned nematic liquid crystal cellused in the experiments. Note that the gap d is highly exaggerated forclarity.

S. Dhara and N. V. Madhusudana

oscillator of the lock-in amplifier is adjusted to a rms valueof 1 V, and applied to the cell which is in series with a 1 ␮Fstandard capacitor 共C兲 as shown in the figure. The amplitude
and phase of the applied voltage are also checked using areference circuit with two standard resistors in series, connected in place of the cell. The phase 共␾兲 and the rms amplitude (V) of the voltage measured across the series 1 ␮F
capacitor are used to calculate the capacitance and parallel
resistance corresponding to the cell by using the following
relations:14
R P⫽

for short兴 which has the phase sequence: crystal 45 °C N
102 °C I. It was obtained from Merck. The lateral cyanogroup of the molecule gives rise to a large negative dielectricanisotropy in the nematic phase.13 The material has a very
wide nematic range and the liquid crystal can be easily supercooled to room temperature. It is also chemically stable.
The cells 共Fig. 1兲 were prepared using 0.5 mm thick indium
tin oxide 共ITO兲 coated glass plates. An appropriate pattern
was etched in the two plates so that the cell had an overlapping ITO 共i.e., electrode兲 area a of 0.6 cm2. The etched plates
were coated with a polyimide layer and cured at 280 °C for 1
h. The coating was then rubbed unidirectionally with a tissue
paper. The plates of the cell were separated by appropriate
spherical spacers and sealed with a glue. The rubbing directions on the two plates were arranged to be parallel to ensure
a planar alignment of the liquid crystal. The thickness d ofthe empty cell was measured using channel spectrum. Two
thicknesses viz, d⫽6.7 ␮ m and d⫽1.2 ␮ m were chosen forthe studies. The cells were filled with the sample in the isotropic phase and on slow cooling to the nematic range, wellaligned samples were obtained. The electrical connections to
the two plates were made through copper wires which were
soldered to the appropriate connection pads using an ultrasonic soldering gun.
The temperature of the sample was maintained and measured 共to ⬃10 mK兲 by placing it in an Instec hot stage. The
dielectric properties of the cell were measured using a DSP
lock-in amplifier 共SRS 830兲. The block diagram of the setup
is shown in Fig. 2. The sinusoidal output of the internalY␻ sin ␣X
C P⫽ ,
Y

where X⫽cos ␣⫺Q, Y ⫽(sin2 ␣⫹X2)/CQ, Q⫽V/V 0 , and␣⫽␺⫺␾ where V 0 and ␺ are the amplitude and phase and ␻
the angular frequency of the applied signal.
We have checked the calibration of the setup by connecting standard capacitors in parallel with standard resistors inplace of the liquid crystal cell. The measured values agreewith the standard values to within ⬃3%. We have also
checked the calibration by measuring the capacitance andresistance using a Wayne–Kerr bridge which operates at 1.6
kHz.
The measurements were completely controlled by acomputer, using a suitable program. The temperature wasmaintained at various values in the nematic range to an accuracy of ⫾0.01 °C, and the measurements were made atseveral frequencies in the range of 10–1100 Hz.
III. RESULTS AND DISCUSSION

In order to bring out the strong influence of the cell
thickness on the measured properties, we have shown theeffective dielectric constant 关which is the ratio of the cell
capacitance with the sample (C p ) to that of the empty cell兴and conductivity at 1.1 kHz as functions of temperature forboth the cells in Figs. 3 and 4, respectively. These measurements correspond to ⑀⬜ and ␴⬜ , i.e., the properties in a
direction orthogonal to the director. In Fig. 3, the dielectric
constant of the 6.7 ␮m thick cell follows the expected trend,
decreasing with increase of temperature, the rate of decrease
becoming rapid near the nematic–isotropic transition point
T NI . It is easily shown that1 the contribution from the liquid
crystal is given by:⑀⬜ ⫽¯⑀ ⫺ 31 ␦ ⑀ S,

FIG. 2. Block diagram of the experimental setup used for the impedance
analysis of the liquid crystal cell. LIA is an SRS 830 lock-in amplifier. The
voltage across the capacitance C(⫽1 ␮ F) is used in the impedance analysis.

共1兲

共2兲

where 关¯⑀ ⫽( ⑀ 储 ⫹2 ⑀⬜ )/3兴 is the average dielectric constant,and ␦⑀ is the value of the anisotropy 共⌬⑀兲 when the orderparameter S⫽1. As S decreases with increase of temperature,and ␦⑀ is negative, ⑀⬜ decreases with temperature. The trend
shown by the thin cell is peculiar. The dielectric constant issomewhat larger than that for the thicker cell even at low
temperatures, but at temperatures above 60 °C, the value
does not decrease as in the thicker cell but more or lessremains constant and decreases slightly only very near T NI .
In both cases, the dielectric constant jumps to a lower value

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J. Appl. Phys., Vol. 90, No. 7, 1 October 2001

FIG. 3. Temperature dependence of the effective dielectric constant of theliquid crystal taken in cells with two different thicknesses, at a frequency of
1.1 kHz. Note the higher value for the thinner cell, which hardly varies withtemperature in the nematic phase above 60 °C.

at T NI . Furthermore, the value for the thinner cell continues
to be significantly larger than that for the thicker one even in
the isotropic phase.
The effective conductivity given by ␴⬜ ⫽d/(aR P ) isshown in Fig. 4 for both the cells. As expected, the conductivity increases with temperature in both cases as the mobility and the number density of ions increase with temperature.
However, it can be noted that the relative variation of ␴⬜ inthe thin cell is somewhat higher than that in the thick cell.
The markedly different response of the thin cell compared to the thick one arises due to the ionic contribution. In
order to investigate this effect in greater detail, we have carried out frequency dependent measurements of the dielectric
properties at different temperatures. The raw data on the effective dielectric constant and conductivity are not convenient for a comparison with the theoretical analysis to follow.
We first present a summary of the theoretical background.
The liquid crystal is separated by the ITO electrodes byaligning polyimide layers as in LCDs 共see Fig. 1兲. This

S. Dhara and N. V. Madhusudana

3485

FIG. 5. Equivalent circuit of the liquid crystal cell with the ‘‘lumped’’ values of the boundary layer capacitance C bl and resistance R bl arising fromboth the boundaries.

boundary layer has a typical thickness of few hundred Å andits effective electrical resistance (R bl ) as well as capacitance(C bl ) can be expected to be quite large, i.e., the layer essentially is an insulator. As the dielectric constant of this layer issimilar to that of the liquid crystal, it can be assumed that
there is no adsorption of ions at the interface between the
two. The dielectric constant ( ⑀⬜ ) of the liquid crystal medium itself, which depends on the induced dipole and orientation polarization contributions of the molecules can be considered to be essentially frequency independent in the range
共⬍1.5 kHz兲 in which the measurements have been made.
This contribution produces an effective bulk capacitance denoted by C B . The medium also has a uniform distribution ofions, which produces a frequency independent conductivity
␴ which in turn produces a bulk resistance of the medium
denoted by R B . The application of an electric field generates
a nonuniform distribution of ions which depends on the frequency and in turn contributes both to the real and imaginaryparts of the dielectric constant. These in turn give rise to acapacitance C I ( ␻ ) and resistance R I ( ␻ ). The equivalent circuit corresponding to the cell in which all these contributions
have been included is shown in Fig. 5. The complex impedance of the equivalent circuit is given by11
1
1
⫹
2R bl
R B R I共 ␻ 兲
Z A⫽⫹
1⫹ 共 ␻ R bl C bl 兲 2
1
1 2⫹⫹ ␻ 2 关 C B ⫹C I 共 ␻ 兲兴 2
R B R I共 ␻ 兲⫺j⫹FIG. 4. Temperature dependence of the effective conductivity of the liquid
crystal in the two cells as described in the caption of Fig. 3.

冋

冋

冋

册

2 ␻ R 2bl C bl1⫹ 共 ␻ R bl C bl 兲 2册␻ 关 C B ⫹C I 共 ␻ 兲兴
.
2
1
1
⫹
⫹ ␻ 2 关 C B ⫹C I 共 ␻ 兲兴 2
R B R B共 ␻ 兲册

共3兲In terms of the measured values C P and R P 关given by
Eq. 共1兲兴, the measured complex impedance is given by

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3486

J. Appl. Phys., Vol. 90, No. 7, 1 October 2001

Z B⫽

冋

册

␻ R 2P C P
RP⫺j.
1⫹ 共 ␻ R P C P 兲 2
1⫹ 共 ␻ R P C P 兲 2

S. Dhara and N. V. Madhusudana

共4兲

We summarize below a grossly simplified method of
evaluation of the ionic contribution which generates C I ( ␻ )and R I ( ␻ ). It is convenient to assume that positive ions arevery much more mobile than the negative ones and the latterdo not contribute to the current. If ␳ (z,t) is the charge density of the mobile ions, the charge continuity equation is
given by⳵␳ ⳵ J 共 z,t 兲
⫹
⫽0,
⳵t
⳵z

共5兲

where J(z,t) is the current density, which is given by
J 共 z,t 兲 ⫽⫺D

⳵␳ 共 z,t 兲⫹ ␮ ␳ 共 z,t 兲 Ee i ␻ t ,⳵z共6兲

where D is the diffusion constant and ␮ is the mobility. As
the polymide is a blocking layer, we can assume that at thetwo boundaries 共z⫽0 and d兲 J (0) ⫽J (d) ⫽0. As our measurements have been made only at the frequency ␻ of the applied
voltage, we assume that

␳ 共 z,t 兲 ⫽ ␳ 0 共 z 兲 ⫹ ␳ 1 共 z 兲 e i ␻ t ,共7兲

where ␳ 0 (z) is the time independent density of ions and␳ 1 (z) is the amplitude of ion density which oscillates at thefrequency of the applied field. The mobile charges move under the action of the external field and the resulting spacecharges produce a polarization which contributes to the total
dielectric constant. The space charges give rise to a nonuniform electric field which can be calculated by using the Poisson relation. The solutions in that case are quite complicatedand cannot be expressed in a simple analytical form. Our
sample had unknown ionic species whose number density 共n兲is ⬃1015/cc. As such, the sample is really a weak electrolyte.
Furthermore, the transit time of ions from one electrode to
the other under a reversal of a voltage V is given by ␶ tr⫽(6 ␲ ␩ r/q)(d 2 /V) where ␩ the effective viscosity is ⯝1
poise at room temperature, and r the ionic radius can be
expected to be ⬃10 Å. However, as the mesogenic molecules
are strongly polar we can also expect that the ion will be
dressed by a few molecules which are attached to the ion.
The effective ‘‘ionic’’ radius can be a few 共say ⬃5兲 times
larger than the bare value. For the thick cell, ␶ tr ⯝2 s while itis about 0.1 s for the thinner one for the applied voltage of 1
V. As such, in the frequency range of interest 共10–1100 Hz兲,to a good approximation, the nonuniformity in ion distribution can be assumed to be small. The field in the cell can thenbe assumed to be equal to the external field. As the temperature is increased, the viscosity reduces and the approximation
becomes less justifiable. However use of the approximation
has the virtue that analytical expressions can be derived forC I ( ␻ ) and R I ( ␻ ). In a recent paper Sawada et al.12 havealso argued that the approximation may be adequate. Moreover, we can fit the experimental data quite well to the solutions by using this approximation, which is hence further
justified a posteriori. Using the boundary condition that J⫽0 it can be shown that ␳ 0 is actually independent of z and

FIG. 6. Temperature dependencies of the real 共filled circles兲 and the imaginary 共filled squares兲 parts of the measured impedance Z B 关see Eq. 共4兲兴 in thecell with thickness 1.2 ␮m, at 50 °C. The lines show the calculated variations of Z A 关see Eq. 共3兲兴 with the fit parameters shown in Table Ia.

␳ 1共 z 兲 ⫽

冉

冊

␮E
e ␬z
e ⫺␬z
␳0
⫺
,
␬D
1⫹e ␬ d 1⫹e ⫺ ␬ d

共8兲

where ␬ ⫽ 冑i ␻ /D and from the Einstein relation, ␮ /D
⫽q/k B T, where k B is the Boltzmann constant. Hence ␳ 1 (z)has components both in phase and ␲/2 out of phase with theapplied ac field. The additional polarization due to the spacecharge formation is 关 ␳ 1 (z)z 兴 and the thickness averaged
value isP̄ 共 t 兲 ⫽e i ␻ t

兰 d0 z ␳ 1 共 z 兲 dz
兰 d0 dz

共9兲.

The corresponding frequency-dependent ionic contributions to the complex dielectric constant are given by:冋册冋

4 ␲ nq 2 D 1⫹2e A sin共 A 兲 ⫺e 2A
C I共 ␻ 兲 d⫽ ⑀ I⬘ 共 ␻ 兲 ⫽⫺⑀ 0a
␻ k B TA 1⫹2e A cos共 A 兲 ⫹e 2A

册

共10兲
and

FIG. 7. Same as in Fig. 6, at a temperature of 90 °C. Note the minimum andmaximum in the frequency dependence of the imaginary part of the impedance.

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J. Appl. Phys., Vol. 90, No. 7, 1 October 2001

S. Dhara and N. V. Madhusudana

FIG. 8. Same as in Fig. 6, for the thicker cell with d⫽6.7 ␮ m, at 50 °C.

冋

4 ␲ nq 2 Dd1⫽ ⑀ I⬙ 共 ␻ 兲 ⫽␻ ⑀ 0R I共 ␻ 兲 a
␻ k BTA⫹2A

册1⫺2e sin A⫺e,
A 共 1⫹e A cos A⫹e 2A 兲

3487

FIG. 9. Same as in Fig. 8 at 90 °C. Note the peak in the frequency dependence of the imaginary part of the impedance.

共11兲

where a is the area of electrodes, nq⫽ ␳ 0 and A⫽d 冑␻ /2D. Equations 共10兲 and 共11兲 agree with those derived
by Sawada.11 Using these in Eq. 共3兲, the real and imaginaryparts of Z A can now be calculated. Comparing them with thecorresponding parts of the experimental complex impedance
Z B given by Eq. 共4兲, a nonlinear least-squares fitting procedure is used to get the best values for R bl , C bl , D, n and thefrequency independent terms R B and C B . Some representative variations of the real and imaginary parts of the impedance are shown in Figs. 6 –9 at two different temperatures forboth the thin and thick samples. It is seen that the agreement
between the calculated and measured variations is quite satisfactory. The relevant fit parameters at different temperatures are collected in Table I for both the thick and thinsamples. From the figures it is clear that the real part of Z B ,which gives the experimental data has a monotonic decrease
as a function of frequency at all temperatures for the cells ofboth thicknesses. The calculated values, which are the realpart of Z A agree well with the data. The imaginary part of
共the measured兲 Z B , and the corresponding calculations basedon Z A , show a peak around 100 Hz, which is more pronounced for the thin cell. The frequency corresponding to thepeak decreases as the temperature is lowered, while the magnitude of the impedance itself increases. In the thin cell there
is also a low frequency minimum above 75 °C and the impedance strongly increases at very low frequencies. The
physical origin of these features can be understood from Eqs.
共3兲, 共4兲, 共10兲 and 共11兲. A⫽d 冑␻ /2D, i.e., A⬀ ␯ 1/2 where␯⫽␻/2␲ and the lowest frequency that we have used is␯⫽10 Hz. As the typical diffusion constant D⬃10⫺7 cm2/s,
A is much larger than 1 even for the thin cell. The boundary
layer capacitance C bl is ⯝10⫺7 F, and the corresponding
resistance R bl ⯝107 ⍀. Using these, the imaginary part of Z A
can be approximated to Z A (imag)⯝C 1 / ␻ ⫹ 关 C 2 ␻ /(1⫹C 3 ␻ 2 ) 兴 , where C 1 ⬃107 , C 2 ⬃104 and C 3 ⬃10⫺6 . It isclear that the second term leads to a maximum when ␻ 2⫽1/C 3 where 冑C 3 is essentially the charge relaxation time⑀ 0 ⑀⬜ / ␴ . On the other hand, C 1 arises from the boundary
layer impedance and the first term becomes comparable tothe second one for ␻⯝100 共i.e., ␯⯝20 Hz兲. Below this fre-

TABLE I. Fit parameters at different temperatures.

Temperature
in °C

Diffusion
constant D
共cm2/s兲(⫻10⫺8 )

105
90
65
50

6.3
4.9
2.5
1.9

Temperature
in °C

Diffusion
constant D
共cm2/s兲(⫻10⫺8 )

105
90
65
50

9.7
8.8
6.1
5.4

共a兲 Sample thickness d⫽1.2 ␮ m
Boundary
Boundary
layer
layer
Numberresistance R bl
capacitancedensity n(cc) ⫺1 (⫻1014)
共M⍀兲
C bl (⫻10⫺7 F)
20
16
6.6
4.5

7
7
7
7

2.7
2.7
2.7
2.7

共b兲 Sample thickness d⫽6.7 ␮ m
Boundary
Boundary
layer
layer
Numberresistance R bl
capacitancedensity n(cc) ⫺1 (⫻1014)
共M⍀兲
C bl (⫻10⫺8 F)
4.8
2.5
0.6
0.2

12
12
12
13

9
9
9
9

Bulk
resistance R B
共M⍀兲

Bulk
capacitanceC B (⫻10⫺9 F)

1.5
3.4
3.2
8.5

2.1
2.1
2.2
2.4

Bulk
resistance R B
共M⍀兲

Bulk
capacitanceC B (⫻10⫺10 F)

12
50
83
145

3.8
4.2
4.4
4.7

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3488

J. Appl. Phys., Vol. 90, No. 7, 1 October 2001

quency the contribution from the boundary layer leads to a
rapid increase in Z A (imag) and hence produces the sharp
minimum seen in Fig. 7.
From Table I, the number density of ions at any given
temperature is much larger in the thin cell compared to thethick one, which arises due to the fact that the two cells wereprepared on different days and the contamination during processing of the cells is likely to be responsible for the different values. One can assume that the bulk conductivity ␴⯝␮␳and the values of the bulk resistivity R B are consistent withthe diffusion constant D and the number density n in both
cells. Both decrease as the temperature is lowered as expected. The bulk value C B is proportional to ⑀⬜ , the dielectric constant of the liquid crystal. From independentmeasurements13 ⑀⬜ ⯝5 in the compound used in the present
study and decreases with increase of temperature as requiredby Eq. 共2兲. On the other hand, the boundary layer values R bl
and C bl which are the lumped values from the polyamide
coatings on both the electrodes hardly vary with temperature.
It is also easy to understand the higher effective dielectric constant exhibited by the thin cells compared to the thickone 共Fig. 3兲. The ions move faster under the higher field
acting in the thin cell, and also as the transit distance isshorter, the space charge contribution to the polarization isrelatively large. Consequently, the effective dielectric constant goes up. This is reflected in Eq. 共10兲 in which A⬀doccurs in the denominator of the expression for ⑀ I⬘ ( ␻ ). Fromthat equation, the ionic contribution is also ⬀nD 3/2. As thenumber density as well as the diffusion coefficient increase
with temperature 共see Table Ia兲, ⑀ I⬘ ( ␻ ) increases with temperature, and compensates for the decrease of ⑀⬜ with temperature reflecting that of the order parameter S as given by
Eq. 共2兲. The consequence is that the effective dielectric constant remains practically independent of temperature over awide range 共see Fig 3兲.
IV. CONCLUSIONSOur experiments have shown that as the thickness of thecell is reduced to ⬃1–2 ␮m, the ionic contribution to the

S. Dhara and N. V. Madhusudana

dielectric response of a liquid crystal cell can be significant
for samples with a conductivity ⬃10⫺10 (⍀ cm) ⫺1 . The effect grows very large 关⬀ ␻ ⫺3/2 for the real part and ⬀ ␻ ⫺1 forthe imaginary part, see Eqs. 共10兲 and 共11兲兴 as the frequency
is reduced. In our simplified analysis, we have used the approximation that the nonuniformity of the electric field inside
the cell arising from space charges can be ignored. The resulting simplified expressions for ⑀ I⬘ ( ␻ ) and ⑀ I⬙ ( ␻ ) reproduce the experimental data semiquantitatively even for thethin cell at high temperatures. Liquid crystals which are usedin very thin display devices have to be highly purified to
avoid the ionic contribution to the dielectric response andhence to the electro-optic characteristics. Indeed the nonuniform field arising from the space charge effects can be expected to alter the Freedericksz threshold voltage for materials with positive dielectric anisotropy. We have conducted
some preliminary experiments to find this to be the case.
These results along with the relevant analysis will be published elsewhere.

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